Subgroups — C₂×D₇×#168 #11764780

Order 1120.

152 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×C₇⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×#168	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	C₇⋊#168	C₂²	1	2	yes	complemented
11	C₇×#168	C₂²	1	2	yes	complemented
12	C₂×C₈×D₇	—	5	1	no	maximal, Hall
13	C₂²×#168	—	7	1	no	maximal, Hall
14	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₅×D₇	C₈	1	4	yes	complemented
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	D₅	1	1	yes	characteristic
21	C₂×C₇×C₈	—	5	1	no	—
22	C₂×C₇⋊C₈	—	5	1	no	—
23	C₈×D₇	—	5	4	no	—
24	C₂²×C₄×C₅	—	7	1	no	—
25	C₂×#168	—	7	1	no	—
26	C₂×#168	—	7	4	no	—
27	C₂×#168	D₇	1	1	yes	characteristic, direct product factor
28	C₂×C₅×C₇	C₂×C₈	1	2	yes	complemented
29	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
30	C₅×D₇	C₂×C₈	1	4	yes	complemented
31	C₂²×D₇	Dic₅	1	1	yes	characteristic
32	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
33	C₂×Dic₇	Dic₅	1	1	yes	characteristic
34	C₄×D₇	Dic₅	1	2	yes	—
35	C₄×D₇	C₂×D₅	1	2	yes	—
36	C₇×C₈	—	5	2	no	—
37	C₇⋊C₈	—	5	2	no	—
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	#168	—	7	2	no	—
44	#168	C₂×D₇	1	2	yes	direct product factor
45	C₅×C₇	C₂²×C₈	1	1	yes	characteristic, complemented, Hall
46	C₂²×C₈	—	35	1	no	Sylow
47	C₂²×C₇	C₂×Dic₅	1	1	yes	characteristic
48	C₂×D₇	#168	1	4	yes	direct product factor
49	C₂×D₇	C₂×Dic₅	1	2	yes	—
50	C₄×C₇	C₂×Dic₅	1	1	yes	characteristic
51	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
52	Dic₇	C₂×Dic₅	1	2	yes	—
53	C₂²×C₅	—	7	4	no	—
54	C₂²×C₅	—	7	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	C₂×C₈	—	5	1	no	—
61	C₂×C₈	—	35	4	no	—
62	C₂×C₈	—	35	1	no	—
63	C₂×C₇	C₂×#168	1	2	yes	complemented
64	C₂×C₇	C₂²×Dic₅	1	1	yes	characteristic
65	D₇	C₂×#168	1	4	yes	direct product factor
66	C₂×C₅	—	7	4	no	—
67	C₂×C₅	C₈×D₇	1	2	yes	complemented
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	D₅×D₇	1	1	yes	characteristic
74	C₈	—	5	2	no	—
75	C₈	—	35	2	no	—
76	C₇	C₂²×#168	1	1	yes	characteristic, complemented, Sylow
77	C₅	C₂×C₈×D₇	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	7	2	no	—
79	C₂²	—	7	4	no	—
80	C₂²	D₇×Dic₅	1	1	yes	characteristic
81	C₄	—	7	2	no	—
82	C₄	D₇×Dic₅	1	1	yes	characteristic
83	C₄	C₂×D₅×D₇	1	1	yes	characteristic
84	C₂	—	7	4	no	—
85	C₂	D₇×#168	1	2	yes	direct product factor
86	C₂	C₂×D₇×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	D₅×D₇
commutator, hypocenter	C₅×C₇	C₂²×C₈
Frattini	C₄	C₂×D₅×D₇
Fitting	C₂×C₄×C₅×C₇	C₂²
socle	C₂²×C₅×C₇	C₂×C₄
prime radical	C₂²×C₅×D₇	C₄
cosocle	C₄×C₅×C₇	C₂³
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₇⋊#168	C₂	semidirect
3. C₂×C₇×#168	C₂	semidirect
4. D₇×#168	C₂	direct (N × H)
10. C₇⋊#168	C₂²	semidirect
11. C₇×#168	C₂²	semidirect
15. C₂×C₅×D₇	C₈	semidirect
27. C₂×#168	D₇	direct (N × H)
28. C₂×C₅×C₇	C₂×C₈	semidirect
30. C₅×D₇	C₂×C₈	semidirect
44. #168	C₂×D₇	direct (N × H)
45. C₅×C₇	C₂²×C₈	semidirect
48. C₂×D₇	#168	direct (N × H)
63. C₂×C₇	C₂×#168	semidirect
65. D₇	C₂×#168	direct (N × H)
67. C₂×C₅	C₈×D₇	semidirect
76. C₇	C₂²×#168	semidirect
77. C₅	C₂×C₈×D₇	semidirect
85. C₂	D₇×#168	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×C₇⋊#168	2
C₂×C₇×#168	2
D₇×#168	2
C₂×C₈×D₇	5
C₂²×#168	7

Subgroups — C2×D7×#168 #11764780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₇×#168 #11764780