Subgroups — C₇×#5140 #13056175

Order 1344.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#131	C₂	1	1	yes	maximal, characteristic
2	C₇×#812	C₂	1	3	yes	maximal, complemented
3	C₂×C₇×C₃⋊C₁₆	C₂	1	3	yes	maximal, complemented
4	C₇×#504	—	3	1	no	maximal, Hall
5	C₃×C₇×#41	C₄	1	1	yes	characteristic
6	C₃×C₇×M₁₆	C₄	1	3	yes	—
7	C₂×C₃×C₇×C₈	C₂²	1	3	yes	—
8	C₇×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₇×C₃⋊C₁₆	C₂²	1	3	yes	—
10	C₇×#131	D₃	1	1	yes	characteristic
11	C₇×M₃₂	—	3	3	no	—
12	C₂×C₇×C₁₆	—	3	3	no	—
13	#5140	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₃×C₇×Q₈	C₈	1	1	yes	characteristic
15	C₃×C₇×D₄	C₈	1	3	yes	—
16	C₂×C₃×C₄×C₇	C₂×C₄	1	3	yes	—
17	C₃×C₇×C₈	C₂×C₄	1	3	yes	—
18	C₃×C₇×C₈	C₂³	1	1	yes	characteristic
19	C₇×#41	Dic₃	1	1	yes	characteristic
20	C₇×M₁₆	Dic₃	1	3	yes	—
21	C₂×C₇×C₈	C₂×D₃	1	3	yes	—
22	C₇×C₁₆	—	3	1	no	—
23	C₇×C₁₆	—	3	3	no	—
24	C₃×#131	C₂×C₇	1	1	yes	characteristic
25	#812	C₂×C₇	1	3	yes	complemented
26	C₂×C₃⋊C₁₆	C₂×C₇	1	3	yes	complemented
27	C₂²×C₃×C₇	C₂×C₈	1	3	yes	—
28	C₃×C₄×C₇	C₂×C₈	1	3	yes	—
29	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
30	#504	—	3	1	no	Sylow
31	C₇×Q₈	C₃⋊C₈	1	1	yes	characteristic
32	C₇×D₄	C₃⋊C₈	1	3	yes	—
33	C₂×C₄×C₇	C₂×Dic₃	1	3	yes	—
34	C₇×C₈	C₂×Dic₃	1	3	yes	—
35	C₇×C₈	C₂²×D₃	1	1	yes	characteristic
36	C₃×#41	C₄×C₇	1	1	yes	characteristic
37	C₃×M₁₆	C₄×C₇	1	3	yes	—
38	C₂×C₃×C₈	C₂²×C₇	1	3	yes	—
39	C₃⋊C₁₆	C₂²×C₇	1	1	yes	characteristic
40	C₃⋊C₁₆	C₂²×C₇	1	3	yes	—
41	C₂×C₃×C₇	—	2	3	no	—
42	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
43	#131	D₃×C₇	1	1	yes	characteristic
44	M₃₂	—	3	3	no	—
45	C₂×C₁₆	—	3	3	no	—
46	C₂²×C₇	C₂×C₃⋊C₈	1	3	yes	—
47	C₄×C₇	C₂×C₃⋊C₈	1	3	yes	—
48	C₄×C₇	C₂²×Dic₃	1	1	yes	characteristic
49	C₃×Q₈	C₇×C₈	1	1	yes	characteristic
50	C₃×D₄	C₇×C₈	1	3	yes	—
51	C₂×C₃×C₄	C₂×C₄×C₇	1	3	yes	—
52	C₃×C₈	C₂×C₄×C₇	1	3	yes	—
53	C₃×C₈	C₂³×C₇	1	1	yes	characteristic
54	C₃×C₇	#504	1	1	yes	characteristic, complemented, Hall
55	#41	C₇×Dic₃	1	1	yes	characteristic
56	M₁₆	C₇×Dic₃	1	3	yes	—
57	C₂×C₈	C₂×D₃×C₇	1	3	yes	—
58	C₁₆	—	3	1	no	—
59	C₁₆	—	3	3	no	—
60	C₂×C₇	—	2	3	no	—
61	C₂×C₇	C₂²×C₃⋊C₈	1	1	yes	characteristic
62	C₂²×C₃	C₂×C₇×C₈	1	3	yes	—
63	C₃×C₄	C₂×C₇×C₈	1	3	yes	—
64	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
65	Q₈	C₇×C₃⋊C₈	1	1	yes	characteristic
66	D₄	C₇×C₃⋊C₈	1	3	yes	—
67	C₂×C₄	C₂×C₇×Dic₃	1	3	yes	—
68	C₈	C₂×C₇×Dic₃	1	3	yes	—
69	C₈	C₂²×D₃×C₇	1	1	yes	characteristic
70	C₇	#5140	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×C₃	—	2	3	no	—
72	C₂×C₃	C₂²×C₇×C₈	1	1	yes	characteristic
73	C₂²	C₂×C₇×C₃⋊C₈	1	3	yes	—
74	C₄	C₂×C₇×C₃⋊C₈	1	3	yes	—
75	C₄	C₂²×C₇×Dic₃	1	1	yes	characteristic
76	C₃	C₇×#504	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	2	3	no	—
78	C₂	C₂²×C₇×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₈	C₂²×D₃
commutator	C₂×C₃	C₂²×C₇×C₈
Frattini	C₈	C₂²×D₃×C₇
Fitting	C₃×C₇×#131	C₂
socle	C₂×C₃×C₇	C₂²×C₈
prime radical	C₃×C₇×#41	C₄
cosocle	C₃×C₈	C₂³×C₇
hypercenter	C₇×#131	D₃
hypocenter	C₃	C₇×#504
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₇×C₈ ◁ C₇×#131

Complements

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

N	H ≅ G/N	type
2. C₇×#812	C₂	semidirect
3. C₂×C₇×C₃⋊C₁₆	C₂	semidirect
13. #5140	C₇	direct (N × H)
25. #812	C₂×C₇	semidirect
26. C₂×C₃⋊C₁₆	C₂×C₇	semidirect
54. C₃×C₇	#504	semidirect
70. C₇	#5140	direct (N × H)
76. C₃	C₇×#504	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×#131	2
C₇×#812	2
C₂×C₇×C₃⋊C₁₆	2
C₇×#504	3
#5140	7

Subgroups — C7×#5140 #13056175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5140 #13056175