Subgroups — C₇×#4680 #13055715

Order 1344.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#869	C₂	1	2	yes	maximal
2	C₃×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#817	C₂	1	4	yes	maximal
4	C₇×#498	—	3	1	no	maximal, Hall
5	C₂×C₇×Dic₆	—	2	2	no	—
6	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
7	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₇×C₄⋊Dic₃	—	2	2	no	—
9	C₇×C₄⋊Dic₃	C₂²	1	4	yes	—
10	C₇×C₄⋊Q₈	—	3	2	no	—
11	C₇×#106	—	3	4	no	—
12	C₄×C₇×C₈	D₃	1	1	yes	characteristic
13	#4680	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
15	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
16	C₂×C₇×Dic₃	—	2	4	no	—
17	C₇×Dic₆	—	4	4	no	—
18	C₃×C₇×C₈	Q₈	1	4	yes	complemented
19	C₂×C₇×Q₈	—	6	2	no	—
20	C₂×C₇×C₈	C₂×D₃	1	2	yes	—
21	C₇×C₄⋊C₄	—	3	4	no	—
22	C₇×C₄⋊C₄	—	6	2	no	—
23	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
24	#869	C₂×C₇	1	2	yes	—
25	C₃×C₄×C₈	C₂×C₇	1	1	yes	characteristic
26	#817	C₂×C₇	1	4	yes	—
27	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
28	C₃×C₄×C₇	SD₁₆	1	4	yes	—
29	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
30	C₇×Dic₃	—	4	4	no	—
31	#498	—	3	1	no	Sylow
32	C₇×Q₈	—	12	4	no	—
33	C₂×C₄×C₇	—	6	4	no	—
34	C₂×C₄×C₇	D₁₂	1	2	yes	—
35	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
36	C₇×C₈	Dic₆	1	4	yes	complemented
37	C₂×Dic₆	—	2	2	no	—
38	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
39	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
40	C₄⋊Dic₃	—	2	2	no	—
41	C₄⋊Dic₃	C₂²×C₇	1	4	yes	—
42	C₂×C₃×C₇	C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₃×C₇	C₂×SD₁₆	1	2	yes	—
44	C₄⋊Q₈	—	3	2	no	—
45	#106	—	3	4	no	—
46	C₄×C₈	D₃×C₇	1	1	yes	characteristic
47	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
48	C₄×C₇	—	12	4	no	—
49	C₄×C₇	#204	1	4	yes	—
50	C₄×C₇	C₂×Dic₆	1	2	yes	—
51	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
52	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
53	C₂×Dic₃	—	2	4	no	—
54	Dic₆	—	4	4	no	—
55	C₃×C₈	C₇×Q₈	1	4	yes	complemented
56	C₃×C₇	#498	1	1	yes	characteristic, complemented, Hall
57	C₂×Q₈	—	6	2	no	—
58	C₂×C₈	C₂×D₃×C₇	1	2	yes	—
59	C₄⋊C₄	—	3	4	no	—
60	C₄⋊C₄	—	6	2	no	—
61	C₄²	C₂×D₃×C₇	1	1	yes	characteristic
62	C₂×C₇	#869	1	1	yes	characteristic
63	C₂×C₇	C₂×#204	1	2	yes	—
64	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
65	C₃×C₄	C₇×SD₁₆	1	4	yes	—
66	C₃×C₄	C₂×C₇×Q₈	1	2	yes	—
67	Dic₃	—	4	4	no	—
68	Q₈	—	12	4	no	—
69	C₂×C₄	—	6	4	no	—
70	C₂×C₄	C₇×D₁₂	1	2	yes	—
71	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
72	C₈	C₇×Dic₆	1	4	yes	complemented
73	C₇	#4680	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×C₃	C₇×C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₃	C₂×C₇×SD₁₆	1	2	yes	—
76	C₂²	C₂×C₇×D₁₂	1	1	yes	characteristic
77	C₄	—	12	4	no	—
78	C₄	C₇×#204	1	4	yes	—
79	C₄	C₂×C₇×Dic₆	1	2	yes	—
80	C₃	C₇×#498	1	1	yes	characteristic, complemented, Sylow
81	C₂	C₇×#869	1	1	yes	characteristic
82	C₂	C₂×C₇×#204	1	2	yes	—

	subgroup	G/H
center	C₂²×C₇	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₄×C₇×C₈	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₄×C₇×C₈	D₃
hypocenter	C₃	C₇×#498
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×C₈

Complements

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

N	H ≅ G/N	type
13. #4680	C₇	direct (N × H)
18. C₃×C₇×C₈	Q₈	semidirect
36. C₇×C₈	Dic₆	semidirect
55. C₃×C₈	C₇×Q₈	semidirect
56. C₃×C₇	#498	semidirect
72. C₈	C₇×Dic₆	semidirect
73. C₇	#4680	direct (N × H)
80. C₃	C₇×#498	semidirect

Maximal subgroups

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

subgroup	index
C₇×#869	2
C₃×C₄×C₇×C₈	2
C₇×#817	2
C₇×#498	3
#4680	7

Subgroups — C7×#4680 #13055715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4680 #13055715