Group #30605826 — subgroups

Subgroups — #30605826

Order 15616.

232 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 92 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	#21461374	C₂	1	1	yes	maximal, characteristic
2	#21461374	C₂	1	1	yes	maximal, characteristic
3	#21460845	C₂	1	4	yes	maximal, complemented
4	C₄×#14519716	C₂	1	1	yes	maximal, characteristic
5	#16840766	—	2	1	no	—
6	#16840766	—	2	1	no	—
7	#16840623	—	2	1	no	—
8	#16840623	—	2	1	no	—
9	#16840623	C₂²	1	2	yes	—
10	#16840623	C₂²	1	2	yes	—
11	C₂×#14519716	C₄	1	2	yes	—
12	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
13	#16839537	C₂²	1	2	yes	—
14	C₄×#11758185	C₄	1	1	yes	characteristic
15	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
16	C₂×#11758213	—	4	1	no	—
17	C₂×#11758213	—	4	1	no	—
18	C₄⋊#92420	—	2	2	no	—
19	C₄⋊#92420	—	2	2	no	—
20	C₄⋊#92420	—	4	1	no	—
21	C₄⋊#92420	—	4	1	no	—
22	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×D₆₁	Q₈	1	1	yes	characteristic
24	C₄²×D₆₁	D₄	1	1	yes	characteristic
25	#14519716	—	2	4	no	—
26	C₂×#11758185	C₂×C₄	1	2	yes	—
27	C₄×#92418	Q₈	1	1	yes	characteristic
28	C₈×Dic₆₁	D₄	1	1	yes	characteristic
29	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₆₁	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×D₆₁	SD₁₆	1	1	yes	characteristic
35	C₂×#92420	—	4	2	no	—
36	C₂×#92420	—	4	2	no	—
37	#11758213	—	8	2	no	—
38	#11758213	—	8	2	no	—
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
43	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
44	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
46	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
47	C₈×D₆₁	—	2	2	no	—
48	C₈×D₆₁	—	2	2	no	—
49	C₁₆×C₆₁	—	2	2	no	—
50	#11758185	M₁₆	1	4	yes	complemented
51	C₂²×D₆₁	#106	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₆₁	#106	1	1	yes	characteristic
56	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
58	C₄×D₆₁	—	2	2	no	—
59	C₄×D₆₁	—	2	2	no	—
60	C₄×D₆₁	—	2	1	no	—
61	C₄×D₆₁	—	2	1	no	—
62	#92420	—	8	2	no	—
63	#92420	—	8	2	no	—
64	C₈×C₆₁	#108	1	2	yes	—
65	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
66	#92418	—	2	1	no	—
67	#92418	—	2	1	no	—
68	#12020	—	61	1	no	maximal, Sylow
69	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
70	C₂×D₆₁	—	2	1	no	—
71	C₂×D₆₁	—	2	1	no	—
72	C₂×D₆₁	—	2	1	no	—
73	C₄×C₆₁	#365	1	2	yes	—
74	C₄×C₆₁	#365	1	2	yes	—
75	C₄×C₆₁	#423	1	1	yes	characteristic
76	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
77	Dic₆₁	—	2	1	no	—
78	Dic₆₁	—	2	1	no	—
79	Dic₆₁	—	2	1	no	—
80	C₄×M₃₂	—	61	1	no	—
81	#1566	—	61	1	no	—
82	#1566	—	61	1	no	—
83	#1368	—	61	4	no	—
84	C₂×C₆₁	#1566	1	1	yes	characteristic
85	C₂×C₆₁	C₂×#365	1	1	yes	characteristic
86	C₂×C₆₁	C₂×#365	1	1	yes	characteristic
87	D₆₁	—	4	1	no	—
88	C₂×M₃₂	—	61	2	no	—
89	#423	—	122	1	no	—
90	#423	—	122	1	no	—
91	C₂×C₄×C₈	—	61	1	no	—
92	#346	—	61	2	no	—
93	C₄×C₁₆	—	61	1	no	—
94	C₄×C₁₆	#6923	1	1	yes	characteristic
95	#334	—	61	2	no	—
96	#334	—	61	2	no	—
97	#334	—	122	1	no	—
98	#334	—	122	1	no	—
99	C₆₁	#12020	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	244	1	no	—
101	C₂×M₁₆	—	244	1	no	—
102	C₂²×C₈	—	61	1	no	—
103	C₂²×C₈	—	61	1	no	—
104	C₂×C₄²	—	61	1	no	—
105	M₃₂	—	122	4	no	—
106	C₂×C₁₆	—	61	2	no	—
107	C₂×C₁₆	C₂×#6923	1	2	yes	—
108	C₄⋊C₈	—	122	2	no	—
109	C₄⋊C₈	—	122	2	no	—
110	C₄⋊C₈	—	244	1	no	—
111	C₄⋊C₈	—	244	1	no	—
112	C₄×C₈	—	61	1	no	—
113	C₄×C₈	—	61	1	no	—
114	C₄×C₈	—	61	1	no	—
115	C₄×C₈	C₂×#6923	1	1	yes	characteristic
116	C₂²×C₄	—	61	1	no	—
117	C₂²×C₄	—	61	1	no	—
118	C₂²×C₄	—	61	1	no	—
119	M₁₆	—	488	2	no	—
120	M₁₆	—	488	2	no	—
121	C₂×C₈	—	61	1	no	—
122	C₂×C₈	—	61	1	no	—
123	C₂×C₈	—	122	2	no	—
124	C₂×C₈	—	122	2	no	—
125	C₂×C₈	—	244	2	no	—
126	C₂×C₈	—	244	2	no	—
127	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
128	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
129	C₄²	—	61	1	no	—
130	C₄²	—	61	1	no	—
131	C₄²	—	61	1	no	—
132	C₄²	C₄⋊#6923	1	1	yes	characteristic
133	C₁₆	—	2	2	no	—
134	C₁₆	—	61	4	no	—
135	C₂³	—	61	1	no	—
136	C₂×C₄	—	61	1	no	—
137	C₂×C₄	—	61	1	no	—
138	C₂×C₄	—	61	1	no	—
139	C₂×C₄	—	122	1	no	—
140	C₂×C₄	—	122	1	no	—
141	C₂×C₄	—	122	2	no	—
142	C₂×C₄	—	122	2	no	—
143	C₂×C₄	#14519845	1	1	yes	characteristic
144	C₂×C₄	#14519845	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
146	C₈	—	122	1	no	—
147	C₈	—	122	1	no	—
148	C₈	—	488	2	no	—
149	C₈	—	488	2	no	—
150	C₈	#14519848	1	2	yes	—
151	C₈	D₆₁⋊M₁₆	1	2	yes	—
152	C₂²	—	122	1	no	—
153	C₂²	—	122	1	no	—
154	C₂²	—	122	1	no	—
155	C₂²	C₂×#14519845	1	1	yes	characteristic
156	C₄	—	122	1	no	—
157	C₄	—	122	1	no	—
158	C₄	—	122	1	no	—
159	C₄	#16840686	1	2	yes	—
160	C₄	#16840687	1	2	yes	—
161	C₄	#16840767	1	1	yes	characteristic
162	C₄	C₂×#14519848	1	1	yes	characteristic
163	C₂	—	244	1	no	—
164	C₂	#21461375	1	1	yes	characteristic
165	C₂	C₂×#16840686	1	1	yes	characteristic
166	C₂	C₂×#16840687	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461374

C₂

yes

maximal, characteristic

#21461374

C₂

yes

maximal, characteristic

#21460845

C₂

yes

maximal, complemented

C₄×#14519716

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#16840623

C₂²

yes

—

C₂×#14519716

C₄

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839537

C₂²

yes

—

C₄×#11758185

C₄

yes

characteristic

C₄×C₁₆×C₆₁

C₄

yes

—

—

—

—

—

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

Q₈

yes

characteristic

C₄²×D₆₁

D₄

yes

characteristic

#14519716

—

C₂×#11758185

C₂×C₄

yes

—

C₄×#92418

Q₈

yes

characteristic

C₈×Dic₆₁

D₄

yes

characteristic

C₈×Dic₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×D₆₁

SD₁₆

yes

characteristic

C₂×C₄×D₆₁

SD₁₆

yes

—

—

—

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

characteristic

C₄×Dic₆₁

C₂×D₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₂×Q₈

yes

characteristic

C₈×D₆₁

—

C₈×D₆₁

—

C₁₆×C₆₁

—

#11758185

M₁₆

yes

C₂²×D₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

#106

yes

characteristic

C₂×Dic₆₁

C₂×SD₁₆

yes

characteristic

C₂×Dic₆₁

C₂×SD₁₆

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

#92420

—

#92420

—

C₈×C₆₁

#108

yes

—

C₈×C₆₁

C₂×M₁₆

yes

—

—

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#365

yes

—

C₄×C₆₁

#365

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

C₄×M₃₂

—

#1566

—

#1566

—

#1368

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₂×M₃₂

—

#423

—

122

—

#423

—

122

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

yes

—

—

—

122

—

#334

—

122

—

C₆₁

#12020

yes

characteristic, complemented, Sylow

100

C₂×M₁₆

—

244

—

101

C₂×M₁₆

—

244

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

M₃₂

—

122

—

106

C₂×C₁₆

—

107

C₂×C₁₆

C₂×#6923

yes

—

108

C₄⋊C₈

—

122

—

109

C₄⋊C₈

—

122

—

110

C₄⋊C₈

—

244

—

111

C₄⋊C₈

—

244

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

C₂×#6923

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

488

—

120

M₁₆

—

488

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

122

—

124

C₂×C₈

—

122

—

125

C₂×C₈

—

244

—

126

C₂×C₈

—

244

—

127

C₂×C₈

C₄⋊#6923

yes

characteristic

128

C₂×C₈

C₂²×#6923

yes

characteristic

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

C₄⋊#6923

yes

characteristic

133

C₁₆

—

134

C₁₆

—

135

C₂³

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

122

—

140

C₂×C₄

—

122

—

141

C₂×C₄

—

122

—

142

C₂×C₄

—

122

—

143

C₂×C₄

#14519845

yes

characteristic

144

C₂×C₄

#14519845

yes

characteristic

145

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

146

C₈

—

122

—

147

C₈

—

122

—

148

C₈

—

488

—

149

C₈

—

488

—

150

C₈

#14519848

yes

—

151

C₈

D₆₁⋊M₁₆

yes

—

152

C₂²

—

122

—

153

C₂²

—

122

—

154

C₂²

—

122

—

155

C₂²

C₂×#14519845

yes

characteristic

156

C₄

—

122

—

157

C₄

—

122

—

158

C₄

—

122

—

159

C₄

#16840686

yes

—

160

C₄

#16840687

yes

—

161

C₄

#16840767

yes

characteristic

162

C₄

C₂×#14519848

yes

characteristic

163

C₂

—

244

—

164

C₂

#21461375

yes

characteristic

165

C₂

C₂×#16840686

yes

characteristic

166

C₂

C₂×#16840687

yes

characteristic

	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	#106
cosocle	C₈×Dic₆₁	C₂³
hypercenter	C₄×C₁₆	#6923
hypocenter	C₆₁	#12020
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

C₂²×C₆₁

C₂²×D₆₁

C₈×Dic₆₁

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
3. #21460845	C₂	semidirect
50. #11758185	M₁₆	semidirect
99. C₆₁	#12020	semidirect

H ≅ G/N

type

3. #21460845

C₂

semidirect

50. #11758185

M₁₆

semidirect

99. C₆₁

#12020

semidirect

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

subgroup	index
#21461374	2
#21460845	2
C₄×#14519716	2
#12020	61

subgroup

index

Subgroups — #30605826

Conjugacy classes of subgroups

Complements

Maximal subgroups