Group #30594679 — subgroups

Subgroups — #30594679

Order 15616.

198 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 66 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	#21461373	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840626	C₂	1	1	yes	maximal, characteristic
3	#21447619	C₂	1	1	yes	maximal, characteristic
4	#16840765	—	2	1	no	—
5	C₂×C₄⋊#92420	—	2	1	no	—
6	#16840626	—	2	1	no	—
7	#16840626	C₄	1	2	yes	—
8	#16840624	—	2	1	no	—
9	#16840623	C₄	1	2	yes	—
10	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
11	#16839580	—	2	1	no	—
12	C₂²⋊#11758185	C₄	1	1	yes	characteristic
13	#16839537	—	2	1	no	—
14	C₆₁×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
15	C₂×#11758217	—	4	1	no	—
16	C₂²×#92420	—	4	1	no	—
17	#14519844	—	4	1	no	—
18	C₄⋊#92420	—	2	2	no	—
19	C₄⋊#92420	—	2	2	no	—
20	C₄⋊#92420	—	4	1	no	—
21	C₂²×#92418	Q₈	1	1	yes	characteristic
22	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
23	C₂×#11758185	—	2	1	no	—
24	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₆₁	D₄	1	1	yes	characteristic
27	C₈×Dic₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₆₁	—	2	1	no	—
30	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×Dic₆₁	SD₁₆	1	1	yes	characteristic
32	C₂²×Dic₆₁	SD₁₆	1	1	yes	characteristic
33	#11758217	—	8	1	no	—
34	#11758217	—	8	1	no	—
35	C₂×#92420	—	4	2	no	—
36	C₂×#92420	—	4	2	no	—
37	C₂×#92420	—	8	2	no	—
38	C₂×C₈×C₆₁	—	4	1	no	—
39	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₆₁	D₈	1	1	yes	characteristic
44	C₄×Dic₆₁	Q₁₆	1	1	yes	characteristic
45	C₂×#92418	—	2	2	no	—
46	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
47	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
48	C₁₆×C₆₁	—	4	1	no	—
49	#11758185	—	4	1	no	—
50	C₂³×C₆₁	#106	1	1	yes	characteristic
51	C₂×C₄×C₆₁	—	2	1	no	—
52	C₂×C₄×C₆₁	—	2	1	no	—
53	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
55	C₂×Dic₆₁	—	2	1	no	—
56	C₂×Dic₆₁	—	2	1	no	—
57	C₂×Dic₆₁	—	2	1	no	—
58	C₂×Dic₆₁	—	2	1	no	—
59	C₂×Dic₆₁	#102	1	1	yes	characteristic
60	C₂×Dic₆₁	#102	1	1	yes	characteristic
61	C₂×Dic₆₁	#103	1	1	yes	characteristic
62	C₂×Dic₆₁	#103	1	1	yes	characteristic
63	#92420	—	8	2	no	—
64	#92420	—	8	2	no	—
65	C₈×C₆₁	—	2	1	no	—
66	C₈×C₆₁	—	2	1	no	—
67	#92418	—	2	1	no	—
68	#92418	#108	1	2	yes	—
69	#7292	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	—	2	1	no	—
71	C₂²×C₆₁	—	2	1	no	—
72	C₂²×C₆₁	#340	1	1	yes	characteristic
73	C₂²×C₆₁	#365	1	1	yes	characteristic
74	C₂²×C₆₁	#365	1	1	yes	characteristic
75	C₄×C₆₁	—	2	1	no	—
76	C₄×C₆₁	#341	1	1	yes	characteristic
77	C₄×C₆₁	#344	1	1	yes	characteristic
78	Dic₆₁	—	2	1	no	—
79	Dic₆₁	—	2	1	no	—
80	Dic₆₁	—	2	1	no	—
81	Dic₆₁	#361	1	1	yes	characteristic
82	Dic₆₁	#362	1	1	yes	characteristic
83	#2163	—	61	1	no	—
84	#1566	—	61	1	no	—
85	C₂×#335	—	61	1	no	—
86	C₂×C₆₁	—	2	1	no	—
87	C₂×C₆₁	—	2	1	no	—
88	C₂×C₆₁	#1301	1	1	yes	characteristic
89	C₂×C₆₁	#1383	1	1	yes	characteristic
90	C₂×C₆₁	#1387	1	1	yes	characteristic
91	#423	—	122	1	no	—
92	C₂×C₄⋊C₈	—	122	1	no	—
93	C₂×C₄×C₈	—	61	1	no	—
94	C₄⋊C₁₆	—	122	1	no	—
95	C₂²⋊C₁₆	—	61	1	no	—
96	C₂²⋊C₁₆	#6923	1	1	yes	characteristic
97	#346	—	122	1	no	—
98	#335	—	61	2	no	—
99	#335	—	122	1	no	—
100	#334	—	61	2	no	—
101	#334	—	122	1	no	—
102	C₆₁	#7292	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	244	1	no	—
104	C₂²×C₈	—	61	1	no	—
105	C₂²×C₈	—	244	1	no	—
106	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
107	C₂×C₄²	—	61	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	122	1	no	—
110	C₄⋊C₈	—	122	2	no	—
111	C₄⋊C₈	—	122	2	no	—
112	C₄⋊C₈	—	244	1	no	—
113	C₄⋊C₈	—	244	1	no	—
114	C₄×C₈	—	61	1	no	—
115	C₄×C₈	—	61	1	no	—
116	C₄×C₈	—	61	1	no	—
117	C₄×C₈	—	61	1	no	—
118	C₂²×C₄	—	61	1	no	—
119	C₂²×C₄	—	61	1	no	—
120	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
121	M₁₆	—	488	1	no	—
122	M₁₆	—	488	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	61	1	no	—
125	C₂×C₈	—	61	1	no	—
126	C₂×C₈	—	122	2	no	—
127	C₂×C₈	—	488	2	no	—
128	C₂×C₈	—	244	2	no	—
129	C₂×C₈	—	244	2	no	—
130	C₂×C₈	C₄×#6923	1	1	yes	characteristic
131	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
132	C₄²	—	61	1	no	—
133	C₄²	—	61	1	no	—
134	C₄²	—	61	1	no	—
135	C₄²	—	61	1	no	—
136	C₁₆	—	4	1	no	—
137	C₁₆	—	244	1	no	—
138	C₂³	#14519845	1	1	yes	characteristic
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	61	1	no	—
142	C₂×C₄	—	61	1	no	—
143	C₂×C₄	—	61	1	no	—
144	C₂×C₄	—	61	1	no	—
145	C₂×C₄	—	122	1	no	—
146	C₂×C₄	—	122	1	no	—
147	C₂×C₄	—	122	1	no	—
148	C₂×C₄	—	122	1	no	—
149	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
150	C₂×C₄	#14519846	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	61	2	no	—
154	C₈	—	122	1	no	—
155	C₈	—	488	2	no	—
156	C₈	—	488	2	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	#16840636	1	1	yes	characteristic
160	C₂²	#16840686	1	1	yes	characteristic
161	C₂²	#16840687	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	61	1	no	—
164	C₄	—	61	1	no	—
165	C₄	—	122	1	no	—
166	C₄	—	122	1	no	—
167	C₄	—	122	1	no	—
168	C₄	#16840637	1	1	yes	characteristic
169	C₄	#16840643	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	#21460710	1	1	yes	characteristic
173	C₂	#21460884	1	1	yes	characteristic
174	C₂	#21460894	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461373

C₂

yes

maximal, characteristic

C₂×#16840626

C₂

yes

maximal, characteristic

#21447619

C₂

yes

maximal, characteristic

#16840765

—

C₂×C₄⋊#92420

—

#16840626

—

#16840626

C₄

yes

—

#16840624

—

#16840623

C₄

yes

—

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

#16839580

—

C₂²⋊#11758185

C₄

yes

characteristic

#16839537

—

C₆₁×C₂²⋊C₁₆

C₄

yes

characteristic

C₂×#11758217

—

C₂²×#92420

—

—

—

—

—

C₂²×#92418

Q₈

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

characteristic

C₂×#11758185

—

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×Dic₆₁

SD₁₆

yes

characteristic

C₂²×Dic₆₁

SD₁₆

yes

—

—

—

—

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₄×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₄×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₄×Dic₆₁

D₈

yes

characteristic

C₄×Dic₆₁

Q₁₆

yes

characteristic

C₂×#92418

—

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

#106

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

yes

C₂×Dic₆₁

yes

C₂×Dic₆₁

yes

C₂×Dic₆₁

yes

—

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

#108

yes

—

#7292

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

yes

Dic₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

122

—

C₂×C₄⋊C₈

—

122

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

122

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

#6923

yes

characteristic

#346

—

122

—

#335

—

#335

—

122

—

100

#334

—

101

#334

—

122

—

102

C₆₁

#7292

yes

characteristic, complemented, Sylow

103

C₂×M₁₆

—

244

—

104

C₂²×C₈

—

105

C₂²×C₈

—

244

—

106

C₂²×C₈

C₂×#6923

yes

characteristic

107

C₂×C₄²

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

122

—

110

C₄⋊C₈

—

122

—

111

C₄⋊C₈

—

122

—

112

C₄⋊C₈

—

244

—

113

C₄⋊C₈

—

244

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₄⋊#6923

yes

characteristic

121

M₁₆

—

488

—

122

M₁₆

—

488

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

122

—

127

C₂×C₈

—

488

—

128

C₂×C₈

—

244

—

129

C₂×C₈

—

244

—

130

C₂×C₈

C₄×#6923

yes

characteristic

131

C₂×C₈

C₂²⋊#6923

yes

characteristic

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₁₆

—

137

C₁₆

—

244

—

138

C₂³

#14519845

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

122

—

146

C₂×C₄

—

122

—

147

C₂×C₄

—

122

—

148

C₂×C₄

—

122

—

149

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

150

C₂×C₄

#14519846

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

122

—

155

C₈

—

488

—

156

C₈

—

488

—

157

C₂²

—

158

C₂²

—

159

C₂²

#16840636

yes

characteristic

160

C₂²

#16840686

yes

characteristic

161

C₂²

#16840687

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

122

—

166

C₄

—

122

—

167

C₄

—

122

—

168

C₄

#16840637

yes

characteristic

169

C₄

#16840643

yes

characteristic

170

C₂

—

171

C₂

—

172

C₂

#21460710

yes

characteristic

173

C₂

#21460884

yes

characteristic

174

C₂

#21460894

yes

characteristic

	subgroup	G/H
center	C₂²	#16840636
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂²×C₈	C₂×#6923
Fitting	C₆₁×C₂²⋊C₁₆	C₄
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#106
cosocle	C₂×C₈×Dic₆₁	C₂²
hypercenter	C₂²⋊C₁₆	#6923
hypocenter	C₆₁	#7292
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

#16840636

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₂²×C₈

C₂×#6923

Fitting

C₆₁×C₂²⋊C₁₆

C₄

C₂²×C₆₁

C₂³×C₆₁

C₂×C₈×Dic₆₁

C₂²

C₂²⋊C₁₆

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
102. C₆₁	#7292	semidirect

H ≅ G/N

type

102. C₆₁

#7292

semidirect

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

subgroup	index
#21461373	2
C₂×#16840626	2
#21447619	2
#7292	61

subgroup

index

Subgroups — #30594679

Conjugacy classes of subgroups

Complements

Maximal subgroups