Group #21446695 — subgroups

Subgroups — #21446695

Order 7808.

376 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×D₄₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14519699	C₂	1	2	yes	maximal, complemented
4	#16839664	C₂	1	2	yes	maximal, complemented
5	C₂×#97×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₄₄	C₂²	1	2	yes	—
7	C₂×D₄₈₈	—	2	2	no	—
8	C₂×D₄₈₈	—	2	2	no	—
9	C₂×D₄₈₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
11	C₂×#11758198	—	2	4	no	—
12	#14519709	—	2	2	no	—
13	#14519699	—	2	4	no	—
14	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	#97×C₆₁	—	2	2	no	—
18	C₂³×D₆₁	—	2	2	no	—
19	C₂²×C₄×C₆₁	D₄	1	2	yes	—
20	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂²×Dic₆₁	—	2	2	no	—
22	C₂×D₂₄₄	—	2	2	no	—
23	C₂×D₂₄₄	—	2	2	no	—
24	C₂×D₂₄₄	—	2	2	no	—
25	C₂×D₂₄₄	—	2	2	no	—
26	C₂×D₂₄₄	—	2	2	no	—
27	C₂×D₂₄₄	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₆₁	—	2	1	no	—
29	C₂×C₈×C₆₁	—	2	1	no	—
30	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₆₁	D₄	1	2	yes	—
32	C₂×C₈×C₆₁	D₄	1	2	yes	—
33	C₂×C₈×C₆₁	D₄	1	2	yes	—
34	C₄²×C₆₁	—	2	1	no	—
35	C₄²×C₆₁	—	2	1	no	—
36	#11758198	—	4	8	no	—
37	C₄⋊Dic₆₁	—	2	2	no	—
38	C₄⋊Dic₆₁	—	2	2	no	—
39	D₄₈₈	—	2	8	no	—
40	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₆₁	—	2	4	no	—
42	C₂²×D₆₁	—	4	2	no	—
43	C₂²×D₆₁	—	4	2	no	—
44	C₂²×D₆₁	—	4	4	no	—
45	C₂²×D₆₁	—	4	4	no	—
46	C₂×C₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	—	2	4	no	—
48	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
54	C₂×Dic₆₁	—	4	2	no	—
55	C₂×Dic₆₁	—	4	4	no	—
56	D₂₄₄	—	2	4	no	—
57	D₂₄₄	—	2	4	no	—
58	D₂₄₄	—	4	2	no	—
59	D₂₄₄	—	4	2	no	—
60	C₈×C₆₁	—	2	2	no	—
61	C₈×C₆₁	C₂²⋊C₄	1	4	yes	complemented
62	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
63	C₂²×C₆₁	#121	1	1	yes	characteristic
64	C₂²×C₆₁	#127	1	1	yes	characteristic
65	C₂²×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
66	C₂×D₆₁	—	4	4	no	—
67	C₂×D₆₁	—	4	8	no	—
68	C₂×D₆₁	—	8	4	no	—
69	C₂×D₆₁	—	8	2	no	—
70	C₂×D₆₁	—	8	2	no	—
71	C₄×C₆₁	—	4	2	no	—
72	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
74	C₄×C₆₁	#121	1	1	yes	characteristic
75	C₄×C₆₁	#124	1	1	yes	characteristic
76	Dic₆₁	—	8	2	no	—
77	#1972	—	61	1	no	maximal, Sylow
78	C₂×C₆₁	#390	1	1	yes	characteristic
79	C₂×C₆₁	#442	1	2	yes	—
80	C₂×C₆₁	#469	1	2	yes	—
81	C₂×C₆₁	#496	1	2	yes	—
82	D₆₁	—	8	4	no	—
83	C₂²×D₈	—	61	1	no	—
84	C₂×#106	—	61	1	no	—
85	C₂×#102	—	61	2	no	—
86	C₂×#97	D₆₁	1	1	yes	characteristic, complemented
87	#390	—	61	2	no	—
88	C₆₁	#1972	1	1	yes	characteristic, complemented, Sylow
89	C₂²×D₄	—	61	2	no	—
90	C₂×D₈	—	61	4	no	—
91	C₂×D₈	—	122	2	no	—
92	C₂×D₈	—	122	2	no	—
93	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	61	2	no	—
96	C₂×C₂²⋊C₄	—	122	4	no	—
97	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
98	#106	—	122	2	no	—
99	#102	—	122	4	no	—
100	#97	—	2	2	no	—
101	C₂⁴	—	122	2	no	—
102	C₂×D₄	—	61	4	no	—
103	C₂×D₄	—	122	2	no	—
104	C₂×D₄	—	122	2	no	—
105	C₂×D₄	—	122	2	no	—
106	C₂×D₄	—	122	2	no	—
107	C₂×D₄	—	122	2	no	—
108	C₂²×C₄	—	122	2	no	—
109	C₂²×C₄	D₂₄₄	1	2	yes	—
110	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
111	D₈	—	122	8	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	C₄×D₆₁	1	2	yes	—
115	C₂×C₈	D₂₄₄	1	2	yes	—
116	C₂×C₈	D₂₄₄	1	2	yes	—
117	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
118	C₄⋊C₄	—	122	2	no	—
119	C₄⋊C₄	—	122	2	no	—
120	C₂²⋊C₄	—	244	8	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	1	no	—
123	C₂³	—	122	4	no	—
124	C₂³	—	244	2	no	—
125	C₂³	—	244	2	no	—
126	C₂³	—	244	4	no	—
127	C₂³	—	244	4	no	—
128	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
129	D₄	—	122	4	no	—
130	D₄	—	122	4	no	—
131	D₄	—	244	2	no	—
132	D₄	—	244	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	244	2	no	—
136	C₂×C₄	—	244	4	no	—
137	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
139	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
140	C₂×C₄	#11758222	1	1	yes	characteristic
141	C₂×C₄	#11758222	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	#11758198	1	4	yes	complemented
145	C₂²	—	488	4	no	—
146	C₂²	—	488	2	no	—
147	C₂²	—	488	2	no	—
148	C₂²	—	244	8	no	—
149	C₂²	—	244	4	no	—
150	C₂²	C₄×D₂₄₄	1	1	yes	characteristic
151	C₂²	#14519751	1	1	yes	characteristic
152	C₂²	#14519774	1	1	yes	characteristic
153	C₂²	#14519805	1	4	yes	—
154	C₄	—	4	2	no	—
155	C₄	—	488	2	no	—
156	C₄	C₂×#11758198	1	1	yes	characteristic
157	C₄	C₄×D₂₄₄	1	1	yes	characteristic
158	C₄	#14519751	1	1	yes	characteristic
159	C₄	#14519765	1	1	yes	characteristic
160	C₂	—	488	4	no	—
161	C₂	#16839664	1	1	yes	characteristic
162	C₂	#16839875	1	2	yes	—
163	C₂	#16840015	1	2	yes	—
164	C₂	#16840151	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₈₈

C₂

yes

maximal, characteristic

C₂×#14519709

C₂

yes

maximal, characteristic, complemented

C₂×#14519699

C₂

yes

maximal, complemented

#16839664

C₂

yes

maximal, complemented

C₂×#97×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₄₄

C₂²

yes

—

C₂×D₄₈₈

—

C₂×D₄₈₈

—

C₂×D₄₈₈

C₄

yes

complemented

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758198

—

#14519709

—

#14519699

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#97×C₆₁

—

C₂³×D₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

D₄₈₈

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊C₄

yes

complemented

C₂²×C₆₁

C₄×D₄

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

C₄×D₄

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#469

yes

—

C₂×C₆₁

#496

yes

—

D₆₁

—

C₂²×D₈

—

C₂×#106

—

C₂×#102

—

C₂×#97

D₆₁

yes

characteristic, complemented

#390

—

C₆₁

#1972

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

122

—

C₂×D₈

—

122

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#106

—

122

—

#102

—

122

—

100

#97

—

101

C₂⁴

—

122

—

102

C₂×D₄

—

103

C₂×D₄

—

122

—

104

C₂×D₄

—

122

—

105

C₂×D₄

—

122

—

106

C₂×D₄

—

122

—

107

C₂×D₄

—

122

—

108

C₂²×C₄

—

122

—

109

C₂²×C₄

D₂₄₄

yes

—

110

C₂²×C₄

C₂²×D₆₁

yes

characteristic

111

D₈

—

122

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₄×D₆₁

yes

—

115

C₂×C₈

D₂₄₄

yes

—

116

C₂×C₈

D₂₄₄

yes

—

117

C₂×C₈

C₂²⋊D₆₁

yes

—

118

C₄⋊C₄

—

122

—

119

C₄⋊C₄

—

122

—

120

C₂²⋊C₄

—

244

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

122

—

124

C₂³

—

244

—

125

C₂³

—

244

—

126

C₂³

—

244

—

127

C₂³

—

244

—

128

C₂³

C₂×D₂₄₄

yes

characteristic

129

D₄

—

122

—

130

D₄

—

122

—

131

D₄

—

244

—

132

D₄

—

244

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

244

—

136

C₂×C₄

—

244

—

137

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

138

C₂×C₄

C₂×D₂₄₄

yes

characteristic

139

C₂×C₄

C₂×D₂₄₄

yes

characteristic

140

C₂×C₄

#11758222

yes

characteristic

141

C₂×C₄

#11758222

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

143

C₈

—

144

C₈

#11758198

yes

complemented

145

C₂²

—

488

—

146

C₂²

—

488

—

147

C₂²

—

488

—

148

C₂²

—

244

—

149

C₂²

—

244

—

150

C₂²

C₄×D₂₄₄

yes

characteristic

151

C₂²

#14519751

yes

characteristic

152

C₂²

#14519774

yes

characteristic

153

C₂²

#14519805

yes

—

154

C₄

—

155

C₄

—

488

—

156

C₄

C₂×#11758198

yes

characteristic

157

C₄

C₄×D₂₄₄

yes

characteristic

158

C₄

#14519751

yes

characteristic

159

C₄

#14519765

yes

characteristic

160

C₂

—

488

—

161

C₂

#16839664

yes

characteristic

162

C₂

#16839875

yes

—

163

C₂

#16840015

yes

—

164

C₂

#16840151

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#97×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#97	D₆₁
hypocenter	C₆₁	#1972
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×#97×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×#97

D₆₁

hypocenter

C₆₁

#1972

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
2. C₂×#14519709	C₂	semidirect
3. C₂×#14519699	C₂	semidirect
4. #16839664	C₂	semidirect
5. C₂×#97×C₆₁	C₂	semidirect
9. C₂×D₄₈₈	C₄	semidirect
61. C₈×C₆₁	C₂²⋊C₄	semidirect
86. C₂×#97	D₆₁	semidirect
88. C₆₁	#1972	semidirect
144. C₈	#11758198	semidirect

H ≅ G/N

type

2. C₂×#14519709

C₂

semidirect

3. C₂×#14519699

C₂

semidirect

4. #16839664

C₂

semidirect

5. C₂×#97×C₆₁

C₂

semidirect

9. C₂×D₄₈₈

C₄

semidirect

61. C₈×C₆₁

C₂²⋊C₄

semidirect

86. C₂×#97

D₆₁

semidirect

88. C₆₁

#1972

semidirect

144. C₈

#11758198

semidirect

Subgroups — #21446695

Conjugacy classes of subgroups

Complements