Subgroups — C₇²×C₃⋊((C₅⋊D₅)⋊C₃) #36348007

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	C₇²×C₃⋊(C₅⋊D₅)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₇²×(C₅⋊D₅)⋊C₃	—	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	C₇×C₃⋊((C₅⋊D₅)⋊C₃)	C₇	1	8	yes	maximal, direct product factor
8	C₇²×C₅⋊D₅	—	3	1	no	Hall
9	C₃×C₇×C₅²⋊C₃	C₂×C₇	1	8	yes	complemented
10	C₇²×D₁₅	—	15	2	no	—
11	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
12	C₇×C₃⋊(C₅⋊D₅)	C₃×C₇	1	8	yes	complemented
13	C₇×(C₅⋊D₅)⋊C₃	—	3	8	no	—
14	C₃×D₃×C₇²	—	25	1	no	maximal, Hall
15	C₃×C₅×C₇²	—	3	2	no	—
16	C₃×C₅²×C₇	C₂×C₃×C₇	1	8	yes	complemented
17	C₇×C₅²⋊C₃	—	2	8	no	—
18	C₇×C₅²⋊C₃	D₃×C₇	1	8	yes	complemented
19	C₇²×D₅	—	45	2	no	—
20	C₃⋊((C₅⋊D₅)⋊C₃)	C₇²	1	1	yes	characteristic, direct product factor, Hall
21	C₃²×C₇²	—	25	1	no	Hall
22	C₇×C₅⋊D₅	—	3	8	no	—
23	C₂×C₃×C₇²	—	75	1	no	—
24	D₃×C₇²	—	25	1	no	—
25	C₅×C₇²	—	3	2	no	—
26	C₃×C₅²⋊C₃	C₂×C₇²	1	1	yes	characteristic, complemented, Hall
27	C₇×D₁₅	—	15	16	no	—
28	C₅²×C₇	C₃×D₃×C₇	1	8	yes	complemented
29	C₃⋊(C₅⋊D₅)	C₃×C₇²	1	1	yes	characteristic, complemented
30	(C₅⋊D₅)⋊C₃	—	3	1	no	—
31	C₃×C₇²	—	25	1	no	—
32	C₃×C₇²	—	50	1	no	—
33	C₃×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
34	C₃×D₃×C₇	—	25	8	no	—
35	C₃×C₅×C₇	—	3	16	no	—
36	C₂×C₇²	—	75	1	no	Hall
37	C₃×C₅²	C₂×C₃×C₇²	1	1	yes	characteristic, complemented
38	C₅²⋊C₃	—	2	1	no	—
39	C₅²⋊C₃	D₃×C₇²	1	1	yes	characteristic, complemented
40	C₇×D₅	—	45	16	no	—
41	C₃²×C₇	—	25	8	no	—
42	C₅⋊D₅	—	3	1	no	Hall
43	C₇²	C₃⋊((C₅⋊D₅)⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
44	C₂×C₃×C₇	—	75	8	no	—
45	D₃×C₇	—	25	8	no	—
46	C₅×C₇	—	3	16	no	—
47	D₁₅	—	15	2	no	—
48	C₅²	C₃×D₃×C₇²	1	1	yes	characteristic, complemented, Sylow
49	C₃×C₇	—	25	8	no	—
50	C₃×C₇	—	50	8	no	—
51	C₃×C₇	C₇×(C₅⋊D₅)⋊C₃	1	8	yes	complemented
52	C₃×D₃	—	25	1	no	Hall
53	C₃×C₅	—	3	2	no	—
54	C₂×C₇	—	75	8	no	—
55	D₅	—	45	2	no	—
56	C₃²	—	25	1	no	Sylow
57	C₇	C₇×C₃⋊((C₅⋊D₅)⋊C₃)	1	8	yes	direct product factor
58	C₂×C₃	—	75	1	no	—
59	D₃	—	25	1	no	—
60	C₅	—	3	2	no	—
61	C₃	—	25	1	no	—
62	C₃	—	50	1	no	—
63	C₃	C₇²×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
64	C₂	—	75	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇²	C₃⋊((C₅⋊D₅)⋊C₃)
commutator, cosocle, hypocenter	C₃×C₅²	C₂×C₃×C₇²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₇²

Complements

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

N	H ≅ G/N	type
1. C₃×C₇²×C₅²⋊C₃	C₂	semidirect
2. C₇²×C₃⋊(C₅⋊D₅)	C₃	semidirect
5. C₇²×C₅²⋊C₃	D₃	semidirect
6. C₃×C₅²×C₇²	C₂×C₃	semidirect
7. C₇×C₃⋊((C₅⋊D₅)⋊C₃)	C₇	direct (N × H)
9. C₃×C₇×C₅²⋊C₃	C₂×C₇	semidirect
11. C₅²×C₇²	C₃×D₃	semidirect
12. C₇×C₃⋊(C₅⋊D₅)	C₃×C₇	semidirect
16. C₃×C₅²×C₇	C₂×C₃×C₇	semidirect
18. C₇×C₅²⋊C₃	D₃×C₇	semidirect
20. C₃⋊((C₅⋊D₅)⋊C₃)	C₇²	direct (N × H)
26. C₃×C₅²⋊C₃	C₂×C₇²	semidirect
28. C₅²×C₇	C₃×D₃×C₇	semidirect
29. C₃⋊(C₅⋊D₅)	C₃×C₇²	semidirect
33. C₃×C₇²	(C₅⋊D₅)⋊C₃	semidirect
37. C₃×C₅²	C₂×C₃×C₇²	semidirect
39. C₅²⋊C₃	D₃×C₇²	semidirect
43. C₇²	C₃⋊((C₅⋊D₅)⋊C₃)	direct (N × H)
48. C₅²	C₃×D₃×C₇²	semidirect
51. C₃×C₇	C₇×(C₅⋊D₅)⋊C₃	semidirect
57. C₇	C₇×C₃⋊((C₅⋊D₅)⋊C₃)	direct (N × H)
63. C₃	C₇²×(C₅⋊D₅)⋊C₃	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
C₇²×C₃⋊(C₅⋊D₅)	3
C₇²×(C₅⋊D₅)⋊C₃	3
C₇×C₃⋊((C₅⋊D₅)⋊C₃)	7
C₃×D₃×C₇²	25

Subgroups — C72×C3⋊((C5⋊D5)⋊C3) #36348007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×C₃⋊((C₅⋊D₅)⋊C₃) #36348007