Group #15778809 — subgroups

Subgroups — #15778809

Order 3328.

784 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 131 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×#11718140	C₂	1	3	yes	maximal
2	#13115230	C₂	1	3	yes	maximal
3	C₂×Q₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
4	Q₈²×C₁₃	—	2	6	no	—
5	Q₈²×C₁₃	C₄	1	4	yes	complemented
6	C₂×C₁₃×C₄⋊Q₈	—	2	3	no	—
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
8	C₂×C₄×Q₈×C₁₃	—	2	3	no	—
9	#11719517	C₂²	1	1	yes	characteristic
10	C₂×#88076	C₂²	1	3	yes	—
11	#11718886	—	2	3	no	—
12	#11718140	—	2	6	no	—
13	#11718115	—	2	3	no	—
14	C₂²×Q₈×C₁₃	—	2	1	no	—
15	C₂²×Q₈×C₁₃	—	2	1	no	—
16	C₂²×Q₈×C₁₃	D₄	1	3	yes	—
17	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₁₃×C₄⋊Q₈	—	2	12	no	—
19	C₁₃×C₄⋊Q₈	—	2	12	no	—
20	C₁₃×C₄⋊Q₈	—	2	6	no	—
21	C₁₃×C₄⋊Q₈	C₂×C₄	1	6	yes	—
22	C₁₃×C₄⋊Q₈	D₄	1	6	yes	—
23	C₄×Q₈×C₁₃	—	2	12	no	—
24	C₄×Q₈×C₁₃	—	2	12	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
27	C₂×C₄²×C₁₃	—	2	3	no	—
28	C₂×C₄²×C₁₃	D₄	1	3	yes	—
29	Q₈×Dic₁₃	—	4	2	no	—
30	#88180	—	2	3	no	—
31	C₂×#6099	—	2	3	no	—
32	#88076	—	2	6	no	—
33	#88075	—	4	6	no	—
34	#12777	—	13	1	no	maximal, Sylow
35	C₂×Q₈×C₁₃	—	2	4	no	—
36	C₂×Q₈×C₁₃	—	2	8	no	—
37	C₂×Q₈×C₁₃	—	2	4	no	—
38	C₂×Q₈×C₁₃	—	2	3	no	—
39	C₂×Q₈×C₁₃	—	4	1	no	—
40	C₂×Q₈×C₁₃	—	4	1	no	—
41	C₂×Q₈×C₁₃	—	4	3	no	—
42	C₂×Q₈×C₁₃	—	4	3	no	—
43	C₂×Q₈×C₁₃	—	8	1	no	—
44	C₂×Q₈×C₁₃	—	8	1	no	—
45	C₂×Q₈×C₁₃	C₂²⋊C₄	1	6	yes	—
46	C₂×Q₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×Q₈×C₁₃	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₁₃	—	2	3	no	—
49	C₂²×C₄×C₁₃	—	2	3	no	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
51	C₁₃×C₄⋊C₄	—	2	12	no	—
52	C₁₃×C₄⋊C₄	—	2	12	no	—
53	C₁₃×C₄⋊C₄	—	2	12	no	—
54	C₄²×C₁₃	—	2	12	no	—
55	C₄²×C₁₃	—	2	3	no	—
56	C₄²×C₁₃	C₂²⋊C₄	1	6	yes	—
57	C₂²⋊Dic₁₃	—	4	1	no	—
58	C₄⋊Dic₁₃	—	2	6	no	—
59	C₄×Dic₁₃	—	4	3	no	—
60	#6099	—	4	3	no	—
61	#6099	—	4	3	no	—
62	C₂×#1055	—	4	3	no	—
63	C₂×Q₈²	D₁₃	1	1	yes	characteristic
64	C₂×#356	—	13	3	no	—
65	#1886	—	13	3	no	—
66	C₂³×C₁₃	#120	1	1	yes	characteristic
67	Q₈×C₁₃	—	2	16	no	—
68	Q₈×C₁₃	—	4	1	no	—
69	Q₈×C₁₃	—	4	3	no	—
70	Q₈×C₁₃	—	4	6	no	—
71	Q₈×C₁₃	—	8	3	no	—
72	Q₈×C₁₃	—	8	4	no	—
73	C₂×C₄×C₁₃	—	2	12	no	—
74	C₂×C₄×C₁₃	—	2	6	no	—
75	C₂×C₄×C₁₃	—	2	3	no	—
76	C₂×C₄×C₁₃	—	2	3	no	—
77	C₂×C₄×C₁₃	—	2	6	no	—
78	C₂×C₄×C₁₃	—	4	3	no	—
79	C₂×C₄×C₁₃	—	4	3	no	—
80	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
81	C₂×C₄×C₁₃	#120	1	3	yes	—
82	C₂×Dic₁₃	—	4	2	no	—
83	#1055	—	8	3	no	—
84	Q₈²	—	2	6	no	—
85	Q₈²	Dic₁₃	1	4	yes	complemented
86	C₂×C₄⋊Q₈	—	2	3	no	—
87	C₂×C₄⋊Q₈	C₂×D₁₃	1	3	yes	—
88	#519	—	13	1	no	—
89	C₂×C₄×Q₈	—	2	3	no	—
90	#419	—	26	3	no	—
91	C₂×#101	—	13	3	no	—
92	#356	—	26	6	no	—
93	#339	—	26	3	no	—
94	C₂²×C₁₃	—	2	2	no	—
95	C₂²×C₁₃	#379	1	1	yes	characteristic
96	C₂²×C₁₃	B(4,𝔽₂)	1	1	yes	characteristic
97	C₂²×C₁₃	#458	1	1	yes	characteristic
98	C₄×C₁₃	—	2	12	no	—
99	C₄×C₁₃	—	2	3	no	—
100	C₄×C₁₃	—	2	3	no	—
101	C₄×C₁₃	—	4	6	no	—
102	Dic₁₃	—	8	2	no	—
103	C₂²×Q₈	—	2	1	no	—
104	C₂²×Q₈	—	2	1	no	—
105	C₂²×Q₈	C₂²⋊D₁₃	1	3	yes	—
106	C₂²×Q₈	C₂²×D₁₃	1	1	yes	characteristic
107	C₂×M₁₆	—	26	3	no	—
108	C₄⋊Q₈	—	2	12	no	—
109	C₄⋊Q₈	—	2	12	no	—
110	C₄⋊Q₈	—	2	6	no	—
111	C₄⋊Q₈	C₂×Dic₁₃	1	6	yes	—
112	C₄⋊Q₈	C₂²⋊D₁₃	1	6	yes	—
113	C₄×Q₈	—	2	12	no	—
114	C₄×Q₈	—	2	12	no	—
115	C₄×Q₈	—	52	2	no	—
116	#117	—	26	3	no	—
117	C₂×C₄⋊C₄	—	2	3	no	—
118	C₂×C₄⋊C₄	—	2	6	no	—
119	C₂×C₄²	—	2	3	no	—
120	C₂×C₄²	C₂²⋊D₁₃	1	3	yes	—
121	#103	—	52	6	no	—
122	#101	—	26	6	no	—
123	C₂×C₁₃	—	2	1	no	—
124	C₂×C₁₃	—	2	1	no	—
125	C₂×C₁₃	#1790	1	1	yes	characteristic
126	C₂×C₁₃	#2205	1	2	yes	—
127	C₂×Q₈	—	2	4	no	—
128	C₂×Q₈	—	2	8	no	—
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	—	2	3	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	4	3	no	—
134	C₂×Q₈	—	4	3	no	—
135	C₂×Q₈	—	8	1	no	—
136	C₂×Q₈	—	8	1	no	—
137	C₂×Q₈	C₂²⋊Dic₁₃	1	6	yes	—
138	C₂×Q₈	C₂²×Dic₁₃	1	1	yes	characteristic
139	C₂×Q₈	C₂×C₂²⋊D₁₃	1	3	yes	—
140	C₂²×C₄	—	2	3	no	—
141	C₂²×C₄	—	2	3	no	—
142	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
143	M₁₆	—	52	3	no	—
144	M₁₆	—	52	3	no	—
145	C₂×C₈	—	52	3	no	—
146	C₄⋊C₄	—	2	12	no	—
147	C₄⋊C₄	—	2	12	no	—
148	C₄⋊C₄	—	2	12	no	—
149	C₄⋊C₄	—	26	6	no	—
150	C₂²⋊C₄	—	52	1	no	—
151	C₄²	—	2	12	no	—
152	C₄²	—	2	3	no	—
153	C₄²	—	52	3	no	—
154	C₄²	C₂²⋊Dic₁₃	1	6	yes	—
155	C₁₃	#12777	1	1	yes	characteristic, complemented, Sylow
156	C₂³	#88207	1	1	yes	characteristic
157	Q₈	—	2	16	no	—
158	Q₈	—	4	1	no	—
159	Q₈	—	4	3	no	—
160	Q₈	—	4	6	no	—
161	Q₈	—	8	3	no	—
162	Q₈	—	8	4	no	—
163	C₂×C₄	—	2	12	no	—
164	C₂×C₄	—	2	3	no	—
165	C₂×C₄	—	2	3	no	—
166	C₂×C₄	—	2	6	no	—
167	C₂×C₄	—	2	6	no	—
168	C₂×C₄	—	4	3	no	—
169	C₂×C₄	—	4	3	no	—
170	C₂×C₄	—	52	2	no	—
171	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	3	yes	—
172	C₂×C₄	#88207	1	3	yes	—
173	C₈	—	104	3	no	—
174	C₂²	—	2	2	no	—
175	C₂²	#11718904	1	1	yes	characteristic
176	C₂²	#11718905	1	1	yes	characteristic
177	C₂²	#11718908	1	1	yes	characteristic
178	C₄	—	2	12	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	—	4	6	no	—
182	C₄	—	104	2	no	—
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	#13114784	1	1	yes	characteristic
186	C₂	#13116866	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718140

C₂

yes

maximal

#13115230

C₂

yes

maximal

C₂×Q₈²×C₁₃

C₂

yes

maximal, characteristic

Q₈²×C₁₃

—

Q₈²×C₁₃

C₄

yes

complemented

C₂×C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊Q₈

C₂²

yes

—

C₂×C₄×Q₈×C₁₃

—

#11719517

C₂²

yes

characteristic

C₂×#88076

C₂²

yes

—

#11718886

—

#11718140

—

#11718115

—

C₂²×Q₈×C₁₃

—

C₂²×Q₈×C₁₃

—

C₂²×Q₈×C₁₃

D₄

yes

—

C₂²×Q₈×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

C₁₃×C₄⋊Q₈

D₄

yes

—

C₄×Q₈×C₁₃

—

C₄×Q₈×C₁₃

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

D₄

yes

—

Q₈×Dic₁₃

—

—

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂²⋊C₄

yes

—

C₂×Q₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×Q₈×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#6099

—

#6099

—

C₂×#1055

—

C₂×Q₈²

D₁₃

yes

characteristic

C₂×#356

—

#1886

—

C₂³×C₁₃

#120

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

#120

yes

—

C₂×Dic₁₃

—

#1055

—

Q₈²

—

Q₈²

Dic₁₃

yes

complemented

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₂×D₁₃

yes

—

#519

—

C₂×C₄×Q₈

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

100

C₄×C₁₃

—

101

C₄×C₁₃

—

102

Dic₁₃

—

103

C₂²×Q₈

—

104

C₂²×Q₈

—

105

C₂²×Q₈

C₂²⋊D₁₃

yes

—

106

C₂²×Q₈

C₂²×D₁₃

yes

characteristic

107

C₂×M₁₆

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

C₂×Dic₁₃

yes

—

112

C₄⋊Q₈

C₂²⋊D₁₃

yes

—

113

C₄×Q₈

—

114

C₄×Q₈

—

115

C₄×Q₈

—

116

#117

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄²

—

120

C₂×C₄²

C₂²⋊D₁₃

yes

—

121

#103

—

122

#101

—

123

C₂×C₁₃

—

124

C₂×C₁₃

—

125

C₂×C₁₃

#1790

yes

characteristic

126

C₂×C₁₃

#2205

yes

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

C₂²⋊Dic₁₃

yes

—

138

C₂×Q₈

C₂²×Dic₁₃

yes

characteristic

139

C₂×Q₈

C₂×C₂²⋊D₁₃

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

143

M₁₆

—

144

M₁₆

—

145

C₂×C₈

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₂²⋊C₄

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

C₂²⋊Dic₁₃

yes

—

155

C₁₃

#12777

yes

characteristic, complemented, Sylow

156

C₂³

#88207

yes

characteristic

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

—

172

C₂×C₄

#88207

yes

—

173

C₈

—

104

—

174

C₂²

—

175

C₂²

#11718904

yes

characteristic

176

C₂²

#11718905

yes

characteristic

177

C₂²

#11718908

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

104

—

183

C₂

—

184

C₂

—

185

C₂

#13114784

yes

characteristic

186

C₂

#13116866

yes

—

	subgroup	G/H
center	C₂²	#11718905
commutator	C₂×Q₈×C₁₃	C₂²×C₄
Frattini	C₂²×Q₈	C₂²×D₁₃
Fitting	C₂×Q₈²×C₁₃	C₂
socle	C₂²×C₁₃	B(4,𝔽₂)
prime radical	C₂³×C₁₃	#120
cosocle	C₂²×Q₈×C₁₃	C₂³
hypercenter	C₂×Q₈²	D₁₃
hypocenter	C₁₃	#12777
LCS	G ▷ C₂×Q₈×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×Q₈²

subgroup

G/H

center

C₂²

#11718905

commutator

C₂×Q₈×C₁₃

C₂²×C₄

Frattini

C₂²×Q₈

C₂²×D₁₃

Fitting

C₂×Q₈²×C₁₃

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂²×Q₈×C₁₃

C₂³

hypercenter

C₂×Q₈²

D₁₃

hypocenter

C₁₃

#12777

LCS

G ▷ C₂×Q₈×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×Q₈²

N	H ≅ G/N	type
5. Q₈²×C₁₃	C₄	semidirect
85. Q₈²	Dic₁₃	semidirect
155. C₁₃	#12777	semidirect

H ≅ G/N

type

5. Q₈²×C₁₃

C₄

semidirect

85. Q₈²

Dic₁₃

semidirect

155. C₁₃

#12777

semidirect

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

subgroup	index
C₂×#11718140	2
#13115230	2
C₂×Q₈²×C₁₃	2
#12777	13

subgroup

index

C₂×#11718140

#13115230

C₂×Q₈²×C₁₃

#12777

Subgroups — #15778809

Conjugacy classes of subgroups

Complements

Maximal subgroups