Subgroups — #45657440

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	#24925972	C₃	1	2	yes	maximal, complemented
3	C₃₇×#64425	C₃	1	1	yes	maximal, characteristic, complemented
4	#24925949	C₃	1	1	yes	maximal, characteristic, 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	AGL(1,𝔽₇)×C₃₇⋊C₃	—	7	2	no	maximal
9	C₃₇×C₇⋊D₇	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	#13095536	—	7	2	no	—
13	#13095535	—	7	2	no	—
14	#13095535	—	21	2	no	—
15	C₃₇×AGL(1,𝔽₇)	—	7	2	no	—
16	D₇×C₃₇⋊C₃	—	7	2	no	—
17	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	—	37	1	no	maximal, Hall
18	#11715913	—	7	2	no	—
19	#11715913	—	21	2	no	—
20	#11715912	AGL(1,𝔽₇)	1	2	yes	complemented
21	C₇×C₃₇⋊C₃	—	7	2	no	—
22	ΓL(1,𝔽₈)×C₃₇	—	7	2	no	—
23	C₂×C₃×C₃₇⋊C₃	—	49	1	no	Hall
24	D₇×C₃₇	—	7	2	no	—
25	D₇×C₃₇	—	21	2	no	—
26	ΓL(1,𝔽₈)²	—	37	1	no	Hall
27	C₃×C₃₇⋊C₃	—	49	1	no	Hall
28	#64425	C₃₇⋊C₃	1	1	yes	characteristic, complemented
29	#64424	—	37	1	no	—
30	ΓL(1,𝔽₈)⋊D₇	—	37	2	no	—
31	C₇×C₃₇	—	3	2	no	—
32	C₇×C₃₇	C₃×AGL(1,𝔽₇)	1	2	yes	complemented
33	C₂×C₃×C₃₇	—	49	1	no	—
34	C₂×C₃₇⋊C₃	—	49	2	no	—
35	C₂×C₃₇⋊C₃	—	49	1	no	—
36	#3862	C₂×C₃₇⋊C₃	1	1	yes	characteristic, complemented
37	#3861	—	37	1	no	—
38	C₇×ΓL(1,𝔽₈)	—	37	2	no	—
39	C₃×AGL(1,𝔽₇)	—	259	2	no	—
40	C₃×C₃₇	—	49	1	no	—
41	C₃₇⋊C₃	—	7	2	no	—
42	C₃₇⋊C₃	—	49	1	no	—
43	C₇⋊D₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented, Hall
44	C₂×C₃₇	—	49	1	no	Hall
45	C₃×ΓL(1,𝔽₈)	—	259	2	no	—
46	C₇²	C₂×C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
47	C₃×D₇	—	259	2	no	—
48	AGL(1,𝔽₇)	—	259	2	no	—
49	AGL(1,𝔽₇)	—	259	2	no	—
50	AGL(1,𝔽₇)	—	7	2	no	—
51	AGL(1,𝔽₇)	—	777	2	no	—
52	C₃₇	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	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	D₇	—	7	2	no	—
60	D₇	—	21	2	no	—
61	C₃²	—	1813	1	no	Sylow
62	C₇	—	3	2	no	—
63	C₇	AGL(1,𝔽₇)×C₃₇⋊C₃	1	2	yes	complemented
64	C₂×C₃	—	1813	2	no	—
65	C₂×C₃	—	1813	1	no	—
66	C₂×C₃	—	49	1	no	—
67	C₃	—	259	2	no	—
68	C₃	—	1813	1	no	—
69	C₃	—	49	1	no	—
70	C₂	—	49	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. #24925972	C₃	semidirect
3. C₃₇×#64425	C₃	semidirect
4. #24925949	C₃	semidirect
5. C₃₇×#3862	C₂×C₃	semidirect
6. #19129588	C₂×C₃	semidirect
7. C₇×#11715912	C₂×C₃	semidirect
9. C₃₇×C₇⋊D₇	C₃²	semidirect
11. C₇²×C₃₇	C₂×C₃²	semidirect
20. #11715912	AGL(1,𝔽₇)	semidirect
28. #64425	C₃₇⋊C₃	semidirect
32. C₇×C₃₇	C₃×AGL(1,𝔽₇)	semidirect
36. #3862	C₂×C₃₇⋊C₃	semidirect
43. C₇⋊D₇	C₃×C₃₇⋊C₃	semidirect
46. C₇²	C₂×C₃×C₃₇⋊C₃	semidirect
52. C₃₇	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	semidirect
63. C₇	AGL(1,𝔽₇)×C₃₇⋊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
#24925972	3
C₃₇×#64425	3
#24925949	3
AGL(1,𝔽₇)×C₃₇⋊C₃	7
ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	37