Subgroups — #13057639

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	Dic₇⋊SL(2,𝔽₃)	C₂	1	2	yes	maximal
3	#90261	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×C₇⋊SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
5	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	—	4	1	no	maximal
6	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic
7	Q₈×Dic₇	C₂×C₃	1	2	yes	—
8	#6453	—	3	2	no	—
9	#6432	—	3	1	no	—
10	C₄×Dic₁₄	—	3	1	no	—
11	#557⋊C₃	—	7	1	no	maximal, Hall
12	C₇⋊SL(2,𝔽₃)	Q₈	1	1	yes	characteristic, complemented
13	C₂×C₄×ΓL(1,𝔽₈)	—	4	1	no	—
14	C₂×ΓL(1,𝔽₈)⋊C₄	—	4	2	no	—
15	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
16	C₂×Dic₁₄	—	3	1	no	—
17	C₇×C₄⋊C₄	—	3	1	no	—
18	C₄²×C₇	—	3	1	no	—
19	C₄⋊Dic₇	—	3	2	no	—
20	C₄⋊Dic₇	—	3	2	no	—
21	C₄⋊Dic₇	A₄	1	1	yes	characteristic
22	Dic₇⋊C₄	—	3	2	no	—
23	C₄×Dic₇	—	3	2	no	—
24	C₄×SL(2,𝔽₃)	—	7	2	no	—
25	C₄×SL(2,𝔽₃)	—	7	1	no	—
26	C₂²×ΓL(1,𝔽₈)	—	4	1	no	—
27	C₄×ΓL(1,𝔽₈)	—	4	2	no	—
28	ΓL(1,𝔽₈)⋊C₄	—	8	2	no	—
29	#557	—	7	1	no	Sylow
30	C₇×Q₈	—	3	1	no	—
31	C₇×Q₈	C₃×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₇	—	3	1	no	—
33	C₂×C₄×C₇	—	3	1	no	—
34	C₂×C₄×C₇	C₂×A₄	1	1	yes	characteristic
35	C₂×Dic₇	—	3	2	no	—
36	C₂×Dic₇	C₂×A₄	1	2	yes	—
37	Dic₁₄	—	6	2	no	—
38	C₂×SL(2,𝔽₃)	—	7	1	no	—
39	C₃×C₄⋊C₄	—	28	1	no	—
40	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
41	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
42	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
43	C₄⋊Q₈	—	21	1	no	—
44	#125	—	21	2	no	—
45	C₄×Q₈	—	7	2	no	—
46	C₄×Q₈	—	21	1	no	—
47	C₄×Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic
48	C₂²×C₇	C₂²×A₄	1	1	yes	characteristic
49	C₄×C₇	—	3	1	no	—
50	C₄×C₇	—	3	1	no	—
51	C₄×C₇	—	6	1	no	—
52	C₄×C₇	#41⋊C₃	1	2	yes	—
53	Dic₇	—	2	2	no	—
54	Dic₇	—	6	2	no	—
55	C₂×C₃×C₄	—	28	2	no	—
56	C₂×C₃×C₄	—	28	1	no	—
57	SL(2,𝔽₃)	—	7	1	no	—
58	ΓL(1,𝔽₈)	—	4	1	no	Hall
59	C₂×Q₈	—	21	1	no	—
60	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
61	C₄⋊C₄	—	3	1	no	—
62	C₄⋊C₄	—	7	1	no	—
63	C₄⋊C₄	—	21	2	no	—
64	C₄⋊C₄	—	21	2	no	—
65	C₄⋊C₄	—	21	2	no	—
66	C₄²	—	3	1	no	—
67	C₄²	—	21	2	no	—
68	C₂×C₇	Q₈×A₄	1	1	yes	characteristic
69	C₂×C₇	C₂×#41⋊C₃	1	1	yes	characteristic
70	C₂×C₇	2₋¹⁺⁴⋊C₃	1	1	yes	characteristic
71	C₂²×C₃	—	28	1	no	—
72	C₃×C₄	—	28	2	no	—
73	C₃×C₄	—	56	2	no	—
74	Q₈	—	3	1	no	—
75	Q₈	—	42	2	no	—
76	Q₈	ΓL(1,𝔽₈)⋊Q₈	1	1	yes	characteristic
77	C₂×C₄	—	3	1	no	—
78	C₂×C₄	—	3	1	no	—
79	C₂×C₄	—	7	2	no	—
80	C₂×C₄	—	21	2	no	—
81	C₂×C₄	D₇⋊A₄	1	1	yes	characteristic
82	C₇	#557⋊C₃	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	—	28	1	no	—
84	C₂×C₃	—	28	1	no	—
85	C₂×C₃	—	28	1	no	—
86	C₂²	C₂×D₇⋊A₄	1	1	yes	characteristic
87	C₄	—	3	1	no	—
88	C₄	—	3	1	no	—
89	C₄	—	6	1	no	—
90	C₄	—	14	2	no	—
91	C₄	—	42	2	no	—
92	C₄	#66665	1	2	yes	—
93	C₃	—	28	1	no	Sylow
94	C₂	Dic₁₄⋊A₄	1	1	yes	characteristic
95	C₂	C₂×#66665	1	1	yes	characteristic
96	C₂	#10621541	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₇⋊A₄
commutator, cosocle	C₂×C₇×Q₈	C₂²×C₃
Fitting	C₄×C₇×Q₈	C₂×C₃
socle	C₂²×C₇	C₂²×A₄
prime radical	C₂×C₇⋊SL(2,𝔽₃)	C₂²
hypercenter	C₂×C₄	D₇⋊A₄
hypocenter	C₇×Q₈	C₃×Q₈
LCS	G ▷ C₂×C₇×Q₈ ▷ C₇×Q₈
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. #90261	C₃	semidirect
12. C₇⋊SL(2,𝔽₃)	Q₈	semidirect
82. C₇	#557⋊C₃	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
Dic₇⋊SL(2,𝔽₃)	2
#90261	3
C₄⋊(ΓL(1,𝔽₈)⋊C₄)	4
#557⋊C₃	7