Subgroups — ΓL(1,𝔽₈)×#376 #13054907

Order 1344.

356 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal
2	ΓL(1,𝔽₈)×#95	C₂	1	4	yes	maximal
3	C₇×#376	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	6	yes	—
5	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₄²×ΓL(1,𝔽₈)	C₄	1	12	yes	complemented
7	C₂×C₄²×C₇	C₂×C₃	1	3	yes	—
8	C₇×#95	C₂×C₃	1	4	yes	—
9	C₃×#376	—	7	1	no	maximal, Hall
10	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
11	C₂×C₄×ΓL(1,𝔽₈)	—	2	12	no	—
12	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	12	yes	—
13	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	6	yes	—
14	C₂²×C₄×C₇	C₂²×C₃	1	6	yes	—
15	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
16	C₄²×C₇	C₃×C₄	1	12	yes	complemented
17	C₂×C₃×C₄²	—	7	3	no	—
18	C₃×#95	—	7	4	no	—
19	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	3	yes	—
20	C₂²×ΓL(1,𝔽₈)	#41	1	4	yes	—
21	C₄×ΓL(1,𝔽₈)	—	2	12	no	—
22	C₄×ΓL(1,𝔽₈)	C₄²	1	4	yes	—
23	#376	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
24	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
25	C₂×C₄×C₇	—	2	12	no	—
26	C₂×C₄×C₇	C₂×C₃×C₄	1	12	yes	—
27	C₂×C₄×C₇	C₂×C₃×C₄	1	6	yes	—
28	C₂²×C₃×C₄	—	7	6	no	—
29	C₂²×C₃×C₄	—	7	1	no	—
30	C₃×C₄²	—	7	12	no	—
31	C₂×ΓL(1,𝔽₈)	C₂×C₄²	1	1	yes	characteristic
32	C₂×ΓL(1,𝔽₈)	#117	1	6	yes	—
33	C₂×C₄²	C₂×ΓL(1,𝔽₈)	1	3	yes	—
34	#95	C₂×ΓL(1,𝔽₈)	1	4	yes	—
35	C₂²×C₇	C₂²×C₃×C₄	1	3	yes	—
36	C₂²×C₇	C₃×#41	1	4	yes	—
37	C₄×C₇	—	2	12	no	—
38	C₄×C₇	C₃×C₄²	1	4	yes	—
39	C₂³×C₃	—	7	1	no	—
40	C₂×C₃×C₄	—	7	6	no	—
41	C₂×C₃×C₄	—	7	12	no	—
42	C₂×C₃×C₄	—	14	12	no	—
43	ΓL(1,𝔽₈)	#376	1	1	yes	characteristic, direct product factor, Hall
44	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	6	yes	—
45	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
46	C₄²	C₄×ΓL(1,𝔽₈)	1	12	yes	complemented
47	C₂×C₇	C₂×C₃×C₄²	1	1	yes	characteristic
48	C₂×C₇	C₃×#117	1	6	yes	—
49	C₂²×C₃	—	7	3	no	—
50	C₂²×C₃	—	7	4	no	—
51	C₃×C₄	—	7	4	no	—
52	C₃×C₄	—	14	12	no	—
53	C₂³	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
54	C₂×C₄	—	2	12	no	—
55	C₂×C₄	C₂×C₄×ΓL(1,𝔽₈)	1	12	yes	—
56	C₂×C₄	C₂×C₄×ΓL(1,𝔽₈)	1	6	yes	—
57	C₇	C₃×#376	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₃	—	7	1	no	—
59	C₂×C₃	—	7	6	no	—
60	C₂²	C₂²×C₄×ΓL(1,𝔽₈)	1	3	yes	—
61	C₂²	#41×ΓL(1,𝔽₈)	1	4	yes	—
62	C₄	—	2	12	no	—
63	C₄	C₄²×ΓL(1,𝔽₈)	1	4	yes	—
64	C₃	—	7	1	no	Sylow
65	C₂	C₂×C₄²×ΓL(1,𝔽₈)	1	1	yes	characteristic
66	C₂	ΓL(1,𝔽₈)×#117	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×ΓL(1,𝔽₈)
commutator	C₂×C₇	C₂×C₃×C₄²
Frattini	C₂³	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#376	C₃
socle, cosocle	C₂³×C₇	C₂³×C₃
prime radical	C₂³×ΓL(1,𝔽₈)	C₂³
hypercenter	#376	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#376
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
3. C₇×#376	C₃	semidirect
6. C₄²×ΓL(1,𝔽₈)	C₄	semidirect
16. C₄²×C₇	C₃×C₄	semidirect
23. #376	ΓL(1,𝔽₈)	direct (N × H)
43. ΓL(1,𝔽₈)	#376	direct (N × H)
46. C₄²	C₄×ΓL(1,𝔽₈)	semidirect
57. C₇	C₃×#376	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×#95	2
C₇×#376	3
C₃×#376	7

Subgroups — ΓL(1,𝔽8)×#376 #13054907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#376 #13054907