Subgroups — #66935

Order 352.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄₄	C₂	1	2	yes	maximal, complemented
4	C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₁₁)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₁	C₂²	1	1	yes	characteristic
7	C₂²×D₁₁	—	2	2	no	—
8	C₂²×D₁₁	—	2	4	no	—
9	C₂²×D₁₁	C₂²	1	2	yes	—
10	C₂²×D₁₁	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₄×C₁₁	C₂²	1	2	yes	complemented
12	C₂²⋊D₁₁	—	2	2	no	—
13	C₂×Dic₁₁	C₂²	1	1	yes	characteristic, complemented
14	D₄₄	—	2	4	no	—
15	C₂²×C₁₁	—	2	2	no	—
16	C₂²×C₁₁	D₄	1	2	yes	complemented
17	C₂²×C₁₁	C₂³	1	1	yes	characteristic
18	C₂×D₁₁	—	2	8	no	—
19	C₂×D₁₁	—	2	2	no	—
20	C₂×D₁₁	—	2	2	no	—
21	C₂×D₁₁	—	2	2	no	—
22	C₂×D₁₁	—	2	1	no	—
23	C₂×D₁₁	D₄	1	4	yes	complemented
24	C₄×C₁₁	—	2	2	no	—
25	Dic₁₁	—	2	1	no	—
26	#120	—	11	1	no	maximal, Sylow
27	C₂×C₁₁	—	2	2	no	—
28	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₁₁	C₂×D₄	1	2	yes	—
30	D₁₁	—	2	4	no	—
31	D₁₁	—	4	1	no	—
32	C₂⁴	—	11	1	no	—
33	C₂×D₄	—	11	1	no	—
34	C₂×D₄	—	11	2	no	—
35	C₂²⋊C₄	—	11	2	no	—
36	C₂²⋊C₄	D₁₁	1	1	yes	characteristic, complemented
37	C₁₁	#120	1	1	yes	characteristic, complemented, Sylow
38	C₂³	—	11	2	no	—
39	C₂³	—	11	1	no	—
40	C₂³	—	22	4	no	—
41	C₂³	—	22	2	no	—
42	C₂³	C₂×D₁₁	1	1	yes	characteristic
43	D₄	—	22	4	no	—
44	D₄	—	22	2	no	—
45	C₂×C₄	—	11	1	no	—
46	C₂×C₄	C₂×D₁₁	1	2	yes	complemented
47	C₂²	—	2	2	no	—
48	C₂²	—	11	4	no	—
49	C₂²	—	22	8	no	—
50	C₂²	—	22	2	no	—
51	C₂²	—	22	2	no	—
52	C₂²	—	22	1	no	—
53	C₂²	—	22	2	no	—
54	C₂²	D₄₄	1	2	yes	complemented
55	C₂²	C₂²×D₁₁	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	22	1	no	—
58	C₂	—	2	2	no	—
59	C₂	—	22	4	no	—
60	C₂	—	44	1	no	—
61	C₂	C₂×D₄₄	1	1	yes	characteristic
62	C₂	D₄×D₁₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₁
commutator, socle, cosocle	C₂²×C₁₁	C₂³
Fitting	C₁₁×C₂²⋊C₄	C₂
hypercenter	C₂²⋊C₄	D₁₁
hypocenter	C₁₁	#120
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₁₁	C₂	semidirect
2. C₂×C₂²⋊D₁₁	C₂	semidirect
3. C₂×D₄₄	C₂	semidirect
4. C₁₁×C₂²⋊C₄	C₂	semidirect
5. (C₂×D₁₁)⋊C₄	C₂	semidirect
10. C₂²×D₁₁	C₂²	semidirect
11. C₂×C₄×C₁₁	C₂²	semidirect
13. C₂×Dic₁₁	C₂²	semidirect
16. C₂²×C₁₁	D₄	semidirect
23. C₂×D₁₁	D₄	semidirect
36. C₂²⋊C₄	D₁₁	semidirect
37. C₁₁	#120	semidirect
46. C₂×C₄	C₂×D₁₁	semidirect
54. C₂²	D₄₄	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₁₁	2
C₂×C₂²⋊D₁₁	2
C₂×D₄₄	2
C₁₁×C₂²⋊C₄	2
(C₂×D₁₁)⋊C₄	2
#120	11