Subgroups — #37582736

Order 24448.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133275	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133143	C₂	1	1	yes	maximal, characteristic
3	#27132999	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132994	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132971	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132964	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132874	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132869	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132827	C₂	1	1	yes	maximal, characteristic
10	#27132827	C₂	1	1	yes	maximal, characteristic
11	#27132771	C₂	1	1	yes	maximal, characteristic, complemented
12	#27132765	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₇₆₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#15690276	C₂	1	1	yes	maximal, characteristic, complemented
15	#433×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15690287	C₂²	1	1	yes	characteristic
17	C₂×Dic₇₆₄	C₂²	1	1	yes	characteristic
18	C₂×Dic₇₆₄	C₂²	1	1	yes	characteristic
19	C₂×#15690276	C₂²	1	1	yes	characteristic
20	C₂×#15690276	C₂²	1	1	yes	characteristic
21	#20259322	C₂²	1	1	yes	characteristic
22	#20259315	—	2	1	no	—
23	#20259315	—	2	1	no	—
24	#20259314	C₂²	1	1	yes	characteristic
25	#20259312	—	2	1	no	—
26	#20259312	—	2	1	no	—
27	#20259301	C₂²	1	1	yes	characteristic
28	#20259301	C₂²	1	1	yes	characteristic
29	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
30	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
31	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
32	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
33	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
34	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
35	(C₂²⋊D₁₉₁)⋊C₄	—	2	1	no	—
36	(C₂²⋊D₁₉₁)⋊C₄	—	2	1	no	—
37	#20259280	—	2	1	no	—
38	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
39	C₄⋊C₄×D₁₉₁	—	2	1	no	—
40	#20259256	C₂²	1	1	yes	characteristic
41	#20259256	C₂²	1	1	yes	characteristic
42	#20259254	C₂²	1	1	yes	characteristic
43	#20259254	C₂²	1	1	yes	characteristic
44	#20259247	C₂²	1	1	yes	characteristic
45	#20259247	C₂²	1	1	yes	characteristic
46	#20259247	C₂²	1	1	yes	characteristic
47	#20259247	C₂²	1	1	yes	characteristic
48	#20259247	C₂²	1	1	yes	characteristic
49	#20259247	C₂²	1	1	yes	characteristic
50	#20259244	C₂²	1	1	yes	characteristic
51	#20259244	C₂²	1	1	yes	characteristic
52	#117×C₁₉₁	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
57	#97×C₁₉₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
59	#15690287	—	2	2	no	—
60	#15690287	—	2	2	no	—
61	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
62	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
63	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
64	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
66	C₂²×Dic₁₉₁	—	2	1	no	—
67	C₂×C₄×D₁₉₁	—	2	1	no	—
68	C₂×C₄×D₁₉₁	—	2	1	no	—
69	C₂×C₄×D₁₉₁	—	2	1	no	—
70	Dic₇₆₄	—	2	2	no	—
71	Dic₇₆₄	—	2	2	no	—
72	#15690276	—	2	2	no	—
73	#15690276	—	2	2	no	—
74	C₄⋊Dic₁₉₁	—	2	1	no	—
75	C₄⋊Dic₁₉₁	—	2	1	no	—
76	C₄⋊Dic₁₉₁	—	2	1	no	—
77	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
81	Dic₁₉₁⋊C₄	—	2	1	no	—
82	Dic₁₉₁⋊C₄	—	2	1	no	—
83	Dic₁₉₁⋊C₄	—	2	1	no	—
84	Dic₁₉₁⋊C₄	—	2	1	no	—
85	Dic₁₉₁⋊C₄	—	2	1	no	—
86	Dic₁₉₁⋊C₄	—	2	1	no	—
87	Dic₁₉₁⋊C₄	—	2	1	no	—
88	Dic₁₉₁⋊C₄	—	2	1	no	—
89	C₂²⋊Dic₁₉₁	—	2	1	no	—
90	C₂²⋊Dic₁₉₁	—	2	1	no	—
91	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
92	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
93	C₄×Dic₁₉₁	—	2	1	no	—
94	C₄×Dic₁₉₁	—	2	1	no	—
95	C₄×Dic₁₉₁	—	2	1	no	—
96	C₄×Dic₁₉₁	—	2	1	no	—
97	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₉₁	D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₉₁	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₉₁	D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₉₁	D₄	1	1	yes	characteristic
106	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
107	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₉₁	—	2	1	no	—
109	Dic₃₈₂	—	2	1	no	—
110	Dic₃₈₂	—	2	1	no	—
111	Dic₃₈₂	—	2	1	no	—
112	Dic₃₈₂	—	2	1	no	—
113	Dic₃₈₂	—	2	1	no	—
114	Dic₃₈₂	#41	1	2	yes	—
115	D₇₆₄	—	2	1	no	—
116	D₇₆₄	#41	1	2	yes	—
117	C₂²⋊D₁₉₁	—	2	4	no	—
118	C₂×Dic₁₉₁	—	2	1	no	—
119	C₂×Dic₁₉₁	—	2	1	no	—
120	C₂×Dic₁₉₁	—	2	1	no	—
121	C₂×Dic₁₉₁	—	2	1	no	—
122	C₂×Dic₁₉₁	—	2	1	no	—
123	C₂×Dic₁₉₁	—	2	1	no	—
124	C₂×Dic₁₉₁	—	2	1	no	—
125	C₂×Dic₁₉₁	—	8	1	no	—
126	C₄×D₁₉₁	—	2	4	no	—
127	C₄×D₁₉₁	—	8	1	no	—
128	C₄×D₁₉₁	—	8	1	no	—
129	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉₁	—	2	1	no	—
131	C₂×C₄×C₁₉₁	—	2	1	no	—
132	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
138	C₈×C₁₉₁	—	2	1	no	—
139	C₈×C₁₉₁	—	2	1	no	—
140	C₈×C₁₉₁	—	2	1	no	—
141	C₈×C₁₉₁	—	2	1	no	—
142	C₂²×C₁₉₁	—	2	1	no	—
143	C₂²×C₁₉₁	—	2	1	no	—
144	C₂²×C₁₉₁	—	2	1	no	—
145	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₉₁	—	2	2	no	—
147	C₂×D₁₉₁	—	4	1	no	—
148	C₂×D₁₉₁	—	4	1	no	—
149	C₄×C₁₉₁	—	2	1	no	—
150	C₄×C₁₉₁	—	2	1	no	—
151	C₄×C₁₉₁	—	2	1	no	—
152	C₄×C₁₉₁	—	2	1	no	—
153	C₄×C₁₉₁	—	2	1	no	—
154	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
155	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
156	Dic₁₉₁	—	2	2	no	—
157	Dic₁₉₁	—	4	1	no	—
158	Dic₁₉₁	—	4	1	no	—
159	Dic₁₉₁	—	4	1	no	—
160	Dic₁₉₁	—	4	1	no	—
161	Dic₁₉₁	—	4	1	no	—
162	Dic₁₉₁	—	4	1	no	—
163	C₂×C₁₉₁	—	4	1	no	—
164	C₂×C₁₉₁	#525	1	1	yes	characteristic
165	C₂×C₁₉₁	#577	1	1	yes	characteristic
166	C₂×C₁₉₁	#578	1	1	yes	characteristic
167	D₁₉₁	—	4	2	no	—
168	C₁₉₁	#3120	1	1	yes	characteristic, complemented, Sylow
169	#3120	—	191	1	no	maximal, Sylow
170	#541	—	191	1	no	—
171	#520	—	191	1	no	—
172	#484	—	191	1	no	—
173	#483	—	191	1	no	—
174	#479	—	191	1	no	—
175	#478	—	191	1	no	—
176	#462	—	191	1	no	—
177	#461	—	191	1	no	—
178	#452	—	191	1	no	—
179	#452	—	191	1	no	—
180	#441	—	191	1	no	—
181	#440	—	191	1	no	—
182	C₄×Q₁₆	—	191	1	no	—
183	C₄×SD₁₆	—	191	1	no	—
184	#433	D₁₉₁	1	1	yes	characteristic, complemented
185	C₂×#41	—	191	1	no	—
186	C₂×Q₁₆	—	191	1	no	—
187	C₂×Q₁₆	—	191	1	no	—
188	C₂×SD₁₆	—	191	1	no	—
189	C₂×SD₁₆	—	191	1	no	—
190	C₄⋊Q₈	—	191	1	no	—
191	#126	—	382	1	no	—
192	#126	—	382	1	no	—
193	#125	—	382	1	no	—
194	#125	—	382	1	no	—
195	#125	—	191	1	no	—
196	C₂²⋊Q₈	—	191	1	no	—
197	C₂²⋊Q₈	—	191	1	no	—
198	C₄×Q₈	—	191	1	no	—
199	C₄×Q₈	—	191	1	no	—
200	C₄×Q₈	—	191	1	no	—
201	C₄×Q₈	—	191	1	no	—
202	C₄×D₄	—	382	1	no	—
203	C₄×D₄	—	382	1	no	—
204	C₄×D₄	—	191	1	no	—
205	C₄×D₄	—	191	1	no	—
206	#117	—	382	1	no	—
207	#117	C₂×D₁₉₁	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	382	1	no	—
209	C₂×C₄⋊C₄	—	191	1	no	—
210	#107	—	191	1	no	—
211	#107	—	191	1	no	—
212	#106	—	191	1	no	—
213	#106	—	191	1	no	—
214	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
216	#103	—	191	1	no	—
217	#103	—	191	1	no	—
218	#103	—	191	1	no	—
219	#103	—	191	1	no	—
220	#103	—	191	1	no	—
221	#103	—	191	1	no	—
222	#102	—	191	1	no	—
223	#102	—	191	1	no	—
224	C₂²⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
225	C₂²⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
226	#97	C₂×D₁₉₁	1	1	yes	characteristic
227	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
228	#41	—	382	2	no	—
229	#41	—	382	2	no	—
230	C₂×Q₈	—	191	1	no	—
231	C₂×Q₈	—	191	1	no	—
232	C₂×Q₈	—	191	1	no	—
233	C₂×D₄	—	382	1	no	—
234	C₂×D₄	—	191	1	no	—
235	C₂²×C₄	—	382	1	no	—
236	C₂²×C₄	—	382	1	no	—
237	C₂²×C₄	—	382	1	no	—
238	C₂²×C₄	—	382	1	no	—
239	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
240	Q₁₆	—	382	2	no	—
241	Q₁₆	—	382	2	no	—
242	SD₁₆	—	382	2	no	—
243	SD₁₆	—	382	2	no	—
244	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
245	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
248	C₄⋊C₄	—	382	1	no	—
249	C₄⋊C₄	—	382	1	no	—
250	C₄⋊C₄	—	382	1	no	—
251	C₄⋊C₄	—	382	1	no	—
252	C₄⋊C₄	—	382	1	no	—
253	C₄⋊C₄	—	382	1	no	—
254	C₄⋊C₄	—	382	1	no	—
255	C₄⋊C₄	—	382	1	no	—
256	C₄⋊C₄	—	382	1	no	—
257	C₄⋊C₄	—	382	1	no	—
258	C₄⋊C₄	—	382	1	no	—
259	C₄⋊C₄	—	191	1	no	—
260	C₄⋊C₄	—	191	1	no	—
261	C₄⋊C₄	—	191	1	no	—
262	C₄⋊C₄	—	191	1	no	—
263	C₄⋊C₄	D₇₆₄	1	1	yes	characteristic
264	C₄⋊C₄	D₇₆₄	1	1	yes	characteristic
265	C₂²⋊C₄	—	382	1	no	—
266	C₂²⋊C₄	—	382	1	no	—
267	C₂²⋊C₄	—	382	1	no	—
268	C₂²⋊C₄	—	382	1	no	—
269	C₂²⋊C₄	D₇₆₄	1	1	yes	characteristic
270	C₂²⋊C₄	D₇₆₄	1	1	yes	characteristic
271	C₄²	—	382	1	no	—
272	C₄²	—	382	1	no	—
273	C₄²	—	382	1	no	—
274	C₄²	—	382	1	no	—
275	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
276	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
277	C₂³	—	382	1	no	—
278	C₂³	C₂×D₇₆₄	1	1	yes	characteristic
279	Q₈	—	382	1	no	—
280	Q₈	—	382	1	no	—
281	Q₈	—	382	1	no	—
282	Q₈	—	382	1	no	—
283	Q₈	—	382	1	no	—
284	Q₈	—	191	2	no	—
285	D₄	—	382	4	no	—
286	D₄	—	382	1	no	—
287	D₄	—	191	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	382	4	no	—
291	C₂×C₄	—	382	1	no	—
292	C₂×C₄	—	382	1	no	—
293	C₂×C₄	—	382	1	no	—
294	C₂×C₄	—	382	1	no	—
295	C₂×C₄	—	382	1	no	—
296	C₂×C₄	—	382	1	no	—
297	C₂×C₄	—	382	1	no	—
298	C₂×C₄	—	1528	1	no	—
299	C₂×C₄	—	1528	1	no	—
300	C₂×C₄	—	1528	1	no	—
301	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
302	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
303	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
304	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
305	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	382	2	no	—
315	C₂²	—	764	1	no	—
316	C₂²	—	764	1	no	—
317	C₂²	C₂²×D₇₆₄	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	382	2	no	—
324	C₄	—	764	1	no	—
325	C₄	—	764	1	no	—
326	C₄	—	764	1	no	—
327	C₄	—	764	1	no	—
328	C₄	—	764	1	no	—
329	C₄	—	764	1	no	—
330	C₄	#41×D₁₉₁	1	1	yes	characteristic
331	C₄	#41×D₁₉₁	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	764	2	no	—
334	C₂	#27133175	1	1	yes	characteristic
335	C₂	#27133475	1	1	yes	characteristic
336	C₂	#27133480	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₆₄
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#433×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂×#15690287	C₂²
hypercenter	#433	D₁₉₁
hypocenter	C₁₉₁	#3120
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
1. #27133275	C₂	semidirect
3. #27132999	C₂	semidirect
4. #27132994	C₂	semidirect
5. #27132971	C₂	semidirect
6. #27132964	C₂	semidirect
7. #27132874	C₂	semidirect
8. #27132869	C₂	semidirect
11. #27132771	C₂	semidirect
12. #27132765	C₂	semidirect
13. C₄×Dic₇₆₄	C₂	semidirect
14. C₄×#15690276	C₂	semidirect
15. #433×C₁₉₁	C₂	semidirect
168. C₁₉₁	#3120	semidirect
184. #433	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133275	2
#27133143	2
#27132999	2
#27132994	2
#27132971	2
#27132964	2
#27132874	2
#27132869	2
#27132827	2
#27132771	2
#27132765	2
C₄×Dic₇₆₄	2
C₄×#15690276	2
#433×C₁₉₁	2
#3120	191