Subgroups — C₇×#809 #10620607

Order 672.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#102	—	3	1	no	maximal, Hall
5	C₂²×D₃×C₇	—	2	1	no	—
6	C₂×C₃×C₄×C₇	—	2	1	no	—
7	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₇×D₁₂	—	2	1	no	—
9	C₇×D₁₂	C₄	1	2	yes	complemented
10	C₇×C₃⋊C₈	—	2	1	no	—
11	C₂×C₇×D₄	—	3	1	no	—
12	C₂×C₇×C₈	—	3	1	no	—
13	C₇×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
14	#809	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₃×C₇	D₄	1	1	yes	characteristic
16	C₂×D₃×C₇	—	2	2	no	—
17	C₂×D₃×C₇	—	4	1	no	—
18	C₂×D₃×C₇	—	4	1	no	—
19	C₃×C₄×C₇	—	4	1	no	—
20	C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₃×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂³×C₇	—	6	1	no	—
23	C₇×D₄	—	3	2	no	—
24	C₇×D₄	—	6	1	no	—
25	C₂×C₄×C₇	—	2	1	no	—
26	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
27	C₇×C₈	—	6	1	no	—
28	C₂×D₁₂	C₂×C₇	1	1	yes	characteristic
29	C₃×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
30	C₂×C₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
31	C₂×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₃×C₇	D₈	1	1	yes	characteristic
33	C₂×C₃×C₇	SD₁₆	1	1	yes	characteristic
34	D₃×C₇	—	4	2	no	—
35	#102	—	3	1	no	Sylow
36	C₂²×C₇	—	6	2	no	—
37	C₂²×C₇	—	12	1	no	—
38	C₂²×C₇	—	12	1	no	—
39	C₂²×C₇	#67	1	1	yes	characteristic
40	C₄×C₇	—	4	1	no	—
41	C₄×C₇	C₄×D₃	1	1	yes	characteristic
42	C₄×C₇	D₁₂	1	1	yes	characteristic
43	C₂²×D₃	—	2	1	no	—
44	C₂×C₃×C₄	—	2	1	no	—
45	C₂×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
46	D₁₂	—	2	1	no	—
47	D₁₂	C₄×C₇	1	2	yes	complemented
48	C₃⋊C₈	—	2	1	no	—
49	C₃×C₇	#102	1	1	yes	characteristic, complemented, Hall
50	C₂×D₄	—	3	1	no	—
51	C₂×C₈	—	3	1	no	—
52	C₄⋊C₄	D₃×C₇	1	1	yes	characteristic, complemented
53	C₂×C₇	—	12	2	no	—
54	C₂×C₇	(C₂×D₃)⋊C₄	1	1	yes	characteristic
55	C₂×C₇	#213	1	1	yes	characteristic
56	C₂×C₇	#215	1	1	yes	characteristic
57	C₂²×C₃	C₇×D₄	1	1	yes	characteristic
58	C₂×D₃	—	2	2	no	—
59	C₂×D₃	—	4	1	no	—
60	C₂×D₃	—	4	1	no	—
61	C₃×C₄	—	4	1	no	—
62	C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
63	C₃×C₄	C₇×D₄	1	1	yes	characteristic
64	C₂³	—	6	1	no	—
65	D₄	—	3	2	no	—
66	D₄	—	6	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	C₂×D₃×C₇	1	1	yes	characteristic
69	C₈	—	6	1	no	—
70	C₇	#809	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×C₃	C₇×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃	C₇×D₈	1	1	yes	characteristic
73	C₂×C₃	C₇×SD₁₆	1	1	yes	characteristic
74	D₃	—	4	2	no	—
75	C₂²	—	6	2	no	—
76	C₂²	—	12	1	no	—
77	C₂²	—	12	1	no	—
78	C₂²	C₇×#67	1	1	yes	characteristic
79	C₄	—	4	1	no	—
80	C₄	C₄×D₃×C₇	1	1	yes	characteristic
81	C₄	C₇×D₁₂	1	1	yes	characteristic
82	C₃	C₇×#102	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	12	2	no	—
84	C₂	C₇×(C₂×D₃)⋊C₄	1	1	yes	characteristic
85	C₂	C₇×#213	1	1	yes	characteristic
86	C₂	C₇×#215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	#67
commutator	C₃×C₄	C₂×C₄×C₇
Frattini	C₂×C₄	C₂×D₃×C₇
Fitting	C₃×C₇×C₄⋊C₄	C₂
socle	C₂²×C₃×C₇	D₄
prime radical	C₂×C₇×D₁₂	C₂
cosocle	C₂×C₃×C₄	C₂²×C₇
hypercenter	C₇×C₄⋊C₄	D₃
hypocenter	C₃	C₇×#102
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₇×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×C₄⋊C₄	C₂	semidirect
3. C₂×C₇×C₃⋊C₈	C₂	semidirect
9. C₇×D₁₂	C₄	semidirect
13. C₇×C₄⋊C₄	D₃	semidirect
14. #809	C₇	direct (N × H)
29. C₃×C₄⋊C₄	C₂×C₇	semidirect
30. C₂×C₃⋊C₈	C₂×C₇	semidirect
47. D₁₂	C₄×C₇	semidirect
49. C₃×C₇	#102	semidirect
52. C₄⋊C₄	D₃×C₇	semidirect
70. C₇	#809	direct (N × H)
82. C₃	C₇×#102	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×D₁₂	2
C₃×C₇×C₄⋊C₄	2
C₂×C₇×C₃⋊C₈	2
C₇×#102	3
#809	7

Subgroups — C7×#809 #10620607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#809 #10620607