Group #29579862 — subgroups

Subgroups — #29579862

Order 15616.

542 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 195 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	#21445976	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₁×#322	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×#14519709	—	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	C₆₁×#322	—	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	#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	—	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	8	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	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	C₄⋊Dic₆₁	—	4	4	no	—
52	C₄⋊Dic₆₁	—	4	8	no	—
53	Dic₂₄₄	—	4	16	no	—
54	C₂³×C₆₁	#127	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₆₁	#121	1	2	yes	—
58	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
60	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
61	C₂×C₄×C₆₁	#137	1	8	yes	—
62	C₂×Dic₆₁	—	4	4	no	—
63	C₂×Dic₆₁	—	8	2	no	—
64	C₂×Dic₆₁	—	8	2	no	—
65	C₂×Dic₆₁	—	8	2	no	—
66	C₂×Dic₆₁	—	8	4	no	—
67	Dic₁₂₂	—	4	8	no	—
68	Dic₁₂₂	—	8	4	no	—
69	C₈×C₆₁	—	4	4	no	—
70	C₈×C₆₁	#102	1	8	yes	—
71	#9839	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	#390	1	1	yes	characteristic
73	C₂²×C₆₁	#492	1	1	yes	characteristic
74	C₂²×C₆₁	#493	1	1	yes	characteristic
75	C₂²×C₆₁	#497	1	4	yes	—
76	C₄×C₆₁	C₂×#102	1	2	yes	—
77	C₄×C₆₁	#441	1	4	yes	—
78	C₄×C₆₁	#470	1	4	yes	—
79	C₄×C₆₁	#486	1	2	yes	—
80	Dic₆₁	—	8	4	no	—
81	Dic₆₁	—	16	2	no	—
82	C₂×#494	—	61	1	no	—
83	#1990	—	61	4	no	—
84	#1844	—	61	1	no	—
85	C₂×#322	D₆₁	1	1	yes	characteristic
86	C₂×C₆₁	#1716	1	2	yes	—
87	C₂×C₆₁	#1722	1	2	yes	—
88	C₂×C₆₁	#1957	1	1	yes	characteristic
89	C₂×C₆₁	#1971	1	2	yes	—
90	C₂²×Q₁₆	—	122	2	no	—
91	C₂×C₄⋊Q₈	—	61	2	no	—
92	#494	—	61	4	no	—
93	#494	—	122	2	no	—
94	C₂×#106	—	122	2	no	—
95	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
97	#384	—	61	2	no	—
98	#340	—	122	8	no	—
99	#322	—	2	2	no	—
100	C₆₁	#9839	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	122	2	no	—
102	C₂×Q₁₆	—	122	8	no	—
103	C₂×Q₁₆	—	244	4	no	—
104	C₂×Q₁₆	—	244	4	no	—
105	C₂²×C₈	D₂₄₄	1	4	yes	—
106	C₂²×C₈	D₂₄₄	1	2	yes	—
107	C₄⋊Q₈	—	61	4	no	—
108	C₄⋊Q₈	—	122	4	no	—
109	C₄⋊Q₈	—	122	2	no	—
110	C₂×C₄⋊C₄	—	122	4	no	—
111	C₂×C₄⋊C₄	—	122	4	no	—
112	C₂×C₄⋊C₄	—	122	2	no	—
113	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
114	#106	—	244	4	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	C₄×D₆₁	1	2	yes	—
118	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
119	C₂×Q₈	—	122	4	no	—
120	C₂×Q₈	—	244	4	no	—
121	C₂×Q₈	—	244	4	no	—
122	C₂×Q₈	—	244	2	no	—
123	C₂²×C₄	—	244	2	no	—
124	C₂²×C₄	—	244	2	no	—
125	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
126	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
127	Q₁₆	—	244	16	no	—
128	C₂×C₈	—	2	8	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	#11758190	1	4	yes	—
131	C₂×C₈	D₄₈₈	1	4	yes	—
132	C₂×C₈	#11758198	1	4	yes	—
133	C₄⋊C₄	—	122	8	no	—
134	C₄⋊C₄	—	244	8	no	—
135	C₄⋊C₄	—	244	4	no	—
136	C₄⋊C₄	—	244	4	no	—
137	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
138	C₄²	#11758222	1	2	yes	—
139	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
140	C₂³	#14519774	1	1	yes	characteristic
141	Q₈	—	488	4	no	—
142	Q₈	—	244	8	no	—
143	C₂×C₄	—	488	2	no	—
144	C₂×C₄	—	488	4	no	—
145	C₂×C₄	—	488	2	no	—
146	C₂×C₄	—	488	2	no	—
147	C₂×C₄	—	244	4	no	—
148	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
149	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
150	C₂×C₄	#14519751	1	2	yes	—
151	C₂×C₄	#14519765	1	1	yes	characteristic
152	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
153	C₂×C₄	C₂×#11758190	1	2	yes	—
154	C₂×C₄	#14519810	1	8	yes	—
155	C₈	—	4	4	no	—
156	C₈	#14519699	1	8	yes	—
157	C₂²	#16839664	1	1	yes	characteristic
158	C₂²	#16840135	1	1	yes	characteristic
159	C₂²	#16840139	1	1	yes	characteristic
160	C₂²	#16840156	1	4	yes	—
161	C₄	—	976	2	no	—
162	C₄	—	488	4	no	—
163	C₄	C₂×#14519699	1	2	yes	—
164	C₄	#16839870	1	4	yes	—
165	C₄	#16840020	1	4	yes	—
166	C₄	#16840109	1	2	yes	—
167	C₂	#21445452	1	2	yes	—
168	C₂	#21445490	1	2	yes	—
169	C₂	#21446625	1	1	yes	characteristic
170	C₂	#21446690	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840142

C₂

yes

maximal, characteristic

#21446780

C₂

yes

maximal

#21445976

C₂

yes

maximal, characteristic

C₂×C₆₁×#322

C₂

yes

maximal, characteristic

C₂²×Dic₂₄₄

—

C₂×#14519777

C₂²

yes

—

#16840142

—

#16840142

C₄

yes

complemented

C₂×#14519709

—

#16839632

C₂²

yes

—

#16839525

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#322

—

C₂²×Dic₁₂₂

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

—

#14519777

—

#14519777

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519709

—

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

—

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

#137

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

—

C₄×C₆₁

C₂×#102

yes

—

C₄×C₆₁

#441

yes

—

C₄×C₆₁

#470

yes

—

C₄×C₆₁

#486

yes

—

Dic₆₁

—

Dic₆₁

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

#1716

yes

—

C₂×C₆₁

#1722

yes

—

C₂×C₆₁

#1957

yes

characteristic

C₂×C₆₁

#1971

yes

—

C₂²×Q₁₆

—

122

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

122

—

C₂×#106

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#384

—

#340

—

122

—

#322

—

100

C₆₁

#9839

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

122

—

102

C₂×Q₁₆

—

122

—

103

C₂×Q₁₆

—

244

—

104

C₂×Q₁₆

—

244

—

105

C₂²×C₈

D₂₄₄

yes

—

106

C₂²×C₈

D₂₄₄

yes

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

—

122

—

109

C₄⋊Q₈

—

122

—

110

C₂×C₄⋊C₄

—

122

—

111

C₂×C₄⋊C₄

—

122

—

112

C₂×C₄⋊C₄

—

122

—

113

C₂×C₄²

C₂²×D₆₁

yes

characteristic

114

#106

—

244

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

C₄×D₆₁

yes

—

118

C₄×C₈

C₂²⋊D₆₁

yes

—

119

C₂×Q₈

—

122

—

120

C₂×Q₈

—

244

—

121

C₂×Q₈

—

244

—

122

C₂×Q₈

—

244

—

123

C₂²×C₄

—

244

—

124

C₂²×C₄

—

244

—

125

C₂²×C₄

C₂×D₂₄₄

yes

—

126

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

127

Q₁₆

—

244

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

#11758190

yes

—

131

C₂×C₈

D₄₈₈

yes

—

132

C₂×C₈

#11758198

yes

—

133

C₄⋊C₄

—

122

—

134

C₄⋊C₄

—

244

—

135

C₄⋊C₄

—

244

—

136

C₄⋊C₄

—

244

—

137

C₄²

C₂×C₄×D₆₁

yes

characteristic

138

C₄²

#11758222

yes

—

139

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

140

C₂³

#14519774

yes

characteristic

141

Q₈

—

488

—

142

Q₈

—

244

—

143

C₂×C₄

—

488

—

144

C₂×C₄

—

488

—

145

C₂×C₄

—

488

—

146

C₂×C₄

—

488

—

147

C₂×C₄

—

244

—

148

C₂×C₄

C₂×#11758198

yes

characteristic

149

C₂×C₄

C₄×D₂₄₄

yes

—

150

C₂×C₄

#14519751

yes

—

151

C₂×C₄

#14519765

yes

characteristic

152

C₂×C₄

C₂×D₄₈₈

yes

—

153

C₂×C₄

C₂×#11758190

yes

—

154

C₂×C₄

#14519810

yes

—

155

C₈

—

156

C₈

#14519699

yes

—

157

C₂²

#16839664

yes

characteristic

158

C₂²

#16840135

yes

characteristic

159

C₂²

#16840139

yes

characteristic

160

C₂²

#16840156

yes

—

161

C₄

—

976

—

162

C₄

—

488

—

163

C₄

C₂×#14519699

yes

—

164

C₄

#16839870

yes

—

165

C₄

#16840020

yes

—

166

C₄

#16840109

yes

—

167

C₂

#21445452

yes

—

168

C₂

#21445490

yes

—

169

C₂

#21446625

yes

characteristic

170

C₂

#21446690

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	#9839
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
8. #16840142	C₄	semidirect
100. C₆₁	#9839	semidirect

H ≅ G/N

type

8. #16840142

C₄

semidirect

100. C₆₁

#9839

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
#21445976	2
C₂×C₆₁×#322	2
#9839	61

subgroup

index

C₂×C₆₁×#322

Subgroups — #29579862

Conjugacy classes of subgroups

Complements

Maximal subgroups