Group #13131029 — subgroups

Subgroups — #13131029

Order 1664.

260 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 76 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×#88142	C₂	1	1	yes	maximal, characteristic
2	#11719129	C₂	1	1	yes	maximal, characteristic
3	C₄×#6122	C₂	1	1	yes	maximal, characteristic
4	#11718291	C₂	1	4	yes	maximal
5	C₂²×Dic₂₆	C₄	1	2	yes	—
6	C₂×#6122	C₂²	1	2	yes	—
7	C₂×#6122	C₂²	1	2	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×#6101	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
12	#88142	—	2	1	no	—
13	#88142	—	2	1	no	—
14	#88142	—	2	1	no	—
15	#88142	—	2	1	no	—
16	#88112	—	2	1	no	—
17	#88110	—	2	1	no	—
18	C₄⋊#1057	—	2	1	no	—
19	C₄×#1057	—	2	1	no	—
20	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
24	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
25	C₂×Dic₂₆	—	2	2	no	—
26	C₂×Dic₂₆	—	2	2	no	—
27	C₂×Dic₂₆	—	4	4	no	—
28	#6122	—	2	2	no	—
29	#6122	—	2	4	no	—
30	#6122	Q₈	1	4	yes	complemented
31	C₂×#1057	—	2	2	no	—
32	C₂×#1057	—	2	2	no	—
33	C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₄⋊Dic₁₃	—	2	1	no	—
36	C₄⋊Dic₁₃	—	2	1	no	—
37	#6101	—	2	2	no	—
38	#6101	—	2	2	no	—
39	C₄×Dic₁₃	D₄	1	1	yes	characteristic
40	C₄×Dic₁₃	D₄	1	1	yes	characteristic
41	C₄×Dic₁₃	D₄	1	1	yes	characteristic
42	C₄×Dic₁₃	D₄	1	1	yes	characteristic
43	#1997	—	13	1	no	maximal, Sylow
44	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃	—	2	2	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	—	4	2	no	—
48	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×Dic₁₃	—	2	2	no	—
51	C₂×Dic₁₃	—	2	2	no	—
52	C₂×Dic₁₃	—	4	2	no	—
53	C₂×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
58	C₂×Dic₁₃	#41	1	2	yes	—
59	Dic₂₆	—	4	4	no	—
60	Dic₂₆	—	4	4	no	—
61	#1057	—	2	4	no	—
62	#1057	—	4	2	no	—
63	C₂×C₄⋊Q₈	—	13	1	no	—
64	#423	—	13	1	no	—
65	C₄×M₁₆	—	13	1	no	—
66	#343	—	13	4	no	—
67	C₂²×C₁₃	—	2	1	no	—
68	C₂²×C₁₃	—	2	1	no	—
69	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₁₃	C₄×Q₈	1	2	yes	—
71	C₄×C₁₃	—	2	1	no	—
72	C₄×C₁₃	—	4	2	no	—
73	C₄×C₁₃	#101	1	2	yes	—
74	Dic₁₃	—	2	1	no	—
75	Dic₁₃	—	4	2	no	—
76	Dic₁₃	#100	1	2	yes	—
77	Dic₁₃	C₂²⋊Q₈	1	2	yes	—
78	Dic₁₃	#124	1	1	yes	characteristic
79	Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂²×Q₈	—	13	2	no	—
81	C₂×M₁₆	—	13	2	no	—
82	C₂×M₁₆	—	13	2	no	—
83	C₄⋊Q₈	—	26	1	no	—
84	C₄⋊Q₈	—	26	1	no	—
85	C₄⋊Q₈	—	26	1	no	—
86	C₄⋊Q₈	—	26	1	no	—
87	C₂×C₄⋊C₄	—	13	2	no	—
88	C₂×C₄⋊C₄	—	13	1	no	—
89	C₂×C₄⋊C₄	#261	1	1	yes	characteristic
90	C₂×C₄²	—	13	1	no	—
91	C₄⋊C₈	—	26	1	no	—
92	C₄⋊C₈	—	26	1	no	—
93	#97	—	26	1	no	—
94	C₄×C₈	—	26	1	no	—
95	C₂×C₁₃	—	2	2	no	—
96	C₂×C₁₃	#391	1	1	yes	characteristic
97	C₂×C₁₃	C₂×#100	1	1	yes	characteristic
98	C₂×C₁₃	C₂×#101	1	1	yes	characteristic
99	C₂×Q₈	—	26	2	no	—
100	C₂×Q₈	—	26	2	no	—
101	C₂×Q₈	—	52	4	no	—
102	C₂²×C₄	—	13	1	no	—
103	C₂²×C₄	—	13	1	no	—
104	C₂²×C₄	—	13	2	no	—
105	C₂²×C₄	C₂×#261	1	2	yes	—
106	C₂²×C₄	C₂×#261	1	1	yes	characteristic
107	M₁₆	—	13	4	no	—
108	M₁₆	—	26	2	no	—
109	M₁₆	—	26	4	no	—
110	C₂×C₈	—	26	2	no	—
111	C₂×C₈	—	26	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	26	2	no	—
115	C₄⋊C₄	—	26	2	no	—
116	C₄⋊C₄	—	26	1	no	—
117	C₄⋊C₄	—	26	1	no	—
118	C₄²	—	13	1	no	—
119	C₄²	—	13	1	no	—
120	C₄²	—	13	1	no	—
121	C₄²	—	13	1	no	—
122	C₁₃	#1997	1	1	yes	characteristic, complemented, Sylow
123	C₂³	C₂²×#261	1	1	yes	characteristic
124	Q₈	—	52	4	no	—
125	Q₈	—	52	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	13	2	no	—
130	C₂×C₄	—	13	2	no	—
131	C₂×C₄	—	13	1	no	—
132	C₂×C₄	—	13	1	no	—
133	C₂×C₄	—	13	1	no	—
134	C₂×C₄	—	13	1	no	—
135	C₂×C₄	—	26	2	no	—
136	C₂×C₄	—	26	2	no	—
137	C₂×C₄	—	52	2	no	—
138	C₂×C₄	C₂²⋊#261	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊#261	1	1	yes	characteristic
140	C₈	—	26	4	no	—
141	C₈	—	52	2	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	Q₈×#261	1	2	yes	—
145	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	4	2	no	—
148	C₄	—	13	2	no	—
149	C₄	—	13	2	no	—
150	C₄	—	13	1	no	—
151	C₄	—	13	1	no	—
152	C₄	—	26	1	no	—
153	C₄	—	52	2	no	—
154	C₄	#88113	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	C₂×#88113	1	1	yes	characteristic
157	C₂	D₁₃⋊#100	1	1	yes	characteristic
158	C₂	(C₂×#261)⋊Q₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88142

C₂

yes

maximal, characteristic

#11719129

C₂

yes

maximal, characteristic

C₄×#6122

C₂

yes

maximal, characteristic

#11718291

C₂

yes

maximal

C₂²×Dic₂₆

C₄

yes

—

C₂×#6122

C₂²

yes

—

C₂×#6122

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

C₄

yes

characteristic

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

—

—

—

—

—

—

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

#6122

—

#6122

—

#6122

Q₈

yes

complemented

C₂×#1057

—

C₂×#1057

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

D₄

yes

characteristic

C₄×Dic₁₃

D₄

yes

characteristic

C₄×Dic₁₃

D₄

yes

characteristic

C₄×Dic₁₃

D₄

yes

characteristic

#1997

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

#41

yes

—

Dic₂₆

—

Dic₂₆

—

#1057

—

#1057

—

C₂×C₄⋊Q₈

—

#423

—

C₄×M₁₆

—

#343

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×Q₈

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#101

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#100

yes

—

Dic₁₃

C₂²⋊Q₈

yes

—

Dic₁₃

#124

yes

characteristic

Dic₁₃

C₄⋊Q₈

yes

characteristic

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#261

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂×#261

yes

—

106

C₂²×C₄

C₂×#261

yes

characteristic

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₁₃

#1997

yes

characteristic, complemented, Sylow

123

C₂³

C₂²×#261

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂²⋊#261

yes

characteristic

139

C₂×C₄

C₂²⋊#261

yes

characteristic

140

C₈

—

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

Q₈×#261

yes

—

145

C₂²

C₂×C₂²⋊#261

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#88113

yes

—

155

C₂

—

156

C₂

C₂×#88113

yes

characteristic

157

C₂

D₁₃⋊#100

yes

characteristic

158

C₂

(C₂×#261)⋊Q₈

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#261
commutator, prime radical	C₂³×C₁₃	C₂²×C₄
Frattini	C₂³	C₂²×#261
Fitting	C₂×C₁₃×C₄⋊C₄	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂²×Dic₁₃	C₂³
hypercenter	C₂×C₄⋊C₄	#261
hypocenter	C₁₃	#1997
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#261

commutator, prime radical

C₂³×C₁₃

C₂²×C₄

Frattini

C₂³

C₂²×#261

Fitting

C₂×C₁₃×C₄⋊C₄

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂²×Dic₁₃

C₂³

C₂×C₄⋊C₄

C₁₃

G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
30. #6122	Q₈	semidirect
122. C₁₃	#1997	semidirect

H ≅ G/N

type

30. #6122

Q₈

semidirect

122. C₁₃

#1997

semidirect

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

subgroup	index
C₂×#88142	2
#11719129	2
C₄×#6122	2
#11718291	2
#1997	13

subgroup

index

Subgroups — #13131029

Conjugacy classes of subgroups

Complements

Maximal subgroups