Subgroups — #13182652

Order 1760.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁⋊AGL(1,𝔽₅)	C₂	1	2	yes	maximal
2	C₂×C₄×D₅×C₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×D₅×C₁₁	C₂²	1	1	yes	characteristic
4	C₂×C₁₁⋊AGL(1,𝔽₅)	—	2	2	no	—
5	C₂×C₁₁⋊AGL(1,𝔽₅)	—	2	2	no	—
6	C₂×C₁₁⋊AGL(1,𝔽₅)	C₄	1	4	yes	complemented
7	C₂×C₄×C₅×C₁₁	C₄	1	1	yes	characteristic, complemented
8	C₂×C₁₁×Dic₅	C₄	1	1	yes	characteristic, complemented
9	C₄×D₅×C₁₁	—	2	2	no	—
10	C₁₁⋊#95	—	5	1	no	maximal, Hall
11	C₂²×C₅×C₁₁	C₂×C₄	1	1	yes	characteristic
12	C₂×D₅×C₁₁	C₂×C₄	1	2	yes	—
13	C₂×D₅×C₁₁	D₄	1	1	yes	characteristic
14	C₂×D₅×C₁₁	D₄	1	2	yes	—
15	C₂×D₅×C₁₁	Q₈	1	1	yes	characteristic
16	C₁₁⋊AGL(1,𝔽₅)	—	2	4	no	—
17	C₄×C₅×C₁₁	—	2	1	no	—
18	C₁₁×Dic₅	—	2	1	no	—
19	C₂²×C₄×C₁₁	—	5	1	no	—
20	C₂²×Dic₁₁	—	5	2	no	—
21	(C₂×AGL(1,𝔽₅))⋊C₄	—	11	1	no	maximal, Hall
22	C₂×C₅×C₁₁	C₄²	1	1	yes	characteristic
23	C₂×C₅×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₅×C₁₁	C₄⋊C₄	1	1	yes	characteristic
25	D₅×C₁₁	C₂²⋊C₄	1	2	yes	—
26	D₅×C₁₁	C₄⋊C₄	1	2	yes	—
27	C₂³×C₁₁	—	5	1	no	—
28	C₂×C₄×C₁₁	—	5	1	no	—
29	C₂×C₄×C₁₁	—	10	2	no	—
30	C₂×C₄×C₁₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
31	C₂×Dic₁₁	—	5	4	no	—
32	C₂×Dic₁₁	—	10	2	no	—
33	C₂×Dic₁₁	—	10	2	no	—
34	C₂²×AGL(1,𝔽₅)	—	11	2	no	—
35	C₂×C₄×D₅	D₁₁	1	1	yes	characteristic
36	C₅×C₁₁	#95	1	1	yes	characteristic, complemented, Hall
37	C₂²×C₁₁	—	5	2	no	—
38	C₂²×C₁₁	—	5	1	no	—
39	C₂²×C₁₁	—	5	1	no	—
40	C₂²×C₁₁	—	5	2	no	—
41	C₂²×C₁₁	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₄×C₁₁	—	2	1	no	—
43	C₄×C₁₁	—	10	1	no	—
44	Dic₁₁	—	10	4	no	—
45	C₂²×D₅	C₂×D₁₁	1	1	yes	characteristic
46	C₂×AGL(1,𝔽₅)	—	11	4	no	—
47	C₂×AGL(1,𝔽₅)	—	22	2	no	—
48	C₂×AGL(1,𝔽₅)	—	22	2	no	—
49	C₂×C₄×C₅	Dic₁₁	1	1	yes	characteristic, complemented
50	C₂×Dic₅	Dic₁₁	1	1	yes	characteristic, complemented
51	C₄×D₅	—	2	2	no	—
52	#95	—	55	1	no	Sylow
53	C₂×C₁₁	—	5	2	no	—
54	C₂×C₁₁	—	5	2	no	—
55	C₂×C₁₁	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₂×C₁₁	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂×C₁₁	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₂²×C₅	C₂×Dic₁₁	1	1	yes	characteristic
59	C₂×D₅	Dic₂₂	1	1	yes	characteristic
60	C₂×D₅	C₄×D₁₁	1	2	yes	—
61	C₂×D₅	D₄₄	1	1	yes	characteristic
62	C₂×D₅	C₂²⋊D₁₁	1	2	yes	—
63	AGL(1,𝔽₅)	—	22	4	no	—
64	C₄×C₅	—	2	1	no	—
65	Dic₅	—	2	1	no	—
66	C₂²×C₄	—	5	1	no	—
67	C₂²×C₄	—	55	2	no	—
68	C₁₁	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₅	C₄×Dic₁₁	1	1	yes	characteristic
70	C₂×C₅	C₄⋊Dic₁₁	1	1	yes	characteristic
71	C₂×C₅	C₂²⋊Dic₁₁	1	1	yes	characteristic
72	D₅	Dic₁₁⋊C₄	1	2	yes	—
73	D₅	(C₂×D₁₁)⋊C₄	1	2	yes	—
74	C₂³	—	5	1	no	—
75	C₂×C₄	—	5	1	no	—
76	C₂×C₄	—	10	2	no	—
77	C₂×C₄	—	55	4	no	—
78	C₂×C₄	—	110	2	no	—
79	C₂×C₄	—	110	2	no	—
80	C₂×C₄	C₁₁⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
81	C₅	C₁₁⋊#95	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	5	2	no	—
83	C₂²	—	5	1	no	—
84	C₂²	—	5	1	no	—
85	C₂²	—	5	2	no	—
86	C₂²	C₂×C₁₁⋊AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	10	1	no	—
89	C₄	—	110	4	no	—
90	C₂	—	5	2	no	—
91	C₂	—	5	2	no	—
92	C₂	C₄×C₁₁⋊AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₂	C₄⋊(C₁₁⋊AGL(1,𝔽₅))	1	1	yes	characteristic
94	C₂	C₂²⋊(C₁₁⋊AGL(1,𝔽₅))	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×C₁₁⋊AGL(1,𝔽₅)
commutator	C₂×C₅×C₁₁	C₄²
Fitting	C₂×C₄×C₅×C₁₁	C₄
socle	C₂²×C₅×C₁₁	C₂×C₄
prime radical, cosocle	C₂²×D₅×C₁₁	C₂²
hypercenter	C₂×C₄	C₁₁⋊AGL(1,𝔽₅)
hypocenter	C₅×C₁₁	#95
LCS	G ▷ C₂×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
6. C₂×C₁₁⋊AGL(1,𝔽₅)	C₄	semidirect
7. C₂×C₄×C₅×C₁₁	C₄	semidirect
8. C₂×C₁₁×Dic₅	C₄	semidirect
30. C₂×C₄×C₁₁	AGL(1,𝔽₅)	semidirect
36. C₅×C₁₁	#95	semidirect
49. C₂×C₄×C₅	Dic₁₁	semidirect
50. C₂×Dic₅	Dic₁₁	semidirect
68. C₁₁	(C₂×AGL(1,𝔽₅))⋊C₄	semidirect
80. C₂×C₄	C₁₁⋊AGL(1,𝔽₅)	semidirect
81. C₅	C₁₁⋊#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₁⋊AGL(1,𝔽₅)	2
C₂×C₄×D₅×C₁₁	2
C₁₁⋊#95	5
(C₂×AGL(1,𝔽₅))⋊C₄	11