Subgroups — #13182881

Order 1760.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89170	C₂	1	2	yes	maximal
2	C₄⋊Dic₅₅	C₂	1	1	yes	maximal, characteristic
3	C₁₁×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅×Dic₁₁	C₂	1	1	yes	maximal, characteristic
5	Dic₁₁⋊Dic₅	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×Dic₅₅	C₂²	1	2	yes	—
8	C₂×C₁₁×Dic₅	C₂²	1	2	yes	—
9	C₂×C₅×Dic₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₅×Dic₁₁	C₂²	1	1	yes	characteristic
11	#89170	—	2	4	no	—
12	#66941	—	5	1	no	maximal, Hall
13	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
14	C₄×C₅×C₁₁	Q₈	1	2	yes	complemented
15	Dic₅₅	—	2	2	no	—
16	C₅×Dic₁₁	D₄	1	2	yes	—
17	C₅×Dic₁₁	Q₈	1	2	yes	complemented
18	C₁₁×Dic₅	—	2	2	no	—
19	C₂×Dic₂₂	—	5	2	no	—
20	C₁₁×C₄⋊C₄	—	5	1	no	—
21	C₄⋊Dic₁₁	—	5	1	no	—
22	Dic₁₁⋊C₄	—	5	2	no	—
23	C₄×Dic₁₁	D₅	1	1	yes	characteristic
24	#4015	—	11	1	no	maximal, Hall
25	C₂×C₅×C₁₁	C₂×D₄	1	1	yes	characteristic
26	C₂×C₅×C₁₁	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₅×C₁₁	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₁₁	—	5	2	no	—
29	C₂×C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
30	C₂×Dic₁₁	—	5	2	no	—
31	C₂×Dic₁₁	C₂×D₅	1	1	yes	characteristic
32	C₂×Dic₁₁	C₂×D₅	1	1	yes	characteristic
33	Dic₂₂	—	10	4	no	—
34	C₂×Dic₁₀	—	11	2	no	—
35	C₄²×C₅	—	11	1	no	—
36	C₄⋊Dic₅	—	11	1	no	—
37	C₄⋊Dic₅	—	11	2	no	—
38	C₄⋊Dic₅	D₁₁	1	1	yes	characteristic
39	C₅×C₁₁	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
40	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
41	C₄×C₁₁	—	10	2	no	—
42	C₄×C₁₁	Dic₁₀	1	2	yes	complemented
43	Dic₁₁	—	10	2	no	—
44	Dic₁₁	Dic₁₀	1	2	yes	complemented
45	Dic₁₁	D₂₀	1	2	yes	—
46	C₂×C₄×C₅	—	11	1	no	—
47	C₂×C₄×C₅	—	11	1	no	—
48	C₂×C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
49	C₂×Dic₅	—	11	2	no	—
50	C₂×Dic₅	C₂×D₁₁	1	2	yes	—
51	Dic₁₀	—	22	4	no	—
52	C₄⋊Q₈	—	55	1	no	Sylow
53	C₂×C₁₁	C₂×Dic₁₀	1	1	yes	characteristic
54	C₂×C₁₁	C₂×Dic₁₀	1	1	yes	characteristic
55	C₂×C₁₁	C₂×D₂₀	1	1	yes	characteristic
56	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
57	C₄×C₅	—	11	2	no	—
58	C₄×C₅	—	11	2	no	—
59	C₄×C₅	Dic₂₂	1	2	yes	complemented
60	Dic₅	—	2	2	no	—
61	Dic₅	—	22	2	no	—
62	C₂×Q₈	—	55	2	no	—
63	C₄⋊C₄	—	5	1	no	—
64	C₄⋊C₄	—	55	2	no	—
65	C₄⋊C₄	—	55	1	no	—
66	C₄²	—	11	1	no	—
67	C₁₁	#4015	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₅	C₂×Dic₂₂	1	1	yes	characteristic
69	C₂×C₅	D₄×D₁₁	1	1	yes	characteristic
70	C₂×C₅	Q₈×D₁₁	1	1	yes	characteristic
71	Q₈	—	110	4	no	—
72	C₂×C₄	—	5	2	no	—
73	C₂×C₄	—	11	1	no	—
74	C₂×C₄	—	11	1	no	—
75	C₂×C₄	—	55	2	no	—
76	C₂×C₄	D₅×D₁₁	1	1	yes	characteristic
77	C₅	#66941	1	1	yes	characteristic, complemented, Sylow
78	C₂²	C₂×D₅×D₁₁	1	1	yes	characteristic
79	C₄	—	10	2	no	—
80	C₄	—	11	2	no	—
81	C₄	—	11	2	no	—
82	C₄	—	110	2	no	—
83	C₄	#89170	1	2	yes	complemented
84	C₂	D₁₁×Dic₁₀	1	1	yes	characteristic
85	C₂	D₁₁×D₂₀	1	1	yes	characteristic
86	C₂	C₂×#89170	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
14. C₄×C₅×C₁₁	Q₈	semidirect
17. C₅×Dic₁₁	Q₈	semidirect
39. C₅×C₁₁	C₄⋊Q₈	semidirect
42. C₄×C₁₁	Dic₁₀	semidirect
44. Dic₁₁	Dic₁₀	semidirect
59. C₄×C₅	Dic₂₂	semidirect
67. C₁₁	#4015	semidirect
77. C₅	#66941	semidirect
83. C₄	#89170	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89170	2
C₄⋊Dic₅₅	2
C₁₁×C₄⋊Dic₅	2
C₄×C₅×Dic₁₁	2
Dic₁₁⋊Dic₅	2
#66941	5
#4015	11