Subgroups — C₂²×C₂²⋊(C₂²⋊ΓL(1,𝔽₈)) #13062425

Order 1344.

2000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	C₂	1	3	yes	maximal, direct product factor
2	C₂⁶×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	C₂²	1	1	yes	characteristic, direct product factor
4	C₂²×C₂²⋊ΓL(1,𝔽₈)	—	4	5	no	maximal
5	C₂⁵×C₇	—	3	15	no	—
6	C₂⁵×C₇	—	3	5	no	—
7	C₂⁵×C₇	C₂×C₃	1	3	yes	complemented
8	C₂²×C₂²⋊A₄	—	7	1	no	maximal, Hall
9	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	15	no	—
10	C₂⁴×C₇	—	3	90	no	—
11	C₂⁴×C₇	—	3	60	no	—
12	C₂⁴×C₇	—	3	10	no	—
13	C₂⁴×C₇	—	3	10	no	—
14	C₂⁴×C₇	—	3	15	no	—
15	C₂⁴×C₇	—	3	15	no	—
16	C₂⁴×C₇	—	3	15	no	—
17	C₂⁴×C₇	A₄	1	5	yes	complemented
18	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
19	C₂×C₂²⋊A₄	—	7	3	no	—
20	C₂²⋊ΓL(1,𝔽₈)	—	4	5	no	—
21	C₂²×ΓL(1,𝔽₈)	—	16	1	no	—
22	C₂⁶	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
23	C₂³×C₇	—	3	15	no	—
24	C₂³×C₇	—	3	90	no	—
25	C₂³×C₇	—	3	90	no	—
26	C₂³×C₇	—	3	120	no	—
27	C₂³×C₇	—	3	90	no	—
28	C₂³×C₇	—	3	30	no	—
29	C₂³×C₇	—	3	15	no	—
30	C₂³×C₇	—	3	5	no	—
31	C₂³×C₇	—	3	5	no	—
32	C₂³×C₇	C₂×A₄	1	15	yes	complemented
33	C₂²⋊A₄	—	7	1	no	—
34	C₂²×A₄	—	28	5	no	—
35	C₂×ΓL(1,𝔽₈)	—	16	3	no	—
36	C₂⁵	—	3	15	no	—
37	C₂⁵	—	3	5	no	—
38	C₂⁵	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
39	C₂²×C₇	—	3	15	no	—
40	C₂²×C₇	—	3	10	no	—
41	C₂²×C₇	—	3	60	no	—
42	C₂²×C₇	—	3	90	no	—
43	C₂²×C₇	—	3	10	no	—
44	C₂²×C₇	—	3	15	no	—
45	C₂²×C₇	—	3	15	no	—
46	C₂²×C₇	C₂²×A₄	1	5	yes	complemented
47	C₂²×C₇	C₂²⋊A₄	1	1	yes	characteristic, complemented
48	C₂×A₄	—	28	15	no	—
49	ΓL(1,𝔽₈)	—	16	1	no	Hall
50	C₂⁴	—	3	15	no	—
51	C₂⁴	—	3	90	no	—
52	C₂⁴	—	3	60	no	—
53	C₂⁴	—	3	10	no	—
54	C₂⁴	—	3	10	no	—
55	C₂⁴	—	3	15	no	—
56	C₂⁴	—	3	15	no	—
57	C₂⁴	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
58	C₂⁴	C₂²⋊ΓL(1,𝔽₈)	1	5	yes	complemented
59	C₂×C₇	—	3	15	no	—
60	C₂×C₇	—	3	5	no	—
61	C₂×C₇	C₂×C₂²⋊A₄	1	3	yes	complemented
62	C₂²×C₃	—	112	1	no	—
63	A₄	—	28	5	no	—
64	C₂³	—	3	15	no	—
65	C₂³	—	3	90	no	—
66	C₂³	—	3	90	no	—
67	C₂³	—	3	120	no	—
68	C₂³	—	3	30	no	—
69	C₂³	—	3	90	no	—
70	C₂³	—	3	15	no	—
71	C₂³	—	3	5	no	—
72	C₂³	—	3	5	no	—
73	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	15	yes	complemented
74	C₇	C₂²×C₂²⋊A₄	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₃	—	112	3	no	—
76	C₂²	—	3	15	no	—
77	C₂²	—	3	10	no	—
78	C₂²	—	3	90	no	—
79	C₂²	—	3	10	no	—
80	C₂²	—	3	60	no	—
81	C₂²	—	3	15	no	—
82	C₂²	—	3	15	no	—
83	C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)	1	5	yes	complemented
84	C₂²	C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	1	1	yes	characteristic, direct product factor
85	C₃	—	112	1	no	Sylow
86	C₂	—	3	5	no	—
87	C₂	—	3	15	no	—
88	C₂	C₂×C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	C₂²⋊(C₂²⋊ΓL(1,𝔽₈))
commutator, cosocle, hypocenter	C₂⁴×C₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂⁶×C₇	C₃
LCS	G ▷ C₂⁴×C₇
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	C₂	direct (N × H)
2. C₂⁶×C₇	C₃	semidirect
3. C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	C₂²	direct (N × H)
7. C₂⁵×C₇	C₂×C₃	semidirect
17. C₂⁴×C₇	A₄	semidirect
18. C₂⁴×C₇	C₂²×C₃	semidirect
22. C₂⁶	ΓL(1,𝔽₈)	semidirect
32. C₂³×C₇	C₂×A₄	semidirect
38. C₂⁵	C₂×ΓL(1,𝔽₈)	semidirect
46. C₂²×C₇	C₂²×A₄	semidirect
47. C₂²×C₇	C₂²⋊A₄	semidirect
57. C₂⁴	C₂²×ΓL(1,𝔽₈)	semidirect
58. C₂⁴	C₂²⋊ΓL(1,𝔽₈)	semidirect
61. C₂×C₇	C₂×C₂²⋊A₄	semidirect
73. C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	semidirect
74. C₇	C₂²×C₂²⋊A₄	semidirect
83. C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)	semidirect
84. C₂²	C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	direct (N × H)
88. C₂	C₂×C₂²⋊(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×C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	2
C₂⁶×C₇	3
C₂²×C₂²⋊ΓL(1,𝔽₈)	4
C₂²×C₂²⋊A₄	7

Subgroups — C22×C22⋊(C22⋊ΓL(1,𝔽8)) #13062425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊(C₂²⋊ΓL(1,𝔽₈)) #13062425