Group #21447142 — subgroups

Subgroups — #21447142

Order 7808.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂²×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519702	C₂	1	2	yes	maximal
3	#16839668	C₂	1	2	yes	maximal
4	#16839525	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	2	yes	—
7	C₂×Dic₂₄₄	—	2	4	no	—
8	C₂×Dic₂₄₄	—	2	2	no	—
9	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
10	#14519702	—	2	4	no	—
11	#14519684	—	2	2	no	—
12	#14519684	—	2	2	no	—
13	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₆₁	—	2	2	no	—
17	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₆₁	—	2	2	no	—
21	C₂²×Dic₆₁	—	2	2	no	—
22	C₂×Dic₁₂₂	—	2	2	no	—
23	C₂×Dic₁₂₂	—	2	2	no	—
24	C₂×Dic₁₂₂	—	2	2	no	—
25	C₂×Dic₁₂₂	—	2	2	no	—
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₂×Dic₁₂₂	D₄	1	4	yes	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₂×C₈×C₆₁	—	2	1	no	—
30	C₂×C₈×C₆₁	—	2	2	no	—
31	C₂×C₈×C₆₁	D₄	1	2	yes	—
32	C₄²×C₆₁	—	2	1	no	—
33	C₄²×C₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	2	no	—
35	C₄⋊Dic₆₁	—	2	2	no	—
36	Dic₂₄₄	—	2	8	no	—
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	—	2	1	no	—
41	C₂×C₄×C₆₁	—	2	2	no	—
42	C₂×C₄×C₆₁	Q₁₆	1	2	yes	complemented
43	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
46	C₂×C₄×C₆₁	#41	1	2	yes	—
47	C₂×Dic₆₁	—	2	4	no	—
48	C₂×Dic₆₁	—	4	2	no	—
49	C₂×Dic₆₁	—	4	2	no	—
50	C₂×Dic₆₁	—	4	2	no	—
51	C₂×Dic₆₁	—	4	2	no	—
52	C₂×Dic₆₁	—	4	2	no	—
53	Dic₁₂₂	—	2	4	no	—
54	Dic₁₂₂	—	2	4	no	—
55	Dic₁₂₂	—	4	2	no	—
56	Dic₁₂₂	—	4	2	no	—
57	C₈×C₆₁	—	2	2	no	—
58	C₈×C₆₁	—	4	1	no	—
59	C₂²×C₆₁	#121	1	1	yes	characteristic
60	C₂²×C₆₁	#121	1	2	yes	—
61	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₆₁	#135	1	1	yes	characteristic
63	C₂²×C₆₁	#137	1	2	yes	—
64	C₄×C₆₁	—	2	2	no	—
65	C₄×C₆₁	—	4	1	no	—
66	C₄×C₆₁	#120	1	1	yes	characteristic
67	C₄×C₆₁	#123	1	2	yes	—
68	C₄×C₆₁	#124	1	1	yes	characteristic
69	Dic₆₁	—	4	4	no	—
70	Dic₆₁	—	8	2	no	—
71	#2056	—	61	1	no	maximal, Sylow
72	C₂×C₆₁	#394	1	1	yes	characteristic
73	C₂×C₆₁	#462	1	2	yes	—
74	C₂×C₆₁	#464	1	2	yes	—
75	C₂×C₆₁	#467	1	1	yes	characteristic
76	C₂×C₆₁	#470	1	1	yes	characteristic
77	C₂²×Q₁₆	—	61	1	no	—
78	C₂×C₄⋊C₈	D₆₁	1	1	yes	characteristic
79	C₂×#103	—	61	2	no	—
80	#391	—	61	2	no	—
81	#340	—	61	1	no	—
82	C₆₁	#2056	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	61	2	no	—
84	C₂×Q₁₆	—	122	4	no	—
85	C₂×Q₁₆	—	122	2	no	—
86	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	61	2	no	—
89	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#103	—	122	4	no	—
92	#95	—	122	2	no	—
93	#95	—	122	2	no	—
94	C₂×Q₈	—	61	4	no	—
95	C₂×Q₈	—	122	2	no	—
96	C₂×Q₈	—	122	2	no	—
97	C₂×Q₈	—	122	2	no	—
98	C₂×Q₈	—	122	2	no	—
99	C₂×Q₈	—	122	2	no	—
100	C₂²×C₄	—	122	2	no	—
101	C₂²×C₄	—	122	2	no	—
102	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
103	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
105	Q₁₆	—	122	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
110	C₄⋊C₄	—	122	2	no	—
111	C₄⋊C₄	—	122	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
115	Q₈	—	122	4	no	—
116	Q₈	—	122	4	no	—
117	Q₈	—	244	2	no	—
118	Q₈	—	244	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	122	4	no	—
124	C₂×C₄	—	244	2	no	—
125	C₂×C₄	—	244	2	no	—
126	C₂×C₄	—	244	2	no	—
127	C₂×C₄	—	244	2	no	—
128	C₂×C₄	—	244	2	no	—
129	C₂×C₄	Dic₂₄₄	1	2	yes	complemented
130	C₂×C₄	#11758222	1	1	yes	characteristic
131	C₂×C₄	D₄×D₆₁	1	2	yes	—
132	C₂×C₄	#11758226	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#14519751	1	1	yes	characteristic
137	C₂²	#14519752	1	2	yes	—
138	C₂²	C₂×Dic₂₄₄	1	1	yes	characteristic
139	C₂²	#14519799	1	1	yes	characteristic
140	C₂²	#14519810	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	488	2	no	—
144	C₄	—	244	4	no	—
145	C₄	#14519745	1	1	yes	characteristic
146	C₄	#14519762	1	2	yes	—
147	C₄	#14519766	1	1	yes	characteristic
148	C₂	#16839679	1	1	yes	characteristic
149	C₂	#16839973	1	2	yes	—
150	C₂	#16839985	1	2	yes	—
151	C₂	#16840003	1	1	yes	characteristic
152	C₂	#16840020	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₄₄

C₂

yes

maximal, characteristic

C₂×#14519702

C₂

yes

maximal

#16839668

C₂

yes

maximal

#16839525

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

#14519702

—

#14519684

—

#14519684

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

D₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

Q₁₆

yes

complemented

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₂²×C₆₁

#121

yes

—

C₂²×C₆₁

C₂×Q₁₆

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#120

yes

characteristic

C₄×C₆₁

#123

yes

—

C₄×C₆₁

#124

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#464

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂²×Q₁₆

—

C₂×C₄⋊C₈

D₆₁

yes

—

—

—

C₆₁

#2056

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

#103

—

122

—

#95

—

122

—

#95

—

122

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

100

C₂²×C₄

—

122

—

101

C₂²×C₄

—

122

—

102

C₂²×C₄

D₂₄₄

yes

characteristic

103

C₂²×C₄

D₂₄₄

yes

characteristic

104

C₂²×C₄

C₂²×D₆₁

yes

characteristic

105

Q₁₆

—

122

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₆₁

yes

—

110

C₄⋊C₄

—

122

—

111

C₄⋊C₄

—

122

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₂₄₄

yes

characteristic

115

Q₈

—

122

—

116

Q₈

—

122

—

117

Q₈

—

244

—

118

Q₈

—

244

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

122

—

124

C₂×C₄

—

244

—

125

C₂×C₄

—

244

—

126

C₂×C₄

—

244

—

127

C₂×C₄

—

244

—

128

C₂×C₄

—

244

—

129

C₂×C₄

Dic₂₄₄

yes

complemented

130

C₂×C₄

#11758222

yes

characteristic

131

C₂×C₄

D₄×D₆₁

yes

—

132

C₂×C₄

#11758226

yes

—

133

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

134

C₈

—

135

C₈

—

136

C₂²

#14519751

yes

characteristic

137

C₂²

#14519752

yes

—

138

C₂²

C₂×Dic₂₄₄

yes

characteristic

139

C₂²

#14519799

yes

characteristic

140

C₂²

#14519810

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

488

—

144

C₄

—

244

—

145

C₄

#14519745

yes

characteristic

146

C₄

#14519762

yes

—

147

C₄

#14519766

yes

characteristic

148

C₂

#16839679

yes

characteristic

149

C₂

#16839973

yes

—

150

C₂

#16839985

yes

—

151

C₂

#16840003

yes

characteristic

152

C₂

#16840020

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₄₄
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×C₄⋊C₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₆₁
hypocenter	C₆₁	#2056
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₄₄

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×C₄⋊C₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₆₁

hypocenter

C₆₁

#2056

LCS

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
42. C₂×C₄×C₆₁	Q₁₆	semidirect
82. C₆₁	#2056	semidirect
129. C₂×C₄	Dic₂₄₄	semidirect

H ≅ G/N

type

42. C₂×C₄×C₆₁

Q₁₆

semidirect

82. C₆₁

#2056

semidirect

129. C₂×C₄

Dic₂₄₄

semidirect

Subgroups — #21447142

Conjugacy classes of subgroups

Complements