Group #21445377 — subgroups

Subgroups — #21445377

Order 7808.

176 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16839964	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	#16839513	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×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758201	—	2	1	no	—
9	C₂×#11758199	—	2	1	no	—
10	C₂×#11758189	—	2	1	no	—
11	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
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	#14519701	—	2	1	no	—
16	#14519698	—	2	1	no	—
17	#14519689	—	2	1	no	—
18	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	#127×C₆₁	C₂²	1	1	yes	characteristic
20	#102×C₆₁	—	2	1	no	—
21	#102×C₆₁	—	2	1	no	—
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	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₄⋊C₄×C₆₁	—	2	1	no	—
33	C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	#11758201	—	4	2	no	—
35	#11758199	—	4	2	no	—
36	#11758198	—	2	1	no	—
37	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
38	C₂×#92418	—	2	1	no	—
39	C₂×#92418	—	2	1	no	—
40	C₂×#92418	—	2	1	no	—
41	#11758189	—	4	2	no	—
42	C₂³×C₆₁	—	4	1	no	—
43	C₂²×D₆₁	—	2	1	no	—
44	Q₈×C₆₁	—	4	1	no	—
45	Q₈×C₆₁	—	4	1	no	—
46	D₄×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	—	8	1	no	—
51	C₂×C₄×C₆₁	—	4	1	no	—
52	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
55	C₂×Dic₆₁	—	2	1	no	—
56	D₂₄₄	—	2	1	no	—
57	D₂₄₄	—	2	1	no	—
58	C₄×D₆₁	—	4	1	no	—
59	C₄×D₆₁	—	4	1	no	—
60	C₈×C₆₁	—	2	1	no	—
61	C₈×C₆₁	—	2	1	no	—
62	#92418	—	4	1	no	—
63	#92418	—	4	1	no	—
64	#92418	—	4	1	no	—
65	C₂²×C₆₁	—	8	1	no	—
66	C₂²×C₆₁	—	8	1	no	—
67	C₂²×C₆₁	—	8	1	no	—
68	C₂²×C₆₁	#121	1	1	yes	characteristic
69	C₂×D₆₁	—	4	1	no	—
70	C₂×D₆₁	—	4	1	no	—
71	C₂×D₆₁	—	4	1	no	—
72	C₄×C₆₁	—	2	1	no	—
73	C₄×C₆₁	—	8	1	no	—
74	C₄×C₆₁	#135	1	1	yes	characteristic
75	C₄×C₆₁	AGL(1,ℤ/8)	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	Dic₆₁	—	4	1	no	—
79	#1701	—	61	1	no	maximal, Sylow
80	C₂×C₆₁	—	16	1	no	—
81	C₂×C₆₁	#463	1	1	yes	characteristic
82	C₂×C₆₁	#469	1	1	yes	characteristic
83	C₂×C₆₁	#471	1	1	yes	characteristic
84	D₆₁	—	8	1	no	—
85	#486	D₆₁	1	1	yes	characteristic, complemented
86	#461	—	61	1	no	—
87	#459	—	61	1	no	—
88	#436	—	61	1	no	—
89	#334	—	61	1	no	—
90	#332	—	61	1	no	—
91	#331	—	61	1	no	—
92	C₆₁	#1701	1	1	yes	characteristic, complemented, Sylow
93	C₂×SD₁₆	—	122	1	no	—
94	C₂×D₈	—	122	1	no	—
95	C₂×M₁₆	—	122	1	no	—
96	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
97	#127	C₂×D₆₁	1	1	yes	characteristic
98	C₄×D₄	—	61	1	no	—
99	C₄⋊C₈	—	61	1	no	—
100	C₄⋊C₈	—	61	1	no	—
101	C₄⋊C₈	—	61	1	no	—
102	#103	—	122	1	no	—
103	#102	—	2	1	no	—
104	#102	—	2	1	no	—
105	#102	—	122	1	no	—
106	C₂²⋊C₈	—	122	1	no	—
107	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
108	C₂×Q₈	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	61	1	no	—
113	C₂²×C₄	—	122	1	no	—
114	SD₁₆	—	244	2	no	—
115	D₈	—	244	2	no	—
116	M₁₆	—	244	2	no	—
117	C₂×C₈	—	122	1	no	—
118	C₂×C₈	—	122	1	no	—
119	C₂×C₈	—	122	1	no	—
120	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	61	1	no	—
125	C₂²⋊C₄	—	122	1	no	—
126	C₄²	C₂²×D₆₁	1	1	yes	characteristic
127	C₂³	—	4	1	no	—
128	C₂³	—	122	1	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	8	1	no	—
136	D₄	—	122	1	no	—
137	D₄	—	122	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	122	1	no	—
140	C₂×C₄	—	244	1	no	—
141	C₂×C₄	—	244	1	no	—
142	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
143	C₂×C₄	#11758224	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	244	1	no	—
148	C₈	—	244	1	no	—
149	C₈	—	244	1	no	—
150	C₂²	—	8	1	no	—
151	C₂²	—	8	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	244	1	no	—
154	C₂²	—	244	1	no	—
155	C₂²	—	244	1	no	—
156	C₂²	#14519748	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	8	1	no	—
159	C₄	—	244	1	no	—
160	C₄	#14519797	1	1	yes	characteristic
161	C₄	#14519801	1	1	yes	characteristic
162	C₄	#14519801	1	1	yes	characteristic
163	C₄	#14519802	1	1	yes	characteristic
164	C₂	—	16	1	no	—
165	C₂	—	488	1	no	—
166	C₂	#16839974	1	1	yes	characteristic
167	C₂	#16840012	1	1	yes	characteristic
168	C₂	#16840022	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839964

C₂

yes

maximal, characteristic, complemented

#16839954

C₂

yes

maximal, characteristic

#16839841

C₂

yes

maximal, characteristic, complemented

#16839513

C₂

yes

maximal, characteristic, complemented

#16839508

C₂

yes

maximal, characteristic, complemented

#16839506

C₂

yes

maximal, characteristic, complemented

C₆₁×#486

C₂

yes

maximal, characteristic, complemented

C₂×#11758201

—

C₂×#11758199

—

C₂×#11758189

—

C₄×D₂₄₄

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

—

—

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#127×C₆₁

C₂²

yes

characteristic

#102×C₆₁

—

#102×C₆₁

—

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

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

—

—

—

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

—

#92418

—

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

#1701

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#486

D₆₁

yes

characteristic, complemented

—

—

—

—

—

—

C₆₁

#1701

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

122

—

C₂×D₈

—

122

—

C₂×M₁₆

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

#127

C₂×D₆₁

yes

characteristic

C₄×D₄

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#103

—

122

—

103

#102

—

104

#102

—

105

#102

—

122

—

106

C₂²⋊C₈

—

122

—

107

C₄×C₈

C₂×D₆₁

yes

characteristic

108

C₂×Q₈

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂²×C₄

—

122

—

114

SD₁₆

—

244

—

115

D₈

—

244

—

116

M₁₆

—

244

—

117

C₂×C₈

—

122

—

118

C₂×C₈

—

122

—

119

C₂×C₈

—

122

—

120

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

121

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

122

—

126

C₄²

C₂²×D₆₁

yes

characteristic

127

C₂³

—

128

C₂³

—

122

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

122

—

137

D₄

—

122

—

138

C₂×C₄

—

139

C₂×C₄

—

122

—

140

C₂×C₄

—

244

—

141

C₂×C₄

—

244

—

142

C₂×C₄

D₄×D₆₁

yes

characteristic

143

C₂×C₄

#11758224

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

244

—

148

C₈

—

244

—

149

C₈

—

244

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

244

—

154

C₂²

—

244

—

155

C₂²

—

244

—

156

C₂²

#14519748

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

244

—

160

C₄

#14519797

yes

characteristic

161

C₄

#14519801

yes

characteristic

162

C₄

#14519801

yes

characteristic

163

C₄

#14519802

yes

characteristic

164

C₂

—

165

C₂

—

488

—

166

C₂

#16839974

yes

characteristic

167

C₂

#16840012

yes

characteristic

168

C₂

#16840022

yes

characteristic

	subgroup	G/H
center	C₂²	#14519748
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#486	C₂
socle	C₂²×C₆₁	#121
hypercenter	#486	D₆₁
hypocenter	C₆₁	#1701
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#14519748

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#486

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #16839964	C₂	semidirect
3. #16839841	C₂	semidirect
4. #16839513	C₂	semidirect
5. #16839508	C₂	semidirect
6. #16839506	C₂	semidirect
7. C₆₁×#486	C₂	semidirect
85. #486	D₆₁	semidirect
92. C₆₁	#1701	semidirect

H ≅ G/N

type

1. #16839964

C₂

semidirect

3. #16839841

C₂

semidirect

4. #16839513

C₂

semidirect

5. #16839508

C₂

semidirect

6. #16839506

C₂

semidirect

7. C₆₁×#486

C₂

semidirect

85. #486

D₆₁

semidirect

92. C₆₁

#1701

semidirect

Subgroups — #21445377

Conjugacy classes of subgroups

Complements