Subgroups — #37574390

Order 24448.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₅₂₈	C₂	1	1	yes	maximal, characteristic
2	#27132757	C₂	1	2	yes	maximal
3	#27132466	C₂	1	2	yes	maximal
4	#27132438	C₂	1	1	yes	maximal, characteristic
5	#367×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₆₄	C₂²	1	2	yes	—
7	#20259287	—	2	2	no	—
8	Dic₁₅₂₈	C₄	1	4	yes	complemented
9	C₂×C₁₉₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	#20259245	—	2	2	no	—
11	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
13	#107×C₁₉₁	C₂²	1	2	yes	—
14	C₂×Dic₃₈₂	—	2	2	no	—
15	Dic₇₆₄	—	2	2	no	—
16	Dic₇₆₄	C₂×C₄	1	4	yes	—
17	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
18	Dic₁₉₁⋊C₄	—	4	2	no	—
19	C₄×Dic₁₉₁	—	4	2	no	—
20	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
21	C₁₉₁⋊C₁₆	—	2	1	no	—
22	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₉₁	—	2	2	no	—
24	C₁₆×C₁₉₁	C₂×C₄	1	2	yes	—
25	Dic₃₈₂	—	2	4	no	—
26	Dic₃₈₂	—	4	2	no	—
27	C₂×Dic₁₉₁	—	4	2	no	—
28	C₂×Dic₁₉₁	D₈	1	2	yes	—
29	C₁₉₁⋊C₈	—	2	1	no	—
30	C₂×C₄×C₁₉₁	—	4	2	no	—
31	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
32	C₈×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₉₁	#41	1	1	yes	characteristic
34	C₂²×C₁₉₁	C₂×D₈	1	1	yes	characteristic
35	C₄×C₁₉₁	—	8	2	no	—
36	C₄×C₁₉₁	C₄×D₄	1	1	yes	characteristic
37	C₄×C₁₉₁	#135	1	1	yes	characteristic
38	Dic₁₉₁	—	2	2	no	—
39	Dic₁₉₁	—	4	4	no	—
40	C₂×C₁₉₁	C₄×D₈	1	1	yes	characteristic
41	C₂×C₁₉₁	#510	1	2	yes	—
42	C₁₉₁	#2176	1	1	yes	characteristic, complemented, Sylow
43	#2176	—	191	1	no	maximal, Sylow
44	C₂×Q₃₂	—	191	1	no	—
45	C₄×Q₁₆	—	191	2	no	—
46	#367	D₁₉₁	1	1	yes	characteristic
47	#358	—	191	2	no	—
48	#346	—	191	1	no	—
49	C₂×Q₁₆	—	191	2	no	—
50	C₄×Q₈	—	382	2	no	—
51	Q₃₂	—	191	4	no	—
52	C₂×C₁₆	—	191	1	no	—
53	C₂×C₁₆	C₂×D₁₉₁	1	1	yes	characteristic
54	#107	C₂×D₁₉₁	1	2	yes	—
55	#103	—	382	2	no	—
56	C₄×C₈	—	191	1	no	—
57	C₂×Q₈	—	382	2	no	—
58	Q₁₆	—	382	2	no	—
59	Q₁₆	—	191	4	no	—
60	C₂×C₈	—	191	1	no	—
61	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	764	2	no	—
64	C₄²	—	191	1	no	—
65	C₄²	—	764	2	no	—
66	C₁₆	—	382	1	no	—
67	C₁₆	C₄×D₁₉₁	1	2	yes	—
68	Q₈	—	382	4	no	—
69	Q₈	—	764	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	191	2	no	—
72	C₂×C₄	—	764	2	no	—
73	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
74	C₈	—	382	1	no	—
75	C₈	C₂×C₄×D₁₉₁	1	1	yes	characteristic
76	C₈	#15690288	1	1	yes	characteristic
77	C₂²	D₈×D₁₉₁	1	1	yes	characteristic
78	C₄	—	8	2	no	—
79	C₄	—	382	2	no	—
80	C₄	—	764	4	no	—
81	C₄	#20259284	1	1	yes	characteristic
82	C₄	#20259345	1	1	yes	characteristic
83	C₂	#27132745	1	1	yes	characteristic
84	C₂	#27133102	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₉₁
commutator	C₈×C₁₉₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₉₁
Fitting	#367×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	C₂×D₈
cosocle	C₂×C₈×C₁₉₁	C₂³
hypercenter	#367	D₁₉₁
hypocenter	C₁₉₁	#2176
LCS	G ▷ C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
8. Dic₁₅₂₈	C₄	semidirect
42. C₁₉₁	#2176	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₅₂₈	2
#27132757	2
#27132466	2
#27132438	2
#367×C₁₉₁	2
#2176	191