Group #21445061 — subgroups

Subgroups — #21445061

Order 7808.

194 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16840165	C₂	1	1	yes	maximal, characteristic
2	#16840047	C₂	1	1	yes	maximal, characteristic
3	#16839972	C₂	1	1	yes	maximal, characteristic
4	#16839508	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#92418	C₂	1	1	yes	maximal, characteristic
6	Q₈⋊#92418	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#557	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758202	—	2	1	no	—
9	#14519777	C₂²	1	1	yes	characteristic
10	#14519710	—	2	2	no	—
11	#14519708	—	2	1	no	—
12	C₄⋊#92418	C₂²	1	1	yes	characteristic
13	C₄⋊#92418	C₂²	1	1	yes	characteristic
14	#14519701	—	2	1	no	—
15	#14519700	—	2	1	no	—
16	C₄×#92418	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₆₁	—	2	1	no	—
18	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	#125×C₆₁	—	2	2	no	—
20	#125×C₆₁	—	2	1	no	—
21	C₄×Q₈×C₆₁	—	2	1	no	—
22	C₄×Q₈×C₆₁	—	2	1	no	—
23	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₆₁	—	2	1	no	—
26	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂×Dic₁₂₂	—	2	1	no	—
29	C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₄⋊C₄×C₆₁	—	2	2	no	—
31	C₄⋊C₄×C₆₁	—	2	2	no	—
32	C₄⋊C₄×C₆₁	—	2	1	no	—
33	C₄⋊C₄×C₆₁	—	2	1	no	—
34	C₄⋊C₄×C₆₁	—	2	1	no	—
35	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
38	C₄²×C₆₁	—	2	2	no	—
39	C₄²×C₆₁	—	2	1	no	—
40	C₄²×C₆₁	—	2	1	no	—
41	C₄²×C₆₁	C₂³	1	1	yes	characteristic
42	#11758202	—	4	2	no	—
43	C₄⋊Dic₆₁	—	2	1	no	—
44	C₄⋊Dic₆₁	—	2	1	no	—
45	C₂×#92418	—	2	1	no	—
46	C₂×#92418	—	2	1	no	—
47	C₂×#92418	—	2	1	no	—
48	Q₈×C₆₁	—	2	1	no	—
49	Q₈×C₆₁	—	2	1	no	—
50	Q₈×C₆₁	—	2	1	no	—
51	Q₈×C₆₁	—	4	1	no	—
52	Q₈×C₆₁	—	4	1	no	—
53	Q₈×C₆₁	Q₁₆	1	2	yes	complemented
54	C₂×C₄×C₆₁	—	2	1	no	—
55	C₂×C₄×C₆₁	—	2	1	no	—
56	C₂×C₄×C₆₁	—	2	1	no	—
57	C₂×C₄×C₆₁	—	2	1	no	—
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₆₁	—	4	1	no	—
63	Dic₁₂₂	—	4	1	no	—
64	Dic₁₂₂	—	4	1	no	—
65	#92418	—	2	2	no	—
66	#92418	—	4	1	no	—
67	#92418	—	4	1	no	—
68	C₂²×C₆₁	#120	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	—	4	2	no	—
73	C₄×C₆₁	—	4	1	no	—
74	C₄×C₆₁	—	4	1	no	—
75	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₄×C₆₁	#135	1	1	yes	characteristic
77	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₆₁	#137	1	1	yes	characteristic
79	Dic₆₁	—	8	1	no	—
80	#1648	—	61	1	no	maximal, Sylow
81	C₂×C₆₁	#449	1	1	yes	characteristic
82	C₂×C₆₁	#451	1	1	yes	characteristic
83	C₂×C₆₁	#456	1	1	yes	characteristic
84	#557	D₆₁	1	1	yes	characteristic
85	#500	—	61	1	no	—
86	#475	—	61	1	no	—
87	#462	—	61	1	no	—
88	#332	—	61	1	no	—
89	Q₈⋊C₈	—	61	1	no	—
90	Q₈⋊C₈	—	61	1	no	—
91	C₆₁	#1648	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	122	1	no	—
93	C₄⋊Q₈	—	2	1	no	—
94	C₄⋊Q₈	—	61	1	no	—
95	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
96	#125	—	2	2	no	—
97	#125	—	2	1	no	—
98	C₄×Q₈	—	2	1	no	—
99	C₄×Q₈	—	2	1	no	—
100	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
101	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
102	#107	—	122	2	no	—
103	#106	—	122	1	no	—
104	C₄⋊C₈	—	61	1	no	—
105	C₄⋊C₈	—	61	1	no	—
106	#103	—	122	1	no	—
107	#103	—	122	1	no	—
108	C₄×C₈	—	61	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	122	1	no	—
111	C₂×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
112	C₂×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
113	Q₁₆	—	244	2	no	—
114	C₂×C₈	—	122	1	no	—
115	C₂×C₈	—	122	1	no	—
116	C₂×C₈	—	122	1	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	122	1	no	—
124	C₄⋊C₄	—	122	1	no	—
125	C₄⋊C₄	C₂²⋊D₆₁	1	1	yes	characteristic
126	C₄⋊C₄	C₂²⋊D₆₁	1	1	yes	characteristic
127	C₄⋊C₄	C₂²⋊D₆₁	1	2	yes	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	C₂²×D₆₁	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	2	1	no	—
134	Q₈	—	2	1	no	—
135	Q₈	—	4	1	no	—
136	Q₈	—	4	1	no	—
137	Q₈	—	244	1	no	—
138	Q₈	—	244	1	no	—
139	Q₈	#11758202	1	2	yes	complemented
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	244	1	no	—
146	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
149	C₈	—	122	2	no	—
150	C₈	—	244	1	no	—
151	C₈	—	244	1	no	—
152	C₂²	#14519746	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	1	no	—
156	C₄	—	4	1	no	—
157	C₄	—	4	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	488	1	no	—
160	C₄	C₂×#11758202	1	1	yes	characteristic
161	C₄	#14519797	1	1	yes	characteristic
162	C₄	#14519802	1	1	yes	characteristic
163	C₄	#14519808	1	1	yes	characteristic
164	C₂	#16839907	1	1	yes	characteristic
165	C₂	#16839918	1	1	yes	characteristic
166	C₂	#16839945	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840165

C₂

yes

maximal, characteristic

#16840047

C₂

yes

maximal, characteristic

#16839972

C₂

yes

maximal, characteristic

#16839508

C₂

yes

maximal, characteristic

Q₈⋊#92418

C₂

yes

maximal, characteristic

Q₈⋊#92418

C₂

yes

maximal, characteristic

C₆₁×#557

C₂

yes

maximal, characteristic

C₂×#11758202

—

#14519777

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#125×C₆₁

—

#125×C₆₁

—

C₄×Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758202

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

Q₁₆

yes

complemented

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

#92418

—

#92418

—

#92418

—

C₂²×C₆₁

#120

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×Q₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

yes

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₆₁

#1648

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

C₂×D₆₁

yes

characteristic

101

C₄×Q₈

C₂×D₆₁

yes

characteristic

102

#107

—

122

—

103

#106

—

122

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#103

—

122

—

107

#103

—

122

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

122

—

111

C₂×Q₈

C₂²⋊D₆₁

yes

characteristic

112

C₂×Q₈

C₂²⋊D₆₁

yes

characteristic

113

Q₁₆

—

244

—

114

C₂×C₈

—

122

—

115

C₂×C₈

—

122

—

116

C₂×C₈

—

122

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

122

—

124

C₄⋊C₄

—

122

—

125

C₄⋊C₄

C₂²⋊D₆₁

yes

characteristic

126

C₄⋊C₄

C₂²⋊D₆₁

yes

characteristic

127

C₄⋊C₄

C₂²⋊D₆₁

yes

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₆₁

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

Q₈

—

244

—

138

Q₈

—

244

—

139

Q₈

#11758202

yes

complemented

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

244

—

146

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

147

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

149

C₈

—

122

—

150

C₈

—

244

—

151

C₈

—

244

—

152

C₂²

#14519746

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

488

—

160

C₄

C₂×#11758202

yes

characteristic

161

C₄

#14519797

yes

characteristic

162

C₄

#14519802

yes

characteristic

163

C₄

#14519808

yes

characteristic

164

C₂

#16839907

yes

characteristic

165

C₂

#16839918

yes

characteristic

166

C₂

#16839945

yes

characteristic

	subgroup	G/H
center	C₂²	#14519746
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#557	C₂
socle	C₂²×C₆₁	#120
hypercenter	#557	D₆₁
hypocenter	C₆₁	#1648
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #557

subgroup

G/H

center

C₂²

#14519746

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#557

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #557

N	H ≅ G/N	type
53. Q₈×C₆₁	Q₁₆	semidirect
91. C₆₁	#1648	semidirect
139. Q₈	#11758202	semidirect

H ≅ G/N

type

53. Q₈×C₆₁

Q₁₆

semidirect

91. C₆₁

#1648

semidirect

139. Q₈

#11758202

semidirect

Subgroups — #21445061

Conjugacy classes of subgroups

Complements