Subgroups — #13183188

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	C₅×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₅₅	C₂	1	1	yes	maximal, characteristic
4	#11724930	C₂	1	2	yes	maximal
5	C₄×Dic₅₅	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅×C₁₁	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×Dic₅₅	C₂²	1	2	yes	—
9	C₂×Dic₅₅	C₂²	1	1	yes	characteristic
10	C₂×Dic₅₅	C₂²	1	1	yes	characteristic
11	Dic₁₁₀	—	2	4	no	—
12	#66941	—	5	1	no	maximal, Hall
13	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
14	C₄×C₅×C₁₁	—	2	2	no	—
15	C₄×C₅×C₁₁	Q₈	1	2	yes	complemented
16	Dic₅₅	—	2	2	no	—
17	Dic₅₅	D₄	1	2	yes	—
18	Dic₅₅	Q₈	1	2	yes	complemented
19	C₂×Dic₂₂	—	5	2	no	—
20	C₁₁×C₄⋊C₄	D₅	1	1	yes	characteristic
21	C₄⋊Dic₁₁	—	5	1	no	—
22	Dic₁₁⋊C₄	—	5	2	no	—
23	C₄×Dic₁₁	—	5	1	no	—
24	#4034	—	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	C₂×D₅	1	2	yes	—
29	C₂×C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
30	C₂×Dic₁₁	—	5	1	no	—
31	C₂×Dic₁₁	—	5	1	no	—
32	C₂×Dic₁₁	—	5	2	no	—
33	Dic₂₂	—	10	4	no	—
34	C₂×Dic₁₀	—	11	2	no	—
35	C₅×C₄⋊C₄	D₁₁	1	1	yes	characteristic
36	C₄⋊Dic₅	—	11	1	no	—
37	#683	—	11	2	no	—
38	C₄×Dic₅	—	11	1	no	—
39	C₅×C₁₁	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
40	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
41	C₄×C₁₁	—	2	2	no	—
42	C₄×C₁₁	Dic₁₀	1	2	yes	complemented
43	Dic₁₁	—	5	2	no	—
44	Dic₁₁	—	5	2	no	—
45	Dic₁₁	—	10	2	no	—
46	C₂×C₄×C₅	C₂×D₁₁	1	2	yes	—
47	C₂×C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
48	C₂×Dic₅	—	11	1	no	—
49	C₂×Dic₅	—	11	1	no	—
50	C₂×Dic₅	—	11	2	no	—
51	Dic₁₀	—	22	4	no	—
52	C₄⋊Q₈	—	55	1	no	Sylow
53	C₂×C₁₁	C₂×Dic₁₀	1	1	yes	characteristic
54	C₂×C₁₁	D₄×D₅	1	1	yes	characteristic
55	C₂×C₁₁	Q₈×D₅	1	1	yes	characteristic
56	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
57	C₄×C₅	—	2	2	no	—
58	C₄×C₅	Dic₂₂	1	2	yes	complemented
59	Dic₅	—	11	2	no	—
60	Dic₅	—	11	2	no	—
61	Dic₅	—	22	2	no	—
62	C₂×Q₈	—	55	2	no	—
63	C₄⋊C₄	—	55	2	no	—
64	C₄⋊C₄	—	55	1	no	—
65	C₄⋊C₄	D₅₅	1	1	yes	characteristic
66	C₄²	—	55	1	no	—
67	C₁₁	#4034	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	—	55	1	no	—
73	C₂×C₄	—	55	1	no	—
74	C₂×C₄	—	55	2	no	—
75	C₂×C₄	C₂×D₅₅	1	2	yes	—
76	C₂×C₄	C₂×D₅₅	1	1	yes	characteristic
77	C₅	#66941	1	1	yes	characteristic, complemented, Sylow
78	C₂²	C₂²×D₅₅	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	55	2	no	—
81	C₄	—	55	2	no	—
82	C₄	—	110	2	no	—
83	C₄	Dic₁₁₀	1	2	yes	complemented
84	C₂	C₂×Dic₁₁₀	1	1	yes	characteristic
85	C₂	D₄×D₅₅	1	1	yes	characteristic
86	C₂	Q₈×D₅₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
15. C₄×C₅×C₁₁	Q₈	semidirect
18. Dic₅₅	Q₈	semidirect
39. C₅×C₁₁	C₄⋊Q₈	semidirect
42. C₄×C₁₁	Dic₁₀	semidirect
58. C₄×C₅	Dic₂₂	semidirect
67. C₁₁	#4034	semidirect
77. C₅	#66941	semidirect
83. 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
C₅×C₁₁×C₄⋊C₄	2
C₄⋊Dic₅₅	2
#11724930	2
C₄×Dic₅₅	2
#66941	5
#4034	11