Subgroups — C₂×C₃×C₂₅×#3862 #36347766

Order 22050.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂₅×#3862	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₂₅×#3862	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×C₇²×C₂₅	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₅×#3862	C₅	1	1	yes	maximal, characteristic
5	C₂₅×#3862	C₂×C₃	1	3	yes	direct product factor
6	C₃×C₇²×C₂₅	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₃×ΓL(1,𝔽₈)×C₂₅	—	7	2	no	maximal
8	C₂×C₇²×C₂₅	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₅×#3862	C₂×C₅	1	1	yes	characteristic
10	C₃×ΓL(1,𝔽₈)×C₂₅	—	7	2	no	—
11	C₂×C₃×C₅×C₇²	C₃×C₅	1	1	yes	characteristic
12	C₂×C₅×#3862	C₃×C₅	1	3	yes	—
13	C₇²×C₂₅	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₇×C₂₅	—	3	2	no	—
15	C₂×C₃×C₇×C₂₅	ΓL(1,𝔽₈)	1	2	yes	complemented
16	C₂×ΓL(1,𝔽₈)×C₂₅	—	7	6	no	—
17	C₂×C₃×#3862	C₂₅	1	1	yes	characteristic, direct product factor, Hall
18	C₃×C₅×C₇²	C₂×C₃×C₅	1	1	yes	characteristic
19	C₅×#3862	C₂×C₃×C₅	1	3	yes	—
20	C₂×C₃×C₅×ΓL(1,𝔽₈)	—	7	2	no	—
21	C₃×C₇×C₂₅	—	3	2	no	—
22	C₃×C₇×C₂₅	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
23	ΓL(1,𝔽₈)×C₂₅	—	7	6	no	—
24	C₂×C₅×C₇²	C₃²×C₅	1	1	yes	characteristic
25	C₂×C₃²×C₂₅	—	49	1	no	Hall
26	C₃×#3862	C₂×C₂₅	1	1	yes	characteristic, direct product factor, Hall
27	C₂×C₇×C₂₅	—	3	2	no	—
28	C₂×C₇×C₂₅	C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
29	C₃×C₅×ΓL(1,𝔽₈)	—	7	2	no	—
30	C₂×C₃×C₇²	C₃×C₂₅	1	1	yes	characteristic, complemented
31	C₂×#3862	C₃×C₂₅	1	3	yes	direct product factor
32	C₅×C₇²	C₂×C₃²×C₅	1	1	yes	characteristic
33	C₃²×C₂₅	—	49	1	no	Hall
34	C₂×C₃×C₅×C₇	—	3	2	no	—
35	C₂×C₃×C₅×C₇	C₅×ΓL(1,𝔽₈)	1	2	yes	—
36	C₂×C₅×ΓL(1,𝔽₈)	—	7	6	no	—
37	C₇×C₂₅	—	3	2	no	—
38	C₇×C₂₅	C₂×C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
39	C₂×C₃×C₂₅	—	49	3	no	—
40	C₂×C₃×C₂₅	#3862	1	1	yes	characteristic, direct product factor
41	C₃×C₇²	C₂×C₃×C₂₅	1	1	yes	characteristic, complemented
42	#3862	C₂×C₃×C₂₅	1	3	yes	direct product factor
43	C₂×C₃×ΓL(1,𝔽₈)	—	7	2	no	—
44	C₃×C₅×C₇	—	3	2	no	—
45	C₃×C₅×C₇	C₂×C₅×ΓL(1,𝔽₈)	1	2	yes	—
46	C₅×ΓL(1,𝔽₈)	—	7	6	no	—
47	C₂×C₇²	C₃²×C₂₅	1	1	yes	characteristic, complemented, Hall
48	C₂×C₃²×C₅	—	49	1	no	—
49	C₃×C₂₅	—	49	3	no	—
50	C₃×C₂₅	C₂×#3862	1	1	yes	characteristic, direct product factor
51	C₂×C₅×C₇	—	3	2	no	—
52	C₂×C₅×C₇	C₃×C₅×ΓL(1,𝔽₈)	1	2	yes	—
53	C₃×ΓL(1,𝔽₈)	—	7	2	no	—
54	C₂×C₂₅	C₃×#3862	1	1	yes	characteristic, direct product factor, Hall
55	C₇²	C₂×C₃²×C₂₅	1	1	yes	characteristic, complemented, Sylow
56	C₃²×C₅	—	49	1	no	—
57	C₂×C₃×C₇	—	3	2	no	—
58	C₂×C₃×C₇	ΓL(1,𝔽₈)×C₂₅	1	2	yes	complemented
59	C₂×ΓL(1,𝔽₈)	—	7	6	no	—
60	C₅×C₇	—	3	2	no	—
61	C₅×C₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	1	2	yes	—
62	C₂×C₃×C₅	—	49	3	no	—
63	C₂×C₃×C₅	C₅×#3862	1	1	yes	characteristic
64	C₂₅	C₂×C₃×#3862	1	1	yes	characteristic, direct product factor, Sylow
65	C₃×C₇	—	3	2	no	—
66	C₃×C₇	C₂×ΓL(1,𝔽₈)×C₂₅	1	2	yes	complemented
67	ΓL(1,𝔽₈)	—	7	6	no	—
68	C₂×C₃²	—	49	1	no	Hall
69	C₃×C₅	—	49	3	no	—
70	C₃×C₅	C₂×C₅×#3862	1	1	yes	characteristic
71	C₂×C₇	—	3	2	no	—
72	C₂×C₇	C₃×ΓL(1,𝔽₈)×C₂₅	1	2	yes	complemented
73	C₂×C₅	C₃×C₅×#3862	1	1	yes	characteristic
74	C₃²	—	49	1	no	Sylow
75	C₇	—	3	2	no	—
76	C₇	C₂×C₃×ΓL(1,𝔽₈)×C₂₅	1	2	yes	complemented
77	C₂×C₃	—	49	3	no	—
78	C₂×C₃	C₂₅×#3862	1	1	yes	characteristic, direct product factor
79	C₅	C₂×C₃×C₅×#3862	1	1	yes	characteristic
80	C₃	—	49	3	no	—
81	C₃	C₂×C₂₅×#3862	1	1	yes	characteristic, direct product factor
82	C₂	C₃×C₂₅×#3862	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₃×C₂₅	#3862
commutator, hypocenter	C₇²	C₂×C₃²×C₂₅
Frattini	C₅	C₂×C₃×C₅×#3862
Fitting	C₂×C₃×C₇²×C₂₅	C₃
socle	C₂×C₃×C₅×C₇²	C₃×C₅
prime radical	C₂×C₃×C₅×#3862	C₅
cosocle	C₅×C₇²	C₂×C₃²×C₅
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₃×C₂₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₂₅×#3862	C₂	direct (N × H)
2. C₂×C₂₅×#3862	C₃	direct (N × H)
3. C₂×C₃×C₇²×C₂₅	C₃	semidirect
5. C₂₅×#3862	C₂×C₃	direct (N × H)
6. C₃×C₇²×C₂₅	C₂×C₃	semidirect
8. C₂×C₇²×C₂₅	C₃²	semidirect
13. C₇²×C₂₅	C₂×C₃²	semidirect
15. C₂×C₃×C₇×C₂₅	ΓL(1,𝔽₈)	semidirect
17. C₂×C₃×#3862	C₂₅	direct (N × H)
22. C₃×C₇×C₂₅	C₂×ΓL(1,𝔽₈)	semidirect
26. C₃×#3862	C₂×C₂₅	direct (N × H)
28. C₂×C₇×C₂₅	C₃×ΓL(1,𝔽₈)	semidirect
30. C₂×C₃×C₇²	C₃×C₂₅	semidirect
31. C₂×#3862	C₃×C₂₅	direct (N × H)
38. C₇×C₂₅	C₂×C₃×ΓL(1,𝔽₈)	semidirect
40. C₂×C₃×C₂₅	#3862	direct (N × H)
41. C₃×C₇²	C₂×C₃×C₂₅	semidirect
42. #3862	C₂×C₃×C₂₅	direct (N × H)
47. C₂×C₇²	C₃²×C₂₅	semidirect
50. C₃×C₂₅	C₂×#3862	direct (N × H)
54. C₂×C₂₅	C₃×#3862	direct (N × H)
55. C₇²	C₂×C₃²×C₂₅	semidirect
58. C₂×C₃×C₇	ΓL(1,𝔽₈)×C₂₅	semidirect
64. C₂₅	C₂×C₃×#3862	direct (N × H)
66. C₃×C₇	C₂×ΓL(1,𝔽₈)×C₂₅	semidirect
72. C₂×C₇	C₃×ΓL(1,𝔽₈)×C₂₅	semidirect
76. C₇	C₂×C₃×ΓL(1,𝔽₈)×C₂₅	semidirect
78. C₂×C₃	C₂₅×#3862	direct (N × H)
81. C₃	C₂×C₂₅×#3862	direct (N × H)
82. C₂	C₃×C₂₅×#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×C₂₅×#3862	2
C₂×C₂₅×#3862	3
C₂×C₃×C₇²×C₂₅	3
C₂×C₃×C₅×#3862	5
C₂×C₃×ΓL(1,𝔽₈)×C₂₅	7

Subgroups — C2×C3×C25×#3862 #36347766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₂₅×#3862 #36347766