Group #21444487 — subgroups

Subgroups — #21444487

Order 7808.

344 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 101 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	#16839815	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839786	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839506	C₂	1	4	yes	maximal, complemented
4	C₂×#127×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758194	—	2	1	no	—
6	C₄⋊#92418	—	2	1	no	—
7	C₄⋊#92418	C₂²	1	2	yes	complemented
8	C₄⋊#92418	C₂²	1	2	yes	complemented
9	C₂²⋊#92418	—	2	1	no	—
10	C₄×#92418	—	2	1	no	—
11	C₂²×D₄×C₆₁	—	2	1	no	—
12	C₂²×D₄×C₆₁	—	2	1	no	—
13	C₂²×D₄×C₆₁	C₄	1	2	yes	—
14	#127×C₆₁	—	2	1	no	—
15	#127×C₆₁	—	2	1	no	—
16	#127×C₆₁	C₄	1	2	yes	—
17	#127×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
19	C₂⁴×C₆₁	—	2	2	no	—
20	C₂×D₄×C₆₁	—	2	2	no	—
21	C₂×D₄×C₆₁	—	2	2	no	—
22	C₂×D₄×C₆₁	—	2	2	no	—
23	C₂×D₄×C₆₁	—	2	2	no	—
24	C₂×D₄×C₆₁	—	2	2	no	—
25	C₂×D₄×C₆₁	—	4	4	no	—
26	C₂×D₄×C₆₁	—	4	4	no	—
27	C₂×D₄×C₆₁	—	4	2	no	—
28	C₂×D₄×C₆₁	—	4	2	no	—
29	C₂×D₄×C₆₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
33	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₆₁	D₄	1	1	yes	characteristic
35	C₄²×C₆₁	D₄	1	1	yes	characteristic
36	C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	#11758194	—	4	2	no	—
38	C₂×#92418	—	2	2	no	—
39	C₂×#92418	—	2	2	no	—
40	C₂³×C₆₁	—	2	4	no	—
41	C₂³×C₆₁	—	4	4	no	—
42	C₂³×C₆₁	—	4	4	no	—
43	C₂³×C₆₁	—	4	4	no	—
44	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
45	D₄×C₆₁	—	4	4	no	—
46	D₄×C₆₁	—	4	4	no	—
47	D₄×C₆₁	—	4	4	no	—
48	D₄×C₆₁	—	4	4	no	—
49	D₄×C₆₁	—	4	2	no	—
50	D₄×C₆₁	—	4	2	no	—
51	D₄×C₆₁	—	4	4	no	—
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	—	2	2	no	—
54	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	D₈	1	2	yes	—
58	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
59	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	#92418	—	4	2	no	—
63	#92418	—	4	2	no	—
64	C₂²×C₆₁	—	2	1	no	—
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	—	4	4	no	—
67	C₂²×C₆₁	—	4	4	no	—
68	C₂²×C₆₁	—	4	4	no	—
69	C₂²×C₆₁	—	8	8	no	—
70	C₂²×C₆₁	#102	1	2	yes	—
71	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₆₁	—	2	1	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	#100	1	2	yes	—
75	C₄×C₆₁	#102	1	2	yes	—
76	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
77	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
79	#1545	—	61	1	no	maximal, Sylow
80	C₂×C₆₁	—	2	2	no	—
81	C₂×C₆₁	—	8	4	no	—
82	C₂×C₆₁	C₂×#100	1	1	yes	characteristic
83	C₂×C₆₁	C₂×#102	1	1	yes	characteristic
84	C₂×C₆₁	#418	1	1	yes	characteristic
85	C₂×#127	D₆₁	1	1	yes	characteristic
86	#431	—	61	1	no	—
87	#423	—	61	1	no	—
88	#331	—	61	4	no	—
89	C₆₁	#1545	1	1	yes	characteristic, complemented, Sylow
90	C₂²×D₄	—	2	1	no	—
91	C₂²×D₄	—	2	1	no	—
92	C₂²×D₄	Dic₆₁	1	2	yes	—
93	C₂×M₁₆	—	122	1	no	—
94	#127	—	2	1	no	—
95	#127	—	2	1	no	—
96	#127	Dic₆₁	1	2	yes	—
97	#127	C₂×D₆₁	1	2	yes	—
98	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
99	C₄⋊C₈	—	61	2	no	—
100	C₄⋊C₈	—	61	2	no	—
101	C₄⋊C₈	—	122	1	no	—
102	C₂²⋊C₈	—	122	1	no	—
103	C₄×C₈	—	122	1	no	—
104	C₂⁴	—	2	2	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	4	4	no	—
111	C₂×D₄	—	4	4	no	—
112	C₂×D₄	—	4	2	no	—
113	C₂×D₄	—	4	2	no	—
114	C₂×D₄	C₂×Dic₆₁	1	4	yes	—
115	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
118	M₁₆	—	244	2	no	—
119	C₂×C₈	—	122	2	no	—
120	C₂×C₈	—	122	2	no	—
121	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
122	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
123	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
124	C₄²	C₂²×D₆₁	1	1	yes	characteristic
125	C₂³	—	2	4	no	—
126	C₂³	—	4	4	no	—
127	C₂³	—	4	4	no	—
128	C₂³	—	4	4	no	—
129	C₂³	C₂²⋊Dic₆₁	1	1	yes	characteristic
130	D₄	—	4	4	no	—
131	D₄	—	4	4	no	—
132	D₄	—	4	4	no	—
133	D₄	—	4	4	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	#11758199	1	2	yes	—
140	C₂×C₄	#11758200	1	2	yes	—
141	C₂×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
147	C₈	—	244	2	no	—
148	C₈	—	244	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	4	4	no	—
152	C₂²	—	4	4	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	8	8	no	—
155	C₂²	#14519698	1	2	yes	—
156	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#14519694	1	2	yes	—
160	C₄	#14519698	1	2	yes	—
161	C₄	C₂×#11758199	1	1	yes	characteristic
162	C₄	C₂×#11758200	1	1	yes	characteristic
163	C₄	#14519803	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	—	8	4	no	—
166	C₂	C₂×#14519694	1	1	yes	characteristic
167	C₂	C₂×#14519698	1	1	yes	characteristic
168	C₂	#16839767	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839815

C₂

yes

maximal, characteristic, complemented

#16839786

C₂

yes

maximal, characteristic, complemented

#16839506

C₂

yes

maximal, complemented

C₂×#127×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

—

C₄⋊#92418

—

C₄⋊#92418

C₂²

yes

complemented

C₄⋊#92418

C₂²

yes

complemented

C₂²⋊#92418

—

C₄×#92418

—

C₂²×D₄×C₆₁

—

C₂²×D₄×C₆₁

—

C₂²×D₄×C₆₁

C₄

yes

—

#127×C₆₁

—

#127×C₆₁

—

#127×C₆₁

C₄

yes

—

#127×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂⁴×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

—

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

D₈

yes

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#102

yes

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#100

yes

—

C₄×C₆₁

#102

yes

—

C₄×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

AGL(1,ℤ/8)

yes

—

#1545

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

yes

—

—

—

C₆₁

#1545

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

Dic₆₁

yes

—

C₂×M₁₆

—

122

—

#127

—

#127

—

#127

Dic₆₁

yes

—

#127

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

122

—

102

C₂²⋊C₈

—

122

—

103

C₄×C₈

—

122

—

104

C₂⁴

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

C₂×Dic₆₁

yes

—

115

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

116

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

117

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

118

M₁₆

—

244

—

119

C₂×C₈

—

122

—

120

C₂×C₈

—

122

—

121

C₄²

C₂×Dic₆₁

yes

characteristic

122

C₄²

C₂²⋊D₆₁

yes

characteristic

123

C₄²

C₂²⋊D₆₁

yes

characteristic

124

C₄²

C₂²×D₆₁

yes

characteristic

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂²⋊Dic₆₁

yes

characteristic

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#11758199

yes

—

140

C₂×C₄

#11758200

yes

—

141

C₂×C₄

C₂²⋊Dic₆₁

yes

characteristic

142

C₂×C₄

C₂²⋊Dic₆₁

yes

characteristic

143

C₂×C₄

C₂²⋊Dic₆₁

yes

characteristic

144

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

145

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

147

C₈

—

244

—

148

C₈

—

244

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#14519698

yes

—

156

C₂²

C₂×C₂²⋊Dic₆₁

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

#14519694

yes

—

160

C₄

#14519698

yes

—

161

C₄

C₂×#11758199

yes

characteristic

162

C₄

C₂×#11758200

yes

characteristic

163

C₄

#14519803

yes

—

164

C₂

—

165

C₂

—

166

C₂

C₂×#14519694

yes

characteristic

167

C₂

C₂×#14519698

yes

characteristic

168

C₂

#16839767

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₂×#127×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₂×#127	D₆₁
hypocenter	C₆₁	#1545
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₂×#127×C₆₁

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #16839815	C₂	semidirect
2. #16839786	C₂	semidirect
3. #16839506	C₂	semidirect
7. C₄⋊#92418	C₂²	semidirect
8. C₄⋊#92418	C₂²	semidirect
89. C₆₁	#1545	semidirect

H ≅ G/N

type

1. #16839815

C₂

semidirect

2. #16839786

C₂

semidirect

3. #16839506

C₂

semidirect

7. C₄⋊#92418

C₂²

semidirect

8. C₄⋊#92418

C₂²

semidirect

89. C₆₁

#1545

semidirect

Subgroups — #21444487

Conjugacy classes of subgroups

Complements