Group #29594319 — subgroups

Subgroups — #29594319

Order 15616.

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

normal subgroups only

This group has 218 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	#21447971	C₂	1	2	yes	maximal
2	#21445437	C₂	1	1	yes	maximal, characteristic
3	#21443889	C₂	1	1	yes	maximal, characteristic
4	#21443878	C₂	1	2	yes	maximal
5	C₈×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840167	—	2	1	no	—
7	#16840167	C₂²	1	2	yes	—
8	#16840142	—	2	1	no	—
9	#16839587	—	2	2	no	—
10	C₄⋊#11758185	C₂²	1	2	yes	—
11	#16839566	—	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	—	4	2	no	—
18	#14519777	—	2	2	no	—
19	C₂×#11758185	—	2	2	no	—
20	#14519711	—	2	2	no	—
21	#14519711	—	2	2	no	—
22	#14519711	—	4	1	no	—
23	#14519711	—	4	1	no	—
24	C₂×Q₁₆×C₆₁	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₆₁	—	2	1	no	—
26	#107×C₆₁	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₆₁	—	2	1	no	—
28	C₄⋊C₈×C₆₁	—	4	1	no	—
29	#103×C₆₁	—	2	1	no	—
30	C₄×C₈×C₆₁	—	4	1	no	—
31	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×Q₈×C₆₁	—	2	1	no	—
35	C₂×Dic₁₂₂	—	4	2	no	—
36	Q₁₆×C₆₁	—	2	1	no	—
37	Q₁₆×C₆₁	Q₁₆	1	2	yes	complemented
38	C₂×C₈×C₆₁	—	2	1	no	—
39	C₂×C₈×C₆₁	—	4	1	no	—
40	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
41	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₆₁	—	2	1	no	—
45	C₄⋊C₄×C₆₁	—	4	1	no	—
46	C₄²×C₆₁	—	4	1	no	—
47	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₄⋊Dic₆₁	—	4	2	no	—
49	C₄⋊Dic₆₁	—	4	2	no	—
50	Dic₂₄₄	—	8	4	no	—
51	#11758185	—	4	2	no	—
52	Q₈×C₆₁	—	2	2	no	—
53	Q₈×C₆₁	—	4	1	no	—
54	C₂×C₄×C₆₁	—	4	1	no	—
55	C₂×C₄×C₆₁	—	4	1	no	—
56	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂×Dic₆₁	—	8	2	no	—
60	Dic₁₂₂	—	8	4	no	—
61	C₈×C₆₁	—	2	2	no	—
62	C₈×C₆₁	—	2	1	no	—
63	C₈×C₆₁	—	2	1	no	—
64	C₈×C₆₁	—	8	1	no	—
65	C₈×C₆₁	Q₃₂	1	2	yes	—
66	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
67	C₈×C₆₁	#135	1	1	yes	characteristic
68	#12448	—	61	1	no	maximal, Sylow
69	C₂²×C₆₁	#459	1	1	yes	characteristic
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	—	4	2	no	—
72	C₄×C₆₁	—	8	1	no	—
73	C₄×C₆₁	#467	1	1	yes	characteristic
74	C₄×C₆₁	#473	1	1	yes	characteristic
75	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
76	C₄×C₆₁	#510	1	1	yes	characteristic
77	Dic₆₁	—	16	2	no	—
78	#2227	—	61	2	no	—
79	#1713	—	61	1	no	—
80	C₈×Q₁₆	D₆₁	1	1	yes	characteristic
81	#1370	—	61	1	no	—
82	#1365	—	61	2	no	—
83	C₂×C₆₁	#1669	1	1	yes	characteristic
84	C₂×C₆₁	#2209	1	1	yes	characteristic
85	C₂×C₆₁	#2213	1	1	yes	characteristic
86	#500	—	61	2	no	—
87	#500	—	122	1	no	—
88	#494	—	122	1	no	—
89	C₈×Q₈	—	2	1	no	—
90	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
91	#366	—	122	2	no	—
92	C₄⋊C₁₆	—	61	2	no	—
93	#358	—	122	2	no	—
94	#335	C₂×D₆₁	1	1	yes	characteristic
95	Q₈⋊C₈	—	2	1	no	—
96	C₈²	C₂×D₆₁	1	1	yes	characteristic
97	C₆₁	#12448	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₁₆	—	244	2	no	—
99	C₂×Q₁₆	D₂₄₄	1	1	yes	characteristic
100	C₄⋊Q₈	—	122	2	no	—
101	C₄×Q₈	—	2	1	no	—
102	C₂×C₁₆	—	122	2	no	—
103	#107	—	122	2	no	—
104	#107	—	122	2	no	—
105	#107	—	244	1	no	—
106	#107	—	244	1	no	—
107	#107	D₂₄₄	1	1	yes	characteristic
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	4	1	no	—
110	#103	—	2	1	no	—
111	C₄×C₈	—	4	1	no	—
112	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	244	2	no	—
117	Q₁₆	—	2	1	no	—
118	Q₁₆	—	488	4	no	—
119	Q₁₆	Dic₂₄₄	1	2	yes	complemented
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	#11758199	1	1	yes	characteristic
123	C₂×C₈	#11758199	1	1	yes	characteristic
124	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
125	C₂×C₈	#11758222	1	1	yes	characteristic
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	244	2	no	—
129	C₄⋊C₄	—	244	2	no	—
130	C₄²	—	4	1	no	—
131	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
132	C₁₆	—	244	2	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	488	4	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	488	2	no	—
139	C₂×C₄	#14519751	1	1	yes	characteristic
140	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
141	C₂×C₄	#14519802	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	8	1	no	—
146	C₈	#14519723	1	2	yes	—
147	C₈	C₂×Dic₂₄₄	1	1	yes	characteristic
148	C₈	#14519799	1	1	yes	characteristic
149	C₂²	#16839957	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	4	2	no	—
152	C₄	—	8	1	no	—
153	C₄	—	976	2	no	—
154	C₄	#16840003	1	1	yes	characteristic
155	C₄	#16840037	1	1	yes	characteristic
156	C₄	C₂×#14519723	1	1	yes	characteristic
157	C₄	#16840202	1	1	yes	characteristic
158	C₂	#21445184	1	1	yes	characteristic
159	C₂	#21447869	1	1	yes	characteristic
160	C₂	#21447895	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447971

C₂

yes

maximal

#21445437

C₂

yes

maximal, characteristic

#21443889

C₂

yes

maximal, characteristic

#21443878

C₂

yes

maximal

C₈×Q₁₆×C₆₁

C₂

yes

maximal, characteristic

#16840167

—

#16840167

C₂²

yes

—

#16840142

—

#16839587

—

C₄⋊#11758185

C₂²

yes

—

#16839566

—

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×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

Q₁₆

yes

complemented

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

—

#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

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

Q₃₂

yes

—

C₈×C₆₁

C₂×Q₁₆

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

—

#2227

—

#1713

—

C₈×Q₁₆

D₆₁

yes

characteristic

#1370

—

#1365

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

122

—

#494

—

122

—

C₈×Q₈

—

C₄×Q₁₆

C₂×D₆₁

yes

characteristic

#366

—

122

—

C₄⋊C₁₆

—

#358

—

122

—

#335

C₂×D₆₁

yes

characteristic

Q₈⋊C₈

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12448

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

244

—

C₂×Q₁₆

D₂₄₄

yes

characteristic

100

C₄⋊Q₈

—

122

—

101

C₄×Q₈

—

102

C₂×C₁₆

—

122

—

103

#107

—

122

—

104

#107

—

122

—

105

#107

—

244

—

106

#107

—

244

—

107

#107

D₂₄₄

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

#103

—

111

C₄×C₈

—

112

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

113

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

114

C₄×C₈

C₂²×D₆₁

yes

characteristic

115

C₂×Q₈

—

116

C₂×Q₈

—

244

—

117

Q₁₆

—

118

Q₁₆

—

488

—

119

Q₁₆

Dic₂₄₄

yes

complemented

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

#11758199

yes

characteristic

123

C₂×C₈

#11758199

yes

characteristic

124

C₂×C₈

C₂×D₂₄₄

yes

characteristic

125

C₂×C₈

#11758222

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

244

—

129

C₄⋊C₄

—

244

—

130

C₄²

—

131

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

132

C₁₆

—

244

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

488

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

488

—

139

C₂×C₄

#14519751

yes

characteristic

140

C₂×C₄

C₂×#11758199

yes

characteristic

141

C₂×C₄

#14519802

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

#14519723

yes

—

147

C₈

C₂×Dic₂₄₄

yes

characteristic

148

C₈

#14519799

yes

characteristic

149

C₂²

#16839957

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

976

—

154

C₄

#16840003

yes

characteristic

155

C₄

#16840037

yes

characteristic

156

C₄

C₂×#14519723

yes

characteristic

157

C₄

#16840202

yes

characteristic

158

C₂

#21445184

yes

characteristic

159

C₂

#21447869

yes

characteristic

160

C₂

#21447895

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	#12448
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

#12448

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×Q₁₆

N	H ≅ G/N	type
37. Q₁₆×C₆₁	Q₁₆	semidirect
97. C₆₁	#12448	semidirect
119. Q₁₆	Dic₂₄₄	semidirect

H ≅ G/N

type

37. Q₁₆×C₆₁

Q₁₆

semidirect

97. C₆₁

#12448

semidirect

119. Q₁₆

Dic₂₄₄

semidirect

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

subgroup	index
#21447971	2
#21445437	2
#21443889	2
#21443878	2
C₈×Q₁₆×C₆₁	2
#12448	61

subgroup

index

C₈×Q₁₆×C₆₁

Subgroups — #29594319

Conjugacy classes of subgroups

Complements

Maximal subgroups