Group #30594527 — subgroups

Subgroups — #30594527

Order 15616.

192 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 64 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	#21462152	C₂	1	1	yes	maximal, characteristic
2	#21461375	C₂	1	1	yes	maximal, characteristic
3	#21444581	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519852	C₄	1	2	yes	—
5	#16840766	—	2	1	no	—
6	#16840683	—	2	1	no	—
7	C₄⋊#11758187	—	2	1	no	—
8	#16840624	—	2	1	no	—
9	#16840624	C₄	1	2	yes	—
10	#16839818	—	2	1	no	—
11	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
12	#16839515	C₄	1	1	yes	characteristic
13	#16839512	—	2	1	no	—
14	C₆₁×#335	C₄	1	1	yes	characteristic
15	C₂×#11758213	—	4	1	no	—
16	#14519852	—	4	2	no	—
17	C₂×#11758187	—	2	2	no	—
18	C₄⋊#92420	—	2	2	no	—
19	C₄⋊#92420	—	4	1	no	—
20	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×D₆₁	Q₈	1	1	yes	characteristic
22	C₄²×D₆₁	D₄	1	1	yes	characteristic
23	C₄⋊#92418	—	2	1	no	—
24	#14519689	—	4	1	no	—
25	#14519687	—	4	1	no	—
26	C₄×#92418	D₄	1	1	yes	characteristic
27	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₆₁	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₆₁	—	2	1	no	—
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
33	C₂×C₄×D₆₁	Q₁₆	1	1	yes	characteristic
34	C₂×#92420	—	4	2	no	—
35	#11758213	—	8	1	no	—
36	#11758213	—	8	1	no	—
37	C₂×C₈×C₆₁	—	4	1	no	—
38	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
43	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
44	C₂×#92418	—	4	1	no	—
45	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
47	C₈×D₆₁	—	2	1	no	—
48	C₈×D₆₁	—	2	1	no	—
49	C₈×D₆₁	—	4	1	no	—
50	#11758187	—	4	2	no	—
51	C₂²×D₆₁	#107	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
55	C₂×Dic₆₁	#102	1	1	yes	characteristic
56	C₂×Dic₆₁	#103	1	1	yes	characteristic
57	C₂×Dic₆₁	#106	1	1	yes	characteristic
58	C₄×D₆₁	—	2	1	no	—
59	C₄×D₆₁	—	2	1	no	—
60	C₄×D₆₁	—	2	1	no	—
61	C₄×D₆₁	—	2	1	no	—
62	C₄×D₆₁	—	4	1	no	—
63	#92420	—	8	2	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	—	8	1	no	—
69	#92418	#100	1	1	yes	characteristic
70	#92418	#101	1	1	yes	characteristic
71	#7244	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	#340	1	1	yes	characteristic
73	C₂×D₆₁	—	2	1	no	—
74	C₂×D₆₁	—	2	1	no	—
75	C₂×D₆₁	—	2	1	no	—
76	C₄×C₆₁	—	2	1	no	—
77	C₄×C₆₁	#338	1	1	yes	characteristic
78	C₄×C₆₁	#343	1	1	yes	characteristic
79	C₄×C₆₁	#360	1	1	yes	characteristic
80	C₄×C₆₁	#360	1	1	yes	characteristic
81	Dic₆₁	—	2	1	no	—
82	Dic₆₁	—	2	1	no	—
83	Dic₆₁	—	2	1	no	—
84	#2150	—	61	1	no	—
85	#1568	—	61	1	no	—
86	#1566	—	61	1	no	—
87	C₂×C₆₁	#1292	1	1	yes	characteristic
88	C₂×C₆₁	#1386	1	1	yes	characteristic
89	C₂×C₆₁	#1388	1	1	yes	characteristic
90	D₆₁	—	4	1	no	—
91	C₂×M₃₂	—	61	2	no	—
92	#432	—	122	1	no	—
93	#423	—	122	1	no	—
94	C₂×C₄×C₈	—	61	1	no	—
95	C₄⋊C₁₆	—	122	1	no	—
96	C₄⋊C₁₆	—	122	1	no	—
97	#335	—	61	1	no	—
98	#335	#6923	1	1	yes	characteristic
99	#334	—	61	2	no	—
100	#334	—	122	1	no	—
101	#334	—	122	1	no	—
102	C₆₁	#7244	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	244	1	no	—
104	C₂²×C₈	—	61	1	no	—
105	C₂²×C₈	—	61	1	no	—
106	C₂×C₄²	—	61	1	no	—
107	M₃₂	—	244	2	no	—
108	C₂×C₁₆	—	122	2	no	—
109	C₄⋊C₈	—	2	1	no	—
110	C₄⋊C₈	—	122	2	no	—
111	C₄⋊C₈	—	122	1	no	—
112	C₄⋊C₈	—	244	1	no	—
113	C₂²⋊C₈	—	244	1	no	—
114	#97	—	244	1	no	—
115	C₄×C₈	—	61	1	no	—
116	C₄×C₈	—	61	1	no	—
117	C₄×C₈	—	61	1	no	—
118	C₄×C₈	C₂×#6923	1	1	yes	characteristic
119	C₂²×C₄	—	61	1	no	—
120	C₂²×C₄	—	61	1	no	—
121	C₂²×C₄	—	61	1	no	—
122	M₁₆	—	488	1	no	—
123	M₁₆	—	488	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	61	1	no	—
126	C₂×C₈	—	61	1	no	—
127	C₂×C₈	—	122	1	no	—
128	C₂×C₈	—	122	1	no	—
129	C₂×C₈	—	244	1	no	—
130	C₂×C₈	—	244	2	no	—
131	C₂×C₈	—	244	1	no	—
132	C₂×C₈	C₄×#6923	1	1	yes	characteristic
133	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
134	C₄²	—	61	1	no	—
135	C₄²	—	61	1	no	—
136	C₄²	—	61	1	no	—
137	C₄²	C₂²⋊#6923	1	1	yes	characteristic
138	C₁₆	—	244	2	no	—
139	C₂³	—	61	1	no	—
140	C₂×C₄	—	61	1	no	—
141	C₂×C₄	—	61	1	no	—
142	C₂×C₄	—	61	1	no	—
143	C₂×C₄	—	122	1	no	—
144	C₂×C₄	—	122	1	no	—
145	C₂×C₄	—	122	1	no	—
146	C₂×C₄	—	122	1	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	—	61	1	no	—
155	C₈	—	61	1	no	—
156	C₈	—	122	1	no	—
157	C₈	—	488	2	no	—
158	C₈	—	488	1	no	—
159	C₂²	—	122	1	no	—
160	C₂²	—	122	1	no	—
161	C₂²	—	122	1	no	—
162	C₂²	#16840635	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	122	1	no	—
165	C₄	—	122	1	no	—
166	C₄	—	122	1	no	—
167	C₄	#16840631	1	1	yes	characteristic
168	C₄	#16840641	1	1	yes	characteristic
169	C₄	#16840678	1	1	yes	characteristic
170	C₄	#16840679	1	1	yes	characteristic
171	C₂	—	244	1	no	—
172	C₂	#21460688	1	1	yes	characteristic
173	C₂	#21460890	1	1	yes	characteristic
174	C₂	#21460898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462152

C₂

yes

maximal, characteristic

#21461375

C₂

yes

maximal, characteristic

#21444581

C₂

yes

maximal, characteristic

C₂×#14519852

C₄

yes

—

—

—

—

—

C₄

yes

—

#16839818

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839515

C₄

yes

characteristic

#16839512

—

C₆₁×#335

C₄

yes

—

—

—

—

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

Q₈

yes

characteristic

C₄²×D₆₁

D₄

yes

—

—

—

D₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×D₆₁

D₈

yes

characteristic

C₂×C₄×D₆₁

Q₁₆

yes

—

—

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₄²×C₆₁

C₂²⋊C₄

yes

characteristic

C₄×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

characteristic

C₂×#92418

—

C₂×#92418

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₈×D₆₁

—

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

—

C₄×D₆₁

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

yes

yes

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#2150

—

#1568

—

#1566

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₂×M₃₂

—

#432

—

122

—

#423

—

122

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

122

—

C₄⋊C₁₆

—

122

—

—

yes

—

100

#334

—

122

—

101

#334

—

122

—

102

C₆₁

#7244

yes

characteristic, complemented, Sylow

103

C₂×M₁₆

—

244

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

M₃₂

—

244

—

108

C₂×C₁₆

—

122

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

122

—

111

C₄⋊C₈

—

122

—

112

C₄⋊C₈

—

244

—

113

C₂²⋊C₈

—

244

—

114

#97

—

244

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

C₂×#6923

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

M₁₆

—

488

—

123

M₁₆

—

488

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

122

—

128

C₂×C₈

—

122

—

129

C₂×C₈

—

244

—

130

C₂×C₈

—

244

—

131

C₂×C₈

—

244

—

132

C₂×C₈

C₄×#6923

yes

characteristic

133

C₂×C₈

C₄⋊#6923

yes

characteristic

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂²⋊#6923

yes

characteristic

138

C₁₆

—

244

—

139

C₂³

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

122

—

144

C₂×C₄

—

122

—

145

C₂×C₄

—

122

—

146

C₂×C₄

—

122

—

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

—

155

C₈

—

156

C₈

—

122

—

157

C₈

—

488

—

158

C₈

—

488

—

159

C₂²

—

122

—

160

C₂²

—

122

—

161

C₂²

—

122

—

162

C₂²

#16840635

yes

characteristic

163

C₄

—

164

C₄

—

122

—

165

C₄

—

122

—

166

C₄

—

122

—

167

C₄

#16840631

yes

characteristic

168

C₄

#16840641

yes

characteristic

169

C₄

#16840678

yes

characteristic

170

C₄

#16840679

yes

characteristic

171

C₂

—

244

—

172

C₂

#21460688

yes

characteristic

173

C₂

#21460890

yes

characteristic

174

C₂

#21460898

yes

characteristic

	subgroup	G/H
center	C₂²	#16840635
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₄×C₈	C₂×#6923
Fitting	C₆₁×#335	C₄
socle	C₂²×C₆₁	#340
prime radical	C₂²×D₆₁	#107
cosocle	C₄×C₈×D₆₁	C₂²
hypercenter	#335	#6923
hypocenter	C₆₁	#7244
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂²

#16840635

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₄×C₈

C₂×#6923

Fitting

C₆₁×#335

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 ◁ #335

N	H ≅ G/N	type
102. C₆₁	#7244	semidirect

H ≅ G/N

type

102. C₆₁

#7244

semidirect

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

subgroup	index
#21462152	2
#21461375	2
#21444581	2
#7244	61

subgroup

index

Subgroups — #30594527

Conjugacy classes of subgroups

Complements

Maximal subgroups