Subgroups — #6435⋊C₃ #10620708

Order 672.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic
2	C₄²×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×AGL(1,𝔽₇))⋊C₄	C₂	1	2	yes	maximal
4	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#6435	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	C₂²×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
8	C₂×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
9	C₂×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
10	C₂×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
11	C₂×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
12	C₄×AGL(1,𝔽₇)	C₄	1	4	yes	complemented
13	C₂×C₄×D₇	C₂×C₃	1	1	yes	characteristic
14	C₄²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
15	(C₂×D₇)⋊C₄	C₂×C₃	1	2	yes	—
16	Dic₇⋊C₄	C₂×C₃	1	2	yes	complemented
17	C₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
18	C₃×#117	—	7	1	no	maximal, Hall
19	C₂²×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
20	C₂×AGL(1,𝔽₇)	—	2	2	no	—
21	C₂×AGL(1,𝔽₇)	C₂×C₄	1	2	yes	—
22	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
23	C₄×ΓL(1,𝔽₈)	C₂×C₄	1	2	yes	—
24	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
25	ΓL(1,𝔽₈)⋊C₄	C₂×C₄	1	2	yes	—
26	C₂²×D₇	C₂²×C₃	1	1	yes	characteristic
27	C₂×C₄×C₇	C₂²×C₃	1	2	yes	—
28	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
29	C₂×Dic₇	C₂²×C₃	1	2	yes	—
30	C₂×Dic₇	C₂²×C₃	1	1	yes	characteristic
31	C₄×D₇	C₃×C₄	1	4	yes	complemented
32	C₂²×C₃×C₄	—	7	1	no	—
33	C₃×C₄⋊C₄	—	7	2	no	—
34	C₃×C₂²⋊C₄	—	7	2	no	—
35	C₃×C₄²	—	7	1	no	—
36	C₃×C₄²	—	7	1	no	—
37	C₂×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
38	C₂×ΓL(1,𝔽₈)	#41	1	2	yes	—
39	AGL(1,𝔽₇)	—	2	2	no	—
40	#117	—	7	1	no	Sylow
41	C₂²×C₇	C₂³×C₃	1	1	yes	characteristic
42	C₂×D₇	—	2	2	no	—
43	C₂×D₇	C₂×C₃×C₄	1	2	yes	—
44	C₄×C₇	—	2	2	no	—
45	C₄×C₇	C₂×C₃×C₄	1	2	yes	—
46	Dic₇	—	2	2	no	—
47	Dic₇	C₂×C₃×C₄	1	2	yes	—
48	C₂³×C₃	—	7	1	no	—
49	C₂×C₃×C₄	—	7	4	no	—
50	C₂×C₃×C₄	—	7	1	no	—
51	C₂×C₃×C₄	—	7	2	no	—
52	C₂×C₃×C₄	—	7	2	no	—
53	C₂×C₃×C₄	—	7	1	no	—
54	ΓL(1,𝔽₈)	#117	1	1	yes	characteristic, complemented, Hall
55	C₂²×C₄	—	7	1	no	—
56	C₄⋊C₄	—	7	2	no	—
57	C₂²⋊C₄	—	7	2	no	—
58	C₄²	—	7	1	no	—
59	C₄²	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
60	C₂×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
61	C₂×C₇	C₃×#41	1	2	yes	—
62	D₇	—	2	2	no	—
63	C₂²×C₃	—	7	2	no	—
64	C₂²×C₃	—	7	1	no	—
65	C₂²×C₃	—	14	2	no	—
66	C₃×C₄	—	7	2	no	—
67	C₃×C₄	—	7	2	no	—
68	C₃×C₄	—	14	2	no	—
69	C₃×C₄	—	14	2	no	—
70	C₂³	—	7	1	no	—
71	C₂×C₄	—	7	1	no	—
72	C₂×C₄	—	7	4	no	—
73	C₂×C₄	—	7	2	no	—
74	C₂×C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
75	C₂×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
76	C₇	C₃×#117	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₃	—	7	1	no	—
78	C₂×C₃	—	7	2	no	—
79	C₂×C₃	—	14	2	no	—
80	C₂²	—	7	2	no	—
81	C₂²	—	14	2	no	—
82	C₂²	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	—	7	2	no	—
85	C₄	—	14	2	no	—
86	C₄	C₄×AGL(1,𝔽₇)	1	2	yes	—
87	C₃	—	7	1	no	Sylow
88	C₂	—	14	2	no	—
89	C₂	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
90	C₂	#1157⋊C₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×AGL(1,𝔽₇)
commutator	C₂×C₇	C₂²×C₃×C₄
Frattini	C₂²	C₂²×AGL(1,𝔽₇)
Fitting	C₄²×C₇	C₂×C₃
socle, cosocle	C₂²×C₇	C₂³×C₃
prime radical	C₂²×AGL(1,𝔽₇)	C₂²
hypercenter	C₄²	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#117
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄²×ΓL(1,𝔽₈)	C₂	semidirect
4. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂	semidirect
5. C₄×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
6. #6435	C₃	semidirect
12. C₄×AGL(1,𝔽₇)	C₄	semidirect
14. C₄²×C₇	C₂×C₃	semidirect
16. Dic₇⋊C₄	C₂×C₃	semidirect
17. C₄×Dic₇	C₂×C₃	semidirect
31. C₄×D₇	C₃×C₄	semidirect
54. ΓL(1,𝔽₈)	#117	semidirect
59. C₄²	AGL(1,𝔽₇)	semidirect
76. C₇	C₃×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×AGL(1,𝔽₇)	2
C₄²×ΓL(1,𝔽₈)	2
(C₂×AGL(1,𝔽₇))⋊C₄	2
(ΓL(1,𝔽₈)⋊C₄)⋊C₄	2
C₄×ΓL(1,𝔽₈)⋊C₄	2
#6435	3
C₃×#117	7

Subgroups — #6435⋊C3 #10620708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #6435⋊C₃ #10620708