Subgroups — C₃⋊((C₅⋊D₅)⋊#3861) #36348013

Order 22050.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²⋊#3861	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇²×C₃⋊(C₅⋊D₅)	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₅⋊D₅)⋊#3861	—	3	1	no	maximal
4	C₅²⋊#3861	—	2	1	no	—
5	C₅²⋊#3861	D₃	1	1	yes	characteristic, complemented
6	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃⋊((C₅⋊D₅)⋊ΓL(1,𝔽₈))	—	7	8	no	maximal
8	C₇²×C₅⋊D₅	—	3	1	no	Hall
9	C₃×C₇⋊(C₅²⋊C₃)	—	7	8	no	—
10	C₇²×D₁₅	—	15	2	no	—
11	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
12	C₇×C₃⋊(C₅⋊D₅)	ΓL(1,𝔽₈)	1	8	yes	complemented
13	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	21	8	no	—
14	D₃×#3861	—	25	1	no	maximal, Hall
15	C₃×C₅×C₇²	—	3	2	no	—
16	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	8	yes	complemented
17	C₇⋊(C₅²⋊C₃)	—	7	8	no	—
18	C₇⋊(C₅²⋊C₃)	—	14	8	no	—
19	C₇²×D₅	—	45	2	no	—
20	C₃⋊((C₅⋊D₅)⋊C₃)	—	49	1	no	Hall
21	C₃×#3861	—	25	1	no	Hall
22	C₇×C₅⋊D₅	—	3	8	no	—
23	D₃×C₇²	—	25	1	no	—
24	C₂×#3861	—	75	1	no	—
25	C₅×C₇²	—	3	2	no	—
26	C₃×C₅²⋊C₃	—	49	1	no	Hall
27	C₇×D₁₅	—	15	16	no	—
28	C₅²×C₇	D₃×ΓL(1,𝔽₈)	1	8	yes	complemented
29	C₃⋊(C₅⋊D₅)	#3861	1	1	yes	characteristic, complemented
30	(C₅⋊D₅)⋊C₃	—	147	1	no	—
31	C₃×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
32	#3861	—	25	1	no	—
33	#3861	—	50	1	no	—
34	D₃×ΓL(1,𝔽₈)	—	175	8	no	—
35	C₃×C₅×C₇	—	3	16	no	—
36	C₂×C₇²	—	75	1	no	Hall
37	C₃×C₅²	C₂×#3861	1	1	yes	characteristic, complemented
38	C₅²⋊C₃	—	49	1	no	—
39	C₅²⋊C₃	—	98	1	no	—
40	C₇×D₅	—	45	16	no	—
41	C₃×ΓL(1,𝔽₈)	—	175	8	no	—
42	C₅⋊D₅	—	3	1	no	Hall
43	C₇²	C₃⋊((C₅⋊D₅)⋊C₃)	1	1	yes	characteristic, complemented, Sylow
44	D₃×C₇	—	25	8	no	—
45	C₂×ΓL(1,𝔽₈)	—	525	8	no	—
46	C₅×C₇	—	3	16	no	—
47	D₁₅	—	15	2	no	—
48	C₅²	D₃×#3861	1	1	yes	characteristic, complemented, Sylow
49	C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	8	yes	complemented
50	ΓL(1,𝔽₈)	—	350	8	no	—
51	ΓL(1,𝔽₈)	—	175	8	no	—
52	C₃×D₃	—	1225	1	no	Hall
53	C₃×C₅	—	3	2	no	—
54	C₂×C₇	—	75	8	no	—
55	D₅	—	45	2	no	—
56	C₃²	—	1225	1	no	Sylow
57	C₇	C₃⋊((C₅⋊D₅)⋊ΓL(1,𝔽₈))	1	8	yes	complemented
58	C₂×C₃	—	3675	1	no	—
59	D₃	—	25	1	no	—
60	C₅	—	3	2	no	—
61	C₃	—	2450	1	no	—
62	C₃	—	1225	1	no	—
63	C₃	(C₅⋊D₅)⋊#3861	1	1	yes	characteristic, complemented
64	C₂	—	75	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₃×C₅²×C₇²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²⋊#3861	C₂	semidirect
2. C₇²×C₃⋊(C₅⋊D₅)	C₃	semidirect
5. C₅²⋊#3861	D₃	semidirect
6. C₃×C₅²×C₇²	C₂×C₃	semidirect
11. C₅²×C₇²	C₃×D₃	semidirect
12. C₇×C₃⋊(C₅⋊D₅)	ΓL(1,𝔽₈)	semidirect
16. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
28. C₅²×C₇	D₃×ΓL(1,𝔽₈)	semidirect
29. C₃⋊(C₅⋊D₅)	#3861	semidirect
31. C₃×C₇²	(C₅⋊D₅)⋊C₃	semidirect
37. C₃×C₅²	C₂×#3861	semidirect
43. C₇²	C₃⋊((C₅⋊D₅)⋊C₃)	semidirect
48. C₅²	D₃×#3861	semidirect
49. C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	semidirect
57. C₇	C₃⋊((C₅⋊D₅)⋊ΓL(1,𝔽₈))	semidirect
63. C₃	(C₅⋊D₅)⋊#3861	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅²⋊#3861	2
C₇²×C₃⋊(C₅⋊D₅)	3
(C₅⋊D₅)⋊#3861	3
C₃⋊((C₅⋊D₅)⋊ΓL(1,𝔽₈))	7
D₃×#3861	25

Subgroups — C3⋊((C5⋊D5)⋊#3861) #36348013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊((C₅⋊D₅)⋊#3861) #36348013