Group #21444682 — subgroups

Subgroups — #21444682

Order 7808.

182 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic
2	#16839839	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839827	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839489	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839489	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839484	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₈₈	C₄	1	1	yes	characteristic
9	C₂×#11758189	—	2	1	no	—
10	C₂×C₈×D₆₁	—	2	1	no	—
11	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
12	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
13	#14519711	C₄	1	1	yes	characteristic
14	#14519699	C₄	1	1	yes	characteristic
15	#14519699	C₄	1	1	yes	characteristic
16	#14519689	—	2	1	no	—
17	#14519689	—	2	1	no	—
18	C₄×#92418	C₂²	1	1	yes	characteristic
19	C₄×#92418	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	—	2	1	no	—
26	C₂×C₄×D₆₁	—	2	1	no	—
27	C₂×C₈×C₆₁	—	2	1	no	—
28	C₂×C₈×C₆₁	—	2	1	no	—
29	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	#11758198	—	2	1	no	—
33	#11758198	—	2	1	no	—
34	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×#92418	—	2	1	no	—
37	C₂×#92418	D₄	1	1	yes	characteristic
38	C₂×#92418	D₄	1	1	yes	characteristic
39	D₄₈₈	—	2	2	no	—
40	#11758189	—	4	2	no	—
41	C₈×D₆₁	—	4	2	no	—
42	C₂²×D₆₁	—	2	1	no	—
43	C₂²×D₆₁	—	2	1	no	—
44	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
47	C₂×Dic₆₁	—	2	1	no	—
48	C₂×Dic₆₁	—	2	1	no	—
49	D₂₄₄	—	2	1	no	—
50	D₂₄₄	—	2	1	no	—
51	D₂₄₄	—	2	1	no	—
52	D₂₄₄	M₁₆	1	2	yes	complemented
53	C₄×D₆₁	—	4	1	no	—
54	C₄×D₆₁	—	4	1	no	—
55	C₄×D₆₁	—	4	1	no	—
56	C₄×D₆₁	—	4	1	no	—
57	C₈×C₆₁	—	2	1	no	—
58	C₈×C₆₁	—	2	1	no	—
59	C₈×C₆₁	—	4	1	no	—
60	C₈×C₆₁	—	4	1	no	—
61	#92418	—	2	2	no	—
62	#92418	—	2	1	no	—
63	#92418	—	2	1	no	—
64	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
65	C₂×D₆₁	—	2	2	no	—
66	C₂×D₆₁	—	4	1	no	—
67	C₂×D₆₁	—	4	1	no	—
68	C₂×D₆₁	—	4	1	no	—
69	C₂×D₆₁	—	4	1	no	—
70	C₂×D₆₁	—	4	1	no	—
71	C₄×C₆₁	—	2	1	no	—
72	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
73	C₄×C₆₁	#131	1	1	yes	characteristic
74	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Dic₆₁	—	4	1	no	—
77	Dic₆₁	—	4	1	no	—
78	#1586	—	61	1	no	maximal, Sylow
79	C₂×C₆₁	#435	1	1	yes	characteristic
80	C₂×C₆₁	#442	1	1	yes	characteristic
81	C₂×C₆₁	#443	1	1	yes	characteristic
82	D₆₁	—	4	2	no	—
83	D₆₁	—	8	1	no	—
84	C₄×D₈	—	61	1	no	—
85	#436	—	61	1	no	—
86	C₈×D₄	—	61	1	no	—
87	#334	D₆₁	1	1	yes	characteristic, complemented
88	D₄⋊C₈	—	61	1	no	—
89	D₄⋊C₈	—	61	1	no	—
90	#322	—	61	1	no	—
91	C₆₁	#1586	1	1	yes	characteristic, complemented, Sylow
92	C₂×D₈	—	61	1	no	—
93	C₂×M₁₆	—	122	1	no	—
94	C₂²×C₈	—	122	1	no	—
95	C₄×D₄	—	61	1	no	—
96	C₄×D₄	—	61	1	no	—
97	#107	—	61	1	no	—
98	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
99	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
100	#102	—	61	1	no	—
101	#102	—	61	1	no	—
102	C₂²⋊C₈	—	122	1	no	—
103	C₂²⋊C₈	—	122	1	no	—
104	C₄×C₈	—	61	1	no	—
105	C₄×C₈	—	61	1	no	—
106	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
107	C₂×D₄	—	61	1	no	—
108	C₂×D₄	—	61	1	no	—
109	C₂²×C₄	—	122	1	no	—
110	C₂²×C₄	—	122	1	no	—
111	D₈	—	122	2	no	—
112	M₁₆	—	244	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	61	1	no	—
116	C₂×C₈	—	61	1	no	—
117	C₂×C₈	—	122	1	no	—
118	C₂×C₈	—	244	2	no	—
119	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
120	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
121	C₄⋊C₄	—	61	1	no	—
122	C₄⋊C₄	—	61	1	no	—
123	C₂²⋊C₄	—	122	1	no	—
124	C₂²⋊C₄	—	122	1	no	—
125	C₄²	C₂²×D₆₁	1	1	yes	characteristic
126	C₂³	—	122	1	no	—
127	C₂³	—	122	1	no	—
128	D₄	—	61	2	no	—
129	D₄	—	122	1	no	—
130	D₄	—	122	1	no	—
131	D₄	—	122	1	no	—
132	C₂×C₄	—	122	1	no	—
133	C₂×C₄	—	122	1	no	—
134	C₂×C₄	—	244	1	no	—
135	C₂×C₄	—	244	1	no	—
136	C₂×C₄	—	244	1	no	—
137	C₂×C₄	—	244	1	no	—
138	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
139	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
140	C₂×C₄	#11758226	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	4	1	no	—
144	C₈	—	4	1	no	—
145	C₈	—	122	2	no	—
146	C₈	—	122	1	no	—
147	C₈	—	122	1	no	—
148	C₂²	—	122	2	no	—
149	C₂²	—	244	1	no	—
150	C₂²	—	244	1	no	—
151	C₂²	—	244	1	no	—
152	C₂²	—	244	1	no	—
153	C₂²	—	244	1	no	—
154	C₂²	#14519739	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	244	1	no	—
157	C₄	—	244	1	no	—
158	C₄	M₁₆×D₆₁	1	1	yes	characteristic
159	C₄	#14519783	1	1	yes	characteristic
160	C₄	#14519804	1	1	yes	characteristic
161	C₄	#14519804	1	1	yes	characteristic
162	C₂	—	488	1	no	—
163	C₂	—	244	2	no	—
164	C₂	#16839832	1	1	yes	characteristic
165	C₂	#16839873	1	1	yes	characteristic
166	C₂	#16839878	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄₈₈

C₂

yes

maximal, characteristic

#16839839

C₂

yes

maximal, characteristic, complemented

#16839827

C₂

yes

maximal, characteristic, complemented

#16839489

C₂

yes

maximal, characteristic, complemented

#16839489

C₂

yes

maximal, characteristic, complemented

#16839484

C₂

yes

maximal, characteristic, complemented

C₆₁×#334

C₂

yes

maximal, characteristic, complemented

C₂×D₄₈₈

C₄

yes

characteristic

C₂×#11758189

—

C₂×C₈×D₆₁

—

C₄×D₂₄₄

C₂²

yes

characteristic

C₄×D₂₄₄

C₂²

yes

characteristic

#14519711

C₄

yes

characteristic

#14519699

C₄

yes

characteristic

#14519699

C₄

yes

—

—

C₂²

yes

characteristic

C₄×#92418

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₂₄₄

C₂×C₄

yes

characteristic

C₂×D₂₄₄

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

#11758198

—

#11758198

—

C₄⋊Dic₆₁

C₂×C₄

yes

characteristic

C₄⋊Dic₆₁

C₂×C₄

yes

characteristic

C₂×#92418

—

C₂×#92418

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

D₄₈₈

—

#11758189

—

C₈×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

M₁₆

yes

complemented

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

—

#92418

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

D₆₁

—

C₄×D₈

—

#436

—

C₈×D₄

—

#334

D₆₁

yes

characteristic, complemented

D₄⋊C₈

—

D₄⋊C₈

—

#322

—

C₆₁

#1586

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

122

—

C₂²×C₈

—

122

—

C₄×D₄

—

C₄×D₄

—

#107

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₈

C₂×D₆₁

yes

characteristic

100

#102

—

101

#102

—

102

C₂²⋊C₈

—

122

—

103

C₂²⋊C₈

—

122

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

C₂×D₆₁

yes

characteristic

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

122

—

110

C₂²×C₄

—

122

—

111

D₈

—

122

—

112

M₁₆

—

244

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

122

—

118

C₂×C₈

—

244

—

119

C₂×C₈

C₄×D₆₁

yes

characteristic

120

C₂×C₈

C₄×D₆₁

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

122

—

124

C₂²⋊C₄

—

122

—

125

C₄²

C₂²×D₆₁

yes

characteristic

126

C₂³

—

122

—

127

C₂³

—

122

—

128

D₄

—

129

D₄

—

122

—

130

D₄

—

122

—

131

D₄

—

122

—

132

C₂×C₄

—

122

—

133

C₂×C₄

—

122

—

134

C₂×C₄

—

244

—

135

C₂×C₄

—

244

—

136

C₂×C₄

—

244

—

137

C₂×C₄

—

244

—

138

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

139

C₂×C₄

D₄×D₆₁

yes

characteristic

140

C₂×C₄

#11758226

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

122

—

146

C₈

—

122

—

147

C₈

—

122

—

148

C₂²

—

122

—

149

C₂²

—

244

—

150

C₂²

—

244

—

151

C₂²

—

244

—

152

C₂²

—

244

—

153

C₂²

—

244

—

154

C₂²

#14519739

yes

characteristic

155

C₄

—

156

C₄

—

244

—

157

C₄

—

244

—

158

C₄

M₁₆×D₆₁

yes

characteristic

159

C₄

#14519783

yes

characteristic

160

C₄

#14519804

yes

characteristic

161

C₄

#14519804

yes

characteristic

162

C₂

—

488

—

163

C₂

—

244

—

164

C₂

#16839832

yes

characteristic

165

C₂

#16839873

yes

characteristic

166

C₂

#16839878

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#334	C₂
socle	C₂²×C₆₁	C₄×D₄
hypercenter	#334	D₆₁
hypocenter	C₆₁	#1586
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#334

C₂

socle

C₂²×C₆₁

C₄×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #16839839	C₂	semidirect
3. #16839827	C₂	semidirect
4. #16839489	C₂	semidirect
5. #16839489	C₂	semidirect
6. #16839484	C₂	semidirect
7. C₆₁×#334	C₂	semidirect
52. D₂₄₄	M₁₆	semidirect
87. #334	D₆₁	semidirect
91. C₆₁	#1586	semidirect

H ≅ G/N

type

2. #16839839

C₂

semidirect

3. #16839827

C₂

semidirect

4. #16839489

C₂

semidirect

5. #16839489

C₂

semidirect

6. #16839484

C₂

semidirect

7. C₆₁×#334

C₂

semidirect

52. D₂₄₄

M₁₆

semidirect

87. #334

D₆₁

semidirect

91. C₆₁

#1586

semidirect

Subgroups — #21444682

Conjugacy classes of subgroups

Complements