Group #13057421 — subgroups

Subgroups — ΓL(1,𝔽₈)⋊#564 #13057421

Order 1344.

344 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#125	C₂	1	1	yes	maximal, characteristic
2	#6453⋊C₃	C₂	1	4	yes	maximal
3	#6452⋊C₃	C₂	1	4	yes	maximal
4	#6452⋊C₃	C₂	1	4	yes	maximal
5	#6433⋊C₃	C₂	1	2	yes	maximal
6	C₇⋊#564	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
8	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
9	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
10	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
11	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
12	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	4	yes	—
13	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
14	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
15	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
16	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
17	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	4	yes	—
18	C₇×#125	C₂×C₃	1	1	yes	characteristic
19	#6453	C₂×C₃	1	4	yes	—
20	#6452	C₂×C₃	1	4	yes	—
21	#6452	C₂×C₃	1	4	yes	—
22	#6433	C₂×C₃	1	2	yes	—
23	C₃×#564	—	7	1	no	maximal, Hall
24	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
25	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
26	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
27	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	4	yes	—
28	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	4	yes	—
29	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
30	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
31	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
32	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
33	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
34	C₄⋊Dic₇	C₂²×C₃	1	4	yes	—
35	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
36	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
37	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
38	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
39	C₄×Dic₇	C₂²×C₃	1	4	yes	—
40	C₃×#125	—	7	2	no	—
41	C₃×#125	—	7	4	no	—
42	C₃×#125	—	7	4	no	—
43	C₃×#125	—	7	4	no	—
44	C₃×#125	—	7	1	no	—
45	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
46	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
47	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
48	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
49	ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
50	ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
51	#564	—	7	1	no	Sylow
52	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
53	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
54	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
55	C₂×Dic₇	C₂³×C₃	1	4	yes	—
56	C₂×Dic₇	C₂³×C₃	1	4	yes	—
57	C₃×C₄⋊C₄	—	7	4	no	—
58	C₃×C₄⋊C₄	—	7	4	no	—
59	C₃×C₄⋊C₄	—	7	4	no	—
60	C₃×C₄⋊C₄	—	7	4	no	—
61	C₃×C₄⋊C₄	—	7	2	no	—
62	C₃×C₄⋊C₄	—	7	2	no	—
63	C₃×C₄⋊C₄	—	7	4	no	—
64	C₃×C₄⋊C₄	—	7	2	no	—
65	C₃×C₄⋊C₄	—	7	4	no	—
66	C₃×C₄²	—	7	4	no	—
67	C₃×C₄²	—	7	1	no	—
68	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
69	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	2	yes	—
70	#125	—	7	4	no	—
71	#125	—	7	4	no	—
72	#125	—	7	4	no	—
73	#125	—	7	2	no	—
74	#125	AGL(1,𝔽₇)	1	1	yes	characteristic
75	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
76	C₄×C₇	—	2	1	no	—
77	C₄×C₇	—	2	4	no	—
78	C₄×C₇	—	2	2	no	—
79	Dic₇	—	2	4	no	—
80	Dic₇	—	2	4	no	—
81	C₂×C₃×C₄	—	7	4	no	—
82	C₂×C₃×C₄	—	7	4	no	—
83	C₂×C₃×C₄	—	7	4	no	—
84	C₂×C₃×C₄	—	7	1	no	—
85	C₂×C₃×C₄	—	7	2	no	—
86	ΓL(1,𝔽₈)	#564	1	1	yes	characteristic, complemented, Hall
87	C₄⋊C₄	—	7	4	no	—
88	C₄⋊C₄	—	7	4	no	—
89	C₄⋊C₄	—	7	4	no	—
90	C₄⋊C₄	—	7	4	no	—
91	C₄⋊C₄	—	7	4	no	—
92	C₄⋊C₄	—	7	2	no	—
93	C₄⋊C₄	—	7	2	no	—
94	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
95	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	4	yes	—
96	C₄²	—	7	4	no	—
97	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
98	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
99	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	—
100	C₂²×C₃	—	7	1	no	—
101	C₃×C₄	—	14	1	no	—
102	C₃×C₄	—	14	4	no	—
103	C₃×C₄	—	14	2	no	—
104	C₃×C₄	—	14	4	no	—
105	C₃×C₄	—	14	4	no	—
106	C₂×C₄	—	7	4	no	—
107	C₂×C₄	—	7	4	no	—
108	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	4	yes	—
109	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
110	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
111	C₇	C₃×#564	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₃	—	7	1	no	—
113	C₂×C₃	—	7	2	no	—
114	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	4	no	—
117	C₄	—	2	2	no	—
118	C₄	—	14	4	no	—
119	C₄	—	14	4	no	—
120	C₃	—	7	1	no	Sylow
121	C₂	#10621516	1	1	yes	characteristic
122	C₂	#10621519	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

ΓL(1,𝔽₈)×#125

C₂

yes

maximal, characteristic

#6453⋊C₃

C₂

yes

maximal

#6452⋊C₃

C₂

yes

maximal

#6452⋊C₃

C₂

yes

maximal

#6433⋊C₃

C₂

yes

maximal

C₇⋊#564

C₃

yes

maximal, characteristic, complemented, Hall

C₄⋊C₄×ΓL(1,𝔽₈)

C₂²

yes

—

C₄⋊C₄×ΓL(1,𝔽₈)

C₂²

yes

—

C₄²×ΓL(1,𝔽₈)

C₂²

yes

characteristic

C₄⋊(ΓL(1,𝔽₈)⋊C₄)

C₂²

yes

—

C₄⋊(ΓL(1,𝔽₈)⋊C₄)

C₂²

yes

—

C₄⋊(ΓL(1,𝔽₈)⋊C₄)

C₂²

yes

—

(ΓL(1,𝔽₈)⋊C₄)⋊C₄

C₂²

yes

—

(ΓL(1,𝔽₈)⋊C₄)⋊C₄

C₂²

yes

—

(ΓL(1,𝔽₈)⋊C₄)⋊C₄

C₂²

yes

—

(ΓL(1,𝔽₈)⋊C₄)⋊C₄

C₂²

yes

—

C₄×ΓL(1,𝔽₈)⋊C₄

C₂²

yes

—

C₇×#125

C₂×C₃

yes

characteristic

#6453

C₂×C₃

yes

—

#6452

C₂×C₃

yes

—

#6452

C₂×C₃

yes

—

#6433

C₂×C₃

yes

—

C₃×#564

—

maximal, Hall

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

characteristic

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

—

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

—

C₂×ΓL(1,𝔽₈)⋊C₄

C₂³

yes

—

C₂×ΓL(1,𝔽₈)⋊C₄

C₂³

yes

—

C₇×C₄⋊C₄

C₂²×C₃

yes

—

C₇×C₄⋊C₄

C₂²×C₃

yes

—

C₄²×C₇

C₂²×C₃

yes

characteristic

C₄⋊Dic₇

C₂²×C₃

yes

—

C₄⋊Dic₇

C₂²×C₃

yes

—

C₄⋊Dic₇

C₂²×C₃

yes

—

Dic₇⋊C₄

C₂²×C₃

yes

—

Dic₇⋊C₄

C₂²×C₃

yes

—

Dic₇⋊C₄

C₂²×C₃

yes

—

Dic₇⋊C₄

C₂²×C₃

yes

—

C₄×Dic₇

C₂²×C₃

yes

—

—

—

—

—

—

C₂²×ΓL(1,𝔽₈)

C₂⁴

yes

—

—

—

—

—

—

C₂×C₄×C₇

C₂³×C₃

yes

—

C₂×C₄×C₇

C₂³×C₃

yes

characteristic

C₂×C₄×C₇

C₂³×C₃

yes

—

C₂×Dic₇

C₂³×C₃

yes

—

C₂×Dic₇

C₂³×C₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₂×ΓL(1,𝔽₈)

2₋¹⁺⁴

yes

characteristic

C₂×ΓL(1,𝔽₈)

2₋¹⁺⁴

yes

—

—

—

—

—

yes

C₂²×C₇

C₂⁴×C₃

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

ΓL(1,𝔽₈)

#564

yes

characteristic, complemented, Hall

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×AGL(1,𝔽₇)

yes

—

C₄⋊C₄

C₂×AGL(1,𝔽₇)

yes

—

C₄²

—

C₄²

C₂×AGL(1,𝔽₇)

yes

characteristic

C₂×C₇

C₃×2₋¹⁺⁴

yes

characteristic

C₂×C₇

C₃×2₋¹⁺⁴

yes

—

100

C₂²×C₃

—

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

—

109

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

characteristic

110

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

—

111

C₇

C₃×#564

yes

characteristic, complemented, Sylow

112

C₂×C₃

—

113

C₂×C₃

—

114

C₂²

C₂³×AGL(1,𝔽₇)

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₃

—

Sylow

121

C₂

#10621516

yes

characteristic

122

C₂

#10621519

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#125	C₂×C₃
prime radical	C₂²×ΓL(1,𝔽₈)	C₂⁴
hypercenter	#125	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#564
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×AGL(1,𝔽₇)

commutator, socle, cosocle

C₂²×C₇

C₂⁴×C₃

Fitting

C₇×#125

C₂×C₃

prime radical

C₂²×ΓL(1,𝔽₈)

C₂⁴

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
6. C₇⋊#564	C₃	semidirect
86. ΓL(1,𝔽₈)	#564	semidirect
111. C₇	C₃×#564	semidirect

H ≅ G/N

type

6. C₇⋊#564

C₃

semidirect

86. ΓL(1,𝔽₈)

#564

semidirect

111. C₇

C₃×#564

semidirect

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

subgroup	index
ΓL(1,𝔽₈)×#125	2
#6453⋊C₃	2
#6452⋊C₃	2
#6433⋊C₃	2
C₇⋊#564	3
C₃×#564	7

subgroup

index

Subgroups — ΓL(1,𝔽₈)⋊#564 #13057421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽8)⋊#564 #13057421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)⋊#564 #13057421