Group #30593985 — subgroups

Subgroups — #30593985

Order 15616.

292 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 102 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	#21461736	C₂	1	2	yes	maximal
2	#21445970	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#92420	—	2	2	no	—
4	#16840628	—	2	2	no	—
5	#16840628	C₄	1	4	yes	—
6	C₂×C₄⋊#92418	C₄	1	1	yes	characteristic
7	C₂×#14519706	—	2	2	no	—
8	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
10	C₂²×#92420	—	2	4	no	—
11	C₄⋊#92420	—	4	2	no	—
12	C₄⋊#92420	—	4	2	no	—
13	C₂²×#92418	—	2	1	no	—
14	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
16	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
17	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
18	C₄⋊#92418	—	4	1	no	—
19	#14519706	—	4	2	no	—
20	#14519706	—	4	2	no	—
21	C₂²×C₈×C₆₁	—	2	1	no	—
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₈×C₆₁	—	4	1	no	—
24	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×Dic₆₁	C₂²⋊C₄	1	2	yes	—
28	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
29	C₂×#92420	—	4	4	no	—
30	C₂×#92420	—	4	4	no	—
31	C₂×#92420	—	4	4	no	—
32	C₂×C₈×C₆₁	—	4	1	no	—
33	C₂×C₈×C₆₁	—	4	2	no	—
34	C₄²×C₆₁	—	2	1	no	—
35	C₄²×C₆₁	—	2	1	no	—
36	C₄×Dic₆₁	—	2	1	no	—
37	C₄×Dic₆₁	—	2	1	no	—
38	C₄×Dic₆₁	—	4	2	no	—
39	C₄×Dic₆₁	—	4	1	no	—
40	C₄×Dic₆₁	D₈	1	2	yes	—
41	C₄×Dic₆₁	SD₁₆	1	2	yes	—
42	C₄×Dic₆₁	SD₁₆	1	2	yes	—
43	C₄×Dic₆₁	Q₁₆	1	2	yes	—
44	C₂×#92418	—	4	1	no	—
45	C₂×#92418	—	4	2	no	—
46	C₂³×C₆₁	#95	1	1	yes	characteristic
47	C₂×C₄×C₆₁	—	2	1	no	—
48	C₂×C₄×C₆₁	—	2	1	no	—
49	C₂×C₄×C₆₁	—	2	1	no	—
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#102	1	2	yes	—
54	C₂×C₄×C₆₁	#103	1	2	yes	—
55	C₂×C₄×C₆₁	#108	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#108	1	1	yes	characteristic
57	C₂×Dic₆₁	—	2	2	no	—
58	C₂×Dic₆₁	—	2	2	no	—
59	C₂×Dic₆₁	—	2	2	no	—
60	C₂×Dic₆₁	—	2	2	no	—
61	C₂×Dic₆₁	#102	1	2	yes	—
62	C₂×Dic₆₁	#103	1	2	yes	—
63	C₂×Dic₆₁	#106	1	2	yes	—
64	C₂×Dic₆₁	#107	1	2	yes	—
65	#92420	—	8	4	no	—
66	C₈×C₆₁	—	8	1	no	—
67	#92418	—	8	1	no	—
68	#7040	—	61	1	no	maximal, Sylow
69	C₂²×C₆₁	H(3,ℤ/4)	1	1	yes	characteristic
70	C₂²×C₆₁	#340	1	2	yes	—
71	C₂²×C₆₁	#341	1	1	yes	characteristic
72	C₂²×C₆₁	#343	1	1	yes	characteristic
73	C₂²×C₆₁	#344	1	2	yes	—
74	C₄×C₆₁	—	2	1	no	—
75	C₄×C₆₁	—	2	1	no	—
76	C₄×C₆₁	—	4	1	no	—
77	C₄×C₆₁	#331	1	1	yes	characteristic
78	C₄×C₆₁	#332	1	2	yes	—
79	C₄×C₆₁	#333	1	1	yes	characteristic
80	Dic₆₁	—	2	2	no	—
81	Dic₆₁	—	2	2	no	—
82	Dic₆₁	—	2	2	no	—
83	Dic₆₁	—	2	2	no	—
84	#1842	—	61	1	no	—
85	#1831	—	61	2	no	—
86	C₂×C₆₁	#1286	1	2	yes	—
87	C₂×C₆₁	#1290	1	1	yes	characteristic
88	C₂×C₆₁	#1296	1	1	yes	characteristic
89	C₂×C₆₁	#1301	1	2	yes	—
90	C₂×C₆₁	#1307	1	1	yes	characteristic
91	C₂×C₄⋊C₈	—	61	1	no	—
92	C₂×C₄⋊C₈	—	122	2	no	—
93	C₂×C₄⋊C₈	—	122	2	no	—
94	C₂×C₄⋊C₈	#6923	1	1	yes	characteristic
95	C₄³	—	61	1	no	—
96	#336	—	61	4	no	—
97	#336	—	122	2	no	—
98	C₆₁	#7040	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	—	122	4	no	—
101	C₂²×C₈	—	122	1	no	—
102	C₂×C₄²	—	61	2	no	—
103	C₂×C₄²	—	61	1	no	—
104	C₂×C₄²	—	61	2	no	—
105	C₂×C₄²	—	61	1	no	—
106	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
107	C₄⋊C₈	—	4	1	no	—
108	C₄⋊C₈	—	244	1	no	—
109	C₄⋊C₈	—	244	2	no	—
110	C₄⋊C₈	—	244	2	no	—
111	C₄⋊C₈	—	244	2	no	—
112	C₄⋊C₈	—	244	2	no	—
113	C₂²×C₄	—	61	2	no	—
114	C₂²×C₄	—	61	2	no	—
115	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
116	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	244	4	no	—
121	C₂×C₈	—	244	4	no	—
122	C₂×C₈	—	244	4	no	—
123	C₂×C₈	—	244	1	no	—
124	C₂×C₈	—	244	2	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	61	2	no	—
128	C₄²	—	61	2	no	—
129	C₄²	—	61	2	no	—
130	C₄²	—	61	2	no	—
131	C₄²	—	122	1	no	—
132	C₄²	—	122	1	no	—
133	C₄²	—	244	1	no	—
134	C₄²	—	244	2	no	—
135	C₂³	(C₂×#6923)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	61	2	no	—
141	C₂×C₄	—	61	2	no	—
142	C₂×C₄	—	61	2	no	—
143	C₂×C₄	—	61	2	no	—
144	C₂×C₄	—	122	2	no	—
145	C₂×C₄	—	122	2	no	—
146	C₂×C₄	—	122	2	no	—
147	C₂×C₄	—	122	2	no	—
148	C₂×C₄	#14519837	1	1	yes	characteristic
149	C₂×C₄	#14519838	1	1	yes	characteristic
150	C₂×C₄	#14519839	1	2	yes	—
151	C₂×C₄	#14519840	1	2	yes	—
152	C₂×C₄	#14519847	1	1	yes	characteristic
153	C₂×C₄	#14519848	1	1	yes	characteristic
154	C₈	—	8	1	no	—
155	C₈	—	488	4	no	—
156	C₈	—	488	1	no	—
157	C₂²	#16840630	1	1	yes	characteristic
158	C₂²	#16840635	1	2	yes	—
159	C₂²	#16840638	1	1	yes	characteristic
160	C₂²	#16840642	1	1	yes	characteristic
161	C₂²	#16840645	1	2	yes	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	122	2	no	—
166	C₄	—	122	2	no	—
167	C₄	—	122	2	no	—
168	C₄	—	122	2	no	—
169	C₄	#16840620	1	1	yes	characteristic
170	C₄	#16840621	1	2	yes	—
171	C₄	#16840622	1	1	yes	characteristic
172	C₂	#21460674	1	2	yes	—
173	C₂	#21460684	1	1	yes	characteristic
174	C₂	#21460697	1	1	yes	characteristic
175	C₂	#21460711	1	2	yes	—
176	C₂	#21460725	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461736

C₂

yes

maximal

#21445970

C₂

yes

maximal, characteristic

C₂×C₄⋊#92420

—

#16840628

—

#16840628

C₄

yes

—

C₂×C₄⋊#92418

C₄

yes

characteristic

C₂×#14519706

—

C₄²×Dic₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₆₁

C₄

yes

characteristic

C₂²×#92420

—

C₄⋊#92420

—

C₄⋊#92420

—

C₂²×#92418

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

D₄

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

—

C₂×C₄×Dic₆₁

Q₈

yes

—

—

—

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²×C₄×C₆₁

C₄²

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×Dic₆₁

C₂²⋊C₄

yes

—

C₂²×Dic₆₁

C₄⋊C₄

yes

—

C₂×#92420

—

C₂×#92420

—

C₂×#92420

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₈

yes

—

C₄×Dic₆₁

SD₁₆

yes

—

C₄×Dic₆₁

SD₁₆

yes

—

C₄×Dic₆₁

Q₁₆

yes

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#95

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#103

yes

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#102

yes

—

C₂×Dic₆₁

#103

yes

—

C₂×Dic₆₁

#106

yes

—

C₂×Dic₆₁

#107

yes

—

#92420

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#331

yes

characteristic

C₄×C₆₁

#332

yes

—

C₄×C₆₁

#333

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#1842

—

#1831

—

C₂×C₆₁

#1286

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#1307

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

122

—

C₂×C₄⋊C₈

—

122

—

C₂×C₄⋊C₈

#6923

yes

characteristic

C₄³

—

#336

—

#336

—

122

—

C₆₁

#7040

yes

characteristic, complemented, Sylow

C₂²×C₈

—

100

C₂²×C₈

—

122

—

101

C₂²×C₈

—

122

—

102

C₂×C₄²

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×#6923

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

244

—

109

C₄⋊C₈

—

244

—

110

C₄⋊C₈

—

244

—

111

C₄⋊C₈

—

244

—

112

C₄⋊C₈

—

244

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₄×#6923

yes

characteristic

116

C₂²×C₄

C₄⋊#6923

yes

characteristic

117

C₂²×C₄

C₂²⋊#6923

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

244

—

121

C₂×C₈

—

244

—

122

C₂×C₈

—

244

—

123

C₂×C₈

—

244

—

124

C₂×C₈

—

244

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

122

—

132

C₄²

—

122

—

133

C₄²

—

244

—

134

C₄²

—

244

—

135

C₂³

(C₂×#6923)⋊C₄

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

122

—

145

C₂×C₄

—

122

—

146

C₂×C₄

—

122

—

147

C₂×C₄

—

122

—

148

C₂×C₄

#14519837

yes

characteristic

149

C₂×C₄

#14519838

yes

characteristic

150

C₂×C₄

#14519839

yes

—

151

C₂×C₄

#14519840

yes

—

152

C₂×C₄

#14519847

yes

characteristic

153

C₂×C₄

#14519848

yes

characteristic

154

C₈

—

155

C₈

—

488

—

156

C₈

—

488

—

157

C₂²

#16840630

yes

characteristic

158

C₂²

#16840635

yes

—

159

C₂²

#16840638

yes

characteristic

160

C₂²

#16840642

yes

characteristic

161

C₂²

#16840645

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

122

—

166

C₄

—

122

—

167

C₄

—

122

—

168

C₄

—

122

—

169

C₄

#16840620

yes

characteristic

170

C₄

#16840621

yes

—

171

C₄

#16840622

yes

characteristic

172

C₂

#21460674

yes

—

173

C₂

#21460684

yes

characteristic

174

C₂

#21460697

yes

characteristic

175

C₂

#21460711

yes

—

176

C₂

#21460725

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×#6923)⋊C₄
commutator	C₂²×C₄×C₆₁	C₄²
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₂×C₄⋊C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	#95
cosocle	C₄²×Dic₆₁	C₂²
hypercenter	C₂×C₄⋊C₈	#6923
hypocenter	C₆₁	#7040
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

(C₂×#6923)⋊C₄

commutator

C₂²×C₄×C₆₁

C₄²

Frattini

C₂×C₄²

C₂×#6923

Fitting

C₂×C₄⋊C₈×C₆₁

C₄

socle, prime radical

C₂³×C₆₁

#95

cosocle

C₄²×Dic₆₁

C₂²

C₂×C₄⋊C₈

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
98. C₆₁	#7040	semidirect

H ≅ G/N

type

98. C₆₁

#7040

semidirect

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

subgroup	index
#21461736	2
#21445970	2
#7040	61

subgroup

index

#21461736

#21445970

#7040

Subgroups — #30593985

Conjugacy classes of subgroups

Complements

Maximal subgroups