Group #29579779 — subgroups

Subgroups — #29579779

Order 15616.

558 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 199 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	#21446779	C₂	1	4	yes	maximal
3	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1845	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×C₄×#92418	C₂²	1	2	yes	—
10	#16839524	—	2	8	no	—
11	C₈×#92418	—	2	2	no	—
12	C₂×#107×C₆₁	—	2	2	no	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#384	C₂²	1	2	yes	—
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	C₂²×#92418	D₄	1	4	yes	—
20	#14519777	—	2	4	no	—
21	#14519777	—	2	2	no	—
22	#14519777	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₆₁	—	2	4	no	—
24	C₄×#92418	—	2	2	no	—
25	C₄×#92418	—	2	2	no	—
26	C₂²×C₈×C₆₁	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
29	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
30	#107×C₆₁	—	4	4	no	—
31	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₆₁	D₄	1	2	yes	—
33	C₂²×C₄×C₆₁	—	4	2	no	—
34	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
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	C₂²⋊C₄	1	4	yes	—
42	C₂×C₈×C₆₁	D₈	1	4	yes	—
43	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
44	C₄⋊C₄×C₆₁	—	4	4	no	—
45	C₄⋊C₄×C₆₁	—	4	8	no	—
46	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₆₁	#41	1	2	yes	—
49	C₄⋊Dic₆₁	—	2	8	no	—
50	C₄⋊Dic₆₁	—	4	4	no	—
51	C₂×#92418	—	2	4	no	—
52	C₂×#92418	—	2	4	no	—
53	C₂×#92418	Q₁₆	1	8	yes	complemented
54	Dic₂₄₄	—	4	16	no	—
55	C₂³×C₆₁	#127	1	1	yes	characteristic
56	C₂×C₄×C₆₁	—	8	2	no	—
57	C₂×C₄×C₆₁	—	8	4	no	—
58	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₆₁	#121	1	2	yes	—
61	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
64	C₂×C₄×C₆₁	C₂×Q₁₆	1	4	yes	—
65	C₂×C₄×C₆₁	#135	1	4	yes	—
66	C₂×Dic₆₁	—	4	4	no	—
67	C₂×Dic₆₁	—	8	2	no	—
68	C₂×Dic₆₁	—	8	2	no	—
69	Dic₁₂₂	—	4	8	no	—
70	Dic₁₂₂	—	8	4	no	—
71	C₈×C₆₁	#102	1	8	yes	—
72	#92418	—	2	8	no	—
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	—	16	2	no	—
80	C₄×C₆₁	C₂×#102	1	2	yes	—
81	C₄×C₆₁	C₄×Q₁₆	1	4	yes	—
82	C₄×C₆₁	#467	1	4	yes	—
83	C₄×C₆₁	#486	1	2	yes	—
84	Dic₆₁	—	8	4	no	—
85	C₂×#494	—	61	1	no	—
86	#1990	—	61	4	no	—
87	#1845	D₆₁	1	1	yes	characteristic
88	C₂×C₈²	—	61	1	no	—
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	—	2	2	no	—
99	C₂×C₄×C₈	—	61	2	no	—
100	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
101	#384	C₂×D₆₁	1	2	yes	—
102	#340	—	122	8	no	—
103	C₈²	—	122	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	—	61	4	no	—
110	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
111	C₄⋊Q₈	—	61	4	no	—
112	C₄⋊Q₈	—	122	2	no	—
113	C₄⋊Q₈	—	122	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	122	4	no	—
117	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
118	#107	—	4	4	no	—
119	C₄×C₈	—	122	2	no	—
120	C₄×C₈	—	122	2	no	—
121	C₄×C₈	C₄×D₆₁	1	2	yes	—
122	C₄×C₈	D₂₄₄	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	—	4	2	no	—
128	C₂²×C₄	—	244	2	no	—
129	C₂²×C₄	D₄×D₆₁	1	2	yes	—
130	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
131	Q₁₆	—	244	16	no	—
132	C₂×C₈	—	61	8	no	—
133	C₂×C₈	—	122	4	no	—
134	C₂×C₈	—	122	4	no	—
135	C₂×C₈	#11758198	1	4	yes	—
136	C₂×C₈	#11758199	1	4	yes	—
137	C₂×C₈	#11758201	1	4	yes	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	—	4	8	no	—
140	C₄⋊C₄	—	122	8	no	—
141	C₄⋊C₄	—	244	4	no	—
142	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
143	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
144	C₄²	#11758226	1	2	yes	—
145	C₂³	#14519773	1	1	yes	characteristic
146	Q₈	—	488	4	no	—
147	Q₈	—	244	8	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	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	#14519739	1	2	yes	—
155	C₂×C₄	#14519752	1	2	yes	—
156	C₂×C₄	#14519766	1	1	yes	characteristic
157	C₂×C₄	C₂×#11758199	1	2	yes	—
158	C₂×C₄	C₂×#11758201	1	2	yes	—
159	C₂×C₄	Q₁₆×D₆₁	1	4	yes	—
160	C₂×C₄	#14519796	1	4	yes	—
161	C₈	—	122	8	no	—
162	C₈	#14519697	1	8	yes	—
163	C₂²	#16839663	1	1	yes	characteristic
164	C₂²	#16840136	1	1	yes	characteristic
165	C₂²	#16840138	1	1	yes	characteristic
166	C₂²	#16840140	1	2	yes	—
167	C₂²	#16840145	1	2	yes	—
168	C₄	—	16	2	no	—
169	C₄	—	488	4	no	—
170	C₄	C₂×#14519697	1	2	yes	—
171	C₄	#16839856	1	4	yes	—
172	C₄	#16840004	1	4	yes	—
173	C₄	#16840110	1	2	yes	—
174	C₂	#21445417	1	2	yes	—
175	C₂	#21445424	1	1	yes	characteristic
176	C₂	#21445435	1	1	yes	characteristic
177	C₂	#21446624	1	1	yes	characteristic
178	C₂	#21446678	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840142

C₂

yes

maximal, characteristic

#21446779

C₂

yes

maximal

C₂×C₈×#92418

C₂

yes

maximal, characteristic

C₆₁×#1845

C₂

yes

maximal, characteristic

C₂²×Dic₂₄₄

—

C₂×#14519777

C₂²

yes

—

#16840142

—

#16840142

C₄

yes

complemented

C₂×C₄×#92418

C₂²

yes

—

#16839524

—

C₈×#92418

—

C₂×#107×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#384

C₂²

yes

—

C₂²×Dic₁₂₂

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂²×#92418

D₄

yes

—

#14519777

—

#14519777

—

#14519777

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₄×#92418

—

C₄×#92418

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#107×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

Q₁₆

yes

complemented

Dic₂₄₄

—

C₂³×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

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

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

#102

yes

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#495

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#102

yes

—

C₄×C₆₁

C₄×Q₁₆

yes

—

C₄×C₆₁

#467

yes

—

C₄×C₆₁

#486

yes

—

Dic₆₁

—

C₂×#494

—

#1990

—

#1845

D₆₁

yes

characteristic

C₂×C₈²

—

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

—

C₂×C₄×C₈

—

100

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

101

#384

C₂×D₆₁

yes

—

102

#340

—

122

—

103

C₈²

—

122

—

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

—

110

C₂²×C₈

C₂²⋊D₆₁

yes

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

122

—

113

C₄⋊Q₈

—

122

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

122

—

117

C₂×C₄²

C₂²×D₆₁

yes

characteristic

118

#107

—

119

C₄×C₈

—

122

—

120

C₄×C₈

—

122

—

121

C₄×C₈

C₄×D₆₁

yes

—

122

C₄×C₈

D₂₄₄

yes

—

123

C₂×Q₈

—

122

—

124

C₂×Q₈

—

244

—

125

C₂×Q₈

—

244

—

126

C₂×Q₈

—

244

—

127

C₂²×C₄

—

128

C₂²×C₄

—

244

—

129

C₂²×C₄

D₄×D₆₁

yes

—

130

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

131

Q₁₆

—

244

—

132

C₂×C₈

—

133

C₂×C₈

—

122

—

134

C₂×C₈

—

122

—

135

C₂×C₈

#11758198

yes

—

136

C₂×C₈

#11758199

yes

—

137

C₂×C₈

#11758201

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

122

—

141

C₄⋊C₄

—

244

—

142

C₄²

C₂×C₄×D₆₁

yes

characteristic

143

C₄²

C₂×D₂₄₄

yes

characteristic

144

C₄²

#11758226

yes

—

145

C₂³

#14519773

yes

characteristic

146

Q₈

—

488

—

147

Q₈

—

244

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

488

—

151

C₂×C₄

—

488

—

152

C₂×C₄

—

244

—

153

C₂×C₄

C₂×#11758198

yes

characteristic

154

C₂×C₄

#14519739

yes

—

155

C₂×C₄

#14519752

yes

—

156

C₂×C₄

#14519766

yes

characteristic

157

C₂×C₄

C₂×#11758199

yes

—

158

C₂×C₄

C₂×#11758201

yes

—

159

C₂×C₄

Q₁₆×D₆₁

yes

—

160

C₂×C₄

#14519796

yes

—

161

C₈

—

122

—

162

C₈

#14519697

yes

—

163

C₂²

#16839663

yes

characteristic

164

C₂²

#16840136

yes

characteristic

165

C₂²

#16840138

yes

characteristic

166

C₂²

#16840140

yes

—

167

C₂²

#16840145

yes

—

168

C₄

—

169

C₄

—

488

—

170

C₄

C₂×#14519697

yes

—

171

C₄

#16839856

yes

—

172

C₄

#16840004

yes

—

173

C₄

#16840110

yes

—

174

C₂

#21445417

yes

—

175

C₂

#21445424

yes

characteristic

176

C₂

#21445435

yes

characteristic

177

C₂

#21446624

yes

characteristic

178

C₂

#21446678

yes

—

	subgroup	G/H
center	C₂³	#14519773
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1845	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1845	D₆₁
hypocenter	C₆₁	#9825
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#14519773

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1845

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 ◁ #1845

N	H ≅ G/N	type
8. #16840142	C₄	semidirect
53. C₂×#92418	Q₁₆	semidirect
104. C₆₁	#9825	semidirect

H ≅ G/N

type

8. #16840142

C₄

semidirect

53. C₂×#92418

Q₁₆

semidirect

104. C₆₁

#9825

semidirect

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

subgroup	index
C₂×#16840142	2
#21446779	2
C₂×C₈×#92418	2
C₆₁×#1845	2
#9825	61

subgroup

index

C₂×#16840142

#21446779

C₂×C₈×#92418

C₆₁×#1845

#9825

Subgroups — #29579779

Conjugacy classes of subgroups

Complements

Maximal subgroups