Subgroups — #13183168

Order 1760.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₁₀	C₂	1	2	yes	maximal
2	C₄²×C₅×C₁₁	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₅₅	C₂	1	4	yes	maximal
4	C₂×C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅×C₁₁	C₂²	1	2	yes	—
6	C₂×Dic₅₅	C₂²	1	4	yes	—
7	Dic₁₁₀	—	2	4	no	—
8	#66922	—	5	1	no	maximal, Hall
9	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
10	C₄×C₅×C₁₁	D₄	1	2	yes	—
11	C₄×C₅×C₁₁	Q₈	1	4	yes	complemented
12	Dic₅₅	—	2	4	no	—
13	C₂×Dic₂₂	—	5	2	no	—
14	C₄²×C₁₁	D₅	1	1	yes	characteristic
15	C₄⋊Dic₁₁	—	5	4	no	—
16	#4015	—	11	1	no	maximal, Hall
17	C₂×C₅×C₁₁	C₂×D₄	1	1	yes	characteristic
18	C₂×C₅×C₁₁	C₂×Q₈	1	2	yes	—
19	C₂×C₄×C₁₁	C₂×D₅	1	2	yes	—
20	C₂×C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
21	C₂×Dic₁₁	—	5	4	no	—
22	Dic₂₂	—	10	4	no	—
23	C₂×Dic₁₀	—	11	2	no	—
24	C₄²×C₅	D₁₁	1	1	yes	characteristic
25	C₄⋊Dic₅	—	11	4	no	—
26	C₅×C₁₁	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
27	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
28	C₄×C₁₁	Dic₁₀	1	4	yes	complemented
29	C₄×C₁₁	D₂₀	1	2	yes	—
30	Dic₁₁	—	10	4	no	—
31	C₂×C₄×C₅	C₂×D₁₁	1	2	yes	—
32	C₂×C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
33	C₂×Dic₅	—	11	4	no	—
34	Dic₁₀	—	22	4	no	—
35	C₄⋊Q₈	—	55	1	no	Sylow
36	C₂×C₁₁	C₂×Dic₁₀	1	2	yes	—
37	C₂×C₁₁	C₂×D₂₀	1	1	yes	characteristic
38	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
39	C₄×C₅	Dic₂₂	1	4	yes	complemented
40	C₄×C₅	D₄₄	1	2	yes	—
41	Dic₅	—	22	4	no	—
42	C₂×Q₈	—	55	2	no	—
43	C₄⋊C₄	—	55	4	no	—
44	C₄²	D₅₅	1	1	yes	characteristic
45	C₁₁	#4015	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₅	C₂×Dic₂₂	1	2	yes	—
47	C₂×C₅	C₂×D₄₄	1	1	yes	characteristic
48	Q₈	—	110	4	no	—
49	C₂×C₄	—	55	4	no	—
50	C₂×C₄	C₂×D₅₅	1	2	yes	—
51	C₂×C₄	C₂×D₅₅	1	1	yes	characteristic
52	C₅	#66922	1	1	yes	characteristic, complemented, Sylow
53	C₂²	C₂²×D₅₅	1	1	yes	characteristic
54	C₄	—	110	4	no	—
55	C₄	Dic₁₁₀	1	4	yes	complemented
56	C₄	D₂₂₀	1	2	yes	—
57	C₂	C₂×Dic₁₁₀	1	2	yes	—
58	C₂	C₂×D₂₂₀	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅₅
commutator, socle, prime radical, cosocle	C₂²×C₅×C₁₁	C₂³
Fitting	C₄²×C₅×C₁₁	C₂
hypercenter	C₄²	D₅₅
hypocenter	C₅×C₁₁	C₄⋊Q₈
LCS	G ▷ C₂²×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
11. C₄×C₅×C₁₁	Q₈	semidirect
26. C₅×C₁₁	C₄⋊Q₈	semidirect
28. C₄×C₁₁	Dic₁₀	semidirect
39. C₄×C₅	Dic₂₂	semidirect
45. C₁₁	#4015	semidirect
52. C₅	#66922	semidirect
55. C₄	Dic₁₁₀	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
C₄²×C₅×C₁₁	2
C₄⋊Dic₅₅	2
#66922	5
#4015	11