Group #30597385 — subgroups

Subgroups — #30597385

Order 15616.

382 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 136 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	#21461534	C₂	1	1	yes	maximal, characteristic
2	#21461533	C₂	1	1	yes	maximal, characteristic
3	#21460641	C₂	1	4	yes	maximal
4	#21445563	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#92420	C₂²	1	2	yes	—
6	C₂×C₄×#92420	C₂²	1	2	yes	—
7	#16840628	C₄	1	4	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	C₄×#92420	—	2	4	no	—
17	C₄×#92420	—	2	4	no	—
18	C₂²×#92418	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
23	C₈×Dic₆₁	—	2	4	no	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
27	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
28	C₂²×Dic₆₁	C₄²	1	2	yes	—
29	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×#92420	—	2	4	no	—
32	C₂×#92420	—	2	4	no	—
33	C₂×#92420	—	2	8	no	—
34	C₂×#92420	—	2	8	no	—
35	C₂×C₈×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	—	2	2	no	—
37	C₂×C₈×C₆₁	—	2	2	no	—
38	C₄²×C₆₁	—	2	2	no	—
39	C₄×Dic₆₁	—	2	1	no	—
40	C₄×Dic₆₁	—	2	1	no	—
41	C₄×Dic₆₁	—	2	4	no	—
42	C₄×Dic₆₁	—	2	4	no	—
43	C₄×Dic₆₁	#41	1	4	yes	—
44	C₂×#92418	—	2	2	no	—
45	C₂×#92418	—	2	2	no	—
46	C₂×#92418	—	2	2	no	—
47	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	#117	1	2	yes	—
52	C₂×C₄×C₆₁	#131	1	2	yes	—
53	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
55	C₂×Dic₆₁	—	2	2	no	—
56	C₂×Dic₆₁	—	2	2	no	—
57	C₂×Dic₆₁	—	2	2	no	—
58	C₂×Dic₆₁	#117	1	2	yes	—
59	C₂×Dic₆₁	#117	1	2	yes	—
60	C₂×Dic₆₁	#131	1	4	yes	—
61	C₂×Dic₆₁	#131	1	4	yes	—
62	#92420	—	4	4	no	—
63	#92420	—	4	4	no	—
64	C₈×C₆₁	—	4	2	no	—
65	#92418	—	4	2	no	—
66	#8135	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	#376	1	1	yes	characteristic
68	C₂²×C₆₁	#405	1	2	yes	—
69	C₂²×C₆₁	#405	1	1	yes	characteristic
70	C₂²×C₆₁	#405	1	1	yes	characteristic
71	C₂²×C₆₁	#405	1	2	yes	—
72	C₄×C₆₁	—	4	2	no	—
73	C₄×C₆₁	#433	1	2	yes	—
74	C₄×C₆₁	#433	1	2	yes	—
75	Dic₆₁	—	4	2	no	—
76	Dic₆₁	#433	1	4	yes	—
77	Dic₆₁	#433	1	4	yes	—
78	#1744	—	61	1	no	—
79	#1744	—	61	1	no	—
80	#1744	—	61	1	no	—
81	#1273	—	61	4	no	—
82	C₂×C₆₁	#1454	1	2	yes	—
83	C₂×C₆₁	#1454	1	2	yes	—
84	C₂×C₆₁	#1746	1	1	yes	characteristic
85	C₂×C₆₁	#1746	1	1	yes	characteristic
86	C₂×C₆₁	#1746	1	1	yes	characteristic
87	C₂×C₄×C₈	—	61	2	no	—
88	C₂×C₄×C₈	—	61	2	no	—
89	C₂×C₄×C₈	—	61	2	no	—
90	C₄³	—	61	1	no	—
91	#336	—	61	4	no	—
92	#336	—	61	4	no	—
93	#336	—	61	2	no	—
94	#336	—	61	1	no	—
95	#336	#6923	1	1	yes	characteristic
96	C₆₁	#8135	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₈	—	61	4	no	—
98	C₂²×C₈	—	61	4	no	—
99	C₂²×C₈	—	61	2	no	—
100	C₂²×C₈	C₂×#6923	1	2	yes	—
101	C₂×C₄²	—	61	1	no	—
102	C₂×C₄²	—	61	2	no	—
103	C₂×C₄²	—	61	2	no	—
104	C₂×C₄²	—	61	1	no	—
105	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
106	C₄×C₈	—	122	4	no	—
107	C₄×C₈	—	122	4	no	—
108	C₄×C₈	—	122	4	no	—
109	C₂²×C₄	—	61	1	no	—
110	C₂²×C₄	—	61	1	no	—
111	C₂²×C₄	—	61	2	no	—
112	C₂²×C₄	C₄×#6923	1	2	yes	—
113	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	122	4	no	—
118	C₂×C₈	—	122	4	no	—
119	C₂×C₈	—	122	8	no	—
120	C₂×C₈	—	122	8	no	—
121	C₂×C₈	—	122	2	no	—
122	C₂×C₈	—	122	2	no	—
123	C₂×C₈	—	122	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	61	4	no	—
126	C₄²	—	122	4	no	—
127	C₄²	—	122	4	no	—
128	C₄²	—	122	1	no	—
129	C₄²	—	122	1	no	—
130	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	61	4	no	—
135	C₂×C₄	—	61	4	no	—
136	C₂×C₄	—	61	2	no	—
137	C₂×C₄	—	61	2	no	—
138	C₂×C₄	—	122	2	no	—
139	C₂×C₄	—	122	2	no	—
140	C₂×C₄	—	122	2	no	—
141	C₂×C₄	#14519856	1	2	yes	—
142	C₂×C₄	#14519864	1	1	yes	characteristic
143	C₂×C₄	#14519864	1	2	yes	—
144	C₂×C₄	#14519865	1	1	yes	characteristic
145	C₈	—	4	2	no	—
146	C₈	—	244	4	no	—
147	C₈	—	244	4	no	—
148	C₈	—	244	2	no	—
149	C₂²	#16840698	1	1	yes	characteristic
150	C₂²	#16840725	1	2	yes	—
151	C₂²	#16840726	1	1	yes	characteristic
152	C₂²	#16840727	1	1	yes	characteristic
153	C₂²	#16840728	1	2	yes	—
154	C₄	—	4	2	no	—
155	C₄	—	61	4	no	—
156	C₄	—	61	4	no	—
157	C₄	—	244	2	no	—
158	C₄	#16840786	1	2	yes	—
159	C₄	#16840787	1	2	yes	—
160	C₂	#21461040	1	2	yes	—
161	C₂	#21461041	1	2	yes	—
162	C₂	#21461537	1	1	yes	characteristic
163	C₂	#21461538	1	1	yes	characteristic
164	C₂	#21461539	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461534

C₂

yes

maximal, characteristic

#21461533

C₂

yes

maximal, characteristic

#21460641

C₂

yes

maximal

#21445563

C₂

yes

maximal, characteristic

C₂×C₄×#92420

C₂²

yes

—

C₂×C₄×#92420

C₂²

yes

—

#16840628

C₄

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

—

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

—

—

—

—

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

#131

yes

—

C₂×Dic₆₁

#131

yes

—

#92420

—

#92420

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

#433

yes

—

C₄×C₆₁

#433

yes

—

Dic₆₁

—

Dic₆₁

#433

yes

—

Dic₆₁

#433

yes

—

—

—

—

—

C₂×C₆₁

#1454

yes

—

C₂×C₆₁

#1454

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

—

—

—

—

yes

C₆₁

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

C₂×#6923

yes

—

101

C₂×C₄²

—

102

C₂×C₄²

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₄²

C₂×#6923

yes

characteristic

106

C₄×C₈

—

122

—

107

C₄×C₈

—

122

—

108

C₄×C₈

—

122

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₄×#6923

yes

—

113

C₂²×C₄

C₂²×#6923

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

122

—

118

C₂×C₈

—

122

—

119

C₂×C₈

—

122

—

120

C₂×C₈

—

122

—

121

C₂×C₈

—

122

—

122

C₂×C₈

—

122

—

123

C₂×C₈

—

122

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

122

—

127

C₄²

—

122

—

128

C₄²

—

122

—

129

C₄²

—

122

—

130

C₂³

C₂×C₄×#6923

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

122

—

139

C₂×C₄

—

122

—

140

C₂×C₄

—

122

—

141

C₂×C₄

#14519856

yes

—

142

C₂×C₄

#14519864

yes

characteristic

143

C₂×C₄

#14519864

yes

—

144

C₂×C₄

#14519865

yes

characteristic

145

C₈

—

146

C₈

—

244

—

147

C₈

—

244

—

148

C₈

—

244

—

149

C₂²

#16840698

yes

characteristic

150

C₂²

#16840725

yes

—

151

C₂²

#16840726

yes

characteristic

152

C₂²

#16840727

yes

characteristic

153

C₂²

#16840728

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

244

—

158

C₄

#16840786

yes

—

159

C₄

#16840787

yes

—

160

C₂

#21461040

yes

—

161

C₂

#21461041

yes

—

162

C₂

#21461537

yes

characteristic

163

C₂

#21461538

yes

characteristic

164

C₂

#21461539

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	#8135
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
96. C₆₁	#8135	semidirect

H ≅ G/N

type

96. C₆₁

#8135

semidirect

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

subgroup	index
#21461534	2
#21461533	2
#21460641	2
#21445563	2
#8135	61

subgroup

index

Subgroups — #30597385

Conjugacy classes of subgroups

Complements

Maximal subgroups