Group #16790228 — subgroups

Subgroups — #16790228

Order 3328.

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

normal subgroups only

This group has 286 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	#13130775	C₂	1	1	yes	maximal, characteristic
2	#13130773	C₂	1	1	yes	maximal, characteristic
3	#13129901	C₂	1	4	yes	maximal
4	#13116429	C₂	1	1	yes	maximal, characteristic
5	C₂×#88103	—	2	1	no	—
6	C₂×#88103	C₂²	1	2	yes	—
7	C₂×#88102	—	2	1	no	—
8	C₂×#88101	C₂²	1	2	yes	—
9	#11719060	—	2	1	no	—
10	#11719052	—	2	1	no	—
11	C₂×#88038	C₂²	1	2	yes	—
12	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×#346	C₄	1	1	yes	characteristic, complemented
14	C₁₆×Dic₁₃	C₄	1	2	yes	complemented
15	#11718053	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	C₂³	1	1	yes	characteristic
20	C₄²×D₁₃	D₄	1	1	yes	characteristic
21	(C₂×#261)⋊C₄	—	4	2	no	—
22	C₄⋊#1057	—	4	2	no	—
23	#88103	—	2	4	no	—
24	#88103	—	4	2	no	—
25	#88102	—	4	2	no	—
26	#88101	—	2	4	no	—
27	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
31	C₂×#6090	C₂×C₄	1	2	yes	—
32	C₄×#1055	Q₈	1	1	yes	characteristic
33	#88038	—	2	2	no	—
34	#88038	—	2	2	no	—
35	#12887	—	13	1	no	maximal, Sylow
36	C₂²×#261	—	4	2	no	—
37	C₂×C₄×D₁₃	SD₁₆	1	2	yes	—
38	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄×Dic₁₃	SD₁₆	1	2	yes	—
49	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
51	C₈×D₁₃	D₈	1	2	yes	—
52	C₈×D₁₃	Q₁₆	1	2	yes	—
53	C₈×D₁₃	#41	1	2	yes	—
54	C₈×D₁₃	#41	1	2	yes	—
55	C₁₃×C₁₆	—	2	2	no	—
56	#6090	—	2	2	no	—
57	C₄×M₃₂	—	13	1	no	—
58	#1848	—	13	1	no	—
59	#1846	—	13	1	no	—
60	#1383	—	13	4	no	—
61	C₂²×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×#261	—	8	4	no	—
63	C₂×#261	—	8	2	no	—
64	C₂×C₄×C₁₃	#106	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×Dic₁₃	#106	1	2	yes	—
67	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×D₁₃	—	4	2	no	—
69	C₄×D₁₃	#124	1	1	yes	characteristic
70	C₄×D₁₃	#124	1	1	yes	characteristic
71	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
72	C₄×D₁₃	C₂×Q₁₆	1	1	yes	characteristic
73	#1057	—	8	2	no	—
74	C₈×C₁₃	#107	1	2	yes	—
75	C₈×C₁₃	#117	1	2	yes	—
76	#1055	#107	1	2	yes	—
77	#1055	#125	1	2	yes	—
78	C₂×M₃₂	—	13	2	no	—
79	C₂×#108	—	13	2	no	—
80	C₂×#108	—	26	1	no	—
81	C₂×#107	—	26	1	no	—
82	C₂×#106	—	13	2	no	—
83	#423	—	26	1	no	—
84	C₂×C₄×C₈	—	13	1	no	—
85	#382	—	26	1	no	—
86	#346	—	13	1	no	—
87	#346	#261	1	1	yes	characteristic, complemented
88	C₄×C₁₆	—	13	2	no	—
89	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
90	C₂×D₁₃	—	2	2	no	—
91	C₂×D₁₃	#486	1	1	yes	characteristic
92	C₂×D₁₃	#487	1	1	yes	characteristic
93	#261	—	16	2	no	—
94	C₄×C₁₃	—	2	2	no	—
95	C₄×C₁₃	#382	1	1	yes	characteristic
96	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
97	Dic₁₃	—	2	2	no	—
98	Dic₁₃	#486	1	1	yes	characteristic
99	Dic₁₃	#487	1	1	yes	characteristic
100	C₂×M₁₆	—	26	2	no	—
101	C₂²×C₈	—	13	1	no	—
102	C₂²×C₈	—	13	1	no	—
103	C₂×C₄⋊C₄	—	26	2	no	—
104	C₂×C₄²	—	13	1	no	—
105	M₃₂	—	26	2	no	—
106	M₃₂	—	26	2	no	—
107	C₂×C₁₆	—	13	2	no	—
108	C₂×C₁₆	C₂×#261	1	2	yes	—
109	#108	—	26	4	no	—
110	#108	—	52	2	no	—
111	#107	—	52	2	no	—
112	#106	—	26	4	no	—
113	C₄⋊C₈	—	52	2	no	—
114	C₄×C₈	—	13	1	no	—
115	C₄×C₈	—	13	1	no	—
116	C₄×C₈	—	13	1	no	—
117	C₄×C₈	C₂×#261	1	1	yes	characteristic
118	#95	—	52	2	no	—
119	C₂×C₁₃	#1834	1	1	yes	characteristic
120	C₂×C₁₃	#2155	1	2	yes	—
121	D₁₃	—	2	1	no	—
122	D₁₃	—	2	1	no	—
123	C₂²×C₄	—	13	1	no	—
124	C₂²×C₄	—	13	2	no	—
125	C₂²×C₄	—	52	2	no	—
126	M₁₆	—	52	2	no	—
127	M₁₆	—	52	2	no	—
128	C₂×C₈	—	13	1	no	—
129	C₂×C₈	—	13	1	no	—
130	C₂×C₈	—	13	2	no	—
131	C₂×C₈	—	13	2	no	—
132	C₂×C₈	—	13	2	no	—
133	C₂×C₈	—	13	2	no	—
134	C₂×C₈	—	52	2	no	—
135	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
136	C₂×C₈	C₂²×#261	1	1	yes	characteristic
137	C₄⋊C₄	—	52	2	no	—
138	C₄⋊C₄	—	52	2	no	—
139	C₄²	—	13	2	no	—
140	C₄²	—	13	1	no	—
141	C₄²	C₄⋊#261	1	1	yes	characteristic
142	C₁₆	—	2	2	no	—
143	C₁₆	—	26	2	no	—
144	C₁₃	#12887	1	1	yes	characteristic, complemented, Sylow
145	C₂³	—	13	1	no	—
146	C₂×C₄	—	13	1	no	—
147	C₂×C₄	—	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	2	no	—
152	C₂×C₄	—	52	2	no	—
153	C₂×C₄	—	104	2	no	—
154	C₂×C₄	—	104	4	no	—
155	C₂×C₄	#88101	1	2	yes	—
156	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
157	C₈	—	13	2	no	—
158	C₈	—	13	2	no	—
159	C₈	—	104	2	no	—
160	C₈	#88102	1	2	yes	—
161	C₈	#88237	1	2	yes	—
162	C₂²	—	13	1	no	—
163	C₂²	—	13	1	no	—
164	C₂²	—	26	2	no	—
165	C₂²	C₂×#88101	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	13	1	no	—
168	C₄	—	13	1	no	—
169	C₄	—	26	2	no	—
170	C₄	—	208	2	no	—
171	C₄	#11719060	1	1	yes	characteristic
172	C₄	C₂×#88102	1	1	yes	characteristic
173	C₂	—	26	1	no	—
174	C₂	—	26	1	no	—
175	C₂	#13130757	1	1	yes	characteristic
176	C₂	#13131184	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13130775

C₂

yes

maximal, characteristic

#13130773

C₂

yes

maximal, characteristic

#13129901

C₂

yes

maximal

#13116429

C₂

yes

maximal, characteristic

C₂×#88103

—

C₂×#88103

C₂²

yes

—

C₂×#88102

—

C₂×#88101

C₂²

yes

—

#11719060

—

#11719052

—

C₂×#88038

C₂²

yes

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

C₁₃×#346

C₄

yes

characteristic, complemented

C₁₆×Dic₁₃

C₄

yes

complemented

#11718053

C₄

yes

characteristic, complemented

C₂×C₄⋊#261

—

C₂×#6118

—

C₂×C₈×D₁₃

D₄

yes

characteristic

C₂×C₈×D₁₃

C₂³

yes

characteristic

C₄²×D₁₃

D₄

yes

characteristic

(C₂×#261)⋊C₄

—

—

—

—

—

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

Q₈

yes

characteristic

C₂×#6090

C₂×C₄

yes

—

C₄×#1055

Q₈

yes

—

—

—

C₂²×#261

—

C₂×C₄×D₁₃

SD₁₆

yes

—

C₂×C₄×D₁₃

C₂×D₄

yes

—

—

—

—

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₄×Dic₁₃

C₄⋊C₄

yes

characteristic

C₄×Dic₁₃

SD₁₆

yes

—

C₂×#1055

C₄⋊C₄

yes

characteristic

C₂×#1055

C₂×Q₈

yes

characteristic

C₈×D₁₃

D₈

yes

—

C₈×D₁₃

Q₁₆

yes

—

C₈×D₁₃

#41

yes

—

C₈×D₁₃

#41

yes

—

C₁₃×C₁₆

—

#6090

—

C₄×M₃₂

—

#1848

—

#1846

—

#1383

—

C₂²×D₁₃

C₂×SD₁₆

yes

characteristic

C₂×#261

—

C₂×#261

—

C₂×C₄×C₁₃

#106

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₃

#106

yes

—

C₂×Dic₁₃

C₂×SD₁₆

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

yes

C₄×D₁₃

yes

C₄×D₁₃

C₂×D₈

yes

characteristic

C₄×D₁₃

C₂×Q₁₆

yes

characteristic

#1057

—

C₈×C₁₃

#107

yes

—

C₈×C₁₃

#117

yes

—

#1055

#107

yes

—

#1055

#125

yes

—

C₂×M₃₂

—

—

—

—

—

—

C₂×C₄×C₈

—

—

—

yes

characteristic, complemented

C₄×C₁₆

—

C₂²×C₁₃

C₂×#106

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

yes

C₂×D₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

yes

Dic₁₃

yes

100

C₂×M₁₆

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄²

—

105

M₃₂

—

106

M₃₂

—

107

C₂×C₁₆

—

108

C₂×C₁₆

C₂×#261

yes

—

109

#108

—

110

#108

—

111

#107

—

112

#106

—

113

C₄⋊C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

C₂×#261

yes

characteristic

118

#95

—

119

C₂×C₁₃

#1834

yes

characteristic

120

C₂×C₁₃

#2155

yes

—

121

D₁₃

—

122

D₁₃

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

M₁₆

—

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

C₄⋊#261

yes

characteristic

136

C₂×C₈

C₂²×#261

yes

characteristic

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

—

140

C₄²

—

141

C₄²

C₄⋊#261

yes

characteristic

142

C₁₆

—

143

C₁₆

—

144

C₁₃

#12887

yes

characteristic, complemented, Sylow

145

C₂³

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

104

—

154

C₂×C₄

—

104

—

155

C₂×C₄

#88101

yes

—

156

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

—

104

—

160

C₈

#88102

yes

—

161

C₈

#88237

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

C₂×#88101

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

208

—

171

C₄

#11719060

yes

characteristic

172

C₄

C₂×#88102

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#13130757

yes

characteristic

176

C₂

#13131184

yes

—

	subgroup	G/H
center	C₂²	C₂×#88101
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×#346	C₄
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂²×D₁₃	C₂×SD₁₆
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	#346	#261
hypocenter	C₁₃	#12887
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×#88101

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#261

Fitting

C₁₃×#346

C₄

socle

C₂²×C₁₃

C₂×#106

prime radical

C₂²×D₁₃

C₂×SD₁₆

cosocle

C₂×C₈×D₁₃

C₂³

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
13. C₁₃×#346	C₄	semidirect
14. C₁₆×Dic₁₃	C₄	semidirect
15. #11718053	C₄	semidirect
87. #346	#261	semidirect
144. C₁₃	#12887	semidirect

H ≅ G/N

type

13. C₁₃×#346

C₄

semidirect

14. C₁₆×Dic₁₃

C₄

semidirect

15. #11718053

C₄

semidirect

87. #346

#261

semidirect

144. C₁₃

#12887

semidirect

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

subgroup	index
#13130775	2
#13130773	2
#13129901	2
#13116429	2
#12887	13

subgroup

index

Subgroups — #16790228

Conjugacy classes of subgroups

Complements

Maximal subgroups