Subgroups — #27132900

Order 12224.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₂₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259297	C₂	1	2	yes	maximal, complemented
4	#20259242	C₂	1	2	yes	maximal, complemented
5	#107×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₆₄	—	2	2	no	—
7	C₂×D₇₆₄	C₂²	1	2	yes	—
8	C₂×C₄×D₁₉₁	C₂²	1	1	yes	characteristic
9	D₁₅₂₈	—	2	2	no	—
10	C₂×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₈×D₁₉₁	—	2	2	no	—
12	(C₂×D₁₉₁)⋊C₄	—	2	2	no	—
13	C₂×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
15	C₂²×D₁₉₁	—	2	2	no	—
16	C₂²×D₁₉₁	D₄	1	1	yes	characteristic
17	D₇₆₄	—	2	2	no	—
18	D₇₆₄	—	2	2	no	—
19	D₇₆₄	—	4	4	no	—
20	C₂×Dic₁₉₁	D₄	1	1	yes	characteristic
21	C₄×D₁₉₁	—	2	1	no	—
22	C₄×D₁₉₁	—	2	1	no	—
23	C₁₉₁⋊C₈	—	2	1	no	—
24	C₂×C₄×C₁₉₁	—	2	2	no	—
25	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
26	C₈×C₁₉₁	D₄	1	2	yes	complemented
27	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
28	C₂×D₁₉₁	—	2	1	no	—
29	C₂×D₁₉₁	—	2	1	no	—
30	C₂×D₁₉₁	—	4	2	no	—
31	C₂×D₁₉₁	—	4	2	no	—
32	C₂×D₁₉₁	—	4	2	no	—
33	C₂×D₁₉₁	D₈	1	2	yes	complemented
34	C₄×C₁₉₁	—	4	2	no	—
35	C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₉₁	#41	1	1	yes	characteristic
37	Dic₁₉₁	—	2	1	no	—
38	C₂×C₁₉₁	#121	1	1	yes	characteristic
39	C₂×C₁₉₁	C₂×D₈	1	1	yes	characteristic
40	C₂×C₁₉₁	#135	1	1	yes	characteristic
41	D₁₉₁	—	2	2	no	—
42	D₁₉₁	—	8	2	no	—
43	C₁₉₁	#466	1	1	yes	characteristic, complemented, Sylow
44	#466	—	191	1	no	maximal, Sylow
45	C₂×D₈	—	191	1	no	—
46	C₂²×C₈	—	191	1	no	—
47	#121	—	191	2	no	—
48	#107	D₁₉₁	1	1	yes	characteristic, complemented
49	#102	—	191	2	no	—
50	C₂×D₄	—	382	2	no	—
51	C₂×D₄	—	191	2	no	—
52	C₂²×C₄	—	191	1	no	—
53	D₈	—	382	2	no	—
54	C₂×C₈	—	382	2	no	—
55	C₂×C₈	—	191	1	no	—
56	C₂×C₈	C₂×D₁₉₁	1	1	yes	characteristic
57	C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
58	C₂²⋊C₄	—	382	2	no	—
59	C₂³	—	382	2	no	—
60	C₂³	—	191	1	no	—
61	D₄	—	382	2	no	—
62	D₄	—	382	2	no	—
63	D₄	—	764	4	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	382	1	no	—
66	C₂×C₄	—	382	1	no	—
67	C₂×C₄	—	191	1	no	—
68	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
69	C₈	—	382	1	no	—
70	C₈	D₇₆₄	1	2	yes	complemented
71	C₂²	—	382	1	no	—
72	C₂²	—	382	1	no	—
73	C₂²	—	191	2	no	—
74	C₂²	—	764	2	no	—
75	C₂²	—	764	2	no	—
76	C₂²	—	764	2	no	—
77	C₂²	D₄×D₁₉₁	1	1	yes	characteristic
78	C₄	—	4	2	no	—
79	C₄	—	382	1	no	—
80	C₄	C₂×D₇₆₄	1	1	yes	characteristic
81	C₄	#15690288	1	1	yes	characteristic
82	C₂	—	382	2	no	—
83	C₂	—	1528	2	no	—
84	C₂	#20259297	1	1	yes	characteristic
85	C₂	D₈×D₁₉₁	1	1	yes	characteristic
86	C₂	#20259345	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	#107×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×D₄
hypercenter	#107	D₁₉₁
hypocenter	C₁₉₁	#466
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₅₂₈	C₂	semidirect
2. C₂×C₈×D₁₉₁	C₂	semidirect
3. #20259297	C₂	semidirect
4. #20259242	C₂	semidirect
5. #107×C₁₉₁	C₂	semidirect
26. C₈×C₁₉₁	D₄	semidirect
33. C₂×D₁₉₁	D₈	semidirect
43. C₁₉₁	#466	semidirect
48. #107	D₁₉₁	semidirect
70. C₈	D₇₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₂₈	2
C₂×C₈×D₁₉₁	2
#20259297	2
#20259242	2
#107×C₁₉₁	2
#466	191