Subgroups — #27391113

Order 13568.

924 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810701	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275330	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275323	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275281	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275274	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275013	C₂	1	1	yes	maximal, characteristic
8	#20274900	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274783	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274590	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
12	#20274499	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274499	C₂	1	1	yes	maximal, characteristic, complemented
14	#20274499	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#123×C₅₃	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11719917	C₂²	1	1	yes	characteristic
17	C₂²×#11719917	C₂²	1	1	yes	characteristic
18	C₂²×#11719917	C₂²	1	1	yes	characteristic
19	#16810701	—	2	4	no	—
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
28	C₄×#11719917	—	2	4	no	—
29	C₂×#13133128	C₂²	1	1	yes	characteristic
30	C₂×#13133128	C₂²	1	1	yes	characteristic
31	C₂×#13133128	C₂²	1	1	yes	characteristic
32	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
33	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
34	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
35	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
36	#16810378	C₂²	1	1	yes	characteristic
37	#16810378	C₂²	1	1	yes	characteristic
38	#16810378	C₂²	1	1	yes	characteristic
39	#16810378	C₂²	1	1	yes	characteristic
40	#16810378	C₂²	1	1	yes	characteristic
41	#16810378	C₂²	1	1	yes	characteristic
42	#16810378	C₂²	1	1	yes	characteristic
43	#16810378	C₂²	1	1	yes	characteristic
44	#16810378	C₂²	1	1	yes	characteristic
45	#16810378	C₂²	1	1	yes	characteristic
46	C₂×#123×C₅₃	C₄	1	1	yes	characteristic
47	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
48	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
49	C₅₃×#388	C₄	1	1	yes	characteristic
50	C₅₃×#388	C₄	1	1	yes	characteristic
51	C₅₃×#387	C₄	1	1	yes	characteristic
52	C₅₃×#387	C₄	1	1	yes	characteristic
53	C₅₃×#386	C₄	1	1	yes	characteristic
54	C₅₃×#385	C₄	1	1	yes	characteristic
55	C₅₃×#381	C₄	1	1	yes	characteristic
56	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
57	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
58	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
59	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
60	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
61	C₂×#11719917	—	2	2	no	—
62	C₂×#11719917	—	2	2	no	—
63	C₂×#11719917	—	2	2	no	—
64	C₂×#11719917	—	2	2	no	—
65	C₂×#11719917	—	2	2	no	—
66	C₂×#11719917	—	2	2	no	—
67	C₂×#11719917	—	2	2	no	—
68	C₂×#11719917	—	2	2	no	—
69	C₂×#11719917	D₄	1	4	yes	—
70	C₂²×#88322	C₂³	1	1	yes	characteristic
71	C₂²×#88322	C₂³	1	1	yes	characteristic
72	C₂²×#88322	C₂³	1	1	yes	characteristic
73	C₂²×#88322	C₂³	1	1	yes	characteristic
74	C₂²×#88322	C₂³	1	1	yes	characteristic
75	C₂²×#88322	C₂³	1	1	yes	characteristic
76	C₂²×#88322	C₂³	1	1	yes	characteristic
77	C₂²×#88322	C₂³	1	1	yes	characteristic
78	C₄⋊#88322	—	2	2	no	—
79	C₄⋊#88322	—	2	2	no	—
80	C₄⋊#88322	—	2	2	no	—
81	C₂²⋊#88322	—	2	2	no	—
82	C₂²⋊#88322	—	2	2	no	—
83	C₂²⋊#88322	—	2	2	no	—
84	C₂²⋊#88322	—	2	2	no	—
85	C₂²⋊#88322	—	2	2	no	—
86	#13133128	—	2	2	no	—
87	#13133128	—	2	2	no	—
88	#13133128	—	2	2	no	—
89	C₄×#88322	—	2	2	no	—
90	C₄×#88322	—	2	2	no	—
91	C₄×#88322	—	2	2	no	—
92	C₄×#88322	—	2	2	no	—
93	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
97	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
98	#123×C₅₃	—	2	4	no	—
99	C₄×D₄×C₅₃	—	2	2	no	—
100	C₄×D₄×C₅₃	—	2	2	no	—
101	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
106	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
114	C₂×C₄²×C₅₃	—	2	2	no	—
115	C₂×C₄²×C₅₃	—	2	2	no	—
116	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
117	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
118	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
123	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
124	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
125	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
126	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
127	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
128	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
129	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
130	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
131	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
132	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
133	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×D₄×C₅₃	—	2	2	no	—
136	C₂×D₄×C₅₃	—	2	2	no	—
137	C₂×D₄×C₅₃	—	4	1	no	—
138	C₂²×C₄×C₅₃	—	2	1	no	—
139	C₂²×C₄×C₅₃	—	2	1	no	—
140	C₂²×C₄×C₅₃	—	2	1	no	—
141	C₂²×C₄×C₅₃	—	2	1	no	—
142	C₂²×C₄×C₅₃	—	2	2	no	—
143	C₂²×C₄×C₅₃	—	2	2	no	—
144	C₂²×C₄×C₅₃	—	2	2	no	—
145	C₂²×C₄×C₅₃	—	2	2	no	—
146	C₂²×C₄×C₅₃	—	2	1	no	—
147	C₂²×C₄×C₅₃	—	2	1	no	—
148	C₂²×C₄×C₅₃	—	2	1	no	—
149	C₂²×C₄×C₅₃	—	2	1	no	—
150	C₂²×C₄×C₅₃	—	2	1	no	—
151	C₂²×C₄×C₅₃	—	2	1	no	—
152	C₂²×C₄×C₅₃	—	4	1	no	—
153	C₂²×C₄×C₅₃	—	4	1	no	—
154	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₅₃	#41	1	2	yes	—
168	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄×C₅₃	—	2	2	no	—
170	C₄⋊C₄×C₅₃	—	2	2	no	—
171	C₄⋊C₄×C₅₃	—	2	2	no	—
172	C₄⋊C₄×C₅₃	—	4	1	no	—
173	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	complemented
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	—	4	1	no	—
181	C₂²⋊C₄×C₅₃	—	4	1	no	—
182	C₄²×C₅₃	—	2	2	no	—
183	C₄²×C₅₃	—	2	2	no	—
184	C₄²×C₅₃	—	2	1	no	—
185	C₄²×C₅₃	—	2	1	no	—
186	C₄²×C₅₃	—	2	1	no	—
187	C₄²×C₅₃	—	2	1	no	—
188	C₄²×C₅₃	—	2	1	no	—
189	C₄²×C₅₃	—	2	1	no	—
190	C₄²×C₅₃	—	2	1	no	—
191	C₄²×C₅₃	—	2	1	no	—
192	C₄²×C₅₃	—	2	1	no	—
193	C₄²×C₅₃	—	2	1	no	—
194	#11719917	—	2	8	no	—
195	#11719917	—	2	8	no	—
196	#11719917	—	4	4	no	—
197	C₂×#88322	—	2	1	no	—
198	C₂×#88322	—	2	1	no	—
199	C₂×#88322	—	2	1	no	—
200	C₂×#88322	—	2	1	no	—
201	C₂×#88322	—	2	1	no	—
202	C₂×#88322	—	2	1	no	—
203	C₂×#88322	—	2	1	no	—
204	C₂×#88322	—	2	1	no	—
205	C₂×#88322	—	2	1	no	—
206	C₂×#88322	—	2	1	no	—
207	C₂×#88322	—	2	1	no	—
208	C₂×#88322	—	2	1	no	—
209	C₂×#88322	—	2	1	no	—
210	C₂×#88322	—	2	1	no	—
211	C₂×#88322	—	2	1	no	—
212	C₂×#88322	—	2	1	no	—
213	C₂×#88322	—	2	1	no	—
214	C₂×#88322	—	2	1	no	—
215	C₂×#88322	C₂×D₄	1	2	yes	—
216	C₂×#88322	C₂×D₄	1	2	yes	—
217	C₂×#88322	#41	1	2	yes	—
218	C₂×#88322	#41	1	2	yes	—
219	C₂×#88322	#41	1	2	yes	—
220	C₂×#88322	#41	1	2	yes	—
221	C₂³×C₅₃	—	2	1	no	—
222	C₂³×C₅₃	—	2	1	no	—
223	C₂³×C₅₃	—	2	1	no	—
224	C₂³×C₅₃	—	2	1	no	—
225	C₂³×C₅₃	—	2	1	no	—
226	C₂³×C₅₃	—	2	1	no	—
227	C₂³×C₅₃	—	2	1	no	—
228	C₂³×C₅₃	—	2	1	no	—
229	C₂³×C₅₃	—	2	1	no	—
230	C₂³×C₅₃	—	2	1	no	—
231	C₂³×C₅₃	—	2	1	no	—
232	C₂³×C₅₃	—	2	1	no	—
233	C₂³×C₅₃	—	2	1	no	—
234	C₂³×C₅₃	C₄×D₄	1	2	yes	—
235	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
236	D₄×C₅₃	—	4	4	no	—
237	C₂×C₄×C₅₃	—	2	1	no	—
238	C₂×C₄×C₅₃	—	2	1	no	—
239	C₂×C₄×C₅₃	—	2	1	no	—
240	C₂×C₄×C₅₃	—	2	1	no	—
241	C₂×C₄×C₅₃	—	2	1	no	—
242	C₂×C₄×C₅₃	—	2	1	no	—
243	C₂×C₄×C₅₃	—	2	1	no	—
244	C₂×C₄×C₅₃	—	2	1	no	—
245	C₂×C₄×C₅₃	—	2	1	no	—
246	C₂×C₄×C₅₃	—	2	1	no	—
247	C₂×C₄×C₅₃	—	2	2	no	—
248	C₂×C₄×C₅₃	—	2	2	no	—
249	C₂×C₄×C₅₃	—	2	2	no	—
250	C₂×C₄×C₅₃	—	2	2	no	—
251	C₂×C₄×C₅₃	—	2	1	no	—
252	C₂×C₄×C₅₃	—	2	1	no	—
253	C₂×C₄×C₅₃	—	2	1	no	—
254	C₂×C₄×C₅₃	—	2	1	no	—
255	C₂×C₄×C₅₃	—	2	1	no	—
256	C₂×C₄×C₅₃	—	2	1	no	—
257	C₂×C₄×C₅₃	—	2	1	no	—
258	C₂×C₄×C₅₃	—	2	1	no	—
259	C₂×C₄×C₅₃	—	2	1	no	—
260	C₂×C₄×C₅₃	—	2	1	no	—
261	C₂×C₄×C₅₃	—	2	1	no	—
262	C₂×C₄×C₅₃	—	2	1	no	—
263	C₂×C₄×C₅₃	—	2	1	no	—
264	C₂×C₄×C₅₃	—	2	1	no	—
265	C₂×C₄×C₅₃	—	2	1	no	—
266	C₂×C₄×C₅₃	—	2	1	no	—
267	C₂×C₄×C₅₃	—	2	1	no	—
268	C₂×C₄×C₅₃	—	2	1	no	—
269	C₂×C₄×C₅₃	—	2	1	no	—
270	C₂×C₄×C₅₃	—	2	1	no	—
271	C₂×C₄×C₅₃	—	2	1	no	—
272	C₂×C₄×C₅₃	—	4	4	no	—
273	C₂×C₄×C₅₃	—	4	4	no	—
274	C₂×C₄×C₅₃	—	4	1	no	—
275	C₂×C₄×C₅₃	—	4	1	no	—
276	C₂×C₄×C₅₃	—	4	1	no	—
277	C₂×C₄×C₅₃	—	4	1	no	—
278	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
279	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
280	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
281	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
282	C₂×C₄×C₅₃	#131	1	2	yes	—
283	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
285	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
287	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
288	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
289	#88322	—	2	2	no	—
290	#88322	—	2	2	no	—
291	#88322	—	2	2	no	—
292	#88322	—	4	1	no	—
293	#88322	—	4	1	no	—
294	#88322	—	4	1	no	—
295	#88322	—	4	1	no	—
296	#88322	#123	1	4	yes	complemented
297	#21565	—	53	1	no	maximal, Sylow
298	C₂²×C₅₃	—	2	2	no	—
299	C₂²×C₅₃	—	2	2	no	—
300	C₂²×C₅₃	—	2	2	no	—
301	C₂²×C₅₃	—	4	1	no	—
302	C₂²×C₅₃	—	4	1	no	—
303	C₂²×C₅₃	—	4	1	no	—
304	C₂²×C₅₃	—	4	1	no	—
305	C₂²×C₅₃	—	4	1	no	—
306	C₂²×C₅₃	—	4	1	no	—
307	C₂²×C₅₃	—	4	1	no	—
308	C₂²×C₅₃	—	4	1	no	—
309	C₂²×C₅₃	—	4	1	no	—
310	C₂²×C₅₃	—	4	1	no	—
311	C₂²×C₅₃	—	4	1	no	—
312	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
313	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
314	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
315	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
316	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
317	C₂²×C₅₃	#568	1	1	yes	characteristic
318	C₂²×C₅₃	#568	1	1	yes	characteristic
319	C₄×C₅₃	—	2	2	no	—
320	C₄×C₅₃	—	2	2	no	—
321	C₄×C₅₃	—	2	2	no	—
322	C₄×C₅₃	—	2	2	no	—
323	C₄×C₅₃	—	2	2	no	—
324	C₄×C₅₃	—	4	1	no	—
325	C₄×C₅₃	—	4	1	no	—
326	C₄×C₅₃	—	4	1	no	—
327	C₄×C₅₃	—	4	1	no	—
328	C₄×C₅₃	—	4	1	no	—
329	C₄×C₅₃	—	4	1	no	—
330	C₄×C₅₃	—	4	1	no	—
331	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
332	C₄×C₅₃	#525	1	1	yes	characteristic
333	C₄×C₅₃	#529	1	1	yes	characteristic
334	C₄×C₅₃	#529	1	1	yes	characteristic
335	C₂×#436	—	53	1	no	—
336	C₂×C₄×M₁₆	—	53	1	no	—
337	C₄×#123	D₅₃	1	1	yes	characteristic
338	#1924	—	53	1	no	—
339	#1923	—	53	1	no	—
340	#1917	—	53	1	no	—
341	#1916	—	53	1	no	—
342	#1870	—	53	1	no	—
343	#1851	—	53	1	no	—
344	#1817	—	53	1	no	—
345	#1771	—	53	1	no	—
346	C₈×C₄⋊C₄	—	53	1	no	—
347	#1753	—	53	1	no	—
348	#1753	—	53	1	no	—
349	#1753	—	53	1	no	—
350	C₂×C₅₃	—	4	2	no	—
351	C₂×C₅₃	—	8	1	no	—
352	C₂×C₅₃	C₄×#123	1	1	yes	characteristic
353	C₂×C₅₃	#2933	1	1	yes	characteristic
354	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
355	C₂×C₅₃	#2965	1	1	yes	characteristic
356	C₂×C₅₃	#2967	1	1	yes	characteristic
357	C₂×C₅₃	#2969	1	1	yes	characteristic
358	C₂×C₅₃	#2971	1	1	yes	characteristic
359	C₂²×M₁₆	—	53	1	no	—
360	C₂²×M₁₆	—	53	1	no	—
361	C₂²×M₁₆	—	53	1	no	—
362	C₂×#123	Dic₅₃	1	1	yes	characteristic
363	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
364	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
365	#436	—	106	4	no	—
366	C₂×C₄⋊C₈	—	53	1	no	—
367	C₂×C₄⋊C₈	—	53	1	no	—
368	C₂×C₄⋊C₈	—	53	1	no	—
369	C₂×C₂²⋊C₈	—	53	1	no	—
370	C₂×C₂²⋊C₈	—	53	1	no	—
371	C₂×C₂²⋊C₈	—	53	1	no	—
372	C₂×C₂²⋊C₈	—	53	1	no	—
373	C₂×C₂²⋊C₈	—	53	1	no	—
374	C₄×M₁₆	—	106	4	no	—
375	C₂×#97	—	53	1	no	—
376	C₂×#97	—	53	1	no	—
377	C₂×#97	—	53	1	no	—
378	C₂×C₄×C₈	—	53	1	no	—
379	C₂×C₄×C₈	—	53	1	no	—
380	C₂×C₄×C₈	—	53	1	no	—
381	C₂×C₄×C₈	—	53	1	no	—
382	#388	Dic₅₃	1	1	yes	characteristic
383	#388	Dic₅₃	1	1	yes	characteristic
384	#387	Dic₅₃	1	1	yes	characteristic
385	#387	Dic₅₃	1	1	yes	characteristic
386	#386	Dic₅₃	1	1	yes	characteristic
387	#385	Dic₅₃	1	1	yes	characteristic
388	#381	Dic₅₃	1	1	yes	characteristic
389	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
390	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
391	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
392	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
393	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
394	#336	—	53	1	no	—
395	#336	—	53	1	no	—
396	#336	—	53	1	no	—
397	#336	—	53	1	no	—
398	#336	—	53	1	no	—
399	#336	—	53	1	no	—
400	#336	—	53	1	no	—
401	#336	—	53	1	no	—
402	#336	—	53	1	no	—
403	#336	—	53	1	no	—
404	C₅₃	#21565	1	1	yes	characteristic, complemented, Sylow
405	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
406	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
407	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
408	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
409	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
410	C₂×M₁₆	—	53	4	no	—
411	C₂×M₁₆	—	106	2	no	—
412	C₂×M₁₆	—	106	2	no	—
413	C₂×M₁₆	—	106	2	no	—
414	C₂×M₁₆	—	106	2	no	—
415	C₂×M₁₆	—	106	2	no	—
416	C₂×M₁₆	—	106	2	no	—
417	C₂×M₁₆	—	106	2	no	—
418	C₂×M₁₆	—	106	2	no	—
419	C₂²×C₈	—	53	1	no	—
420	C₂²×C₈	—	53	1	no	—
421	C₂²×C₈	—	53	1	no	—
422	C₂²×C₈	—	53	1	no	—
423	C₂²×C₈	—	53	1	no	—
424	C₂²×C₈	—	53	1	no	—
425	C₂²×C₈	—	53	1	no	—
426	C₂²×C₈	—	53	1	no	—
427	#123	—	2	4	no	—
428	C₄×D₄	—	2	2	no	—
429	C₄×D₄	—	2	2	no	—
430	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
435	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
436	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
438	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
439	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
440	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
441	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
442	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
443	C₂×C₄²	—	2	2	no	—
444	C₂×C₄²	—	2	2	no	—
445	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
446	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
447	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
448	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
449	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
450	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
451	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
452	C₄⋊C₈	—	106	2	no	—
453	C₄⋊C₈	—	106	2	no	—
454	C₄⋊C₈	—	106	2	no	—
455	C₂²⋊C₈	—	106	2	no	—
456	C₂²⋊C₈	—	106	2	no	—
457	C₂²⋊C₈	—	106	2	no	—
458	C₂²⋊C₈	—	106	2	no	—
459	C₂²⋊C₈	—	106	2	no	—
460	#97	—	106	2	no	—
461	#97	—	106	2	no	—
462	#97	—	106	2	no	—
463	C₄×C₈	—	106	2	no	—
464	C₄×C₈	—	106	2	no	—
465	C₄×C₈	—	106	2	no	—
466	C₄×C₈	—	106	2	no	—
467	#95	C₂×Dic₅₃	1	1	yes	characteristic
468	#95	C₂×Dic₅₃	1	1	yes	characteristic
469	#95	C₂×Dic₅₃	1	1	yes	characteristic
470	#95	C₂×Dic₅₃	1	1	yes	characteristic
471	#95	C₂×Dic₅₃	1	1	yes	characteristic
472	#95	C₂×Dic₅₃	1	1	yes	characteristic
473	#95	C₂×Dic₅₃	1	1	yes	characteristic
474	#95	C₂×Dic₅₃	1	1	yes	characteristic
475	#95	C₂×Dic₅₃	1	1	yes	characteristic
476	#95	C₂×Dic₅₃	1	1	yes	characteristic
477	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
478	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	4	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	4	1	no	—
497	C₂²×C₄	—	4	1	no	—
498	C₂²×C₄	D₄×D₅₃	1	2	yes	—
499	C₂²×C₄	#11719947	1	2	yes	—
500	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
501	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
502	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
503	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
504	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
505	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
506	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
507	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
508	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
509	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
510	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
511	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
512	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
513	M₁₆	—	106	8	no	—
514	M₁₆	—	106	8	no	—
515	M₁₆	—	212	4	no	—
516	C₂×C₈	—	53	2	no	—
517	C₂×C₈	—	53	2	no	—
518	C₂×C₈	—	53	2	no	—
519	C₂×C₈	—	53	2	no	—
520	C₂×C₈	—	53	2	no	—
521	C₂×C₈	—	53	2	no	—
522	C₂×C₈	—	106	1	no	—
523	C₂×C₈	—	106	1	no	—
524	C₂×C₈	—	106	1	no	—
525	C₂×C₈	—	106	1	no	—
526	C₂×C₈	—	106	1	no	—
527	C₂×C₈	—	106	1	no	—
528	C₂×C₈	—	106	1	no	—
529	C₂×C₈	—	106	1	no	—
530	C₂×C₈	—	106	1	no	—
531	C₂×C₈	—	106	1	no	—
532	C₂×C₈	—	106	1	no	—
533	C₂×C₈	—	106	1	no	—
534	C₂×C₈	—	106	1	no	—
535	C₂×C₈	—	106	1	no	—
536	C₂×C₈	—	106	1	no	—
537	C₂×C₈	—	106	1	no	—
538	C₂×C₈	—	106	1	no	—
539	C₂×C₈	—	106	1	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	—	4	1	no	—
544	C₄⋊C₄	#11719917	1	4	yes	complemented
545	C₂²⋊C₄	—	2	2	no	—
546	C₂²⋊C₄	—	2	2	no	—
547	C₂²⋊C₄	—	2	2	no	—
548	C₂²⋊C₄	—	2	2	no	—
549	C₂²⋊C₄	—	2	2	no	—
550	C₂²⋊C₄	—	2	2	no	—
551	C₂²⋊C₄	—	4	1	no	—
552	C₂²⋊C₄	—	4	1	no	—
553	C₄²	—	2	2	no	—
554	C₄²	—	2	2	no	—
555	C₄²	—	2	1	no	—
556	C₄²	—	2	1	no	—
557	C₄²	—	2	1	no	—
558	C₄²	—	2	1	no	—
559	C₄²	—	2	1	no	—
560	C₄²	—	2	1	no	—
561	C₄²	—	2	1	no	—
562	C₄²	—	2	1	no	—
563	C₄²	—	2	1	no	—
564	C₄²	—	2	1	no	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	1	no	—
568	C₂³	—	2	1	no	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	1	no	—
571	C₂³	—	2	1	no	—
572	C₂³	—	2	1	no	—
573	C₂³	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	—	2	1	no	—
576	C₂³	—	2	1	no	—
577	C₂³	—	2	1	no	—
578	C₂³	D₄×Dic₅₃	1	2	yes	—
579	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
580	D₄	—	4	4	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	4	4	no	—
617	C₂×C₄	—	4	4	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
623	C₂×C₄	C₂×#11719917	1	2	yes	—
624	C₂×C₄	C₂×#11719917	1	2	yes	—
625	C₂×C₄	C₂×#11719917	1	2	yes	—
626	C₂×C₄	#13133224	1	2	yes	—
627	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
628	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
629	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
630	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
631	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
632	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
633	C₈	—	53	4	no	—
634	C₈	—	106	2	no	—
635	C₈	—	106	2	no	—
636	C₈	—	106	2	no	—
637	C₈	—	212	1	no	—
638	C₈	—	212	1	no	—
639	C₈	—	212	1	no	—
640	C₈	—	212	1	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	2	2	no	—
643	C₂²	—	2	2	no	—
644	C₂²	—	4	1	no	—
645	C₂²	—	4	1	no	—
646	C₂²	—	4	1	no	—
647	C₂²	—	4	1	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	1	no	—
651	C₂²	—	4	1	no	—
652	C₂²	—	4	1	no	—
653	C₂²	—	4	1	no	—
654	C₂²	—	4	1	no	—
655	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
656	C₂²	#41×Dic₅₃	1	1	yes	characteristic
657	C₂²	#41×Dic₅₃	1	1	yes	characteristic
658	C₂²	C₂²×#11719917	1	1	yes	characteristic
659	C₂²	C₂×#13133224	1	1	yes	characteristic
660	C₂²	#16811390	1	1	yes	characteristic
661	C₂²	#16811390	1	1	yes	characteristic
662	C₄	—	2	2	no	—
663	C₄	—	2	2	no	—
664	C₄	—	2	2	no	—
665	C₄	—	2	2	no	—
666	C₄	—	2	2	no	—
667	C₄	—	4	1	no	—
668	C₄	—	4	1	no	—
669	C₄	—	4	1	no	—
670	C₄	—	4	1	no	—
671	C₄	—	4	1	no	—
672	C₄	—	4	1	no	—
673	C₄	—	4	1	no	—
674	C₄	#123×D₅₃	1	1	yes	characteristic
675	C₄	#16811131	1	1	yes	characteristic
676	C₄	#16811150	1	1	yes	characteristic
677	C₄	#16811150	1	1	yes	characteristic
678	C₂	—	4	2	no	—
679	C₂	—	8	1	no	—
680	C₂	#123×Dic₅₃	1	1	yes	characteristic
681	C₂	#20283442	1	1	yes	characteristic
682	C₂	D₄×#11719917	1	1	yes	characteristic
683	C₂	#20283682	1	1	yes	characteristic
684	C₂	#20283702	1	1	yes	characteristic
685	C₂	#20283720	1	1	yes	characteristic
686	C₂	#20283735	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₄×#123×C₅₃	C₂
prime radical	C₂²×D₄×C₅₃	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#123	D₅₃
hypocenter	C₅₃	#21565
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#123

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11719917	C₂	semidirect
3. #20275330	C₂	semidirect
4. #20275323	C₂	semidirect
5. #20275281	C₂	semidirect
6. #20275274	C₂	semidirect
8. #20274900	C₂	semidirect
9. #20274783	C₂	semidirect
10. #20274590	C₂	semidirect
11. C₄⋊C₄×#88322	C₂	semidirect
12. #20274499	C₂	semidirect
13. #20274499	C₂	semidirect
14. #20274499	C₂	semidirect
173. C₄⋊C₄×C₅₃	M₁₆	semidirect
296. #88322	#123	semidirect
404. C₅₃	#21565	semidirect
544. C₄⋊C₄	#11719917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810701	2
C₂×C₄×#11719917	2
#20275330	2
#20275323	2
#20275281	2
#20275274	2
#20275013	2
#20274900	2
#20274783	2
#20274590	2
C₄⋊C₄×#88322	2
#20274499	2
C₄×#123×C₅₃	2
#21565	53