Group #21446957 — subgroups

Subgroups — #21446957

Order 7808.

290 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 95 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×#14519700	C₂	1	2	yes	maximal
2	#16839668	C₂	1	1	yes	maximal, characteristic
3	#16839524	C₂	1	2	yes	maximal
4	#16839517	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
7	C₂²×#92418	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
9	#14519700	—	2	4	no	—
10	#14519684	—	2	1	no	—
11	#14519684	—	2	1	no	—
12	C₂²×Q₈×C₆₁	—	2	1	no	—
13	C₄⋊Q₈×C₆₁	—	2	4	no	—
14	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
15	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₆₁	—	2	4	no	—
18	C₂×Q₈×C₆₁	—	4	2	no	—
19	C₂×Q₈×C₆₁	—	4	2	no	—
20	C₂²×C₄×C₆₁	—	2	2	no	—
21	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₆₁	—	2	1	no	—
25	C₂²×Dic₆₁	—	2	1	no	—
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₂×Dic₁₂₂	—	2	2	no	—
28	C₂×Dic₁₂₂	—	2	1	no	—
29	C₂×Dic₁₂₂	Q₈	1	2	yes	complemented
30	C₄⋊C₄×C₆₁	—	2	4	no	—
31	C₄⋊C₄×C₆₁	—	2	2	no	—
32	C₄⋊C₄×C₆₁	D₄	1	4	yes	—
33	C₄²×C₆₁	—	2	2	no	—
34	C₄⋊Dic₆₁	—	2	2	no	—
35	C₂×#92418	—	2	2	no	—
36	C₂×#92418	—	2	2	no	—
37	C₂×#92418	—	2	2	no	—
38	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₆₁	—	4	8	no	—
40	C₂×C₄×C₆₁	—	2	1	no	—
41	C₂×C₄×C₆₁	—	2	1	no	—
42	C₂×C₄×C₆₁	—	2	1	no	—
43	C₂×C₄×C₆₁	—	2	1	no	—
44	C₂×C₄×C₆₁	—	2	4	no	—
45	C₂×C₄×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	—	4	2	no	—
47	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
48	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
49	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#41	1	2	yes	—
52	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
53	C₂×Dic₆₁	—	2	2	no	—
54	C₂×Dic₆₁	—	4	1	no	—
55	C₂×Dic₆₁	—	4	1	no	—
56	C₂×Dic₆₁	—	4	1	no	—
57	C₂×Dic₆₁	—	4	1	no	—
58	C₂×Dic₆₁	—	4	1	no	—
59	Dic₁₂₂	—	2	4	no	—
60	Dic₁₂₂	—	4	2	no	—
61	#92418	—	4	2	no	—
62	C₂²×C₆₁	#120	1	1	yes	characteristic
63	C₂²×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₆₁	#124	1	1	yes	characteristic
65	C₂²×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₆₁	#137	1	1	yes	characteristic
69	C₄×C₆₁	—	2	2	no	—
70	C₄×C₆₁	—	2	2	no	—
71	C₄×C₆₁	—	4	4	no	—
72	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₆₁	#124	1	2	yes	—
74	Dic₆₁	—	4	2	no	—
75	Dic₆₁	—	8	1	no	—
76	#2024	—	61	1	no	maximal, Sylow
77	C₂×C₆₁	#393	1	1	yes	characteristic
78	C₂×C₆₁	#450	1	1	yes	characteristic
79	C₂×C₆₁	#451	1	1	yes	characteristic
80	C₂×C₆₁	#475	1	1	yes	characteristic
81	C₂×C₆₁	#477	1	1	yes	characteristic
82	C₂×C₆₁	#487	1	1	yes	characteristic
83	C₂×C₆₁	#489	1	1	yes	characteristic
84	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
85	C₂×#103	—	61	2	no	—
86	#391	—	61	1	no	—
87	#340	—	61	2	no	—
88	#336	—	61	1	no	—
89	C₆₁	#2024	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	2	1	no	—
91	C₂²×Q₈	—	61	1	no	—
92	C₂²×C₈	—	61	2	no	—
93	C₄⋊Q₈	—	2	4	no	—
94	C₂×C₄⋊C₄	—	2	1	no	—
95	C₂×C₄⋊C₄	—	61	1	no	—
96	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
97	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
98	#103	—	122	4	no	—
99	#95	—	122	1	no	—
100	#95	—	122	1	no	—
101	C₂×Q₈	—	2	4	no	—
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	4	2	no	—
104	C₂×Q₈	—	61	2	no	—
105	C₂×Q₈	—	122	1	no	—
106	C₂×Q₈	—	122	2	no	—
107	C₂×Q₈	—	122	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	122	1	no	—
110	C₂²×C₄	—	122	1	no	—
111	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
114	C₂×C₈	—	122	2	no	—
115	C₂×C₈	—	122	2	no	—
116	C₂×C₈	—	122	2	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	122	2	no	—
120	C₄⋊C₄	C₂²⋊D₆₁	1	4	yes	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
123	Q₈	—	4	8	no	—
124	Q₈	—	122	4	no	—
125	Q₈	—	244	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	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	—	244	1	no	—
135	C₂×C₄	—	244	1	no	—
136	C₂×C₄	—	244	1	no	—
137	C₂×C₄	—	244	1	no	—
138	C₂×C₄	—	244	1	no	—
139	C₂×C₄	#11758200	1	2	yes	—
140	C₂×C₄	#11758202	1	2	yes	—
141	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
142	C₂×C₄	#11758226	1	2	yes	—
143	C₂×C₄	#11758226	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
145	C₈	—	244	2	no	—
146	C₂²	#14519746	1	1	yes	characteristic
147	C₂²	#14519753	1	1	yes	characteristic
148	C₂²	#14519766	1	1	yes	characteristic
149	C₂²	C₂×#11758200	1	1	yes	characteristic
150	C₂²	C₂×#11758202	1	1	yes	characteristic
151	C₂²	#14519803	1	1	yes	characteristic
152	C₂²	#14519808	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	4	no	—
156	C₄	—	488	1	no	—
157	C₄	—	244	2	no	—
158	C₄	#14519753	1	2	yes	—
159	C₄	#14519766	1	2	yes	—
160	C₂	#16839673	1	1	yes	characteristic
161	C₂	#16839913	1	1	yes	characteristic
162	C₂	#16839918	1	1	yes	characteristic
163	C₂	#16840044	1	1	yes	characteristic
164	C₂	#16840054	1	1	yes	characteristic
165	C₂	#16840114	1	1	yes	characteristic
166	C₂	#16840124	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519700

C₂

yes

maximal

#16839668

C₂

yes

maximal, characteristic

#16839524

C₂

yes

maximal

#16839517

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

characteristic

C₂²×#92418

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

—

—

C₂²×Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×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×Dic₁₂₂

Q₈

yes

complemented

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

#92418

—

C₂²×C₆₁

#120

yes

characteristic

C₂²×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₁

#124

yes

characteristic

C₂²×C₆₁

C₂×SD₁₆

yes

characteristic

C₂²×C₆₁

C₂×Q₁₆

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

#124

yes

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄⋊Q₈

D₆₁

yes

—

—

—

—

C₆₁

#2024

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#103

—

122

—

#95

—

122

—

100

#95

—

122

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

122

—

106

C₂×Q₈

—

122

—

107

C₂×Q₈

—

122

—

108

C₂²×C₄

—

109

C₂²×C₄

—

122

—

110

C₂²×C₄

—

122

—

111

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

112

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

113

C₂²×C₄

C₂²×D₆₁

yes

characteristic

114

C₂×C₈

—

122

—

115

C₂×C₈

—

122

—

116

C₂×C₈

—

122

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

122

—

120

C₄⋊C₄

C₂²⋊D₆₁

yes

—

121

C₄²

—

122

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

123

Q₈

—

124

Q₈

—

122

—

125

Q₈

—

244

—

126

C₂×C₄

—

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

—

244

—

135

C₂×C₄

—

244

—

136

C₂×C₄

—

244

—

137

C₂×C₄

—

244

—

138

C₂×C₄

—

244

—

139

C₂×C₄

#11758200

yes

—

140

C₂×C₄

#11758202

yes

—

141

C₂×C₄

Q₈×D₆₁

yes

characteristic

142

C₂×C₄

#11758226

yes

—

143

C₂×C₄

#11758226

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

145

C₈

—

244

—

146

C₂²

#14519746

yes

characteristic

147

C₂²

#14519753

yes

characteristic

148

C₂²

#14519766

yes

characteristic

149

C₂²

C₂×#11758200

yes

characteristic

150

C₂²

C₂×#11758202

yes

characteristic

151

C₂²

#14519803

yes

characteristic

152

C₂²

#14519808

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

488

—

157

C₄

—

244

—

158

C₄

#14519753

yes

—

159

C₄

#14519766

yes

—

160

C₂

#16839673

yes

characteristic

161

C₂

#16839913

yes

characteristic

162

C₂

#16839918

yes

characteristic

163

C₂

#16840044

yes

characteristic

164

C₂

#16840054

yes

characteristic

165

C₂

#16840114

yes

characteristic

166

C₂

#16840124

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×C₄⋊Q₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	#2024
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×C₄⋊Q₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₁

hypocenter

C₆₁

#2024

LCS

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
29. C₂×Dic₁₂₂	Q₈	semidirect
89. C₆₁	#2024	semidirect

H ≅ G/N

type

29. C₂×Dic₁₂₂

Q₈

semidirect

89. C₆₁

#2024

semidirect

Subgroups — #21446957

Conjugacy classes of subgroups

Complements