Group #15780967 — subgroups

Subgroups — #15780967

Order 3328.

286 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 115 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	#13116620	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13115582	C₂	1	2	yes	maximal
4	#13112933	C₂	1	2	yes	maximal
5	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#108	C₂²	1	2	yes	—
8	#11718780	—	2	2	no	—
9	#11718767	C₂²	1	2	yes	—
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂²×#6090	C₂²	1	1	yes	characteristic
12	C₁₃×#368	—	2	2	no	—
13	#11718147	—	2	2	no	—
14	#11718097	C₄	1	2	yes	complemented
15	#11718096	C₄	1	2	yes	complemented
16	C₁₆×Dic₁₃	—	2	2	no	—
17	C₂×C₁₃×M₁₆	—	2	2	no	—
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	#88180	—	2	2	no	—
20	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
21	C₂²×#1055	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
25	C₂×C₁₃×C₁₆	Q₈	1	2	yes	—
26	C₁₃×#108	—	2	2	no	—
27	C₁₃×#108	—	2	2	no	—
28	#88058	C₂×C₄	1	4	yes	—
29	#88057	—	2	2	no	—
30	#88054	—	4	4	no	—
31	C₈×Dic₁₃	—	2	1	no	—
32	C₈×Dic₁₃	—	2	1	no	—
33	C₂×#6090	—	2	1	no	—
34	C₂×#6090	—	2	1	no	—
35	C₂×#6090	Q₈	1	2	yes	—
36	#13169	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₃	—	2	1	no	—
39	C₁₃×M₁₆	—	4	2	no	—
40	C₁₃×M₁₆	—	4	2	no	—
41	C₂×C₈×C₁₃	—	4	2	no	—
42	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
47	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
48	C₂²⋊Dic₁₃	—	4	2	no	—
49	C₄⋊Dic₁₃	—	2	4	no	—
50	C₄×Dic₁₃	—	2	1	no	—
51	C₄×Dic₁₃	—	4	2	no	—
52	C₄×Dic₁₃	Q₁₆	1	2	yes	—
53	C₂×#1055	—	2	1	no	—
54	C₂×#1055	—	2	1	no	—
55	C₂×#1055	D₈	1	2	yes	—
56	C₁₃×C₁₆	C₄⋊C₄	1	2	yes	—
57	C₁₃×C₁₆	C₄⋊C₄	1	2	yes	—
58	#6090	—	2	2	no	—
59	C₂×#368	D₁₃	1	1	yes	characteristic
60	#2158	—	13	1	no	—
61	C₂×C₄×C₁₆	—	13	1	no	—
62	#1945	—	13	2	no	—
63	#1382	—	13	2	no	—
64	C₂³×C₁₃	#135	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
71	C₂×Dic₁₃	—	2	4	no	—
72	C₂×Dic₁₃	—	2	2	no	—
73	C₂×Dic₁₃	—	4	4	no	—
74	C₈×C₁₃	—	8	2	no	—
75	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₈×C₁₃	#125	1	1	yes	characteristic
79	#1055	—	2	2	no	—
80	C₂²×C₁₆	—	13	1	no	—
81	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
82	C₂×#108	C₂×D₁₃	1	2	yes	—
83	#427	—	13	2	no	—
84	#424	—	26	2	no	—
85	C₂×C₄×C₈	—	13	1	no	—
86	#368	—	2	2	no	—
87	#367	—	13	2	no	—
88	#366	—	13	2	no	—
89	C₄×C₁₆	—	26	2	no	—
90	#339	—	26	2	no	—
91	C₂²×C₁₃	—	2	2	no	—
92	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
93	C₂²×C₁₃	#467	1	1	yes	characteristic
94	C₂²×C₁₃	#499	1	1	yes	characteristic
95	C₄×C₁₃	#389	1	1	yes	characteristic
96	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
97	C₄×C₁₃	#466	1	1	yes	characteristic
98	C₄×C₁₃	#500	1	1	yes	characteristic
99	Dic₁₃	—	2	4	no	—
100	Dic₁₃	—	8	4	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	—	13	1	no	—
103	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
104	#117	—	26	2	no	—
105	C₂×C₄²	—	13	1	no	—
106	C₂×C₁₆	—	13	2	no	—
107	C₂×C₁₆	—	26	1	no	—
108	C₂×C₁₆	—	26	1	no	—
109	C₂×C₁₆	Dic₂₆	1	2	yes	—
110	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
111	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
112	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
113	#108	—	2	2	no	—
114	#108	—	2	2	no	—
115	#107	—	13	4	no	—
116	#106	—	26	2	no	—
117	C₄⋊C₈	—	52	4	no	—
118	C₄×C₈	—	26	1	no	—
119	C₄×C₈	—	26	1	no	—
120	C₂×C₁₃	—	2	1	no	—
121	C₂×C₁₃	—	2	1	no	—
122	C₂×C₁₃	#1942	1	1	yes	characteristic
123	C₂×C₁₃	#2178	1	2	yes	—
124	C₂²×C₄	—	26	1	no	—
125	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
126	M₁₆	—	4	2	no	—
127	M₁₆	—	4	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	13	2	no	—
130	C₂×C₈	—	26	1	no	—
131	C₂×C₈	—	26	1	no	—
132	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
134	C₂×C₈	#6129	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
136	C₂×C₈	#6131	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
138	C₄⋊C₄	—	26	4	no	—
139	C₂²⋊C₄	—	52	2	no	—
140	C₄²	—	13	2	no	—
141	C₄²	—	26	1	no	—
142	C₄²	—	52	2	no	—
143	C₁₆	—	26	2	no	—
144	C₁₆	#6101	1	2	yes	—
145	C₁₆	#6101	1	2	yes	—
146	C₁₃	#13169	1	1	yes	characteristic, complemented, Sylow
147	C₂³	#88173	1	1	yes	characteristic
148	C₂×C₄	—	26	4	no	—
149	C₂×C₄	—	26	2	no	—
150	C₂×C₄	—	52	4	no	—
151	C₂×C₄	#88142	1	1	yes	characteristic
152	C₂×C₄	#88147	1	1	yes	characteristic
153	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
154	C₂×C₄	#88166	1	1	yes	characteristic
155	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
156	C₂×C₄	#88192	1	1	yes	characteristic
157	C₈	—	8	2	no	—
158	C₈	—	26	2	no	—
159	C₈	#88141	1	1	yes	characteristic
160	C₈	#88144	1	1	yes	characteristic
161	C₈	C₂×#6101	1	1	yes	characteristic
162	C₈	#88200	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#11718533	1	1	yes	characteristic
165	C₂²	#11718539	1	1	yes	characteristic
166	C₂²	#11718849	1	1	yes	characteristic
167	C₄	—	26	4	no	—
168	C₄	—	104	4	no	—
169	C₄	#11718534	1	1	yes	characteristic
170	C₄	#11718535	1	1	yes	characteristic
171	C₄	#11718634	1	1	yes	characteristic
172	C₄	#11718818	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#13115567	1	1	yes	characteristic
176	C₂	#13116716	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13116620

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₁₃

C₂

yes

maximal, characteristic

#13115582

C₂

yes

maximal

#13112933

C₂

yes

maximal

C₂×C₁₃×#368

C₂

yes

maximal, characteristic

C₂²×C₁₃×C₁₆

C₂²

yes

characteristic

C₂×C₁₃×#108

C₂²

yes

—

#11718780

—

#11718767

C₂²

yes

—

C₂×C₈×Dic₁₃

C₂²

yes

characteristic

C₂²×#6090

C₂²

yes

characteristic

C₁₃×#368

—

#11718147

—

#11718097

C₄

yes

complemented

#11718096

C₄

yes

complemented

C₁₆×Dic₁₃

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

C₂³

yes

characteristic

#88180

—

C₂×C₄×Dic₁₃

D₄

yes

characteristic

C₂²×#1055

D₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

—

C₂×C₁₃×C₁₆

Q₈

yes

—

C₁₃×#108

—

C₁₃×#108

—

#88058

C₂×C₄

yes

—

#88057

—

#88054

—

C₈×Dic₁₃

—

C₈×Dic₁₃

—

C₂×#6090

—

C₂×#6090

—

C₂×#6090

Q₈

yes

—

#13169

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

C₂×C₈×C₁₃

yes

C₂×C₈×C₁₃

yes

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

Q₁₆

yes

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

D₈

yes

—

C₁₃×C₁₆

C₄⋊C₄

yes

—

C₁₃×C₁₆

C₄⋊C₄

yes

—

#6090

—

C₂×#368

D₁₃

yes

characteristic

#2158

—

C₂×C₄×C₁₆

—

#1945

—

#1382

—

C₂³×C₁₃

#135

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₃

C₄×Q₈

yes

characteristic

C₈×C₁₃

C₂²⋊Q₈

yes

C₈×C₁₃

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₁₃

yes

characteristic

C₂×#108

C₂×D₁₃

yes

—

#427

—

#424

—

C₂×C₄×C₈

—

#368

—

#367

—

#366

—

C₄×C₁₆

—

#339

—

C₂²×C₁₃

—

C₂²×C₁₃

C₄×D₈

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₄×Q₁₆

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

100

Dic₁₃

—

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂²×D₁₃

yes

characteristic

104

#117

—

105

C₂×C₄²

—

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

Dic₂₆

yes

—

110

C₂×C₁₆

C₄×D₁₃

yes

characteristic

111

C₂×C₁₆

C₄×D₁₃

yes

characteristic

112

C₂×C₁₆

C₂²⋊D₁₃

yes

—

113

#108

—

114

#108

—

115

#107

—

116

#106

—

117

C₄⋊C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₂×C₁₃

—

121

C₂×C₁₃

—

122

C₂×C₁₃

#1942

yes

characteristic

123

C₂×C₁₃

#2178

yes

—

124

C₂²×C₄

—

125

C₂²×C₄

D₄×D₁₃

yes

characteristic

126

M₁₆

—

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

C₂×Dic₂₆

yes

characteristic

133

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

134

C₂×C₈

#6129

yes

characteristic

135

C₂×C₈

Q₈×D₁₃

yes

characteristic

136

C₂×C₈

#6131

yes

characteristic

137

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

138

C₄⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₁₆

—

144

C₁₆

#6101

yes

—

145

C₁₆

#6101

yes

—

146

C₁₃

#13169

yes

characteristic, complemented, Sylow

147

C₂³

#88173

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#88142

yes

characteristic

152

C₂×C₄

#88147

yes

characteristic

153

C₂×C₄

D₈×D₁₃

yes

characteristic

154

C₂×C₄

#88166

yes

characteristic

155

C₂×C₄

Q₁₆×D₁₃

yes

characteristic

156

C₂×C₄

#88192

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

#88141

yes

characteristic

160

C₈

#88144

yes

characteristic

161

C₈

C₂×#6101

yes

characteristic

162

C₈

#88200

yes

characteristic

163

C₂²

—

164

C₂²

#11718533

yes

characteristic

165

C₂²

#11718539

yes

characteristic

166

C₂²

#11718849

yes

characteristic

167

C₄

—

168

C₄

—

104

—

169

C₄

#11718534

yes

characteristic

170

C₄

#11718535

yes

characteristic

171

C₄

#11718634

yes

characteristic

172

C₄

#11718818

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#13115567

yes

characteristic

176

C₂

#13116716

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#368	C₂
socle	C₂²×C₁₃	C₄×D₈
prime radical	C₂³×C₁₃	#135
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#368	D₁₃
hypocenter	C₁₃	#13169
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₁₃

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₂×C₁₃×#368

C₂

socle

C₂²×C₁₃

C₄×D₈

prime radical

C₂³×C₁₃

#135

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
14. #11718097	C₄	semidirect
15. #11718096	C₄	semidirect
146. C₁₃	#13169	semidirect

H ≅ G/N

type

14. #11718097

C₄

semidirect

15. #11718096

C₄

semidirect

146. C₁₃

#13169

semidirect

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

subgroup	index
#13116620	2
C₂×C₁₆×Dic₁₃	2
#13115582	2
#13112933	2
C₂×C₁₃×#368	2
#13169	13

subgroup

index

C₂×C₁₆×Dic₁₃

C₂×C₁₃×#368

Subgroups — #15780967

Conjugacy classes of subgroups

Complements

Maximal subgroups