Subgroups — #89343⋊C₃ #13050818

Order 1344.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#6410⋊C₃	C₂	1	2	yes	maximal
3	#89343	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×Q₈×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
5	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₄⋊C₄×ΓL(1,𝔽₈)	—	2	2	no	—
7	C₄²×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
8	#1136⋊C₃	—	2	2	no	—
9	C₇×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
10	#6410	C₂×C₃	1	2	yes	—
11	C₃×#356	—	7	1	no	maximal, Hall
12	Q₈×ΓL(1,𝔽₈)	—	4	1	no	—
13	Q₈×ΓL(1,𝔽₈)	—	4	1	no	—
14	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
15	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
16	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	2	yes	—
18	ΓL(1,𝔽₈)⋊C₈	—	4	2	no	—
19	C₂×C₇×Q₈	C₃×C₄	1	1	yes	characteristic
20	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
21	C₇×C₄⋊C₄	—	2	2	no	—
22	C₄²×C₇	C₃×C₄	1	1	yes	characteristic
23	#1136	—	2	2	no	—
24	C₃×C₄⋊Q₈	—	7	1	no	—
25	C₃×#101	—	7	2	no	—
26	C₂²×ΓL(1,𝔽₈)	C₂²⋊C₄	1	1	yes	characteristic
27	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
28	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
29	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
30	C₄×ΓL(1,𝔽₈)	—	4	1	no	—
31	#356	—	7	1	no	Sylow
32	C₇×Q₈	—	4	1	no	—
33	C₇×Q₈	—	4	1	no	—
34	C₂×C₄×C₇	—	2	1	no	—
35	C₂×C₄×C₇	—	2	1	no	—
36	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
38	C₇⋊C₈	—	4	2	no	—
39	C₂×C₃×Q₈	—	7	1	no	—
40	C₂×C₃×Q₈	—	7	1	no	—
41	C₃×M₁₆	—	14	2	no	—
42	C₃×C₄⋊C₄	—	14	2	no	—
43	C₃×C₄²	—	7	1	no	—
44	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
45	C₂×ΓL(1,𝔽₈)	#99	1	1	yes	characteristic
46	C₄⋊Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic
47	#101	—	7	2	no	—
48	C₂²×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
49	C₄×C₇	—	2	2	no	—
50	C₄×C₇	—	2	2	no	—
51	C₄×C₇	—	2	1	no	—
52	C₄×C₇	—	4	1	no	—
53	C₃×Q₈	—	28	1	no	—
54	C₃×Q₈	—	28	1	no	—
55	C₂×C₃×C₄	—	7	1	no	—
56	C₂×C₃×C₄	—	7	2	no	—
57	C₂×C₃×C₄	—	14	1	no	—
58	C₂×C₃×C₄	—	14	1	no	—
59	C₃×C₈	—	28	2	no	—
60	ΓL(1,𝔽₈)	#356	1	1	yes	characteristic, complemented, Hall
61	C₂×Q₈	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
62	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
63	M₁₆	—	14	2	no	—
64	C₄⋊C₄	—	2	2	no	—
65	C₄²	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
66	C₂×C₇	—	2	1	no	—
67	C₂×C₇	C₃×#99	1	1	yes	characteristic
68	C₂²×C₃	—	7	1	no	—
69	C₃×C₄	—	14	2	no	—
70	C₃×C₄	—	14	1	no	—
71	C₃×C₄	—	14	2	no	—
72	C₃×C₄	—	28	1	no	—
73	Q₈	—	4	1	no	—
74	Q₈	—	4	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	C₂×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
78	C₂×C₄	(C₂²⋊D₇)⋊C₃	1	2	yes	—
79	C₈	—	28	2	no	—
80	C₇	C₃×#356	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃	—	7	1	no	—
82	C₂×C₃	—	14	1	no	—
83	C₂²	(C₂²⋊Dic₇)⋊C₃	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	2	2	no	—
86	C₄	—	2	1	no	—
87	C₄	—	4	1	no	—
88	C₃	—	7	1	no	Sylow
89	C₂	—	2	1	no	—
90	C₂	#6408⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6408⋊C₃
commutator	C₂×C₄×C₇	C₂×C₃×C₄
Frattini	C₂×Q₈	C₂×AGL(1,𝔽₇)
Fitting	C₇×C₄⋊Q₈	C₂×C₃
socle	C₂×C₇	C₃×#99
prime radical	C₂²×ΓL(1,𝔽₈)	C₂²⋊C₄
cosocle	C₂×C₇×Q₈	C₂²×C₃
hypercenter	C₄⋊Q₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#356
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #89343	C₃	semidirect
60. ΓL(1,𝔽₈)	#356	semidirect
80. C₇	C₃×#356	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)×C₄⋊Q₈	2
#6410⋊C₃	2
#89343	3
C₃×#356	7

Subgroups — #89343⋊C3 #13050818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #89343⋊C₃ #13050818