Group #21445281 — subgroups

Subgroups — #21445281

Order 7808.

182 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16840039	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839954	C₂	1	1	yes	maximal, characteristic
3	#16839841	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839515	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839508	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839506	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758199	—	2	1	no	—
9	C₂×#11758189	—	2	1	no	—
10	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
11	#14519710	—	2	1	no	—
12	C₄⋊#92418	C₂²	1	1	yes	characteristic
13	C₄⋊#92418	C₂²	1	1	yes	characteristic
14	C₄⋊#92418	C₂²	1	1	yes	characteristic
15	#14519698	—	2	1	no	—
16	#14519697	—	2	1	no	—
17	#14519689	—	2	1	no	—
18	C₂×SD₁₆×C₆₁	—	2	1	no	—
19	C₂×SD₁₆×C₆₁	—	2	1	no	—
20	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	#127×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	—	2	1	no	—
24	C₂×D₄×C₆₁	—	2	1	no	—
25	C₂×D₄×C₆₁	—	2	1	no	—
26	C₂×D₄×C₆₁	—	2	1	no	—
27	C₂×D₂₄₄	D₄	1	1	yes	characteristic
28	C₂×C₄×D₆₁	—	2	1	no	—
29	SD₁₆×C₆₁	—	4	2	no	—
30	SD₁₆×C₆₁	—	4	2	no	—
31	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₆₁	—	2	1	no	—
34	C₄⋊C₄×C₆₁	—	2	1	no	—
35	C₄²×C₆₁	C₂³	1	1	yes	characteristic
36	#11758199	—	4	2	no	—
37	#11758198	—	2	1	no	—
38	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
39	C₂×#92418	—	2	1	no	—
40	C₂×#92418	—	2	1	no	—
41	C₂×#92418	—	2	1	no	—
42	#11758189	—	4	2	no	—
43	C₂³×C₆₁	—	4	1	no	—
44	C₂²×D₆₁	—	2	1	no	—
45	Q₈×C₆₁	—	4	1	no	—
46	Q₈×C₆₁	—	4	1	no	—
47	D₄×C₆₁	—	4	1	no	—
48	D₄×C₆₁	—	4	1	no	—
49	D₄×C₆₁	—	4	1	no	—
50	D₄×C₆₁	—	4	1	no	—
51	D₄×C₆₁	—	8	1	no	—
52	C₂×C₄×C₆₁	—	4	1	no	—
53	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
56	C₂×Dic₆₁	—	2	1	no	—
57	D₂₄₄	—	2	1	no	—
58	D₂₄₄	—	2	1	no	—
59	C₄×D₆₁	—	4	1	no	—
60	C₄×D₆₁	—	4	1	no	—
61	C₈×C₆₁	—	2	1	no	—
62	C₈×C₆₁	D₈	1	2	yes	complemented
63	#92418	—	4	1	no	—
64	#92418	—	4	1	no	—
65	#92418	—	4	1	no	—
66	C₂²×C₆₁	—	8	1	no	—
67	C₂²×C₆₁	—	8	1	no	—
68	C₂²×C₆₁	—	8	1	no	—
69	C₂²×C₆₁	#121	1	1	yes	characteristic
70	C₂×D₆₁	—	4	1	no	—
71	C₂×D₆₁	—	4	1	no	—
72	C₂×D₆₁	—	4	1	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	—	8	1	no	—
75	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
76	C₄×C₆₁	AGL(1,ℤ/8)	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	—	4	1	no	—
80	#1685	—	61	1	no	maximal, Sylow
81	C₂×C₆₁	—	16	1	no	—
82	C₂×C₆₁	#459	1	1	yes	characteristic
83	C₂×C₆₁	#468	1	1	yes	characteristic
84	C₂×C₆₁	#471	1	1	yes	characteristic
85	D₆₁	—	8	1	no	—
86	#492	D₆₁	1	1	yes	characteristic, complemented
87	#474	—	61	1	no	—
88	#459	—	61	1	no	—
89	#436	—	61	1	no	—
90	#335	—	61	1	no	—
91	#332	—	61	1	no	—
92	#331	—	61	1	no	—
93	C₆₁	#1685	1	1	yes	characteristic, complemented, Sylow
94	C₂×SD₁₆	—	2	1	no	—
95	C₂×SD₁₆	—	2	1	no	—
96	C₂×D₈	—	122	1	no	—
97	C₂×M₁₆	—	122	1	no	—
98	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
99	#127	C₂×D₆₁	1	1	yes	characteristic
100	C₄×D₄	—	61	1	no	—
101	#107	—	122	1	no	—
102	C₄⋊C₈	—	61	1	no	—
103	C₄⋊C₈	—	61	1	no	—
104	C₄⋊C₈	—	61	1	no	—
105	#102	—	122	1	no	—
106	#102	—	122	1	no	—
107	C₂²⋊C₈	—	122	1	no	—
108	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
109	C₂×Q₈	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	61	1	no	—
114	C₂²×C₄	—	122	1	no	—
115	SD₁₆	—	4	2	no	—
116	SD₁₆	—	4	2	no	—
117	D₈	—	244	2	no	—
118	M₁₆	—	244	2	no	—
119	C₂×C₈	—	122	1	no	—
120	C₂×C₈	—	122	1	no	—
121	C₂×C₈	—	122	1	no	—
122	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	61	1	no	—
127	C₂²⋊C₄	—	122	1	no	—
128	C₄²	C₂²×D₆₁	1	1	yes	characteristic
129	C₂³	—	4	1	no	—
130	C₂³	—	122	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	8	1	no	—
138	D₄	—	122	1	no	—
139	D₄	—	122	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	122	1	no	—
142	C₂×C₄	—	244	1	no	—
143	C₂×C₄	—	244	1	no	—
144	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
145	C₂×C₄	#11758224	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	244	1	no	—
149	C₈	—	244	1	no	—
150	C₈	—	244	1	no	—
151	C₈	#11758199	1	2	yes	complemented
152	C₂²	—	8	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	244	1	no	—
156	C₂²	—	244	1	no	—
157	C₂²	—	244	1	no	—
158	C₂²	#14519748	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	8	1	no	—
161	C₄	—	244	1	no	—
162	C₄	C₂×#11758199	1	1	yes	characteristic
163	C₄	#14519803	1	1	yes	characteristic
164	C₄	#14519804	1	1	yes	characteristic
165	C₄	#14519806	1	1	yes	characteristic
166	C₂	—	16	1	no	—
167	C₂	—	488	1	no	—
168	C₂	#16839954	1	1	yes	characteristic
169	C₂	#16840005	1	1	yes	characteristic
170	C₂	#16840022	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840039

C₂

yes

maximal, characteristic, complemented

#16839954

C₂

yes

maximal, characteristic

#16839841

C₂

yes

maximal, characteristic, complemented

#16839515

C₂

yes

maximal, characteristic, complemented

#16839508

C₂

yes

maximal, characteristic, complemented

#16839506

C₂

yes

maximal, characteristic, complemented

C₆₁×#492

C₂

yes

maximal, characteristic, complemented

C₂×#11758199

—

C₂×#11758189

—

C₄×D₂₄₄

C₂²

yes

characteristic

#14519710

—

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

—

—

C₂×SD₁₆×C₆₁

—

C₂×SD₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#127×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₂₄₄

D₄

yes

characteristic

C₂×C₄×D₆₁

—

SD₁₆×C₆₁

—

SD₁₆×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758199

—

#11758198

—

C₄⋊Dic₆₁

D₄

yes

—

—

—

—

C₂³×C₆₁

—

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

D₈

yes

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

#1685

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#492

D₆₁

yes

characteristic, complemented

—

—

—

—

—

—

C₆₁

#1685

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

122

—

C₂×M₁₆

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

#127

C₂×D₆₁

yes

characteristic

100

C₄×D₄

—

101

#107

—

122

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#102

—

122

—

106

#102

—

122

—

107

C₂²⋊C₈

—

122

—

108

C₄×C₈

C₂×D₆₁

yes

characteristic

109

C₂×Q₈

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

122

—

115

SD₁₆

—

116

SD₁₆

—

117

D₈

—

244

—

118

M₁₆

—

244

—

119

C₂×C₈

—

122

—

120

C₂×C₈

—

122

—

121

C₂×C₈

—

122

—

122

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

123

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

122

—

128

C₄²

C₂²×D₆₁

yes

characteristic

129

C₂³

—

130

C₂³

—

122

—

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

122

—

139

D₄

—

122

—

140

C₂×C₄

—

141

C₂×C₄

—

122

—

142

C₂×C₄

—

244

—

143

C₂×C₄

—

244

—

144

C₂×C₄

D₄×D₆₁

yes

characteristic

145

C₂×C₄

#11758224

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

147

C₈

—

148

C₈

—

244

—

149

C₈

—

244

—

150

C₈

—

244

—

151

C₈

#11758199

yes

complemented

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

244

—

156

C₂²

—

244

—

157

C₂²

—

244

—

158

C₂²

#14519748

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

244

—

162

C₄

C₂×#11758199

yes

characteristic

163

C₄

#14519803

yes

characteristic

164

C₄

#14519804

yes

characteristic

165

C₄

#14519806

yes

characteristic

166

C₂

—

167

C₂

—

488

—

168

C₂

#16839954

yes

characteristic

169

C₂

#16840005

yes

characteristic

170

C₂

#16840022

yes

characteristic

	subgroup	G/H
center	C₂²	#14519748
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#492	C₂
socle	C₂²×C₆₁	#121
hypercenter	#492	D₆₁
hypocenter	C₆₁	#1685
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#14519748

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#492

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #16840039	C₂	semidirect
3. #16839841	C₂	semidirect
4. #16839515	C₂	semidirect
5. #16839508	C₂	semidirect
6. #16839506	C₂	semidirect
7. C₆₁×#492	C₂	semidirect
62. C₈×C₆₁	D₈	semidirect
86. #492	D₆₁	semidirect
93. C₆₁	#1685	semidirect
151. C₈	#11758199	semidirect

H ≅ G/N

type

1. #16840039

C₂

semidirect

3. #16839841

C₂

semidirect

4. #16839515

C₂

semidirect

5. #16839508

C₂

semidirect

6. #16839506

C₂

semidirect

7. C₆₁×#492

C₂

semidirect

62. C₈×C₆₁

D₈

semidirect

86. #492

D₆₁

semidirect

93. C₆₁

#1685

semidirect

151. C₈

#11758199

semidirect

Subgroups — #21445281

Conjugacy classes of subgroups

Complements