Subgroups — #45657459

Order 32634.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654579	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×(C₃₇⋊C₃)⋊D₇	C₃	1	1	yes	maximal, characteristic
3	C₇²×C₃₇⋊C₃	C₂×C₃	1	1	yes	characteristic
4	#17987958	—	7	1	no	maximal
5	#17987952	—	7	1	no	maximal
6	C₇×D₂₅₉	C₉	1	1	yes	characteristic, complemented, Hall
7	#14556422	—	7	1	no	—
8	#14556422	—	7	1	no	—
9	#14556422	—	14	3	no	—
10	C₇²×C₃₇	C₂×C₉	1	1	yes	characteristic, complemented, Hall
11	(C₃₇⋊C₃)⋊D₇	ΓL(1,𝔽₈)	1	1	yes	characteristic
12	C₇×D₃₇⋊C₃	—	7	1	no	—
13	#11725076	—	37	1	no	maximal, Hall
14	C₇×C₃₇⋊C₃	—	2	3	no	—
15	C₇×C₃₇⋊C₃	AGL(1,𝔽₇)	1	1	yes	characteristic
16	C₇×C₃₇⋊C₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
17	#66526⋊C₂	—	49	1	no	Hall
18	D₂₅₉	C₇⋊C₉	1	1	yes	characteristic, complemented
19	C₇×D₃₇	—	7	1	no	—
20	#89204	—	37	1	no	Hall
21	#66526	—	49	1	no	Hall
22	C₃×C₇×D₇	—	37	1	no	—
23	C₇×C₃₇	—	2	3	no	—
24	C₇×C₃₇	D₇⋊C₉	1	1	yes	characteristic, complemented
25	C₇×C₃₇	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented
26	D₃₇⋊C₃	—	7	1	no	—
27	C₃×C₇²	—	37	1	no	—
28	C₂×C₇⋊C₉	—	259	1	no	—
29	D₇⋊C₉	—	259	1	no	—
30	C₃₇⋊C₃	#64423	1	1	yes	characteristic
31	C₇×D₇	—	37	1	no	Hall
32	D₃₇	—	7	1	no	Hall
33	C₇⋊C₉	—	259	1	no	—
34	C₇⋊C₉	—	259	1	no	—
35	C₇⋊C₉	—	518	3	no	—
36	C₇²	#66526⋊C₂	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃×C₇	—	259	1	no	—
38	C₃×D₇	—	37	1	no	—
39	C₃₇	#11725076	1	1	yes	characteristic, complemented, Sylow
40	C₃×C₇	—	37	1	no	—
41	C₃×C₇	—	37	1	no	—
42	C₃×C₇	—	74	3	no	—
43	C₂×C₉	—	1813	1	no	Hall
44	C₂×C₇	—	259	1	no	—
45	D₇	—	37	1	no	—
46	C₉	—	1813	1	no	Sylow
47	C₇	—	2	3	no	—
48	C₇	#17987952	1	1	yes	characteristic, complemented
49	C₇	#17987958	1	1	yes	characteristic, complemented
50	C₂×C₃	—	259	1	no	—
51	C₃	—	37	1	no	—
52	C₂	—	259	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇²×C₃₇	C₂×C₉
prime radical	C₇×(C₃₇⋊C₃)⋊D₇	C₃
cosocle	C₇²×C₃₇⋊C₃	C₂×C₃
LCS	G ▷ C₇²×C₃₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #30654579	C₂	semidirect
6. C₇×D₂₅₉	C₉	semidirect
10. C₇²×C₃₇	C₂×C₉	semidirect
18. D₂₅₉	C₇⋊C₉	semidirect
24. C₇×C₃₇	D₇⋊C₉	semidirect
25. C₇×C₃₇	C₂×C₇⋊C₉	semidirect
36. C₇²	#66526⋊C₂	semidirect
39. C₃₇	#11725076	semidirect
48. C₇	#17987952	semidirect
49. C₇	#17987958	semidirect

Maximal subgroups

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

subgroup	index
#30654579	2
C₇×(C₃₇⋊C₃)⋊D₇	3
#17987958	7
#17987952	7
#11725076	37