Subgroups — #13183218

Order 1760.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11725035	C₂	1	2	yes	maximal, complemented
2	C₅×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11724935	C₂	1	2	yes	maximal, complemented
4	D₄₄₀	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₅₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×Q₈×C₁₁	C₂²	1	2	yes	—
7	D₂₂₀	—	2	2	no	—
8	D₂₂₀	C₂²	1	2	yes	—
9	C₄×D₅₅	—	2	2	no	—
10	C₄×D₅₅	C₂²	1	1	yes	characteristic
11	C₅×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₁₁⋊#168	C₂²	1	1	yes	characteristic
13	#66972	—	5	1	no	maximal, Hall
14	C₂×D₅₅	—	2	2	no	—
15	C₂×D₅₅	D₄	1	1	yes	characteristic
16	C₄×C₅×C₁₁	—	2	2	no	—
17	C₄×C₅×C₁₁	C₂³	1	1	yes	characteristic
18	Dic₅₅	D₄	1	1	yes	characteristic
19	#4343	—	5	2	no	—
20	C₁₁×Q₁₆	D₅	1	1	yes	characteristic, complemented
21	#4325	—	5	2	no	—
22	D₈₈	—	5	1	no	—
23	C₈×D₁₁	—	5	1	no	—
24	#4065	—	11	1	no	maximal, Hall
25	C₂×C₅×C₁₁	C₂×D₄	1	1	yes	characteristic
26	D₅₅	—	2	1	no	—
27	D₅₅	—	4	2	no	—
28	Q₈×C₁₁	C₂×D₅	1	2	yes	—
29	D₄₄	—	5	2	no	—
30	D₄₄	—	10	2	no	—
31	C₄×D₁₁	—	5	1	no	—
32	C₄×D₁₁	—	10	2	no	—
33	C₈×C₁₁	C₂×D₅	1	1	yes	characteristic
34	C₁₁⋊C₈	—	5	1	no	—
35	#713	—	11	2	no	—
36	C₅×Q₁₆	D₁₁	1	1	yes	characteristic, complemented
37	#688	—	11	2	no	—
38	D₄₀	—	11	1	no	—
39	C₈×D₅	—	11	1	no	—
40	C₅×C₁₁	#135	1	1	yes	characteristic, complemented, Hall
41	C₂×D₁₁	—	5	1	no	—
42	C₂×D₁₁	—	10	2	no	—
43	C₄×C₁₁	—	2	2	no	—
44	C₄×C₁₁	C₂²×D₅	1	1	yes	characteristic
45	Dic₁₁	—	5	1	no	—
46	C₅×Q₈	C₂×D₁₁	1	2	yes	—
47	D₂₀	—	11	2	no	—
48	D₂₀	—	22	2	no	—
49	C₄×D₅	—	11	1	no	—
50	C₄×D₅	—	22	2	no	—
51	C₅×C₈	C₂×D₁₁	1	1	yes	characteristic
52	#168	—	11	1	no	—
53	#135	—	55	1	no	Sylow
54	C₂×C₁₁	D₄×D₅	1	1	yes	characteristic
55	D₁₁	—	10	1	no	—
56	D₁₁	—	20	2	no	—
57	C₂×D₅	—	11	1	no	—
58	C₂×D₅	—	22	2	no	—
59	C₄×C₅	—	2	2	no	—
60	C₄×C₅	C₂²×D₁₁	1	1	yes	characteristic
61	Dic₅	—	11	1	no	—
62	#41	—	55	2	no	—
63	Q₁₆	D₅₅	1	1	yes	characteristic, complemented
64	SD₁₆	—	55	2	no	—
65	D₈	—	55	1	no	—
66	C₂×C₈	—	55	1	no	—
67	C₁₁	#4065	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₅	D₄×D₁₁	1	1	yes	characteristic
69	D₅	—	22	1	no	—
70	D₅	—	44	2	no	—
71	Q₈	C₂×D₅₅	1	2	yes	—
72	D₄	—	55	2	no	—
73	D₄	—	110	2	no	—
74	C₂×C₄	—	55	1	no	—
75	C₂×C₄	—	110	2	no	—
76	C₈	—	55	1	no	—
77	C₈	C₂×D₅₅	1	1	yes	characteristic
78	C₅	#66972	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	55	1	no	—
80	C₂²	—	110	2	no	—
81	C₄	—	2	2	no	—
82	C₄	—	55	1	no	—
83	C₄	C₂²×D₅₅	1	1	yes	characteristic
84	C₂	—	110	1	no	—
85	C₂	—	220	2	no	—
86	C₂	D₄×D₅₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₅₅
commutator, cosocle	C₄×C₅×C₁₁	C₂³
Frattini	C₄	C₂²×D₅₅
Fitting	C₅×C₁₁×Q₁₆	C₂
socle	C₂×C₅×C₁₁	C₂×D₄
hypercenter	Q₁₆	D₅₅
hypocenter	C₅×C₁₁	#135
LCS	G ▷ C₄×C₅×C₁₁ ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #11725035	C₂	semidirect
2. C₅×C₁₁×Q₁₆	C₂	semidirect
3. #11724935	C₂	semidirect
4. D₄₄₀	C₂	semidirect
5. C₈×D₅₅	C₂	semidirect
20. C₁₁×Q₁₆	D₅	semidirect
36. C₅×Q₁₆	D₁₁	semidirect
40. C₅×C₁₁	#135	semidirect
63. Q₁₆	D₅₅	semidirect
67. C₁₁	#4065	semidirect
78. C₅	#66972	semidirect

Maximal subgroups

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

subgroup	index
#11725035	2
C₅×C₁₁×Q₁₆	2
#11724935	2
D₄₄₀	2
C₈×D₅₅	2
#66972	5
#4065	11