Group #30599500 — subgroups

Subgroups — #30599500

Order 15616.

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

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 212 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	#21461751	C₂	1	2	yes	maximal
2	C₂×#16840624	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#92420	—	2	4	no	—
5	C₂×C₄⋊#92420	C₂²	1	4	yes	—
6	#16840624	—	2	8	no	—
7	C₂×C₄×#92418	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
9	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
10	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
12	C₂²×#92420	—	2	4	no	—
13	C₄⋊#92420	—	2	8	no	—
14	C₄⋊#92420	—	4	8	no	—
15	C₂²×#92418	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
18	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
20	C₄×#92418	C₈	1	4	yes	complemented
21	C₈×Dic₆₁	—	2	4	no	—
22	C₈×Dic₆₁	D₄	1	4	yes	—
23	C₈×Dic₆₁	Q₈	1	4	yes	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₁	—	2	2	no	—
27	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
29	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
30	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
31	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×#92420	—	4	4	no	—
33	C₂×#92420	—	4	8	no	—
34	C₂×C₈×C₆₁	—	2	2	no	—
35	C₂×C₈×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
37	C₄²×C₆₁	C₂×C₈	1	2	yes	—
38	C₄²×C₆₁	M₁₆	1	2	yes	—
39	C₄×Dic₆₁	—	2	4	no	—
40	C₄×Dic₆₁	—	2	2	no	—
41	C₄×Dic₆₁	SD₁₆	1	8	yes	—
42	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
43	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
44	C₂×#92418	C₄⋊C₄	1	4	yes	—
45	C₂×#92418	C₂×C₈	1	4	yes	—
46	C₂×#92418	M₁₆	1	4	yes	—
47	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
49	C₂×C₄×C₆₁	#106	1	4	yes	—
50	C₂×C₄×C₆₁	#108	1	4	yes	—
51	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
54	C₂×C₄×C₆₁	C₂×M₁₆	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	#106	1	4	yes	—
59	C₂×Dic₆₁	#127	1	1	yes	characteristic
60	C₂×Dic₆₁	C₄⋊Q₈	1	2	yes	—
61	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×Dic₆₁	C₂×SD₁₆	1	4	yes	—
63	#92420	—	8	4	no	—
64	C₈×C₆₁	—	2	4	no	—
65	#92418	C₄⋊C₈	1	8	yes	complemented
66	#8805	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	#384	1	1	yes	characteristic
68	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₆₁	#423	1	1	yes	characteristic
70	C₂²×C₆₁	C₂×#106	1	2	yes	—
71	C₂²×C₆₁	C₂×#108	1	2	yes	—
72	C₄×C₆₁	#334	1	8	yes	complemented
73	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
74	C₄×C₆₁	#423	1	2	yes	—
75	Dic₆₁	—	2	4	no	—
76	Dic₆₁	—	2	2	no	—
77	Dic₆₁	#492	1	2	yes	—
78	Dic₆₁	#498	1	2	yes	—
79	#1842	—	61	2	no	—
80	C₄²×C₈	—	61	1	no	—
81	C₂×#334	—	61	4	no	—
82	C₂×C₆₁	C₂×#334	1	2	yes	—
83	C₂×C₆₁	#1566	1	2	yes	—
84	C₂×C₆₁	#1842	1	1	yes	characteristic
85	C₂×C₆₁	#1844	1	1	yes	characteristic
86	C₂×C₆₁	#1848	1	1	yes	characteristic
87	C₂×C₄⋊C₈	—	61	4	no	—
88	C₂×C₄⋊C₈	—	122	4	no	—
89	C₂×C₄×C₈	—	61	2	no	—
90	C₂×C₄×C₈	—	61	1	no	—
91	C₂×C₄×C₈	—	61	2	no	—
92	C₂×C₄×C₈	#6923	1	1	yes	characteristic
93	C₄³	—	61	1	no	—
94	#334	—	122	8	no	—
95	C₆₁	#8805	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	61	2	no	—
97	C₂²×C₈	—	122	4	no	—
98	C₂²×C₈	C₂×#6923	1	2	yes	—
99	C₂×C₄²	—	61	2	no	—
100	C₂×C₄²	—	61	1	no	—
101	C₂×C₄²	—	61	2	no	—
102	C₂×C₄²	—	61	1	no	—
103	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
104	C₄⋊C₈	—	122	8	no	—
105	C₄⋊C₈	—	244	8	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	61	4	no	—
108	C₄×C₈	—	61	4	no	—
109	C₄×C₈	—	61	4	no	—
110	C₄×C₈	—	122	4	no	—
111	C₂²×C₄	—	61	1	no	—
112	C₂²×C₄	—	61	2	no	—
113	C₂²×C₄	—	61	1	no	—
114	C₂²×C₄	C₄⋊#6923	1	2	yes	—
115	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	61	4	no	—
119	C₂×C₈	—	61	4	no	—
120	C₂×C₈	—	61	4	no	—
121	C₂×C₈	—	244	4	no	—
122	C₂×C₈	—	244	8	no	—
123	C₂×C₈	C₄⋊#6923	1	4	yes	—
124	C₄²	—	61	8	no	—
125	C₄²	—	61	2	no	—
126	C₄²	—	61	2	no	—
127	C₄²	—	122	4	no	—
128	C₄²	—	122	2	no	—
129	C₄²	D₆₁⋊C₈	1	2	yes	—
130	C₄²	#11758217	1	2	yes	—
131	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
132	C₂×C₄	—	61	1	no	—
133	C₂×C₄	—	61	1	no	—
134	C₂×C₄	—	61	4	no	—
135	C₂×C₄	—	61	2	no	—
136	C₂×C₄	—	61	4	no	—
137	C₂×C₄	—	122	2	no	—
138	C₂×C₄	—	122	2	no	—
139	C₂×C₄	—	122	2	no	—
140	C₂×C₄	#14519844	1	4	yes	—
141	C₂×C₄	#14519845	1	4	yes	—
142	C₂×C₄	#14519848	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
144	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
145	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
146	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	61	8	no	—
149	C₈	—	488	4	no	—
150	C₂²	#16840708	1	1	yes	characteristic
151	C₂²	C₂×#14519844	1	1	yes	characteristic
152	C₂²	#16840767	1	1	yes	characteristic
153	C₂²	C₂×#14519845	1	2	yes	—
154	C₂²	C₂×#14519848	1	2	yes	—
155	C₄	—	61	2	no	—
156	C₄	—	61	2	no	—
157	C₄	—	122	2	no	—
158	C₄	—	122	4	no	—
159	C₄	#16840624	1	8	yes	complemented
160	C₄	D₆₁⋊(C₄⋊C₈)	1	2	yes	—
161	C₄	#16840765	1	2	yes	—
162	C₂	C₂×#16840624	1	2	yes	—
163	C₂	#21461375	1	2	yes	—
164	C₂	#21461752	1	1	yes	characteristic
165	C₂	#21461754	1	1	yes	characteristic
166	C₂	#21461760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461751

C₂

yes

maximal

C₂×#16840624

C₂

yes

maximal

C₄×C₈×Dic₆₁

C₂

yes

maximal, characteristic

C₂×C₄⋊#92420

—

C₂×C₄⋊#92420

C₂²

yes

—

#16840624

—

C₂×C₄×#92418

C₄

yes

characteristic

C₂×C₈×Dic₆₁

C₄

yes

—

C₂×C₈×Dic₆₁

C₂²

yes

—

C₄²×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₁

C₄

yes

characteristic

C₂²×#92420

—

C₄⋊#92420

—

C₄⋊#92420

—

C₂²×#92418

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

—

C₂×C₄×Dic₆₁

Q₈

yes

—

C₂×C₄×Dic₆₁

C₂³

yes

characteristic

C₄×#92418

C₈

yes

complemented

C₈×Dic₆₁

—

C₈×Dic₆₁

D₄

yes

—

C₈×Dic₆₁

Q₈

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

C₄⋊C₄

yes

—

C₂²×Dic₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

C₂×Q₈

yes

characteristic

C₂×#92420

—

C₂×#92420

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

M₁₆

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

SD₁₆

yes

—

C₄×Dic₆₁

C₂×D₄

yes

—

C₄×Dic₆₁

C₂×Q₈

yes

—

C₂×#92418

C₄⋊C₄

yes

—

C₂×#92418

C₂×C₈

yes

—

C₂×#92418

M₁₆

yes

—

C₂³×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

#106

yes

—

C₂×C₄×C₆₁

#108

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#106

yes

—

C₂×Dic₆₁

#127

yes

characteristic

C₂×Dic₆₁

C₄⋊Q₈

yes

—

C₂×Dic₆₁

C₄⋊Q₈

yes

characteristic

C₂×Dic₆₁

C₂×SD₁₆

yes

—

#92420

—

C₈×C₆₁

—

#92418

C₄⋊C₈

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

C₂×C₄⋊C₈

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

C₂×#108

yes

—

C₄×C₆₁

#334

yes

complemented

C₄×C₆₁

C₂×C₄⋊C₈

yes

—

C₄×C₆₁

#423

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#492

yes

—

Dic₆₁

#498

yes

—

#1842

—

C₄²×C₈

—

C₂×#334

—

C₂×C₆₁

C₂×#334

yes

—

C₂×C₆₁

#1566

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

122

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#6923

yes

characteristic

C₄³

—

#334

—

122

—

C₆₁

#8805

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

122

—

C₂²×C₈

C₂×#6923

yes

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

—

103

C₂×C₄²

C₂×#6923

yes

characteristic

104

C₄⋊C₈

—

122

—

105

C₄⋊C₈

—

244

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

122

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₄⋊#6923

yes

—

115

C₂²×C₄

C₂²×#6923

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

244

—

122

C₂×C₈

—

244

—

123

C₂×C₈

C₄⋊#6923

yes

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

122

—

128

C₄²

—

122

—

129

C₄²

D₆₁⋊C₈

yes

—

130

C₄²

#11758217

yes

—

131

C₂³

C₂×C₄⋊#6923

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

122

—

138

C₂×C₄

—

122

—

139

C₂×C₄

—

122

—

140

C₂×C₄

#14519844

yes

—

141

C₂×C₄

#14519845

yes

—

142

C₂×C₄

#14519848

yes

—

143

C₂×C₄

C₂×C₄⋊#6923

yes

—

144

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

145

C₂×C₄

C₂×#11758217

yes

characteristic

146

C₂×C₄

D₆₁⋊M₁₆

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

488

—

150

C₂²

#16840708

yes

characteristic

151

C₂²

C₂×#14519844

yes

characteristic

152

C₂²

#16840767

yes

characteristic

153

C₂²

C₂×#14519845

yes

—

154

C₂²

C₂×#14519848

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

122

—

158

C₄

—

122

—

159

C₄

#16840624

yes

complemented

160

C₄

D₆₁⋊(C₄⋊C₈)

yes

—

161

C₄

#16840765

yes

—

162

C₂

C₂×#16840624

yes

—

163

C₂

#21461375

yes

—

164

C₂

#21461752

yes

characteristic

165

C₂

#21461754

yes

characteristic

166

C₂

#21461760

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#8805
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×C₄⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₂×C₄×C₈×C₆₁

C₄

socle, prime radical

C₂³×C₆₁

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₆₁

C₂³

C₂×C₄×C₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
20. C₄×#92418	C₈	semidirect
65. #92418	C₄⋊C₈	semidirect
72. C₄×C₆₁	#334	semidirect
95. C₆₁	#8805	semidirect
159. C₄	#16840624	semidirect

H ≅ G/N

type

20. C₄×#92418

C₈

semidirect

65. #92418

C₄⋊C₈

semidirect

72. C₄×C₆₁

#334

semidirect

95. C₆₁

#8805

semidirect

159. C₄

#16840624

semidirect

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

subgroup	index
#21461751	2
C₂×#16840624	2
C₄×C₈×Dic₆₁	2
#8805	61

subgroup

index

#21461751

C₂×#16840624

C₄×C₈×Dic₆₁

#8805

Subgroups — #30599500

Conjugacy classes of subgroups

Complements

Maximal subgroups