Group #13123554 — subgroups

Subgroups — #13123554

Order 1664.

484 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 221 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	C₂×#88202	C₂	1	2	yes	maximal, complemented
3	#11718895	C₂	1	2	yes	maximal, complemented
4	#11718672	C₂	1	8	yes	maximal, complemented
5	#11718585	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88043	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#41	C₂²	1	2	yes	—
8	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₁₃×#127	C₄	1	1	yes	characteristic
10	C₁₃×#124	C₄	1	2	yes	—
11	C₁₃×#121	C₄	1	4	yes	—
12	C₄×D₄×C₁₃	C₂²	1	4	yes	—
13	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	#88202	—	2	8	no	—
15	C₂×#6099	C₂²	1	4	yes	—
16	C₂×#6099	C₂²	1	2	yes	—
17	C₂²×#1055	C₂²	1	4	yes	—
18	C₂²×#1055	C₂²	1	1	yes	characteristic
19	C₂²×#1055	C₂²	1	1	yes	characteristic
20	C₂²⋊#1055	C₂²	1	8	yes	—
21	C₄⋊#1055	C₂²	1	4	yes	—
22	#88043	C₂²	1	4	yes	—
23	C₁₃×#41	—	2	4	no	—
24	C₁₃×#41	—	2	4	no	—
25	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂×D₄×C₁₃	C₂×C₄	1	4	yes	—
27	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₁₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₁₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
34	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₁₃	C₂³	1	2	yes	—
37	#6099	—	2	4	no	—
38	#6099	—	2	8	no	—
39	C₂×#1055	—	2	2	no	—
40	C₂×#1055	—	2	8	no	—
41	C₂×#1055	D₄	1	4	yes	complemented
42	C₂×#1055	C₂³	1	2	yes	—
43	C₂×#1055	C₂³	1	4	yes	—
44	C₂×#1055	C₂³	1	2	yes	—
45	#2932	—	13	1	no	maximal, Sylow
46	C₂³×C₁₃	C₂²×C₄	1	4	yes	—
47	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	Q₈×C₁₃	—	2	4	no	—
49	D₄×C₁₃	—	2	8	no	—
50	D₄×C₁₃	—	2	4	no	—
51	D₄×C₁₃	—	4	4	no	—
52	C₂×C₄×C₁₃	—	2	4	no	—
53	C₂×C₄×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	—	4	2	no	—
55	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃	#41	1	2	yes	—
61	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
62	#1055	—	2	4	no	—
63	#1055	—	2	2	no	—
64	#1055	C₂×D₄	1	4	yes	—
65	C₂×#131	—	13	2	no	—
66	#532	D₁₃	1	1	yes	characteristic
67	#435	—	13	8	no	—
68	#423	—	13	1	no	—
69	#408	—	13	2	no	—
70	C₂×#97	—	13	1	no	—
71	C₂²×C₁₃	—	2	2	no	—
72	C₂²×C₁₃	—	2	4	no	—
73	C₂²×C₁₃	—	2	8	no	—
74	C₂²×C₁₃	C₄×D₄	1	2	yes	—
75	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
76	C₄×C₁₃	—	2	4	no	—
77	C₄×C₁₃	—	2	2	no	—
78	C₄×C₁₃	—	2	2	no	—
79	C₄×C₁₃	C₄×D₄	1	2	yes	—
80	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
81	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
82	C₂×#41	C₂×D₁₃	1	2	yes	—
83	#131	—	26	8	no	—
84	C₂×M₁₆	—	13	2	no	—
85	C₂×M₁₆	—	13	4	no	—
86	C₂²×C₈	—	13	1	no	—
87	C₂²×C₈	—	13	4	no	—
88	C₂²×C₈	—	13	1	no	—
89	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
90	#127	Dic₁₃	1	1	yes	characteristic
91	#124	Dic₁₃	1	2	yes	—
92	#121	Dic₁₃	1	4	yes	—
93	C₄×D₄	C₂×D₁₃	1	4	yes	—
94	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
95	C₄⋊C₈	—	13	4	no	—
96	C₂²⋊C₈	—	13	8	no	—
97	#97	—	13	4	no	—
98	C₂×C₁₃	—	2	2	no	—
99	C₂×C₁₃	—	4	4	no	—
100	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
101	C₂×C₁₃	#568	1	2	yes	—
102	#41	—	2	4	no	—
103	#41	—	2	4	no	—
104	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
105	C₂×D₄	C₂×Dic₁₃	1	4	yes	—
106	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
107	C₂×D₄	C₂×Dic₁₃	1	4	yes	—
108	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
110	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
111	M₁₆	—	26	8	no	—
112	M₁₆	—	26	4	no	—
113	C₂×C₈	—	13	4	no	—
114	C₂×C₈	—	13	2	no	—
115	C₂×C₈	—	13	2	no	—
116	C₂×C₈	—	13	4	no	—
117	C₂×C₈	—	26	8	no	—
118	C₂×C₈	—	26	2	no	—
119	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
120	C₂²⋊C₄	C₂×Dic₁₃	1	8	yes	—
121	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
122	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
123	C₄²	C₂²×D₁₃	1	2	yes	—
124	C₁₃	#2932	1	1	yes	characteristic, complemented, Sylow
125	C₂³	C₂²×Dic₁₃	1	4	yes	—
126	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	D₄	—	2	8	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	D₄×D₁₃	1	2	yes	—
135	C₂×C₄	#6129	1	2	yes	—
136	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
137	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
138	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
139	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
140	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
141	C₈	—	13	4	no	—
142	C₈	—	26	4	no	—
143	C₈	—	26	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	D₄×Dic₁₃	1	2	yes	—
148	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	D₄×Dic₁₃	1	2	yes	—
153	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
154	C₄	C₂×#6129	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	4	4	no	—
157	C₂	C₂×D₄×Dic₁₃	1	1	yes	characteristic
158	C₂	#11719524	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#532

C₂

yes

maximal, characteristic

C₂×#88202

C₂

yes

maximal, complemented

#11718895

C₂

yes

maximal, complemented

#11718672

C₂

yes

maximal, complemented

#11718585

C₂

yes

maximal, characteristic, complemented

C₂×#88043

C₂

yes

maximal, characteristic, complemented

C₂×C₁₃×#41

C₂²

yes

—

C₁₃×C₄⋊Q₈

C₄

yes

characteristic

C₁₃×#127

C₄

yes

characteristic

C₁₃×#124

C₄

yes

—

C₁₃×#121

C₄

yes

—

C₄×D₄×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88202

—

C₂×#6099

C₂²

yes

—

C₂×#6099

C₂²

yes

—

C₂²×#1055

C₂²

yes

—

C₂²×#1055

C₂²

yes

characteristic

C₂²×#1055

C₂²

yes

characteristic

C₂²⋊#1055

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

#88043

C₂²

yes

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₂²⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂³

yes

—

—

—

—

—

D₄

yes

complemented

C₂×#1055

C₂³

yes

—

C₂×#1055

C₂³

yes

—

C₂×#1055

C₂³

yes

—

#2932

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

—

C₂³×C₁₃

C₂²×C₄

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

—

—

C₂×D₄

yes

—

C₂×#131

—

#532

D₁₃

yes

—

—

—

C₂×#97

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₄×D₄

yes

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

C₂×#41

yes

characteristic

C₂×#41

C₂×D₁₃

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₁₃

yes

characteristic

#127

Dic₁₃

yes

characteristic

#124

Dic₁₃

yes

—

#121

Dic₁₃

yes

—

C₄×D₄

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂×C₁₃

—

C₂×C₁₃

—

100

C₂×C₁₃

C₂×C₄×D₄

yes

characteristic

101

C₂×C₁₃

#568

yes

—

102

#41

—

103

#41

—

104

C₂×Q₈

C₂×Dic₁₃

yes

—

105

C₂×D₄

C₂×Dic₁₃

yes

—

106

C₂×D₄

C₂×Dic₁₃

yes

—

107

C₂×D₄

C₂×Dic₁₃

yes

—

108

C₂²×C₄

C₂×Dic₁₃

yes

—

109

C₂²×C₄

C₂²×D₁₃

yes

—

110

C₂²×C₄

C₂²×D₁₃

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₄⋊C₄

C₂×Dic₁₃

yes

—

120

C₂²⋊C₄

C₂×Dic₁₃

yes

—

121

C₄²

C₂×Dic₁₃

yes

characteristic

122

C₄²

C₂×Dic₁₃

yes

characteristic

123

C₄²

C₂²×D₁₃

yes

—

124

C₁₃

#2932

yes

characteristic, complemented, Sylow

125

C₂³

C₂²×Dic₁₃

yes

—

126

C₂³

C₂²×Dic₁₃

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×D₁₃

yes

—

135

C₂×C₄

#6129

yes

—

136

C₂×C₄

C₂²×Dic₁₃

yes

—

137

C₂×C₄

C₂²×Dic₁₃

yes

—

138

C₂×C₄

C₂²×Dic₁₃

yes

—

139

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

140

C₂×C₄

C₂³×D₁₃

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

D₄×Dic₁₃

yes

—

148

C₂²

C₂³×Dic₁₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

D₄×Dic₁₃

yes

—

153

C₄

C₂×D₄×D₁₃

yes

characteristic

154

C₄

C₂×#6129

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

C₂×D₄×Dic₁₃

yes

characteristic

158

C₂

#11719524

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting, prime radical	C₁₃×#532	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#532	D₁₃
hypocenter	C₁₃	#2932
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting, prime radical

C₁₃×#532

C₂

cosocle

C₂×C₄×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. #11718895	C₂	semidirect
4. #11718672	C₂	semidirect
5. #11718585	C₂	semidirect
6. C₂×#88043	C₂	semidirect
41. C₂×#1055	D₄	semidirect
124. C₁₃	#2932	semidirect

H ≅ G/N

type

2. C₂×#88202

C₂

semidirect

3. #11718895

C₂

semidirect

4. #11718672

C₂

semidirect

5. #11718585

C₂

semidirect

6. C₂×#88043

C₂

semidirect

41. C₂×#1055

D₄

semidirect

124. C₁₃

#2932

semidirect

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

subgroup	index
C₁₃×#532	2
C₂×#88202	2
#11718895	2
#11718672	2
#11718585	2
C₂×#88043	2
#2932	13

subgroup

index

C₁₃×#532

Subgroups — #13123554

Conjugacy classes of subgroups

Complements

Maximal subgroups