Subgroups — #46802018

Order 34100.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#13133724	C₂	1	1	yes	maximal, characteristic
2	C₅×#13133724	C₄	1	1	yes	characteristic, complemented, Hall
3	#20291885	—	5	1	no	maximal
4	Dic₁₇₀₅	C₅	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133724	—	2	2	no	—
6	C₂×#13133724	D₅	1	1	yes	characteristic
7	C₂×C₅×C₁₁×C₃₁	C₂×C₅	1	1	yes	characteristic
8	ΓL(1,𝔽₃₂)⋊Dic₅	—	11	1	no	maximal, Hall
9	C₅×C₁₁×C₃₁	C₄×C₅	1	1	yes	characteristic, complemented
10	#13133724	—	2	2	no	—
11	#13133724	Dic₅	1	1	yes	characteristic, complemented
12	C₂×C₅×ΓL(1,𝔽₃₂)	—	11	1	no	—
13	Dic₃₄₁	—	5	1	no	Hall
14	(C₁₁⋊C₅)⋊Dic₅	—	31	1	no	maximal, Hall
15	C₅×ΓL(1,𝔽₃₂)	—	11	1	no	Hall
16	C₂×C₁₁×C₃₁	C₅×D₅	1	1	yes	characteristic
17	Dic₁₅₅	—	11	1	no	—
18	Dic₃₁⋊C₅	—	55	1	no	—
19	C₂×C₅×C₁₁⋊C₅	—	31	1	no	—
20	C₁₁×C₃₁	C₅×Dic₅	1	1	yes	characteristic, complemented, Hall
21	C₂×C₅×C₃₁	AGL(1,𝔽₁₁)	1	1	yes	characteristic
22	C₂×ΓL(1,𝔽₃₂)	—	11	1	no	—
23	C₂×ΓL(1,𝔽₃₂)	—	22	2	no	—
24	C₅×C₁₁⋊C₅	—	31	1	no	Hall
25	Dic₅₅	—	31	1	no	—
26	Dic₁₁⋊C₅	—	155	1	no	—
27	C₅×C₃₁	Dic₁₁⋊C₅	1	1	yes	characteristic, complemented
28	ΓL(1,𝔽₃₂)	—	11	1	no	—
29	ΓL(1,𝔽₃₂)	—	22	2	no	—
30	Dic₃₁	—	55	1	no	Hall
31	C₂×C₅×C₁₁	D₃₁⋊C₅	1	1	yes	characteristic
32	C₂×C₁₁⋊C₅	—	31	1	no	—
33	C₂×C₁₁⋊C₅	—	62	2	no	—
34	C₅×Dic₅	—	341	1	no	Hall
35	C₂×C₃₁	C₅⋊AGL(1,𝔽₁₁)	1	1	yes	characteristic
36	C₅×C₁₁	Dic₃₁⋊C₅	1	1	yes	characteristic, complemented
37	C₁₁⋊C₅	—	31	1	no	—
38	C₁₁⋊C₅	—	62	2	no	—
39	C₂×C₅²	—	341	1	no	—
40	Dic₁₁	—	155	1	no	Hall
41	C₃₁	(C₁₁⋊C₅)⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
42	C₅²	—	341	1	no	Sylow
43	C₂×C₁₁	C₅⋊(D₃₁⋊C₅)	1	1	yes	characteristic
44	C₄×C₅	—	1705	1	no	—
45	Dic₅	—	341	1	no	—
46	C₁₁	ΓL(1,𝔽₃₂)⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₅	—	341	1	no	—
48	C₂×C₅	—	682	2	no	—
49	C₂×C₅	#16811786	1	1	yes	characteristic
50	C₅	—	341	1	no	—
51	C₅	—	682	2	no	—
52	C₅	#20291885	1	1	yes	characteristic, complemented
53	C₄	—	1705	1	no	Sylow
54	C₂	C₅⋊#16811786	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₅⋊#16811786
commutator, hypocenter	C₅×C₁₁×C₃₁	C₄×C₅
Fitting, socle, cosocle	C₂×C₅×C₁₁×C₃₁	C₂×C₅
prime radical	C₂×C₅×#13133724	C₂
LCS	G ▷ C₅×C₁₁×C₃₁
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₅×#13133724	C₄	semidirect
4. Dic₁₇₀₅	C₅	semidirect
9. C₅×C₁₁×C₃₁	C₄×C₅	semidirect
11. #13133724	Dic₅	semidirect
20. C₁₁×C₃₁	C₅×Dic₅	semidirect
27. C₅×C₃₁	Dic₁₁⋊C₅	semidirect
36. C₅×C₁₁	Dic₃₁⋊C₅	semidirect
41. C₃₁	(C₁₁⋊C₅)⋊Dic₅	semidirect
46. C₁₁	ΓL(1,𝔽₃₂)⋊Dic₅	semidirect
52. C₅	#20291885	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#13133724	2
#20291885	5
Dic₁₇₀₅	5
ΓL(1,𝔽₃₂)⋊Dic₅	11
(C₁₁⋊C₅)⋊Dic₅	31