Group #15749984 — subgroups

Subgroups — #15749984

Order 3328.

180 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 59 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	#13114985	C₂	1	1	yes	maximal, characteristic
2	#13113133	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2165	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
5	#11718586	—	2	1	no	—
6	C₁₃×#364	—	2	1	no	—
7	C₁₃×#349	—	2	1	no	—
8	C₁₃×#347	C₄	1	2	yes	complemented
9	#11718152	C₄	1	1	yes	characteristic
10	#11718149	C₄	1	1	yes	characteristic
11	#11718148	—	2	1	no	—
12	#11718115	—	2	1	no	—
13	#11718047	C₄	1	1	yes	characteristic, complemented
14	#11718046	—	2	1	no	—
15	C₈×#1055	C₄	1	1	yes	characteristic, complemented
16	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	#88180	—	2	1	no	—
20	C₂×#6099	—	2	1	no	—
21	C₁₃×M₃₂	—	2	2	no	—
22	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
23	C₁₃×#97	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²⋊#1055	—	2	2	no	—
27	C₄⋊#1055	—	2	2	no	—
28	C₄×#1055	—	2	1	no	—
29	C₄×#1055	—	2	1	no	—
30	#7226	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	—	2	1	no	—
32	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₁₃×M₁₆	—	4	1	no	—
34	C₁₃×M₁₆	—	4	1	no	—
35	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₃	—	2	1	no	—
40	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²⋊Dic₁₃	—	4	1	no	—
43	C₄⋊Dic₁₃	—	4	1	no	—
44	C₄×Dic₁₃	—	4	1	no	—
45	#6099	—	4	1	no	—
46	#6099	—	4	1	no	—
47	C₂×#1055	—	2	2	no	—
48	C₂×#1055	—	2	2	no	—
49	C₂×#1055	—	4	1	no	—
50	C₁₃×C₁₆	—	4	2	no	—
51	#2165	D₁₃	1	1	yes	characteristic
52	#1841	—	13	1	no	—
53	#1453	—	13	1	no	—
54	C₂³×C₁₃	#104	1	1	yes	characteristic
55	C₂×C₄×C₁₃	—	2	1	no	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	—	2	1	no	—
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	—	4	2	no	—
60	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
62	C₂×Dic₁₃	—	8	1	no	—
63	C₈×C₁₃	—	2	1	no	—
64	C₈×C₁₃	—	2	1	no	—
65	C₈×C₁₃	—	2	1	no	—
66	C₈×C₁₃	—	2	1	no	—
67	#1055	—	4	2	no	—
68	#1055	—	4	2	no	—
69	#1055	—	8	1	no	—
70	#431	—	26	1	no	—
71	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
72	#364	—	2	1	no	—
73	#349	—	2	1	no	—
74	#347	Dic₁₃	1	2	yes	complemented
75	#344	—	26	1	no	—
76	#339	—	26	1	no	—
77	#338	—	13	1	no	—
78	H(3,ℤ/4)	—	13	1	no	—
79	#322	—	13	1	no	—
80	#322	—	26	1	no	—
81	C₈²	—	13	1	no	—
82	C₂²×C₁₃	—	2	1	no	—
83	C₂²×C₁₃	—	4	1	no	—
84	C₂²×C₁₃	#339	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	H(3,ℤ/4)	1	1	yes	characteristic
89	C₄×C₁₃	#344	1	1	yes	characteristic
90	Dic₁₃	—	16	1	no	—
91	C₂×M₁₆	—	26	1	no	—
92	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
93	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
94	#117	—	26	1	no	—
95	C₂×C₄²	D₅₂	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	C₄⋊C₈	—	26	2	no	—
98	C₂²⋊C₈	—	26	2	no	—
99	#97	Dic₂₆	1	1	yes	characteristic
100	#97	C₂×Dic₁₃	1	1	yes	characteristic
101	C₄×C₈	—	26	1	no	—
102	C₄×C₈	—	26	1	no	—
103	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
105	C₂×C₁₃	—	2	1	no	—
106	C₂×C₁₃	—	4	1	no	—
107	C₂×C₁₃	#1288	1	1	yes	characteristic
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	#6103	1	1	yes	characteristic
110	M₁₆	—	4	1	no	—
111	M₁₆	—	4	1	no	—
112	M₁₆	—	52	1	no	—
113	M₁₆	—	52	1	no	—
114	C₂×C₈	—	26	2	no	—
115	C₂×C₈	—	26	2	no	—
116	C₂×C₈	—	52	1	no	—
117	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
118	C₂×C₈	#6101	1	1	yes	characteristic
119	C₂×C₈	#6101	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
121	C₄⋊C₄	—	52	1	no	—
122	C₂²⋊C₄	—	52	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	52	1	no	—
125	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
126	C₄²	#6103	1	1	yes	characteristic
127	C₁₆	—	4	2	no	—
128	C₁₃	#7226	1	1	yes	characteristic, complemented, Sylow
129	C₂³	#88065	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	104	1	no	—
136	C₂×C₄	#88065	1	1	yes	characteristic
137	C₂×C₄	#88071	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	52	2	no	—
143	C₈	—	52	2	no	—
144	C₈	—	104	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	4	1	no	—
147	C₂²	#11718147	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	208	1	no	—
152	C₄	#11718123	1	1	yes	characteristic
153	C₄	#11718151	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	4	1	no	—
156	C₂	#13112702	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114985

C₂

yes

maximal, characteristic

#13113133

C₂

yes

maximal, characteristic

C₁₃×#2165

C₂

yes

maximal, characteristic

C₄×C₁₃×M₁₆

C₂²

yes

characteristic

#11718586

—

C₁₃×#364

—

C₁₃×#349

—

C₁₃×#347

C₄

yes

complemented

#11718152

C₄

yes

characteristic

#11718149

C₄

yes

—

—

C₄

yes

characteristic, complemented

#11718046

—

C₈×#1055

C₄

yes

characteristic, complemented

C₂×C₁₃×M₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×M₁₆

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

#88180

—

C₂×#6099

—

C₁₃×M₃₂

—

C₁₃×#97

C₂×C₄

yes

characteristic

C₁₃×#97

Q₈

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₂²⋊#1055

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

—

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

—

C₁₃×C₁₆

—

#2165

D₁₃

yes

characteristic

#1841

—

#1453

—

C₂³×C₁₃

#104

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

C₄×M₁₆

C₂×D₁₃

yes

—

—

Dic₁₃

yes

—

—

—

—

—

—

C₈²

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#339

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₁₃

yes

characteristic

C₂×M₁₆

C₂²⋊D₁₃

yes

characteristic

#117

—

C₂×C₄²

D₅₂

yes

characteristic

M₃₂

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

Dic₂₆

yes

characteristic

100

#97

C₂×Dic₁₃

yes

characteristic

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₄×D₁₃

yes

characteristic

104

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

105

C₂×C₁₃

—

106

C₂×C₁₃

—

107

C₂×C₁₃

#1288

yes

characteristic

108

C₂²×C₄

—

109

C₂²×C₄

#6103

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₄×Dic₁₃

yes

characteristic

118

C₂×C₈

#6101

yes

characteristic

119

C₂×C₈

#6101

yes

characteristic

120

C₂×C₈

C₂²⋊Dic₁₃

yes

characteristic

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₄⋊Dic₁₃

yes

characteristic

126

C₄²

#6103

yes

characteristic

127

C₁₆

—

128

C₁₃

#7226

yes

characteristic, complemented, Sylow

129

C₂³

#88065

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

104

—

136

C₂×C₄

#88065

yes

characteristic

137

C₂×C₄

#88071

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

104

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11718147

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

208

—

152

C₄

#11718123

yes

characteristic

153

C₄

#11718151

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#13112702

yes

characteristic

	subgroup	G/H
center	C₄	#11718123
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₄×M₁₆	C₂×D₁₃
Fitting	C₁₃×#2165	C₂
socle	C₂×C₁₃	#1288
prime radical	C₂³×C₁₃	#104
cosocle	C₄×C₁₃×M₁₆	C₂²
hypercenter	#2165	D₁₃
hypocenter	C₁₃	#7226
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2165

subgroup

G/H

center

C₄

#11718123

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₄×M₁₆

C₂×D₁₃

Fitting

C₁₃×#2165

C₂

C₂×C₁₃

C₂³×C₁₃

C₄×C₁₃×M₁₆

C₂²

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2165

N	H ≅ G/N	type
8. C₁₃×#347	C₄	semidirect
13. #11718047	C₄	semidirect
15. C₈×#1055	C₄	semidirect
74. #347	Dic₁₃	semidirect
128. C₁₃	#7226	semidirect

H ≅ G/N

type

8. C₁₃×#347

C₄

semidirect

13. #11718047

C₄

semidirect

15. C₈×#1055

C₄

semidirect

74. #347

Dic₁₃

semidirect

128. C₁₃

#7226

semidirect

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

subgroup	index
#13114985	2
#13113133	2
C₁₃×#2165	2
#7226	13

subgroup

index

#13114985

#13113133

C₁₃×#2165

#7226

Subgroups — #15749984

Conjugacy classes of subgroups

Complements

Maximal subgroups