Subgroups — C₂×C₇×#676 #11765144

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#676	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₇×D₅	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₇×Dic₅	C₄	1	1	yes	characteristic
8	C₄×C₇×D₅	C₄	1	2	yes	—
9	C₄×C₇×D₅	C₂²	1	2	yes	—
10	C₅×C₇×C₈	C₂²	1	2	yes	complemented
11	C₇×#168	C₂²	1	2	yes	complemented
12	C₂×C₇×M₁₆	—	5	1	no	maximal, Hall
13	C₂×#676	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₇×D₅	—	2	2	no	—
16	C₂×C₇×D₅	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	C₇×Dic₅	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₇	—	5	1	no	—
21	C₇×M₁₆	—	5	4	no	—
22	C₂×C₇×C₈	—	5	1	no	—
23	C₂×C₇×C₈	D₅	1	1	yes	characteristic, complemented
24	C₂×C₄×D₅	C₂×C₇	1	1	yes	characteristic
25	C₂×C₅×C₈	C₂×C₇	1	1	yes	characteristic, complemented
26	C₂×#168	C₂×C₇	1	1	yes	characteristic, complemented
27	#676	C₂×C₇	1	4	yes	direct product factor
28	C₂×C₅×C₇	M₁₆	1	2	yes	complemented
29	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
30	C₇×D₅	—	2	2	no	—
31	C₂³×C₇	—	5	1	no	—
32	C₂×C₄×C₇	—	5	1	no	—
33	C₂×C₄×C₇	—	5	2	no	—
34	C₂×C₄×C₇	—	5	2	no	—
35	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
36	C₇×C₈	—	5	2	no	—
37	C₇×C₈	C₂×D₅	1	2	yes	complemented
38	C₂²×D₅	C₄×C₇	1	1	yes	characteristic
39	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
40	C₂×Dic₅	C₄×C₇	1	1	yes	characteristic
41	C₄×D₅	C₄×C₇	1	2	yes	—
42	C₄×D₅	C₂²×C₇	1	2	yes	—
43	C₅×C₈	C₂²×C₇	1	2	yes	complemented
44	#168	C₂²×C₇	1	2	yes	complemented
45	C₅×C₇	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
46	C₂×M₁₆	—	5	1	no	Sylow
47	C₂²×C₇	—	5	2	no	—
48	C₂²×C₇	—	10	2	no	—
49	C₂²×C₇	C₄×D₅	1	1	yes	characteristic
50	C₄×C₇	—	5	2	no	—
51	C₄×C₇	C₄×D₅	1	1	yes	characteristic
52	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
53	C₂²×C₅	C₂×C₄×C₇	1	1	yes	characteristic
54	C₂×D₅	—	2	2	no	—
55	C₂×D₅	C₂×C₄×C₇	1	2	yes	—
56	C₄×C₅	C₂×C₄×C₇	1	1	yes	characteristic
57	C₄×C₅	C₂³×C₇	1	1	yes	characteristic
58	Dic₅	C₂×C₄×C₇	1	2	yes	—
59	C₂²×C₄	—	5	1	no	—
60	M₁₆	—	5	4	no	—
61	C₂×C₈	—	5	1	no	—
62	C₂×C₈	C₇×D₅	1	1	yes	characteristic, complemented
63	C₂×C₇	—	10	2	no	—
64	C₂×C₇	#676	1	2	yes	direct product factor
65	C₂×C₇	C₂×C₄×D₅	1	1	yes	characteristic
66	C₂×C₅	C₇×M₁₆	1	2	yes	complemented
67	C₂×C₅	C₂²×C₄×C₇	1	1	yes	characteristic
68	D₅	—	2	2	no	—
69	C₂³	—	5	1	no	—
70	C₂×C₄	—	5	1	no	—
71	C₂×C₄	—	5	2	no	—
72	C₂×C₄	—	5	2	no	—
73	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
74	C₈	—	5	2	no	—
75	C₈	C₂×C₇×D₅	1	2	yes	complemented
76	C₇	C₂×#676	1	1	yes	characteristic, direct product factor, Sylow
77	C₅	C₂×C₇×M₁₆	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	5	2	no	—
79	C₂²	—	10	2	no	—
80	C₂²	C₄×C₇×D₅	1	1	yes	characteristic
81	C₄	—	5	2	no	—
82	C₄	C₄×C₇×D₅	1	1	yes	characteristic
83	C₄	C₂²×C₇×D₅	1	1	yes	characteristic
84	C₂	—	10	2	no	—
85	C₂	C₇×#676	1	2	yes	direct product factor
86	C₂	C₂×C₄×C₇×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₅
commutator	C₂×C₅	C₂²×C₄×C₇
Frattini	C₄	C₂²×C₇×D₅
Fitting	C₂×C₅×C₇×C₈	C₂
socle	C₂²×C₅×C₇	C₂×C₄
prime radical	C₂²×C₇×D₅	C₄
cosocle	C₄×C₅	C₂³×C₇
hypercenter	C₂×C₇×C₈	D₅
hypocenter	C₅	C₂×C₇×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×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×C₇×#168	C₂	semidirect
4. C₇×#676	C₂	direct (N × H)
10. C₅×C₇×C₈	C₂²	semidirect
11. C₇×#168	C₂²	semidirect
13. C₂×#676	C₇	direct (N × H)
23. C₂×C₇×C₈	D₅	semidirect
25. C₂×C₅×C₈	C₂×C₇	semidirect
26. C₂×#168	C₂×C₇	semidirect
27. #676	C₂×C₇	direct (N × H)
28. C₂×C₅×C₇	M₁₆	semidirect
37. C₇×C₈	C₂×D₅	semidirect
43. C₅×C₈	C₂²×C₇	semidirect
44. #168	C₂²×C₇	semidirect
45. C₅×C₇	C₂×M₁₆	semidirect
62. C₂×C₈	C₇×D₅	semidirect
64. C₂×C₇	#676	direct (N × H)
66. C₂×C₅	C₇×M₁₆	semidirect
75. C₈	C₂×C₇×D₅	semidirect
76. C₇	C₂×#676	direct (N × H)
77. C₅	C₂×C₇×M₁₆	semidirect
85. C₂	C₇×#676	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×C₇×D₅	2
C₂×C₅×C₇×C₈	2
C₂×C₇×#168	2
C₇×#676	2
C₂×C₇×M₁₆	5
C₂×#676	7

Subgroups — C2×C7×#676 #11765144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#676 #11765144