Group #21447144 — subgroups

Subgroups — #21447144

Order 7808.

282 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 99 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×#14519710	C₂	1	2	yes	maximal
2	#16839660	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×#92418	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
9	C₂×#11758196	—	2	1	no	—
10	#14519710	—	2	4	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	C₄⋊C₄×C₆₁	—	2	4	no	—
27	C₄⋊C₄×C₆₁	—	2	2	no	—
28	C₄⋊C₄×C₆₁	D₄	1	4	yes	—
29	C₄²×C₆₁	—	2	2	no	—
30	C₄⋊Dic₆₁	—	2	1	no	—
31	C₄⋊Dic₆₁	—	2	2	no	—
32	C₄⋊Dic₆₁	Q₈	1	2	yes	complemented
33	#11758196	—	4	2	no	—
34	C₂×#92418	—	2	2	no	—
35	C₂×#92418	—	2	2	no	—
36	C₂×#92418	Q₈	1	4	yes	complemented
37	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₆₁	—	4	8	no	—
39	C₂×C₄×C₆₁	—	2	1	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	4	no	—
44	C₂×C₄×C₆₁	—	4	2	no	—
45	C₂×C₄×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	D₈	1	2	yes	—
47	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
48	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
52	C₂×Dic₆₁	—	2	2	no	—
53	C₂×Dic₆₁	—	4	1	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	#92418	—	2	4	no	—
59	C₂²×C₆₁	#120	1	1	yes	characteristic
60	C₂²×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₆₁	#137	1	1	yes	characteristic
66	C₄×C₆₁	—	2	2	no	—
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	—	4	4	no	—
69	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
70	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
71	Dic₆₁	—	4	2	no	—
72	Dic₆₁	—	8	1	no	—
73	#2057	—	61	1	no	maximal, Sylow
74	C₂×C₆₁	#395	1	1	yes	characteristic
75	C₂×C₆₁	#447	1	1	yes	characteristic
76	C₂×C₆₁	#451	1	1	yes	characteristic
77	C₂×C₆₁	#474	1	1	yes	characteristic
78	C₂×C₆₁	#477	1	1	yes	characteristic
79	C₂×C₆₁	#500	1	1	yes	characteristic
80	C₂×C₆₁	#501	1	1	yes	characteristic
81	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
82	C₂×#107	—	61	2	no	—
83	#389	—	61	1	no	—
84	#340	—	61	2	no	—
85	#336	—	61	1	no	—
86	C₆₁	#2057	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₈	—	2	1	no	—
88	C₂²×C₈	—	61	2	no	—
89	C₄⋊Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	61	1	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	C₂×D₆₁	1	1	yes	characteristic
96	#107	—	122	4	no	—
97	#95	—	122	1	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	122	1	no	—
103	C₂²×C₄	—	122	1	no	—
104	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
107	C₂×C₈	—	61	4	no	—
108	C₂×C₈	—	122	2	no	—
109	C₂×C₈	—	122	2	no	—
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	61	2	no	—
113	C₄⋊C₄	—	122	1	no	—
114	C₄⋊C₄	—	122	2	no	—
115	C₄⋊C₄	—	244	2	no	—
116	C₄⋊C₄	C₂²⋊D₆₁	1	4	yes	—
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
119	Q₈	—	4	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	122	2	no	—
128	C₂×C₄	—	244	1	no	—
129	C₂×C₄	—	244	1	no	—
130	C₂×C₄	—	244	1	no	—
131	C₂×C₄	—	244	1	no	—
132	C₂×C₄	—	244	1	no	—
133	C₂×C₄	#11758199	1	2	yes	—
134	C₂×C₄	#11758202	1	2	yes	—
135	C₂×C₄	Q₈×D₆₁	1	2	yes	—
136	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
137	C₂×C₄	#11758226	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
139	C₈	—	122	4	no	—
140	C₂²	#14519746	1	1	yes	characteristic
141	C₂²	#14519753	1	1	yes	characteristic
142	C₂²	#14519775	1	1	yes	characteristic
143	C₂²	C₂×#11758199	1	1	yes	characteristic
144	C₂²	C₂×#11758202	1	1	yes	characteristic
145	C₂²	#14519803	1	1	yes	characteristic
146	C₂²	#14519808	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	488	1	no	—
151	C₄	—	244	2	no	—
152	C₄	#14519753	1	2	yes	—
153	C₄	#14519775	1	2	yes	—
154	C₂	#16839681	1	1	yes	characteristic
155	C₂	#16839896	1	1	yes	characteristic
156	C₂	#16839918	1	1	yes	characteristic
157	C₂	#16840039	1	1	yes	characteristic
158	C₂	#16840054	1	1	yes	characteristic
159	C₂	#16840165	1	1	yes	characteristic
160	C₂	#16840168	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519710

C₂

yes

maximal

#16839660

C₂

yes

maximal, characteristic

#16839524

C₂

yes

maximal

#16839517

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

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

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

Q₈

yes

—

—

—

Q₈

yes

complemented

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

D₈

yes

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#92418

—

C₂²×C₆₁

#120

yes

characteristic

C₂²×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₁

C₄⋊Q₈

yes

characteristic

C₂²×C₆₁

C₂×D₈

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

C₄⋊Q₈

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

#2057

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#107

—

122

—

#95

—

122

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

122

—

103

C₂²×C₄

—

122

—

104

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

105

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

106

C₂²×C₄

C₂²×D₆₁

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

122

—

109

C₂×C₈

—

122

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

122

—

114

C₄⋊C₄

—

122

—

115

C₄⋊C₄

—

244

—

116

C₄⋊C₄

C₂²⋊D₆₁

yes

—

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

122

—

128

C₂×C₄

—

244

—

129

C₂×C₄

—

244

—

130

C₂×C₄

—

244

—

131

C₂×C₄

—

244

—

132

C₂×C₄

—

244

—

133

C₂×C₄

#11758199

yes

—

134

C₂×C₄

#11758202

yes

—

135

C₂×C₄

Q₈×D₆₁

yes

—

136

C₂×C₄

Q₈×D₆₁

yes

characteristic

137

C₂×C₄

#11758226

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

139

C₈

—

122

—

140

C₂²

#14519746

yes

characteristic

141

C₂²

#14519753

yes

characteristic

142

C₂²

#14519775

yes

characteristic

143

C₂²

C₂×#11758199

yes

characteristic

144

C₂²

C₂×#11758202

yes

characteristic

145

C₂²

#14519803

yes

characteristic

146

C₂²

#14519808

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

488

—

151

C₄

—

244

—

152

C₄

#14519753

yes

—

153

C₄

#14519775

yes

—

154

C₂

#16839681

yes

characteristic

155

C₂

#16839896

yes

characteristic

156

C₂

#16839918

yes

characteristic

157

C₂

#16840039

yes

characteristic

158

C₂

#16840054

yes

characteristic

159

C₂

#16840165

yes

characteristic

160

C₂

#16840168

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	#2057
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

#2057

LCS

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
32. C₄⋊Dic₆₁	Q₈	semidirect
36. C₂×#92418	Q₈	semidirect
86. C₆₁	#2057	semidirect

H ≅ G/N

type

32. C₄⋊Dic₆₁

Q₈

semidirect

36. C₂×#92418

Q₈

semidirect

86. C₆₁

#2057

semidirect

Subgroups — #21447144

Conjugacy classes of subgroups

Complements