Subgroups — #27133379

Order 12224.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259311	C₂	1	4	yes	maximal, complemented
2	#20259297	C₂	1	4	yes	maximal, complemented
3	C₄×D₇₆₄	C₂	1	2	yes	maximal, complemented
4	#20259280	C₂	1	4	yes	maximal, complemented
5	C₄⋊Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₆₄	C₂²	1	2	yes	—
7	C₂×D₇₆₄	C₂²	1	4	yes	—
8	C₂×C₄×D₁₉₁	C₂²	1	4	yes	—
9	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
10	(C₂×D₁₉₁)⋊C₄	C₂²	1	8	yes	—
11	(C₂×D₁₉₁)⋊C₄	C₂²	1	4	yes	—
12	C₄×Dic₁₉₁	C₂²	1	4	yes	—
13	C₂×Q₈×C₁₉₁	C₂²	1	2	yes	—
14	C₄⋊C₄×C₁₉₁	C₂²	1	4	yes	—
15	C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₂²×D₁₉₁	C₂³	1	4	yes	—
17	D₇₆₄	—	2	4	no	—
18	D₇₆₄	—	4	4	no	—
19	C₂×Dic₁₉₁	C₂³	1	4	yes	—
20	C₄×D₁₉₁	—	2	8	no	—
21	Q₈×C₁₉₁	—	4	2	no	—
22	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₁₉₁	C₂³	1	4	yes	—
24	C₂×C₄×C₁₉₁	C₂³	1	2	yes	—
25	C₂²×C₁₉₁	C₂⁴	1	1	yes	characteristic
26	C₂×D₁₉₁	—	2	4	no	—
27	C₂×D₁₉₁	—	2	4	no	—
28	C₂×D₁₉₁	—	2	4	no	—
29	C₄×C₁₉₁	—	2	1	no	—
30	C₄×C₁₉₁	—	2	4	no	—
31	C₄×C₁₉₁	#41	1	4	yes	—
32	Dic₁₉₁	—	2	4	no	—
33	C₂×C₁₉₁	C₂×#41	1	2	yes	—
34	C₂×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
35	D₁₉₁	—	4	4	no	—
36	C₁₉₁	#555	1	1	yes	characteristic, complemented, Sylow
37	#555	—	191	1	no	maximal, Sylow
38	C₄⋊Q₈	D₁₉₁	1	1	yes	characteristic, complemented
39	#124	—	191	4	no	—
40	#121	—	191	4	no	—
41	C₄×D₄	—	191	2	no	—
42	#117	—	191	4	no	—
43	C₂×Q₈	C₂×D₁₉₁	1	2	yes	—
44	C₂×D₄	—	191	2	no	—
45	C₂×D₄	—	191	4	no	—
46	C₂²×C₄	—	191	4	no	—
47	C₄⋊C₄	—	191	2	no	—
48	C₄⋊C₄	C₂×D₁₉₁	1	4	yes	—
49	C₂²⋊C₄	—	191	8	no	—
50	C₂²⋊C₄	—	191	4	no	—
51	C₄²	—	191	4	no	—
52	C₄²	C₂×D₁₉₁	1	1	yes	characteristic
53	C₂³	—	191	4	no	—
54	Q₈	—	4	2	no	—
55	D₄	—	382	4	no	—
56	D₄	—	764	4	no	—
57	C₂×C₄	—	382	8	no	—
58	C₂×C₄	—	191	4	no	—
59	C₂×C₄	C₂²×D₁₉₁	1	4	yes	—
60	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₁₉₁	1	2	yes	—
62	C₂²	—	382	4	no	—
63	C₂²	—	382	4	no	—
64	C₂²	—	382	4	no	—
65	C₂²	C₂³×D₁₉₁	1	1	yes	characteristic
66	C₄	—	2	4	no	—
67	C₄	—	2	1	no	—
68	C₄	—	382	4	no	—
69	C₄	#15690288	1	4	yes	—
70	C₂	—	764	4	no	—
71	C₂	C₂×#15690288	1	2	yes	—
72	C₂	#20259367	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₁
commutator, socle, cosocle	C₂²×C₁₉₁	C₂⁴
Fitting	C₄⋊Q₈×C₁₉₁	C₂
hypercenter	C₄⋊Q₈	D₁₉₁
hypocenter	C₁₉₁	#555
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20259311	C₂	semidirect
2. #20259297	C₂	semidirect
3. C₄×D₇₆₄	C₂	semidirect
4. #20259280	C₂	semidirect
5. C₄⋊Q₈×C₁₉₁	C₂	semidirect
36. C₁₉₁	#555	semidirect
38. C₄⋊Q₈	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#20259311	2
#20259297	2
C₄×D₇₆₄	2
#20259280	2
C₄⋊Q₈×C₁₉₁	2
#555	191