Group #28508338 — subgroups

Subgroups — #28508338

Order 15104.

356 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#16837800	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837225	C₂	1	2	yes	maximal
3	#21424844	C₂	1	1	yes	maximal, characteristic
4	#21422202	C₂	1	2	yes	maximal
5	C₅₉×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273727	—	2	1	no	—
7	#16837800	—	2	4	no	—
8	C₂×#14273663	C₂²	1	1	yes	characteristic
9	#16837440	C₂²	1	1	yes	characteristic
10	C₂×#14273659	—	2	1	no	—
11	C₂×#14273659	C₂²	1	2	yes	—
12	#16837428	—	2	1	no	—
13	#16837225	—	2	4	no	—
14	#16837162	—	2	1	no	—
15	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₁₈	—	4	1	no	—
18	C₂×#11745664	—	2	1	no	—
19	#14273727	—	2	4	no	—
20	#14273727	—	4	2	no	—
21	#14273686	—	2	1	no	—
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	C₂×C₄⋊Dic₅₉	—	4	1	no	—
24	#14273663	—	2	2	no	—
25	#14273661	Q₈	1	2	yes	complemented
26	#14273659	—	2	4	no	—
27	#14273659	—	2	2	no	—
28	#14273659	—	2	1	no	—
29	#14273659	D₄	1	4	yes	—
30	Dic₅₉⋊C₈	—	4	2	no	—
31	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
34	M₃₂×C₅₉	—	2	2	no	—
35	M₃₂×C₅₉	Q₈	1	4	yes	complemented
36	C₂×C₁₆×C₅₉	—	2	2	no	—
37	C₄×C₈×C₅₉	—	2	2	no	—
38	C₂²×C₄×C₅₉	—	2	1	no	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₉	—	4	2	no	—
41	C₂×Dic₁₁₈	—	4	4	no	—
42	C₂×Dic₁₁₈	—	8	4	no	—
43	C₂×C₈×C₅₉	—	2	1	no	—
44	C₂×C₈×C₅₉	—	2	1	no	—
45	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
46	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
50	C₄²×C₅₉	—	2	1	no	—
51	C₄²×C₅₉	SD₁₆	1	2	yes	—
52	C₂²⋊Dic₅₉	—	4	1	no	—
53	C₄⋊Dic₅₉	—	2	4	no	—
54	C₄⋊Dic₅₉	—	2	2	no	—
55	C₄⋊Dic₅₉	—	4	4	no	—
56	C₄⋊Dic₅₉	—	4	2	no	—
57	C₄×Dic₅₉	—	4	1	no	—
58	#11745664	—	4	1	no	—
59	#11745664	—	4	1	no	—
60	C₂×C₅₉⋊C₈	—	4	1	no	—
61	C₁₆×C₅₉	—	2	4	no	—
62	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	4	no	—
64	C₂×C₄×C₅₉	—	2	2	no	—
65	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
71	C₂×Dic₅₉	—	4	4	no	—
72	C₂×Dic₅₉	—	4	2	no	—
73	C₂×Dic₅₉	—	8	4	no	—
74	Dic₁₁₈	—	8	8	no	—
75	C₈×C₅₉	—	2	2	no	—
76	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
78	C₅₉⋊C₈	—	8	1	no	—
79	#13457	—	59	1	no	maximal, Sylow
80	C₂²×C₅₉	—	2	2	no	—
81	C₂²×C₅₉	#450	1	1	yes	characteristic
82	C₂²×C₅₉	#476	1	1	yes	characteristic
83	C₂²×C₅₉	#501	1	1	yes	characteristic
84	C₄×C₅₉	—	2	4	no	—
85	C₄×C₅₉	#395	1	1	yes	characteristic
86	C₄×C₅₉	#448	1	1	yes	characteristic
87	C₄×C₅₉	#475	1	1	yes	characteristic
88	C₄×C₅₉	#498	1	1	yes	characteristic
89	Dic₅₉	—	8	4	no	—
90	Dic₅₉	—	8	2	no	—
91	C₂×#498	—	59	1	no	—
92	C₂×#365	—	59	2	no	—
93	#1945	—	59	1	no	—
94	#1386	—	59	2	no	—
95	#1376	D₅₉	1	1	yes	characteristic
96	C₂×C₅₉	—	2	1	no	—
97	C₂×C₅₉	—	2	1	no	—
98	C₂×C₅₉	#2058	1	1	yes	characteristic
99	C₂×C₅₉	#2230	1	2	yes	—
100	C₂×C₄⋊Q₈	—	118	1	no	—
101	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
102	#498	—	118	4	no	—
103	C₂×#108	—	59	1	no	—
104	#427	—	59	1	no	—
105	C₂×#106	—	59	2	no	—
106	C₂×#106	—	118	1	no	—
107	#424	—	118	1	no	—
108	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	#365	—	118	4	no	—
110	#339	—	118	1	no	—
111	C₅₉	#13457	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	236	1	no	—
113	C₂×M₁₆	—	118	1	no	—
114	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
116	C₄⋊Q₈	—	118	4	no	—
117	C₄⋊Q₈	—	236	2	no	—
118	#117	—	118	1	no	—
119	C₂×C₄⋊C₄	—	118	2	no	—
120	C₂×C₄⋊C₄	—	236	1	no	—
121	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	Dic₁₁₈	1	4	yes	complemented
124	C₂×C₁₆	—	2	2	no	—
125	#108	—	118	2	no	—
126	#107	—	59	2	no	—
127	#106	—	59	4	no	—
128	#106	—	118	4	no	—
129	#106	—	118	2	no	—
130	#106	—	118	1	no	—
131	C₄⋊C₈	—	236	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₂×Q₈	—	472	4	no	—
134	C₂×Q₈	—	236	4	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	236	2	no	—
137	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
138	M₁₆	—	236	1	no	—
139	M₁₆	—	236	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	236	1	no	—
143	C₂×C₈	#11745660	1	2	yes	—
144	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
145	C₂×C₈	D₄×D₅₉	1	2	yes	—
146	C₂×C₈	#11745687	1	1	yes	characteristic
147	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
148	C₄⋊C₄	—	118	4	no	—
149	C₄⋊C₄	—	118	2	no	—
150	C₄⋊C₄	—	236	4	no	—
151	C₄⋊C₄	—	236	2	no	—
152	C₂²⋊C₄	—	236	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	236	1	no	—
155	C₄²	#11745660	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	#14273755	1	1	yes	characteristic
158	Q₈	—	472	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	472	4	no	—
162	C₂×C₄	—	236	2	no	—
163	C₂×C₄	—	236	4	no	—
164	C₂×C₄	#14273697	1	1	yes	characteristic
165	C₂×C₄	#14273707	1	1	yes	characteristic
166	C₂×C₄	#14273727	1	1	yes	characteristic
167	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
168	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
169	C₂×C₄	#14273760	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	472	1	no	—
172	C₈	#14273704	1	2	yes	—
173	C₈	#14273726	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	#16837555	1	1	yes	characteristic
176	C₂²	#16837693	1	1	yes	characteristic
177	C₂²	#16837811	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	472	4	no	—
180	C₄	—	472	2	no	—
181	C₄	#16837323	1	1	yes	characteristic
182	C₄	#16837543	1	1	yes	characteristic
183	C₄	#16837688	1	1	yes	characteristic
184	C₄	#16837800	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#21425430	1	1	yes	characteristic
188	C₂	#21426268	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837800

C₂

yes

maximal, characteristic

C₂×#16837225

C₂

yes

maximal

#21424844

C₂

yes

maximal, characteristic

#21422202

C₂

yes

maximal

C₅₉×#1376

C₂

yes

maximal, characteristic

C₂×#14273727

—

#16837800

—

C₂×#14273663

C₂²

yes

characteristic

#16837440

C₂²

yes

characteristic

C₂×#14273659

—

C₂×#14273659

C₂²

yes

—

#16837428

—

#16837225

—

#16837162

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂²×Dic₁₁₈

—

—

—

—

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273663

—

#14273661

Q₈

yes

—

—

—

D₄

yes

—

Dic₅₉⋊C₈

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

M₃₂×C₅₉

—

M₃₂×C₅₉

Q₈

yes

complemented

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

SD₁₆

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₂³×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#120

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

#137

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂²⋊Q₈

yes

—

C₈×C₅₉

C₄⋊Q₈

yes

—

C₅₉⋊C₈

—

#13457

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#2058

yes

characteristic

C₂×C₅₉

#2230

yes

—

100

C₂×C₄⋊Q₈

—

118

—

101

C₂×M₃₂

C₂×D₅₉

yes

—

102

#498

—

118

—

103

C₂×#108

—

104

#427

—

105

C₂×#106

—

106

C₂×#106

—

118

—

107

#424

—

118

—

108

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

109

#365

—

118

—

110

#339

—

118

—

111

C₅₉

#13457

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

236

—

113

C₂×M₁₆

—

118

—

114

C₂²×C₈

D₂₃₆

yes

characteristic

115

C₂²×C₈

C₂²×D₅₉

yes

characteristic

116

C₄⋊Q₈

—

118

—

117

C₄⋊Q₈

—

236

—

118

#117

—

118

—

119

C₂×C₄⋊C₄

—

118

—

120

C₂×C₄⋊C₄

—

236

—

121

C₂×C₄²

D₂₃₆

yes

characteristic

122

M₃₂

—

123

M₃₂

Dic₁₁₈

yes

complemented

124

C₂×C₁₆

—

125

#108

—

118

—

126

#107

—

127

#106

—

128

#106

—

118

—

129

#106

—

118

—

130

#106

—

118

—

131

C₄⋊C₈

—

236

—

132

C₄×C₈

—

133

C₂×Q₈

—

472

—

134

C₂×Q₈

—

236

—

135

C₂²×C₄

—

136

C₂²×C₄

—

236

—

137

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

138

M₁₆

—

236

—

139

M₁₆

—

236

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

236

—

143

C₂×C₈

#11745660

yes

—

144

C₂×C₈

C₂×Dic₁₁₈

yes

—

145

C₂×C₈

D₄×D₅₉

yes

—

146

C₂×C₈

#11745687

yes

characteristic

147

C₂×C₈

Q₈×D₅₉

yes

characteristic

148

C₄⋊C₄

—

118

—

149

C₄⋊C₄

—

118

—

150

C₄⋊C₄

—

236

—

151

C₄⋊C₄

—

236

—

152

C₂²⋊C₄

—

236

—

153

C₄²

—

154

C₄²

—

236

—

155

C₄²

#11745660

yes

—

156

C₁₆

—

157

C₂³

#14273755

yes

characteristic

158

Q₈

—

472

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

472

—

162

C₂×C₄

—

236

—

163

C₂×C₄

—

236

—

164

C₂×C₄

#14273697

yes

characteristic

165

C₂×C₄

#14273707

yes

characteristic

166

C₂×C₄

#14273727

yes

characteristic

167

C₂×C₄

C₂×#11745660

yes

characteristic

168

C₂×C₄

C₂×#11745660

yes

characteristic

169

C₂×C₄

#14273760

yes

characteristic

170

C₈

—

171

C₈

—

472

—

172

C₈

#14273704

yes

—

173

C₈

#14273726

yes

—

174

C₂²

—

175

C₂²

#16837555

yes

characteristic

176

C₂²

#16837693

yes

characteristic

177

C₂²

#16837811

yes

characteristic

178

C₄

—

179

C₄

—

472

—

180

C₄

—

472

—

181

C₄

#16837323

yes

characteristic

182

C₄

#16837543

yes

characteristic

183

C₄

#16837688

yes

characteristic

184

C₄

#16837800

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#21425430

yes

characteristic

188

C₂

#21426268

yes

—

	subgroup	G/H
center	C₂²	#16837693
commutator, cosocle	C₂²×C₈×C₅₉	C₂³
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1376	C₂
socle	C₂²×C₅₉	#476
prime radical	C₂³×C₅₉	AGL(1,ℤ/8)
hypercenter	#1376	D₅₉
hypocenter	C₅₉	#13457
LCS	G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#16837693

commutator, cosocle

C₂²×C₈×C₅₉

C₂³

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1376

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
25. #14273661	Q₈	semidirect
35. M₃₂×C₅₉	Q₈	semidirect
111. C₅₉	#13457	semidirect
123. M₃₂	Dic₁₁₈	semidirect

H ≅ G/N

type

25. #14273661

Q₈

semidirect

35. M₃₂×C₅₉

Q₈

semidirect

111. C₅₉

#13457

semidirect

123. M₃₂

Dic₁₁₈

semidirect

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

subgroup	index
C₂×#16837800	2
C₂×#16837225	2
#21424844	2
#21422202	2
C₅₉×#1376	2
#13457	59

subgroup

index

C₅₉×#1376

Subgroups — #28508338

Conjugacy classes of subgroups

Complements

Maximal subgroups