Group #21447229 — subgroups

Subgroups — #21447229

Order 7808.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14519702	C₂	1	2	yes	maximal
2	#16839668	C₂	1	1	yes	maximal, characteristic
3	#16839632	C₂	1	1	yes	maximal, characteristic
4	#16839525	C₂	1	2	yes	maximal
5	C₆₁×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₁	—	2	1	no	—
8	C₂×C₄⋊Dic₆₁	—	2	1	no	—
9	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
11	#14519702	—	2	4	no	—
12	#14519684	—	2	1	no	—
13	#14519684	—	2	1	no	—
14	C₂²×C₈×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
19	C₂²×Dic₆₁	—	2	1	no	—
20	C₂²×Dic₆₁	—	2	1	no	—
21	C₂²×Dic₆₁	—	2	2	no	—
22	C₂×Dic₁₂₂	—	2	2	no	—
23	C₂×Dic₁₂₂	—	2	2	no	—
24	C₂×Dic₁₂₂	—	2	1	no	—
25	C₂×Dic₁₂₂	D₄	1	2	yes	—
26	C₂×C₈×C₆₁	—	2	2	no	—
27	C₂×C₈×C₆₁	—	2	2	no	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₄²×C₆₁	—	2	2	no	—
30	C₄⋊Dic₆₁	—	2	2	no	—
31	C₄⋊Dic₆₁	—	2	2	no	—
32	C₄⋊Dic₆₁	—	2	2	no	—
33	C₄⋊Dic₆₁	—	4	2	no	—
34	C₄⋊Dic₆₁	—	4	2	no	—
35	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₁	—	2	1	no	—
37	C₂×C₄×C₆₁	—	2	1	no	—
38	C₂×C₄×C₆₁	—	2	1	no	—
39	C₂×C₄×C₆₁	—	2	1	no	—
40	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
41	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
42	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	#41	1	2	yes	—
44	C₂×C₄×C₆₁	#41	1	2	yes	—
45	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
46	C₂×Dic₆₁	—	2	2	no	—
47	C₂×Dic₆₁	—	4	1	no	—
48	C₂×Dic₆₁	—	4	1	no	—
49	C₂×Dic₆₁	—	4	2	no	—
50	C₂×Dic₆₁	—	4	1	no	—
51	C₂×Dic₆₁	—	4	2	no	—
52	C₂×Dic₆₁	—	4	2	no	—
53	C₂×Dic₆₁	—	4	1	no	—
54	C₂×Dic₆₁	—	4	1	no	—
55	Dic₁₂₂	—	2	4	no	—
56	Dic₁₂₂	—	4	2	no	—
57	C₈×C₆₁	—	4	2	no	—
58	C₂²×C₆₁	#121	1	1	yes	characteristic
59	C₂²×C₆₁	#123	1	1	yes	characteristic
60	C₂²×C₆₁	#124	1	1	yes	characteristic
61	C₂²×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₆₁	#137	1	1	yes	characteristic
65	C₄×C₆₁	—	2	2	no	—
66	C₄×C₆₁	—	2	2	no	—
67	C₄×C₆₁	#123	1	2	yes	—
68	C₄×C₆₁	#126	1	2	yes	—
69	Dic₆₁	—	4	2	no	—
70	Dic₆₁	—	8	2	no	—
71	Dic₆₁	—	8	1	no	—
72	#2072	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	#397	1	1	yes	characteristic
74	C₂×C₆₁	#460	1	1	yes	characteristic
75	C₂×C₆₁	#462	1	1	yes	characteristic
76	C₂×C₆₁	#483	1	1	yes	characteristic
77	C₂×C₆₁	#484	1	1	yes	characteristic
78	C₂×C₆₁	#487	1	1	yes	characteristic
79	C₂×C₆₁	#489	1	1	yes	characteristic
80	C₂×#103	—	61	2	no	—
81	#391	—	61	1	no	—
82	#384	—	61	1	no	—
83	#340	—	61	2	no	—
84	#336	D₆₁	1	1	yes	characteristic
85	C₆₁	#2072	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	61	1	no	—
87	C₂²×C₈	C₂×D₆₁	1	2	yes	—
88	C₂×C₄⋊C₄	—	61	1	no	—
89	C₂×C₄⋊C₄	—	61	2	no	—
90	C₂×C₄⋊C₄	—	122	1	no	—
91	C₂×C₄⋊C₄	—	122	1	no	—
92	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
93	#103	—	122	4	no	—
94	#95	—	122	1	no	—
95	#95	—	122	1	no	—
96	C₂×Q₈	—	61	2	no	—
97	C₂×Q₈	—	122	1	no	—
98	C₂×Q₈	—	122	2	no	—
99	C₂×Q₈	—	122	2	no	—
100	C₂²×C₄	—	122	1	no	—
101	C₂²×C₄	—	122	1	no	—
102	C₂²×C₄	—	122	2	no	—
103	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
104	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₄⋊C₄	—	122	2	no	—
110	C₄⋊C₄	—	122	2	no	—
111	C₄⋊C₄	—	122	2	no	—
112	C₄⋊C₄	—	244	2	no	—
113	C₄⋊C₄	—	244	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
116	Q₈	—	122	4	no	—
117	Q₈	—	244	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	122	2	no	—
123	C₂×C₄	—	244	1	no	—
124	C₂×C₄	—	244	2	no	—
125	C₂×C₄	—	244	1	no	—
126	C₂×C₄	—	244	2	no	—
127	C₂×C₄	—	244	2	no	—
128	C₂×C₄	—	244	1	no	—
129	C₂×C₄	—	244	1	no	—
130	C₂×C₄	—	244	1	no	—
131	C₂×C₄	#11758190	1	2	yes	—
132	C₂×C₄	Dic₂₄₄	1	2	yes	—
133	C₂×C₄	#11758222	1	2	yes	—
134	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
135	C₂×C₄	#11758224	1	2	yes	—
136	C₂×C₄	#11758226	1	1	yes	characteristic
137	C₈	—	4	2	no	—
138	C₂²	#14519752	1	1	yes	characteristic
139	C₂²	#14519761	1	1	yes	characteristic
140	C₂²	#14519765	1	1	yes	characteristic
141	C₂²	C₂×#11758190	1	1	yes	characteristic
142	C₂²	C₂×Dic₂₄₄	1	1	yes	characteristic
143	C₂²	#14519805	1	1	yes	characteristic
144	C₂²	#14519810	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	488	2	no	—
148	C₄	—	488	1	no	—
149	C₄	—	244	2	no	—
150	C₄	#14519759	1	2	yes	—
151	C₄	#14519772	1	2	yes	—
152	C₂	#16839689	1	1	yes	characteristic
153	C₂	#16839963	1	1	yes	characteristic
154	C₂	#16839973	1	1	yes	characteristic
155	C₂	#16840093	1	1	yes	characteristic
156	C₂	#16840098	1	1	yes	characteristic
157	C₂	#16840113	1	1	yes	characteristic
158	C₂	#16840122	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519702

C₂

yes

maximal

#16839668

C₂

yes

maximal, characteristic

#16839632

C₂

yes

maximal, characteristic

#16839525

C₂

yes

maximal

C₆₁×#336

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

—

—

C₂²×C₈×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

D₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

C₂×SD₁₆

yes

characteristic

C₂²×C₆₁

C₂×Q₁₆

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#123

yes

—

C₄×C₆₁

#126

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

—

D₆₁

yes

characteristic

C₆₁

#2072

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#103

—

122

—

#95

—

122

—

#95

—

122

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

100

C₂²×C₄

—

122

—

101

C₂²×C₄

—

122

—

102

C₂²×C₄

—

122

—

103

C₂²×C₄

D₂₄₄

yes

characteristic

104

C₂²×C₄

D₂₄₄

yes

characteristic

105

C₂²×C₄

C₂²×D₆₁

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

122

—

110

C₄⋊C₄

—

122

—

111

C₄⋊C₄

—

122

—

112

C₄⋊C₄

—

244

—

113

C₄⋊C₄

—

244

—

114

C₄²

—

115

C₂³

C₂×D₂₄₄

yes

characteristic

116

Q₈

—

122

—

117

Q₈

—

244

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

122

—

123

C₂×C₄

—

244

—

124

C₂×C₄

—

244

—

125

C₂×C₄

—

244

—

126

C₂×C₄

—

244

—

127

C₂×C₄

—

244

—

128

C₂×C₄

—

244

—

129

C₂×C₄

—

244

—

130

C₂×C₄

—

244

—

131

C₂×C₄

#11758190

yes

—

132

C₂×C₄

Dic₂₄₄

yes

—

133

C₂×C₄

#11758222

yes

—

134

C₂×C₄

D₄×D₆₁

yes

characteristic

135

C₂×C₄

#11758224

yes

—

136

C₂×C₄

#11758226

yes

characteristic

137

C₈

—

138

C₂²

#14519752

yes

characteristic

139

C₂²

#14519761

yes

characteristic

140

C₂²

#14519765

yes

characteristic

141

C₂²

C₂×#11758190

yes

characteristic

142

C₂²

C₂×Dic₂₄₄

yes

characteristic

143

C₂²

#14519805

yes

characteristic

144

C₂²

#14519810

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

488

—

148

C₄

—

488

—

149

C₄

—

244

—

150

C₄

#14519759

yes

—

151

C₄

#14519772

yes

—

152

C₂

#16839689

yes

characteristic

153

C₂

#16839963

yes

characteristic

154

C₂

#16839973

yes

characteristic

155

C₂

#16840093

yes

characteristic

156

C₂

#16840098

yes

characteristic

157

C₂

#16840113

yes

characteristic

158

C₂

#16840122

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₄₄
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#336	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	#336	D₆₁
hypocenter	C₆₁	#2072
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₄₄

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#336

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
85. C₆₁	#2072	semidirect

H ≅ G/N

type

85. C₆₁

#2072

semidirect

Subgroups — #21447229

Conjugacy classes of subgroups

Complements