Group #13115983 — subgroups

Subgroups — #13115983

Order 1664.

294 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×#88061	C₂	1	2	yes	maximal
3	#11718600	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718143	C₂	1	2	yes	maximal, complemented
6	C₂²×D₅₂	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#6103	—	2	2	no	—
10	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×#6101	—	2	2	no	—
13	#88124	—	2	4	no	—
14	#88061	—	2	4	no	—
15	C₂³×D₁₃	—	2	1	no	—
16	C₂²×C₄×C₁₃	D₄	1	2	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₃	—	2	2	no	—
19	C₂²×Dic₁₃	—	2	1	no	—
20	C₂×D₅₂	—	2	2	no	—
21	C₂×D₅₂	—	2	2	no	—
22	C₂×D₅₂	—	2	1	no	—
23	C₂×D₅₂	Q₈	1	2	yes	—
24	C₂×C₈×C₁₃	—	2	2	no	—
25	C₂×C₈×C₁₃	—	2	2	no	—
26	C₂×C₈×C₁₃	—	2	2	no	—
27	C₄²×C₁₃	—	2	2	no	—
28	#6103	—	4	4	no	—
29	C₄⋊Dic₁₃	—	2	2	no	—
30	C₄⋊Dic₁₃	—	2	2	no	—
31	C₄⋊Dic₁₃	D₄	1	4	yes	—
32	#6101	—	2	8	no	—
33	#2025	—	13	1	no	maximal, Sylow
34	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×D₁₃	—	2	2	no	—
36	C₂²×D₁₃	—	4	1	no	—
37	C₂²×D₁₃	—	4	1	no	—
38	C₂²×D₁₃	—	4	2	no	—
39	C₂²×D₁₃	—	4	2	no	—
40	C₂×C₄×C₁₃	—	2	2	no	—
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	—	2	2	no	—
43	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₃	#41	1	2	yes	—
46	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₂×Dic₁₃	—	2	4	no	—
48	C₂×Dic₁₃	—	4	2	no	—
49	C₂×Dic₁₃	—	4	2	no	—
50	C₂×Dic₁₃	—	4	1	no	—
51	C₂×Dic₁₃	—	4	2	no	—
52	D₅₂	—	2	4	no	—
53	D₅₂	—	4	2	no	—
54	C₈×C₁₃	—	4	2	no	—
55	C₂×#125	—	13	1	no	—
56	C₂×#102	—	13	2	no	—
57	#390	—	13	1	no	—
58	#340	—	13	2	no	—
59	#336	D₁₃	1	1	yes	characteristic, complemented
60	C₂²×C₁₃	#120	1	1	yes	characteristic
61	C₂²×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂²×C₁₃	#124	1	1	yes	characteristic
63	C₂²×C₁₃	#135	1	2	yes	—
64	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
65	C₂×D₁₃	—	4	2	no	—
66	C₂×D₁₃	—	4	4	no	—
67	C₂×D₁₃	—	8	2	no	—
68	C₂×D₁₃	—	8	1	no	—
69	C₂×D₁₃	—	8	1	no	—
70	C₄×C₁₃	—	4	2	no	—
71	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₁₃	#124	1	2	yes	—
73	Dic₁₃	—	4	4	no	—
74	Dic₁₃	—	8	1	no	—
75	C₂²×D₄	—	13	1	no	—
76	C₂²×C₈	C₂×D₁₃	1	2	yes	—
77	#125	—	26	4	no	—
78	C₂×C₄⋊C₄	—	13	2	no	—
79	C₂×C₄⋊C₄	—	13	1	no	—
80	C₂×C₄⋊C₄	—	26	2	no	—
81	C₂×C₂²⋊C₄	—	26	2	no	—
82	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
83	#102	—	26	4	no	—
84	C₂×C₁₃	#393	1	1	yes	characteristic
85	C₂×C₁₃	#449	1	2	yes	—
86	C₂×C₁₃	#478	1	2	yes	—
87	C₂×C₁₃	#488	1	1	yes	characteristic
88	C₂×C₁₃	#490	1	1	yes	characteristic
89	D₁₃	—	8	2	no	—
90	C₂⁴	—	26	1	no	—
91	C₂×D₄	—	13	2	no	—
92	C₂×D₄	—	26	2	no	—
93	C₂×D₄	—	26	2	no	—
94	C₂×D₄	—	26	1	no	—
95	C₂²×C₄	—	26	2	no	—
96	C₂²×C₄	—	26	1	no	—
97	C₂²×C₄	D₅₂	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₄⋊C₄	—	13	4	no	—
103	C₄⋊C₄	—	26	8	no	—
104	C₄⋊C₄	—	26	2	no	—
105	C₄⋊C₄	—	26	2	no	—
106	C₂²⋊C₄	—	52	4	no	—
107	C₄²	—	2	2	no	—
108	C₁₃	#2025	1	1	yes	characteristic, complemented, Sylow
109	C₂³	—	26	2	no	—
110	C₂³	—	52	1	no	—
111	C₂³	—	52	1	no	—
112	C₂³	—	52	2	no	—
113	C₂³	—	52	2	no	—
114	C₂³	C₂×D₅₂	1	1	yes	characteristic
115	D₄	—	26	4	no	—
116	D₄	—	52	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	26	4	no	—
121	C₂×C₄	—	52	2	no	—
122	C₂×C₄	—	52	2	no	—
123	C₂×C₄	—	52	1	no	—
124	C₂×C₄	—	52	2	no	—
125	C₂×C₄	#6127	1	2	yes	—
126	C₂×C₄	D₄×D₁₃	1	2	yes	—
127	C₂×C₄	#6129	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₃	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₂²	—	52	4	no	—
131	C₂²	—	52	2	no	—
132	C₂²	—	104	1	no	—
133	C₂²	—	104	1	no	—
134	C₂²	—	104	2	no	—
135	C₂²	#88129	1	1	yes	characteristic
136	C₂²	#88136	1	1	yes	characteristic
137	C₂²	#88151	1	1	yes	characteristic
138	C₂²	#88158	1	2	yes	—
139	C₂²	#88162	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	52	4	no	—
142	C₄	—	104	1	no	—
143	C₄	#88139	1	2	yes	—
144	C₄	#88150	1	2	yes	—
145	C₂	—	104	2	no	—
146	C₂	#11718334	1	1	yes	characteristic
147	C₂	#11718358	1	1	yes	characteristic
148	C₂	#11718374	1	1	yes	characteristic
149	C₂	#11718395	1	2	yes	—
150	C₂	#11718508	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88124

C₂

yes

maximal, characteristic, complemented

C₂×#88061

C₂

yes

maximal

#11718600

C₂

yes

maximal, characteristic

C₁₃×#336

C₂

yes

maximal, characteristic, complemented

#11718143

C₂

yes

maximal, complemented

C₂²×D₅₂

C₂²

yes

characteristic

C₂²×C₈×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6103

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

—

#88124

—

#88061

—

C₂³×D₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

Q₈

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

#6103

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

#6101

—

#2025

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

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

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₈×C₁₃

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

#120

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

#124

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×D₄

—

C₂²×C₈

C₂×D₁₃

yes

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#102

—

C₂×C₁₃

#393

yes

characteristic

C₂×C₁₃

#449

yes

—

C₂×C₁₃

#478

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

D₁₃

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₅₂

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×D₅₂

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

#6127

yes

—

126

C₂×C₄

D₄×D₁₃

yes

—

127

C₂×C₄

#6129

yes

characteristic

128

C₂×C₄

Q₈×D₁₃

yes

characteristic

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

104

—

133

C₂²

—

104

—

134

C₂²

—

104

—

135

C₂²

#88129

yes

characteristic

136

C₂²

#88136

yes

characteristic

137

C₂²

#88151

yes

characteristic

138

C₂²

#88158

yes

—

139

C₂²

#88162

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

104

—

143

C₄

#88139

yes

—

144

C₄

#88150

yes

—

145

C₂

—

104

—

146

C₂

#11718334

yes

characteristic

147

C₂

#11718358

yes

characteristic

148

C₂

#11718374

yes

characteristic

149

C₂

#11718395

yes

—

150

C₂

#11718508

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₂
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#336	C₂
socle	C₂³×C₁₃	C₂×D₄
prime radical	C₂²×D₅₂	C₂²
hypercenter	#336	D₁₃
hypocenter	C₁₃	#2025
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₅₂

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#336

C₂

socle

C₂³×C₁₃

C₂×D₄

prime radical

C₂²×D₅₂

C₂²

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
1. C₂×#88124	C₂	semidirect
4. C₁₃×#336	C₂	semidirect
5. #11718143	C₂	semidirect
59. #336	D₁₃	semidirect
108. C₁₃	#2025	semidirect

H ≅ G/N

type

1. C₂×#88124

C₂

semidirect

4. C₁₃×#336

C₂

semidirect

5. #11718143

C₂

semidirect

59. #336

D₁₃

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
C₂×#88061	2
#11718600	2
C₁₃×#336	2
#11718143	2
#2025	13

subgroup

index

C₁₃×#336

Subgroups — #13115983

Conjugacy classes of subgroups

Complements

Maximal subgroups