Group #15777671 — subgroups

Subgroups — #15777671

Order 3328.

340 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 91 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	#13117086	C₂	1	2	yes	maximal, complemented
2	#13117060	C₂	1	2	yes	maximal, complemented
3	#13114451	C₂	1	1	yes	maximal, characteristic, complemented
4	#13114428	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₀₈	—	2	4	no	—
7	#11718356	—	2	1	no	—
8	#11718356	—	2	1	no	—
9	#11718356	C₂²	1	1	yes	characteristic
10	#11718356	C₂²	1	2	yes	—
11	#11718355	—	2	1	no	—
12	#11718354	—	2	1	no	—
13	#11718343	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
15	C₈²×C₁₃	C₂²	1	1	yes	characteristic
16	#11718101	—	2	4	no	—
17	C₂×D₁₀₄	—	2	2	no	—
18	C₂×D₁₀₄	—	2	2	no	—
19	C₂×D₁₀₄	—	2	2	no	—
20	C₂×D₁₀₄	—	4	2	no	—
21	C₂×D₁₀₄	—	4	2	no	—
22	C₂×#6095	—	4	2	no	—
23	#88128	—	2	1	no	—
24	#88128	—	2	2	no	—
25	#88123	—	2	1	no	—
26	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
27	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	#88061	—	4	2	no	—
31	#88058	—	2	2	no	—
32	D₂₀₈	—	4	8	no	—
33	#12525	—	13	1	no	maximal, Sylow
34	C₂×D₅₂	—	4	1	no	—
35	C₂×D₅₂	—	4	2	no	—
36	C₂×D₅₂	—	4	1	no	—
37	C₂×D₅₂	—	4	2	no	—
38	C₂×D₅₂	—	4	2	no	—
39	C₂×D₅₂	—	4	1	no	—
40	C₂×Dic₂₆	—	4	1	no	—
41	C₂×C₈×C₁₃	D₈	1	2	yes	—
42	C₂×C₈×C₁₃	D₈	1	2	yes	—
43	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
44	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Dic₁₃	—	4	1	no	—
46	C₄⋊Dic₁₃	—	4	1	no	—
47	D₁₀₄	—	4	4	no	—
48	D₁₀₄	—	4	4	no	—
49	D₁₀₄	—	4	4	no	—
50	D₁₀₄	—	8	4	no	—
51	D₁₀₄	—	8	4	no	—
52	#6095	—	8	4	no	—
53	C₁₃×C₁₆	—	2	4	no	—
54	#2246	—	13	2	no	—
55	#2240	—	13	2	no	—
56	#1712	—	13	1	no	—
57	#1706	—	13	1	no	—
58	#1312	D₁₃	1	1	yes	characteristic, complemented
59	C₂²×D₁₃	—	8	1	no	—
60	C₂²×D₁₃	—	8	2	no	—
61	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
64	C₂×Dic₁₃	—	8	1	no	—
65	D₅₂	—	8	2	no	—
66	D₅₂	—	8	2	no	—
67	D₅₂	—	8	2	no	—
68	D₅₂	—	8	2	no	—
69	D₅₂	—	8	4	no	—
70	D₅₂	—	8	4	no	—
71	D₅₂	—	8	2	no	—
72	Dic₂₆	—	8	2	no	—
73	C₈×C₁₃	—	2	2	no	—
74	C₈×C₁₃	D₁₆	1	4	yes	complemented
75	C₈×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
76	C₂×D₁₆	—	26	4	no	—
77	#500	—	13	1	no	—
78	#493	—	13	1	no	—
79	#493	—	13	2	no	—
80	#493	—	26	1	no	—
81	#493	—	26	1	no	—
82	#492	—	26	1	no	—
83	#486	—	26	1	no	—
84	#357	—	26	4	no	—
85	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
86	C₈²	C₂×D₁₃	1	1	yes	characteristic
87	C₂²×C₁₃	#493	1	1	yes	characteristic
88	C₂×D₁₃	—	16	2	no	—
89	C₂×D₁₃	—	16	1	no	—
90	C₂×D₁₃	—	16	1	no	—
91	C₂×D₁₃	—	16	2	no	—
92	C₂×D₁₃	—	16	2	no	—
93	C₂×D₁₃	—	16	1	no	—
94	C₄×C₁₃	#496	1	2	yes	—
95	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
96	C₄×C₁₃	#509	1	2	yes	—
97	Dic₁₃	—	16	1	no	—
98	C₂×SD₁₆	—	52	2	no	—
99	C₂×D₈	—	26	2	no	—
100	C₂×D₈	—	26	2	no	—
101	C₂×D₈	—	26	2	no	—
102	C₂×D₈	—	52	2	no	—
103	C₂×D₈	—	52	2	no	—
104	C₄⋊Q₈	—	26	1	no	—
105	#127	—	26	1	no	—
106	#127	—	26	2	no	—
107	D₁₆	—	52	8	no	—
108	C₂×C₁₆	D₅₂	1	4	yes	—
109	#107	—	26	2	no	—
110	#102	—	52	2	no	—
111	C₄×C₈	D₅₂	1	1	yes	characteristic
112	C₄×C₈	D₅₂	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
114	C₂×C₁₃	#1721	1	1	yes	characteristic
115	C₂×C₁₃	#2246	1	1	yes	characteristic
116	C₂×C₁₃	#2250	1	1	yes	characteristic
117	D₁₃	—	32	1	no	—
118	D₁₃	—	32	2	no	—
119	C₂×Q₈	—	52	1	no	—
120	C₂×D₄	—	52	1	no	—
121	C₂×D₄	—	52	2	no	—
122	C₂×D₄	—	52	1	no	—
123	C₂×D₄	—	52	2	no	—
124	C₂×D₄	—	52	2	no	—
125	C₂×D₄	—	52	1	no	—
126	SD₁₆	—	104	4	no	—
127	D₈	—	52	4	no	—
128	D₈	—	52	4	no	—
129	D₈	—	52	4	no	—
130	D₈	—	104	4	no	—
131	D₈	—	104	4	no	—
132	C₂×C₈	D₁₀₄	1	2	yes	—
133	C₂×C₈	D₁₀₄	1	2	yes	—
134	C₂×C₈	C₂×D₅₂	1	2	yes	—
135	C₄⋊C₄	—	52	1	no	—
136	C₄⋊C₄	—	52	1	no	—
137	C₄²	C₂×D₅₂	1	1	yes	characteristic
138	C₁₆	—	2	4	no	—
139	C₁₃	#12525	1	1	yes	characteristic, complemented, Sylow
140	C₂³	—	104	1	no	—
141	C₂³	—	104	2	no	—
142	Q₈	—	104	2	no	—
143	D₄	—	104	4	no	—
144	D₄	—	104	2	no	—
145	D₄	—	104	4	no	—
146	D₄	—	104	2	no	—
147	D₄	—	104	2	no	—
148	D₄	—	104	2	no	—
149	D₄	—	104	2	no	—
150	C₂×C₄	—	104	1	no	—
151	C₂×C₄	#88128	1	1	yes	characteristic
152	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
153	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	D₂₀₈	1	4	yes	complemented
156	C₈	#88162	1	4	yes	—
157	C₂²	—	208	1	no	—
158	C₂²	—	208	2	no	—
159	C₂²	—	208	2	no	—
160	C₂²	—	208	1	no	—
161	C₂²	—	208	2	no	—
162	C₂²	—	208	1	no	—
163	C₂²	#11718356	1	1	yes	characteristic
164	C₄	—	208	1	no	—
165	C₄	#11718373	1	2	yes	—
166	C₄	C₂×D₂₀₈	1	2	yes	—
167	C₄	#11718569	1	2	yes	—
168	C₂	—	416	2	no	—
169	C₂	—	416	1	no	—
170	C₂	#13114502	1	1	yes	characteristic
171	C₂	#13117086	1	1	yes	characteristic
172	C₂	#13117102	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117086

C₂

yes

maximal, complemented

#13117060

C₂

yes

maximal, complemented

#13114451

C₂

yes

maximal, characteristic, complemented

#13114428

C₂

yes

maximal, characteristic, complemented

C₁₃×#1312

C₂

yes

maximal, characteristic, complemented

C₂×D₂₀₈

—

#11718356

—

#11718356

—

#11718356

C₂²

yes

characteristic

#11718356

C₂²

yes

—

#11718355

—

#11718354

—

#11718343

C₂²

yes

characteristic

C₄×C₁₃×C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718101

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

—

—

—

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88061

—

#88058

—

D₂₀₈

—

#12525

—

maximal, Sylow

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

#6095

—

C₁₃×C₁₆

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

D₁₆

yes

complemented

C₈×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×D₁₆

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₃

yes

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

#496

yes

—

C₄×C₁₃

C₂×D₁₆

yes

—

C₄×C₁₃

#509

yes

—

Dic₁₃

—

C₂×SD₁₆

—

C₂×D₈

—

100

C₂×D₈

—

101

C₂×D₈

—

102

C₂×D₈

—

103

C₂×D₈

—

104

C₄⋊Q₈

—

105

#127

—

106

#127

—

107

D₁₆

—

108

C₂×C₁₆

D₅₂

yes

—

109

#107

—

110

#102

—

111

C₄×C₈

D₅₂

yes

characteristic

112

C₄×C₈

D₅₂

yes

characteristic

113

C₄×C₈

C₂²×D₁₃

yes

characteristic

114

C₂×C₁₃

#1721

yes

characteristic

115

C₂×C₁₃

#2246

yes

characteristic

116

C₂×C₁₃

#2250

yes

characteristic

117

D₁₃

—

118

D₁₃

—

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

SD₁₆

—

104

—

127

D₈

—

128

D₈

—

129

D₈

—

130

D₈

—

104

—

131

D₈

—

104

—

132

C₂×C₈

D₁₀₄

yes

—

133

C₂×C₈

D₁₀₄

yes

—

134

C₂×C₈

C₂×D₅₂

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

C₂×D₅₂

yes

characteristic

138

C₁₆

—

139

C₁₃

#12525

yes

characteristic, complemented, Sylow

140

C₂³

—

104

—

141

C₂³

—

104

—

142

Q₈

—

104

—

143

D₄

—

104

—

144

D₄

—

104

—

145

D₄

—

104

—

146

D₄

—

104

—

147

D₄

—

104

—

148

D₄

—

104

—

149

D₄

—

104

—

150

C₂×C₄

—

104

—

151

C₂×C₄

#88128

yes

characteristic

152

C₂×C₄

C₂×D₁₀₄

yes

characteristic

153

C₂×C₄

C₂×D₁₀₄

yes

characteristic

154

C₈

—

155

C₈

D₂₀₈

yes

complemented

156

C₈

#88162

yes

—

157

C₂²

—

208

—

158

C₂²

—

208

—

159

C₂²

—

208

—

160

C₂²

—

208

—

161

C₂²

—

208

—

162

C₂²

—

208

—

163

C₂²

#11718356

yes

characteristic

164

C₄

—

208

—

165

C₄

#11718373

yes

—

166

C₄

C₂×D₂₀₈

yes

—

167

C₄

#11718569

yes

—

168

C₂

—

416

—

169

C₂

—

416

—

170

C₂

#13114502

yes

characteristic

171

C₂

#13117086

yes

characteristic

172

C₂

#13117102

yes

characteristic

	subgroup	G/H
center	C₂²	#11718356
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1312	C₂
socle	C₂²×C₁₃	#493
hypercenter	#1312	D₁₃
hypocenter	C₁₃	#12525
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#11718356

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1312

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #13117086	C₂	semidirect
2. #13117060	C₂	semidirect
3. #13114451	C₂	semidirect
4. #13114428	C₂	semidirect
5. C₁₃×#1312	C₂	semidirect
58. #1312	D₁₃	semidirect
74. C₈×C₁₃	D₁₆	semidirect
139. C₁₃	#12525	semidirect
155. C₈	D₂₀₈	semidirect

H ≅ G/N

type

1. #13117086

C₂

semidirect

2. #13117060

C₂

semidirect

3. #13114451

C₂

semidirect

4. #13114428

C₂

semidirect

5. C₁₃×#1312

C₂

semidirect

58. #1312

D₁₃

semidirect

74. C₈×C₁₃

D₁₆

semidirect

139. C₁₃

#12525

semidirect

155. C₈

D₂₀₈

semidirect

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

subgroup	index
#13117086	2
#13117060	2
#13114451	2
#13114428	2
C₁₃×#1312	2
#12525	13

subgroup

index

C₁₃×#1312

Subgroups — #15777671

Conjugacy classes of subgroups

Complements

Maximal subgroups