Group #30607215 — subgroups

Subgroups — #30607215

Order 15616.

434 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 124 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	#21462046	C₂	1	2	yes	maximal
2	#21460866	C₂	1	4	yes	maximal, complemented
3	C₂×#16839718	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519851	C₂²	1	4	yes	—
5	C₂²⋊#11758187	—	2	4	no	—
6	C₂²×#11758194	C₄	1	1	yes	characteristic
7	C₂³×#92418	C₂²	1	1	yes	characteristic
8	#16839718	—	2	4	no	—
9	C₂×#14519688	C₂²	1	2	yes	—
10	C₂×C₄×#92418	C₄	1	2	yes	—
11	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
12	#14519851	—	2	8	no	—
13	C₂×#11758187	—	2	4	no	—
14	C₂×#11758194	—	2	2	no	—
15	C₂×#11758194	—	2	4	no	—
16	C₂²×#92418	—	2	4	no	—
17	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
18	C₂²×#92418	C₂×C₄	1	2	yes	—
19	C₂²×#92418	D₄	1	4	yes	—
20	C₂²×#92418	C₂³	1	1	yes	characteristic
21	#14519688	—	2	4	no	—
22	C₄×#92418	—	2	4	no	—
23	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
24	#117×C₆₁	—	2	2	no	—
25	#117×C₆₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
27	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
28	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
29	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	—	2	2	no	—
32	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₆₁	—	2	4	no	—
38	C₂²⋊C₄×C₆₁	—	2	4	no	—
39	C₄²×C₆₁	—	2	2	no	—
40	C₄²×C₆₁	—	2	2	no	—
41	#11758194	—	2	8	no	—
42	C₂×#92418	—	2	2	no	—
43	C₂×#92418	—	2	2	no	—
44	C₂×#92418	—	2	2	no	—
45	C₂×#92418	—	2	8	no	—
46	C₂×#92418	C₂²⋊C₄	1	2	yes	—
47	C₂×#92418	M₁₆	1	4	yes	—
48	C₂×#92418	C₂×D₄	1	2	yes	—
49	C₂×#92418	#41	1	4	yes	—
50	#11758187	—	4	4	no	—
51	C₂³×C₆₁	—	2	2	no	—
52	C₂³×C₆₁	—	4	2	no	—
53	C₂³×C₆₁	C₂²⋊C₈	1	2	yes	—
54	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₁	—	2	2	no	—
56	C₂×C₄×C₆₁	—	2	2	no	—
57	C₂×C₄×C₆₁	—	2	2	no	—
58	C₂×C₄×C₆₁	—	2	4	no	—
59	C₂×C₄×C₆₁	—	2	4	no	—
60	C₂×C₄×C₆₁	—	4	4	no	—
61	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#117	1	2	yes	—
64	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
66	#92418	—	2	2	no	—
67	#92418	—	2	4	no	—
68	#92418	#123	1	4	yes	—
69	#12847	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	—	2	2	no	—
71	C₂²×C₆₁	—	2	2	no	—
72	C₂²×C₆₁	—	2	4	no	—
73	C₂²×C₆₁	—	4	4	no	—
74	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₂²×C₆₁	#431	1	2	yes	—
76	C₄×C₆₁	—	2	2	no	—
77	C₄×C₆₁	—	4	4	no	—
78	C₄×C₆₁	#381	1	1	yes	characteristic
79	C₄×C₆₁	#407	1	1	yes	characteristic
80	C₄×C₆₁	#432	1	2	yes	—
81	C₂×#405	—	61	1	no	—
82	#2112	—	61	2	no	—
83	#1377	—	61	4	no	—
84	C₂×C₆₁	—	2	2	no	—
85	C₂×C₆₁	—	4	2	no	—
86	C₂×C₆₁	#1817	1	1	yes	characteristic
87	C₂×C₆₁	#2115	1	2	yes	—
88	C₂²×M₁₆	—	61	1	no	—
89	C₂³×C₈	—	61	1	no	—
90	C₂×#117	#6923	1	1	yes	characteristic
91	C₂×M₃₂	—	61	4	no	—
92	#405	—	122	4	no	—
93	C₂×#97	—	61	2	no	—
94	C₂×C₄×C₈	—	61	2	no	—
95	C₂²⋊C₁₆	—	122	4	no	—
96	C₆₁	#12847	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
98	C₂×M₁₆	—	122	4	no	—
99	C₂×M₁₆	—	122	2	no	—
100	C₂²×C₈	—	61	1	no	—
101	C₂²×C₈	—	61	4	no	—
102	C₂²×C₈	—	61	1	no	—
103	C₂²×C₈	—	61	2	no	—
104	C₂²×C₈	—	122	4	no	—
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	C₂×C₄⋊C₄	#92420	1	2	yes	—
108	C₂×C₂²⋊C₄	#92420	1	2	yes	—
109	C₂×C₄²	C₂×#6923	1	2	yes	—
110	M₃₂	—	122	8	no	—
111	C₂×C₁₆	—	122	4	no	—
112	#97	—	122	4	no	—
113	C₄×C₈	—	122	4	no	—
114	C₂⁴	C₂×#92420	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
118	C₂²×C₄	C₂×#92420	1	4	yes	—
119	C₂²×C₄	#11758217	1	2	yes	—
120	C₂²×C₄	C₂²⋊#6923	1	2	yes	—
121	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
122	M₁₆	—	122	8	no	—
123	C₂×C₈	—	61	2	no	—
124	C₂×C₈	—	61	4	no	—
125	C₂×C₈	—	61	2	no	—
126	C₂×C₈	—	61	4	no	—
127	C₂×C₈	—	122	8	no	—
128	C₂×C₈	—	122	2	no	—
129	C₂×C₈	—	122	2	no	—
130	C₂×C₈	—	122	2	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₁₆	—	244	4	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	4	2	no	—
138	C₂³	C₂²⋊#92420	1	2	yes	—
139	C₂³	C₂²×#92420	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₂²⋊#92420	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
148	C₂×C₄	#14519856	1	2	yes	—
149	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
150	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
151	C₈	—	61	4	no	—
152	C₈	—	122	2	no	—
153	C₈	—	122	4	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	4	no	—
158	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
159	C₂²	#16840778	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#16840705	1	1	yes	characteristic
163	C₄	#16840734	1	1	yes	characteristic
164	C₄	#16840782	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	—	4	2	no	—
167	C₂	#21461709	1	1	yes	characteristic
168	C₂	#21462055	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21462046

C₂

yes

maximal

#21460866

C₂

yes

maximal, complemented

C₂×#16839718

C₂

yes

maximal, characteristic

C₂×#14519851

C₂²

yes

—

C₂²⋊#11758187

—

C₂²×#11758194

C₄

yes

characteristic

C₂³×#92418

C₂²

yes

characteristic

#16839718

—

C₂×#14519688

C₂²

yes

—

C₂×C₄×#92418

C₄

yes

—

C₂×#117×C₆₁

C₄

yes

—

—

—

—

C₂²×#92418

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

—

C₂²×#92418

D₄

yes

—

C₂²×#92418

C₂³

yes

characteristic

#14519688

—

C₄×#92418

—

C₂³×C₄×C₆₁

C₂×C₄

yes

characteristic

#117×C₆₁

—

#117×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₈

yes

—

C₂×C₂²⋊C₄×C₆₁

C₈

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂⁴×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₈

yes

—

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂×#92418

M₁₆

yes

—

C₂×#92418

C₂×D₄

yes

—

C₂×#92418

#41

yes

—

#11758187

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂²⋊C₈

yes

—

C₂³×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

—

yes

—

#12847

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₆₁

#431

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₂×#405

—

#2112

—

#1377

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1817

yes

characteristic

C₂×C₆₁

#2115

yes

—

C₂²×M₁₆

—

C₂³×C₈

—

C₂×#117

#6923

yes

characteristic

C₂×M₃₂

—

#405

—

122

—

C₂×#97

—

C₂×C₄×C₈

—

C₂²⋊C₁₆

—

122

—

C₆₁

#12847

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×#6923

yes

characteristic

C₂×M₁₆

—

122

—

C₂×M₁₆

—

122

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

—

122

—

105

#117

—

106

#117

—

107

C₂×C₄⋊C₄

#92420

yes

—

108

C₂×C₂²⋊C₄

#92420

yes

—

109

C₂×C₄²

C₂×#6923

yes

—

110

M₃₂

—

122

—

111

C₂×C₁₆

—

122

—

112

#97

—

122

—

113

C₄×C₈

—

122

—

114

C₂⁴

C₂×#92420

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂×#92420

yes

characteristic

118

C₂²×C₄

C₂×#92420

yes

—

119

C₂²×C₄

#11758217

yes

—

120

C₂²×C₄

C₂²⋊#6923

yes

—

121

C₂²×C₄

C₂²×#6923

yes

characteristic

122

M₁₆

—

122

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

122

—

128

C₂×C₈

—

122

—

129

C₂×C₈

—

122

—

130

C₂×C₈

—

122

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₁₆

—

244

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂²⋊#92420

yes

—

139

C₂³

C₂²×#92420

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊#92420

yes

—

147

C₂×C₄

C₂×C₂²⋊#6923

yes

characteristic

148

C₂×C₄

#14519856

yes

—

149

C₂×C₄

C₂×#11758217

yes

characteristic

150

C₂×C₄

D₆₁⋊M₁₆

yes

—

151

C₈

—

152

C₈

—

122

—

153

C₈

—

122

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×C₂²⋊#92420

yes

characteristic

159

C₂²

#16840778

yes

—

160

C₄

—

161

C₄

—

162

C₄

#16840705

yes

characteristic

163

C₄

#16840734

yes

characteristic

164

C₄

#16840782

yes

—

165

C₂

—

166

C₂

—

167

C₂

#21461709

yes

characteristic

168

C₂

#21462055

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6923
commutator	C₂³×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#117×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×#117	#6923
hypocenter	C₆₁	#12847
LCS	G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#6923

commutator

C₂³×C₆₁

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₂×#117×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₆₁

C₂×C₈

cosocle

C₂²×#92418

C₂³

C₆₁

G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. #21460866	C₂	semidirect
96. C₆₁	#12847	semidirect

H ≅ G/N

type

2. #21460866

C₂

semidirect

96. C₆₁

#12847

semidirect

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

subgroup	index
#21462046	2
#21460866	2
C₂×#16839718	2
#12847	61

subgroup

index

Subgroups — #30607215

Conjugacy classes of subgroups

Complements

Maximal subgroups