Group #15756732 — subgroups

Subgroups — #15756732

Order 3328.

338 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 147 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	#13115007	C₂	1	2	yes	maximal
2	#13113595	C₂	1	1	yes	maximal, characteristic
3	#13113594	C₂	1	2	yes	maximal
4	C₂×#11718046	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1567	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#423	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88062	—	2	4	no	—
9	#11718585	C₂²	1	2	yes	—
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	C₁₃×#335	—	2	2	no	—
13	#11718071	—	2	4	no	—
14	#11718050	—	2	2	no	—
15	#11718046	C₄	1	4	yes	—
16	C₂×C₁₃×M₁₆	—	2	2	no	—
17	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×#6099	—	2	2	no	—
20	C₂²×#1055	D₄	1	2	yes	—
21	C₂²×#1055	Q₈	1	2	yes	—
22	C₁₃×C₄⋊C₈	—	2	2	no	—
23	C₁₃×C₄⋊C₈	—	2	2	no	—
24	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₃	D₄	1	2	yes	—
26	#88062	—	4	8	no	—
27	C₄⋊#1055	—	2	2	no	—
28	C₄⋊#1055	—	2	2	no	—
29	C₄×#1055	C₂×C₄	1	2	yes	—
30	C₄×#1055	C₂×C₄	1	2	yes	—
31	C₄×#1055	D₄	1	2	yes	—
32	C₄×#1055	Q₈	1	2	yes	—
33	#8826	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
37	C₁₃×M₁₆	—	4	4	no	—
38	C₂×C₈×C₁₃	—	4	2	no	—
39	C₂×C₈×C₁₃	—	4	2	no	—
40	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
41	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	#6099	—	4	4	no	—
46	C₂×#1055	—	4	2	no	—
47	C₂×#1055	—	4	2	no	—
48	C₂×#1055	C₄⋊C₄	1	4	yes	—
49	C₂×#1055	C₄⋊C₄	1	4	yes	—
50	C₂×#1055	D₈	1	4	yes	—
51	C₂×#1055	Q₁₆	1	4	yes	—
52	#1848	—	13	2	no	—
53	#1567	—	13	2	no	—
54	#1567	—	13	1	no	—
55	#1567	D₁₃	1	1	yes	characteristic
56	C₂×#322	—	13	1	no	—
57	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
68	C₈×C₁₃	—	2	4	no	—
69	C₈×C₁₃	—	8	2	no	—
70	#1055	—	2	4	no	—
71	#1055	—	8	2	no	—
72	#1055	#107	1	8	yes	—
73	C₂×#108	—	26	4	no	—
74	#423	—	13	2	no	—
75	#423	C₂×D₁₃	1	2	yes	—
76	C₂×C₄×C₈	—	13	1	no	—
77	C₂×C₄×C₈	—	13	1	no	—
78	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
79	#335	—	2	2	no	—
80	#335	—	26	4	no	—
81	#335	—	26	2	no	—
82	#322	—	13	4	no	—
83	C₂²×C₁₃	—	2	1	no	—
84	C₂²×C₁₃	—	2	1	no	—
85	C₂²×C₁₃	#384	1	1	yes	characteristic
86	C₂²×C₁₃	#500	1	2	yes	—
87	C₄×C₁₃	—	2	1	no	—
88	C₄×C₁₃	—	2	1	no	—
89	C₄×C₁₃	C₂×#107	1	2	yes	—
90	C₄×C₁₃	#430	1	2	yes	—
91	C₄×C₁₃	#430	1	2	yes	—
92	C₄×C₁₃	#493	1	1	yes	characteristic
93	C₄×C₁₃	#494	1	1	yes	characteristic
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	26	2	no	—
96	C₂²×C₈	—	13	2	no	—
97	C₂²×C₈	—	13	2	no	—
98	C₂²×C₈	Dic₂₆	1	2	yes	—
99	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
100	#108	—	52	8	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	26	2	no	—
104	C₄⋊C₈	—	26	2	no	—
105	C₄×C₈	—	13	2	no	—
106	C₄×C₈	—	13	2	no	—
107	C₄×C₈	—	13	2	no	—
108	C₄×C₈	—	13	2	no	—
109	C₄×C₈	C₄×D₁₃	1	2	yes	—
110	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
111	C₂×C₁₃	—	2	2	no	—
112	C₂×C₁₃	#1845	1	1	yes	characteristic
113	C₂×C₁₃	#1849	1	1	yes	characteristic
114	C₂×C₁₃	#1849	1	1	yes	characteristic
115	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
116	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
117	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
118	M₁₆	—	4	4	no	—
119	M₁₆	—	52	4	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	13	4	no	—
123	C₂×C₈	—	13	4	no	—
124	C₂×C₈	—	13	4	no	—
125	C₂×C₈	—	13	4	no	—
126	C₂×C₈	—	52	2	no	—
127	C₂×C₈	—	52	2	no	—
128	C₂×C₈	#6101	1	4	yes	—
129	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
130	C₄²	D₄×D₁₃	1	1	yes	characteristic
131	C₄²	Q₈×D₁₃	1	1	yes	characteristic
132	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
133	C₁₃	#8826	1	1	yes	characteristic, complemented, Sylow
134	C₂³	#88142	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#88142	1	1	yes	characteristic
138	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
139	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
140	C₂×C₄	D₈×D₁₃	1	2	yes	—
141	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
142	C₂×C₄	C₂×#6101	1	1	yes	characteristic
143	C₂×C₄	#88194	1	1	yes	characteristic
144	C₂×C₄	#88198	1	1	yes	characteristic
145	C₈	—	2	4	no	—
146	C₈	—	8	2	no	—
147	C₈	—	13	8	no	—
148	C₈	—	26	4	no	—
149	C₈	—	104	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#11718535	1	2	yes	—
153	C₂²	#11718637	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	D₁₃×#107	1	2	yes	—
157	C₄	#11718554	1	2	yes	—
158	C₄	#11718785	1	2	yes	—
159	C₄	#11718812	1	1	yes	characteristic
160	C₄	#11718842	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	#13114994	1	1	yes	characteristic
163	C₂	#13115009	1	1	yes	characteristic
164	C₂	#13115010	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115007

C₂

yes

maximal

#13113595

C₂

yes

maximal, characteristic

#13113594

C₂

yes

maximal

C₂×#11718046

C₂

yes

maximal, characteristic

C₁₃×#1567

C₂

yes

maximal, characteristic

C₁₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88062

—

#11718585

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

C₁₃×#335

—

#11718071

—

#11718050

—

#11718046

C₄

yes

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₂²×#1055

D₄

yes

—

C₂²×#1055

Q₈

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

D₄

yes

—

C₄×#1055

Q₈

yes

—

#8826

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

C₄⋊C₄

yes

—

C₂×#1055

C₄⋊C₄

yes

—

C₂×#1055

D₈

yes

—

C₂×#1055

Q₁₆

yes

—

—

—

—

D₁₃

yes

characteristic

C₂×#322

—

C₂³×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

yes

—

C₂×#108

—

#423

—

#423

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#384

yes

characteristic

C₂²×C₁₃

#500

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#107

yes

—

C₄×C₁₃

#430

yes

—

C₄×C₁₃

#430

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

Dic₂₆

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

100

#108

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₄×D₁₃

yes

—

110

C₄×C₈

C₂²⋊D₁₃

yes

—

111

C₂×C₁₃

—

112

C₂×C₁₃

#1845

yes

characteristic

113

C₂×C₁₃

#1849

yes

characteristic

114

C₂×C₁₃

#1849

yes

characteristic

115

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

116

C₂²×C₄

D₄×D₁₃

yes

characteristic

117

C₂²×C₄

Q₈×D₁₃

yes

characteristic

118

M₁₆

—

119

M₁₆

—

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

#6101

yes

—

129

C₄²

C₂×C₄×D₁₃

yes

characteristic

130

C₄²

D₄×D₁₃

yes

characteristic

131

C₄²

Q₈×D₁₃

yes

characteristic

132

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

133

C₁₃

#8826

yes

characteristic, complemented, Sylow

134

C₂³

#88142

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#88142

yes

characteristic

138

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

139

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

140

C₂×C₄

D₈×D₁₃

yes

—

141

C₂×C₄

Q₁₆×D₁₃

yes

—

142

C₂×C₄

C₂×#6101

yes

characteristic

143

C₂×C₄

#88194

yes

characteristic

144

C₂×C₄

#88198

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

104

—

150

C₂²

—

151

C₂²

—

152

C₂²

#11718535

yes

—

153

C₂²

#11718637

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

D₁₃×#107

yes

—

157

C₄

#11718554

yes

—

158

C₄

#11718785

yes

—

159

C₄

#11718812

yes

characteristic

160

C₄

#11718842

yes

characteristic

161

C₂

—

162

C₂

#13114994

yes

characteristic

163

C₂

#13115009

yes

characteristic

164

C₂

#13115010

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1567	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1567	D₁₃
hypocenter	C₁₃	#8826
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#88194

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1567

C₂

socle

C₂²×C₁₃

#384

prime radical

C₂³×C₁₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
133. C₁₃	#8826	semidirect

H ≅ G/N

type

133. C₁₃

#8826

semidirect

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

subgroup	index
#13115007	2
#13113595	2
#13113594	2
C₂×#11718046	2
C₁₃×#1567	2
#8826	13

subgroup

index

C₁₃×#1567

Subgroups — #15756732

Conjugacy classes of subgroups

Complements

Maximal subgroups