Group #21458241 — subgroups

Subgroups — #21458241

Order 7808.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16840478	C₂	1	2	yes	maximal, complemented
2	#16840089	C₂	1	4	yes	maximal, complemented
3	#16840049	C₂	1	2	yes	maximal, complemented
4	#16839999	C₂	1	2	yes	maximal, complemented
5	C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839799	C₂	1	2	yes	maximal, complemented
8	C₆₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₆₁	C₂²	1	2	yes	—
11	#14519766	—	2	4	no	—
12	#14519752	—	2	2	no	—
13	#14519752	C₂²	1	4	yes	—
14	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
15	C₄×D₂₄₄	C₂²	1	2	yes	—
16	#14519735	—	2	2	no	—
17	#14519735	C₂²	1	4	yes	—
18	#14519711	C₂²	1	1	yes	characteristic
19	#14519708	C₂²	1	4	yes	—
20	C₄⋊#92418	C₂²	1	1	yes	characteristic
21	#14519699	C₂²	1	2	yes	—
22	#14519697	C₂²	1	4	yes	—
23	#14519689	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
25	#107×C₆₁	C₂²	1	2	yes	—
26	#107×C₆₁	C₂²	1	2	yes	—
27	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₆₁	—	2	2	no	—
29	C₂×D₂₄₄	—	2	4	no	—
30	C₂×D₂₄₄	D₄	1	1	yes	characteristic
31	C₂×D₂₄₄	C₂³	1	2	yes	—
32	C₂×C₄×D₆₁	—	2	2	no	—
33	C₂×C₄×D₆₁	C₂³	1	2	yes	—
34	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
38	C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	#11758198	—	2	4	no	—
40	#11758198	—	2	2	no	—
41	#11758198	—	2	4	no	—
42	#11758198	D₄	1	2	yes	—
43	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
44	C₄⋊Dic₆₁	C₂³	1	2	yes	—
45	C₄×Dic₆₁	—	2	4	no	—
46	C₂×#92418	C₂³	1	2	yes	—
47	D₄₈₈	—	2	2	no	—
48	C₈×D₆₁	—	2	4	no	—
49	C₂²×D₆₁	—	2	2	no	—
50	C₂²×D₆₁	C₂×D₄	1	2	yes	—
51	Q₈×C₆₁	—	8	2	no	—
52	C₂×C₄×C₆₁	—	2	4	no	—
53	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₆₁	—	2	2	no	—
57	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
58	D₂₄₄	—	2	2	no	—
59	D₂₄₄	—	2	2	no	—
60	D₂₄₄	—	2	2	no	—
61	D₂₄₄	—	8	4	no	—
62	C₄×D₆₁	—	2	2	no	—
63	C₄×D₆₁	—	2	2	no	—
64	C₄×D₆₁	—	4	4	no	—
65	C₈×C₆₁	—	2	1	no	—
66	C₈×C₆₁	#41	1	2	yes	—
67	#92418	—	2	2	no	—
68	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₆₁	—	2	2	no	—
70	C₂×D₆₁	—	2	2	no	—
71	C₂×D₆₁	—	2	2	no	—
72	C₂×D₆₁	—	4	2	no	—
73	C₂×D₆₁	—	4	2	no	—
74	C₂×D₆₁	—	4	2	no	—
75	C₄×C₆₁	—	2	1	no	—
76	C₄×C₆₁	—	4	4	no	—
77	C₄×C₆₁	#135	1	2	yes	—
78	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
79	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
80	Dic₆₁	—	2	2	no	—
81	Dic₆₁	—	4	2	no	—
82	#3341	—	61	1	no	maximal, Sylow
83	C₂×C₆₁	#547	1	1	yes	characteristic
84	C₂×C₆₁	C₂×#135	1	1	yes	characteristic
85	C₂×C₆₁	#576	1	1	yes	characteristic
86	D₆₁	—	4	2	no	—
87	D₆₁	—	8	2	no	—
88	#555	—	61	2	no	—
89	#500	D₆₁	1	1	yes	characteristic, complemented
90	#482	—	61	4	no	—
91	#476	—	61	2	no	—
92	#466	—	61	2	no	—
93	C₄×D₈	—	61	1	no	—
94	C₈×D₄	—	61	1	no	—
95	#427	—	61	2	no	—
96	C₆₁	#3341	1	1	yes	characteristic, complemented, Sylow
97	C₂×D₈	—	61	1	no	—
98	C₂²×C₈	—	61	2	no	—
99	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
100	#124	—	122	4	no	—
101	#121	—	61	4	no	—
102	#121	—	122	2	no	—
103	C₄×D₄	—	61	1	no	—
104	C₄×D₄	—	61	2	no	—
105	#117	—	61	4	no	—
106	#117	—	122	2	no	—
107	#107	—	61	1	no	—
108	#107	C₂×D₆₁	1	2	yes	—
109	#107	C₂×D₆₁	1	2	yes	—
110	#106	—	61	4	no	—
111	C₄⋊C₈	—	61	1	no	—
112	#102	—	61	2	no	—
113	#102	—	61	4	no	—
114	C₂²⋊C₈	—	61	2	no	—
115	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
116	C₂×Q₈	—	2	2	no	—
117	C₂×D₄	—	61	1	no	—
118	C₂×D₄	—	61	2	no	—
119	C₂×D₄	—	122	4	no	—
120	C₂²×C₄	—	61	2	no	—
121	C₂²×C₄	—	122	2	no	—
122	D₈	—	122	2	no	—
123	C₂×C₈	—	61	2	no	—
124	C₂×C₈	—	122	4	no	—
125	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
126	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	61	1	no	—
129	C₄⋊C₄	—	61	2	no	—
130	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
131	C₂²⋊C₄	—	61	2	no	—
132	C₂²⋊C₄	—	122	4	no	—
133	C₂²⋊C₄	—	122	4	no	—
134	C₂²⋊C₄	—	122	2	no	—
135	C₄²	—	122	4	no	—
136	C₄²	C₂²×D₆₁	1	1	yes	characteristic
137	C₂³	—	61	2	no	—
138	C₂³	—	122	2	no	—
139	Q₈	—	8	2	no	—
140	D₄	—	122	2	no	—
141	D₄	—	122	2	no	—
142	D₄	—	122	2	no	—
143	D₄	—	488	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	61	2	no	—
146	C₂×C₄	—	122	2	no	—
147	C₂×C₄	—	122	2	no	—
148	C₂×C₄	—	122	2	no	—
149	C₂×C₄	—	244	4	no	—
150	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
151	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
152	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	122	2	no	—
155	C₈	#11758226	1	2	yes	—
156	C₂²	—	122	2	no	—
157	C₂²	—	122	2	no	—
158	C₂²	—	122	2	no	—
159	C₂²	—	244	2	no	—
160	C₂²	—	244	2	no	—
161	C₂²	—	244	2	no	—
162	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	4	no	—
165	C₄	—	122	2	no	—
166	C₄	—	244	2	no	—
167	C₄	#14519800	1	2	yes	—
168	C₄	C₂×#11758226	1	1	yes	characteristic
169	C₄	#14519825	1	1	yes	characteristic
170	C₂	—	488	2	no	—
171	C₂	—	244	2	no	—
172	C₂	#16840415	1	1	yes	characteristic
173	C₂	C₂×#14519800	1	1	yes	characteristic
174	C₂	#16840563	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840478

C₂

yes

maximal, complemented

#16840089

C₂

yes

maximal, complemented

#16840049

C₂

yes

maximal, complemented

#16839999

C₂

yes

maximal, complemented

C₄×D₄₈₈

C₂

yes

maximal, characteristic, complemented

C₈×D₂₄₄

C₂

yes

maximal, characteristic, complemented

#16839799

C₂

yes

maximal, complemented

C₆₁×#500

C₂

yes

maximal, characteristic, complemented

C₂×D₄₈₈

C₂²

yes

characteristic

C₂×C₈×D₆₁

C₂²

yes

—

#14519766

—

#14519752

—

#14519752

C₂²

yes

—

C₄×D₂₄₄

C₂²

yes

characteristic

C₄×D₂₄₄

C₂²

yes

—

#14519735

—

#14519735

C₂²

yes

—

#14519711

C₂²

yes

characteristic

#14519708

C₂²

yes

—

C₄⋊#92418

C₂²

yes

characteristic

#14519699

C₂²

yes

—

#14519697

C₂²

yes

—

#14519689

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

#107×C₆₁

C₂²

yes

—

#107×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

D₄

yes

characteristic

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

—

—

—

D₄

yes

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

—

C₂×#92418

C₂³

yes

—

D₄₈₈

—

C₈×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

C₂×D₄

yes

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#41

yes

—

#92418

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#135

yes

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

D₆₁

—

#555

—

#500

D₆₁

yes

characteristic, complemented

#482

—

#476

—

#466

—

C₄×D₈

—

C₈×D₄

—

#427

—

C₆₁

#3341

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₆₁

yes

—

100

#124

—

122

—

101

#121

—

102

#121

—

122

—

103

C₄×D₄

—

104

C₄×D₄

—

105

#117

—

106

#117

—

122

—

107

#107

—

108

#107

C₂×D₆₁

yes

—

109

#107

C₂×D₆₁

yes

—

110

#106

—

111

C₄⋊C₈

—

112

#102

—

113

#102

—

114

C₂²⋊C₈

—

115

C₄×C₈

C₂×D₆₁

yes

characteristic

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

122

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

—

122

D₈

—

122

—

123

C₂×C₈

—

124

C₂×C₈

—

122

—

125

C₂×C₈

C₂²×D₆₁

yes

characteristic

126

C₂×C₈

C₂²×D₆₁

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

C₂²×D₆₁

yes

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

122

—

133

C₂²⋊C₄

—

122

—

134

C₂²⋊C₄

—

122

—

135

C₄²

—

122

—

136

C₄²

C₂²×D₆₁

yes

characteristic

137

C₂³

—

138

C₂³

—

122

—

139

Q₈

—

140

D₄

—

122

—

141

D₄

—

122

—

142

D₄

—

122

—

143

D₄

—

488

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

122

—

147

C₂×C₄

—

122

—

148

C₂×C₄

—

122

—

149

C₂×C₄

—

244

—

150

C₂×C₄

D₄×D₆₁

yes

characteristic

151

C₂×C₄

D₄×D₆₁

yes

characteristic

152

C₂×C₄

C₂³×D₆₁

yes

characteristic

153

C₈

—

154

C₈

—

122

—

155

C₈

#11758226

yes

—

156

C₂²

—

122

—

157

C₂²

—

122

—

158

C₂²

—

122

—

159

C₂²

—

244

—

160

C₂²

—

244

—

161

C₂²

—

244

—

162

C₂²

C₂×D₄×D₆₁

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

122

—

166

C₄

—

244

—

167

C₄

#14519800

yes

—

168

C₄

C₂×#11758226

yes

characteristic

169

C₄

#14519825

yes

characteristic

170

C₂

—

488

—

171

C₂

—

244

—

172

C₂

#16840415

yes

characteristic

173

C₂

C₂×#14519800

yes

characteristic

174

C₂

#16840563

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#500	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#500	D₆₁
hypocenter	C₆₁	#3341
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#500

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #16840478	C₂	semidirect
2. #16840089	C₂	semidirect
3. #16840049	C₂	semidirect
4. #16839999	C₂	semidirect
5. C₄×D₄₈₈	C₂	semidirect
6. C₈×D₂₄₄	C₂	semidirect
7. #16839799	C₂	semidirect
8. C₆₁×#500	C₂	semidirect
89. #500	D₆₁	semidirect
96. C₆₁	#3341	semidirect

H ≅ G/N

type

1. #16840478

C₂

semidirect

2. #16840089

C₂

semidirect

3. #16840049

C₂

semidirect

4. #16839999

C₂

semidirect

5. C₄×D₄₈₈

C₂

semidirect

6. C₈×D₂₄₄

C₂

semidirect

7. #16839799

C₂

semidirect

8. C₆₁×#500

C₂

semidirect

89. #500

D₆₁

semidirect

96. C₆₁

#3341

semidirect

Subgroups — #21458241

Conjugacy classes of subgroups

Complements