Subgroups — C₇×#926 #10621323

Order 672.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₇×#217	C₂	1	2	yes	maximal
3	C₇×C₄⋊Dic₃	C₂	1	2	yes	maximal, complemented
4	C₄×C₇×Dic₃	C₂	1	2	yes	maximal, complemented
5	C₇×#117	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₇	C₄	1	4	yes	complemented
8	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₇×Dic₃	C₂²	1	4	yes	—
11	C₂²×C₄×C₇	D₃	1	1	yes	characteristic
12	C₇×C₄⋊C₄	—	3	2	no	—
13	C₇×C₂²⋊C₄	—	3	2	no	—
14	C₄²×C₇	—	3	2	no	—
15	#926	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₃×C₇	—	2	2	no	—
17	C₂²×C₃×C₇	C₂×C₄	1	2	yes	—
18	C₂²×C₃×C₇	C₂³	1	1	yes	characteristic
19	C₃×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₃×C₄×C₇	C₂×C₄	1	2	yes	—
21	C₇×Dic₃	—	2	4	no	—
22	C₂³×C₇	C₂×D₃	1	1	yes	characteristic
23	C₂×C₄×C₇	—	3	4	no	—
24	C₂×C₄×C₇	Dic₃	1	4	yes	complemented
25	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
26	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
27	C₂²×C₃×C₄	C₂×C₇	1	1	yes	characteristic
28	#217	C₂×C₇	1	2	yes	—
29	C₄⋊Dic₃	C₂×C₇	1	2	yes	complemented
30	C₄×Dic₃	C₂×C₇	1	2	yes	complemented
31	C₂×C₃×C₇	—	2	2	no	—
32	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₃×C₇	#41	1	2	yes	—
34	#117	—	3	1	no	Sylow
35	C₂²×C₇	—	2	2	no	—
36	C₂²×C₇	C₂×Dic₃	1	2	yes	—
37	C₂²×C₇	C₂²×D₃	1	1	yes	characteristic
38	C₄×C₇	—	6	4	no	—
39	C₄×C₇	C₂×Dic₃	1	2	yes	—
40	C₄×C₇	C₂×Dic₃	1	2	yes	—
41	C₂³×C₃	C₂²×C₇	1	1	yes	characteristic
42	C₂×C₃×C₄	C₄×C₇	1	4	yes	complemented
43	C₂×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
45	C₂×Dic₃	C₂²×C₇	1	4	yes	—
46	C₃×C₇	#117	1	1	yes	characteristic, complemented, Hall
47	C₂²×C₄	D₃×C₇	1	1	yes	characteristic
48	C₄⋊C₄	—	3	2	no	—
49	C₂²⋊C₄	—	3	2	no	—
50	C₄²	—	3	2	no	—
51	C₂×C₇	—	2	2	no	—
52	C₂×C₇	#235	1	2	yes	—
53	C₂×C₇	C₂²×Dic₃	1	1	yes	characteristic
54	C₂²×C₃	—	2	2	no	—
55	C₂²×C₃	C₂×C₄×C₇	1	2	yes	—
56	C₂²×C₃	C₂³×C₇	1	1	yes	characteristic
57	C₃×C₄	C₂×C₄×C₇	1	2	yes	—
58	C₃×C₄	C₂×C₄×C₇	1	2	yes	—
59	Dic₃	—	2	4	no	—
60	C₂³	C₂×D₃×C₇	1	1	yes	characteristic
61	C₂×C₄	—	3	4	no	—
62	C₂×C₄	C₇×Dic₃	1	4	yes	complemented
63	C₂×C₄	C₂×D₃×C₇	1	1	yes	characteristic
64	C₂×C₄	C₂×D₃×C₇	1	1	yes	characteristic
65	C₇	#926	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
68	C₂×C₃	C₇×#41	1	2	yes	—
69	C₂²	—	2	2	no	—
70	C₂²	C₂×C₇×Dic₃	1	2	yes	—
71	C₂²	C₂²×D₃×C₇	1	1	yes	characteristic
72	C₄	—	6	4	no	—
73	C₄	C₂×C₇×Dic₃	1	2	yes	—
74	C₄	C₂×C₇×Dic₃	1	2	yes	—
75	C₃	C₇×#117	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	2	2	no	—
77	C₂	C₇×#235	1	2	yes	—
78	C₂	C₂²×C₇×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₃
commutator	C₂×C₃	C₂²×C₄×C₇
Frattini	C₂²	C₂²×D₃×C₇
Fitting	C₂²×C₃×C₄×C₇	C₂
socle	C₂²×C₃×C₇	C₂³
prime radical	C₂³×C₃×C₇	C₂²
cosocle	C₂²×C₃	C₂³×C₇
hypercenter	C₂²×C₄×C₇	D₃
hypocenter	C₃	C₇×#117
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇

Complements

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

N	H ≅ G/N	type
3. C₇×C₄⋊Dic₃	C₂	semidirect
4. C₄×C₇×Dic₃	C₂	semidirect
7. C₂×C₃×C₄×C₇	C₄	semidirect
15. #926	C₇	direct (N × H)
24. C₂×C₄×C₇	Dic₃	semidirect
29. C₄⋊Dic₃	C₂×C₇	semidirect
30. C₄×Dic₃	C₂×C₇	semidirect
42. C₂×C₃×C₄	C₄×C₇	semidirect
46. C₃×C₇	#117	semidirect
62. C₂×C₄	C₇×Dic₃	semidirect
65. C₇	#926	direct (N × H)
75. C₃	C₇×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×C₇	2
C₇×#217	2
C₇×C₄⋊Dic₃	2
C₄×C₇×Dic₃	2
C₇×#117	3
#926	7

Subgroups — C7×#926 #10621323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#926 #10621323