Group #29594200 — subgroups

Subgroups — #29594200

Order 15616.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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	#21447869	C₂	1	2	yes	maximal
2	#21445437	C₂	1	1	yes	maximal, characteristic
3	#21443889	C₂	1	1	yes	maximal, characteristic
4	#21443867	C₂	1	2	yes	maximal
5	C₈×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519723	—	2	2	no	—
7	#16840167	—	2	1	no	—
8	#16840167	—	2	1	no	—
9	#16840142	C₂²	1	2	yes	—
10	C₄⋊#11758185	C₂²	1	2	yes	—
11	#16839565	—	2	2	no	—
12	C₈×Q₈×C₆₁	—	2	1	no	—
13	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#335	C₂²	1	1	yes	characteristic
15	C₆₁×Q₈⋊C₈	—	2	1	no	—
16	C₈²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₄₄	—	2	2	no	—
18	C₂×Dic₂₄₄	—	2	2	no	—
19	#14519777	—	2	2	no	—
20	#14519723	—	4	4	no	—
21	C₂×#11758185	—	2	2	no	—
22	#14519711	—	4	2	no	—
23	#14519711	—	4	1	no	—
24	#14519711	—	4	1	no	—
25	C₂×Q₁₆×C₆₁	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₆₁	—	2	1	no	—
27	#107×C₆₁	D₄	1	1	yes	characteristic
28	C₄⋊C₈×C₆₁	—	2	1	no	—
29	C₄⋊C₈×C₆₁	—	4	1	no	—
30	#103×C₆₁	—	2	1	no	—
31	C₄×C₈×C₆₁	—	4	1	no	—
32	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
34	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×Q₈×C₆₁	—	2	1	no	—
36	C₂×Dic₁₂₂	—	4	2	no	—
37	Q₁₆×C₆₁	—	2	1	no	—
38	Q₁₆×C₆₁	D₈	1	2	yes	—
39	C₂×C₈×C₆₁	—	2	1	no	—
40	C₂×C₈×C₆₁	—	4	1	no	—
41	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
42	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₆₁	—	2	1	no	—
46	C₄⋊C₄×C₆₁	—	4	1	no	—
47	C₄²×C₆₁	—	4	1	no	—
48	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₄⋊Dic₆₁	—	4	2	no	—
50	C₄⋊Dic₆₁	—	4	2	no	—
51	Dic₂₄₄	—	4	2	no	—
52	Dic₂₄₄	—	4	2	no	—
53	Dic₂₄₄	—	8	2	no	—
54	#11758185	—	4	2	no	—
55	Q₈×C₆₁	—	2	2	no	—
56	Q₈×C₆₁	—	4	1	no	—
57	C₂×C₄×C₆₁	—	4	1	no	—
58	C₂×C₄×C₆₁	—	4	1	no	—
59	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₂×Dic₆₁	—	8	2	no	—
63	Dic₁₂₂	—	8	2	no	—
64	Dic₁₂₂	—	8	2	no	—
65	C₈×C₆₁	—	2	2	no	—
66	C₈×C₆₁	—	2	1	no	—
67	C₈×C₆₁	—	2	1	no	—
68	C₈×C₆₁	—	8	1	no	—
69	C₈×C₆₁	Q₃₂	1	2	yes	complemented
70	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
71	C₈×C₆₁	#135	1	1	yes	characteristic
72	#12427	—	61	1	no	maximal, Sylow
73	C₂²×C₆₁	#459	1	1	yes	characteristic
74	C₄×C₆₁	—	2	1	no	—
75	C₄×C₆₁	—	4	2	no	—
76	C₄×C₆₁	—	8	1	no	—
77	C₄×C₆₁	#466	1	1	yes	characteristic
78	C₄×C₆₁	#472	1	1	yes	characteristic
79	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
80	C₄×C₆₁	#510	1	1	yes	characteristic
81	Dic₆₁	—	16	2	no	—
82	#2209	—	61	2	no	—
83	#1713	—	61	1	no	—
84	C₈×Q₁₆	D₆₁	1	1	yes	characteristic
85	#1370	—	61	1	no	—
86	#1362	—	61	2	no	—
87	C₂×C₆₁	#1667	1	1	yes	characteristic
88	C₂×C₆₁	#2209	1	1	yes	characteristic
89	C₂×C₆₁	#2214	1	1	yes	characteristic
90	C₂×Q₃₂	—	122	2	no	—
91	#500	—	122	1	no	—
92	#500	—	122	1	no	—
93	#494	—	61	2	no	—
94	C₈×Q₈	—	2	1	no	—
95	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
96	C₄⋊C₁₆	—	61	2	no	—
97	#358	—	122	2	no	—
98	#335	C₂×D₆₁	1	1	yes	characteristic
99	Q₈⋊C₈	—	2	1	no	—
100	C₈²	C₂×D₆₁	1	1	yes	characteristic
101	C₆₁	#12427	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₁₆	—	122	2	no	—
103	C₂×Q₁₆	—	122	2	no	—
104	C₂×Q₁₆	D₂₄₄	1	1	yes	characteristic
105	C₄⋊Q₈	—	122	2	no	—
106	C₄×Q₈	—	2	1	no	—
107	Q₃₂	—	244	4	no	—
108	C₂×C₁₆	—	122	2	no	—
109	#107	—	244	2	no	—
110	#107	—	244	1	no	—
111	#107	—	244	1	no	—
112	#107	D₂₄₄	1	1	yes	characteristic
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	4	1	no	—
115	#103	—	2	1	no	—
116	C₄×C₈	—	4	1	no	—
117	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
118	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	—	244	2	no	—
122	Q₁₆	—	2	1	no	—
123	Q₁₆	—	488	2	no	—
124	Q₁₆	—	244	2	no	—
125	Q₁₆	—	244	2	no	—
126	Q₁₆	D₄₈₈	1	2	yes	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	#11758199	1	1	yes	characteristic
130	C₂×C₈	#11758199	1	1	yes	characteristic
131	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
132	C₂×C₈	#11758222	1	1	yes	characteristic
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	244	2	no	—
136	C₄⋊C₄	—	244	2	no	—
137	C₄²	—	4	1	no	—
138	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
139	C₁₆	—	244	2	no	—
140	Q₈	—	2	2	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	488	2	no	—
143	Q₈	—	488	2	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	488	2	no	—
147	C₂×C₄	#14519751	1	1	yes	characteristic
148	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
149	C₂×C₄	#14519802	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	8	1	no	—
154	C₈	#14519723	1	2	yes	complemented
155	C₈	C₂×D₄₈₈	1	1	yes	characteristic
156	C₈	#14519799	1	1	yes	characteristic
157	C₂²	#16839957	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	4	2	no	—
160	C₄	—	8	1	no	—
161	C₄	—	976	2	no	—
162	C₄	#16839998	1	1	yes	characteristic
163	C₄	#16840032	1	1	yes	characteristic
164	C₄	C₂×#14519723	1	1	yes	characteristic
165	C₄	#16840202	1	1	yes	characteristic
166	C₂	#21445174	1	1	yes	characteristic
167	C₂	#21447869	1	1	yes	characteristic
168	C₂	#21447900	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447869

C₂

yes

maximal

#21445437

C₂

yes

maximal, characteristic

#21443889

C₂

yes

maximal, characteristic

#21443867

C₂

yes

maximal

C₈×Q₁₆×C₆₁

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄⋊#11758185

C₂²

yes

—

#16839565

—

C₈×Q₈×C₆₁

—

C₄×Q₁₆×C₆₁

C₂²

yes

characteristic

C₆₁×#335

C₂²

yes

characteristic

C₆₁×Q₈⋊C₈

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

—

—

—

—

—

—

C₂×Q₁₆×C₆₁

D₄

yes

characteristic

C₄×Q₈×C₆₁

—

#107×C₆₁

D₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

#103×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

characteristic

C₂×C₈×C₆₁

D₈

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

#11758185

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

Q₃₂

yes

complemented

C₈×C₆₁

C₂×D₈

yes

C₈×C₆₁

yes

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂×Q₃₂

yes

characteristic

C₄×C₆₁

#510

yes

characteristic

Dic₆₁

—

#2209

—

#1713

—

C₈×Q₁₆

D₆₁

yes

characteristic

#1370

—

#1362

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×Q₃₂

—

122

—

#500

—

122

—

#500

—

122

—

#494

—

C₈×Q₈

—

C₄×Q₁₆

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

#358

—

122

—

#335

C₂×D₆₁

yes

characteristic

Q₈⋊C₈

—

100

C₈²

C₂×D₆₁

yes

characteristic

101

C₆₁

#12427

yes

characteristic, complemented, Sylow

102

C₂×Q₁₆

—

122

—

103

C₂×Q₁₆

—

122

—

104

C₂×Q₁₆

D₂₄₄

yes

characteristic

105

C₄⋊Q₈

—

122

—

106

C₄×Q₈

—

107

Q₃₂

—

244

—

108

C₂×C₁₆

—

122

—

109

#107

—

244

—

110

#107

—

244

—

111

#107

—

244

—

112

#107

D₂₄₄

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

#103

—

116

C₄×C₈

—

117

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

118

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

119

C₄×C₈

C₂²×D₆₁

yes

characteristic

120

C₂×Q₈

—

121

C₂×Q₈

—

244

—

122

Q₁₆

—

123

Q₁₆

—

488

—

124

Q₁₆

—

244

—

125

Q₁₆

—

244

—

126

Q₁₆

D₄₈₈

yes

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

#11758199

yes

characteristic

130

C₂×C₈

#11758199

yes

characteristic

131

C₂×C₈

C₂×D₂₄₄

yes

characteristic

132

C₂×C₈

#11758222

yes

characteristic

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

244

—

136

C₄⋊C₄

—

244

—

137

C₄²

—

138

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

139

C₁₆

—

244

—

140

Q₈

—

141

Q₈

—

142

Q₈

—

488

—

143

Q₈

—

488

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

488

—

147

C₂×C₄

#14519751

yes

characteristic

148

C₂×C₄

C₂×#11758199

yes

characteristic

149

C₂×C₄

#14519802

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

#14519723

yes

complemented

155

C₈

C₂×D₄₈₈

yes

characteristic

156

C₈

#14519799

yes

characteristic

157

C₂²

#16839957

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

976

—

162

C₄

#16839998

yes

characteristic

163

C₄

#16840032

yes

characteristic

164

C₄

C₂×#14519723

yes

characteristic

165

C₄

#16840202

yes

characteristic

166

C₂

#21445174

yes

characteristic

167

C₂

#21447869

yes

characteristic

168

C₂

#21447900

yes

characteristic

	subgroup	G/H
center	C₂²	#16839957
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₈×Q₁₆×C₆₁	C₂
socle, prime radical	C₂²×C₆₁	#459
hypercenter	C₈×Q₁₆	D₆₁
hypocenter	C₆₁	#12427
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×Q₁₆

subgroup

G/H

center

C₂²

#16839957

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₈×Q₁₆×C₆₁

C₂

socle, prime radical

C₂²×C₆₁

#459

hypercenter

C₈×Q₁₆

D₆₁

hypocenter

C₆₁

#12427

LCS

G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×Q₁₆

N	H ≅ G/N	type
69. C₈×C₆₁	Q₃₂	semidirect
101. C₆₁	#12427	semidirect
154. C₈	#14519723	semidirect

H ≅ G/N

type

69. C₈×C₆₁

Q₃₂

semidirect

101. C₆₁

#12427

semidirect

154. C₈

#14519723

semidirect

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

subgroup	index
#21447869	2
#21445437	2
#21443889	2
#21443867	2
C₈×Q₁₆×C₆₁	2
#12427	61

subgroup

index

C₈×Q₁₆×C₆₁

Subgroups — #29594200

Conjugacy classes of subgroups

Complements

Maximal subgroups