Group #30607321 — subgroups

Subgroups — #30607321

Order 15616.

286 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 100 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	#21461759	C₂	1	1	yes	maximal, characteristic
2	#21461756	C₂	1	1	yes	maximal, characteristic
3	#21460884	C₂	1	4	yes	maximal
4	C₄×#14519716	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519848	—	2	1	no	—
6	C₂×#14519847	C₂²	1	2	yes	—
7	C₂×#14519846	—	2	1	no	—
8	C₂×#14519845	C₂²	1	2	yes	—
9	#16840767	—	2	1	no	—
10	#16840706	—	2	1	no	—
11	C₂×#14519716	C₂²	1	2	yes	—
12	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
13	#16839537	C₄	1	2	yes	complemented
14	C₄×#11758185	C₄	1	1	yes	characteristic, complemented
15	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
16	C₂×#11758213	—	2	2	no	—
17	C₂×C₄⋊#6923	—	2	2	no	—
18	#14519848	—	4	2	no	—
19	#14519847	—	2	4	no	—
20	#14519846	—	4	2	no	—
21	#14519845	—	2	4	no	—
22	#14519844	—	4	2	no	—
23	(C₂×#6923)⋊C₄	—	4	2	no	—
24	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
25	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
26	C₄²×D₆₁	D₄	1	1	yes	characteristic
27	#14519716	—	2	2	no	—
28	#14519716	—	2	2	no	—
29	C₂×#11758185	C₂×C₄	1	2	yes	—
30	C₄×#92418	Q₈	1	1	yes	characteristic
31	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
33	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×#6923	—	4	2	no	—
36	C₂×C₄×D₆₁	SD₁₆	1	2	yes	—
37	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×#92420	—	4	2	no	—
39	C₄⋊#6923	—	4	2	no	—
40	C₄⋊#6923	—	4	2	no	—
41	#11758213	—	4	2	no	—
42	#11758213	—	4	2	no	—
43	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
46	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
47	C₄×Dic₆₁	SD₁₆	1	2	yes	—
48	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
50	C₈×D₆₁	D₈	1	2	yes	—
51	C₈×D₆₁	Q₁₆	1	2	yes	—
52	C₈×D₆₁	#41	1	2	yes	—
53	C₈×D₆₁	#41	1	2	yes	—
54	C₁₆×C₆₁	—	2	2	no	—
55	#11758185	—	2	2	no	—
56	C₂²×D₆₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×#6923	—	8	4	no	—
58	C₂×#6923	—	8	2	no	—
59	C₂×C₄×C₆₁	#106	1	2	yes	—
60	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×Dic₆₁	#106	1	2	yes	—
62	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×D₆₁	—	4	2	no	—
64	C₄×D₆₁	#124	1	1	yes	characteristic
65	C₄×D₆₁	#124	1	1	yes	characteristic
66	C₄×D₆₁	C₂×D₈	1	1	yes	characteristic
67	C₄×D₆₁	C₂×Q₁₆	1	1	yes	characteristic
68	#92420	—	8	2	no	—
69	C₈×C₆₁	#107	1	2	yes	—
70	C₈×C₆₁	#117	1	2	yes	—
71	#92418	#107	1	2	yes	—
72	#92418	#125	1	2	yes	—
73	#12887	—	61	1	no	maximal, Sylow
74	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
75	C₂×D₆₁	—	2	2	no	—
76	C₂×D₆₁	#486	1	1	yes	characteristic
77	C₂×D₆₁	#487	1	1	yes	characteristic
78	#6923	—	16	2	no	—
79	C₄×C₆₁	—	2	2	no	—
80	C₄×C₆₁	#382	1	1	yes	characteristic
81	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
82	Dic₆₁	—	2	2	no	—
83	Dic₆₁	#486	1	1	yes	characteristic
84	Dic₆₁	#487	1	1	yes	characteristic
85	C₄×M₃₂	—	61	1	no	—
86	#1848	—	61	1	no	—
87	#1846	—	61	1	no	—
88	#1383	—	61	4	no	—
89	C₂×C₆₁	#1834	1	1	yes	characteristic
90	C₂×C₆₁	#2155	1	2	yes	—
91	D₆₁	—	2	1	no	—
92	D₆₁	—	2	1	no	—
93	C₂×M₃₂	—	61	2	no	—
94	C₂×#108	—	61	2	no	—
95	C₂×#108	—	122	1	no	—
96	C₂×#107	—	122	1	no	—
97	C₂×#106	—	61	2	no	—
98	#423	—	122	1	no	—
99	C₂×C₄×C₈	—	61	1	no	—
100	#382	—	122	1	no	—
101	#346	—	61	2	no	—
102	C₄×C₁₆	—	61	1	no	—
103	C₄×C₁₆	#6923	1	1	yes	characteristic, complemented
104	C₆₁	#12887	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₁₆	—	122	2	no	—
106	C₂²×C₈	—	61	1	no	—
107	C₂²×C₈	—	61	1	no	—
108	C₂×C₄⋊C₄	—	122	2	no	—
109	C₂×C₄²	—	61	1	no	—
110	M₃₂	—	122	2	no	—
111	M₃₂	—	122	2	no	—
112	C₂×C₁₆	—	61	2	no	—
113	C₂×C₁₆	C₂×#6923	1	2	yes	—
114	#108	—	122	4	no	—
115	#108	—	244	2	no	—
116	#107	—	244	2	no	—
117	#106	—	122	4	no	—
118	C₄⋊C₈	—	244	2	no	—
119	C₄×C₈	—	61	1	no	—
120	C₄×C₈	—	61	1	no	—
121	C₄×C₈	—	61	1	no	—
122	C₄×C₈	C₂×#6923	1	1	yes	characteristic
123	#95	—	244	2	no	—
124	C₂²×C₄	—	61	1	no	—
125	C₂²×C₄	—	61	2	no	—
126	C₂²×C₄	—	244	2	no	—
127	M₁₆	—	244	2	no	—
128	M₁₆	—	244	2	no	—
129	C₂×C₈	—	61	1	no	—
130	C₂×C₈	—	61	1	no	—
131	C₂×C₈	—	61	2	no	—
132	C₂×C₈	—	61	2	no	—
133	C₂×C₈	—	61	2	no	—
134	C₂×C₈	—	61	2	no	—
135	C₂×C₈	—	244	2	no	—
136	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
137	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
138	C₄⋊C₄	—	244	2	no	—
139	C₄⋊C₄	—	244	2	no	—
140	C₄²	—	61	2	no	—
141	C₄²	—	61	1	no	—
142	C₄²	C₄⋊#6923	1	1	yes	characteristic
143	C₁₆	—	2	2	no	—
144	C₁₆	—	122	2	no	—
145	C₂³	—	61	1	no	—
146	C₂×C₄	—	61	1	no	—
147	C₂×C₄	—	61	1	no	—
148	C₂×C₄	—	61	1	no	—
149	C₂×C₄	—	61	1	no	—
150	C₂×C₄	—	61	1	no	—
151	C₂×C₄	—	61	2	no	—
152	C₂×C₄	—	488	2	no	—
153	C₂×C₄	—	488	4	no	—
154	C₂×C₄	—	244	2	no	—
155	C₂×C₄	#14519845	1	2	yes	—
156	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
157	C₈	—	61	2	no	—
158	C₈	—	61	2	no	—
159	C₈	—	488	2	no	—
160	C₈	#14519846	1	2	yes	—
161	C₈	#14519856	1	2	yes	—
162	C₂²	—	61	1	no	—
163	C₂²	—	61	1	no	—
164	C₂²	—	122	2	no	—
165	C₂²	C₂×#14519845	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	61	1	no	—
168	C₄	—	61	1	no	—
169	C₄	—	122	2	no	—
170	C₄	—	976	2	no	—
171	C₄	#16840706	1	1	yes	characteristic
172	C₄	C₂×#14519846	1	1	yes	characteristic
173	C₂	—	122	1	no	—
174	C₂	—	122	1	no	—
175	C₂	#21461740	1	1	yes	characteristic
176	C₂	#21462166	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461759

C₂

yes

maximal, characteristic

#21461756

C₂

yes

maximal, characteristic

#21460884

C₂

yes

maximal

C₄×#14519716

C₂

yes

maximal, characteristic

C₂×#14519848

—

C₂×#14519847

C₂²

yes

—

C₂×#14519846

—

C₂×#14519845

C₂²

yes

—

#16840767

—

#16840706

—

C₂×#14519716

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839537

C₄

yes

complemented

C₄×#11758185

C₄

yes

characteristic, complemented

C₄×C₁₆×C₆₁

C₄

yes

characteristic, complemented

C₂×#11758213

—

C₂×C₄⋊#6923

—

—

—

—

—

—

—

C₂×C₈×D₆₁

D₄

yes

characteristic

C₂×C₈×D₆₁

C₂³

yes

characteristic

C₄²×D₆₁

D₄

yes

—

—

C₂×C₄

yes

—

C₄×#92418

Q₈

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

Q₈

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×#6923

—

C₂×C₄×D₆₁

SD₁₆

yes

—

C₂×C₄×D₆₁

C₂×D₄

yes

—

—

—

—

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

—

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₂×Q₈

yes

characteristic

C₈×D₆₁

D₈

yes

—

C₈×D₆₁

Q₁₆

yes

—

C₈×D₆₁

#41

yes

—

C₈×D₆₁

#41

yes

—

C₁₆×C₆₁

—

#11758185

—

C₂²×D₆₁

C₂×SD₁₆

yes

characteristic

C₂×#6923

—

C₂×#6923

—

C₂×C₄×C₆₁

#106

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

#106

yes

—

C₂×Dic₆₁

C₂×SD₁₆

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

yes

C₄×D₆₁

yes

C₄×D₆₁

C₂×D₈

yes

characteristic

C₄×D₆₁

C₂×Q₁₆

yes

characteristic

#92420

—

C₈×C₆₁

#107

yes

—

C₈×C₆₁

#117

yes

—

#92418

#107

yes

—

#92418

#125

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

yes

C₂×D₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

yes

Dic₆₁

yes

C₄×M₃₂

—

#1848

—

#1846

—

#1383

—

C₂×C₆₁

#1834

yes

characteristic

C₂×C₆₁

#2155

yes

—

D₆₁

—

D₆₁

—

C₂×M₃₂

—

C₂×#108

—

C₂×#108

—

122

—

C₂×#107

—

122

—

C₂×#106

—

#423

—

122

—

C₂×C₄×C₈

—

100

#382

—

122

—

101

#346

—

102

C₄×C₁₆

—

103

C₄×C₁₆

#6923

yes

characteristic, complemented

104

C₆₁

#12887

yes

characteristic, complemented, Sylow

105

C₂×M₁₆

—

122

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂×C₄⋊C₄

—

122

—

109

C₂×C₄²

—

110

M₃₂

—

122

—

111

M₃₂

—

122

—

112

C₂×C₁₆

—

113

C₂×C₁₆

C₂×#6923

yes

—

114

#108

—

122

—

115

#108

—

244

—

116

#107

—

244

—

117

#106

—

122

—

118

C₄⋊C₈

—

244

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₂×#6923

yes

characteristic

123

#95

—

244

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

244

—

127

M₁₆

—

244

—

128

M₁₆

—

244

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

244

—

136

C₂×C₈

C₄⋊#6923

yes

characteristic

137

C₂×C₈

C₂²×#6923

yes

characteristic

138

C₄⋊C₄

—

244

—

139

C₄⋊C₄

—

244

—

140

C₄²

—

141

C₄²

—

142

C₄²

C₄⋊#6923

yes

characteristic

143

C₁₆

—

144

C₁₆

—

122

—

145

C₂³

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

488

—

153

C₂×C₄

—

488

—

154

C₂×C₄

—

244

—

155

C₂×C₄

#14519845

yes

—

156

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

—

488

—

160

C₈

#14519846

yes

—

161

C₈

#14519856

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

122

—

165

C₂²

C₂×#14519845

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

122

—

170

C₄

—

976

—

171

C₄

#16840706

yes

characteristic

172

C₄

C₂×#14519846

yes

characteristic

173

C₂

—

122

—

174

C₂

—

122

—

175

C₂

#21461740

yes

characteristic

176

C₂

#21462166

yes

—

	subgroup	G/H
center	C₂²	C₂×#14519845
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₄×C₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×#106
prime radical	C₂²×D₆₁	C₂×SD₁₆
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	C₄×C₁₆	#6923
hypocenter	C₆₁	#12887
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#14519845

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₄×C₁₆×C₆₁

C₄

socle

C₂²×C₆₁

C₂×#106

prime radical

C₂²×D₆₁

C₂×SD₁₆

cosocle

C₂×C₈×D₆₁

C₂³

C₄×C₁₆

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
13. #16839537	C₄	semidirect
14. C₄×#11758185	C₄	semidirect
15. C₄×C₁₆×C₆₁	C₄	semidirect
103. C₄×C₁₆	#6923	semidirect
104. C₆₁	#12887	semidirect

H ≅ G/N

type

13. #16839537

C₄

semidirect

14. C₄×#11758185

C₄

semidirect

15. C₄×C₁₆×C₆₁

C₄

semidirect

103. C₄×C₁₆

#6923

semidirect

104. C₆₁

#12887

semidirect

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

subgroup	index
#21461759	2
#21461756	2
#21460884	2
C₄×#14519716	2
#12887	61

subgroup

index

Subgroups — #30607321

Conjugacy classes of subgroups

Complements

Maximal subgroups