Subgroups — #72247347

Order 3960.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×A₄×C₁₁⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	A₄⋊AGL(1,𝔽₁₁)	—	3	3	no	maximal
3	#63159⋊C₅	—	3	1	no	maximal
4	C₃⋊(C₃⋊AGL(1,𝔽₁₁))	—	4	1	no	maximal
5	C₃⋊(C₁₁⋊S₄)	C₅	1	1	yes	maximal, characteristic, complemented, Hall
6	C₂²×C₃×C₁₁⋊C₅	D₃	1	1	yes	characteristic, complemented
7	C₂×C₃⋊AGL(1,𝔽₁₁)	—	3	1	no	—
8	A₄×C₁₁⋊C₅	D₃	1	3	yes	complemented
9	C₃⋊(Dic₁₁⋊C₅)	—	3	1	no	—
10	C₃²×C₁₁⋊C₅	—	4	1	no	Hall
11	C₂²⋊AGL(1,𝔽₁₁)	—	9	1	no	Hall
12	C₃×C₁₁×A₄	C₂×C₅	1	1	yes	characteristic, complemented
13	C₅×C₃⋊S₄	—	11	1	no	maximal, Hall
14	C₂×C₃×C₁₁⋊C₅	—	3	1	no	—
15	C₃⋊AGL(1,𝔽₁₁)	—	6	1	no	—
16	C₃⋊AGL(1,𝔽₁₁)	—	12	3	no	—
17	C₁₁⋊S₄	—	3	3	no	—
18	#63159	—	3	1	no	—
19	C₂²×C₁₁⋊C₅	C₃⋊D₃	1	1	yes	characteristic, complemented
20	C₂×AGL(1,𝔽₁₁)	—	9	1	no	—
21	Dic₁₁⋊C₅	—	9	1	no	—
22	C₃⋊D₃₃	—	4	1	no	—
23	C₃×C₅×A₄	—	11	1	no	—
24	C₃×C₁₁⋊C₅	—	4	3	no	—
25	C₃×C₁₁⋊C₅	S₄	1	1	yes	characteristic, complemented
26	C₂²×C₃×C₁₁	C₅×D₃	1	1	yes	characteristic, complemented
27	C₂×D₃₃	—	3	1	no	—
28	C₁₁×A₄	C₅×D₃	1	3	yes	complemented
29	Dic₃₃	—	3	1	no	—
30	C₅×S₄	—	33	3	no	—
31	C₅×#67	—	33	1	no	—
32	C₂×C₁₁⋊C₅	—	3	1	no	—
33	AGL(1,𝔽₁₁)	—	18	1	no	—
34	C₃²×C₁₁	—	4	1	no	Hall
35	C₅×C₃⋊D₃	—	44	1	no	—
36	C₂²⋊D₁₁	—	9	1	no	Hall
37	C₃⋊S₄	—	55	1	no	Hall
38	C₂×C₃×C₁₁	—	3	1	no	—
39	D₃₃	—	6	1	no	—
40	D₃₃	—	12	3	no	—
41	C₂²×C₃×C₅	—	11	1	no	—
42	C₂×C₅×D₃	—	33	1	no	—
43	C₅×A₄	—	11	3	no	—
44	C₅×Dic₃	—	33	1	no	—
45	C₁₁⋊C₅	C₃⋊S₄	1	1	yes	characteristic, complemented, Hall
46	C₃²×C₅	—	44	1	no	Hall
47	C₂²×C₁₁	C₅×C₃⋊D₃	1	1	yes	characteristic, complemented
48	C₂×D₁₁	—	9	1	no	—
49	Dic₁₁	—	9	1	no	—
50	C₅×D₄	—	99	1	no	Hall
51	C₃×A₄	AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented
52	C₃×C₁₁	—	4	3	no	—
53	C₃×C₁₁	C₅×S₄	1	1	yes	characteristic, complemented
54	C₂×C₃×C₅	—	33	1	no	—
55	C₅×D₃	—	66	1	no	—
56	C₅×D₃	—	132	3	no	—
57	S₄	—	33	3	no	—
58	#67	—	165	1	no	—
59	C₂×C₁₁	—	3	1	no	—
60	D₁₁	—	18	1	no	—
61	C₂²×C₅	—	11	1	no	—
62	C₂²×C₅	—	99	1	no	—
63	C₄×C₅	—	99	1	no	—
64	C₃⋊D₃	—	44	1	no	—
65	C₃×C₅	—	11	1	no	—
66	C₃×C₅	—	44	3	no	—
67	C₂²×C₃	C₃⋊AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented
68	C₂×D₃	—	33	1	no	—
69	A₄	C₃⋊AGL(1,𝔽₁₁)	1	3	yes	complemented
70	Dic₃	—	165	1	no	—
71	C₁₁	C₅×C₃⋊S₄	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₅	—	33	1	no	—
73	C₂×C₅	—	198	1	no	—
74	C₃²	—	4	1	no	Sylow
75	D₄	—	495	1	no	Sylow
76	C₂×C₃	—	3	1	no	—
77	D₃	—	330	1	no	—
78	D₃	—	660	3	no	—
79	C₅	—	11	1	no	Sylow
80	C₂²	—	495	1	no	—
81	C₂²	C₃⋊(C₃⋊AGL(1,𝔽₁₁))	1	1	yes	characteristic, complemented
82	C₄	—	495	1	no	—
83	C₃	—	4	3	no	—
84	C₃	A₄⋊AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented
85	C₂	—	3	1	no	—
86	C₂	—	990	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₁₁×A₄	C₂×C₅
Fitting, socle	C₂²×C₃×C₁₁	C₅×D₃
LCS	G ▷ C₃×C₁₁×A₄
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×A₄×C₁₁⋊C₅	C₂	semidirect
5. C₃⋊(C₁₁⋊S₄)	C₅	semidirect
6. C₂²×C₃×C₁₁⋊C₅	D₃	semidirect
8. A₄×C₁₁⋊C₅	D₃	semidirect
12. C₃×C₁₁×A₄	C₂×C₅	semidirect
19. C₂²×C₁₁⋊C₅	C₃⋊D₃	semidirect
25. C₃×C₁₁⋊C₅	S₄	semidirect
26. C₂²×C₃×C₁₁	C₅×D₃	semidirect
28. C₁₁×A₄	C₅×D₃	semidirect
45. C₁₁⋊C₅	C₃⋊S₄	semidirect
47. C₂²×C₁₁	C₅×C₃⋊D₃	semidirect
51. C₃×A₄	AGL(1,𝔽₁₁)	semidirect
53. C₃×C₁₁	C₅×S₄	semidirect
67. C₂²×C₃	C₃⋊AGL(1,𝔽₁₁)	semidirect
69. A₄	C₃⋊AGL(1,𝔽₁₁)	semidirect
71. C₁₁	C₅×C₃⋊S₄	semidirect
81. C₂²	C₃⋊(C₃⋊AGL(1,𝔽₁₁))	semidirect
84. C₃	A₄⋊AGL(1,𝔽₁₁)	semidirect

Maximal subgroups

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

subgroup	index
C₃×A₄×C₁₁⋊C₅	2
A₄⋊AGL(1,𝔽₁₁)	3
#63159⋊C₅	3
C₃⋊(C₃⋊AGL(1,𝔽₁₁))	4
C₃⋊(C₁₁⋊S₄)	5
C₅×C₃⋊S₄	11