Subgroups — #38760345

Order 25284.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#11744748	C₂	1	1	yes	maximal, characteristic
2	C₇×Dic₃₀₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×#11744748	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇²×C₄₃	C₂×C₃	1	1	yes	characteristic
5	#16813400	—	7	1	no	maximal
6	(C₄₃⋊C₃)⋊Dic₇	—	7	1	no	maximal
7	C₇²×C₄₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₂×#11744748	D₇	1	1	yes	characteristic
9	C₂×C₇×C₄₃⋊C₃	—	7	1	no	—
10	Dic₃₀₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
11	C₇×Dic₄₃	—	7	1	no	—
12	#11744748	Dic₇	1	1	yes	characteristic, complemented
13	C₇×C₄₃⋊C₃	—	7	1	no	—
14	C₂×C₇×C₄₃	—	6	1	no	—
15	C₂×C₇×C₄₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
16	C₂×C₇×C₄₃	C₃×D₇	1	1	yes	characteristic
17	ΓL(1,𝔽₈)×Dic₇	—	43	1	no	maximal, Hall
18	(C₄₃⋊C₃)⋊C₄	—	49	1	no	Hall
19	C₇×C₄₃	—	6	1	no	—
20	C₇×C₄₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₇×C₄₃	C₃×Dic₇	1	1	yes	characteristic, complemented
22	C₂×C₇×ΓL(1,𝔽₈)	—	86	1	no	—
23	C₂×C₄₃⋊C₃	—	7	1	no	—
24	C₇×Dic₇	—	43	1	no	Hall
25	Dic₄₃	—	7	1	no	Hall
26	C₇×ΓL(1,𝔽₈)	—	86	1	no	Hall
27	C₄₃⋊C₃	—	7	1	no	Hall
28	C₂×C₇²	D₄₃⋊C₃	1	1	yes	characteristic
29	C₂×C₄₃	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
30	C₃×Dic₇	—	301	1	no	—
31	C₄×ΓL(1,𝔽₈)	—	301	1	no	—
32	C₇²	(C₄₃⋊C₃)⋊C₄	1	1	yes	characteristic, complemented, Sylow
33	C₄₃	ΓL(1,𝔽₈)×Dic₇	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃×C₇	—	602	1	no	—
35	C₂×ΓL(1,𝔽₈)	—	86	1	no	—
36	C₄×C₇	—	301	1	no	—
37	Dic₇	—	43	1	no	—
38	C₃×C₇	—	602	1	no	—
39	ΓL(1,𝔽₈)	—	86	1	no	—
40	C₂×C₇	—	6	1	no	—
41	C₂×C₇	(C₄₃⋊C₃)⋊D₇	1	1	yes	characteristic
42	C₂×C₇	#14268587	1	1	yes	characteristic
43	C₃×C₄	—	2107	1	no	Hall
44	C₇	—	6	1	no	—
45	C₇	(C₄₃⋊C₃)⋊Dic₇	1	1	yes	characteristic, complemented
46	C₇	#16813400	1	1	yes	characteristic, complemented
47	C₂×C₃	—	602	1	no	—
48	C₄	—	301	1	no	Sylow
49	C₃	—	602	1	no	Sylow
50	C₂	#27162438	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#27162438
commutator, hypocenter	C₇²×C₄₃	C₃×C₄
Fitting, socle, cosocle	C₂×C₇²×C₄₃	C₂×C₃
prime radical	C₂×C₇×#11744748	C₂
LCS	G ▷ C₇²×C₄₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇×Dic₃₀₁	C₃	semidirect
3. C₇×#11744748	C₄	semidirect
7. C₇²×C₄₃	C₃×C₄	semidirect
10. Dic₃₀₁	ΓL(1,𝔽₈)	semidirect
12. #11744748	Dic₇	semidirect
20. C₇×C₄₃	C₄×ΓL(1,𝔽₈)	semidirect
21. C₇×C₄₃	C₃×Dic₇	semidirect
32. C₇²	(C₄₃⋊C₃)⋊C₄	semidirect
33. C₄₃	ΓL(1,𝔽₈)×Dic₇	semidirect
45. C₇	(C₄₃⋊C₃)⋊Dic₇	semidirect
46. C₇	#16813400	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#11744748	2
C₇×Dic₃₀₁	3
#16813400	7
(C₄₃⋊C₃)⋊Dic₇	7
ΓL(1,𝔽₈)×Dic₇	43