Group #29573798 — subgroups

Subgroups — #29573798

Order 15616.

438 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 227 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	#21445979	C₂	1	1	yes	maximal, characteristic
2	#21445976	C₂	1	2	yes	maximal
3	#21445976	C₂	1	1	yes	maximal, characteristic
4	#21445947	C₂	1	2	yes	maximal
5	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519711	—	2	2	no	—
7	C₂×#14519709	—	2	4	no	—
8	C₂×#14519709	—	2	2	no	—
9	#16839632	C₂²	1	1	yes	characteristic
10	#16839632	C₂²	1	2	yes	—
11	#16839632	C₂²	1	1	yes	characteristic
12	#16839525	—	2	4	no	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#322	C₄	1	4	yes	complemented
16	C₂×C₄⋊Dic₆₁	—	2	2	no	—
17	C₂×C₄⋊Dic₆₁	—	2	2	no	—
18	C₂×C₄⋊Dic₆₁	—	2	2	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	2	1	no	—
22	C₂×C₄⋊Dic₆₁	—	2	1	no	—
23	#14519711	—	4	4	no	—
24	#14519709	—	4	8	no	—
25	#14519709	—	4	4	no	—
26	C₂²×C₈×C₆₁	D₄	1	4	yes	—
27	C₂²×C₈×C₆₁	D₄	1	2	yes	—
28	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₁	Q₈	1	4	yes	—
32	C₄×C₈×C₆₁	Q₈	1	2	yes	—
33	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
35	C₂²×Dic₆₁	—	4	1	no	—
36	C₂²×Dic₆₁	—	4	1	no	—
37	C₂²×Dic₆₁	—	4	2	no	—
38	C₂×C₈×C₆₁	—	2	4	no	—
39	C₂×C₈×C₆₁	—	2	4	no	—
40	C₂×C₈×C₆₁	C₄⋊C₄	1	8	yes	—
41	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
42	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
43	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
44	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
47	C₄⋊Dic₆₁	—	4	4	no	—
48	C₄⋊Dic₆₁	—	4	2	no	—
49	C₄⋊Dic₆₁	—	4	2	no	—
50	C₄⋊Dic₆₁	—	4	4	no	—
51	C₄⋊Dic₆₁	—	4	4	no	—
52	C₄⋊Dic₆₁	—	4	4	no	—
53	C₄⋊Dic₆₁	—	4	4	no	—
54	C₂³×C₆₁	#127	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
61	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
62	C₂×C₄×C₆₁	#137	1	4	yes	—
63	C₂×Dic₆₁	—	8	2	no	—
64	C₂×Dic₆₁	—	8	1	no	—
65	C₂×Dic₆₁	—	8	1	no	—
66	C₂×Dic₆₁	—	8	2	no	—
67	C₂×Dic₆₁	—	8	2	no	—
68	C₂×Dic₆₁	—	8	2	no	—
69	C₂×Dic₆₁	—	8	2	no	—
70	C₈×C₆₁	—	2	8	no	—
71	C₈×C₆₁	#106	1	8	yes	complemented
72	#8819	—	61	1	no	maximal, Sylow
73	C₂²×C₆₁	#384	1	1	yes	characteristic
74	C₂²×C₆₁	#492	1	1	yes	characteristic
75	C₂²×C₆₁	#492	1	1	yes	characteristic
76	C₂²×C₆₁	#496	1	2	yes	—
77	C₂²×C₆₁	#497	1	2	yes	—
78	C₄×C₆₁	C₂×#106	1	2	yes	—
79	C₄×C₆₁	#428	1	4	yes	—
80	C₄×C₆₁	#498	1	2	yes	—
81	C₄×C₆₁	#501	1	4	yes	—
82	Dic₆₁	—	16	2	no	—
83	Dic₆₁	—	16	1	no	—
84	Dic₆₁	—	16	1	no	—
85	#1845	—	61	1	no	—
86	#1844	—	61	1	no	—
87	#1844	—	61	2	no	—
88	#1834	—	61	2	no	—
89	C₂×#322	D₆₁	1	1	yes	characteristic
90	C₂×C₆₁	#1714	1	2	yes	—
91	C₂×C₆₁	#1715	1	1	yes	characteristic
92	C₂×C₆₁	#1716	1	1	yes	characteristic
93	C₂×C₆₁	#1844	1	1	yes	characteristic
94	C₂×C₆₁	#1847	1	2	yes	—
95	C₂×#107	—	122	2	no	—
96	C₂×#106	—	122	4	no	—
97	C₂×#106	—	122	2	no	—
98	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
99	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
100	#384	—	61	1	no	—
101	#384	—	61	2	no	—
102	#384	—	61	1	no	—
103	#340	—	122	4	no	—
104	#322	Dic₆₁	1	4	yes	complemented
105	C₆₁	#8819	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	D₂₄₄	1	4	yes	—
107	C₂²×C₈	D₂₄₄	1	2	yes	—
108	C₂×C₄⋊C₄	—	122	2	no	—
109	C₂×C₄⋊C₄	—	122	2	no	—
110	C₂×C₄⋊C₄	—	122	2	no	—
111	C₂×C₄⋊C₄	—	122	2	no	—
112	C₂×C₄⋊C₄	—	122	2	no	—
113	C₂×C₄⋊C₄	—	122	1	no	—
114	C₂×C₄⋊C₄	—	122	1	no	—
115	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
116	#107	—	244	4	no	—
117	#106	—	244	8	no	—
118	#106	—	244	4	no	—
119	C₄×C₈	Dic₁₂₂	1	4	yes	—
120	C₄×C₈	Dic₁₂₂	1	2	yes	—
121	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
122	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
123	C₂²×C₄	—	244	1	no	—
124	C₂²×C₄	—	244	1	no	—
125	C₂²×C₄	—	244	2	no	—
126	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
127	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	#11758190	1	4	yes	—
131	C₂×C₈	#11758190	1	4	yes	—
132	C₂×C₈	C₄⋊Dic₆₁	1	8	yes	—
133	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
134	C₄⋊C₄	—	244	4	no	—
135	C₄⋊C₄	—	244	4	no	—
136	C₄⋊C₄	—	244	2	no	—
137	C₄⋊C₄	—	244	2	no	—
138	C₄⋊C₄	—	244	4	no	—
139	C₄⋊C₄	—	244	4	no	—
140	C₄⋊C₄	—	244	4	no	—
141	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
142	C₄²	C₂×Dic₁₂₂	1	2	yes	—
143	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
144	C₂³	#14519774	1	1	yes	characteristic
145	C₂×C₄	—	488	2	no	—
146	C₂×C₄	—	488	1	no	—
147	C₂×C₄	—	488	1	no	—
148	C₂×C₄	—	488	2	no	—
149	C₂×C₄	—	488	2	no	—
150	C₂×C₄	—	488	2	no	—
151	C₂×C₄	—	488	2	no	—
152	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
154	C₂×C₄	#14519777	1	1	yes	characteristic
155	C₂×C₄	#14519777	1	2	yes	—
156	C₂×C₄	C₂×#11758190	1	2	yes	—
157	C₂×C₄	C₂×#11758190	1	2	yes	—
158	C₂×C₄	#14519805	1	4	yes	—
159	C₂×C₄	#14519810	1	4	yes	—
160	C₈	—	2	8	no	—
161	C₈	#14519709	1	8	yes	complemented
162	C₂²	#16839632	1	1	yes	characteristic
163	C₂²	#16840135	1	1	yes	characteristic
164	C₂²	#16840135	1	1	yes	characteristic
165	C₂²	#16840151	1	2	yes	—
166	C₂²	#16840156	1	2	yes	—
167	C₄	—	976	2	no	—
168	C₄	—	976	1	no	—
169	C₄	—	976	1	no	—
170	C₄	C₂×#14519709	1	2	yes	—
171	C₄	#16839804	1	4	yes	—
172	C₄	#16840160	1	2	yes	—
173	C₄	#16840171	1	4	yes	—
174	C₂	#21445441	1	2	yes	—
175	C₂	#21445447	1	1	yes	characteristic
176	C₂	#21445452	1	1	yes	characteristic
177	C₂	#21445976	1	1	yes	characteristic
178	C₂	#21445987	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445979

C₂

yes

maximal, characteristic

#21445976

C₂

yes

maximal

#21445976

C₂

yes

maximal, characteristic

#21445947

C₂

yes

maximal

C₂×C₆₁×#322

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

#16839632

C₂²

yes

—

#16839632

C₂²

yes

characteristic

#16839525

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#322

C₄

yes

complemented

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519711

—

#14519709

—

#14519709

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂³×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#497

yes

—

C₄×C₆₁

C₂×#106

yes

—

C₄×C₆₁

#428

yes

—

C₄×C₆₁

#498

yes

—

C₄×C₆₁

#501

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

#1714

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×#107

—

122

—

C₂×#106

—

122

—

C₂×#106

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

100

#384

—

101

#384

—

102

#384

—

103

#340

—

122

—

104

#322

Dic₆₁

yes

complemented

105

C₆₁

#8819

yes

characteristic, complemented, Sylow

106

C₂²×C₈

D₂₄₄

yes

—

107

C₂²×C₈

D₂₄₄

yes

—

108

C₂×C₄⋊C₄

—

122

—

109

C₂×C₄⋊C₄

—

122

—

110

C₂×C₄⋊C₄

—

122

—

111

C₂×C₄⋊C₄

—

122

—

112

C₂×C₄⋊C₄

—

122

—

113

C₂×C₄⋊C₄

—

122

—

114

C₂×C₄⋊C₄

—

122

—

115

C₂×C₄²

C₂²×D₆₁

yes

characteristic

116

#107

—

244

—

117

#106

—

244

—

118

#106

—

244

—

119

C₄×C₈

Dic₁₂₂

yes

—

120

C₄×C₈

Dic₁₂₂

yes

—

121

C₄×C₈

C₂×Dic₆₁

yes

—

122

C₄×C₈

C₂×Dic₆₁

yes

—

123

C₂²×C₄

—

244

—

124

C₂²×C₄

—

244

—

125

C₂²×C₄

—

244

—

126

C₂²×C₄

C₂×D₂₄₄

yes

—

127

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

#11758190

yes

—

131

C₂×C₈

#11758190

yes

—

132

C₂×C₈

C₄⋊Dic₆₁

yes

—

133

C₂×C₈

C₄⋊Dic₆₁

yes

—

134

C₄⋊C₄

—

244

—

135

C₄⋊C₄

—

244

—

136

C₄⋊C₄

—

244

—

137

C₄⋊C₄

—

244

—

138

C₄⋊C₄

—

244

—

139

C₄⋊C₄

—

244

—

140

C₄⋊C₄

—

244

—

141

C₄²

C₂×Dic₁₂₂

yes

characteristic

142

C₄²

C₂×Dic₁₂₂

yes

—

143

C₄²

C₂²×Dic₆₁

yes

characteristic

144

C₂³

#14519774

yes

characteristic

145

C₂×C₄

—

488

—

146

C₂×C₄

—

488

—

147

C₂×C₄

—

488

—

148

C₂×C₄

—

488

—

149

C₂×C₄

—

488

—

150

C₂×C₄

—

488

—

151

C₂×C₄

—

488

—

152

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

153

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

—

154

C₂×C₄

#14519777

yes

characteristic

155

C₂×C₄

#14519777

yes

—

156

C₂×C₄

C₂×#11758190

yes

—

157

C₂×C₄

C₂×#11758190

yes

—

158

C₂×C₄

#14519805

yes

—

159

C₂×C₄

#14519810

yes

—

160

C₈

—

161

C₈

#14519709

yes

complemented

162

C₂²

#16839632

yes

characteristic

163

C₂²

#16840135

yes

characteristic

164

C₂²

#16840135

yes

characteristic

165

C₂²

#16840151

yes

—

166

C₂²

#16840156

yes

—

167

C₄

—

976

—

168

C₄

—

976

—

169

C₄

—

976

—

170

C₄

C₂×#14519709

yes

—

171

C₄

#16839804

yes

—

172

C₄

#16840160

yes

—

173

C₄

#16840171

yes

—

174

C₂

#21445441

yes

—

175

C₂

#21445447

yes

characteristic

176

C₂

#21445452

yes

characteristic

177

C₂

#21445976

yes

characteristic

178

C₂

#21445987

yes

—

	subgroup	G/H
center	C₂³	#14519774
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#322	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#322	D₆₁
hypocenter	C₆₁	#8819
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#14519774

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₆₁×#322

C₂

socle, prime radical

C₂³×C₆₁

#127

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
15. C₆₁×#322	C₄	semidirect
71. C₈×C₆₁	#106	semidirect
104. #322	Dic₆₁	semidirect
105. C₆₁	#8819	semidirect
161. C₈	#14519709	semidirect

H ≅ G/N

type

15. C₆₁×#322

C₄

semidirect

71. C₈×C₆₁

#106

semidirect

104. #322

Dic₆₁

semidirect

105. C₆₁

#8819

semidirect

161. C₈

#14519709

semidirect

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

subgroup	index
#21445979	2
#21445976	2
#21445947	2
C₂×C₆₁×#322	2
#8819	61

subgroup

index

C₂×C₆₁×#322

Subgroups — #29573798

Conjugacy classes of subgroups

Complements

Maximal subgroups