Subgroups — C₂²×#63308 #11765416

Order 1120.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#63308	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂×Dic₃₅	C₂²	1	6	yes	—
7	C₂×C₇×Dic₅	C₂²	1	6	yes	—
8	C₂×C₅×Dic₇	C₂²	1	6	yes	—
9	#63308	C₂²	1	16	yes	direct product factor
10	C₂²×Dic₁₄	—	5	1	no	maximal, Hall
11	C₂²×Dic₁₀	—	7	1	no	maximal, Hall
12	C₂²×C₅×C₇	Q₈	1	4	yes	complemented
13	C₂²×C₅×C₇	C₂³	1	3	yes	—
14	Dic₃₅	C₂³	1	4	yes	—
15	C₅×Dic₇	C₂³	1	4	yes	—
16	C₇×Dic₅	C₂³	1	4	yes	—
17	C₂²×C₄×C₇	—	5	1	no	—
18	C₂²×Dic₇	—	5	1	no	—
19	C₂²×Dic₇	D₅	1	1	yes	characteristic
20	C₂×Dic₁₄	—	5	12	no	—
21	C₂²×C₄×C₅	—	7	1	no	—
22	C₂²×Dic₅	—	7	1	no	—
23	C₂²×Dic₅	D₇	1	1	yes	characteristic
24	C₂×Dic₁₀	—	7	12	no	—
25	C₂×C₅×C₇	C₂×Q₈	1	6	yes	complemented
26	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
27	C₂³×C₇	C₂×D₅	1	1	yes	characteristic
28	C₂×C₄×C₇	—	5	6	no	—
29	C₂×Dic₇	—	5	6	no	—
30	C₂×Dic₇	C₂×D₅	1	6	yes	—
31	Dic₁₄	—	5	16	no	—
32	C₂³×C₅	C₂×D₇	1	1	yes	characteristic
33	C₂×C₄×C₅	—	7	6	no	—
34	C₂×Dic₅	—	7	6	no	—
35	C₂×Dic₅	C₂×D₇	1	6	yes	—
36	Dic₁₀	—	7	16	no	—
37	C₅×C₇	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
38	C₂²×Q₈	—	35	1	no	Sylow
39	C₂²×C₇	Dic₁₀	1	4	yes	complemented
40	C₂²×C₇	C₂²×D₅	1	3	yes	—
41	C₄×C₇	—	5	4	no	—
42	Dic₇	—	5	4	no	—
43	Dic₇	C₂²×D₅	1	4	yes	—
44	C₂²×C₅	Dic₁₄	1	4	yes	complemented
45	C₂²×C₅	C₂²×D₇	1	3	yes	—
46	C₄×C₅	—	7	4	no	—
47	Dic₅	—	7	4	no	—
48	Dic₅	C₂²×D₇	1	4	yes	—
49	C₂×Q₈	—	35	12	no	—
50	C₂²×C₄	—	5	1	no	—
51	C₂²×C₄	—	7	1	no	—
52	C₂²×C₄	—	35	1	no	—
53	C₂×C₇	C₂×Dic₁₀	1	6	yes	complemented
54	C₂×C₇	C₂³×D₅	1	1	yes	characteristic
55	C₂×C₅	C₂×Dic₁₄	1	6	yes	complemented
56	C₂×C₅	C₂³×D₇	1	1	yes	characteristic
57	C₂³	D₅×D₇	1	1	yes	characteristic
58	Q₈	—	35	16	no	—
59	C₂×C₄	—	5	6	no	—
60	C₂×C₄	—	7	6	no	—
61	C₂×C₄	—	35	6	no	—
62	C₇	C₂²×Dic₁₀	1	1	yes	characteristic, complemented, Sylow
63	C₅	C₂²×Dic₁₄	1	1	yes	characteristic, complemented, Sylow
64	C₂²	#63308	1	4	yes	direct product factor
65	C₂²	C₂×D₅×D₇	1	3	yes	—
66	C₄	—	5	4	no	—
67	C₄	—	7	4	no	—
68	C₄	—	35	4	no	—
69	C₂	C₂×#63308	1	6	yes	direct product factor
70	C₂	C₂²×D₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₅×D₇
commutator, cosocle	C₂×C₅×C₇	C₂⁴
Frattini	C₂	C₂²×D₅×D₇
Fitting, socle, prime radical	C₂³×C₅×C₇	C₂²
hypocenter	C₅×C₇	C₂²×Q₈
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
4. C₂×#63308	C₂	direct (N × H)
9. #63308	C₂²	direct (N × H)
12. C₂²×C₅×C₇	Q₈	semidirect
25. C₂×C₅×C₇	C₂×Q₈	semidirect
37. C₅×C₇	C₂²×Q₈	semidirect
39. C₂²×C₇	Dic₁₀	semidirect
44. C₂²×C₅	Dic₁₄	semidirect
53. C₂×C₇	C₂×Dic₁₀	semidirect
55. C₂×C₅	C₂×Dic₁₄	semidirect
62. C₇	C₂²×Dic₁₀	semidirect
63. C₅	C₂²×Dic₁₄	semidirect
64. C₂²	#63308	direct (N × H)
69. C₂	C₂×#63308	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₅	2
C₂²×C₇×Dic₅	2
C₂²×C₅×Dic₇	2
C₂×#63308	2
C₂²×Dic₁₄	5
C₂²×Dic₁₀	7

Subgroups — C22×#63308 #11765416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63308 #11765416