Subgroups — #13132897

Order 1680.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄₂₀	C₂	1	3	yes	maximal, complemented
3	C₄×D₁₀₅	C₂	1	3	yes	maximal, complemented
4	#10587991	—	3	1	no	maximal, Hall
5	C₂×D₁₀₅	C₂²	1	3	yes	—
6	C₃×C₄×C₅×C₇	C₂²	1	3	yes	—
7	Dic₁₀₅	C₂²	1	1	yes	characteristic
8	#66732	—	5	1	no	maximal, Hall
9	C₅×C₇×Q₈	D₃	1	1	yes	characteristic, complemented
10	D₁₄₀	—	3	3	no	—
11	C₄×D₃₅	—	3	3	no	—
12	#6821	—	7	1	no	maximal, Hall
13	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
14	D₁₀₅	—	2	3	no	—
15	C₃×C₇×Q₈	D₅	1	1	yes	characteristic, complemented
16	D₈₄	—	5	3	no	—
17	C₄×D₂₁	—	5	3	no	—
18	C₂×D₃₅	—	3	3	no	—
19	C₄×C₅×C₇	C₂×D₃	1	3	yes	—
20	Dic₃₅	—	3	1	no	—
21	C₃×C₅×Q₈	D₇	1	1	yes	characteristic, complemented
22	D₆₀	—	7	3	no	—
23	C₄×D₁₅	—	7	3	no	—
24	#1161	—	15	1	no	Hall
25	C₃×C₅×C₇	#41	1	1	yes	characteristic, complemented, Hall
26	C₂×D₂₁	—	5	3	no	—
27	C₃×C₄×C₇	C₂×D₅	1	3	yes	—
28	Dic₂₁	—	5	1	no	—
29	#713	—	21	1	no	Hall
30	C₂×C₅×C₇	C₂²×D₃	1	1	yes	characteristic
31	D₃₅	—	6	3	no	—
32	C₂×D₁₅	—	7	3	no	—
33	C₃×C₄×C₅	C₂×D₇	1	3	yes	—
34	Dic₁₅	—	7	1	no	—
35	C₇×Q₈	D₁₅	1	1	yes	characteristic, complemented
36	D₂₈	—	15	3	no	—
37	C₄×D₇	—	15	3	no	—
38	#239	—	35	1	no	Hall
39	C₂×C₃×C₇	C₂²×D₅	1	1	yes	characteristic
40	D₂₁	—	10	3	no	—
41	C₅×Q₈	D₂₁	1	1	yes	characteristic, complemented
42	D₂₀	—	21	3	no	—
43	C₄×D₅	—	21	3	no	—
44	C₅×C₇	#239	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₅	C₂²×D₇	1	1	yes	characteristic
46	D₁₅	—	14	3	no	—
47	C₂×D₇	—	15	3	no	—
48	C₄×C₇	C₂×D₁₅	1	3	yes	—
49	Dic₇	—	15	1	no	—
50	C₃×Q₈	D₃₅	1	1	yes	characteristic, complemented
51	D₁₂	—	35	3	no	—
52	C₄×D₃	—	35	3	no	—
53	C₃×C₇	#713	1	1	yes	characteristic, complemented, Hall
54	C₂×D₅	—	21	3	no	—
55	C₄×C₅	C₂×D₂₁	1	3	yes	—
56	Dic₅	—	21	1	no	—
57	#41	—	105	1	no	Sylow
58	C₃×C₅	#1161	1	1	yes	characteristic, complemented, Hall
59	C₂×C₇	C₂²×D₁₅	1	1	yes	characteristic
60	D₇	—	30	3	no	—
61	C₂×D₃	—	35	3	no	—
62	C₃×C₄	C₂×D₃₅	1	3	yes	—
63	Dic₃	—	35	1	no	—
64	C₂×C₅	C₂²×D₂₁	1	1	yes	characteristic
65	D₅	—	42	3	no	—
66	Q₈	D₁₀₅	1	1	yes	characteristic, complemented
67	D₄	—	105	3	no	—
68	C₂×C₄	—	105	3	no	—
69	C₇	#6821	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	C₂²×D₃₅	1	1	yes	characteristic
71	D₃	—	70	3	no	—
72	C₅	#66732	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	105	3	no	—
74	C₄	—	105	1	no	—
75	C₄	C₂×D₁₀₅	1	3	yes	—
76	C₃	#10587991	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	210	3	no	—
78	C₂	C₂²×D₁₀₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₁₀₅
commutator, socle, cosocle	C₂×C₃×C₅×C₇	C₂³
Fitting	C₃×C₅×C₇×Q₈	C₂
hypercenter	Q₈	D₁₀₅
hypocenter	C₃×C₅×C₇	#41
LCS	G ▷ C₂×C₃×C₅×C₇ ▷ C₃×C₅×C₇
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₇×Q₈	C₂	semidirect
2. D₄₂₀	C₂	semidirect
3. C₄×D₁₀₅	C₂	semidirect
9. C₅×C₇×Q₈	D₃	semidirect
15. C₃×C₇×Q₈	D₅	semidirect
21. C₃×C₅×Q₈	D₇	semidirect
25. C₃×C₅×C₇	#41	semidirect
35. C₇×Q₈	D₁₅	semidirect
41. C₅×Q₈	D₂₁	semidirect
44. C₅×C₇	#239	semidirect
50. C₃×Q₈	D₃₅	semidirect
53. C₃×C₇	#713	semidirect
58. C₃×C₅	#1161	semidirect
66. Q₈	D₁₀₅	semidirect
69. C₇	#6821	semidirect
72. C₅	#66732	semidirect
76. C₃	#10587991	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₇×Q₈	2
D₄₂₀	2
C₄×D₁₀₅	2
#10587991	3
#66732	5
#6821	7