Group #21445166 — subgroups

Subgroups — #21445166

Order 7808.

172 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16840134	C₂	1	1	yes	maximal, characteristic
2	#16840048	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839959	C₂	1	1	yes	maximal, characteristic
4	#16839886	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839512	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839509	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#331	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758200	—	2	1	no	—
9	C₂×#11758200	—	2	1	no	—
10	C₂×#11758200	—	2	1	no	—
11	#14519777	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
13	#14519709	—	2	1	no	—
14	C₄⋊#92418	C₂²	1	1	yes	characteristic
15	#14519706	—	2	1	no	—
16	#14519702	—	2	1	no	—
17	#14519698	—	2	1	no	—
18	C₄×#92418	C₂²	1	1	yes	characteristic
19	C₈×Dic₆₁	—	2	1	no	—
20	#127×C₆₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₆₁	—	2	1	no	—
24	C₂×D₄×C₆₁	—	2	1	no	—
25	C₂×D₄×C₆₁	—	2	1	no	—
26	C₂×Dic₁₂₂	—	2	1	no	—
27	C₂×Dic₁₂₂	D₄	1	1	yes	characteristic
28	C₂×C₈×C₆₁	—	2	1	no	—
29	C₂×C₈×C₆₁	—	2	1	no	—
30	C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	#11758200	—	4	2	no	—
32	#11758200	—	4	2	no	—
33	#11758200	—	4	2	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
37	#11758196	—	2	1	no	—
38	C₄×Dic₆₁	—	2	1	no	—
39	C₂×#92418	—	2	1	no	—
40	C₂×#92418	D₄	1	1	yes	characteristic
41	C₂×#92418	D₄	1	1	yes	characteristic
42	C₂³×C₆₁	—	4	1	no	—
43	D₄×C₆₁	—	4	1	no	—
44	D₄×C₆₁	—	4	1	no	—
45	D₄×C₆₁	—	4	1	no	—
46	D₄×C₆₁	—	4	1	no	—
47	D₄×C₆₁	—	8	1	no	—
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
51	C₂×Dic₆₁	—	2	1	no	—
52	C₂×Dic₆₁	—	2	1	no	—
53	C₂×Dic₆₁	—	4	1	no	—
54	Dic₁₂₂	—	2	1	no	—
55	Dic₁₂₂	—	4	1	no	—
56	Dic₁₂₂	—	4	1	no	—
57	Dic₁₂₂	SD₁₆	1	2	yes	complemented
58	C₈×C₆₁	—	4	1	no	—
59	C₈×C₆₁	—	4	1	no	—
60	#92418	—	2	1	no	—
61	#92418	—	4	1	no	—
62	#92418	SD₁₆	1	2	yes	complemented
63	C₂²×C₆₁	—	8	1	no	—
64	C₂²×C₆₁	—	8	1	no	—
65	C₂²×C₆₁	—	8	1	no	—
66	C₂²×C₆₁	#121	1	1	yes	characteristic
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₆₁	#135	1	1	yes	characteristic
71	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₆₁	—	2	2	no	—
73	Dic₆₁	—	4	1	no	—
74	Dic₆₁	—	8	1	no	—
75	#1666	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	—	16	1	no	—
77	C₂×C₆₁	#460	1	1	yes	characteristic
78	C₂×C₆₁	#465	1	1	yes	characteristic
79	C₂×C₆₁	#471	1	1	yes	characteristic
80	#492	—	61	1	no	—
81	#475	—	61	1	no	—
82	#460	—	61	1	no	—
83	C₈×Q₈	—	61	1	no	—
84	#334	—	61	1	no	—
85	#332	—	61	1	no	—
86	#331	D₆₁	1	1	yes	characteristic
87	C₆₁	#1666	1	1	yes	characteristic, complemented, Sylow
88	C₂×SD₁₆	—	122	1	no	—
89	C₂×SD₁₆	—	122	1	no	—
90	C₂×SD₁₆	—	122	1	no	—
91	C₄⋊Q₈	—	61	1	no	—
92	#127	C₂×D₆₁	1	1	yes	characteristic
93	C₄×Q₈	—	61	1	no	—
94	#106	—	122	1	no	—
95	C₄⋊C₈	—	61	1	no	—
96	C₄⋊C₈	—	122	1	no	—
97	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
98	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
99	#103	—	122	1	no	—
100	#102	—	122	1	no	—
101	C₄×C₈	—	61	1	no	—
102	C₄×C₈	—	122	1	no	—
103	C₂×Q₈	—	61	1	no	—
104	C₂×Q₈	—	122	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	SD₁₆	—	244	2	no	—
109	SD₁₆	—	244	2	no	—
110	SD₁₆	—	244	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	61	1	no	—
114	C₂×C₈	—	61	1	no	—
115	C₂×C₈	—	122	1	no	—
116	C₄⋊C₄	—	61	1	no	—
117	C₄⋊C₄	—	122	1	no	—
118	C₄⋊C₄	—	122	1	no	—
119	C₄⋊C₄	—	122	1	no	—
120	C₄²	—	122	1	no	—
121	C₄²	C₂²×D₆₁	1	1	yes	characteristic
122	C₂³	—	4	1	no	—
123	Q₈	—	61	2	no	—
124	Q₈	—	122	1	no	—
125	Q₈	—	244	1	no	—
126	Q₈	—	244	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	8	1	no	—
132	C₂×C₄	—	122	1	no	—
133	C₂×C₄	—	122	1	no	—
134	C₂×C₄	—	244	1	no	—
135	C₂×C₄	#11758222	1	1	yes	characteristic
136	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
137	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
138	C₈	—	4	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	61	2	no	—
141	C₈	—	122	1	no	—
142	C₈	—	244	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	#14519750	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	122	2	no	—
149	C₄	—	488	1	no	—
150	C₄	—	244	1	no	—
151	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
152	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
153	C₄	#14519796	1	1	yes	characteristic
154	C₄	#14519801	1	1	yes	characteristic
155	C₂	—	16	1	no	—
156	C₂	#16839961	1	1	yes	characteristic
157	C₂	#16839990	1	1	yes	characteristic
158	C₂	#16840024	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840134

C₂

yes

maximal, characteristic

#16840048

C₂

yes

maximal, characteristic, complemented

#16839959

C₂

yes

maximal, characteristic

#16839886

C₂

yes

maximal, characteristic, complemented

#16839512

C₂

yes

maximal, characteristic, complemented

#16839509

C₂

yes

maximal, characteristic, complemented

C₆₁×#331

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₂₂

C₂²

yes

characteristic

#14519709

—

C₄⋊#92418

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₈×Dic₆₁

—

#127×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

D₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂³

yes

—

—

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

C₂³×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

SD₁₆

yes

complemented

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

—

#92418

SD₁₆

yes

complemented

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#1666

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

C₈×Q₈

—

#334

—

#332

—

#331

D₆₁

yes

characteristic

C₆₁

#1666

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

122

—

C₂×SD₁₆

—

122

—

C₂×SD₁₆

—

122

—

C₄⋊Q₈

—

#127

C₂×D₆₁

yes

characteristic

C₄×Q₈

—

#106

—

122

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#103

—

122

—

100

#102

—

122

—

101

C₄×C₈

—

102

C₄×C₈

—

122

—

103

C₂×Q₈

—

104

C₂×Q₈

—

122

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

SD₁₆

—

244

—

109

SD₁₆

—

244

—

110

SD₁₆

—

244

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

122

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

122

—

118

C₄⋊C₄

—

122

—

119

C₄⋊C₄

—

122

—

120

C₄²

—

122

—

121

C₄²

C₂²×D₆₁

yes

characteristic

122

C₂³

—

123

Q₈

—

124

Q₈

—

122

—

125

Q₈

—

244

—

126

Q₈

—

244

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

122

—

133

C₂×C₄

—

122

—

134

C₂×C₄

—

244

—

135

C₂×C₄

#11758222

yes

characteristic

136

C₂×C₄

D₄×D₆₁

yes

characteristic

137

C₂×C₄

D₄×D₆₁

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

122

—

142

C₈

—

244

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#14519750

yes

characteristic

147

C₄

—

148

C₄

—

122

—

149

C₄

—

488

—

150

C₄

—

244

—

151

C₄

SD₁₆×D₆₁

yes

characteristic

152

C₄

SD₁₆×D₆₁

yes

characteristic

153

C₄

#14519796

yes

characteristic

154

C₄

#14519801

yes

characteristic

155

C₂

—

156

C₂

#16839961

yes

characteristic

157

C₂

#16839990

yes

characteristic

158

C₂

#16840024

yes

characteristic

	subgroup	G/H
center	C₂²	#14519750
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#331	C₂
socle	C₂²×C₆₁	#121
hypercenter	#331	D₆₁
hypocenter	C₆₁	#1666
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14519750

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#331

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #16840048	C₂	semidirect
4. #16839886	C₂	semidirect
5. #16839512	C₂	semidirect
6. #16839509	C₂	semidirect
57. Dic₁₂₂	SD₁₆	semidirect
62. #92418	SD₁₆	semidirect
87. C₆₁	#1666	semidirect

H ≅ G/N

type

2. #16840048

C₂

semidirect

4. #16839886

C₂

semidirect

5. #16839512

C₂

semidirect

6. #16839509

C₂

semidirect

57. Dic₁₂₂

SD₁₆

semidirect

62. #92418

SD₁₆

semidirect

87. C₆₁

#1666

semidirect

Subgroups — #21445166

Conjugacy classes of subgroups

Complements