Group #21445178 — subgroups

Subgroups — #21445178

Order 7808.

188 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16840138	C₂	1	1	yes	maximal, characteristic
2	#16839963	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839959	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839886	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839514	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839505	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758190	—	2	1	no	—
9	C₂×#11758200	—	2	1	no	—
10	C₂×#11758199	—	2	1	no	—
11	C₂×#11758199	—	2	1	no	—
12	#14519774	C₂²	1	1	yes	characteristic
13	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
14	C₄⋊#92418	C₂²	1	1	yes	characteristic
15	#14519706	—	2	1	no	—
16	#14519699	—	2	1	no	—
17	#14519698	—	2	1	no	—
18	C₄×#92418	C₂²	1	1	yes	characteristic
19	C₈×Dic₆₁	—	2	1	no	—
20	#127×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×D₄×C₆₁	—	2	1	no	—
24	C₂×D₄×C₆₁	—	2	1	no	—
25	C₂×D₄×C₆₁	—	2	1	no	—
26	C₂×D₂₄₄	—	2	1	no	—
27	C₂×D₂₄₄	—	2	1	no	—
28	C₂×D₂₄₄	—	2	1	no	—
29	C₂×Dic₁₂₂	D₄	1	1	yes	characteristic
30	C₂×C₈×C₆₁	—	2	1	no	—
31	C₂×C₈×C₆₁	—	2	1	no	—
32	C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	#11758200	—	4	2	no	—
34	#11758199	—	4	2	no	—
35	#11758199	—	4	2	no	—
36	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
37	#11758196	—	2	1	no	—
38	C₄×Dic₆₁	—	2	1	no	—
39	C₂×#92418	—	2	1	no	—
40	C₂×#92418	D₄	1	1	yes	characteristic
41	C₂×#92418	D₄	1	1	yes	characteristic
42	#11758190	—	4	2	no	—
43	C₂³×C₆₁	—	4	1	no	—
44	C₂²×D₆₁	—	4	1	no	—
45	D₄×C₆₁	—	4	1	no	—
46	D₄×C₆₁	—	4	1	no	—
47	D₄×C₆₁	—	4	1	no	—
48	D₄×C₆₁	—	4	1	no	—
49	D₄×C₆₁	—	8	1	no	—
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×Dic₆₁	—	2	1	no	—
54	C₂×Dic₆₁	—	2	1	no	—
55	D₂₄₄	—	4	1	no	—
56	D₂₄₄	—	4	1	no	—
57	D₂₄₄	—	4	1	no	—
58	D₂₄₄	—	4	1	no	—
59	D₂₄₄	—	8	1	no	—
60	Dic₁₂₂	—	2	1	no	—
61	Dic₁₂₂	D₈	1	2	yes	complemented
62	C₈×C₆₁	—	4	1	no	—
63	C₈×C₆₁	—	4	1	no	—
64	#92418	—	2	1	no	—
65	#92418	—	4	1	no	—
66	#92418	SD₁₆	1	2	yes	complemented
67	C₂²×C₆₁	—	8	1	no	—
68	C₂²×C₆₁	—	8	1	no	—
69	C₂²×C₆₁	—	8	1	no	—
70	C₂²×C₆₁	#121	1	1	yes	characteristic
71	C₂×D₆₁	—	8	1	no	—
72	C₂×D₆₁	—	8	1	no	—
73	C₂×D₆₁	—	8	1	no	—
74	C₄×C₆₁	—	2	1	no	—
75	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
76	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₆₁	#135	1	1	yes	characteristic
78	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Dic₆₁	—	2	2	no	—
80	Dic₆₁	—	4	1	no	—
81	#1668	—	61	1	no	maximal, Sylow
82	C₂×C₆₁	—	16	1	no	—
83	C₂×C₆₁	#460	1	1	yes	characteristic
84	C₂×C₆₁	#466	1	1	yes	characteristic
85	C₂×C₆₁	#472	1	1	yes	characteristic
86	D₆₁	—	16	1	no	—
87	#493	—	61	1	no	—
88	#460	—	61	1	no	—
89	#460	—	61	1	no	—
90	C₈×Q₈	—	61	1	no	—
91	#335	—	61	1	no	—
92	#331	—	61	1	no	—
93	#331	D₆₁	1	1	yes	characteristic, complemented
94	C₆₁	#1668	1	1	yes	characteristic, complemented, Sylow
95	C₂×SD₁₆	—	122	1	no	—
96	C₂×SD₁₆	—	122	1	no	—
97	C₂×D₈	—	122	1	no	—
98	C₂×D₈	—	122	1	no	—
99	#127	—	61	1	no	—
100	#127	C₂×D₆₁	1	1	yes	characteristic
101	C₄×Q₈	—	61	1	no	—
102	C₄⋊C₈	—	61	1	no	—
103	C₄⋊C₈	—	122	1	no	—
104	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
106	#102	—	122	1	no	—
107	#102	—	122	1	no	—
108	C₄×C₈	—	61	1	no	—
109	C₄×C₈	—	122	1	no	—
110	C₂×Q₈	—	61	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	122	1	no	—
115	C₂×D₄	—	122	1	no	—
116	C₂×D₄	—	122	1	no	—
117	SD₁₆	—	244	2	no	—
118	SD₁₆	—	244	2	no	—
119	D₈	—	244	2	no	—
120	D₈	—	244	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	61	1	no	—
124	C₂×C₈	—	61	1	no	—
125	C₂×C₈	—	122	1	no	—
126	C₄⋊C₄	—	61	1	no	—
127	C₄⋊C₄	—	122	1	no	—
128	C₄²	—	122	1	no	—
129	C₄²	C₂²×D₆₁	1	1	yes	characteristic
130	C₂³	—	4	1	no	—
131	C₂³	—	244	1	no	—
132	Q₈	—	61	2	no	—
133	Q₈	—	122	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	8	1	no	—
139	D₄	—	488	1	no	—
140	D₄	—	244	1	no	—
141	D₄	—	244	1	no	—
142	D₄	—	244	1	no	—
143	D₄	—	244	1	no	—
144	C₂×C₄	—	122	1	no	—
145	C₂×C₄	—	122	1	no	—
146	C₂×C₄	#11758222	1	1	yes	characteristic
147	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
148	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	61	2	no	—
152	C₈	—	122	1	no	—
153	C₈	—	244	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	488	1	no	—
158	C₂²	—	488	1	no	—
159	C₂²	—	488	1	no	—
160	C₂²	#14519750	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	122	2	no	—
163	C₄	—	244	1	no	—
164	C₄	D₈×D₆₁	1	1	yes	characteristic
165	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
166	C₄	#14519798	1	1	yes	characteristic
167	C₄	#14519801	1	1	yes	characteristic
168	C₂	—	16	1	no	—
169	C₂	—	976	1	no	—
170	C₂	#16839961	1	1	yes	characteristic
171	C₂	#16839997	1	1	yes	characteristic
172	C₂	#16840031	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840138

C₂

yes

maximal, characteristic

#16839963

C₂

yes

maximal, characteristic, complemented

#16839959

C₂

yes

maximal, characteristic, complemented

#16839886

C₂

yes

maximal, characteristic, complemented

#16839514

C₂

yes

maximal, characteristic, complemented

#16839505

C₂

yes

maximal, characteristic, complemented

C₆₁×#331

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₂₂

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₈×Dic₆₁

—

#127×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

D₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂³

yes

—

—

—

C₄⋊Dic₆₁

D₄

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

#11758190

—

C₂³×C₆₁

—

C₂²×D₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

Dic₁₂₂

D₈

yes

complemented

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

—

#92418

SD₁₆

yes

complemented

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

#1668

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#493

—

#460

—

#460

—

C₈×Q₈

—

#335

—

#331

—

#331

D₆₁

yes

characteristic, complemented

C₆₁

#1668

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

122

—

C₂×SD₁₆

—

122

—

C₂×D₈

—

122

—

C₂×D₈

—

122

—

#127

—

100

#127

C₂×D₆₁

yes

characteristic

101

C₄×Q₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

122

—

104

C₄⋊C₈

C₂×D₆₁

yes

characteristic

105

C₄⋊C₈

C₂×D₆₁

yes

characteristic

106

#102

—

122

—

107

#102

—

122

—

108

C₄×C₈

—

109

C₄×C₈

—

122

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

122

—

115

C₂×D₄

—

122

—

116

C₂×D₄

—

122

—

117

SD₁₆

—

244

—

118

SD₁₆

—

244

—

119

D₈

—

244

—

120

D₈

—

244

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

122

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

122

—

128

C₄²

—

122

—

129

C₄²

C₂²×D₆₁

yes

characteristic

130

C₂³

—

131

C₂³

—

244

—

132

Q₈

—

133

Q₈

—

122

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

488

—

140

D₄

—

244

—

141

D₄

—

244

—

142

D₄

—

244

—

143

D₄

—

244

—

144

C₂×C₄

—

122

—

145

C₂×C₄

—

122

—

146

C₂×C₄

#11758222

yes

characteristic

147

C₂×C₄

D₄×D₆₁

yes

characteristic

148

C₂×C₄

D₄×D₆₁

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

122

—

153

C₈

—

244

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

488

—

158

C₂²

—

488

—

159

C₂²

—

488

—

160

C₂²

#14519750

yes

characteristic

161

C₄

—

162

C₄

—

122

—

163

C₄

—

244

—

164

C₄

D₈×D₆₁

yes

characteristic

165

C₄

SD₁₆×D₆₁

yes

characteristic

166

C₄

#14519798

yes

characteristic

167

C₄

#14519801

yes

characteristic

168

C₂

—

169

C₂

—

976

—

170

C₂

#16839961

yes

characteristic

171

C₂

#16839997

yes

characteristic

172

C₂

#16840031

yes

characteristic

	subgroup	G/H
center	C₂²	#14519750
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#331	C₂
socle	C₂²×C₆₁	#121
hypercenter	#331	D₆₁
hypocenter	C₆₁	#1668
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14519750

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#331

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #16839963	C₂	semidirect
3. #16839959	C₂	semidirect
4. #16839886	C₂	semidirect
5. #16839514	C₂	semidirect
6. #16839505	C₂	semidirect
7. C₆₁×#331	C₂	semidirect
61. Dic₁₂₂	D₈	semidirect
66. #92418	SD₁₆	semidirect
93. #331	D₆₁	semidirect
94. C₆₁	#1668	semidirect

H ≅ G/N

type

2. #16839963

C₂

semidirect

3. #16839959

C₂

semidirect

4. #16839886

C₂

semidirect

5. #16839514

C₂

semidirect

6. #16839505

C₂

semidirect

7. C₆₁×#331

C₂

semidirect

61. Dic₁₂₂

D₈

semidirect

66. #92418

SD₁₆

semidirect

93. #331

D₆₁

semidirect

94. C₆₁

#1668

semidirect

Subgroups — #21445178

Conjugacy classes of subgroups

Complements