Group #21447523 — subgroups

Subgroups — #21447523

Order 7808.

228 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂×#14519796	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839799	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839567	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839565	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839563	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758224	—	2	1	no	—
9	#14519796	—	2	2	no	—
10	#14519796	C₄	1	4	yes	complemented
11	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
12	C₂×#11758200	—	2	1	no	—
13	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
14	#14519735	—	2	1	no	—
15	C₂×#11758185	C₂²	1	1	yes	characteristic
16	C₁₆×D₆₁	—	2	2	no	—
17	#14519711	C₂²	1	1	yes	characteristic
18	#14519708	—	2	1	no	—
19	C₂×D₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
21	#107×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₆₁	—	2	1	no	—
23	C₂×C₂²⋊D₆₁	—	4	1	no	—
24	C₂²×Dic₆₁	—	4	1	no	—
25	#11758224	—	2	4	no	—
26	#11758224	—	4	2	no	—
27	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
28	C₂×Dic₁₂₂	—	2	1	no	—
29	D₈×C₆₁	—	2	1	no	—
30	D₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₆₁	—	2	1	no	—
33	#11758200	—	2	4	no	—
34	#11758198	—	4	1	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄×Dic₆₁	—	4	1	no	—
37	C₂×#92418	D₄	1	1	yes	characteristic
38	Dic₂₄₄	—	2	1	no	—
39	Dic₂₄₄	C₂×C₄	1	2	yes	—
40	C₈×D₆₁	C₂×C₄	1	2	yes	—
41	C₈×D₆₁	D₄	1	2	yes	—
42	C₁₆×C₆₁	—	2	1	no	—
43	#11758185	—	2	1	no	—
44	C₂³×C₆₁	—	4	1	no	—
45	C₂²×D₆₁	D₈	1	1	yes	characteristic
46	D₄×C₆₁	—	2	2	no	—
47	D₄×C₆₁	—	4	1	no	—
48	C₂×C₄×C₆₁	—	4	1	no	—
49	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂²⋊D₆₁	—	4	4	no	—
51	C₂×Dic₆₁	—	4	1	no	—
52	C₂×Dic₆₁	—	4	4	no	—
53	C₂×Dic₆₁	—	4	1	no	—
54	C₂×Dic₆₁	D₈	1	1	yes	characteristic
55	C₄×D₆₁	C₂²⋊C₄	1	2	yes	—
56	C₄×D₆₁	SD₁₆	1	2	yes	—
57	Dic₁₂₂	—	2	2	no	—
58	Dic₁₂₂	—	4	1	no	—
59	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
61	#92418	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₆₁	—	4	2	no	—
63	C₂²×C₆₁	—	8	1	no	—
64	C₂²×C₆₁	—	8	1	no	—
65	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
66	C₂×D₆₁	—	2	1	no	—
67	C₂×D₆₁	—	2	1	no	—
68	C₂×D₆₁	#102	1	2	yes	—
69	C₄×C₆₁	—	8	1	no	—
70	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
72	Dic₆₁	—	4	2	no	—
73	Dic₆₁	—	8	1	no	—
74	Dic₆₁	#102	1	2	yes	—
75	#2138	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	—	8	2	no	—
77	C₂×C₆₁	C₂×#102	1	1	yes	characteristic
78	C₂×C₆₁	#508	1	1	yes	characteristic
79	C₂×C₆₁	#508	1	1	yes	characteristic
80	D₆₁	—	2	2	no	—
81	C₂×#135	—	61	1	no	—
82	C₂²×C₁₆	—	61	1	no	—
83	#427	—	61	1	no	—
84	#358	—	61	1	no	—
85	#358	—	61	1	no	—
86	#357	—	61	1	no	—
87	#357	D₆₁	1	1	yes	characteristic, complemented
88	C₆₁	#2138	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	—	122	1	no	—
90	#135	—	61	4	no	—
91	#135	—	122	2	no	—
92	C₂×Q₁₆	—	61	1	no	—
93	C₂×SD₁₆	—	122	1	no	—
94	C₂×D₈	C₂×D₆₁	1	1	yes	characteristic
95	C₂²×C₈	—	61	1	no	—
96	#117	—	122	1	no	—
97	C₂×C₁₆	—	61	1	no	—
98	C₂×C₁₆	—	122	2	no	—
99	C₂×C₁₆	C₂×D₆₁	1	1	yes	characteristic
100	#107	—	61	1	no	—
101	#107	C₂×D₆₁	1	1	yes	characteristic
102	#106	—	122	1	no	—
103	#41	—	122	4	no	—
104	#41	—	244	2	no	—
105	C₂×Q₈	—	122	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	244	1	no	—
108	C₂²×C₄	—	61	1	no	—
109	C₂²×C₄	—	244	1	no	—
110	Q₁₆	—	61	2	no	—
111	Q₁₆	—	122	1	no	—
112	SD₁₆	—	122	4	no	—
113	D₈	—	2	1	no	—
114	D₈	C₄×D₆₁	1	2	yes	—
115	C₂×C₈	—	61	1	no	—
116	C₂×C₈	—	61	2	no	—
117	C₂×C₈	—	61	2	no	—
118	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	122	1	no	—
121	C₂²⋊C₄	—	244	1	no	—
122	C₄²	—	244	1	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	—	122	1	no	—
125	C₂³	—	4	1	no	—
126	C₂³	—	61	1	no	—
127	Q₈	—	122	2	no	—
128	Q₈	—	244	1	no	—
129	D₄	—	2	2	no	—
130	D₄	—	4	1	no	—
131	D₄	—	244	4	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	61	1	no	—
134	C₂×C₄	—	61	2	no	—
135	C₂×C₄	—	61	2	no	—
136	C₂×C₄	—	244	4	no	—
137	C₂×C₄	—	244	1	no	—
138	C₂×C₄	—	244	1	no	—
139	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
140	C₈	—	61	2	no	—
141	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
142	C₈	D₄×D₆₁	1	1	yes	characteristic
143	C₂²	—	4	2	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	61	2	no	—
147	C₂²	—	122	1	no	—
148	C₂²	—	122	1	no	—
149	C₂²	D₈×D₆₁	1	1	yes	characteristic
150	C₄	—	8	1	no	—
151	C₄	—	61	2	no	—
152	C₄	—	488	1	no	—
153	C₄	—	244	2	no	—
154	C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
155	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
156	C₂	—	8	2	no	—
157	C₂	—	122	2	no	—
158	C₂	#102×D₆₁	1	1	yes	characteristic
159	C₂	#16840191	1	1	yes	characteristic
160	C₂	#16840193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519796

C₂

yes

maximal, characteristic

C₂×C₁₆×D₆₁

C₂

yes

maximal, characteristic, complemented

#16839799

C₂

yes

maximal, characteristic, complemented

#16839567

C₂

yes

maximal, characteristic, complemented

#16839565

C₂

yes

maximal, characteristic, complemented

#16839563

C₂

yes

maximal, characteristic, complemented

C₆₁×#357

C₂

yes

maximal, characteristic, complemented

C₂×#11758224

—

#14519796

—

#14519796

C₄

yes

complemented

C₂×Dic₂₄₄

C₂²

yes

characteristic

C₂×#11758200

—

C₂×C₈×D₆₁

C₂²

yes

characteristic

#14519735

—

C₂×#11758185

C₂²

yes

characteristic

C₁₆×D₆₁

—

#14519711

C₂²

yes

characteristic

#14519708

—

C₂×D₈×C₆₁

C₂²

yes

characteristic

C₂×C₁₆×C₆₁

C₂²

yes

characteristic

#107×C₆₁

C₂²

yes

characteristic

C₂×D₄×C₆₁

—

C₂×C₂²⋊D₆₁

—

C₂²×Dic₆₁

—

#11758224

—

#11758224

—

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

D₈×C₆₁

—

D₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

#11758200

—

#11758198

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

D₄

yes

characteristic

Dic₂₄₄

—

Dic₂₄₄

C₂×C₄

yes

—

C₈×D₆₁

C₂×C₄

yes

—

C₈×D₆₁

D₄

yes

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

—

C₂²×D₆₁

D₈

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

D₈

yes

characteristic

C₄×D₆₁

C₂²⋊C₄

yes

—

C₄×D₆₁

SD₁₆

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂²×C₄

yes

characteristic

C₈×C₆₁

C₂×D₄

yes

characteristic

#92418

C₂²⋊C₄

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#102

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#102

yes

—

#2138

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₂×#135

—

C₂²×C₁₆

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₆₁

#2138

yes

characteristic, complemented, Sylow

C₂×#41

—

122

—

#135

—

#135

—

122

—

C₂×Q₁₆

—

C₂×SD₁₆

—

122

—

C₂×D₈

C₂×D₆₁

yes

characteristic

C₂²×C₈

—

#117

—

122

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₂×D₆₁

yes

characteristic

100

#107

—

101

#107

C₂×D₆₁

yes

characteristic

102

#106

—

122

—

103

#41

—

122

—

104

#41

—

244

—

105

C₂×Q₈

—

122

—

106

C₂×D₄

—

107

C₂×D₄

—

244

—

108

C₂²×C₄

—

109

C₂²×C₄

—

244

—

110

Q₁₆

—

111

Q₁₆

—

122

—

112

SD₁₆

—

122

—

113

D₈

—

114

D₈

C₄×D₆₁

yes

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂²×D₆₁

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

122

—

121

C₂²⋊C₄

—

244

—

122

C₄²

—

244

—

123

C₁₆

—

124

C₁₆

—

122

—

125

C₂³

—

126

C₂³

—

127

Q₈

—

122

—

128

Q₈

—

244

—

129

D₄

—

130

D₄

—

131

D₄

—

244

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

244

—

137

C₂×C₄

—

244

—

138

C₂×C₄

—

244

—

139

C₂×C₄

D₄×D₆₁

yes

characteristic

140

C₈

—

141

C₈

C₂×C₄×D₆₁

yes

characteristic

142

C₈

D₄×D₆₁

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

122

—

148

C₂²

—

122

—

149

C₂²

D₈×D₆₁

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

488

—

153

C₄

—

244

—

154

C₄

C₂²⋊C₄×D₆₁

yes

characteristic

155

C₄

SD₁₆×D₆₁

yes

characteristic

156

C₂

—

157

C₂

—

122

—

158

C₂

#102×D₆₁

yes

characteristic

159

C₂

#16840191

yes

characteristic

160

C₂

#16840193

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₁
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#357	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	#357	D₆₁
hypocenter	C₆₁	#2138
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₆₁

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×#357

C₂

socle

C₂²×C₆₁

C₂×D₈

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
2. C₂×C₁₆×D₆₁	C₂	semidirect
3. #16839799	C₂	semidirect
4. #16839567	C₂	semidirect
5. #16839565	C₂	semidirect
6. #16839563	C₂	semidirect
7. C₆₁×#357	C₂	semidirect
10. #14519796	C₄	semidirect
87. #357	D₆₁	semidirect
88. C₆₁	#2138	semidirect

H ≅ G/N

type

2. C₂×C₁₆×D₆₁

C₂

semidirect

3. #16839799

C₂

semidirect

4. #16839567

C₂

semidirect

5. #16839565

C₂

semidirect

6. #16839563

C₂

semidirect

7. C₆₁×#357

C₂

semidirect

10. #14519796

C₄

semidirect

87. #357

D₆₁

semidirect

88. C₆₁

#2138

semidirect

Subgroups — #21447523

Conjugacy classes of subgroups

Complements