Group #21461685 — subgroups

Subgroups — #21461685

Order 7808.

272 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 80 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	#16840765	C₂	1	2	yes	maximal
2	C₂×#14519838	C₂	1	2	yes	maximal
3	#16840642	C₂	1	2	yes	maximal
4	C₂×C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758217	C₂²	1	4	yes	—
6	#14519844	—	2	2	no	—
7	C₄⋊#92420	—	2	2	no	—
8	#14519838	—	2	4	no	—
9	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₂₂	—	2	2	no	—
11	C₄×Dic₁₂₂	—	2	2	no	—
12	C₂×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
13	C₂×#11758196	C₄	1	1	yes	characteristic
14	C₂×#11758196	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₆₁	C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₆₁	C₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₂₂	—	2	2	no	—
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₂×Dic₁₂₂	D₄	1	2	yes	—
28	C₂×Dic₁₂₂	Q₈	1	2	yes	—
29	#11758217	—	2	4	no	—
30	#11758217	—	2	4	no	—
31	C₂×#92420	—	2	4	no	—
32	C₄²×C₆₁	—	2	1	no	—
33	C₄²×C₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	2	no	—
35	#11758196	—	2	2	no	—
36	#11758196	—	2	2	no	—
37	C₄×Dic₆₁	—	2	1	no	—
38	C₄×Dic₆₁	—	2	1	no	—
39	C₄×Dic₆₁	D₄	1	2	yes	—
40	C₄×Dic₆₁	D₄	1	2	yes	—
41	C₂³×C₆₁	C₂²×C₄	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	1	no	—
46	C₂×C₄×C₆₁	—	4	1	no	—
47	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
50	C₂×Dic₆₁	—	2	1	no	—
51	C₂×Dic₆₁	—	2	1	no	—
52	C₂×Dic₆₁	—	2	1	no	—
53	C₂×Dic₆₁	—	4	1	no	—
54	C₂×Dic₆₁	—	4	1	no	—
55	C₂×Dic₆₁	C₂²⋊C₄	1	2	yes	—
56	C₂×Dic₆₁	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
59	C₂×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×Dic₆₁	#41	1	2	yes	—
61	Dic₁₂₂	—	2	4	no	—
62	Dic₁₂₂	—	2	4	no	—
63	#92420	—	4	4	no	—
64	C₂²×C₆₁	—	2	1	no	—
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₆₁	#117	1	1	yes	characteristic
69	C₄×C₆₁	—	2	2	no	—
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	—	4	1	no	—
72	C₄×C₆₁	#101	1	2	yes	complemented
73	Dic₆₁	—	2	2	no	—
74	Dic₆₁	—	2	2	no	—
75	Dic₆₁	—	4	1	no	—
76	Dic₆₁	#121	1	2	yes	—
77	Dic₆₁	C₂²⋊Q₈	1	2	yes	—
78	#1801	—	61	1	no	maximal, Sylow
79	C₂×C₆₁	—	2	1	no	—
80	C₂×C₆₁	—	2	1	no	—
81	C₂×C₆₁	#380	1	1	yes	characteristic
82	C₂×C₆₁	C₂×#101	1	1	yes	characteristic
83	C₂×C₆₁	#413	1	1	yes	characteristic
84	C₂×C₄×Q₈	—	61	1	no	—
85	#423	—	61	2	no	—
86	C₂×#101	—	61	2	no	—
87	#343	—	61	2	no	—
88	C₆₁	#1801	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	—	61	1	no	—
90	C₂×M₁₆	—	61	4	no	—
91	C₄×Q₈	—	122	2	no	—
92	C₄×Q₈	—	122	2	no	—
93	C₂×C₄⋊C₄	—	61	1	no	—
94	C₂×C₄⋊C₄	—	61	1	no	—
95	C₂×C₄⋊C₄	—	61	1	no	—
96	C₂×C₄²	—	61	1	no	—
97	C₂×C₄²	—	61	1	no	—
98	C₂×C₄²	#6923	1	1	yes	characteristic
99	C₄⋊C₈	—	122	2	no	—
100	C₄⋊C₈	—	122	2	no	—
101	#101	—	122	4	no	—
102	C₂×Q₈	—	61	2	no	—
103	C₂×Q₈	—	61	2	no	—
104	C₂×Q₈	—	122	2	no	—
105	C₂×Q₈	—	122	2	no	—
106	C₂²×C₄	—	61	1	no	—
107	C₂²×C₄	—	61	1	no	—
108	C₂²×C₄	—	61	1	no	—
109	C₂²×C₄	—	61	1	no	—
110	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
111	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
112	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
113	M₁₆	—	122	4	no	—
114	M₁₆	—	122	4	no	—
115	C₂×C₈	—	122	4	no	—
116	C₄⋊C₄	—	122	2	no	—
117	C₄⋊C₄	—	122	2	no	—
118	C₄⋊C₄	—	122	2	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	61	2	no	—
122	C₄²	—	61	2	no	—
123	C₄²	—	122	1	no	—
124	C₄²	—	122	1	no	—
125	C₂³	C₂²×#6923	1	1	yes	characteristic
126	Q₈	—	122	4	no	—
127	Q₈	—	122	4	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	61	2	no	—
134	C₂×C₄	—	61	1	no	—
135	C₂×C₄	—	61	1	no	—
136	C₂×C₄	—	61	2	no	—
137	C₂×C₄	—	61	2	no	—
138	C₂×C₄	—	61	2	no	—
139	C₂×C₄	—	122	1	no	—
140	C₂×C₄	—	122	1	no	—
141	C₂×C₄	—	122	1	no	—
142	C₂×C₄	—	244	1	no	—
143	C₂×C₄	—	244	1	no	—
144	C₂×C₄	C₄⋊#6923	1	2	yes	—
145	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
146	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
147	C₈	—	244	4	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
151	C₂²	C₂×C₄⋊#6923	1	1	yes	characteristic
152	C₂²	#14519856	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	61	2	no	—
157	C₄	—	61	2	no	—
158	C₄	—	122	2	no	—
159	C₄	—	122	2	no	—
160	C₄	—	244	1	no	—
161	C₄	#14519838	1	2	yes	complemented
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#16840704	1	1	yes	characteristic
165	C₂	C₂×#14519838	1	1	yes	characteristic
166	C₂	#16840748	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840765

C₂

yes

maximal

C₂×#14519838

C₂

yes

maximal

#16840642

C₂

yes

maximal

C₂×C₄×Dic₁₂₂

C₂

yes

maximal, characteristic

C₂×#11758217

C₂²

yes

—

#14519844

—

C₄⋊#92420

—

#14519838

—

C₂²×Dic₁₂₂

C₂²

yes

characteristic

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

—

C₂×C₄⋊Dic₆₁

C₄

yes

characteristic

C₂×#11758196

C₄

yes

characteristic

C₂×#11758196

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

D₄

yes

—

C₂×Dic₁₂₂

Q₈

yes

—

#11758217

—

#11758217

—

C₂×#92420

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

D₄

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

—

C₂×Dic₆₁

C₄⋊C₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×Q₈

yes

characteristic

C₂×Dic₆₁

#41

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

#117

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#101

yes

complemented

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#121

yes

—

Dic₆₁

C₂²⋊Q₈

yes

—

#1801

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄×Q₈

—

#423

—

C₂×#101

—

#343

—

C₆₁

#1801

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄×Q₈

—

122

—

C₄×Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#6923

yes

characteristic

C₄⋊C₈

—

122

—

100

C₄⋊C₈

—

122

—

101

#101

—

122

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

122

—

105

C₂×Q₈

—

122

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂×#6923

yes

characteristic

111

C₂²×C₄

C₂×#6923

yes

characteristic

112

C₂²×C₄

C₂×#6923

yes

characteristic

113

M₁₆

—

122

—

114

M₁₆

—

122

—

115

C₂×C₈

—

122

—

116

C₄⋊C₄

—

122

—

117

C₄⋊C₄

—

122

—

118

C₄⋊C₄

—

122

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

122

—

124

C₄²

—

122

—

125

C₂³

C₂²×#6923

yes

characteristic

126

Q₈

—

122

—

127

Q₈

—

122

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

122

—

140

C₂×C₄

—

122

—

141

C₂×C₄

—

122

—

142

C₂×C₄

—

244

—

143

C₂×C₄

—

244

—

144

C₂×C₄

C₄⋊#6923

yes

—

145

C₂×C₄

C₂²⋊#6923

yes

characteristic

146

C₂×C₄

C₂²⋊#6923

yes

characteristic

147

C₈

—

244

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×C₂²⋊#6923

yes

characteristic

151

C₂²

C₂×C₄⋊#6923

yes

characteristic

152

C₂²

#14519856

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

122

—

159

C₄

—

122

—

160

C₄

—

244

—

161

C₄

#14519838

yes

complemented

162

C₂

—

163

C₂

—

164

C₂

#16840704

yes

characteristic

165

C₂

C₂×#14519838

yes

characteristic

166

C₂

#16840748

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂³×C₆₁	C₂²×C₄
Frattini	C₂³	C₂²×#6923
Fitting	C₂×C₄²×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₂×C₄²	#6923
hypocenter	C₆₁	#1801
LCS	G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂³×C₆₁

C₂²×C₄

Frattini

C₂³

C₂²×#6923

Fitting

C₂×C₄²×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₂×C₄²

C₆₁

G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
72. C₄×C₆₁	#101	semidirect
88. C₆₁	#1801	semidirect
161. C₄	#14519838	semidirect

H ≅ G/N

type

72. C₄×C₆₁

#101

semidirect

88. C₆₁

#1801

semidirect

161. C₄

#14519838

semidirect

Subgroups — #21461685

Conjugacy classes of subgroups

Complements