Group #29579780 — subgroups

Subgroups — #29579780

Order 15616.

558 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 211 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	C₂×#16840142	C₂	1	1	yes	maximal, characteristic
2	#21446780	C₂	1	4	yes	maximal
3	#21445979	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₄₄	—	2	2	no	—
6	C₂×#14519777	C₂²	1	2	yes	—
7	#16840142	—	2	2	no	—
8	#16840142	C₄	1	4	yes	complemented
9	C₂×#14519711	—	2	2	no	—
10	#16839632	C₂²	1	2	yes	—
11	#16839525	—	2	8	no	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₈²×C₆₁	—	2	2	no	—
15	C₂²×Dic₁₂₂	—	2	2	no	—
16	C₂×Dic₂₄₄	—	2	8	no	—
17	C₂×Dic₂₄₄	—	4	4	no	—
18	C₂×Dic₂₄₄	—	4	4	no	—
19	#14519777	—	2	4	no	—
20	#14519777	—	2	2	no	—
21	#14519777	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊Dic₆₁	—	2	4	no	—
23	C₂×C₄⋊Dic₆₁	—	2	4	no	—
24	C₂×C₄⋊Dic₆₁	—	2	2	no	—
25	#14519711	—	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	—	2	2	no	—
30	C₄×C₈×C₆₁	—	2	2	no	—
31	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₆₁	D₄	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	2	no	—
36	C₂²×Dic₆₁	—	4	2	no	—
37	C₂×Dic₁₂₂	—	2	4	no	—
38	C₂×Dic₁₂₂	—	4	4	no	—
39	C₂×Dic₁₂₂	—	4	4	no	—
40	C₂×Dic₁₂₂	—	4	2	no	—
41	C₂×C₈×C₆₁	—	2	4	no	—
42	C₂×C₈×C₆₁	—	2	4	no	—
43	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
44	C₂×C₈×C₆₁	D₈	1	4	yes	—
45	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
46	C₂×C₈×C₆₁	Q₁₆	1	8	yes	complemented
47	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₆₁	#41	1	2	yes	—
50	C₄⋊Dic₆₁	—	2	8	no	—
51	C₄⋊Dic₆₁	—	4	4	no	—
52	C₄⋊Dic₆₁	—	4	4	no	—
53	C₄⋊Dic₆₁	—	4	8	no	—
54	Dic₂₄₄	—	4	16	no	—
55	C₂³×C₆₁	#127	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
58	C₂×C₄×C₆₁	#121	1	2	yes	—
59	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
61	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×Q₁₆	1	4	yes	—
63	C₂×C₄×C₆₁	#135	1	4	yes	—
64	C₂×Dic₆₁	—	4	4	no	—
65	C₂×Dic₆₁	—	8	2	no	—
66	C₂×Dic₆₁	—	8	2	no	—
67	C₂×Dic₆₁	—	8	2	no	—
68	C₂×Dic₆₁	—	8	4	no	—
69	Dic₁₂₂	—	4	8	no	—
70	Dic₁₂₂	—	8	4	no	—
71	C₈×C₆₁	—	2	8	no	—
72	C₈×C₆₁	#102	1	8	yes	—
73	#9825	—	61	1	no	maximal, Sylow
74	C₂²×C₆₁	#390	1	1	yes	characteristic
75	C₂²×C₆₁	#492	1	1	yes	characteristic
76	C₂²×C₆₁	#493	1	1	yes	characteristic
77	C₂²×C₆₁	#494	1	2	yes	—
78	C₂²×C₆₁	#495	1	2	yes	—
79	C₄×C₆₁	C₂×#102	1	2	yes	—
80	C₄×C₆₁	C₄×Q₁₆	1	4	yes	—
81	C₄×C₆₁	#467	1	4	yes	—
82	C₄×C₆₁	#486	1	2	yes	—
83	Dic₆₁	—	8	4	no	—
84	Dic₆₁	—	16	2	no	—
85	C₂×#494	—	61	1	no	—
86	#1990	—	61	4	no	—
87	#1845	—	61	1	no	—
88	C₂×C₈²	D₆₁	1	1	yes	characteristic
89	C₂×C₆₁	#1707	1	2	yes	—
90	C₂×C₆₁	#1709	1	1	yes	characteristic
91	C₂×C₆₁	#1713	1	1	yes	characteristic
92	C₂×C₆₁	#1957	1	1	yes	characteristic
93	C₂×C₆₁	#1969	1	2	yes	—
94	C₂²×Q₁₆	—	122	2	no	—
95	C₂×C₄⋊Q₈	—	61	2	no	—
96	#494	—	61	4	no	—
97	#494	—	122	2	no	—
98	C₂×#107	—	122	2	no	—
99	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
101	#384	—	61	2	no	—
102	#340	—	122	8	no	—
103	C₈²	—	2	2	no	—
104	C₆₁	#9825	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	122	2	no	—
106	C₂×Q₁₆	—	122	8	no	—
107	C₂×Q₁₆	—	244	4	no	—
108	C₂×Q₁₆	—	244	4	no	—
109	C₂²×C₈	D₂₄₄	1	4	yes	—
110	C₂²×C₈	D₂₄₄	1	2	yes	—
111	C₄⋊Q₈	—	61	4	no	—
112	C₄⋊Q₈	—	122	4	no	—
113	C₄⋊Q₈	—	122	2	no	—
114	C₂×C₄⋊C₄	—	122	4	no	—
115	C₂×C₄⋊C₄	—	122	4	no	—
116	C₂×C₄⋊C₄	—	122	2	no	—
117	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
118	#107	—	244	4	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	C₄×D₆₁	1	2	yes	—
122	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
123	C₂×Q₈	—	122	4	no	—
124	C₂×Q₈	—	244	4	no	—
125	C₂×Q₈	—	244	4	no	—
126	C₂×Q₈	—	244	2	no	—
127	C₂²×C₄	—	244	2	no	—
128	C₂²×C₄	—	244	2	no	—
129	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
131	Q₁₆	—	244	16	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	#11758190	1	4	yes	—
135	C₂×C₈	D₄₈₈	1	4	yes	—
136	C₂×C₈	Dic₂₄₄	1	8	yes	complemented
137	C₂×C₈	#11758198	1	4	yes	—
138	C₄⋊C₄	—	122	8	no	—
139	C₄⋊C₄	—	244	4	no	—
140	C₄⋊C₄	—	244	8	no	—
141	C₄⋊C₄	—	244	4	no	—
142	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
143	C₄²	#11758222	1	2	yes	—
144	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
145	C₂³	#14519774	1	1	yes	characteristic
146	Q₈	—	488	4	no	—
147	Q₈	—	244	8	no	—
148	C₂×C₄	—	488	2	no	—
149	C₂×C₄	—	488	4	no	—
150	C₂×C₄	—	488	2	no	—
151	C₂×C₄	—	488	2	no	—
152	C₂×C₄	—	244	4	no	—
153	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
154	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
155	C₂×C₄	#14519751	1	2	yes	—
156	C₂×C₄	#14519765	1	1	yes	characteristic
157	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
158	C₂×C₄	C₂×#11758190	1	2	yes	—
159	C₂×C₄	C₂×Dic₂₄₄	1	4	yes	—
160	C₂×C₄	#14519799	1	4	yes	—
161	C₈	—	2	8	no	—
162	C₈	#14519699	1	8	yes	—
163	C₂²	#16839664	1	1	yes	characteristic
164	C₂²	#16840135	1	1	yes	characteristic
165	C₂²	#16840139	1	1	yes	characteristic
166	C₂²	#16840142	1	2	yes	—
167	C₂²	#16840146	1	2	yes	—
168	C₄	—	976	2	no	—
169	C₄	—	488	4	no	—
170	C₄	C₂×#14519699	1	2	yes	—
171	C₄	C₄×Dic₂₄₄	1	4	yes	—
172	C₄	#16840003	1	4	yes	—
173	C₄	#16840109	1	2	yes	—
174	C₂	#21445416	1	2	yes	—
175	C₂	#21445425	1	1	yes	characteristic
176	C₂	#21445437	1	1	yes	characteristic
177	C₂	#21446625	1	1	yes	characteristic
178	C₂	#21446680	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840142

C₂

yes

maximal, characteristic

#21446780

C₂

yes

maximal

#21445979

C₂

yes

maximal, characteristic

C₂×C₈²×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₄₄

—

C₂×#14519777

C₂²

yes

—

#16840142

—

#16840142

C₄

yes

complemented

C₂×#14519711

—

#16839632

C₂²

yes

—

#16839525

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₈²×C₆₁

—

C₂²×Dic₁₂₂

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

—

#14519777

—

#14519777

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519711

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

Q₁₆

yes

complemented

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

C₂³×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×Q₁₆

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

#102

yes

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#495

yes

—

C₄×C₆₁

C₂×#102

yes

—

C₄×C₆₁

C₄×Q₁₆

yes

—

C₄×C₆₁

#467

yes

—

C₄×C₆₁

#486

yes

—

Dic₆₁

—

Dic₆₁

—

C₂×#494

—

#1990

—

#1845

—

C₂×C₈²

D₆₁

yes

characteristic

C₂×C₆₁

#1707

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂²×Q₁₆

—

122

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

122

—

C₂×#107

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

—

100

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

101

#384

—

102

#340

—

122

—

103

C₈²

—

104

C₆₁

#9825

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

122

—

106

C₂×Q₁₆

—

122

—

107

C₂×Q₁₆

—

244

—

108

C₂×Q₁₆

—

244

—

109

C₂²×C₈

D₂₄₄

yes

—

110

C₂²×C₈

D₂₄₄

yes

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

122

—

113

C₄⋊Q₈

—

122

—

114

C₂×C₄⋊C₄

—

122

—

115

C₂×C₄⋊C₄

—

122

—

116

C₂×C₄⋊C₄

—

122

—

117

C₂×C₄²

C₂²×D₆₁

yes

characteristic

118

#107

—

244

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

C₄×D₆₁

yes

—

122

C₄×C₈

C₂²⋊D₆₁

yes

—

123

C₂×Q₈

—

122

—

124

C₂×Q₈

—

244

—

125

C₂×Q₈

—

244

—

126

C₂×Q₈

—

244

—

127

C₂²×C₄

—

244

—

128

C₂²×C₄

—

244

—

129

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

130

C₂²×C₄

C₂×D₂₄₄

yes

—

131

Q₁₆

—

244

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

#11758190

yes

—

135

C₂×C₈

D₄₈₈

yes

—

136

C₂×C₈

Dic₂₄₄

yes

complemented

137

C₂×C₈

#11758198

yes

—

138

C₄⋊C₄

—

122

—

139

C₄⋊C₄

—

244

—

140

C₄⋊C₄

—

244

—

141

C₄⋊C₄

—

244

—

142

C₄²

C₂×C₄×D₆₁

yes

characteristic

143

C₄²

#11758222

yes

—

144

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

145

C₂³

#14519774

yes

characteristic

146

Q₈

—

488

—

147

Q₈

—

244

—

148

C₂×C₄

—

488

—

149

C₂×C₄

—

488

—

150

C₂×C₄

—

488

—

151

C₂×C₄

—

488

—

152

C₂×C₄

—

244

—

153

C₂×C₄

C₂×#11758198

yes

characteristic

154

C₂×C₄

C₄×D₂₄₄

yes

—

155

C₂×C₄

#14519751

yes

—

156

C₂×C₄

#14519765

yes

characteristic

157

C₂×C₄

C₂×D₄₈₈

yes

—

158

C₂×C₄

C₂×#11758190

yes

—

159

C₂×C₄

C₂×Dic₂₄₄

yes

—

160

C₂×C₄

#14519799

yes

—

161

C₈

—

162

C₈

#14519699

yes

—

163

C₂²

#16839664

yes

characteristic

164

C₂²

#16840135

yes

characteristic

165

C₂²

#16840139

yes

characteristic

166

C₂²

#16840142

yes

—

167

C₂²

#16840146

yes

—

168

C₄

—

976

—

169

C₄

—

488

—

170

C₄

C₂×#14519699

yes

—

171

C₄

C₄×Dic₂₄₄

yes

—

172

C₄

#16840003

yes

—

173

C₄

#16840109

yes

—

174

C₂

#21445416

yes

—

175

C₂

#21445425

yes

characteristic

176

C₂

#21445437

yes

characteristic

177

C₂

#21446625

yes

characteristic

178

C₂

#21446680

yes

—

	subgroup	G/H
center	C₂³	#14519774
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈²×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈²	D₆₁
hypocenter	C₆₁	#9825
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#14519774

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₈²×C₆₁

C₂

socle, prime radical

C₂³×C₆₁

#127

cosocle

C₂×C₄²×C₆₁

C₂³

hypercenter

C₂×C₈²

D₆₁

hypocenter

C₆₁

#9825

LCS

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
8. #16840142	C₄	semidirect
46. C₂×C₈×C₆₁	Q₁₆	semidirect
104. C₆₁	#9825	semidirect
136. C₂×C₈	Dic₂₄₄	semidirect

H ≅ G/N

type

8. #16840142

C₄

semidirect

46. C₂×C₈×C₆₁

Q₁₆

semidirect

104. C₆₁

#9825

semidirect

136. C₂×C₈

Dic₂₄₄

semidirect

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

subgroup	index
C₂×#16840142	2
#21446780	2
#21445979	2
C₂×C₈²×C₆₁	2
#9825	61

subgroup

index

C₂×C₈²×C₆₁

Subgroups — #29579780

Conjugacy classes of subgroups

Complements

Maximal subgroups