Group #15774116 — subgroups

Subgroups — #15774116

Order 3328.

274 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 79 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	#13116648	C₂	1	2	yes	maximal
2	#13112871	C₂	1	4	yes	maximal, complemented
3	C₁₃×#3139	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#533	C₄	1	1	yes	characteristic
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₁₃×#491	C₄	1	2	yes	—
7	C₁₃×#491	C₂²	1	2	yes	—
8	C₁₃×#487	—	2	2	no	—
9	C₁₃×#419	—	2	2	no	—
10	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	#11718145	—	2	2	no	—
12	#11718058	C₂²	1	4	yes	—
13	#11718055	—	2	2	no	—
14	C₂²×Q₈×C₁₃	—	2	1	no	—
15	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
16	C₁₃×C₄⋊Q₈	—	2	1	no	—
17	C₁₃×C₄⋊Q₈	—	2	1	no	—
18	C₁₃×C₄⋊Q₈	—	4	1	no	—
19	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
20	C₁₃×#125	C₂×C₄	1	2	yes	—
21	C₁₃×C₂²⋊Q₈	—	4	1	no	—
22	C₄×Q₈×C₁₃	—	4	1	no	—
23	C₁₃×#117	—	2	1	no	—
24	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₁₃×#103	—	2	4	no	—
27	C₁₃×#103	—	2	4	no	—
28	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
29	C₁₃×#97	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₁₃	D₄	1	2	yes	—
31	#88052	—	2	4	no	—
32	#11916	—	13	1	no	maximal, Sylow
33	C₂×Q₈×C₁₃	—	2	2	no	—
34	C₂×Q₈×C₁₃	—	4	1	no	—
35	C₂×Q₈×C₁₃	—	4	4	no	—
36	C₂×Q₈×C₁₃	—	4	1	no	—
37	C₂²×C₄×C₁₃	—	2	1	no	—
38	C₂²×C₄×C₁₃	—	4	1	no	—
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₁₃×M₁₆	—	4	2	no	—
41	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
43	C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₁₃×C₄⋊C₄	—	4	2	no	—
45	C₁₃×C₄⋊C₄	—	4	2	no	—
46	C₁₃×C₄⋊C₄	—	4	2	no	—
47	C₁₃×C₄⋊C₄	—	4	1	no	—
48	C₁₃×C₂²⋊C₄	—	4	1	no	—
49	C₄²×C₁₃	—	2	1	no	—
50	C₄²×C₁₃	—	4	1	no	—
51	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	#6090	—	4	4	no	—
54	#3139	D₁₃	1	1	yes	characteristic
55	#2165	—	13	2	no	—
56	#1356	—	13	4	no	—
57	C₂³×C₁₃	#100	1	1	yes	characteristic
58	Q₈×C₁₃	—	4	4	no	—
59	Q₈×C₁₃	—	8	2	no	—
60	Q₈×C₁₃	—	8	2	no	—
61	C₂×C₄×C₁₃	—	2	1	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	4	2	no	—
65	C₂×C₄×C₁₃	—	4	2	no	—
66	C₂×C₄×C₁₃	—	4	2	no	—
67	C₂×C₄×C₁₃	—	8	2	no	—
68	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₈×C₁₃	—	2	2	no	—
71	C₈×C₁₃	—	2	2	no	—
72	#533	Dic₁₃	1	1	yes	characteristic
73	C₂×C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
74	#491	Dic₁₃	1	2	yes	—
75	#491	C₂×D₁₃	1	2	yes	—
76	#487	—	2	2	no	—
77	#419	—	2	2	no	—
78	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
79	#364	—	26	2	no	—
80	#349	—	26	2	no	—
81	#347	—	13	4	no	—
82	C₂²×C₁₃	—	2	1	no	—
83	C₂²×C₁₃	—	4	1	no	—
84	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
85	C₄×C₁₃	—	2	2	no	—
86	C₄×C₁₃	—	2	1	no	—
87	C₄×C₁₃	—	4	1	no	—
88	C₄×C₁₃	—	8	2	no	—
89	C₄×C₁₃	—	8	2	no	—
90	C₄×C₁₃	#421	1	2	yes	—
91	C₂²×Q₈	—	2	1	no	—
92	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
93	C₄⋊Q₈	—	2	1	no	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	—	4	1	no	—
96	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
97	#125	C₂×Dic₁₃	1	2	yes	—
98	C₂²⋊Q₈	—	4	1	no	—
99	C₄×Q₈	—	4	1	no	—
100	#117	—	2	1	no	—
101	C₂×C₄⋊C₄	—	4	1	no	—
102	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
103	M₃₂	—	26	4	no	—
104	#103	—	2	4	no	—
105	#103	—	2	4	no	—
106	#97	C₂×Dic₁₃	1	1	yes	characteristic
107	#97	C₂²×D₁₃	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
109	C₂×C₁₃	—	2	1	no	—
110	C₂×C₁₃	—	4	1	no	—
111	C₂×C₁₃	#1533	1	1	yes	characteristic
112	C₂×Q₈	—	2	2	no	—
113	C₂×Q₈	—	4	1	no	—
114	C₂×Q₈	—	4	4	no	—
115	C₂×Q₈	—	4	1	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	4	1	no	—
118	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
121	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	1	no	—
127	C₂²⋊C₄	—	4	1	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	4	1	no	—
130	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
131	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
132	C₁₆	—	52	4	no	—
133	C₁₃	#11916	1	1	yes	characteristic, complemented, Sylow
134	C₂³	#88073	1	1	yes	characteristic
135	Q₈	—	4	4	no	—
136	Q₈	—	8	2	no	—
137	Q₈	—	8	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#88073	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	C₂×#88073	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	8	2	no	—
156	C₄	—	8	2	no	—
157	C₄	#11718897	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	4	1	no	—
160	C₂	#13113460	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116648

C₂

yes

maximal

#13112871

C₂

yes

maximal, complemented

C₁₃×#3139

C₂

yes

maximal, characteristic

C₁₃×#533

C₄

yes

characteristic

C₂×C₁₃×C₄⋊Q₈

C₄

yes

characteristic

C₁₃×#491

C₄

yes

—

C₁₃×#491

C₂²

yes

—

C₁₃×#487

—

C₁₃×#419

—

C₄×C₁₃×M₁₆

C₂²

yes

characteristic

#11718145

—

#11718058

C₂²

yes

—

#11718055

—

C₂²×Q₈×C₁₃

—

C₂×C₁₃×M₁₆

D₄

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

C₁₃×#125

C₂×C₄

yes

—

C₁₃×C₂²⋊Q₈

—

C₄×Q₈×C₁₃

—

C₁₃×#117

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₁₃×#103

—

C₁₃×#103

—

C₁₃×#97

C₂×C₄

yes

characteristic

C₁₃×#97

C₂³

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

—

#88052

—

#11916

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

#6090

—

#3139

D₁₃

yes

characteristic

#2165

—

#1356

—

C₂³×C₁₃

#100

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#100

yes

characteristic

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

—

#533

Dic₁₃

yes

characteristic

C₂×C₄⋊Q₈

Dic₁₃

yes

characteristic

#491

Dic₁₃

yes

—

#491

C₂×D₁₃

yes

—

#487

—

#419

—

C₄×M₁₆

C₂×D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×#100

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#421

yes

—

C₂²×Q₈

—

C₂×M₁₆

C₂²⋊D₁₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×Dic₁₃

yes

—

#125

C₂×Dic₁₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

100

#117

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

103

M₃₂

—

104

#103

—

105

#103

—

106

#97

C₂×Dic₁₃

yes

characteristic

107

#97

C₂²×D₁₃

yes

characteristic

108

C₄×C₈

C₂²⋊D₁₃

yes

—

109

C₂×C₁₃

—

110

C₂×C₁₃

—

111

C₂×C₁₃

#1533

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂²⋊Dic₁₃

yes

characteristic

119

M₁₆

—

120

C₂×C₈

C₂²⋊Dic₁₃

yes

—

121

C₂×C₈

C₂×C₂²⋊D₁₃

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂²⋊Dic₁₃

yes

characteristic

131

C₄²

C₂²×Dic₁₃

yes

characteristic

132

C₁₆

—

133

C₁₃

#11916

yes

characteristic, complemented, Sylow

134

C₂³

#88073

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#88073

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

characteristic

147

C₈

—

148

C₈

—

149

C₂²

—

150

C₂²

—

151

C₂²

C₂×#88073

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#11718897

yes

—

158

C₂

—

159

C₂

—

160

C₂

#13113460

yes

characteristic

	subgroup	G/H
center	C₂	#13113460
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	#97	C₂²×D₁₃
Fitting	C₁₃×#3139	C₂
socle	C₂×C₁₃	#1533
prime radical	C₂³×C₁₃	#100
cosocle	C₁₃×#97	C₂³
hypercenter	#3139	D₁₃
hypocenter	C₁₃	#11916
LCS	G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3139

subgroup

G/H

center

C₂

#13113460

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

#97

C₂²×D₁₃

Fitting

C₁₃×#3139

C₂

C₂×C₁₃

C₂³×C₁₃

C₁₃×#97

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3139

N	H ≅ G/N	type
2. #13112871	C₂	semidirect
133. C₁₃	#11916	semidirect

H ≅ G/N

type

2. #13112871

C₂

semidirect

133. C₁₃

#11916

semidirect

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

subgroup	index
#13116648	2
#13112871	2
C₁₃×#3139	2
#11916	13

subgroup

index

#13116648

#13112871

C₁₃×#3139

#11916

Subgroups — #15774116

Conjugacy classes of subgroups

Complements

Maximal subgroups