Group #13115982 — subgroups

Subgroups — #13115982

Order 1664.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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×#88124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#390	C₂	1	1	yes	maximal, characteristic
3	C₂×#88072	C₂	1	2	yes	maximal
4	#11718121	C₂	1	2	yes	maximal, complemented
5	#11718116	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×#6101	—	2	2	no	—
12	C₂²×#1055	C₂²	1	2	yes	—
13	#88124	—	2	4	no	—
14	#88072	—	2	4	no	—
15	C₂⁴×C₁₃	—	2	1	no	—
16	C₂×D₄×C₁₃	—	2	2	no	—
17	C₂×D₄×C₁₃	—	2	2	no	—
18	C₂×D₄×C₁₃	—	2	1	no	—
19	C₂×D₄×C₁₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₁₃	—	2	1	no	—
21	C₂²×C₄×C₁₃	D₄	1	2	yes	—
22	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₃	—	2	2	no	—
24	C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₁₃×C₂²⋊C₄	—	4	4	no	—
26	C₄²×C₁₃	—	2	2	no	—
27	C₄⋊Dic₁₃	—	2	2	no	—
28	C₄⋊Dic₁₃	D₄	1	4	yes	—
29	#6101	—	2	8	no	—
30	C₂×#1055	—	2	2	no	—
31	C₂×#1055	—	2	2	no	—
32	C₂×#1055	—	2	2	no	—
33	#2025	—	13	1	no	maximal, Sylow
34	C₂³×C₁₃	—	2	2	no	—
35	C₂³×C₁₃	—	4	1	no	—
36	C₂³×C₁₃	—	4	1	no	—
37	C₂³×C₁₃	—	4	2	no	—
38	C₂³×C₁₃	—	4	2	no	—
39	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
40	D₄×C₁₃	—	2	4	no	—
41	D₄×C₁₃	—	4	2	no	—
42	C₂×C₄×C₁₃	—	2	2	no	—
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	4	1	no	—
46	C₂×C₄×C₁₃	—	4	2	no	—
47	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₃	#41	1	2	yes	—
50	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
51	C₂×Dic₁₃	—	2	4	no	—
52	C₂×Dic₁₃	—	4	2	no	—
53	C₂×Dic₁₃	—	4	2	no	—
54	#1055	—	4	2	no	—
55	C₂×#125	—	13	1	no	—
56	C₂×#102	—	13	2	no	—
57	#390	D₁₃	1	1	yes	characteristic
58	#340	—	13	2	no	—
59	#336	—	13	1	no	—
60	C₂²×C₁₃	—	4	2	no	—
61	C₂²×C₁₃	—	4	4	no	—
62	C₂²×C₁₃	—	8	2	no	—
63	C₂²×C₁₃	—	8	1	no	—
64	C₂²×C₁₃	—	8	1	no	—
65	C₂²×C₁₃	#120	1	1	yes	characteristic
66	C₂²×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂²×C₁₃	#124	1	1	yes	characteristic
68	C₂²×C₁₃	#135	1	2	yes	—
69	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
70	C₄×C₁₃	—	4	2	no	—
71	C₄×C₁₃	—	8	1	no	—
72	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₁₃	#124	1	2	yes	—
74	Dic₁₃	—	4	4	no	—
75	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
76	C₂²×C₈	—	13	2	no	—
77	#125	—	26	4	no	—
78	C₂×C₄⋊C₄	—	13	2	no	—
79	C₂×C₄⋊C₄	—	26	2	no	—
80	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	—	2	2	no	—
82	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
83	#102	—	26	4	no	—
84	C₂×C₁₃	—	8	2	no	—
85	C₂×C₁₃	#393	1	1	yes	characteristic
86	C₂×C₁₃	#449	1	2	yes	—
87	C₂×C₁₃	#478	1	2	yes	—
88	C₂×C₁₃	#488	1	1	yes	characteristic
89	C₂×C₁₃	#490	1	1	yes	characteristic
90	C₂⁴	—	2	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	Dic₂₆	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	26	2	no	—
97	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
99	C₂×C₈	—	26	2	no	—
100	C₂×C₈	—	26	2	no	—
101	C₂×C₈	—	26	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	13	4	no	—
104	C₄⋊C₄	—	26	8	no	—
105	C₄⋊C₄	—	26	2	no	—
106	C₂²⋊C₄	—	4	4	no	—
107	C₄²	—	2	2	no	—
108	C₁₃	#2025	1	1	yes	characteristic, complemented, Sylow
109	C₂³	—	2	2	no	—
110	C₂³	—	4	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	1	no	—
113	C₂³	—	4	2	no	—
114	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
115	D₄	—	2	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	26	4	no	—
123	C₂×C₄	—	52	2	no	—
124	C₂×C₄	—	52	2	no	—
125	C₂×C₄	C₂×Dic₂₆	1	1	yes	characteristic
126	C₂×C₄	#6127	1	1	yes	characteristic
127	C₂×C₄	D₄×D₁₃	1	2	yes	—
128	C₂×C₄	#6129	1	2	yes	—
129	C₈	—	52	2	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	8	2	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	#88178	1	1	yes	characteristic
136	C₂²	#88186	1	2	yes	—
137	C₂²	#88190	1	1	yes	characteristic
138	C₂²	#88191	1	1	yes	characteristic
139	C₂²	#88204	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	8	1	no	—
142	C₄	—	52	4	no	—
143	C₄	#88132	1	2	yes	—
144	C₄	#88139	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	#11718608	1	1	yes	characteristic
147	C₂	#11718670	1	2	yes	—
148	C₂	#11718698	1	2	yes	—
149	C₂	#11718717	1	1	yes	characteristic
150	C₂	#11718732	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88124

C₂

yes

maximal, characteristic, complemented

C₁₃×#390

C₂

yes

maximal, characteristic

C₂×#88072

C₂

yes

maximal

#11718121

C₂

yes

maximal, complemented

#11718116

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₃×C₂²⋊C₄

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

—

C₂²×#1055

C₂²

yes

—

#88124

—

#88072

—

C₂⁴×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

Q₈

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

—

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

—

—

—

D₁₃

yes

characteristic

#340

—

#336

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#120

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

#124

yes

—

Dic₁₃

—

C₂²×D₄

C₂×D₁₃

yes

characteristic

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#102

—

C₂×C₁₃

—

C₂×C₁₃

#393

yes

characteristic

C₂×C₁₃

#449

yes

—

C₂×C₁₃

#478

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₂₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₁₃

#2025

yes

characteristic, complemented, Sylow

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×Dic₂₆

yes

characteristic

126

C₂×C₄

#6127

yes

characteristic

127

C₂×C₄

D₄×D₁₃

yes

—

128

C₂×C₄

#6129

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#88178

yes

characteristic

136

C₂²

#88186

yes

—

137

C₂²

#88190

yes

characteristic

138

C₂²

#88191

yes

characteristic

139

C₂²

#88204

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#88132

yes

—

144

C₄

#88139

yes

—

145

C₂

—

146

C₂

#11718608

yes

characteristic

147

C₂

#11718670

yes

—

148

C₂

#11718698

yes

—

149

C₂

#11718717

yes

characteristic

150

C₂

#11718732

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#390	C₂
socle	C₂³×C₁₃	C₂×D₄
prime radical	C₂²×D₄×C₁₃	C₂²
hypercenter	#390	D₁₃
hypocenter	C₁₃	#2025
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#390

C₂

socle

C₂³×C₁₃

C₂×D₄

prime radical

C₂²×D₄×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#88124	C₂	semidirect
4. #11718121	C₂	semidirect
5. #11718116	C₂	semidirect
108. C₁₃	#2025	semidirect

H ≅ G/N

type

1. C₂×#88124

C₂

semidirect

4. #11718121

C₂

semidirect

5. #11718116

C₂

semidirect

108. C₁₃

#2025

semidirect

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

subgroup	index
C₂×#88124	2
C₁₃×#390	2
C₂×#88072	2
#11718121	2
#11718116	2
#2025	13

subgroup

index

C₁₃×#390

Subgroups — #13115982

Conjugacy classes of subgroups

Complements

Maximal subgroups