Subgroups — C₂×C₅×#1136 #11765079

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₅×#1136	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₅×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂³×C₅×C₇	C₄	1	1	yes	characteristic
5	C₂×C₄×C₅×C₇	C₄	1	2	yes	—
6	C₂×C₄×C₅×C₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₇	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
9	C₅×C₇⋊C₈	C₂²	1	4	yes	complemented
10	C₂×#1136	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₅×M₁₆	—	7	1	no	maximal, Hall
12	C₂²×C₅×C₇	—	2	2	no	—
13	C₂²×C₅×C₇	C₂×C₄	1	2	yes	—
14	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
15	C₄×C₅×C₇	C₂×C₄	1	2	yes	—
16	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
17	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₇	C₂×C₅	1	1	yes	characteristic
19	#1136	C₂×C₅	1	4	yes	direct product factor
20	C₂×C₇⋊C₈	C₂×C₅	1	2	yes	complemented
21	C₂²×C₄×C₅	D₇	1	1	yes	characteristic
22	C₅×M₁₆	—	7	4	no	—
23	C₂×C₅×C₈	—	7	2	no	—
24	C₂×C₅×C₇	—	2	2	no	—
25	C₂×C₅×C₇	M₁₆	1	2	yes	complemented
26	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
27	C₂³×C₇	C₄×C₅	1	1	yes	characteristic
28	C₂×C₄×C₇	C₄×C₅	1	2	yes	—
29	C₂×C₄×C₇	C₄×C₅	1	1	yes	characteristic
30	C₂×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
31	C₂×C₄×C₇	C₂²×C₅	1	2	yes	—
32	C₇⋊C₈	C₂²×C₅	1	4	yes	complemented
33	C₂³×C₅	Dic₇	1	1	yes	characteristic
34	C₂×C₄×C₅	Dic₇	1	1	yes	characteristic
35	C₂×C₄×C₅	Dic₇	1	2	yes	—
36	C₂×C₄×C₅	C₂×D₇	1	2	yes	—
37	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
38	C₅×C₈	—	7	4	no	—
39	C₅×C₇	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
40	C₂×M₁₆	—	7	1	no	Sylow
41	C₂²×C₇	—	2	2	no	—
42	C₂²×C₇	C₂×C₄×C₅	1	2	yes	—
43	C₂²×C₇	C₂×C₄×C₅	1	1	yes	characteristic
44	C₄×C₇	C₂×C₄×C₅	1	1	yes	characteristic
45	C₄×C₇	C₂×C₄×C₅	1	2	yes	—
46	C₄×C₇	C₂³×C₅	1	1	yes	characteristic
47	C₂²×C₅	—	2	2	no	—
48	C₂²×C₅	C₂×Dic₇	1	2	yes	—
49	C₂²×C₅	C₂×Dic₇	1	1	yes	characteristic
50	C₄×C₅	C₂×Dic₇	1	2	yes	—
51	C₄×C₅	C₂×Dic₇	1	1	yes	characteristic
52	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
53	C₂²×C₄	C₅×D₇	1	1	yes	characteristic
54	M₁₆	—	7	4	no	—
55	C₂×C₈	—	7	2	no	—
56	C₂×C₇	—	2	2	no	—
57	C₂×C₇	C₅×M₁₆	1	2	yes	complemented
58	C₂×C₇	C₂²×C₄×C₅	1	1	yes	characteristic
59	C₂×C₅	—	2	2	no	—
60	C₂×C₅	#1136	1	2	yes	direct product factor
61	C₂×C₅	C₂²×Dic₇	1	1	yes	characteristic
62	C₂³	C₅×Dic₇	1	1	yes	characteristic
63	C₂×C₄	C₅×Dic₇	1	2	yes	—
64	C₂×C₄	C₅×Dic₇	1	1	yes	characteristic
65	C₂×C₄	C₂×C₅×D₇	1	1	yes	characteristic
66	C₂×C₄	C₂×C₅×D₇	1	2	yes	—
67	C₈	—	7	4	no	—
68	C₇	C₂×C₅×M₁₆	1	1	yes	characteristic, complemented, Sylow
69	C₅	C₂×#1136	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²	—	2	2	no	—
71	C₂²	C₂×C₅×Dic₇	1	2	yes	—
72	C₂²	C₂×C₅×Dic₇	1	1	yes	characteristic
73	C₄	C₂×C₅×Dic₇	1	1	yes	characteristic
74	C₄	C₂×C₅×Dic₇	1	2	yes	—
75	C₄	C₂²×C₅×D₇	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₅×#1136	1	2	yes	direct product factor
78	C₂	C₂²×C₅×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₇
commutator	C₂×C₇	C₂²×C₄×C₅
Frattini	C₄	C₂²×C₅×D₇
Fitting	C₂²×C₄×C₅×C₇	C₂
socle	C₂²×C₅×C₇	C₂×C₄
prime radical	C₂³×C₅×C₇	C₄
cosocle	C₄×C₇	C₂³×C₅
hypercenter	C₂²×C₄×C₅	D₇
hypocenter	C₇	C₂×C₅×M₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×#1136	C₂	direct (N × H)
3. C₂×C₅×C₇⋊C₈	C₂	semidirect
9. C₅×C₇⋊C₈	C₂²	semidirect
10. C₂×#1136	C₅	direct (N × H)
19. #1136	C₂×C₅	direct (N × H)
20. C₂×C₇⋊C₈	C₂×C₅	semidirect
25. C₂×C₅×C₇	M₁₆	semidirect
32. C₇⋊C₈	C₂²×C₅	semidirect
39. C₅×C₇	C₂×M₁₆	semidirect
57. C₂×C₇	C₅×M₁₆	semidirect
60. C₂×C₅	#1136	direct (N × H)
68. C₇	C₂×C₅×M₁₆	semidirect
69. C₅	C₂×#1136	direct (N × H)
77. C₂	C₅×#1136	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×C₄×C₅×C₇	2
C₅×#1136	2
C₂×C₅×C₇⋊C₈	2
C₂×#1136	5
C₂×C₅×M₁₆	7

Subgroups — C2×C5×#1136 #11765079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1136 #11765079