Subgroups — #6383⋊C₃ #10620437

Order 672.

100 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	C₄⋊C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×ΓL(1,𝔽₈)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6383	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂²×AGL(1,𝔽₇)	—	2	1	no	—
6	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
7	C₂×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
8	C₄⋊AGL(1,𝔽₇)	—	2	1	no	—
9	C₄⋊AGL(1,𝔽₇)	C₄	1	2	yes	complemented
10	ΓL(1,𝔽₈)⋊C₈	—	2	1	no	—
11	C₂×D₂₈	C₂×C₃	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
14	C₃×#102	—	7	1	no	maximal, Hall
15	C₂²×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
16	C₂×AGL(1,𝔽₇)	—	2	2	no	—
17	C₂×AGL(1,𝔽₇)	—	4	1	no	—
18	C₂×AGL(1,𝔽₇)	—	4	1	no	—
19	C₄×ΓL(1,𝔽₈)	—	4	1	no	—
20	C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
21	C₄×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
22	C₂²×D₇	—	2	1	no	—
23	C₂×C₄×C₇	—	2	1	no	—
24	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
25	D₂₈	—	2	1	no	—
26	D₂₈	C₃×C₄	1	2	yes	complemented
27	C₇⋊C₈	—	2	1	no	—
28	C₂×C₃×D₄	—	7	1	no	—
29	C₂×C₃×C₈	—	7	1	no	—
30	C₃×C₄⋊C₄	—	7	1	no	—
31	C₂×ΓL(1,𝔽₈)	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×ΓL(1,𝔽₈)	D₈	1	1	yes	characteristic
33	C₂×ΓL(1,𝔽₈)	SD₁₆	1	1	yes	characteristic
34	AGL(1,𝔽₇)	—	4	2	no	—
35	#102	—	7	1	no	Sylow
36	C₂²×C₇	C₃×D₄	1	1	yes	characteristic
37	C₂×D₇	—	2	2	no	—
38	C₂×D₇	—	4	1	no	—
39	C₂×D₇	—	4	1	no	—
40	C₄×C₇	—	4	1	no	—
41	C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
42	C₄×C₇	C₃×D₄	1	1	yes	characteristic
43	C₂³×C₃	—	14	1	no	—
44	C₃×D₄	—	7	2	no	—
45	C₃×D₄	—	14	1	no	—
46	C₂×C₃×C₄	—	7	1	no	—
47	C₂×C₃×C₄	—	14	1	no	—
48	C₃×C₈	—	14	1	no	—
49	ΓL(1,𝔽₈)	#102	1	1	yes	characteristic, complemented, Hall
50	C₂×D₄	—	7	1	no	—
51	C₂×C₈	—	7	1	no	—
52	C₄⋊C₄	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
53	C₂×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₇	C₃×D₈	1	1	yes	characteristic
55	C₂×C₇	C₃×SD₁₆	1	1	yes	characteristic
56	D₇	—	4	2	no	—
57	C₂²×C₃	—	7	1	no	—
58	C₂²×C₃	—	14	2	no	—
59	C₂²×C₃	—	28	1	no	—
60	C₂²×C₃	—	28	1	no	—
61	C₃×C₄	—	7	1	no	—
62	C₃×C₄	—	7	1	no	—
63	C₃×C₄	—	28	1	no	—
64	C₂³	—	14	1	no	—
65	D₄	—	7	2	no	—
66	D₄	—	14	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
69	C₈	—	14	1	no	—
70	C₇	C₃×#102	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	7	1	no	—
72	C₂×C₃	—	7	1	no	—
73	C₂×C₃	—	7	1	no	—
74	C₂×C₃	—	28	2	no	—
75	C₂²	—	14	2	no	—
76	C₂²	—	28	1	no	—
77	C₂²	—	28	1	no	—
78	C₂²	(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic
79	C₄	—	4	1	no	—
80	C₄	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
81	C₄	C₄⋊AGL(1,𝔽₇)	1	1	yes	characteristic
82	C₃	—	7	1	no	Sylow
83	C₂	—	28	2	no	—
84	C₂	(C₂×AGL(1,𝔽₇))⋊C₄	1	1	yes	characteristic
85	C₂	#1141⋊C₃	1	1	yes	characteristic
86	C₂	#1143⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂²⋊D₇)⋊C₃
commutator	C₄×C₇	C₂×C₃×C₄
Frattini	C₂×C₄	C₂×AGL(1,𝔽₇)
Fitting	C₇×C₄⋊C₄	C₂×C₃
socle	C₂²×C₇	C₃×D₄
prime radical	C₂×C₄⋊AGL(1,𝔽₇)	C₂
cosocle	C₂×C₄×C₇	C₂²×C₃
hypercenter	C₄⋊C₄	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#102
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×ΓL(1,𝔽₈)	C₂	semidirect
3. C₂×ΓL(1,𝔽₈)⋊C₈	C₂	semidirect
4. #6383	C₃	semidirect
9. C₄⋊AGL(1,𝔽₇)	C₄	semidirect
12. C₇×C₄⋊C₄	C₂×C₃	semidirect
13. C₂×C₇⋊C₈	C₂×C₃	semidirect
26. D₂₈	C₃×C₄	semidirect
49. ΓL(1,𝔽₈)	#102	semidirect
52. C₄⋊C₄	AGL(1,𝔽₇)	semidirect
70. C₇	C₃×#102	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
C₄⋊C₄×ΓL(1,𝔽₈)	2
C₂×ΓL(1,𝔽₈)⋊C₈	2
#6383	3
C₃×#102	7

Subgroups — #6383⋊C3 #10620437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #6383⋊C₃ #10620437