Subgroups — #44528409

Order 31164.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃×#3861	C₂	1	1	yes	maximal, characteristic
2	C₇×C₇⋊#6157	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅₃×#3861	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇²×D₅₃	C₂×C₃	1	1	yes	characteristic
5	ΓL(1,𝔽₈)×#6157	—	7	1	no	maximal
6	ΓL(1,𝔽₈)⋊#6157	—	7	1	no	maximal
7	C₇²×C₅₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	ΓL(1,𝔽₈)×D₅₃	—	7	1	no	—
9	ΓL(1,𝔽₈)×D₅₃	—	7	1	no	—
10	ΓL(1,𝔽₈)×D₅₃	—	14	3	no	—
11	C₇⋊#6157	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
12	C₇×#6157	—	7	1	no	—
13	ΓL(1,𝔽₈)×C₅₃	—	7	1	no	—
14	ΓL(1,𝔽₈)×C₅₃	—	7	1	no	—
15	ΓL(1,𝔽₈)×C₅₃	—	14	3	no	—
16	C₇×D₅₃	—	2	3	no	—
17	C₇×D₅₃	AGL(1,𝔽₇)	1	1	yes	characteristic
18	C₇×D₅₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
19	C₃×#6157	—	49	1	no	Hall
20	#10596873	—	53	1	no	maximal, Hall
21	C₇×C₅₃	—	2	3	no	—
22	C₇×C₅₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
23	C₇×C₅₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	C₃×D₅₃	—	49	1	no	—
25	C₂×#3861	—	53	1	no	—
26	#6157	—	7	1	no	Hall
27	C₇×Dic₇	—	53	1	no	Hall
28	C₃×C₅₃	—	49	1	no	Hall
29	#3861	#6157	1	1	yes	characteristic, complemented, Hall
30	D₅₃	#64423	1	1	yes	characteristic
31	C₂×C₇²	—	53	1	no	—
32	C₄×ΓL(1,𝔽₈)	—	371	1	no	—
33	ΓL(1,𝔽₈)⋊C₄	—	371	1	no	—
34	C₅₃	#10596873	1	1	yes	characteristic, complemented, Sylow
35	C₇²	C₃×#6157	1	1	yes	characteristic, complemented, Sylow
36	C₂×ΓL(1,𝔽₈)	—	371	1	no	—
37	C₂×ΓL(1,𝔽₈)	—	371	1	no	—
38	C₂×ΓL(1,𝔽₈)	—	742	3	no	—
39	C₄×C₇	—	371	1	no	—
40	Dic₇	—	53	1	no	—
41	ΓL(1,𝔽₈)	—	7	1	no	—
42	ΓL(1,𝔽₈)	—	7	1	no	—
43	ΓL(1,𝔽₈)	—	14	3	no	—
44	C₂×C₇	—	53	1	no	—
45	C₂×C₇	—	53	1	no	—
46	C₂×C₇	—	106	3	no	—
47	C₃×C₄	—	2597	1	no	Hall
48	C₇	—	2	3	no	—
49	C₇	ΓL(1,𝔽₈)⋊#6157	1	1	yes	characteristic, complemented
50	C₇	ΓL(1,𝔽₈)×#6157	1	1	yes	characteristic, complemented
51	C₂×C₃	—	2597	1	no	—
52	C₄	—	371	1	no	Sylow
53	C₃	—	49	1	no	Sylow
54	C₂	—	53	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇²×C₅₃	C₃×C₄
prime radical	D₅₃×#3861	C₂
cosocle	C₇²×D₅₃	C₂×C₃
LCS	G ▷ C₇²×C₅₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇×C₇⋊#6157	C₃	semidirect
3. C₅₃×#3861	C₄	semidirect
7. C₇²×C₅₃	C₃×C₄	semidirect
11. C₇⋊#6157	ΓL(1,𝔽₈)	semidirect
22. C₇×C₅₃	ΓL(1,𝔽₈)⋊C₄	semidirect
23. C₇×C₅₃	C₄×ΓL(1,𝔽₈)	semidirect
29. #3861	#6157	semidirect
34. C₅₃	#10596873	semidirect
35. C₇²	C₃×#6157	semidirect
49. C₇	ΓL(1,𝔽₈)⋊#6157	semidirect
50. C₇	ΓL(1,𝔽₈)×#6157	semidirect

Maximal subgroups

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

subgroup	index
D₅₃×#3861	2
C₇×C₇⋊#6157	3
ΓL(1,𝔽₈)×#6157	7
ΓL(1,𝔽₈)⋊#6157	7
#10596873	53