Subgroups — C₂×#10587991 #11765459

Order 1120.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#10587991	C₂	1	8	yes	maximal, direct product factor
3	C₂×D₁₄₀	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×D₃₅	C₂	1	3	yes	maximal, complemented
5	C₂²×D₃₅	C₂²	1	3	yes	—
6	C₅×C₇×Q₈	C₂²	1	4	yes	complemented
7	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
8	C₂×Dic₃₅	C₂²	1	1	yes	characteristic
9	D₁₄₀	C₂²	1	12	yes	complemented
10	C₄×D₃₅	C₂²	1	12	yes	complemented
11	C₂×#1161	—	5	1	no	maximal, Hall
12	C₂×#713	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₂×D₃₅	—	2	6	no	—
15	C₂×D₃₅	C₂³	1	6	yes	—
16	C₄×C₅×C₇	C₂³	1	6	yes	—
17	Dic₃₅	C₂³	1	2	yes	—
18	C₂×C₇×Q₈	D₅	1	1	yes	characteristic, complemented
19	#1161	—	5	8	no	—
20	C₂×D₂₈	—	5	3	no	—
21	C₂×C₄×D₇	—	5	3	no	—
22	C₂×C₅×Q₈	D₇	1	1	yes	characteristic, complemented
23	#713	—	7	8	no	—
24	C₂×D₂₀	—	7	3	no	—
25	C₂×C₄×D₅	—	7	3	no	—
26	C₂×C₅×C₇	#41	1	2	yes	complemented
27	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
28	D₃₅	—	2	6	no	—
29	C₂²×D₇	—	5	3	no	—
30	C₇×Q₈	C₂×D₅	1	4	yes	complemented
31	C₂×C₄×C₇	C₂×D₅	1	3	yes	—
32	C₂×Dic₇	—	5	1	no	—
33	D₂₈	—	5	12	no	—
34	C₄×D₇	—	5	12	no	—
35	C₂²×D₅	—	7	3	no	—
36	C₅×Q₈	C₂×D₇	1	4	yes	complemented
37	C₂×C₄×C₅	C₂×D₇	1	3	yes	—
38	C₂×Dic₅	—	7	1	no	—
39	D₂₀	—	7	12	no	—
40	C₄×D₅	—	7	12	no	—
41	C₅×C₇	C₂×#41	1	1	yes	characteristic, complemented, Hall
42	C₂×#41	—	35	1	no	Sylow
43	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
44	C₂×D₇	—	5	6	no	—
45	C₂×D₇	—	10	6	no	—
46	C₄×C₇	C₂²×D₅	1	6	yes	—
47	Dic₇	—	5	2	no	—
48	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
49	C₂×D₅	—	7	6	no	—
50	C₂×D₅	—	14	6	no	—
51	C₄×C₅	C₂²×D₇	1	6	yes	—
52	Dic₅	—	7	2	no	—
53	#41	—	35	8	no	—
54	C₂×Q₈	D₃₅	1	1	yes	characteristic, complemented
55	C₂×D₄	—	35	3	no	—
56	C₂²×C₄	—	35	3	no	—
57	C₂×C₇	#713	1	2	yes	complemented
58	C₂×C₇	C₂³×D₅	1	1	yes	characteristic
59	D₇	—	10	6	no	—
60	C₂×C₅	#1161	1	2	yes	complemented
61	C₂×C₅	C₂³×D₇	1	1	yes	characteristic
62	D₅	—	14	6	no	—
63	C₂³	—	35	3	no	—
64	Q₈	C₂×D₃₅	1	4	yes	complemented
65	D₄	—	35	12	no	—
66	C₂×C₄	—	35	1	no	—
67	C₂×C₄	—	35	12	no	—
68	C₂×C₄	C₂×D₃₅	1	3	yes	—
69	C₇	C₂×#713	1	1	yes	characteristic, complemented, Sylow
70	C₅	C₂×#1161	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	35	6	no	—
72	C₂²	—	70	6	no	—
73	C₂²	C₂²×D₃₅	1	1	yes	characteristic
74	C₄	—	35	2	no	—
75	C₄	C₂²×D₃₅	1	6	yes	—
76	C₂	—	70	6	no	—
77	C₂	#10587991	1	2	yes	direct product factor
78	C₂	C₂³×D₃₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₅
commutator, cosocle	C₂×C₅×C₇	C₂⁴
Frattini	C₂	C₂³×D₃₅
Fitting	C₂×C₅×C₇×Q₈	C₂
socle	C₂²×C₅×C₇	C₂³
hypercenter	C₂×Q₈	D₃₅
hypocenter	C₅×C₇	C₂×#41
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₇×Q₈	C₂	semidirect
2. #10587991	C₂	direct (N × H)
3. C₂×D₁₄₀	C₂	semidirect
4. C₂×C₄×D₃₅	C₂	semidirect
6. C₅×C₇×Q₈	C₂²	semidirect
9. D₁₄₀	C₂²	semidirect
10. C₄×D₃₅	C₂²	semidirect
18. C₂×C₇×Q₈	D₅	semidirect
22. C₂×C₅×Q₈	D₇	semidirect
26. C₂×C₅×C₇	#41	semidirect
30. C₇×Q₈	C₂×D₅	semidirect
36. C₅×Q₈	C₂×D₇	semidirect
41. C₅×C₇	C₂×#41	semidirect
54. C₂×Q₈	D₃₅	semidirect
57. C₂×C₇	#713	semidirect
60. C₂×C₅	#1161	semidirect
64. Q₈	C₂×D₃₅	semidirect
69. C₇	C₂×#713	semidirect
70. C₅	C₂×#1161	semidirect
77. C₂	#10587991	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₇×Q₈	2
#10587991	2
C₂×D₁₄₀	2
C₂×C₄×D₃₅	2
C₂×#1161	5
C₂×#713	7

Subgroups — C2×#10587991 #11765459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10587991 #11765459