Subgroups — C₅²⋊C₃×C₃⋊(C₇⋊D₇) #36348006

Order 22050.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×C₅²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅²×C₃⋊(C₇⋊D₇)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅²⋊C₃×C₇⋊D₇	—	3	1	no	maximal
4	C₇²×C₅²⋊C₃	—	2	1	no	—
5	C₇²×C₅²⋊C₃	D₃	1	1	yes	characteristic, complemented
6	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	D₂₁×C₅²⋊C₃	—	7	8	no	maximal
8	C₅²×C₇⋊D₇	—	3	1	no	Hall
9	C₃×C₇×C₅²⋊C₃	D₇	1	8	yes	complemented
10	C₅×C₃⋊(C₇⋊D₇)	—	3	2	no	—
11	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
12	C₅²×D₂₁	—	7	8	no	—
13	D₇×C₅²⋊C₃	—	21	8	no	—
14	C₃×C₃⋊(C₇⋊D₇)	—	25	1	no	maximal, Hall
15	C₃×C₅×C₇²	—	3	2	no	—
16	C₃×C₅²×C₇	C₃×D₇	1	8	yes	complemented
17	C₇×C₅²⋊C₃	—	2	8	no	—
18	C₇×C₅²⋊C₃	D₂₁	1	8	yes	complemented
19	C₅×C₇⋊D₇	—	9	2	no	—
20	D₃×C₅²⋊C₃	—	49	1	no	Hall
21	C₃²×C₇²	—	25	1	no	Hall
22	C₅²×D₇	—	21	8	no	—
23	C₃⋊(C₇⋊D₇)	C₅²⋊C₃	1	1	yes	characteristic, direct product factor
24	C₃×C₇⋊D₇	—	75	1	no	—
25	C₅×C₇²	—	3	2	no	—
26	C₃×C₅²⋊C₃	C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
27	C₅×D₂₁	—	21	16	no	—
28	C₅²×C₇	C₃×D₂₁	1	8	yes	complemented
29	C₅²×D₃	—	49	1	no	—
30	C₂×C₅²⋊C₃	—	147	1	no	—
31	C₃×C₇²	—	25	1	no	—
32	C₃×C₇²	—	50	1	no	—
33	C₃×C₇²	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
34	C₃×D₂₁	—	175	8	no	—
35	C₃×C₅×C₇	—	3	16	no	—
36	C₇⋊D₇	—	3	1	no	Hall
37	C₃×C₅²	C₃×C₇⋊D₇	1	1	yes	characteristic, complemented
38	C₅²⋊C₃	—	2	1	no	—
39	C₅²⋊C₃	C₃⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor
40	C₅×D₇	—	63	16	no	—
41	C₃²×C₇	—	25	8	no	—
42	C₂×C₅²	—	147	1	no	Hall
43	C₇²	D₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
44	D₂₁	—	7	8	no	—
45	C₃×D₇	—	525	8	no	—
46	C₅×C₇	—	3	16	no	—
47	C₅×D₃	—	147	2	no	—
48	C₅²	C₃×C₃⋊(C₇⋊D₇)	1	1	yes	characteristic, complemented, Sylow
49	C₃×C₇	—	25	8	no	—
50	C₃×C₇	—	50	8	no	—
51	C₃×C₇	D₇×C₅²⋊C₃	1	8	yes	complemented
52	C₃×D₃	—	1225	1	no	Hall
53	C₃×C₅	—	3	2	no	—
54	D₇	—	21	8	no	—
55	C₂×C₅	—	441	2	no	—
56	C₃²	—	25	1	no	Sylow
57	C₇	D₂₁×C₅²⋊C₃	1	8	yes	complemented
58	C₂×C₃	—	3675	1	no	—
59	D₃	—	49	1	no	—
60	C₅	—	3	2	no	—
61	C₃	—	25	1	no	—
62	C₃	—	50	1	no	—
63	C₃	C₅²⋊C₃×C₇⋊D₇	1	1	yes	characteristic, complemented
64	C₂	—	147	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₃×C₅²×C₇²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₇²×C₅²⋊C₃	C₂	semidirect
2. C₅²×C₃⋊(C₇⋊D₇)	C₃	semidirect
5. C₇²×C₅²⋊C₃	D₃	semidirect
6. C₃×C₅²×C₇²	C₂×C₃	semidirect
9. C₃×C₇×C₅²⋊C₃	D₇	semidirect
11. C₅²×C₇²	C₃×D₃	semidirect
16. C₃×C₅²×C₇	C₃×D₇	semidirect
18. C₇×C₅²⋊C₃	D₂₁	semidirect
23. C₃⋊(C₇⋊D₇)	C₅²⋊C₃	direct (N × H)
26. C₃×C₅²⋊C₃	C₇⋊D₇	semidirect
28. C₅²×C₇	C₃×D₂₁	semidirect
33. C₃×C₇²	C₂×C₅²⋊C₃	semidirect
37. C₃×C₅²	C₃×C₇⋊D₇	semidirect
39. C₅²⋊C₃	C₃⋊(C₇⋊D₇)	direct (N × H)
43. C₇²	D₃×C₅²⋊C₃	semidirect
48. C₅²	C₃×C₃⋊(C₇⋊D₇)	semidirect
51. C₃×C₇	D₇×C₅²⋊C₃	semidirect
57. C₇	D₂₁×C₅²⋊C₃	semidirect
63. C₃	C₅²⋊C₃×C₇⋊D₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇²×C₅²⋊C₃	2
C₅²×C₃⋊(C₇⋊D₇)	3
C₅²⋊C₃×C₇⋊D₇	3
D₂₁×C₅²⋊C₃	7
C₃×C₃⋊(C₇⋊D₇)	25

Subgroups — C52⋊C3×C3⋊(C7⋊D7) #36348006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²⋊C₃×C₃⋊(C₇⋊D₇) #36348006