Subgroups — C₄×C₂²⋊(C₂²⋊ΓL(1,𝔽₈)) #13062323

Order 1344.

524 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	1	yes	maximal, characteristic
2	C₂⁴×C₄×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	C₄	1	1	yes	characteristic, direct product factor
4	C₄×C₂²⋊ΓL(1,𝔽₈)	—	4	5	no	maximal
5	C₂⁵×C₇	C₂×C₃	1	1	yes	characteristic
6	C₂³×C₄×C₇	—	3	5	no	—
7	C₂³×C₄×C₇	—	3	5	no	—
8	C₄×C₂²⋊A₄	—	7	1	no	maximal, Hall
9	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	5	no	—
10	C₂⁴×C₇	—	3	5	no	—
11	C₂⁴×C₇	—	3	5	no	—
12	C₂⁴×C₇	C₃×C₄	1	1	yes	characteristic, complemented
13	C₂²×C₄×C₇	—	3	30	no	—
14	C₂²×C₄×C₇	—	3	10	no	—
15	C₂²×C₄×C₇	—	3	5	no	—
16	C₂²×C₄×C₇	A₄	1	5	yes	complemented
17	C₂×C₂²⋊A₄	—	7	1	no	—
18	C₂²⋊ΓL(1,𝔽₈)	—	4	5	no	—
19	C₄×ΓL(1,𝔽₈)	—	16	1	no	—
20	C₂⁴×C₄	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
21	C₂³×C₇	—	3	10	no	—
22	C₂³×C₇	—	3	30	no	—
23	C₂³×C₇	—	3	5	no	—
24	C₂³×C₇	—	3	5	no	—
25	C₂³×C₇	C₂×A₄	1	5	yes	—
26	C₂×C₄×C₇	—	3	30	no	—
27	C₂×C₄×C₇	—	3	5	no	—
28	C₂×C₄×C₇	—	3	5	no	—
29	C₂²⋊A₄	—	7	1	no	—
30	C₄×A₄	—	28	5	no	—
31	C₂×ΓL(1,𝔽₈)	—	16	1	no	—
32	C₂⁵	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
33	C₂³×C₄	—	3	5	no	—
34	C₂³×C₄	—	3	5	no	—
35	C₂²×C₇	—	3	5	no	—
36	C₂²×C₇	—	3	30	no	—
37	C₂²×C₇	—	3	10	no	—
38	C₂²×C₇	—	3	5	no	—
39	C₂²×C₇	C₄×A₄	1	5	yes	complemented
40	C₄×C₇	—	3	5	no	—
41	C₄×C₇	C₂²⋊A₄	1	1	yes	characteristic, complemented
42	C₂×A₄	—	28	5	no	—
43	ΓL(1,𝔽₈)	—	16	1	no	Hall
44	C₂⁴	—	3	5	no	—
45	C₂⁴	—	3	5	no	—
46	C₂⁴	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
47	C₂²×C₄	—	3	30	no	—
48	C₂²×C₄	—	3	10	no	—
49	C₂²×C₄	—	3	5	no	—
50	C₂²×C₄	C₂²⋊ΓL(1,𝔽₈)	1	5	yes	complemented
51	C₂×C₇	—	3	5	no	—
52	C₂×C₇	—	3	5	no	—
53	C₂×C₇	C₂×C₂²⋊A₄	1	1	yes	characteristic
54	A₄	—	28	5	no	—
55	C₃×C₄	—	112	1	no	—
56	C₂³	—	3	10	no	—
57	C₂³	—	3	30	no	—
58	C₂³	—	3	5	no	—
59	C₂³	—	3	5	no	—
60	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	5	yes	—
61	C₂×C₄	—	3	30	no	—
62	C₂×C₄	—	3	5	no	—
63	C₂×C₄	—	3	5	no	—
64	C₇	C₄×C₂²⋊A₄	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	112	1	no	—
66	C₂²	—	3	5	no	—
67	C₂²	—	3	10	no	—
68	C₂²	—	3	30	no	—
69	C₂²	—	3	5	no	—
70	C₂²	C₄×C₂²⋊ΓL(1,𝔽₈)	1	5	yes	complemented
71	C₄	—	3	5	no	—
72	C₄	C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	1	1	yes	characteristic, direct product factor
73	C₃	—	112	1	no	Sylow
74	C₂	—	3	5	no	—
75	C₂	—	3	5	no	—
76	C₂	C₂×C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	C₂²⋊(C₂²⋊ΓL(1,𝔽₈))
commutator, hypocenter	C₂⁴×C₇	C₃×C₄
Frattini	C₂	C₂×C₂²⋊(C₂²⋊ΓL(1,𝔽₈))
Fitting	C₂⁴×C₄×C₇	C₃
socle, cosocle	C₂⁵×C₇	C₂×C₃
prime radical	C₂×C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	C₂
LCS	G ▷ C₂⁴×C₇
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×C₇	C₃	semidirect
3. C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	C₄	direct (N × H)
12. C₂⁴×C₇	C₃×C₄	semidirect
16. C₂²×C₄×C₇	A₄	semidirect
20. C₂⁴×C₄	ΓL(1,𝔽₈)	semidirect
39. C₂²×C₇	C₄×A₄	semidirect
41. C₄×C₇	C₂²⋊A₄	semidirect
46. C₂⁴	C₄×ΓL(1,𝔽₈)	semidirect
50. C₂²×C₄	C₂²⋊ΓL(1,𝔽₈)	semidirect
64. C₇	C₄×C₂²⋊A₄	semidirect
70. C₂²	C₄×C₂²⋊ΓL(1,𝔽₈)	semidirect
72. C₄	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
C₂⁴×C₄×C₇	3
C₄×C₂²⋊ΓL(1,𝔽₈)	4
C₄×C₂²⋊A₄	7

Subgroups — C4×C22⋊(C22⋊ΓL(1,𝔽8)) #13062323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂²⋊(C₂²⋊ΓL(1,𝔽₈)) #13062323