← C₃⋊((C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))⋊C₃) #36345201

Subgroups — C₃⋊((C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))⋊C₃) #36345201

Order 21780.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×C₁₁²⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))⋊C₃	—	3	1	no	maximal
4	C₃×C₅×C₁₁²⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃⋊((C₁₁⋊Dic₁₁)⋊C₃)	—	5	1	no	maximal, Hall
6	C₃×D₅×C₁₁²	C₂×C₃	1	1	yes	characteristic
7	D₅×C₁₁²⋊C₃	—	2	1	no	—
8	D₅×C₁₁²⋊C₃	D₃	1	1	yes	characteristic
9	C₁₁⋊(C₁₁⋊AGL(1,𝔽₅))	—	3	1	no	Hall
10	C₂×C₃×C₁₁²⋊C₃	—	5	1	no	—
11	C₃×C₅×C₁₁²	C₃×C₄	1	1	yes	characteristic, complemented
12	C₅×C₁₁²⋊C₃	—	2	1	no	—
13	C₅×C₁₁²⋊C₃	Dic₃	1	1	yes	characteristic, complemented
14	C₃⋊(C₁₁⋊Dic₁₁)	—	5	1	no	—
15	(C₁₁⋊Dic₁₁)⋊C₃	—	15	1	no	—
16	D₅×C₁₁²	C₃×D₃	1	1	yes	characteristic
17	C₃×C₁₁²⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃×C₁₁²	—	5	1	no	—
19	C₂×C₁₁²⋊C₃	—	5	1	no	—
20	C₂×C₁₁²⋊C₃	—	10	1	no	—
21	C₃⋊(C₁₁⋊AGL(1,𝔽₅))	—	33	4	no	—
22	C₅×C₁₁²	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
23	C₁₁⋊Dic₁₁	—	15	1	no	Hall
24	C₃×C₁₁²	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
25	C₁₁²⋊C₃	—	2	1	no	—
26	C₁₁²⋊C₃	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
27	C₃×D₅×C₁₁	—	3	4	no	—
28	C₂×C₁₁²	—	5	1	no	—
29	C₁₁⋊AGL(1,𝔽₅)	—	99	4	no	—
30	C₃×ΓL(1,𝔽₁₆)	—	121	1	no	maximal, Hall
31	C₃×C₅×C₁₁	—	3	4	no	—
32	Dic₃₃	—	165	4	no	—
33	C₁₁²	C₃×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
34	D₅×C₁₁	—	3	4	no	—
35	C₃²×D₅	—	121	1	no	—
36	C₂×C₃×C₁₁	—	15	4	no	—
37	ΓL(1,𝔽₁₆)	—	121	1	no	—
38	C₃×AGL(1,𝔽₅)	—	363	1	no	—
39	C₅×C₁₁	—	3	4	no	—
40	C₃²×C₅	—	121	1	no	Hall
41	Dic₁₁	—	495	4	no	—
42	C₃×Dic₃	—	605	1	no	Hall
43	C₃×C₁₁	—	3	4	no	—
44	C₃×D₅	—	121	1	no	—
45	C₃×D₅	—	242	1	no	—
46	C₃×D₅	(C₁₁⋊D₁₁)⋊C₃	1	1	yes	characteristic
47	C₂×C₁₁	—	15	4	no	—
48	AGL(1,𝔽₅)	—	363	1	no	Hall
49	C₂×C₃²	—	605	1	no	—
50	C₃×C₅	—	121	1	no	—
51	C₃×C₅	—	242	1	no	—
52	C₃×C₅	(C₁₁⋊Dic₁₁)⋊C₃	1	1	yes	characteristic, complemented
53	C₃×C₄	—	1815	1	no	—
54	Dic₃	—	605	1	no	—
55	C₁₁	—	3	4	no	—
56	D₅	C₃⋊((C₁₁⋊D₁₁)⋊C₃)	1	1	yes	characteristic
57	C₃²	—	121	1	no	Sylow
58	C₂×C₃	—	5	1	no	—
59	C₂×C₃	—	605	1	no	—
60	C₂×C₃	—	1210	1	no	—
61	C₅	C₃⋊((C₁₁⋊Dic₁₁)⋊C₃)	1	1	yes	characteristic, complemented, Sylow
62	C₄	—	1815	1	no	Sylow
63	C₃	—	121	1	no	—
64	C₃	—	242	1	no	—
65	C₃	(C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))⋊C₃	1	1	yes	characteristic, complemented
66	C₂	—	5	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₅×C₁₁²	C₃×C₄
prime radical	C₃×D₅×C₁₁²⋊C₃	C₂
cosocle	C₃×D₅×C₁₁²	C₂×C₃
LCS	G ▷ C₃×C₅×C₁₁²
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))	C₃	semidirect
4. C₃×C₅×C₁₁²⋊C₃	C₄	semidirect
11. C₃×C₅×C₁₁²	C₃×C₄	semidirect
13. C₅×C₁₁²⋊C₃	Dic₃	semidirect
17. C₃×C₁₁²⋊C₃	AGL(1,𝔽₅)	semidirect
22. C₅×C₁₁²	C₃×Dic₃	semidirect
24. C₃×C₁₁²	C₃×AGL(1,𝔽₅)	semidirect
26. C₁₁²⋊C₃	ΓL(1,𝔽₁₆)	semidirect
33. C₁₁²	C₃×ΓL(1,𝔽₁₆)	semidirect
52. C₃×C₅	(C₁₁⋊Dic₁₁)⋊C₃	semidirect
61. C₅	C₃⋊((C₁₁⋊Dic₁₁)⋊C₃)	semidirect
65. C₃	(C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₅×C₁₁²⋊C₃	2
C₃⋊(C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))	3
(C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))⋊C₃	3
C₃⋊((C₁₁⋊Dic₁₁)⋊C₃)	5
C₃×ΓL(1,𝔽₁₆)	121

Subgroups — C3⋊((C11⋊(C11⋊AGL(1,𝔽5)))⋊C3) #36345201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊((C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))⋊C₃) #36345201