Group #21454913 — subgroups

Subgroups — #21454913

Order 7808.

322 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 161 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	#16840319	C₂	1	1	yes	maximal, characteristic
2	#16839892	C₂	1	2	yes	maximal, complemented
3	#16839891	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₁₂₂	C₂	1	2	yes	maximal, complemented
5	#16839820	C₂	1	4	yes	maximal
6	#16839819	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#431	C₂	1	1	yes	maximal, characteristic
8	#14519777	C₄	1	1	yes	characteristic
9	#14519777	C₄	1	1	yes	characteristic
10	#14519769	C₄	1	2	yes	—
11	#14519756	C₄	1	2	yes	—
12	#14519756	C₄	1	2	yes	—
13	C₄×Dic₁₂₂	C₂²	1	2	yes	—
14	C₄×Dic₁₂₂	C₂²	1	2	yes	—
15	#14519736	C₂²	1	2	yes	—
16	C₄⋊#92418	C₂²	1	2	yes	—
17	#14519706	C₂²	1	4	yes	—
18	#14519706	C₂²	1	4	yes	—
19	C₂²⋊#92418	C₂²	1	2	yes	—
20	#14519688	C₂²	1	2	yes	—
21	#14519687	C₂²	1	4	yes	—
22	C₈×Dic₆₁	C₂²	1	4	yes	—
23	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
25	C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
26	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
32	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₆₁	C₂³	1	2	yes	—
34	C₂×C₈×C₆₁	C₂³	1	2	yes	—
35	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₆₁	C₂³	1	1	yes	characteristic
38	C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₆₁	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
44	#11758196	C₂×C₄	1	4	yes	—
45	#11758196	C₂×C₄	1	4	yes	—
46	C₄×Dic₆₁	C₂³	1	4	yes	—
47	C₂×#92418	C₂³	1	4	yes	—
48	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	—	4	1	no	—
52	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
59	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
60	Dic₁₂₂	—	2	4	no	—
61	Dic₁₂₂	—	4	2	no	—
62	C₈×C₆₁	—	2	2	no	—
63	C₈×C₆₁	—	2	2	no	—
64	#92418	—	2	4	no	—
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	—	2	1	no	—
67	C₂²×C₆₁	—	2	1	no	—
68	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₆₁	—	2	1	no	—
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	—	2	1	no	—
72	C₄×C₆₁	—	2	1	no	—
73	C₄×C₆₁	#131	1	2	yes	—
74	C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
75	Dic₆₁	—	2	4	no	—
76	Dic₆₁	—	2	4	no	—
77	#2985	—	61	1	no	maximal, Sylow
78	C₂×C₆₁	—	4	1	no	—
79	C₂×C₆₁	#519	1	1	yes	characteristic
80	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
81	C₂×C₆₁	#568	1	1	yes	characteristic
82	#533	—	61	1	no	—
83	#446	—	61	4	no	—
84	#446	—	61	2	no	—
85	C₈×Q₈	—	61	2	no	—
86	#433	—	61	4	no	—
87	#432	—	61	1	no	—
88	#431	D₆₁	1	1	yes	characteristic
89	C₆₁	#2985	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	61	1	no	—
91	C₄⋊Q₈	—	61	1	no	—
92	#125	—	61	2	no	—
93	C₂²⋊Q₈	—	61	2	no	—
94	C₂²⋊Q₈	—	61	2	no	—
95	C₄×Q₈	—	61	2	no	—
96	C₄×Q₈	—	61	2	no	—
97	#117	—	61	2	no	—
98	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
99	C₄⋊C₈	—	61	4	no	—
100	C₄⋊C₈	—	61	4	no	—
101	C₄⋊C₈	—	61	2	no	—
102	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
103	C₂²⋊C₈	—	61	2	no	—
104	C₂²⋊C₈	C₂×D₆₁	1	2	yes	—
105	#97	—	61	4	no	—
106	#97	—	61	2	no	—
107	C₄×C₈	—	61	4	no	—
108	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
109	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
110	C₂×Q₈	—	61	2	no	—
111	C₂×Q₈	—	61	2	no	—
112	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
115	C₂×C₈	—	61	4	no	—
116	C₂×C₈	C₂²×D₆₁	1	2	yes	—
117	C₂×C₈	C₂²×D₆₁	1	2	yes	—
118	C₄⋊C₄	—	61	4	no	—
119	C₄⋊C₄	—	61	4	no	—
120	C₄⋊C₄	—	61	2	no	—
121	C₄⋊C₄	—	61	2	no	—
122	C₄⋊C₄	—	61	2	no	—
123	C₄⋊C₄	—	61	2	no	—
124	C₂²⋊C₄	—	61	4	no	—
125	C₄²	—	61	4	no	—
126	C₄²	C₄×D₆₁	1	1	yes	characteristic
127	C₄²	C₄×D₆₁	1	1	yes	characteristic
128	C₄²	C₂²×D₆₁	1	1	yes	characteristic
129	C₄²	C₂²×D₆₁	1	1	yes	characteristic
130	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
131	Q₈	—	122	4	no	—
132	Q₈	—	244	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	61	4	no	—
137	C₂×C₄	—	61	4	no	—
138	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	122	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	122	4	no	—
156	C₄	—	122	4	no	—
157	C₄	#14519783	1	2	yes	—
158	C₄	#14519824	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#16840239	1	1	yes	characteristic
161	C₂	C₂×#14519783	1	1	yes	characteristic
162	C₂	#16840528	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal, characteristic

#16839892

C₂

yes

maximal, complemented

#16839891

C₂

yes

maximal, complemented

C₈×Dic₁₂₂

C₂

yes

maximal, complemented

#16839820

C₂

yes

maximal

#16839819

C₂

yes

maximal, characteristic

C₆₁×#431

C₂

yes

maximal, characteristic

#14519777

C₄

yes

characteristic

#14519777

C₄

yes

characteristic

#14519769

C₄

yes

—

#14519756

C₄

yes

—

#14519756

C₄

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

#14519736

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519706

C₂²

yes

—

#14519706

C₂²

yes

—

C₂²⋊#92418

C₂²

yes

—

#14519688

C₂²

yes

—

#14519687

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₂²⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#131

yes

—

C₄×C₆₁

2₋¹⁺⁴

yes

—

Dic₆₁

—

Dic₆₁

—

#2985

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

C₈×Q₈

—

#433

—

#432

—

#431

D₆₁

yes

characteristic

C₆₁

#2985

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

C₂×D₆₁

yes

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

C₂×D₆₁

yes

—

105

#97

—

106

#97

—

107

C₄×C₈

—

108

C₄×C₈

C₂×D₆₁

yes

characteristic

109

C₄×C₈

C₂×D₆₁

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

C₄×D₆₁

yes

characteristic

113

C₂²×C₄

C₄×D₆₁

yes

characteristic

114

C₂²×C₄

C₂²×D₆₁

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

C₂²×D₆₁

yes

—

117

C₂×C₈

C₂²×D₆₁

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

C₄×D₆₁

yes

characteristic

127

C₄²

C₄×D₆₁

yes

characteristic

128

C₄²

C₂²×D₆₁

yes

characteristic

129

C₄²

C₂²×D₆₁

yes

characteristic

130

C₂³

C₂×C₄×D₆₁

yes

characteristic

131

Q₈

—

122

—

132

Q₈

—

244

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

139

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

140

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

141

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

142

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

143

C₂×C₄

C₂³×D₆₁

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

122

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂²×C₄×D₆₁

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

122

—

156

C₄

—

122

—

157

C₄

#14519783

yes

—

158

C₄

#14519824

yes

—

159

C₂

—

160

C₂

#16840239

yes

characteristic

161

C₂

C₂×#14519783

yes

characteristic

162

C₂

#16840528

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#431	C₂
hypercenter	#431	D₆₁
hypocenter	C₆₁	#2985
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₆₁×#431

C₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #16839892	C₂	semidirect
3. #16839891	C₂	semidirect
4. C₈×Dic₁₂₂	C₂	semidirect
89. C₆₁	#2985	semidirect

H ≅ G/N

type

2. #16839892

C₂

semidirect

3. #16839891

C₂

semidirect

4. C₈×Dic₁₂₂

C₂

semidirect

89. C₆₁

#2985

semidirect

Subgroups — #21454913

Conjugacy classes of subgroups

Complements