Subgroups — #37583902

Order 24448.

386 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133215	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133208	C₂	1	1	yes	maximal, characteristic
3	#27132957	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132956	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132942	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132941	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132911	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	#27132867	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132866	C₂	1	1	yes	maximal, characteristic, complemented
12	#27132862	C₂	1	1	yes	maximal, characteristic, complemented
13	#27132861	C₂	1	1	yes	maximal, characteristic, complemented
14	#27132686	C₂	1	1	yes	maximal, characteristic, complemented
15	#468×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×#15690274	C₂²	1	1	yes	characteristic
20	C₂×#15690271	C₂²	1	1	yes	characteristic
21	#20259321	C₂²	1	1	yes	characteristic
22	#20259321	C₂²	1	1	yes	characteristic
23	#20259320	C₂²	1	1	yes	characteristic
24	#20259318	C₂²	1	1	yes	characteristic
25	#20259312	—	2	1	no	—
26	#20259308	—	2	1	no	—
27	#20259301	C₂²	1	1	yes	characteristic
28	#20259299	—	2	1	no	—
29	#20259296	C₂²	1	1	yes	characteristic
30	#20259294	—	2	1	no	—
31	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
32	#20259287	—	2	1	no	—
33	#20259287	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	#20259255	C₂²	1	1	yes	characteristic
42	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
46	#20259247	C₂²	1	1	yes	characteristic
47	#20259246	C₂²	1	1	yes	characteristic
48	#20259245	C₂²	1	1	yes	characteristic
49	#20259244	C₂²	1	1	yes	characteristic
50	#20259243	C₂²	1	1	yes	characteristic
51	#20259242	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
55	#121×C₁₉₁	C₂²	1	1	yes	characteristic
56	#107×C₁₉₁	C₂²	1	1	yes	characteristic
57	#103×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	—	2	1	no	—
62	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
63	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
65	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
66	C₂×C₂²⋊D₁₉₁	—	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	—	4	1	no	—
72	#15690275	—	4	1	no	—
73	#15690274	—	4	1	no	—
74	#15690271	—	4	1	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	Dic₁₉₁⋊C₄	—	2	1	no	—
85	Dic₁₉₁⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₁₉₁	—	2	1	no	—
87	C₂²⋊Dic₁₉₁	—	2	1	no	—
88	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
90	C₄×Dic₁₉₁	—	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×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₉₁	—	2	1	no	—
98	C₂×D₄×C₁₉₁	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
100	SD₁₆×C₁₉₁	—	4	1	no	—
101	M₁₆×C₁₉₁	—	4	1	no	—
102	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₉₁	—	2	1	no	—
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	Dic₃₈₂	—	4	2	no	—
114	D₇₆₄	—	2	1	no	—
115	D₇₆₄	—	2	1	no	—
116	C₂²⋊D₁₉₁	—	4	2	no	—
117	C₂²⋊D₁₉₁	—	4	2	no	—
118	C₂²⋊D₁₉₁	—	4	2	no	—
119	C₂×Dic₁₉₁	—	2	2	no	—
120	C₂×Dic₁₉₁	—	2	2	no	—
121	C₂×Dic₁₉₁	—	2	1	no	—
122	C₂×Dic₁₉₁	—	2	1	no	—
123	C₂×Dic₁₉₁	—	2	1	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	—	2	1	no	—
132	C₂³×C₁₉₁	—	2	1	no	—
133	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
134	Q₈×C₁₉₁	—	2	1	no	—
135	Q₈×C₁₉₁	—	2	1	no	—
136	D₄×C₁₉₁	—	2	1	no	—
137	D₄×C₁₉₁	—	2	1	no	—
138	D₄×C₁₉₁	—	4	2	no	—
139	C₂×C₄×C₁₉₁	—	2	1	no	—
140	C₂×C₄×C₁₉₁	—	2	1	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	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
146	C₈×C₁₉₁	—	2	1	no	—
147	C₈×C₁₉₁	—	2	1	no	—
148	C₂²×C₁₉₁	—	2	1	no	—
149	C₂²×C₁₉₁	—	2	1	no	—
150	C₂²×C₁₉₁	—	2	1	no	—
151	C₂²×C₁₉₁	—	4	1	no	—
152	C₂²×C₁₉₁	—	4	1	no	—
153	C₂²×C₁₉₁	—	4	1	no	—
154	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₉₁	—	4	1	no	—
156	C₂×D₁₉₁	—	4	1	no	—
157	C₂×D₁₉₁	—	4	1	no	—
158	C₄×C₁₉₁	—	2	1	no	—
159	C₄×C₁₉₁	—	4	1	no	—
160	C₄×C₁₉₁	—	4	1	no	—
161	C₄×C₁₉₁	—	4	1	no	—
162	C₄×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
163	C₄×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
164	Dic₁₉₁	—	2	1	no	—
165	Dic₁₉₁	—	2	1	no	—
166	Dic₁₉₁	—	4	1	no	—
167	Dic₁₉₁	—	4	1	no	—
168	Dic₁₉₁	—	4	1	no	—
169	Dic₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
170	Dic₁₉₁	#137	1	2	yes	—
171	C₂×C₁₉₁	—	4	1	no	—
172	C₂×C₁₉₁	—	8	1	no	—
173	C₂×C₁₉₁	#537	1	1	yes	characteristic
174	C₂×C₁₉₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×C₁₉₁	C₂×#137	1	1	yes	characteristic
176	D₁₉₁	—	8	1	no	—
177	C₁₉₁	#3254	1	1	yes	characteristic, complemented, Sylow
178	#3254	—	191	1	no	maximal, Sylow
179	#533	—	191	1	no	—
180	#532	—	191	1	no	—
181	#477	—	191	1	no	—
182	#477	—	191	1	no	—
183	#474	—	191	1	no	—
184	#474	—	191	1	no	—
185	#468	—	191	1	no	—
186	#468	—	191	1	no	—
187	#468	—	191	1	no	—
188	#468	D₁₉₁	1	1	yes	characteristic, complemented
189	#461	—	191	1	no	—
190	#461	—	191	1	no	—
191	#460	—	191	1	no	—
192	#460	—	191	1	no	—
193	#423	—	191	1	no	—
194	C₂×#41	—	382	1	no	—
195	C₂×SD₁₆	—	191	1	no	—
196	C₂×SD₁₆	—	191	1	no	—
197	C₂×SD₁₆	—	191	1	no	—
198	C₂×SD₁₆	C₂×D₁₉₁	1	1	yes	characteristic
199	C₂×M₁₆	—	191	1	no	—
200	C₂×M₁₆	C₂×D₁₉₁	1	1	yes	characteristic
201	C₄⋊Q₈	—	191	1	no	—
202	C₄⋊Q₈	—	191	1	no	—
203	C₄⋊Q₈	—	191	1	no	—
204	#127	—	191	1	no	—
205	#125	—	382	1	no	—
206	#124	—	382	1	no	—
207	C₂²⋊Q₈	—	382	1	no	—
208	C₂²⋊Q₈	—	191	1	no	—
209	C₂²⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
210	#121	—	382	1	no	—
211	#121	—	191	1	no	—
212	#121	C₂×D₁₉₁	1	1	yes	characteristic
213	C₄×Q₈	—	382	1	no	—
214	C₄×Q₈	—	191	1	no	—
215	C₄×Q₈	—	191	1	no	—
216	C₄×D₄	—	382	1	no	—
217	C₄×D₄	—	191	1	no	—
218	C₄×D₄	—	191	1	no	—
219	#117	—	382	1	no	—
220	C₂×C₄²	—	191	1	no	—
221	#107	—	191	1	no	—
222	#107	—	191	1	no	—
223	#107	—	191	1	no	—
224	#107	C₂×D₁₉₁	1	1	yes	characteristic
225	C₄⋊C₈	—	191	1	no	—
226	C₄⋊C₈	—	191	1	no	—
227	C₄⋊C₈	—	191	1	no	—
228	C₄⋊C₈	—	191	1	no	—
229	#103	—	191	1	no	—
230	#103	—	191	1	no	—
231	#103	—	191	1	no	—
232	#103	C₂×D₁₉₁	1	1	yes	characteristic
233	#102	—	191	1	no	—
234	#102	—	191	1	no	—
235	#102	—	191	1	no	—
236	#102	C₂×D₁₉₁	1	1	yes	characteristic
237	#41	—	764	4	no	—
238	C₂×Q₈	—	382	1	no	—
239	C₂×Q₈	—	382	1	no	—
240	C₂×Q₈	—	191	1	no	—
241	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	382	1	no	—
244	C₂×D₄	—	382	1	no	—
245	C₂×D₄	—	382	1	no	—
246	C₂×D₄	—	191	1	no	—
247	C₂×D₄	C₂²×D₁₉₁	1	1	yes	characteristic
248	C₂²×C₄	—	382	1	no	—
249	C₂²×C₄	—	382	1	no	—
250	C₂²×C₄	—	191	1	no	—
251	C₂²×C₄	—	191	1	no	—
252	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
253	SD₁₆	—	4	1	no	—
254	SD₁₆	—	764	1	no	—
255	SD₁₆	—	764	1	no	—
256	SD₁₆	—	764	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	764	1	no	—
259	C₂×C₈	—	191	1	no	—
260	C₂×C₈	—	191	1	no	—
261	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	382	1	no	—
265	C₄⋊C₄	—	382	1	no	—
266	C₄⋊C₄	—	382	1	no	—
267	C₄⋊C₄	—	382	1	no	—
268	C₄⋊C₄	—	382	1	no	—
269	C₄⋊C₄	—	382	1	no	—
270	C₄⋊C₄	—	382	1	no	—
271	C₄⋊C₄	—	191	1	no	—
272	C₄⋊C₄	—	191	1	no	—
273	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	382	1	no	—
278	C₂²⋊C₄	—	382	1	no	—
279	C₂²⋊C₄	—	382	1	no	—
280	C₂²⋊C₄	—	382	1	no	—
281	C₄²	—	382	1	no	—
282	C₄²	—	382	1	no	—
283	C₄²	—	191	1	no	—
284	C₄²	—	191	1	no	—
285	C₄²	—	191	1	no	—
286	C₄²	—	191	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	382	1	no	—
289	C₂³	D₄×D₁₉₁	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	382	1	no	—
293	Q₈	—	382	1	no	—
294	Q₈	—	764	2	no	—
295	Q₈	—	764	2	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	4	2	no	—
299	D₄	—	382	1	no	—
300	D₄	—	382	1	no	—
301	D₄	—	764	2	no	—
302	D₄	—	764	2	no	—
303	D₄	—	764	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	382	2	no	—
310	C₂×C₄	—	382	2	no	—
311	C₂×C₄	—	382	1	no	—
312	C₂×C₄	—	382	1	no	—
313	C₂×C₄	—	382	1	no	—
314	C₂×C₄	—	191	1	no	—
315	C₂×C₄	—	191	1	no	—
316	C₂×C₄	—	191	1	no	—
317	C₂×C₄	—	191	1	no	—
318	C₂×C₄	—	764	2	no	—
319	C₂×C₄	—	764	2	no	—
320	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	382	1	no	—
325	C₈	—	382	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	764	1	no	—
333	C₂²	—	764	1	no	—
334	C₂²	—	764	1	no	—
335	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	382	1	no	—
341	C₄	—	382	1	no	—
342	C₄	—	191	2	no	—
343	C₄	—	191	2	no	—
344	C₄	—	764	1	no	—
345	C₄	—	764	1	no	—
346	C₄	—	764	1	no	—
347	C₄	#20259367	1	1	yes	characteristic
348	C₄	#20259369	1	1	yes	characteristic
349	C₂	—	4	1	no	—
350	C₂	—	8	1	no	—
351	C₂	—	1528	1	no	—
352	C₂	#27133240	1	1	yes	characteristic
353	C₂	AGL(1,ℤ/8)×D₁₉₁	1	1	yes	characteristic
354	C₂	#137×D₁₉₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#468×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	#27133208	C₂
hypercenter	#468	D₁₉₁
hypocenter	C₁₉₁	#3254
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. #27133215	C₂	semidirect
3. #27132957	C₂	semidirect
4. #27132956	C₂	semidirect
5. #27132942	C₂	semidirect
6. #27132941	C₂	semidirect
7. #27132911	C₂	semidirect
8. #27132909	C₂	semidirect
9. #27132907	C₂	semidirect
10. #27132867	C₂	semidirect
11. #27132866	C₂	semidirect
12. #27132862	C₂	semidirect
13. #27132861	C₂	semidirect
14. #27132686	C₂	semidirect
15. #468×C₁₉₁	C₂	semidirect
177. C₁₉₁	#3254	semidirect
188. #468	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133215	2
#27133208	2
#27132957	2
#27132956	2
#27132942	2
#27132941	2
#27132911	2
#27132909	2
#27132907	2
#27132867	2
#27132866	2
#27132862	2
#27132861	2
#27132686	2
#468×C₁₉₁	2
#3254	191