Subgroups — C₅₃×#34233 #27241745

Order 13568.

2826 conjugacy classes of subgroups.

The subgroup lattice has 776 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#525	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2361	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2360	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2358	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2353	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2349	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2347	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2341	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2339	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2335	C₂	1	4	yes	maximal, complemented
12	C₅₃×#2318	C₂	1	2	yes	maximal, complemented
13	C₄×C₂²⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
14	C₄×#121×C₅₃	C₂	1	2	yes	maximal, complemented
15	C₂×C₅₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅₃×#2275	C₂	1	2	yes	maximal, complemented
17	C₂×C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#525	C₄	1	16	yes	complemented
21	C₂×#123×C₅₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#121×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#121×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#120×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
28	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
29	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
30	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
33	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
37	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
38	C₅₃×#391	C₂²	1	2	yes	complemented
39	C₅₃×#390	C₂²	1	4	yes	—
40	C₅₃×#390	C₂²	1	2	yes	complemented
41	C₅₃×#389	C₂²	1	2	yes	complemented
42	C₅₃×#389	C₂²	1	2	yes	complemented
43	C₅₃×#389	C₂²	1	4	yes	complemented
44	C₅₃×#388	C₂²	1	4	yes	—
45	C₅₃×#388	C₂²	1	4	yes	—
46	C₅₃×#388	C₂²	1	4	yes	complemented
47	C₅₃×#388	C₂²	1	4	yes	complemented
48	C₅₃×#387	C₂²	1	4	yes	complemented
49	C₅₃×#387	C₂²	1	4	yes	complemented
50	C₅₃×#386	C₂²	1	4	yes	—
51	C₅₃×#386	C₂²	1	4	yes	—
52	C₅₃×#386	C₂²	1	4	yes	—
53	C₅₃×#385	C₂²	1	4	yes	—
54	C₅₃×#385	C₂²	1	4	yes	—
55	C₅₃×#385	C₂²	1	4	yes	complemented
56	C₅₃×#383	C₂²	1	2	yes	complemented
57	C₅₃×#382	C₂²	1	2	yes	complemented
58	C₅₃×#381	C₂²	1	2	yes	complemented
59	C₅₃×#380	C₂²	1	2	yes	—
60	C₅₃×#380	C₂²	1	2	yes	—
61	C₅₃×#380	C₂²	1	1	yes	characteristic
62	C₅₃×#380	C₂²	1	1	yes	characteristic
63	C₅₃×#380	C₂²	1	2	yes	—
64	C₅₃×#380	C₂²	1	2	yes	complemented
65	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
66	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
67	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
68	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
69	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
70	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
71	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
72	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
73	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
74	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
75	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
76	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
77	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
78	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
79	C₅₃×#376	C₂²	1	2	yes	complemented
80	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
81	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
82	C₂×#95×C₅₃	C₂²	1	2	yes	—
83	C₂×#95×C₅₃	C₂²	1	2	yes	complemented
84	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
85	C₂×#41×C₅₃	—	4	4	no	—
86	C₂×#41×C₅₃	C₂×C₄	1	8	yes	complemented
87	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
89	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
90	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
91	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
92	C₂³×C₄×C₅₃	—	4	4	no	—
93	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	complemented
94	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
95	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
96	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
97	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
98	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
99	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
100	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
101	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
102	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
103	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
104	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
105	#123×C₅₃	C₂×C₄	1	16	yes	complemented
106	C₂²⋊Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
107	C₂²⋊Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
108	#121×C₅₃	C₂×C₄	1	8	yes	complemented
109	#121×C₅₃	C₂×C₄	1	8	yes	complemented
110	#120×C₅₃	C₂×C₄	1	16	yes	complemented
111	C₄×Q₈×C₅₃	—	4	4	no	—
112	C₄×D₄×C₅₃	—	2	16	no	—
113	C₄×D₄×C₅₃	—	4	8	no	—
114	C₄×D₄×C₅₃	—	4	4	no	—
115	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
116	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
117	#117×C₅₃	C₂×C₄	1	8	yes	complemented
118	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
119	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
122	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
124	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
125	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
126	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
128	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
129	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
130	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
131	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
132	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
133	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
134	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
135	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
136	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
137	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
138	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
139	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
140	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
141	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
142	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
143	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
144	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
145	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
146	C₂×C₄²×C₅₃	—	4	4	no	—
147	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
148	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
149	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
150	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
151	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
152	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
153	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
154	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
155	#95×C₅₃	C₂³	1	1	yes	characteristic
156	#95×C₅₃	C₂³	1	2	yes	—
157	#95×C₅₃	C₂³	1	1	yes	characteristic
158	#95×C₅₃	C₂³	1	1	yes	characteristic
159	#95×C₅₃	C₂³	1	1	yes	characteristic
160	#95×C₅₃	C₂³	1	2	yes	—
161	#95×C₅₃	C₂³	1	2	yes	—
162	#95×C₅₃	C₂³	1	2	yes	—
163	#95×C₅₃	C₂³	1	4	yes	—
164	#95×C₅₃	C₂³	1	4	yes	—
165	#95×C₅₃	C₂³	1	4	yes	—
166	#95×C₅₃	C₂³	1	4	yes	—
167	#95×C₅₃	C₂³	1	4	yes	—
168	#95×C₅₃	C₂³	1	4	yes	—
169	#95×C₅₃	C₂³	1	4	yes	—
170	C₂⁴×C₅₃	—	2	4	no	—
171	C₂⁴×C₅₃	—	2	2	no	—
172	C₂⁴×C₅₃	—	2	2	no	—
173	C₂⁴×C₅₃	—	4	2	no	—
174	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
175	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
176	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
177	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
178	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
179	#41×C₅₃	—	4	16	no	—
180	C₂×Q₈×C₅₃	—	2	4	no	—
181	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
182	C₂×D₄×C₅₃	—	2	8	no	—
183	C₂×D₄×C₅₃	—	2	4	no	—
184	C₂×D₄×C₅₃	—	2	4	no	—
185	C₂×D₄×C₅₃	—	2	4	no	—
186	C₂×D₄×C₅₃	—	4	4	no	—
187	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
188	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
189	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
190	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
191	C₂²×C₄×C₅₃	—	2	4	no	—
192	C₂²×C₄×C₅₃	—	2	4	no	—
193	C₂²×C₄×C₅₃	—	2	4	no	—
194	C₂²×C₄×C₅₃	—	2	4	no	—
195	C₂²×C₄×C₅₃	—	2	4	no	—
196	C₂²×C₄×C₅₃	—	2	8	no	—
197	C₂²×C₄×C₅₃	—	2	4	no	—
198	C₂²×C₄×C₅₃	—	2	4	no	—
199	C₂²×C₄×C₅₃	—	2	8	no	—
200	C₂²×C₄×C₅₃	—	4	16	no	—
201	C₂²×C₄×C₅₃	—	4	2	no	—
202	C₂²×C₄×C₅₃	—	4	2	no	—
203	C₂²×C₄×C₅₃	—	4	2	no	—
204	C₂²×C₄×C₅₃	—	4	2	no	—
205	C₂²×C₄×C₅₃	—	4	2	no	—
206	C₂²×C₄×C₅₃	—	4	4	no	—
207	C₂²×C₄×C₅₃	—	4	4	no	—
208	C₂²×C₄×C₅₃	—	4	4	no	—
209	C₂²×C₄×C₅₃	—	4	4	no	—
210	C₂²×C₄×C₅₃	—	4	4	no	—
211	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
212	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
213	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
214	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
215	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
216	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
217	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
218	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
219	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
220	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
221	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
222	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
223	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
224	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
225	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
226	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
227	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
228	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
229	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
230	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
231	C₄⋊C₄×C₅₃	—	2	8	no	—
232	C₄⋊C₄×C₅₃	—	2	4	no	—
233	C₄⋊C₄×C₅₃	—	2	4	no	—
234	C₄⋊C₄×C₅₃	—	4	4	no	—
235	C₄⋊C₄×C₅₃	—	4	4	no	—
236	C₄⋊C₄×C₅₃	—	4	4	no	—
237	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
238	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
239	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
240	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
241	C₂²⋊C₄×C₅₃	—	2	4	no	—
242	C₂²⋊C₄×C₅₃	—	2	8	no	—
243	C₂²⋊C₄×C₅₃	—	2	4	no	—
244	C₂²⋊C₄×C₅₃	—	2	4	no	—
245	C₂²⋊C₄×C₅₃	—	2	8	no	—
246	C₂²⋊C₄×C₅₃	—	4	2	no	—
247	C₂²⋊C₄×C₅₃	—	4	2	no	—
248	C₂²⋊C₄×C₅₃	—	4	4	no	—
249	C₂²⋊C₄×C₅₃	—	4	4	no	—
250	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
251	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
252	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
253	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
254	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
255	C₂²⋊C₄×C₅₃	#41	1	8	yes	complemented
256	C₄²×C₅₃	—	2	8	no	—
257	C₄²×C₅₃	—	2	4	no	—
258	C₄²×C₅₃	—	2	4	no	—
259	C₄²×C₅₃	—	4	2	no	—
260	C₄²×C₅₃	—	4	2	no	—
261	C₄²×C₅₃	—	4	4	no	—
262	C₄²×C₅₃	—	4	4	no	—
263	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
264	C₂³×C₅₃	—	2	2	no	—
265	C₂³×C₅₃	—	2	2	no	—
266	C₂³×C₅₃	—	2	2	no	—
267	C₂³×C₅₃	—	2	1	no	—
268	C₂³×C₅₃	—	2	1	no	—
269	C₂³×C₅₃	—	2	8	no	—
270	C₂³×C₅₃	—	2	8	no	—
271	C₂³×C₅₃	—	2	4	no	—
272	C₂³×C₅₃	—	2	2	no	—
273	C₂³×C₅₃	—	2	2	no	—
274	C₂³×C₅₃	—	2	4	no	—
275	C₂³×C₅₃	—	2	2	no	—
276	C₂³×C₅₃	—	2	1	no	—
277	C₂³×C₅₃	—	2	2	no	—
278	C₂³×C₅₃	—	2	1	no	—
279	C₂³×C₅₃	—	4	4	no	—
280	C₂³×C₅₃	—	4	8	no	—
281	C₂³×C₅₃	—	4	8	no	—
282	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
283	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
284	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
285	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
286	C₂³×C₅₃	C₂×#41	1	4	yes	—
287	C₂³×C₅₃	H(5,𝔽₂)	1	2	yes	—
288	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
289	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
290	Q₈×C₅₃	—	2	8	no	—
291	D₄×C₅₃	—	2	16	no	—
292	D₄×C₅₃	—	2	16	no	—
293	D₄×C₅₃	—	2	8	no	—
294	D₄×C₅₃	—	4	8	no	—
295	C₂×C₄×C₅₃	—	2	4	no	—
296	C₂×C₄×C₅₃	—	2	2	no	—
297	C₂×C₄×C₅₃	—	2	2	no	—
298	C₂×C₄×C₅₃	—	2	2	no	—
299	C₂×C₄×C₅₃	—	2	2	no	—
300	C₂×C₄×C₅₃	—	2	2	no	—
301	C₂×C₄×C₅₃	—	2	4	no	—
302	C₂×C₄×C₅₃	—	2	2	no	—
303	C₂×C₄×C₅₃	—	2	2	no	—
304	C₂×C₄×C₅₃	—	2	2	no	—
305	C₂×C₄×C₅₃	—	2	2	no	—
306	C₂×C₄×C₅₃	—	2	8	no	—
307	C₂×C₄×C₅₃	—	2	8	no	—
308	C₂×C₄×C₅₃	—	2	8	no	—
309	C₂×C₄×C₅₃	—	2	4	no	—
310	C₂×C₄×C₅₃	—	2	2	no	—
311	C₂×C₄×C₅₃	—	2	2	no	—
312	C₂×C₄×C₅₃	—	2	4	no	—
313	C₂×C₄×C₅₃	—	2	2	no	—
314	C₂×C₄×C₅₃	—	2	4	no	—
315	C₂×C₄×C₅₃	—	2	4	no	—
316	C₂×C₄×C₅₃	—	2	2	no	—
317	C₂×C₄×C₅₃	—	2	2	no	—
318	C₂×C₄×C₅₃	—	2	4	no	—
319	C₂×C₄×C₅₃	—	2	4	no	—
320	C₂×C₄×C₅₃	—	2	2	no	—
321	C₂×C₄×C₅₃	—	4	4	no	—
322	C₂×C₄×C₅₃	—	4	4	no	—
323	C₂×C₄×C₅₃	—	4	4	no	—
324	C₂×C₄×C₅₃	—	4	8	no	—
325	C₂×C₄×C₅₃	—	4	8	no	—
326	C₂×C₄×C₅₃	—	4	8	no	—
327	C₂×C₄×C₅₃	—	8	8	no	—
328	C₂×C₄×C₅₃	—	8	8	no	—
329	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
330	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
331	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
332	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
333	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
334	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
335	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
336	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
337	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
338	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
339	#34233	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
340	C₂²×C₅₃	—	2	2	no	—
341	C₂²×C₅₃	—	2	2	no	—
342	C₂²×C₅₃	—	2	2	no	—
343	C₂²×C₅₃	—	2	2	no	—
344	C₂²×C₅₃	—	2	4	no	—
345	C₂²×C₅₃	—	2	4	no	—
346	C₂²×C₅₃	—	2	4	no	—
347	C₂²×C₅₃	—	2	4	no	—
348	C₂²×C₅₃	—	2	4	no	—
349	C₂²×C₅₃	—	2	2	no	—
350	C₂²×C₅₃	—	2	2	no	—
351	C₂²×C₅₃	—	2	4	no	—
352	C₂²×C₅₃	—	2	4	no	—
353	C₂²×C₅₃	—	2	2	no	—
354	C₂²×C₅₃	—	4	16	no	—
355	C₂²×C₅₃	—	4	4	no	—
356	C₂²×C₅₃	—	4	2	no	—
357	C₂²×C₅₃	—	4	4	no	—
358	C₂²×C₅₃	—	4	2	no	—
359	C₂²×C₅₃	C₄×#41	1	4	yes	—
360	C₂²×C₅₃	#520	1	4	yes	—
361	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
362	C₂²×C₅₃	C₂²×#41	1	2	yes	—
363	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
364	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
365	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
366	C₄×C₅₃	—	2	4	no	—
367	C₄×C₅₃	—	2	2	no	—
368	C₄×C₅₃	—	2	2	no	—
369	C₄×C₅₃	—	2	2	no	—
370	C₄×C₅₃	—	2	4	no	—
371	C₄×C₅₃	—	2	4	no	—
372	C₄×C₅₃	—	2	4	no	—
373	C₄×C₅₃	—	2	4	no	—
374	C₄×C₅₃	—	4	2	no	—
375	C₄×C₅₃	—	4	4	no	—
376	C₄×C₅₃	—	4	4	no	—
377	C₄×C₅₃	—	4	2	no	—
378	C₄×C₅₃	#518	1	4	yes	—
379	C₂×#525	C₂×C₅₃	1	1	yes	characteristic
380	#2361	C₂×C₅₃	1	2	yes	complemented
381	#2360	C₂×C₅₃	1	2	yes	complemented
382	#2358	C₂×C₅₃	1	4	yes	complemented
383	#2353	C₂×C₅₃	1	2	yes	complemented
384	#2349	C₂×C₅₃	1	2	yes	complemented
385	#2347	C₂×C₅₃	1	1	yes	characteristic, complemented
386	#2346	C₂×C₅₃	1	1	yes	characteristic, complemented
387	#2341	C₂×C₅₃	1	1	yes	characteristic, complemented
388	#2339	C₂×C₅₃	1	1	yes	characteristic, complemented
389	#2335	C₂×C₅₃	1	4	yes	complemented
390	#2318	C₂×C₅₃	1	2	yes	complemented
391	C₄×C₂²⋊Q₈	C₂×C₅₃	1	2	yes	complemented
392	C₄×#121	C₂×C₅₃	1	2	yes	complemented
393	C₂×#380	C₂×C₅₃	1	1	yes	characteristic, complemented
394	#2275	C₂×C₅₃	1	2	yes	complemented
395	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
396	C₂×C₅₃	—	2	4	no	—
397	C₂×C₅₃	—	2	4	no	—
398	C₂×C₅₃	—	4	4	no	—
399	C₂×C₅₃	—	4	2	no	—
400	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
401	C₂×C₅₃	C₂×#518	1	1	yes	characteristic
402	C₂×C₅₃	C₂×#520	1	1	yes	characteristic
403	C₂×C₅₃	#3519	1	2	yes	—
404	C₂×C₅₃	#3520	1	2	yes	—
405	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
406	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
407	#525	C₄×C₅₃	1	16	yes	complemented
408	C₂×#123	C₂²×C₅₃	1	2	yes	—
409	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
410	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
411	C₂×#121	C₂²×C₅₃	1	1	yes	characteristic
412	C₂×#121	C₂²×C₅₃	1	1	yes	characteristic
413	C₂×#120	C₂²×C₅₃	1	2	yes	—
414	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
415	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
416	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
417	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
418	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
419	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
420	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
421	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
422	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
423	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
424	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
425	#391	C₂²×C₅₃	1	2	yes	complemented
426	#390	C₂²×C₅₃	1	4	yes	—
427	#390	C₂²×C₅₃	1	2	yes	complemented
428	#389	C₂²×C₅₃	1	4	yes	complemented
429	#389	C₂²×C₅₃	1	2	yes	complemented
430	#389	C₂²×C₅₃	1	2	yes	complemented
431	#388	C₂²×C₅₃	1	4	yes	—
432	#388	C₂²×C₅₃	1	4	yes	—
433	#388	C₂²×C₅₃	1	4	yes	complemented
434	#388	C₂²×C₅₃	1	4	yes	complemented
435	#387	C₂²×C₅₃	1	4	yes	complemented
436	#387	C₂²×C₅₃	1	4	yes	complemented
437	#386	C₂²×C₅₃	1	4	yes	—
438	#386	C₂²×C₅₃	1	4	yes	—
439	#386	C₂²×C₅₃	1	4	yes	—
440	#385	C₂²×C₅₃	1	4	yes	—
441	#385	C₂²×C₅₃	1	4	yes	—
442	#385	C₂²×C₅₃	1	4	yes	complemented
443	#383	C₂²×C₅₃	1	2	yes	complemented
444	#382	C₂²×C₅₃	1	2	yes	complemented
445	#381	C₂²×C₅₃	1	2	yes	complemented
446	#380	C₂²×C₅₃	1	2	yes	—
447	#380	C₂²×C₅₃	1	1	yes	characteristic
448	#380	C₂²×C₅₃	1	1	yes	characteristic
449	#380	C₂²×C₅₃	1	2	yes	—
450	#380	C₂²×C₅₃	1	2	yes	—
451	#380	C₂²×C₅₃	1	2	yes	complemented
452	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
453	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
454	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
455	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
456	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
457	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
458	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
459	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
460	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
461	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
462	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
463	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
464	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
465	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	complemented
466	#376	C₂²×C₅₃	1	2	yes	complemented
467	C₂×#95	C₂²×C₅₃	1	2	yes	—
468	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
469	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
470	C₂×#95	C₂²×C₅₃	1	2	yes	complemented
471	C₅₃	#34233	1	1	yes	characteristic, direct product factor, Sylow
472	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
473	C₂×#41	—	4	4	no	—
474	C₂×#41	C₂×C₄×C₅₃	1	8	yes	complemented
475	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
476	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
477	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
478	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
479	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
480	C₂³×C₄	—	4	4	no	—
481	C₂³×C₄	C₂×C₄×C₅₃	1	8	yes	complemented
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	1	yes	characteristic
486	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
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	1	yes	characteristic
491	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
492	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
493	#123	C₂×C₄×C₅₃	1	16	yes	complemented
494	C₂²⋊Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
495	C₂²⋊Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
496	#121	C₂×C₄×C₅₃	1	8	yes	complemented
497	#121	C₂×C₄×C₅₃	1	8	yes	complemented
498	#120	C₂×C₄×C₅₃	1	16	yes	complemented
499	C₄×Q₈	—	4	4	no	—
500	C₄×D₄	—	2	16	no	—
501	C₄×D₄	—	4	4	no	—
502	C₄×D₄	—	4	8	no	—
503	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
504	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
505	#117	C₂×C₄×C₅₃	1	8	yes	complemented
506	C₂×C₄⋊C₄	—	4	4	no	—
507	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
508	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
509	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
510	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
511	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
512	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
513	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
514	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
515	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
516	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
517	C₂×C₂²⋊C₄	—	2	8	no	—
518	C₂×C₂²⋊C₄	—	4	4	no	—
519	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
520	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
521	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
522	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
523	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
524	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
525	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
526	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
527	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
528	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
529	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
530	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
531	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
532	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
533	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
534	C₂×C₄²	—	4	4	no	—
535	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
536	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
537	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
538	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
539	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
540	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
541	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
542	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
543	#95	C₂³×C₅₃	1	1	yes	characteristic
544	#95	C₂³×C₅₃	1	2	yes	—
545	#95	C₂³×C₅₃	1	1	yes	characteristic
546	#95	C₂³×C₅₃	1	1	yes	characteristic
547	#95	C₂³×C₅₃	1	1	yes	characteristic
548	#95	C₂³×C₅₃	1	2	yes	—
549	#95	C₂³×C₅₃	1	2	yes	—
550	#95	C₂³×C₅₃	1	2	yes	—
551	#95	C₂³×C₅₃	1	4	yes	—
552	#95	C₂³×C₅₃	1	4	yes	—
553	#95	C₂³×C₅₃	1	4	yes	—
554	#95	C₂³×C₅₃	1	4	yes	—
555	#95	C₂³×C₅₃	1	4	yes	—
556	#95	C₂³×C₅₃	1	4	yes	—
557	#95	C₂³×C₅₃	1	4	yes	—
558	C₂⁴	—	2	4	no	—
559	C₂⁴	—	2	2	no	—
560	C₂⁴	—	2	2	no	—
561	C₂⁴	—	4	2	no	—
562	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
563	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
564	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
565	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
566	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
567	#41	—	4	16	no	—
568	C₂×Q₈	—	2	4	no	—
569	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
570	C₂×D₄	—	2	8	no	—
571	C₂×D₄	—	2	4	no	—
572	C₂×D₄	—	2	4	no	—
573	C₂×D₄	—	2	4	no	—
574	C₂×D₄	—	4	4	no	—
575	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
576	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
577	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
578	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
579	C₂²×C₄	—	2	4	no	—
580	C₂²×C₄	—	2	4	no	—
581	C₂²×C₄	—	2	4	no	—
582	C₂²×C₄	—	2	4	no	—
583	C₂²×C₄	—	2	4	no	—
584	C₂²×C₄	—	2	8	no	—
585	C₂²×C₄	—	2	4	no	—
586	C₂²×C₄	—	2	4	no	—
587	C₂²×C₄	—	2	8	no	—
588	C₂²×C₄	—	4	16	no	—
589	C₂²×C₄	—	4	2	no	—
590	C₂²×C₄	—	4	2	no	—
591	C₂²×C₄	—	4	2	no	—
592	C₂²×C₄	—	4	2	no	—
593	C₂²×C₄	—	4	2	no	—
594	C₂²×C₄	—	4	4	no	—
595	C₂²×C₄	—	4	4	no	—
596	C₂²×C₄	—	4	4	no	—
597	C₂²×C₄	—	4	4	no	—
598	C₂²×C₄	—	4	4	no	—
599	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
600	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
601	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
602	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
603	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
604	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
605	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
606	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
607	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
608	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
609	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
610	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
611	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
612	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
613	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
614	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
615	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
616	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
617	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
618	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
619	C₄⋊C₄	—	2	8	no	—
620	C₄⋊C₄	—	2	4	no	—
621	C₄⋊C₄	—	2	4	no	—
622	C₄⋊C₄	—	4	4	no	—
623	C₄⋊C₄	—	4	4	no	—
624	C₄⋊C₄	—	4	4	no	—
625	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
626	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
627	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
628	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
629	C₂²⋊C₄	—	2	4	no	—
630	C₂²⋊C₄	—	2	8	no	—
631	C₂²⋊C₄	—	2	4	no	—
632	C₂²⋊C₄	—	2	4	no	—
633	C₂²⋊C₄	—	2	8	no	—
634	C₂²⋊C₄	—	4	2	no	—
635	C₂²⋊C₄	—	4	2	no	—
636	C₂²⋊C₄	—	4	4	no	—
637	C₂²⋊C₄	—	4	4	no	—
638	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
639	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
640	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
641	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
642	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
643	C₂²⋊C₄	#41×C₅₃	1	8	yes	complemented
644	C₄²	—	2	8	no	—
645	C₄²	—	2	4	no	—
646	C₄²	—	2	4	no	—
647	C₄²	—	4	2	no	—
648	C₄²	—	4	2	no	—
649	C₄²	—	4	4	no	—
650	C₄²	—	4	4	no	—
651	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
652	C₂³	—	2	2	no	—
653	C₂³	—	2	2	no	—
654	C₂³	—	2	2	no	—
655	C₂³	—	2	1	no	—
656	C₂³	—	2	1	no	—
657	C₂³	—	2	8	no	—
658	C₂³	—	2	8	no	—
659	C₂³	—	2	2	no	—
660	C₂³	—	2	4	no	—
661	C₂³	—	2	1	no	—
662	C₂³	—	2	2	no	—
663	C₂³	—	2	4	no	—
664	C₂³	—	2	2	no	—
665	C₂³	—	2	2	no	—
666	C₂³	—	2	1	no	—
667	C₂³	—	4	4	no	—
668	C₂³	—	4	8	no	—
669	C₂³	—	4	8	no	—
670	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
671	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
672	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
673	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
674	C₂³	C₂×#41×C₅₃	1	4	yes	—
675	C₂³	H(5,𝔽₂)×C₅₃	1	2	yes	—
676	C₂³	2₋¹⁺⁴×C₅₃	1	2	yes	—
677	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
678	Q₈	—	2	8	no	—
679	D₄	—	2	16	no	—
680	D₄	—	2	16	no	—
681	D₄	—	2	8	no	—
682	D₄	—	4	8	no	—
683	C₂×C₄	—	2	4	no	—
684	C₂×C₄	—	2	2	no	—
685	C₂×C₄	—	2	2	no	—
686	C₂×C₄	—	2	2	no	—
687	C₂×C₄	—	2	2	no	—
688	C₂×C₄	—	2	2	no	—
689	C₂×C₄	—	2	4	no	—
690	C₂×C₄	—	2	2	no	—
691	C₂×C₄	—	2	2	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	2	8	no	—
695	C₂×C₄	—	2	8	no	—
696	C₂×C₄	—	2	8	no	—
697	C₂×C₄	—	2	4	no	—
698	C₂×C₄	—	2	2	no	—
699	C₂×C₄	—	2	2	no	—
700	C₂×C₄	—	2	4	no	—
701	C₂×C₄	—	2	2	no	—
702	C₂×C₄	—	2	2	no	—
703	C₂×C₄	—	2	4	no	—
704	C₂×C₄	—	2	4	no	—
705	C₂×C₄	—	2	4	no	—
706	C₂×C₄	—	2	2	no	—
707	C₂×C₄	—	2	4	no	—
708	C₂×C₄	—	2	2	no	—
709	C₂×C₄	—	4	4	no	—
710	C₂×C₄	—	4	4	no	—
711	C₂×C₄	—	4	4	no	—
712	C₂×C₄	—	4	8	no	—
713	C₂×C₄	—	4	8	no	—
714	C₂×C₄	—	4	8	no	—
715	C₂×C₄	—	8	8	no	—
716	C₂×C₄	—	8	8	no	—
717	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
718	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
719	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
720	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
721	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
722	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
723	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
724	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
725	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
726	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
727	C₂²	—	2	4	no	—
728	C₂²	—	2	4	no	—
729	C₂²	—	2	4	no	—
730	C₂²	—	2	4	no	—
731	C₂²	—	2	2	no	—
732	C₂²	—	2	2	no	—
733	C₂²	—	2	2	no	—
734	C₂²	—	2	2	no	—
735	C₂²	—	2	4	no	—
736	C₂²	—	2	2	no	—
737	C₂²	—	2	2	no	—
738	C₂²	—	2	4	no	—
739	C₂²	—	2	4	no	—
740	C₂²	—	2	2	no	—
741	C₂²	—	4	16	no	—
742	C₂²	—	4	4	no	—
743	C₂²	—	4	2	no	—
744	C₂²	—	4	2	no	—
745	C₂²	—	4	4	no	—
746	C₂²	C₄×#41×C₅₃	1	4	yes	—
747	C₂²	C₅₃×#520	1	4	yes	—
748	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
749	C₂²	C₂²×#41×C₅₃	1	2	yes	—
750	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
751	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
752	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
753	C₄	—	2	4	no	—
754	C₄	—	2	4	no	—
755	C₄	—	2	4	no	—
756	C₄	—	2	2	no	—
757	C₄	—	2	2	no	—
758	C₄	—	2	2	no	—
759	C₄	—	2	4	no	—
760	C₄	—	2	4	no	—
761	C₄	—	4	2	no	—
762	C₄	—	4	4	no	—
763	C₄	—	4	4	no	—
764	C₄	—	4	2	no	—
765	C₄	C₅₃×#518	1	4	yes	—
766	C₂	—	2	4	no	—
767	C₂	—	2	4	no	—
768	C₂	—	4	2	no	—
769	C₂	—	4	4	no	—
770	C₂	C₂×C₄×#41×C₅₃	1	1	yes	characteristic
771	C₂	C₂×C₅₃×#518	1	1	yes	characteristic
772	C₂	C₂×C₅₃×#520	1	1	yes	characteristic
773	C₂	C₅₃×#3519	1	2	yes	—
774	C₂	C₅₃×#3520	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂×C₅₃×#525	C₂
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
2. C₅₃×#2361	C₂	semidirect
3. C₅₃×#2360	C₂	semidirect
4. C₅₃×#2358	C₂	semidirect
5. C₅₃×#2353	C₂	semidirect
6. C₅₃×#2349	C₂	semidirect
7. C₅₃×#2347	C₂	semidirect
8. C₅₃×#2346	C₂	semidirect
9. C₅₃×#2341	C₂	semidirect
10. C₅₃×#2339	C₂	semidirect
11. C₅₃×#2335	C₂	semidirect
12. C₅₃×#2318	C₂	semidirect
13. C₄×C₂²⋊Q₈×C₅₃	C₂	semidirect
14. C₄×#121×C₅₃	C₂	semidirect
15. C₂×C₅₃×#380	C₂	semidirect
16. C₅₃×#2275	C₂	semidirect
17. C₂×C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
20. C₅₃×#525	C₄	semidirect
27. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
32. C₂×C₄×D₄×C₅₃	C₂²	semidirect
38. C₅₃×#391	C₂²	semidirect
40. C₅₃×#390	C₂²	semidirect
41. C₅₃×#389	C₂²	semidirect
42. C₅₃×#389	C₂²	semidirect
43. C₅₃×#389	C₂²	semidirect
46. C₅₃×#388	C₂²	semidirect
47. C₅₃×#388	C₂²	semidirect
48. C₅₃×#387	C₂²	semidirect
49. C₅₃×#387	C₂²	semidirect
55. C₅₃×#385	C₂²	semidirect
56. C₅₃×#383	C₂²	semidirect
57. C₅₃×#382	C₂²	semidirect
58. C₅₃×#381	C₂²	semidirect
64. C₅₃×#380	C₂²	semidirect
65. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
66. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
67. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
68. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
77. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
78. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
79. C₅₃×#376	C₂²	semidirect
83. C₂×#95×C₅₃	C₂²	semidirect
86. C₂×#41×C₅₃	C₂×C₄	semidirect
93. C₂³×C₄×C₅₃	C₂×C₄	semidirect
105. #123×C₅₃	C₂×C₄	semidirect
106. C₂²⋊Q₈×C₅₃	C₂×C₄	semidirect
107. C₂²⋊Q₈×C₅₃	C₂×C₄	semidirect
108. #121×C₅₃	C₂×C₄	semidirect
109. #121×C₅₃	C₂×C₄	semidirect
110. #120×C₅₃	C₂×C₄	semidirect
115. C₄×D₄×C₅₃	C₂×C₄	semidirect
116. C₄×D₄×C₅₃	C₂×C₄	semidirect
117. #117×C₅₃	C₂×C₄	semidirect
255. C₂²⋊C₄×C₅₃	#41	semidirect
339. #34233	C₅₃	direct (N × H)
380. #2361	C₂×C₅₃	semidirect
381. #2360	C₂×C₅₃	semidirect
382. #2358	C₂×C₅₃	semidirect
383. #2353	C₂×C₅₃	semidirect
384. #2349	C₂×C₅₃	semidirect
385. #2347	C₂×C₅₃	semidirect
386. #2346	C₂×C₅₃	semidirect
387. #2341	C₂×C₅₃	semidirect
388. #2339	C₂×C₅₃	semidirect
389. #2335	C₂×C₅₃	semidirect
390. #2318	C₂×C₅₃	semidirect
391. C₄×C₂²⋊Q₈	C₂×C₅₃	semidirect
392. C₄×#121	C₂×C₅₃	semidirect
393. C₂×#380	C₂×C₅₃	semidirect
394. #2275	C₂×C₅₃	semidirect
395. C₂×C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
407. #525	C₄×C₅₃	semidirect
414. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
419. C₂×C₄×D₄	C₂²×C₅₃	semidirect
425. #391	C₂²×C₅₃	semidirect
427. #390	C₂²×C₅₃	semidirect
428. #389	C₂²×C₅₃	semidirect
429. #389	C₂²×C₅₃	semidirect
430. #389	C₂²×C₅₃	semidirect
433. #388	C₂²×C₅₃	semidirect
434. #388	C₂²×C₅₃	semidirect
435. #387	C₂²×C₅₃	semidirect
436. #387	C₂²×C₅₃	semidirect
442. #385	C₂²×C₅₃	semidirect
443. #383	C₂²×C₅₃	semidirect
444. #382	C₂²×C₅₃	semidirect
445. #381	C₂²×C₅₃	semidirect
451. #380	C₂²×C₅₃	semidirect
452. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
453. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
454. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
455. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
464. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
465. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
466. #376	C₂²×C₅₃	semidirect
470. C₂×#95	C₂²×C₅₃	semidirect
471. C₅₃	#34233	direct (N × H)
474. C₂×#41	C₂×C₄×C₅₃	semidirect
481. C₂³×C₄	C₂×C₄×C₅₃	semidirect
493. #123	C₂×C₄×C₅₃	semidirect
494. C₂²⋊Q₈	C₂×C₄×C₅₃	semidirect
495. C₂²⋊Q₈	C₂×C₄×C₅₃	semidirect
496. #121	C₂×C₄×C₅₃	semidirect
497. #121	C₂×C₄×C₅₃	semidirect
498. #120	C₂×C₄×C₅₃	semidirect
503. C₄×D₄	C₂×C₄×C₅₃	semidirect
504. C₄×D₄	C₂×C₄×C₅₃	semidirect
505. #117	C₂×C₄×C₅₃	semidirect
643. C₂²⋊C₄	#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#525	2
C₅₃×#2361	2
C₅₃×#2360	2
C₅₃×#2358	2
C₅₃×#2353	2
C₅₃×#2349	2
C₅₃×#2347	2
C₅₃×#2346	2
C₅₃×#2341	2
C₅₃×#2339	2
C₅₃×#2335	2
C₅₃×#2318	2
C₄×C₂²⋊Q₈×C₅₃	2
C₄×#121×C₅₃	2
C₂×C₅₃×#380	2
C₅₃×#2275	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
#34233	53

Subgroups — C53×#34233 #27241745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34233 #27241745