Group #28482478 — subgroups

Subgroups — #28482478

Order 15104.

438 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 227 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	#21424256	C₂	1	2	yes	maximal
2	#21424256	C₂	1	1	yes	maximal, characteristic
3	#21424253	C₂	1	1	yes	maximal, characteristic
4	#21424224	C₂	1	2	yes	maximal
5	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273661	—	2	4	no	—
7	C₂×#14273661	—	2	2	no	—
8	C₂×#14273659	—	2	2	no	—
9	#16837272	C₂²	1	1	yes	characteristic
10	#16837272	C₂²	1	2	yes	—
11	#16837272	C₂²	1	1	yes	characteristic
12	#16837165	—	2	4	no	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#322	C₄	1	4	yes	complemented
16	C₂×C₄⋊Dic₅₉	—	2	2	no	—
17	C₂×C₄⋊Dic₅₉	—	2	2	no	—
18	C₂×C₄⋊Dic₅₉	—	2	1	no	—
19	C₂×C₄⋊Dic₅₉	—	2	1	no	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	2	2	no	—
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	#14273661	—	4	8	no	—
24	#14273661	—	4	4	no	—
25	#14273659	—	4	4	no	—
26	C₂²×C₈×C₅₉	D₄	1	4	yes	—
27	C₂²×C₈×C₅₉	D₄	1	2	yes	—
28	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₉	Q₈	1	4	yes	—
32	C₄×C₈×C₅₉	Q₈	1	2	yes	—
33	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
35	C₂²×Dic₅₉	—	4	2	no	—
36	C₂²×Dic₅₉	—	4	1	no	—
37	C₂²×Dic₅₉	—	4	1	no	—
38	C₂×C₈×C₅₉	—	2	4	no	—
39	C₂×C₈×C₅₉	—	2	4	no	—
40	C₂×C₈×C₅₉	C₄⋊C₄	1	8	yes	—
41	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
42	C₂×C₈×C₅₉	D₈	1	4	yes	—
43	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
44	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
47	C₄⋊Dic₅₉	—	4	2	no	—
48	C₄⋊Dic₅₉	—	4	2	no	—
49	C₄⋊Dic₅₉	—	4	4	no	—
50	C₄⋊Dic₅₉	—	4	4	no	—
51	C₄⋊Dic₅₉	—	4	4	no	—
52	C₄⋊Dic₅₉	—	4	4	no	—
53	C₄⋊Dic₅₉	—	4	4	no	—
54	C₂³×C₅₉	#127	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
60	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
61	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
62	C₂×C₄×C₅₉	#137	1	4	yes	—
63	C₂×Dic₅₉	—	8	1	no	—
64	C₂×Dic₅₉	—	8	2	no	—
65	C₂×Dic₅₉	—	8	1	no	—
66	C₂×Dic₅₉	—	8	2	no	—
67	C₂×Dic₅₉	—	8	2	no	—
68	C₂×Dic₅₉	—	8	2	no	—
69	C₂×Dic₅₉	—	8	2	no	—
70	C₈×C₅₉	—	2	8	no	—
71	C₈×C₅₉	#107	1	8	yes	complemented
72	#8820	—	59	1	no	maximal, Sylow
73	C₂²×C₅₉	#384	1	1	yes	characteristic
74	C₂²×C₅₉	#493	1	1	yes	characteristic
75	C₂²×C₅₉	#494	1	1	yes	characteristic
76	C₂²×C₅₉	#496	1	2	yes	—
77	C₂²×C₅₉	#497	1	2	yes	—
78	C₄×C₅₉	C₂×#107	1	2	yes	—
79	C₄×C₅₉	#428	1	4	yes	—
80	C₄×C₅₉	#500	1	2	yes	—
81	C₄×C₅₉	#501	1	4	yes	—
82	Dic₅₉	—	16	1	no	—
83	Dic₅₉	—	16	2	no	—
84	Dic₅₉	—	16	1	no	—
85	#1845	—	59	1	no	—
86	#1845	—	59	2	no	—
87	#1844	—	59	1	no	—
88	#1834	—	59	2	no	—
89	C₂×#322	D₅₉	1	1	yes	characteristic
90	C₂×C₅₉	#1721	1	1	yes	characteristic
91	C₂×C₅₉	#1722	1	1	yes	characteristic
92	C₂×C₅₉	#1723	1	2	yes	—
93	C₂×C₅₉	#1845	1	1	yes	characteristic
94	C₂×C₅₉	#1847	1	2	yes	—
95	C₂×#107	—	118	4	no	—
96	C₂×#107	—	118	2	no	—
97	C₂×#106	—	118	2	no	—
98	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
99	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
100	#384	—	59	1	no	—
101	#384	—	59	2	no	—
102	#384	—	59	1	no	—
103	#340	—	118	4	no	—
104	#322	Dic₅₉	1	4	yes	complemented
105	C₅₉	#8820	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	D₂₃₆	1	4	yes	—
107	C₂²×C₈	D₂₃₆	1	2	yes	—
108	C₂×C₄⋊C₄	—	118	2	no	—
109	C₂×C₄⋊C₄	—	118	2	no	—
110	C₂×C₄⋊C₄	—	118	1	no	—
111	C₂×C₄⋊C₄	—	118	1	no	—
112	C₂×C₄⋊C₄	—	118	2	no	—
113	C₂×C₄⋊C₄	—	118	2	no	—
114	C₂×C₄⋊C₄	—	118	2	no	—
115	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
116	#107	—	236	8	no	—
117	#107	—	236	4	no	—
118	#106	—	236	4	no	—
119	C₄×C₈	Dic₁₁₈	1	4	yes	—
120	C₄×C₈	Dic₁₁₈	1	2	yes	—
121	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
122	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
123	C₂²×C₄	—	236	2	no	—
124	C₂²×C₄	—	236	1	no	—
125	C₂²×C₄	—	236	1	no	—
126	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
127	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	D₄₇₂	1	4	yes	—
131	C₂×C₈	Dic₂₃₆	1	4	yes	—
132	C₂×C₈	C₄⋊Dic₅₉	1	8	yes	—
133	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
134	C₄⋊C₄	—	236	2	no	—
135	C₄⋊C₄	—	236	2	no	—
136	C₄⋊C₄	—	236	4	no	—
137	C₄⋊C₄	—	236	4	no	—
138	C₄⋊C₄	—	236	4	no	—
139	C₄⋊C₄	—	236	4	no	—
140	C₄⋊C₄	—	236	4	no	—
141	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
142	C₄²	C₂×Dic₁₁₈	1	2	yes	—
143	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
144	C₂³	#14273724	1	1	yes	characteristic
145	C₂×C₄	—	472	1	no	—
146	C₂×C₄	—	472	2	no	—
147	C₂×C₄	—	472	1	no	—
148	C₂×C₄	—	472	2	no	—
149	C₂×C₄	—	472	2	no	—
150	C₂×C₄	—	472	2	no	—
151	C₂×C₄	—	472	2	no	—
152	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
154	C₂×C₄	#14273727	1	1	yes	characteristic
155	C₂×C₄	#14273727	1	2	yes	—
156	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
157	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
158	C₂×C₄	#14273755	1	4	yes	—
159	C₂×C₄	#14273760	1	4	yes	—
160	C₈	—	2	8	no	—
161	C₈	#14273661	1	8	yes	complemented
162	C₂²	#16837272	1	1	yes	characteristic
163	C₂²	#16837779	1	1	yes	characteristic
164	C₂²	#16837782	1	1	yes	characteristic
165	C₂²	#16837791	1	2	yes	—
166	C₂²	#16837796	1	2	yes	—
167	C₄	—	944	1	no	—
168	C₄	—	944	2	no	—
169	C₄	—	944	1	no	—
170	C₄	C₂×#14273661	1	2	yes	—
171	C₄	#16837444	1	4	yes	—
172	C₄	#16837807	1	2	yes	—
173	C₄	#16837811	1	4	yes	—
174	C₂	#21423762	1	1	yes	characteristic
175	C₂	#21423767	1	1	yes	characteristic
176	C₂	#21423773	1	2	yes	—
177	C₂	#21424256	1	1	yes	characteristic
178	C₂	#21424264	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424256

C₂

yes

maximal

#21424256

C₂

yes

maximal, characteristic

#21424253

C₂

yes

maximal, characteristic

#21424224

C₂

yes

maximal

C₂×C₅₉×#322

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

#16837272

C₂²

yes

—

#16837272

C₂²

yes

characteristic

#16837165

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

C₄

yes

complemented

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273661

—

#14273661

—

#14273659

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

yes

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#497

yes

—

C₄×C₅₉

C₂×#107

yes

—

C₄×C₅₉

#428

yes

—

C₄×C₅₉

#500

yes

—

C₄×C₅₉

#501

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

—

—

—

D₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#1845

yes

characteristic

C₂×C₅₉

#1847

yes

—

C₂×#107

—

118

—

C₂×#107

—

118

—

C₂×#106

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

100

#384

—

101

#384

—

102

#384

—

103

#340

—

118

—

104

#322

Dic₅₉

yes

complemented

105

C₅₉

#8820

yes

characteristic, complemented, Sylow

106

C₂²×C₈

D₂₃₆

yes

—

107

C₂²×C₈

D₂₃₆

yes

—

108

C₂×C₄⋊C₄

—

118

—

109

C₂×C₄⋊C₄

—

118

—

110

C₂×C₄⋊C₄

—

118

—

111

C₂×C₄⋊C₄

—

118

—

112

C₂×C₄⋊C₄

—

118

—

113

C₂×C₄⋊C₄

—

118

—

114

C₂×C₄⋊C₄

—

118

—

115

C₂×C₄²

C₂²×D₅₉

yes

characteristic

116

#107

—

236

—

117

#107

—

236

—

118

#106

—

236

—

119

C₄×C₈

Dic₁₁₈

yes

—

120

C₄×C₈

Dic₁₁₈

yes

—

121

C₄×C₈

C₂×Dic₅₉

yes

—

122

C₄×C₈

C₂×Dic₅₉

yes

—

123

C₂²×C₄

—

236

—

124

C₂²×C₄

—

236

—

125

C₂²×C₄

—

236

—

126

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

127

C₂²×C₄

C₂×D₂₃₆

yes

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

D₄₇₂

yes

—

131

C₂×C₈

Dic₂₃₆

yes

—

132

C₂×C₈

C₄⋊Dic₅₉

yes

—

133

C₂×C₈

C₄⋊Dic₅₉

yes

—

134

C₄⋊C₄

—

236

—

135

C₄⋊C₄

—

236

—

136

C₄⋊C₄

—

236

—

137

C₄⋊C₄

—

236

—

138

C₄⋊C₄

—

236

—

139

C₄⋊C₄

—

236

—

140

C₄⋊C₄

—

236

—

141

C₄²

C₂×Dic₁₁₈

yes

characteristic

142

C₄²

C₂×Dic₁₁₈

yes

—

143

C₄²

C₂²×Dic₅₉

yes

characteristic

144

C₂³

#14273724

yes

characteristic

145

C₂×C₄

—

472

—

146

C₂×C₄

—

472

—

147

C₂×C₄

—

472

—

148

C₂×C₄

—

472

—

149

C₂×C₄

—

472

—

150

C₂×C₄

—

472

—

151

C₂×C₄

—

472

—

152

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

153

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

—

154

C₂×C₄

#14273727

yes

characteristic

155

C₂×C₄

#14273727

yes

—

156

C₂×C₄

C₂×D₄₇₂

yes

—

157

C₂×C₄

C₂×Dic₂₃₆

yes

—

158

C₂×C₄

#14273755

yes

—

159

C₂×C₄

#14273760

yes

—

160

C₈

—

161

C₈

#14273661

yes

complemented

162

C₂²

#16837272

yes

characteristic

163

C₂²

#16837779

yes

characteristic

164

C₂²

#16837782

yes

characteristic

165

C₂²

#16837791

yes

—

166

C₂²

#16837796

yes

—

167

C₄

—

944

—

168

C₄

—

944

—

169

C₄

—

944

—

170

C₄

C₂×#14273661

yes

—

171

C₄

#16837444

yes

—

172

C₄

#16837807

yes

—

173

C₄

#16837811

yes

—

174

C₂

#21423762

yes

characteristic

175

C₂

#21423767

yes

characteristic

176

C₂

#21423773

yes

—

177

C₂

#21424256

yes

characteristic

178

C₂

#21424264

yes

—

	subgroup	G/H
center	C₂³	#14273724
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#322	C₂
socle, prime radical	C₂³×C₅₉	#127
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#322	D₅₉
hypocenter	C₅₉	#8820
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#14273724

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₅₉×#322

C₂

socle, prime radical

C₂³×C₅₉

#127

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
15. C₅₉×#322	C₄	semidirect
71. C₈×C₅₉	#107	semidirect
104. #322	Dic₅₉	semidirect
105. C₅₉	#8820	semidirect
161. C₈	#14273661	semidirect

H ≅ G/N

type

15. C₅₉×#322

C₄

semidirect

71. C₈×C₅₉

#107

semidirect

104. #322

Dic₅₉

semidirect

105. C₅₉

#8820

semidirect

161. C₈

#14273661

semidirect

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

subgroup	index
#21424256	2
#21424253	2
#21424224	2
C₂×C₅₉×#322	2
#8820	59

subgroup

index

C₂×C₅₉×#322

Subgroups — #28482478

Conjugacy classes of subgroups

Complements

Maximal subgroups