Group #21446580 — subgroups

Subgroups — #21446580

Order 7808.

256 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 121 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×#14519708	C₂	1	2	yes	maximal
2	C₂×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	#16839524	C₂	1	2	yes	maximal
5	C₆₁×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×#92418	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₆₁	—	2	1	no	—
10	#14519708	—	2	4	no	—
11	C₄⋊#92418	C₄	1	4	yes	complemented
12	#14519684	—	2	1	no	—
13	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
15	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	—	2	2	no	—
18	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂²×Dic₆₁	—	2	1	no	—
22	C₂²×Dic₆₁	—	2	1	no	—
23	C₄⋊C₄×C₆₁	—	2	2	no	—
24	C₄⋊C₄×C₆₁	—	2	2	no	—
25	C₄⋊C₄×C₆₁	—	4	2	no	—
26	C₄⋊C₄×C₆₁	—	4	2	no	—
27	C₄²×C₆₁	C₂×C₄	1	2	yes	—
28	C₄²×C₆₁	Q₈	1	2	yes	—
29	C₄⋊Dic₆₁	—	2	1	no	—
30	C₄⋊Dic₆₁	—	2	1	no	—
31	C₄⋊Dic₆₁	D₄	1	2	yes	—
32	C₄⋊Dic₆₁	Q₈	1	2	yes	—
33	C₄×Dic₆₁	—	4	2	no	—
34	C₂×#92418	—	2	2	no	—
35	C₂×#92418	—	2	2	no	—
36	C₂×#92418	C₂×C₄	1	4	yes	—
37	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	—	2	1	no	—
39	C₂×C₄×C₆₁	—	2	1	no	—
40	C₂×C₄×C₆₁	—	4	2	no	—
41	C₂×C₄×C₆₁	—	4	2	no	—
42	C₂×C₄×C₆₁	—	4	2	no	—
43	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
46	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
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×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
53	C₂×Dic₆₁	—	2	2	no	—
54	C₂×Dic₆₁	—	2	2	no	—
55	C₂×Dic₆₁	—	4	1	no	—
56	C₂×Dic₆₁	—	4	1	no	—
57	C₂×Dic₆₁	—	4	1	no	—
58	C₂×Dic₆₁	—	4	1	no	—
59	#92418	—	2	4	no	—
60	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₆₁	#121	1	1	yes	characteristic
62	C₂²×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₆₁	#137	1	1	yes	characteristic
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	—	8	2	no	—
69	C₄×C₆₁	#106	1	4	yes	complemented
70	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
71	C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
72	C₄×C₆₁	#125	1	1	yes	characteristic
73	C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
74	Dic₆₁	—	4	2	no	—
75	Dic₆₁	—	4	2	no	—
76	#1948	—	61	1	no	maximal, Sylow
77	C₂×C₆₁	#389	1	1	yes	characteristic
78	C₂×C₆₁	C₂×#106	1	1	yes	characteristic
79	C₂×C₆₁	#428	1	1	yes	characteristic
80	C₂×C₆₁	#460	1	1	yes	characteristic
81	C₂×C₆₁	#461	1	1	yes	characteristic
82	C₂×C₆₁	#475	1	1	yes	characteristic
83	C₂×C₆₁	#476	1	1	yes	characteristic
84	C₂×#106	—	61	2	no	—
85	C₂×C₄⋊C₈	—	61	1	no	—
86	#384	D₆₁	1	1	yes	characteristic
87	C₄×C₄⋊C₄	—	61	1	no	—
88	#340	—	61	2	no	—
89	C₆₁	#1948	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	61	2	no	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄⋊C₄	—	61	1	no	—
94	C₂×C₄⋊C₄	—	61	1	no	—
95	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
96	C₂×C₄²	—	122	1	no	—
97	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
98	#106	—	122	4	no	—
99	C₄⋊C₈	—	61	4	no	—
100	#95	—	122	1	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	122	1	no	—
103	C₂²×C₄	—	122	1	no	—
104	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
107	C₂×C₈	—	61	4	no	—
108	C₂×C₈	—	122	2	no	—
109	C₂×C₈	—	122	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	61	2	no	—
115	C₄⋊C₄	—	61	2	no	—
116	C₄⋊C₄	—	122	1	no	—
117	C₄⋊C₄	—	122	1	no	—
118	C₄²	—	244	2	no	—
119	C₄²	Dic₁₂₂	1	2	yes	—
120	C₄²	C₄×D₆₁	1	2	yes	—
121	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	122	2	no	—
128	C₂×C₄	—	122	2	no	—
129	C₂×C₄	—	244	1	no	—
130	C₂×C₄	—	244	1	no	—
131	C₂×C₄	—	244	1	no	—
132	C₂×C₄	—	244	1	no	—
133	C₂×C₄	#11758196	1	2	yes	—
134	C₂×C₄	#11758196	1	2	yes	—
135	C₂×C₄	#11758200	1	2	yes	—
136	C₂×C₄	#11758201	1	2	yes	—
137	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
139	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
140	C₂×C₄	#11758224	1	1	yes	characteristic
141	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
142	C₂×C₄	#11758226	1	1	yes	characteristic
143	C₈	—	122	4	no	—
144	C₂²	C₂×#11758196	1	1	yes	characteristic
145	C₂²	#14519748	1	1	yes	characteristic
146	C₂²	#14519753	1	1	yes	characteristic
147	C₂²	C₂×#11758201	1	1	yes	characteristic
148	C₂²	C₂×#11758200	1	1	yes	characteristic
149	C₂²	#14519803	1	1	yes	characteristic
150	C₂²	#14519808	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	244	2	no	—
154	C₄	—	244	2	no	—
155	C₄	#14519708	1	4	yes	complemented
156	C₄	#14519739	1	1	yes	characteristic
157	C₄	#14519742	1	1	yes	characteristic
158	C₄	#14519768	1	1	yes	characteristic
159	C₄	#14519776	1	1	yes	characteristic
160	C₂	#16839658	1	1	yes	characteristic
161	C₂	C₂×#14519708	1	1	yes	characteristic
162	C₂	#16839802	1	1	yes	characteristic
163	C₂	#16839959	1	1	yes	characteristic
164	C₂	#16839964	1	1	yes	characteristic
165	C₂	#16840044	1	1	yes	characteristic
166	C₂	#16840049	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519708

C₂

yes

maximal

C₂×C₄⋊#92418

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₆₁

C₂

yes

maximal, characteristic

#16839524

C₂

yes

maximal

C₆₁×#384

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

—

#14519708

—

C₄⋊#92418

C₄

yes

complemented

#14519684

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

Q₈

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

—

C₄⋊Dic₆₁

Q₈

yes

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₂×C₄

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#92418

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

#121

yes

characteristic

C₂²×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₁

C₂×SD₁₆

yes

characteristic

C₂²×C₆₁

C₂×SD₁₆

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#106

yes

complemented

C₄×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

C₄×Q₈

yes

characteristic

C₄×C₆₁

#125

yes

characteristic

C₄×C₆₁

C₄⋊Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₄⋊C₈

—

#384

D₆₁

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₆₁

#1948

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#106

—

122

—

C₄⋊C₈

—

100

#95

—

122

—

101

C₂²×C₄

—

102

C₂²×C₄

—

122

—

103

C₂²×C₄

—

122

—

104

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

105

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

106

C₂²×C₄

C₂²×D₆₁

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

122

—

109

C₂×C₈

—

122

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

122

—

117

C₄⋊C₄

—

122

—

118

C₄²

—

244

—

119

C₄²

Dic₁₂₂

yes

—

120

C₄²

C₄×D₆₁

yes

—

121

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

122

—

128

C₂×C₄

—

122

—

129

C₂×C₄

—

244

—

130

C₂×C₄

—

244

—

131

C₂×C₄

—

244

—

132

C₂×C₄

—

244

—

133

C₂×C₄

#11758196

yes

—

134

C₂×C₄

#11758196

yes

—

135

C₂×C₄

#11758200

yes

—

136

C₂×C₄

#11758201

yes

—

137

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

138

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

139

C₂×C₄

D₄×D₆₁

yes

characteristic

140

C₂×C₄

#11758224

yes

characteristic

141

C₂×C₄

Q₈×D₆₁

yes

characteristic

142

C₂×C₄

#11758226

yes

characteristic

143

C₈

—

122

—

144

C₂²

C₂×#11758196

yes

characteristic

145

C₂²

#14519748

yes

characteristic

146

C₂²

#14519753

yes

characteristic

147

C₂²

C₂×#11758201

yes

characteristic

148

C₂²

C₂×#11758200

yes

characteristic

149

C₂²

#14519803

yes

characteristic

150

C₂²

#14519808

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

244

—

154

C₄

—

244

—

155

C₄

#14519708

yes

complemented

156

C₄

#14519739

yes

characteristic

157

C₄

#14519742

yes

characteristic

158

C₄

#14519768

yes

characteristic

159

C₄

#14519776

yes

characteristic

160

C₂

#16839658

yes

characteristic

161

C₂

C₂×#14519708

yes

characteristic

162

C₂

#16839802

yes

characteristic

163

C₂

#16839959

yes

characteristic

164

C₂

#16839964

yes

characteristic

165

C₂

#16840044

yes

characteristic

166

C₂

#16840049

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#384	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	#384	D₆₁
hypocenter	C₆₁	#1948
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#384

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. C₄⋊#92418	C₄	semidirect
69. C₄×C₆₁	#106	semidirect
89. C₆₁	#1948	semidirect
155. C₄	#14519708	semidirect

H ≅ G/N

type

11. C₄⋊#92418

C₄

semidirect

69. C₄×C₆₁

#106

semidirect

89. C₆₁

#1948

semidirect

155. C₄

#14519708

semidirect

Subgroups — #21446580

Conjugacy classes of subgroups

Complements