Subgroups — #28319498

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291193	C₂	1	2	yes	maximal
2	#20290653	C₂	1	2	yes	maximal, complemented
3	#20290637	C₂	1	1	yes	maximal, characteristic, complemented
4	#20290636	C₂	1	1	yes	maximal, characteristic, complemented
5	#20290593	C₂	1	2	yes	maximal
6	#20290592	C₂	1	1	yes	maximal, characteristic
7	#20290591	C₂	1	1	yes	maximal, characteristic
8	#20290529	C₂	1	2	yes	maximal, complemented
9	C₄×#13133286	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13133285	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×D₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133286	C₂²	1	2	yes	—
13	C₂×#13133286	C₂²	1	2	yes	—
14	C₂×#13133285	C₂²	1	2	yes	—
15	C₂×#13133285	C₂²	1	2	yes	—
16	#16811573	—	2	2	no	—
17	#16811573	—	2	2	no	—
18	#16811572	—	2	2	no	—
19	#16811572	—	2	2	no	—
20	#16811569	C₂²	1	2	yes	—
21	#16811567	C₂²	1	2	yes	—
22	#16811565	C₂²	1	2	yes	—
23	#16811565	C₂²	1	2	yes	—
24	C₄×C₄⋊#6157	C₂²	1	2	yes	—
25	C₄×C₄⋊#6157	C₂²	1	2	yes	—
26	#16811497	C₂²	1	1	yes	characteristic
27	#16811497	C₂²	1	1	yes	characteristic
28	#16811497	C₂²	1	1	yes	characteristic
29	#16811497	C₂²	1	1	yes	characteristic
30	#16811497	C₂²	1	2	yes	—
31	#16811497	C₂²	1	2	yes	—
32	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	#117×D₅₃	C₂²	1	1	yes	characteristic
35	#16810581	C₄	1	1	yes	characteristic, complemented
36	#16810580	C₄	1	1	yes	characteristic, complemented
37	#16810580	C₄	1	1	yes	characteristic, complemented
38	#16810579	C₄	1	1	yes	characteristic, complemented
39	#16810578	C₄	1	1	yes	characteristic, complemented
40	#16810577	C₄	1	1	yes	characteristic, complemented
41	#16810577	C₄	1	1	yes	characteristic, complemented
42	#97×D₅₃	C₂²	1	1	yes	characteristic
43	#97×D₅₃	C₂²	1	1	yes	characteristic
44	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
45	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
46	C₅₃×#405	C₄	1	1	yes	characteristic, complemented
47	C₂×C₄⋊#6157	—	2	2	no	—
48	C₂×C₄⋊#6157	—	2	2	no	—
49	C₂×C₄⋊#6157	C₂³	1	2	yes	—
50	C₂×C₄⋊#6157	C₂³	1	2	yes	—
51	C₂×C₄⋊#6157	C₂³	1	2	yes	—
52	C₂×C₄⋊#6157	C₂³	1	2	yes	—
53	C₂×C₂²⋊#6157	—	2	2	no	—
54	C₂×C₂²⋊#6157	—	2	2	no	—
55	C₂×C₄×#6157	—	2	2	no	—
56	C₂×C₄×#6157	—	2	2	no	—
57	#13133286	—	2	4	no	—
58	#13133286	—	2	4	no	—
59	#13133285	—	2	4	no	—
60	#13133285	—	2	4	no	—
61	(C₂×#6157)⋊C₄	—	2	2	no	—
62	(C₂×#6157)⋊C₄	—	2	2	no	—
63	(C₂×#6157)⋊C₄	—	2	2	no	—
64	(C₂×#6157)⋊C₄	—	2	2	no	—
65	(C₂×#6157)⋊C₄	—	2	2	no	—
66	(C₂×#6157)⋊C₄	—	2	2	no	—
67	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
68	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
69	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
70	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
71	M₁₆×D₅₃	—	2	4	no	—
72	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
73	C₂×C₈×D₅₃	—	2	4	no	—
74	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
79	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
80	#13133176	C₂×C₄	1	1	yes	characteristic
81	#13133175	Q₈	1	1	yes	characteristic
82	#13133175	Q₈	1	1	yes	characteristic
83	#13133174	C₂×C₄	1	1	yes	characteristic
84	#13133174	C₂×C₄	1	1	yes	characteristic
85	#13133173	Q₈	1	1	yes	characteristic
86	#13133173	Q₈	1	1	yes	characteristic
87	C₄⋊C₄×D₅₃	D₄	1	1	yes	characteristic
88	C₄⋊C₄×D₅₃	D₄	1	1	yes	characteristic
89	C₂²⋊C₄×D₅₃	D₄	1	1	yes	characteristic
90	C₂²⋊C₄×D₅₃	D₄	1	1	yes	characteristic
91	C₄²×D₅₃	C₂³	1	1	yes	characteristic
92	C₄²×D₅₃	C₂³	1	1	yes	characteristic
93	#13133128	C₂×C₄	1	1	yes	characteristic
94	#13133128	C₂×C₄	1	1	yes	characteristic
95	#13133127	C₂×C₄	1	1	yes	characteristic
96	#13133127	C₂×C₄	1	1	yes	characteristic
97	#13133127	C₂×C₄	1	1	yes	characteristic
98	#13133127	C₂×C₄	1	1	yes	characteristic
99	C₄×#88322	C₂×C₄	1	1	yes	characteristic
100	C₄×#88322	C₂×C₄	1	1	yes	characteristic
101	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
102	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
103	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
104	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
105	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
106	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
107	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
108	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
109	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
112	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×#6157	—	2	2	no	—
114	C₂²×#6157	—	2	2	no	—
115	C₂²×#6157	—	2	2	no	—
116	C₂²×#6157	—	2	2	no	—
117	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₄×D₅₃	—	2	2	no	—
120	C₂×C₄×D₅₃	—	2	2	no	—
121	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
128	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
129	C₂²⋊#6157	—	8	2	no	—
130	C₂²⋊#6157	—	8	2	no	—
131	C₄⋊#6157	—	2	2	no	—
132	C₄⋊#6157	—	2	2	no	—
133	C₄⋊#6157	—	2	2	no	—
134	C₄⋊#6157	—	2	2	no	—
135	C₄⋊#6157	—	2	2	no	—
136	C₄⋊#6157	—	2	2	no	—
137	C₄⋊#6157	—	2	2	no	—
138	C₄⋊#6157	—	2	2	no	—
139	C₄⋊#6157	—	8	2	no	—
140	C₄⋊#6157	—	8	2	no	—
141	C₄×#6157	—	8	2	no	—
142	C₄×#6157	—	8	2	no	—
143	M₁₆×C₅₃	—	2	2	no	—
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
150	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
151	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
152	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
153	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
156	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
157	#11719921	C₄⋊C₄	1	1	yes	characteristic
158	#11719921	C₄⋊C₄	1	1	yes	characteristic
159	#11719921	C₄⋊C₄	1	1	yes	characteristic
160	#11719921	C₄⋊C₄	1	1	yes	characteristic
161	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
162	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
163	#11719919	C₄⋊C₄	1	1	yes	characteristic
164	#11719919	C₄⋊C₄	1	1	yes	characteristic
165	#11719919	C₄⋊C₄	1	1	yes	characteristic
166	#11719919	C₄⋊C₄	1	1	yes	characteristic
167	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
168	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
169	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
170	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
171	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
172	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
173	#11719917	—	2	2	no	—
174	C₂×#88322	—	2	2	no	—
175	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
176	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
177	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
178	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
179	#11719912	—	2	2	no	—
180	#11719912	—	2	2	no	—
181	C₈×D₅₃	—	2	2	no	—
182	C₈×D₅₃	—	2	2	no	—
183	C₈×D₅₃	—	2	2	no	—
184	C₈×D₅₃	—	2	2	no	—
185	C₈×D₅₃	#41	1	4	yes	—
186	C₈×D₅₃	#41	1	4	yes	—
187	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂²×D₅₃	—	2	2	no	—
189	C₂²×D₅₃	—	2	1	no	—
190	C₂²×D₅₃	—	2	2	no	—
191	C₂²×D₅₃	—	2	1	no	—
192	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
194	C₂×#6157	—	4	2	no	—
195	C₂×#6157	—	4	2	no	—
196	C₂×#6157	—	4	2	no	—
197	C₂×#6157	—	4	2	no	—
198	C₂×#6157	—	4	2	no	—
199	C₂×#6157	—	4	2	no	—
200	C₂×#6157	—	4	2	no	—
201	C₂×#6157	—	4	2	no	—
202	C₂×#6157	—	4	2	no	—
203	C₂×#6157	—	4	2	no	—
204	C₂×#6157	—	4	2	no	—
205	C₂×#6157	—	4	2	no	—
206	C₂×C₄×C₅₃	—	2	1	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
214	C₂×Dic₅₃	—	2	2	no	—
215	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
216	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
217	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
218	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
219	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
220	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
221	C₄×D₅₃	—	2	2	no	—
222	C₄×D₅₃	—	2	1	no	—
223	C₄×D₅₃	—	2	1	no	—
224	C₄×D₅₃	—	2	2	no	—
225	C₄×D₅₃	—	2	2	no	—
226	C₄×D₅₃	—	2	1	no	—
227	C₄×D₅₃	—	2	1	no	—
228	C₄×D₅₃	—	2	1	no	—
229	C₄×D₅₃	—	2	1	no	—
230	C₄×D₅₃	—	2	2	no	—
231	C₄×D₅₃	C₂×#41	1	2	yes	—
232	C₄×D₅₃	C₂×#41	1	2	yes	—
233	C₈×C₅₃	—	2	1	no	—
234	C₈×C₅₃	—	2	1	no	—
235	C₈×C₅₃	#117	1	2	yes	—
236	C₈×C₅₃	#117	1	2	yes	—
237	#88322	—	2	1	no	—
238	#88322	—	2	1	no	—
239	#88322	#117	1	2	yes	—
240	#88322	#117	1	2	yes	—
241	#20822	—	53	1	no	maximal, Sylow
242	C₂²×C₅₃	—	2	1	no	—
243	C₂²×C₅₃	—	2	1	no	—
244	C₂²×C₅₃	—	2	1	no	—
245	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
246	C₂×D₅₃	—	2	1	no	—
247	C₂×D₅₃	—	2	1	no	—
248	C₂×D₅₃	—	2	1	no	—
249	C₂×D₅₃	—	4	2	no	—
250	C₂×D₅₃	—	4	2	no	—
251	C₂×D₅₃	#532	1	2	yes	—
252	C₂×D₅₃	#576	1	1	yes	characteristic
253	C₂×D₅₃	#577	1	2	yes	—
254	C₂×D₅₃	#578	1	1	yes	characteristic
255	#6157	—	8	2	no	—
256	#6157	—	8	2	no	—
257	#6157	—	8	2	no	—
258	#6157	—	8	2	no	—
259	C₄×C₅₃	—	2	1	no	—
260	C₄×C₅₃	—	2	1	no	—
261	C₄×C₅₃	—	2	1	no	—
262	C₄×C₅₃	—	2	1	no	—
263	C₄×C₅₃	—	2	1	no	—
264	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
265	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
266	Dic₅₃	—	2	1	no	—
267	Dic₅₃	—	2	1	no	—
268	Dic₅₃	—	2	1	no	—
269	Dic₅₃	—	2	1	no	—
270	Dic₅₃	—	2	1	no	—
271	Dic₅₃	#533	1	2	yes	—
272	C₂×#405	—	53	1	no	—
273	#2314	—	53	2	no	—
274	#1847	—	53	2	no	—
275	#1835	—	53	1	no	—
276	#1834	—	53	1	no	—
277	#1809	—	53	2	no	—
278	#1808	—	53	1	no	—
279	#1807	—	53	1	no	—
280	#1776	—	53	2	no	—
281	C₄×#107	—	53	1	no	—
282	C₄×#106	—	53	1	no	—
283	C₂×C₅₃	—	4	1	no	—
284	C₂×C₅₃	#2285	1	1	yes	characteristic
285	C₂×C₅₃	#2912	1	1	yes	characteristic
286	C₂×C₅₃	#2913	1	1	yes	characteristic
287	D₅₃	—	4	1	no	—
288	D₅₃	#3178	1	2	yes	—
289	D₅₃	#3179	1	2	yes	—
290	C₂²×M₁₆	—	53	1	no	—
291	C₂³×C₈	—	53	1	no	—
292	C₂×#117	—	53	1	no	—
293	C₂×#107	—	53	2	no	—
294	C₂×#107	—	53	2	no	—
295	C₂×#106	—	53	2	no	—
296	C₂×#106	—	53	2	no	—
297	#405	—	53	1	no	—
298	#405	—	53	1	no	—
299	#405	—	53	1	no	—
300	#405	—	53	1	no	—
301	#405	—	53	1	no	—
302	#405	—	53	1	no	—
303	#405	—	53	1	no	—
304	#405	#6157	1	1	yes	characteristic, complemented
305	C₂×#97	—	53	1	no	—
306	C₂×#97	—	53	1	no	—
307	C₂×C₄×C₈	—	53	1	no	—
308	C₂×C₄×C₈	—	53	1	no	—
309	#388	—	106	2	no	—
310	#388	—	106	2	no	—
311	#387	—	106	2	no	—
312	#387	—	106	2	no	—
313	#384	—	53	2	no	—
314	#382	—	53	2	no	—
315	#380	—	53	2	no	—
316	#380	—	53	2	no	—
317	C₄×C₄⋊C₄	—	53	2	no	—
318	C₄×C₄⋊C₄	—	53	2	no	—
319	#340	—	53	1	no	—
320	#340	—	53	1	no	—
321	#340	—	53	1	no	—
322	#340	—	53	1	no	—
323	#340	—	53	2	no	—
324	#340	—	53	2	no	—
325	C₅₃	#20822	1	1	yes	characteristic, complemented, Sylow
326	C₂³×C₄	—	53	1	no	—
327	C₂×M₁₆	—	53	1	no	—
328	C₂×M₁₆	—	53	1	no	—
329	C₂×M₁₆	—	53	1	no	—
330	C₂×M₁₆	—	106	4	no	—
331	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
332	C₂²×C₈	—	53	1	no	—
333	C₂²×C₈	—	53	1	no	—
334	C₂²×C₈	—	53	1	no	—
335	C₂²×C₈	—	53	1	no	—
336	C₂²×C₈	—	53	1	no	—
337	C₂²×C₈	—	53	1	no	—
338	C₂²×C₈	—	53	1	no	—
339	C₂²×C₈	—	106	4	no	—
340	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
341	#117	—	53	1	no	—
342	#117	—	53	1	no	—
343	#117	—	53	1	no	—
344	#117	—	53	1	no	—
345	#117	—	53	1	no	—
346	#117	—	53	1	no	—
347	#117	—	53	1	no	—
348	#117	C₂×#6157	1	1	yes	characteristic
349	C₂×C₄⋊C₄	—	53	1	no	—
350	C₂×C₄⋊C₄	—	53	1	no	—
351	C₂×C₄⋊C₄	—	53	2	no	—
352	C₂×C₄⋊C₄	—	53	2	no	—
353	C₂×C₄⋊C₄	—	53	2	no	—
354	C₂×C₄⋊C₄	—	53	2	no	—
355	C₂×C₄⋊C₄	—	106	2	no	—
356	C₂×C₄⋊C₄	—	106	2	no	—
357	C₂×C₂²⋊C₄	—	53	1	no	—
358	C₂×C₂²⋊C₄	—	53	1	no	—
359	C₂×C₂²⋊C₄	—	106	2	no	—
360	C₂×C₂²⋊C₄	—	106	2	no	—
361	C₂×C₄²	—	53	1	no	—
362	C₂×C₄²	—	53	1	no	—
363	C₂×C₄²	—	106	2	no	—
364	C₂×C₄²	—	106	2	no	—
365	#107	—	106	4	no	—
366	#107	—	106	4	no	—
367	#106	—	106	4	no	—
368	#106	—	106	4	no	—
369	#97	—	53	1	no	—
370	#97	—	53	1	no	—
371	#97	—	53	1	no	—
372	#97	—	53	1	no	—
373	#97	—	53	1	no	—
374	#97	—	53	1	no	—
375	#97	C₂×#6157	1	1	yes	characteristic
376	#97	C₂×#6157	1	1	yes	characteristic
377	C₄×C₈	—	53	1	no	—
378	C₄×C₈	—	53	1	no	—
379	C₄×C₈	—	53	1	no	—
380	C₄×C₈	—	53	1	no	—
381	C₄×C₈	—	53	1	no	—
382	C₄×C₈	—	53	1	no	—
383	C₄×C₈	C₂×#6157	1	1	yes	characteristic
384	C₄×C₈	C₂×#6157	1	1	yes	characteristic
385	#95	—	106	2	no	—
386	#95	—	106	2	no	—
387	#95	—	106	2	no	—
388	#95	—	106	2	no	—
389	#95	—	106	2	no	—
390	#95	—	106	2	no	—
391	C₂⁴	—	53	1	no	—
392	C₂²×C₄	—	53	1	no	—
393	C₂²×C₄	—	53	1	no	—
394	C₂²×C₄	—	53	1	no	—
395	C₂²×C₄	—	53	1	no	—
396	C₂²×C₄	—	53	1	no	—
397	C₂²×C₄	—	53	1	no	—
398	C₂²×C₄	—	53	1	no	—
399	C₂²×C₄	—	53	1	no	—
400	C₂²×C₄	—	53	1	no	—
401	C₂²×C₄	—	106	2	no	—
402	C₂²×C₄	—	106	2	no	—
403	C₂²×C₄	—	106	2	no	—
404	C₂²×C₄	—	106	2	no	—
405	C₂²×C₄	—	106	2	no	—
406	C₂²×C₄	—	106	2	no	—
407	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
408	M₁₆	—	2	2	no	—
409	M₁₆	—	106	2	no	—
410	M₁₆	—	106	2	no	—
411	M₁₆	—	106	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	53	1	no	—
414	C₂×C₈	—	53	1	no	—
415	C₂×C₈	—	53	4	no	—
416	C₂×C₈	—	53	4	no	—
417	C₂×C₈	—	53	1	no	—
418	C₂×C₈	—	53	1	no	—
419	C₂×C₈	—	106	2	no	—
420	C₂×C₈	—	106	2	no	—
421	C₂×C₈	—	106	2	no	—
422	C₂×C₈	—	106	2	no	—
423	C₂×C₈	—	106	2	no	—
424	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
425	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
426	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
427	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
428	C₄⋊C₄	—	53	1	no	—
429	C₄⋊C₄	—	53	1	no	—
430	C₄⋊C₄	—	53	1	no	—
431	C₄⋊C₄	—	53	1	no	—
432	C₄⋊C₄	—	53	1	no	—
433	C₄⋊C₄	—	53	1	no	—
434	C₄⋊C₄	—	106	2	no	—
435	C₄⋊C₄	—	106	2	no	—
436	C₄⋊C₄	—	106	2	no	—
437	C₄⋊C₄	—	106	2	no	—
438	C₄⋊C₄	—	106	2	no	—
439	C₄⋊C₄	—	106	2	no	—
440	C₄⋊C₄	—	106	2	no	—
441	C₄⋊C₄	—	106	2	no	—
442	C₄⋊C₄	—	424	2	no	—
443	C₄⋊C₄	—	424	2	no	—
444	C₄⋊C₄	C₄⋊#6157	1	1	yes	characteristic
445	C₄⋊C₄	C₄⋊#6157	1	1	yes	characteristic
446	C₂²⋊C₄	—	53	1	no	—
447	C₂²⋊C₄	—	53	1	no	—
448	C₂²⋊C₄	—	53	1	no	—
449	C₂²⋊C₄	—	53	1	no	—
450	C₂²⋊C₄	—	53	1	no	—
451	C₂²⋊C₄	—	53	1	no	—
452	C₂²⋊C₄	—	424	2	no	—
453	C₂²⋊C₄	—	424	2	no	—
454	C₂²⋊C₄	C₄⋊#6157	1	1	yes	characteristic
455	C₂²⋊C₄	C₄⋊#6157	1	1	yes	characteristic
456	C₄²	—	53	1	no	—
457	C₄²	—	53	1	no	—
458	C₄²	—	53	1	no	—
459	C₄²	—	53	1	no	—
460	C₄²	—	53	1	no	—
461	C₄²	—	53	1	no	—
462	C₄²	—	424	2	no	—
463	C₄²	—	424	2	no	—
464	C₄²	C₂²×#6157	1	1	yes	characteristic
465	C₄²	C₂²×#6157	1	1	yes	characteristic
466	C₂³	—	53	1	no	—
467	C₂³	—	53	1	no	—
468	C₂³	—	106	1	no	—
469	C₂³	—	106	2	no	—
470	C₂³	—	106	2	no	—
471	C₂³	—	106	1	no	—
472	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	53	1	no	—
476	C₂×C₄	—	53	2	no	—
477	C₂×C₄	—	53	2	no	—
478	C₂×C₄	—	53	1	no	—
479	C₂×C₄	—	53	1	no	—
480	C₂×C₄	—	53	1	no	—
481	C₂×C₄	—	53	1	no	—
482	C₂×C₄	—	53	1	no	—
483	C₂×C₄	—	106	2	no	—
484	C₂×C₄	—	106	1	no	—
485	C₂×C₄	—	106	2	no	—
486	C₂×C₄	—	106	1	no	—
487	C₂×C₄	—	106	2	no	—
488	C₂×C₄	—	106	1	no	—
489	C₂×C₄	—	106	1	no	—
490	C₂×C₄	—	106	2	no	—
491	C₂×C₄	—	106	2	no	—
492	C₂×C₄	—	106	1	no	—
493	C₂×C₄	—	106	1	no	—
494	C₂×C₄	—	212	2	no	—
495	C₂×C₄	—	212	2	no	—
496	C₂×C₄	—	212	2	no	—
497	C₂×C₄	—	212	2	no	—
498	C₂×C₄	—	212	2	no	—
499	C₂×C₄	—	212	2	no	—
500	C₂×C₄	—	212	2	no	—
501	C₂×C₄	—	212	2	no	—
502	C₂×C₄	—	212	2	no	—
503	C₂×C₄	—	212	2	no	—
504	C₂×C₄	—	212	2	no	—
505	C₂×C₄	—	212	2	no	—
506	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
507	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
508	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
509	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
510	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
511	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
512	C₈	—	2	1	no	—
513	C₈	—	2	1	no	—
514	C₈	—	53	2	no	—
515	C₈	—	53	2	no	—
516	C₈	—	106	1	no	—
517	C₈	—	106	1	no	—
518	C₈	#13133296	1	2	yes	—
519	C₈	#13133296	1	2	yes	—
520	C₂²	—	2	1	no	—
521	C₂²	—	2	1	no	—
522	C₂²	—	2	1	no	—
523	C₂²	—	53	2	no	—
524	C₂²	—	53	1	no	—
525	C₂²	—	53	1	no	—
526	C₂²	—	53	2	no	—
527	C₂²	—	106	1	no	—
528	C₂²	—	106	1	no	—
529	C₂²	—	106	1	no	—
530	C₂²	—	212	2	no	—
531	C₂²	—	212	2	no	—
532	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	2	1	no	—
536	C₄	—	2	1	no	—
537	C₄	—	2	1	no	—
538	C₄	—	53	2	no	—
539	C₄	—	106	1	no	—
540	C₄	—	106	1	no	—
541	C₄	—	106	1	no	—
542	C₄	—	106	1	no	—
543	C₄	—	106	1	no	—
544	C₄	—	424	2	no	—
545	C₄	—	424	2	no	—
546	C₄	—	424	2	no	—
547	C₄	—	424	2	no	—
548	C₄	C₂×#13133296	1	1	yes	characteristic
549	C₄	C₂×#13133296	1	1	yes	characteristic
550	C₂	—	4	1	no	—
551	C₂	—	53	2	no	—
552	C₂	—	53	2	no	—
553	C₂	—	212	1	no	—
554	C₂	#20291164	1	1	yes	characteristic
555	C₂	#20291369	1	1	yes	characteristic
556	C₂	#20291370	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#405	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#405	#6157
hypocenter	C₅₃	#20822
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20290653	C₂	semidirect
3. #20290637	C₂	semidirect
4. #20290636	C₂	semidirect
8. #20290529	C₂	semidirect
9. C₄×#13133286	C₂	semidirect
10. C₄×#13133285	C₂	semidirect
35. #16810581	C₄	semidirect
36. #16810580	C₄	semidirect
37. #16810580	C₄	semidirect
38. #16810579	C₄	semidirect
39. #16810578	C₄	semidirect
40. #16810577	C₄	semidirect
41. #16810577	C₄	semidirect
46. C₅₃×#405	C₄	semidirect
304. #405	#6157	semidirect
325. C₅₃	#20822	semidirect

Maximal subgroups

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

subgroup	index
#20291193	2
#20290653	2
#20290637	2
#20290636	2
#20290593	2
#20290592	2
#20290591	2
#20290529	2
C₄×#13133286	2
C₄×#13133285	2
#405×D₅₃	2
#20822	53