Subgroups — #13057159

Order 1344.

118 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	#89793	—	3	1	no	maximal, Hall
4	C₂×C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
5	C₄⋊Dic₂₁	—	4	1	no	maximal
6	C₄×C₇×Q₈	D₃	1	1	yes	characteristic
7	#6432	—	3	1	no	—
8	#6409	—	3	2	no	—
9	#6381	—	3	2	no	—
10	C₄⋊(C₇⋊C₈)	—	3	1	no	—
11	#5391	—	7	1	no	maximal, Hall
12	C₂×C₃×C₄×C₇	—	4	1	no	—
13	C₂×Dic₂₁	—	4	2	no	—
14	C₇×SL(2,𝔽₃)	Q₈	1	1	yes	characteristic, complemented
15	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
16	C₂×Dic₁₄	—	6	1	no	—
17	C₇×C₄⋊C₄	—	3	1	no	—
18	C₄²×C₇	—	3	1	no	—
19	C₄⋊Dic₇	—	3	2	no	—
20	C₄⋊Dic₇	—	6	1	no	—
21	C₂×C₇⋊C₈	—	3	2	no	—
22	C₄×SL(2,𝔽₃)	D₇	1	1	yes	characteristic
23	#859	—	7	2	no	—
24	C₂²×C₃×C₇	—	4	1	no	—
25	C₃×C₄×C₇	—	4	2	no	—
26	Dic₂₁	—	8	2	no	—
27	#477	—	21	1	no	Sylow
28	C₇×Q₈	—	3	1	no	—
29	C₇×Q₈	Dic₆	1	1	yes	characteristic
30	C₂×C₄×C₇	—	3	1	no	—
31	C₂×C₄×C₇	—	3	1	no	—
32	C₂×C₄×C₇	S₄	1	1	yes	characteristic
33	C₂×Dic₇	—	6	2	no	—
34	Dic₁₄	—	12	2	no	—
35	C₇⋊C₈	—	6	2	no	—
36	C₂×SL(2,𝔽₃)	C₂×D₇	1	1	yes	characteristic
37	C₄⋊Dic₃	—	28	1	no	—
38	C₂×C₃×C₇	—	4	1	no	—
39	C₂×C₃×C₇	—	4	1	no	—
40	C₂×C₃×C₇	—	4	1	no	—
41	C₄⋊Q₈	—	21	1	no	—
42	C₄×Q₈	D₂₁	1	1	yes	characteristic
43	#107	—	21	2	no	—
44	C₄⋊C₈	—	21	1	no	—
45	#103	—	21	2	no	—
46	C₂²×C₇	C₂×S₄	1	1	yes	characteristic
47	C₄×C₇	—	3	1	no	—
48	C₄×C₇	—	3	1	no	—
49	C₄×C₇	—	6	1	no	—
50	C₄×C₇	#226	1	2	yes	—
51	Dic₇	—	12	2	no	—
52	C₂×C₃×C₄	—	4	1	no	—
53	C₂×Dic₃	—	28	2	no	—
54	SL(2,𝔽₃)	Dic₁₄	1	1	yes	characteristic, complemented
55	C₃×C₇	—	4	1	no	Hall
56	C₂×Q₈	—	42	1	no	—
57	C₂×Q₈	C₂×D₂₁	1	1	yes	characteristic
58	C₂×C₈	—	21	2	no	—
59	C₄⋊C₄	—	3	1	no	—
60	C₄⋊C₄	—	21	2	no	—
61	C₄⋊C₄	—	42	1	no	—
62	C₄²	—	3	1	no	—
63	C₂×C₇	A₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₇	C₂×#226	1	1	yes	characteristic
65	C₂×C₇	#986	1	1	yes	characteristic
66	C₂²×C₃	—	4	1	no	—
67	C₃×C₄	—	4	2	no	—
68	Dic₃	—	56	2	no	—
69	Q₈	—	3	1	no	—
70	Q₈	—	84	2	no	—
71	Q₈	Dic₄₂	1	1	yes	characteristic
72	C₂×C₄	—	3	1	no	—
73	C₂×C₄	—	3	1	no	—
74	C₂×C₄	—	42	2	no	—
75	C₂×C₄	C₇⋊S₄	1	1	yes	characteristic
76	C₈	—	42	2	no	—
77	C₇	#5391	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₃	—	4	1	no	—
79	C₂×C₃	—	4	1	no	—
80	C₂×C₃	—	4	1	no	—
81	C₂²	C₂×C₇⋊S₄	1	1	yes	characteristic
82	C₄	—	3	1	no	—
83	C₄	—	3	1	no	—
84	C₄	—	6	1	no	—
85	C₄	—	84	2	no	—
86	C₄	C₇⋊#226	1	2	yes	—
87	C₃	—	4	1	no	Sylow
88	C₂	#10621496	1	1	yes	characteristic
89	C₂	C₂×C₇⋊#226	1	1	yes	characteristic
90	C₂	#10621504	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
14. C₇×SL(2,𝔽₃)	Q₈	semidirect
54. SL(2,𝔽₃)	Dic₁₄	semidirect
77. C₇	#5391	semidirect

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
#89793	3
C₄⋊Dic₂₁	4
#5391	7