Group #29598606 — subgroups

Subgroups — #29598606

Order 15616.

334 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16840164	C₂	1	1	yes	maximal, characteristic
2	#21447575	C₂	1	1	yes	maximal, characteristic
3	#21443929	C₂	1	4	yes	maximal
4	C₄×#108×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519769	—	2	1	no	—
6	C₂×#14519717	C₂²	1	2	yes	—
7	#16840164	—	2	2	no	—
8	#16840164	C₄	1	4	yes	—
9	C₂×#14519711	C₂²	1	2	yes	—
10	C₂×#14519709	—	2	1	no	—
11	C₄⋊#11758185	—	2	2	no	—
12	C₂×#108×C₆₁	C₂²	1	2	yes	—
13	C₄×M₁₆×C₆₁	—	2	1	no	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#341	—	2	1	no	—
16	#14519769	—	2	4	no	—
17	#14519769	—	4	1	no	—
18	#14519769	—	4	1	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×#11758196	—	4	2	no	—
21	#14519717	—	2	4	no	—
22	C₂×#11758185	—	2	2	no	—
23	#14519711	—	2	2	no	—
24	#14519711	C₂×C₄	1	4	yes	—
25	#14519709	—	2	4	no	—
26	C₂×M₁₆×C₆₁	—	2	2	no	—
27	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
30	#108×C₆₁	—	2	2	no	—
31	#108×C₆₁	Q₈	1	4	yes	—
32	#97×C₆₁	—	4	1	no	—
33	C₄×C₈×C₆₁	—	4	1	no	—
34	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₆₁	D₄	1	2	yes	—
36	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
37	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₆₁	—	4	2	no	—
39	M₁₆×C₆₁	—	2	4	no	—
40	M₁₆×C₆₁	—	4	2	no	—
41	C₂×C₈×C₆₁	—	2	2	no	—
42	C₂×C₈×C₆₁	—	4	2	no	—
43	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₆₁	#41	1	2	yes	—
48	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
49	C₄²×C₆₁	D₈	1	1	yes	characteristic
50	C₄²×C₆₁	D₈	1	1	yes	characteristic
51	C₄⋊Dic₆₁	—	2	4	no	—
52	C₄⋊Dic₆₁	—	4	2	no	—
53	#11758196	—	4	4	no	—
54	#11758196	—	4	4	no	—
55	#11758185	—	4	2	no	—
56	C₂³×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₆₁	—	4	2	no	—
58	C₂×C₄×C₆₁	#102	1	2	yes	—
59	C₂×C₄×C₆₁	#102	1	2	yes	—
60	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#137	1	2	yes	—
64	C₂×Dic₆₁	—	4	4	no	—
65	C₂×Dic₆₁	—	8	4	no	—
66	C₈×C₆₁	—	2	2	no	—
67	C₈×C₆₁	—	4	4	no	—
68	C₈×C₆₁	C₄×Q₈	1	2	yes	—
69	C₈×C₆₁	#124	1	1	yes	characteristic
70	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
71	#13280	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	—	2	2	no	—
73	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
74	C₂²×C₆₁	#475	1	2	yes	—
75	C₄×C₆₁	—	2	2	no	—
76	C₄×C₆₁	—	2	2	no	—
77	C₄×C₆₁	#391	1	1	yes	characteristic
78	C₄×C₆₁	#419	1	1	yes	characteristic
79	C₄×C₆₁	#474	1	2	yes	—
80	Dic₆₁	—	8	4	no	—
81	C₂×#499	—	61	1	no	—
82	#2150	—	61	1	no	—
83	C₄×#108	D₆₁	1	1	yes	characteristic
84	#1387	—	61	4	no	—
85	C₂×C₆₁	—	2	2	no	—
86	C₂×C₆₁	#1988	1	1	yes	characteristic
87	C₂×C₆₁	#2148	1	2	yes	—
88	C₂×#125	—	122	1	no	—
89	C₂×M₃₂	—	61	2	no	—
90	#499	—	61	4	no	—
91	#499	—	122	2	no	—
92	C₂×#108	C₂×D₆₁	1	2	yes	—
93	C₂×#107	—	61	2	no	—
94	C₂×#106	—	122	1	no	—
95	C₄×M₁₆	—	2	1	no	—
96	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
97	C₄⋊C₁₆	—	122	2	no	—
98	#341	—	2	1	no	—
99	C₆₁	#13280	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
103	#125	—	122	4	no	—
104	#125	—	244	1	no	—
105	#125	—	244	1	no	—
106	C₂×C₄⋊C₄	—	122	2	no	—
107	C₂×C₄⋊C₄	—	244	2	no	—
108	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
109	M₃₂	—	122	4	no	—
110	C₂×C₁₆	—	122	2	no	—
111	#108	—	2	2	no	—
112	#108	Dic₁₂₂	1	4	yes	—
113	#107	—	61	4	no	—
114	#107	—	122	2	no	—
115	#106	—	122	4	no	—
116	#97	—	4	1	no	—
117	C₄×C₈	—	4	1	no	—
118	C₄×C₈	C₄×D₆₁	1	2	yes	—
119	C₄×C₈	D₂₄₄	1	2	yes	—
120	C₂²×C₄	—	244	2	no	—
121	C₂²×C₄	#11758201	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	#11758198	1	2	yes	—
128	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
131	C₂×C₈	#11758222	1	2	yes	—
132	C₄⋊C₄	—	122	4	no	—
133	C₄⋊C₄	—	244	4	no	—
134	C₄⋊C₄	—	244	4	no	—
135	C₄⋊C₄	—	244	2	no	—
136	C₄²	#11758198	1	2	yes	—
137	C₄²	#11758199	1	1	yes	characteristic
138	C₄²	#11758199	1	1	yes	characteristic
139	C₁₆	—	244	2	no	—
140	C₂³	C₂×#11758201	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	488	4	no	—
143	C₂×C₄	—	244	4	no	—
144	C₂×C₄	#14519697	1	2	yes	—
145	C₂×C₄	#14519697	1	2	yes	—
146	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
147	C₂×C₄	#14519756	1	2	yes	—
148	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
149	C₂×C₄	#14519807	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	4	4	no	—
152	C₈	C₄×Dic₁₂₂	1	2	yes	—
153	C₈	#14519765	1	1	yes	characteristic
154	C₈	#14519777	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#14519697	1	1	yes	characteristic
157	C₂²	#16840047	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	488	4	no	—
161	C₄	#16839668	1	1	yes	characteristic
162	C₄	#16839770	1	1	yes	characteristic
163	C₄	#16840042	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#21446771	1	1	yes	characteristic
166	C₂	#21447566	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840164

C₂

yes

maximal, characteristic

#21447575

C₂

yes

maximal, characteristic

#21443929

C₂

yes

maximal

C₄×#108×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519769

—

C₂×#14519717

C₂²

yes

—

#16840164

—

#16840164

C₄

yes

—

C₂×#14519711

C₂²

yes

—

C₂×#14519709

—

C₄⋊#11758185

—

C₂×#108×C₆₁

C₂²

yes

—

C₄×M₁₆×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#341

—

#14519769

—

#14519769

—

#14519769

—

C₂×C₄⋊Dic₆₁

—

—

—

—

—

C₂×C₄

yes

—

#14519709

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

#108×C₆₁

—

#108×C₆₁

Q₈

yes

—

#97×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

SD₁₆

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄²×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

D₈

yes

characteristic

C₄²×C₆₁

D₈

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

#11758196

—

#11758185

—

C₂³×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#102

yes

—

C₂×C₄×C₆₁

#102

yes

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₄×Q₈

yes

—

C₈×C₆₁

#124

yes

characteristic

C₈×C₆₁

C₄⋊Q₈

yes

characteristic

#13280

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#102

yes

characteristic

C₂²×C₆₁

#475

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

C₂×#499

—

#2150

—

C₄×#108

D₆₁

yes

characteristic

#1387

—

C₂×C₆₁

—

C₂×C₆₁

#1988

yes

characteristic

C₂×C₆₁

#2148

yes

—

C₂×#125

—

122

—

C₂×M₃₂

—

#499

—

#499

—

122

—

C₂×#108

C₂×D₆₁

yes

—

C₂×#107

—

C₂×#106

—

122

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

122

—

#341

—

C₆₁

#13280

yes

characteristic, complemented, Sylow

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

102

C₂²×C₈

C₂²×D₆₁

yes

characteristic

103

#125

—

122

—

104

#125

—

244

—

105

#125

—

244

—

106

C₂×C₄⋊C₄

—

122

—

107

C₂×C₄⋊C₄

—

244

—

108

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

109

M₃₂

—

122

—

110

C₂×C₁₆

—

122

—

111

#108

—

112

#108

Dic₁₂₂

yes

—

113

#107

—

114

#107

—

122

—

115

#106

—

122

—

116

#97

—

117

C₄×C₈

—

118

C₄×C₈

C₄×D₆₁

yes

—

119

C₄×C₈

D₂₄₄

yes

—

120

C₂²×C₄

—

244

—

121

C₂²×C₄

#11758201

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

#11758198

yes

—

128

C₂×C₈

C₂×Dic₁₂₂

yes

—

129

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

130

C₂×C₈

C₂×D₂₄₄

yes

characteristic

131

C₂×C₈

#11758222

yes

—

132

C₄⋊C₄

—

122

—

133

C₄⋊C₄

—

244

—

134

C₄⋊C₄

—

244

—

135

C₄⋊C₄

—

244

—

136

C₄²

#11758198

yes

—

137

C₄²

#11758199

yes

characteristic

138

C₄²

#11758199

yes

characteristic

139

C₁₆

—

244

—

140

C₂³

C₂×#11758201

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

488

—

143

C₂×C₄

—

244

—

144

C₂×C₄

#14519697

yes

—

145

C₂×C₄

#14519697

yes

—

146

C₂×C₄

C₂×#11758198

yes

characteristic

147

C₂×C₄

#14519756

yes

—

148

C₂×C₄

C₂×#11758199

yes

characteristic

149

C₂×C₄

#14519807

yes

—

150

C₈

—

151

C₈

—

152

C₈

C₄×Dic₁₂₂

yes

—

153

C₈

#14519765

yes

characteristic

154

C₈

#14519777

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#14519697

yes

characteristic

157

C₂²

#16840047

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

488

—

161

C₄

#16839668

yes

characteristic

162

C₄

#16839770

yes

characteristic

163

C₄

#16840042

yes

—

164

C₂

—

165

C₂

#21446771

yes

characteristic

166

C₂

#21447566

yes

—

	subgroup	G/H
center	C₂²	C₂×#14519697
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₄×#108×C₆₁	C₂
socle	C₂²×C₆₁	C₂×#102
prime radical	C₂³×C₆₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₄×#108	D₆₁
hypocenter	C₆₁	#13280
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂²

C₂×#14519697

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₄×#108×C₆₁

C₂

socle

C₂²×C₆₁

C₂×#102

prime radical

C₂³×C₆₁

C₂×SD₁₆

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
99. C₆₁	#13280	semidirect

H ≅ G/N

type

99. C₆₁

#13280

semidirect

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

subgroup	index
C₂×#16840164	2
#21447575	2
#21443929	2
C₄×#108×C₆₁	2
#13280	61

subgroup

index

C₄×#108×C₆₁

Subgroups — #29598606

Conjugacy classes of subgroups

Complements

Maximal subgroups