Subgroups — C₂×C₇×#688 #11765187

Order 1120.

136 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	C₂×C₇×D₂₀	C₂	1	1	yes	maximal, characteristic
3	C₇×#688	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₇×D₅	—	2	1	no	—
6	C₅×C₇×Q₈	—	2	1	no	—
7	C₅×C₇×Q₈	C₂²	1	2	yes	complemented
8	C₂×C₄×C₅×C₇	—	2	1	no	—
9	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
10	C₇×D₂₀	—	2	1	no	—
11	C₇×D₂₀	C₂²	1	2	yes	—
12	C₇×#168	C₂²	1	2	yes	complemented
13	C₂×C₇×SD₁₆	—	5	1	no	maximal, Hall
14	C₂×#688	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₅×C₇	D₄	1	1	yes	characteristic
16	C₂×C₇×D₅	—	2	2	no	—
17	C₂×C₇×D₅	—	4	2	no	—
18	C₄×C₅×C₇	—	2	2	no	—
19	C₄×C₅×C₇	D₄	1	1	yes	characteristic
20	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
21	C₂×C₇×Q₈	D₅	1	1	yes	characteristic, complemented
22	C₂×C₇×D₄	—	5	1	no	—
23	C₇×SD₁₆	—	5	4	no	—
24	C₂×C₇×C₈	—	5	1	no	—
25	C₂×C₅×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
26	C₂×D₂₀	C₂×C₇	1	1	yes	characteristic
27	#688	C₂×C₇	1	4	yes	direct product factor
28	C₂×#168	C₂×C₇	1	1	yes	characteristic, complemented
29	C₂×C₅×C₇	SD₁₆	1	2	yes	complemented
30	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
31	C₇×D₅	—	4	2	no	—
32	C₂³×C₇	—	10	1	no	—
33	C₇×Q₈	—	2	1	no	—
34	C₇×Q₈	C₂×D₅	1	2	yes	complemented
35	C₇×D₄	—	5	2	no	—
36	C₇×D₄	—	10	1	no	—
37	C₂×C₄×C₇	—	2	1	no	—
38	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
39	C₇×C₈	—	5	2	no	—
40	C₂²×D₅	—	2	1	no	—
41	C₅×Q₈	—	2	1	no	—
42	C₅×Q₈	C₂²×C₇	1	2	yes	complemented
43	C₂×C₄×C₅	—	2	1	no	—
44	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
45	D₂₀	—	2	1	no	—
46	D₂₀	C₂²×C₇	1	2	yes	—
47	#168	C₂²×C₇	1	2	yes	complemented
48	C₅×C₇	C₂×SD₁₆	1	1	yes	characteristic, complemented, Hall
49	C₂×SD₁₆	—	5	1	no	Sylow
50	C₂²×C₇	—	10	2	no	—
51	C₂²×C₇	—	20	2	no	—
52	C₂²×C₇	C₂²⋊D₅	1	1	yes	characteristic
53	C₄×C₇	—	2	2	no	—
54	C₄×C₇	C₂²⋊D₅	1	1	yes	characteristic
55	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
56	C₂²×C₅	C₇×D₄	1	1	yes	characteristic
57	C₂×D₅	—	2	2	no	—
58	C₂×D₅	—	4	2	no	—
59	C₄×C₅	—	2	2	no	—
60	C₄×C₅	C₇×D₄	1	1	yes	characteristic
61	C₄×C₅	C₂³×C₇	1	1	yes	characteristic
62	C₂×Q₈	C₇×D₅	1	1	yes	characteristic, complemented
63	C₂×D₄	—	5	1	no	—
64	SD₁₆	—	5	4	no	—
65	C₂×C₈	—	5	1	no	—
66	C₂×C₇	—	20	2	no	—
67	C₂×C₇	#688	1	2	yes	direct product factor
68	C₂×C₇	C₂×C₂²⋊D₅	1	1	yes	characteristic
69	C₂×C₅	C₇×SD₁₆	1	2	yes	complemented
70	C₂×C₅	C₂×C₇×D₄	1	1	yes	characteristic
71	D₅	—	4	2	no	—
72	C₂³	—	10	1	no	—
73	Q₈	—	2	1	no	—
74	Q₈	C₂×C₇×D₅	1	2	yes	complemented
75	D₄	—	5	2	no	—
76	D₄	—	10	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
79	C₈	—	5	2	no	—
80	C₇	C₂×#688	1	1	yes	characteristic, direct product factor, Sylow
81	C₅	C₂×C₇×SD₁₆	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	10	2	no	—
83	C₂²	—	20	2	no	—
84	C₂²	C₇×C₂²⋊D₅	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	C₇×C₂²⋊D₅	1	1	yes	characteristic
87	C₄	C₂²×C₇×D₅	1	1	yes	characteristic
88	C₂	—	20	2	no	—
89	C₂	C₇×#688	1	2	yes	direct product factor
90	C₂	C₂×C₇×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂³×C₇
Frattini	C₄	C₂²×C₇×D₅
Fitting	C₂×C₅×C₇×Q₈	C₂
socle	C₂²×C₅×C₇	D₄
prime radical	C₂×C₇×D₂₀	C₂
hypercenter	C₂×C₇×Q₈	D₅
hypocenter	C₅	C₂×C₇×SD₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₇×Q₈	C₂	semidirect
3. C₇×#688	C₂	direct (N × H)
4. C₂×C₇×#168	C₂	semidirect
7. C₅×C₇×Q₈	C₂²	semidirect
12. C₇×#168	C₂²	semidirect
14. C₂×#688	C₇	direct (N × H)
21. C₂×C₇×Q₈	D₅	semidirect
25. C₂×C₅×Q₈	C₂×C₇	semidirect
27. #688	C₂×C₇	direct (N × H)
28. C₂×#168	C₂×C₇	semidirect
29. C₂×C₅×C₇	SD₁₆	semidirect
34. C₇×Q₈	C₂×D₅	semidirect
42. C₅×Q₈	C₂²×C₇	semidirect
47. #168	C₂²×C₇	semidirect
48. C₅×C₇	C₂×SD₁₆	semidirect
62. C₂×Q₈	C₇×D₅	semidirect
67. C₂×C₇	#688	direct (N × H)
69. C₂×C₅	C₇×SD₁₆	semidirect
74. Q₈	C₂×C₇×D₅	semidirect
80. C₇	C₂×#688	direct (N × H)
81. C₅	C₂×C₇×SD₁₆	semidirect
89. C₂	C₇×#688	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
C₂×C₇×D₂₀	2
C₇×#688	2
C₂×C₇×#168	2
C₂×C₇×SD₁₆	5
C₂×#688	7

Subgroups — C2×C7×#688 #11765187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#688 #11765187