Group #21454889 — subgroups

Subgroups — #21454889

Order 7808.

394 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 157 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	#16840312	C₂	1	1	yes	maximal, characteristic
2	#16839835	C₂	1	4	yes	maximal, complemented
3	#16839832	C₂	1	4	yes	maximal, complemented
4	#16839819	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839727	C₂	1	4	yes	maximal, complemented
6	C₆₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11758222	C₂²	1	2	yes	—
8	C₂×#11758189	C₂²	1	4	yes	—
9	C₂×C₈×D₆₁	C₂²	1	4	yes	—
10	#14519777	C₄	1	1	yes	characteristic
11	#14519774	C₄	1	1	yes	characteristic
12	#14519765	C₄	1	2	yes	—
13	#14519751	C₄	1	2	yes	—
14	#14519751	C₄	1	2	yes	—
15	C₄×D₂₄₄	C₂²	1	2	yes	—
16	C₄×D₂₄₄	C₂²	1	2	yes	—
17	C₄⋊#92418	C₂²	1	2	yes	—
18	C₂²⋊#92418	C₂²	1	2	yes	—
19	#14519689	C₂²	1	8	yes	—
20	#14519688	C₂²	1	2	yes	—
21	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
23	C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
24	#97×C₆₁	C₂²	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₆₁	C₂×C₄	1	4	yes	—
29	#11758222	—	4	2	no	—
30	#11758222	—	4	2	no	—
31	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
32	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
33	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₆₁	C₂³	1	4	yes	—
35	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
36	C₂×C₈×C₆₁	C₂³	1	4	yes	—
37	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	#11758198	C₂×C₄	1	4	yes	—
42	#11758198	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
44	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
45	C₂×#92418	C₂³	1	4	yes	—
46	#11758189	—	4	4	no	—
47	C₈×D₆₁	—	4	4	no	—
48	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂²×D₆₁	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	—	4	1	no	—
52	C₂×C₄×C₆₁	—	4	1	no	—
53	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₆₁	—	4	4	no	—
60	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
61	D₂₄₄	—	4	2	no	—
62	D₂₄₄	—	4	2	no	—
63	D₂₄₄	—	4	2	no	—
64	C₄×D₆₁	—	2	4	no	—
65	C₄×D₆₁	—	2	4	no	—
66	Dic₁₂₂	—	4	2	no	—
67	C₈×C₆₁	—	2	4	no	—
68	#92418	—	2	4	no	—
69	C₂²×C₆₁	—	2	1	no	—
70	C₂²×C₆₁	—	2	1	no	—
71	C₂²×C₆₁	—	2	1	no	—
72	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
73	C₂×D₆₁	—	2	4	no	—
74	C₂×D₆₁	—	2	4	no	—
75	C₂×D₆₁	—	2	4	no	—
76	C₄×C₆₁	—	2	1	no	—
77	C₄×C₆₁	—	2	1	no	—
78	C₄×C₆₁	—	2	1	no	—
79	C₄×C₆₁	—	2	1	no	—
80	C₄×C₆₁	—	2	1	no	—
81	C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
82	Dic₆₁	—	2	4	no	—
83	#2980	—	61	1	no	maximal, Sylow
84	C₂×C₆₁	—	4	1	no	—
85	C₂×C₆₁	#518	1	1	yes	characteristic
86	C₂×C₆₁	#568	1	1	yes	characteristic
87	C₂×C₆₁	#568	1	1	yes	characteristic
88	D₆₁	—	4	4	no	—
89	#532	—	61	1	no	—
90	#435	—	61	4	no	—
91	#435	—	61	4	no	—
92	#432	—	61	1	no	—
93	#432	D₆₁	1	1	yes	characteristic, complemented
94	#408	—	61	4	no	—
95	C₆₁	#2980	1	1	yes	characteristic, complemented, Sylow
96	C₂×#41	—	61	2	no	—
97	C₂×M₁₆	—	61	4	no	—
98	C₂²×C₈	—	61	4	no	—
99	C₄⋊Q₈	—	61	1	no	—
100	#127	—	61	1	no	—
101	#124	—	61	2	no	—
102	#121	—	61	2	no	—
103	#121	—	61	2	no	—
104	C₄×D₄	—	61	2	no	—
105	C₄×D₄	—	61	2	no	—
106	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
107	C₄⋊C₈	—	61	2	no	—
108	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
109	C₂²⋊C₈	—	61	2	no	—
110	C₂²⋊C₈	—	61	8	no	—
111	C₂²⋊C₈	C₂×D₆₁	1	2	yes	—
112	#97	—	61	2	no	—
113	#97	C₂×D₆₁	1	2	yes	—
114	#41	—	244	2	no	—
115	#41	—	244	2	no	—
116	C₂×Q₈	—	61	2	no	—
117	C₂×D₄	—	61	4	no	—
118	C₂×D₄	—	61	2	no	—
119	C₂×D₄	—	61	2	no	—
120	C₂×D₄	—	61	2	no	—
121	C₂²×C₄	—	61	4	no	—
122	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
123	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
125	M₁₆	—	244	4	no	—
126	C₂×C₈	—	61	4	no	—
127	C₂×C₈	—	244	4	no	—
128	C₂×C₈	C₂²×D₆₁	1	4	yes	—
129	C₄⋊C₄	—	61	2	no	—
130	C₄⋊C₄	—	61	2	no	—
131	C₂²⋊C₄	—	61	4	no	—
132	C₂²⋊C₄	—	61	4	no	—
133	C₄²	C₄×D₆₁	1	1	yes	characteristic
134	C₄²	C₄×D₆₁	1	1	yes	characteristic
135	C₄²	C₂²×D₆₁	1	1	yes	characteristic
136	C₄²	C₂²×D₆₁	1	1	yes	characteristic
137	C₂³	—	61	4	no	—
138	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
139	Q₈	—	244	2	no	—
140	D₄	—	244	2	no	—
141	D₄	—	244	2	no	—
142	D₄	—	244	2	no	—
143	D₄	—	244	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	61	4	no	—
148	C₂×C₄	—	122	4	no	—
149	C₂×C₄	—	122	4	no	—
150	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
155	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
156	C₈	—	2	4	no	—
157	C₈	—	122	4	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	122	4	no	—
162	C₂²	—	122	4	no	—
163	C₂²	—	122	4	no	—
164	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	122	4	no	—
171	C₄	#14519823	1	2	yes	—
172	C₂	—	4	1	no	—
173	C₂	—	244	4	no	—
174	C₂	#16840236	1	1	yes	characteristic
175	C₂	#16840528	1	1	yes	characteristic
176	C₂	#16840528	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840312

C₂

yes

maximal, characteristic

#16839835

C₂

yes

maximal, complemented

#16839832

C₂

yes

maximal, complemented

#16839819

C₂

yes

maximal, characteristic, complemented

#16839727

C₂

yes

maximal, complemented

C₆₁×#432

C₂

yes

maximal, characteristic, complemented

C₂×#11758222

C₂²

yes

—

C₂×#11758189

C₂²

yes

—

C₂×C₈×D₆₁

C₂²

yes

—

#14519777

C₄

yes

characteristic

#14519774

C₄

yes

characteristic

#14519765

C₄

yes

—

#14519751

C₄

yes

—

#14519751

C₄

yes

—

C₄×D₂₄₄

C₂²

yes

—

C₄×D₂₄₄

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

C₂²⋊#92418

C₂²

yes

—

#14519689

C₂²

yes

—

#14519688

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₂²⋊C₈×C₆₁

C₂²

yes

—

#97×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₂²⋊D₆₁

C₂×C₄

yes

—

#11758222

—

#11758222

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

#11758198

C₂×C₄

yes

—

#11758198

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₂×#92418

C₂³

yes

—

#11758189

—

C₈×D₆₁

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂²×D₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

C₂²×C₄

yes

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

H(5,𝔽₂)

yes

—

Dic₆₁

—

#2980

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

—

—

—

—

D₆₁

yes

characteristic, complemented

#408

—

C₆₁

#2980

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

100

#127

—

101

#124

—

102

#121

—

103

#121

—

104

C₄×D₄

—

105

C₄×D₄

—

106

C₂×C₄²

C₂×D₆₁

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

C₂×D₆₁

yes

—

109

C₂²⋊C₈

—

110

C₂²⋊C₈

—

111

C₂²⋊C₈

C₂×D₆₁

yes

—

112

#97

—

113

#97

C₂×D₆₁

yes

—

114

#41

—

244

—

115

#41

—

244

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₄×D₆₁

yes

characteristic

123

C₂²×C₄

C₄×D₆₁

yes

characteristic

124

C₂²×C₄

C₂²×D₆₁

yes

characteristic

125

M₁₆

—

244

—

126

C₂×C₈

—

127

C₂×C₈

—

244

—

128

C₂×C₈

C₂²×D₆₁

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

C₄×D₆₁

yes

characteristic

134

C₄²

C₄×D₆₁

yes

characteristic

135

C₄²

C₂²×D₆₁

yes

characteristic

136

C₄²

C₂²×D₆₁

yes

characteristic

137

C₂³

—

138

C₂³

C₂×C₄×D₆₁

yes

characteristic

139

Q₈

—

244

—

140

D₄

—

244

—

141

D₄

—

244

—

142

D₄

—

244

—

143

D₄

—

244

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

122

—

149

C₂×C₄

—

122

—

150

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

151

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

152

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

153

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

154

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

155

C₂×C₄

C₂³×D₆₁

yes

characteristic

156

C₈

—

157

C₈

—

122

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

122

—

162

C₂²

—

122

—

163

C₂²

—

122

—

164

C₂²

C₂²×C₄×D₆₁

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

122

—

171

C₄

#14519823

yes

—

172

C₂

—

173

C₂

—

244

—

174

C₂

#16840236

yes

characteristic

175

C₂

#16840528

yes

characteristic

176

C₂

#16840528

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#432	C₂
hypercenter	#432	D₆₁
hypocenter	C₆₁	#2980
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₆₁×#432

C₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #16839835	C₂	semidirect
3. #16839832	C₂	semidirect
4. #16839819	C₂	semidirect
5. #16839727	C₂	semidirect
6. C₆₁×#432	C₂	semidirect
93. #432	D₆₁	semidirect
95. C₆₁	#2980	semidirect

H ≅ G/N

type

2. #16839835

C₂

semidirect

3. #16839832

C₂

semidirect

4. #16839819

C₂

semidirect

5. #16839727

C₂

semidirect

6. C₆₁×#432

C₂

semidirect

93. #432

D₆₁

semidirect

95. C₆₁

#2980

semidirect

Subgroups — #21454889

Conjugacy classes of subgroups

Complements