Group #15777672 — subgroups

Subgroups — #15777672

Order 3328.

340 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13117085	C₂	1	2	yes	maximal, complemented
2	#13117059	C₂	1	2	yes	maximal, complemented
3	#13114428	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112754	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1712	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#493	—	2	1	no	—
7	C₁₃×#493	—	2	1	no	—
8	C₁₃×#493	C₂²	1	2	yes	—
9	C₂×#88066	—	2	4	no	—
10	#11718356	C₂²	1	1	yes	characteristic
11	#11718355	—	2	1	no	—
12	#11718354	—	2	1	no	—
13	#11718343	C₂²	1	1	yes	characteristic
14	C₈²×C₁₃	C₂²	1	1	yes	characteristic
15	#11718154	—	2	4	no	—
16	C₄×#6090	C₂²	1	2	yes	—
17	C₂×C₁₃×D₈	—	2	2	no	—
18	C₂×C₁₃×D₈	—	2	2	no	—
19	C₂×C₁₃×D₈	—	4	2	no	—
20	C₂×C₁₃×D₈	—	4	2	no	—
21	C₁₃×#127	—	2	2	no	—
22	C₂×D₁₀₄	—	2	2	no	—
23	C₂×#6095	—	4	2	no	—
24	#88128	—	2	1	no	—
25	#88123	—	2	1	no	—
26	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	#88066	—	4	8	no	—
30	#88061	—	4	2	no	—
31	#88058	—	2	2	no	—
32	C₂×#6090	D₄	1	4	yes	—
33	#12525	—	13	1	no	maximal, Sylow
34	C₂×D₄×C₁₃	—	4	2	no	—
35	C₂×D₄×C₁₃	—	4	2	no	—
36	C₂×D₄×C₁₃	—	4	2	no	—
37	C₂×D₅₂	—	4	1	no	—
38	C₂×D₅₂	—	4	1	no	—
39	C₂×D₅₂	—	4	1	no	—
40	C₂×Dic₂₆	—	4	1	no	—
41	C₁₃×D₈	—	4	4	no	—
42	C₁₃×D₈	—	4	4	no	—
43	C₁₃×D₈	—	8	4	no	—
44	C₁₃×D₈	—	8	4	no	—
45	C₂×C₈×C₁₃	D₈	1	2	yes	—
46	C₂×C₈×C₁₃	D₈	1	2	yes	—
47	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
48	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₄⋊Dic₁₃	—	4	1	no	—
50	C₄⋊Dic₁₃	—	4	1	no	—
51	D₁₀₄	—	4	4	no	—
52	#6095	—	8	4	no	—
53	#6090	—	2	4	no	—
54	#2246	—	13	2	no	—
55	#2240	—	13	2	no	—
56	#1712	D₁₃	1	1	yes	characteristic, complemented
57	#1706	—	13	1	no	—
58	#1312	—	13	1	no	—
59	C₂³×C₁₃	—	8	2	no	—
60	C₂²×D₁₃	—	8	1	no	—
61	D₄×C₁₃	—	8	2	no	—
62	D₄×C₁₃	—	8	2	no	—
63	D₄×C₁₃	—	8	4	no	—
64	D₄×C₁₃	—	8	4	no	—
65	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
68	C₂×Dic₁₃	—	8	1	no	—
69	D₅₂	—	8	2	no	—
70	D₅₂	—	8	2	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	—	2	1	no	—
79	#493	—	2	1	no	—
80	#493	—	13	1	no	—
81	#493	C₂×D₁₃	1	2	yes	—
82	#492	—	26	1	no	—
83	#486	—	26	1	no	—
84	#357	—	26	4	no	—
85	C₄×C₁₆	—	13	2	no	—
86	C₈²	C₂×D₁₃	1	1	yes	characteristic
87	C₂²×C₁₃	—	16	2	no	—
88	C₂²×C₁₃	—	16	2	no	—
89	C₂²×C₁₃	—	16	2	no	—
90	C₂²×C₁₃	#493	1	1	yes	characteristic
91	C₂×D₁₃	—	16	1	no	—
92	C₂×D₁₃	—	16	1	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	—	2	2	no	—
100	C₂×D₈	—	2	2	no	—
101	C₂×D₈	—	4	2	no	—
102	C₂×D₈	—	4	2	no	—
103	C₂×D₈	—	26	2	no	—
104	C₄⋊Q₈	—	26	1	no	—
105	#127	—	2	2	no	—
106	#127	—	26	1	no	—
107	D₁₆	—	52	8	no	—
108	C₂×C₁₆	—	13	4	no	—
109	#107	—	26	2	no	—
110	#102	—	52	2	no	—
111	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
114	C₂×C₁₃	—	32	2	no	—
115	C₂×C₁₃	#1721	1	1	yes	characteristic
116	C₂×C₁₃	#2246	1	1	yes	characteristic
117	C₂×C₁₃	#2250	1	1	yes	characteristic
118	D₁₃	—	32	1	no	—
119	C₂×Q₈	—	52	1	no	—
120	C₂×D₄	—	4	2	no	—
121	C₂×D₄	—	4	2	no	—
122	C₂×D₄	—	4	2	no	—
123	C₂×D₄	—	52	1	no	—
124	C₂×D₄	—	52	1	no	—
125	C₂×D₄	—	52	1	no	—
126	SD₁₆	—	104	4	no	—
127	D₈	—	4	4	no	—
128	D₈	—	4	4	no	—
129	D₈	—	8	4	no	—
130	D₈	—	8	4	no	—
131	D₈	—	52	4	no	—
132	C₂×C₈	#6104	1	2	yes	—
133	C₂×C₈	#6104	1	2	yes	—
134	C₂×C₈	D₄×D₁₃	1	2	yes	—
135	C₄⋊C₄	—	52	1	no	—
136	C₄⋊C₄	—	52	1	no	—
137	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
138	C₁₆	—	26	4	no	—
139	C₁₃	#12525	1	1	yes	characteristic, complemented, Sylow
140	C₂³	—	8	2	no	—
141	C₂³	—	104	1	no	—
142	Q₈	—	104	2	no	—
143	D₄	—	8	4	no	—
144	D₄	—	8	4	no	—
145	D₄	—	8	2	no	—
146	D₄	—	8	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	C₂×#6104	1	1	yes	characteristic
152	C₂×C₄	C₂×#6104	1	1	yes	characteristic
153	C₂×C₄	#88194	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	#88066	1	4	yes	complemented
156	C₈	#88165	1	4	yes	—
157	C₂²	—	16	2	no	—
158	C₂²	—	16	2	no	—
159	C₂²	—	16	2	no	—
160	C₂²	—	208	1	no	—
161	C₂²	—	208	1	no	—
162	C₂²	—	208	1	no	—
163	C₂²	#11718734	1	1	yes	characteristic
164	C₄	—	208	1	no	—
165	C₄	C₂×#88066	1	2	yes	—
166	C₄	#11718808	1	2	yes	—
167	C₄	#11718815	1	2	yes	—
168	C₂	—	32	2	no	—
169	C₂	—	416	1	no	—
170	C₂	#13114503	1	1	yes	characteristic
171	C₂	#13117085	1	1	yes	characteristic
172	C₂	#13117100	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117085

C₂

yes

maximal, complemented

#13117059

C₂

yes

maximal, complemented

#13114428

C₂

yes

maximal, characteristic, complemented

#13112754

C₂

yes

maximal, characteristic, complemented

C₁₃×#1712

C₂

yes

maximal, characteristic, complemented

C₁₃×#493

—

C₁₃×#493

—

C₁₃×#493

C₂²

yes

—

C₂×#88066

—

#11718356

C₂²

yes

—

—

C₂²

yes

characteristic

C₈²×C₁₃

C₂²

yes

characteristic

#11718154

—

C₄×#6090

C₂²

yes

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₁₃×#127

—

C₂×D₁₀₄

—

C₂×#6095

—

#88128

—

#88123

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

D₄

yes

—

#12525

—

maximal, Sylow

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₁₃×D₈

—

C₁₃×D₈

—

C₁₃×D₈

—

C₁₃×D₈

—

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

—

—

—

—

—

D₁₃

yes

characteristic, complemented

#1706

—

#1312

—

C₂³×C₁₃

—

C₂²×D₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

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

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

D₁₆

yes

complemented

C₈×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×D₁₆

—

—

—

—

—

C₂×D₁₃

yes

—

#492

—

#486

—

#357

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#493

yes

characteristic

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

—

109

#107

—

110

#102

—

111

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

112

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

113

C₄×C₈

C₂²×D₁₃

yes

characteristic

114

C₂×C₁₃

—

115

C₂×C₁₃

#1721

yes

characteristic

116

C₂×C₁₃

#2246

yes

characteristic

117

C₂×C₁₃

#2250

yes

characteristic

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

—

131

D₈

—

132

C₂×C₈

#6104

yes

—

133

C₂×C₈

#6104

yes

—

134

C₂×C₈

D₄×D₁₃

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

138

C₁₆

—

139

C₁₃

#12525

yes

characteristic, complemented, Sylow

140

C₂³

—

141

C₂³

—

104

—

142

Q₈

—

104

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

147

D₄

—

104

—

148

D₄

—

104

—

149

D₄

—

104

—

150

C₂×C₄

—

104

—

151

C₂×C₄

C₂×#6104

yes

characteristic

152

C₂×C₄

C₂×#6104

yes

characteristic

153

C₂×C₄

#88194

yes

characteristic

154

C₈

—

155

C₈

#88066

yes

complemented

156

C₈

#88165

yes

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

208

—

161

C₂²

—

208

—

162

C₂²

—

208

—

163

C₂²

#11718734

yes

characteristic

164

C₄

—

208

—

165

C₄

C₂×#88066

yes

—

166

C₄

#11718808

yes

—

167

C₄

#11718815

yes

—

168

C₂

—

169

C₂

—

416

—

170

C₂

#13114503

yes

characteristic

171

C₂

#13117085

yes

characteristic

172

C₂

#13117100

yes

characteristic

	subgroup	G/H
center	C₂²	#11718734
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1712	C₂
socle	C₂²×C₁₃	#493
hypercenter	#1712	D₁₃
hypocenter	C₁₃	#12525
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#11718734

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1712

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #13117085	C₂	semidirect
2. #13117059	C₂	semidirect
3. #13114428	C₂	semidirect
4. #13112754	C₂	semidirect
5. C₁₃×#1712	C₂	semidirect
56. #1712	D₁₃	semidirect
74. C₈×C₁₃	D₁₆	semidirect
139. C₁₃	#12525	semidirect
155. C₈	#88066	semidirect

H ≅ G/N

type

1. #13117085

C₂

semidirect

2. #13117059

C₂

semidirect

3. #13114428

C₂

semidirect

4. #13112754

C₂

semidirect

5. C₁₃×#1712

C₂

semidirect

56. #1712

D₁₃

semidirect

74. C₈×C₁₃

D₁₆

semidirect

139. C₁₃

#12525

semidirect

155. C₈

#88066

semidirect

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

subgroup	index
#13117085	2
#13117059	2
#13114428	2
#13112754	2
C₁₃×#1712	2
#12525	13

subgroup

index

C₁₃×#1712

Subgroups — #15777672

Conjugacy classes of subgroups

Complements

Maximal subgroups