Subgroups — #13182657

Order 1760.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₁₁⋊AGL(1,𝔽₅))	C₂	1	2	yes	maximal, complemented
3	C₂³×C₅×C₁₁	C₄	1	1	yes	characteristic, complemented
4	C₂²×D₅×C₁₁	C₂²	1	1	yes	characteristic
5	C₂×C₁₁⋊AGL(1,𝔽₅)	—	2	2	no	—
6	C₁₁×C₂²⋊D₅	—	2	2	no	—
7	C₂×C₁₁×Dic₅	C₄	1	1	yes	characteristic, complemented
8	#66898	—	5	1	no	maximal, Hall
9	C₂²×C₅×C₁₁	—	2	1	no	—
10	C₂²×C₅×C₁₁	—	4	1	no	—
11	C₂²×C₅×C₁₁	C₂×C₄	1	1	yes	characteristic
12	C₂×D₅×C₁₁	—	4	1	no	—
13	C₂×D₅×C₁₁	D₄	1	2	yes	—
14	C₁₁⋊AGL(1,𝔽₅)	—	4	2	no	—
15	C₁₁×Dic₅	—	2	1	no	—
16	C₂×D₄×C₁₁	—	5	1	no	—
17	C₂²⋊Dic₁₁	—	5	2	no	—
18	#4011	—	11	1	no	maximal, Hall
19	C₂×C₅×C₁₁	—	2	1	no	—
20	C₂×C₅×C₁₁	—	4	1	no	—
21	C₂×C₅×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
22	D₅×C₁₁	—	2	2	no	—
23	C₂³×C₁₁	—	5	1	no	—
24	C₂³×C₁₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
25	D₄×C₁₁	—	10	2	no	—
26	C₂×C₄×C₁₁	—	5	1	no	—
27	C₂×Dic₁₁	—	10	2	no	—
28	C₂×C₂²⋊D₅	D₁₁	1	1	yes	characteristic
29	C₂²⋊AGL(1,𝔽₅)	—	11	2	no	—
30	C₅×C₁₁	#99	1	1	yes	characteristic, complemented, Hall
31	C₂²×C₁₁	—	2	1	no	—
32	C₂²×C₁₁	—	4	1	no	—
33	C₂²×C₁₁	—	5	2	no	—
34	C₂²×C₁₁	—	20	1	no	—
35	C₂²×C₁₁	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
36	C₄×C₁₁	—	10	1	no	—
37	Dic₁₁	—	20	2	no	—
38	C₂³×C₅	Dic₁₁	1	1	yes	characteristic, complemented
39	C₂²×D₅	C₂×D₁₁	1	1	yes	characteristic
40	C₂×AGL(1,𝔽₅)	—	22	2	no	—
41	C₂²⋊D₅	—	2	2	no	—
42	C₂×Dic₅	Dic₁₁	1	1	yes	characteristic, complemented
43	#99	—	55	1	no	Sylow
44	C₂×C₁₁	—	2	1	no	—
45	C₂×C₁₁	—	4	1	no	—
46	C₂×C₁₁	—	10	2	no	—
47	C₂×C₁₁	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
48	C₂²×C₅	—	2	1	no	—
49	C₂²×C₅	—	4	1	no	—
50	C₂²×C₅	C₂×Dic₁₁	1	1	yes	characteristic
51	C₂×D₅	—	4	1	no	—
52	C₂×D₅	C₂²⋊D₁₁	1	2	yes	—
53	AGL(1,𝔽₅)	—	44	2	no	—
54	Dic₅	—	2	1	no	—
55	C₂×D₄	—	5	1	no	—
56	C₂²⋊C₄	—	55	2	no	—
57	C₁₁	#4011	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₅	—	2	1	no	—
59	C₂×C₅	—	4	1	no	—
60	C₂×C₅	C₂²⋊Dic₁₁	1	1	yes	characteristic
61	D₅	—	2	2	no	—
62	C₂³	—	5	1	no	—
63	C₂³	C₁₁⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
64	D₄	—	10	2	no	—
65	C₂×C₄	—	5	1	no	—
66	C₂×C₄	—	110	2	no	—
67	C₅	#66898	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	2	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	5	2	no	—
71	C₂²	—	20	1	no	—
72	C₂²	C₂×C₁₁⋊AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₄	—	10	1	no	—
74	C₄	—	220	2	no	—
75	C₂	—	2	1	no	—
76	C₂	—	4	1	no	—
77	C₂	—	10	2	no	—
78	C₂	C₂²⋊(C₁₁⋊AGL(1,𝔽₅))	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊(C₁₁⋊AGL(1,𝔽₅))
commutator	C₂²×C₅×C₁₁	C₂×C₄
Frattini	C₂²	C₂×C₁₁⋊AGL(1,𝔽₅)
Fitting	C₂³×C₅×C₁₁	C₄
socle	C₂×C₅×C₁₁	C₂²⋊C₄
prime radical	C₂×C₁₁×C₂²⋊D₅	C₂
cosocle	C₂²×D₅×C₁₁	C₂²
hypercenter	C₂³	C₁₁⋊AGL(1,𝔽₅)
hypocenter	C₅×C₁₁	#99
LCS	G ▷ C₂²×C₅×C₁₁ ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂²⋊(C₁₁⋊AGL(1,𝔽₅))	C₂	semidirect
3. C₂³×C₅×C₁₁	C₄	semidirect
7. C₂×C₁₁×Dic₅	C₄	semidirect
24. C₂³×C₁₁	AGL(1,𝔽₅)	semidirect
30. C₅×C₁₁	#99	semidirect
38. C₂³×C₅	Dic₁₁	semidirect
42. C₂×Dic₅	Dic₁₁	semidirect
57. C₁₁	#4011	semidirect
63. C₂³	C₁₁⋊AGL(1,𝔽₅)	semidirect
67. C₅	#66898	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₁×C₂²⋊D₅	2
C₂²⋊(C₁₁⋊AGL(1,𝔽₅))	2
#66898	5
#4011	11