Group #21446367 — subgroups

Subgroups — #21446367

Order 7808.

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

normal subgroups only

This group has 340 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	#16840529	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839746	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14519694	C₂	1	2	yes	maximal, complemented
4	#16839532	C₂	1	2	yes	maximal, complemented
5	C₆₁×#535	C₂	1	1	yes	maximal, characteristic
6	#14519784	—	2	4	no	—
7	C₂×#11758194	—	2	2	no	—
8	C₂×#11758194	C₂²	1	2	yes	—
9	C₂×#11758194	C₂²	1	2	yes	—
10	#14519696	—	2	1	no	—
11	#14519694	—	2	1	no	—
12	#14519694	—	2	4	no	—
13	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₆₁	C₂²	1	1	yes	characteristic
15	#127×C₆₁	—	2	1	no	—
16	#124×C₆₁	—	2	1	no	—
17	#121×C₆₁	—	2	2	no	—
18	#120×C₆₁	C₄	1	4	yes	—
19	#117×C₆₁	C₂²	1	1	yes	characteristic
20	C₂⁴×C₆₁	C₂×C₄	1	2	yes	—
21	#41×C₆₁	—	2	2	no	—
22	#41×C₆₁	—	2	2	no	—
23	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
24	C₂×D₄×C₆₁	—	2	4	no	—
25	C₂×D₄×C₆₁	—	2	1	no	—
26	C₂×D₄×C₆₁	—	2	1	no	—
27	C₂×D₄×C₆₁	—	2	2	no	—
28	C₂×D₄×C₆₁	—	2	2	no	—
29	C₂×D₄×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	D₄	1	2	yes	—
32	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₆₁	—	2	1	no	—
34	C₂²⋊C₄×C₆₁	—	2	4	no	—
35	C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
36	C₄²×C₆₁	—	2	1	no	—
37	#11758194	—	2	4	no	—
38	#11758194	—	2	2	no	—
39	#11758194	—	2	2	no	—
40	#11758194	—	2	2	no	—
41	#11758194	D₄	1	4	yes	complemented
42	C₂×#92418	—	2	4	no	—
43	C₂×#92418	—	2	2	no	—
44	C₂×#92418	—	2	2	no	—
45	C₂³×C₆₁	—	2	2	no	—
46	C₂³×C₆₁	—	2	4	no	—
47	C₂³×C₆₁	—	8	2	no	—
48	C₂³×C₆₁	C₂²⋊C₄	1	2	yes	—
49	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	Q₈×C₆₁	—	2	2	no	—
51	D₄×C₆₁	—	2	4	no	—
52	D₄×C₆₁	—	2	2	no	—
53	D₄×C₆₁	—	4	1	no	—
54	D₄×C₆₁	—	4	2	no	—
55	D₄×C₆₁	—	4	1	no	—
56	D₄×C₆₁	—	8	4	no	—
57	C₂×C₄×C₆₁	—	2	2	no	—
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#41	1	2	yes	—
65	#92418	—	2	4	no	—
66	#92418	—	4	2	no	—
67	C₂²×C₆₁	—	2	2	no	—
68	C₂²×C₆₁	—	4	1	no	—
69	C₂²×C₆₁	—	4	2	no	—
70	C₂²×C₆₁	—	4	4	no	—
71	C₂²×C₆₁	—	4	2	no	—
72	C₂²×C₆₁	—	8	4	no	—
73	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₆₁	C₄×D₄	1	2	yes	—
75	C₄×C₆₁	—	2	2	no	—
76	C₄×C₆₁	—	4	2	no	—
77	C₄×C₆₁	#120	1	1	yes	characteristic
78	C₄×C₆₁	#121	1	2	yes	—
79	C₄×C₆₁	#123	1	1	yes	characteristic
80	#1914	—	61	1	no	maximal, Sylow
81	C₂×C₆₁	—	2	1	no	—
82	C₂×C₆₁	—	2	2	no	—
83	C₂×C₆₁	—	4	2	no	—
84	C₂×C₆₁	—	8	2	no	—
85	C₂×C₆₁	#386	1	1	yes	characteristic
86	#568	—	61	1	no	—
87	#535	D₆₁	1	1	yes	characteristic
88	#413	—	61	1	no	—
89	C₂×#100	—	61	2	no	—
90	#344	—	61	2	no	—
91	C₆₁	#1914	1	1	yes	characteristic, complemented, Sylow
92	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
93	C₂²×D₄	C₂×D₆₁	1	1	yes	characteristic
94	#131	—	122	4	no	—
95	C₂×M₁₆	—	61	2	no	—
96	C₂×M₁₆	—	61	2	no	—
97	C₂×M₁₆	—	122	2	no	—
98	#127	—	2	1	no	—
99	#124	—	2	1	no	—
100	#121	—	2	2	no	—
101	#120	Dic₆₁	1	4	yes	—
102	#117	C₂×D₆₁	1	1	yes	characteristic
103	#101	—	122	1	no	—
104	#100	—	122	4	no	—
105	#100	—	122	1	no	—
106	C₂⁴	C₂×Dic₆₁	1	2	yes	—
107	#41	—	2	2	no	—
108	#41	—	2	2	no	—
109	C₂×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
110	C₂×D₄	—	2	4	no	—
111	C₂×D₄	—	2	2	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	2	1	no	—
115	C₂×D₄	C₂×Dic₆₁	1	2	yes	—
116	C₂×D₄	C₂²⋊D₆₁	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
118	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
119	M₁₆	—	61	4	no	—
120	M₁₆	—	122	4	no	—
121	M₁₆	—	122	2	no	—
122	M₁₆	—	122	2	no	—
123	M₁₆	—	122	2	no	—
124	C₂×C₈	—	122	4	no	—
125	C₂×C₈	—	122	2	no	—
126	C₂×C₈	—	122	2	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	2	4	no	—
129	C₂²⋊C₄	C₂×Dic₆₁	1	2	yes	—
130	C₄²	—	2	1	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	4	no	—
133	C₂³	—	8	2	no	—
134	C₂³	C₂²⋊Dic₆₁	1	2	yes	—
135	C₂³	C₂²×Dic₆₁	1	1	yes	characteristic
136	Q₈	—	2	2	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	2	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	2	no	—
141	D₄	—	4	1	no	—
142	D₄	—	8	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	C₂²⋊Dic₆₁	1	2	yes	—
147	C₂×C₄	D₄×D₆₁	1	2	yes	—
148	C₂×C₄	#11758224	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
151	C₈	—	122	4	no	—
152	C₈	—	244	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	8	4	no	—
159	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
160	C₂²	D₄×Dic₆₁	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	—	4	2	no	—
163	C₄	#14519745	1	1	yes	characteristic
164	C₄	#14519749	1	2	yes	—
165	C₄	#14519758	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	—	4	2	no	—
169	C₂	—	8	2	no	—
170	C₂	#16839642	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840529

C₂

yes

maximal, characteristic, complemented

#16839746

C₂

yes

maximal, characteristic, complemented

C₂×#14519694

C₂

yes

maximal, complemented

#16839532

C₂

yes

maximal, complemented

C₆₁×#535

C₂

yes

maximal, characteristic

#14519784

—

C₂×#11758194

—

C₂×#11758194

C₂²

yes

—

C₂×#11758194

C₂²

yes

—

#14519696

—

#14519694

—

#14519694

—

C₂×#41×C₆₁

C₂²

yes

characteristic

C₂²×D₄×C₆₁

C₂²

yes

characteristic

#127×C₆₁

—

#124×C₆₁

—

#121×C₆₁

—

#120×C₆₁

C₄

yes

—

#117×C₆₁

C₂²

yes

characteristic

C₂⁴×C₆₁

C₂×C₄

yes

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂×D₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

—

—

—

—

—

D₄

yes

—

—

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂²⋊C₄

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#120

yes

characteristic

C₄×C₆₁

#121

yes

—

C₄×C₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

—

D₆₁

yes

—

—

—

C₆₁

#1914

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₆₁

yes

characteristic

C₂²×D₄

C₂×D₆₁

yes

characteristic

#131

—

122

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

122

—

#127

—

#124

—

100

#121

—

101

#120

Dic₆₁

yes

—

102

#117

C₂×D₆₁

yes

characteristic

103

#101

—

122

—

104

#100

—

122

—

105

#100

—

122

—

106

C₂⁴

C₂×Dic₆₁

yes

—

107

#41

—

108

#41

—

109

C₂×Q₈

C₂²⋊D₆₁

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

C₂×Dic₆₁

yes

—

116

C₂×D₄

C₂²⋊D₆₁

yes

characteristic

117

C₂²×C₄

C₂²⋊D₆₁

yes

—

118

C₂²×C₄

C₂²×D₆₁

yes

characteristic

119

M₁₆

—

120

M₁₆

—

122

—

121

M₁₆

—

122

—

122

M₁₆

—

122

—

123

M₁₆

—

122

—

124

C₂×C₈

—

122

—

125

C₂×C₈

—

122

—

126

C₂×C₈

—

122

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

C₂×Dic₆₁

yes

—

130

C₄²

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂²⋊Dic₆₁

yes

—

135

C₂³

C₂²×Dic₆₁

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

141

D₄

—

142

D₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊Dic₆₁

yes

—

147

C₂×C₄

D₄×D₆₁

yes

—

148

C₂×C₄

#11758224

yes

—

149

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

151

C₈

—

122

—

152

C₈

—

244

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

C₂×C₂²⋊Dic₆₁

yes

characteristic

160

C₂²

D₄×Dic₆₁

yes

—

161

C₄

—

162

C₄

—

163

C₄

#14519745

yes

characteristic

164

C₄

#14519749

yes

—

165

C₄

#14519758

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

170

C₂

#16839642

yes

characteristic

	subgroup	G/H
center	C₂	#16839642
commutator	C₂³×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#535	C₂
socle	C₂×C₆₁	#386
hypercenter	#535	D₆₁
hypocenter	C₆₁	#1914
LCS	G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

subgroup

G/H

center

C₂

#16839642

commutator

C₂³×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#535

C₂

C₂×C₆₁

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

N	H ≅ G/N	type
1. #16840529	C₂	semidirect
2. #16839746	C₂	semidirect
3. C₂×#14519694	C₂	semidirect
4. #16839532	C₂	semidirect
41. #11758194	D₄	semidirect
91. C₆₁	#1914	semidirect

H ≅ G/N

type

1. #16840529

C₂

semidirect

2. #16839746

C₂

semidirect

3. C₂×#14519694

C₂

semidirect

4. #16839532

C₂

semidirect

41. #11758194

D₄

semidirect

91. C₆₁

#1914

semidirect

Subgroups — #21446367

Conjugacy classes of subgroups

Complements