Group #30594460 — subgroups

Subgroups — #30594460

Order 15616.

206 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 60 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	#21462183	C₂	1	1	yes	maximal, characteristic
2	#21461741	C₂	1	1	yes	maximal, characteristic
3	#21444573	C₂	1	1	yes	maximal, characteristic
4	#16840706	—	2	1	no	—
5	C₄⋊#11758187	—	2	1	no	—
6	#16840657	—	2	1	no	—
7	#16840651	C₄	1	2	yes	complemented
8	#16840636	—	2	1	no	—
9	#16840636	C₄	1	2	yes	—
10	#16839784	—	2	1	no	—
11	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
12	#16839513	C₄	1	1	yes	characteristic, complemented
13	#16839512	—	2	1	no	—
14	C₆₁×#334	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#6923	—	2	2	no	—
16	C₂×#11758187	—	2	2	no	—
17	(C₂×#6923)⋊C₄	—	4	1	no	—
18	(C₂×#6923)⋊C₄	—	4	1	no	—
19	C₂×#11758189	—	4	1	no	—
20	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
22	C₄²×D₆₁	D₄	1	1	yes	characteristic
23	C₄⋊#92418	—	2	1	no	—
24	#14519706	—	4	1	no	—
25	C₄×#92418	D₄	1	1	yes	characteristic
26	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
28	C₄⋊C₈×C₆₁	—	2	1	no	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×#6923	—	4	2	no	—
31	C₂×C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
33	C₂×C₄×D₆₁	SD₁₆	1	1	yes	characteristic
34	C₄⋊#6923	—	4	2	no	—
35	C₄⋊#6923	—	4	2	no	—
36	C₂×C₈×C₆₁	—	4	1	no	—
37	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
42	C₄×Dic₆₁	Q₁₆	1	1	yes	characteristic
43	C₂×#92418	—	4	1	no	—
44	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
46	#11758189	—	8	2	no	—
47	C₈×D₆₁	—	4	1	no	—
48	C₈×D₆₁	—	4	1	no	—
49	#11758187	—	4	2	no	—
50	C₂²×D₆₁	#102	1	1	yes	characteristic
51	C₂×#6923	—	8	2	no	—
52	C₂×#6923	—	8	2	no	—
53	C₂×#6923	—	8	2	no	—
54	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
57	C₂×Dic₆₁	#103	1	1	yes	characteristic
58	C₂×Dic₆₁	#106	1	1	yes	characteristic
59	C₂×Dic₆₁	#107	1	1	yes	characteristic
60	C₄×D₆₁	—	2	2	no	—
61	C₄×D₆₁	—	2	1	no	—
62	C₄×D₆₁	—	2	1	no	—
63	C₄×D₆₁	—	4	1	no	—
64	C₈×C₆₁	—	2	1	no	—
65	C₈×C₆₁	—	2	1	no	—
66	C₈×C₆₁	—	8	1	no	—
67	#92418	—	2	1	no	—
68	#92418	—	2	1	no	—
69	#92418	—	8	1	no	—
70	#7221	—	61	1	no	maximal, Sylow
71	C₂²×C₆₁	#340	1	1	yes	characteristic
72	C₂×D₆₁	—	2	1	no	—
73	C₂×D₆₁	—	2	1	no	—
74	C₂×D₆₁	#360	1	2	yes	—
75	#6923	—	16	2	no	—
76	C₄×C₆₁	—	2	1	no	—
77	C₄×C₆₁	—	2	1	no	—
78	C₄×C₆₁	H(3,ℤ/4)	1	1	yes	characteristic
79	C₄×C₆₁	#344	1	1	yes	characteristic
80	Dic₆₁	—	2	1	no	—
81	Dic₆₁	—	2	1	no	—
82	Dic₆₁	—	2	1	no	—
83	#2163	—	61	1	no	—
84	#1835	—	61	1	no	—
85	#1566	—	61	1	no	—
86	C₂×C₆₁	#1286	1	1	yes	characteristic
87	C₂×C₆₁	#1386	1	1	yes	characteristic
88	C₂×C₆₁	#1389	1	1	yes	characteristic
89	D₆₁	—	2	2	no	—
90	#423	—	122	1	no	—
91	C₂×C₄×C₈	—	61	1	no	—
92	#382	—	122	1	no	—
93	C₄⋊C₁₆	—	122	1	no	—
94	C₂²⋊C₁₆	—	122	1	no	—
95	#346	—	61	2	no	—
96	#340	—	61	2	no	—
97	#340	—	122	1	no	—
98	#334	—	61	1	no	—
99	#334	—	122	1	no	—
100	#334	#6923	1	1	yes	characteristic, complemented
101	C₆₁	#7221	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	244	1	no	—
103	C₂²×C₈	—	61	1	no	—
104	C₂²×C₈	—	61	1	no	—
105	C₂×C₄⋊C₄	—	122	2	no	—
106	C₂×C₄²	—	61	1	no	—
107	C₂×C₁₆	—	122	2	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	122	1	no	—
110	C₄⋊C₈	—	244	1	no	—
111	C₄×C₈	—	61	1	no	—
112	C₄×C₈	—	61	1	no	—
113	C₄×C₈	—	61	1	no	—
114	C₄×C₈	C₂×#6923	1	1	yes	characteristic
115	#95	—	244	1	no	—
116	#95	—	244	1	no	—
117	C₂²×C₄	—	61	1	no	—
118	C₂²×C₄	—	61	1	no	—
119	C₂²×C₄	—	61	1	no	—
120	C₂²×C₄	—	244	2	no	—
121	M₁₆	—	488	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	61	1	no	—
124	C₂×C₈	—	61	1	no	—
125	C₂×C₈	—	244	1	no	—
126	C₂×C₈	—	244	1	no	—
127	C₂×C₈	—	244	1	no	—
128	C₂×C₈	C₄×#6923	1	1	yes	characteristic
129	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
130	C₄⋊C₄	—	244	2	no	—
131	C₄⋊C₄	—	244	2	no	—
132	C₄²	—	61	1	no	—
133	C₄²	—	61	1	no	—
134	C₄²	—	61	1	no	—
135	C₄²	C₂²⋊#6923	1	1	yes	characteristic
136	C₁₆	—	244	2	no	—
137	C₂³	—	61	1	no	—
138	C₂×C₄	—	61	1	no	—
139	C₂×C₄	—	61	1	no	—
140	C₂×C₄	—	61	1	no	—
141	C₂×C₄	—	122	2	no	—
142	C₂×C₄	—	122	1	no	—
143	C₂×C₄	—	122	1	no	—
144	C₂×C₄	—	488	2	no	—
145	C₂×C₄	—	488	2	no	—
146	C₂×C₄	—	488	2	no	—
147	C₂×C₄	—	244	1	no	—
148	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#14519839	1	1	yes	characteristic
150	C₂×C₄	#14519840	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	8	1	no	—
154	C₈	—	122	1	no	—
155	C₈	—	122	1	no	—
156	C₈	—	488	1	no	—
157	C₂²	—	61	2	no	—
158	C₂²	—	122	1	no	—
159	C₂²	—	122	1	no	—
160	C₂²	#16840635	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	122	1	no	—
164	C₄	—	122	1	no	—
165	C₄	—	122	1	no	—
166	C₄	—	976	2	no	—
167	C₄	#16840630	1	1	yes	characteristic
168	C₄	#16840645	1	1	yes	characteristic
169	C₂	—	122	2	no	—
170	C₂	#21460674	1	1	yes	characteristic
171	C₂	#21460892	1	1	yes	characteristic
172	C₂	#21460899	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462183

C₂

yes

maximal, characteristic

#21461741

C₂

yes

maximal, characteristic

#21444573

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

complemented

#16840636

—

#16840636

C₄

yes

—

#16839784

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839513

C₄

yes

characteristic, complemented

#16839512

—

C₆₁×#334

C₄

yes

characteristic, complemented

C₂×C₄⋊#6923

—

—

—

—

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

D₄

yes

characteristic

C₄²×D₆₁

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

Q₈

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×#6923

—

C₂×C₄×D₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₆₁

D₈

yes

characteristic

C₂×C₄×D₆₁

SD₁₆

yes

characteristic

C₄⋊#6923

—

C₄⋊#6923

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₄²×C₆₁

C₂²⋊C₄

yes

characteristic

C₄×Dic₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

characteristic

C₄×Dic₆₁

Q₁₆

yes

characteristic

C₂×#92418

—

C₂×#92418

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

#11758189

—

C₈×D₆₁

—

C₈×D₆₁

—

#11758187

—

C₂²×D₆₁

yes

—

—

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

yes

C₂×Dic₆₁

yes

C₂×Dic₆₁

yes

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#360

yes

—

#6923

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#2163

—

#1835

—

#1566

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#423

—

122

—

C₂×C₄×C₈

—

#382

—

122

—

C₄⋊C₁₆

—

122

—

C₂²⋊C₁₆

—

122

—

#346

—

#340

—

#340

—

122

—

#334

—

#334

—

122

—

100

#334

#6923

yes

characteristic, complemented

101

C₆₁

#7221

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

244

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂×C₄⋊C₄

—

122

—

106

C₂×C₄²

—

107

C₂×C₁₆

—

122

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

122

—

110

C₄⋊C₈

—

244

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₂×#6923

yes

characteristic

115

#95

—

244

—

116

#95

—

244

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

244

—

121

M₁₆

—

488

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

244

—

126

C₂×C₈

—

244

—

127

C₂×C₈

—

244

—

128

C₂×C₈

C₄×#6923

yes

characteristic

129

C₂×C₈

C₄⋊#6923

yes

characteristic

130

C₄⋊C₄

—

244

—

131

C₄⋊C₄

—

244

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

C₂²⋊#6923

yes

characteristic

136

C₁₆

—

244

—

137

C₂³

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

122

—

142

C₂×C₄

—

122

—

143

C₂×C₄

—

122

—

144

C₂×C₄

—

488

—

145

C₂×C₄

—

488

—

146

C₂×C₄

—

488

—

147

C₂×C₄

—

244

—

148

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

149

C₂×C₄

#14519839

yes

characteristic

150

C₂×C₄

#14519840

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

122

—

155

C₈

—

122

—

156

C₈

—

488

—

157

C₂²

—

158

C₂²

—

122

—

159

C₂²

—

122

—

160

C₂²

#16840635

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

122

—

164

C₄

—

122

—

165

C₄

—

122

—

166

C₄

—

976

—

167

C₄

#16840630

yes

characteristic

168

C₄

#16840645

yes

characteristic

169

C₂

—

122

—

170

C₂

#21460674

yes

characteristic

171

C₂

#21460892

yes

characteristic

172

C₂

#21460899

yes

characteristic

	subgroup	G/H
center	C₂²	#16840635
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₄×C₈	C₂×#6923
Fitting	C₆₁×#334	C₄
socle	C₂²×C₆₁	#340
prime radical	C₂²×D₆₁	#102
cosocle	C₄×C₈×D₆₁	C₂²
hypercenter	#334	#6923
hypocenter	C₆₁	#7221
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#16840635

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₄×C₈

C₂×#6923

Fitting

C₆₁×#334

C₄

C₂²×C₆₁

C₂²×D₆₁

C₄×C₈×D₆₁

C₂²

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
7. #16840651	C₄	semidirect
12. #16839513	C₄	semidirect
14. C₆₁×#334	C₄	semidirect
100. #334	#6923	semidirect
101. C₆₁	#7221	semidirect

H ≅ G/N

type

7. #16840651

C₄

semidirect

12. #16839513

C₄

semidirect

14. C₆₁×#334

C₄

semidirect

100. #334

#6923

semidirect

101. C₆₁

#7221

semidirect

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

subgroup	index
#21462183	2
#21461741	2
#21444573	2
#7221	61

subgroup

index

Subgroups — #30594460

Conjugacy classes of subgroups

Complements

Maximal subgroups