Group #13113305 — subgroups

Subgroups — #13113305

Order 1664.

246 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₁₃×#532	C₂	1	1	yes	maximal, characteristic
2	#11718587	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊#1055	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#41	—	2	1	no	—
6	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₁₃×#127	C₄	1	1	yes	characteristic
8	C₁₃×#124	—	2	1	no	—
9	C₁₃×#121	—	2	1	no	—
10	C₁₃×#121	C₄	1	2	yes	—
11	C₄×D₄×C₁₃	—	2	1	no	—
12	C₄×D₄×C₁₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²⋊#1055	—	2	1	no	—
15	C₂²⋊#1055	—	2	1	no	—
16	C₄⋊#1055	—	2	1	no	—
17	C₄⋊#1055	C₂²	1	2	yes	—
18	#88043	—	2	1	no	—
19	C₄×#1055	C₂²	1	2	yes	—
20	C₁₃×#41	—	4	4	no	—
21	C₂×Q₈×C₁₃	—	2	1	no	—
22	C₂×D₄×C₁₃	—	2	2	no	—
23	C₂×D₄×C₁₃	—	2	1	no	—
24	C₂×D₄×C₁₃	—	2	1	no	—
25	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₃	—	2	2	no	—
27	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
30	C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₁₃×C₂²⋊C₄	—	2	2	no	—
33	C₁₃×C₂²⋊C₄	—	2	2	no	—
34	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×#1055	—	2	2	no	—
39	C₂×#1055	—	2	2	no	—
40	#1496	—	13	1	no	maximal, Sylow
41	C₂³×C₁₃	—	2	2	no	—
42	C₂³×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
43	Q₈×C₁₃	—	4	2	no	—
44	D₄×C₁₃	—	2	4	no	—
45	D₄×C₁₃	—	4	2	no	—
46	D₄×C₁₃	—	4	2	no	—
47	D₄×C₁₃	—	4	4	no	—
48	C₂×C₄×C₁₃	—	2	2	no	—
49	C₂×C₄×C₁₃	—	2	2	no	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	4	1	no	—
52	C₂×C₄×C₁₃	—	4	4	no	—
53	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	#1055	—	4	2	no	—
60	#1055	—	4	2	no	—
61	#532	D₁₃	1	1	yes	characteristic
62	#432	—	13	1	no	—
63	#431	—	13	1	no	—
64	D₄⋊C₈	—	13	4	no	—
65	C₂²×C₁₃	—	2	1	no	—
66	C₂²×C₁₃	—	2	1	no	—
67	C₂²×C₁₃	—	2	1	no	—
68	C₂²×C₁₃	—	4	2	no	—
69	C₂²×C₁₃	—	4	2	no	—
70	C₂²×C₁₃	—	4	2	no	—
71	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₁₃	—	2	1	no	—
73	C₄×C₁₃	—	2	1	no	—
74	C₄×C₁₃	—	2	1	no	—
75	C₄×C₁₃	—	2	1	no	—
76	C₄×C₁₃	—	4	2	no	—
77	C₄×C₁₃	#131	1	2	yes	—
78	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×#41	—	2	1	no	—
80	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
81	#127	Dic₁₃	1	1	yes	characteristic
82	#124	—	2	1	no	—
83	#121	—	2	1	no	—
84	#121	Dic₁₃	1	2	yes	—
85	C₄×D₄	—	2	1	no	—
86	C₄×D₄	C₂×D₁₃	1	2	yes	—
87	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
88	C₄⋊C₈	—	13	2	no	—
89	C₄⋊C₈	—	26	1	no	—
90	C₂²⋊C₈	—	26	1	no	—
91	C₂²⋊C₈	—	26	1	no	—
92	#97	—	26	1	no	—
93	C₄×C₈	—	13	2	no	—
94	C₂×C₁₃	—	4	1	no	—
95	C₂×C₁₃	—	8	2	no	—
96	C₂×C₁₃	#408	1	1	yes	characteristic
97	C₂×C₁₃	#418	1	1	yes	characteristic
98	C₂×C₁₃	#421	1	1	yes	characteristic
99	#41	—	4	4	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
107	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
108	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
109	C₂×C₈	—	26	2	no	—
110	C₂×C₈	—	26	2	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
116	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
117	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
118	C₄²	C₂²×D₁₃	1	1	yes	characteristic
119	C₁₃	#1496	1	1	yes	characteristic, complemented, Sylow
120	C₂³	—	2	2	no	—
121	C₂³	C₂²⋊Dic₁₃	1	1	yes	characteristic
122	Q₈	—	4	2	no	—
123	D₄	—	2	4	no	—
124	D₄	—	4	2	no	—
125	D₄	—	4	2	no	—
126	D₄	—	4	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
135	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
138	C₈	—	52	2	no	—
139	C₈	—	52	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	2	no	—
146	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	2	no	—
152	C₄	#88186	1	2	yes	—
153	C₄	#88202	1	2	yes	—
154	C₂	—	4	1	no	—
155	C₂	—	8	2	no	—
156	C₂	#11718876	1	1	yes	characteristic
157	C₂	#11718895	1	1	yes	characteristic
158	C₂	#11718897	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#532

C₂

yes

maximal, characteristic

#11718587

C₂

yes

maximal, characteristic, complemented

#11718586

C₂

yes

maximal, characteristic, complemented

D₄⋊#1055

C₂

yes

maximal, complemented

C₂×C₁₃×#41

—

C₁₃×C₄⋊Q₈

C₄

yes

characteristic

C₁₃×#127

C₄

yes

characteristic

C₁₃×#124

—

C₁₃×#121

—

C₁₃×#121

C₄

yes

—

C₄×D₄×C₁₃

—

C₄×D₄×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂²⋊#1055

—

C₂²⋊#1055

—

C₄⋊#1055

—

C₄⋊#1055

C₂²

yes

—

#88043

—

C₄×#1055

C₂²

yes

—

C₁₃×#41

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

C₂³

yes

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₄

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

—

D₁₃

yes

characteristic

#432

—

#431

—

D₄⋊C₈

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#131

yes

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×#41

—

C₄⋊Q₈

Dic₁₃

yes

characteristic

#127

Dic₁₃

yes

—

—

Dic₁₃

yes

—

C₄×D₄

—

C₄×D₄

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

C₂×Dic₁₃

yes

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂×Dic₁₃

yes

characteristic

107

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

108

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

C₂×Dic₁₃

yes

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

C₂×Dic₁₃

yes

characteristic

116

C₄²

C₂²⋊D₁₃

yes

characteristic

117

C₄²

C₂²⋊D₁₃

yes

characteristic

118

C₄²

C₂²×D₁₃

yes

characteristic

119

C₁₃

#1496

yes

characteristic, complemented, Sylow

120

C₂³

—

121

C₂³

C₂²⋊Dic₁₃

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₂²⋊Dic₁₃

yes

characteristic

133

C₂×C₄

C₂²⋊Dic₁₃

yes

characteristic

134

C₂×C₄

C₂²⋊Dic₁₃

yes

characteristic

135

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

138

C₈

—

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#88186

yes

—

153

C₄

#88202

yes

—

154

C₂

—

155

C₂

—

156

C₂

#11718876

yes

characteristic

157

C₂

#11718895

yes

characteristic

158

C₂

#11718897

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃
Fitting, prime radical	C₁₃×#532	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
cosocle	C₄²×C₁₃	C₂³
hypercenter	#532	D₁₃
hypocenter	C₁₃	#1496
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃

Fitting, prime radical

C₁₃×#532

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

cosocle

C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
2. #11718587	C₂	semidirect
3. #11718586	C₂	semidirect
4. D₄⋊#1055	C₂	semidirect
119. C₁₃	#1496	semidirect

H ≅ G/N

type

2. #11718587

C₂

semidirect

3. #11718586

C₂

semidirect

4. D₄⋊#1055

C₂

semidirect

119. C₁₃

#1496

semidirect

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

subgroup	index
C₁₃×#532	2
#11718587	2
#11718586	2
D₄⋊#1055	2
#1496	13

subgroup

index

C₁₃×#532

Subgroups — #13113305

Conjugacy classes of subgroups

Complements

Maximal subgroups