Group #21461985 — subgroups

Subgroups — #21461985

Order 7808.

324 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 88 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	D₆₁⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519840	C₂	1	4	yes	maximal
3	C₄×C₄⋊#6923	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×D₆₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#6923	C₂²	1	2	yes	—
7	C₂×C₄×#6923	—	2	1	no	—
8	#14519844	—	2	1	no	—
9	C₄⋊#92420	—	2	1	no	—
10	#14519840	—	2	8	no	—
11	(C₂×#6923)⋊C₄	—	2	1	no	—
12	C₂×Q₈×D₆₁	C₂²	1	2	yes	—
13	#14519777	C₄	1	1	yes	characteristic, complemented
14	#14519776	—	2	1	no	—
15	#14519776	—	2	1	no	—
16	#14519775	C₄	1	2	yes	complemented
17	C₄⋊C₄×D₆₁	—	2	1	no	—
18	C₄⋊C₄×D₆₁	—	2	1	no	—
19	C₄²×D₆₁	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic, complemented
21	C₂²×#6923	—	2	2	no	—
22	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
23	Q₈×D₆₁	—	2	4	no	—
24	Q₈×D₆₁	—	2	4	no	—
25	C₂×C₄×D₆₁	—	2	2	no	—
26	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
27	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
28	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₂₂	—	2	2	no	—
30	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
31	C₂×#92420	—	2	2	no	—
32	C₄⋊#6923	—	2	2	no	—
33	C₄⋊#6923	D₄	1	4	yes	—
34	C₄×#6923	—	4	2	no	—
35	D₆₁⋊C₈	—	2	4	no	—
36	C₄⋊C₄×C₆₁	—	2	1	no	—
37	C₄⋊C₄×C₆₁	—	2	1	no	—
38	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₄⋊Dic₆₁	—	2	1	no	—
40	C₄⋊Dic₆₁	—	2	1	no	—
41	#11758196	—	2	2	no	—
42	#11758196	—	2	2	no	—
43	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₆₁	D₄	1	1	yes	characteristic
45	C₄×Dic₆₁	D₄	1	1	yes	characteristic
46	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×#6923	—	2	4	no	—
48	C₂×#6923	—	4	2	no	—
49	C₂×#6923	—	4	2	no	—
50	Q₈×C₆₁	—	2	4	no	—
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₆₁	—	2	2	no	—
56	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
59	C₄×D₆₁	—	2	2	no	—
60	C₄×D₆₁	—	4	4	no	—
61	C₄×D₆₁	SD₁₆	1	2	yes	—
62	C₄×D₆₁	Q₁₆	1	2	yes	—
63	C₄×D₆₁	C₂×D₄	1	2	yes	—
64	C₄×D₆₁	#41	1	2	yes	—
65	Dic₁₂₂	—	2	4	no	—
66	Dic₁₂₂	—	4	2	no	—
67	Dic₁₂₂	—	4	2	no	—
68	#92420	—	4	2	no	—
69	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×D₆₁	#121	1	2	yes	—
71	C₂×D₆₁	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×D₆₁	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×D₆₁	#137	1	2	yes	—
74	#6923	—	4	4	no	—
75	C₄×C₆₁	—	2	1	no	—
76	C₄×C₆₁	—	4	2	no	—
77	C₄×C₆₁	#103	1	2	yes	complemented
78	C₄×C₆₁	C₄×D₄	1	2	yes	—
79	Dic₆₁	—	2	1	no	—
80	Dic₆₁	—	4	2	no	—
81	Dic₆₁	#103	1	2	yes	complemented
82	Dic₆₁	#124	1	1	yes	characteristic
83	Dic₆₁	#127	1	1	yes	characteristic
84	#1981	—	61	1	no	maximal, Sylow
85	C₂×C₆₁	#390	1	1	yes	characteristic
86	C₂×C₆₁	C₂×#103	1	1	yes	characteristic
87	C₂×C₆₁	#419	1	1	yes	characteristic
88	D₆₁	#461	1	2	yes	—
89	D₆₁	#462	1	2	yes	—
90	C₂×C₄⋊Q₈	—	61	1	no	—
91	C₂×C₄⋊C₈	—	61	1	no	—
92	C₂×#103	—	61	4	no	—
93	C₄×C₄⋊C₄	—	61	1	no	—
94	C₆₁	#1981	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	—	61	2	no	—
96	C₂²×C₈	—	61	2	no	—
97	C₄⋊Q₈	—	61	2	no	—
98	C₄⋊Q₈	—	61	1	no	—
99	C₄⋊Q₈	—	122	1	no	—
100	C₄⋊Q₈	—	122	1	no	—
101	C₄⋊Q₈	#6923	1	1	yes	characteristic, complemented
102	C₂×C₄⋊C₄	—	61	2	no	—
103	C₂×C₄⋊C₄	—	122	1	no	—
104	C₂×C₄⋊C₄	—	122	1	no	—
105	C₂×C₄²	—	61	1	no	—
106	C₂×C₄²	—	122	1	no	—
107	C₄⋊C₈	—	122	1	no	—
108	C₄⋊C₈	—	122	1	no	—
109	#103	—	122	8	no	—
110	#95	—	122	1	no	—
111	C₂×Q₈	—	61	2	no	—
112	C₂×Q₈	—	122	2	no	—
113	C₂×Q₈	—	122	4	no	—
114	C₂×Q₈	—	122	4	no	—
115	C₂×Q₈	C₂×#6923	1	2	yes	—
116	C₂²×C₄	—	61	1	no	—
117	C₂²×C₄	—	61	1	no	—
118	C₂²×C₄	—	61	1	no	—
119	C₂²×C₄	—	122	2	no	—
120	C₂²×C₄	—	122	2	no	—
121	C₂×C₈	—	122	4	no	—
122	C₂×C₈	—	122	2	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	61	4	no	—
126	C₄⋊C₄	—	122	2	no	—
127	C₄⋊C₄	—	122	2	no	—
128	C₄⋊C₄	—	122	2	no	—
129	C₄⋊C₄	—	122	1	no	—
130	C₄⋊C₄	—	122	1	no	—
131	C₄²	—	61	1	no	—
132	C₄²	—	61	1	no	—
133	C₄²	—	61	1	no	—
134	C₄²	—	244	2	no	—
135	C₄²	C₂×#6923	1	1	yes	characteristic
136	C₂³	—	61	1	no	—
137	Q₈	—	2	4	no	—
138	Q₈	—	122	4	no	—
139	Q₈	—	244	2	no	—
140	Q₈	—	244	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	61	1	no	—
143	C₂×C₄	—	61	1	no	—
144	C₂×C₄	—	61	2	no	—
145	C₂×C₄	—	61	2	no	—
146	C₂×C₄	—	61	2	no	—
147	C₂×C₄	—	61	2	no	—
148	C₂×C₄	—	61	1	no	—
149	C₂×C₄	—	122	2	no	—
150	C₂×C₄	—	122	4	no	—
151	C₂×C₄	—	122	2	no	—
152	C₂×C₄	—	244	4	no	—
153	C₂×C₄	—	244	2	no	—
154	C₂×C₄	—	244	2	no	—
155	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
156	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
157	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
158	C₈	—	244	2	no	—
159	C₂²	—	61	2	no	—
160	C₂²	—	61	1	no	—
161	C₂²	—	61	1	no	—
162	C₂²	—	61	2	no	—
163	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
164	C₄	—	2	1	no	—
165	C₄	—	4	2	no	—
166	C₄	—	61	1	no	—
167	C₄	—	61	1	no	—
168	C₄	—	61	2	no	—
169	C₄	—	122	1	no	—
170	C₄	—	244	4	no	—
171	C₄	—	244	2	no	—
172	C₄	#14519840	1	2	yes	complemented
173	C₄	D₄×#6923	1	2	yes	—
174	C₂	—	61	2	no	—
175	C₂	—	61	2	no	—
176	C₂	#16840714	1	1	yes	characteristic
177	C₂	C₂×#14519840	1	1	yes	characteristic
178	C₂	#16840755	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₆₁⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14519840

C₂

yes

maximal

C₄×C₄⋊#6923

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₆₁

C₂

yes

maximal, characteristic

C₂×D₆₁⋊C₈

C₂²

yes

—

C₂×C₄⋊#6923

C₂²

yes

—

C₂×C₄×#6923

—

—

—

—

—

C₂×Q₈×D₆₁

C₂²

yes

—

#14519777

C₄

yes

characteristic, complemented

#14519776

—

#14519776

—

#14519775

C₄

yes

complemented

C₄⋊C₄×D₆₁

—

C₄⋊C₄×D₆₁

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₄

yes

characteristic, complemented

C₂²×#6923

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

Q₈×D₆₁

—

Q₈×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×#92420

—

C₄⋊#6923

—

C₄⋊#6923

D₄

yes

—

C₄×#6923

—

D₆₁⋊C₈

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

#11758196

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₂²×D₆₁

C₂×D₄

yes

—

—

—

Q₈×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×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

SD₁₆

yes

—

C₄×D₆₁

Q₁₆

yes

—

C₄×D₆₁

C₂×D₄

yes

—

C₄×D₆₁

#41

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

#121

yes

—

C₂×D₆₁

C₂×SD₁₆

yes

characteristic

C₂×D₆₁

C₂×Q₁₆

yes

characteristic

C₂×D₆₁

#137

yes

—

#6923

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#103

yes

complemented

C₄×C₆₁

C₄×D₄

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

yes

Dic₆₁

yes

Dic₆₁

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

yes

—

D₆₁

#462

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₄×C₄⋊C₄

—

C₆₁

#1981

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

100

C₄⋊Q₈

—

122

—

101

C₄⋊Q₈

#6923

yes

characteristic, complemented

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

122

—

104

C₂×C₄⋊C₄

—

122

—

105

C₂×C₄²

—

106

C₂×C₄²

—

122

—

107

C₄⋊C₈

—

122

—

108

C₄⋊C₈

—

122

—

109

#103

—

122

—

110

#95

—

122

—

111

C₂×Q₈

—

112

C₂×Q₈

—

122

—

113

C₂×Q₈

—

122

—

114

C₂×Q₈

—

122

—

115

C₂×Q₈

C₂×#6923

yes

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

122

—

120

C₂²×C₄

—

122

—

121

C₂×C₈

—

122

—

122

C₂×C₈

—

122

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

122

—

127

C₄⋊C₄

—

122

—

128

C₄⋊C₄

—

122

—

129

C₄⋊C₄

—

122

—

130

C₄⋊C₄

—

122

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

244

—

135

C₄²

C₂×#6923

yes

characteristic

136

C₂³

—

137

Q₈

—

138

Q₈

—

122

—

139

Q₈

—

244

—

140

Q₈

—

244

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

122

—

150

C₂×C₄

—

122

—

151

C₂×C₄

—

122

—

152

C₂×C₄

—

244

—

153

C₂×C₄

—

244

—

154

C₂×C₄

—

244

—

155

C₂×C₄

C₂²⋊#6923

yes

characteristic

156

C₂×C₄

C₂²⋊#6923

yes

characteristic

157

C₂×C₄

C₂²×#6923

yes

characteristic

158

C₈

—

244

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

C₂×C₂²⋊#6923

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

122

—

170

C₄

—

244

—

171

C₄

—

244

—

172

C₄

#14519840

yes

complemented

173

C₄

D₄×#6923

yes

—

174

C₂

—

175

C₂

—

176

C₂

#16840714

yes

characteristic

177

C₂

C₂×#14519840

yes

characteristic

178

C₂

#16840755

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄⋊Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6923
hypocenter	C₆₁	#1981
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₄⋊Q₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₄⋊Q₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
13. #14519777	C₄	semidirect
16. #14519775	C₄	semidirect
20. C₄⋊Q₈×C₆₁	C₄	semidirect
77. C₄×C₆₁	#103	semidirect
81. Dic₆₁	#103	semidirect
94. C₆₁	#1981	semidirect
101. C₄⋊Q₈	#6923	semidirect
172. C₄	#14519840	semidirect

H ≅ G/N

type

13. #14519777

C₄

semidirect

16. #14519775

C₄

semidirect

20. C₄⋊Q₈×C₆₁

C₄

semidirect

77. C₄×C₆₁

#103

semidirect

81. Dic₆₁

#103

semidirect

94. C₆₁

#1981

semidirect

101. C₄⋊Q₈

#6923

semidirect

172. C₄

#14519840

semidirect

Subgroups — #21461985

Conjugacy classes of subgroups

Complements