Subgroups — C₂×C₅₃×#3539 #27270239

Order 13568.

7354 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3539	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₅₃×#560	C₂	1	4	yes	maximal, complemented
3	C₂×C₅₃×#553	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#553	C₂	1	4	yes	maximal, complemented
5	C₂×C₅₃×#552	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#550	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#547	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅₃×#540	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#540	C₂	1	4	yes	maximal, complemented
11	C₂×C₅₃×#539	C₂	1	4	yes	maximal, complemented
12	C₂×C₅₃×#537	C₂	1	2	yes	maximal, complemented
13	C₂×C₅₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₅₃×#525	C₂	1	2	yes	maximal, complemented
15	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#41×C₅₃	C₂²	1	2	yes	—
17	C₂²×#41×C₅₃	C₂²	1	2	yes	complemented
18	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#560	C₂²	1	64	yes	complemented
20	C₅₃×#553	C₂²	1	32	yes	complemented
21	C₅₃×#553	C₂²	1	64	yes	complemented
22	C₅₃×#552	C₂²	1	32	yes	complemented
23	C₅₃×#550	C₂²	1	32	yes	complemented
24	C₅₃×#547	C₂²	1	32	yes	complemented
25	C₅₃×#541	C₂²	1	16	yes	complemented
26	C₅₃×#540	C₂²	1	16	yes	complemented
27	C₅₃×#540	C₂²	1	64	yes	complemented
28	C₅₃×#539	C₂²	1	64	yes	complemented
29	C₅₃×#537	C₂²	1	32	yes	complemented
30	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
31	C₂×#127×C₅₃	C₂²	1	4	yes	—
32	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic, complemented
33	C₂×#126×C₅₃	C₂²	1	4	yes	—
34	C₂×#126×C₅₃	C₂²	1	4	yes	—
35	C₂×#126×C₅₃	C₂²	1	4	yes	complemented
36	C₂×#125×C₅₃	C₂²	1	4	yes	complemented
37	C₂×#125×C₅₃	C₂²	1	4	yes	complemented
38	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₅₃×#525	C₂²	1	16	yes	complemented
41	C₅₃×#525	C₂²	1	32	yes	complemented
42	C₂×#123×C₅₃	C₂²	1	4	yes	—
43	C₂×#123×C₅₃	C₂²	1	4	yes	—
44	C₂×#123×C₅₃	C₂²	1	4	yes	—
45	C₂×#123×C₅₃	C₂²	1	4	yes	—
46	C₂×#123×C₅₃	C₂²	1	2	yes	—
47	C₂×#123×C₅₃	C₂²	1	4	yes	complemented
48	C₂×#123×C₅₃	C₂²	1	4	yes	complemented
49	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
50	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
51	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
52	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
53	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
54	C₂×#121×C₅₃	C₂²	1	2	yes	—
55	C₂×#121×C₅₃	C₂²	1	4	yes	—
56	C₂×#121×C₅₃	C₂²	1	4	yes	—
57	C₂×#121×C₅₃	C₂²	1	4	yes	—
58	C₂×#121×C₅₃	C₂²	1	2	yes	—
59	C₂×#121×C₅₃	C₂²	1	4	yes	—
60	C₂×#121×C₅₃	C₂²	1	4	yes	—
61	C₂×#121×C₅₃	C₂²	1	4	yes	—
62	C₂×#121×C₅₃	C₂²	1	2	yes	—
63	C₂×#121×C₅₃	C₂²	1	4	yes	complemented
64	C₂×#121×C₅₃	C₂²	1	2	yes	complemented
65	C₂×#121×C₅₃	C₂²	1	4	yes	complemented
66	C₂×#120×C₅₃	C₂²	1	2	yes	—
67	C₂×#120×C₅₃	C₂²	1	4	yes	—
68	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
69	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
70	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
71	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
72	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
73	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
74	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
75	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
76	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
77	C₂×#117×C₅₃	C₂²	1	2	yes	—
78	C₂×#117×C₅₃	C₂²	1	4	yes	complemented
79	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
80	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
81	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×#41×C₅₃	—	4	4	no	—
83	C₂×#41×C₅₃	—	4	8	no	—
84	C₂×#41×C₅₃	—	4	8	no	—
85	C₂×#41×C₅₃	C₂³	1	8	yes	—
86	C₂×#41×C₅₃	C₂³	1	16	yes	—
87	C₂×#41×C₅₃	C₂³	1	16	yes	complemented
88	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
89	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
91	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
92	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
93	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
94	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
95	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
96	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
97	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
98	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
99	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
101	C₂³×C₄×C₅₃	—	4	4	no	—
102	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
103	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
104	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
105	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
106	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
107	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
108	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
109	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
110	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
111	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
112	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	complemented
113	#127×C₅₃	C₂³	1	32	yes	—
114	#127×C₅₃	C₂³	1	8	yes	complemented
115	#126×C₅₃	C₂³	1	32	yes	—
116	#126×C₅₃	C₂³	1	32	yes	—
117	#126×C₅₃	C₂³	1	32	yes	complemented
118	#125×C₅₃	C₂³	1	32	yes	complemented
119	#125×C₅₃	C₂³	1	32	yes	complemented
120	#125×C₅₃	C₂³	1	8	yes	complemented
121	#125×C₅₃	C₂³	1	8	yes	complemented
122	#123×C₅₃	C₂³	1	32	yes	—
123	#123×C₅₃	C₂³	1	16	yes	—
124	#123×C₅₃	C₂³	1	32	yes	—
125	#123×C₅₃	C₂³	1	32	yes	—
126	#123×C₅₃	C₂³	1	32	yes	—
127	#123×C₅₃	C₂³	1	32	yes	complemented
128	#123×C₅₃	C₂³	1	32	yes	complemented
129	C₂²⋊Q₈×C₅₃	C₂³	1	16	yes	—
130	C₂²⋊Q₈×C₅₃	C₂³	1	16	yes	—
131	C₂²⋊Q₈×C₅₃	C₂³	1	16	yes	—
132	C₂²⋊Q₈×C₅₃	C₂³	1	32	yes	complemented
133	C₂²⋊Q₈×C₅₃	C₂³	1	16	yes	complemented
134	#121×C₅₃	C₂³	1	16	yes	—
135	#121×C₅₃	C₂³	1	16	yes	—
136	#121×C₅₃	C₂³	1	16	yes	—
137	#121×C₅₃	C₂³	1	32	yes	—
138	#121×C₅₃	C₂³	1	32	yes	—
139	#121×C₅₃	C₂³	1	32	yes	—
140	#121×C₅₃	C₂³	1	32	yes	—
141	#121×C₅₃	C₂³	1	32	yes	—
142	#121×C₅₃	C₂³	1	32	yes	—
143	#121×C₅₃	C₂³	1	32	yes	complemented
144	#121×C₅₃	C₂³	1	16	yes	complemented
145	#121×C₅₃	C₂³	1	32	yes	complemented
146	#120×C₅₃	C₂³	1	32	yes	—
147	#120×C₅₃	C₂³	1	16	yes	—
148	C₄×Q₈×C₅₃	C₂³	1	16	yes	complemented
149	C₄×D₄×C₅₃	C₂³	1	32	yes	—
150	C₄×D₄×C₅₃	C₂³	1	16	yes	—
151	C₄×D₄×C₅₃	C₂³	1	16	yes	—
152	C₄×D₄×C₅₃	C₂³	1	32	yes	—
153	C₄×D₄×C₅₃	C₂³	1	32	yes	complemented
154	C₄×D₄×C₅₃	C₂³	1	16	yes	complemented
155	C₄×D₄×C₅₃	C₂³	1	32	yes	complemented
156	#117×C₅₃	C₂³	1	8	yes	—
157	#117×C₅₃	C₂³	1	16	yes	—
158	#117×C₅₃	C₂³	1	32	yes	complemented
159	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
160	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
161	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
162	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
163	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
164	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
165	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
166	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
167	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
168	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
169	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
170	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
171	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
172	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	complemented
173	C₂×C₄⋊C₄×C₅₃	C₂³	1	32	yes	complemented
174	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
175	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
176	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
177	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
178	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
179	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
180	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
181	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
182	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
183	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
184	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
185	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
186	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
187	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
188	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
189	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
190	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
191	C₂⁴×C₅₃	—	4	2	no	—
192	C₂⁴×C₅₃	—	4	4	no	—
193	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
194	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
195	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
196	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
197	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
198	#41×C₅₃	—	4	16	no	—
199	#41×C₅₃	—	4	32	no	—
200	#41×C₅₃	—	4	32	no	—
201	C₂×Q₈×C₅₃	—	2	8	no	—
202	C₂×Q₈×C₅₃	—	4	2	no	—
203	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
204	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
205	C₂×D₄×C₅₃	—	2	8	no	—
206	C₂×D₄×C₅₃	—	4	4	no	—
207	C₂×D₄×C₅₃	—	4	8	no	—
208	C₂×D₄×C₅₃	—	4	8	no	—
209	C₂×D₄×C₅₃	—	4	8	no	—
210	C₂×D₄×C₅₃	—	4	4	no	—
211	C₂×D₄×C₅₃	—	4	8	no	—
212	C₂×D₄×C₅₃	—	4	4	no	—
213	C₂×D₄×C₅₃	—	4	2	no	—
214	C₂×D₄×C₅₃	—	4	8	no	—
215	C₂×D₄×C₅₃	—	4	8	no	—
216	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
217	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
218	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
219	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
220	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
221	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
222	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
223	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
224	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
225	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
226	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
227	C₂²×C₄×C₅₃	—	2	8	no	—
228	C₂²×C₄×C₅₃	—	2	4	no	—
229	C₂²×C₄×C₅₃	—	2	8	no	—
230	C₂²×C₄×C₅₃	—	4	16	no	—
231	C₂²×C₄×C₅₃	—	4	2	no	—
232	C₂²×C₄×C₅₃	—	4	4	no	—
233	C₂²×C₄×C₅₃	—	4	8	no	—
234	C₂²×C₄×C₅₃	—	4	8	no	—
235	C₂²×C₄×C₅₃	—	4	8	no	—
236	C₂²×C₄×C₅₃	—	4	8	no	—
237	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
238	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
239	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
240	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
241	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
242	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
243	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
244	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
245	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
246	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
247	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
248	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
249	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
250	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
251	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
252	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
253	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
254	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
255	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
256	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
257	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
258	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
259	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
260	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
261	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
262	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
263	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
264	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
265	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
266	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
267	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
268	C₂²⋊C₄×C₅₃	C₂⁴	1	8	yes	—
269	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
270	C₂²⋊C₄×C₅₃	C₂⁴	1	8	yes	—
271	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
272	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
273	C₂²⋊C₄×C₅₃	C₂⁴	1	8	yes	—
274	C₂²⋊C₄×C₅₃	C₂⁴	1	8	yes	—
275	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
276	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
277	C₂²⋊C₄×C₅₃	C₂⁴	1	8	yes	—
278	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
279	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
280	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
281	C₄²×C₅₃	C₂⁴	1	16	yes	—
282	C₄²×C₅₃	C₂⁴	1	4	yes	—
283	C₄²×C₅₃	C₂⁴	1	4	yes	—
284	C₄²×C₅₃	C₂⁴	1	16	yes	—
285	C₂³×C₅₃	—	2	4	no	—
286	C₂³×C₅₃	—	2	2	no	—
287	C₂³×C₅₃	—	2	4	no	—
288	C₂³×C₅₃	—	2	8	no	—
289	C₂³×C₅₃	—	2	4	no	—
290	C₂³×C₅₃	—	2	8	no	—
291	C₂³×C₅₃	—	2	4	no	—
292	C₂³×C₅₃	—	2	8	no	—
293	C₂³×C₅₃	—	2	4	no	—
294	C₂³×C₅₃	—	2	4	no	—
295	C₂³×C₅₃	—	2	4	no	—
296	C₂³×C₅₃	—	4	4	no	—
297	C₂³×C₅₃	—	4	8	no	—
298	C₂³×C₅₃	—	4	16	no	—
299	C₂³×C₅₃	C₂⁵	1	4	yes	—
300	C₂³×C₅₃	C₂⁵	1	8	yes	—
301	C₂³×C₅₃	C₂⁵	1	4	yes	—
302	C₂³×C₅₃	C₂⁵	1	2	yes	—
303	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
304	Q₈×C₅₃	—	2	16	no	—
305	Q₈×C₅₃	—	4	4	no	—
306	D₄×C₅₃	—	2	16	no	—
307	D₄×C₅₃	—	4	8	no	—
308	D₄×C₅₃	—	4	16	no	—
309	D₄×C₅₃	—	4	16	no	—
310	D₄×C₅₃	—	4	16	no	—
311	D₄×C₅₃	—	4	8	no	—
312	D₄×C₅₃	—	4	16	no	—
313	D₄×C₅₃	—	4	8	no	—
314	D₄×C₅₃	—	4	4	no	—
315	D₄×C₅₃	—	4	16	no	—
316	D₄×C₅₃	—	4	16	no	—
317	C₂×C₄×C₅₃	—	2	4	no	—
318	C₂×C₄×C₅₃	—	2	2	no	—
319	C₂×C₄×C₅₃	—	2	4	no	—
320	C₂×C₄×C₅₃	—	2	8	no	—
321	C₂×C₄×C₅₃	—	2	4	no	—
322	C₂×C₄×C₅₃	—	2	8	no	—
323	C₂×C₄×C₅₃	—	2	8	no	—
324	C₂×C₄×C₅₃	—	2	4	no	—
325	C₂×C₄×C₅₃	—	2	16	no	—
326	C₂×C₄×C₅₃	—	2	8	no	—
327	C₂×C₄×C₅₃	—	2	16	no	—
328	C₂×C₄×C₅₃	—	4	4	no	—
329	C₂×C₄×C₅₃	—	4	8	no	—
330	C₂×C₄×C₅₃	—	4	16	no	—
331	C₂×C₄×C₅₃	—	4	16	no	—
332	C₂×C₄×C₅₃	—	4	16	no	—
333	C₂×C₄×C₅₃	—	4	16	no	—
334	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
335	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
336	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
337	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
338	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
339	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
340	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
341	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
342	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
343	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
344	C₂×#3539	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
345	C₂²×C₅₃	—	2	4	no	—
346	C₂²×C₅₃	—	2	2	no	—
347	C₂²×C₅₃	—	2	4	no	—
348	C₂²×C₅₃	—	2	8	no	—
349	C₂²×C₅₃	—	2	4	no	—
350	C₂²×C₅₃	—	2	8	no	—
351	C₂²×C₅₃	—	2	4	no	—
352	C₂²×C₅₃	—	2	8	no	—
353	C₂²×C₅₃	—	2	4	no	—
354	C₂²×C₅₃	—	2	4	no	—
355	C₂²×C₅₃	—	2	4	no	—
356	C₂²×C₅₃	—	4	16	no	—
357	C₂²×C₅₃	—	4	8	no	—
358	C₂²×C₅₃	—	4	16	no	—
359	C₂²×C₅₃	—	4	8	no	—
360	C₂²×C₅₃	—	4	4	no	—
361	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
362	C₂²×C₅₃	#585	1	4	yes	—
363	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
364	C₄×C₅₃	—	2	4	no	—
365	C₄×C₅₃	—	2	2	no	—
366	C₄×C₅₃	—	2	4	no	—
367	C₄×C₅₃	—	2	8	no	—
368	C₄×C₅₃	—	2	4	no	—
369	C₄×C₅₃	—	2	8	no	—
370	C₄×C₅₃	—	2	8	no	—
371	C₄×C₅₃	—	2	4	no	—
372	C₄×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
373	#3539	C₂×C₅₃	1	32	yes	direct product factor
374	C₂×#560	C₂×C₅₃	1	4	yes	complemented
375	C₂×#553	C₂×C₅₃	1	2	yes	complemented
376	C₂×#553	C₂×C₅₃	1	4	yes	complemented
377	C₂×#552	C₂×C₅₃	1	2	yes	complemented
378	C₂×#550	C₂×C₅₃	1	2	yes	complemented
379	C₂×#547	C₂×C₅₃	1	2	yes	complemented
380	C₂×#541	C₂×C₅₃	1	1	yes	characteristic, complemented
381	C₂×#540	C₂×C₅₃	1	1	yes	characteristic, complemented
382	C₂×#540	C₂×C₅₃	1	4	yes	complemented
383	C₂×#539	C₂×C₅₃	1	4	yes	complemented
384	C₂×#537	C₂×C₅₃	1	2	yes	complemented
385	C₂×#525	C₂×C₅₃	1	1	yes	characteristic, complemented
386	C₂×#525	C₂×C₅₃	1	2	yes	complemented
387	C₂×C₅₃	—	4	4	no	—
388	C₂×C₅₃	—	4	8	no	—
389	C₂×C₅₃	—	4	4	no	—
390	C₂×C₅₃	—	4	2	no	—
391	C₂×C₅₃	#3539	1	4	yes	direct product factor
392	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
393	C₂×C₅₃	C₂×#585	1	2	yes	—
394	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
395	C₂²×#41	C₂²×C₅₃	1	2	yes	—
396	C₂²×#41	C₂²×C₅₃	1	2	yes	complemented
397	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
398	#560	C₂²×C₅₃	1	64	yes	complemented
399	#553	C₂²×C₅₃	1	32	yes	complemented
400	#553	C₂²×C₅₃	1	64	yes	complemented
401	#552	C₂²×C₅₃	1	32	yes	complemented
402	#550	C₂²×C₅₃	1	32	yes	complemented
403	#547	C₂²×C₅₃	1	32	yes	complemented
404	#541	C₂²×C₅₃	1	16	yes	complemented
405	#540	C₂²×C₅₃	1	16	yes	complemented
406	#540	C₂²×C₅₃	1	64	yes	complemented
407	#539	C₂²×C₅₃	1	64	yes	complemented
408	#537	C₂²×C₅₃	1	32	yes	complemented
409	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
410	C₂×#127	C₂²×C₅₃	1	4	yes	—
411	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic, complemented
412	C₂×#126	C₂²×C₅₃	1	4	yes	—
413	C₂×#126	C₂²×C₅₃	1	4	yes	—
414	C₂×#126	C₂²×C₅₃	1	4	yes	complemented
415	C₂×#125	C₂²×C₅₃	1	4	yes	complemented
416	C₂×#125	C₂²×C₅₃	1	4	yes	complemented
417	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic, complemented
418	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic, complemented
419	#525	C₂²×C₅₃	1	16	yes	complemented
420	#525	C₂²×C₅₃	1	32	yes	complemented
421	C₂×#123	C₂²×C₅₃	1	4	yes	—
422	C₂×#123	C₂²×C₅₃	1	2	yes	—
423	C₂×#123	C₂²×C₅₃	1	4	yes	—
424	C₂×#123	C₂²×C₅₃	1	4	yes	—
425	C₂×#123	C₂²×C₅₃	1	4	yes	—
426	C₂×#123	C₂²×C₅₃	1	4	yes	complemented
427	C₂×#123	C₂²×C₅₃	1	4	yes	complemented
428	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
429	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
430	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
431	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
432	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
433	C₂×#121	C₂²×C₅₃	1	2	yes	—
434	C₂×#121	C₂²×C₅₃	1	2	yes	—
435	C₂×#121	C₂²×C₅₃	1	2	yes	—
436	C₂×#121	C₂²×C₅₃	1	4	yes	—
437	C₂×#121	C₂²×C₅₃	1	4	yes	—
438	C₂×#121	C₂²×C₅₃	1	4	yes	—
439	C₂×#121	C₂²×C₅₃	1	4	yes	—
440	C₂×#121	C₂²×C₅₃	1	4	yes	—
441	C₂×#121	C₂²×C₅₃	1	4	yes	—
442	C₂×#121	C₂²×C₅₃	1	4	yes	complemented
443	C₂×#121	C₂²×C₅₃	1	2	yes	complemented
444	C₂×#121	C₂²×C₅₃	1	4	yes	complemented
445	C₂×#120	C₂²×C₅₃	1	2	yes	—
446	C₂×#120	C₂²×C₅₃	1	4	yes	—
447	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
448	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
449	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
450	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
451	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
452	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
453	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
454	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
455	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
456	C₂×#117	C₂²×C₅₃	1	2	yes	—
457	C₂×#117	C₂²×C₅₃	1	4	yes	complemented
458	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
459	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
460	C₅₃	C₂×#3539	1	1	yes	characteristic, direct product factor, Sylow
461	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
462	C₂×#41	—	4	4	no	—
463	C₂×#41	—	4	8	no	—
464	C₂×#41	—	4	8	no	—
465	C₂×#41	C₂³×C₅₃	1	8	yes	—
466	C₂×#41	C₂³×C₅₃	1	16	yes	—
467	C₂×#41	C₂³×C₅₃	1	16	yes	complemented
468	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
469	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
470	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
471	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
472	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
473	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
474	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
475	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
476	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
477	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
478	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
479	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
480	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
481	C₂³×C₄	—	4	4	no	—
482	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
483	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
484	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
485	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
486	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
487	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
488	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
489	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
490	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
491	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
492	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	complemented
493	#127	C₂³×C₅₃	1	32	yes	—
494	#127	C₂³×C₅₃	1	8	yes	complemented
495	#126	C₂³×C₅₃	1	32	yes	—
496	#126	C₂³×C₅₃	1	32	yes	—
497	#126	C₂³×C₅₃	1	32	yes	complemented
498	#125	C₂³×C₅₃	1	32	yes	complemented
499	#125	C₂³×C₅₃	1	32	yes	complemented
500	#125	C₂³×C₅₃	1	8	yes	complemented
501	#125	C₂³×C₅₃	1	8	yes	complemented
502	#123	C₂³×C₅₃	1	32	yes	—
503	#123	C₂³×C₅₃	1	16	yes	—
504	#123	C₂³×C₅₃	1	32	yes	—
505	#123	C₂³×C₅₃	1	32	yes	—
506	#123	C₂³×C₅₃	1	32	yes	—
507	#123	C₂³×C₅₃	1	32	yes	complemented
508	#123	C₂³×C₅₃	1	32	yes	complemented
509	C₂²⋊Q₈	C₂³×C₅₃	1	16	yes	—
510	C₂²⋊Q₈	C₂³×C₅₃	1	16	yes	—
511	C₂²⋊Q₈	C₂³×C₅₃	1	16	yes	—
512	C₂²⋊Q₈	C₂³×C₅₃	1	32	yes	complemented
513	C₂²⋊Q₈	C₂³×C₅₃	1	16	yes	complemented
514	#121	C₂³×C₅₃	1	16	yes	—
515	#121	C₂³×C₅₃	1	16	yes	—
516	#121	C₂³×C₅₃	1	16	yes	—
517	#121	C₂³×C₅₃	1	32	yes	—
518	#121	C₂³×C₅₃	1	32	yes	—
519	#121	C₂³×C₅₃	1	32	yes	—
520	#121	C₂³×C₅₃	1	32	yes	—
521	#121	C₂³×C₅₃	1	32	yes	—
522	#121	C₂³×C₅₃	1	32	yes	—
523	#121	C₂³×C₅₃	1	32	yes	complemented
524	#121	C₂³×C₅₃	1	16	yes	complemented
525	#121	C₂³×C₅₃	1	32	yes	complemented
526	#120	C₂³×C₅₃	1	16	yes	—
527	#120	C₂³×C₅₃	1	32	yes	—
528	C₄×Q₈	C₂³×C₅₃	1	16	yes	complemented
529	C₄×D₄	C₂³×C₅₃	1	32	yes	—
530	C₄×D₄	C₂³×C₅₃	1	16	yes	—
531	C₄×D₄	C₂³×C₅₃	1	16	yes	—
532	C₄×D₄	C₂³×C₅₃	1	32	yes	—
533	C₄×D₄	C₂³×C₅₃	1	32	yes	complemented
534	C₄×D₄	C₂³×C₅₃	1	16	yes	complemented
535	C₄×D₄	C₂³×C₅₃	1	32	yes	complemented
536	#117	C₂³×C₅₃	1	8	yes	—
537	#117	C₂³×C₅₃	1	16	yes	—
538	#117	C₂³×C₅₃	1	32	yes	complemented
539	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
540	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
541	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
542	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
543	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
544	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
545	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
546	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
547	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
548	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
549	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
550	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
551	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
552	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	complemented
553	C₂×C₄⋊C₄	C₂³×C₅₃	1	32	yes	complemented
554	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
555	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
556	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
557	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
558	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
559	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
560	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
561	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
562	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
563	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
564	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
565	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
566	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
567	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
568	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
569	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
570	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
571	C₂⁴	—	4	2	no	—
572	C₂⁴	—	4	4	no	—
573	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
574	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
575	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
576	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
577	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
578	#41	—	4	16	no	—
579	#41	—	4	32	no	—
580	#41	—	4	32	no	—
581	C₂×Q₈	—	2	8	no	—
582	C₂×Q₈	—	4	2	no	—
583	C₂×Q₈	C₂⁴×C₅₃	1	8	yes	—
584	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
585	C₂×D₄	—	2	8	no	—
586	C₂×D₄	—	4	4	no	—
587	C₂×D₄	—	4	8	no	—
588	C₂×D₄	—	4	8	no	—
589	C₂×D₄	—	4	8	no	—
590	C₂×D₄	—	4	4	no	—
591	C₂×D₄	—	4	8	no	—
592	C₂×D₄	—	4	4	no	—
593	C₂×D₄	—	4	2	no	—
594	C₂×D₄	—	4	8	no	—
595	C₂×D₄	—	4	8	no	—
596	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
597	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
598	C₂×D₄	C₂⁴×C₅₃	1	16	yes	—
599	C₂×D₄	C₂⁴×C₅₃	1	16	yes	—
600	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
601	C₂×D₄	C₂⁴×C₅₃	1	16	yes	—
602	C₂×D₄	C₂⁴×C₅₃	1	16	yes	—
603	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
604	C₂×D₄	C₂⁴×C₅₃	1	16	yes	—
605	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
606	C₂×D₄	C₂⁴×C₅₃	1	16	yes	—
607	C₂²×C₄	—	2	8	no	—
608	C₂²×C₄	—	2	4	no	—
609	C₂²×C₄	—	2	8	no	—
610	C₂²×C₄	—	4	16	no	—
611	C₂²×C₄	—	4	2	no	—
612	C₂²×C₄	—	4	4	no	—
613	C₂²×C₄	—	4	8	no	—
614	C₂²×C₄	—	4	8	no	—
615	C₂²×C₄	—	4	8	no	—
616	C₂²×C₄	—	4	8	no	—
617	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
618	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
619	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
620	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
621	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
622	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
623	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
624	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
625	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
626	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
627	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
628	C₂²×C₄	C₂⁴×C₅₃	1	16	yes	—
629	C₂²×C₄	C₂⁴×C₅₃	1	16	yes	—
630	C₂²×C₄	C₂⁴×C₅₃	1	16	yes	—
631	C₂²×C₄	C₂⁴×C₅₃	1	16	yes	—
632	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
633	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
634	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
635	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
636	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
637	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
638	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
639	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
640	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
641	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
642	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
643	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
644	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
645	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
646	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
647	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
648	C₂²⋊C₄	C₂⁴×C₅₃	1	8	yes	—
649	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	—
650	C₂²⋊C₄	C₂⁴×C₅₃	1	8	yes	—
651	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	—
652	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	—
653	C₂²⋊C₄	C₂⁴×C₅₃	1	8	yes	—
654	C₂²⋊C₄	C₂⁴×C₅₃	1	8	yes	—
655	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	—
656	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	—
657	C₂²⋊C₄	C₂⁴×C₅₃	1	8	yes	—
658	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	—
659	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	—
660	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	—
661	C₄²	C₂⁴×C₅₃	1	16	yes	—
662	C₄²	C₂⁴×C₅₃	1	4	yes	—
663	C₄²	C₂⁴×C₅₃	1	4	yes	—
664	C₄²	C₂⁴×C₅₃	1	16	yes	—
665	C₂³	—	2	4	no	—
666	C₂³	—	2	2	no	—
667	C₂³	—	2	4	no	—
668	C₂³	—	2	8	no	—
669	C₂³	—	2	4	no	—
670	C₂³	—	2	8	no	—
671	C₂³	—	2	4	no	—
672	C₂³	—	2	8	no	—
673	C₂³	—	2	4	no	—
674	C₂³	—	2	4	no	—
675	C₂³	—	2	4	no	—
676	C₂³	—	4	4	no	—
677	C₂³	—	4	8	no	—
678	C₂³	—	4	16	no	—
679	C₂³	C₂⁵×C₅₃	1	2	yes	—
680	C₂³	C₂⁵×C₅₃	1	4	yes	—
681	C₂³	C₂⁵×C₅₃	1	4	yes	—
682	C₂³	C₂⁵×C₅₃	1	8	yes	—
683	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
684	Q₈	—	2	16	no	—
685	Q₈	—	4	4	no	—
686	D₄	—	2	16	no	—
687	D₄	—	4	8	no	—
688	D₄	—	4	16	no	—
689	D₄	—	4	16	no	—
690	D₄	—	4	16	no	—
691	D₄	—	4	8	no	—
692	D₄	—	4	16	no	—
693	D₄	—	4	8	no	—
694	D₄	—	4	4	no	—
695	D₄	—	4	16	no	—
696	D₄	—	4	16	no	—
697	C₂×C₄	—	2	4	no	—
698	C₂×C₄	—	2	2	no	—
699	C₂×C₄	—	2	4	no	—
700	C₂×C₄	—	2	8	no	—
701	C₂×C₄	—	2	4	no	—
702	C₂×C₄	—	2	8	no	—
703	C₂×C₄	—	2	8	no	—
704	C₂×C₄	—	2	4	no	—
705	C₂×C₄	—	2	16	no	—
706	C₂×C₄	—	2	8	no	—
707	C₂×C₄	—	2	16	no	—
708	C₂×C₄	—	4	4	no	—
709	C₂×C₄	—	4	8	no	—
710	C₂×C₄	—	4	16	no	—
711	C₂×C₄	—	4	16	no	—
712	C₂×C₄	—	4	16	no	—
713	C₂×C₄	—	4	16	no	—
714	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
715	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
716	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
717	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
718	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
719	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
720	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
721	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
722	C₂×C₄	C₂⁵×C₅₃	1	2	yes	—
723	C₂×C₄	C₂⁵×C₅₃	1	2	yes	—
724	C₂²	—	2	4	no	—
725	C₂²	—	2	2	no	—
726	C₂²	—	2	4	no	—
727	C₂²	—	2	8	no	—
728	C₂²	—	2	4	no	—
729	C₂²	—	2	8	no	—
730	C₂²	—	2	4	no	—
731	C₂²	—	2	8	no	—
732	C₂²	—	2	4	no	—
733	C₂²	—	2	4	no	—
734	C₂²	—	2	4	no	—
735	C₂²	—	4	16	no	—
736	C₂²	—	4	8	no	—
737	C₂²	—	4	4	no	—
738	C₂²	—	4	8	no	—
739	C₂²	—	4	16	no	—
740	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
741	C₂²	C₅₃×#585	1	4	yes	—
742	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
743	C₄	—	2	4	no	—
744	C₄	—	2	2	no	—
745	C₄	—	2	4	no	—
746	C₄	—	2	8	no	—
747	C₄	—	2	4	no	—
748	C₄	—	2	8	no	—
749	C₄	—	2	8	no	—
750	C₄	—	2	4	no	—
751	C₄	C₂×H(5,𝔽₂)×C₅₃	1	4	yes	—
752	C₂	—	4	4	no	—
753	C₂	—	4	2	no	—
754	C₂	—	4	4	no	—
755	C₂	—	4	8	no	—
756	C₂	C₅₃×#3539	1	4	yes	direct product factor
757	C₂	C₂²×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
758	C₂	C₂×C₅₃×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₅₃×#3539	C₂	direct (N × H)
2. C₂×C₅₃×#560	C₂	semidirect
3. C₂×C₅₃×#553	C₂	semidirect
4. C₂×C₅₃×#553	C₂	semidirect
5. C₂×C₅₃×#552	C₂	semidirect
6. C₂×C₅₃×#550	C₂	semidirect
7. C₂×C₅₃×#547	C₂	semidirect
8. C₂×C₅₃×#541	C₂	semidirect
9. C₂×C₅₃×#540	C₂	semidirect
10. C₂×C₅₃×#540	C₂	semidirect
11. C₂×C₅₃×#539	C₂	semidirect
12. C₂×C₅₃×#537	C₂	semidirect
13. C₂×C₅₃×#525	C₂	semidirect
14. C₂×C₅₃×#525	C₂	semidirect
17. C₂²×#41×C₅₃	C₂²	semidirect
19. C₅₃×#560	C₂²	semidirect
20. C₅₃×#553	C₂²	semidirect
21. C₅₃×#553	C₂²	semidirect
22. C₅₃×#552	C₂²	semidirect
23. C₅₃×#550	C₂²	semidirect
24. C₅₃×#547	C₂²	semidirect
25. C₅₃×#541	C₂²	semidirect
26. C₅₃×#540	C₂²	semidirect
27. C₅₃×#540	C₂²	semidirect
28. C₅₃×#539	C₂²	semidirect
29. C₅₃×#537	C₂²	semidirect
30. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
32. C₂×#127×C₅₃	C₂²	semidirect
35. C₂×#126×C₅₃	C₂²	semidirect
36. C₂×#125×C₅₃	C₂²	semidirect
37. C₂×#125×C₅₃	C₂²	semidirect
38. C₂×#125×C₅₃	C₂²	semidirect
39. C₂×#125×C₅₃	C₂²	semidirect
40. C₅₃×#525	C₂²	semidirect
41. C₅₃×#525	C₂²	semidirect
47. C₂×#123×C₅₃	C₂²	semidirect
48. C₂×#123×C₅₃	C₂²	semidirect
52. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
53. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
63. C₂×#121×C₅₃	C₂²	semidirect
64. C₂×#121×C₅₃	C₂²	semidirect
65. C₂×#121×C₅₃	C₂²	semidirect
68. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
73. C₂×C₄×D₄×C₅₃	C₂²	semidirect
74. C₂×C₄×D₄×C₅₃	C₂²	semidirect
75. C₂×C₄×D₄×C₅₃	C₂²	semidirect
78. C₂×#117×C₅₃	C₂²	semidirect
79. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
80. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
87. C₂×#41×C₅₃	C₂³	semidirect
112. C₄⋊Q₈×C₅₃	C₂³	semidirect
114. #127×C₅₃	C₂³	semidirect
117. #126×C₅₃	C₂³	semidirect
118. #125×C₅₃	C₂³	semidirect
119. #125×C₅₃	C₂³	semidirect
120. #125×C₅₃	C₂³	semidirect
121. #125×C₅₃	C₂³	semidirect
127. #123×C₅₃	C₂³	semidirect
128. #123×C₅₃	C₂³	semidirect
132. C₂²⋊Q₈×C₅₃	C₂³	semidirect
133. C₂²⋊Q₈×C₅₃	C₂³	semidirect
143. #121×C₅₃	C₂³	semidirect
144. #121×C₅₃	C₂³	semidirect
145. #121×C₅₃	C₂³	semidirect
148. C₄×Q₈×C₅₃	C₂³	semidirect
153. C₄×D₄×C₅₃	C₂³	semidirect
154. C₄×D₄×C₅₃	C₂³	semidirect
155. C₄×D₄×C₅₃	C₂³	semidirect
158. #117×C₅₃	C₂³	semidirect
172. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
173. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
344. C₂×#3539	C₅₃	direct (N × H)
373. #3539	C₂×C₅₃	direct (N × H)
374. C₂×#560	C₂×C₅₃	semidirect
375. C₂×#553	C₂×C₅₃	semidirect
376. C₂×#553	C₂×C₅₃	semidirect
377. C₂×#552	C₂×C₅₃	semidirect
378. C₂×#550	C₂×C₅₃	semidirect
379. C₂×#547	C₂×C₅₃	semidirect
380. C₂×#541	C₂×C₅₃	semidirect
381. C₂×#540	C₂×C₅₃	semidirect
382. C₂×#540	C₂×C₅₃	semidirect
383. C₂×#539	C₂×C₅₃	semidirect
384. C₂×#537	C₂×C₅₃	semidirect
385. C₂×#525	C₂×C₅₃	semidirect
386. C₂×#525	C₂×C₅₃	semidirect
391. C₂×C₅₃	#3539	direct (N × H)
396. C₂²×#41	C₂²×C₅₃	semidirect
398. #560	C₂²×C₅₃	semidirect
399. #553	C₂²×C₅₃	semidirect
400. #553	C₂²×C₅₃	semidirect
401. #552	C₂²×C₅₃	semidirect
402. #550	C₂²×C₅₃	semidirect
403. #547	C₂²×C₅₃	semidirect
404. #541	C₂²×C₅₃	semidirect
405. #540	C₂²×C₅₃	semidirect
406. #540	C₂²×C₅₃	semidirect
407. #539	C₂²×C₅₃	semidirect
408. #537	C₂²×C₅₃	semidirect
409. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
411. C₂×#127	C₂²×C₅₃	semidirect
414. C₂×#126	C₂²×C₅₃	semidirect
415. C₂×#125	C₂²×C₅₃	semidirect
416. C₂×#125	C₂²×C₅₃	semidirect
417. C₂×#125	C₂²×C₅₃	semidirect
418. C₂×#125	C₂²×C₅₃	semidirect
419. #525	C₂²×C₅₃	semidirect
420. #525	C₂²×C₅₃	semidirect
426. C₂×#123	C₂²×C₅₃	semidirect
427. C₂×#123	C₂²×C₅₃	semidirect
431. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
432. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
442. C₂×#121	C₂²×C₅₃	semidirect
443. C₂×#121	C₂²×C₅₃	semidirect
444. C₂×#121	C₂²×C₅₃	semidirect
447. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
452. C₂×C₄×D₄	C₂²×C₅₃	semidirect
453. C₂×C₄×D₄	C₂²×C₅₃	semidirect
454. C₂×C₄×D₄	C₂²×C₅₃	semidirect
457. C₂×#117	C₂²×C₅₃	semidirect
458. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
459. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
460. C₅₃	C₂×#3539	direct (N × H)
467. C₂×#41	C₂³×C₅₃	semidirect
492. C₄⋊Q₈	C₂³×C₅₃	semidirect
494. #127	C₂³×C₅₃	semidirect
497. #126	C₂³×C₅₃	semidirect
498. #125	C₂³×C₅₃	semidirect
499. #125	C₂³×C₅₃	semidirect
500. #125	C₂³×C₅₃	semidirect
501. #125	C₂³×C₅₃	semidirect
507. #123	C₂³×C₅₃	semidirect
508. #123	C₂³×C₅₃	semidirect
512. C₂²⋊Q₈	C₂³×C₅₃	semidirect
513. C₂²⋊Q₈	C₂³×C₅₃	semidirect
523. #121	C₂³×C₅₃	semidirect
524. #121	C₂³×C₅₃	semidirect
525. #121	C₂³×C₅₃	semidirect
528. C₄×Q₈	C₂³×C₅₃	semidirect
533. C₄×D₄	C₂³×C₅₃	semidirect
534. C₄×D₄	C₂³×C₅₃	semidirect
535. C₄×D₄	C₂³×C₅₃	semidirect
538. #117	C₂³×C₅₃	semidirect
552. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
553. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
756. C₂	C₅₃×#3539	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅₃×#3539	2
C₂×C₅₃×#560	2
C₂×C₅₃×#553	2
C₂×C₅₃×#552	2
C₂×C₅₃×#550	2
C₂×C₅₃×#547	2
C₂×C₅₃×#541	2
C₂×C₅₃×#540	2
C₂×C₅₃×#539	2
C₂×C₅₃×#537	2
C₂×C₅₃×#525	2
C₂×#3539	53

Subgroups — C2×C53×#3539 #27270239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3539 #27270239