Group #16782434 — subgroups

Subgroups — #16782434

Order 3328.

354 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 140 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	#13130777	C₂	1	2	yes	maximal
2	#13130392	C₂	1	2	yes	maximal
3	#13130391	C₂	1	2	yes	maximal
4	C₈²×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#88104	—	2	4	no	—
6	#11719052	C₂²	1	4	yes	—
7	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	#11718255	—	2	4	no	—
11	#11718253	—	2	4	no	—
12	C₈²×C₁₃	C₄	1	1	yes	characteristic
13	C₈×#1055	C₄	1	1	yes	characteristic
14	C₈×#1055	C₄	1	1	yes	characteristic
15	C₈×#1055	C₄	1	1	yes	characteristic
16	C₂×#6118	—	2	4	no	—
17	C₂×C₈×D₁₃	D₄	1	2	yes	—
18	C₂×C₈×D₁₃	Q₈	1	2	yes	—
19	C₂×C₈×D₁₃	Q₈	1	2	yes	—
20	C₄²×D₁₃	C₂³	1	1	yes	characteristic
21	C₄⋊#1057	—	2	4	no	—
22	C₄⋊#1057	—	2	4	no	—
23	#88104	—	4	8	no	—
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₃	D₄	1	2	yes	—
28	C₈×Dic₁₃	D₄	1	2	yes	—
29	C₈×Dic₁₃	Q₈	1	2	yes	—
30	C₄×#1055	C₂×C₄	1	1	yes	characteristic
31	C₄×#1055	C₂×C₄	1	1	yes	characteristic
32	C₄×#1055	C₂×C₄	1	1	yes	characteristic
33	#8822	—	13	1	no	maximal, Sylow
34	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×#1057	—	4	4	no	—
38	#6118	—	4	8	no	—
39	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
42	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
44	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
45	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×#1055	C₄⋊C₄	1	2	yes	—
47	C₂×#1055	C₄⋊C₄	1	2	yes	—
48	C₂×#1055	C₄⋊C₄	1	2	yes	—
49	C₈×D₁₃	—	2	4	no	—
50	C₈×D₁₃	—	2	4	no	—
51	C₈×D₁₃	SD₁₆	1	8	yes	—
52	#1848	—	13	2	no	—
53	#1566	—	13	2	no	—
54	#1566	—	13	2	no	—
55	C₂×C₈²	—	13	1	no	—
56	C₂²×D₁₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×Dic₁₃	#127	1	1	yes	characteristic
61	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₄×D₁₃	—	2	2	no	—
64	C₄×D₁₃	—	2	2	no	—
65	C₄×D₁₃	C₂×SD₁₆	1	4	yes	—
66	#1057	—	8	4	no	—
67	C₈×C₁₃	—	2	2	no	—
68	C₈×C₁₃	#106	1	4	yes	—
69	C₈×C₁₃	#108	1	4	yes	complemented
70	#1055	—	2	2	no	—
71	#1055	#106	1	4	yes	—
72	#1055	#108	1	4	yes	complemented
73	C₂×#108	—	26	4	no	—
74	#423	—	13	4	no	—
75	C₂×C₄×C₈	—	13	1	no	—
76	C₂×C₄×C₈	—	13	1	no	—
77	C₂×C₄×C₈	—	13	1	no	—
78	#334	—	26	4	no	—
79	#334	—	26	4	no	—
80	C₈²	—	13	1	no	—
81	C₈²	—	13	1	no	—
82	C₈²	—	13	1	no	—
83	C₈²	#261	1	1	yes	characteristic
84	C₂²×C₁₃	#384	1	1	yes	characteristic
85	C₂×D₁₃	—	2	1	no	—
86	C₂×D₁₃	—	2	1	no	—
87	C₂×D₁₃	#498	1	2	yes	—
88	C₄×C₁₃	C₂×#106	1	2	yes	—
89	C₄×C₁₃	C₂×#108	1	2	yes	—
90	C₄×C₁₃	C₂×#108	1	2	yes	—
91	Dic₁₃	—	2	1	no	—
92	Dic₁₃	—	2	1	no	—
93	Dic₁₃	#492	1	2	yes	—
94	C₂×M₁₆	—	26	4	no	—
95	C₂²×C₈	—	13	2	no	—
96	C₂²×C₈	—	13	2	no	—
97	C₂²×C₈	—	13	2	no	—
98	C₂×C₄²	—	13	1	no	—
99	#108	—	52	8	no	—
100	C₄⋊C₈	—	26	4	no	—
101	C₄⋊C₈	—	26	4	no	—
102	C₄×C₈	—	13	2	no	—
103	C₄×C₈	—	13	2	no	—
104	C₄×C₈	—	13	2	no	—
105	C₄×C₈	—	13	1	no	—
106	C₄×C₈	—	13	1	no	—
107	C₄×C₈	—	13	1	no	—
108	C₄×C₈	C₂×#261	1	1	yes	characteristic
109	C₄×C₈	C₂×#261	1	1	yes	characteristic
110	C₄×C₈	C₂×#261	1	1	yes	characteristic
111	C₂×C₁₃	#1844	1	1	yes	characteristic
112	C₂×C₁₃	#1848	1	1	yes	characteristic
113	C₂×C₁₃	#1848	1	1	yes	characteristic
114	D₁₃	—	2	2	no	—
115	C₂²×C₄	—	13	1	no	—
116	C₂²×C₄	—	13	1	no	—
117	C₂²×C₄	—	13	1	no	—
118	M₁₆	—	52	8	no	—
119	C₂×C₈	—	13	2	no	—
120	C₂×C₈	—	13	2	no	—
121	C₂×C₈	—	13	8	no	—
122	C₂×C₈	—	13	2	no	—
123	C₂×C₈	—	26	4	no	—
124	C₂×C₈	—	26	4	no	—
125	C₂×C₈	—	52	4	no	—
126	C₂×C₈	C₄⋊#261	1	2	yes	—
127	C₂×C₈	C₄⋊#261	1	2	yes	—
128	C₂×C₈	C₄⋊#261	1	2	yes	—
129	C₄²	—	13	1	no	—
130	C₄²	—	13	1	no	—
131	C₄²	—	13	1	no	—
132	C₄²	C₂²×#261	1	1	yes	characteristic
133	C₁₃	#8822	1	1	yes	characteristic, complemented, Sylow
134	C₂³	—	13	1	no	—
135	C₂×C₄	—	13	1	no	—
136	C₂×C₄	—	13	4	no	—
137	C₂×C₄	—	13	1	no	—
138	C₂×C₄	—	13	1	no	—
139	C₂×C₄	—	26	2	no	—
140	C₂×C₄	—	26	2	no	—
141	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	13	4	no	—
146	C₈	—	13	4	no	—
147	C₈	—	26	2	no	—
148	C₈	—	104	4	no	—
149	C₈	#88101	1	4	yes	—
150	C₈	#88104	1	4	yes	complemented
151	C₂²	—	13	2	no	—
152	C₂²	—	26	1	no	—
153	C₂²	—	26	1	no	—
154	C₂²	#11719059	1	1	yes	characteristic
155	C₄	—	13	2	no	—
156	C₄	—	26	1	no	—
157	C₄	—	26	1	no	—
158	C₄	C₂×#88101	1	2	yes	—
159	C₄	C₂×#88104	1	2	yes	—
160	C₄	D₁₃⋊#108	1	2	yes	—
161	C₂	—	26	2	no	—
162	C₂	#13130771	1	1	yes	characteristic
163	C₂	#13130776	1	1	yes	characteristic
164	C₂	#13130777	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130777

C₂

yes

maximal

#13130392

C₂

yes

maximal

#13130391

C₂

yes

maximal

C₈²×D₁₃

C₂

yes

maximal, characteristic

C₂×#88104

—

#11719052

C₂²

yes

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

C₄×C₈×D₁₃

C₂²

yes

characteristic

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718255

—

#11718253

—

C₈²×C₁₃

C₄

yes

characteristic

C₈×#1055

C₄

yes

characteristic

C₈×#1055

C₄

yes

characteristic

C₈×#1055

C₄

yes

characteristic

C₂×#6118

—

C₂×C₈×D₁₃

D₄

yes

—

C₂×C₈×D₁₃

Q₈

yes

—

C₂×C₈×D₁₃

Q₈

yes

—

C₄²×D₁₃

C₂³

yes

—

—

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

D₄

yes

—

C₈×Dic₁₃

D₄

yes

—

C₈×Dic₁₃

Q₈

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

C₄×#1055

C₂×C₄

yes

characteristic

C₄×#1055

C₂×C₄

yes

characteristic

#8822

—

maximal, Sylow

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×D₁₃

C₂×Q₈

yes

characteristic

C₂×#1057

—

#6118

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

C₂×Q₈

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

—

C₂×#1055

C₄⋊C₄

yes

—

C₂×#1055

C₄⋊C₄

yes

—

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

SD₁₆

yes

—

#1848

—

#1566

—

#1566

—

C₂×C₈²

—

C₂²×D₁₃

C₄⋊Q₈

yes

characteristic

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×Dic₁₃

#127

yes

characteristic

C₂×Dic₁₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₁₃

C₄⋊Q₈

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂×SD₁₆

yes

—

#1057

—

C₈×C₁₃

—

C₈×C₁₃

#106

yes

—

C₈×C₁₃

yes

—

yes

—

yes

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#334

—

#334

—

C₈²

—

C₈²

—

C₈²

—

C₈²

yes

C₂²×C₁₃

yes

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

#498

yes

—

C₄×C₁₃

C₂×#106

yes

—

C₄×C₁₃

C₂×#108

yes

—

C₄×C₁₃

C₂×#108

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#492

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₂×#261

yes

characteristic

109

C₄×C₈

C₂×#261

yes

characteristic

110

C₄×C₈

C₂×#261

yes

characteristic

111

C₂×C₁₃

#1844

yes

characteristic

112

C₂×C₁₃

#1848

yes

characteristic

113

C₂×C₁₃

#1848

yes

characteristic

114

D₁₃

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

C₄⋊#261

yes

—

127

C₂×C₈

C₄⋊#261

yes

—

128

C₂×C₈

C₄⋊#261

yes

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

C₂²×#261

yes

characteristic

133

C₁₃

#8822

yes

characteristic, complemented, Sylow

134

C₂³

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

142

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

143

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

104

—

149

C₈

#88101

yes

—

150

C₈

#88104

yes

complemented

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

#11719059

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

C₂×#88101

yes

—

159

C₄

C₂×#88104

yes

—

160

C₄

D₁₃⋊#108

yes

—

161

C₂

—

162

C₂

#13130771

yes

characteristic

163

C₂

#13130776

yes

characteristic

164

C₂

#13130777

yes

characteristic

	subgroup	G/H
center	C₂²	#11719059
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₈²×C₁₃	C₄
socle	C₂²×C₁₃	#384
prime radical	C₂²×D₁₃	C₄⋊Q₈
cosocle	C₄²×D₁₃	C₂³
hypercenter	C₈²	#261
hypocenter	C₁₃	#8822
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#11719059

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₈²×C₁₃

C₄

socle

C₂²×C₁₃

#384

prime radical

C₂²×D₁₃

C₄⋊Q₈

cosocle

C₄²×D₁₃

C₂³

C₈²

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
69. C₈×C₁₃	#108	semidirect
72. #1055	#108	semidirect
133. C₁₃	#8822	semidirect
150. C₈	#88104	semidirect

H ≅ G/N

type

69. C₈×C₁₃

#108

semidirect

72. #1055

#108

semidirect

133. C₁₃

#8822

semidirect

150. C₈

#88104

semidirect

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

subgroup	index
#13130777	2
#13130392	2
#13130391	2
C₈²×D₁₃	2
#8822	13

subgroup

index

C₈²×D₁₃

Subgroups — #16782434

Conjugacy classes of subgroups

Complements

Maximal subgroups