Group #13125181 — subgroups

Subgroups — #13125181

Order 1664.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#11719175	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718894	C₂	1	4	yes	maximal, complemented
4	#11718743	C₂	1	2	yes	maximal, complemented
5	#11718732	C₂	1	2	yes	maximal, complemented
6	#11718716	C₂	1	4	yes	maximal, complemented
7	C₂×#88043	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#41	C₂²	1	2	yes	—
9	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₁₃×#127	C₂²	1	1	yes	characteristic
11	C₁₃×#124	C₂²	1	2	yes	—
12	C₁₃×#121	—	2	2	no	—
13	C₄×D₄×C₁₃	—	2	2	no	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	#88180	C₂²	1	2	yes	—
16	#88178	—	2	2	no	—
17	C₂²×#1055	C₂²	1	2	yes	—
18	#88124	C₂²	1	2	yes	—
19	#88123	C₂²	1	2	yes	—
20	C₄×Dic₂₆	—	2	2	no	—
21	#88075	C₂²	1	8	yes	—
22	#88072	C₂²	1	8	yes	—
23	#88043	C₂²	1	2	yes	—
24	#88043	C₂²	1	2	yes	—
25	C₁₃×#41	—	2	8	no	—
26	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
27	C₂×D₄×C₁₃	—	2	2	no	—
28	C₂×D₄×C₁₃	—	2	2	no	—
29	C₂×D₄×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₃	—	2	2	no	—
31	C₂²×C₄×C₁₃	D₄	1	2	yes	—
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₂₆	—	2	2	no	—
34	C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₁₃×C₂²⋊C₄	—	2	4	no	—
36	C₄²×C₁₃	D₄	1	2	yes	—
37	C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₁₃	—	2	2	no	—
40	C₄⋊Dic₁₃	—	2	2	no	—
41	C₄⋊Dic₁₃	C₂³	1	4	yes	—
42	#6101	—	2	4	no	—
43	C₄×Dic₁₃	—	2	2	no	—
44	C₂×#1055	—	2	4	no	—
45	C₂×#1055	C₂³	1	4	yes	—
46	#3135	—	13	1	no	maximal, Sylow
47	C₂³×C₁₃	—	2	2	no	—
48	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
49	Q₈×C₁₃	—	2	4	no	—
50	D₄×C₁₃	—	2	4	no	—
51	D₄×C₁₃	—	4	4	no	—
52	D₄×C₁₃	—	4	4	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	4	2	no	—
55	C₂×C₄×C₁₃	—	4	4	no	—
56	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#41	1	4	yes	—
60	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
61	C₂×Dic₁₃	—	2	4	no	—
62	Dic₂₆	—	4	4	no	—
63	#1055	—	2	4	no	—
64	#533	—	13	1	no	—
65	#532	D₁₃	1	1	yes	characteristic
66	#491	—	13	2	no	—
67	#490	—	13	2	no	—
68	#489	—	13	4	no	—
69	#416	—	13	4	no	—
70	C₂×#97	—	13	1	no	—
71	C₂²×C₁₃	—	2	2	no	—
72	C₂²×C₁₃	—	4	2	no	—
73	C₂²×C₁₃	—	4	4	no	—
74	C₂²×C₁₃	#124	1	2	yes	—
75	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₁₃	—	2	2	no	—
77	C₄×C₁₃	—	2	2	no	—
78	C₄×C₁₃	—	4	2	no	—
79	C₄×C₁₃	#124	1	2	yes	—
80	C₄×C₁₃	C₂×#41	1	2	yes	—
81	Dic₁₃	—	4	4	no	—
82	C₂×#41	C₂×D₁₃	1	2	yes	—
83	C₂²×C₈	—	13	2	no	—
84	C₄⋊Q₈	—	13	2	no	—
85	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
86	#127	C₂×D₁₃	1	1	yes	characteristic
87	#125	—	13	2	no	—
88	#124	C₂×D₁₃	1	2	yes	—
89	C₂²⋊Q₈	—	26	2	no	—
90	#121	—	2	2	no	—
91	C₄×Q₈	—	26	2	no	—
92	C₄×D₄	—	2	2	no	—
93	#117	—	13	2	no	—
94	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
95	#103	—	13	8	no	—
96	#102	—	13	8	no	—
97	#97	—	13	2	no	—
98	#97	—	13	2	no	—
99	C₂×C₁₃	—	2	2	no	—
100	C₂×C₁₃	—	8	2	no	—
101	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
102	C₂×C₁₃	#575	1	2	yes	—
103	#41	—	2	8	no	—
104	C₂×Q₈	—	26	2	no	—
105	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	C₂²×D₁₃	1	2	yes	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
112	C₂×C₈	—	13	4	no	—
113	C₂×C₈	—	26	4	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	13	4	no	—
116	C₄⋊C₄	—	26	4	no	—
117	C₄⋊C₄	—	26	2	no	—
118	C₂²⋊C₄	—	2	4	no	—
119	C₂²⋊C₄	—	26	2	no	—
120	C₄²	—	26	2	no	—
121	C₄²	C₂²⋊D₁₃	1	2	yes	—
122	C₄²	C₂²×D₁₃	1	1	yes	characteristic
123	C₄²	C₂²×D₁₃	1	1	yes	characteristic
124	C₁₃	#3135	1	1	yes	characteristic, complemented, Sylow
125	C₂³	—	2	2	no	—
126	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	52	4	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	D₄	—	4	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	26	4	no	—
136	C₂×C₄	#6129	1	4	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
140	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
141	C₈	—	26	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	4	no	—
145	C₂²	#88190	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	52	4	no	—
151	C₄	#88190	1	2	yes	—
152	C₄	C₂×#6129	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	8	2	no	—
155	C₂	C₂×#88190	1	1	yes	characteristic
156	C₂	#11719527	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#532

C₂

yes

maximal, characteristic

#11719175

C₂

yes

maximal, characteristic, complemented

#11718894

C₂

yes

maximal, complemented

#11718743

C₂

yes

maximal, complemented

#11718732

C₂

yes

maximal, complemented

#11718716

C₂

yes

maximal, complemented

C₂×#88043

C₂

yes

maximal, characteristic, complemented

C₂×C₁₃×#41

C₂²

yes

—

C₁₃×C₄⋊Q₈

C₂²

yes

characteristic

C₁₃×#127

C₂²

yes

characteristic

C₁₃×#124

C₂²

yes

—

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88180

C₂²

yes

—

#88178

—

C₂²×#1055

C₂²

yes

—

#88124

C₂²

yes

—

#88123

C₂²

yes

—

C₄×Dic₂₆

—

#88075

C₂²

yes

—

#88072

C₂²

yes

—

#88043

C₂²

yes

—

#88043

C₂²

yes

—

C₁₃×#41

—

C₂×Q₈×C₁₃

C₂³

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₄²×C₁₃

C₂³

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂³

yes

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

C₂×#1055

C₂³

yes

—

#3135

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

Dic₂₆

—

#1055

—

#533

—

#532

D₁₃

yes

—

—

—

—

C₂×#97

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#124

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#124

yes

—

C₄×C₁₃

C₂×#41

yes

—

Dic₁₃

—

C₂×#41

C₂×D₁₃

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₃

yes

characteristic

#127

C₂×D₁₃

yes

characteristic

#125

—

#124

C₂×D₁₃

yes

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

—

C₂×C₁₃

—

100

C₂×C₁₃

—

101

C₂×C₁₃

C₂×#124

yes

characteristic

102

C₂×C₁₃

#575

yes

—

103

#41

—

104

C₂×Q₈

—

105

C₂×Q₈

C₂²×D₁₃

yes

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

C₂²×D₁₃

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₁₃

yes

—

111

C₂²×C₄

C₂²×D₁₃

yes

characteristic

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂²⋊D₁₃

yes

—

122

C₄²

C₂²×D₁₃

yes

characteristic

123

C₄²

C₂²×D₁₃

yes

characteristic

124

C₁₃

#3135

yes

characteristic, complemented, Sylow

125

C₂³

—

126

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

127

Q₈

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#6129

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

139

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

140

C₂×C₄

C₂³×D₁₃

yes

characteristic

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#88190

yes

—

146

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#88190

yes

—

152

C₄

C₂×#6129

yes

—

153

C₂

—

154

C₂

—

155

C₂

C₂×#88190

yes

characteristic

156

C₂

#11719527

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting, prime radical	C₁₃×#532	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	#532	D₁₃
hypocenter	C₁₃	#3135
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, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting, prime radical

C₁₃×#532

C₂

socle

C₂²×C₁₃

C₂²×D₄

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
2. #11719175	C₂	semidirect
3. #11718894	C₂	semidirect
4. #11718743	C₂	semidirect
5. #11718732	C₂	semidirect
6. #11718716	C₂	semidirect
7. C₂×#88043	C₂	semidirect
124. C₁₃	#3135	semidirect

H ≅ G/N

type

2. #11719175

C₂

semidirect

3. #11718894

C₂

semidirect

4. #11718743

C₂

semidirect

5. #11718732

C₂

semidirect

6. #11718716

C₂

semidirect

7. C₂×#88043

C₂

semidirect

124. C₁₃

#3135

semidirect

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

subgroup	index
C₁₃×#532	2
#11719175	2
#11718894	2
#11718743	2
#11718732	2
#11718716	2
C₂×#88043	2
#3135	13

subgroup

index

C₁₃×#532

Subgroups — #13125181

Conjugacy classes of subgroups

Complements

Maximal subgroups