Subgroups — C₇⋊D₇×D₃₁⋊C₃ #32912183

Order 18228.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁⋊C₃×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₃₁⋊C₃)⋊(C₇⋊D₇)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇²×D₃₁⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃₁×C₇⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₇²×C₃₁⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇⋊D₂₁₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃₁×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇²×D₃₁	C₂×C₃	1	1	yes	characteristic, complemented
9	D₇×D₃₁⋊C₃	—	7	8	no	maximal
10	C₇²×C₃₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
11	(C₃₁⋊C₃)⋊D₇	—	7	8	no	—
12	D₇×C₃₁⋊C₃	—	7	8	no	—
13	C₇×D₃₁⋊C₃	D₇	1	8	yes	complemented
14	D₇×D₃₁	—	7	8	no	—
15	C₇×C₃₁⋊C₃	C₂×D₇	1	8	yes	complemented
16	C₂×C₃×C₇⋊D₇	—	31	1	no	maximal, Hall
17	D₂₁₇	—	7	8	no	—
18	C₇×D₃₁	C₃×D₇	1	8	yes	complemented
19	D₇×C₃₁	—	7	8	no	—
20	C₂×D₃₁⋊C₃	—	49	1	no	Hall
21	C₂×C₃×C₇²	—	31	1	no	—
22	C₃×C₇⋊D₇	—	31	1	no	—
23	C₃×C₇⋊D₇	—	62	1	no	—
24	C₇×C₃₁	C₂×C₃×D₇	1	8	yes	complemented
25	C₂×C₇⋊D₇	—	31	1	no	Hall
26	C₂×C₃₁⋊C₃	—	49	1	no	—
27	D₃₁⋊C₃	—	49	1	no	—
28	D₃₁⋊C₃	C₇⋊D₇	1	1	yes	characteristic, direct product factor
29	C₃×C₇²	—	62	1	no	Hall
30	C₂×D₃₁	—	49	1	no	Hall
31	C₂×C₇²	—	31	1	no	—
32	C₇⋊D₇	—	31	1	no	—
33	C₇⋊D₇	D₃₁⋊C₃	1	1	yes	characteristic, direct product factor
34	C₃₁⋊C₃	C₂×C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×D₇	—	217	8	no	—
36	C₂×C₃₁	—	49	1	no	—
37	D₃₁	—	49	1	no	—
38	D₃₁	C₃×C₇⋊D₇	1	1	yes	characteristic, complemented
39	C₇²	C₂×D₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃×C₇	—	31	8	no	—
41	C₃×D₇	—	434	8	no	—
42	C₃×D₇	—	217	8	no	—
43	C₃₁	C₂×C₃×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
44	C₂×D₇	—	217	8	no	—
45	C₃×C₇	—	62	8	no	—
46	C₂×C₇	—	31	8	no	—
47	D₇	—	7	8	no	—
48	D₇	—	217	8	no	—
49	C₂²×C₃	—	1519	1	no	Hall
50	C₇	D₇×D₃₁⋊C₃	1	8	yes	complemented
51	C₂×C₃	—	31	1	no	—
52	C₂×C₃	—	3038	1	no	—
53	C₂×C₃	—	1519	1	no	—
54	C₂²	—	1519	1	no	Sylow
55	C₃	—	62	1	no	Sylow
56	C₂	—	31	1	no	—
57	C₂	—	49	1	no	—
58	C₂	—	1519	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₃₁	C₂²×C₃
LCS	G ▷ C₇²×C₃₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃₁⋊C₃×C₇⋊D₇	C₂	semidirect
2. (C₃₁⋊C₃)⋊(C₇⋊D₇)	C₂	semidirect
3. C₇²×D₃₁⋊C₃	C₂	semidirect
4. D₃₁×C₇⋊D₇	C₃	semidirect
5. C₇²×C₃₁⋊C₃	C₂²	semidirect
6. C₇⋊D₂₁₇	C₂×C₃	semidirect
7. C₃₁×C₇⋊D₇	C₂×C₃	semidirect
8. C₇²×D₃₁	C₂×C₃	semidirect
10. C₇²×C₃₁	C₂²×C₃	semidirect
13. C₇×D₃₁⋊C₃	D₇	semidirect
15. C₇×C₃₁⋊C₃	C₂×D₇	semidirect
18. C₇×D₃₁	C₃×D₇	semidirect
24. C₇×C₃₁	C₂×C₃×D₇	semidirect
28. D₃₁⋊C₃	C₇⋊D₇	direct (N × H)
33. C₇⋊D₇	D₃₁⋊C₃	direct (N × H)
34. C₃₁⋊C₃	C₂×C₇⋊D₇	semidirect
38. D₃₁	C₃×C₇⋊D₇	semidirect
39. C₇²	C₂×D₃₁⋊C₃	semidirect
43. C₃₁	C₂×C₃×C₇⋊D₇	semidirect
50. C₇	D₇×D₃₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃₁⋊C₃×C₇⋊D₇	2
(C₃₁⋊C₃)⋊(C₇⋊D₇)	2
C₇²×D₃₁⋊C₃	2
D₃₁×C₇⋊D₇	3
D₇×D₃₁⋊C₃	7
C₂×C₃×C₇⋊D₇	31

Subgroups — C7⋊D7×D31⋊C3 #32912183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇⋊D₇×D₃₁⋊C₃ #32912183