Group #21446705 — subgroups

Subgroups — #21446705

Order 7808.

300 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14519799	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519703	C₂	1	2	yes	maximal, complemented
3	#16839730	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#108×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11758222	C₂²	1	2	yes	—
7	#14519799	—	2	2	no	—
8	#14519799	—	2	2	no	—
9	C₂×Dic₂₄₄	C₄	1	1	yes	characteristic, complemented
10	C₂×#11758190	C₄	1	2	yes	complemented
11	C₂×D₄₈₈	C₄	1	1	yes	characteristic, complemented
12	C₂²×#92418	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
14	#14519703	—	2	4	no	—
15	#14519689	—	2	4	no	—
16	C₈×Dic₆₁	—	2	2	no	—
17	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
18	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	#108×C₆₁	—	2	2	no	—
20	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂×C₂²⋊D₆₁	—	2	2	no	—
22	C₂²×Dic₆₁	—	2	1	no	—
23	#11758222	—	2	2	no	—
24	#11758222	—	2	2	no	—
25	#11758222	—	2	2	no	—
26	#11758222	—	2	2	no	—
27	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₆₁	—	2	2	no	—
29	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
30	M₁₆×C₆₁	—	2	2	no	—
31	M₁₆×C₆₁	—	2	2	no	—
32	C₂×C₈×C₆₁	—	2	2	no	—
33	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	D₄	1	2	yes	—
36	C₂×C₈×C₆₁	D₄	1	2	yes	—
37	C₄×Dic₆₁	—	2	1	no	—
38	C₄×Dic₆₁	D₄	1	2	yes	—
39	C₂×#92418	—	2	1	no	—
40	C₂×#92418	—	2	1	no	—
41	C₂×#92418	D₄	1	2	yes	—
42	Dic₂₄₄	—	2	2	no	—
43	D₄₈₈	—	2	2	no	—
44	#11758190	—	2	4	no	—
45	C₂³×C₆₁	#41	1	1	yes	characteristic
46	C₂²×D₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₄	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	C₂²⋊D₆₁	—	4	2	no	—
54	C₂²⋊D₆₁	—	4	2	no	—
55	C₂×Dic₆₁	—	2	4	no	—
56	C₂×Dic₆₁	—	2	2	no	—
57	C₂×Dic₆₁	—	2	2	no	—
58	D₂₄₄	—	2	2	no	—
59	D₂₄₄	—	2	2	no	—
60	C₄×D₆₁	—	4	2	no	—
61	C₄×D₆₁	—	4	2	no	—
62	Dic₁₂₂	—	2	2	no	—
63	Dic₁₂₂	—	2	2	no	—
64	C₈×C₆₁	—	4	2	no	—
65	C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
66	C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
67	#92418	—	2	2	no	—
68	C₂²×C₆₁	—	2	2	no	—
69	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
70	C₂²×C₆₁	#121	1	1	yes	characteristic
71	C₂²×C₆₁	#124	1	1	yes	characteristic
72	C₂×D₆₁	—	4	2	no	—
73	C₂×D₆₁	—	4	4	no	—
74	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
76	C₄×C₆₁	#121	1	1	yes	characteristic
77	C₄×C₆₁	#127	1	1	yes	characteristic
78	Dic₆₁	—	2	4	no	—
79	Dic₆₁	—	4	2	no	—
80	#1974	—	61	1	no	maximal, Sylow
81	C₂×C₆₁	—	2	1	no	—
82	C₂×C₆₁	—	2	1	no	—
83	C₂×C₆₁	#390	1	1	yes	characteristic
84	C₂×C₆₁	#443	1	2	yes	—
85	D₆₁	—	8	2	no	—
86	C₂×#135	—	61	1	no	—
87	C₂×#108	D₆₁	1	1	yes	characteristic, complemented
88	C₂×#104	—	61	2	no	—
89	#408	—	61	2	no	—
90	C₂×C₄×C₈	—	61	1	no	—
91	C₆₁	#1974	1	1	yes	characteristic, complemented, Sylow
92	C₂×#41	—	61	2	no	—
93	#135	—	122	2	no	—
94	#135	—	122	2	no	—
95	C₂×Q₁₆	—	61	1	no	—
96	C₂×SD₁₆	—	61	2	no	—
97	C₂×D₈	—	61	1	no	—
98	C₂×M₁₆	C₂×D₆₁	1	2	yes	—
99	C₂²×C₈	—	61	1	no	—
100	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
101	C₂×C₄²	—	61	1	no	—
102	#108	—	2	2	no	—
103	#104	—	122	4	no	—
104	C₂²⋊C₈	—	122	4	no	—
105	C₄×C₈	—	122	2	no	—
106	#41	—	122	2	no	—
107	#41	—	122	2	no	—
108	#41	—	122	2	no	—
109	#41	—	122	2	no	—
110	C₂×Q₈	—	61	2	no	—
111	C₂×D₄	—	61	2	no	—
112	C₂×D₄	—	122	2	no	—
113	C₂²×C₄	—	122	1	no	—
114	C₂²×C₄	—	122	2	no	—
115	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
116	Q₁₆	—	122	2	no	—
117	SD₁₆	—	122	4	no	—
118	D₈	—	122	2	no	—
119	M₁₆	—	2	2	no	—
120	M₁₆	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	61	2	no	—
123	C₂×C₈	—	122	1	no	—
124	C₂×C₈	—	122	1	no	—
125	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
126	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
127	C₂×C₈	D₂₄₄	1	2	yes	—
128	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
129	C₄²	—	61	2	no	—
130	C₄²	—	122	1	no	—
131	C₂³	—	122	2	no	—
132	C₂³	#11758226	1	1	yes	characteristic
133	Q₈	—	122	2	no	—
134	Q₈	—	122	2	no	—
135	D₄	—	122	2	no	—
136	D₄	—	122	2	no	—
137	D₄	—	244	2	no	—
138	D₄	—	244	2	no	—
139	C₂×C₄	—	122	4	no	—
140	C₂×C₄	—	122	2	no	—
141	C₂×C₄	—	122	2	no	—
142	C₂×C₄	—	244	2	no	—
143	C₂×C₄	—	244	2	no	—
144	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
145	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
146	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
147	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
148	C₂×C₄	#11758226	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
150	C₈	—	4	2	no	—
151	C₈	—	122	2	no	—
152	C₈	#11758198	1	2	yes	—
153	C₈	#11758198	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	244	4	no	—
156	C₂²	—	244	2	no	—
157	C₂²	#14519739	1	1	yes	characteristic
158	C₂²	#14519752	1	1	yes	characteristic
159	C₂²	#14519766	1	1	yes	characteristic
160	C₄	—	122	4	no	—
161	C₄	—	244	2	no	—
162	C₄	C₂×#11758198	1	1	yes	characteristic
163	C₄	#14519739	1	1	yes	characteristic
164	C₄	#14519752	1	1	yes	characteristic
165	C₄	#14519773	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	488	2	no	—
169	C₂	#16839663	1	1	yes	characteristic
170	C₂	#16839878	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519799

C₂

yes

maximal, characteristic

C₂×#14519703

C₂

yes

maximal, complemented

#16839730

C₂

yes

maximal, complemented

C₂×C₈×Dic₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#108×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#11758222

C₂²

yes

—

#14519799

—

#14519799

—

C₂×Dic₂₄₄

C₄

yes

characteristic, complemented

C₂×#11758190

C₄

yes

complemented

C₂×D₄₈₈

C₄

yes

characteristic, complemented

C₂²×#92418

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519703

—

#14519689

—

C₈×Dic₆₁

—

C₂×M₁₆×C₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

#108×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₂²⋊D₆₁

—

C₂²×Dic₆₁

—

—

—

—

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

D₄

yes

—

Dic₂₄₄

—

D₄₈₈

—

#11758190

—

C₂³×C₆₁

#41

yes

characteristic

C₂²×D₆₁

—

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

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂²⋊D₆₁

—

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊C₄

yes

—

C₈×C₆₁

C₂²⋊C₄

yes

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×D₄

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

C₄×D₄

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

#1974

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#390

yes

characteristic

C₂×C₆₁

#443

yes

—

D₆₁

—

C₂×#135

—

C₂×#108

D₆₁

yes

characteristic, complemented

C₂×#104

—

#408

—

C₂×C₄×C₈

—

C₆₁

#1974

yes

characteristic, complemented, Sylow

C₂×#41

—

#135

—

122

—

#135

—

122

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

C₂×D₆₁

yes

—

C₂²×C₈

—

100

C₂²×C₈

C₂×D₆₁

yes

characteristic

101

C₂×C₄²

—

102

#108

—

103

#104

—

122

—

104

C₂²⋊C₈

—

122

—

105

C₄×C₈

—

122

—

106

#41

—

122

—

107

#41

—

122

—

108

#41

—

122

—

109

#41

—

122

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

122

—

113

C₂²×C₄

—

122

—

114

C₂²×C₄

—

122

—

115

C₂²×C₄

C₂²×D₆₁

yes

characteristic

116

Q₁₆

—

122

—

117

SD₁₆

—

122

—

118

D₈

—

122

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

122

—

124

C₂×C₈

—

122

—

125

C₂×C₈

C₄×D₆₁

yes

characteristic

126

C₂×C₈

C₄×D₆₁

yes

characteristic

127

C₂×C₈

D₂₄₄

yes

—

128

C₂×C₈

C₂²⋊D₆₁

yes

—

129

C₄²

—

130

C₄²

—

122

—

131

C₂³

—

122

—

132

C₂³

#11758226

yes

characteristic

133

Q₈

—

122

—

134

Q₈

—

122

—

135

D₄

—

122

—

136

D₄

—

122

—

137

D₄

—

244

—

138

D₄

—

244

—

139

C₂×C₄

—

122

—

140

C₂×C₄

—

122

—

141

C₂×C₄

—

122

—

142

C₂×C₄

—

244

—

143

C₂×C₄

—

244

—

144

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

145

C₂×C₄

C₂×D₂₄₄

yes

characteristic

146

C₂×C₄

D₄×D₆₁

yes

characteristic

147

C₂×C₄

D₄×D₆₁

yes

characteristic

148

C₂×C₄

#11758226

yes

characteristic

149

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

150

C₈

—

151

C₈

—

122

—

152

C₈

#11758198

yes

—

153

C₈

#11758198

yes

—

154

C₂²

—

155

C₂²

—

244

—

156

C₂²

—

244

—

157

C₂²

#14519739

yes

characteristic

158

C₂²

#14519752

yes

characteristic

159

C₂²

#14519766

yes

characteristic

160

C₄

—

122

—

161

C₄

—

244

—

162

C₄

C₂×#11758198

yes

characteristic

163

C₄

#14519739

yes

characteristic

164

C₄

#14519752

yes

characteristic

165

C₄

#14519773

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

488

—

169

C₂

#16839663

yes

characteristic

170

C₂

#16839878

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#108×C₆₁	C₂
socle	C₂²×C₆₁	C₄×D₄
hypercenter	C₂×#108	D₆₁
hypocenter	C₆₁	#1974
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×#108×C₆₁

C₂

socle

C₂²×C₆₁

C₄×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
2. C₂×#14519703	C₂	semidirect
3. #16839730	C₂	semidirect
4. C₂×C₈×Dic₆₁	C₂	semidirect
5. C₂×#108×C₆₁	C₂	semidirect
9. C₂×Dic₂₄₄	C₄	semidirect
10. C₂×#11758190	C₄	semidirect
11. C₂×D₄₈₈	C₄	semidirect
87. C₂×#108	D₆₁	semidirect
91. C₆₁	#1974	semidirect

H ≅ G/N

type

2. C₂×#14519703

C₂

semidirect

3. #16839730

C₂

semidirect

4. C₂×C₈×Dic₆₁

C₂

semidirect

5. C₂×#108×C₆₁

C₂

semidirect

9. C₂×Dic₂₄₄

C₄

semidirect

10. C₂×#11758190

C₄

semidirect

11. C₂×D₄₈₈

C₄

semidirect

87. C₂×#108

D₆₁

semidirect

91. C₆₁

#1974

semidirect

Subgroups — #21446705

Conjugacy classes of subgroups

Complements