Subgroups — C₇×#5403 #13057123

Order 1344.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#984	C₂	1	2	yes	maximal
2	C₇×#131⋊C₃	C₂	1	1	yes	maximal, characteristic
3	C₇×#473	—	3	1	no	maximal, Hall
4	C₇×#41⋊C₃	C₂²	1	1	yes	characteristic
5	C₇×#226	—	2	2	no	—
6	C₇×Dic₁₂	—	4	1	no	maximal
7	C₇×#137	—	3	2	no	—
8	C₂×C₇×Q₁₆	—	3	1	no	—
9	C₇×#131	D₃	1	1	yes	characteristic
10	C₇×#108	—	3	1	no	—
11	C₇×#101	—	3	2	no	—
12	#5403	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₇×Dic₆	—	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×Q₈	—	3	2	no	—
18	C₇×Q₁₆	—	6	1	no	—
19	C₇×Q₁₆	—	6	2	no	—
20	C₇×Q₁₆	—	6	1	no	—
21	C₇×SD₁₆	—	6	2	no	—
22	C₇×M₁₆	—	3	1	no	—
23	C₇×M₁₆	—	3	2	no	—
24	C₂×C₇×C₈	—	3	1	no	—
25	#984	C₂×C₇	1	2	yes	—
26	#131⋊C₃	C₂×C₇	1	1	yes	characteristic
27	C₃×C₄×C₇	—	4	1	no	—
28	C₇×Dic₃	—	8	2	no	—
29	#473	—	3	1	no	Sylow
30	C₇×Q₈	—	6	2	no	—
31	C₇×Q₈	—	6	2	no	—
32	C₇×Q₈	D₁₂	1	1	yes	characteristic
33	C₇×D₄	—	3	1	no	—
34	C₂×C₄×C₇	—	3	1	no	—
35	C₂×C₄×C₇	—	6	2	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	#226	—	2	2	no	—
41	Dic₁₂	—	4	1	no	—
42	C₂×C₃×C₇	—	4	1	no	—
43	#137	—	3	2	no	—
44	C₂×Q₁₆	—	3	1	no	—
45	#131	D₃×C₇	1	1	yes	characteristic
46	#108	—	3	1	no	—
47	#101	—	3	2	no	—
48	C₂²×C₇	—	3	1	no	—
49	C₄×C₇	—	3	1	no	—
50	C₄×C₇	—	12	2	no	—
51	C₄×C₇	C₂×S₄	1	1	yes	characteristic
52	Dic₆	—	4	2	no	—
53	SL(2,𝔽₃)	C₇×D₄	1	1	yes	characteristic
54	C₃×C₈	—	4	1	no	—
55	C₃×C₇	—	4	1	no	Hall
56	#41	C₂×D₃×C₇	1	1	yes	characteristic
57	C₂×Q₈	—	3	2	no	—
58	Q₁₆	—	6	1	no	—
59	Q₁₆	—	6	2	no	—
60	Q₁₆	—	6	1	no	—
61	SD₁₆	—	6	2	no	—
62	M₁₆	—	3	1	no	—
63	M₁₆	—	3	2	no	—
64	C₂×C₈	—	3	1	no	—
65	C₂×C₇	—	6	1	no	—
66	C₂×C₇	C₄⋊S₄	1	1	yes	characteristic
67	C₃×C₄	—	4	1	no	—
68	Dic₃	—	8	2	no	—
69	Q₈	—	6	2	no	—
70	Q₈	—	6	2	no	—
71	Q₈	C₇×D₁₂	1	1	yes	characteristic
72	D₄	—	3	1	no	—
73	C₂×C₄	—	3	1	no	—
74	C₂×C₄	—	6	2	no	—
75	C₈	—	3	1	no	—
76	C₈	—	6	2	no	—
77	C₈	C₇×S₄	1	1	yes	characteristic
78	C₇	#5403	1	1	yes	characteristic, direct product factor, Sylow
79	C₂×C₃	—	4	1	no	—
80	C₂²	—	3	1	no	—
81	C₄	—	3	1	no	—
82	C₄	—	12	2	no	—
83	C₄	C₂×C₇×S₄	1	1	yes	characteristic
84	C₃	—	4	1	no	Sylow
85	C₂	—	6	1	no	—
86	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
prime radical	C₇×#41⋊C₃	C₂²
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
12. #5403	C₇	direct (N × H)
78. C₇	#5403	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×#984	2
C₇×#131⋊C₃	2
C₇×#473	3
C₇×Dic₁₂	4
#5403	7

Subgroups — C7×#5403 #13057123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5403 #13057123