Subgroups — #13182656

Order 1760.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₅×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₁⋊AGL(1,𝔽₅))	C₂	1	1	yes	maximal, characteristic
3	D₅⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₁⋊AGL(1,𝔽₅)	—	2	1	no	—
5	C₅×Q₈×C₁₁	C₄	1	1	yes	characteristic, complemented
6	C₄×D₅×C₁₁	—	2	1	no	—
7	C₄×D₅×C₁₁	C₂²	1	1	yes	characteristic
8	C₁₁×Dic₁₀	—	2	1	no	—
9	C₁₁×Dic₁₀	C₄	1	1	yes	characteristic, complemented
10	C₁₁⋊#170	—	2	1	no	—
11	#66899	—	5	1	no	maximal, Hall
12	C₂×D₅×C₁₁	D₄	1	1	yes	characteristic
13	C₁₁⋊AGL(1,𝔽₅)	—	4	1	no	—
14	C₄×C₅×C₁₁	—	2	1	no	—
15	C₄×C₅×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₁₁×Dic₅	—	2	1	no	—
17	C₁₁×Dic₅	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₁₁	—	5	1	no	—
19	C₄⋊Dic₁₁	—	5	1	no	—
20	C₂×C₁₁⋊C₈	—	5	1	no	—
21	#4009	—	11	1	no	maximal, Hall
22	C₂×C₅×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
23	D₅×C₁₁	SD₁₆	1	1	yes	characteristic
24	D₅×C₁₁	Q₁₆	1	1	yes	characteristic
25	Q₈×C₁₁	—	5	1	no	—
26	Q₈×C₁₁	—	10	1	no	—
27	Q₈×C₁₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
28	C₂×C₄×C₁₁	—	5	1	no	—
29	C₂×C₄×C₁₁	—	10	1	no	—
30	C₂×Dic₁₁	—	10	1	no	—
31	C₁₁⋊C₈	—	10	1	no	—
32	Q₈×D₅	D₁₁	1	1	yes	characteristic
33	C₄⋊AGL(1,𝔽₅)	—	11	1	no	—
34	D₅⋊C₈	—	11	1	no	—
35	C₅×C₁₁	#103	1	1	yes	characteristic, complemented, Hall
36	C₂²×C₁₁	—	5	1	no	—
37	C₄×C₁₁	—	2	1	no	—
38	C₄×C₁₁	—	5	1	no	—
39	C₄×C₁₁	—	10	1	no	—
40	C₄×C₁₁	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
41	Dic₁₁	—	20	1	no	—
42	C₂×AGL(1,𝔽₅)	—	22	1	no	—
43	C₅×Q₈	Dic₁₁	1	1	yes	characteristic, complemented
44	C₄×D₅	—	2	1	no	—
45	C₄×D₅	C₂×D₁₁	1	1	yes	characteristic
46	Dic₁₀	—	2	1	no	—
47	Dic₁₀	Dic₁₁	1	1	yes	characteristic, complemented
48	#170	—	22	1	no	—
49	#103	—	55	1	no	Sylow
50	C₂×C₁₁	—	5	1	no	—
51	C₂×C₁₁	—	5	1	no	—
52	C₂×C₁₁	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₂×D₅	C₂²⋊D₁₁	1	1	yes	characteristic
54	AGL(1,𝔽₅)	—	44	1	no	—
55	C₄×C₅	—	2	1	no	—
56	C₄×C₅	C₂×Dic₁₁	1	1	yes	characteristic
57	Dic₅	—	2	1	no	—
58	Dic₅	C₂²⋊D₁₁	1	1	yes	characteristic
59	C₂×Q₈	—	5	1	no	—
60	C₂×C₈	—	55	1	no	—
61	C₄⋊C₄	—	55	1	no	—
62	C₁₁	#4009	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₅	C₂²⋊Dic₁₁	1	1	yes	characteristic
64	D₅	#4325	1	1	yes	characteristic
65	D₅	#4326	1	1	yes	characteristic
66	Q₈	—	5	1	no	—
67	Q₈	—	10	1	no	—
68	Q₈	C₁₁⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
69	C₂×C₄	—	5	1	no	—
70	C₂×C₄	—	10	1	no	—
71	C₂×C₄	—	110	1	no	—
72	C₈	—	110	1	no	—
73	C₅	#66899	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	5	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	5	1	no	—
77	C₄	—	10	1	no	—
78	C₄	—	220	1	no	—
79	C₄	C₂×C₁₁⋊AGL(1,𝔽₅)	1	1	yes	characteristic
80	C₂	—	5	1	no	—
81	C₂	—	5	1	no	—
82	C₂	C₂²⋊(C₁₁⋊AGL(1,𝔽₅))	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊(C₁₁⋊AGL(1,𝔽₅))
commutator	C₄×C₅×C₁₁	C₂×C₄
Frattini	C₄	C₂×C₁₁⋊AGL(1,𝔽₅)
Fitting	C₅×Q₈×C₁₁	C₄
socle	C₂×C₅×C₁₁	C₂²⋊C₄
prime radical	C₂×D₅×C₁₁	D₄
cosocle	C₄×D₅×C₁₁	C₂²
hypercenter	Q₈	C₁₁⋊AGL(1,𝔽₅)
hypocenter	C₅×C₁₁	#103
LCS	G ▷ C₄×C₅×C₁₁ ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
5. C₅×Q₈×C₁₁	C₄	semidirect
9. C₁₁×Dic₁₀	C₄	semidirect
27. Q₈×C₁₁	AGL(1,𝔽₅)	semidirect
35. C₅×C₁₁	#103	semidirect
43. C₅×Q₈	Dic₁₁	semidirect
47. Dic₁₀	Dic₁₁	semidirect
62. C₁₁	#4009	semidirect
68. Q₈	C₁₁⋊AGL(1,𝔽₅)	semidirect
73. C₅	#66899	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₅×C₁₁	2
C₄⋊(C₁₁⋊AGL(1,𝔽₅))	2
D₅⋊(C₁₁⋊C₈)	2
#66899	5
#4009	11