Group #30594704 — subgroups

Subgroups — #30594704

Order 15616.

212 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 62 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	#21462181	C₂	1	1	yes	maximal, characteristic
2	#21461373	C₂	1	1	yes	maximal, characteristic
3	#21445720	C₂	1	1	yes	maximal, characteristic
4	#16840765	—	2	1	no	—
5	#16840683	C₄	1	2	yes	—
6	C₂²⋊#11758187	—	2	1	no	—
7	#16840651	—	2	1	no	—
8	#16840624	—	2	1	no	—
9	#16840623	C₄	1	2	yes	—
10	C₂×#14519712	—	2	1	no	—
11	M₁₆×Dic₆₁	—	2	1	no	—
12	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
13	#16839527	C₄	1	1	yes	characteristic
14	C₆₁×#341	C₄	1	1	yes	characteristic
15	C₂×#11758217	—	4	1	no	—
16	C₂×#11758187	—	2	2	no	—
17	#14519844	—	4	1	no	—
18	C₄⋊#92420	—	2	2	no	—
19	C₂×#11758194	—	2	1	no	—
20	C₂²×#92418	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
22	#14519712	—	2	4	no	—
23	#14519687	—	4	1	no	—
24	C₈×Dic₆₁	—	4	1	no	—
25	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₆₁	D₄	1	1	yes	characteristic
28	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
29	C₂×M₁₆×C₆₁	—	2	1	no	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×Dic₆₁	SD₁₆	1	1	yes	characteristic
33	C₂²×Dic₆₁	Q₁₆	1	1	yes	characteristic
34	#11758217	—	8	2	no	—
35	C₂×#92420	—	4	2	no	—
36	M₁₆×C₆₁	—	2	2	no	—
37	M₁₆×C₆₁	—	4	1	no	—
38	C₂×C₈×C₆₁	—	4	1	no	—
39	C₂×C₈×C₆₁	—	4	1	no	—
40	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₆₁	D₈	1	1	yes	characteristic
45	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
46	#11758194	—	2	2	no	—
47	#11758194	—	4	1	no	—
48	C₂×#92418	—	2	2	no	—
49	C₂×#92418	—	4	1	no	—
50	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
52	#11758187	—	4	2	no	—
53	C₂³×C₆₁	#103	1	1	yes	characteristic
54	C₂×C₄×C₆₁	—	2	1	no	—
55	C₂×C₄×C₆₁	—	2	1	no	—
56	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
58	C₂×Dic₆₁	—	2	2	no	—
59	C₂×Dic₆₁	—	4	1	no	—
60	C₂×Dic₆₁	#102	1	1	yes	characteristic
61	C₂×Dic₆₁	#103	1	1	yes	characteristic
62	C₂×Dic₆₁	#106	1	1	yes	characteristic
63	C₂×Dic₆₁	#107	1	1	yes	characteristic
64	#92420	—	8	2	no	—
65	C₈×C₆₁	—	2	1	no	—
66	C₈×C₆₁	—	2	1	no	—
67	C₈×C₆₁	—	4	2	no	—
68	#92418	—	2	1	no	—
69	#92418	—	4	2	no	—
70	#92418	#104	1	2	yes	—
71	#7300	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	—	2	1	no	—
73	C₂²×C₆₁	—	2	1	no	—
74	C₂²×C₆₁	—	2	1	no	—
75	C₂²×C₆₁	#340	1	1	yes	characteristic
76	C₄×C₆₁	—	2	1	no	—
77	C₄×C₆₁	#339	1	1	yes	characteristic
78	C₄×C₆₁	#344	1	1	yes	characteristic
79	Dic₆₁	—	2	1	no	—
80	Dic₆₁	—	2	1	no	—
81	Dic₆₁	—	2	1	no	—
82	Dic₆₁	#365	1	2	yes	—
83	#2163	—	61	1	no	—
84	C₄×#108	—	61	1	no	—
85	#1566	—	61	1	no	—
86	C₂×C₆₁	—	4	1	no	—
87	C₂×C₆₁	#1303	1	1	yes	characteristic
88	C₂×C₆₁	#1383	1	1	yes	characteristic
89	C₂×C₆₁	#1388	1	1	yes	characteristic
90	C₂×#108	—	122	1	no	—
91	#423	—	122	1	no	—
92	C₄×M₁₆	—	122	1	no	—
93	C₂×C₄×C₈	—	61	1	no	—
94	C₄⋊C₁₆	—	61	2	no	—
95	C₂²⋊C₁₆	—	122	1	no	—
96	#346	—	122	1	no	—
97	#341	—	61	1	no	—
98	#341	#6923	1	1	yes	characteristic
99	#334	—	61	2	no	—
100	#334	—	122	1	no	—
101	C₆₁	#7300	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	—	122	1	no	—
104	C₂×M₁₆	—	244	1	no	—
105	C₂²×C₈	—	61	1	no	—
106	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
107	C₂×C₄²	—	61	1	no	—
108	C₂×C₁₆	—	122	2	no	—
109	#108	—	122	4	no	—
110	C₄⋊C₈	—	122	2	no	—
111	C₄⋊C₈	—	244	1	no	—
112	#97	—	244	1	no	—
113	C₄×C₈	—	61	1	no	—
114	C₄×C₈	—	61	1	no	—
115	C₄×C₈	—	61	1	no	—
116	C₄×C₈	—	61	1	no	—
117	C₄×C₈	—	244	1	no	—
118	C₂²×C₄	—	61	1	no	—
119	C₂²×C₄	—	61	1	no	—
120	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
121	M₁₆	—	2	2	no	—
122	M₁₆	—	4	1	no	—
123	M₁₆	—	122	2	no	—
124	M₁₆	—	488	2	no	—
125	M₁₆	—	244	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	—	61	1	no	—
129	C₂×C₈	—	61	1	no	—
130	C₂×C₈	—	122	2	no	—
131	C₂×C₈	—	244	2	no	—
132	C₂×C₈	—	244	1	no	—
133	C₂×C₈	C₄×#6923	1	1	yes	characteristic
134	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
135	C₄²	—	61	1	no	—
136	C₄²	—	61	1	no	—
137	C₄²	—	61	1	no	—
138	C₄²	—	61	1	no	—
139	C₁₆	—	244	2	no	—
140	C₂³	#14519840	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	61	1	no	—
144	C₂×C₄	—	61	1	no	—
145	C₂×C₄	—	61	1	no	—
146	C₂×C₄	—	61	1	no	—
147	C₂×C₄	—	122	2	no	—
148	C₂×C₄	—	244	1	no	—
149	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
150	C₂×C₄	#14519839	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	4	2	no	—
154	C₈	—	61	2	no	—
155	C₈	—	122	1	no	—
156	C₈	—	488	2	no	—
157	C₈	—	244	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	#16840635	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	61	2	no	—
164	C₄	—	122	1	no	—
165	C₄	—	122	1	no	—
166	C₄	—	122	1	no	—
167	C₄	#16840633	1	1	yes	characteristic
168	C₄	#16840643	1	1	yes	characteristic
169	C₂	—	4	1	no	—
170	C₂	#21460717	1	1	yes	characteristic
171	C₂	#21460885	1	1	yes	characteristic
172	C₂	#21460896	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462181

C₂

yes

maximal, characteristic

#21461373

C₂

yes

maximal, characteristic

#21445720

C₂

yes

maximal, characteristic

#16840765

—

#16840683

C₄

yes

—

—

—

—

C₄

yes

—

C₂×#14519712

—

M₁₆×Dic₆₁

—

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

#16839527

C₄

yes

characteristic

C₆₁×#341

C₄

yes

—

—

—

—

—

C₂²×#92418

D₄

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

characteristic

#14519712

—

#14519687

—

C₈×Dic₆₁

—

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

characteristic

C₈×Dic₆₁

Q₈

yes

characteristic

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×Dic₆₁

SD₁₆

yes

characteristic

C₂²×Dic₆₁

Q₁₆

yes

characteristic

#11758217

—

C₂×#92420

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₄×Dic₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

D₈

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

#11758187

—

C₂³×C₆₁

#103

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

yes

C₂×Dic₆₁

yes

C₂×Dic₆₁

yes

C₂×Dic₆₁

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

yes

—

#7300

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#340

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#365

yes

—

#2163

—

C₄×#108

—

#1566

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

122

—

#423

—

122

—

C₄×M₁₆

—

122

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

122

—

#346

—

122

—

—

yes

—

100

#334

—

122

—

101

C₆₁

#7300

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

103

C₂×M₁₆

—

122

—

104

C₂×M₁₆

—

244

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×#6923

yes

characteristic

107

C₂×C₄²

—

108

C₂×C₁₆

—

122

—

109

#108

—

122

—

110

C₄⋊C₈

—

122

—

111

C₄⋊C₈

—

244

—

112

#97

—

244

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

244

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂²⋊#6923

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

122

—

124

M₁₆

—

488

—

125

M₁₆

—

244

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

122

—

131

C₂×C₈

—

244

—

132

C₂×C₈

—

244

—

133

C₂×C₈

C₄×#6923

yes

characteristic

134

C₂×C₈

C₄⋊#6923

yes

characteristic

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₁₆

—

244

—

140

C₂³

#14519840

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

122

—

148

C₂×C₄

—

244

—

149

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

150

C₂×C₄

#14519839

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

122

—

156

C₈

—

488

—

157

C₈

—

244

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

#16840635

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

122

—

165

C₄

—

122

—

166

C₄

—

122

—

167

C₄

#16840633

yes

characteristic

168

C₄

#16840643

yes

characteristic

169

C₂

—

170

C₂

#21460717

yes

characteristic

171

C₂

#21460885

yes

characteristic

172

C₂

#21460896

yes

characteristic

	subgroup	G/H
center	C₂²	#16840635
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂²×C₈	C₂×#6923
Fitting	C₆₁×#341	C₄
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#103
cosocle	C₂×C₈×Dic₆₁	C₂²
hypercenter	#341	#6923
hypocenter	C₆₁	#7300
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #341

subgroup

G/H

center

C₂²

#16840635

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₂²×C₈

C₂×#6923

Fitting

C₆₁×#341

C₄

C₂²×C₆₁

C₂³×C₆₁

C₂×C₈×Dic₆₁

C₂²

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #341

N	H ≅ G/N	type
101. C₆₁	#7300	semidirect

H ≅ G/N

type

101. C₆₁

#7300

semidirect

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

subgroup	index
#21462181	2
#21461373	2
#21445720	2
#7300	61

subgroup

index

Subgroups — #30594704

Conjugacy classes of subgroups

Complements

Maximal subgroups