Subgroups — #27132600

Order 12224.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₁₉₁	C₂	1	3	yes	maximal, complemented
2	C₂×(C₂×D₁₉₁)⋊C₄	C₂	1	3	yes	maximal
3	#95×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₁₉₁	C₂²	1	1	yes	characteristic
5	C₂²×Dic₁₉₁	C₂²	1	3	yes	—
6	C₄⋊Dic₁₉₁	—	2	6	no	—
7	(C₂×D₁₉₁)⋊C₄	—	2	6	no	—
8	C₂²×C₄×C₁₉₁	C₂²	1	3	yes	—
9	C₂²×D₁₉₁	—	2	3	no	—
10	C₂²×D₁₉₁	—	2	3	no	—
11	C₂²×D₁₉₁	Q₈	1	2	yes	—
12	C₂×Dic₁₉₁	—	2	3	no	—
13	C₂×Dic₁₉₁	—	2	6	no	—
14	C₂³×C₁₉₁	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₁₉₁	—	2	6	no	—
16	C₂×C₄×C₁₉₁	D₄	1	6	yes	—
17	C₂²×C₁₉₁	C₂×D₄	1	3	yes	—
18	C₂²×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
19	C₂²×C₁₉₁	#41	1	3	yes	—
20	C₂×D₁₉₁	—	2	6	no	—
21	C₂×D₁₉₁	—	4	3	no	—
22	C₂×D₁₉₁	—	4	1	no	—
23	C₄×C₁₉₁	—	2	6	no	—
24	Dic₁₉₁	—	4	3	no	—
25	C₂×C₁₉₁	C₂²⋊Q₈	1	3	yes	—
26	C₂×C₁₉₁	#123	1	3	yes	—
27	C₂×C₁₉₁	#127	1	1	yes	characteristic
28	D₁₉₁	—	4	2	no	—
29	C₁₉₁	#399	1	1	yes	characteristic, complemented, Sylow
30	#399	—	191	1	no	maximal, Sylow
31	C₂×C₄⋊C₄	—	191	3	no	—
32	C₂×C₂²⋊C₄	—	191	3	no	—
33	#95	D₁₉₁	1	1	yes	characteristic, complemented
34	C₂⁴	—	191	1	no	—
35	C₂²×C₄	—	191	3	no	—
36	C₂²×C₄	C₂×D₁₉₁	1	3	yes	—
37	C₄⋊C₄	—	382	6	no	—
38	C₂²⋊C₄	—	382	6	no	—
39	C₂³	—	382	3	no	—
40	C₂³	—	382	3	no	—
41	C₂³	—	191	2	no	—
42	C₂³	C₂²×D₁₉₁	1	1	yes	characteristic
43	C₂×C₄	—	2	6	no	—
44	C₂×C₄	—	382	3	no	—
45	C₂×C₄	—	382	6	no	—
46	C₂×C₄	D₇₆₄	1	6	yes	—
47	C₂²	—	382	6	no	—
48	C₂²	—	764	3	no	—
49	C₂²	—	764	1	no	—
50	C₂²	C₂×D₇₆₄	1	3	yes	—
51	C₂²	Q₈×D₁₉₁	1	1	yes	characteristic
52	C₂²	#15690286	1	3	yes	—
53	C₄	—	2	6	no	—
54	C₄	—	764	3	no	—
55	C₂	—	764	2	no	—
56	C₂	#20259302	1	3	yes	—
57	C₂	#20259306	1	3	yes	—
58	C₂	#20259319	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₉₁
commutator, socle, cosocle	C₂³×C₁₉₁	C₂³
Fitting	#95×C₁₉₁	C₂
prime radical	C₂³×D₁₉₁	C₂²
hypercenter	#95	D₁₉₁
hypocenter	C₁₉₁	#399
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₁₉₁	C₂	semidirect
3. #95×C₁₉₁	C₂	semidirect
29. C₁₉₁	#399	semidirect
33. #95	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₁₉₁	2
C₂×(C₂×D₁₉₁)⋊C₄	2
#95×C₁₉₁	2
#399	191