Subgroups — ΓL(1,𝔽₈)×#627 #13094722

Order 1512.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₃×Dic₃×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal
3	C₇×#627	C₃	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₈)×Dic₆	—	3	2	no	maximal
5	C₂×C₃²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₇×C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic
7	C₃×C₇×Dic₃	C₂×C₃	1	2	yes	—
8	C₃×C₄×ΓL(1,𝔽₈)	—	3	2	no	—
9	Dic₃×ΓL(1,𝔽₈)	—	2	2	no	—
10	Dic₃×ΓL(1,𝔽₈)	—	3	2	no	—
11	Dic₃×ΓL(1,𝔽₈)	—	6	1	no	—
12	Dic₃×ΓL(1,𝔽₈)	D₃	1	2	yes	—
13	C₃×#627	—	7	1	no	maximal, Hall
14	C₃²×ΓL(1,𝔽₈)	Q₈	1	1	yes	characteristic, complemented, Hall
15	C₇×Dic₆	—	3	2	no	—
16	Q₈×ΓL(1,𝔽₈)	—	9	1	no	—
17	C₂×C₃²×C₇	C₂²×C₃	1	1	yes	characteristic
18	C₂×C₃×ΓL(1,𝔽₈)	—	2	2	no	—
19	C₂×C₃×ΓL(1,𝔽₈)	—	2	1	no	—
20	C₂×C₃×ΓL(1,𝔽₈)	—	4	1	no	—
21	C₂×C₃×ΓL(1,𝔽₈)	C₂×D₃	1	2	yes	—
22	C₃×C₃⋊Dic₃	—	7	1	no	—
23	C₃²×Dic₃	—	7	2	no	—
24	C₃×C₄×C₇	—	3	2	no	—
25	C₇×Dic₃	—	3	2	no	—
26	C₇×Dic₃	—	6	1	no	—
27	C₇×Dic₃	C₃×D₃	1	2	yes	—
28	C₄×ΓL(1,𝔽₈)	—	3	2	no	—
29	C₄×ΓL(1,𝔽₈)	—	6	2	no	—
30	C₄×ΓL(1,𝔽₈)	—	9	1	no	—
31	C₃×Dic₆	—	21	2	no	—
32	#627	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
33	C₃²×C₇	C₃×Q₈	1	1	yes	characteristic, complemented
34	C₃×ΓL(1,𝔽₈)	—	2	2	no	—
35	C₃×ΓL(1,𝔽₈)	—	2	1	no	—
36	C₃×ΓL(1,𝔽₈)	—	4	1	no	—
37	C₃×ΓL(1,𝔽₈)	Dic₆	1	2	yes	complemented
38	C₇×Q₈	—	9	1	no	Hall
39	C₂×C₃³	—	7	1	no	—
40	C₂×C₃×C₇	—	2	1	no	—
41	C₂×C₃×C₇	C₂×C₃×D₃	1	2	yes	—
42	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
43	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
44	C₂×ΓL(1,𝔽₈)	D₃²	1	1	yes	characteristic
45	C₃²×C₄	—	21	2	no	—
46	C₃⋊Dic₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
47	C₃×Dic₃	—	7	2	no	—
48	C₃×Dic₃	—	14	2	no	—
49	C₃×Dic₃	—	21	2	no	—
50	C₃×Dic₃	—	42	1	no	—
51	C₃×Dic₃	C₂×ΓL(1,𝔽₈)	1	2	yes	—
52	C₄×C₇	—	3	2	no	—
53	C₄×C₇	—	9	1	no	—
54	C₃³	—	7	1	no	Sylow
55	C₃×Q₈	—	63	1	no	—
56	Dic₆	—	3	2	no	—
57	C₃×C₇	—	2	1	no	—
58	C₃×C₇	C₃×Dic₆	1	2	yes	complemented
59	ΓL(1,𝔽₈)	—	2	2	no	—
60	ΓL(1,𝔽₈)	—	4	1	no	—
61	ΓL(1,𝔽₈)	#627	1	1	yes	characteristic, direct product factor
62	C₂×C₃²	—	7	2	no	—
63	C₂×C₃²	—	14	1	no	—
64	C₂×C₃²	—	14	2	no	—
65	C₂×C₃²	—	28	1	no	—
66	C₂×C₃²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
67	C₂×C₇	C₃×D₃²	1	1	yes	characteristic
68	C₃×C₄	—	3	2	no	—
69	C₃×C₄	—	21	2	no	—
70	C₃×C₄	—	42	2	no	—
71	C₃×C₄	—	63	1	no	—
72	Dic₃	—	3	2	no	—
73	Dic₃	—	6	1	no	—
74	Dic₃	D₃×ΓL(1,𝔽₈)	1	2	yes	—
75	C₃²	—	7	2	no	—
76	C₃²	—	14	1	no	—
77	C₃²	—	14	2	no	—
78	C₃²	—	28	1	no	—
79	C₃²	Q₈×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
80	Q₈	—	9	1	no	Sylow
81	C₇	C₃×#627	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₃	—	2	1	no	—
83	C₂×C₃	—	7	1	no	—
84	C₂×C₃	—	14	2	no	—
85	C₂×C₃	—	28	1	no	—
86	C₂×C₃	C₂×D₃×ΓL(1,𝔽₈)	1	2	yes	—
87	C₄	—	3	2	no	—
88	C₄	—	9	1	no	—
89	C₃	—	2	1	no	—
90	C₃	—	7	1	no	—
91	C₃	—	14	2	no	—
92	C₃	—	28	1	no	—
93	C₃	ΓL(1,𝔽₈)×Dic₆	1	2	yes	complemented
94	C₂	D₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃²×ΓL(1,𝔽₈)
commutator, Fitting, socle, cosocle	C₂×C₃²×C₇	C₂²×C₃
prime radical	C₂×C₃²×ΓL(1,𝔽₈)	C₂²
hypocenter	C₃²×C₇	C₃×Q₈
LCS	G ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
3. C₇×#627	C₃	semidirect
14. C₃²×ΓL(1,𝔽₈)	Q₈	semidirect
32. #627	ΓL(1,𝔽₈)	direct (N × H)
33. C₃²×C₇	C₃×Q₈	semidirect
37. C₃×ΓL(1,𝔽₈)	Dic₆	semidirect
58. C₃×C₇	C₃×Dic₆	semidirect
61. ΓL(1,𝔽₈)	#627	direct (N × H)
79. C₃²	Q₈×ΓL(1,𝔽₈)	semidirect
81. C₇	C₃×#627	semidirect
93. C₃	ΓL(1,𝔽₈)×Dic₆	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×C₃⋊Dic₃	2
C₃×Dic₃×ΓL(1,𝔽₈)	2
C₇×#627	3
ΓL(1,𝔽₈)×Dic₆	3
C₃×#627	7

Subgroups — ΓL(1,𝔽8)×#627 #13094722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#627 #13094722