Subgroups — #28413530

Order 14700.

416 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₃⋊(C₅⋊D₃₅)	C₂	1	2	yes	maximal, complemented
2	C₃×C₅²×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₇⋊D₇)⋊(C₅⋊D₅)	—	3	1	no	maximal, Hall
4	C₃×C₅²×C₇²	C₂²	1	1	yes	characteristic, complemented, Hall
5	#15668837	—	5	6	no	maximal
6	C₇×C₅⋊D₃₅	—	3	2	no	—
7	C₅²×C₇⋊D₇	D₃	1	1	yes	characteristic, complemented
8	D₇×C₃⋊(C₅⋊D₅)	—	7	2	no	maximal
9	C₇×D₁₀₅	—	5	12	no	—
10	C₃×C₅×C₇⋊D₇	D₅	1	6	yes	complemented
11	C₅²×C₇²	C₂×D₃	1	1	yes	characteristic, complemented, Hall
12	C₃⋊(C₅⋊D₃₅)	D₇	1	2	yes	complemented
13	C₇×C₃⋊(C₅⋊D₅)	—	7	2	no	—
14	C₃×C₅²×D₇	—	7	2	no	—
15	C₃×C₅²×D₇	—	14	3	no	—
16	D₃₅⋊D₇	—	15	6	no	—
17	C₃×C₅×C₇²	C₂×D₅	1	6	yes	complemented
18	D₇×C₅⋊D₅	—	21	2	no	—
19	D₂₁⋊D₇	—	25	1	no	Hall
20	C₃×C₅²×C₇	—	2	3	no	—
21	C₃×C₅²×C₇	C₂×D₇	1	2	yes	complemented
22	C₇×D₃₅	—	15	12	no	—
23	C₅×C₇⋊D₇	D₁₅	1	6	yes	complemented
24	D₇×D₁₅	—	35	12	no	—
25	C₅⋊D₃₅	—	3	2	no	—
26	C₇×C₅⋊D₅	—	21	2	no	—
27	C₅²×D₇	—	7	2	no	—
28	C₅²×D₇	—	14	3	no	—
29	C₂×C₃⋊(C₅⋊D₅)	—	49	1	no	Hall
30	C₇×D₂₁	—	25	2	no	—
31	C₃×C₇⋊D₇	C₅⋊D₅	1	1	yes	characteristic, complemented
32	C₅×C₇²	C₂×D₁₅	1	6	yes	complemented
33	D₁₀₅	—	5	12	no	—
34	C₇×D₁₅	—	35	12	no	—
35	C₃×C₅×D₇	—	7	12	no	—
36	C₃×C₅×D₇	—	14	18	no	—
37	D₇²	—	75	1	no	Hall
38	C₅²×C₇	—	2	3	no	—
39	C₅²×C₇	D₃×D₇	1	2	yes	complemented
40	C₂×C₃×C₅²	—	49	1	no	—
41	C₃⋊(C₅⋊D₅)	—	7	2	no	—
42	C₃×C₇²	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
43	D₅×D₇	—	105	12	no	—
44	C₃×C₅×C₇	—	2	18	no	—
45	C₃×C₅×C₇	D₅×D₇	1	12	yes	complemented
46	C₂×C₅⋊D₅	—	147	1	no	Hall
47	C₇⋊D₇	C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented
48	C₇×D₇	—	75	2	no	—
49	D₃×D₇	—	175	2	no	—
50	C₃×C₅²	D₇²	1	1	yes	characteristic, complemented, Hall
51	D₃₅	—	15	12	no	—
52	C₅×D₇	—	7	12	no	—
53	C₅×D₇	—	14	18	no	—
54	C₇×D₅	—	105	12	no	—
55	C₂×D₁₅	—	245	6	no	—
56	C₂×C₅²	—	49	1	no	—
57	C₅⋊D₅	—	21	2	no	—
58	C₇²	C₂×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented, Sylow
59	D₂₁	—	25	2	no	—
60	C₃×D₇	—	7	2	no	—
61	C₃×D₇	—	14	3	no	—
62	D₃×C₇	—	175	2	no	—
63	C₅×C₇	—	2	18	no	—
64	C₅×C₇	D₇×D₁₅	1	12	yes	complemented
65	C₂×C₃×C₅	—	49	6	no	—
66	D₁₅	—	35	12	no	—
67	C₂×D₇	—	525	2	no	—
68	C₅²	D₂₁⋊D₇	1	1	yes	characteristic, complemented, Sylow
69	C₃×C₇	—	2	3	no	—
70	C₃×C₇	D₇×C₅⋊D₅	1	2	yes	complemented
71	C₂×D₅	—	735	6	no	—
72	C₃×C₅	D₃₅⋊D₇	1	6	yes	complemented
73	C₂×C₇	—	525	2	no	—
74	D₇	—	7	2	no	—
75	D₇	—	14	3	no	—
76	D₇	—	75	2	no	—
77	C₂×D₃	—	1225	1	no	Hall
78	C₂×C₅	—	49	6	no	—
79	D₅	—	105	12	no	—
80	C₇	—	2	3	no	—
81	C₇	D₇×C₃⋊(C₅⋊D₅)	1	2	yes	complemented
82	C₂×C₃	—	49	1	no	—
83	D₃	—	175	2	no	—
84	C₅	#15668837	1	6	yes	complemented
85	C₂²	—	3675	1	no	Sylow
86	C₃	(C₇⋊D₇)⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented, Sylow
87	C₂	—	525	2	no	—
88	C₂	—	49	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²×C₇²	C₂²
LCS	G ▷ C₃×C₅²×C₇²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₇×C₃⋊(C₅⋊D₃₅)	C₂	semidirect
2. C₃×C₅²×C₇⋊D₇	C₂	semidirect
4. C₃×C₅²×C₇²	C₂²	semidirect
7. C₅²×C₇⋊D₇	D₃	semidirect
10. C₃×C₅×C₇⋊D₇	D₅	semidirect
11. C₅²×C₇²	C₂×D₃	semidirect
12. C₃⋊(C₅⋊D₃₅)	D₇	semidirect
17. C₃×C₅×C₇²	C₂×D₅	semidirect
21. C₃×C₅²×C₇	C₂×D₇	semidirect
23. C₅×C₇⋊D₇	D₁₅	semidirect
31. C₃×C₇⋊D₇	C₅⋊D₅	semidirect
32. C₅×C₇²	C₂×D₁₅	semidirect
39. C₅²×C₇	D₃×D₇	semidirect
42. C₃×C₇²	C₂×C₅⋊D₅	semidirect
45. C₃×C₅×C₇	D₅×D₇	semidirect
47. C₇⋊D₇	C₃⋊(C₅⋊D₅)	semidirect
50. C₃×C₅²	D₇²	semidirect
58. C₇²	C₂×C₃⋊(C₅⋊D₅)	semidirect
64. C₅×C₇	D₇×D₁₅	semidirect
68. C₅²	D₂₁⋊D₇	semidirect
70. C₃×C₇	D₇×C₅⋊D₅	semidirect
72. C₃×C₅	D₃₅⋊D₇	semidirect
81. C₇	D₇×C₃⋊(C₅⋊D₅)	semidirect
84. C₅	#15668837	semidirect
86. C₃	(C₇⋊D₇)⋊(C₅⋊D₅)	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₃⋊(C₅⋊D₃₅)	2
C₃×C₅²×C₇⋊D₇	2
(C₇⋊D₇)⋊(C₅⋊D₅)	3
#15668837	5
D₇×C₃⋊(C₅⋊D₅)	7