Subgroups — C₂²×ΓL(1,𝔽₈)⋊C₁₆ #13051996

Order 1344.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×ΓL(1,𝔽₈)⋊C₁₆	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₇⋊C₁₆	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₄×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
5	C₂×C₈×ΓL(1,𝔽₈)	C₄	1	3	yes	—
6	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
7	ΓL(1,𝔽₈)⋊C₁₆	C₂²	1	4	yes	direct product factor
8	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
9	C₂×C₇⋊C₁₆	C₂×C₃	1	6	yes	complemented
10	C₂²×C₃×C₁₆	—	7	1	no	maximal, Hall
11	C₂³×ΓL(1,𝔽₈)	C₈	1	1	yes	characteristic
12	C₂×C₄×ΓL(1,𝔽₈)	C₈	1	3	yes	—
13	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	3	yes	—
14	C₈×ΓL(1,𝔽₈)	C₂×C₄	1	3	yes	—
15	C₈×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
16	C₂²×C₄×C₇	C₃×C₄	1	1	yes	characteristic
17	C₂×C₇×C₈	C₃×C₄	1	3	yes	—
18	C₂×C₇×C₈	C₂²×C₃	1	3	yes	—
19	C₇⋊C₁₆	C₂²×C₃	1	4	yes	complemented
20	C₂²×C₃×C₈	—	7	1	no	—
21	C₂×C₃×C₁₆	—	7	6	no	—
22	C₂²×ΓL(1,𝔽₈)	C₁₆	1	4	yes	complemented
23	C₂²×ΓL(1,𝔽₈)	C₂×C₈	1	3	yes	—
24	C₄×ΓL(1,𝔽₈)	C₂×C₈	1	3	yes	—
25	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₁₆	—	7	1	no	Sylow
27	C₂³×C₇	C₃×C₈	1	1	yes	characteristic
28	C₂×C₄×C₇	C₃×C₈	1	3	yes	—
29	C₂×C₄×C₇	C₂×C₃×C₄	1	3	yes	—
30	C₇×C₈	C₂×C₃×C₄	1	3	yes	—
31	C₇×C₈	C₂³×C₃	1	1	yes	characteristic
32	C₂²×C₃×C₄	—	7	1	no	—
33	C₂×C₃×C₈	—	7	3	no	—
34	C₂×C₃×C₈	—	7	3	no	—
35	C₃×C₁₆	—	7	4	no	—
36	C₂×ΓL(1,𝔽₈)	C₂×C₁₆	1	6	yes	complemented
37	C₂×ΓL(1,𝔽₈)	C₂²×C₈	1	1	yes	characteristic
38	C₂²×C₈	AGL(1,𝔽₇)	1	1	yes	characteristic
39	C₂×C₁₆	—	7	6	no	—
40	C₂²×C₇	C₃×C₁₆	1	4	yes	complemented
41	C₂²×C₇	C₂×C₃×C₈	1	3	yes	—
42	C₄×C₇	C₂×C₃×C₈	1	3	yes	—
43	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
44	C₂³×C₃	—	7	1	no	—
45	C₂×C₃×C₄	—	7	3	no	—
46	C₂×C₃×C₄	—	7	3	no	—
47	C₃×C₈	—	7	3	no	—
48	C₃×C₈	—	7	1	no	—
49	ΓL(1,𝔽₈)	C₂²×C₁₆	1	1	yes	characteristic, complemented, Hall
50	C₂²×C₄	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
51	C₂×C₈	ΓL(1,𝔽₈)⋊C₄	1	3	yes	—
52	C₂×C₈	C₂×AGL(1,𝔽₇)	1	3	yes	—
53	C₁₆	—	7	4	no	—
54	C₂×C₇	C₂×C₃×C₁₆	1	6	yes	complemented
55	C₂×C₇	C₂²×C₃×C₈	1	1	yes	characteristic
56	C₂²×C₃	—	7	4	no	—
57	C₂²×C₃	—	7	3	no	—
58	C₃×C₄	—	7	1	no	—
59	C₃×C₄	—	7	3	no	—
60	C₂³	ΓL(1,𝔽₈)⋊C₈	1	1	yes	characteristic
61	C₂×C₄	ΓL(1,𝔽₈)⋊C₈	1	3	yes	—
62	C₂×C₄	C₂×ΓL(1,𝔽₈)⋊C₄	1	3	yes	—
63	C₈	C₂×ΓL(1,𝔽₈)⋊C₄	1	3	yes	—
64	C₈	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
65	C₇	C₂²×C₃×C₁₆	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	7	6	no	—
67	C₂×C₃	—	7	1	no	—
68	C₂²	ΓL(1,𝔽₈)⋊C₁₆	1	4	yes	direct product factor
69	C₂²	C₂×ΓL(1,𝔽₈)⋊C₈	1	3	yes	—
70	C₄	C₂×ΓL(1,𝔽₈)⋊C₈	1	3	yes	—
71	C₄	C₂²×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
72	C₃	—	7	1	no	Sylow
73	C₂	C₂×ΓL(1,𝔽₈)⋊C₁₆	1	6	yes	direct product factor
74	C₂	C₂²×ΓL(1,𝔽₈)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂²×C₃×C₁₆
Frattini	C₈	C₂²×AGL(1,𝔽₇)
Fitting	C₂²×C₇×C₈	C₂×C₃
socle	C₂³×C₇	C₃×C₈
prime radical	C₂³×ΓL(1,𝔽₈)	C₈
cosocle	C₇×C₈	C₂³×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×ΓL(1,𝔽₈)⋊C₁₆	C₂	direct (N × H)
3. C₂²×C₇⋊C₁₆	C₃	semidirect
7. ΓL(1,𝔽₈)⋊C₁₆	C₂²	direct (N × H)
9. C₂×C₇⋊C₁₆	C₂×C₃	semidirect
19. C₇⋊C₁₆	C₂²×C₃	semidirect
22. C₂²×ΓL(1,𝔽₈)	C₁₆	semidirect
36. C₂×ΓL(1,𝔽₈)	C₂×C₁₆	semidirect
40. C₂²×C₇	C₃×C₁₆	semidirect
49. ΓL(1,𝔽₈)	C₂²×C₁₆	semidirect
54. C₂×C₇	C₂×C₃×C₁₆	semidirect
65. C₇	C₂²×C₃×C₁₆	semidirect
68. C₂²	ΓL(1,𝔽₈)⋊C₁₆	direct (N × H)
73. C₂	C₂×ΓL(1,𝔽₈)⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₈×ΓL(1,𝔽₈)	2
C₂×ΓL(1,𝔽₈)⋊C₁₆	2
C₂²×C₇⋊C₁₆	3
C₂²×C₃×C₁₆	7

Subgroups — C22×ΓL(1,𝔽8)⋊C16 #13051996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×ΓL(1,𝔽₈)⋊C₁₆ #13051996