Subgroups — C₇×#5934 #13062131

Order 1344.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#997	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×B(4,𝔽₂)	—	3	1	no	maximal, Hall
3	C₂×C₇×S₄	—	4	3	no	maximal
4	C₇×H(5,𝔽₂)	D₃	1	1	yes	characteristic, complemented
5	C₇×#120	—	3	3	no	—
6	C₇×#99	—	3	3	no	—
7	#5934	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₇×A₄	—	4	3	no	—
9	C₇×S₄	—	4	6	no	—
10	C₇×SL(2,𝔽₃)	—	8	1	no	—
11	C₂⁴×C₇	—	3	1	no	—
12	C₇×#41	—	6	1	no	—
13	C₂×C₇×D₄	—	3	3	no	—
14	C₂×C₇×D₄	—	6	3	no	—
15	C₇×C₂²⋊C₄	—	3	3	no	—
16	C₇×C₂²⋊C₄	—	6	3	no	—
17	#997	C₂×C₇	1	1	yes	characteristic, complemented
18	C₂×D₃×C₇	—	16	1	no	—
19	C₇×A₄	—	4	3	no	—
20	B(4,𝔽₂)	—	3	1	no	Sylow
21	C₂³×C₇	—	3	1	no	—
22	C₂³×C₇	—	6	3	no	—
23	C₂³×C₇	—	12	2	no	—
24	C₂³×C₇	S₄	1	3	yes	complemented
25	C₇×Q₈	—	2	1	no	—
26	C₇×D₄	—	6	3	no	—
27	C₇×D₄	—	12	6	no	—
28	C₂×C₄×C₇	—	3	3	no	—
29	C₂×C₄×C₇	—	6	3	no	—
30	C₂×S₄	—	4	3	no	—
31	C₂×C₃×C₇	—	16	1	no	—
32	D₃×C₇	—	16	2	no	—
33	H(5,𝔽₂)	D₃×C₇	1	1	yes	characteristic, complemented
34	#120	—	3	3	no	—
35	#99	—	3	3	no	—
36	C₂²×C₇	—	3	3	no	—
37	C₂²×C₇	—	4	3	no	—
38	C₂²×C₇	—	6	6	no	—
39	C₂²×C₇	—	12	1	no	—
40	C₂²×C₇	—	12	3	no	—
41	C₂²×C₇	—	12	1	no	—
42	C₄×C₇	—	6	1	no	—
43	C₄×C₇	—	12	3	no	—
44	C₂×A₄	—	4	3	no	—
45	S₄	—	4	6	no	—
46	SL(2,𝔽₃)	—	8	1	no	—
47	C₃×C₇	—	16	1	no	Hall
48	C₂⁴	—	3	1	no	—
49	#41	—	6	1	no	—
50	C₂×D₄	—	3	3	no	—
51	C₂×D₄	—	6	3	no	—
52	C₂²⋊C₄	—	3	3	no	—
53	C₂²⋊C₄	—	6	3	no	—
54	C₂×C₇	—	6	3	no	—
55	C₂×C₇	—	12	2	no	—
56	C₂×C₇	#1020	1	1	yes	characteristic
57	C₂×D₃	—	16	1	no	—
58	A₄	—	4	3	no	—
59	C₂³	—	3	1	no	—
60	C₂³	—	6	3	no	—
61	C₂³	—	12	2	no	—
62	C₂³	C₇×S₄	1	3	yes	complemented
63	Q₈	—	2	1	no	—
64	D₄	—	6	3	no	—
65	D₄	—	12	6	no	—
66	C₂×C₄	—	3	3	no	—
67	C₂×C₄	—	6	3	no	—
68	C₇	#5934	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃	—	16	1	no	—
70	D₃	—	16	2	no	—
71	C₂²	—	3	3	no	—
72	C₂²	—	4	3	no	—
73	C₂²	—	6	6	no	—
74	C₂²	—	12	1	no	—
75	C₂²	—	12	3	no	—
76	C₂²	—	12	1	no	—
77	C₄	—	6	1	no	—
78	C₄	—	12	3	no	—
79	C₃	—	16	1	no	Sylow
80	C₂	—	6	3	no	—
81	C₂	—	12	2	no	—
82	C₂	C₇×#1020	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂×C₇	#1020
commutator, cosocle, hypocenter	#997	C₂×C₇
Frattini	C₂	C₇×#1020
Fitting	C₇×H(5,𝔽₂)	D₃
LCS	G ▷ #997
UCS	C₁ ◁ C₂×C₇

Complements

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

N	H ≅ G/N	type
1. C₇×#997	C₂	semidirect
4. C₇×H(5,𝔽₂)	D₃	semidirect
7. #5934	C₇	direct (N × H)
17. #997	C₂×C₇	semidirect
24. C₂³×C₇	S₄	semidirect
33. H(5,𝔽₂)	D₃×C₇	semidirect
62. C₂³	C₇×S₄	semidirect
68. C₇	#5934	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#997	2
C₇×B(4,𝔽₂)	3
C₂×C₇×S₄	4
#5934	7

Subgroups — C7×#5934 #13062131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5934 #13062131