Group #13057457 — subgroups

Subgroups — #90413⋊C₃ #13057457

Order 1344.

424 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 206 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)×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#6507⋊C₃	C₂	1	4	yes	maximal
3	#6451⋊C₃	C₂	1	4	yes	maximal
4	#6450⋊C₃	C₂	1	4	yes	maximal
5	C₄×ΓL(1,𝔽₈)⋊Q₈	C₂	1	2	yes	maximal
6	#90413	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
8	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	4	yes	—
9	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
10	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
11	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
13	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
14	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	8	yes	—
15	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	4	yes	—
16	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	4	yes	—
17	C₇×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
18	#6507	C₂×C₃	1	4	yes	—
19	#6451	C₂×C₃	1	4	yes	—
20	#6450	C₂×C₃	1	4	yes	—
21	C₄×Dic₁₄	C₂×C₃	1	2	yes	—
22	C₃×Q₈²	—	7	1	no	maximal, Hall
23	Q₈×ΓL(1,𝔽₈)	—	4	2	no	—
24	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
25	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
26	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
27	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	4	yes	—
28	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	4	yes	—
29	ΓL(1,𝔽₈)⋊Q₈	—	4	4	no	—
30	ΓL(1,𝔽₈)⋊Q₈	Q₈	1	8	yes	complemented
31	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
32	C₂×Dic₁₄	C₂²×C₃	1	4	yes	—
33	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
34	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
35	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
36	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
37	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
38	Dic₇⋊C₄	C₂²×C₃	1	8	yes	—
39	C₄×Dic₇	C₂²×C₃	1	4	yes	—
40	C₄×Dic₇	C₂²×C₃	1	4	yes	—
41	C₃×C₄⋊Q₈	—	7	4	no	—
42	C₃×C₄⋊Q₈	—	7	4	no	—
43	C₃×C₄⋊Q₈	—	7	1	no	—
44	C₃×C₄×Q₈	—	7	2	no	—
45	C₃×C₄×Q₈	—	7	4	no	—
46	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
47	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
48	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
49	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	4	yes	—
50	ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
51	ΓL(1,𝔽₈)⋊C₄	C₂×Q₈	1	8	yes	—
52	Q₈²	—	7	1	no	Sylow
53	C₇×Q₈	—	4	2	no	—
54	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
55	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
56	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
57	C₂×Dic₇	C₂³×C₃	1	4	yes	—
58	C₂×Dic₇	C₂³×C₃	1	4	yes	—
59	Dic₁₄	—	4	4	no	—
60	Dic₁₄	C₃×Q₈	1	8	yes	complemented
61	C₂×C₃×Q₈	—	7	2	no	—
62	C₂×C₃×Q₈	—	7	4	no	—
63	C₂×C₃×Q₈	—	7	2	no	—
64	C₃×C₄⋊C₄	—	7	4	no	—
65	C₃×C₄⋊C₄	—	7	8	no	—
66	C₃×C₄⋊C₄	—	7	2	no	—
67	C₃×C₄⋊C₄	—	7	4	no	—
68	C₃×C₄²	—	7	4	no	—
69	C₃×C₄²	—	7	4	no	—
70	C₃×C₄²	—	7	1	no	—
71	C₂×ΓL(1,𝔽₈)	C₂²×Q₈	1	2	yes	—
72	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
73	C₄⋊Q₈	—	7	4	no	—
74	C₄⋊Q₈	—	7	4	no	—
75	C₄⋊Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic
76	C₄×Q₈	—	7	4	no	—
77	C₄×Q₈	—	7	2	no	—
78	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
79	C₄×C₇	—	2	4	no	—
80	C₄×C₇	—	2	1	no	—
81	C₄×C₇	C₂×C₃×Q₈	1	4	yes	—
82	Dic₇	—	2	4	no	—
83	Dic₇	C₂×C₃×Q₈	1	8	yes	—
84	C₃×Q₈	—	7	8	no	—
85	C₃×Q₈	—	28	2	no	—
86	C₃×Q₈	—	28	4	no	—
87	C₂×C₃×C₄	—	7	4	no	—
88	C₂×C₃×C₄	—	7	4	no	—
89	C₂×C₃×C₄	—	7	2	no	—
90	C₂×C₃×C₄	—	7	4	no	—
91	C₂×C₃×C₄	—	7	1	no	—
92	ΓL(1,𝔽₈)	Q₈²	1	1	yes	characteristic, complemented, Hall
93	C₂×Q₈	—	7	2	no	—
94	C₂×Q₈	—	7	4	no	—
95	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	2	yes	—
96	C₄⋊C₄	—	7	8	no	—
97	C₄⋊C₄	—	7	4	no	—
98	C₄⋊C₄	—	7	2	no	—
99	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	4	yes	—
100	C₄²	—	7	4	no	—
101	C₄²	—	7	4	no	—
102	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
103	C₂×C₇	C₂²×C₃×Q₈	1	2	yes	—
104	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
105	C₂²×C₃	—	7	1	no	—
106	C₃×C₄	—	7	8	no	—
107	C₃×C₄	—	7	4	no	—
108	C₃×C₄	—	14	1	no	—
109	C₃×C₄	—	14	4	no	—
110	C₃×C₄	—	14	4	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	7	8	no	—
113	Q₈	—	28	4	no	—
114	C₂×C₄	—	7	4	no	—
115	C₂×C₄	—	7	4	no	—
116	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	4	yes	—
117	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
118	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
119	C₇	C₃×Q₈²	1	1	yes	characteristic, complemented, Sylow
120	C₂×C₃	—	7	1	no	—
121	C₂×C₃	—	7	2	no	—
122	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	2	4	no	—
125	C₄	—	7	8	no	—
126	C₄	—	14	4	no	—
127	C₄	Q₈×AGL(1,𝔽₇)	1	4	yes	—
128	C₃	—	7	1	no	Sylow
129	C₂	#6548⋊C₃	1	1	yes	characteristic
130	C₂	C₂×Q₈×AGL(1,𝔽₇)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

ΓL(1,𝔽₈)×C₄⋊Q₈

C₂

yes

maximal, characteristic

#6507⋊C₃

C₂

yes

maximal

#6451⋊C₃

C₂

yes

maximal

#6450⋊C₃

C₂

yes

maximal

C₄×ΓL(1,𝔽₈)⋊Q₈

C₂

yes

maximal

#90413

C₃

yes

maximal, characteristic, complemented, Hall

C₂×Q₈×ΓL(1,𝔽₈)

C₂²

yes

—

C₂×ΓL(1,𝔽₈)⋊Q₈

C₂²

yes

—

C₂×ΓL(1,𝔽₈)⋊Q₈

C₂²

yes

—

C₄⋊C₄×ΓL(1,𝔽₈)

C₂²

yes

—

C₄²×ΓL(1,𝔽₈)

C₂²

yes

characteristic

C₄⋊(ΓL(1,𝔽₈)⋊C₄)

C₂²

yes

—

(ΓL(1,𝔽₈)⋊C₄)⋊C₄

C₂²

yes

—

(ΓL(1,𝔽₈)⋊C₄)⋊C₄

C₂²

yes

—

C₄×ΓL(1,𝔽₈)⋊C₄

C₂²

yes

—

C₄×ΓL(1,𝔽₈)⋊C₄

C₂²

yes

—

C₇×C₄⋊Q₈

C₂×C₃

yes

characteristic

#6507

C₂×C₃

yes

—

#6451

C₂×C₃

yes

—

#6450

C₂×C₃

yes

—

C₄×Dic₁₄

C₂×C₃

yes

—

C₃×Q₈²

—

maximal, Hall

Q₈×ΓL(1,𝔽₈)

—

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

—

ΓL(1,𝔽₈)⋊Q₈

—

ΓL(1,𝔽₈)⋊Q₈

Q₈

yes

complemented

C₂×C₇×Q₈

C₂²×C₃

yes

—

C₂×Dic₁₄

C₂²×C₃

yes

—

C₂×Dic₁₄

C₂²×C₃

yes

—

C₇×C₄⋊C₄

C₂²×C₃

yes

—

C₄²×C₇

C₂²×C₃

yes

characteristic

C₄⋊Dic₇

C₂²×C₃

yes

—

Dic₇⋊C₄

C₂²×C₃

yes

—

Dic₇⋊C₄

C₂²×C₃

yes

—

C₄×Dic₇

C₂²×C₃

yes

—

C₄×Dic₇

C₂²×C₃

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

—

C₂²×ΓL(1,𝔽₈)

C₂⁴

yes

—

—

C₂×Q₈

yes

—

ΓL(1,𝔽₈)⋊C₄

—

ΓL(1,𝔽₈)⋊C₄

C₂×Q₈

yes

—

Q₈²

—

Sylow

C₇×Q₈

—

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

—

Dic₁₄

—

Dic₁₄

C₃×Q₈

yes

complemented

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₂×ΓL(1,𝔽₈)

C₂²×Q₈

yes

—

C₂×ΓL(1,𝔽₈)

H(5,𝔽₂)

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

AGL(1,𝔽₇)

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂²×C₇

C₂⁴×C₃

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×C₃×Q₈

yes

—

Dic₇

—

Dic₇

C₂×C₃×Q₈

yes

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

ΓL(1,𝔽₈)

Q₈²

yes

characteristic, complemented, Hall

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×AGL(1,𝔽₇)

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×AGL(1,𝔽₇)

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂×AGL(1,𝔽₇)

yes

characteristic

103

C₂×C₇

C₂²×C₃×Q₈

yes

—

104

C₂×C₇

C₃×H(5,𝔽₂)

yes

characteristic

105

C₂²×C₃

—

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

—

117

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

characteristic

118

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

—

119

C₇

C₃×Q₈²

yes

characteristic, complemented, Sylow

120

C₂×C₃

—

121

C₂×C₃

—

122

C₂²

C₂³×AGL(1,𝔽₇)

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

Q₈×AGL(1,𝔽₇)

yes

—

128

C₃

—

Sylow

129

C₂

#6548⋊C₃

yes

characteristic

130

C₂

C₂×Q₈×AGL(1,𝔽₇)

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×C₄⋊Q₈	C₂×C₃
prime radical	C₂²×ΓL(1,𝔽₈)	C₂⁴
hypercenter	C₄⋊Q₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×Q₈²
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×AGL(1,𝔽₇)

commutator, socle, cosocle

C₂²×C₇

C₂⁴×C₃

Fitting

C₇×C₄⋊Q₈

C₂×C₃

prime radical

C₂²×ΓL(1,𝔽₈)

C₂⁴

hypercenter

C₄⋊Q₈

AGL(1,𝔽₇)

hypocenter

C₇

C₃×Q₈²

LCS

G ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
6. #90413	C₃	semidirect
30. ΓL(1,𝔽₈)⋊Q₈	Q₈	semidirect
60. Dic₁₄	C₃×Q₈	semidirect
92. ΓL(1,𝔽₈)	Q₈²	semidirect
119. C₇	C₃×Q₈²	semidirect

H ≅ G/N

type

6. #90413

C₃

semidirect

30. ΓL(1,𝔽₈)⋊Q₈

Q₈

semidirect

60. Dic₁₄

C₃×Q₈

semidirect

92. ΓL(1,𝔽₈)

Q₈²

semidirect

119. C₇

C₃×Q₈²

semidirect

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
#6507⋊C₃	2
#6451⋊C₃	2
#6450⋊C₃	2
C₄×ΓL(1,𝔽₈)⋊Q₈	2
#90413	3
C₃×Q₈²	7

subgroup

index

ΓL(1,𝔽₈)×C₄⋊Q₈

C₄×ΓL(1,𝔽₈)⋊Q₈

C₃×Q₈²

Subgroups — #90413⋊C₃ #13057457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90413⋊C3 #13057457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90413⋊C₃ #13057457