Group #16782374 — subgroups

Subgroups — #16782374

Order 3328.

294 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 100 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	#13130776	C₂	1	1	yes	maximal, characteristic
2	#13130773	C₂	1	1	yes	maximal, characteristic
3	#13129690	C₂	1	4	yes	maximal
4	#13113118	C₂	1	1	yes	maximal, characteristic
5	C₂×#88104	—	2	1	no	—
6	C₂×#88103	—	2	1	no	—
7	C₂×#88102	—	2	1	no	—
8	C₂×#88101	—	2	1	no	—
9	#11719060	C₂²	1	2	yes	—
10	#11719053	C₂²	1	2	yes	—
11	#11718346	C₂²	1	2	yes	—
12	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
13	C₈²×C₁₃	C₄	1	1	yes	characteristic, complemented
14	#11718046	C₄	1	2	yes	complemented
15	C₈×#1055	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄⋊#261	—	2	2	no	—
17	C₂×#6118	—	2	2	no	—
18	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
19	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
20	C₄²×D₁₃	C₂³	1	1	yes	characteristic
21	(C₂×#261)⋊C₄	—	2	4	no	—
22	#88112	—	2	4	no	—
23	#88104	—	4	2	no	—
24	#88103	—	4	2	no	—
25	#88102	—	4	2	no	—
26	#88101	—	4	2	no	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	#88060	—	2	2	no	—
30	#88054	—	2	2	no	—
31	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
32	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
33	C₄×#1055	C₂×C₄	1	2	yes	—
34	C₄×#1055	C₂×C₄	1	1	yes	characteristic
35	#8793	—	13	1	no	maximal, Sylow
36	C₂²×#261	—	4	2	no	—
37	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×D₁₃	#41	1	2	yes	—
39	C₂×#1057	—	4	2	no	—
40	C₄⋊#261	—	4	2	no	—
41	C₄⋊#261	—	4	2	no	—
42	#6118	—	4	2	no	—
43	#6118	—	4	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
49	C₄×Dic₁₃	#41	1	2	yes	—
50	C₂×#1055	—	2	2	no	—
51	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
52	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
53	C₈×D₁₃	D₈	1	2	yes	—
54	C₈×D₁₃	SD₁₆	1	2	yes	—
55	C₈×D₁₃	SD₁₆	1	2	yes	—
56	C₈×D₁₃	Q₁₆	1	2	yes	—
57	#1848	—	13	1	no	—
58	#1846	—	13	1	no	—
59	#1450	—	13	1	no	—
60	#1286	—	13	4	no	—
61	C₂²×D₁₃	#124	1	1	yes	characteristic
62	C₂×#261	—	8	4	no	—
63	C₂×#261	—	8	2	no	—
64	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	#117	1	2	yes	—
66	C₂×Dic₁₃	#124	1	1	yes	characteristic
67	C₂×Dic₁₃	#125	1	2	yes	—
68	C₄×D₁₃	—	4	2	no	—
69	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
70	C₄×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₄×D₁₃	C₂×Q₁₆	1	1	yes	characteristic
73	#1057	—	8	2	no	—
74	C₈×C₁₃	—	4	2	no	—
75	C₈×C₁₃	#106	1	2	yes	—
76	C₈×C₁₃	#107	1	2	yes	—
77	#1055	—	4	2	no	—
78	#1055	#106	1	2	yes	—
79	#1055	#107	1	2	yes	—
80	#431	—	13	2	no	—
81	C₂×#108	—	26	1	no	—
82	C₂×#108	—	26	1	no	—
83	C₂×#107	—	26	1	no	—
84	C₂×#106	—	26	1	no	—
85	#423	—	13	2	no	—
86	C₂×C₄×C₈	—	13	1	no	—
87	#382	—	13	2	no	—
88	#322	—	13	2	no	—
89	C₈²	—	13	1	no	—
90	C₈²	#261	1	1	yes	characteristic, complemented
91	C₂²×C₁₃	#382	1	1	yes	characteristic
92	C₂×D₁₃	—	2	2	no	—
93	C₂×D₁₃	#486	1	1	yes	characteristic
94	C₂×D₁₃	#487	1	1	yes	characteristic
95	#261	—	16	2	no	—
96	C₄×C₁₃	—	2	2	no	—
97	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
98	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
99	Dic₁₃	—	2	2	no	—
100	Dic₁₃	#486	1	1	yes	characteristic
101	Dic₁₃	#487	1	1	yes	characteristic
102	C₂×M₁₆	—	26	2	no	—
103	C₂²×C₈	—	13	1	no	—
104	C₂²×C₈	—	13	1	no	—
105	C₂×C₄⋊C₄	—	26	2	no	—
106	C₂×C₄²	—	13	1	no	—
107	#108	—	52	2	no	—
108	#108	—	52	2	no	—
109	#107	—	52	2	no	—
110	#106	—	52	2	no	—
111	C₄⋊C₈	—	26	2	no	—
112	C₄⋊C₈	—	26	4	no	—
113	C₂²⋊C₈	—	26	2	no	—
114	C₄×C₈	—	13	1	no	—
115	C₄×C₈	—	13	1	no	—
116	C₄×C₈	—	13	1	no	—
117	C₄×C₈	—	13	2	no	—
118	C₄×C₈	C₂×#261	1	2	yes	—
119	C₄×C₈	C₂×#261	1	1	yes	characteristic
120	#95	—	26	4	no	—
121	C₂×C₁₃	#1833	1	2	yes	—
122	C₂×C₁₃	#1834	1	1	yes	characteristic
123	D₁₃	—	2	1	no	—
124	D₁₃	—	2	1	no	—
125	C₂²×C₄	—	13	1	no	—
126	C₂²×C₄	—	13	2	no	—
127	C₂²×C₄	—	52	2	no	—
128	M₁₆	—	52	2	no	—
129	M₁₆	—	52	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	13	1	no	—
132	C₂×C₈	—	13	1	no	—
133	C₂×C₈	—	13	2	no	—
134	C₂×C₈	—	13	2	no	—
135	C₂×C₈	—	13	2	no	—
136	C₂×C₈	—	13	2	no	—
137	C₂×C₈	—	26	2	no	—
138	C₂×C₈	—	52	2	no	—
139	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
140	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
141	C₄⋊C₄	—	52	2	no	—
142	C₄⋊C₄	—	52	2	no	—
143	C₄²	—	13	1	no	—
144	C₄²	—	13	2	no	—
145	C₄²	C₂²×#261	1	1	yes	characteristic
146	C₁₃	#8793	1	1	yes	characteristic, complemented, Sylow
147	C₂³	—	13	1	no	—
148	C₂×C₄	—	13	1	no	—
149	C₂×C₄	—	13	1	no	—
150	C₂×C₄	—	13	1	no	—
151	C₂×C₄	—	13	1	no	—
152	C₂×C₄	—	13	1	no	—
153	C₂×C₄	—	13	2	no	—
154	C₂×C₄	—	52	2	no	—
155	C₂×C₄	—	104	2	no	—
156	C₂×C₄	—	104	4	no	—
157	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
158	C₂×C₄	#88237	1	2	yes	—
159	C₈	—	4	2	no	—
160	C₈	—	13	2	no	—
161	C₈	—	13	2	no	—
162	C₈	—	52	2	no	—
163	C₈	—	104	2	no	—
164	C₈	#88101	1	2	yes	—
165	C₈	#88102	1	2	yes	—
166	C₂²	—	13	1	no	—
167	C₂²	—	13	1	no	—
168	C₂²	—	26	2	no	—
169	C₂²	#11719060	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	13	1	no	—
172	C₄	—	13	1	no	—
173	C₄	—	26	2	no	—
174	C₄	—	208	2	no	—
175	C₄	C₂×#88101	1	1	yes	characteristic
176	C₄	C₂×#88102	1	1	yes	characteristic
177	C₂	—	26	1	no	—
178	C₂	—	26	1	no	—
179	C₂	#13130755	1	2	yes	—
180	C₂	#13130757	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130776

C₂

yes

maximal, characteristic

#13130773

C₂

yes

maximal, characteristic

#13129690

C₂

yes

maximal

#13113118

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#11719053

C₂²

yes

—

#11718346

C₂²

yes

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

C₈²×C₁₃

C₄

yes

characteristic, complemented

#11718046

C₄

yes

complemented

C₈×#1055

C₄

yes

characteristic, complemented

C₂×C₄⋊#261

—

C₂×#6118

—

C₂×C₈×D₁₃

D₄

yes

characteristic

C₂×C₈×D₁₃

D₄

yes

characteristic

C₄²×D₁₃

C₂³

yes

characteristic

(C₂×#261)⋊C₄

—

—

—

—

—

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88060

—

#88054

—

C₈×Dic₁₃

Q₈

yes

characteristic

C₈×Dic₁₃

Q₈

yes

characteristic

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#8793

—

maximal, Sylow

C₂²×#261

—

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

#41

yes

—

—

—

—

—

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄×Dic₁₃

C₂×Q₈

yes

characteristic

C₄×Dic₁₃

#41

yes

—

C₂×#1055

—

C₂×#1055

C₄⋊C₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₈×D₁₃

D₈

yes

—

C₈×D₁₃

SD₁₆

yes

—

C₈×D₁₃

SD₁₆

yes

—

C₈×D₁₃

Q₁₆

yes

—

—

—

—

—

C₂²×D₁₃

yes

—

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×Dic₁₃

#124

yes

characteristic

C₂×Dic₁₃

#125

yes

—

C₄×D₁₃

—

C₄×D₁₃

C₂×D₈

yes

characteristic

C₄×D₁₃

C₂×SD₁₆

yes

characteristic

C₄×D₁₃

C₂×SD₁₆

yes

characteristic

C₄×D₁₃

C₂×Q₁₆

yes

characteristic

#1057

—

C₈×C₁₃

—

C₈×C₁₃

#106

yes

—

C₈×C₁₃

#107

yes

—

#1055

—

#1055

#106

yes

—

#1055

#107

yes

—

—

—

—

—

—

—

C₂×C₄×C₈

—

#382

—

#322

—

C₈²

—

C₈²

#261

yes

characteristic, complemented

C₂²×C₁₃

#382

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

yes

C₂×D₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

100

Dic₁₃

#486

yes

characteristic

101

Dic₁₃

#487

yes

characteristic

102

C₂×M₁₆

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

—

107

#108

—

108

#108

—

109

#107

—

110

#106

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₂²⋊C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

C₂×#261

yes

—

119

C₄×C₈

C₂×#261

yes

characteristic

120

#95

—

121

C₂×C₁₃

#1833

yes

—

122

C₂×C₁₃

#1834

yes

characteristic

123

D₁₃

—

124

D₁₃

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

M₁₆

—

129

M₁₆

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

C₄⋊#261

yes

characteristic

140

C₂×C₈

C₄⋊#261

yes

characteristic

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₄²

C₂²×#261

yes

characteristic

146

C₁₃

#8793

yes

characteristic, complemented, Sylow

147

C₂³

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

104

—

156

C₂×C₄

—

104

—

157

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

158

C₂×C₄

#88237

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

104

—

164

C₈

#88101

yes

—

165

C₈

#88102

yes

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

#11719060

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

208

—

175

C₄

C₂×#88101

yes

characteristic

176

C₄

C₂×#88102

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#13130755

yes

—

180

C₂

#13130757

yes

characteristic

	subgroup	G/H
center	C₂²	#11719060
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₈²×C₁₃	C₄
socle	C₂²×C₁₃	#382
prime radical	C₂²×D₁₃	#124
cosocle	C₄²×D₁₃	C₂³
hypercenter	C₈²	#261
hypocenter	C₁₃	#8793
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#11719060

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₈²×C₁₃

C₄

C₂²×C₁₃

C₂²×D₁₃

C₄²×D₁₃

C₂³

C₈²

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
13. C₈²×C₁₃	C₄	semidirect
14. #11718046	C₄	semidirect
15. C₈×#1055	C₄	semidirect
90. C₈²	#261	semidirect
146. C₁₃	#8793	semidirect

H ≅ G/N

type

13. C₈²×C₁₃

C₄

semidirect

14. #11718046

C₄

semidirect

15. C₈×#1055

C₄

semidirect

90. C₈²

#261

semidirect

146. C₁₃

#8793

semidirect

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

subgroup	index
#13130776	2
#13130773	2
#13129690	2
#13113118	2
#8793	13

subgroup

index

Subgroups — #16782374

Conjugacy classes of subgroups

Complements

Maximal subgroups