Subgroups — #13182733

Order 1760.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₁₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×Dic₁₁	C₂	1	1	yes	maximal, characteristic
4	Dic₁₁⋊Dic₅	C₂	1	4	yes	maximal
5	C₂×C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×Dic₅₅	C₂²	1	2	yes	—
7	Dic₁₁₀	—	2	2	no	—
8	C₂×C₁₁×Dic₅	C₂²	1	2	yes	—
9	C₁₁×Dic₁₀	—	2	2	no	—
10	C₂×C₅×Dic₁₁	C₂²	1	2	yes	—
11	#66997	—	5	1	no	maximal, Hall
12	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
13	C₄×C₅×C₁₁	D₄	1	2	yes	—
14	Dic₅₅	—	2	2	no	—
15	C₅×Dic₁₁	Q₈	1	4	yes	complemented
16	C₁₁×Dic₅	—	2	2	no	—
17	C₂×Q₈×C₁₁	—	5	1	no	—
18	C₂×Dic₂₂	—	5	1	no	—
19	Dic₁₁⋊C₄	—	5	4	no	—
20	C₄×Dic₁₁	D₅	1	1	yes	characteristic
21	#4015	—	11	1	no	maximal, Hall
22	C₂×C₅×C₁₁	C₂×D₄	1	1	yes	characteristic
23	C₂×C₅×C₁₁	C₂×Q₈	1	2	yes	—
24	Q₈×C₁₁	—	10	2	no	—
25	C₂×C₄×C₁₁	—	5	2	no	—
26	C₂×C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
27	C₂×Dic₁₁	—	5	2	no	—
28	C₂×Dic₁₁	C₂×D₅	1	2	yes	—
29	Dic₂₂	—	10	2	no	—
30	C₂×Dic₁₀	—	11	1	no	—
31	C₂×Dic₁₀	D₁₁	1	1	yes	characteristic
32	C₄²×C₅	—	11	1	no	—
33	C₄⋊Dic₅	—	11	4	no	—
34	C₅×C₁₁	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
35	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
36	C₄×C₁₁	—	10	2	no	—
37	C₄×C₁₁	D₂₀	1	2	yes	—
38	Dic₁₁	—	10	2	no	—
39	Dic₁₁	Dic₁₀	1	4	yes	complemented
40	C₂×C₄×C₅	—	11	2	no	—
41	C₂×C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
42	C₂×Dic₅	—	11	2	no	—
43	C₂×Dic₅	C₂×D₁₁	1	2	yes	—
44	Dic₁₀	—	2	2	no	—
45	Dic₁₀	—	22	2	no	—
46	C₄⋊Q₈	—	55	1	no	Sylow
47	C₂×C₁₁	C₂×Dic₁₀	1	2	yes	—
48	C₂×C₁₁	C₂×D₂₀	1	1	yes	characteristic
49	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
50	C₄×C₅	—	11	4	no	—
51	C₄×C₅	C₂²⋊D₁₁	1	2	yes	—
52	Dic₅	—	2	2	no	—
53	Dic₅	—	22	2	no	—
54	C₂×Q₈	—	5	1	no	—
55	C₂×Q₈	—	55	1	no	—
56	C₄⋊C₄	—	55	4	no	—
57	C₄²	—	11	1	no	—
58	C₁₁	#4015	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₅	Q₈×D₁₁	1	2	yes	—
60	C₂×C₅	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
61	Q₈	—	10	2	no	—
62	Q₈	—	110	2	no	—
63	C₂×C₄	—	5	2	no	—
64	C₂×C₄	—	11	2	no	—
65	C₂×C₄	—	55	2	no	—
66	C₂×C₄	D₅×D₁₁	1	1	yes	characteristic
67	C₅	#66997	1	1	yes	characteristic, complemented, Sylow
68	C₂²	C₂×D₅×D₁₁	1	1	yes	characteristic
69	C₄	—	10	2	no	—
70	C₄	—	11	4	no	—
71	C₄	—	110	2	no	—
72	C₄	#89169	1	2	yes	—
73	C₂	D₁₁×Dic₁₀	1	2	yes	—
74	C₂	C₂×#89169	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×D₁₁
commutator, socle, prime radical, cosocle	C₂²×C₅×C₁₁	C₂³
Fitting	C₂×C₄×C₅×C₁₁	C₂²
hypercenter	C₂×C₄	D₅×D₁₁
hypocenter	C₅×C₁₁	C₄⋊Q₈
LCS	G ▷ C₂²×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
15. C₅×Dic₁₁	Q₈	semidirect
34. C₅×C₁₁	C₄⋊Q₈	semidirect
39. Dic₁₁	Dic₁₀	semidirect
58. C₁₁	#4015	semidirect
67. C₅	#66997	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₁₀	2
C₂×C₁₁×Dic₁₀	2
C₄×C₅×Dic₁₁	2
Dic₁₁⋊Dic₅	2
#66997	5
#4015	11