Group #30594290 — subgroups

Subgroups — #30594290

Order 15616.

246 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₄×#14519848	C₂	1	1	yes	maximal, characteristic
2	#21461369	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
4	#16840781	—	2	1	no	—
5	C₂×#14519848	C₄	1	2	yes	—
6	C₄×#11758213	—	2	1	no	—
7	#16840638	—	2	1	no	—
8	#16840625	C₄	1	2	yes	—
9	#16840624	—	2	1	no	—
10	C₂×C₁₆×D₆₁	—	2	1	no	—
11	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
12	C₄×#11758185	C₄	1	1	yes	characteristic
13	C₁₆×Dic₆₁	—	2	1	no	—
14	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
15	C₂×#11758213	—	2	2	no	—
16	#14519848	—	2	2	no	—
17	#14519848	C₈	1	4	yes	complemented
18	C₄⋊#92420	—	2	2	no	—
19	#14519834	—	4	1	no	—
20	#14519832	—	4	1	no	—
21	C₄×#92420	—	4	1	no	—
22	C₄×#92420	—	4	1	no	—
23	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
25	C₄²×D₆₁	D₄	1	1	yes	characteristic
26	C₂×#11758185	—	2	1	no	—
27	C₁₆×D₆₁	—	2	4	no	—
28	C₄×#92418	Q₈	1	1	yes	characteristic
29	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₆₁	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₆₁	—	2	1	no	—
32	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×D₆₁	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×D₆₁	Q₁₆	1	1	yes	characteristic
36	C₂×#92420	—	4	2	no	—
37	C₂×#92420	—	4	2	no	—
38	#11758213	—	2	4	no	—
39	#11758213	—	4	2	no	—
40	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×Dic₆₁	D₈	1	1	yes	characteristic
45	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
46	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
47	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
48	C₈×D₆₁	—	2	2	no	—
49	C₈×D₆₁	C₂×C₈	1	2	yes	—
50	C₈×D₆₁	M₁₆	1	2	yes	—
51	C₁₆×C₆₁	—	2	2	no	—
52	#11758185	—	2	2	no	—
53	C₂²×D₆₁	#103	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
57	C₂×Dic₆₁	#102	1	1	yes	characteristic
58	C₂×Dic₆₁	#102	1	1	yes	characteristic
59	C₂×Dic₆₁	#103	1	1	yes	characteristic
60	C₄×D₆₁	—	2	1	no	—
61	C₄×D₆₁	—	2	1	no	—
62	C₄×D₆₁	—	2	2	no	—
63	C₄×D₆₁	C₂²⋊C₈	1	2	yes	—
64	C₄×D₆₁	#104	1	2	yes	—
65	#92420	—	4	4	no	—
66	#92420	—	8	2	no	—
67	C₈×C₆₁	—	2	1	no	—
68	C₈×C₆₁	C₄×C₈	1	1	yes	characteristic
69	C₈×C₆₁	#97	1	1	yes	characteristic
70	#92418	—	2	1	no	—
71	#92418	C₄⋊C₈	1	2	yes	—
72	#7172	—	61	1	no	maximal, Sylow
73	C₂²×C₆₁	#340	1	1	yes	characteristic
74	C₂×D₆₁	—	2	1	no	—
75	C₂×D₆₁	—	2	1	no	—
76	C₂×D₆₁	Q₈⋊C₈	1	2	yes	—
77	C₄×C₆₁	—	2	1	no	—
78	C₄×C₆₁	#336	1	1	yes	characteristic
79	C₄×C₆₁	#339	1	1	yes	characteristic
80	C₄×C₆₁	#368	1	1	yes	characteristic
81	C₄×C₆₁	#368	1	1	yes	characteristic
82	Dic₆₁	—	2	1	no	—
83	Dic₆₁	D₄⋊C₈	1	2	yes	—
84	Dic₆₁	#359	1	2	yes	—
85	C₂×C₄×C₁₆	—	61	1	no	—
86	C₄×#108	—	61	1	no	—
87	#1564	—	61	1	no	—
88	C₂×C₆₁	#1279	1	1	yes	characteristic
89	C₂×C₆₁	#1381	1	1	yes	characteristic
90	C₂×C₆₁	#1382	1	1	yes	characteristic
91	D₆₁	—	2	2	no	—
92	C₂²×C₁₆	—	122	1	no	—
93	#431	—	122	1	no	—
94	C₂×#108	—	61	2	no	—
95	C₄×M₁₆	—	122	1	no	—
96	C₂×C₄×C₈	—	61	1	no	—
97	C₄×C₁₆	—	61	1	no	—
98	C₄×C₁₆	—	122	1	no	—
99	C₄×C₁₆	#6923	1	1	yes	characteristic
100	#341	—	122	1	no	—
101	#335	—	61	2	no	—
102	#334	—	122	1	no	—
103	C₆₁	#7172	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₁₆	—	122	2	no	—
105	C₂²×C₈	—	61	1	no	—
106	C₂²×C₈	—	61	1	no	—
107	C₂×C₄²	—	61	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	122	4	no	—
110	C₂×C₁₆	—	122	1	no	—
111	#108	—	61	4	no	—
112	#108	—	122	2	no	—
113	C₄⋊C₈	—	122	2	no	—
114	C₂²⋊C₈	—	244	1	no	—
115	#97	—	244	1	no	—
116	C₄×C₈	—	61	1	no	—
117	C₄×C₈	—	61	1	no	—
118	C₄×C₈	—	61	1	no	—
119	C₄×C₈	—	244	1	no	—
120	C₄×C₈	—	244	1	no	—
121	C₄×C₈	C₂×#6923	1	1	yes	characteristic
122	C₂²×C₄	—	61	1	no	—
123	C₂²×C₄	—	61	1	no	—
124	C₂²×C₄	—	61	1	no	—
125	M₁₆	—	122	4	no	—
126	M₁₆	—	244	2	no	—
127	C₂×C₈	—	61	1	no	—
128	C₂×C₈	—	61	2	no	—
129	C₂×C₈	—	61	2	no	—
130	C₂×C₈	—	61	1	no	—
131	C₂×C₈	—	122	2	no	—
132	C₂×C₈	—	244	2	no	—
133	C₂×C₈	—	244	2	no	—
134	C₂×C₈	C₄×#6923	1	1	yes	characteristic
135	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
136	C₄²	—	61	1	no	—
137	C₄²	—	61	1	no	—
138	C₄²	—	61	1	no	—
139	C₄²	C₄⋊#6923	1	1	yes	characteristic
140	C₁₆	—	2	2	no	—
141	C₁₆	—	122	2	no	—
142	C₂³	—	61	1	no	—
143	C₂×C₄	—	61	1	no	—
144	C₂×C₄	—	61	1	no	—
145	C₂×C₄	—	61	2	no	—
146	C₂×C₄	—	61	2	no	—
147	C₂×C₄	—	61	1	no	—
148	C₂×C₄	—	122	2	no	—
149	C₂×C₄	—	122	1	no	—
150	C₂×C₄	—	122	1	no	—
151	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
152	C₂×C₄	#14519845	1	1	yes	characteristic
153	C₂×C₄	#14519846	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	61	2	no	—
156	C₈	—	122	1	no	—
157	C₈	—	488	2	no	—
158	C₈	—	244	4	no	—
159	C₈	C₈×#6923	1	1	yes	characteristic
160	C₈	#14519831	1	1	yes	characteristic
161	C₂²	—	61	2	no	—
162	C₂²	—	122	1	no	—
163	C₂²	—	122	1	no	—
164	C₂²	#16840636	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	61	2	no	—
167	C₄	—	61	2	no	—
168	C₄	—	122	1	no	—
169	C₄	(C₂×#6923)⋊C₈	1	1	yes	characteristic
170	C₄	#16840633	1	1	yes	characteristic
171	C₄	#16840690	1	1	yes	characteristic
172	C₄	#16840691	1	1	yes	characteristic
173	C₂	—	122	2	no	—
174	C₂	#21460654	1	1	yes	characteristic
175	C₂	#21460879	1	1	yes	characteristic
176	C₂	#21460881	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14519848

C₂

yes

maximal, characteristic

#21461369

C₂

yes

maximal, characteristic

C₄×C₁₆×D₆₁

C₂

yes

maximal, characteristic

#16840781

—

C₂×#14519848

C₄

yes

—

C₄×#11758213

—

#16840638

—

#16840625

C₄

yes

—

#16840624

—

C₂×C₁₆×D₆₁

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

C₄×#11758185

C₄

yes

characteristic

C₁₆×Dic₆₁

—

C₄×C₁₆×C₆₁

C₄

yes

—

—

C₈

yes

—

—

—

—

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

D₄

yes

characteristic

C₄²×D₆₁

D₄

yes

characteristic

C₂×#11758185

—

C₁₆×D₆₁

—

C₄×#92418

Q₈

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₆₁

SD₁₆

yes

characteristic

C₂×C₄×D₆₁

Q₁₆

yes

—

—

—

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₄×Dic₆₁

D₈

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₈×D₆₁

—

C₈×D₆₁

C₂×C₈

yes

—

C₈×D₆₁

M₁₆

yes

—

C₁₆×C₆₁

—

#11758185

—

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

C₂²⋊C₈

yes

—

C₄×D₆₁

#104

yes

—

#92420

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

C₄×C₈

yes

C₈×C₆₁

yes

—

C₄⋊C₈

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

Q₈⋊C₈

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

D₄⋊C₈

yes

—

Dic₆₁

#359

yes

—

C₂×C₄×C₁₆

—

C₄×#108

—

#1564

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₂²×C₁₆

—

122

—

#431

—

122

—

C₂×#108

—

C₄×M₁₆

—

122

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

122

—

C₄×C₁₆

#6923

yes

characteristic

100

#341

—

122

—

101

#335

—

102

#334

—

122

—

103

C₆₁

#7172

yes

characteristic, complemented, Sylow

104

C₂×M₁₆

—

122

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄²

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

122

—

110

C₂×C₁₆

—

122

—

111

#108

—

112

#108

—

122

—

113

C₄⋊C₈

—

122

—

114

C₂²⋊C₈

—

244

—

115

#97

—

244

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

244

—

120

C₄×C₈

—

244

—

121

C₄×C₈

C₂×#6923

yes

characteristic

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

M₁₆

—

122

—

126

M₁₆

—

244

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

122

—

132

C₂×C₈

—

244

—

133

C₂×C₈

—

244

—

134

C₂×C₈

C₄×#6923

yes

characteristic

135

C₂×C₈

C₂²⋊#6923

yes

characteristic

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₄⋊#6923

yes

characteristic

140

C₁₆

—

141

C₁₆

—

122

—

142

C₂³

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

122

—

149

C₂×C₄

—

122

—

150

C₂×C₄

—

122

—

151

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

152

C₂×C₄

#14519845

yes

characteristic

153

C₂×C₄

#14519846

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

—

122

—

157

C₈

—

488

—

158

C₈

—

244

—

159

C₈

C₈×#6923

yes

characteristic

160

C₈

#14519831

yes

characteristic

161

C₂²

—

162

C₂²

—

122

—

163

C₂²

—

122

—

164

C₂²

#16840636

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

122

—

169

C₄

(C₂×#6923)⋊C₈

yes

characteristic

170

C₄

#16840633

yes

characteristic

171

C₄

#16840690

yes

characteristic

172

C₄

#16840691

yes

characteristic

173

C₂

—

122

—

174

C₂

#21460654

yes

characteristic

175

C₂

#21460879

yes

characteristic

176

C₂

#21460881

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14519846
commutator	C₈×C₆₁	C₄×C₈
Frattini	C₄×C₈	C₂×#6923
Fitting	C₄×C₁₆×C₆₁	C₄
socle	C₂²×C₆₁	#340
prime radical	C₂²×D₆₁	#103
cosocle	C₄×C₈×D₆₁	C₂²
hypercenter	C₄×C₁₆	#6923
hypocenter	C₆₁	#7172
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂×C₄

#14519846

commutator

C₈×C₆₁

C₄×C₈

Frattini

C₄×C₈

C₂×#6923

Fitting

C₄×C₁₆×C₆₁

C₄

C₂²×C₆₁

C₂²×D₆₁

C₄×C₈×D₆₁

C₂²

C₄×C₁₆

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
17. #14519848	C₈	semidirect
103. C₆₁	#7172	semidirect

H ≅ G/N

type

17. #14519848

C₈

semidirect

103. C₆₁

#7172

semidirect

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

subgroup	index
C₄×#14519848	2
#21461369	2
C₄×C₁₆×D₆₁	2
#7172	61

subgroup

index

C₄×#14519848

#21461369

C₄×C₁₆×D₆₁

#7172

Subgroups — #30594290

Conjugacy classes of subgroups

Complements

Maximal subgroups