Group #21445924 — subgroups

Subgroups — #21445924

Order 7808.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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	#41×Dic₆₁	C₂	1	1	yes	maximal, characteristic
2	#16839736	C₂	1	1	yes	maximal, characteristic
3	#16839523	C₂	1	2	yes	maximal, complemented
4	#16839519	C₂	1	2	yes	maximal, complemented
5	C₆₁×#413	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₆₁	—	2	1	no	—
7	D₄×Dic₆₁	—	2	2	no	—
8	D₄×Dic₆₁	—	2	1	no	—
9	#14519736	C₄	1	2	yes	complemented
10	#14519736	C₂²	1	2	yes	—
11	#14519736	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₆₁	C₄	1	2	yes	complemented
13	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
14	#14519692	—	2	2	no	—
15	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
17	#101×C₆₁	—	2	1	no	—
18	#100×C₆₁	—	2	1	no	—
19	#41×C₆₁	—	2	2	no	—
20	#41×C₆₁	—	2	2	no	—
21	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
22	C₂×D₄×C₆₁	D₄	1	2	yes	—
23	C₂×D₄×C₆₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₆₁	—	2	2	no	—
27	C₂²×Dic₆₁	—	2	1	no	—
28	M₁₆×C₆₁	—	2	2	no	—
29	M₁₆×C₆₁	—	2	2	no	—
30	C₂×C₈×C₆₁	—	2	2	no	—
31	C₂²⋊Dic₆₁	—	2	2	no	—
32	C₂²⋊Dic₆₁	—	2	1	no	—
33	C₂²⋊Dic₆₁	—	2	2	no	—
34	C₄⋊Dic₆₁	—	2	2	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	—	2	2	no	—
37	C₄×Dic₆₁	—	2	2	no	—
38	C₄×Dic₆₁	—	2	1	no	—
39	C₄×Dic₆₁	—	2	2	no	—
40	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₆₁	C₂×C₄	1	2	yes	—
42	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₆₁	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₆₁	—	2	2	no	—
46	D₄×C₆₁	—	2	4	no	—
47	D₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#41	1	2	yes	—
54	C₂×C₄×C₆₁	#41	1	2	yes	—
55	C₂×Dic₆₁	—	2	1	no	—
56	C₂×Dic₆₁	—	2	2	no	—
57	C₂×Dic₆₁	—	2	1	no	—
58	C₂×Dic₆₁	—	4	2	no	—
59	C₂×Dic₆₁	—	4	4	no	—
60	C₂×Dic₆₁	—	4	2	no	—
61	C₈×C₆₁	—	4	2	no	—
62	C₂²×C₆₁	—	2	2	no	—
63	C₂²×C₆₁	—	4	1	no	—
64	C₂²×C₆₁	—	4	2	no	—
65	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₆₁	#123	1	2	yes	—
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	#117	1	1	yes	characteristic
69	C₄×C₆₁	#117	1	1	yes	characteristic
70	C₄×C₆₁	#123	1	2	yes	—
71	Dic₆₁	—	2	2	no	—
72	Dic₆₁	—	4	2	no	—
73	Dic₆₁	—	4	1	no	—
74	Dic₆₁	—	8	2	no	—
75	#1827	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	—	2	1	no	—
77	C₂×C₆₁	—	2	2	no	—
78	C₂×C₆₁	—	4	2	no	—
79	C₂×C₆₁	#381	1	1	yes	characteristic
80	C₄×#41	—	61	1	no	—
81	#413	D₆₁	1	1	yes	characteristic
82	#410	—	61	1	no	—
83	#339	—	61	2	no	—
84	H(3,ℤ/4)	—	61	2	no	—
85	C₆₁	#1827	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
87	C₂×M₁₆	C₂×D₆₁	1	2	yes	—
88	C₄×Q₈	—	122	1	no	—
89	C₄×D₄	—	122	2	no	—
90	C₄×D₄	—	122	1	no	—
91	#117	—	61	2	no	—
92	#117	—	61	1	no	—
93	#117	—	61	2	no	—
94	C₂×C₄²	—	61	2	no	—
95	C₂×C₄²	—	61	1	no	—
96	#101	—	2	1	no	—
97	#100	—	2	1	no	—
98	#99	—	122	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
102	C₂×D₄	D₂₄₄	1	2	yes	—
103	C₂×D₄	C₂²⋊D₆₁	1	1	yes	characteristic
104	C₂²×C₄	—	122	2	no	—
105	C₂²×C₄	—	122	1	no	—
106	C₂²×C₄	C₄×D₆₁	1	2	yes	—
107	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₄⋊C₄	—	122	2	no	—
112	C₄⋊C₄	—	122	1	no	—
113	C₄⋊C₄	—	122	2	no	—
114	C₂²⋊C₄	—	122	2	no	—
115	C₂²⋊C₄	—	122	1	no	—
116	C₂²⋊C₄	—	122	2	no	—
117	C₄²	—	61	1	no	—
118	C₄²	—	61	2	no	—
119	C₄²	—	61	1	no	—
120	C₄²	—	122	2	no	—
121	C₄²	—	122	1	no	—
122	C₄²	—	122	2	no	—
123	C₂³	#11758198	1	2	yes	—
124	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	122	1	no	—
131	C₂×C₄	—	122	2	no	—
132	C₂×C₄	—	122	1	no	—
133	C₂×C₄	—	244	2	no	—
134	C₂×C₄	—	244	4	no	—
135	C₂×C₄	—	244	2	no	—
136	C₂×C₄	#11758198	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
138	C₂×C₄	#11758224	1	2	yes	—
139	C₂×C₄	#11758224	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
141	C₈	—	4	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂×#11758198	1	1	yes	characteristic
146	C₂²	#14519761	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	122	2	no	—
149	C₄	—	488	2	no	—
150	C₄	—	244	2	no	—
151	C₄	—	244	1	no	—
152	C₄	#14519733	1	1	yes	characteristic
153	C₄	#14519733	1	1	yes	characteristic
154	C₄	#14519758	1	2	yes	—
155	C₂	—	2	1	no	—
156	C₂	—	2	2	no	—
157	C₂	—	4	2	no	—
158	C₂	#16839623	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×Dic₆₁

C₂

yes

maximal, characteristic

#16839736

C₂

yes

maximal, characteristic

#16839523

C₂

yes

maximal, complemented

#16839519

C₂

yes

maximal, complemented

C₆₁×#413

C₂

yes

maximal, characteristic

Q₈×Dic₆₁

—

D₄×Dic₆₁

—

D₄×Dic₆₁

—

#14519736

C₄

yes

complemented

#14519736

C₂²

yes

—

#14519736

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₄

yes

complemented

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519692

—

C₂×#41×C₆₁

C₂²

yes

characteristic

C₂×M₁₆×C₆₁

C₂²

yes

—

#101×C₆₁

—

#100×C₆₁

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×D₄×C₆₁

D₄

yes

—

C₂×D₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂²⋊Dic₆₁

—

C₂²⋊Dic₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂³×C₆₁

C₂²⋊C₄

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×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

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

#123

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#1827

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#381

yes

characteristic

C₄×#41

—

#413

D₆₁

yes

—

—

—

C₆₁

#1827

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₆₁

yes

characteristic

C₂×M₁₆

C₂×D₆₁

yes

—

C₄×Q₈

—

122

—

C₄×D₄

—

122

—

C₄×D₄

—

122

—

#117

—

#117

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

#101

—

#100

—

#99

—

122

—

#41

—

100

#41

—

101

C₂×Q₈

C₂²⋊D₆₁

yes

characteristic

102

C₂×D₄

D₂₄₄

yes

—

103

C₂×D₄

C₂²⋊D₆₁

yes

characteristic

104

C₂²×C₄

—

122

—

105

C₂²×C₄

—

122

—

106

C₂²×C₄

C₄×D₆₁

yes

—

107

C₂²×C₄

C₂²×D₆₁

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

122

—

112

C₄⋊C₄

—

122

—

113

C₄⋊C₄

—

122

—

114

C₂²⋊C₄

—

122

—

115

C₂²⋊C₄

—

122

—

116

C₂²⋊C₄

—

122

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

122

—

121

C₄²

—

122

—

122

C₄²

—

122

—

123

C₂³

#11758198

yes

—

124

C₂³

C₂×D₂₄₄

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

122

—

131

C₂×C₄

—

122

—

132

C₂×C₄

—

122

—

133

C₂×C₄

—

244

—

134

C₂×C₄

—

244

—

135

C₂×C₄

—

244

—

136

C₂×C₄

#11758198

yes

—

137

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

138

C₂×C₄

#11758224

yes

—

139

C₂×C₄

#11758224

yes

—

140

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×#11758198

yes

characteristic

146

C₂²

#14519761

yes

—

147

C₄

—

148

C₄

—

122

—

149

C₄

—

488

—

150

C₄

—

244

—

151

C₄

—

244

—

152

C₄

#14519733

yes

characteristic

153

C₄

#14519733

yes

characteristic

154

C₄

#14519758

yes

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

#16839623

yes

characteristic

	subgroup	G/H
center	C₄	#14519733
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#413	C₂
socle	C₂×C₆₁	#381
hypercenter	#413	D₆₁
hypocenter	C₆₁	#1827
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#14519733

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#413

C₂

C₂×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. #16839523	C₂	semidirect
4. #16839519	C₂	semidirect
9. #14519736	C₄	semidirect
12. C₂×C₄×Dic₆₁	C₄	semidirect
85. C₆₁	#1827	semidirect

H ≅ G/N

type

3. #16839523

C₂

semidirect

4. #16839519

C₂

semidirect

9. #14519736

C₄

semidirect

12. C₂×C₄×Dic₆₁

C₄

semidirect

85. C₆₁

#1827

semidirect

Subgroups — #21445924

Conjugacy classes of subgroups

Complements