Group #13052084 — subgroups

Subgroups — #89963⋊C₃ #13052084

Order 1344.

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 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	C₂×Q₁₆×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	#6510⋊C₃	C₂	1	2	yes	maximal, complemented
3	C₂×#1143⋊C₃	C₂	1	2	yes	maximal, complemented
4	C₂×D₅₆⋊C₃	C₂	1	1	yes	maximal, characteristic
5	C₈×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#89963	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	2	yes	—
9	Q₁₆×ΓL(1,𝔽₈)	—	2	2	no	—
10	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
11	#1143⋊C₃	—	2	4	no	—
12	(C₂×AGL(1,𝔽₇))⋊C₄	—	2	4	no	—
13	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	1	yes	characteristic
15	D₅₆⋊C₃	—	2	2	no	—
16	C₂×C₇×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
17	#6510	C₂×C₃	1	2	yes	complemented
18	C₂×#1143	C₂×C₃	1	2	yes	complemented
19	C₂×D₅₆	C₂×C₃	1	1	yes	characteristic
20	C₈×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃×#495	—	7	1	no	maximal, Hall
22	C₂²×AGL(1,𝔽₇)	—	2	2	no	—
23	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
24	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
25	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
26	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
27	C₂×ΓL(1,𝔽₈)⋊C₄	D₄	1	2	yes	—
28	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
29	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
30	C₈×ΓL(1,𝔽₈)	D₄	1	2	yes	—
31	ΓL(1,𝔽₈)⋊C₈	D₄	1	2	yes	—
32	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
33	C₂×D₂₈	C₂²×C₃	1	2	yes	—
34	C₇×Q₁₆	—	2	2	no	—
35	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
36	#1143	—	2	4	no	—
37	(C₂×D₇)⋊C₄	—	2	4	no	—
38	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
39	C₂×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
40	D₅₆	—	2	2	no	—
41	C₂×C₃×Q₁₆	—	7	1	no	—
42	C₂×C₃×SD₁₆	—	7	2	no	—
43	C₂×C₃×D₈	—	7	1	no	—
44	C₃×#124	—	7	2	no	—
45	C₃×C₄×C₈	—	7	1	no	—
46	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
47	C₂×AGL(1,𝔽₇)	—	4	2	no	—
48	C₂×AGL(1,𝔽₇)	—	4	4	no	—
49	C₄×ΓL(1,𝔽₈)	—	4	2	no	—
50	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
51	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
52	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
53	#495	—	7	1	no	Sylow
54	C₂²×D₇	—	2	2	no	—
55	C₇×Q₈	—	2	2	no	—
56	C₇×Q₈	—	2	2	no	—
57	C₂×C₄×C₇	—	2	2	no	—
58	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
59	C₂×Dic₇	C₃×D₄	1	2	yes	—
60	D₂₈	—	2	2	no	—
61	D₂₈	—	2	2	no	—
62	C₇×C₈	C₃×D₄	1	2	yes	—
63	C₇⋊C₈	C₃×D₄	1	2	yes	—
64	C₂×C₃×Q₈	—	7	2	no	—
65	C₂×C₃×D₄	—	7	2	no	—
66	C₃×Q₁₆	—	14	2	no	—
67	C₃×SD₁₆	—	14	4	no	—
68	C₃×D₈	—	14	2	no	—
69	C₂×C₃×C₈	—	7	1	no	—
70	C₂×C₃×C₈	—	7	1	no	—
71	C₃×C₂²⋊C₄	—	14	4	no	—
72	C₃×C₄²	—	7	1	no	—
73	C₂×ΓL(1,𝔽₈)	#127	1	1	yes	characteristic
74	C₂×ΓL(1,𝔽₈)	#135	1	2	yes	—
75	AGL(1,𝔽₇)	—	8	2	no	—
76	C₂×Q₁₆	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
77	C₂×SD₁₆	—	7	2	no	—
78	C₂×D₈	—	7	1	no	—
79	#124	—	7	2	no	—
80	C₄×C₈	—	7	1	no	—
81	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
82	C₂×D₇	—	4	2	no	—
83	C₂×D₇	—	4	4	no	—
84	C₄×C₇	—	4	2	no	—
85	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
86	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
87	Dic₇	—	2	2	no	—
88	C₂³×C₃	—	14	2	no	—
89	C₃×Q₈	—	14	2	no	—
90	C₃×Q₈	—	14	2	no	—
91	C₃×D₄	—	14	2	no	—
92	C₃×D₄	—	14	2	no	—
93	C₂×C₃×C₄	—	7	2	no	—
94	C₂×C₃×C₄	—	7	1	no	—
95	C₂×C₃×C₄	—	14	2	no	—
96	C₃×C₈	—	7	2	no	—
97	C₃×C₈	—	7	2	no	—
98	ΓL(1,𝔽₈)	#495	1	1	yes	characteristic, complemented, Hall
99	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	2	yes	—
100	C₂×D₄	—	7	2	no	—
101	Q₁₆	—	2	2	no	—
102	SD₁₆	—	14	4	no	—
103	D₈	—	14	2	no	—
104	C₂×C₈	—	7	1	no	—
105	C₂×C₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
106	C₂²⋊C₄	—	14	4	no	—
107	C₄²	—	7	1	no	—
108	C₂×C₇	C₃×#127	1	1	yes	characteristic
109	C₂×C₇	C₃×#135	1	2	yes	—
110	D₇	—	8	2	no	—
111	C₂²×C₃	—	7	1	no	—
112	C₂²×C₃	—	28	2	no	—
113	C₂²×C₃	—	28	4	no	—
114	C₃×C₄	—	7	1	no	—
115	C₃×C₄	—	7	1	no	—
116	C₃×C₄	—	14	2	no	—
117	C₃×C₄	—	28	2	no	—
118	C₂³	—	14	2	no	—
119	Q₈	—	2	2	no	—
120	Q₈	—	2	2	no	—
121	D₄	—	14	2	no	—
122	D₄	—	14	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	7	2	no	—
125	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
126	C₈	—	7	2	no	—
127	C₈	(C₂²⋊D₇)⋊C₃	1	2	yes	—
128	C₇	C₃×#495	1	1	yes	characteristic, complemented, Sylow
129	C₂×C₃	—	7	2	no	—
130	C₂×C₃	—	7	1	no	—
131	C₂×C₃	—	56	2	no	—
132	C₂²	—	28	4	no	—
133	C₂²	—	28	2	no	—
134	C₂²	D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	14	2	no	—
137	C₄	D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
138	C₄	C₂×(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic
139	C₃	—	7	1	no	Sylow
140	C₂	—	56	2	no	—
141	C₂	#6482⋊C₃	1	2	yes	—
142	C₂	#6503⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×ΓL(1,𝔽₈)

C₂

yes

maximal, characteristic, complemented

#6510⋊C₃

C₂

yes

maximal, complemented

C₂×#1143⋊C₃

C₂

yes

maximal, complemented

C₂×D₅₆⋊C₃

C₂

yes

maximal, characteristic

C₈×ΓL(1,𝔽₈)⋊C₄

C₂

yes

maximal, characteristic, complemented

#89963

C₃

yes

maximal, characteristic, complemented, Hall

C₂×Q₈×ΓL(1,𝔽₈)

C₂²

yes

—

C₂×C₄⋊AGL(1,𝔽₇)

C₂²

yes

—

Q₁₆×ΓL(1,𝔽₈)

—

C₂×C₈×ΓL(1,𝔽₈)

C₂²

yes

characteristic

#1143⋊C₃

—

(C₂×AGL(1,𝔽₇))⋊C₄

—

C₄×ΓL(1,𝔽₈)⋊C₄

C₂²

yes

characteristic

C₂×ΓL(1,𝔽₈)⋊C₈

C₂²

yes

characteristic

D₅₆⋊C₃

—

C₂×C₇×Q₁₆

C₂×C₃

yes

characteristic, complemented

#6510

C₂×C₃

yes

complemented

C₂×#1143

C₂×C₃

yes

complemented

C₂×D₅₆

C₂×C₃

yes

characteristic

C₈×Dic₇

C₂×C₃

yes

characteristic, complemented

C₃×#495

—

maximal, Hall

C₂²×AGL(1,𝔽₇)

—

Q₈×ΓL(1,𝔽₈)

—

Q₈×ΓL(1,𝔽₈)

—

C₂×C₄×ΓL(1,𝔽₈)

—

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

characteristic

C₂×ΓL(1,𝔽₈)⋊C₄

D₄

yes

—

C₄⋊AGL(1,𝔽₇)

—

C₄⋊AGL(1,𝔽₇)

—

C₈×ΓL(1,𝔽₈)

D₄

yes

—

ΓL(1,𝔽₈)⋊C₈

D₄

yes

—

C₂×C₇×Q₈

C₂²×C₃

yes

—

C₂×D₂₈

C₂²×C₃

yes

—

C₇×Q₁₆

—

C₂×C₇×C₈

C₂²×C₃

yes

characteristic

#1143

—

(C₂×D₇)⋊C₄

—

C₄×Dic₇

C₂²×C₃

yes

characteristic

C₂×C₇⋊C₈

C₂²×C₃

yes

characteristic

D₅₆

—

C₂×C₃×Q₁₆

—

C₂×C₃×SD₁₆

—

C₂×C₃×D₈

—

C₃×#124

—

C₃×C₄×C₈

—

C₂²×ΓL(1,𝔽₈)

C₂×D₄

yes

—

—

—

C₂×D₄

yes

characteristic

C₄×ΓL(1,𝔽₈)

C₂×D₄

yes

—

—

C₂²×D₇

—

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³×C₃

yes

characteristic

C₂×Dic₇

C₃×D₄

yes

—

D₂₈

—

D₂₈

—

C₇×C₈

C₃×D₄

yes

—

C₇⋊C₈

C₃×D₄

yes

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₃×Q₁₆

—

C₃×SD₁₆

—

C₃×D₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

yes

yes

—

AGL(1,𝔽₇)

—

C₂×Q₁₆

AGL(1,𝔽₇)

yes

characteristic, complemented

C₂×SD₁₆

—

C₂×D₈

—

#124

—

C₄×C₈

—

C₂²×C₇

C₂×C₃×D₄

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

C₂×C₃×D₄

yes

characteristic

C₄×C₇

C₂×C₃×D₄

yes

characteristic

Dic₇

—

C₂³×C₃

—

C₃×Q₈

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃×C₈

—

C₃×C₈

—

ΓL(1,𝔽₈)

#495

yes

characteristic, complemented, Hall

C₂×Q₈

C₂×AGL(1,𝔽₇)

yes

—

100

C₂×D₄

—

101

Q₁₆

—

102

SD₁₆

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂×AGL(1,𝔽₇)

yes

characteristic

106

C₂²⋊C₄

—

107

C₄²

—

108

C₂×C₇

C₃×#127

yes

characteristic

109

C₂×C₇

C₃×#135

yes

—

110

D₇

—

111

C₂²×C₃

—

112

C₂²×C₃

—

113

C₂²×C₃

—

114

C₃×C₄

—

115

C₃×C₄

—

116

C₃×C₄

—

117

C₃×C₄

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

characteristic

126

C₈

—

127

C₈

(C₂²⋊D₇)⋊C₃

yes

—

128

C₇

C₃×#495

yes

characteristic, complemented, Sylow

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

—

132

C₂²

—

133

C₂²

—

134

C₂²

D₄×AGL(1,𝔽₇)

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

D₄×AGL(1,𝔽₇)

yes

characteristic

138

C₄

C₂×(C₂²⋊D₇)⋊C₃

yes

characteristic

139

C₃

—

Sylow

140

C₂

—

141

C₂

#6482⋊C₃

yes

—

142

C₂

#6503⋊C₃

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×AGL(1,𝔽₇)
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₇)
Fitting	C₂×C₇×Q₁₆	C₂×C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂×D₅₆⋊C₃	C₂
hypercenter	C₂×Q₁₆	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#495
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×AGL(1,𝔽₇)

commutator, cosocle

C₂×C₄×C₇

C₂³×C₃

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₇)

Fitting

C₂×C₇×Q₁₆

C₂×C₃

socle

C₂²×C₇

C₂×C₃×D₄

prime radical

C₂×D₅₆⋊C₃

C₂

C₂×Q₁₆

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
1. C₂×Q₁₆×ΓL(1,𝔽₈)	C₂	semidirect
2. #6510⋊C₃	C₂	semidirect
3. C₂×#1143⋊C₃	C₂	semidirect
5. C₈×ΓL(1,𝔽₈)⋊C₄	C₂	semidirect
6. #89963	C₃	semidirect
16. C₂×C₇×Q₁₆	C₂×C₃	semidirect
17. #6510	C₂×C₃	semidirect
18. C₂×#1143	C₂×C₃	semidirect
20. C₈×Dic₇	C₂×C₃	semidirect
76. C₂×Q₁₆	AGL(1,𝔽₇)	semidirect
98. ΓL(1,𝔽₈)	#495	semidirect
128. C₇	C₃×#495	semidirect

H ≅ G/N

type

1. C₂×Q₁₆×ΓL(1,𝔽₈)

C₂

semidirect

2. #6510⋊C₃

C₂

semidirect

3. C₂×#1143⋊C₃

C₂

semidirect

5. C₈×ΓL(1,𝔽₈)⋊C₄

C₂

semidirect

6. #89963

C₃

semidirect

16. C₂×C₇×Q₁₆

C₂×C₃

semidirect

17. #6510

C₂×C₃

semidirect

18. C₂×#1143

C₂×C₃

semidirect

20. C₈×Dic₇

C₂×C₃

semidirect

76. C₂×Q₁₆

AGL(1,𝔽₇)

semidirect

98. ΓL(1,𝔽₈)

#495

semidirect

128. C₇

C₃×#495

semidirect

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

subgroup	index
C₂×Q₁₆×ΓL(1,𝔽₈)	2
#6510⋊C₃	2
C₂×#1143⋊C₃	2
C₂×D₅₆⋊C₃	2
C₈×ΓL(1,𝔽₈)⋊C₄	2
#89963	3
C₃×#495	7

subgroup

index

C₂×Q₁₆×ΓL(1,𝔽₈)

#6510⋊C₃

C₂×#1143⋊C₃

C₂×D₅₆⋊C₃

C₈×ΓL(1,𝔽₈)⋊C₄

#89963

C₃×#495

Subgroups — #89963⋊C₃ #13052084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #89963⋊C3 #13052084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #89963⋊C₃ #13052084