Group #30597383 — subgroups

Subgroups — #30597383

Order 15616.

414 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 144 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	#21461533	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519832	C₂	1	1	yes	maximal, characteristic
3	#21460641	C₂	1	4	yes	maximal
4	#21445563	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519832	C₄	1	4	yes	—
6	C₂×C₄×#92420	C₂²	1	2	yes	—
7	C₂×C₄×#92420	C₂²	1	2	yes	—
8	#16840628	C₄	1	4	yes	—
9	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
10	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
11	#16839517	C₄	1	1	yes	characteristic
12	#16839516	C₄	1	2	yes	—
13	C₆₁×#336	C₄	1	1	yes	characteristic
14	C₂²×#92420	C₂×C₄	1	4	yes	—
15	C₂²×#92420	C₂×C₄	1	4	yes	—
16	#14519832	—	2	8	no	—
17	C₄×#92420	—	2	4	no	—
18	C₄×#92420	—	2	4	no	—
19	C₂²×#92418	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
24	C₈×Dic₆₁	—	2	4	no	—
25	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
28	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
29	C₂²×Dic₆₁	C₄²	1	2	yes	—
30	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×#92420	—	2	4	no	—
33	C₂×#92420	—	2	4	no	—
34	C₂×#92420	—	2	4	no	—
35	C₂×#92420	—	2	8	no	—
36	C₂×#92420	M₁₆	1	8	yes	—
37	C₂×C₈×C₆₁	—	2	2	no	—
38	C₂×C₈×C₆₁	—	2	2	no	—
39	C₂×C₈×C₆₁	—	2	2	no	—
40	C₄²×C₆₁	—	2	2	no	—
41	C₄×Dic₆₁	—	2	1	no	—
42	C₄×Dic₆₁	—	2	1	no	—
43	C₄×Dic₆₁	—	2	4	no	—
44	C₄×Dic₆₁	—	2	4	no	—
45	C₄×Dic₆₁	#41	1	4	yes	—
46	C₂×#92418	—	2	2	no	—
47	C₂×#92418	—	2	2	no	—
48	C₂×#92418	—	2	2	no	—
49	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	#117	1	2	yes	—
54	C₂×C₄×C₆₁	#131	1	2	yes	—
55	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
57	C₂×Dic₆₁	—	2	2	no	—
58	C₂×Dic₆₁	—	2	2	no	—
59	C₂×Dic₆₁	—	2	2	no	—
60	C₂×Dic₆₁	#117	1	2	yes	—
61	C₂×Dic₆₁	#117	1	2	yes	—
62	C₂×Dic₆₁	C₂×M₁₆	1	4	yes	—
63	C₂×Dic₆₁	#131	1	4	yes	—
64	#92420	—	2	8	no	—
65	#92420	—	4	4	no	—
66	C₈×C₆₁	—	4	2	no	—
67	#92418	—	4	2	no	—
68	#8134	—	61	1	no	maximal, Sylow
69	C₂²×C₆₁	#376	1	1	yes	characteristic
70	C₂²×C₆₁	C₄×M₁₆	1	2	yes	—
71	C₂²×C₆₁	#405	1	2	yes	—
72	C₂²×C₆₁	#405	1	1	yes	characteristic
73	C₂²×C₆₁	#405	1	1	yes	characteristic
74	C₄×C₆₁	—	4	2	no	—
75	C₄×C₆₁	#433	1	2	yes	—
76	C₄×C₆₁	#433	1	2	yes	—
77	Dic₆₁	—	4	2	no	—
78	Dic₆₁	#432	1	4	yes	—
79	Dic₆₁	#433	1	4	yes	—
80	#1744	—	61	1	no	—
81	#1744	—	61	1	no	—
82	C₄×#97	—	61	1	no	—
83	#1273	—	61	4	no	—
84	C₂×C₆₁	#1453	1	2	yes	—
85	C₂×C₆₁	#1453	1	2	yes	—
86	C₂×C₆₁	#1745	1	1	yes	characteristic
87	C₂×C₆₁	#1746	1	1	yes	characteristic
88	C₂×C₆₁	#1746	1	1	yes	characteristic
89	C₂×#97	—	61	4	no	—
90	C₂×C₄×C₈	—	61	2	no	—
91	C₂×C₄×C₈	—	61	2	no	—
92	C₂×C₄×C₈	—	61	2	no	—
93	C₄³	—	61	1	no	—
94	#336	—	61	4	no	—
95	#336	—	61	2	no	—
96	#336	—	61	1	no	—
97	#336	#6923	1	1	yes	characteristic
98	C₆₁	#8134	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	61	4	no	—
100	C₂²×C₈	—	61	4	no	—
101	C₂²×C₈	—	61	2	no	—
102	C₂²×C₈	C₂×#6923	1	2	yes	—
103	C₂×C₄²	—	61	1	no	—
104	C₂×C₄²	—	61	2	no	—
105	C₂×C₄²	—	61	2	no	—
106	C₂×C₄²	—	61	1	no	—
107	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
108	#97	—	122	8	no	—
109	C₄×C₈	—	122	4	no	—
110	C₄×C₈	—	122	4	no	—
111	C₄×C₈	—	122	4	no	—
112	C₂²×C₄	—	61	1	no	—
113	C₂²×C₄	—	61	1	no	—
114	C₂²×C₄	—	61	2	no	—
115	C₂²×C₄	C₄×#6923	1	2	yes	—
116	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	61	8	no	—
121	C₂×C₈	—	122	4	no	—
122	C₂×C₈	—	122	4	no	—
123	C₂×C₈	—	122	4	no	—
124	C₂×C₈	—	122	8	no	—
125	C₂×C₈	—	122	2	no	—
126	C₂×C₈	—	122	2	no	—
127	C₂×C₈	—	122	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	61	4	no	—
130	C₄²	—	122	4	no	—
131	C₄²	—	122	4	no	—
132	C₄²	—	122	1	no	—
133	C₄²	—	122	1	no	—
134	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	61	4	no	—
139	C₂×C₄	—	61	4	no	—
140	C₂×C₄	—	61	2	no	—
141	C₂×C₄	—	61	2	no	—
142	C₂×C₄	—	122	2	no	—
143	C₂×C₄	—	122	2	no	—
144	C₂×C₄	—	122	2	no	—
145	C₂×C₄	#14519856	1	2	yes	—
146	C₂×C₄	#14519864	1	1	yes	characteristic
147	C₂×C₄	#14519864	1	2	yes	—
148	C₂×C₄	#14519865	1	1	yes	characteristic
149	C₈	—	4	2	no	—
150	C₈	—	122	8	no	—
151	C₈	—	244	4	no	—
152	C₈	—	244	2	no	—
153	C₂²	#16840698	1	1	yes	characteristic
154	C₂²	M₁₆×#6923	1	2	yes	—
155	C₂²	#16840725	1	2	yes	—
156	C₂²	#16840726	1	1	yes	characteristic
157	C₂²	#16840727	1	1	yes	characteristic
158	C₄	—	4	2	no	—
159	C₄	—	61	4	no	—
160	C₄	—	61	4	no	—
161	C₄	—	244	2	no	—
162	C₄	#16840786	1	2	yes	—
163	C₄	#16840787	1	2	yes	—
164	C₂	#21461036	1	2	yes	—
165	C₂	#21461037	1	2	yes	—
166	C₂	#21461535	1	1	yes	characteristic
167	C₂	#21461537	1	1	yes	characteristic
168	C₂	#21461538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461533

C₂

yes

maximal, characteristic

C₄×#14519832

C₂

yes

maximal, characteristic

#21460641

C₂

yes

maximal

#21445563

C₂

yes

maximal, characteristic

C₂×#14519832

C₄

yes

—

C₂×C₄×#92420

C₂²

yes

—

C₂×C₄×#92420

C₂²

yes

—

#16840628

C₄

yes

—

C₂×C₈×Dic₆₁

C₂²

yes

—

C₄²×Dic₆₁

C₂²

yes

characteristic

#16839517

C₄

yes

characteristic

#16839516

C₄

yes

—

C₆₁×#336

C₄

yes

characteristic

C₂²×#92420

C₂×C₄

yes

—

C₂²×#92420

C₂×C₄

yes

—

#14519832

—

C₄×#92420

—

C₄×#92420

—

C₂²×#92418

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂³

yes

characteristic

C₈×Dic₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₄²

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

C₄²

yes

—

C₂²×Dic₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

C₂²×C₄

yes

—

—

—

—

M₁₆

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

#41

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#117

yes

—

C₂×Dic₆₁

#117

yes

—

C₂×Dic₆₁

C₂×M₁₆

yes

—

C₂×Dic₆₁

#131

yes

—

#92420

—

#92420

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

C₄×M₁₆

yes

—

C₂²×C₆₁

#405

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

#433

yes

—

C₄×C₆₁

#433

yes

—

Dic₆₁

—

Dic₆₁

#432

yes

—

Dic₆₁

#433

yes

—

#1744

—

#1744

—

C₄×#97

—

#1273

—

C₂×C₆₁

#1453

yes

—

C₂×C₆₁

#1453

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

—

—

—

yes

C₆₁

yes

characteristic, complemented, Sylow

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

C₂×#6923

yes

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

—

107

C₂×C₄²

C₂×#6923

yes

characteristic

108

#97

—

122

—

109

C₄×C₈

—

122

—

110

C₄×C₈

—

122

—

111

C₄×C₈

—

122

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₄×#6923

yes

—

116

C₂²×C₄

C₂²×#6923

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

—

122

C₂×C₈

—

122

—

123

C₂×C₈

—

122

—

124

C₂×C₈

—

122

—

125

C₂×C₈

—

122

—

126

C₂×C₈

—

122

—

127

C₂×C₈

—

122

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

122

—

131

C₄²

—

122

—

132

C₄²

—

122

—

133

C₄²

—

122

—

134

C₂³

C₂×C₄×#6923

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

122

—

143

C₂×C₄

—

122

—

144

C₂×C₄

—

122

—

145

C₂×C₄

#14519856

yes

—

146

C₂×C₄

#14519864

yes

characteristic

147

C₂×C₄

#14519864

yes

—

148

C₂×C₄

#14519865

yes

characteristic

149

C₈

—

150

C₈

—

122

—

151

C₈

—

244

—

152

C₈

—

244

—

153

C₂²

#16840698

yes

characteristic

154

C₂²

M₁₆×#6923

yes

—

155

C₂²

#16840725

yes

—

156

C₂²

#16840726

yes

characteristic

157

C₂²

#16840727

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

244

—

162

C₄

#16840786

yes

—

163

C₄

#16840787

yes

—

164

C₂

#21461036

yes

—

165

C₂

#21461037

yes

—

166

C₂

#21461535

yes

characteristic

167

C₂

#21461537

yes

characteristic

168

C₂

#21461538

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator, socle, prime radical	C₂³×C₆₁	C₂×C₄²
Fitting	C₆₁×#336	C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	#336	#6923
hypocenter	C₆₁	#8134
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#6923

commutator, socle, prime radical

C₂³×C₆₁

C₂×C₄²

Fitting

C₆₁×#336

C₄

cosocle

C₂×C₄×Dic₆₁

C₂³

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
98. C₆₁	#8134	semidirect

H ≅ G/N

type

98. C₆₁

#8134

semidirect

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

subgroup	index
#21461533	2
C₄×#14519832	2
#21460641	2
#21445563	2
#8134	61

subgroup

index

Subgroups — #30597383

Conjugacy classes of subgroups

Complements

Maximal subgroups