Subgroups — C₅⋊(C₅⋊#10624391) #31827682

Order 18150.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₁₁²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₁×C₅⋊D₅₅	—	3	1	no	maximal, Hall
3	C₅⋊#10624391	—	5	1	no	maximal
4	C₅⋊#10624391	—	5	2	no	maximal
5	C₅⋊#10624391	—	5	3	no	maximal
6	C₅²×C₁₁²	D₃	1	1	yes	characteristic, complemented, Hall
7	C₅×C₁₁²⋊C₃	D₅	1	1	yes	characteristic, complemented
8	C₅×C₁₁²⋊C₃	D₅	1	2	yes	complemented
9	C₅×C₁₁²⋊C₃	D₅	1	3	yes	complemented
10	C₁₁×D₅₅	—	15	2	no	—
11	C₁₁×D₅₅	—	15	1	no	—
12	C₁₁×D₅₅	—	15	3	no	—
13	#10624391	—	25	1	no	Hall
14	C₅×C₁₁²	D₁₅	1	2	yes	complemented
15	C₅×C₁₁²	D₁₅	1	1	yes	characteristic, complemented
16	C₅×C₁₁²	D₁₅	1	3	yes	complemented
17	C₅⋊D₅₅	—	3	1	no	—
18	C₁₁×C₅⋊D₅	—	33	1	no	—
19	C₁₁²⋊C₃	C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
20	C₅²×C₁₁	—	3	1	no	—
21	C₅²×C₁₁	—	3	1	no	—
22	C₅²×C₁₁	—	6	1	no	—
23	C₁₁×D₁₁	—	75	1	no	Hall
24	C₃⋊(C₅⋊D₅)	—	121	1	no	maximal, Hall
25	C₁₁²	C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented, Sylow
26	D₅₅	—	15	2	no	—
27	D₅₅	—	15	1	no	—
28	D₅₅	—	15	3	no	—
29	D₅×C₁₁	—	165	2	no	—
30	D₅×C₁₁	—	165	1	no	—
31	D₅×C₁₁	—	165	3	no	—
32	C₃×C₅²	—	242	1	no	Hall
33	C₅×C₁₁	—	3	2	no	—
34	C₅×C₁₁	—	6	2	no	—
35	C₅×C₁₁	—	6	1	no	—
36	C₅×C₁₁	—	6	3	no	—
37	C₅×C₁₁	—	6	1	no	—
38	C₅×C₁₁	—	6	3	no	—
39	C₅×C₁₁	—	6	2	no	—
40	C₅×C₁₁	—	6	1	no	—
41	C₅×C₁₁	—	6	3	no	—
42	C₅⋊D₅	—	33	1	no	Hall
43	D₁₅	—	605	1	no	—
44	D₁₅	—	605	2	no	—
45	D₁₅	—	605	3	no	—
46	C₅²	#10624391	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₁₁	—	825	1	no	—
48	D₁₁	—	75	1	no	—
49	C₃×C₅	—	121	1	no	—
50	C₃×C₅	—	242	2	no	—
51	C₃×C₅	—	242	3	no	—
52	C₁₁	—	3	1	no	—
53	C₁₁	—	6	1	no	—
54	C₁₁	—	6	1	no	—
55	D₅	—	165	2	no	—
56	D₅	—	165	1	no	—
57	D₅	—	165	3	no	—
58	D₃	—	3025	1	no	Hall
59	C₅	C₅⋊#10624391	1	2	yes	complemented
60	C₅	C₅⋊#10624391	1	1	yes	characteristic, complemented
61	C₅	C₅⋊#10624391	1	3	yes	complemented
62	C₃	—	242	1	no	Sylow
63	C₂	—	825	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₅²×C₁₁²⋊C₃	C₂
Fitting, socle	C₅²×C₁₁²	D₃
LCS	G ▷ C₅²×C₁₁²⋊C₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅²×C₁₁²⋊C₃	C₂	semidirect
6. C₅²×C₁₁²	D₃	semidirect
7. C₅×C₁₁²⋊C₃	D₅	semidirect
8. C₅×C₁₁²⋊C₃	D₅	semidirect
9. C₅×C₁₁²⋊C₃	D₅	semidirect
14. C₅×C₁₁²	D₁₅	semidirect
15. C₅×C₁₁²	D₁₅	semidirect
16. C₅×C₁₁²	D₁₅	semidirect
19. C₁₁²⋊C₃	C₅⋊D₅	semidirect
25. C₁₁²	C₃⋊(C₅⋊D₅)	semidirect
46. C₅²	#10624391	semidirect
59. C₅	C₅⋊#10624391	semidirect
60. C₅	C₅⋊#10624391	semidirect
61. C₅	C₅⋊#10624391	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₁₁²⋊C₃	2
C₁₁×C₅⋊D₅₅	3
C₅⋊#10624391	5
C₃⋊(C₅⋊D₅)	121

Subgroups — C5⋊(C5⋊#10624391) #31827682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅⋊(C₅⋊#10624391) #31827682