Subgroups — #3861×D₇₅ #36347760

Order 22050.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂₅×#3861	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇²×D₇₅	C₃	1	1	yes	maximal, characteristic, complemented
3	D₂₅×#3861	—	3	1	no	maximal
4	D₁₅×#3861	—	5	1	no	maximal
5	C₂₅×#3861	—	2	1	no	—
6	C₂₅×#3861	D₃	1	1	yes	characteristic, complemented
7	C₃×C₇²×C₂₅	C₂×C₃	1	1	yes	characteristic, complemented
8	ΓL(1,𝔽₈)×D₇₅	—	7	8	no	maximal
9	C₇²×D₂₅	—	3	1	no	Hall
10	C₃×C₅×#3861	D₅	1	1	yes	characteristic
11	C₃×ΓL(1,𝔽₈)×C₂₅	—	7	8	no	—
12	C₇²×D₁₅	—	5	1	no	—
13	D₅×#3861	—	15	1	no	—
14	C₇²×C₂₅	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₇×D₇₅	ΓL(1,𝔽₈)	1	8	yes	complemented
16	ΓL(1,𝔽₈)×D₂₅	—	21	8	no	—
17	D₃×#3861	—	25	1	no	Hall
18	C₃×C₅×C₇²	C₃×D₅	1	1	yes	characteristic
19	C₅×#3861	—	2	1	no	—
20	C₅×#3861	D₁₅	1	1	yes	characteristic
21	ΓL(1,𝔽₈)×D₁₅	—	35	8	no	—
22	C₃×C₇×C₂₅	C₂×ΓL(1,𝔽₈)	1	8	yes	complemented
23	ΓL(1,𝔽₈)×C₂₅	—	7	8	no	—
24	ΓL(1,𝔽₈)×C₂₅	—	14	8	no	—
25	C₇²×D₅	—	15	1	no	—
26	C₃×D₇₅	—	49	1	no	Hall
27	C₃×#3861	D₂₅	1	1	yes	characteristic, complemented, Hall
28	C₇×D₂₅	—	3	8	no	—
29	C₃×C₅×ΓL(1,𝔽₈)	—	7	8	no	—
30	D₃×C₇²	—	25	1	no	—
31	C₂×#3861	—	75	1	no	—
32	C₅×C₇²	C₃×D₁₅	1	1	yes	characteristic
33	C₃²×C₂₅	—	49	1	no	Hall
34	C₇×D₁₅	—	5	8	no	—
35	D₅×ΓL(1,𝔽₈)	—	105	8	no	—
36	C₇×C₂₅	D₃×ΓL(1,𝔽₈)	1	8	yes	complemented
37	D₇₅	#3861	1	1	yes	characteristic, direct product factor
38	C₃×D₂₅	—	147	1	no	—
39	C₃×C₇²	C₃×D₂₅	1	1	yes	characteristic, complemented
40	#3861	—	2	1	no	—
41	#3861	D₇₅	1	1	yes	characteristic, direct product factor
42	D₃×ΓL(1,𝔽₈)	—	175	8	no	—
43	C₃×C₅×C₇	D₅×ΓL(1,𝔽₈)	1	8	yes	—
44	C₅×ΓL(1,𝔽₈)	—	7	8	no	—
45	C₅×ΓL(1,𝔽₈)	—	14	8	no	—
46	C₂×C₇²	—	75	1	no	Hall
47	C₃×D₁₅	—	245	1	no	—
48	C₃×C₂₅	—	49	1	no	—
49	C₃×C₂₅	—	98	1	no	—
50	C₃×C₂₅	C₂×#3861	1	1	yes	characteristic, complemented
51	C₇×D₅	—	15	8	no	—
52	C₃×ΓL(1,𝔽₈)	—	7	8	no	—
53	D₂₅	—	3	1	no	Hall
54	C₇²	C₃×D₇₅	1	1	yes	characteristic, complemented, Sylow
55	C₃²×C₅	—	49	1	no	—
56	D₃×C₇	—	25	8	no	—
57	C₂×ΓL(1,𝔽₈)	—	525	8	no	—
58	C₅×C₇	ΓL(1,𝔽₈)×D₁₅	1	8	yes	—
59	D₁₅	—	5	1	no	—
60	C₃×D₅	—	735	1	no	—
61	C₂₅	D₃×#3861	1	1	yes	characteristic, complemented, Sylow
62	C₃×C₇	ΓL(1,𝔽₈)×D₂₅	1	8	yes	complemented
63	ΓL(1,𝔽₈)	—	7	8	no	—
64	ΓL(1,𝔽₈)	—	14	8	no	—
65	C₃×D₃	—	1225	1	no	Hall
66	C₃×C₅	—	49	1	no	—
67	C₃×C₅	—	98	1	no	—
68	C₃×C₅	D₅×#3861	1	1	yes	characteristic
69	C₂×C₇	—	75	8	no	—
70	D₅	—	15	1	no	—
71	C₃²	—	49	1	no	Sylow
72	C₇	ΓL(1,𝔽₈)×D₇₅	1	8	yes	complemented
73	C₂×C₃	—	3675	1	no	—
74	D₃	—	25	1	no	—
75	C₅	D₁₅×#3861	1	1	yes	characteristic
76	C₃	—	49	1	no	—
77	C₃	—	98	1	no	—
78	C₃	D₂₅×#3861	1	1	yes	characteristic, complemented
79	C₂	—	75	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₇²×C₂₅	C₂×C₃
Frattini	C₅	D₁₅×#3861
socle	C₃×C₅×C₇²	C₃×D₅
LCS	G ▷ C₃×C₇²×C₂₅
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₂₅×#3861	C₂	semidirect
2. C₇²×D₇₅	C₃	semidirect
6. C₂₅×#3861	D₃	semidirect
7. C₃×C₇²×C₂₅	C₂×C₃	semidirect
14. C₇²×C₂₅	C₃×D₃	semidirect
15. C₇×D₇₅	ΓL(1,𝔽₈)	semidirect
22. C₃×C₇×C₂₅	C₂×ΓL(1,𝔽₈)	semidirect
27. C₃×#3861	D₂₅	semidirect
36. C₇×C₂₅	D₃×ΓL(1,𝔽₈)	semidirect
37. D₇₅	#3861	direct (N × H)
39. C₃×C₇²	C₃×D₂₅	semidirect
41. #3861	D₇₅	direct (N × H)
50. C₃×C₂₅	C₂×#3861	semidirect
54. C₇²	C₃×D₇₅	semidirect
61. C₂₅	D₃×#3861	semidirect
62. C₃×C₇	ΓL(1,𝔽₈)×D₂₅	semidirect
72. C₇	ΓL(1,𝔽₈)×D₇₅	semidirect
78. C₃	D₂₅×#3861	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₂₅×#3861	2
C₇²×D₇₅	3
D₂₅×#3861	3
D₁₅×#3861	5
ΓL(1,𝔽₈)×D₇₅	7

Subgroups — #3861×D75 #36347760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3861×D₇₅ #36347760