Subgroups — #45657410

Order 32634.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654589	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₃₇×#3862	C₃	1	1	yes	maximal, characteristic, complemented
3	#24925974	C₃	1	1	yes	maximal, characteristic, complemented
4	C₇×#13095532	C₃	1	2	yes	maximal, complemented
5	C₃₇×#3862	C₂×C₃	1	1	yes	characteristic, complemented
6	#19129588	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×#11715912	C₂×C₃	1	2	yes	complemented
8	ΓL(1,𝔽₈)×D₃₇⋊C₃	—	7	2	no	maximal
9	C₇²×D₃₇	C₃²	1	1	yes	characteristic, complemented, Hall
10	ΓL(1,𝔽₈)×C₃₇⋊C₃	—	7	2	no	—
11	C₇²×C₃₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
12	#13095532	ΓL(1,𝔽₈)	1	2	yes	complemented
13	#13095531	—	7	2	no	—
14	#13095531	—	21	2	no	—
15	ΓL(1,𝔽₈)×D₃₇	—	7	2	no	—
16	C₇×D₃₇⋊C₃	—	7	2	no	—
17	C₂×ΓL(1,𝔽₈)²	—	37	1	no	maximal, Hall
18	#11715913	—	7	2	no	—
19	#11715913	—	21	2	no	—
20	#11715912	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
21	C₇×C₃₇⋊C₃	—	7	2	no	—
22	ΓL(1,𝔽₈)×C₃₇	—	7	2	no	—
23	C₃×D₃₇⋊C₃	—	49	1	no	Hall
24	C₇×D₃₇	—	3	2	no	—
25	C₇×D₃₇	C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
26	ΓL(1,𝔽₈)²	—	37	1	no	Hall
27	C₃×C₃₇⋊C₃	—	49	1	no	Hall
28	C₂×#3862	—	37	1	no	—
29	C₂×#3861	—	37	1	no	—
30	C₂×C₇×ΓL(1,𝔽₈)	—	37	2	no	—
31	C₇×C₃₇	—	3	2	no	—
32	C₇×C₃₇	C₂×C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
33	C₃×D₃₇	—	49	1	no	—
34	D₃₇⋊C₃	—	7	2	no	—
35	D₃₇⋊C₃	—	49	1	no	—
36	#3862	D₃₇⋊C₃	1	1	yes	characteristic, complemented
37	#3861	—	37	1	no	—
38	C₇×ΓL(1,𝔽₈)	—	37	2	no	—
39	C₂×C₃×ΓL(1,𝔽₈)	—	259	2	no	—
40	C₃×C₃₇	—	49	1	no	—
41	C₃₇⋊C₃	—	7	2	no	—
42	C₃₇⋊C₃	—	49	1	no	—
43	C₂×C₇²	—	37	1	no	Hall
44	D₃₇	ΓL(1,𝔽₈)²	1	1	yes	characteristic, complemented, Hall
45	C₃×ΓL(1,𝔽₈)	—	259	2	no	—
46	C₇²	C₃×D₃₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃×C₇	—	259	2	no	—
48	C₂×ΓL(1,𝔽₈)	—	259	2	no	—
49	C₂×ΓL(1,𝔽₈)	—	259	2	no	—
50	C₂×ΓL(1,𝔽₈)	—	777	2	no	—
51	C₂×ΓL(1,𝔽₈)	—	37	2	no	—
52	C₃₇	C₂×ΓL(1,𝔽₈)²	1	1	yes	characteristic, complemented, Sylow
53	C₃×C₇	—	259	2	no	—
54	ΓL(1,𝔽₈)	—	259	2	no	—
55	ΓL(1,𝔽₈)	—	7	2	no	—
56	ΓL(1,𝔽₈)	—	777	2	no	—
57	ΓL(1,𝔽₈)	—	37	2	no	—
58	C₂×C₃²	—	1813	1	no	Hall
59	C₂×C₇	—	37	2	no	—
60	C₂×C₇	—	111	2	no	—
61	C₃²	—	1813	1	no	Sylow
62	C₇	—	3	2	no	—
63	C₇	ΓL(1,𝔽₈)×D₃₇⋊C₃	1	2	yes	complemented
64	C₂×C₃	—	259	2	no	—
65	C₂×C₃	—	1813	1	no	—
66	C₂×C₃	—	1813	1	no	—
67	C₃	—	259	2	no	—
68	C₃	—	1813	1	no	—
69	C₃	—	49	1	no	—
70	C₂	—	37	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₃₇	C₂×C₃²
LCS	G ▷ C₇²×C₃₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #30654589	C₂	semidirect
2. D₃₇×#3862	C₃	semidirect
3. #24925974	C₃	semidirect
4. C₇×#13095532	C₃	semidirect
5. C₃₇×#3862	C₂×C₃	semidirect
6. #19129588	C₂×C₃	semidirect
7. C₇×#11715912	C₂×C₃	semidirect
9. C₇²×D₃₇	C₃²	semidirect
11. C₇²×C₃₇	C₂×C₃²	semidirect
12. #13095532	ΓL(1,𝔽₈)	semidirect
20. #11715912	C₂×ΓL(1,𝔽₈)	semidirect
25. C₇×D₃₇	C₃×ΓL(1,𝔽₈)	semidirect
32. C₇×C₃₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
36. #3862	D₃₇⋊C₃	semidirect
44. D₃₇	ΓL(1,𝔽₈)²	semidirect
46. C₇²	C₃×D₃₇⋊C₃	semidirect
52. C₃₇	C₂×ΓL(1,𝔽₈)²	semidirect
63. C₇	ΓL(1,𝔽₈)×D₃₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#30654589	2
D₃₇×#3862	3
#24925974	3
C₇×#13095532	3
ΓL(1,𝔽₈)×D₃₇⋊C₃	7
C₂×ΓL(1,𝔽₈)²	37