Group #21446357 — subgroups

Subgroups — #21446357

Order 7808.

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

normal subgroups only

This group has 314 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	#16840240	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839779	C₂	1	2	yes	maximal
3	C₂×#14519696	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839518	C₂	1	2	yes	maximal, complemented
5	C₆₁×#536	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758194	C₂²	1	2	yes	—
7	Q₈×Dic₆₁	—	2	4	no	—
8	#14519736	—	2	2	no	—
9	#14519736	C₂²	1	2	yes	—
10	#14519704	—	2	4	no	—
11	#14519696	—	2	2	no	—
12	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₆₁	C₄	1	2	yes	complemented
15	#124×C₆₁	C₄	1	2	yes	complemented
16	#123×C₆₁	—	2	2	no	—
17	C₂²⋊Q₈×C₆₁	—	2	2	no	—
18	#117×C₆₁	C₂²	1	1	yes	characteristic
19	#41×C₆₁	—	2	2	no	—
20	#41×C₆₁	—	2	2	no	—
21	C₂×Q₈×C₆₁	—	2	2	no	—
22	C₂×Q₈×C₆₁	—	2	2	no	—
23	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₆₁	D₄	1	2	yes	—
26	C₂×D₄×C₆₁	—	2	1	no	—
27	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	—	2	1	no	—
29	C₂²×C₄×C₆₁	D₄	1	2	yes	—
30	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₆₁	—	2	4	no	—
32	C₄⋊C₄×C₆₁	—	2	1	no	—
33	C₂²⋊C₄×C₆₁	—	2	1	no	—
34	C₂²⋊C₄×C₆₁	—	2	4	no	—
35	C₄²×C₆₁	C₂×C₄	1	2	yes	—
36	C₂²⋊Dic₆₁	—	2	2	no	—
37	C₄⋊Dic₆₁	—	2	4	no	—
38	C₄⋊Dic₆₁	—	2	2	no	—
39	C₄×Dic₆₁	—	2	4	no	—
40	C₄×Dic₆₁	D₄	1	4	yes	—
41	#11758194	—	2	2	no	—
42	#11758194	—	2	2	no	—
43	C₂×#92418	—	2	2	no	—
44	C₂³×C₆₁	—	2	1	no	—
45	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₆₁	—	2	4	no	—
47	Q₈×C₆₁	—	2	4	no	—
48	Q₈×C₆₁	—	2	2	no	—
49	D₄×C₆₁	—	2	2	no	—
50	D₄×C₆₁	—	4	1	no	—
51	D₄×C₆₁	—	4	1	no	—
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	—	2	2	no	—
54	C₂×C₄×C₆₁	—	2	2	no	—
55	C₂×C₄×C₆₁	—	2	1	no	—
56	C₂×C₄×C₆₁	—	4	2	no	—
57	C₂×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#41	1	2	yes	—
62	C₂×Dic₆₁	—	2	4	no	—
63	#92418	—	4	2	no	—
64	C₂²×C₆₁	—	4	1	no	—
65	C₂²×C₆₁	—	4	2	no	—
66	C₂²×C₆₁	—	4	1	no	—
67	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₆₁	#120	1	1	yes	characteristic
69	C₂²×C₆₁	#123	1	1	yes	characteristic
70	C₄×C₆₁	—	2	4	no	—
71	C₄×C₆₁	—	4	2	no	—
72	C₄×C₆₁	—	4	1	no	—
73	C₄×C₆₁	C₄×D₄	1	2	yes	—
74	C₄×C₆₁	#121	1	2	yes	—
75	Dic₆₁	—	4	4	no	—
76	#1912	—	61	1	no	maximal, Sylow
77	C₂×C₆₁	—	2	1	no	—
78	C₂×C₆₁	—	2	1	no	—
79	C₂×C₆₁	—	2	1	no	—
80	C₂×C₆₁	—	8	1	no	—
81	C₂×C₆₁	#386	1	1	yes	characteristic
82	#536	D₆₁	1	1	yes	characteristic
83	#519	—	61	1	no	—
84	#421	—	61	2	no	—
85	C₂×#101	—	61	1	no	—
86	H(3,ℤ/4)	—	61	2	no	—
87	C₆₁	#1912	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
89	C₂²×Q₈	C₂×D₆₁	1	1	yes	characteristic
90	C₂×M₁₆	—	61	2	no	—
91	C₄⋊Q₈	Dic₆₁	1	2	yes	complemented
92	#124	Dic₆₁	1	2	yes	complemented
93	#123	—	2	2	no	—
94	C₂²⋊Q₈	—	2	2	no	—
95	C₄×Q₈	—	122	4	no	—
96	#117	—	61	2	no	—
97	#117	—	122	2	no	—
98	#117	C₂×D₆₁	1	1	yes	characteristic
99	#104	—	122	4	no	—
100	#101	—	122	2	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
106	C₂×Q₈	C₂×Dic₆₁	1	2	yes	—
107	C₂×Q₈	C₂²⋊D₆₁	1	2	yes	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	C₂×Dic₆₁	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
112	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
113	M₁₆	—	122	2	no	—
114	M₁₆	—	122	2	no	—
115	C₂×C₈	—	122	2	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	122	4	no	—
119	C₄⋊C₄	—	122	2	no	—
120	C₂²⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	2	4	no	—
122	C₂²⋊C₄	—	122	2	no	—
123	C₄²	—	61	4	no	—
124	C₄²	—	122	4	no	—
125	C₄²	C₂×Dic₆₁	1	2	yes	—
126	C₂³	—	2	1	no	—
127	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	Q₈	—	2	2	no	—
131	D₄	—	2	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	122	4	no	—
140	C₂×C₄	C₂²⋊Dic₆₁	1	4	yes	—
141	C₂×C₄	D₄×D₆₁	1	2	yes	—
142	C₂×C₄	#11758224	1	2	yes	—
143	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
145	C₈	—	244	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	—	4	1	no	—
149	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
150	C₂²	#14519745	1	1	yes	characteristic
151	C₂²	#14519758	1	1	yes	characteristic
152	C₄	—	2	4	no	—
153	C₄	—	4	2	no	—
154	C₄	—	4	1	no	—
155	C₄	—	244	4	no	—
156	C₄	D₄×Dic₆₁	1	2	yes	—
157	C₄	#14519749	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	8	1	no	—
162	C₂	#16839642	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840240

C₂

yes

maximal, characteristic, complemented

#16839779

C₂

yes

maximal

C₂×#14519696

C₂

yes

maximal, characteristic, complemented

#16839518

C₂

yes

maximal, complemented

C₆₁×#536

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

Q₈×Dic₆₁

—

#14519736

—

#14519736

C₂²

yes

—

#14519704

—

#14519696

—

C₂×#41×C₆₁

C₂²

yes

characteristic

C₂²×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₄

yes

complemented

#124×C₆₁

C₄

yes

complemented

#123×C₆₁

—

C₂²⋊Q₈×C₆₁

—

#117×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×Q₈×C₆₁

D₄

yes

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

—

#11758194

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

#121

yes

—

Dic₆₁

—

#1912

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#386

yes

characteristic

#536

D₆₁

yes

—

—

—

—

C₆₁

#1912

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₆₁

yes

characteristic

C₂²×Q₈

C₂×D₆₁

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

Dic₆₁

yes

complemented

#124

Dic₆₁

yes

complemented

#123

—

C₂²⋊Q₈

—

C₄×Q₈

—

122

—

#117

—

#117

—

122

—

#117

C₂×D₆₁

yes

characteristic

#104

—

122

—

100

#101

—

122

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₂×Dic₆₁

yes

characteristic

106

C₂×Q₈

C₂×Dic₆₁

yes

—

107

C₂×Q₈

C₂²⋊D₆₁

yes

—

108

C₂×D₄

—

109

C₂×D₄

C₂×Dic₆₁

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₆₁

yes

—

112

C₂²×C₄

C₂²×D₆₁

yes

characteristic

113

M₁₆

—

122

—

114

M₁₆

—

122

—

115

C₂×C₈

—

122

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

122

—

119

C₄⋊C₄

—

122

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

122

—

123

C₄²

—

124

C₄²

—

122

—

125

C₄²

C₂×Dic₆₁

yes

—

126

C₂³

—

127

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

122

—

140

C₂×C₄

C₂²⋊Dic₆₁

yes

—

141

C₂×C₄

D₄×D₆₁

yes

—

142

C₂×C₄

#11758224

yes

—

143

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

145

C₈

—

244

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊Dic₆₁

yes

characteristic

150

C₂²

#14519745

yes

characteristic

151

C₂²

#14519758

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

244

—

156

C₄

D₄×Dic₆₁

yes

—

157

C₄

#14519749

yes

—

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

—

162

C₂

#16839642

yes

characteristic

	subgroup	G/H
center	C₂	#16839642
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#536	C₂
socle	C₂×C₆₁	#386
hypercenter	#536	D₆₁
hypocenter	C₆₁	#1912
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#16839642

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#536

C₂

C₂×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
1. #16840240	C₂	semidirect
3. C₂×#14519696	C₂	semidirect
4. #16839518	C₂	semidirect
14. C₄⋊Q₈×C₆₁	C₄	semidirect
15. #124×C₆₁	C₄	semidirect
87. C₆₁	#1912	semidirect
91. C₄⋊Q₈	Dic₆₁	semidirect
92. #124	Dic₆₁	semidirect

H ≅ G/N

type

1. #16840240

C₂

semidirect

3. C₂×#14519696

C₂

semidirect

4. #16839518

C₂

semidirect

14. C₄⋊Q₈×C₆₁

C₄

semidirect

15. #124×C₆₁

C₄

semidirect

87. C₆₁

#1912

semidirect

91. C₄⋊Q₈

Dic₆₁

semidirect

92. #124

Dic₆₁

semidirect

Subgroups — #21446357

Conjugacy classes of subgroups

Complements