Subgroups — C₇×#6712 #13132345

Order 1680.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Dic₁₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×Dic₁₅	C₂	1	1	yes	maximal, characteristic
4	C₇×#683	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₅×C₇	—	2	1	no	—
7	C₇×Dic₁₅	—	2	1	no	—
8	C₇×Dic₁₅	C₄	1	2	yes	complemented
9	C₇×Dic₃⋊C₄	—	5	1	no	maximal, Hall
10	C₂×C₄×C₅×C₇	D₃	1	1	yes	characteristic
11	C₂×C₇×Dic₅	—	3	1	no	—
12	C₂×C₇×Dic₅	—	3	1	no	—
13	#6712	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₅×C₇	D₄	1	1	yes	characteristic
16	C₂×C₃×C₅×C₇	Q₈	1	1	yes	characteristic
17	C₂×C₃×C₄×C₇	D₅	1	1	yes	characteristic
18	C₂×C₇×Dic₃	—	5	1	no	—
19	C₂×C₇×Dic₃	—	5	1	no	—
20	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
21	C₄×C₅×C₇	—	2	1	no	—
22	C₇×Dic₅	—	3	2	no	—
23	C₇×Dic₅	—	6	1	no	—
24	C₂×C₃×C₄×C₅	C₂×C₇	1	1	yes	characteristic
25	C₂×Dic₁₅	C₂×C₇	1	1	yes	characteristic
26	C₂×Dic₁₅	C₂×C₇	1	1	yes	characteristic
27	C₇×C₄⋊C₄	—	15	1	no	Hall
28	C₃×C₅×C₇	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×C₃×C₇	C₂×D₅	1	1	yes	characteristic
30	C₃×C₄×C₇	—	2	1	no	—
31	C₇×Dic₃	—	5	2	no	—
32	C₇×Dic₃	—	10	1	no	—
33	#683	—	3	1	no	Hall
34	C₂×C₅×C₇	Dic₆	1	1	yes	characteristic
35	C₂×C₅×C₇	C₄×D₃	1	1	yes	characteristic
36	C₂×C₅×C₇	#67	1	1	yes	characteristic
37	C₂²×C₃×C₅	C₂²×C₇	1	1	yes	characteristic
38	C₃×C₄×C₅	—	2	1	no	—
39	Dic₁₅	—	2	1	no	—
40	Dic₁₅	C₄×C₇	1	2	yes	complemented
41	C₂×C₄×C₇	—	15	1	no	—
42	C₂×C₄×C₇	—	15	1	no	—
43	C₂×C₄×C₇	D₁₅	1	1	yes	characteristic
44	Dic₃⋊C₄	—	5	1	no	Hall
45	C₂×C₃×C₇	Dic₁₀	1	1	yes	characteristic
46	C₂×C₃×C₇	C₄×D₅	1	1	yes	characteristic
47	C₂×C₃×C₇	C₂²⋊D₅	1	1	yes	characteristic
48	C₂×C₄×C₅	D₃×C₇	1	1	yes	characteristic
49	C₂×Dic₅	—	3	1	no	—
50	C₂×Dic₅	—	3	1	no	—
51	C₅×C₇	Dic₃⋊C₄	1	1	yes	characteristic, complemented, Hall
52	C₂×C₃×C₅	C₂×C₄×C₇	1	1	yes	characteristic
53	C₂×C₃×C₅	C₇×D₄	1	1	yes	characteristic
54	C₂×C₃×C₅	C₇×Q₈	1	1	yes	characteristic
55	C₂²×C₇	C₂×D₁₅	1	1	yes	characteristic
56	C₄×C₇	—	2	1	no	—
57	C₄×C₇	—	15	2	no	—
58	C₄×C₇	—	30	1	no	—
59	C₂×C₃×C₄	C₇×D₅	1	1	yes	characteristic
60	C₂×Dic₃	—	5	1	no	—
61	C₂×Dic₃	—	5	1	no	—
62	C₃×C₇	#683	1	1	yes	characteristic, complemented, Hall
63	C₂²×C₅	C₂×D₃×C₇	1	1	yes	characteristic
64	C₄×C₅	—	2	1	no	—
65	Dic₅	—	3	2	no	—
66	Dic₅	—	6	1	no	—
67	C₄⋊C₄	—	15	1	no	Sylow
68	C₃×C₅	C₇×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
69	C₂×C₇	Dic₃₀	1	1	yes	characteristic
70	C₂×C₇	C₄×D₁₅	1	1	yes	characteristic
71	C₂×C₇	#1220	1	1	yes	characteristic
72	C₂²×C₃	C₂×C₇×D₅	1	1	yes	characteristic
73	C₃×C₄	—	2	1	no	—
74	Dic₃	—	5	2	no	—
75	Dic₃	—	10	1	no	—
76	C₂×C₅	C₇×Dic₆	1	1	yes	characteristic
77	C₂×C₅	C₄×D₃×C₇	1	1	yes	characteristic
78	C₂×C₅	C₇×#67	1	1	yes	characteristic
79	C₂×C₄	—	15	1	no	—
80	C₂×C₄	—	15	1	no	—
81	C₂×C₄	C₇×D₁₅	1	1	yes	characteristic
82	C₇	#6712	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₃	C₇×Dic₁₀	1	1	yes	characteristic
84	C₂×C₃	C₄×C₇×D₅	1	1	yes	characteristic
85	C₂×C₃	C₇×C₂²⋊D₅	1	1	yes	characteristic
86	C₅	C₇×Dic₃⋊C₄	1	1	yes	characteristic, complemented, Sylow
87	C₂²	C₂×C₇×D₁₅	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	15	2	no	—
90	C₄	—	30	1	no	—
91	C₃	C₇×#683	1	1	yes	characteristic, complemented, Sylow
92	C₂	C₇×Dic₃₀	1	1	yes	characteristic
93	C₂	C₄×C₇×D₁₅	1	1	yes	characteristic
94	C₂	C₇×#1220	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
8. C₇×Dic₁₅	C₄	semidirect
13. #6712	C₇	direct (N × H)
28. C₃×C₅×C₇	C₄⋊C₄	semidirect
40. Dic₁₅	C₄×C₇	semidirect
51. C₅×C₇	Dic₃⋊C₄	semidirect
62. C₃×C₇	#683	semidirect
68. C₃×C₅	C₇×C₄⋊C₄	semidirect
82. C₇	#6712	direct (N × H)
86. C₅	C₇×Dic₃⋊C₄	semidirect
91. C₃	C₇×#683	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×C₇	2
C₂×C₇×Dic₁₅	2
C₇×#683	3
C₇×Dic₃⋊C₄	5
#6712	7

Subgroups — C7×#6712 #13132345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6712 #13132345