Group #15777148 — subgroups

Subgroups — #15777148

Order 3328.

230 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 67 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	#13116669	C₂	1	1	yes	maximal, characteristic
3	#13114202	C₂	1	2	yes	maximal
4	#13112884	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1372	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₁₀₄	—	2	1	no	—
7	C₂×Dic₁₀₄	—	2	1	no	—
8	C₂×#88038	—	2	1	no	—
9	#11718507	—	2	2	no	—
10	#11718360	C₂²	1	2	yes	—
11	C₈×D₅₂	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₁₃×#335	C₂²	1	2	yes	—
14	#11718099	—	2	1	no	—
15	#11718078	—	2	2	no	—
16	C₄⋊#6090	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₂	—	2	2	no	—
18	C₂×Dic₅₂	—	2	2	no	—
19	C₂×C₈×D₁₃	—	2	1	no	—
20	C₄×D₅₂	D₄	1	1	yes	characteristic
21	#88123	—	2	2	no	—
22	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
24	C₁₃×C₄⋊C₈	—	2	2	no	—
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	#88061	—	4	2	no	—
27	#88060	—	2	1	no	—
28	#88058	—	4	2	no	—
29	C₂×#6090	—	2	1	no	—
30	C₄⋊#1055	D₄	1	1	yes	characteristic
31	Dic₁₀₄	—	4	2	no	—
32	Dic₁₀₄	—	4	2	no	—
33	#88038	—	4	2	no	—
34	#12434	—	13	1	no	maximal, Sylow
35	C₂×D₅₂	D₈	1	1	yes	characteristic
36	C₂×C₄×D₁₃	—	2	1	no	—
37	C₂×Dic₂₆	—	4	2	no	—
38	C₂×C₈×C₁₃	—	4	2	no	—
39	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
41	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
42	#6103	—	2	1	no	—
43	C₄⋊Dic₁₃	—	4	2	no	—
44	C₄⋊Dic₁₃	—	4	2	no	—
45	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
46	C₂×#1055	—	2	1	no	—
47	Dic₅₂	—	4	2	no	—
48	Dic₅₂	—	4	2	no	—
49	Dic₅₂	—	8	2	no	—
50	C₈×D₁₃	—	4	1	no	—
51	C₈×D₁₃	—	4	1	no	—
52	C₁₃×C₁₆	—	2	1	no	—
53	C₁₃×C₁₆	D₈	1	2	yes	—
54	#6090	—	4	1	no	—
55	#2247	—	13	1	no	—
56	#2169	—	13	1	no	—
57	#1669	—	13	2	no	—
58	#1372	D₁₃	1	1	yes	characteristic, complemented
59	#1362	—	13	2	no	—
60	C₂²×D₁₃	—	2	1	no	—
61	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
64	C₂×Dic₁₃	—	2	1	no	—
65	C₂×Dic₁₃	—	8	2	no	—
66	D₅₂	—	2	2	no	—
67	C₄×D₁₃	—	4	1	no	—
68	C₄×D₁₃	—	4	1	no	—
69	Dic₂₆	—	8	2	no	—
70	Dic₂₆	—	8	2	no	—
71	C₈×C₁₃	—	2	1	no	—
72	C₈×C₁₃	—	8	2	no	—
73	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
74	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	#1055	—	4	1	no	—
76	C₂×Q₃₂	—	26	1	no	—
77	C₂×Q₃₂	—	26	1	no	—
78	C₂×M₃₂	—	26	1	no	—
79	#494	—	13	2	no	—
80	#474	—	26	2	no	—
81	C₈×D₄	—	13	1	no	—
82	C₄⋊C₁₆	—	13	1	no	—
83	C₂²⋊C₁₆	—	26	1	no	—
84	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
85	#335	C₂×D₁₃	1	2	yes	—
86	#332	—	26	2	no	—
87	C₂²×C₁₃	#466	1	1	yes	characteristic
88	C₂×D₁₃	—	4	1	no	—
89	C₂×D₁₃	—	4	1	no	—
90	C₂×D₁₃	—	4	1	no	—
91	C₄×C₁₃	—	2	1	no	—
92	C₄×C₁₃	#459	1	1	yes	characteristic
93	C₄×C₁₃	#472	1	1	yes	characteristic
94	C₄×C₁₃	#510	1	2	yes	—
95	Dic₁₃	—	4	1	no	—
96	Dic₁₃	—	16	2	no	—
97	C₂×Q₁₆	—	26	2	no	—
98	C₂×Q₁₆	—	26	2	no	—
99	C₂²×C₈	—	26	1	no	—
100	C₄⋊Q₈	—	26	2	no	—
101	C₄×D₄	—	13	1	no	—
102	Q₃₂	—	52	2	no	—
103	Q₃₂	—	52	2	no	—
104	M₃₂	—	52	2	no	—
105	C₂×C₁₆	—	26	1	no	—
106	C₂×C₁₆	D₅₂	1	1	yes	characteristic
107	C₂×C₁₆	D₅₂	1	1	yes	characteristic
108	#107	—	52	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	13	1	no	—
111	#102	—	52	2	no	—
112	C₂²⋊C₈	—	26	1	no	—
113	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
114	C₂×C₁₃	#1667	1	1	yes	characteristic
115	C₂×C₁₃	#2219	1	1	yes	characteristic
116	C₂×C₁₃	#2222	1	1	yes	characteristic
117	D₁₃	—	8	1	no	—
118	C₂×Q₈	—	52	2	no	—
119	C₂×D₄	—	13	1	no	—
120	C₂²×C₄	—	26	1	no	—
121	Q₁₆	—	52	2	no	—
122	Q₁₆	—	52	2	no	—
123	Q₁₆	—	104	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	26	1	no	—
126	C₂×C₈	—	52	1	no	—
127	C₂×C₈	—	52	1	no	—
128	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
129	C₂×C₈	#6131	1	1	yes	characteristic
130	C₄⋊C₄	—	13	1	no	—
131	C₄⋊C₄	—	52	2	no	—
132	C₄⋊C₄	—	52	2	no	—
133	C₂²⋊C₄	—	26	1	no	—
134	C₄²	D₄×D₁₃	1	1	yes	characteristic
135	C₁₆	—	2	1	no	—
136	C₁₆	—	52	1	no	—
137	C₁₆	D₁₀₄	1	2	yes	—
138	C₁₃	#12434	1	1	yes	characteristic, complemented, Sylow
139	C₂³	—	26	1	no	—
140	Q₈	—	104	2	no	—
141	Q₈	—	104	2	no	—
142	D₄	—	26	2	no	—
143	C₂×C₄	—	26	1	no	—
144	C₂×C₄	—	52	1	no	—
145	C₂×C₄	—	52	1	no	—
146	C₂×C₄	—	104	2	no	—
147	C₂×C₄	#88149	1	1	yes	characteristic
148	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
149	C₂×C₄	#88173	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	52	1	no	—
153	C₈	C₂×D₁₀₄	1	1	yes	characteristic
154	C₈	#88162	1	1	yes	characteristic
155	C₂²	—	52	1	no	—
156	C₂²	—	52	1	no	—
157	C₂²	—	52	1	no	—
158	C₂²	#11718544	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	52	1	no	—
161	C₄	—	208	2	no	—
162	C₄	#11718504	1	1	yes	characteristic
163	C₄	#11718558	1	1	yes	characteristic
164	C₄	#11718576	1	2	yes	—
165	C₂	—	104	1	no	—
166	C₂	#13114192	1	1	yes	characteristic
167	C₂	#13116945	1	1	yes	characteristic
168	C₂	#13116962	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117089

C₂

yes

maximal, characteristic, complemented

#13116669

C₂

yes

maximal, characteristic

#13114202

C₂

yes

maximal

#13112884

C₂

yes

maximal, complemented

C₁₃×#1372

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₀₄

—

C₂×Dic₁₀₄

—

C₂×#88038

—

#11718507

—

#11718360

C₂²

yes

—

C₈×D₅₂

C₂²

yes

characteristic

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#335

C₂²

yes

—

#11718099

—

#11718078

—

C₄⋊#6090

C₂²

yes

characteristic

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×C₈×D₁₃

—

C₄×D₅₂

D₄

yes

characteristic

#88123

—

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

D₄

yes

characteristic

Dic₁₀₄

—

Dic₁₀₄

—

#88038

—

#12434

—

maximal, Sylow

C₂×D₅₂

D₈

yes

characteristic

C₂×C₄×D₁₃

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

#6103

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₈

yes

characteristic

C₂×#1055

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

C₈×D₁₃

—

C₈×D₁₃

—

C₁₃×C₁₆

—

C₁₃×C₁₆

D₈

yes

—

—

—

—

—

D₁₃

yes

characteristic, complemented

#1362

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×D₈

yes

C₈×C₁₃

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×M₃₂

—

#494

—

#474

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

#335

C₂×D₁₃

yes

—

#332

—

C₂²×C₁₃

#466

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

100

C₄⋊Q₈

—

101

C₄×D₄

—

102

Q₃₂

—

103

Q₃₂

—

104

M₃₂

—

105

C₂×C₁₆

—

106

C₂×C₁₆

D₅₂

yes

characteristic

107

C₂×C₁₆

D₅₂

yes

characteristic

108

#107

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#102

—

112

C₂²⋊C₈

—

113

C₄×C₈

C₂²×D₁₃

yes

characteristic

114

C₂×C₁₃

#1667

yes

characteristic

115

C₂×C₁₃

#2219

yes

characteristic

116

C₂×C₁₃

#2222

yes

characteristic

117

D₁₃

—

118

C₂×Q₈

—

119

C₂×D₄

—

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

C₂×D₅₂

yes

characteristic

129

C₂×C₈

#6131

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

D₄×D₁₃

yes

characteristic

135

C₁₆

—

136

C₁₆

—

137

C₁₆

D₁₀₄

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

#88149

yes

characteristic

148

C₂×C₄

D₈×D₁₃

yes

characteristic

149

C₂×C₄

#88173

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

C₂×D₁₀₄

yes

characteristic

154

C₈

#88162

yes

characteristic

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#11718544

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

208

—

162

C₄

#11718504

yes

characteristic

163

C₄

#11718558

yes

characteristic

164

C₄

#11718576

yes

—

165

C₂

—

104

—

166

C₂

#13114192

yes

characteristic

167

C₂

#13116945

yes

characteristic

168

C₂

#13116962

yes

characteristic

	subgroup	G/H
center	C₂²	#11718544
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1372	C₂
socle	C₂²×C₁₃	#466
prime radical	C₂×D₅₂	D₈
hypercenter	#1372	D₁₃
hypocenter	C₁₃	#12434
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂²

#11718544

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1372

C₂

socle

C₂²×C₁₃

#466

prime radical

C₂×D₅₂

D₈

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
1. #13117089	C₂	semidirect
4. #13112884	C₂	semidirect
5. C₁₃×#1372	C₂	semidirect
58. #1372	D₁₃	semidirect
138. C₁₃	#12434	semidirect

H ≅ G/N

type

1. #13117089

C₂

semidirect

4. #13112884

C₂

semidirect

5. C₁₃×#1372

C₂

semidirect

58. #1372

D₁₃

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
#13116669	2
#13114202	2
#13112884	2
C₁₃×#1372	2
#12434	13

subgroup

index

C₁₃×#1372

Subgroups — #15777148

Conjugacy classes of subgroups

Complements

Maximal subgroups