Subgroups — #37583110

Order 24448.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133208	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133194	C₂	1	1	yes	maximal, characteristic, complemented
3	#27133070	C₂	1	1	yes	maximal, characteristic, complemented
4	#27133051	C₂	1	1	yes	maximal, characteristic, complemented
5	#27133026	C₂	1	1	yes	maximal, characteristic, complemented
6	#27133021	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132914	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132909	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132907	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132754	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132750	C₂	1	1	yes	maximal, characteristic, complemented
12	D₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#27132743	C₂	1	1	yes	maximal, characteristic, complemented
14	M₁₆×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#469×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15690286	—	2	1	no	—
17	C₂×#15690276	C₂²	1	1	yes	characteristic
18	C₂×#15690275	C₂²	1	1	yes	characteristic
19	C₂×#15690272	C₂²	1	1	yes	characteristic
20	C₂×#15690271	C₂²	1	1	yes	characteristic
21	#20259321	C₂²	1	1	yes	characteristic
22	#20259318	C₂²	1	1	yes	characteristic
23	#20259315	—	2	1	no	—
24	#20259313	C₂²	1	1	yes	characteristic
25	#20259309	C₂²	1	1	yes	characteristic
26	#20259308	—	2	1	no	—
27	#20259303	—	2	1	no	—
28	#20259301	C₂²	1	1	yes	characteristic
29	#20259296	C₂²	1	1	yes	characteristic
30	#20259294	—	2	1	no	—
31	#20259287	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₉₁	—	2	1	no	—
33	D₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
34	D₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
35	#20259284	C₂²	1	1	yes	characteristic
36	(C₂²⋊D₁₉₁)⋊C₄	—	2	1	no	—
37	#20259278	—	2	1	no	—
38	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
39	#20259256	C₂²	1	1	yes	characteristic
40	#20259255	C₂²	1	1	yes	characteristic
41	#20259253	C₂²	1	1	yes	characteristic
42	#20259247	C₂²	1	1	yes	characteristic
43	#20259245	C₂²	1	1	yes	characteristic
44	#20259244	C₂²	1	1	yes	characteristic
45	#20259243	C₂²	1	1	yes	characteristic
46	#20259243	C₂²	1	1	yes	characteristic
47	#20259242	C₂²	1	1	yes	characteristic
48	#20259232	C₂²	1	1	yes	characteristic
49	#20259232	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₁₉₁	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
54	#121×C₁₉₁	C₂²	1	1	yes	characteristic
55	#121×C₁₉₁	C₂²	1	1	yes	characteristic
56	#106×C₁₉₁	C₂²	1	1	yes	characteristic
57	#102×C₁₉₁	C₂²	1	1	yes	characteristic
58	#102×C₁₉₁	C₂²	1	1	yes	characteristic
59	#15690286	—	4	4	no	—
60	C₂×Dic₃₈₂	—	2	1	no	—
61	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
62	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
64	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
65	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
66	C₂²×Dic₁₉₁	—	2	1	no	—
67	C₂²×Dic₁₉₁	—	2	1	no	—
68	C₂²×Dic₁₉₁	D₄	1	1	yes	characteristic
69	C₂²×Dic₁₉₁	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₉₁	—	2	1	no	—
71	#15690276	—	2	2	no	—
72	#15690275	—	2	2	no	—
73	#15690272	—	2	2	no	—
74	#15690271	—	2	2	no	—
75	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₉₁	—	2	1	no	—
78	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
80	Dic₁₉₁⋊C₄	—	2	1	no	—
81	Dic₁₉₁⋊C₄	—	2	1	no	—
82	Dic₁₉₁⋊C₄	—	2	1	no	—
83	Dic₁₉₁⋊C₄	—	2	1	no	—
84	C₂²⋊Dic₁₉₁	—	2	1	no	—
85	C₂²⋊Dic₁₉₁	—	2	1	no	—
86	C₂²⋊Dic₁₉₁	—	2	1	no	—
87	C₂²⋊Dic₁₉₁	—	2	1	no	—
88	C₂²⋊Dic₁₉₁	—	2	1	no	—
89	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
91	C₄×Dic₁₉₁	—	2	1	no	—
92	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
93	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
94	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
95	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₉₁	—	2	1	no	—
97	C₂×D₄×C₁₉₁	—	2	1	no	—
98	C₂×D₄×C₁₉₁	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₉₁	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
101	D₈×C₁₉₁	—	2	2	no	—
102	M₁₆×C₁₉₁	—	2	2	no	—
103	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
108	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
109	C₂²×D₁₉₁	—	2	1	no	—
110	Dic₃₈₂	—	2	1	no	—
111	Dic₃₈₂	—	2	1	no	—
112	Dic₃₈₂	—	4	2	no	—
113	D₇₆₄	—	2	1	no	—
114	D₇₆₄	—	2	1	no	—
115	C₂²⋊D₁₉₁	—	4	2	no	—
116	C₂²⋊D₁₉₁	—	4	2	no	—
117	C₂²⋊D₁₉₁	—	4	2	no	—
118	C₂×Dic₁₉₁	—	2	2	no	—
119	C₂×Dic₁₉₁	—	2	1	no	—
120	C₂×Dic₁₉₁	—	2	1	no	—
121	C₂×Dic₁₉₁	—	2	1	no	—
122	C₂×Dic₁₉₁	—	4	1	no	—
123	C₂×Dic₁₉₁	—	4	2	no	—
124	C₂×Dic₁₉₁	—	4	2	no	—
125	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
129	C₄×D₁₉₁	—	4	2	no	—
130	C₁₉₁⋊C₈	—	2	1	no	—
131	C₁₉₁⋊C₈	#41	1	2	yes	complemented
132	C₂³×C₁₉₁	—	2	1	no	—
133	C₂³×C₁₉₁	—	2	1	no	—
134	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
135	D₄×C₁₉₁	—	2	1	no	—
136	D₄×C₁₉₁	—	2	1	no	—
137	D₄×C₁₉₁	—	2	1	no	—
138	D₄×C₁₉₁	—	2	1	no	—
139	D₄×C₁₉₁	—	4	2	no	—
140	D₄×C₁₉₁	—	4	2	no	—
141	C₂×C₄×C₁₉₁	—	2	1	no	—
142	C₂×C₄×C₁₉₁	—	2	1	no	—
143	C₂×C₄×C₁₉₁	—	2	1	no	—
144	C₂×C₄×C₁₉₁	—	2	1	no	—
145	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
147	C₈×C₁₉₁	—	2	1	no	—
148	C₈×C₁₉₁	#41	1	2	yes	complemented
149	C₂²×C₁₉₁	—	2	1	no	—
150	C₂²×C₁₉₁	—	2	1	no	—
151	C₂²×C₁₉₁	—	2	1	no	—
152	C₂²×C₁₉₁	—	4	1	no	—
153	C₂²×C₁₉₁	—	4	1	no	—
154	C₂²×C₁₉₁	—	4	1	no	—
155	C₂²×C₁₉₁	—	4	1	no	—
156	C₂²×C₁₉₁	—	4	1	no	—
157	C₂²×C₁₉₁	—	4	1	no	—
158	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₁₉₁	—	4	1	no	—
160	C₂×D₁₉₁	—	4	1	no	—
161	C₂×D₁₉₁	—	4	1	no	—
162	C₄×C₁₉₁	—	2	1	no	—
163	C₄×C₁₉₁	—	4	1	no	—
164	C₄×C₁₉₁	—	4	1	no	—
165	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
166	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
167	Dic₁₉₁	—	2	1	no	—
168	Dic₁₉₁	—	2	1	no	—
169	Dic₁₉₁	—	2	1	no	—
170	Dic₁₉₁	—	4	1	no	—
171	Dic₁₉₁	—	4	1	no	—
172	Dic₁₉₁	—	4	1	no	—
173	Dic₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₁₉₁	—	4	1	no	—
175	C₂×C₁₉₁	—	8	1	no	—
176	C₂×C₁₉₁	—	8	1	no	—
177	C₂×C₁₉₁	#532	1	1	yes	characteristic
178	C₂×C₁₉₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₂×C₁₉₁	#575	1	1	yes	characteristic
180	D₁₉₁	—	8	1	no	—
181	C₁₉₁	#3171	1	1	yes	characteristic, complemented, Sylow
182	#3171	—	191	1	no	maximal, Sylow
183	#532	—	191	1	no	—
184	#529	—	191	1	no	—
185	#501	—	191	1	no	—
186	#496	—	191	1	no	—
187	#490	—	191	1	no	—
188	#489	—	191	1	no	—
189	#469	—	191	1	no	—
190	#469	D₁₉₁	1	1	yes	characteristic, complemented
191	#468	—	191	1	no	—
192	#468	—	191	1	no	—
193	C₄×SD₁₆	—	191	1	no	—
194	C₄×SD₁₆	—	191	1	no	—
195	C₄×D₈	—	191	1	no	—
196	C₄×D₈	—	191	1	no	—
197	C₄×M₁₆	—	191	1	no	—
198	C₂×#41	—	382	1	no	—
199	C₂×SD₁₆	—	191	1	no	—
200	C₂×SD₁₆	—	191	1	no	—
201	C₂×D₈	—	191	1	no	—
202	C₂×D₈	C₂×D₁₉₁	1	1	yes	characteristic
203	C₂×M₁₆	—	191	1	no	—
204	C₂×M₁₆	C₂×D₁₉₁	1	1	yes	characteristic
205	C₄⋊Q₈	—	191	1	no	—
206	#127	—	191	1	no	—
207	#126	—	382	1	no	—
208	#125	—	191	1	no	—
209	#124	—	382	1	no	—
210	#124	—	191	1	no	—
211	#123	—	382	1	no	—
212	C₂²⋊Q₈	—	191	1	no	—
213	#121	—	382	1	no	—
214	#121	—	191	1	no	—
215	#121	C₂×D₁₉₁	1	1	yes	characteristic
216	#121	C₂×D₁₉₁	1	1	yes	characteristic
217	C₄×Q₈	—	191	1	no	—
218	C₄×D₄	—	382	1	no	—
219	C₄×D₄	—	382	1	no	—
220	C₄×D₄	—	191	1	no	—
221	C₄×D₄	—	191	1	no	—
222	C₄×D₄	—	191	1	no	—
223	#117	—	382	1	no	—
224	C₂×C₄²	—	191	1	no	—
225	#107	—	191	1	no	—
226	#107	—	191	1	no	—
227	#106	—	191	1	no	—
228	#106	C₂×D₁₉₁	1	1	yes	characteristic
229	#103	—	191	1	no	—
230	#103	—	191	1	no	—
231	#102	—	191	1	no	—
232	#102	—	191	1	no	—
233	#102	—	191	1	no	—
234	#102	—	191	1	no	—
235	#102	C₂×D₁₉₁	1	1	yes	characteristic
236	#102	C₂×D₁₉₁	1	1	yes	characteristic
237	#97	—	191	1	no	—
238	#97	—	191	1	no	—
239	C₄×C₈	—	191	1	no	—
240	C₄×C₈	—	191	1	no	—
241	#41	—	764	4	no	—
242	C₂×Q₈	—	382	1	no	—
243	C₂×Q₈	—	191	1	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	382	1	no	—
247	C₂×D₄	—	382	1	no	—
248	C₂×D₄	—	382	1	no	—
249	C₂×D₄	—	191	1	no	—
250	C₂×D₄	C₂²×D₁₉₁	1	1	yes	characteristic
251	C₂×D₄	C₂²×D₁₉₁	1	1	yes	characteristic
252	C₂²×C₄	—	382	1	no	—
253	C₂²×C₄	—	382	1	no	—
254	C₂²×C₄	—	382	1	no	—
255	C₂²×C₄	—	191	1	no	—
256	C₂²×C₄	—	191	1	no	—
257	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
258	SD₁₆	—	382	2	no	—
259	SD₁₆	—	382	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	382	2	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	382	2	no	—
264	C₂×C₈	—	191	1	no	—
265	C₂×C₈	—	191	1	no	—
266	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
268	C₄⋊C₄	—	382	1	no	—
269	C₄⋊C₄	—	382	1	no	—
270	C₄⋊C₄	—	382	1	no	—
271	C₄⋊C₄	—	382	1	no	—
272	C₄⋊C₄	—	382	1	no	—
273	C₄⋊C₄	—	191	1	no	—
274	C₄⋊C₄	—	191	1	no	—
275	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	382	1	no	—
280	C₂²⋊C₄	—	382	1	no	—
281	C₂²⋊C₄	—	382	1	no	—
282	C₂²⋊C₄	—	382	1	no	—
283	C₂²⋊C₄	—	382	1	no	—
284	C₂²⋊C₄	—	382	1	no	—
285	C₂²⋊C₄	—	382	1	no	—
286	C₄²	—	382	1	no	—
287	C₄²	—	191	1	no	—
288	C₄²	—	191	1	no	—
289	C₄²	—	191	1	no	—
290	C₄²	—	191	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	382	1	no	—
294	C₂³	D₄×D₁₉₁	1	1	yes	characteristic
295	Q₈	—	382	1	no	—
296	Q₈	—	382	1	no	—
297	Q₈	—	764	2	no	—
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	382	1	no	—
305	D₄	—	382	1	no	—
306	D₄	—	764	2	no	—
307	D₄	—	764	2	no	—
308	D₄	—	764	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	382	2	no	—
314	C₂×C₄	—	382	1	no	—
315	C₂×C₄	—	382	1	no	—
316	C₂×C₄	—	382	1	no	—
317	C₂×C₄	—	191	1	no	—
318	C₂×C₄	—	191	1	no	—
319	C₂×C₄	—	191	1	no	—
320	C₂×C₄	—	191	1	no	—
321	C₂×C₄	—	764	2	no	—
322	C₂×C₄	—	764	1	no	—
323	C₂×C₄	—	764	2	no	—
324	C₂×C₄	—	764	2	no	—
325	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	382	1	no	—
329	C₈	—	191	2	no	—
330	C₈	#15690286	1	2	yes	complemented
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	764	1	no	—
341	C₂²	—	764	1	no	—
342	C₂²	—	764	1	no	—
343	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	382	1	no	—
348	C₄	—	382	1	no	—
349	C₄	—	382	1	no	—
350	C₄	—	191	2	no	—
351	C₄	—	764	1	no	—
352	C₄	—	764	1	no	—
353	C₄	—	764	1	no	—
354	C₄	#41×D₁₉₁	1	1	yes	characteristic
355	C₄	C₂×#15690286	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	8	1	no	—
358	C₂	—	8	1	no	—
359	C₂	—	1528	1	no	—
360	C₂	#27133208	1	1	yes	characteristic
361	C₂	AGL(1,ℤ/8)×D₁₉₁	1	1	yes	characteristic
362	C₂	#27133459	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#469×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
hypercenter	#469	D₁₉₁
hypocenter	C₁₉₁	#3171
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #27133208	C₂	semidirect
2. #27133194	C₂	semidirect
3. #27133070	C₂	semidirect
4. #27133051	C₂	semidirect
5. #27133026	C₂	semidirect
6. #27133021	C₂	semidirect
7. #27132914	C₂	semidirect
8. #27132909	C₂	semidirect
9. #27132907	C₂	semidirect
10. #27132754	C₂	semidirect
11. #27132750	C₂	semidirect
12. D₈×Dic₁₉₁	C₂	semidirect
13. #27132743	C₂	semidirect
14. M₁₆×Dic₁₉₁	C₂	semidirect
15. #469×C₁₉₁	C₂	semidirect
131. C₁₉₁⋊C₈	#41	semidirect
148. C₈×C₁₉₁	#41	semidirect
181. C₁₉₁	#3171	semidirect
190. #469	D₁₉₁	semidirect
330. C₈	#15690286	semidirect

Maximal subgroups

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

subgroup	index
#27133208	2
#27133194	2
#27133070	2
#27133051	2
#27133026	2
#27133021	2
#27132914	2
#27132909	2
#27132907	2
#27132754	2
#27132750	2
D₈×Dic₁₉₁	2
#27132743	2
M₁₆×Dic₁₉₁	2
#469×C₁₉₁	2
#3171	191