Subgroups — #37576043

Order 24448.

500 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259298	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132600	C₂	1	2	yes	maximal
4	#27132595	C₂	1	2	yes	maximal, complemented
5	#27132562	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132555	C₂	1	2	yes	maximal
7	#27132548	C₂	1	2	yes	maximal, complemented
8	#27132533	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132520	C₂	1	1	yes	maximal, characteristic
10	C₄×(C₂×D₁₉₁)⋊C₄	C₂	1	1	yes	maximal, characteristic
11	#391×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₉₁	C₂²	1	1	yes	characteristic
13	#20259298	—	2	4	no	—
14	Q₈×Dic₁₉₁	—	2	4	no	—
15	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
21	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	2	yes	—
25	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
28	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
29	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
30	C₁₉₁⋊#95	C₂²	1	2	yes	—
31	C₁₉₁⋊#95	C₂²	1	2	yes	—
32	C₂²×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
35	#95×C₁₉₁	C₂²	1	2	yes	—
36	#95×C₁₉₁	C₂²	1	2	yes	—
37	C₂³×D₁₉₁	C₂³	1	1	yes	characteristic
38	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
39	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
43	C₂×C₄×D₁₉₁	—	2	2	no	—
44	C₂×C₄×D₁₉₁	—	2	2	no	—
45	C₂×C₄×D₁₉₁	—	2	2	no	—
46	C₄⋊Dic₁₉₁	—	2	4	no	—
47	C₄⋊Dic₁₉₁	—	2	2	no	—
48	C₄⋊Dic₁₉₁	—	2	2	no	—
49	C₄⋊Dic₁₉₁	—	2	4	no	—
50	C₄⋊Dic₁₉₁	—	4	2	no	—
51	C₄⋊Dic₁₉₁	—	4	2	no	—
52	Dic₁₉₁⋊C₄	—	2	4	no	—
53	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
54	(C₂×D₁₉₁)⋊C₄	—	2	2	no	—
55	(C₂×D₁₉₁)⋊C₄	—	4	2	no	—
56	(C₂×D₁₉₁)⋊C₄	Q₈	1	4	yes	complemented
57	C₄×Dic₁₉₁	—	2	4	no	—
58	C₄×Dic₁₉₁	—	2	2	no	—
59	C₂×Q₈×C₁₉₁	—	2	2	no	—
60	C₂×Q₈×C₁₉₁	—	2	2	no	—
61	C₂×Q₈×C₁₉₁	D₄	1	4	yes	—
62	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₉₁	—	2	2	no	—
68	C₄²×C₁₉₁	—	2	2	no	—
69	C₂²×D₁₉₁	—	2	1	no	—
70	C₂²×D₁₉₁	—	2	1	no	—
71	C₂²×D₁₉₁	—	2	1	no	—
72	C₂²×D₁₉₁	—	2	1	no	—
73	C₂²×D₁₉₁	—	2	1	no	—
74	C₂²×D₁₉₁	—	2	1	no	—
75	C₂²×D₁₉₁	C₂×Q₈	1	2	yes	—
76	C₂×Dic₁₉₁	—	2	1	no	—
77	C₂×Dic₁₉₁	—	2	1	no	—
78	C₂×Dic₁₉₁	—	2	1	no	—
79	C₂×Dic₁₉₁	—	2	2	no	—
80	C₂×Dic₁₉₁	—	2	2	no	—
81	C₂×Dic₁₉₁	—	2	1	no	—
82	C₂×Dic₁₉₁	—	2	1	no	—
83	C₂×Dic₁₉₁	—	2	2	no	—
84	C₂×Dic₁₉₁	—	2	2	no	—
85	C₂×Dic₁₉₁	—	2	1	no	—
86	C₂×Dic₁₉₁	—	2	1	no	—
87	C₂×Dic₁₉₁	—	2	2	no	—
88	C₂×Dic₁₉₁	—	2	1	no	—
89	C₂×Dic₁₉₁	—	2	2	no	—
90	C₂×Dic₁₉₁	C₂×Q₈	1	2	yes	—
91	C₄×D₁₉₁	—	4	4	no	—
92	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
93	Q₈×C₁₉₁	—	2	8	no	—
94	C₂×C₄×C₁₉₁	—	2	1	no	—
95	C₂×C₄×C₁₉₁	—	2	1	no	—
96	C₂×C₄×C₁₉₁	—	2	2	no	—
97	C₂×C₄×C₁₉₁	—	2	2	no	—
98	C₂×C₄×C₁₉₁	—	2	1	no	—
99	C₂×C₄×C₁₉₁	—	2	1	no	—
100	C₂×C₄×C₁₉₁	—	2	2	no	—
101	C₂×C₄×C₁₉₁	—	2	2	no	—
102	C₂×C₄×C₁₉₁	—	2	2	no	—
103	C₂×C₄×C₁₉₁	C₂×D₄	1	4	yes	—
104	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
106	C₂×C₄×C₁₉₁	#41	1	2	yes	—
107	C₂×C₄×C₁₉₁	#41	1	2	yes	—
108	C₂×C₄×C₁₉₁	#41	1	2	yes	—
109	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
110	C₂²×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
111	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
112	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
113	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
114	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×D₁₉₁	—	2	2	no	—
117	C₂×D₁₉₁	—	2	2	no	—
118	C₂×D₁₉₁	—	2	2	no	—
119	C₂×D₁₉₁	—	4	1	no	—
120	C₂×D₁₉₁	—	4	1	no	—
121	C₂×D₁₉₁	—	4	1	no	—
122	C₂×D₁₉₁	—	4	1	no	—
123	C₄×C₁₉₁	—	2	4	no	—
124	C₄×C₁₉₁	—	2	2	no	—
125	C₄×C₁₉₁	—	2	2	no	—
126	C₄×C₁₉₁	—	4	2	no	—
127	C₄×C₁₉₁	#123	1	4	yes	—
128	Dic₁₉₁	—	2	2	no	—
129	Dic₁₉₁	—	4	1	no	—
130	Dic₁₉₁	—	4	1	no	—
131	Dic₁₉₁	—	4	2	no	—
132	Dic₁₉₁	—	4	2	no	—
133	C₂×C₁₉₁	C₂×#123	1	1	yes	characteristic
134	C₂×C₁₉₁	#533	1	1	yes	characteristic
135	C₂×C₁₉₁	#542	1	1	yes	characteristic
136	C₂×C₁₉₁	D₄×Q₈	1	1	yes	characteristic
137	C₂×C₁₉₁	#550	1	1	yes	characteristic
138	C₂×C₁₉₁	#552	1	1	yes	characteristic
139	C₂×C₁₉₁	#554	1	1	yes	characteristic
140	D₁₉₁	—	4	2	no	—
141	C₁₉₁	#2423	1	1	yes	characteristic, complemented, Sylow
142	#2423	—	191	1	no	maximal, Sylow
143	C₂×C₂²⋊Q₈	—	191	1	no	—
144	C₂×C₄×Q₈	—	191	1	no	—
145	#399	—	191	2	no	—
146	#398	—	191	2	no	—
147	#391	D₁₉₁	1	1	yes	characteristic, complemented
148	#389	—	191	1	no	—
149	#388	—	191	2	no	—
150	#387	—	191	2	no	—
151	#384	—	191	1	no	—
152	#380	—	191	1	no	—
153	C₄×C₂²⋊C₄	—	191	1	no	—
154	C₂²×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
155	C₂³×C₄	—	191	1	no	—
156	C₂²⋊Q₈	—	382	4	no	—
157	C₄×Q₈	—	382	4	no	—
158	C₂×C₄⋊C₄	—	191	1	no	—
159	C₂×C₄⋊C₄	—	191	2	no	—
160	C₂×C₄⋊C₄	—	191	1	no	—
161	C₂×C₄⋊C₄	—	191	2	no	—
162	C₂×C₄⋊C₄	—	191	2	no	—
163	C₂×C₄⋊C₄	—	191	2	no	—
164	C₂×C₄⋊C₄	—	191	2	no	—
165	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	—	191	2	no	—
167	C₂×C₂²⋊C₄	—	191	1	no	—
168	C₂×C₂²⋊C₄	—	191	1	no	—
169	C₂×C₂²⋊C₄	—	191	2	no	—
170	C₂×C₄²	—	191	1	no	—
171	C₂×C₄²	—	191	2	no	—
172	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
173	#95	—	191	1	no	—
174	#95	—	191	1	no	—
175	#95	—	191	2	no	—
176	#95	—	191	2	no	—
177	#95	C₂×D₁₉₁	1	2	yes	—
178	#95	C₂×D₁₉₁	1	2	yes	—
179	C₂⁴	—	191	1	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	D₇₆₄	1	4	yes	—
183	C₂²×C₄	—	382	2	no	—
184	C₂²×C₄	—	382	2	no	—
185	C₂²×C₄	—	382	2	no	—
186	C₂²×C₄	—	191	1	no	—
187	C₂²×C₄	—	191	1	no	—
188	C₂²×C₄	—	191	2	no	—
189	C₂²×C₄	—	191	2	no	—
190	C₂²×C₄	—	191	1	no	—
191	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
192	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
193	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	382	4	no	—
198	C₄⋊C₄	—	382	4	no	—
199	C₄⋊C₄	—	382	2	no	—
200	C₄⋊C₄	—	382	2	no	—
201	C₄⋊C₄	—	382	4	no	—
202	C₄⋊C₄	—	764	2	no	—
203	C₄⋊C₄	—	764	2	no	—
204	C₂²⋊C₄	—	382	4	no	—
205	C₂²⋊C₄	—	382	2	no	—
206	C₂²⋊C₄	—	191	4	no	—
207	C₂²⋊C₄	—	764	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	382	4	no	—
210	C₄²	—	382	2	no	—
211	C₂³	—	382	1	no	—
212	C₂³	—	382	1	no	—
213	C₂³	—	382	1	no	—
214	C₂³	—	382	1	no	—
215	C₂³	—	382	1	no	—
216	C₂³	—	382	1	no	—
217	C₂³	—	191	2	no	—
218	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
219	Q₈	—	2	8	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	382	1	no	—
230	C₂×C₄	—	382	1	no	—
231	C₂×C₄	—	382	1	no	—
232	C₂×C₄	—	382	2	no	—
233	C₂×C₄	—	382	2	no	—
234	C₂×C₄	—	382	1	no	—
235	C₂×C₄	—	382	1	no	—
236	C₂×C₄	—	382	1	no	—
237	C₂×C₄	—	382	1	no	—
238	C₂×C₄	—	382	2	no	—
239	C₂×C₄	—	382	2	no	—
240	C₂×C₄	—	382	2	no	—
241	C₂×C₄	—	382	1	no	—
242	C₂×C₄	—	382	2	no	—
243	C₂×C₄	—	191	2	no	—
244	C₂×C₄	—	764	4	no	—
245	C₂×C₄	C₂×D₇₆₄	1	4	yes	—
246	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
247	C₂×C₄	Q₈×D₁₉₁	1	2	yes	—
248	C₂×C₄	#15690286	1	2	yes	—
249	C₂×C₄	#15690286	1	2	yes	—
250	C₂×C₄	#15690288	1	2	yes	—
251	C₂²	—	382	2	no	—
252	C₂²	—	382	2	no	—
253	C₂²	—	382	2	no	—
254	C₂²	—	764	1	no	—
255	C₂²	—	764	1	no	—
256	C₂²	—	764	1	no	—
257	C₂²	—	764	1	no	—
258	C₂²	C₂²×D₇₆₄	1	1	yes	characteristic
259	C₂²	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
260	C₂²	C₂×#15690286	1	1	yes	characteristic
261	C₂²	C₂×#15690286	1	1	yes	characteristic
262	C₂²	C₂×#15690288	1	1	yes	characteristic
263	C₂²	#20259368	1	1	yes	characteristic
264	C₂²	#20259369	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	4	no	—
268	C₄	—	4	2	no	—
269	C₄	—	382	2	no	—
270	C₄	—	764	1	no	—
271	C₄	—	764	2	no	—
272	C₄	—	764	2	no	—
273	C₄	—	764	1	no	—
274	C₄	#20259306	1	4	yes	—
275	C₂	—	764	2	no	—
276	C₂	C₂×#20259306	1	1	yes	characteristic
277	C₂	#27133218	1	1	yes	characteristic
278	C₂	#27133282	1	1	yes	characteristic
279	C₂	Q₈×D₇₆₄	1	1	yes	characteristic
280	C₂	#27133335	1	1	yes	characteristic
281	C₂	#27133355	1	1	yes	characteristic
282	C₂	#27133371	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator, socle, cosocle	C₂³×C₁₉₁	C₂⁴
Fitting	#391×C₁₉₁	C₂
prime radical	C₂³×D₁₉₁	C₂³
hypercenter	#391	D₁₉₁
hypocenter	C₁₉₁	#2423
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₁₉₁	C₂	semidirect
4. #27132595	C₂	semidirect
5. #27132562	C₂	semidirect
7. #27132548	C₂	semidirect
8. #27132533	C₂	semidirect
11. #391×C₁₉₁	C₂	semidirect
56. (C₂×D₁₉₁)⋊C₄	Q₈	semidirect
141. C₁₉₁	#2423	semidirect
147. #391	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259298	2
C₂×Q₈×Dic₁₉₁	2
#27132600	2
#27132595	2
#27132562	2
#27132555	2
#27132548	2
#27132533	2
#27132520	2
C₄×(C₂×D₁₉₁)⋊C₄	2
#391×C₁₉₁	2
#2423	191