Group #21461997 — subgroups

Subgroups — #21461997

Order 7808.

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

normal subgroups only

This group has 292 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	C₄×C₄⋊#6923	C₂	1	1	yes	maximal, characteristic
3	#16840635	C₂	1	4	yes	maximal
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	(C₂×#6923)⋊C₄	—	2	1	no	—
11	C₂×Q₈×D₆₁	—	2	1	no	—
12	#14519777	C₄	1	1	yes	characteristic, complemented
13	#14519776	—	2	1	no	—
14	#14519776	C₄	1	2	yes	complemented
15	#14519775	—	2	1	no	—
16	C₄⋊C₄×D₆₁	—	2	1	no	—
17	C₄⋊C₄×D₆₁	C₂²	1	2	yes	—
18	C₄²×D₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic, complemented
20	C₂²×#6923	—	2	2	no	—
21	C₂×Q₈×C₆₁	—	2	1	no	—
22	Q₈×D₆₁	—	4	4	no	—
23	C₂×C₄×D₆₁	—	2	2	no	—
24	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
26	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₂₂	—	2	1	no	—
28	C₂×Dic₁₂₂	—	2	2	no	—
29	C₂×#92420	—	2	2	no	—
30	C₄⋊#6923	—	2	2	no	—
31	C₄⋊#6923	Q₈	1	4	yes	complemented
32	C₄×#6923	—	4	2	no	—
33	D₆₁⋊C₈	—	2	4	no	—
34	C₄⋊C₄×C₆₁	—	2	1	no	—
35	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
36	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄⋊Dic₆₁	—	2	1	no	—
38	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
39	#11758196	—	2	2	no	—
40	#11758196	—	2	2	no	—
41	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₆₁	D₄	1	1	yes	characteristic
43	C₄×Dic₆₁	D₄	1	1	yes	characteristic
44	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×#6923	—	2	4	no	—
46	C₂×#6923	—	4	2	no	—
47	C₂×#6923	—	4	2	no	—
48	Q₈×C₆₁	—	4	2	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₆₁	—	2	2	no	—
54	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
57	C₄×D₆₁	—	2	2	no	—
58	C₄×D₆₁	—	4	4	no	—
59	C₄×D₆₁	SD₁₆	1	2	yes	—
60	C₄×D₆₁	Q₁₆	1	2	yes	—
61	C₄×D₆₁	C₂×Q₈	1	2	yes	—
62	C₄×D₆₁	#41	1	2	yes	—
63	Dic₁₂₂	—	4	4	no	—
64	Dic₁₂₂	—	4	2	no	—
65	#92420	—	4	2	no	—
66	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×D₆₁	C₂²⋊Q₈	1	2	yes	—
68	C₂×D₆₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×D₆₁	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×D₆₁	AGL(1,ℤ/8)	1	2	yes	—
71	#6923	—	4	4	no	—
72	C₄×C₆₁	—	2	1	no	—
73	C₄×C₆₁	—	4	2	no	—
74	C₄×C₆₁	#103	1	2	yes	—
75	C₄×C₆₁	C₄×Q₈	1	2	yes	—
76	Dic₆₁	—	2	1	no	—
77	Dic₆₁	—	4	2	no	—
78	Dic₆₁	#103	1	2	yes	—
79	Dic₆₁	#124	1	1	yes	characteristic
80	Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
81	#1987	—	61	1	no	maximal, Sylow
82	C₂×C₆₁	#391	1	1	yes	characteristic
83	C₂×C₆₁	C₂×#103	1	1	yes	characteristic
84	C₂×C₆₁	#418	1	1	yes	characteristic
85	D₆₁	#476	1	2	yes	—
86	D₆₁	#477	1	2	yes	—
87	C₂×C₄⋊Q₈	—	61	1	no	—
88	C₂×C₄⋊C₈	—	61	1	no	—
89	C₄×C₄⋊C₄	—	61	1	no	—
90	#340	—	61	4	no	—
91	C₆₁	#1987	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	—	122	1	no	—
93	C₂²×C₈	—	61	2	no	—
94	C₄⋊Q₈	—	61	2	no	—
95	C₄⋊Q₈	—	61	1	no	—
96	C₄⋊Q₈	—	122	1	no	—
97	C₄⋊Q₈	—	122	1	no	—
98	C₄⋊Q₈	#6923	1	1	yes	characteristic, complemented
99	C₂×C₄⋊C₄	—	61	2	no	—
100	C₂×C₄⋊C₄	—	61	2	no	—
101	C₂×C₄⋊C₄	—	122	1	no	—
102	C₂×C₄²	—	61	1	no	—
103	C₂×C₄²	—	122	1	no	—
104	C₄⋊C₈	—	122	1	no	—
105	C₄⋊C₈	—	122	1	no	—
106	#95	—	122	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	122	2	no	—
109	C₂×Q₈	—	122	1	no	—
110	C₂×Q₈	—	244	4	no	—
111	C₂²×C₄	—	61	1	no	—
112	C₂²×C₄	—	61	1	no	—
113	C₂²×C₄	—	61	1	no	—
114	C₂²×C₄	—	122	2	no	—
115	C₂²×C₄	—	122	2	no	—
116	C₂×C₈	—	122	2	no	—
117	C₂×C₈	—	122	4	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	61	4	no	—
120	C₄⋊C₄	—	61	2	no	—
121	C₄⋊C₄	—	122	2	no	—
122	C₄⋊C₄	—	122	2	no	—
123	C₄⋊C₄	—	122	2	no	—
124	C₄⋊C₄	—	122	1	no	—
125	C₄⋊C₄	C₂×#6923	1	2	yes	—
126	C₄²	—	61	1	no	—
127	C₄²	—	61	1	no	—
128	C₄²	—	61	1	no	—
129	C₄²	—	244	2	no	—
130	C₄²	C₂×#6923	1	1	yes	characteristic
131	C₂³	—	61	1	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	244	2	no	—
134	Q₈	—	244	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	61	1	no	—
137	C₂×C₄	—	61	1	no	—
138	C₂×C₄	—	61	2	no	—
139	C₂×C₄	—	61	2	no	—
140	C₂×C₄	—	61	2	no	—
141	C₂×C₄	—	61	2	no	—
142	C₂×C₄	—	61	1	no	—
143	C₂×C₄	—	122	2	no	—
144	C₂×C₄	—	122	4	no	—
145	C₂×C₄	—	122	2	no	—
146	C₂×C₄	—	244	4	no	—
147	C₂×C₄	—	244	2	no	—
148	C₂×C₄	—	244	2	no	—
149	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
150	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
151	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
152	C₈	—	244	2	no	—
153	C₂²	—	61	2	no	—
154	C₂²	—	61	1	no	—
155	C₂²	—	61	1	no	—
156	C₂²	—	61	2	no	—
157	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	4	2	no	—
160	C₄	—	61	1	no	—
161	C₄	—	61	1	no	—
162	C₄	—	61	2	no	—
163	C₄	—	122	1	no	—
164	C₄	—	244	4	no	—
165	C₄	—	244	2	no	—
166	C₄	#14519840	1	2	yes	—
167	C₄	Q₈×#6923	1	2	yes	—
168	C₂	—	61	2	no	—
169	C₂	—	61	2	no	—
170	C₂	(C₂×#6923)⋊Q₈	1	1	yes	characteristic
171	C₂	C₂×#14519840	1	1	yes	characteristic
172	C₂	#16840754	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₆₁⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊#6923

C₂

yes

maximal, characteristic

#16840635

C₂

yes

maximal

C₄⋊Q₈×D₆₁

C₂

yes

maximal, characteristic

C₂×D₆₁⋊C₈

C₂²

yes

—

C₂×C₄⋊#6923

C₂²

yes

—

C₂×C₄×#6923

—

#14519844

—

C₄⋊#92420

—

(C₂×#6923)⋊C₄

—

C₂×Q₈×D₆₁

—

#14519777

C₄

yes

characteristic, complemented

#14519776

—

#14519776

C₄

yes

complemented

#14519775

—

C₄⋊C₄×D₆₁

—

C₄⋊C₄×D₆₁

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₄

yes

characteristic, complemented

C₂²×#6923

—

C₂×Q₈×C₆₁

—

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×#92420

—

C₄⋊#6923

—

C₄⋊#6923

Q₈

yes

complemented

C₄×#6923

—

D₆₁⋊C₈

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#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×Q₈

yes

—

C₄×D₆₁

#41

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

C₂²⋊Q₈

yes

—

C₂×D₆₁

C₂×SD₁₆

yes

characteristic

C₂×D₆₁

C₂×Q₁₆

yes

characteristic

C₂×D₆₁

AGL(1,ℤ/8)

yes

—

#6923

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#103

yes

—

C₄×C₆₁

C₄×Q₈

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#103

yes

—

Dic₆₁

#124

yes

characteristic

Dic₆₁

C₄⋊Q₈

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

yes

—

D₆₁

#477

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₆₁

#1987

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

#6923

yes

characteristic, complemented

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

122

—

102

C₂×C₄²

—

103

C₂×C₄²

—

122

—

104

C₄⋊C₈

—

122

—

105

C₄⋊C₈

—

122

—

106

#95

—

122

—

107

C₂×Q₈

—

108

C₂×Q₈

—

122

—

109

C₂×Q₈

—

122

—

110

C₂×Q₈

—

244

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

122

—

115

C₂²×C₄

—

122

—

116

C₂×C₈

—

122

—

117

C₂×C₈

—

122

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

—

122

C₄⋊C₄

—

122

—

123

C₄⋊C₄

—

122

—

124

C₄⋊C₄

—

122

—

125

C₄⋊C₄

C₂×#6923

yes

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

244

—

130

C₄²

C₂×#6923

yes

characteristic

131

C₂³

—

132

Q₈

—

133

Q₈

—

244

—

134

Q₈

—

244

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

122

—

144

C₂×C₄

—

122

—

145

C₂×C₄

—

122

—

146

C₂×C₄

—

244

—

147

C₂×C₄

—

244

—

148

C₂×C₄

—

244

—

149

C₂×C₄

C₂²⋊#6923

yes

characteristic

150

C₂×C₄

C₂²⋊#6923

yes

characteristic

151

C₂×C₄

C₂²×#6923

yes

characteristic

152

C₈

—

244

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₂²⋊#6923

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

122

—

164

C₄

—

244

—

165

C₄

—

244

—

166

C₄

#14519840

yes

—

167

C₄

Q₈×#6923

yes

—

168

C₂

—

169

C₂

—

170

C₂

(C₂×#6923)⋊Q₈

yes

characteristic

171

C₂

C₂×#14519840

yes

characteristic

172

C₂

#16840754

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	#1987
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
12. #14519777	C₄	semidirect
14. #14519776	C₄	semidirect
19. C₄⋊Q₈×C₆₁	C₄	semidirect
31. C₄⋊#6923	Q₈	semidirect
91. C₆₁	#1987	semidirect
98. C₄⋊Q₈	#6923	semidirect

H ≅ G/N

type

12. #14519777

C₄

semidirect

14. #14519776

C₄

semidirect

19. C₄⋊Q₈×C₆₁

C₄

semidirect

31. C₄⋊#6923

Q₈

semidirect

91. C₆₁

#1987

semidirect

98. C₄⋊Q₈

#6923

semidirect

Subgroups — #21461997

Conjugacy classes of subgroups

Complements