Subgroups — D₇×C₃⋊#168 #13132122

Order 1680.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₇	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₇⋊#168)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×#168	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅×D₇	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₃×C₅×Dic₇	C₄	1	1	yes	characteristic
8	D₇×C₃⋊C₈	—	5	1	no	maximal, Hall
9	C₄×C₅×D₇	D₃	1	1	yes	characteristic
10	C₇⋊#168	—	3	1	no	—
11	C₇×#168	—	3	1	no	—
12	C₂×C₃⋊#168	—	7	1	no	maximal, Hall
13	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
14	C₃×C₅×D₇	C₈	1	2	yes	complemented
15	C₃×C₄×D₇	D₅	1	1	yes	characteristic
16	C₃⋊(C₇⋊C₈)	—	5	1	no	—
17	C₇×C₃⋊C₈	—	5	1	no	—
18	C₂×C₅×D₇	Dic₃	1	1	yes	characteristic
19	C₄×C₅×C₇	C₂×D₃	1	1	yes	characteristic
20	C₅×Dic₇	Dic₃	1	1	yes	characteristic
21	C₂×C₃×C₄×C₅	—	7	1	no	—
22	C₃⋊#168	—	7	1	no	—
23	C₃⋊#168	D₇	1	1	yes	characteristic, direct product factor
24	C₈×D₇	—	15	1	no	Hall
25	C₃×C₅×C₇	C₂×C₈	1	1	yes	characteristic, complemented, Hall
26	C₂×C₃×D₇	Dic₅	1	1	yes	characteristic
27	C₃×C₄×C₇	C₂×D₅	1	1	yes	characteristic
28	C₃×Dic₇	Dic₅	1	1	yes	characteristic
29	C₂×#168	—	21	1	no	Hall
30	C₂×C₅×C₇	C₂×Dic₃	1	1	yes	characteristic
31	C₅×D₇	C₃⋊C₈	1	2	yes	complemented
32	C₂²×C₃×C₅	—	7	1	no	—
33	C₃×C₄×C₅	—	7	1	no	—
34	C₃×C₄×C₅	C₂×D₇	1	1	yes	characteristic
35	C₄×D₇	D₁₅	1	1	yes	characteristic
36	C₇×C₈	—	15	1	no	—
37	C₇⋊C₈	—	15	1	no	—
38	C₂×C₃⋊C₈	—	35	1	no	Hall
39	C₂×C₃×C₇	C₂×Dic₅	1	1	yes	characteristic
40	C₃×D₇	#168	1	2	yes	complemented
41	C₂×C₄×C₅	—	7	1	no	—
42	#168	—	3	1	no	—
43	#168	—	21	1	no	—
44	C₅×C₇	C₂×C₃⋊C₈	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₅	—	7	2	no	—
46	C₂×C₃×C₅	C₄×D₇	1	1	yes	characteristic
47	C₂×D₇	Dic₁₅	1	1	yes	characteristic
48	C₄×C₇	C₂×D₁₅	1	1	yes	characteristic
49	Dic₇	Dic₁₅	1	1	yes	characteristic
50	C₂×C₃×C₄	—	7	1	no	—
51	C₃⋊C₈	—	5	1	no	—
52	C₃⋊C₈	—	35	1	no	—
53	C₃×C₇	C₂×#168	1	1	yes	characteristic, complemented, Hall
54	C₂²×C₅	—	7	1	no	—
55	C₄×C₅	—	7	1	no	—
56	C₄×C₅	D₃×D₇	1	1	yes	characteristic
57	C₂×C₈	—	105	1	no	Sylow
58	C₃×C₅	C₈×D₇	1	1	yes	characteristic, complemented, Hall
59	C₂×C₇	C₂×Dic₁₅	1	1	yes	characteristic
60	D₇	C₃⋊#168	1	2	yes	direct product factor
61	C₂²×C₃	—	7	1	no	—
62	C₃×C₄	—	7	1	no	—
63	C₃×C₄	D₅×D₇	1	1	yes	characteristic
64	C₂×C₅	—	7	2	no	—
65	C₂×C₅	Dic₃×D₇	1	1	yes	characteristic
66	C₂×C₄	—	7	1	no	—
67	C₈	—	15	1	no	—
68	C₈	—	105	1	no	—
69	C₇	C₂×C₃⋊#168	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	7	2	no	—
71	C₂×C₃	D₇×Dic₅	1	1	yes	characteristic
72	C₅	D₇×C₃⋊C₈	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	7	1	no	—
74	C₄	—	7	1	no	—
75	C₄	D₇×D₁₅	1	1	yes	characteristic
76	C₃	D₇×#168	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	7	2	no	—
78	C₂	D₇×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₇×D₁₅
commutator, hypocenter	C₃×C₅×C₇	C₂×C₈
Fitting, cosocle	C₃×C₄×C₅×C₇	C₂²
socle	C₂×C₃×C₅×C₇	C₂×C₄
prime radical	C₂×C₃×C₅×D₇	C₄
LCS	G ▷ C₃×C₅×C₇
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₇⋊#168)	C₂	semidirect
3. C₇×C₃⋊#168	C₂	semidirect
14. C₃×C₅×D₇	C₈	semidirect
23. C₃⋊#168	D₇	direct (N × H)
25. C₃×C₅×C₇	C₂×C₈	semidirect
31. C₅×D₇	C₃⋊C₈	semidirect
40. C₃×D₇	#168	semidirect
44. C₅×C₇	C₂×C₃⋊C₈	semidirect
53. C₃×C₇	C₂×#168	semidirect
58. C₃×C₅	C₈×D₇	semidirect
60. D₇	C₃⋊#168	direct (N × H)
69. C₇	C₂×C₃⋊#168	semidirect
72. C₅	D₇×C₃⋊C₈	semidirect
76. C₃	D₇×#168	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₄×C₅×D₇	2
C₃⋊(C₇⋊#168)	2
C₇×C₃⋊#168	2
D₇×#168	3
D₇×C₃⋊C₈	5
C₂×C₃⋊#168	7

Subgroups — D7×C3⋊#168 #13132122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×C₃⋊#168 #13132122