Group #21446550 — subgroups

Subgroups — #21446550

Order 7808.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14519711	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839658	C₂	1	2	yes	maximal
4	#16839524	C₂	1	2	yes	maximal
5	C₂×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#92418	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
8	C₂×#11758196	—	2	2	no	—
9	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
10	#14519711	C₄	1	4	yes	complemented
11	C₈×Dic₆₁	—	2	2	no	—
12	#14519684	—	2	2	no	—
13	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
15	#107×C₆₁	—	2	2	no	—
16	C₂²×C₄×C₆₁	—	2	2	no	—
17	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₆₁	—	2	2	no	—
19	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
20	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₆₁	D₄	1	2	yes	—
23	C₂×C₈×C₆₁	Q₈	1	2	yes	—
24	C₄⋊C₄×C₆₁	—	2	2	no	—
25	C₄⋊C₄×C₆₁	—	2	2	no	—
26	C₄⋊Dic₆₁	—	2	2	no	—
27	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
28	#11758196	—	4	4	no	—
29	C₄×Dic₆₁	—	2	1	no	—
30	C₄×Dic₆₁	—	2	1	no	—
31	C₂×#92418	—	2	1	no	—
32	C₂×#92418	—	2	1	no	—
33	C₂×#92418	Q₈	1	2	yes	—
34	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₆₁	—	4	2	no	—
36	C₂×C₄×C₆₁	—	4	2	no	—
37	C₂×C₄×C₆₁	—	4	2	no	—
38	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
43	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
44	C₂×Dic₆₁	—	2	1	no	—
45	C₂×Dic₆₁	—	2	1	no	—
46	C₂×Dic₆₁	—	2	1	no	—
47	C₂×Dic₆₁	—	2	1	no	—
48	C₂×Dic₆₁	—	2	4	no	—
49	C₂×Dic₆₁	—	4	2	no	—
50	C₂×Dic₆₁	—	4	2	no	—
51	C₂×Dic₆₁	D₈	1	2	yes	—
52	C₂×Dic₆₁	Q₁₆	1	2	yes	—
53	C₈×C₆₁	C₄⋊C₄	1	4	yes	complemented
54	#92418	—	2	2	no	—
55	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
56	C₂²×C₆₁	#121	1	1	yes	characteristic
57	C₂²×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₆₁	#135	1	1	yes	characteristic
61	C₂²×C₆₁	#135	1	1	yes	characteristic
62	C₄×C₆₁	—	8	2	no	—
63	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
66	C₄×C₆₁	#125	1	1	yes	characteristic
67	Dic₆₁	—	2	2	no	—
68	Dic₆₁	—	2	2	no	—
69	Dic₆₁	—	4	4	no	—
70	#1942	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	#389	1	1	yes	characteristic
72	C₂×C₆₁	C₄×D₈	1	1	yes	characteristic
73	C₂×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
74	C₂×C₆₁	#466	1	1	yes	characteristic
75	C₂×C₆₁	#467	1	1	yes	characteristic
76	C₂×C₆₁	#499	1	1	yes	characteristic
77	C₂×C₆₁	#500	1	1	yes	characteristic
78	C₂×#107	—	61	1	no	—
79	C₂×#107	D₆₁	1	1	yes	characteristic
80	C₂×C₄×C₈	—	61	1	no	—
81	#389	—	61	2	no	—
82	#340	—	61	2	no	—
83	C₆₁	#1942	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	61	1	no	—
85	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	61	2	no	—
87	C₂×C₄⋊C₄	—	122	2	no	—
88	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
89	C₂×C₄²	—	61	1	no	—
90	#107	—	2	2	no	—
91	#107	—	61	4	no	—
92	C₄×C₈	—	122	2	no	—
93	#95	—	122	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	61	1	no	—
96	C₂²×C₄	—	61	1	no	—
97	C₂²×C₄	—	122	2	no	—
98	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
99	C₂×C₈	—	61	2	no	—
100	C₂×C₈	—	122	1	no	—
101	C₂×C₈	—	122	1	no	—
102	C₂×C₈	Dic₁₂₂	1	2	yes	—
103	C₂×C₈	C₄×D₆₁	1	2	yes	—
104	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	61	4	no	—
108	C₄⋊C₄	—	122	2	no	—
109	C₄⋊C₄	—	244	4	no	—
110	C₄²	—	122	1	no	—
111	C₄²	—	122	1	no	—
112	C₂³	D₄×D₆₁	1	1	yes	characteristic
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	61	2	no	—
117	C₂×C₄	—	61	2	no	—
118	C₂×C₄	—	122	1	no	—
119	C₂×C₄	—	122	1	no	—
120	C₂×C₄	—	122	1	no	—
121	C₂×C₄	—	122	1	no	—
122	C₂×C₄	—	122	4	no	—
123	C₂×C₄	—	244	2	no	—
124	C₂×C₄	—	244	2	no	—
125	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
127	C₂×C₄	#11758224	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
129	C₂×C₄	#11758226	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
131	C₈	—	122	2	no	—
132	C₈	#11758196	1	4	yes	complemented
133	C₂²	#14519739	1	1	yes	characteristic
134	C₂²	#14519748	1	1	yes	characteristic
135	C₂²	#14519776	1	1	yes	characteristic
136	C₂²	D₈×D₆₁	1	1	yes	characteristic
137	C₂²	Q₁₆×D₆₁	1	1	yes	characteristic
138	C₂²	#14519796	1	1	yes	characteristic
139	C₂²	#14519800	1	1	yes	characteristic
140	C₄	—	8	2	no	—
141	C₄	—	122	2	no	—
142	C₄	—	122	2	no	—
143	C₄	—	244	4	no	—
144	C₄	C₂×#11758196	1	1	yes	characteristic
145	C₄	#14519742	1	1	yes	characteristic
146	C₄	#14519753	1	1	yes	characteristic
147	C₄	#14519768	1	1	yes	characteristic
148	C₂	#16839658	1	1	yes	characteristic
149	C₂	#16839844	1	1	yes	characteristic
150	C₂	#16839856	1	1	yes	characteristic
151	C₂	#16839995	1	1	yes	characteristic
152	C₂	#16840000	1	1	yes	characteristic
153	C₂	#16840163	1	1	yes	characteristic
154	C₂	#16840166	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519711

C₂

yes

maximal, characteristic

C₂×C₈×Dic₆₁

C₂

yes

maximal, characteristic

#16839658

C₂

yes

maximal

#16839524

C₂

yes

maximal

C₂×#107×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519711

C₄

yes

complemented

C₈×Dic₆₁

—

#14519684

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

#107×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

characteristic

C₂²×Dic₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

Q₈

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

Q₈

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

D₈

yes

—

C₂×Dic₆₁

Q₁₆

yes

—

C₈×C₆₁

C₄⋊C₄

yes

complemented

#92418

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₂²×C₆₁

#121

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

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

C₄×Q₈

yes

characteristic

C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₄×C₆₁

#125

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₄×D₈

yes

characteristic

C₂×C₆₁

C₄×Q₁₆

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

yes

characteristic

C₂×C₄×C₈

—

#389

—

#340

—

C₆₁

#1942

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

#107

—

#107

—

C₄×C₈

—

122

—

#95

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

122

—

101

C₂×C₈

—

122

—

102

C₂×C₈

Dic₁₂₂

yes

—

103

C₂×C₈

C₄×D₆₁

yes

—

104

C₂×C₈

C₂²⋊D₆₁

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

122

—

109

C₄⋊C₄

—

244

—

110

C₄²

—

122

—

111

C₄²

—

122

—

112

C₂³

D₄×D₆₁

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

122

—

119

C₂×C₄

—

122

—

120

C₂×C₄

—

122

—

121

C₂×C₄

—

122

—

122

C₂×C₄

—

122

—

123

C₂×C₄

—

244

—

124

C₂×C₄

—

244

—

125

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

126

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

127

C₂×C₄

#11758224

yes

characteristic

128

C₂×C₄

Q₈×D₆₁

yes

characteristic

129

C₂×C₄

#11758226

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

131

C₈

—

122

—

132

C₈

#11758196

yes

complemented

133

C₂²

#14519739

yes

characteristic

134

C₂²

#14519748

yes

characteristic

135

C₂²

#14519776

yes

characteristic

136

C₂²

D₈×D₆₁

yes

characteristic

137

C₂²

Q₁₆×D₆₁

yes

characteristic

138

C₂²

#14519796

yes

characteristic

139

C₂²

#14519800

yes

characteristic

140

C₄

—

141

C₄

—

122

—

142

C₄

—

122

—

143

C₄

—

244

—

144

C₄

C₂×#11758196

yes

characteristic

145

C₄

#14519742

yes

characteristic

146

C₄

#14519753

yes

characteristic

147

C₄

#14519768

yes

characteristic

148

C₂

#16839658

yes

characteristic

149

C₂

#16839844

yes

characteristic

150

C₂

#16839856

yes

characteristic

151

C₂

#16839995

yes

characteristic

152

C₂

#16840000

yes

characteristic

153

C₂

#16840163

yes

characteristic

154

C₂

#16840166

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#107×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#107	D₆₁
hypocenter	C₆₁	#1942
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×#107×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
10. #14519711	C₄	semidirect
53. C₈×C₆₁	C₄⋊C₄	semidirect
83. C₆₁	#1942	semidirect
132. C₈	#11758196	semidirect

H ≅ G/N

type

10. #14519711

C₄

semidirect

53. C₈×C₆₁

C₄⋊C₄

semidirect

83. C₆₁

#1942

semidirect

132. C₈

#11758196

semidirect

Subgroups — #21446550

Conjugacy classes of subgroups

Complements