Group #28497642 — subgroups

Subgroups — #28497642

Order 15104.

842 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 235 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	#21423835	C₂	1	3	yes	maximal
2	#21423793	C₂	1	3	yes	maximal
3	C₅₉×#3458	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11745664	C₂²	1	3	yes	—
5	C₂³×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
6	#16837171	—	2	6	no	—
7	#16837157	—	2	6	no	—
8	C₅₉×#544	—	2	6	no	—
9	C₅₉×#543	C₄	1	4	yes	—
10	C₅₉×#533	—	2	6	no	—
11	C₂×#117×C₅₉	C₂²	1	3	yes	—
12	C₂×#11745664	—	2	3	no	—
13	C₂×#11745664	—	2	6	no	—
14	C₂×#11745664	—	2	6	no	—
15	C₂×#11745664	Q₈	1	6	yes	complemented
16	C₂²×C₅₉⋊C₈	—	2	3	no	—
17	C₂²×C₅₉⋊C₈	—	2	3	no	—
18	C₂²×C₅₉⋊C₈	—	2	3	no	—
19	C₂²×C₅₉⋊C₈	Q₈	1	2	yes	complemented
20	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈×C₅₉	—	2	12	no	—
22	#125×C₅₉	—	2	12	no	—
23	C₂²⋊Q₈×C₅₉	—	2	24	no	—
24	C₄×Q₈×C₅₉	—	2	8	no	—
25	#117×C₅₉	—	2	6	no	—
26	#117×C₅₉	—	2	6	no	—
27	C₂×C₄⋊C₄×C₅₉	D₄	1	6	yes	—
28	C₂×C₂²⋊C₄×C₅₉	C₂×C₄	1	6	yes	—
29	C₂×C₄²×C₅₉	D₄	1	6	yes	—
30	C₂⁴×C₅₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×Q₈×C₅₉	—	2	8	no	—
32	C₂²×C₄×C₅₉	C₂²⋊C₄	1	12	yes	—
33	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
34	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
35	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂×Q₈	1	3	yes	—
37	C₂²×C₄×C₅₉	#41	1	3	yes	—
38	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
39	C₄⋊C₄×C₅₉	—	2	24	no	—
40	C₄⋊C₄×C₅₉	—	2	12	no	—
41	C₂²⋊C₄×C₅₉	—	2	12	no	—
42	C₄²×C₅₉	—	2	6	no	—
43	C₄²×C₅₉	—	2	6	no	—
44	#11745664	—	2	12	no	—
45	#11745664	—	4	6	no	—
46	C₂×C₅₉⋊C₈	—	2	6	no	—
47	C₂×C₅₉⋊C₈	—	2	6	no	—
48	C₂×C₅₉⋊C₈	—	4	1	no	—
49	C₂×C₅₉⋊C₈	—	4	3	no	—
50	C₂×C₅₉⋊C₈	—	4	3	no	—
51	C₂×C₅₉⋊C₈	—	4	3	no	—
52	C₂³×C₅₉	—	4	2	no	—
53	C₂³×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
54	C₂³×C₅₉	C₄×Q₈	1	1	yes	characteristic
55	C₂³×C₅₉	C₄×Q₈	1	3	yes	—
56	Q₈×C₅₉	—	8	8	no	—
57	C₂×C₄×C₅₉	—	2	12	no	—
58	C₂×C₄×C₅₉	—	4	12	no	—
59	C₂×C₄×C₅₉	#120	1	3	yes	—
60	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂²⋊Q₈	1	3	yes	—
62	C₂×C₄×C₅₉	C₂²⋊Q₈	1	3	yes	—
63	C₂×C₄×C₅₉	C₂²⋊Q₈	1	6	yes	—
64	C₂×C₄×C₅₉	#124	1	3	yes	—
65	C₂×C₄×C₅₉	#124	1	3	yes	—
66	C₂×C₄×C₅₉	C₄⋊Q₈	1	3	yes	—
67	C₂×C₄×C₅₉	C₄⋊Q₈	1	3	yes	—
68	C₅₉⋊C₈	—	4	2	no	—
69	C₅₉⋊C₈	—	4	6	no	—
70	#11391	—	59	1	no	maximal, Sylow
71	C₂²×C₅₉	—	2	6	no	—
72	C₂²×C₅₉	—	4	1	no	—
73	C₂²×C₅₉	—	4	3	no	—
74	C₂²×C₅₉	#379	1	1	yes	characteristic
75	C₂²×C₅₉	#391	1	3	yes	—
76	C₂²×C₅₉	#391	1	3	yes	—
77	C₄×C₅₉	—	4	12	no	—
78	C₄×C₅₉	#393	1	3	yes	—
79	C₄×C₅₉	#393	1	1	yes	characteristic
80	C₄×C₅₉	#395	1	3	yes	—
81	C₄×C₅₉	#395	1	1	yes	characteristic
82	#3458	D₅₉	1	1	yes	characteristic
83	#1742	—	59	3	no	—
84	#1729	—	59	3	no	—
85	C₂×C₅₉	—	2	3	no	—
86	C₂×C₅₉	—	2	3	no	—
87	C₂×C₅₉	#1446	1	1	yes	characteristic
88	C₂×C₅₉	#1791	1	2	yes	—
89	C₂²×M₁₆	—	59	3	no	—
90	C₂³×C₈	—	59	1	no	—
91	#544	—	2	6	no	—
92	#543	Dic₅₉	1	4	yes	—
93	#533	—	2	6	no	—
94	C₂×#117	C₂×D₅₉	1	3	yes	—
95	#343	—	118	6	no	—
96	#336	—	118	6	no	—
97	C₅₉	#11391	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
99	C₂×M₁₆	—	59	6	no	—
100	C₂×M₁₆	—	118	3	no	—
101	C₂×M₁₆	—	118	6	no	—
102	C₂×M₁₆	—	118	6	no	—
103	C₂²×C₈	—	59	2	no	—
104	C₂²×C₈	—	118	3	no	—
105	C₂²×C₈	—	118	3	no	—
106	C₂²×C₈	—	118	3	no	—
107	C₄⋊Q₈	—	2	12	no	—
108	#125	—	2	12	no	—
109	C₂²⋊Q₈	—	2	24	no	—
110	C₄×Q₈	—	2	8	no	—
111	#117	—	2	6	no	—
112	#117	—	2	6	no	—
113	C₂×C₄⋊C₄	C₂²⋊D₅₉	1	6	yes	—
114	C₂×C₂²⋊C₄	C₂×Dic₅₉	1	6	yes	—
115	C₂×C₄²	C₂²⋊D₅₉	1	6	yes	—
116	C₂⁴	C₂²×Dic₅₉	1	1	yes	characteristic
117	C₂×Q₈	—	2	8	no	—
118	C₂²×C₄	C₂²⋊Dic₅₉	1	12	yes	—
119	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
120	C₂²×C₄	Q₈×D₅₉	1	3	yes	—
121	C₂²×C₄	#11745689	1	3	yes	—
122	C₂²×C₄	#11745689	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
125	M₁₆	—	118	12	no	—
126	M₁₆	—	236	6	no	—
127	C₂×C₈	—	118	6	no	—
128	C₂×C₈	—	118	6	no	—
129	C₂×C₈	—	236	3	no	—
130	C₂×C₈	—	236	1	no	—
131	C₂×C₈	—	236	3	no	—
132	C₂×C₈	—	236	3	no	—
133	C₄⋊C₄	—	2	24	no	—
134	C₄⋊C₄	—	2	12	no	—
135	C₂²⋊C₄	—	2	12	no	—
136	C₄²	—	2	6	no	—
137	C₄²	—	2	6	no	—
138	C₂³	—	4	2	no	—
139	C₂³	C₂×C₂²⋊Dic₅₉	1	3	yes	—
140	C₂³	Q₈×Dic₅₉	1	1	yes	characteristic
141	C₂³	Q₈×Dic₅₉	1	3	yes	—
142	Q₈	—	8	8	no	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	—	4	12	no	—
145	C₂×C₄	#14273696	1	3	yes	—
146	C₂×C₄	#14273696	1	1	yes	characteristic
147	C₂×C₄	#14273703	1	3	yes	—
148	C₂×C₄	#14273703	1	3	yes	—
149	C₂×C₄	#14273703	1	6	yes	—
150	C₂×C₄	#14273716	1	3	yes	—
151	C₂×C₄	#14273716	1	3	yes	—
152	C₂×C₄	#14273725	1	3	yes	—
153	C₂×C₄	#14273725	1	3	yes	—
154	C₈	—	236	2	no	—
155	C₈	—	236	6	no	—
156	C₂²	—	2	6	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	3	no	—
159	C₂²	#16837258	1	1	yes	characteristic
160	C₂²	#16837309	1	3	yes	—
161	C₂²	#16837309	1	3	yes	—
162	C₄	—	4	12	no	—
163	C₄	#16837313	1	3	yes	—
164	C₄	#16837313	1	1	yes	characteristic
165	C₄	#16837321	1	3	yes	—
166	C₄	#16837321	1	1	yes	characteristic
167	C₂	—	2	3	no	—
168	C₂	—	2	3	no	—
169	C₂	#21422367	1	1	yes	characteristic
170	C₂	#21424047	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21423835

C₂

yes

maximal

#21423793

C₂

yes

maximal

C₅₉×#3458

C₂

yes

maximal, characteristic

C₂²×#11745664

C₂²

yes

—

C₂³×C₅₉⋊C₈

C₂²

yes

characteristic

#16837171

—

#16837157

—

C₅₉×#544

—

C₅₉×#543

C₄

yes

—

C₅₉×#533

—

C₂×#117×C₅₉

C₂²

yes

—

—

—

—

Q₈

yes

complemented

C₂²×C₅₉⋊C₈

—

C₂²×C₅₉⋊C₈

—

C₂²×C₅₉⋊C₈

—

C₂²×C₅₉⋊C₈

Q₈

yes

complemented

C₂³×C₄×C₅₉

C₂³

yes

characteristic

C₄⋊Q₈×C₅₉

—

#125×C₅₉

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

#117×C₅₉

—

#117×C₅₉

—

C₂×C₄⋊C₄×C₅₉

D₄

yes

—

C₂×C₂²⋊C₄×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

D₄

yes

—

C₂⁴×C₅₉

C₂²×C₄

yes

characteristic

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

—

C₂²×C₄×C₅₉

#41

yes

—

C₂²×C₄×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×C₂²⋊C₄

yes

—

C₂³×C₅₉

C₄×Q₈

yes

characteristic

C₂³×C₅₉

C₄×Q₈

yes

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#120

yes

—

C₂×C₄×C₅₉

#120

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

#124

yes

—

C₂×C₄×C₅₉

#124

yes

—

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#11391

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#379

yes

characteristic

C₂²×C₅₉

#391

yes

—

C₂²×C₅₉

#391

yes

—

C₄×C₅₉

—

C₄×C₅₉

#393

yes

—

C₄×C₅₉

#393

yes

characteristic

C₄×C₅₉

#395

yes

—

C₄×C₅₉

#395

yes

characteristic

#3458

D₅₉

yes

characteristic

#1742

—

#1729

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1446

yes

characteristic

C₂×C₅₉

#1791

yes

—

C₂²×M₁₆

—

C₂³×C₈

—

#544

—

#543

Dic₅₉

yes

—

#533

—

C₂×#117

C₂×D₅₉

yes

—

#343

—

118

—

#336

—

118

—

C₅₉

#11391

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂²×D₅₉

yes

characteristic

C₂×M₁₆

—

100

C₂×M₁₆

—

118

—

101

C₂×M₁₆

—

118

—

102

C₂×M₁₆

—

118

—

103

C₂²×C₈

—

104

C₂²×C₈

—

118

—

105

C₂²×C₈

—

118

—

106

C₂²×C₈

—

118

—

107

C₄⋊Q₈

—

108

#125

—

109

C₂²⋊Q₈

—

110

C₄×Q₈

—

111

#117

—

112

#117

—

113

C₂×C₄⋊C₄

C₂²⋊D₅₉

yes

—

114

C₂×C₂²⋊C₄

C₂×Dic₅₉

yes

—

115

C₂×C₄²

C₂²⋊D₅₉

yes

—

116

C₂⁴

C₂²×Dic₅₉

yes

characteristic

117

C₂×Q₈

—

118

C₂²×C₄

C₂²⋊Dic₅₉

yes

—

119

C₂²×C₄

Q₈×D₅₉

yes

characteristic

120

C₂²×C₄

Q₈×D₅₉

yes

—

121

C₂²×C₄

#11745689

yes

—

122

C₂²×C₄

#11745689

yes

characteristic

123

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

—

125

M₁₆

—

118

—

126

M₁₆

—

236

—

127

C₂×C₈

—

118

—

128

C₂×C₈

—

118

—

129

C₂×C₈

—

236

—

130

C₂×C₈

—

236

—

131

C₂×C₈

—

236

—

132

C₂×C₈

—

236

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₂³

—

139

C₂³

C₂×C₂²⋊Dic₅₉

yes

—

140

C₂³

Q₈×Dic₅₉

yes

characteristic

141

C₂³

Q₈×Dic₅₉

yes

—

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#14273696

yes

—

146

C₂×C₄

#14273696

yes

characteristic

147

C₂×C₄

#14273703

yes

—

148

C₂×C₄

#14273703

yes

—

149

C₂×C₄

#14273703

yes

—

150

C₂×C₄

#14273716

yes

—

151

C₂×C₄

#14273716

yes

—

152

C₂×C₄

#14273725

yes

—

153

C₂×C₄

#14273725

yes

—

154

C₈

—

236

—

155

C₈

—

236

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#16837258

yes

characteristic

160

C₂²

#16837309

yes

—

161

C₂²

#16837309

yes

—

162

C₄

—

163

C₄

#16837313

yes

—

164

C₄

#16837313

yes

characteristic

165

C₄

#16837321

yes

—

166

C₄

#16837321

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#21422367

yes

characteristic

170

C₂

#21424047

yes

—

	subgroup	G/H
center	C₂×C₄	#14273696
commutator, prime radical	C₂⁴×C₅₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₉
Fitting	C₅₉×#3458	C₂
socle	C₂²×C₅₉	#379
cosocle	C₂³×C₄×C₅₉	C₂³
hypercenter	#3458	D₅₉
hypocenter	C₅₉	#11391
LCS	G ▷ C₂⁴×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3458

subgroup

G/H

center

C₂×C₄

#14273696

commutator, prime radical

C₂⁴×C₅₉

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₅₉

Fitting

C₅₉×#3458

C₂

socle

C₂²×C₅₉

#379

cosocle

C₂³×C₄×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂⁴×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3458

N	H ≅ G/N	type
15. C₂×#11745664	Q₈	semidirect
19. C₂²×C₅₉⋊C₈	Q₈	semidirect
97. C₅₉	#11391	semidirect

H ≅ G/N

type

15. C₂×#11745664

Q₈

semidirect

19. C₂²×C₅₉⋊C₈

Q₈

semidirect

97. C₅₉

#11391

semidirect

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

subgroup	index
#21423835	2
#21423793	2
C₅₉×#3458	2
#11391	59

subgroup

index

#21423835

#21423793

C₅₉×#3458

#11391

Subgroups — #28497642

Conjugacy classes of subgroups

Complements

Maximal subgroups