Subgroups — #28413177

Order 14700.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₇⋊(C₅²⋊C₃)	C₂	1	1	yes	maximal, characteristic
2	C₇×C₇⋊#1039	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₇⋊(C₅²⋊C₃)	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic
5	#1039⋊ΓL(1,𝔽₈)	—	7	1	no	maximal
6	#14523572	—	7	1	no	maximal
7	C₅²×C₇²	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₂×C₇⋊(C₅²⋊C₃)	D₇	1	1	yes	characteristic
9	C₂×C₇×C₅²⋊C₃	—	7	1	no	—
10	C₇×Dic₃₅	—	15	2	no	—
11	C₇⋊#1039	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
12	C₇×#1039	—	7	1	no	—
13	ΓL(1,𝔽₈)×Dic₇	—	25	1	no	maximal, Hall
14	C₇⋊(C₅²⋊C₃)	Dic₇	1	1	yes	characteristic, complemented
15	C₇×C₅²⋊C₃	—	7	1	no	—
16	C₂×C₅×C₇²	—	3	2	no	—
17	C₂×C₅²×C₇	—	6	1	no	—
18	C₂×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
19	C₂×C₅²×C₇	C₃×D₇	1	1	yes	characteristic
20	#1039⋊C₃	—	49	1	no	Hall
21	C₂×C₇×ΓL(1,𝔽₈)	—	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	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
26	C₅²×C₇	C₃×Dic₇	1	1	yes	characteristic, complemented
27	C₂×C₅²⋊C₃	—	7	1	no	—
28	C₇×ΓL(1,𝔽₈)	—	25	1	no	Hall
29	Dic₃₅	—	15	2	no	—
30	C₇×Dic₅	—	105	2	no	—
31	#1039	—	7	1	no	Hall
32	C₂×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic
33	C₃×Dic₇	—	175	1	no	—
34	C₄×ΓL(1,𝔽₈)	—	175	1	no	—
35	C₅²⋊C₃	—	7	1	no	Hall
36	C₂×C₅×C₇	—	3	2	no	—
37	C₂×C₅×C₇	—	3	2	no	—
38	C₂×C₅×C₇	—	6	6	no	—
39	C₂×C₅²	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
40	C₇²	#1039⋊C₃	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃×C₇	—	175	1	no	—
42	C₂×ΓL(1,𝔽₈)	—	25	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	ΓL(1,𝔽₈)×Dic₇	1	1	yes	characteristic, complemented, Sylow
49	C₃×C₇	—	175	1	no	—
50	ΓL(1,𝔽₈)	—	25	1	no	—
51	Dic₅	—	105	2	no	—
52	C₂×C₇	—	6	1	no	—
53	C₂×C₇	#11760384	1	1	yes	characteristic
54	C₂×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
55	C₃×C₄	—	1225	1	no	Hall
56	C₂×C₅	—	3	2	no	—
57	C₇	—	6	1	no	—
58	C₇	#14523572	1	1	yes	characteristic, complemented
59	C₇	#1039⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
60	C₂×C₃	—	175	1	no	—
61	C₅	—	3	2	no	—
62	C₄	—	175	1	no	Sylow
63	C₃	—	175	1	no	Sylow
64	C₂	#20301298	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#20301298
commutator, hypocenter	C₅²×C₇²	C₃×C₄
Fitting, socle, cosocle	C₂×C₅²×C₇²	C₂×C₃
prime radical	C₂×C₇×C₇⋊(C₅²⋊C₃)	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. C₇×C₇⋊(C₅²⋊C₃)	C₄	semidirect
7. C₅²×C₇²	C₃×C₄	semidirect
11. C₇⋊#1039	ΓL(1,𝔽₈)	semidirect
14. C₇⋊(C₅²⋊C₃)	Dic₇	semidirect
25. C₅²×C₇	C₄×ΓL(1,𝔽₈)	semidirect
26. C₅²×C₇	C₃×Dic₇	semidirect
40. C₇²	#1039⋊C₃	semidirect
48. C₅²	ΓL(1,𝔽₈)×Dic₇	semidirect
58. C₇	#14523572	semidirect
59. C₇	#1039⋊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
C₂×C₇×C₇⋊(C₅²⋊C₃)	2
C₇×C₇⋊#1039	3
#1039⋊ΓL(1,𝔽₈)	7
#14523572	7
ΓL(1,𝔽₈)×Dic₇	25