Subgroups — #28413156

Order 14700.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#3862	C₂	1	1	yes	maximal, characteristic
2	C₇×C₇⋊#1039	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²×#3862	C₄	1	1	yes	characteristic, complemented, Hall
4	#15668737	—	5	6	no	maximal
5	C₂×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic
6	ΓL(1,𝔽₈)⋊#1039	—	7	1	no	maximal
7	ΓL(1,𝔽₈)×#1039	—	7	1	no	maximal
8	C₂×C₅×#3862	D₅	1	6	yes	—
9	C₅²×C₇²	C₃×C₄	1	1	yes	characteristic, complemented, Hall
10	C₂×C₅²×ΓL(1,𝔽₈)	—	7	1	no	—
11	C₂×C₅²×ΓL(1,𝔽₈)	—	7	1	no	—
12	C₇×Dic₃₅	—	5	6	no	—
13	C₅×#3862	Dic₅	1	6	yes	complemented
14	C₇⋊#1039	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
15	C₇×#1039	—	7	1	no	—
16	#10596872	—	25	1	no	Hall
17	C₅²×ΓL(1,𝔽₈)	—	7	1	no	—
18	C₅²×ΓL(1,𝔽₈)	—	7	1	no	—
19	C₂×C₅×C₇²	C₃×D₅	1	6	yes	—
20	ΓL(1,𝔽₈)⋊Dic₅	—	35	6	no	—
21	Dic₅×ΓL(1,𝔽₈)	—	35	6	no	—
22	C₂×C₅²×C₇	—	6	1	no	—
23	C₂×C₅²×C₇	AGL(1,𝔽₇)	1	1	yes	characteristic
24	C₂×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
25	C₃×#1039	—	49	1	no	Hall
26	C₂×#3862	C₅⋊D₅	1	1	yes	characteristic
27	C₅×C₇²	C₃×Dic₅	1	6	yes	complemented
28	C₂×C₅×ΓL(1,𝔽₈)	—	7	6	no	—
29	C₂×C₅×ΓL(1,𝔽₈)	—	7	6	no	—
30	C₇×Dic₇	—	25	1	no	Hall
31	C₅²×C₇	—	6	1	no	—
32	C₅²×C₇	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
33	C₅²×C₇	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅²	—	49	1	no	—
35	#3862	#1039	1	1	yes	characteristic, complemented, Hall
36	Dic₃₅	—	5	6	no	—
37	C₇×Dic₅	—	35	6	no	—
38	C₅×ΓL(1,𝔽₈)	—	7	6	no	—
39	C₅×ΓL(1,𝔽₈)	—	7	6	no	—
40	#1039	—	7	1	no	Hall
41	C₂×C₇²	C₃×C₅⋊D₅	1	1	yes	characteristic
42	C₄×ΓL(1,𝔽₈)	—	175	1	no	—
43	ΓL(1,𝔽₈)⋊C₄	—	175	1	no	—
44	C₃×C₅²	—	49	1	no	Hall
45	C₂×C₅×C₇	—	6	6	no	—
46	C₂×C₅×C₇	D₅×ΓL(1,𝔽₈)	1	6	yes	—
47	C₂×C₅×C₇	C₅⋊AGL(1,𝔽₇)	1	6	yes	—
48	C₃×Dic₅	—	245	6	no	—
49	C₂×C₅²	#64422	1	1	yes	characteristic
50	C₇²	C₃×#1039	1	1	yes	characteristic, complemented, Sylow
51	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
52	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
53	C₅×C₇	—	6	6	no	—
54	C₅×C₇	Dic₅×ΓL(1,𝔽₈)	1	6	yes	complemented
55	C₅×C₇	ΓL(1,𝔽₈)⋊Dic₅	1	6	yes	complemented
56	C₂×C₃×C₅	—	49	6	no	—
57	C₄×C₇	—	175	1	no	—
58	Dic₇	—	25	1	no	—
59	C₅²	#10596872	1	1	yes	characteristic, complemented, Sylow
60	ΓL(1,𝔽₈)	—	7	1	no	—
61	ΓL(1,𝔽₈)	—	7	1	no	—
62	Dic₅	—	35	6	no	—
63	C₃×C₅	—	49	6	no	—
64	C₂×C₇	—	6	1	no	—
65	C₂×C₇	ΓL(1,𝔽₈)×C₅⋊D₅	1	1	yes	characteristic
66	C₂×C₇	C₅⋊(C₅⋊AGL(1,𝔽₇))	1	1	yes	characteristic
67	C₃×C₄	—	1225	1	no	Hall
68	C₂×C₅	C₅⋊#64422	1	6	yes	—
69	C₇	—	6	1	no	—
70	C₇	ΓL(1,𝔽₈)×#1039	1	1	yes	characteristic, complemented
71	C₇	ΓL(1,𝔽₈)⋊#1039	1	1	yes	characteristic, complemented
72	C₂×C₃	—	49	1	no	—
73	C₅	#15668737	1	6	yes	complemented
74	C₄	—	175	1	no	Sylow
75	C₃	—	49	1	no	Sylow
76	C₂	C₅⋊(C₅⋊#64422)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₅⋊(C₅⋊#64422)
commutator, hypocenter	C₅²×C₇²	C₃×C₄
Fitting, socle, cosocle	C₂×C₅²×C₇²	C₂×C₃
prime radical	C₂×C₅²×#3862	C₂
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇×C₇⋊#1039	C₃	semidirect
3. C₅²×#3862	C₄	semidirect
9. C₅²×C₇²	C₃×C₄	semidirect
13. C₅×#3862	Dic₅	semidirect
14. C₇⋊#1039	ΓL(1,𝔽₈)	semidirect
27. C₅×C₇²	C₃×Dic₅	semidirect
32. C₅²×C₇	ΓL(1,𝔽₈)⋊C₄	semidirect
33. C₅²×C₇	C₄×ΓL(1,𝔽₈)	semidirect
35. #3862	#1039	semidirect
50. C₇²	C₃×#1039	semidirect
54. C₅×C₇	Dic₅×ΓL(1,𝔽₈)	semidirect
55. C₅×C₇	ΓL(1,𝔽₈)⋊Dic₅	semidirect
59. C₅²	#10596872	semidirect
70. C₇	ΓL(1,𝔽₈)×#1039	semidirect
71. C₇	ΓL(1,𝔽₈)⋊#1039	semidirect
73. C₅	#15668737	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×#3862	2
C₇×C₇⋊#1039	3
#15668737	5
ΓL(1,𝔽₈)⋊#1039	7
ΓL(1,𝔽₈)×#1039	7