Group #21454687 — subgroups

Subgroups — #21454687

Order 7808.

456 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 221 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	#16840240	C₂	1	2	yes	maximal
2	#16839869	C₂	1	4	yes	maximal, complemented
3	Q₁₆×Dic₆₁	C₂	1	4	yes	maximal, complemented
4	#16839800	C₂	1	1	yes	maximal, characteristic
5	#16839771	C₂	1	2	yes	maximal
6	#16839717	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758194	C₂²	1	1	yes	characteristic
9	Q₈×Dic₆₁	—	2	4	no	—
10	Q₈×Dic₆₁	C₂²	1	8	yes	—
11	#14519736	—	2	4	no	—
12	#14519736	C₂²	1	2	yes	—
13	#14519736	C₂²	1	1	yes	characteristic
14	#14519711	C₂²	1	2	yes	—
15	#14519709	C₂²	1	2	yes	—
16	#14519701	C₂²	1	8	yes	—
17	#14519687	C₂²	1	2	yes	—
18	C₈×Dic₆₁	C₂²	1	2	yes	—
19	C₂²×Q₈×C₆₁	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₆₁	C₄	1	8	yes	complemented
21	C₂×Q₁₆×C₆₁	C₂²	1	2	yes	—
22	C₂×Q₁₆×C₆₁	C₂²	1	2	yes	—
23	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₆₁	—	2	2	no	—
25	C₂×Q₈×C₆₁	—	2	4	no	—
26	C₂×Q₈×C₆₁	C₂×C₄	1	8	yes	—
27	C₂×Q₈×C₆₁	C₂³	1	4	yes	—
28	C₂²×C₄×C₆₁	—	2	2	no	—
29	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	Q₁₆×C₆₁	C₂×C₄	1	8	yes	—
31	Q₁₆×C₆₁	C₂×C₄	1	8	yes	—
32	C₂×C₈×C₆₁	C₂×C₄	1	4	yes	—
33	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₆₁	—	2	2	no	—
36	C₂²⋊Dic₆₁	D₄	1	2	yes	—
37	C₄⋊Dic₆₁	—	2	8	no	—
38	C₄⋊Dic₆₁	D₄	1	2	yes	—
39	C₄⋊Dic₆₁	C₂³	1	4	yes	—
40	C₄×Dic₆₁	—	2	2	no	—
41	C₄×Dic₆₁	—	2	8	no	—
42	C₄×Dic₆₁	C₂³	1	2	yes	—
43	#11758194	—	2	2	no	—
44	C₂×#92418	C₂³	1	2	yes	—
45	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₆₁	—	2	8	no	—
47	Q₈×C₆₁	—	2	4	no	—
48	Q₈×C₆₁	C₂²×C₄	1	8	yes	—
49	C₂×C₄×C₆₁	—	2	8	no	—
50	C₂×C₄×C₆₁	—	2	4	no	—
51	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#41	1	2	yes	—
54	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₆₁	—	2	4	no	—
56	C₂×Dic₆₁	C₂×D₄	1	4	yes	—
57	C₈×C₆₁	C₂²×C₄	1	2	yes	—
58	C₈×C₆₁	C₂²×C₄	1	2	yes	—
59	#92418	—	2	2	no	—
60	C₂²×C₆₁	—	2	2	no	—
61	C₂²×C₆₁	C₄×D₄	1	2	yes	—
62	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₆₁	—	2	8	no	—
64	C₄×C₆₁	C₄×D₄	1	2	yes	—
65	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
67	Dic₆₁	—	2	4	no	—
68	Dic₆₁	—	4	4	no	—
69	#2950	—	61	1	no	maximal, Sylow
70	C₂×C₆₁	—	2	2	no	—
71	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
72	C₂×C₆₁	#578	1	2	yes	—
73	C₂²×Q₁₆	D₆₁	1	1	yes	characteristic
74	#519	—	61	2	no	—
75	#441	—	61	4	no	—
76	C₄×Q₁₆	—	61	4	no	—
77	#427	—	61	1	no	—
78	#419	—	61	2	no	—
79	#405	—	61	1	no	—
80	C₆₁	#2950	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	C₂×D₆₁	1	2	yes	—
82	C₂×Q₁₆	Dic₆₁	1	8	yes	complemented
83	C₂×Q₁₆	C₂×D₆₁	1	2	yes	—
84	C₂×Q₁₆	C₂×D₆₁	1	2	yes	—
85	C₂×M₁₆	—	61	1	no	—
86	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
87	C₄×Q₈	—	61	8	no	—
88	C₄×Q₈	—	122	4	no	—
89	#117	—	61	1	no	—
90	#117	—	61	2	no	—
91	#117	—	122	4	no	—
92	#107	—	61	2	no	—
93	#106	—	61	2	no	—
94	#103	—	61	8	no	—
95	#97	—	61	2	no	—
96	C₄×C₈	—	61	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	C₂×Dic₆₁	1	8	yes	—
100	C₂×Q₈	C₂²×D₆₁	1	4	yes	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
103	Q₁₆	C₂×Dic₆₁	1	8	yes	—
104	Q₁₆	C₂×Dic₆₁	1	8	yes	—
105	M₁₆	—	122	2	no	—
106	C₂×C₈	—	61	2	no	—
107	C₂×C₈	C₂×Dic₆₁	1	4	yes	—
108	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
110	C₄⋊C₄	—	61	2	no	—
111	C₄⋊C₄	—	61	4	no	—
112	C₄⋊C₄	—	122	8	no	—
113	C₂²⋊C₄	—	61	2	no	—
114	C₂²⋊C₄	—	122	2	no	—
115	C₄²	—	61	2	no	—
116	C₄²	—	122	8	no	—
117	C₄²	—	122	2	no	—
118	C₂³	D₄×D₆₁	1	1	yes	characteristic
119	Q₈	—	2	4	no	—
120	Q₈	—	2	8	no	—
121	Q₈	C₂²×Dic₆₁	1	8	yes	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	61	4	no	—
125	C₂×C₄	—	122	4	no	—
126	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
127	C₂×C₄	#11758224	1	2	yes	—
128	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
129	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
130	C₈	—	122	2	no	—
131	C₈	C₂²×Dic₆₁	1	2	yes	—
132	C₈	C₂²×Dic₆₁	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	D₄×Dic₆₁	1	2	yes	—
135	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	—	122	4	no	—
138	C₄	—	244	4	no	—
139	C₄	D₄×Dic₆₁	1	2	yes	—
140	C₄	C₂³×Dic₆₁	1	1	yes	characteristic
141	C₄	C₂×#11758224	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×D₄×Dic₆₁	1	1	yes	characteristic
144	C₂	#16840575	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840240

C₂

yes

maximal

#16839869

C₂

yes

maximal, complemented

Q₁₆×Dic₆₁

C₂

yes

maximal, complemented

#16839800

C₂

yes

maximal, characteristic

#16839771

C₂

yes

maximal

#16839717

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

characteristic

Q₈×Dic₆₁

—

Q₈×Dic₆₁

C₂²

yes

—

#14519736

—

#14519736

C₂²

yes

—

#14519736

C₂²

yes

characteristic

#14519711

C₂²

yes

—

#14519709

C₂²

yes

—

#14519701

C₂²

yes

—

#14519687

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂²×Q₈×C₆₁

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₄

yes

complemented

C₂×Q₁₆×C₆₁

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×Q₈×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

Q₁₆×C₆₁

C₂×C₄

yes

—

Q₁₆×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₂²⋊Dic₆₁

D₄

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

—

C₄⋊Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂³

yes

—

#11758194

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₈×C₆₁

C₂²×C₄

yes

—

C₈×C₆₁

C₂²×C₄

yes

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×D₄

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#2950

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄×D₄

yes

characteristic

C₂×C₆₁

#578

yes

—

C₂²×Q₁₆

D₆₁

yes

characteristic

#519

—

#441

—

C₄×Q₁₆

—

#427

—

#419

—

#405

—

C₆₁

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₆₁

yes

—

C₂×Q₁₆

Dic₆₁

yes

complemented

C₂×Q₁₆

C₂×D₆₁

yes

—

C₂×Q₁₆

C₂×D₆₁

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

122

—

#117

—

#117

—

#117

—

122

—

—

—

—

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₆₁

yes

—

100

C₂×Q₈

C₂²×D₆₁

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₆₁

yes

characteristic

103

Q₁₆

C₂×Dic₆₁

yes

—

104

Q₁₆

C₂×Dic₆₁

yes

—

105

M₁₆

—

122

—

106

C₂×C₈

—

107

C₂×C₈

C₂×Dic₆₁

yes

—

108

C₂×C₈

C₂²×D₆₁

yes

characteristic

109

C₂×C₈

C₂²×D₆₁

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

122

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

122

—

115

C₄²

—

116

C₄²

—

122

—

117

C₄²

—

122

—

118

C₂³

D₄×D₆₁

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

C₂²×Dic₆₁

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

122

—

126

C₂×C₄

D₄×D₆₁

yes

characteristic

127

C₂×C₄

#11758224

yes

—

128

C₂×C₄

C₂²×Dic₆₁

yes

—

129

C₂×C₄

C₂³×D₆₁

yes

characteristic

130

C₈

—

122

—

131

C₈

C₂²×Dic₆₁

yes

—

132

C₈

C₂²×Dic₆₁

yes

—

133

C₂²

—

134

C₂²

D₄×Dic₆₁

yes

—

135

C₂²

C₂×D₄×D₆₁

yes

characteristic

136

C₄

—

137

C₄

—

122

—

138

C₄

—

244

—

139

C₄

D₄×Dic₆₁

yes

—

140

C₄

C₂³×Dic₆₁

yes

characteristic

141

C₄

C₂×#11758224

yes

characteristic

142

C₂

—

143

C₂

C₂×D₄×Dic₆₁

yes

characteristic

144

C₂

#16840575

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂²×Q₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₂²×Q₁₆	D₆₁
hypocenter	C₆₁	#2950
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₄×C₆₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂²×Q₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

C₂²×Q₁₆

D₆₁

hypocenter

C₆₁

#2950

LCS

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. #16839869	C₂	semidirect
3. Q₁₆×Dic₆₁	C₂	semidirect
20. C₂×Q₁₆×C₆₁	C₄	semidirect
80. C₆₁	#2950	semidirect
82. C₂×Q₁₆	Dic₆₁	semidirect

H ≅ G/N

type

2. #16839869

C₂

semidirect

3. Q₁₆×Dic₆₁

C₂

semidirect

20. C₂×Q₁₆×C₆₁

C₄

semidirect

80. C₆₁

#2950

semidirect

82. C₂×Q₁₆

Dic₆₁

semidirect

Subgroups — #21454687

Conjugacy classes of subgroups

Complements