Group #21444235 — subgroups

Subgroups — #21444235

Order 7808.

262 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₄×#11758194	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839489	C₂	1	4	yes	maximal, complemented
4	C₆₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11758222	—	2	1	no	—
6	C₂×#11758194	—	2	1	no	—
7	#14519777	C₄	1	1	yes	characteristic
8	#14519774	C₄	1	1	yes	characteristic
9	#14519765	—	2	1	no	—
10	#14519751	—	2	1	no	—
11	#14519751	C₄	1	2	yes	—
12	C₄×D₂₄₄	—	2	1	no	—
13	C₄×D₂₄₄	C₂²	1	2	yes	—
14	#14519688	—	2	1	no	—
15	C₄×#92418	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
18	C₂²⋊C₈×C₆₁	—	2	1	no	—
19	C₄×C₈×C₆₁	—	2	1	no	—
20	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₆₁	—	2	2	no	—
24	#11758222	—	4	4	no	—
25	C₂×D₂₄₄	—	2	1	no	—
26	C₂×D₂₄₄	—	2	1	no	—
27	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₆₁	—	2	2	no	—
29	C₂×Dic₁₂₂	—	2	1	no	—
30	C₂×C₈×C₆₁	—	2	2	no	—
31	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₆₁	D₄	1	1	yes	characteristic
33	C₄²×C₆₁	D₄	1	1	yes	characteristic
34	C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	#11758198	—	2	2	no	—
36	#11758198	—	2	2	no	—
37	C₄⋊Dic₆₁	—	2	1	no	—
38	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
39	#11758194	—	2	4	no	—
40	C₂×#92418	—	2	2	no	—
41	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×D₆₁	—	2	2	no	—
43	C₂×C₄×C₆₁	—	2	2	no	—
44	C₂×C₄×C₆₁	—	2	2	no	—
45	C₂×C₄×C₆₁	—	2	2	no	—
46	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂²×C₄	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	C₂²⋊D₆₁	—	4	4	no	—
53	C₂×Dic₆₁	—	2	2	no	—
54	D₂₄₄	—	2	4	no	—
55	D₂₄₄	—	4	2	no	—
56	D₂₄₄	—	4	2	no	—
57	C₄×D₆₁	—	4	4	no	—
58	Dic₁₂₂	—	4	2	no	—
59	C₈×C₆₁	—	4	2	no	—
60	#92418	—	2	4	no	—
61	C₂²×C₆₁	—	2	1	no	—
62	C₂²×C₆₁	—	2	1	no	—
63	C₂²×C₆₁	—	2	1	no	—
64	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₆₁	—	4	2	no	—
66	C₂×D₆₁	—	4	2	no	—
67	C₂×D₆₁	—	4	2	no	—
68	C₄×C₆₁	—	2	1	no	—
69	C₄×C₆₁	—	2	1	no	—
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	#104	1	2	yes	—
72	C₄×C₆₁	#131	1	2	yes	—
73	C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
74	Dic₆₁	—	4	2	no	—
75	#1484	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	—	4	1	no	—
77	C₂×C₆₁	#408	1	1	yes	characteristic
78	C₂×C₆₁	#418	1	1	yes	characteristic
79	C₂×C₆₁	C₂×#104	1	1	yes	characteristic
80	D₆₁	—	8	2	no	—
81	#532	—	61	1	no	—
82	#431	D₆₁	1	1	yes	characteristic, complemented
83	C₄×M₁₆	—	61	1	no	—
84	D₄⋊C₈	—	61	4	no	—
85	C₆₁	#1484	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	122	1	no	—
87	C₂×M₁₆	—	122	1	no	—
88	C₄⋊Q₈	—	61	1	no	—
89	#127	—	61	1	no	—
90	#124	—	122	1	no	—
91	#121	—	61	2	no	—
92	#121	—	122	1	no	—
93	C₄×D₄	—	61	2	no	—
94	C₄×D₄	—	122	1	no	—
95	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
96	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
97	C₂²⋊C₈	—	2	1	no	—
98	#97	—	122	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	61	2	no	—
101	#41	—	244	4	no	—
102	C₂×Q₈	—	122	1	no	—
103	C₂×D₄	—	61	2	no	—
104	C₂×D₄	—	122	2	no	—
105	C₂×D₄	—	122	1	no	—
106	C₂×D₄	—	122	1	no	—
107	C₂²×C₄	—	122	2	no	—
108	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
109	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
111	M₁₆	—	122	4	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	122	2	no	—
114	C₄⋊C₄	—	61	2	no	—
115	C₄⋊C₄	—	122	1	no	—
116	C₂²⋊C₄	—	122	2	no	—
117	C₂²⋊C₄	—	122	2	no	—
118	C₄²	C₄×D₆₁	1	1	yes	characteristic
119	C₄²	D₂₄₄	1	1	yes	characteristic
120	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
121	C₄²	C₂²×D₆₁	1	1	yes	characteristic
122	C₂³	—	122	2	no	—
123	C₂³	#11758198	1	1	yes	characteristic
124	Q₈	—	244	2	no	—
125	D₄	—	122	4	no	—
126	D₄	—	244	2	no	—
127	D₄	—	244	4	no	—
128	D₄	—	244	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	122	2	no	—
133	C₂×C₄	—	244	4	no	—
134	C₂×C₄	#11758198	1	1	yes	characteristic
135	C₂×C₄	#11758198	1	1	yes	characteristic
136	C₂×C₄	#11758198	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
140	C₈	—	4	2	no	—
141	C₈	—	122	4	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	244	2	no	—
146	C₂²	—	244	2	no	—
147	C₂²	—	244	2	no	—
148	C₂²	C₂×#11758198	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	244	2	no	—
153	C₄	#14519703	1	2	yes	—
154	C₄	#14519783	1	2	yes	—
155	C₄	#14519802	1	2	yes	—
156	C₂	—	4	1	no	—
157	C₂	—	488	2	no	—
158	C₂	#16839728	1	1	yes	characteristic
159	C₂	#16839766	1	1	yes	characteristic
160	C₂	C₂×#14519703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840312

C₂

yes

maximal, characteristic

C₄×#11758194

C₂

yes

maximal, characteristic, complemented

#16839489

C₂

yes

maximal, complemented

C₆₁×#431

C₂

yes

maximal, characteristic, complemented

C₂×#11758222

—

C₂×#11758194

—

#14519777

C₄

yes

characteristic

#14519774

C₄

yes

—

—

C₄

yes

—

C₄×D₂₄₄

—

C₄×D₂₄₄

C₂²

yes

—

#14519688

—

C₄×#92418

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂×C₂²⋊D₆₁

—

#11758222

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

#11758198

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×D₆₁

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#104

yes

—

C₄×C₆₁

#131

yes

—

C₄×C₆₁

AGL(1,ℤ/8)

yes

—

Dic₆₁

—

#1484

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#532

—

#431

D₆₁

yes

characteristic, complemented

C₄×M₁₆

—

D₄⋊C₈

—

C₆₁

#1484

yes

characteristic, complemented, Sylow

C₂×#41

—

122

—

C₂×M₁₆

—

122

—

C₄⋊Q₈

—

#127

—

#124

—

122

—

#121

—

#121

—

122

—

C₄×D₄

—

C₄×D₄

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

C₂×D₆₁

yes

—

C₂²⋊C₈

—

#97

—

122

—

C₄×C₈

—

100

C₄×C₈

—

101

#41

—

244

—

102

C₂×Q₈

—

122

—

103

C₂×D₄

—

104

C₂×D₄

—

122

—

105

C₂×D₄

—

122

—

106

C₂×D₄

—

122

—

107

C₂²×C₄

—

122

—

108

C₂²×C₄

C₄×D₆₁

yes

characteristic

109

C₂²×C₄

D₂₄₄

yes

characteristic

110

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

111

M₁₆

—

122

—

112

C₂×C₈

—

113

C₂×C₈

—

122

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

122

—

116

C₂²⋊C₄

—

122

—

117

C₂²⋊C₄

—

122

—

118

C₄²

C₄×D₆₁

yes

characteristic

119

C₄²

D₂₄₄

yes

characteristic

120

C₄²

C₂²⋊D₆₁

yes

characteristic

121

C₄²

C₂²×D₆₁

yes

characteristic

122

C₂³

—

122

—

123

C₂³

#11758198

yes

characteristic

124

Q₈

—

244

—

125

D₄

—

122

—

126

D₄

—

244

—

127

D₄

—

244

—

128

D₄

—

244

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

122

—

133

C₂×C₄

—

244

—

134

C₂×C₄

#11758198

yes

characteristic

135

C₂×C₄

#11758198

yes

characteristic

136

C₂×C₄

#11758198

yes

characteristic

137

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

138

C₂×C₄

C₂×D₂₄₄

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

140

C₈

—

141

C₈

—

122

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

244

—

146

C₂²

—

244

—

147

C₂²

—

244

—

148

C₂²

C₂×#11758198

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

244

—

153

C₄

#14519703

yes

—

154

C₄

#14519783

yes

—

155

C₄

#14519802

yes

—

156

C₂

—

157

C₂

—

488

—

158

C₂

#16839728

yes

characteristic

159

C₂

#16839766

yes

characteristic

160

C₂

C₂×#14519703

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#431	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	#431	D₆₁
hypocenter	C₆₁	#1484
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#431

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
2. C₄×#11758194	C₂	semidirect
3. #16839489	C₂	semidirect
4. C₆₁×#431	C₂	semidirect
82. #431	D₆₁	semidirect
85. C₆₁	#1484	semidirect

H ≅ G/N

type

2. C₄×#11758194

C₂

semidirect

3. #16839489

C₂

semidirect

4. C₆₁×#431

C₂

semidirect

82. #431

D₆₁

semidirect

85. C₆₁

#1484

semidirect

Subgroups — #21444235

Conjugacy classes of subgroups

Complements