Group #28506760 — subgroups

Subgroups — #28506760

Order 15104.

262 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16837232	C₂	1	1	yes	maximal, characteristic
2	#21425692	C₂	1	1	yes	maximal, characteristic
3	#21424887	C₂	1	2	yes	maximal
4	#21422209	C₂	1	2	yes	maximal
5	C₂×C₅₉×#368	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	1	yes	characteristic
7	#16837451	—	2	2	no	—
8	C₂×#14273663	C₂²	1	2	yes	—
9	#16837357	C₂²	1	1	yes	characteristic
10	#16837232	—	2	1	no	—
11	#16837232	—	2	1	no	—
12	#16837177	—	2	1	no	—
13	C₁₆×Dic₅₉	—	2	1	no	—
14	#16837174	—	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	—	2	2	no	—
19	C₂×#11745664	D₄	1	1	yes	characteristic
20	#14273686	D₄	1	1	yes	characteristic
21	#14273667	—	2	1	no	—
22	#14273667	Q₈	1	2	yes	—
23	C₂×C₅₉⋊C₁₆	—	2	1	no	—
24	#14273663	—	2	2	no	—
25	#14273663	—	2	2	no	—
26	#14273663	—	4	4	no	—
27	#14273637	—	2	1	no	—
28	C₈×Dic₅₉	—	2	1	no	—
29	C₂×M₁₆×C₅₉	—	2	2	no	—
30	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
35	#108×C₅₉	—	2	2	no	—
36	#108×C₅₉	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₅₉	—	2	4	no	—
39	M₁₆×C₅₉	—	4	2	no	—
40	C₂×C₈×C₅₉	—	4	2	no	—
41	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
46	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
47	C₂²⋊Dic₅₉	Q₁₆	1	2	yes	—
48	C₄⋊Dic₅₉	—	2	1	no	—
49	C₄×Dic₅₉	—	2	1	no	—
50	#11745664	—	2	1	no	—
51	#11745664	—	4	2	no	—
52	#11745664	—	4	2	no	—
53	#11745664	D₈	1	2	yes	—
54	C₂×C₅₉⋊C₈	—	2	1	no	—
55	C₂×C₅₉⋊C₈	—	4	2	no	—
56	C₁₆×C₅₉	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₅₉	C₄⋊C₄	1	2	yes	—
58	C₅₉⋊C₁₆	—	2	2	no	—
59	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
66	C₂×Dic₅₉	—	2	2	no	—
67	C₈×C₅₉	—	4	4	no	—
68	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
70	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₈×C₅₉	#125	1	1	yes	characteristic
72	C₅₉⋊C₈	—	2	2	no	—
73	C₅₉⋊C₈	—	8	2	no	—
74	#13173	—	59	1	no	maximal, Sylow
75	C₂²×C₅₉	—	4	1	no	—
76	C₂²×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
77	C₂²×C₅₉	#467	1	1	yes	characteristic
78	C₂²×C₅₉	#500	1	1	yes	characteristic
79	C₄×C₅₉	#389	1	1	yes	characteristic
80	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
81	C₄×C₅₉	#466	1	1	yes	characteristic
82	C₄×C₅₉	#499	1	1	yes	characteristic
83	Dic₅₉	—	4	2	no	—
84	C₂×#368	—	59	1	no	—
85	C₂×#368	D₅₉	1	1	yes	characteristic
86	#2108	—	59	1	no	—
87	#1953	—	59	2	no	—
88	#1388	—	59	2	no	—
89	C₂×C₅₉	—	2	1	no	—
90	C₂×C₅₉	—	2	1	no	—
91	C₂×C₅₉	—	2	1	no	—
92	C₂×C₅₉	#1942	1	1	yes	characteristic
93	C₂×M₃₂	—	59	1	no	—
94	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
95	#430	—	118	2	no	—
96	C₂×#108	—	59	2	no	—
97	C₂×#108	C₂×D₅₉	1	2	yes	—
98	#405	—	59	1	no	—
99	#368	—	118	1	no	—
100	#368	—	118	1	no	—
101	#368	Dic₅₉	1	4	yes	complemented
102	#346	—	118	1	no	—
103	C₄×C₁₆	—	118	1	no	—
104	#344	—	118	2	no	—
105	C₅₉	#13173	1	1	yes	characteristic, complemented, Sylow
106	C₂×M₁₆	—	2	2	no	—
107	C₂×M₁₆	—	59	1	no	—
108	C₂×M₁₆	—	118	2	no	—
109	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
110	#117	—	59	1	no	—
111	M₃₂	—	59	2	no	—
112	M₃₂	—	118	1	no	—
113	C₂×C₁₆	—	118	1	no	—
114	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
115	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
116	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
117	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
118	#108	—	2	2	no	—
119	#108	—	118	2	no	—
120	#108	—	118	2	no	—
121	#108	—	236	4	no	—
122	#108	C₂×Dic₅₉	1	4	yes	—
123	#97	—	118	1	no	—
124	C₄×C₈	—	118	1	no	—
125	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	M₁₆	—	4	2	no	—
128	M₁₆	—	59	2	no	—
129	M₁₆	—	118	1	no	—
130	M₁₆	—	236	2	no	—
131	M₁₆	—	236	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	118	1	no	—
134	C₂×C₈	—	236	2	no	—
135	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
136	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
137	C₂×C₈	#11745687	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
139	C₂×C₈	#11745689	1	1	yes	characteristic
140	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
141	C₄⋊C₄	—	118	1	no	—
142	C₂²⋊C₄	—	59	2	no	—
143	C₄²	—	118	1	no	—
144	C₁₆	—	118	2	no	—
145	C₁₆	C₄⋊Dic₅₉	1	2	yes	—
146	C₁₆	C₄⋊Dic₅₉	1	2	yes	—
147	C₂³	Q₁₆×D₅₉	1	1	yes	characteristic
148	C₂×C₄	—	118	2	no	—
149	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
150	C₂×C₄	#14273702	1	1	yes	characteristic
151	C₂×C₄	#14273726	1	1	yes	characteristic
152	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
153	C₂×C₄	#14273746	1	1	yes	characteristic
154	C₂×C₄	#14273750	1	1	yes	characteristic
155	C₈	—	4	4	no	—
156	C₈	—	118	2	no	—
157	C₈	—	472	2	no	—
158	C₈	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
159	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
160	C₈	#14273707	1	1	yes	characteristic
161	C₈	#14273718	1	1	yes	characteristic
162	C₂²	—	4	1	no	—
163	C₂²	Q₁₆×Dic₅₉	1	1	yes	characteristic
164	C₂²	#16837644	1	1	yes	characteristic
165	C₂²	#16837806	1	1	yes	characteristic
166	C₄	—	236	2	no	—
167	C₄	#16837301	1	1	yes	characteristic
168	C₄	D₈×Dic₅₉	1	1	yes	characteristic
169	C₄	#16837639	1	1	yes	characteristic
170	C₄	#16837803	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#21424830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837232

C₂

yes

maximal, characteristic

#21425692

C₂

yes

maximal, characteristic

#21424887

C₂

yes

maximal

#21422209

C₂

yes

maximal

C₂×C₅₉×#368

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

characteristic

#16837451

—

C₂×#14273663

C₂²

yes

—

#16837357

C₂²

yes

—

—

—

C₁₆×Dic₅₉

—

#16837174

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#108×C₅₉

C₂²

yes

—

C₅₉×#368

C₄

yes

complemented

C₂×#11745664

—

C₂×#11745664

D₄

yes

characteristic

#14273686

D₄

yes

characteristic

#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

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

Q₁₆

yes

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

—

—

—

D₈

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

C₄⋊C₄

yes

—

C₁₆×C₅₉

C₄⋊C₄

yes

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×Q₁₆

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

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

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

—

C₅₉⋊C₈

—

#13173

—

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

—

C₂×#368

—

C₂×#368

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1942

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

#430

—

118

—

C₂×#108

—

C₂×#108

C₂×D₅₉

yes

—

#405

—

#368

—

118

—

100

#368

—

118

—

101

#368

Dic₅₉

yes

complemented

102

#346

—

118

—

103

C₄×C₁₆

—

118

—

104

#344

—

118

—

105

C₅₉

#13173

yes

characteristic, complemented, Sylow

106

C₂×M₁₆

—

107

C₂×M₁₆

—

108

C₂×M₁₆

—

118

—

109

C₂²×C₈

C₂²×D₅₉

yes

characteristic

110

#117

—

111

M₃₂

—

112

M₃₂

—

118

—

113

C₂×C₁₆

—

118

—

114

C₂×C₁₆

Dic₁₁₈

yes

—

115

C₂×C₁₆

D₂₃₆

yes

characteristic

116

C₂×C₁₆

D₂₃₆

yes

characteristic

117

C₂×C₁₆

C₂×Dic₅₉

yes

—

118

#108

—

119

#108

—

118

—

120

#108

—

118

—

121

#108

—

236

—

122

#108

C₂×Dic₅₉

yes

—

123

#97

—

118

—

124

C₄×C₈

—

118

—

125

C₂²×C₄

D₄×D₅₉

yes

characteristic

126

M₁₆

—

127

M₁₆

—

128

M₁₆

—

129

M₁₆

—

118

—

130

M₁₆

—

236

—

131

M₁₆

—

236

—

132

C₂×C₈

—

133

C₂×C₈

—

118

—

134

C₂×C₈

—

236

—

135

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

136

C₂×C₈

C₂×D₂₃₆

yes

characteristic

137

C₂×C₈

#11745687

yes

characteristic

138

C₂×C₈

Q₈×D₅₉

yes

characteristic

139

C₂×C₈

#11745689

yes

characteristic

140

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

141

C₄⋊C₄

—

118

—

142

C₂²⋊C₄

—

143

C₄²

—

118

—

144

C₁₆

—

118

—

145

C₁₆

C₄⋊Dic₅₉

yes

—

146

C₁₆

C₄⋊Dic₅₉

yes

—

147

C₂³

Q₁₆×D₅₉

yes

characteristic

148

C₂×C₄

—

118

—

149

C₂×C₄

D₄×Dic₅₉

yes

characteristic

150

C₂×C₄

#14273702

yes

characteristic

151

C₂×C₄

#14273726

yes

characteristic

152

C₂×C₄

D₈×D₅₉

yes

characteristic

153

C₂×C₄

#14273746

yes

characteristic

154

C₂×C₄

#14273750

yes

characteristic

155

C₈

—

156

C₈

—

118

—

157

C₈

—

472

—

158

C₈

C₂×C₄⋊Dic₅₉

yes

characteristic

159

C₈

Q₈×Dic₅₉

yes

characteristic

160

C₈

#14273707

yes

characteristic

161

C₈

#14273718

yes

characteristic

162

C₂²

—

163

C₂²

Q₁₆×Dic₅₉

yes

characteristic

164

C₂²

#16837644

yes

characteristic

165

C₂²

#16837806

yes

characteristic

166

C₄

—

236

—

167

C₄

#16837301

yes

characteristic

168

C₄

D₈×Dic₅₉

yes

characteristic

169

C₄

#16837639

yes

characteristic

170

C₄

#16837803

yes

characteristic

171

C₂

—

172

C₂

—

173

C₂

—

174

C₂

#21424830

yes

characteristic

	subgroup	G/H
center	C₄	#16837639
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#368	C₂
socle	C₂×C₅₉	#1942
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#368	D₅₉
hypocenter	C₅₉	#13173
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₄

#16837639

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#368

C₂

socle

C₂×C₅₉

#1942

prime radical

C₂³×C₅₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
17. C₅₉×#368	C₄	semidirect
101. #368	Dic₅₉	semidirect
105. C₅₉	#13173	semidirect

H ≅ G/N

type

17. C₅₉×#368

C₄

semidirect

101. #368

Dic₅₉

semidirect

105. C₅₉

#13173

semidirect

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

subgroup	index
C₂×#16837232	2
#21425692	2
#21424887	2
#21422209	2
C₂×C₅₉×#368	2
#13173	59

subgroup

index

C₂×C₅₉×#368

Subgroups — #28506760

Conjugacy classes of subgroups

Complements

Maximal subgroups