Subgroups — #384⋊ΓL(1,𝔽₈) #13055065

Order 1344.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇⋊(C₄²⋊C₃)	C₂	1	1	yes	maximal, characteristic
2	C₇×#384	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₇⋊(C₄²⋊C₃)	C₄	1	1	yes	characteristic, complemented
5	C₂×C₇×C₄⋊C₄	—	3	1	no	—
6	C₂×C₇×C₄⋊C₄	—	3	1	no	—
7	C₂×C₄²×C₇	C₂×C₃	1	1	yes	characteristic
8	#384⋊C₃	—	7	1	no	maximal, Hall
9	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
10	C₂²×C₄×C₇	—	3	1	no	—
11	C₂²×C₄×C₇	—	3	1	no	—
12	C₂²×C₄×C₇	A₄	1	1	yes	characteristic
13	C₇×C₄⋊C₄	—	6	2	no	—
14	C₇×C₄⋊C₄	—	6	2	no	—
15	C₄²×C₇	—	3	1	no	—
16	C₄²×C₇	C₃×C₄	1	1	yes	characteristic, complemented
17	C₂×C₄²⋊C₃	—	7	1	no	—
18	C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
19	C₄×ΓL(1,𝔽₈)	—	16	1	no	—
20	#384	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
21	C₂³×C₇	C₂×A₄	1	1	yes	characteristic
22	C₂×C₄×C₇	—	3	2	no	—
23	C₂×C₄×C₇	—	3	2	no	—
24	C₂×C₄×C₇	—	3	1	no	—
25	C₂×C₄×C₇	—	3	1	no	—
26	C₂×C₄×C₇	—	6	1	no	—
27	C₂×C₄×C₇	—	6	1	no	—
28	C₂×C₄×C₇	—	6	2	no	—
29	C₄×A₄	—	28	1	no	—
30	C₄²⋊C₃	—	7	1	no	—
31	C₂×ΓL(1,𝔽₈)	—	16	1	no	—
32	C₂×C₄⋊C₄	—	3	1	no	—
33	C₂×C₄⋊C₄	—	3	1	no	—
34	C₂×C₄²	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
35	C₂²×C₇	—	3	1	no	—
36	C₂²×C₇	—	3	1	no	—
37	C₂²×C₇	C₄×A₄	1	1	yes	characteristic
38	C₄×C₇	—	3	2	no	—
39	C₄×C₇	—	3	2	no	—
40	C₄×C₇	—	4	1	no	—
41	C₄×C₇	—	12	1	no	—
42	C₂×A₄	—	28	1	no	—
43	ΓL(1,𝔽₈)	—	16	1	no	Hall
44	C₂²×C₄	—	3	1	no	—
45	C₂²×C₄	—	3	1	no	—
46	C₂²×C₄	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
47	C₄⋊C₄	—	6	2	no	—
48	C₄⋊C₄	—	6	2	no	—
49	C₄²	—	3	1	no	—
50	C₄²	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
51	C₂×C₇	—	3	1	no	—
52	C₂×C₇	—	3	1	no	—
53	C₂×C₇	#127⋊C₃	1	1	yes	characteristic
54	A₄	—	28	1	no	—
55	C₃×C₄	—	112	1	no	—
56	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
57	C₂×C₄	—	3	2	no	—
58	C₂×C₄	—	3	2	no	—
59	C₂×C₄	—	3	1	no	—
60	C₂×C₄	—	3	1	no	—
61	C₂×C₄	—	6	1	no	—
62	C₂×C₄	—	6	1	no	—
63	C₂×C₄	—	6	2	no	—
64	C₇	#384⋊C₃	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	112	1	no	—
66	C₂²	—	3	1	no	—
67	C₂²	—	3	1	no	—
68	C₂²	C₄×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
69	C₄	—	3	2	no	—
70	C₄	—	3	2	no	—
71	C₄	—	4	1	no	—
72	C₄	—	12	1	no	—
73	C₃	—	112	1	no	Sylow
74	C₂	—	3	1	no	—
75	C₂	—	3	1	no	—
76	C₂	#127⋊ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	#127⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₄²×C₇	C₃×C₄
Frattini	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×#384	C₃
socle	C₂³×C₇	C₂×A₄
prime radical	C₂×C₇⋊(C₄²⋊C₃)	C₂
cosocle	C₂×C₄²×C₇	C₂×C₃
LCS	G ▷ C₄²×C₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇×#384	C₃	semidirect
4. C₇⋊(C₄²⋊C₃)	C₄	semidirect
16. C₄²×C₇	C₃×C₄	semidirect
20. #384	ΓL(1,𝔽₈)	semidirect
50. C₄²	C₄×ΓL(1,𝔽₈)	semidirect
64. C₇	#384⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇⋊(C₄²⋊C₃)	2
C₇×#384	3
C₄×C₂²⋊ΓL(1,𝔽₈)	4
#384⋊C₃	7

Subgroups — #384⋊ΓL(1,𝔽8) #13055065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #384⋊ΓL(1,𝔽₈) #13055065