Subgroups — C₄²×C₂²⋊ΓL(1,𝔽₈) #13060769

Order 1344.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂²⋊ΓL(1,𝔽₈)	C₂	1	3	yes	maximal
2	C₂²×C₄²×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²×C₂²⋊ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
4	C₄×C₂²⋊ΓL(1,𝔽₈)	C₄	1	6	yes	direct product factor
5	C₄²×ΓL(1,𝔽₈)	—	4	1	no	maximal
6	C₂³×C₄×C₇	C₂×C₃	1	3	yes	—
7	C₂×C₄²×C₇	—	3	1	no	—
8	C₂×C₄²×C₇	—	3	3	no	—
9	C₄²×A₄	—	7	1	no	maximal, Hall
10	C₂×C₂²⋊ΓL(1,𝔽₈)	C₂×C₄	1	3	yes	—
11	C₂×C₄×ΓL(1,𝔽₈)	—	4	3	no	—
12	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
13	C₂²×C₄×C₇	—	3	3	no	—
14	C₂²×C₄×C₇	—	3	3	no	—
15	C₂²×C₄×C₇	—	3	6	no	—
16	C₂²×C₄×C₇	C₃×C₄	1	6	yes	complemented
17	C₄²×C₇	—	3	2	no	—
18	C₄²×C₇	—	3	3	no	—
19	C₄²×C₇	A₄	1	1	yes	characteristic, complemented
20	C₂×C₄×A₄	—	7	3	no	—
21	C₂²⋊ΓL(1,𝔽₈)	C₄²	1	1	yes	characteristic, direct product factor
22	C₂²×ΓL(1,𝔽₈)	—	4	1	no	—
23	C₄×ΓL(1,𝔽₈)	—	4	6	no	—
24	C₂²×C₄²	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
25	C₂³×C₇	—	3	3	no	—
26	C₂³×C₇	—	3	1	no	—
27	C₂³×C₇	C₂×C₃×C₄	1	3	yes	—
28	C₂×C₄×C₇	—	3	6	no	—
29	C₂×C₄×C₇	—	3	3	no	—
30	C₂×C₄×C₇	—	3	6	no	—
31	C₂×C₄×C₇	—	3	6	no	—
32	C₂×C₄×C₇	—	3	6	no	—
33	C₂×C₄×C₇	C₂×A₄	1	3	yes	—
34	C₂²×A₄	—	7	1	no	—
35	C₄×A₄	—	7	6	no	—
36	C₃×C₄²	—	28	1	no	—
37	C₂×ΓL(1,𝔽₈)	—	4	3	no	—
38	C₂³×C₄	C₂×ΓL(1,𝔽₈)	1	3	yes	—
39	C₂×C₄²	—	3	1	no	—
40	C₂×C₄²	—	3	3	no	—
41	C₂²×C₇	—	3	3	no	—
42	C₂²×C₇	—	3	2	no	—
43	C₂²×C₇	—	3	3	no	—
44	C₂²×C₇	—	3	3	no	—
45	C₂²×C₇	C₃×C₄²	1	1	yes	characteristic, complemented
46	C₂²×C₇	C₂²×A₄	1	1	yes	characteristic
47	C₄×C₇	—	3	6	no	—
48	C₄×C₇	C₄×A₄	1	6	yes	complemented
49	C₂×A₄	—	7	3	no	—
50	C₂×C₃×C₄	—	28	3	no	—
51	ΓL(1,𝔽₈)	—	4	1	no	Hall
52	C₂⁴	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
53	C₂²×C₄	—	3	3	no	—
54	C₂²×C₄	—	3	3	no	—
55	C₂²×C₄	—	3	6	no	—
56	C₂²×C₄	C₄×ΓL(1,𝔽₈)	1	6	yes	complemented
57	C₄²	—	3	2	no	—
58	C₄²	—	3	3	no	—
59	C₄²	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
60	C₂×C₇	—	3	3	no	—
61	C₂×C₇	—	3	1	no	—
62	C₂×C₇	C₂×C₄×A₄	1	3	yes	—
63	C₂²×C₃	—	28	1	no	—
64	A₄	—	7	1	no	—
65	C₃×C₄	—	28	6	no	—
66	C₂³	—	3	3	no	—
67	C₂³	—	3	1	no	—
68	C₂³	C₂×C₄×ΓL(1,𝔽₈)	1	3	yes	—
69	C₂×C₄	—	3	6	no	—
70	C₂×C₄	—	3	3	no	—
71	C₂×C₄	—	3	6	no	—
72	C₂×C₄	—	3	6	no	—
73	C₂×C₄	—	3	6	no	—
74	C₂×C₄	C₂×C₂²⋊ΓL(1,𝔽₈)	1	3	yes	—
75	C₇	C₄²×A₄	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₃	—	28	3	no	—
77	C₂²	—	3	3	no	—
78	C₂²	—	3	2	no	—
79	C₂²	—	3	3	no	—
80	C₂²	—	3	3	no	—
81	C₂²	C₄²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
82	C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
83	C₄	—	3	6	no	—
84	C₄	C₄×C₂²⋊ΓL(1,𝔽₈)	1	6	yes	direct product factor
85	C₃	—	28	1	no	Sylow
86	C₂	—	3	1	no	—
87	C₂	—	3	3	no	—
88	C₂	C₂×C₄×C₂²⋊ΓL(1,𝔽₈)	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²	C₂²⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₂²×C₇	C₃×C₄²
Frattini	C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₂²×C₄²×C₇	C₃
socle, cosocle	C₂⁴×C₇	C₂²×C₃
prime radical	C₂²×C₂²⋊ΓL(1,𝔽₈)	C₂²
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄²×C₇	C₃	semidirect
4. C₄×C₂²⋊ΓL(1,𝔽₈)	C₄	direct (N × H)
16. C₂²×C₄×C₇	C₃×C₄	semidirect
19. C₄²×C₇	A₄	semidirect
21. C₂²⋊ΓL(1,𝔽₈)	C₄²	direct (N × H)
24. C₂²×C₄²	ΓL(1,𝔽₈)	semidirect
45. C₂²×C₇	C₃×C₄²	semidirect
48. C₄×C₇	C₄×A₄	semidirect
56. C₂²×C₄	C₄×ΓL(1,𝔽₈)	semidirect
59. C₄²	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
75. C₇	C₄²×A₄	semidirect
81. C₂²	C₄²×ΓL(1,𝔽₈)	semidirect
84. C₄	C₄×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂²⋊ΓL(1,𝔽₈)	2
C₂²×C₄²×C₇	3
C₄²×ΓL(1,𝔽₈)	4
C₄²×A₄	7

Subgroups — C42×C22⋊ΓL(1,𝔽8) #13060769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₂²⋊ΓL(1,𝔽₈) #13060769