Group #30594702 — subgroups

Subgroups — #30594702

Order 15616.

212 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 62 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	#21462182	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519848	C₂	1	1	yes	maximal, characteristic
3	#21444573	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519848	—	2	1	no	—
5	C₄×#11758213	—	2	1	no	—
6	#16840683	C₄	1	2	yes	—
7	#16840657	—	2	1	no	—
8	#16840650	—	2	1	no	—
9	#16840638	C₄	1	2	yes	—
10	#16839784	—	2	1	no	—
11	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
12	#16839513	C₄	1	1	yes	characteristic
13	#16839512	—	2	1	no	—
14	C₆₁×#334	C₄	1	1	yes	characteristic
15	C₂×#11758213	—	2	2	no	—
16	C₂×#11758187	—	2	2	no	—
17	#14519848	—	2	4	no	—
18	#14519832	—	4	1	no	—
19	C₄×#92420	—	4	1	no	—
20	C₂×#11758189	—	4	1	no	—
21	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
23	C₄²×D₆₁	D₄	1	1	yes	characteristic
24	C₄⋊#92418	—	2	1	no	—
25	#14519706	—	4	1	no	—
26	C₄×#92418	D₄	1	1	yes	characteristic
27	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
29	C₄⋊C₈×C₆₁	—	2	1	no	—
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×D₆₁	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×D₆₁	Q₁₆	1	1	yes	characteristic
34	C₂×#92420	—	4	2	no	—
35	#11758213	—	2	4	no	—
36	#11758213	—	4	2	no	—
37	C₂×C₈×C₆₁	—	4	1	no	—
38	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₆₁	D₈	1	1	yes	characteristic
43	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
44	C₂×#92418	—	4	1	no	—
45	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
47	#11758189	—	8	2	no	—
48	C₈×D₆₁	—	2	2	no	—
49	C₈×D₆₁	—	4	1	no	—
50	#11758187	—	4	2	no	—
51	C₂²×D₆₁	#103	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
55	C₂×Dic₆₁	#102	1	1	yes	characteristic
56	C₂×Dic₆₁	#106	1	1	yes	characteristic
57	C₂×Dic₆₁	#107	1	1	yes	characteristic
58	C₄×D₆₁	—	2	1	no	—
59	C₄×D₆₁	—	2	1	no	—
60	C₄×D₆₁	—	2	2	no	—
61	C₄×D₆₁	—	4	1	no	—
62	#92420	—	4	4	no	—
63	C₈×C₆₁	—	2	1	no	—
64	C₈×C₆₁	—	2	1	no	—
65	C₈×C₆₁	—	8	1	no	—
66	#92418	—	2	1	no	—
67	#92418	—	8	1	no	—
68	#92418	#104	1	2	yes	—
69	#7300	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	#340	1	1	yes	characteristic
71	C₂×D₆₁	—	2	1	no	—
72	C₂×D₆₁	—	2	1	no	—
73	C₂×D₆₁	—	2	1	no	—
74	C₄×C₆₁	—	2	1	no	—
75	C₄×C₆₁	—	2	1	no	—
76	C₄×C₆₁	#339	1	1	yes	characteristic
77	C₄×C₆₁	#344	1	1	yes	characteristic
78	Dic₆₁	—	2	1	no	—
79	Dic₆₁	—	2	1	no	—
80	Dic₆₁	#365	1	2	yes	—
81	#2163	—	61	1	no	—
82	C₄×#108	—	61	1	no	—
83	#1566	—	61	1	no	—
84	C₂×C₆₁	#1303	1	1	yes	characteristic
85	C₂×C₆₁	#1383	1	1	yes	characteristic
86	C₂×C₆₁	#1388	1	1	yes	characteristic
87	D₆₁	—	4	1	no	—
88	C₂×#108	—	122	1	no	—
89	#423	—	122	1	no	—
90	C₄×M₁₆	—	122	1	no	—
91	C₂×C₄×C₈	—	61	1	no	—
92	C₄⋊C₁₆	—	61	2	no	—
93	C₂²⋊C₁₆	—	122	1	no	—
94	#346	—	122	1	no	—
95	#341	—	61	2	no	—
96	#334	—	61	1	no	—
97	#334	—	122	1	no	—
98	#334	#6923	1	1	yes	characteristic
99	C₆₁	#7300	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	122	2	no	—
101	C₂×M₁₆	—	244	1	no	—
102	C₂²×C₈	—	61	1	no	—
103	C₂²×C₈	—	61	1	no	—
104	C₂×C₄²	—	61	1	no	—
105	C₂×C₁₆	—	122	2	no	—
106	#108	—	122	4	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	122	1	no	—
109	C₄⋊C₈	—	244	1	no	—
110	#97	—	244	1	no	—
111	C₄×C₈	—	61	1	no	—
112	C₄×C₈	—	61	1	no	—
113	C₄×C₈	—	61	1	no	—
114	C₄×C₈	—	244	1	no	—
115	C₄×C₈	C₂×#6923	1	1	yes	characteristic
116	C₂²×C₄	—	61	1	no	—
117	C₂²×C₄	—	61	1	no	—
118	C₂²×C₄	—	61	1	no	—
119	M₁₆	—	122	4	no	—
120	M₁₆	—	488	2	no	—
121	M₁₆	—	244	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	61	1	no	—
124	C₂×C₈	—	61	1	no	—
125	C₂×C₈	—	122	2	no	—
126	C₂×C₈	—	244	1	no	—
127	C₂×C₈	—	244	1	no	—
128	C₂×C₈	—	244	2	no	—
129	C₂×C₈	C₄×#6923	1	1	yes	characteristic
130	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
131	C₄²	—	61	1	no	—
132	C₄²	—	61	1	no	—
133	C₄²	—	61	1	no	—
134	C₄²	C₂²⋊#6923	1	1	yes	characteristic
135	C₁₆	—	244	2	no	—
136	C₂³	—	61	1	no	—
137	C₂×C₄	—	61	1	no	—
138	C₂×C₄	—	61	1	no	—
139	C₂×C₄	—	61	1	no	—
140	C₂×C₄	—	122	2	no	—
141	C₂×C₄	—	122	1	no	—
142	C₂×C₄	—	122	1	no	—
143	C₂×C₄	—	244	1	no	—
144	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
145	C₂×C₄	#14519839	1	1	yes	characteristic
146	C₂×C₄	#14519840	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	8	1	no	—
150	C₈	—	61	2	no	—
151	C₈	—	122	1	no	—
152	C₈	—	488	1	no	—
153	C₈	—	244	4	no	—
154	C₂²	—	122	1	no	—
155	C₂²	—	122	1	no	—
156	C₂²	—	122	1	no	—
157	C₂²	#16840635	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	61	2	no	—
161	C₄	—	122	1	no	—
162	C₄	—	122	1	no	—
163	C₄	#16840634	1	1	yes	characteristic
164	C₄	#16840644	1	1	yes	characteristic
165	C₂	—	244	1	no	—
166	C₂	#21460715	1	1	yes	characteristic
167	C₂	#21460883	1	1	yes	characteristic
168	C₂	#21460898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462182

C₂

yes

maximal, characteristic

C₄×#14519848

C₂

yes

maximal, characteristic

#21444573

C₂

yes

maximal, characteristic

C₂×#14519848

—

C₄×#11758213

—

#16840683

C₄

yes

—

#16840657

—

#16840650

—

#16840638

C₄

yes

—

#16839784

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839513

C₄

yes

characteristic

#16839512

—

C₆₁×#334

C₄

yes

—

—

—

—

—

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

D₄

yes

characteristic

C₄²×D₆₁

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

Q₈

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₆₁

SD₁₆

yes

characteristic

C₂×C₄×D₆₁

Q₁₆

yes

—

—

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₄²×C₆₁

C₂²⋊C₄

yes

characteristic

C₄×Dic₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

D₈

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

characteristic

C₂×#92418

—

C₂×#92418

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

#11758189

—

C₈×D₆₁

—

C₈×D₆₁

—

#11758187

—

C₂²×D₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

yes

C₂×Dic₆₁

yes

C₂×Dic₆₁

yes

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#365

yes

—

#2163

—

C₄×#108

—

#1566

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₂×#108

—

122

—

#423

—

122

—

C₄×M₁₆

—

122

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

122

—

#346

—

122

—

#341

—

#334

—

#334

—

122

—

yes

C₆₁

yes

characteristic, complemented, Sylow

100

C₂×M₁₆

—

122

—

101

C₂×M₁₆

—

244

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

C₂×C₁₆

—

122

—

106

#108

—

122

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

122

—

109

C₄⋊C₈

—

244

—

110

#97

—

244

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

244

—

115

C₄×C₈

C₂×#6923

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

122

—

120

M₁₆

—

488

—

121

M₁₆

—

244

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

122

—

126

C₂×C₈

—

244

—

127

C₂×C₈

—

244

—

128

C₂×C₈

—

244

—

129

C₂×C₈

C₄×#6923

yes

characteristic

130

C₂×C₈

C₄⋊#6923

yes

characteristic

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

C₂²⋊#6923

yes

characteristic

135

C₁₆

—

244

—

136

C₂³

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

122

—

141

C₂×C₄

—

122

—

142

C₂×C₄

—

122

—

143

C₂×C₄

—

244

—

144

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

145

C₂×C₄

#14519839

yes

characteristic

146

C₂×C₄

#14519840

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

122

—

152

C₈

—

488

—

153

C₈

—

244

—

154

C₂²

—

122

—

155

C₂²

—

122

—

156

C₂²

—

122

—

157

C₂²

#16840635

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

122

—

162

C₄

—

122

—

163

C₄

#16840634

yes

characteristic

164

C₄

#16840644

yes

characteristic

165

C₂

—

244

—

166

C₂

#21460715

yes

characteristic

167

C₂

#21460883

yes

characteristic

168

C₂

#21460898

yes

characteristic

	subgroup	G/H
center	C₂²	#16840635
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₄×C₈	C₂×#6923
Fitting	C₆₁×#334	C₄
socle	C₂²×C₆₁	#340
prime radical	C₂²×D₆₁	#103
cosocle	C₄×C₈×D₆₁	C₂²
hypercenter	#334	#6923
hypocenter	C₆₁	#7300
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#16840635

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₄×C₈

C₂×#6923

Fitting

C₆₁×#334

C₄

C₂²×C₆₁

C₂²×D₆₁

C₄×C₈×D₆₁

C₂²

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
99. C₆₁	#7300	semidirect

H ≅ G/N

type

99. C₆₁

#7300

semidirect

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

subgroup	index
#21462182	2
C₄×#14519848	2
#21444573	2
#7300	61

subgroup

index

Subgroups — #30594702

Conjugacy classes of subgroups

Complements

Maximal subgroups