Group #21447242 — subgroups

Subgroups — #21447242

Order 7808.

256 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 85 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×#14519711	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839622	C₂	1	2	yes	maximal
3	#16839525	C₂	1	2	yes	maximal, complemented
4	#16839525	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
9	#14519711	—	2	2	no	—
10	#14519684	—	2	2	no	—
11	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₆₁	—	2	2	no	—
15	C₂⁴×C₆₁	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	—	2	1	no	—
17	C₂²×C₄×C₆₁	—	2	1	no	—
18	C₂²×C₄×C₆₁	—	2	2	no	—
19	C₂²×C₄×C₆₁	—	2	2	no	—
20	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₆₁	—	2	2	no	—
23	C₂²×Dic₆₁	—	2	2	no	—
24	C₂×C₈×C₆₁	—	2	2	no	—
25	C₂×C₈×C₆₁	—	2	1	no	—
26	C₂×C₈×C₆₁	—	2	2	no	—
27	C₂×C₈×C₆₁	D₄	1	2	yes	—
28	C₂²⋊Dic₆₁	—	4	4	no	—
29	C₄⋊Dic₆₁	—	2	2	no	—
30	C₄⋊Dic₆₁	—	2	2	no	—
31	C₄⋊Dic₆₁	—	2	2	no	—
32	C₄⋊Dic₆₁	—	2	2	no	—
33	C₂³×C₆₁	—	2	1	no	—
34	C₂³×C₆₁	—	2	2	no	—
35	C₂³×C₆₁	—	2	1	no	—
36	C₂³×C₆₁	—	2	2	no	—
37	C₂³×C₆₁	D₈	1	2	yes	—
38	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	2	1	no	—
40	C₂×C₄×C₆₁	—	2	2	no	—
41	C₂×C₄×C₆₁	—	4	1	no	—
42	C₂×C₄×C₆₁	—	4	1	no	—
43	C₂×C₄×C₆₁	—	4	2	no	—
44	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	#41	1	2	yes	—
46	C₂×C₄×C₆₁	#41	1	2	yes	—
47	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
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	C₂×Dic₆₁	—	4	2	no	—
54	C₈×C₆₁	—	2	2	no	—
55	C₈×C₆₁	—	4	1	no	—
56	C₂²×C₆₁	—	2	2	no	—
57	C₂²×C₆₁	—	2	4	no	—
58	C₂²×C₆₁	—	4	1	no	—
59	C₂²×C₆₁	—	4	1	no	—
60	C₂²×C₆₁	—	4	2	no	—
61	C₂²×C₆₁	#121	1	1	yes	characteristic
62	C₂²×C₆₁	#123	1	2	yes	—
63	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
64	C₂²×C₆₁	#135	1	1	yes	characteristic
65	C₂²×C₆₁	#137	1	2	yes	—
66	C₄×C₆₁	—	4	1	no	—
67	C₄×C₆₁	#123	1	1	yes	characteristic
68	C₄×C₆₁	#124	1	1	yes	characteristic
69	C₄×C₆₁	#126	1	2	yes	—
70	Dic₆₁	—	8	2	no	—
71	Dic₆₁	—	8	2	no	—
72	#2075	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	—	4	2	no	—
74	C₂×C₆₁	#397	1	1	yes	characteristic
75	C₂×C₆₁	#466	1	1	yes	characteristic
76	C₂×C₆₁	#470	1	1	yes	characteristic
77	C₂×C₆₁	#480	1	2	yes	—
78	C₂×C₆₁	#485	1	2	yes	—
79	C₂×#107	—	61	1	no	—
80	C₂×C₂²⋊C₈	D₆₁	1	1	yes	characteristic
81	#380	—	61	2	no	—
82	#340	—	61	1	no	—
83	#340	—	61	2	no	—
84	C₆₁	#2075	1	1	yes	characteristic, complemented, Sylow
85	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
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×C₄⋊C₄	—	61	2	no	—
90	C₂×C₂²⋊C₄	—	122	2	no	—
91	#107	—	122	2	no	—
92	C₂²⋊C₈	—	2	2	no	—
93	#95	—	122	2	no	—
94	C₂⁴	D₂₄₄	1	1	yes	characteristic
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	122	2	no	—
100	C₂²×C₄	—	122	2	no	—
101	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
107	C₄⋊C₄	—	122	2	no	—
108	C₄⋊C₄	—	122	2	no	—
109	C₄⋊C₄	—	122	2	no	—
110	C₄⋊C₄	—	122	2	no	—
111	C₂²⋊C₄	—	244	4	no	—
112	C₂³	—	2	2	no	—
113	C₂³	—	2	1	no	—
114	C₂³	—	2	2	no	—
115	C₂³	—	2	1	no	—
116	C₂³	D₄₈₈	1	2	yes	—
117	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	244	2	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	#11758222	1	1	yes	characteristic
130	C₂×C₄	#11758224	1	2	yes	—
131	C₂×C₄	#11758224	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	1	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	4	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	2	no	—
140	C₂²	#14519751	1	1	yes	characteristic
141	C₂²	#14519761	1	2	yes	—
142	C₂²	C₂×D₄₈₈	1	1	yes	characteristic
143	C₂²	#14519799	1	1	yes	characteristic
144	C₂²	#14519810	1	2	yes	—
145	C₄	—	4	1	no	—
146	C₄	—	488	2	no	—
147	C₄	—	488	2	no	—
148	C₄	#14519758	1	1	yes	characteristic
149	C₄	#14519764	1	1	yes	characteristic
150	C₄	#14519770	1	2	yes	—
151	C₂	—	4	2	no	—
152	C₂	#16839687	1	1	yes	characteristic
153	C₂	#16839998	1	1	yes	characteristic
154	C₂	#16840020	1	1	yes	characteristic
155	C₂	#16840076	1	2	yes	—
156	C₂	#16840104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519711

C₂

yes

maximal, characteristic, complemented

#16839622

C₂

yes

maximal

#16839525

C₂

yes

maximal, complemented

#16839525

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₈×C₆₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₂²⋊Dic₆₁

—

#14519711

—

#14519684

—

C₂³×C₄×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²⋊C₈×C₆₁

—

C₂⁴×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

D₈

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

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

—

C₈×C₆₁

—

C₈×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₂²×C₆₁

#123

yes

—

C₂²×C₆₁

C₂×D₈

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

Dic₆₁

—

#2075

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#485

yes

—

C₂×#107

—

C₂×C₂²⋊C₈

D₆₁

yes

—

—

—

C₆₁

#2075

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

122

—

#107

—

122

—

C₂²⋊C₈

—

#95

—

122

—

C₂⁴

D₂₄₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

100

C₂²×C₄

—

122

—

101

C₂²×C₄

D₂₄₄

yes

characteristic

102

C₂²×C₄

C₂²×D₆₁

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²⋊D₆₁

yes

—

107

C₄⋊C₄

—

122

—

108

C₄⋊C₄

—

122

—

109

C₄⋊C₄

—

122

—

110

C₄⋊C₄

—

122

—

111

C₂²⋊C₄

—

244

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

D₄₈₈

yes

—

117

C₂³

C₂×D₂₄₄

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

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

#11758222

yes

characteristic

130

C₂×C₄

#11758224

yes

—

131

C₂×C₄

#11758224

yes

—

132

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#14519751

yes

characteristic

141

C₂²

#14519761

yes

—

142

C₂²

C₂×D₄₈₈

yes

characteristic

143

C₂²

#14519799

yes

characteristic

144

C₂²

#14519810

yes

—

145

C₄

—

146

C₄

—

488

—

147

C₄

—

488

—

148

C₄

#14519758

yes

characteristic

149

C₄

#14519764

yes

characteristic

150

C₄

#14519770

yes

—

151

C₂

—

152

C₂

#16839687

yes

characteristic

153

C₂

#16839998

yes

characteristic

154

C₂

#16840020

yes

characteristic

155

C₂

#16840076

yes

—

156

C₂

#16840104

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₄₄
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×C₂²⋊C₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×C₂²⋊C₈	D₆₁
hypocenter	C₆₁	#2075
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₄₄

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×C₂²⋊C₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×C₂²⋊C₈

D₆₁

hypocenter

C₆₁

#2075

LCS

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
1. C₂×#14519711	C₂	semidirect
3. #16839525	C₂	semidirect
4. #16839525	C₂	semidirect
84. C₆₁	#2075	semidirect

H ≅ G/N

type

1. C₂×#14519711

C₂

semidirect

3. #16839525

C₂

semidirect

4. #16839525

C₂

semidirect

84. C₆₁

#2075

semidirect

Subgroups — #21447242

Conjugacy classes of subgroups

Complements