Group #15776320 — subgroups

Subgroups — #15776320

Order 3328.

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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	#13126635	C₂	1	1	yes	maximal, characteristic
2	#13116914	C₂	1	2	yes	maximal, complemented
3	#13112873	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112810	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1709	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719462	—	2	2	no	—
7	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
8	C₁₃×#493	C₂²	1	1	yes	characteristic
9	C₁₃×#486	—	2	2	no	—
10	#11718840	—	2	2	no	—
11	#11718821	—	2	2	no	—
12	#11718738	—	2	2	no	—
13	#11718361	C₂²	1	2	yes	—
14	C₈²×C₁₃	C₂²	1	1	yes	characteristic, complemented
15	#11718157	—	2	2	no	—
16	#11718055	C₂²	1	2	yes	complemented
17	#88251	—	2	2	no	—
18	C₂×C₁₃×Q₁₆	—	2	2	no	—
19	C₂×C₁₃×D₈	—	2	2	no	—
20	C₁₃×C₄⋊Q₈	—	2	1	no	—
21	C₁₃×#127	D₄	1	2	yes	—
22	#88186	—	2	4	no	—
23	#88166	—	4	2	no	—
24	#88165	—	4	4	no	—
25	D₈×D₁₃	—	4	2	no	—
26	#88158	D₄	1	2	yes	—
27	#88155	—	2	2	no	—
28	C₁₃×#102	—	4	4	no	—
29	C₄×C₈×C₁₃	—	2	1	no	—
30	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	#88073	—	2	2	no	—
32	#88069	—	4	2	no	—
33	#88068	—	4	2	no	—
34	#88059	D₄	1	2	yes	—
35	#88052	—	2	2	no	—
36	#88045	—	2	2	no	—
37	#12312	—	13	1	no	maximal, Sylow
38	C₂×Q₈×C₁₃	—	4	1	no	—
39	C₂×D₄×C₁₃	—	2	2	no	—
40	C₂×D₄×C₁₃	—	4	2	no	—
41	C₂×C₂²⋊D₁₃	—	8	2	no	—
42	#6129	—	4	4	no	—
43	D₄×D₁₃	—	4	4	no	—
44	#6127	—	2	2	no	—
45	C₁₃×Q₁₆	—	4	2	no	—
46	C₁₃×Q₁₆	—	4	2	no	—
47	C₁₃×D₈	—	4	2	no	—
48	C₁₃×D₈	—	4	2	no	—
49	C₂×C₈×C₁₃	—	2	2	no	—
50	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
51	C₁₃×C₄⋊C₄	—	4	2	no	—
52	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
53	#6105	—	4	4	no	—
54	#6104	—	4	4	no	—
55	#6099	—	2	2	no	—
56	Dic₅₂	D₈	1	2	yes	—
57	D₁₀₄	D₈	1	2	yes	—
58	#6095	—	2	2	no	—
59	#6094	—	4	2	no	—
60	C₈×D₁₃	—	4	2	no	—
61	#6090	—	4	2	no	—
62	#3292	—	13	1	no	—
63	#2214	—	13	2	no	—
64	#1709	D₁₃	1	1	yes	characteristic, complemented
65	#1357	—	13	1	no	—
66	#1335	—	13	2	no	—
67	C₂³×C₁₃	—	4	2	no	—
68	C₂²×D₁₃	—	8	2	no	—
69	Q₈×C₁₃	—	8	2	no	—
70	D₄×C₁₃	—	4	4	no	—
71	D₄×C₁₃	—	8	4	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	8	1	no	—
74	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
75	C₂²⋊D₁₃	—	4	2	no	—
76	C₂²⋊D₁₃	—	8	4	no	—
77	C₂×Dic₁₃	—	8	2	no	—
78	D₅₂	—	2	2	no	—
79	C₄×D₁₃	—	4	2	no	—
80	Dic₂₆	—	2	2	no	—
81	C₈×C₁₃	—	4	2	no	—
82	C₈×C₁₃	C₂×D₈	1	2	yes	—
83	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
84	#1055	—	4	2	no	—
85	#576	—	26	2	no	—
86	#510	—	26	2	no	—
87	#494	C₂×D₁₃	1	1	yes	characteristic, complemented
88	#493	C₂×D₁₃	1	1	yes	characteristic
89	#486	—	2	2	no	—
90	#472	—	26	2	no	—
91	#453	—	26	2	no	—
92	#443	—	13	2	no	—
93	#364	—	13	2	no	—
94	#362	—	26	2	no	—
95	C₈²	C₂×D₁₃	1	1	yes	characteristic, complemented
96	C₂²×C₁₃	—	8	2	no	—
97	C₂²×C₁₃	—	16	2	no	—
98	C₂²×C₁₃	#453	1	1	yes	characteristic
99	C₂×D₁₃	—	4	2	no	—
100	C₂×D₁₃	—	32	2	no	—
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	—	16	1	no	—
103	C₄×C₁₃	#447	1	2	yes	—
104	Dic₁₃	—	4	2	no	—
105	H(5,𝔽₂)	—	26	2	no	—
106	AGL(1,ℤ/8)	—	26	4	no	—
107	AGL(1,ℤ/8)	—	52	4	no	—
108	#135	—	13	2	no	—
109	#135	—	52	2	no	—
110	C₂×Q₁₆	—	2	2	no	—
111	C₂×D₈	—	2	2	no	—
112	C₂×D₈	—	52	2	no	—
113	#131	—	26	2	no	—
114	C₄⋊Q₈	—	2	1	no	—
115	#127	C₂²⋊D₁₃	1	2	yes	—
116	Q₃₂	—	52	2	no	—
117	SD₃₂	—	52	2	no	—
118	M₃₂	—	26	2	no	—
119	#108	—	13	2	no	—
120	#104	—	26	2	no	—
121	#102	—	4	4	no	—
122	#100	—	26	2	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
125	C₂×C₁₃	—	2	1	no	—
126	C₂×C₁₃	—	16	2	no	—
127	C₂×C₁₃	#1619	1	1	yes	characteristic
128	D₁₃	—	8	2	no	—
129	#41	—	26	2	no	—
130	#41	—	52	4	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×D₄	—	2	2	no	—
133	C₂×D₄	—	4	2	no	—
134	C₂×D₄	—	52	4	no	—
135	C₂×D₄	—	104	2	no	—
136	Q₁₆	—	4	2	no	—
137	Q₁₆	—	4	2	no	—
138	Q₁₆	—	13	2	no	—
139	SD₁₆	—	26	2	no	—
140	SD₁₆	—	52	4	no	—
141	D₈	—	4	2	no	—
142	D₈	—	4	2	no	—
143	D₈	—	13	2	no	—
144	D₈	—	52	4	no	—
145	M₁₆	—	26	2	no	—
146	M₁₆	—	52	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	52	2	no	—
149	C₂×C₈	D₄×D₁₃	1	2	yes	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
152	C₁₆	—	52	2	no	—
153	C₁₃	#12312	1	1	yes	characteristic, complemented, Sylow
154	C₂³	—	4	2	no	—
155	C₂³	—	104	2	no	—
156	Q₈	—	8	2	no	—
157	Q₈	—	26	2	no	—
158	D₄	—	4	4	no	—
159	D₄	—	8	4	no	—
160	D₄	—	26	2	no	—
161	D₄	—	52	2	no	—
162	D₄	—	104	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	52	2	no	—
166	C₂×C₄	—	104	2	no	—
167	C₂×C₄	#88191	1	1	yes	characteristic
168	C₈	—	4	2	no	—
169	C₈	—	52	2	no	—
170	C₈	D₈×D₁₃	1	2	yes	—
171	C₈	#88165	1	2	yes	—
172	C₂²	—	8	2	no	—
173	C₂²	—	16	2	no	—
174	C₂²	—	52	2	no	—
175	C₂²	—	416	2	no	—
176	C₂²	#11718738	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	16	1	no	—
179	C₄	—	52	2	no	—
180	C₄	#11718817	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	16	2	no	—
183	C₂	—	104	2	no	—
184	C₂	#13113888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13126635

C₂

yes

maximal, characteristic

#13116914

C₂

yes

maximal, complemented

#13112873

C₂

yes

maximal, characteristic, complemented

#13112810

C₂

yes

maximal, complemented

C₁₃×#1709

C₂

yes

maximal, characteristic, complemented

#11719462

—

C₁₃×#494

C₂²

yes

characteristic, complemented

C₁₃×#493

C₂²

yes

characteristic

C₁₃×#486

—

—

—

—

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic, complemented

#11718157

—

#11718055

C₂²

yes

complemented

#88251

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×D₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×#127

D₄

yes

—

#88186

—

#88166

—

#88165

—

D₈×D₁₃

—

#88158

D₄

yes

—

#88155

—

C₁₃×#102

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

D₄

yes

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×C₂²⋊D₁₃

—

#6129

—

D₄×D₁₃

—

#6127

—

C₁₃×Q₁₆

—

C₁₃×Q₁₆

—

C₁₃×D₈

—

C₁₃×D₈

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

—

—

—

Dic₅₂

D₈

yes

—

D₁₀₄

D₈

yes

—

#6095

—

#6094

—

C₈×D₁₃

—

—

—

—

D₁₃

yes

characteristic, complemented

#1357

—

#1335

—

C₂³×C₁₃

—

C₂²×D₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#120

yes

characteristic

C₂²⋊D₁₃

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

D₅₂

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

C₂×D₈

yes

—

C₈×C₁₃

AGL(1,ℤ/8)

yes

—

—

—

—

C₂×D₁₃

yes

characteristic, complemented

#493

C₂×D₁₃

yes

—

—

—

—

—

—

C₈²

C₂×D₁₃

yes

characteristic, complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#453

yes

characteristic

C₂×D₁₃

—

100

C₂×D₁₃

—

101

C₄×C₁₃

—

102

C₄×C₁₃

—

103

C₄×C₁₃

#447

yes

—

104

Dic₁₃

—

105

H(5,𝔽₂)

—

106

AGL(1,ℤ/8)

—

107

AGL(1,ℤ/8)

—

108

#135

—

109

#135

—

110

C₂×Q₁₆

—

111

C₂×D₈

—

112

C₂×D₈

—

113

#131

—

114

C₄⋊Q₈

—

115

#127

C₂²⋊D₁₃

yes

—

116

Q₃₂

—

117

SD₃₂

—

118

M₃₂

—

119

#108

—

120

#104

—

121

#102

—

122

#100

—

123

C₄×C₈

—

124

C₄×C₈

C₂²×D₁₃

yes

characteristic

125

C₂×C₁₃

—

126

C₂×C₁₃

—

127

C₂×C₁₃

#1619

yes

characteristic

128

D₁₃

—

129

#41

—

130

#41

—

131

C₂×Q₈

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

104

—

136

Q₁₆

—

137

Q₁₆

—

138

Q₁₆

—

139

SD₁₆

—

140

SD₁₆

—

141

D₈

—

142

D₈

—

143

D₈

—

144

D₈

—

145

M₁₆

—

146

M₁₆

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

D₄×D₁₃

yes

—

150

C₄⋊C₄

—

151

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

152

C₁₆

—

153

C₁₃

#12312

yes

characteristic, complemented, Sylow

154

C₂³

—

155

C₂³

—

104

—

156

Q₈

—

157

Q₈

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

104

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

104

—

167

C₂×C₄

#88191

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

D₈×D₁₃

yes

—

171

C₈

#88165

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

416

—

176

C₂²

#11718738

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#11718817

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

104

—

184

C₂

#13113888

yes

characteristic

	subgroup	G/H
center	C₂	#13113888
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1709	C₂
socle	C₂×C₁₃	#1619
prime radical	#13126635	C₂
hypercenter	#1709	D₁₃
hypocenter	C₁₃	#12312
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1709

subgroup

G/H

center

C₂

#13113888

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1709

C₂

C₂×C₁₃

C₂

D₁₃

C₁₃

G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1709

N	H ≅ G/N	type
2. #13116914	C₂	semidirect
3. #13112873	C₂	semidirect
4. #13112810	C₂	semidirect
5. C₁₃×#1709	C₂	semidirect
7. C₁₃×#494	C₂²	semidirect
14. C₈²×C₁₃	C₂²	semidirect
16. #11718055	C₂²	semidirect
64. #1709	D₁₃	semidirect
87. #494	C₂×D₁₃	semidirect
95. C₈²	C₂×D₁₃	semidirect
153. C₁₃	#12312	semidirect

H ≅ G/N

type

2. #13116914

C₂

semidirect

3. #13112873

C₂

semidirect

4. #13112810

C₂

semidirect

5. C₁₃×#1709

C₂

semidirect

7. C₁₃×#494

C₂²

semidirect

14. C₈²×C₁₃

C₂²

semidirect

16. #11718055

C₂²

semidirect

64. #1709

D₁₃

semidirect

87. #494

C₂×D₁₃

semidirect

95. C₈²

C₂×D₁₃

semidirect

153. C₁₃

#12312

semidirect

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

subgroup	index
#13126635	2
#13116914	2
#13112873	2
#13112810	2
C₁₃×#1709	2
#12312	13

subgroup

index

C₁₃×#1709

Subgroups — #15776320

Conjugacy classes of subgroups

Complements

Maximal subgroups