Subgroups — C₅²×C₃⋊(C₇⋊#261) #41031963

Order 27300.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×C₇×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₅²×C₇⋊#261	—	3	1	no	maximal, Hall
3	C₃×C₅²×C₇×C₁₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₅×C₃⋊(C₇⋊#261)	C₅	1	6	yes	maximal, direct product factor
5	C₅²×C₇×D₁₃	D₃	1	1	yes	characteristic
6	C₅²×C₃⋊#261	—	7	1	no	maximal, Hall
7	C₃×C₅×C₇×D₁₃	C₂×C₅	1	6	yes	—
8	C₅²×C₇×C₁₃	Dic₃	1	1	yes	characteristic, complemented, Hall
9	C₅²×Dic₂₁	—	13	1	no	maximal, Hall
10	C₃×C₅²×D₁₃	D₇	1	1	yes	characteristic
11	C₅×C₇⋊#261	—	3	6	no	—
12	C₃×C₅×C₇×C₁₃	C₄×C₅	1	6	yes	complemented
13	C₅²×#261	—	21	1	no	Hall
14	C₃⋊(C₇⋊#261)	C₅²	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₃×C₅²×C₇	—	13	1	no	—
16	C₃×C₅²×C₁₃	Dic₇	1	1	yes	characteristic, complemented, Hall
17	C₅×C₇×D₁₃	C₅×D₃	1	6	yes	—
18	C₅×C₃⋊#261	—	7	6	no	—
19	C₅²×Dic₇	—	39	1	no	Hall
20	C₅²×D₁₃	D₂₁	1	1	yes	characteristic
21	C₃×C₇×D₁₃	C₂×C₅²	1	1	yes	characteristic
22	C₃×C₅²×C₇	#261	1	1	yes	characteristic, complemented, Hall
23	C₅×C₇×C₁₃	C₅×Dic₃	1	6	yes	complemented
24	C₅×Dic₂₁	—	13	6	no	—
25	C₃×C₅×D₁₃	C₅×D₇	1	6	yes	—
26	C₇⋊#261	—	3	1	no	Hall
27	C₂×C₅²×C₇	—	13	1	no	—
28	C₅²×C₁₃	Dic₂₁	1	1	yes	characteristic, complemented, Hall
29	C₅²×Dic₃	—	91	1	no	Hall
30	C₃×C₇×C₁₃	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
31	C₅×#261	—	21	6	no	—
32	C₂×C₃×C₅×C₇	—	13	6	no	—
33	C₃×C₅×C₁₃	C₅×Dic₇	1	6	yes	complemented
34	C₇×D₁₃	C₅²×D₃	1	1	yes	characteristic
35	C₅²×C₇	C₃⋊#261	1	1	yes	characteristic, complemented, Hall
36	C₃⋊#261	—	7	1	no	Hall
37	C₂×C₃×C₅²	—	13	1	no	—
38	C₅×Dic₇	—	39	6	no	—
39	C₅×D₁₃	C₅×D₂₁	1	6	yes	—
40	C₃×C₅×C₇	C₅×#261	1	6	yes	complemented
41	C₄×C₅²	—	273	1	no	Hall
42	C₇×C₁₃	C₅²×Dic₃	1	1	yes	characteristic, complemented, Hall
43	Dic₂₁	—	13	1	no	Hall
44	C₃×D₁₃	C₅²×D₇	1	1	yes	characteristic
45	C₃×C₅²	C₇⋊#261	1	1	yes	characteristic, complemented, Hall
46	C₂×C₅×C₇	—	13	6	no	—
47	C₅×C₁₃	C₅×Dic₂₁	1	6	yes	complemented
48	C₅×Dic₃	—	91	6	no	—
49	#261	—	21	1	no	Hall
50	C₂×C₅²	—	13	1	no	—
51	C₂×C₃×C₇	—	13	1	no	—
52	C₃×C₁₃	C₅²×Dic₇	1	1	yes	characteristic, complemented, Hall
53	C₅×C₇	C₅×C₃⋊#261	1	6	yes	complemented
54	C₂×C₃×C₅	—	13	6	no	—
55	Dic₇	—	39	1	no	Hall
56	D₁₃	C₅²×D₂₁	1	1	yes	characteristic
57	C₅²	C₃⋊(C₇⋊#261)	1	1	yes	characteristic, direct product factor, Sylow
58	C₃×C₇	C₅²×#261	1	1	yes	characteristic, complemented, Hall
59	C₄×C₅	—	273	6	no	—
60	C₃×C₅	C₅×C₇⋊#261	1	6	yes	complemented
61	C₂×C₇	—	13	1	no	—
62	C₁₃	C₅²×Dic₂₁	1	1	yes	characteristic, complemented, Sylow
63	Dic₃	—	91	1	no	Hall
64	C₂×C₅	—	13	6	no	—
65	C₇	C₅²×C₃⋊#261	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	13	1	no	—
67	C₅	C₅×C₃⋊(C₇⋊#261)	1	6	yes	direct product factor
68	C₄	—	273	1	no	Sylow
69	C₃	C₅²×C₇⋊#261	1	1	yes	characteristic, complemented, Sylow
70	C₂	—	13	1	no	—

	subgroup	G/H
center, hypercenter	C₅²	C₃⋊(C₇⋊#261)
commutator, hypocenter	C₃×C₇×C₁₃	C₄×C₅²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇×C₁₃	C₄
prime radical	C₃×C₅²×C₇×D₁₃	C₂
cosocle	C₃×C₇×D₁₃	C₂×C₅²
LCS	G ▷ C₃×C₇×C₁₃
UCS	C₁ ◁ C₅²

Complements

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

N	H ≅ G/N	type
3. C₃×C₅²×C₇×C₁₃	C₄	semidirect
4. C₅×C₃⋊(C₇⋊#261)	C₅	direct (N × H)
8. C₅²×C₇×C₁₃	Dic₃	semidirect
12. C₃×C₅×C₇×C₁₃	C₄×C₅	semidirect
14. C₃⋊(C₇⋊#261)	C₅²	direct (N × H)
16. C₃×C₅²×C₁₃	Dic₇	semidirect
22. C₃×C₅²×C₇	#261	semidirect
23. C₅×C₇×C₁₃	C₅×Dic₃	semidirect
28. C₅²×C₁₃	Dic₂₁	semidirect
30. C₃×C₇×C₁₃	C₄×C₅²	semidirect
33. C₃×C₅×C₁₃	C₅×Dic₇	semidirect
35. C₅²×C₇	C₃⋊#261	semidirect
40. C₃×C₅×C₇	C₅×#261	semidirect
42. C₇×C₁₃	C₅²×Dic₃	semidirect
45. C₃×C₅²	C₇⋊#261	semidirect
47. C₅×C₁₃	C₅×Dic₂₁	semidirect
52. C₃×C₁₃	C₅²×Dic₇	semidirect
53. C₅×C₇	C₅×C₃⋊#261	semidirect
57. C₅²	C₃⋊(C₇⋊#261)	direct (N × H)
58. C₃×C₇	C₅²×#261	semidirect
60. C₃×C₅	C₅×C₇⋊#261	semidirect
62. C₁₃	C₅²×Dic₂₁	semidirect
65. C₇	C₅²×C₃⋊#261	semidirect
67. C₅	C₅×C₃⋊(C₇⋊#261)	direct (N × H)
69. C₃	C₅²×C₇⋊#261	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅²×C₇×D₁₃	2
C₅²×C₇⋊#261	3
C₅×C₃⋊(C₇⋊#261)	5
C₅²×C₃⋊#261	7
C₅²×Dic₂₁	13

Subgroups — C52×C3⋊(C7⋊#261) #41031963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₃⋊(C₇⋊#261) #41031963