Group #21444269 — subgroups

Subgroups — #21444269

Order 7808.

276 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×C₄×#92418	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839489	C₂	1	4	yes	maximal, complemented
4	C₆₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11758222	—	2	1	no	—
6	C₂²×#92418	—	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	C₄×#92418	—	2	1	no	—
15	C₄×#92418	C₂²	1	2	yes	—
16	C₂×M₁₆×C₆₁	—	2	1	no	—
17	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₆₁	—	2	1	no	—
19	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
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	2	no	—
25	#11758222	—	4	2	no	—
26	C₂×D₂₄₄	—	2	1	no	—
27	C₂×D₂₄₄	—	2	1	no	—
28	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₆₁	—	2	2	no	—
30	C₂×Dic₁₂₂	—	2	1	no	—
31	M₁₆×C₆₁	—	4	2	no	—
32	C₂×C₈×C₆₁	—	2	2	no	—
33	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₆₁	D₄	1	1	yes	characteristic
35	C₄²×C₆₁	D₄	1	1	yes	characteristic
36	C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	#11758198	—	2	2	no	—
38	#11758198	—	2	2	no	—
39	C₄⋊Dic₆₁	—	2	1	no	—
40	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
41	C₂×#92418	—	2	4	no	—
42	C₂×#92418	—	2	2	no	—
43	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×D₆₁	—	2	2	no	—
45	C₂×C₄×C₆₁	—	2	2	no	—
46	C₂×C₄×C₆₁	—	2	2	no	—
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	D₈	1	2	yes	—
51	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
52	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂²⋊D₆₁	—	4	4	no	—
56	C₂×Dic₆₁	—	2	2	no	—
57	D₂₄₄	—	2	4	no	—
58	D₂₄₄	—	4	2	no	—
59	D₂₄₄	—	4	2	no	—
60	C₄×D₆₁	—	4	2	no	—
61	C₄×D₆₁	—	4	2	no	—
62	Dic₁₂₂	—	4	2	no	—
63	C₈×C₆₁	—	4	2	no	—
64	#92418	—	2	4	no	—
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	—	2	1	no	—
67	C₂²×C₆₁	#102	1	2	yes	—
68	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×D₆₁	—	4	2	no	—
70	C₂×D₆₁	—	4	2	no	—
71	C₂×D₆₁	—	4	2	no	—
72	C₄×C₆₁	—	2	1	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	#102	1	2	yes	—
75	C₄×C₆₁	#104	1	2	yes	—
76	C₄×C₆₁	#131	1	2	yes	—
77	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
78	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
79	Dic₆₁	—	4	2	no	—
80	#1492	—	61	1	no	maximal, Sylow
81	C₂×C₆₁	—	2	2	no	—
82	C₂×C₆₁	#408	1	1	yes	characteristic
83	C₂×C₆₁	C₂×#102	1	1	yes	characteristic
84	C₂×C₆₁	C₂×#104	1	1	yes	characteristic
85	D₆₁	—	8	2	no	—
86	#532	—	61	1	no	—
87	#423	D₆₁	1	1	yes	characteristic, complemented
88	C₂×C₄×C₈	—	61	1	no	—
89	D₄⋊C₈	—	61	4	no	—
90	C₆₁	#1492	1	1	yes	characteristic, complemented, Sylow
91	C₂×#41	—	122	1	no	—
92	C₂×M₁₆	—	2	1	no	—
93	C₂²×C₈	—	122	1	no	—
94	C₄⋊Q₈	—	61	1	no	—
95	#127	—	61	1	no	—
96	#124	—	122	1	no	—
97	#121	—	61	2	no	—
98	#121	—	122	1	no	—
99	C₄×D₄	—	61	2	no	—
100	C₄×D₄	—	122	1	no	—
101	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
102	C₄⋊C₈	—	2	1	no	—
103	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
104	C₄×C₈	—	61	2	no	—
105	C₄×C₈	—	122	1	no	—
106	#41	—	244	2	no	—
107	#41	—	244	2	no	—
108	C₂×Q₈	—	122	1	no	—
109	C₂×D₄	—	61	2	no	—
110	C₂×D₄	—	122	2	no	—
111	C₂×D₄	—	122	1	no	—
112	C₂×D₄	—	122	1	no	—
113	C₂²×C₄	—	122	2	no	—
114	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
115	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	122	4	no	—
120	C₂×C₈	—	122	2	no	—
121	C₄⋊C₄	—	61	2	no	—
122	C₄⋊C₄	—	122	1	no	—
123	C₂²⋊C₄	—	122	2	no	—
124	C₂²⋊C₄	—	122	2	no	—
125	C₄²	C₄×D₆₁	1	1	yes	characteristic
126	C₄²	D₂₄₄	1	1	yes	characteristic
127	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
128	C₄²	C₂²×D₆₁	1	1	yes	characteristic
129	C₂³	—	122	2	no	—
130	C₂³	#11758198	1	1	yes	characteristic
131	Q₈	—	244	2	no	—
132	D₄	—	122	4	no	—
133	D₄	—	244	4	no	—
134	D₄	—	244	2	no	—
135	D₄	—	244	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	122	2	no	—
139	C₂×C₄	—	244	2	no	—
140	C₂×C₄	—	244	2	no	—
141	C₂×C₄	#11758198	1	1	yes	characteristic
142	C₂×C₄	#11758198	1	1	yes	characteristic
143	C₂×C₄	#11758198	1	1	yes	characteristic
144	C₂×C₄	#11758199	1	2	yes	—
145	C₂×C₄	#11758201	1	2	yes	—
146	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
147	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
149	C₈	—	4	2	no	—
150	C₈	—	122	4	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	244	2	no	—
154	C₂²	—	244	2	no	—
155	C₂²	—	244	2	no	—
156	C₂²	#14519697	1	2	yes	—
157	C₂²	C₂×#11758198	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	244	2	no	—
161	C₄	#14519697	1	2	yes	—
162	C₄	#14519703	1	2	yes	—
163	C₄	#14519785	1	2	yes	—
164	C₄	C₂×#11758199	1	1	yes	characteristic
165	C₄	C₂×#11758201	1	1	yes	characteristic
166	C₂	—	2	2	no	—
167	C₂	—	488	2	no	—
168	C₂	#16839730	1	1	yes	characteristic
169	C₂	C₂×#14519697	1	1	yes	characteristic
170	C₂	C₂×#14519703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840312

C₂

yes

maximal, characteristic

C₂×C₄×#92418

C₂

yes

maximal, characteristic, complemented

#16839489

C₂

yes

maximal, complemented

C₆₁×#423

C₂

yes

maximal, characteristic, complemented

C₂×#11758222

—

C₂²×#92418

—

#14519777

C₄

yes

characteristic

#14519774

C₄

yes

—

—

C₄

yes

—

C₄×D₂₄₄

—

C₄×D₂₄₄

C₂²

yes

—

C₄×#92418

—

C₄×#92418

C₂²

yes

—

C₂×M₁₆×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂×C₂²⋊D₆₁

—

#11758222

—

#11758222

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

M₁₆×C₆₁

—

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

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×D₆₁

—

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

D₈

yes

—

C₂×C₄×C₆₁

SD₁₆

yes

—

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

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#102

yes

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#102

yes

—

C₄×C₆₁

#104

yes

—

C₄×C₆₁

#131

yes

—

C₄×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

Dic₆₁

—

#1492

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#532

—

#423

D₆₁

yes

characteristic, complemented

C₂×C₄×C₈

—

D₄⋊C₈

—

C₆₁

#1492

yes

characteristic, complemented, Sylow

C₂×#41

—

122

—

C₂×M₁₆

—

C₂²×C₈

—

122

—

C₄⋊Q₈

—

#127

—

#124

—

122

—

#121

—

#121

—

122

—

C₄×D₄

—

100

C₄×D₄

—

122

—

101

C₂×C₄²

C₂×D₆₁

yes

characteristic

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₆₁

yes

—

104

C₄×C₈

—

105

C₄×C₈

—

122

—

106

#41

—

244

—

107

#41

—

244

—

108

C₂×Q₈

—

122

—

109

C₂×D₄

—

110

C₂×D₄

—

122

—

111

C₂×D₄

—

122

—

112

C₂×D₄

—

122

—

113

C₂²×C₄

—

122

—

114

C₂²×C₄

C₄×D₆₁

yes

characteristic

115

C₂²×C₄

D₂₄₄

yes

characteristic

116

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

122

—

120

C₂×C₈

—

122

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

122

—

123

C₂²⋊C₄

—

122

—

124

C₂²⋊C₄

—

122

—

125

C₄²

C₄×D₆₁

yes

characteristic

126

C₄²

D₂₄₄

yes

characteristic

127

C₄²

C₂²⋊D₆₁

yes

characteristic

128

C₄²

C₂²×D₆₁

yes

characteristic

129

C₂³

—

122

—

130

C₂³

#11758198

yes

characteristic

131

Q₈

—

244

—

132

D₄

—

122

—

133

D₄

—

244

—

134

D₄

—

244

—

135

D₄

—

244

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

122

—

139

C₂×C₄

—

244

—

140

C₂×C₄

—

244

—

141

C₂×C₄

#11758198

yes

characteristic

142

C₂×C₄

#11758198

yes

characteristic

143

C₂×C₄

#11758198

yes

characteristic

144

C₂×C₄

#11758199

yes

—

145

C₂×C₄

#11758201

yes

—

146

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

147

C₂×C₄

C₂×D₂₄₄

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

149

C₈

—

150

C₈

—

122

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

244

—

154

C₂²

—

244

—

155

C₂²

—

244

—

156

C₂²

#14519697

yes

—

157

C₂²

C₂×#11758198

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

244

—

161

C₄

#14519697

yes

—

162

C₄

#14519703

yes

—

163

C₄

#14519785

yes

—

164

C₄

C₂×#11758199

yes

characteristic

165

C₄

C₂×#11758201

yes

characteristic

166

C₂

—

167

C₂

—

488

—

168

C₂

#16839730

yes

characteristic

169

C₂

C₂×#14519697

yes

characteristic

170

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×#423	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	#423	D₆₁
hypocenter	C₆₁	#1492
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#423

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×C₄×#92418	C₂	semidirect
3. #16839489	C₂	semidirect
4. C₆₁×#423	C₂	semidirect
87. #423	D₆₁	semidirect
90. C₆₁	#1492	semidirect

H ≅ G/N

type

2. C₂×C₄×#92418

C₂

semidirect

3. #16839489

C₂

semidirect

4. C₆₁×#423

C₂

semidirect

87. #423

D₆₁

semidirect

90. C₆₁

#1492

semidirect

Subgroups — #21444269

Conjugacy classes of subgroups

Complements