Subgroups — C₇×#5386 #13057109

Order 1344.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₇×#859	C₂	1	2	yes	maximal
3	C₇×#475	—	3	1	no	maximal, Hall
4	C₂×C₇×Dic₆	—	4	1	no	maximal
5	C₂×C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₇×C₄⋊Q₈	—	3	1	no	—
7	C₄×C₇×Q₈	D₃	1	1	yes	characteristic
8	C₇×#106	—	3	2	no	—
9	C₇×C₄⋊C₈	—	3	1	no	—
10	C₇×#103	—	3	2	no	—
11	#5386	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₃×C₄×C₇	—	4	1	no	—
13	C₂×C₇×Dic₃	—	4	2	no	—
14	C₇×Dic₆	—	4	4	no	—
15	C₇×SL(2,𝔽₃)	Q₈	1	1	yes	characteristic, complemented
16	C₂×C₇×Q₈	—	6	1	no	—
17	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
18	C₂×C₇×C₈	—	3	2	no	—
19	C₇×C₄⋊C₄	—	3	1	no	—
20	C₇×C₄⋊C₄	—	3	2	no	—
21	C₇×C₄⋊C₄	—	6	1	no	—
22	C₄²×C₇	—	3	1	no	—
23	C₄×SL(2,𝔽₃)	C₂×C₇	1	1	yes	characteristic
24	#859	C₂×C₇	1	2	yes	—
25	C₂²×C₃×C₇	—	4	1	no	—
26	C₃×C₄×C₇	—	4	2	no	—
27	C₇×Dic₃	—	4	4	no	—
28	#475	—	3	1	no	Sylow
29	C₇×Q₈	—	3	1	no	—
30	C₇×Q₈	—	12	2	no	—
31	C₇×Q₈	Dic₆	1	1	yes	characteristic, complemented
32	C₂×C₄×C₇	—	3	1	no	—
33	C₂×C₄×C₇	—	3	1	no	—
34	C₂×C₄×C₇	—	6	2	no	—
35	C₂×C₄×C₇	S₄	1	1	yes	characteristic
36	C₇×C₈	—	6	2	no	—
37	C₂×Dic₆	—	4	1	no	—
38	C₂×SL(2,𝔽₃)	C₂²×C₇	1	1	yes	characteristic
39	C₂×C₃×C₇	—	4	1	no	—
40	C₂×C₃×C₇	—	4	1	no	—
41	C₂×C₃×C₇	—	4	1	no	—
42	C₄⋊Q₈	—	3	1	no	—
43	C₄×Q₈	D₃×C₇	1	1	yes	characteristic
44	#106	—	3	2	no	—
45	C₄⋊C₈	—	3	1	no	—
46	#103	—	3	2	no	—
47	C₂²×C₇	C₂×S₄	1	1	yes	characteristic
48	C₄×C₇	—	3	1	no	—
49	C₄×C₇	—	3	1	no	—
50	C₄×C₇	—	6	1	no	—
51	C₄×C₇	—	12	2	no	—
52	C₄×C₇	GL(2,𝔽₃)	1	2	yes	—
53	C₂×C₃×C₄	—	4	1	no	—
54	C₂×Dic₃	—	4	2	no	—
55	Dic₆	—	4	4	no	—
56	SL(2,𝔽₃)	C₇×Q₈	1	1	yes	characteristic, complemented
57	C₃×C₇	—	4	1	no	Hall
58	C₂×Q₈	—	6	1	no	—
59	C₂×Q₈	C₂×D₃×C₇	1	1	yes	characteristic
60	C₂×C₈	—	3	2	no	—
61	C₄⋊C₄	—	3	1	no	—
62	C₄⋊C₄	—	3	2	no	—
63	C₄⋊C₄	—	6	1	no	—
64	C₄²	—	3	1	no	—
65	C₂×C₇	A₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₇	C₂×GL(2,𝔽₃)	1	1	yes	characteristic
67	C₂×C₇	#984	1	1	yes	characteristic
68	C₂²×C₃	—	4	1	no	—
69	C₃×C₄	—	4	2	no	—
70	Dic₃	—	4	4	no	—
71	Q₈	—	3	1	no	—
72	Q₈	—	12	2	no	—
73	Q₈	C₇×Dic₆	1	1	yes	characteristic, complemented
74	C₂×C₄	—	3	1	no	—
75	C₂×C₄	—	3	1	no	—
76	C₂×C₄	—	6	2	no	—
77	C₂×C₄	C₇×S₄	1	1	yes	characteristic
78	C₈	—	6	2	no	—
79	C₇	#5386	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×C₃	—	4	1	no	—
81	C₂×C₃	—	4	1	no	—
82	C₂×C₃	—	4	1	no	—
83	C₂²	C₂×C₇×S₄	1	1	yes	characteristic
84	C₄	—	3	1	no	—
85	C₄	—	3	1	no	—
86	C₄	—	6	1	no	—
87	C₄	—	12	2	no	—
88	C₄	C₇×GL(2,𝔽₃)	1	2	yes	—
89	C₃	—	4	1	no	Sylow
90	C₂	C₇×A₄⋊Q₈	1	1	yes	characteristic
91	C₂	C₂×C₇×GL(2,𝔽₃)	1	1	yes	characteristic
92	C₂	C₇×#984	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂²×C₇
Frattini	C₂²	C₂×C₇×S₄
Fitting	C₄×C₇×Q₈	D₃
prime radical	C₂×C₇×SL(2,𝔽₃)	C₂²
hypercenter	C₂×C₄×C₇	S₄
hypocenter	SL(2,𝔽₃)	C₇×Q₈
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇

Complements

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

N	H ≅ G/N	type
11. #5386	C₇	direct (N × H)
15. C₇×SL(2,𝔽₃)	Q₈	semidirect
31. C₇×Q₈	Dic₆	semidirect
56. SL(2,𝔽₃)	C₇×Q₈	semidirect
73. Q₈	C₇×Dic₆	semidirect
79. C₇	#5386	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×SL(2,𝔽₃)	2
C₇×#859	2
C₇×#475	3
C₂×C₇×Dic₆	4
#5386	7

Subgroups — C7×#5386 #13057109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5386 #13057109