Group #21447139 — subgroups

Subgroups — #21447139

Order 7808.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14519706	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519701	C₂	1	2	yes	maximal
3	#16839669	C₂	1	2	yes	maximal
4	#16839525	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×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×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
9	#14519706	—	2	2	no	—
10	#14519701	—	2	4	no	—
11	#14519684	—	2	2	no	—
12	#14519684	—	2	2	no	—
13	C₂²×Q₈×C₆₁	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₆₁	—	2	4	no	—
15	C₂×Q₁₆×C₆₁	—	2	2	no	—
16	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₆₁	—	2	2	no	—
18	C₂×Q₈×C₆₁	—	2	2	no	—
19	C₂×Q₈×C₆₁	—	2	2	no	—
20	C₂×Q₈×C₆₁	—	2	2	no	—
21	C₂×Q₈×C₆₁	—	2	2	no	—
22	C₂×Q₈×C₆₁	D₄	1	4	yes	—
23	C₂²×C₄×C₆₁	—	2	2	no	—
24	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
25	C₂²×Dic₆₁	—	2	2	no	—
26	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
27	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
28	Q₁₆×C₆₁	—	2	8	no	—
29	C₂×C₈×C₆₁	—	2	2	no	—
30	C₂×C₈×C₆₁	D₄	1	2	yes	—
31	C₄⋊Dic₆₁	—	2	2	no	—
32	C₄⋊Dic₆₁	—	2	2	no	—
33	C₄×Dic₆₁	—	2	1	no	—
34	C₄×Dic₆₁	—	2	1	no	—
35	C₂×#92418	—	2	2	no	—
36	C₂×#92418	—	2	1	no	—
37	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₆₁	—	2	4	no	—
39	Q₈×C₆₁	—	2	4	no	—
40	Q₈×C₆₁	—	4	2	no	—
41	Q₈×C₆₁	—	4	2	no	—
42	C₂×C₄×C₆₁	—	2	4	no	—
43	C₂×C₄×C₆₁	—	4	2	no	—
44	C₂×C₄×C₆₁	—	4	2	no	—
45	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
48	C₂×C₄×C₆₁	#41	1	2	yes	—
49	C₂×Dic₆₁	—	2	1	no	—
50	C₂×Dic₆₁	—	2	1	no	—
51	C₂×Dic₆₁	—	2	1	no	—
52	C₂×Dic₆₁	—	2	2	no	—
53	C₂×Dic₆₁	—	4	2	no	—
54	C₂×Dic₆₁	—	4	2	no	—
55	C₂×Dic₆₁	—	4	2	no	—
56	C₂×Dic₆₁	Q₁₆	1	2	yes	complemented
57	C₈×C₆₁	—	2	2	no	—
58	#92418	—	4	1	no	—
59	C₂²×C₆₁	#121	1	1	yes	characteristic
60	C₂²×C₆₁	#121	1	2	yes	—
61	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₆₁	#135	1	1	yes	characteristic
63	C₂²×C₆₁	#137	1	2	yes	—
64	C₄×C₆₁	—	4	4	no	—
65	C₄×C₆₁	#120	1	1	yes	characteristic
66	C₄×C₆₁	#123	1	2	yes	—
67	C₄×C₆₁	#124	1	1	yes	characteristic
68	Dic₆₁	—	2	2	no	—
69	Dic₆₁	—	4	1	no	—
70	Dic₆₁	—	8	2	no	—
71	#2056	—	61	1	no	maximal, Sylow
72	C₂×C₆₁	#394	1	1	yes	characteristic
73	C₂×C₆₁	#462	1	2	yes	—
74	C₂×C₆₁	#464	1	2	yes	—
75	C₂×C₆₁	#467	1	1	yes	characteristic
76	C₂×C₆₁	#470	1	1	yes	characteristic
77	C₂²×Q₁₆	D₆₁	1	1	yes	characteristic
78	C₂×C₄⋊C₈	—	61	1	no	—
79	C₂×#103	—	61	2	no	—
80	#391	—	61	2	no	—
81	#340	—	61	1	no	—
82	C₆₁	#2056	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	C₂×D₆₁	1	2	yes	—
84	C₂×Q₁₆	—	2	4	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₂²×C₈	—	61	1	no	—
87	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	61	2	no	—
89	C₂×C₄²	—	61	1	no	—
90	C₄⋊C₈	—	122	2	no	—
91	#103	—	122	4	no	—
92	#95	—	122	2	no	—
93	#95	—	122	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	C₂²⋊D₆₁	1	4	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	61	1	no	—
102	C₂²×C₄	—	61	1	no	—
103	C₂²×C₄	—	122	2	no	—
104	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
105	Q₁₆	—	2	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	122	2	no	—
108	C₂×C₈	—	122	1	no	—
109	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
110	C₄⋊C₄	—	122	2	no	—
111	C₄⋊C₄	—	122	2	no	—
112	C₄²	—	122	1	no	—
113	C₄²	—	122	1	no	—
114	C₂³	D₄×D₆₁	1	1	yes	characteristic
115	Q₈	—	2	4	no	—
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	61	2	no	—
123	C₂×C₄	—	122	1	no	—
124	C₂×C₄	—	122	1	no	—
125	C₂×C₄	—	122	2	no	—
126	C₂×C₄	—	122	1	no	—
127	C₂×C₄	—	244	2	no	—
128	C₂×C₄	—	244	2	no	—
129	C₂×C₄	—	244	2	no	—
130	C₂×C₄	#11758224	1	1	yes	characteristic
131	C₂×C₄	#11758224	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	244	1	no	—
136	C₂²	#14519748	1	1	yes	characteristic
137	C₂²	#14519749	1	2	yes	—
138	C₂²	Q₁₆×D₆₁	1	1	yes	characteristic
139	C₂²	#14519800	1	1	yes	characteristic
140	C₂²	#14519809	1	2	yes	—
141	C₄	—	4	4	no	—
142	C₄	—	122	2	no	—
143	C₄	—	488	2	no	—
144	C₄	—	244	1	no	—
145	C₄	#14519746	1	1	yes	characteristic
146	C₄	#14519758	1	2	yes	—
147	C₄	#14519764	1	1	yes	characteristic
148	C₂	#16839676	1	1	yes	characteristic
149	C₂	#16839970	1	2	yes	—
150	C₂	#16839986	1	2	yes	—
151	C₂	#16840000	1	1	yes	characteristic
152	C₂	#16840017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519706

C₂

yes

maximal, characteristic

C₂×#14519701

C₂

yes

maximal

#16839669

C₂

yes

maximal

#16839525

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

—

—

—

C₂²×Q₈×C₆₁

C₂²

yes

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

characteristic

C₂²×Dic₆₁

D₄

yes

characteristic

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

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

Q₁₆

yes

complemented

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

#121

yes

characteristic

C₂²×C₆₁

#121

yes

—

C₂²×C₆₁

C₂×Q₁₆

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

#120

yes

characteristic

C₄×C₆₁

#123

yes

—

C₄×C₆₁

#124

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#464

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂²×Q₁₆

D₆₁

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#103

—

#391

—

#340

—

C₆₁

#2056

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₆₁

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

122

—

#103

—

122

—

#95

—

122

—

#95

—

122

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₆₁

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

122

—

104

C₂²×C₄

C₂²×D₆₁

yes

characteristic

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

122

—

108

C₂×C₈

—

122

—

109

C₂×C₈

C₂²⋊D₆₁

yes

—

110

C₄⋊C₄

—

122

—

111

C₄⋊C₄

—

122

—

112

C₄²

—

122

—

113

C₄²

—

122

—

114

C₂³

D₄×D₆₁

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

122

—

124

C₂×C₄

—

122

—

125

C₂×C₄

—

122

—

126

C₂×C₄

—

122

—

127

C₂×C₄

—

244

—

128

C₂×C₄

—

244

—

129

C₂×C₄

—

244

—

130

C₂×C₄

#11758224

yes

characteristic

131

C₂×C₄

#11758224

yes

—

132

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

133

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

134

C₈

—

135

C₈

—

244

—

136

C₂²

#14519748

yes

characteristic

137

C₂²

#14519749

yes

—

138

C₂²

Q₁₆×D₆₁

yes

characteristic

139

C₂²

#14519800

yes

characteristic

140

C₂²

#14519809

yes

—

141

C₄

—

142

C₄

—

122

—

143

C₄

—

488

—

144

C₄

—

244

—

145

C₄

#14519746

yes

characteristic

146

C₄

#14519758

yes

—

147

C₄

#14519764

yes

characteristic

148

C₂

#16839676

yes

characteristic

149

C₂

#16839970

yes

—

150

C₂

#16839986

yes

—

151

C₂

#16840000

yes

characteristic

152

C₂

#16840017

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₁
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂²×Q₁₆×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂²×Q₁₆	D₆₁
hypocenter	C₆₁	#2056
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₆₁

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂²×Q₁₆×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂²×Q₁₆

D₆₁

hypocenter

C₆₁

#2056

LCS

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
56. C₂×Dic₆₁	Q₁₆	semidirect
82. C₆₁	#2056	semidirect

H ≅ G/N

type

56. C₂×Dic₆₁

Q₁₆

semidirect

82. C₆₁

#2056

semidirect

Subgroups — #21447139

Conjugacy classes of subgroups

Complements