Group #15777147 — subgroups

Subgroups — #15777147

Order 3328.

230 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 71 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	#13117089	C₂	1	1	yes	maximal, characteristic, complemented
2	#13114201	C₂	1	2	yes	maximal
3	#13112910	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112886	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2169	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	—	2	1	no	—
7	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	#11718668	—	2	2	no	—
9	C₂×Dic₁₀₄	—	2	1	no	—
10	C₂×Dic₁₀₄	—	2	1	no	—
11	#11718360	C₂²	1	2	yes	—
12	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
14	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	#11718074	—	2	2	no	—
16	#11718050	C₂²	1	2	yes	—
17	C₂²×C₈×C₁₃	—	2	1	no	—
18	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
19	C₂×Dic₅₂	—	2	2	no	—
20	C₂×Dic₅₂	—	2	2	no	—
21	#88123	—	2	2	no	—
22	C₁₃×M₃₂	—	4	2	no	—
23	C₂×C₁₃×C₁₆	—	2	1	no	—
24	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
25	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
26	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
27	C₁₃×C₂²⋊C₈	—	2	1	no	—
28	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	#88072	—	4	2	no	—
30	#88047	—	4	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	Dic₁₀₄	—	4	2	no	—
33	Dic₁₀₄	—	4	2	no	—
34	#12434	—	13	1	no	maximal, Sylow
35	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	—	2	1	no	—
37	C₂×Dic₂₆	—	4	2	no	—
38	C₂×C₈×C₁₃	—	2	1	no	—
39	C₂×C₈×C₁₃	—	4	1	no	—
40	C₂×C₈×C₁₃	—	4	1	no	—
41	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
43	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
44	C₁₃×C₂²⋊C₄	—	2	1	no	—
45	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₄⋊Dic₁₃	—	4	2	no	—
47	C₄⋊Dic₁₃	—	4	2	no	—
48	C₂×#1055	—	4	2	no	—
49	Dic₅₂	—	4	2	no	—
50	Dic₅₂	—	4	2	no	—
51	Dic₅₂	—	8	2	no	—
52	C₁₃×C₁₆	—	2	1	no	—
53	C₁₃×C₁₆	—	4	1	no	—
54	C₁₃×C₁₆	D₈	1	2	yes	—
55	#2247	—	13	1	no	—
56	#2169	D₁₃	1	1	yes	characteristic
57	#1669	—	13	2	no	—
58	#1372	—	13	1	no	—
59	#1362	—	13	2	no	—
60	C₂³×C₁₃	—	2	1	no	—
61	D₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	4	1	no	—
64	C₂×C₄×C₁₃	—	4	1	no	—
65	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
68	C₂×Dic₁₃	—	8	2	no	—
69	Dic₂₆	—	8	2	no	—
70	Dic₂₆	—	8	2	no	—
71	C₈×C₁₃	—	2	1	no	—
72	C₈×C₁₃	—	4	1	no	—
73	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
74	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	#1055	—	8	2	no	—
76	C₂×Q₃₂	—	26	1	no	—
77	C₂×Q₃₂	—	26	1	no	—
78	C₂×M₃₂	—	2	1	no	—
79	#494	—	13	2	no	—
80	#474	—	26	2	no	—
81	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
82	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
83	C₂²⋊C₁₆	—	2	1	no	—
84	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
85	#335	—	13	2	no	—
86	#332	—	26	2	no	—
87	C₂²×C₁₃	—	4	1	no	—
88	C₂²×C₁₃	—	4	1	no	—
89	C₂²×C₁₃	—	4	1	no	—
90	C₂²×C₁₃	#466	1	1	yes	characteristic
91	C₄×C₁₃	—	2	1	no	—
92	C₄×C₁₃	—	4	1	no	—
93	C₄×C₁₃	#459	1	1	yes	characteristic
94	C₄×C₁₃	#472	1	1	yes	characteristic
95	C₄×C₁₃	#510	1	2	yes	—
96	Dic₁₃	—	16	2	no	—
97	C₂×Q₁₆	—	26	2	no	—
98	C₂×Q₁₆	—	26	2	no	—
99	C₂²×C₈	—	2	1	no	—
100	C₄⋊Q₈	—	26	2	no	—
101	C₄×D₄	D₅₂	1	1	yes	characteristic
102	Q₃₂	—	52	2	no	—
103	Q₃₂	—	52	2	no	—
104	M₃₂	—	4	2	no	—
105	C₂×C₁₆	—	2	1	no	—
106	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
107	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
108	#107	—	52	2	no	—
109	C₄⋊C₈	—	26	2	no	—
110	C₄⋊C₈	D₅₂	1	1	yes	characteristic
111	#102	—	52	2	no	—
112	C₂²⋊C₈	—	2	1	no	—
113	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
114	C₂×C₁₃	—	8	1	no	—
115	C₂×C₁₃	#1667	1	1	yes	characteristic
116	C₂×C₁₃	#2219	1	1	yes	characteristic
117	C₂×C₁₃	#2222	1	1	yes	characteristic
118	C₂×Q₈	—	52	2	no	—
119	C₂×D₄	D₁₀₄	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	Q₁₆	—	52	2	no	—
122	Q₁₆	—	52	2	no	—
123	Q₁₆	—	104	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	52	2	no	—
128	C₂×C₈	#6127	1	1	yes	characteristic
129	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
130	C₄⋊C₄	—	52	2	no	—
131	C₄⋊C₄	—	52	2	no	—
132	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
133	C₂²⋊C₄	—	2	1	no	—
134	C₄²	C₂×D₅₂	1	1	yes	characteristic
135	C₁₆	—	2	1	no	—
136	C₁₆	—	4	1	no	—
137	C₁₆	#6104	1	2	yes	—
138	C₁₃	#12434	1	1	yes	characteristic, complemented, Sylow
139	C₂³	—	2	1	no	—
140	Q₈	—	104	2	no	—
141	Q₈	—	104	2	no	—
142	D₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	104	2	no	—
147	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
148	C₂×C₄	#88158	1	1	yes	characteristic
149	C₂×C₄	#88184	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	4	1	no	—
152	C₈	—	104	2	no	—
153	C₈	C₂×#6104	1	1	yes	characteristic
154	C₈	#88203	1	1	yes	characteristic
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	—	4	1	no	—
158	C₂²	#11718776	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	208	2	no	—
162	C₄	#11718570	1	2	yes	—
163	C₄	#11718676	1	1	yes	characteristic
164	C₄	#11718803	1	1	yes	characteristic
165	C₂	—	8	1	no	—
166	C₂	#13114191	1	1	yes	characteristic
167	C₂	#13116944	1	1	yes	characteristic
168	C₂	#13116961	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117089

C₂

yes

maximal, characteristic, complemented

#13114201

C₂

yes

maximal

#13112910

C₂

yes

maximal, characteristic, complemented

#13112886

C₂

yes

maximal, complemented

C₁₃×#2169

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

—

C₈×D₄×C₁₃

C₂²

yes

characteristic

#11718668

—

C₂×Dic₁₀₄

—

C₂×Dic₁₀₄

—

#11718360

C₂²

yes

—

C₁₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₁₃×C₂²⋊C₁₆

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

#11718074

—

#11718050

C₂²

yes

—

C₂²×C₈×C₁₃

—

C₄×D₄×C₁₃

D₄

yes

characteristic

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88123

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₁₃×C₄⋊C₈

D₄

yes

characteristic

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

Dic₁₀₄

—

Dic₁₀₄

—

#12434

—

maximal, Sylow

C₂×D₄×C₁₃

D₈

yes

characteristic

C₂²×C₄×C₁₃

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₁₃×C₄⋊C₄

D₈

yes

characteristic

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

D₈

yes

—

#2247

—

#2169

D₁₃

yes

—

—

—

C₂³×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×D₈

yes

C₈×C₁₃

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×M₃₂

—

#494

—

#474

—

C₈×D₄

C₂×D₁₃

yes

characteristic

C₄⋊C₁₆

C₂×D₁₃

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

#335

—

#332

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#466

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

100

C₄⋊Q₈

—

101

C₄×D₄

D₅₂

yes

characteristic

102

Q₃₂

—

103

Q₃₂

—

104

M₃₂

—

105

C₂×C₁₆

—

106

C₂×C₁₆

C₂²⋊D₁₃

yes

characteristic

107

C₂×C₁₆

C₂²⋊D₁₃

yes

characteristic

108

#107

—

109

C₄⋊C₈

—

110

C₄⋊C₈

D₅₂

yes

characteristic

111

#102

—

112

C₂²⋊C₈

—

113

C₄×C₈

C₂²×D₁₃

yes

characteristic

114

C₂×C₁₃

—

115

C₂×C₁₃

#1667

yes

characteristic

116

C₂×C₁₃

#2219

yes

characteristic

117

C₂×C₁₃

#2222

yes

characteristic

118

C₂×Q₈

—

119

C₂×D₄

D₁₀₄

yes

characteristic

120

C₂²×C₄

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

104

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

#6127

yes

characteristic

129

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

D₁₀₄

yes

characteristic

133

C₂²⋊C₄

—

134

C₄²

C₂×D₅₂

yes

characteristic

135

C₁₆

—

136

C₁₆

—

137

C₁₆

#6104

yes

—

138

C₁₃

#12434

yes

characteristic, complemented, Sylow

139

C₂³

—

140

Q₈

—

104

—

141

Q₈

—

104

—

142

D₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

104

—

147

C₂×C₄

C₂×D₁₀₄

yes

characteristic

148

C₂×C₄

#88158

yes

characteristic

149

C₂×C₄

#88184

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

104

—

153

C₈

C₂×#6104

yes

characteristic

154

C₈

#88203

yes

characteristic

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#11718776

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

208

—

162

C₄

#11718570

yes

—

163

C₄

#11718676

yes

characteristic

164

C₄

#11718803

yes

characteristic

165

C₂

—

166

C₂

#13114191

yes

characteristic

167

C₂

#13116944

yes

characteristic

168

C₂

#13116961

yes

characteristic

	subgroup	G/H
center	C₂²	#11718776
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2169	C₂
socle	C₂²×C₁₃	#466
prime radical	C₂×D₄×C₁₃	D₈
hypercenter	#2169	D₁₃
hypocenter	C₁₃	#12434
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

subgroup

G/H

center

C₂²

#11718776

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#2169

C₂

socle

C₂²×C₁₃

#466

prime radical

C₂×D₄×C₁₃

D₈

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

N	H ≅ G/N	type
1. #13117089	C₂	semidirect
3. #13112910	C₂	semidirect
4. #13112886	C₂	semidirect
138. C₁₃	#12434	semidirect

H ≅ G/N

type

1. #13117089

C₂

semidirect

3. #13112910

C₂

semidirect

4. #13112886

C₂

semidirect

138. C₁₃

#12434

semidirect

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

subgroup	index
#13117089	2
#13114201	2
#13112910	2
#13112886	2
C₁₃×#2169	2
#12434	13

subgroup

index

C₁₃×#2169

Subgroups — #15777147

Conjugacy classes of subgroups

Complements

Maximal subgroups