Subgroups — C₇×#5407 #13057127

Order 1344.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#986	C₂	1	2	yes	maximal, complemented
2	C₇×#131⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#472	—	3	1	no	maximal, Hall
4	C₇×#41⋊C₃	C₂²	1	1	yes	characteristic
5	C₇×GL(2,𝔽₃)	—	2	2	no	—
6	C₇×D₂₄	—	4	1	no	maximal
7	C₇×AGL(1,ℤ/8)	—	3	2	no	—
8	C₂×C₇×D₈	—	3	1	no	—
9	C₇×#131	D₃	1	1	yes	characteristic, complemented
10	C₇×#108	—	3	1	no	—
11	C₇×#100	—	3	2	no	—
12	#5407	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₇×D₁₂	—	4	2	no	—
14	C₇×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
15	C₃×C₇×C₈	—	4	1	no	—
16	C₇×#41	C₂×D₃	1	1	yes	characteristic
17	C₂×C₇×D₄	—	3	2	no	—
18	C₇×SD₁₆	—	6	2	no	—
19	C₇×D₈	—	6	1	no	—
20	C₇×D₈	—	6	2	no	—
21	C₇×D₈	—	6	1	no	—
22	C₇×M₁₆	—	3	1	no	—
23	C₇×M₁₆	—	3	2	no	—
24	C₂×C₇×C₈	—	3	1	no	—
25	#986	C₂×C₇	1	2	yes	complemented
26	#131⋊C₃	C₂×C₇	1	1	yes	characteristic, complemented
27	C₂×D₃×C₇	—	8	2	no	—
28	C₃×C₄×C₇	—	4	1	no	—
29	#472	—	3	1	no	Sylow
30	C₂³×C₇	—	6	2	no	—
31	C₇×Q₈	D₁₂	1	1	yes	characteristic
32	C₇×D₄	—	3	1	no	—
33	C₇×D₄	—	6	2	no	—
34	C₇×D₄	—	6	2	no	—
35	C₂×C₄×C₇	—	3	1	no	—
36	C₇×C₈	—	3	1	no	—
37	C₇×C₈	—	6	2	no	—
38	C₇×C₈	S₄	1	1	yes	characteristic
39	#41⋊C₃	C₂²×C₇	1	1	yes	characteristic
40	GL(2,𝔽₃)	—	2	2	no	—
41	D₂₄	—	4	1	no	—
42	C₂×C₃×C₇	—	4	1	no	—
43	D₃×C₇	—	16	2	no	—
44	AGL(1,ℤ/8)	—	3	2	no	—
45	C₂×D₈	—	3	1	no	—
46	#131	D₃×C₇	1	1	yes	characteristic, complemented
47	#108	—	3	1	no	—
48	#100	—	3	2	no	—
49	C₂²×C₇	—	3	1	no	—
50	C₂²×C₇	—	12	2	no	—
51	C₂²×C₇	—	24	2	no	—
52	C₄×C₇	—	3	1	no	—
53	C₄×C₇	C₂×S₄	1	1	yes	characteristic
54	D₁₂	—	4	2	no	—
55	SL(2,𝔽₃)	C₇×D₄	1	1	yes	characteristic
56	C₃×C₈	—	4	1	no	—
57	C₃×C₇	—	4	1	no	Hall
58	#41	C₂×D₃×C₇	1	1	yes	characteristic
59	C₂×D₄	—	3	2	no	—
60	SD₁₆	—	6	2	no	—
61	D₈	—	6	1	no	—
62	D₈	—	6	2	no	—
63	D₈	—	6	1	no	—
64	M₁₆	—	3	1	no	—
65	M₁₆	—	3	2	no	—
66	C₂×C₈	—	3	1	no	—
67	C₂×C₇	—	6	1	no	—
68	C₂×C₇	—	24	2	no	—
69	C₂×C₇	C₄⋊S₄	1	1	yes	characteristic
70	C₂×D₃	—	8	2	no	—
71	C₃×C₄	—	4	1	no	—
72	C₂³	—	6	2	no	—
73	Q₈	C₇×D₁₂	1	1	yes	characteristic
74	D₄	—	3	1	no	—
75	D₄	—	6	2	no	—
76	D₄	—	6	2	no	—
77	C₂×C₄	—	3	1	no	—
78	C₈	—	3	1	no	—
79	C₈	—	6	2	no	—
80	C₈	C₇×S₄	1	1	yes	characteristic
81	C₇	#5407	1	1	yes	characteristic, direct product factor, Sylow
82	C₂×C₃	—	4	1	no	—
83	D₃	—	16	2	no	—
84	C₂²	—	3	1	no	—
85	C₂²	—	12	2	no	—
86	C₂²	—	24	2	no	—
87	C₄	—	3	1	no	—
88	C₄	C₂×C₇×S₄	1	1	yes	characteristic
89	C₃	—	4	1	no	Sylow
90	C₂	—	6	1	no	—
91	C₂	—	24	2	no	—
92	C₂	C₇×C₄⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	C₄⋊S₄
commutator, cosocle	#41⋊C₃	C₂²×C₇
Frattini	C₄	C₂×C₇×S₄
Fitting	C₇×#131	D₃
hypercenter	C₇×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₇×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈

Complements

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

N	H ≅ G/N	type
1. C₇×#986	C₂	semidirect
2. C₇×#131⋊C₃	C₂	semidirect
9. C₇×#131	D₃	semidirect
12. #5407	C₇	direct (N × H)
25. #986	C₂×C₇	semidirect
26. #131⋊C₃	C₂×C₇	semidirect
46. #131	D₃×C₇	semidirect
81. C₇	#5407	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×#986	2
C₇×#131⋊C₃	2
C₇×#472	3
C₇×D₂₄	4
#5407	7

Subgroups — C7×#5407 #13057127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5407 #13057127