Group #28506755 — subgroups

Subgroups — #28506755

Order 15104.

274 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 123 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×#16837230	C₂	1	1	yes	maximal, characteristic
2	#21425692	C₂	1	1	yes	maximal, characteristic
3	#21424882	C₂	1	2	yes	maximal
4	#21422206	C₂	1	2	yes	maximal
5	C₂×C₅₉×#368	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	1	yes	characteristic
7	#16837444	—	2	2	no	—
8	C₂×#14273661	C₂²	1	2	yes	—
9	#16837357	C₂²	1	1	yes	characteristic
10	#16837230	—	2	1	no	—
11	#16837230	—	2	1	no	—
12	#16837177	—	2	1	no	—
13	C₁₆×Dic₅₉	—	2	1	no	—
14	#16837170	—	2	2	no	—
15	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×#108×C₅₉	C₂²	1	2	yes	—
17	C₅₉×#368	C₄	1	4	yes	complemented
18	C₂×#11745664	D₄	1	1	yes	characteristic
19	#14273686	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	#14273667	—	2	1	no	—
22	#14273667	Q₈	1	2	yes	—
23	C₂×C₅₉⋊C₁₆	—	2	1	no	—
24	#14273661	—	2	2	no	—
25	#14273661	—	2	2	no	—
26	#14273661	—	4	2	no	—
27	#14273659	—	4	2	no	—
28	#14273637	—	2	1	no	—
29	C₈×Dic₅₉	—	2	1	no	—
30	C₂×M₁₆×C₅₉	—	2	2	no	—
31	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
36	#108×C₅₉	—	2	2	no	—
37	#108×C₅₉	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₅₉	—	4	2	no	—
40	M₁₆×C₅₉	—	2	4	no	—
41	M₁₆×C₅₉	—	4	2	no	—
42	C₂×C₈×C₅₉	—	4	2	no	—
43	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
48	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
49	C₂²⋊Dic₅₉	—	2	1	no	—
50	C₄⋊Dic₅₉	—	4	2	no	—
51	C₄⋊Dic₅₉	—	4	2	no	—
52	C₄⋊Dic₅₉	D₈	1	2	yes	—
53	C₄×Dic₅₉	—	2	1	no	—
54	#11745664	—	2	1	no	—
55	#11745664	Q₁₆	1	2	yes	—
56	C₂×C₅₉⋊C₈	—	2	1	no	—
57	C₁₆×C₅₉	C₄⋊C₄	1	2	yes	—
58	C₁₆×C₅₉	C₄⋊C₄	1	2	yes	—
59	C₅₉⋊C₁₆	—	2	2	no	—
60	C₂³×C₅₉	#135	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
67	C₂×Dic₅₉	—	2	2	no	—
68	C₂×Dic₅₉	—	8	2	no	—
69	C₂×Dic₅₉	—	16	2	no	—
70	C₈×C₅₉	—	4	4	no	—
71	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
74	C₈×C₅₉	#125	1	1	yes	characteristic
75	C₅₉⋊C₈	—	2	2	no	—
76	#13172	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	—	2	2	no	—
78	C₂²×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
79	C₂²×C₅₉	#466	1	1	yes	characteristic
80	C₂²×C₅₉	#499	1	1	yes	characteristic
81	C₄×C₅₉	#389	1	1	yes	characteristic
82	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
83	C₄×C₅₉	#467	1	1	yes	characteristic
84	C₄×C₅₉	#500	1	1	yes	characteristic
85	Dic₅₉	—	4	2	no	—
86	Dic₅₉	—	16	2	no	—
87	C₂×#368	D₅₉	1	1	yes	characteristic
88	C₂×#367	—	59	1	no	—
89	#2108	—	59	1	no	—
90	#1952	—	59	2	no	—
91	#1387	—	59	2	no	—
92	C₂×C₅₉	—	2	1	no	—
93	C₂×C₅₉	—	2	1	no	—
94	C₂×C₅₉	#1942	1	1	yes	characteristic
95	C₂×C₅₉	#2223	1	2	yes	—
96	C₂×M₃₂	—	59	1	no	—
97	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
98	C₂×#108	C₂×D₅₉	1	2	yes	—
99	#428	—	118	2	no	—
100	C₂×#107	—	59	2	no	—
101	#405	—	59	1	no	—
102	#368	Dic₅₉	1	4	yes	complemented
103	#367	—	118	1	no	—
104	#367	—	118	1	no	—
105	#346	—	118	1	no	—
106	C₄×C₁₆	—	118	1	no	—
107	#343	—	118	2	no	—
108	C₅₉	#13172	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	59	1	no	—
111	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
112	#117	—	59	1	no	—
113	C₂×C₄⋊C₄	—	118	2	no	—
114	M₃₂	—	59	2	no	—
115	M₃₂	—	118	1	no	—
116	C₂×C₁₆	—	118	1	no	—
117	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
118	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
119	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
120	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
121	#108	—	2	2	no	—
122	#108	C₂×Dic₅₉	1	4	yes	—
123	#107	—	118	2	no	—
124	#107	—	118	2	no	—
125	#107	—	236	2	no	—
126	#106	—	236	2	no	—
127	#97	—	118	1	no	—
128	C₄×C₈	—	118	1	no	—
129	C₂²×C₄	—	236	2	no	—
130	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
131	M₁₆	—	2	4	no	—
132	M₁₆	—	4	2	no	—
133	M₁₆	—	59	2	no	—
134	M₁₆	—	118	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	118	1	no	—
137	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
138	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
139	C₂×C₈	#11745687	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
141	C₂×C₈	#11745689	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
143	C₄⋊C₄	—	59	2	no	—
144	C₄⋊C₄	—	236	2	no	—
145	C₄⋊C₄	—	236	2	no	—
146	C₂²⋊C₄	—	118	1	no	—
147	C₄²	—	118	1	no	—
148	C₁₆	—	118	2	no	—
149	C₁₆	C₄⋊Dic₅₉	1	2	yes	—
150	C₁₆	C₄⋊Dic₅₉	1	2	yes	—
151	C₂³	#14273750	1	1	yes	characteristic
152	C₂×C₄	—	118	2	no	—
153	C₂×C₄	—	944	2	no	—
154	C₂×C₄	—	472	2	no	—
155	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
156	C₂×C₄	#14273702	1	1	yes	characteristic
157	C₂×C₄	#14273726	1	1	yes	characteristic
158	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
159	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
160	C₂×C₄	#14273746	1	1	yes	characteristic
161	C₈	—	4	4	no	—
162	C₈	—	118	2	no	—
163	C₈	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
164	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
165	C₈	#14273707	1	1	yes	characteristic
166	C₈	#14273718	1	1	yes	characteristic
167	C₂²	—	2	2	no	—
168	C₂²	Q₁₆×Dic₅₉	1	1	yes	characteristic
169	C₂²	#16837639	1	1	yes	characteristic
170	C₂²	#16837803	1	1	yes	characteristic
171	C₄	—	944	2	no	—
172	C₄	—	236	2	no	—
173	C₄	#16837301	1	1	yes	characteristic
174	C₄	D₈×Dic₅₉	1	1	yes	characteristic
175	C₄	#16837644	1	1	yes	characteristic
176	C₄	#16837806	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#21424830	1	1	yes	characteristic
180	C₂	#21426227	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837230

C₂

yes

maximal, characteristic

#21425692

C₂

yes

maximal, characteristic

#21424882

C₂

yes

maximal

#21422206

C₂

yes

maximal

C₂×C₅₉×#368

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

characteristic

#16837444

—

C₂×#14273661

C₂²

yes

—

#16837357

C₂²

yes

—

—

—

C₁₆×Dic₅₉

—

#16837170

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#108×C₅₉

C₂²

yes

—

C₅₉×#368

C₄

yes

complemented

C₂×#11745664

D₄

yes

characteristic

#14273686

D₄

yes

characteristic

C₂×C₄⋊Dic₅₉

—

#14273667

—

#14273667

Q₈

yes

—

C₂×C₅₉⋊C₁₆

—

—

—

—

—

—

C₈×Dic₅₉

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

Q₈

yes

—

#108×C₅₉

—

#108×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

C₂×C₈×C₅₉

yes

C₂×C₈×C₅₉

yes

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₈

yes

—

C₄×Dic₅₉

—

#11745664

—

#11745664

Q₁₆

yes

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

C₄⋊C₄

yes

—

C₁₆×C₅₉

C₄⋊C₄

yes

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

#135

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₉

C₄×Q₈

yes

characteristic

C₈×C₅₉

C₂²⋊Q₈

yes

characteristic

C₈×C₅₉

#125

yes

characteristic

C₅₉⋊C₈

—

#13172

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₄×Q₁₆

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₄×D₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

C₂×#368

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1942

yes

characteristic

C₂×C₅₉

#2223

yes

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

C₂×#108

C₂×D₅₉

yes

—

#428

—

118

—

100

C₂×#107

—

101

#405

—

102

#368

Dic₅₉

yes

complemented

103

#367

—

118

—

104

#367

—

118

—

105

#346

—

118

—

106

C₄×C₁₆

—

118

—

107

#343

—

118

—

108

C₅₉

#13172

yes

characteristic, complemented, Sylow

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂²×C₈

C₂²×D₅₉

yes

characteristic

112

#117

—

113

C₂×C₄⋊C₄

—

118

—

114

M₃₂

—

115

M₃₂

—

118

—

116

C₂×C₁₆

—

118

—

117

C₂×C₁₆

Dic₁₁₈

yes

—

118

C₂×C₁₆

D₂₃₆

yes

characteristic

119

C₂×C₁₆

D₂₃₆

yes

characteristic

120

C₂×C₁₆

C₂×Dic₅₉

yes

—

121

#108

—

122

#108

C₂×Dic₅₉

yes

—

123

#107

—

118

—

124

#107

—

118

—

125

#107

—

236

—

126

#106

—

236

—

127

#97

—

118

—

128

C₄×C₈

—

118

—

129

C₂²×C₄

—

236

—

130

C₂²×C₄

D₄×D₅₉

yes

characteristic

131

M₁₆

—

132

M₁₆

—

133

M₁₆

—

134

M₁₆

—

118

—

135

C₂×C₈

—

136

C₂×C₈

—

118

—

137

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

138

C₂×C₈

C₂×D₂₃₆

yes

characteristic

139

C₂×C₈

#11745687

yes

characteristic

140

C₂×C₈

Q₈×D₅₉

yes

characteristic

141

C₂×C₈

#11745689

yes

characteristic

142

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

236

—

145

C₄⋊C₄

—

236

—

146

C₂²⋊C₄

—

118

—

147

C₄²

—

118

—

148

C₁₆

—

118

—

149

C₁₆

C₄⋊Dic₅₉

yes

—

150

C₁₆

C₄⋊Dic₅₉

yes

—

151

C₂³

#14273750

yes

characteristic

152

C₂×C₄

—

118

—

153

C₂×C₄

—

944

—

154

C₂×C₄

—

472

—

155

C₂×C₄

D₄×Dic₅₉

yes

characteristic

156

C₂×C₄

#14273702

yes

characteristic

157

C₂×C₄

#14273726

yes

characteristic

158

C₂×C₄

D₈×D₅₉

yes

characteristic

159

C₂×C₄

Q₁₆×D₅₉

yes

characteristic

160

C₂×C₄

#14273746

yes

characteristic

161

C₈

—

162

C₈

—

118

—

163

C₈

C₂×C₄⋊Dic₅₉

yes

characteristic

164

C₈

Q₈×Dic₅₉

yes

characteristic

165

C₈

#14273707

yes

characteristic

166

C₈

#14273718

yes

characteristic

167

C₂²

—

168

C₂²

Q₁₆×Dic₅₉

yes

characteristic

169

C₂²

#16837639

yes

characteristic

170

C₂²

#16837803

yes

characteristic

171

C₄

—

944

—

172

C₄

—

236

—

173

C₄

#16837301

yes

characteristic

174

C₄

D₈×Dic₅₉

yes

characteristic

175

C₄

#16837644

yes

characteristic

176

C₄

#16837806

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#21424830

yes

characteristic

180

C₂

#21426227

yes

—

	subgroup	G/H
center	C₂²	#16837639
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#368	C₂
socle	C₂²×C₅₉	#466
prime radical	C₂³×C₅₉	#135
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#368	D₅₉
hypocenter	C₅₉	#13172
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂²

#16837639

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#368

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
17. C₅₉×#368	C₄	semidirect
102. #368	Dic₅₉	semidirect
108. C₅₉	#13172	semidirect

H ≅ G/N

type

17. C₅₉×#368

C₄

semidirect

102. #368

Dic₅₉

semidirect

108. C₅₉

#13172

semidirect

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

subgroup	index
C₂×#16837230	2
#21425692	2
#21424882	2
#21422206	2
C₂×C₅₉×#368	2
#13172	59

subgroup

index

C₂×C₅₉×#368

Subgroups — #28506755

Conjugacy classes of subgroups

Complements

Maximal subgroups