Subgroups — C₇×#66564 #72269293

Order 2352.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₃⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1142	—	3	1	no	maximal, Hall
5	C₂×D₃×C₇²	—	2	1	no	—
6	C₃×C₄×C₇²	C₂²	1	1	yes	characteristic
7	C₃×C₇×Dic₇	—	2	1	no	—
8	C₇²×D₄	—	3	1	no	—
9	C₇×Dic₁₄	D₃	1	1	yes	characteristic, complemented
10	C₇×C₇⋊C₈	—	3	1	no	—
11	C₇×#215	—	7	1	no	maximal
12	#66564	C₇	1	1	yes	maximal, characteristic, direct product factor
13	C₂×C₃×C₇²	D₄	1	1	yes	characteristic
14	D₃×C₇²	—	4	1	no	—
15	C₂²×C₇²	—	6	1	no	—
16	C₄×C₇²	C₂×D₃	1	1	yes	characteristic
17	C₇×Dic₇	—	2	1	no	—
18	C₃×C₇×Q₈	—	7	1	no	—
19	C₇×D₁₂	—	2	3	no	—
20	C₇×D₁₂	D₇	1	1	yes	characteristic
21	C₇×D₁₂	C₂×C₇	1	1	yes	characteristic
22	C₃×Dic₁₄	C₂×C₇	1	1	yes	characteristic, complemented
23	C₃⋊(C₇⋊C₈)	C₂×C₇	1	1	yes	characteristic, complemented
24	C₇×C₃⋊C₈	—	7	1	no	—
25	C₃×C₇²	SD₁₆	1	1	yes	characteristic, complemented, Hall
26	C₇×SD₁₆	—	21	1	no	—
27	#1142	—	3	1	no	—
28	C₂×C₇²	—	12	1	no	—
29	C₂×C₇²	#67	1	1	yes	characteristic
30	C₂×D₃×C₇	—	2	6	no	—
31	C₂×D₃×C₇	—	2	1	no	—
32	C₂×D₃×C₇	—	2	1	no	—
33	C₃×C₄×C₇	—	2	3	no	—
34	C₃×C₄×C₇	—	14	1	no	—
35	C₃×C₄×C₇	C₂×D₇	1	1	yes	characteristic
36	C₃×C₄×C₇	C₂²×C₇	1	1	yes	characteristic
37	C₃×Dic₇	—	2	1	no	—
38	C₇×Q₈	—	7	1	no	—
39	C₇×D₄	—	3	1	no	—
40	C₇×D₄	—	3	1	no	—
41	C₇×D₄	—	6	3	no	—
42	Dic₁₄	D₃×C₇	1	1	yes	characteristic, complemented
43	C₇×C₈	—	21	1	no	—
44	C₇⋊C₈	—	3	1	no	—
45	C₇²	#215	1	1	yes	characteristic, complemented, Sylow
46	#215	—	7	1	no	Hall
47	C₂×C₃×C₇	—	2	3	no	—
48	C₂×C₃×C₇	C₂²⋊D₇	1	1	yes	characteristic
49	C₂×C₃×C₇	C₇×D₄	1	1	yes	characteristic
50	D₃×C₇	—	4	6	no	—
51	D₃×C₇	—	4	1	no	—
52	D₃×C₇	—	4	1	no	—
53	C₂²×C₇	—	6	6	no	—
54	C₂²×C₇	—	6	1	no	—
55	C₂²×C₇	—	6	1	no	—
56	C₄×C₇	—	2	3	no	—
57	C₄×C₇	—	14	1	no	—
58	C₄×C₇	D₃×D₇	1	1	yes	characteristic
59	C₄×C₇	C₂×D₃×C₇	1	1	yes	characteristic
60	Dic₇	—	2	1	no	—
61	C₃×Q₈	—	7	1	no	—
62	D₁₂	C₇×D₇	1	1	yes	characteristic
63	C₃⋊C₈	—	7	1	no	—
64	C₃×C₇	—	2	3	no	—
65	C₃×C₇	#1142	1	1	yes	characteristic, complemented
66	C₃×C₇	C₇×SD₁₆	1	1	yes	characteristic, complemented
67	SD₁₆	—	21	1	no	Sylow
68	C₂×C₇	—	2	3	no	—
69	C₂×C₇	—	12	6	no	—
70	C₂×C₇	—	12	1	no	—
71	C₂×C₇	—	12	1	no	—
72	C₂×C₇	#4245	1	1	yes	characteristic
73	C₂×C₇	C₇×#67	1	1	yes	characteristic
74	C₂×D₃	—	2	1	no	—
75	C₃×C₄	—	14	1	no	—
76	C₃×C₄	C₂×C₇×D₇	1	1	yes	characteristic
77	Q₈	—	7	1	no	—
78	D₄	—	3	1	no	—
79	C₈	—	21	1	no	—
80	C₇	—	2	3	no	—
81	C₇	#66564	1	1	yes	characteristic, direct product factor
82	C₇	C₇×#215	1	1	yes	characteristic, complemented
83	C₂×C₃	C₇×C₂²⋊D₇	1	1	yes	characteristic
84	D₃	—	4	1	no	—
85	C₂²	—	6	1	no	—
86	C₄	—	14	1	no	—
87	C₄	D₃×C₇×D₇	1	1	yes	characteristic
88	C₃	C₇×#1142	1	1	yes	characteristic, complemented, Sylow
89	C₂	—	12	1	no	—
90	C₂	C₇×#4245	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	#4245
commutator, cosocle	C₃×C₄×C₇	C₂²×C₇
Frattini	C₄	D₃×C₇×D₇
Fitting	C₃×C₄×C₇²	C₂²
socle	C₂×C₃×C₇²	D₄
prime radical	C₇²×D₁₂	C₂
hypercenter	C₄×C₇	D₃×D₇
hypocenter	C₃×C₇	C₇×SD₁₆
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×Dic₁₄	C₂	semidirect
3. C₇×C₃⋊(C₇⋊C₈)	C₂	semidirect
9. C₇×Dic₁₄	D₃	semidirect
12. #66564	C₇	direct (N × H)
22. C₃×Dic₁₄	C₂×C₇	semidirect
23. C₃⋊(C₇⋊C₈)	C₂×C₇	semidirect
25. C₃×C₇²	SD₁₆	semidirect
42. Dic₁₄	D₃×C₇	semidirect
45. C₇²	#215	semidirect
65. C₃×C₇	#1142	semidirect
66. C₃×C₇	C₇×SD₁₆	semidirect
81. C₇	#66564	direct (N × H)
82. C₇	C₇×#215	semidirect
88. C₃	C₇×#1142	semidirect

Maximal subgroups

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

subgroup	index
C₇²×D₁₂	2
C₃×C₇×Dic₁₄	2
C₇×C₃⋊(C₇⋊C₈)	2
C₇×#1142	3
C₇×#215	7
#66564	7

Subgroups — C7×#66564 #72269293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#66564 #72269293