Subgroups — #28413360

Order 14700.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(C₅⋊D₅)⋊#3862	C₂	1	1	yes	maximal, characteristic
2	C₇×C₇⋊#1043	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²⋊#3862	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇²×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic
5	#1043⋊ΓL(1,𝔽₈)	—	7	1	no	maximal
6	#14523608	—	7	1	no	maximal
7	C₅²×C₇²	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	7	1	no	—
9	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	7	1	no	—
10	C₇×C₇⋊AGL(1,𝔽₅)	—	15	2	no	—
11	C₇⋊#1043	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
12	C₇×#1043	—	7	1	no	—
13	#10596872	—	25	1	no	maximal, Hall
14	C₇⋊(C₅²⋊C₃)	—	7	1	no	—
15	C₇⋊(C₅²⋊C₃)	—	7	1	no	—
16	C₇²×D₅	—	15	2	no	—
17	C₇×C₅⋊D₅	—	6	1	no	—
18	C₇×C₅⋊D₅	AGL(1,𝔽₇)	1	1	yes	characteristic
19	C₇×C₅⋊D₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
20	#1043⋊C₃	—	49	1	no	Hall
21	C₂×#3862	—	25	1	no	—
22	C₅×C₇²	—	3	2	no	—
23	C₇×Dic₇	—	25	1	no	Hall
24	C₅²×C₇	—	6	1	no	—
25	C₅²×C₇	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
26	C₅²×C₇	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
27	(C₅⋊D₅)⋊C₃	—	49	1	no	—
28	#3862	—	25	1	no	Hall
29	C₇⋊AGL(1,𝔽₅)	—	15	2	no	—
30	C₇×AGL(1,𝔽₅)	—	105	2	no	—
31	#1043	—	7	1	no	Hall
32	C₂×C₇²	—	25	1	no	—
33	C₄×ΓL(1,𝔽₈)	—	175	1	no	—
34	ΓL(1,𝔽₈)⋊C₄	—	175	1	no	—
35	C₅²⋊C₃	—	49	1	no	Hall
36	C₇×D₅	—	15	2	no	—
37	C₇×D₅	—	15	2	no	—
38	C₇×D₅	—	30	6	no	—
39	C₅⋊D₅	#64422	1	1	yes	characteristic
40	C₇²	#1043⋊C₃	1	1	yes	characteristic, complemented, Sylow
41	C₂×ΓL(1,𝔽₈)	—	175	1	no	—
42	C₂×ΓL(1,𝔽₈)	—	175	1	no	—
43	C₅×C₇	—	3	2	no	—
44	C₅×C₇	—	3	2	no	—
45	C₅×C₇	—	6	6	no	—
46	C₄×C₇	—	175	1	no	—
47	Dic₇	—	25	1	no	—
48	C₅²	#10596872	1	1	yes	characteristic, complemented, Sylow
49	ΓL(1,𝔽₈)	—	175	1	no	—
50	ΓL(1,𝔽₈)	—	175	1	no	—
51	AGL(1,𝔽₅)	—	105	2	no	—
52	C₂×C₇	—	25	1	no	—
53	C₂×C₇	—	25	1	no	—
54	C₂×C₇	—	150	1	no	—
55	C₃×C₄	—	1225	1	no	Hall
56	D₅	—	15	2	no	—
57	C₇	—	6	1	no	—
58	C₇	#14523608	1	1	yes	characteristic, complemented
59	C₇	#1043⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
60	C₂×C₃	—	1225	1	no	—
61	C₅	—	3	2	no	—
62	C₄	—	175	1	no	Sylow
63	C₃	—	1225	1	no	Sylow
64	C₂	—	25	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₇²	C₃×C₄
prime radical	(C₅⋊D₅)⋊#3862	C₂
cosocle	C₇²×C₅⋊D₅	C₂×C₃
LCS	G ▷ C₅²×C₇²
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇×C₇⋊#1043	C₃	semidirect
3. C₅²⋊#3862	C₄	semidirect
7. C₅²×C₇²	C₃×C₄	semidirect
11. C₇⋊#1043	ΓL(1,𝔽₈)	semidirect
25. C₅²×C₇	ΓL(1,𝔽₈)⋊C₄	semidirect
26. C₅²×C₇	C₄×ΓL(1,𝔽₈)	semidirect
40. C₇²	#1043⋊C₃	semidirect
48. C₅²	#10596872	semidirect
58. C₇	#14523608	semidirect
59. C₇	#1043⋊ΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
(C₅⋊D₅)⋊#3862	2
C₇×C₇⋊#1043	3
#1043⋊ΓL(1,𝔽₈)	7
#14523608	7
#10596872	25