Group #28488456 — subgroups

Subgroups — #28488456

Order 15104.

558 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 211 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×#16837782	C₂	1	1	yes	maximal, characteristic
2	#21425057	C₂	1	4	yes	maximal
3	#21424256	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₃₆	—	2	2	no	—
6	C₂×#14273727	C₂²	1	2	yes	—
7	#16837782	—	2	2	no	—
8	#16837782	C₄	1	4	yes	complemented
9	C₂×#14273661	—	2	2	no	—
10	#16837272	C₂²	1	2	yes	—
11	#16837165	—	2	8	no	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₈²×C₅₉	—	2	2	no	—
15	C₂²×Dic₁₁₈	—	2	2	no	—
16	C₂×Dic₂₃₆	—	2	8	no	—
17	C₂×Dic₂₃₆	—	4	4	no	—
18	C₂×Dic₂₃₆	—	4	4	no	—
19	#14273727	—	2	4	no	—
20	#14273727	—	2	2	no	—
21	#14273727	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊Dic₅₉	—	2	4	no	—
23	C₂×C₄⋊Dic₅₉	—	2	4	no	—
24	C₂×C₄⋊Dic₅₉	—	2	2	no	—
25	#14273661	—	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	—	2	2	no	—
30	C₄×C₈×C₅₉	—	2	2	no	—
31	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₉	D₄	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	2	no	—
37	C₂×Dic₁₁₈	—	2	4	no	—
38	C₂×Dic₁₁₈	—	4	4	no	—
39	C₂×Dic₁₁₈	—	4	4	no	—
40	C₂×Dic₁₁₈	—	4	2	no	—
41	C₂×C₈×C₅₉	—	2	4	no	—
42	C₂×C₈×C₅₉	—	2	4	no	—
43	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
44	C₂×C₈×C₅₉	D₈	1	4	yes	—
45	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
46	C₂×C₈×C₅₉	Q₁₆	1	8	yes	complemented
47	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₉	#41	1	2	yes	—
50	C₄⋊Dic₅₉	—	2	8	no	—
51	C₄⋊Dic₅₉	—	4	4	no	—
52	C₄⋊Dic₅₉	—	4	4	no	—
53	C₄⋊Dic₅₉	—	4	8	no	—
54	Dic₂₃₆	—	4	16	no	—
55	C₂³×C₅₉	#127	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
58	C₂×C₄×C₅₉	#121	1	2	yes	—
59	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
61	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
63	C₂×C₄×C₅₉	#135	1	4	yes	—
64	C₂×Dic₅₉	—	4	4	no	—
65	C₂×Dic₅₉	—	8	2	no	—
66	C₂×Dic₅₉	—	8	2	no	—
67	C₂×Dic₅₉	—	8	2	no	—
68	C₂×Dic₅₉	—	8	4	no	—
69	Dic₁₁₈	—	4	8	no	—
70	Dic₁₁₈	—	8	4	no	—
71	C₈×C₅₉	—	2	8	no	—
72	C₈×C₅₉	#102	1	8	yes	—
73	#9825	—	59	1	no	maximal, Sylow
74	C₂²×C₅₉	#390	1	1	yes	characteristic
75	C₂²×C₅₉	#492	1	1	yes	characteristic
76	C₂²×C₅₉	#493	1	1	yes	characteristic
77	C₂²×C₅₉	#494	1	2	yes	—
78	C₂²×C₅₉	#495	1	2	yes	—
79	C₄×C₅₉	C₂×#102	1	2	yes	—
80	C₄×C₅₉	C₄×Q₁₆	1	4	yes	—
81	C₄×C₅₉	#467	1	4	yes	—
82	C₄×C₅₉	#486	1	2	yes	—
83	Dic₅₉	—	8	4	no	—
84	Dic₅₉	—	16	2	no	—
85	C₂×#494	—	59	1	no	—
86	#1990	—	59	4	no	—
87	#1845	—	59	1	no	—
88	C₂×C₈²	D₅₉	1	1	yes	characteristic
89	C₂×C₅₉	#1707	1	2	yes	—
90	C₂×C₅₉	#1709	1	1	yes	characteristic
91	C₂×C₅₉	#1713	1	1	yes	characteristic
92	C₂×C₅₉	#1957	1	1	yes	characteristic
93	C₂×C₅₉	#1969	1	2	yes	—
94	C₂²×Q₁₆	—	118	2	no	—
95	C₂×C₄⋊Q₈	—	59	2	no	—
96	#494	—	59	4	no	—
97	#494	—	118	2	no	—
98	C₂×#107	—	118	2	no	—
99	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
101	#384	—	59	2	no	—
102	#340	—	118	8	no	—
103	C₈²	—	2	2	no	—
104	C₅₉	#9825	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	118	2	no	—
106	C₂×Q₁₆	—	118	8	no	—
107	C₂×Q₁₆	—	236	4	no	—
108	C₂×Q₁₆	—	236	4	no	—
109	C₂²×C₈	D₂₃₆	1	4	yes	—
110	C₂²×C₈	D₂₃₆	1	2	yes	—
111	C₄⋊Q₈	—	59	4	no	—
112	C₄⋊Q₈	—	118	4	no	—
113	C₄⋊Q₈	—	118	2	no	—
114	C₂×C₄⋊C₄	—	118	4	no	—
115	C₂×C₄⋊C₄	—	118	4	no	—
116	C₂×C₄⋊C₄	—	118	2	no	—
117	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
118	#107	—	236	4	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	C₄×D₅₉	1	2	yes	—
122	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
123	C₂×Q₈	—	118	4	no	—
124	C₂×Q₈	—	236	4	no	—
125	C₂×Q₈	—	236	4	no	—
126	C₂×Q₈	—	236	2	no	—
127	C₂²×C₄	—	236	2	no	—
128	C₂²×C₄	—	236	2	no	—
129	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
131	Q₁₆	—	236	16	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	#11745660	1	4	yes	—
135	C₂×C₈	D₄₇₂	1	4	yes	—
136	C₂×C₈	Dic₂₃₆	1	8	yes	complemented
137	C₂×C₈	(C₂×D₅₉)⋊C₄	1	4	yes	—
138	C₄⋊C₄	—	118	8	no	—
139	C₄⋊C₄	—	236	4	no	—
140	C₄⋊C₄	—	236	8	no	—
141	C₄⋊C₄	—	236	4	no	—
142	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
143	C₄²	#11745685	1	2	yes	—
144	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
145	C₂³	#14273724	1	1	yes	characteristic
146	Q₈	—	472	4	no	—
147	Q₈	—	236	8	no	—
148	C₂×C₄	—	472	2	no	—
149	C₂×C₄	—	472	4	no	—
150	C₂×C₄	—	472	2	no	—
151	C₂×C₄	—	472	2	no	—
152	C₂×C₄	—	236	4	no	—
153	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
154	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
155	C₂×C₄	#14273701	1	2	yes	—
156	C₂×C₄	#14273715	1	1	yes	characteristic
157	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
158	C₂×C₄	C₂×#11745660	1	2	yes	—
159	C₂×C₄	C₂×Dic₂₃₆	1	4	yes	—
160	C₂×C₄	#14273749	1	4	yes	—
161	C₈	—	2	8	no	—
162	C₈	#14273649	1	8	yes	—
163	C₂²	#16837304	1	1	yes	characteristic
164	C₂²	#16837775	1	1	yes	characteristic
165	C₂²	#16837779	1	1	yes	characteristic
166	C₂²	#16837782	1	2	yes	—
167	C₂²	#16837786	1	2	yes	—
168	C₄	—	944	2	no	—
169	C₄	—	472	4	no	—
170	C₄	C₂×#14273649	1	2	yes	—
171	C₄	C₄×Dic₂₃₆	1	4	yes	—
172	C₄	#16837643	1	4	yes	—
173	C₄	#16837749	1	2	yes	—
174	C₂	#21423693	1	2	yes	—
175	C₂	#21423702	1	1	yes	characteristic
176	C₂	#21423714	1	1	yes	characteristic
177	C₂	#21424902	1	1	yes	characteristic
178	C₂	#21424957	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837782

C₂

yes

maximal, characteristic

#21425057

C₂

yes

maximal

#21424256

C₂

yes

maximal, characteristic

C₂×C₈²×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₃₆

—

C₂×#14273727

C₂²

yes

—

#16837782

—

#16837782

C₄

yes

complemented

C₂×#14273661

—

#16837272

C₂²

yes

—

#16837165

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₈²×C₅₉

—

C₂²×Dic₁₁₈

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

#14273727

—

#14273727

—

#14273727

C₂×C₄

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273661

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

complemented

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₂₃₆

—

C₂³×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#102

yes

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#495

yes

—

C₄×C₅₉

C₂×#102

yes

—

C₄×C₅₉

C₄×Q₁₆

yes

—

C₄×C₅₉

#467

yes

—

C₄×C₅₉

#486

yes

—

Dic₅₉

—

Dic₅₉

—

C₂×#494

—

#1990

—

#1845

—

C₂×C₈²

D₅₉

yes

characteristic

C₂×C₅₉

#1707

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂²×Q₁₆

—

118

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

118

—

C₂×#107

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

100

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

101

#384

—

102

#340

—

118

—

103

C₈²

—

104

C₅₉

#9825

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

118

—

106

C₂×Q₁₆

—

118

—

107

C₂×Q₁₆

—

236

—

108

C₂×Q₁₆

—

236

—

109

C₂²×C₈

D₂₃₆

yes

—

110

C₂²×C₈

D₂₃₆

yes

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

118

—

113

C₄⋊Q₈

—

118

—

114

C₂×C₄⋊C₄

—

118

—

115

C₂×C₄⋊C₄

—

118

—

116

C₂×C₄⋊C₄

—

118

—

117

C₂×C₄²

C₂²×D₅₉

yes

characteristic

118

#107

—

236

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

C₄×D₅₉

yes

—

122

C₄×C₈

C₂²⋊D₅₉

yes

—

123

C₂×Q₈

—

118

—

124

C₂×Q₈

—

236

—

125

C₂×Q₈

—

236

—

126

C₂×Q₈

—

236

—

127

C₂²×C₄

—

236

—

128

C₂²×C₄

—

236

—

129

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

130

C₂²×C₄

C₂×D₂₃₆

yes

—

131

Q₁₆

—

236

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

#11745660

yes

—

135

C₂×C₈

D₄₇₂

yes

—

136

C₂×C₈

Dic₂₃₆

yes

complemented

137

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

138

C₄⋊C₄

—

118

—

139

C₄⋊C₄

—

236

—

140

C₄⋊C₄

—

236

—

141

C₄⋊C₄

—

236

—

142

C₄²

C₂×C₄×D₅₉

yes

characteristic

143

C₄²

#11745685

yes

—

144

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

145

C₂³

#14273724

yes

characteristic

146

Q₈

—

472

—

147

Q₈

—

236

—

148

C₂×C₄

—

472

—

149

C₂×C₄

—

472

—

150

C₂×C₄

—

472

—

151

C₂×C₄

—

472

—

152

C₂×C₄

—

236

—

153

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

154

C₂×C₄

C₄×D₂₃₆

yes

—

155

C₂×C₄

#14273701

yes

—

156

C₂×C₄

#14273715

yes

characteristic

157

C₂×C₄

C₂×D₄₇₂

yes

—

158

C₂×C₄

C₂×#11745660

yes

—

159

C₂×C₄

C₂×Dic₂₃₆

yes

—

160

C₂×C₄

#14273749

yes

—

161

C₈

—

162

C₈

#14273649

yes

—

163

C₂²

#16837304

yes

characteristic

164

C₂²

#16837775

yes

characteristic

165

C₂²

#16837779

yes

characteristic

166

C₂²

#16837782

yes

—

167

C₂²

#16837786

yes

—

168

C₄

—

944

—

169

C₄

—

472

—

170

C₄

C₂×#14273649

yes

—

171

C₄

C₄×Dic₂₃₆

yes

—

172

C₄

#16837643

yes

—

173

C₄

#16837749

yes

—

174

C₂

#21423693

yes

—

175

C₂

#21423702

yes

characteristic

176

C₂

#21423714

yes

characteristic

177

C₂

#21424902

yes

characteristic

178

C₂

#21424957

yes

—

	subgroup	G/H
center	C₂³	#14273724
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₈²×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	#127
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₈²	D₅₉
hypocenter	C₅₉	#9825
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#14273724

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₈²×C₅₉

C₂

socle, prime radical

C₂³×C₅₉

#127

cosocle

C₂×C₄²×C₅₉

C₂³

hypercenter

C₂×C₈²

D₅₉

hypocenter

C₅₉

#9825

LCS

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
8. #16837782	C₄	semidirect
46. C₂×C₈×C₅₉	Q₁₆	semidirect
104. C₅₉	#9825	semidirect
136. C₂×C₈	Dic₂₃₆	semidirect

H ≅ G/N

type

8. #16837782

C₄

semidirect

46. C₂×C₈×C₅₉

Q₁₆

semidirect

104. C₅₉

#9825

semidirect

136. C₂×C₈

Dic₂₃₆

semidirect

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

subgroup	index
C₂×#16837782	2
#21425057	2
#21424256	2
C₂×C₈²×C₅₉	2
#9825	59

subgroup

index

C₂×C₈²×C₅₉

Subgroups — #28488456

Conjugacy classes of subgroups

Complements

Maximal subgroups