Group #15753371 — subgroups

Subgroups — #15753371

Order 3328.

330 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 131 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	#13114524	C₂	1	1	yes	maximal, characteristic
2	#13113129	C₂	1	2	yes	maximal
3	#13112667	C₂	1	1	yes	maximal, characteristic
4	#13112664	C₂	1	2	yes	maximal
5	C₁₃×#1275	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#431	C₂²	1	2	yes	—
7	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
8	C₂×#88055	C₄	1	2	yes	—
9	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
10	#11718593	C₂²	1	1	yes	characteristic
11	#11718586	C₂²	1	2	yes	—
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	#11718047	C₄	1	4	yes	complemented
14	#11718046	C₄	1	4	yes	complemented
15	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
18	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
19	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
20	C₁₃×C₄⋊C₈	—	2	2	no	—
21	C₁₃×C₂²⋊C₈	—	2	2	no	—
22	C₁₃×#97	—	2	2	no	—
23	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
24	#88071	—	2	1	no	—
25	#88071	—	2	1	no	—
26	C₂²⋊#1055	—	2	2	no	—
27	#88055	—	2	4	no	—
28	C₈×Dic₁₃	—	2	4	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄×#1055	C₂×C₄	1	4	yes	—
31	C₄×#1055	C₂×C₄	1	2	yes	—
32	C₄×#1055	C₂×C₄	1	2	yes	—
33	#8140	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
37	C₂²×Dic₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	—	2	2	no	—
39	C₂×C₈×C₁₃	—	2	2	no	—
40	C₂×C₈×C₁₃	—	2	4	no	—
41	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
42	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
43	C₄²×C₁₃	#41	1	2	yes	—
44	C₄×Dic₁₃	—	2	2	no	—
45	C₄×Dic₁₃	M₁₆	1	4	yes	—
46	C₂×#1055	—	2	2	no	—
47	C₂×#1055	—	2	2	no	—
48	C₂×#1055	—	2	4	no	—
49	C₂×#1055	C₄²	1	1	yes	characteristic
50	C₂×#1055	C₄²	1	1	yes	characteristic
51	C₂×#1055	C₄²	1	2	yes	—
52	#1726	—	13	1	no	—
53	#1452	—	13	2	no	—
54	#1275	—	13	1	no	—
55	#1275	D₁₃	1	1	yes	characteristic
56	#1274	—	13	2	no	—
57	C₂³×C₁₃	#117	1	1	yes	characteristic
58	C₂×C₄×C₁₃	—	4	1	no	—
59	C₂×C₄×C₁₃	—	4	1	no	—
60	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#117	1	2	yes	—
62	C₂×C₄×C₁₃	#117	1	2	yes	—
63	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
66	C₂×Dic₁₃	—	2	4	no	—
67	C₂×Dic₁₃	—	4	2	no	—
68	C₂×Dic₁₃	—	4	2	no	—
69	C₈×C₁₃	—	2	4	no	—
70	C₈×C₁₃	—	4	4	no	—
71	#1055	—	2	1	no	—
72	#1055	—	2	1	no	—
73	#1055	—	2	2	no	—
74	#1055	—	4	4	no	—
75	#431	—	13	2	no	—
76	#431	C₂×D₁₃	1	2	yes	—
77	C₂×#97	—	13	2	no	—
78	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
79	C₂×C₄×C₈	—	13	1	no	—
80	C₂×C₄×C₈	—	13	2	no	—
81	#376	—	13	1	no	—
82	#322	—	13	4	no	—
83	#322	—	13	4	no	—
84	C₂²×C₁₃	—	2	1	no	—
85	C₂²×C₁₃	—	2	1	no	—
86	C₂²×C₁₃	#376	1	1	yes	characteristic
87	C₂²×C₁₃	#432	1	2	yes	—
88	C₄×C₁₃	—	2	2	no	—
89	C₄×C₁₃	—	2	2	no	—
90	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
91	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
92	C₄×C₁₃	#432	1	2	yes	—
93	Dic₁₃	—	4	4	no	—
94	C₂²×C₈	—	13	1	no	—
95	C₂²×C₈	—	13	1	no	—
96	C₂²×C₈	C₄×D₁₃	1	2	yes	—
97	C₂×C₄²	—	13	2	no	—
98	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
99	C₄⋊C₈	—	2	2	no	—
100	C₄⋊C₈	—	26	2	no	—
101	C₂²⋊C₈	—	2	2	no	—
102	C₂²⋊C₈	—	26	2	no	—
103	#97	—	2	2	no	—
104	#97	—	26	4	no	—
105	C₄×C₈	—	13	2	no	—
106	C₄×C₈	—	13	2	no	—
107	C₄×C₈	—	13	4	no	—
108	C₄×C₈	—	26	4	no	—
109	C₄×C₈	C₄×D₁₃	1	4	yes	—
110	#95	—	26	1	no	—
111	#95	—	26	1	no	—
112	C₂×C₁₃	—	2	2	no	—
113	C₂×C₁₃	#1745	1	1	yes	characteristic
114	C₂×C₁₃	#1747	1	1	yes	characteristic
115	C₂×C₁₃	#1747	1	1	yes	characteristic
116	C₂²×C₄	—	26	2	no	—
117	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
118	C₂²×C₄	#6129	1	1	yes	characteristic
119	C₂²×C₄	#6131	1	1	yes	characteristic
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	4	no	—
123	C₂×C₈	—	13	1	no	—
124	C₂×C₈	—	13	1	no	—
125	C₂×C₈	—	13	2	no	—
126	C₂×C₈	—	26	2	no	—
127	C₂×C₈	—	26	2	no	—
128	C₂×C₈	—	26	4	no	—
129	C₂×C₈	#6094	1	4	yes	—
130	C₄²	—	13	4	no	—
131	C₄²	—	26	2	no	—
132	C₄²	C₂×C₄×D₁₃	1	2	yes	—
133	C₄²	#6129	1	2	yes	—
134	C₁₃	#8140	1	1	yes	characteristic, complemented, Sylow
135	C₂³	#88146	1	1	yes	characteristic
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	26	4	no	—
139	C₂×C₄	—	52	2	no	—
140	C₂×C₄	—	52	2	no	—
141	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
142	C₂×C₄	#88131	1	2	yes	—
143	C₂×C₄	#88131	1	1	yes	characteristic
144	C₂×C₄	#88146	1	2	yes	—
145	C₂×C₄	C₂×#6094	1	2	yes	—
146	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	4	4	no	—
149	C₈	—	26	1	no	—
150	C₈	—	26	1	no	—
151	C₈	—	26	2	no	—
152	C₈	—	52	4	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#11718309	1	1	yes	characteristic
156	C₂²	#11718496	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	52	4	no	—
160	C₄	C₄×#6094	1	1	yes	characteristic
161	C₄	#11718368	1	1	yes	characteristic
162	C₄	#11718380	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	#13114583	1	1	yes	characteristic
165	C₂	#13114590	1	1	yes	characteristic
166	C₂	#13114592	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114524

C₂

yes

maximal, characteristic

#13113129

C₂

yes

maximal

#13112667

C₂

yes

maximal, characteristic

#13112664

C₂

yes

maximal

C₁₃×#1275

C₂

yes

maximal, characteristic

C₁₃×#431

C₂²

yes

—

C₂×C₁₃×#97

C₂²

yes

characteristic

C₂×#88055

C₄

yes

—

C₂×C₈×Dic₁₃

C₄

yes

—

#11718593

C₂²

yes

characteristic

#11718586

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

#11718047

C₄

yes

complemented

#11718046

C₄

yes

complemented

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄×Dic₁₃

C₂×C₄

yes

—

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂×C₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₁₃×#97

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

#88071

—

#88071

—

C₂²⋊#1055

—

#88055

—

C₈×Dic₁₃

—

C₄⋊#1055

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

—

#8140

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂²×C₄

yes

C₂²×C₄×C₁₃

yes

C₂²×C₄×C₁₃

yes

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

M₁₆

yes

—

C₄²×C₁₃

C₂²×C₄

yes

—

C₄²×C₁₃

#41

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

M₁₆

yes

—

—

—

—

C₄²

yes

characteristic

C₂×#1055

C₄²

yes

characteristic

C₂×#1055

C₄²

yes

—

—

—

—

D₁₃

yes

characteristic

#1274

—

C₂³×C₁₃

#117

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#117

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

—

C₂×D₁₃

yes

—

C₂×#97

—

C₂×#97

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

#322

—

#322

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#376

yes

characteristic

C₂²×C₁₃

#432

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×M₁₆

yes

characteristic

C₄×C₁₃

C₄×M₁₆

yes

characteristic

C₄×C₁₃

#432

yes

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

C₂²⋊C₈

—

103

#97

—

104

#97

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₄×D₁₃

yes

—

110

#95

—

111

#95

—

112

C₂×C₁₃

—

113

C₂×C₁₃

#1745

yes

characteristic

114

C₂×C₁₃

#1747

yes

characteristic

115

C₂×C₁₃

#1747

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

118

C₂²×C₄

#6129

yes

characteristic

119

C₂²×C₄

#6131

yes

characteristic

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

#6094

yes

—

130

C₄²

—

131

C₄²

—

132

C₄²

C₂×C₄×D₁₃

yes

—

133

C₄²

#6129

yes

—

134

C₁₃

#8140

yes

characteristic, complemented, Sylow

135

C₂³

#88146

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₄²×D₁₃

yes

characteristic

142

C₂×C₄

#88131

yes

—

143

C₂×C₄

#88131

yes

characteristic

144

C₂×C₄

#88146

yes

—

145

C₂×C₄

C₂×#6094

yes

—

146

C₂×C₄

M₁₆×D₁₃

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₂²

—

154

C₂²

—

155

C₂²

#11718309

yes

characteristic

156

C₂²

#11718496

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

C₄×#6094

yes

characteristic

161

C₄

#11718368

yes

characteristic

162

C₄

#11718380

yes

—

163

C₂

—

164

C₂

#13114583

yes

characteristic

165

C₂

#13114590

yes

characteristic

166

C₂

#13114592

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1275	C₂
socle	C₂²×C₁₃	#376
prime radical	C₂³×C₁₃	#117
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1275	D₁₃
hypocenter	C₁₃	#8140
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #1275

subgroup

G/H

center

C₂×C₄

C₄²×D₁₃

commutator

C₂×C₄×C₁₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1275

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #1275

N	H ≅ G/N	type
13. #11718047	C₄	semidirect
14. #11718046	C₄	semidirect
134. C₁₃	#8140	semidirect

H ≅ G/N

type

13. #11718047

C₄

semidirect

14. #11718046

C₄

semidirect

134. C₁₃

#8140

semidirect

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

subgroup	index
#13114524	2
#13113129	2
#13112667	2
#13112664	2
C₁₃×#1275	2
#8140	13

subgroup

index

C₁₃×#1275

Subgroups — #15753371

Conjugacy classes of subgroups

Complements

Maximal subgroups