Group #21445652 — subgroups

Subgroups — #21445652

Order 7808.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519700	C₂	1	2	yes	maximal
3	C₂×#14519688	C₂	1	1	yes	maximal, characteristic
4	#16839524	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
7	C₂²×#92418	C₂²	1	2	yes	—
8	C₄×Dic₁₂₂	—	2	2	no	—
9	C₄×Dic₁₂₂	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×#11758196	—	2	1	no	—
12	C₂×C₄×Dic₆₁	—	2	1	no	—
13	#14519700	C₄	1	8	yes	complemented
14	#14519688	—	2	2	no	—
15	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₁	—	2	1	no	—
17	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	#95×C₆₁	—	2	1	no	—
19	C₂²×C₄×C₆₁	—	2	2	no	—
20	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₆₁	—	2	1	no	—
24	C₂²×Dic₆₁	—	2	1	no	—
25	C₂×Dic₁₂₂	—	2	1	no	—
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₂×Dic₁₂₂	C₂×C₄	1	4	yes	—
28	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
29	C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
31	C₄²×C₆₁	—	4	2	no	—
32	C₄²×C₆₁	C₂×C₄	1	2	yes	—
33	C₄²×C₆₁	D₄	1	2	yes	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
36	#11758196	—	2	4	no	—
37	C₄×Dic₆₁	—	2	4	no	—
38	C₂×#92418	—	2	2	no	—
39	C₂×#92418	—	2	2	no	—
40	C₂×#92418	C₂×C₄	1	4	yes	—
41	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	—	2	1	no	—
43	C₂×C₄×C₆₁	—	2	1	no	—
44	C₂×C₄×C₆₁	—	2	2	no	—
45	C₂×C₄×C₆₁	—	2	4	no	—
46	C₂×C₄×C₆₁	—	4	2	no	—
47	C₂×C₄×C₆₁	—	4	2	no	—
48	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#41	1	2	yes	—
54	C₂×Dic₆₁	—	2	2	no	—
55	C₂×Dic₆₁	—	2	4	no	—
56	C₂×Dic₆₁	—	2	2	no	—
57	C₂×Dic₆₁	—	4	1	no	—
58	C₂×Dic₆₁	—	4	1	no	—
59	Dic₁₂₂	—	2	2	no	—
60	Dic₁₂₂	—	2	4	no	—
61	Dic₁₂₂	C₄²	1	4	yes	complemented
62	#92418	—	2	4	no	—
63	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₆₁	C₄×D₄	1	2	yes	—
65	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₆₁	#137	1	1	yes	characteristic
67	C₂²×C₆₁	#137	1	2	yes	—
68	C₄×C₆₁	—	2	2	no	—
69	C₄×C₆₁	—	2	2	no	—
70	C₄×C₆₁	—	4	4	no	—
71	C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₆₁	#117	1	1	yes	characteristic
73	C₄×C₆₁	C₄×D₄	1	2	yes	—
74	Dic₆₁	—	2	4	no	—
75	Dic₆₁	—	4	2	no	—
76	#1763	—	61	1	no	maximal, Sylow
77	C₂×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₆₁	#417	1	1	yes	characteristic
79	C₂×C₆₁	#419	1	1	yes	characteristic
80	C₂×C₆₁	#440	1	2	yes	—
81	C₂×C₆₁	#441	1	2	yes	—
82	C₂×C₄×Q₈	—	61	1	no	—
83	C₂×#103	—	61	2	no	—
84	C₂×#97	—	61	1	no	—
85	C₄×C₄⋊C₄	D₆₁	1	1	yes	characteristic
86	#340	—	61	2	no	—
87	C₆₁	#1763	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	61	1	no	—
89	C₂²×C₈	—	61	2	no	—
90	C₄×Q₈	—	61	4	no	—
91	C₄×Q₈	—	122	2	no	—
92	C₂×C₄⋊C₄	—	61	1	no	—
93	C₂×C₄⋊C₄	—	122	1	no	—
94	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
95	C₂×C₄²	—	2	1	no	—
96	C₂×C₄²	—	122	1	no	—
97	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
98	#103	—	61	8	no	—
99	#97	—	122	2	no	—
100	#95	—	2	1	no	—
101	C₂×Q₈	—	61	4	no	—
102	C₂×Q₈	—	61	2	no	—
103	C₂×Q₈	—	122	1	no	—
104	C₂×Q₈	—	122	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	122	1	no	—
107	C₂²×C₄	—	122	1	no	—
108	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
111	C₂×C₈	—	61	4	no	—
112	C₂×C₈	—	122	2	no	—
113	C₂×C₈	—	122	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	61	2	no	—
116	C₄⋊C₄	—	122	4	no	—
117	C₄⋊C₄	—	122	1	no	—
118	C₄⋊C₄	C₄×D₆₁	1	4	yes	—
119	C₄²	—	4	2	no	—
120	C₄²	—	122	4	no	—
121	C₄²	C₄×D₆₁	1	2	yes	—
122	C₄²	D₂₄₄	1	2	yes	—
123	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
124	Q₈	—	61	4	no	—
125	Q₈	—	122	2	no	—
126	Q₈	—	122	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	122	2	no	—
134	C₂×C₄	—	122	4	no	—
135	C₂×C₄	—	122	2	no	—
136	C₂×C₄	—	244	1	no	—
137	C₂×C₄	—	244	1	no	—
138	C₂×C₄	#11758198	1	2	yes	—
139	C₂×C₄	#11758198	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
142	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
143	C₂×C₄	#11758222	1	2	yes	—
144	C₈	—	122	4	no	—
145	C₂²	C₂×#11758198	1	1	yes	characteristic
146	C₂²	C₄×C₂²⋊D₆₁	1	2	yes	—
147	C₂²	#14519803	1	1	yes	characteristic
148	C₂²	#14519807	1	2	yes	—
149	C₂²	#14519808	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	122	4	no	—
154	C₄	—	244	2	no	—
155	C₄	C₄²×D₆₁	1	1	yes	characteristic
156	C₄	#14519734	1	1	yes	characteristic
157	C₄	C₄×D₂₄₄	1	2	yes	—
158	C₂	C₄×#11758198	1	1	yes	characteristic
159	C₂	#16839761	1	1	yes	characteristic
160	C₂	#16839770	1	1	yes	characteristic
161	C₂	#16839862	1	2	yes	—
162	C₂	#16839867	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₂₂

C₂

yes

maximal, characteristic

C₂×#14519700

C₂

yes

maximal

C₂×#14519688

C₂

yes

maximal, characteristic

#16839524

C₂

yes

maximal

C₄×C₄⋊C₄×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

characteristic

C₂²×#92418

C₂²

yes

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

C₄

yes

complemented

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

C₂×#11758196

—

C₂×C₄×Dic₆₁

—

#14519700

C₄

yes

complemented

#14519688

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#95×C₆₁

—

C₂²×C₄×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×C₄

yes

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

D₄

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₂×C₄

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×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

C₄²

yes

complemented

#92418

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₄×D₄

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄²

yes

characteristic

C₄×C₆₁

#117

yes

characteristic

C₄×C₆₁

C₄×D₄

yes

—

Dic₆₁

—

Dic₆₁

—

#1763

—

maximal, Sylow

C₂×C₆₁

C₄×C₂²⋊C₄

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#441

yes

—

C₂×C₄×Q₈

—

C₂×#103

—

C₂×#97

—

C₄×C₄⋊C₄

D₆₁

yes

characteristic

#340

—

C₆₁

#1763

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#103

—

#97

—

122

—

100

#95

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

122

—

104

C₂×Q₈

—

122

—

105

C₂²×C₄

—

106

C₂²×C₄

—

122

—

107

C₂²×C₄

—

122

—

108

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

109

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

110

C₂²×C₄

C₂²×D₆₁

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

122

—

113

C₂×C₈

—

122

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

122

—

117

C₄⋊C₄

—

122

—

118

C₄⋊C₄

C₄×D₆₁

yes

—

119

C₄²

—

120

C₄²

—

122

—

121

C₄²

C₄×D₆₁

yes

—

122

C₄²

D₂₄₄

yes

—

123

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

124

Q₈

—

125

Q₈

—

122

—

126

Q₈

—

122

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

122

—

134

C₂×C₄

—

122

—

135

C₂×C₄

—

122

—

136

C₂×C₄

—

244

—

137

C₂×C₄

—

244

—

138

C₂×C₄

#11758198

yes

—

139

C₂×C₄

#11758198

yes

—

140

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

141

C₂×C₄

C₂×C₄×D₆₁

yes

—

142

C₂×C₄

C₂×D₂₄₄

yes

characteristic

143

C₂×C₄

#11758222

yes

—

144

C₈

—

122

—

145

C₂²

C₂×#11758198

yes

characteristic

146

C₂²

C₄×C₂²⋊D₆₁

yes

—

147

C₂²

#14519803

yes

characteristic

148

C₂²

#14519807

yes

—

149

C₂²

#14519808

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

122

—

154

C₄

—

244

—

155

C₄

C₄²×D₆₁

yes

characteristic

156

C₄

#14519734

yes

characteristic

157

C₄

C₄×D₂₄₄

yes

—

158

C₂

C₄×#11758198

yes

characteristic

159

C₂

#16839761

yes

characteristic

160

C₂

#16839770

yes

characteristic

161

C₂

#16839862

yes

—

162

C₂

#16839867

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₄×C₄⋊C₄×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₄×C₄⋊C₄	D₆₁
hypocenter	C₆₁	#1763
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₄×C₆₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₄×C₄⋊C₄×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₄×C₄⋊C₄

D₆₁

hypocenter

C₆₁

#1763

LCS

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
9. C₄×Dic₁₂₂	C₄	semidirect
13. #14519700	C₄	semidirect
61. Dic₁₂₂	C₄²	semidirect
87. C₆₁	#1763	semidirect

H ≅ G/N

type

9. C₄×Dic₁₂₂

C₄

semidirect

13. #14519700

C₄

semidirect

61. Dic₁₂₂

C₄²

semidirect

87. C₆₁

#1763

semidirect

Subgroups — #21445652

Conjugacy classes of subgroups

Complements