Subgroups — C₃⋊((C₅⋊D₅)⋊#3862) #36348017

Order 22050.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²⋊#3862	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇²×C₃⋊(C₅⋊D₅)	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₅⋊D₅)⋊#3862	—	3	1	no	maximal
4	C₅²⋊#3862	—	2	1	no	—
5	C₅²⋊#3862	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	2	no	maximal
8	C₇²×C₅⋊D₅	—	3	1	no	Hall
9	C₃×C₇⋊(C₅²⋊C₃)	—	7	2	no	—
10	C₇²×D₁₅	—	15	2	no	—
11	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
12	C₇×C₃⋊(C₅⋊D₅)	—	3	2	no	—
13	C₇×C₃⋊(C₅⋊D₅)	ΓL(1,𝔽₈)	1	2	yes	complemented
14	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	21	2	no	—
15	D₃×#3862	—	25	1	no	maximal, Hall
16	C₃×C₅×C₇²	—	3	2	no	—
17	C₃×C₅²×C₇	—	3	2	no	—
18	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
19	C₇⋊(C₅²⋊C₃)	—	7	2	no	—
20	C₇⋊(C₅²⋊C₃)	—	14	2	no	—
21	C₇²×D₅	—	45	2	no	—
22	C₃⋊((C₅⋊D₅)⋊C₃)	—	49	1	no	Hall
23	C₃×#3862	—	25	1	no	Hall
24	C₇×C₅⋊D₅	—	3	2	no	—
25	C₇×C₅⋊D₅	—	9	2	no	—
26	D₃×C₇²	—	25	1	no	—
27	C₂×#3862	—	75	1	no	—
28	C₅×C₇²	—	3	2	no	—
29	C₃×C₅²⋊C₃	—	49	1	no	Hall
30	C₇×D₁₅	—	15	4	no	—
31	C₇×D₁₅	—	15	12	no	—
32	C₅²×C₇	—	3	2	no	—
33	C₅²×C₇	D₃×ΓL(1,𝔽₈)	1	2	yes	complemented
34	C₃⋊(C₅⋊D₅)	#3862	1	1	yes	characteristic, complemented
35	(C₅⋊D₅)⋊C₃	—	147	1	no	—
36	C₃×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
37	#3862	—	25	1	no	—
38	#3862	—	50	1	no	—
39	D₃×ΓL(1,𝔽₈)	—	175	2	no	—
40	C₃×C₅×C₇	—	3	12	no	—
41	C₃×C₅×C₇	—	3	4	no	—
42	C₂×C₇²	—	75	1	no	Hall
43	C₃×C₅²	C₂×#3862	1	1	yes	characteristic, complemented
44	C₅²⋊C₃	—	49	1	no	—
45	C₅²⋊C₃	—	98	1	no	—
46	C₇×D₅	—	45	4	no	—
47	C₇×D₅	—	45	12	no	—
48	C₃×ΓL(1,𝔽₈)	—	175	2	no	—
49	C₅⋊D₅	—	3	1	no	Hall
50	C₇²	C₃⋊((C₅⋊D₅)⋊C₃)	1	1	yes	characteristic, complemented, Sylow
51	D₃×C₇	—	25	2	no	—
52	D₃×C₇	—	75	2	no	—
53	C₂×ΓL(1,𝔽₈)	—	525	2	no	—
54	C₅×C₇	—	3	12	no	—
55	C₅×C₇	—	3	4	no	—
56	D₁₅	—	15	2	no	—
57	C₅²	D₃×#3862	1	1	yes	characteristic, complemented, Sylow
58	C₃×C₇	—	3	2	no	—
59	C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	2	yes	complemented
60	ΓL(1,𝔽₈)	—	350	2	no	—
61	ΓL(1,𝔽₈)	—	175	2	no	—
62	C₃×D₃	—	1225	1	no	Hall
63	C₃×C₅	—	3	2	no	—
64	C₂×C₇	—	75	2	no	—
65	C₂×C₇	—	225	2	no	—
66	D₅	—	45	2	no	—
67	C₃²	—	1225	1	no	Sylow
68	C₇	—	3	2	no	—
69	C₇	C₃⋊((C₅⋊D₅)⋊ΓL(1,𝔽₈))	1	2	yes	complemented
70	C₂×C₃	—	3675	1	no	—
71	D₃	—	25	1	no	—
72	C₅	—	3	2	no	—
73	C₃	—	2450	1	no	—
74	C₃	—	1225	1	no	—
75	C₃	(C₅⋊D₅)⋊#3862	1	1	yes	characteristic, complemented
76	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⋊#3862	C₂	semidirect
2. C₇²×C₃⋊(C₅⋊D₅)	C₃	semidirect
5. C₅²⋊#3862	D₃	semidirect
6. C₃×C₅²×C₇²	C₂×C₃	semidirect
11. C₅²×C₇²	C₃×D₃	semidirect
13. C₇×C₃⋊(C₅⋊D₅)	ΓL(1,𝔽₈)	semidirect
18. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
33. C₅²×C₇	D₃×ΓL(1,𝔽₈)	semidirect
34. C₃⋊(C₅⋊D₅)	#3862	semidirect
36. C₃×C₇²	(C₅⋊D₅)⋊C₃	semidirect
43. C₃×C₅²	C₂×#3862	semidirect
50. C₇²	C₃⋊((C₅⋊D₅)⋊C₃)	semidirect
57. C₅²	D₃×#3862	semidirect
59. C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	semidirect
69. C₇	C₃⋊((C₅⋊D₅)⋊ΓL(1,𝔽₈))	semidirect
75. C₃	(C₅⋊D₅)⋊#3862	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅²⋊#3862	2
C₇²×C₃⋊(C₅⋊D₅)	3
(C₅⋊D₅)⋊#3862	3
C₃⋊((C₅⋊D₅)⋊ΓL(1,𝔽₈))	7
D₃×#3862	25

Subgroups — C3⋊((C5⋊D5)⋊#3862) #36348017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊((C₅⋊D₅)⋊#3862) #36348017