Subgroups — Q₈×ΓL(1,𝔽₈)⋊C₈ #13051923

Order 1344.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(ΓL(1,𝔽₈)⋊C₈)	C₂	1	3	yes	maximal
3	C₄×ΓL(1,𝔽₈)⋊C₈	C₂	1	3	yes	maximal
4	Q₈×C₇⋊C₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×Q₈×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
6	C₄⋊C₄×ΓL(1,𝔽₈)	C₄	1	3	yes	—
7	C₄²×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
8	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	3	yes	—
9	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic
11	C₄⋊(C₇⋊C₈)	C₂×C₃	1	3	yes	—
12	C₄×C₇⋊C₈	C₂×C₃	1	3	yes	—
13	C₃×C₈×Q₈	—	7	1	no	maximal, Hall
14	Q₈×ΓL(1,𝔽₈)	C₈	1	4	yes	complemented
15	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	3	yes	—
16	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	3	yes	—
17	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
18	ΓL(1,𝔽₈)⋊C₈	—	2	3	no	—
19	ΓL(1,𝔽₈)⋊C₈	Q₈	1	2	yes	direct product factor
20	C₂×C₇×Q₈	C₃×C₄	1	1	yes	characteristic
21	C₇×C₄⋊C₄	C₃×C₄	1	3	yes	—
22	C₄²×C₇	C₂²×C₃	1	3	yes	—
23	C₂×C₇⋊C₈	C₂²×C₃	1	3	yes	—
24	C₂×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
25	C₃×C₄×Q₈	—	7	1	no	—
26	C₃×C₄⋊C₈	—	7	3	no	—
27	C₃×C₄×C₈	—	7	3	no	—
28	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
29	C₄×ΓL(1,𝔽₈)	—	2	3	no	—
30	C₄×ΓL(1,𝔽₈)	C₂×C₈	1	6	yes	—
31	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	1	yes	characteristic
32	C₄×ΓL(1,𝔽₈)	#41	1	1	yes	characteristic
33	C₈×Q₈	—	7	1	no	Sylow
34	C₇×Q₈	C₃×C₈	1	4	yes	complemented
35	C₂×C₄×C₇	C₂×C₃×C₄	1	3	yes	—
36	C₂×C₄×C₇	C₂×C₃×C₄	1	3	yes	—
37	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
38	C₇⋊C₈	—	2	3	no	—
39	C₇⋊C₈	C₃×Q₈	1	2	yes	complemented
40	C₂×C₃×Q₈	—	7	1	no	—
41	C₂×C₃×C₈	—	7	1	no	—
42	C₂×C₃×C₈	—	7	3	no	—
43	C₃×C₄⋊C₄	—	7	3	no	—
44	C₃×C₄²	—	7	3	no	—
45	C₂×ΓL(1,𝔽₈)	C₄×Q₈	1	1	yes	characteristic
46	C₂×ΓL(1,𝔽₈)	C₂²×C₈	1	1	yes	characteristic
47	C₂×ΓL(1,𝔽₈)	#131	1	1	yes	characteristic
48	C₄×Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic
49	C₄⋊C₈	—	7	3	no	—
50	C₄×C₈	—	7	3	no	—
51	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
52	C₄×C₇	—	2	3	no	—
53	C₄×C₇	C₂×C₃×C₈	1	6	yes	—
54	C₄×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
55	C₄×C₇	C₃×#41	1	1	yes	characteristic
56	C₃×Q₈	—	7	4	no	—
57	C₂×C₃×C₄	—	7	3	no	—
58	C₂×C₃×C₄	—	7	3	no	—
59	C₂×C₃×C₄	—	7	1	no	—
60	C₃×C₈	—	7	2	no	—
61	C₃×C₈	—	14	3	no	—
62	ΓL(1,𝔽₈)	C₈×Q₈	1	1	yes	characteristic, complemented, Hall
63	C₂×Q₈	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
64	C₂×C₈	—	7	1	no	—
65	C₂×C₈	—	7	3	no	—
66	C₄⋊C₄	ΓL(1,𝔽₈)⋊C₄	1	3	yes	—
67	C₄²	C₂×AGL(1,𝔽₇)	1	3	yes	—
68	C₂×C₇	C₃×C₄×Q₈	1	1	yes	characteristic
69	C₂×C₇	C₂²×C₃×C₈	1	1	yes	characteristic
70	C₂×C₇	C₃×#131	1	1	yes	characteristic
71	C₂²×C₃	—	7	1	no	—
72	C₃×C₄	—	7	1	no	—
73	C₃×C₄	—	7	1	no	—
74	C₃×C₄	—	7	6	no	—
75	C₃×C₄	—	14	3	no	—
76	Q₈	ΓL(1,𝔽₈)⋊C₈	1	4	yes	direct product factor
77	C₂×C₄	C₂×ΓL(1,𝔽₈)⋊C₄	1	3	yes	—
78	C₂×C₄	C₂×ΓL(1,𝔽₈)⋊C₄	1	3	yes	—
79	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
80	C₈	—	7	2	no	—
81	C₈	—	14	3	no	—
82	C₇	C₃×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	—	7	1	no	—
84	C₂×C₃	—	7	1	no	—
85	C₂×C₃	—	7	1	no	—
86	C₂²	C₂²×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
87	C₄	—	2	3	no	—
88	C₄	C₂×ΓL(1,𝔽₈)⋊C₈	1	6	yes	—
89	C₄	Q₈×AGL(1,𝔽₇)	1	1	yes	characteristic
90	C₄	#66659	1	1	yes	characteristic
91	C₃	—	7	1	no	Sylow
92	C₂	C₂²×ΓL(1,𝔽₈)⋊C₈	1	1	yes	characteristic
93	C₂	Q₈×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
94	C₂	#10620784	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×AGL(1,𝔽₇)
commutator	C₂×C₇	C₂²×C₃×C₈
Fitting	C₄×C₇×Q₈	C₂×C₃
socle	C₂²×C₇	C₂²×C₃×C₄
prime radical	C₂²×ΓL(1,𝔽₈)	C₂²×C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₄×Q₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×C₈×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
4. Q₈×C₇⋊C₈	C₃	semidirect
14. Q₈×ΓL(1,𝔽₈)	C₈	semidirect
19. ΓL(1,𝔽₈)⋊C₈	Q₈	direct (N × H)
34. C₇×Q₈	C₃×C₈	semidirect
39. C₇⋊C₈	C₃×Q₈	semidirect
62. ΓL(1,𝔽₈)	C₈×Q₈	semidirect
76. Q₈	ΓL(1,𝔽₈)⋊C₈	direct (N × H)
82. C₇	C₃×C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×ΓL(1,𝔽₈)	2
C₄⋊(ΓL(1,𝔽₈)⋊C₈)	2
C₄×ΓL(1,𝔽₈)⋊C₈	2
Q₈×C₇⋊C₈	3
C₃×C₈×Q₈	7

Subgroups — Q8×ΓL(1,𝔽8)⋊C8 #13051923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×ΓL(1,𝔽₈)⋊C₈ #13051923