Subgroups — #11927643

Order 1232.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₇₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₇×C₁₁	C₂²	1	1	yes	characteristic
5	C₂×D₇₇	—	2	1	no	—
6	C₂×D₇₇	—	2	1	no	—
7	C₂×D₇₇	C₄	1	2	yes	complemented
8	C₇×Dic₁₁	—	2	1	no	—
9	C₁₁×Dic₇	—	2	1	no	—
10	(C₂×D₁₁)⋊C₄	—	7	1	no	maximal, Hall
11	C₂×C₇×C₁₁	C₂×C₄	1	1	yes	characteristic
12	C₂×C₇×C₁₁	D₄	1	1	yes	characteristic
13	C₂×C₇×C₁₁	D₄	1	1	yes	characteristic
14	D₇₇	—	2	2	no	—
15	(C₂×D₇)⋊C₄	—	11	1	no	maximal, Hall
16	C₂²×D₁₁	—	7	1	no	—
17	C₂×C₄×C₁₁	—	7	1	no	—
18	C₂×Dic₁₁	D₇	1	1	yes	characteristic, complemented
19	C₇×C₁₁	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
20	C₂²×D₇	—	11	1	no	—
21	C₂×C₄×C₇	—	11	1	no	—
22	C₂×Dic₇	D₁₁	1	1	yes	characteristic, complemented
23	C₂²×C₁₁	C₂×D₇	1	1	yes	characteristic
24	C₂×D₁₁	—	7	2	no	—
25	C₂×D₁₁	—	14	1	no	—
26	C₂×D₁₁	—	14	1	no	—
27	C₄×C₁₁	—	14	1	no	—
28	Dic₁₁	—	2	1	no	—
29	C₂²×C₇	C₂×D₁₁	1	1	yes	characteristic
30	C₂×D₇	—	11	2	no	—
31	C₂×D₇	—	22	1	no	—
32	C₂×D₇	—	22	1	no	—
33	C₄×C₇	—	22	1	no	—
34	Dic₇	—	2	1	no	—
35	C₂×C₁₁	C₄×D₇	1	1	yes	characteristic
36	C₂×C₁₁	D₂₈	1	1	yes	characteristic
37	C₂×C₁₁	C₂²⋊D₇	1	1	yes	characteristic
38	D₁₁	—	14	2	no	—
39	C₂²⋊C₄	—	77	1	no	Sylow
40	C₂×C₇	C₄×D₁₁	1	1	yes	characteristic
41	C₂×C₇	D₄₄	1	1	yes	characteristic
42	C₂×C₇	C₂²⋊D₁₁	1	1	yes	characteristic
43	D₇	—	22	2	no	—
44	C₁₁	(C₂×D₇)⋊C₄	1	1	yes	characteristic, complemented, Sylow
45	C₂³	—	77	1	no	—
46	C₂×C₄	—	7	1	no	—
47	C₂×C₄	—	11	1	no	—
48	C₇	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	77	2	no	—
50	C₂²	—	154	1	no	—
51	C₂²	—	154	1	no	—
52	C₂²	D₇×D₁₁	1	1	yes	characteristic
53	C₄	—	14	1	no	—
54	C₄	—	22	1	no	—
55	C₂	—	154	2	no	—
56	C₂	D₇₇⋊C₄	1	1	yes	characteristic
57	C₂	#10597506	1	1	yes	characteristic
58	C₂	#10597507	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₇×D₁₁
commutator	C₂×C₇×C₁₁	C₂×C₄
Fitting, socle, cosocle	C₂²×C₇×C₁₁	C₂²
prime radical	C₂²×D₇₇	C₂
hypocenter	C₇×C₁₁	C₂²⋊C₄
LCS	G ▷ C₂×C₇×C₁₁ ▷ C₇×C₁₁
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₁×Dic₇	C₂	semidirect
3. C₂×C₇×Dic₁₁	C₂	semidirect
7. C₂×D₇₇	C₄	semidirect
18. C₂×Dic₁₁	D₇	semidirect
19. C₇×C₁₁	C₂²⋊C₄	semidirect
22. C₂×Dic₇	D₁₁	semidirect
44. C₁₁	(C₂×D₇)⋊C₄	semidirect
48. C₇	(C₂×D₁₁)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₇₇	2
C₂×C₁₁×Dic₇	2
C₂×C₇×Dic₁₁	2
(C₂×D₁₁)⋊C₄	7
(C₂×D₇)⋊C₄	11