Subgroups — C₂×C₅₃×#2335 #27241590

Order 13568.

2982 conjugacy classes of subgroups.

The subgroup lattice has 796 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2335	C₂	direct (N × H)
4. C₂×C₅₃×#389	C₂	semidirect
5. C₂×C₅₃×#388	C₂	semidirect
6. C₂×C₅₃×#388	C₂	semidirect
7. C₂×C₅₃×#388	C₂	semidirect
8. C₂×C₅₃×#388	C₂	semidirect
9. C₂×C₅₃×#387	C₂	semidirect
10. C₂×C₅₃×#387	C₂	semidirect
12. C₂×C₅₃×#385	C₂	semidirect
13. C₂×C₅₃×#380	C₂	semidirect
14. C₂×C₄×C₄⋊C₄×C₅₃	C₂	semidirect
15. C₂×C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
16. C₂×C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
20. C₂×#123×C₅₃	C₄	semidirect
45. C₅₃×#389	C₂²	semidirect
46. C₅₃×#388	C₂²	semidirect
47. C₅₃×#388	C₂²	semidirect
48. C₅₃×#388	C₂²	semidirect
49. C₅₃×#388	C₂²	semidirect
50. C₅₃×#387	C₂²	semidirect
51. C₅₃×#387	C₂²	semidirect
53. C₅₃×#385	C₂²	semidirect
54. C₅₃×#380	C₂²	semidirect
55. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
56. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
57. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
79. C₂³×C₄×C₅₃	C₂×C₄	semidirect
95. #123×C₅₃	C₂×C₄	semidirect
101. C₂×C₄⋊C₄×C₅₃	C₂×C₄	semidirect
102. C₂×C₄⋊C₄×C₅₃	C₂×C₄	semidirect
119. C₂×C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
215. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
236. C₄⋊C₄×C₅₃	C₂²×C₄	semidirect
237. C₄⋊C₄×C₅₃	C₂²×C₄	semidirect
247. C₂²⋊C₄×C₅₃	C₂²×C₄	semidirect
344. C₂×#2335	C₅₃	direct (N × H)
389. #2335	C₂×C₅₃	direct (N × H)
391. C₂×#389	C₂×C₅₃	semidirect
392. C₂×#388	C₂×C₅₃	semidirect
393. C₂×#388	C₂×C₅₃	semidirect
394. C₂×#388	C₂×C₅₃	semidirect
395. C₂×#388	C₂×C₅₃	semidirect
396. C₂×#387	C₂×C₅₃	semidirect
397. C₂×#387	C₂×C₅₃	semidirect
399. C₂×#385	C₂×C₅₃	semidirect
400. C₂×#380	C₂×C₅₃	semidirect
401. C₂×C₄×C₄⋊C₄	C₂×C₅₃	semidirect
402. C₂×C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
403. C₂×C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
406. C₂×C₅₃	#2335	direct (N × H)
417. C₂×#123	C₄×C₅₃	semidirect
442. #389	C₂²×C₅₃	semidirect
443. #388	C₂²×C₅₃	semidirect
444. #388	C₂²×C₅₃	semidirect
445. #388	C₂²×C₅₃	semidirect
446. #388	C₂²×C₅₃	semidirect
447. #387	C₂²×C₅₃	semidirect
448. #387	C₂²×C₅₃	semidirect
450. #385	C₂²×C₅₃	semidirect
451. #380	C₂²×C₅₃	semidirect
452. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
453. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
454. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
465. C₅₃	C₂×#2335	direct (N × H)
477. C₂³×C₄	C₂×C₄×C₅₃	semidirect
493. #123	C₂×C₄×C₅₃	semidirect
499. C₂×C₄⋊C₄	C₂×C₄×C₅₃	semidirect
500. C₂×C₄⋊C₄	C₂×C₄×C₅₃	semidirect
517. C₂×C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
613. C₂²×C₄	C₂²×C₄×C₅₃	semidirect
634. C₄⋊C₄	C₂²×C₄×C₅₃	semidirect
635. C₄⋊C₄	C₂²×C₄×C₅₃	semidirect
645. C₂²⋊C₄	C₂²×C₄×C₅₃	semidirect
787. C₂	C₅₃×#2335	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#123×C₅₃	2
C₅₃×#2335	2
C₂×C₅₃×#390	2
C₂×C₅₃×#389	2
C₂×C₅₃×#388	2
C₂×C₅₃×#387	2
C₂×C₅₃×#386	2
C₂×C₅₃×#385	2
C₂×C₅₃×#380	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
C₂×#2335	53

Subgroups — C2×C53×#2335 #27241590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2335 #27241590