Subgroups — C₂×C₅×#1131 #11765058

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1131	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅×D₇	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₅×Dic₇	C₄	1	1	yes	characteristic
8	C₄×C₅×D₇	C₄	1	2	yes	—
9	C₄×C₅×D₇	C₂²	1	2	yes	—
10	C₅×C₇×C₈	C₂²	1	2	yes	complemented
11	C₅×C₇⋊C₈	C₂²	1	2	yes	complemented
12	C₂×#1131	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₅×M₁₆	—	7	1	no	maximal, Hall
14	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₅×D₇	—	2	2	no	—
16	C₂×C₅×D₇	C₂×C₄	1	2	yes	—
17	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
18	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
19	C₅×Dic₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₇	C₂×C₅	1	1	yes	characteristic
21	C₂×C₇×C₈	C₂×C₅	1	1	yes	characteristic, complemented
22	C₂×C₇⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
23	#1131	C₂×C₅	1	4	yes	direct product factor
24	C₂²×C₄×C₅	—	7	1	no	—
25	C₅×M₁₆	—	7	4	no	—
26	C₂×C₅×C₈	—	7	1	no	—
27	C₂×C₅×C₈	D₇	1	1	yes	characteristic, complemented
28	C₂×C₅×C₇	M₁₆	1	2	yes	complemented
29	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
30	C₅×D₇	—	2	2	no	—
31	C₂²×D₇	C₄×C₅	1	1	yes	characteristic
32	C₂×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
33	C₂×Dic₇	C₄×C₅	1	1	yes	characteristic
34	C₄×D₇	C₄×C₅	1	2	yes	—
35	C₄×D₇	C₂²×C₅	1	2	yes	—
36	C₇×C₈	C₂²×C₅	1	2	yes	complemented
37	C₇⋊C₈	C₂²×C₅	1	2	yes	complemented
38	C₂³×C₅	—	7	1	no	—
39	C₂×C₄×C₅	—	7	1	no	—
40	C₂×C₄×C₅	—	7	2	no	—
41	C₂×C₄×C₅	—	7	2	no	—
42	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
43	C₅×C₈	—	7	2	no	—
44	C₅×C₈	C₂×D₇	1	2	yes	complemented
45	C₅×C₇	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
46	C₂×M₁₆	—	7	1	no	Sylow
47	C₂²×C₇	C₂×C₄×C₅	1	1	yes	characteristic
48	C₂×D₇	—	2	2	no	—
49	C₂×D₇	C₂×C₄×C₅	1	2	yes	—
50	C₄×C₇	C₂×C₄×C₅	1	1	yes	characteristic
51	C₄×C₇	C₂³×C₅	1	1	yes	characteristic
52	Dic₇	C₂×C₄×C₅	1	2	yes	—
53	C₂²×C₅	—	7	2	no	—
54	C₂²×C₅	—	14	2	no	—
55	C₂²×C₅	C₄×D₇	1	1	yes	characteristic
56	C₄×C₅	—	7	2	no	—
57	C₄×C₅	C₄×D₇	1	1	yes	characteristic
58	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
59	C₂²×C₄	—	7	1	no	—
60	M₁₆	—	7	4	no	—
61	C₂×C₈	—	7	1	no	—
62	C₂×C₈	C₅×D₇	1	1	yes	characteristic, complemented
63	C₂×C₇	C₅×M₁₆	1	2	yes	complemented
64	C₂×C₇	C₂²×C₄×C₅	1	1	yes	characteristic
65	D₇	—	2	2	no	—
66	C₂×C₅	—	14	2	no	—
67	C₂×C₅	#1131	1	2	yes	direct product factor
68	C₂×C₅	C₂×C₄×D₇	1	1	yes	characteristic
69	C₂³	—	7	1	no	—
70	C₂×C₄	—	7	1	no	—
71	C₂×C₄	—	7	2	no	—
72	C₂×C₄	—	7	2	no	—
73	C₂×C₄	C₂×C₅×D₇	1	1	yes	characteristic
74	C₈	—	7	2	no	—
75	C₈	C₂×C₅×D₇	1	2	yes	complemented
76	C₇	C₂×C₅×M₁₆	1	1	yes	characteristic, complemented, Sylow
77	C₅	C₂×#1131	1	1	yes	characteristic, direct product factor, Sylow
78	C₂²	—	7	2	no	—
79	C₂²	—	14	2	no	—
80	C₂²	C₄×C₅×D₇	1	1	yes	characteristic
81	C₄	—	7	2	no	—
82	C₄	C₄×C₅×D₇	1	1	yes	characteristic
83	C₄	C₂²×C₅×D₇	1	1	yes	characteristic
84	C₂	—	14	2	no	—
85	C₂	C₅×#1131	1	2	yes	direct product factor
86	C₂	C₂×C₄×C₅×D₇	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	C₂×C₅×C₇×C₈	C₂
socle	C₂²×C₅×C₇	C₂×C₄
prime radical	C₂²×C₅×D₇	C₄
cosocle	C₄×C₇	C₂³×C₅
hypercenter	C₂×C₅×C₈	D₇
hypocenter	C₇	C₂×C₅×M₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₇×C₈	C₂	semidirect
3. C₂×C₅×C₇⋊C₈	C₂	semidirect
4. C₅×#1131	C₂	direct (N × H)
10. C₅×C₇×C₈	C₂²	semidirect
11. C₅×C₇⋊C₈	C₂²	semidirect
12. C₂×#1131	C₅	direct (N × H)
21. C₂×C₇×C₈	C₂×C₅	semidirect
22. C₂×C₇⋊C₈	C₂×C₅	semidirect
23. #1131	C₂×C₅	direct (N × H)
27. C₂×C₅×C₈	D₇	semidirect
28. C₂×C₅×C₇	M₁₆	semidirect
36. C₇×C₈	C₂²×C₅	semidirect
37. C₇⋊C₈	C₂²×C₅	semidirect
44. C₅×C₈	C₂×D₇	semidirect
45. C₅×C₇	C₂×M₁₆	semidirect
62. C₂×C₈	C₅×D₇	semidirect
63. C₂×C₇	C₅×M₁₆	semidirect
67. C₂×C₅	#1131	direct (N × H)
75. C₈	C₂×C₅×D₇	semidirect
76. C₇	C₂×C₅×M₁₆	semidirect
77. C₅	C₂×#1131	direct (N × H)
85. C₂	C₅×#1131	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×C₄×C₅×D₇	2
C₂×C₅×C₇×C₈	2
C₂×C₅×C₇⋊C₈	2
C₅×#1131	2
C₂×#1131	5
C₂×C₅×M₁₆	7

Subgroups — C2×C5×#1131 #11765058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1131 #11765058