Group #15756708 — subgroups

Subgroups — #15756708

Order 3328.

438 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 227 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	#13114996	C₂	1	2	yes	maximal
2	#13114996	C₂	1	1	yes	maximal, characteristic
3	#13114993	C₂	1	1	yes	maximal, characteristic
4	#13114964	C₂	1	2	yes	maximal
5	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88058	—	2	4	no	—
9	C₂×#88058	—	2	2	no	—
10	C₂×#88057	—	2	2	no	—
11	#11718596	C₂²	1	1	yes	characteristic
12	#11718596	C₂²	1	2	yes	—
13	#11718596	C₂²	1	1	yes	characteristic
14	C₁₃×#322	C₄	1	4	yes	complemented
15	#11718143	—	2	4	no	—
16	C₂²×C₈×C₁₃	D₄	1	4	yes	—
17	C₂²×C₈×C₁₃	D₄	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	—	2	1	no	—
20	C₂×C₄⋊Dic₁₃	—	2	1	no	—
21	C₂×C₄⋊Dic₁₃	—	2	2	no	—
22	C₂×C₄⋊Dic₁₃	—	2	2	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	Q₈	1	4	yes	—
29	C₄×C₈×C₁₃	Q₈	1	2	yes	—
30	#88058	—	4	8	no	—
31	#88058	—	4	4	no	—
32	#88057	—	4	4	no	—
33	#8820	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
36	C₂²×Dic₁₃	—	4	1	no	—
37	C₂²×Dic₁₃	—	4	2	no	—
38	C₂²×Dic₁₃	—	4	1	no	—
39	C₂×C₈×C₁₃	—	2	4	no	—
40	C₂×C₈×C₁₃	—	2	4	no	—
41	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
42	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
43	C₂×C₈×C₁₃	D₈	1	4	yes	—
44	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
45	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
48	C₄⋊Dic₁₃	—	4	2	no	—
49	C₄⋊Dic₁₃	—	4	2	no	—
50	C₄⋊Dic₁₃	—	4	4	no	—
51	C₄⋊Dic₁₃	—	4	4	no	—
52	C₄⋊Dic₁₃	—	4	4	no	—
53	C₄⋊Dic₁₃	—	4	4	no	—
54	C₄⋊Dic₁₃	—	4	4	no	—
55	#1845	—	13	1	no	—
56	#1845	—	13	2	no	—
57	#1844	—	13	1	no	—
58	#1834	—	13	2	no	—
59	C₂×#322	D₁₃	1	1	yes	characteristic
60	C₂³×C₁₃	#127	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
67	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
68	C₂×C₄×C₁₃	#137	1	4	yes	—
69	C₂×Dic₁₃	—	8	1	no	—
70	C₂×Dic₁₃	—	8	2	no	—
71	C₂×Dic₁₃	—	8	1	no	—
72	C₂×Dic₁₃	—	8	2	no	—
73	C₂×Dic₁₃	—	8	2	no	—
74	C₂×Dic₁₃	—	8	2	no	—
75	C₂×Dic₁₃	—	8	2	no	—
76	C₈×C₁₃	—	2	8	no	—
77	C₈×C₁₃	#107	1	8	yes	complemented
78	C₂×#107	—	26	4	no	—
79	C₂×#107	—	26	2	no	—
80	C₂×#106	—	26	2	no	—
81	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
83	#384	—	13	1	no	—
84	#384	—	13	2	no	—
85	#384	—	13	1	no	—
86	#340	—	26	4	no	—
87	#322	Dic₁₃	1	4	yes	complemented
88	C₂²×C₁₃	#384	1	1	yes	characteristic
89	C₂²×C₁₃	#493	1	1	yes	characteristic
90	C₂²×C₁₃	#494	1	1	yes	characteristic
91	C₂²×C₁₃	#496	1	2	yes	—
92	C₂²×C₁₃	#497	1	2	yes	—
93	C₄×C₁₃	C₂×#107	1	2	yes	—
94	C₄×C₁₃	#428	1	4	yes	—
95	C₄×C₁₃	#500	1	2	yes	—
96	C₄×C₁₃	#501	1	4	yes	—
97	Dic₁₃	—	16	1	no	—
98	Dic₁₃	—	16	2	no	—
99	Dic₁₃	—	16	1	no	—
100	C₂²×C₈	D₅₂	1	4	yes	—
101	C₂²×C₈	D₅₂	1	2	yes	—
102	C₂×C₄⋊C₄	—	26	2	no	—
103	C₂×C₄⋊C₄	—	26	2	no	—
104	C₂×C₄⋊C₄	—	26	1	no	—
105	C₂×C₄⋊C₄	—	26	1	no	—
106	C₂×C₄⋊C₄	—	26	2	no	—
107	C₂×C₄⋊C₄	—	26	2	no	—
108	C₂×C₄⋊C₄	—	26	2	no	—
109	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
110	#107	—	52	8	no	—
111	#107	—	52	4	no	—
112	#106	—	52	4	no	—
113	C₄×C₈	Dic₂₆	1	4	yes	—
114	C₄×C₈	Dic₂₆	1	2	yes	—
115	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
116	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
117	C₂×C₁₃	#1721	1	1	yes	characteristic
118	C₂×C₁₃	#1722	1	1	yes	characteristic
119	C₂×C₁₃	#1723	1	2	yes	—
120	C₂×C₁₃	#1845	1	1	yes	characteristic
121	C₂×C₁₃	#1847	1	2	yes	—
122	C₂²×C₄	—	52	2	no	—
123	C₂²×C₄	—	52	1	no	—
124	C₂²×C₄	—	52	1	no	—
125	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₅₂	1	2	yes	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	D₁₀₄	1	4	yes	—
130	C₂×C₈	Dic₅₂	1	4	yes	—
131	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
132	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
133	C₄⋊C₄	—	52	2	no	—
134	C₄⋊C₄	—	52	2	no	—
135	C₄⋊C₄	—	52	4	no	—
136	C₄⋊C₄	—	52	4	no	—
137	C₄⋊C₄	—	52	4	no	—
138	C₄⋊C₄	—	52	4	no	—
139	C₄⋊C₄	—	52	4	no	—
140	C₄²	C₂×Dic₂₆	1	2	yes	—
141	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
142	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
143	C₁₃	#8820	1	1	yes	characteristic, complemented, Sylow
144	C₂³	#88128	1	1	yes	characteristic
145	C₂×C₄	—	104	1	no	—
146	C₂×C₄	—	104	2	no	—
147	C₂×C₄	—	104	1	no	—
148	C₂×C₄	—	104	2	no	—
149	C₂×C₄	—	104	2	no	—
150	C₂×C₄	—	104	2	no	—
151	C₂×C₄	—	104	2	no	—
152	C₂×C₄	#88123	1	1	yes	characteristic
153	C₂×C₄	#88123	1	2	yes	—
154	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
155	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
156	C₂×C₄	#88162	1	4	yes	—
157	C₂×C₄	#88163	1	4	yes	—
158	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
160	C₈	—	2	8	no	—
161	C₈	#88058	1	8	yes	complemented
162	C₂²	#11718356	1	1	yes	characteristic
163	C₂²	#11718360	1	1	yes	characteristic
164	C₂²	#11718373	1	2	yes	—
165	C₂²	#11718376	1	2	yes	—
166	C₂²	#11718596	1	1	yes	characteristic
167	C₄	—	208	1	no	—
168	C₄	—	208	2	no	—
169	C₄	—	208	1	no	—
170	C₄	#11718343	1	2	yes	—
171	C₄	#11718363	1	4	yes	—
172	C₄	C₂×#88058	1	2	yes	—
173	C₄	#11718782	1	4	yes	—
174	C₂	#13114502	1	1	yes	characteristic
175	C₂	#13114507	1	1	yes	characteristic
176	C₂	#13114513	1	2	yes	—
177	C₂	#13114996	1	1	yes	characteristic
178	C₂	#13115004	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114996

C₂

yes

maximal

#13114996

C₂

yes

maximal, characteristic

#13114993

C₂

yes

maximal, characteristic

#13114964

C₂

yes

maximal

C₂×C₁₃×#322

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

—

—

C₂²

yes

characteristic

#11718596

C₂²

yes

—

#11718596

C₂²

yes

characteristic

C₁₃×#322

C₄

yes

complemented

#11718143

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

yes

—

—

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

—

—

Dic₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#497

yes

—

C₄×C₁₃

C₂×#107

yes

—

C₄×C₁₃

#428

yes

—

C₄×C₁₃

#500

yes

—

C₄×C₁₃

#501

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

100

C₂²×C₈

D₅₂

yes

—

101

C₂²×C₈

D₅₂

yes

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

C₂²×D₁₃

yes

characteristic

110

#107

—

111

#107

—

112

#106

—

113

C₄×C₈

Dic₂₆

yes

—

114

C₄×C₈

Dic₂₆

yes

—

115

C₄×C₈

C₂×Dic₁₃

yes

—

116

C₄×C₈

C₂×Dic₁₃

yes

—

117

C₂×C₁₃

#1721

yes

characteristic

118

C₂×C₁₃

#1722

yes

characteristic

119

C₂×C₁₃

#1723

yes

—

120

C₂×C₁₃

#1845

yes

characteristic

121

C₂×C₁₃

#1847

yes

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂×D₅₂

yes

characteristic

126

C₂²×C₄

C₂×D₅₂

yes

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

D₁₀₄

yes

—

130

C₂×C₈

Dic₅₂

yes

—

131

C₂×C₈

C₄⋊Dic₁₃

yes

—

132

C₂×C₈

C₄⋊Dic₁₃

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄²

C₂×Dic₂₆

yes

—

141

C₄²

C₂×Dic₂₆

yes

characteristic

142

C₄²

C₂²×Dic₁₃

yes

characteristic

143

C₁₃

#8820

yes

characteristic, complemented, Sylow

144

C₂³

#88128

yes

characteristic

145

C₂×C₄

—

104

—

146

C₂×C₄

—

104

—

147

C₂×C₄

—

104

—

148

C₂×C₄

—

104

—

149

C₂×C₄

—

104

—

150

C₂×C₄

—

104

—

151

C₂×C₄

—

104

—

152

C₂×C₄

#88123

yes

characteristic

153

C₂×C₄

#88123

yes

—

154

C₂×C₄

C₂×D₁₀₄

yes

—

155

C₂×C₄

C₂×Dic₅₂

yes

—

156

C₂×C₄

#88162

yes

—

157

C₂×C₄

#88163

yes

—

158

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

159

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

—

160

C₈

—

161

C₈

#88058

yes

complemented

162

C₂²

#11718356

yes

characteristic

163

C₂²

#11718360

yes

characteristic

164

C₂²

#11718373

yes

—

165

C₂²

#11718376

yes

—

166

C₂²

#11718596

yes

characteristic

167

C₄

—

208

—

168

C₄

—

208

—

169

C₄

—

208

—

170

C₄

#11718343

yes

—

171

C₄

#11718363

yes

—

172

C₄

C₂×#88058

yes

—

173

C₄

#11718782

yes

—

174

C₂

#13114502

yes

characteristic

175

C₂

#13114507

yes

characteristic

176

C₂

#13114513

yes

—

177

C₂

#13114996

yes

characteristic

178

C₂

#13115004

yes

—

	subgroup	G/H
center	C₂³	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle, prime radical	C₂³×C₁₃	#127
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#8820
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#88128

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₁₃×#322

C₂

socle, prime radical

C₂³×C₁₃

#127

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
14. C₁₃×#322	C₄	semidirect
77. C₈×C₁₃	#107	semidirect
87. #322	Dic₁₃	semidirect
143. C₁₃	#8820	semidirect
161. C₈	#88058	semidirect

H ≅ G/N

type

14. C₁₃×#322

C₄

semidirect

77. C₈×C₁₃

#107

semidirect

87. #322

Dic₁₃

semidirect

143. C₁₃

#8820

semidirect

161. C₈

#88058

semidirect

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

subgroup	index
#13114996	2
#13114993	2
#13114964	2
C₂×C₁₃×#322	2
#8820	13

subgroup

index

C₂×C₁₃×#322

Subgroups — #15756708

Conjugacy classes of subgroups

Complements

Maximal subgroups