Group #21458832 — subgroups

Subgroups — #21458832

Order 7808.

384 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16840497	C₂	1	2	yes	maximal, complemented
2	#16840146	C₂	1	2	yes	maximal, complemented
3	#16840139	C₂	1	1	yes	maximal, characteristic
4	#16839978	C₂	1	4	yes	maximal, complemented
5	#16839963	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₄₈₈	C₂	1	2	yes	maximal, complemented
8	C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
10	C₂×#11758190	C₂²	1	4	yes	—
11	C₂×D₄₈₈	C₂²	1	2	yes	—
12	C₂×D₄₈₈	C₂²	1	2	yes	—
13	#14519774	C₂²	1	2	yes	—
14	#14519766	—	2	2	no	—
15	#14519765	C₂²	1	4	yes	—
16	#14519762	—	2	4	no	—
17	C₄×Dic₁₂₂	C₂²	1	2	yes	—
18	C₄×D₂₄₄	C₂²	1	4	yes	—
19	#14519739	—	2	2	no	—
20	#14519711	C₂²	1	1	yes	characteristic
21	#14519702	C₂²	1	2	yes	—
22	#14519699	C₂²	1	4	yes	—
23	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
26	C₄×C₈×C₆₁	C₂²	1	2	yes	—
27	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
29	C₂×D₂₄₄	—	2	2	no	—
30	C₂×D₂₄₄	—	2	2	no	—
31	C₂×D₂₄₄	C₂³	1	4	yes	—
32	C₂×C₄×D₆₁	—	2	4	no	—
33	C₂×Dic₁₂₂	C₂³	1	2	yes	—
34	C₂×C₈×C₆₁	C₂³	1	2	yes	—
35	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
39	C₄²×C₆₁	C₂³	1	2	yes	—
40	C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	#11758198	—	2	4	no	—
42	#11758198	—	2	4	no	—
43	#11758198	—	2	4	no	—
44	C₄⋊Dic₆₁	C₂³	1	2	yes	—
45	#11758196	—	2	2	no	—
46	C₄×Dic₆₁	—	2	2	no	—
47	Dic₂₄₄	—	2	2	no	—
48	D₄₈₈	—	2	4	no	—
49	D₄₈₈	—	4	2	no	—
50	#11758190	—	4	4	no	—
51	C₂²×D₆₁	—	2	4	no	—
52	Q₈×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₆₁	—	2	2	no	—
59	C₂×Dic₆₁	—	2	2	no	—
60	D₂₄₄	—	2	4	no	—
61	D₂₄₄	—	2	4	no	—
62	D₂₄₄	—	4	4	no	—
63	D₂₄₄	—	8	2	no	—
64	C₄×D₆₁	—	8	4	no	—
65	Dic₁₂₂	—	2	2	no	—
66	Dic₁₂₂	—	2	2	no	—
67	C₈×C₆₁	—	2	2	no	—
68	C₈×C₆₁	—	2	1	no	—
69	C₈×C₆₁	#41	1	2	yes	—
70	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
71	C₂×D₆₁	—	4	4	no	—
72	C₂×D₆₁	—	4	4	no	—
73	C₂×D₆₁	—	4	4	no	—
74	C₄×C₆₁	—	2	2	no	—
75	C₄×C₆₁	—	2	2	no	—
76	C₄×C₆₁	—	2	1	no	—
77	C₄×C₆₁	#135	1	2	yes	—
78	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
79	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
80	Dic₆₁	—	4	2	no	—
81	Dic₆₁	—	4	2	no	—
82	#3396	—	61	1	no	maximal, Sylow
83	C₂×C₆₁	#550	1	1	yes	characteristic
84	C₂×C₆₁	C₂×#135	1	1	yes	characteristic
85	C₂×C₆₁	#576	1	1	yes	characteristic
86	D₆₁	—	8	4	no	—
87	#559	—	61	2	no	—
88	#495	—	61	2	no	—
89	#493	—	61	1	no	—
90	#463	—	61	4	no	—
91	#460	—	61	2	no	—
92	C₈×Q₈	D₆₁	1	1	yes	characteristic, complemented
93	C₄×Q₁₆	—	61	1	no	—
94	C₄×D₈	—	61	2	no	—
95	C₆₁	#3396	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₁₆	—	61	1	no	—
97	C₂×SD₁₆	—	61	4	no	—
98	C₂×D₈	—	61	2	no	—
99	C₂×D₈	—	61	2	no	—
100	#127	—	61	2	no	—
101	#124	—	61	4	no	—
102	#124	—	122	2	no	—
103	#123	—	122	4	no	—
104	C₄×Q₈	—	61	2	no	—
105	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
106	C₄×D₄	—	61	4	no	—
107	C₄×D₄	—	122	2	no	—
108	#107	—	61	1	no	—
109	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
110	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
111	#103	—	61	2	no	—
112	#102	—	61	4	no	—
113	C₄×C₈	C₂×D₆₁	1	2	yes	—
114	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
115	C₂×Q₈	—	61	2	no	—
116	C₂×Q₈	D₂₄₄	1	1	yes	characteristic
117	C₂×D₄	—	61	4	no	—
118	C₂×D₄	—	122	2	no	—
119	C₂×D₄	—	122	2	no	—
120	C₂²×C₄	—	122	4	no	—
121	Q₁₆	—	122	2	no	—
122	SD₁₆	—	244	4	no	—
123	D₈	—	122	4	no	—
124	D₈	—	244	2	no	—
125	C₂×C₈	C₂²×D₆₁	1	2	yes	—
126	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
127	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
128	C₄⋊C₄	—	61	2	no	—
129	C₄⋊C₄	—	122	2	no	—
130	C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
131	C₄⋊C₄	D₂₄₄	1	2	yes	—
132	C₂²⋊C₄	—	122	4	no	—
133	C₂²⋊C₄	—	122	4	no	—
134	C₂²⋊C₄	—	122	4	no	—
135	C₄²	—	122	2	no	—
136	C₄²	C₂²×D₆₁	1	2	yes	—
137	C₄²	C₂²×D₆₁	1	1	yes	characteristic
138	C₂³	—	122	4	no	—
139	Q₈	—	2	2	no	—
140	Q₈	—	122	2	no	—
141	Q₈	—	122	2	no	—
142	D₄	—	122	4	no	—
143	D₄	—	122	4	no	—
144	D₄	—	488	2	no	—
145	D₄	—	244	4	no	—
146	C₂×C₄	—	122	2	no	—
147	C₂×C₄	—	122	2	no	—
148	C₂×C₄	—	488	4	no	—
149	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
150	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
151	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
152	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	2	no	—
156	C₈	#11758226	1	2	yes	—
157	C₂²	—	244	4	no	—
158	C₂²	—	244	4	no	—
159	C₂²	—	244	4	no	—
160	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	1	no	—
164	C₄	—	244	2	no	—
165	C₄	—	244	2	no	—
166	C₄	#14519799	1	2	yes	—
167	C₄	C₂×#11758226	1	1	yes	characteristic
168	C₄	#14519823	1	1	yes	characteristic
169	C₂	—	488	4	no	—
170	C₂	#16840434	1	1	yes	characteristic
171	C₂	C₂×#14519799	1	1	yes	characteristic
172	C₂	#16840567	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840497

C₂

yes

maximal, complemented

#16840146

C₂

yes

maximal, complemented

#16840139

C₂

yes

maximal, characteristic

#16839978

C₂

yes

maximal, complemented

#16839963

C₂

yes

maximal, complemented

C₄×Dic₂₄₄

C₂

yes

maximal, characteristic, complemented

C₄×D₄₈₈

C₂

yes

maximal, complemented

C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₄₄

C₂²

yes

characteristic

C₂×#11758190

C₂²

yes

—

C₂×D₄₈₈

C₂²

yes

—

C₂×D₄₈₈

C₂²

yes

—

#14519774

C₂²

yes

—

#14519766

—

#14519765

C₂²

yes

—

#14519762

—

C₄×Dic₁₂₂

C₂²

yes

—

C₄×D₂₄₄

C₂²

yes

—

#14519739

—

#14519711

C₂²

yes

characteristic

#14519702

C₂²

yes

—

#14519699

C₂²

yes

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

—

—

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

—

Dic₂₄₄

—

D₄₈₈

—

D₄₈₈

—

#11758190

—

C₂²×D₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#41

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#135

yes

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

H(5,𝔽₂)

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

—

—

—

—

—

C₈×Q₈

D₆₁

yes

characteristic, complemented

C₄×Q₁₆

—

C₄×D₈

—

C₆₁

#3396

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

100

#127

—

101

#124

—

102

#124

—

122

—

103

#123

—

122

—

104

C₄×Q₈

—

105

C₄×Q₈

C₂×D₆₁

yes

characteristic

106

C₄×D₄

—

107

C₄×D₄

—

122

—

108

#107

—

109

C₄⋊C₈

C₂×D₆₁

yes

characteristic

110

C₄⋊C₈

C₂×D₆₁

yes

—

111

#103

—

112

#102

—

113

C₄×C₈

C₂×D₆₁

yes

—

114

C₄×C₈

C₂×D₆₁

yes

characteristic

115

C₂×Q₈

—

116

C₂×Q₈

D₂₄₄

yes

characteristic

117

C₂×D₄

—

118

C₂×D₄

—

122

—

119

C₂×D₄

—

122

—

120

C₂²×C₄

—

122

—

121

Q₁₆

—

122

—

122

SD₁₆

—

244

—

123

D₈

—

122

—

124

D₈

—

244

—

125

C₂×C₈

C₂²×D₆₁

yes

—

126

C₂×C₈

C₂²×D₆₁

yes

characteristic

127

C₂×C₈

C₂²×D₆₁

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

122

—

130

C₄⋊C₄

D₂₄₄

yes

characteristic

131

C₄⋊C₄

D₂₄₄

yes

—

132

C₂²⋊C₄

—

122

—

133

C₂²⋊C₄

—

122

—

134

C₂²⋊C₄

—

122

—

135

C₄²

—

122

—

136

C₄²

C₂²×D₆₁

yes

—

137

C₄²

C₂²×D₆₁

yes

characteristic

138

C₂³

—

122

—

139

Q₈

—

140

Q₈

—

122

—

141

Q₈

—

122

—

142

D₄

—

122

—

143

D₄

—

122

—

144

D₄

—

488

—

145

D₄

—

244

—

146

C₂×C₄

—

122

—

147

C₂×C₄

—

122

—

148

C₂×C₄

—

488

—

149

C₂×C₄

C₂×D₂₄₄

yes

—

150

C₂×C₄

C₂×D₂₄₄

yes

—

151

C₂×C₄

C₂×D₂₄₄

yes

characteristic

152

C₂×C₄

C₂×D₂₄₄

yes

characteristic

153

C₂×C₄

C₂³×D₆₁

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

#11758226

yes

—

157

C₂²

—

244

—

158

C₂²

—

244

—

159

C₂²

—

244

—

160

C₂²

C₂²×D₂₄₄

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

244

—

165

C₄

—

244

—

166

C₄

#14519799

yes

—

167

C₄

C₂×#11758226

yes

characteristic

168

C₄

#14519823

yes

characteristic

169

C₂

—

488

—

170

C₂

#16840434

yes

characteristic

171

C₂

C₂×#14519799

yes

characteristic

172

C₂

#16840567

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₈×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₈×Q₈	D₆₁
hypocenter	C₆₁	#3396
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₈×Q₈×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

C₈×Q₈

D₆₁

hypocenter

C₆₁

#3396

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #16840497	C₂	semidirect
2. #16840146	C₂	semidirect
4. #16839978	C₂	semidirect
5. #16839963	C₂	semidirect
6. C₄×Dic₂₄₄	C₂	semidirect
7. C₄×D₄₈₈	C₂	semidirect
8. C₈×Q₈×C₆₁	C₂	semidirect
92. C₈×Q₈	D₆₁	semidirect
95. C₆₁	#3396	semidirect

H ≅ G/N

type

1. #16840497

C₂

semidirect

2. #16840146

C₂

semidirect

4. #16839978

C₂

semidirect

5. #16839963

C₂

semidirect

6. C₄×Dic₂₄₄

C₂

semidirect

7. C₄×D₄₈₈

C₂

semidirect

8. C₈×Q₈×C₆₁

C₂

semidirect

92. C₈×Q₈

D₆₁

semidirect

95. C₆₁

#3396

semidirect

Subgroups — #21458832

Conjugacy classes of subgroups

Complements