Group #15756729 — subgroups

Subgroups — #15756729

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	#13113592	C₂	1	1	yes	maximal, characteristic
3	#13113591	C₂	1	2	yes	maximal
4	C₂×#11718046	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1566	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×#334	—	2	2	no	—
13	#11718072	—	2	4	no	—
14	#11718049	—	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	#8825	—	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	SD₁₆	1	8	yes	—
51	#1848	—	13	2	no	—
52	#1566	—	13	2	no	—
53	#1566	—	13	1	no	—
54	#1566	D₁₃	1	1	yes	characteristic
55	C₂×#322	—	13	1	no	—
56	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#127	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
66	C₈×C₁₃	—	2	4	no	—
67	C₈×C₁₃	—	8	2	no	—
68	#1055	—	2	4	no	—
69	#1055	—	8	2	no	—
70	#1055	#106	1	8	yes	—
71	C₂×#108	—	26	4	no	—
72	#423	—	13	2	no	—
73	#423	C₂×D₁₃	1	2	yes	—
74	C₂×C₄×C₈	—	13	1	no	—
75	C₂×C₄×C₈	—	13	1	no	—
76	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
77	#334	—	2	2	no	—
78	#334	—	26	4	no	—
79	#334	—	26	2	no	—
80	#322	—	13	4	no	—
81	C₂²×C₁₃	—	2	1	no	—
82	C₂²×C₁₃	—	2	1	no	—
83	C₂²×C₁₃	#384	1	1	yes	characteristic
84	C₂²×C₁₃	#498	1	2	yes	—
85	C₄×C₁₃	—	2	1	no	—
86	C₄×C₁₃	—	2	1	no	—
87	C₄×C₁₃	C₂×#106	1	2	yes	—
88	C₄×C₁₃	#430	1	2	yes	—
89	C₄×C₁₃	#430	1	2	yes	—
90	C₄×C₁₃	#492	1	2	yes	—
91	C₂×M₁₆	—	2	2	no	—
92	C₂×M₁₆	—	26	2	no	—
93	C₂²×C₈	—	13	2	no	—
94	C₂²×C₈	—	13	2	no	—
95	C₂²×C₈	Dic₂₆	1	2	yes	—
96	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
97	#108	—	52	8	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	2	2	no	—
100	C₄⋊C₈	—	26	2	no	—
101	C₄⋊C₈	—	26	2	no	—
102	C₄×C₈	—	13	2	no	—
103	C₄×C₈	—	13	2	no	—
104	C₄×C₈	—	13	2	no	—
105	C₄×C₈	—	13	2	no	—
106	C₄×C₈	C₄×D₁₃	1	2	yes	—
107	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
108	C₂×C₁₃	—	2	2	no	—
109	C₂×C₁₃	#1844	1	1	yes	characteristic
110	C₂×C₁₃	#1849	1	1	yes	characteristic
111	C₂×C₁₃	#1849	1	1	yes	characteristic
112	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
113	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
114	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
115	M₁₆	—	4	4	no	—
116	M₁₆	—	52	4	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	13	4	no	—
120	C₂×C₈	—	13	8	no	—
121	C₂×C₈	—	13	4	no	—
122	C₂×C₈	—	52	2	no	—
123	C₂×C₈	—	52	2	no	—
124	C₂×C₈	#6101	1	4	yes	—
125	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
126	C₄²	D₄×D₁₃	1	1	yes	characteristic
127	C₄²	Q₈×D₁₃	1	1	yes	characteristic
128	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
129	C₁₃	#8825	1	1	yes	characteristic, complemented, Sylow
130	C₂³	#88142	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	#88142	1	1	yes	characteristic
134	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
135	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
136	C₂×C₄	SD₁₆×D₁₃	1	4	yes	—
137	C₂×C₄	C₂×#6101	1	1	yes	characteristic
138	C₂×C₄	#88194	1	1	yes	characteristic
139	C₂×C₄	#88198	1	1	yes	characteristic
140	C₈	—	2	4	no	—
141	C₈	—	8	2	no	—
142	C₈	—	13	8	no	—
143	C₈	—	26	4	no	—
144	C₈	—	104	2	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#11718516	1	2	yes	—
148	C₂²	#11718637	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	D₁₃×#106	1	2	yes	—
152	C₄	#11718554	1	2	yes	—
153	C₄	#11718785	1	2	yes	—
154	C₄	#11718836	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	#13114991	1	1	yes	characteristic
157	C₂	#13115009	1	1	yes	characteristic
158	C₂	#13115010	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115007

C₂

yes

maximal

#13113592

C₂

yes

maximal, characteristic

#13113591

C₂

yes

maximal

C₂×#11718046

C₂

yes

maximal, characteristic

C₁₃×#1566

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×#334

—

#11718072

—

#11718049

—

#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

—

#8825

—

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

SD₁₆

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×SD₁₆

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

#498

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#106

yes

—

C₄×C₁₃

#430

yes

—

C₄×C₁₃

#430

yes

—

C₄×C₁₃

#492

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

Dic₂₆

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

C₄×D₁₃

yes

—

107

C₄×C₈

C₂²⋊D₁₃

yes

—

108

C₂×C₁₃

—

109

C₂×C₁₃

#1844

yes

characteristic

110

C₂×C₁₃

#1849

yes

characteristic

111

C₂×C₁₃

#1849

yes

characteristic

112

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

113

C₂²×C₄

D₄×D₁₃

yes

characteristic

114

C₂²×C₄

Q₈×D₁₃

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

#6101

yes

—

125

C₄²

C₂×C₄×D₁₃

yes

characteristic

126

C₄²

D₄×D₁₃

yes

characteristic

127

C₄²

Q₈×D₁₃

yes

characteristic

128

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

129

C₁₃

#8825

yes

characteristic, complemented, Sylow

130

C₂³

#88142

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#88142

yes

characteristic

134

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

135

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

136

C₂×C₄

SD₁₆×D₁₃

yes

—

137

C₂×C₄

C₂×#6101

yes

characteristic

138

C₂×C₄

#88194

yes

characteristic

139

C₂×C₄

#88198

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

104

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11718516

yes

—

148

C₂²

#11718637

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

D₁₃×#106

yes

—

152

C₄

#11718554

yes

—

153

C₄

#11718785

yes

—

154

C₄

#11718836

yes

—

155

C₂

—

156

C₂

#13114991

yes

characteristic

157

C₂

#13115009

yes

characteristic

158

C₂

#13115010

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1566	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1566	D₁₃
hypocenter	C₁₃	#8825
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#88194

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1566

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 ◁ #1566

N	H ≅ G/N	type
129. C₁₃	#8825	semidirect

H ≅ G/N

type

129. C₁₃

#8825

semidirect

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

subgroup	index
#13115007	2
#13113592	2
#13113591	2
C₂×#11718046	2
C₁₃×#1566	2
#8825	13

subgroup

index

C₁₃×#1566

Subgroups — #15756729

Conjugacy classes of subgroups

Complements

Maximal subgroups