Subgroups — C₃×(C₅⋊D₅)⋊#3862 #36347957

Order 22050.

192 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	C₃×C₇²×C₅⋊D₅	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₅⋊D₅)⋊#3862	C₃	1	3	yes	maximal, direct product factor
4	C₅²⋊#3862	C₂×C₃	1	3	yes	complemented
5	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	7	2	no	maximal
7	C₇²×C₅⋊D₅	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇⋊(C₅²⋊C₃)	—	7	2	no	—
9	C₃×C₇²×D₅	—	15	2	no	—
10	C₅²×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
11	C₃×C₇×C₅⋊D₅	—	3	2	no	—
12	C₃×C₇×C₅⋊D₅	ΓL(1,𝔽₈)	1	2	yes	complemented
13	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	7	6	no	—
14	C₂×C₃×#3862	—	25	1	no	maximal, Hall
15	C₃×C₅×C₇²	—	3	2	no	—
16	C₃×C₅²×C₇	—	3	2	no	—
17	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
18	C₇⋊(C₅²⋊C₃)	—	7	6	no	—
19	C₇²×D₅	—	15	2	no	—
20	C₃×(C₅⋊D₅)⋊C₃	—	49	1	no	Hall
21	C₃×#3862	—	25	1	no	Hall
22	C₇×C₅⋊D₅	—	3	2	no	—
23	C₇×C₅⋊D₅	C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
24	C₂×C₃×C₇²	—	75	1	no	—
25	C₂×#3862	—	25	3	no	—
26	C₅×C₇²	—	3	2	no	—
27	C₃×C₅²⋊C₃	—	49	1	no	Hall
28	C₃×C₇×D₅	—	15	12	no	—
29	C₃×C₇×D₅	—	15	4	no	—
30	C₅²×C₇	—	3	2	no	—
31	C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
32	C₃×C₅⋊D₅	#3862	1	1	yes	characteristic, complemented
33	(C₅⋊D₅)⋊C₃	—	49	3	no	—
34	C₃×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
35	#3862	—	25	3	no	—
36	C₂×C₃×ΓL(1,𝔽₈)	—	175	2	no	—
37	C₃×C₅×C₇	—	3	12	no	—
38	C₃×C₅×C₇	—	3	4	no	—
39	C₂×C₇²	—	75	1	no	Hall
40	C₃×C₅²	C₂×#3862	1	1	yes	characteristic, complemented
41	C₅²⋊C₃	—	49	3	no	—
42	C₇×D₅	—	15	12	no	—
43	C₇×D₅	—	15	4	no	—
44	C₃×ΓL(1,𝔽₈)	—	175	2	no	—
45	C₅⋊D₅	C₃×#3862	1	1	yes	characteristic, complemented, Hall
46	C₇²	C₃×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃×C₇	—	75	2	no	—
48	C₂×C₃×C₇	—	75	2	no	—
49	C₂×ΓL(1,𝔽₈)	—	175	6	no	—
50	C₅×C₇	—	3	12	no	—
51	C₅×C₇	—	3	4	no	—
52	C₃×D₅	—	15	2	no	—
53	C₅²	C₂×C₃×#3862	1	1	yes	characteristic, complemented, Sylow
54	C₃×C₇	—	3	2	no	—
55	C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	2	yes	complemented
56	ΓL(1,𝔽₈)	—	175	6	no	—
57	C₂×C₃²	—	1225	1	no	Hall
58	C₃×C₅	—	3	2	no	—
59	C₂×C₇	—	75	2	no	—
60	C₂×C₇	—	75	2	no	—
61	D₅	—	15	2	no	—
62	C₃²	—	1225	1	no	Sylow
63	C₇	—	3	2	no	—
64	C₇	C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	2	yes	complemented
65	C₂×C₃	—	75	1	no	—
66	C₂×C₃	—	1225	3	no	—
67	C₅	—	3	2	no	—
68	C₃	—	1225	3	no	—
69	C₃	(C₅⋊D₅)⋊#3862	1	1	yes	characteristic, direct product factor
70	C₂	—	75	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	(C₅⋊D₅)⋊#3862
commutator, cosocle, hypocenter	C₅²×C₇²	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²⋊#3862	C₂	semidirect
2. C₃×C₇²×C₅⋊D₅	C₃	semidirect
3. (C₅⋊D₅)⋊#3862	C₃	direct (N × H)
4. C₅²⋊#3862	C₂×C₃	semidirect
5. C₃×C₅²×C₇²	C₂×C₃	semidirect
7. C₇²×C₅⋊D₅	C₃²	semidirect
10. C₅²×C₇²	C₂×C₃²	semidirect
12. C₃×C₇×C₅⋊D₅	ΓL(1,𝔽₈)	semidirect
17. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
23. C₇×C₅⋊D₅	C₃×ΓL(1,𝔽₈)	semidirect
31. C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
32. C₃×C₅⋊D₅	#3862	semidirect
34. C₃×C₇²	(C₅⋊D₅)⋊C₃	semidirect
40. C₃×C₅²	C₂×#3862	semidirect
45. C₅⋊D₅	C₃×#3862	semidirect
46. C₇²	C₃×(C₅⋊D₅)⋊C₃	semidirect
53. C₅²	C₂×C₃×#3862	semidirect
55. C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	semidirect
64. C₇	C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	semidirect
69. C₃	(C₅⋊D₅)⋊#3862	direct (N × H)

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
C₃×C₇²×C₅⋊D₅	3
(C₅⋊D₅)⋊#3862	3
C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	7
C₂×C₃×#3862	25

Subgroups — C3×(C5⋊D5)⋊#3862 #36347957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×(C₅⋊D₅)⋊#3862 #36347957