Group #29584825 — subgroups

Subgroups — #29584825

Order 15616.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	#21446639	C₂	1	1	yes	maximal, characteristic
2	#21445960	C₂	1	1	yes	maximal, characteristic
3	#21444490	C₂	1	2	yes	maximal
4	#21443678	C₂	1	2	yes	maximal
5	C₆₁×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519777	C₂²	1	1	yes	characteristic
7	#16839786	C₂²	1	2	yes	—
8	C₂×#14519702	—	2	2	no	—
9	#16839628	C₂²	1	1	yes	characteristic
10	#16839527	—	2	2	no	—
11	#16839507	—	2	4	no	—
12	C₆₁×#431	C₂²	1	2	yes	—
13	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₂₂	—	2	1	no	—
15	C₂×#11758194	—	2	2	no	—
16	#14519777	—	2	2	no	—
17	#14519777	—	2	1	no	—
18	#14519777	D₄	1	2	yes	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	1	no	—
21	C₄⋊#92418	—	2	2	no	—
22	C₄⋊#92418	—	2	2	no	—
23	#14519702	—	4	4	no	—
24	#14519684	—	2	2	no	—
25	C₂²×C₈×C₆₁	D₄	1	2	yes	—
26	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₆₁	—	2	2	no	—
28	C₂²⋊C₈×C₆₁	—	2	2	no	—
29	#97×C₆₁	—	2	2	no	—
30	C₄×C₈×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	#41	1	2	yes	—
33	C₂²×Dic₆₁	—	4	1	no	—
34	C₂²×Dic₆₁	—	4	1	no	—
35	C₂×Dic₁₂₂	—	2	2	no	—
36	C₂×Dic₁₂₂	—	4	2	no	—
37	C₂×Dic₁₂₂	—	4	2	no	—
38	C₂×Dic₁₂₂	—	4	1	no	—
39	C₂×C₈×C₆₁	—	2	2	no	—
40	C₂×C₈×C₆₁	—	2	2	no	—
41	C₂×C₈×C₆₁	—	2	2	no	—
42	C₂×C₈×C₆₁	—	4	2	no	—
43	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₆₁	#41	1	1	yes	characteristic
45	C₄²×C₆₁	#41	1	2	yes	—
46	C₄⋊Dic₆₁	—	2	4	no	—
47	C₄⋊Dic₆₁	—	4	2	no	—
48	C₄⋊Dic₆₁	—	4	2	no	—
49	#11758194	—	4	2	no	—
50	#11758194	—	4	2	no	—
51	C₂×#92418	—	4	2	no	—
52	C₂³×C₆₁	#124	1	1	yes	characteristic
53	C₂×C₄×C₆₁	—	4	2	no	—
54	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#123	1	2	yes	—
56	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#126	1	2	yes	—
58	C₂×C₄×C₆₁	#135	1	2	yes	—
59	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×Dic₆₁	—	4	2	no	—
61	C₂×Dic₆₁	—	8	1	no	—
62	C₂×Dic₆₁	—	8	1	no	—
63	C₂×Dic₆₁	—	8	2	no	—
64	C₂×Dic₆₁	—	8	1	no	—
65	Dic₁₂₂	—	4	4	no	—
66	Dic₁₂₂	—	8	2	no	—
67	C₈×C₆₁	—	4	2	no	—
68	C₈×C₆₁	—	8	2	no	—
69	#92418	—	8	2	no	—
70	#10635	—	61	1	no	maximal, Sylow
71	C₂²×C₆₁	—	2	2	no	—
72	C₂²×C₆₁	#397	1	1	yes	characteristic
73	C₂²×C₆₁	#488	1	1	yes	characteristic
74	C₂²×C₆₁	#490	1	1	yes	characteristic
75	C₄×C₆₁	—	2	2	no	—
76	C₄×C₆₁	—	2	2	no	—
77	C₄×C₆₁	#463	1	2	yes	—
78	C₄×C₆₁	#482	1	2	yes	—
79	Dic₆₁	—	8	2	no	—
80	Dic₆₁	—	16	1	no	—
81	#1960	—	61	1	no	—
82	#1838	—	61	1	no	—
83	#1546	—	61	2	no	—
84	#1286	—	61	2	no	—
85	#1281	D₆₁	1	1	yes	characteristic
86	C₂×C₆₁	—	2	1	no	—
87	C₂×C₆₁	—	2	1	no	—
88	C₂×C₆₁	#2073	1	1	yes	characteristic
89	C₂×C₆₁	#2082	1	2	yes	—
90	C₂×C₄⋊Q₈	—	61	1	no	—
91	#431	C₂×D₆₁	1	2	yes	—
92	#423	—	61	2	no	—
93	C₂×#103	—	122	2	no	—
94	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
95	#382	—	61	1	no	—
96	#341	—	122	2	no	—
97	#332	—	122	4	no	—
98	C₆₁	#10635	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	122	1	no	—
100	C₂×M₁₆	—	122	2	no	—
101	C₂²×C₈	D₂₄₄	1	2	yes	—
102	C₄⋊Q₈	—	61	2	no	—
103	C₄⋊Q₈	—	122	2	no	—
104	C₄⋊Q₈	—	122	1	no	—
105	C₂×C₄⋊C₄	—	122	2	no	—
106	C₂×C₄⋊C₄	—	122	1	no	—
107	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	122	2	no	—
110	C₄⋊C₈	—	122	2	no	—
111	#103	—	244	4	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	—	2	2	no	—
115	#95	—	122	2	no	—
116	C₂×Q₈	—	122	2	no	—
117	C₂×Q₈	—	244	2	no	—
118	C₂×Q₈	—	244	2	no	—
119	C₂×Q₈	—	244	1	no	—
120	C₂²×C₄	—	244	1	no	—
121	C₂²×C₄	—	244	1	no	—
122	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
123	C₂²×C₄	#11758222	1	2	yes	—
124	M₁₆	—	244	2	no	—
125	M₁₆	—	244	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	244	2	no	—
131	C₄⋊C₄	—	122	4	no	—
132	C₄⋊C₄	—	244	2	no	—
133	C₄⋊C₄	—	244	2	no	—
134	C₄²	D₄×D₆₁	1	1	yes	characteristic
135	C₄²	#11758224	1	2	yes	—
136	C₄²	#11758226	1	1	yes	characteristic
137	C₂³	#14519765	1	1	yes	characteristic
138	Q₈	—	488	2	no	—
139	Q₈	—	244	4	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	488	2	no	—
142	C₂×C₄	—	488	1	no	—
143	C₂×C₄	—	488	1	no	—
144	C₂×C₄	—	488	1	no	—
145	C₂×C₄	—	244	2	no	—
146	C₂×C₄	#14519752	1	1	yes	characteristic
147	C₂×C₄	#14519759	1	2	yes	—
148	C₂×C₄	#14519761	1	1	yes	characteristic
149	C₂×C₄	#14519772	1	2	yes	—
150	C₂×C₄	#14519799	1	2	yes	—
151	C₂×C₄	#14519805	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	8	2	no	—
154	C₈	—	488	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#16839689	1	1	yes	characteristic
157	C₂²	#16840117	1	1	yes	characteristic
158	C₂²	#16840127	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	976	1	no	—
162	C₄	—	488	2	no	—
163	C₄	#16839978	1	2	yes	—
164	C₄	#16840087	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#21447234	1	1	yes	characteristic
168	C₂	#21447285	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446639

C₂

yes

maximal, characteristic

#21445960

C₂

yes

maximal, characteristic

#21444490

C₂

yes

maximal

#21443678

C₂

yes

maximal

C₆₁×#1281

C₂

yes

maximal, characteristic

C₂×#14519777

C₂²

yes

characteristic

#16839786

C₂²

yes

—

C₂×#14519702

—

#16839628

C₂²

yes

characteristic

#16839527

—

#16839507

—

C₆₁×#431

C₂²

yes

—

C₂×#97×C₆₁

C₂²

yes

characteristic

C₂²×Dic₁₂₂

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

—

—

—

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

#97×C₆₁

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

C₄²×C₆₁

yes

C₄²×C₆₁

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758194

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

#123

yes

characteristic

C₂×C₄×C₆₁

#126

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#10635

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#463

yes

—

C₄×C₆₁

#482

yes

—

Dic₆₁

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#2073

yes

characteristic

C₂×C₆₁

#2082

yes

—

C₂×C₄⋊Q₈

—

#431

C₂×D₆₁

yes

—

#423

—

C₂×#103

—

122

—

C₂×#97

C₂×D₆₁

yes

characteristic

#382

—

#341

—

122

—

#332

—

122

—

C₆₁

#10635

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

100

C₂×M₁₆

—

122

—

101

C₂²×C₈

D₂₄₄

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

122

—

104

C₄⋊Q₈

—

122

—

105

C₂×C₄⋊C₄

—

122

—

106

C₂×C₄⋊C₄

—

122

—

107

C₂×C₄²

C₂²×D₆₁

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

122

—

110

C₄⋊C₈

—

122

—

111

#103

—

244

—

112

C₂²⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

#95

—

122

—

116

C₂×Q₈

—

122

—

117

C₂×Q₈

—

244

—

118

C₂×Q₈

—

244

—

119

C₂×Q₈

—

244

—

120

C₂²×C₄

—

244

—

121

C₂²×C₄

—

244

—

122

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

123

C₂²×C₄

#11758222

yes

—

124

M₁₆

—

244

—

125

M₁₆

—

244

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

244

—

131

C₄⋊C₄

—

122

—

132

C₄⋊C₄

—

244

—

133

C₄⋊C₄

—

244

—

134

C₄²

D₄×D₆₁

yes

characteristic

135

C₄²

#11758224

yes

—

136

C₄²

#11758226

yes

characteristic

137

C₂³

#14519765

yes

characteristic

138

Q₈

—

488

—

139

Q₈

—

244

—

140

C₂×C₄

—

141

C₂×C₄

—

488

—

142

C₂×C₄

—

488

—

143

C₂×C₄

—

488

—

144

C₂×C₄

—

488

—

145

C₂×C₄

—

244

—

146

C₂×C₄

#14519752

yes

characteristic

147

C₂×C₄

#14519759

yes

—

148

C₂×C₄

#14519761

yes

characteristic

149

C₂×C₄

#14519772

yes

—

150

C₂×C₄

#14519799

yes

—

151

C₂×C₄

#14519805

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

488

—

155

C₂²

—

156

C₂²

#16839689

yes

characteristic

157

C₂²

#16840117

yes

characteristic

158

C₂²

#16840127

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

976

—

162

C₄

—

488

—

163

C₄

#16839978

yes

—

164

C₄

#16840087

yes

—

165

C₂

—

166

C₂

—

167

C₂

#21447234

yes

characteristic

168

C₂

#21447285

yes

—

	subgroup	G/H
center	C₂²	#16839689
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1281	C₂
socle	C₂²×C₆₁	#397
prime radical	C₂³×C₆₁	#124
hypercenter	#1281	D₆₁
hypocenter	C₆₁	#10635
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂²

#16839689

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1281

C₂

C₂²×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
98. C₆₁	#10635	semidirect

H ≅ G/N

type

98. C₆₁

#10635

semidirect

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

subgroup	index
#21446639	2
#21445960	2
#21444490	2
#21443678	2
C₆₁×#1281	2
#10635	61

subgroup

index

C₆₁×#1281

Subgroups — #29584825

Conjugacy classes of subgroups

Complements

Maximal subgroups