Group #21447867 — subgroups

Subgroups — #21447867

Order 7808.

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

normal subgroups only

This group has 154 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	C₂×#14519723	C₂	1	1	yes	maximal, characteristic
2	#16840140	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	#16839580	C₂	1	1	yes	maximal, characteristic
5	#16839567	C₂	1	1	yes	maximal, characteristic
6	#16839566	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
9	C₂×#11758202	—	2	1	no	—
10	#14519775	—	2	1	no	—
11	Q₈×Dic₆₁	—	2	1	no	—
12	#14519723	—	2	2	no	—
13	C₂×#11758185	C₂²	1	1	yes	characteristic
14	#14519711	C₂²	1	1	yes	characteristic
15	#14519701	—	2	1	no	—
16	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
19	Q₃₂×C₆₁	—	2	2	no	—
20	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₆₁	—	2	1	no	—
22	C₂×Q₈×C₆₁	—	2	1	no	—
23	C₂×Dic₁₂₂	—	2	1	no	—
24	Q₁₆×C₆₁	—	2	1	no	—
25	Q₁₆×C₆₁	—	2	1	no	—
26	Q₁₆×C₆₁	—	2	1	no	—
27	Q₁₆×C₆₁	D₄	1	2	yes	—
28	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	#11758202	—	4	2	no	—
30	C₄⋊Dic₆₁	—	2	1	no	—
31	C₄⋊Dic₆₁	—	4	1	no	—
32	#11758196	—	4	1	no	—
33	#11758196	—	4	1	no	—
34	C₄×Dic₆₁	—	4	1	no	—
35	C₄×Dic₆₁	D₄	1	1	yes	characteristic
36	C₂×#92418	D₄	1	1	yes	characteristic
37	Dic₂₄₄	—	2	1	no	—
38	Dic₂₄₄	—	2	1	no	—
39	C₁₆×C₆₁	—	2	1	no	—
40	#11758185	—	2	1	no	—
41	Q₈×C₆₁	—	2	2	no	—
42	Q₈×C₆₁	—	4	1	no	—
43	Q₈×C₆₁	—	4	1	no	—
44	Q₈×C₆₁	—	4	1	no	—
45	C₂×C₄×C₆₁	—	4	1	no	—
46	C₂×C₄×C₆₁	—	4	1	no	—
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₆₁	—	4	1	no	—
49	C₂×Dic₆₁	—	4	1	no	—
50	C₂×Dic₆₁	D₈	1	1	yes	characteristic
51	C₂×Dic₆₁	D₈	1	1	yes	characteristic
52	Dic₁₂₂	—	4	1	no	—
53	Dic₁₂₂	—	4	1	no	—
54	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₈×C₆₁	#41	1	1	yes	characteristic
56	#92418	—	2	1	no	—
57	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
58	C₄×C₆₁	—	4	2	no	—
59	C₄×C₆₁	—	8	1	no	—
60	C₄×C₆₁	#121	1	1	yes	characteristic
61	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₆₁	—	2	1	no	—
63	Dic₆₁	—	8	1	no	—
64	Dic₆₁	—	8	1	no	—
65	Dic₆₁	Q₃₂	1	2	yes	complemented
66	#2209	—	61	1	no	maximal, Sylow
67	C₂×C₆₁	#459	1	1	yes	characteristic
68	C₂×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
69	C₂×C₆₁	#510	1	1	yes	characteristic
70	C₂×Q₃₂	—	61	1	no	—
71	C₂×Q₃₂	D₆₁	1	1	yes	characteristic
72	#494	—	61	1	no	—
73	C₄×Q₁₆	—	61	1	no	—
74	C₄⋊C₁₆	—	61	1	no	—
75	#358	—	61	1	no	—
76	#358	—	61	1	no	—
77	C₆₁	#2209	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	61	1	no	—
79	C₂×Q₁₆	—	122	1	no	—
80	C₂×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
81	C₂×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
82	C₄⋊Q₈	—	122	1	no	—
83	C₄×Q₈	—	122	1	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	122	2	no	—
86	C₂×C₁₆	—	61	1	no	—
87	C₂×C₁₆	C₂×D₆₁	1	1	yes	characteristic
88	#107	—	61	1	no	—
89	#103	—	122	1	no	—
90	C₄×C₈	—	61	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	122	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	122	1	no	—
98	Q₁₆	—	122	1	no	—
99	Q₁₆	—	244	2	no	—
100	Q₁₆	C₂²⋊D₆₁	1	2	yes	—
101	C₂×C₈	—	61	1	no	—
102	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
103	C₄⋊C₄	—	122	1	no	—
104	C₄⋊C₄	—	244	1	no	—
105	C₄⋊C₄	—	244	1	no	—
106	C₄⋊C₄	—	244	1	no	—
107	C₄²	—	61	1	no	—
108	C₄²	—	244	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	122	1	no	—
111	Q₈	—	2	2	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	4	1	no	—
115	Q₈	—	244	1	no	—
116	Q₈	—	244	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	61	1	no	—
120	C₂×C₄	—	61	1	no	—
121	C₂×C₄	—	244	1	no	—
122	C₂×C₄	—	244	1	no	—
123	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
124	C₈	—	122	1	no	—
125	C₈	#11758224	1	1	yes	characteristic
126	C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
127	C₂²	D₈×D₆₁	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	—	8	1	no	—
130	C₄	—	61	2	no	—
131	C₄	—	122	1	no	—
132	C₄	—	488	1	no	—
133	C₄	—	488	1	no	—
134	C₄	#14519749	1	1	yes	characteristic
135	C₄	#14519801	1	1	yes	characteristic
136	C₂	#16839955	1	1	yes	characteristic
137	C₂	Q₃₂×D₆₁	1	1	yes	characteristic
138	C₂	#16840204	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519723

C₂

yes

maximal, characteristic

#16840140

C₂

yes

maximal, characteristic

Q₁₆×Dic₆₁

C₂

yes

maximal, characteristic

#16839580

C₂

yes

maximal, characteristic

#16839567

C₂

yes

maximal, characteristic

#16839566

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₆₁

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

C₂²

yes

characteristic

C₂×#11758202

—

#14519775

—

Q₈×Dic₆₁

—

#14519723

—

C₂×#11758185

C₂²

yes

characteristic

#14519711

C₂²

yes

characteristic

#14519701

—

C₈×Dic₆₁

C₂²

yes

characteristic

C₂×Q₁₆×C₆₁

C₂²

yes

characteristic

C₂×Q₁₆×C₆₁

C₂²

yes

characteristic

Q₃₂×C₆₁

—

C₂×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

#11758202

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

—

#11758185

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

D₈

yes

characteristic

C₂×Dic₆₁

D₈

yes

characteristic

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂×D₄

yes

C₈×C₆₁

yes

—

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

Q₃₂

yes

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×Q₃₂

yes

characteristic

C₂×C₆₁

#510

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

D₆₁

yes

characteristic

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#358

—

C₆₁

#2209

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

122

—

C₂×Q₁₆

C₂×D₆₁

yes

characteristic

C₂×Q₁₆

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

122

—

C₄×Q₈

—

122

—

Q₃₂

—

Q₃₂

—

122

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₁

yes

characteristic

#107

—

#103

—

122

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

122

—

Q₁₆

—

122

—

Q₁₆

—

244

—

100

Q₁₆

C₂²⋊D₆₁

yes

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₆₁

yes

characteristic

103

C₄⋊C₄

—

122

—

104

C₄⋊C₄

—

244

—

105

C₄⋊C₄

—

244

—

106

C₄⋊C₄

—

244

—

107

C₄²

—

108

C₄²

—

244

—

109

C₁₆

—

110

C₁₆

—

122

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

244

—

116

Q₈

—

244

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

244

—

122

C₂×C₄

—

244

—

123

C₂×C₄

D₄×D₆₁

yes

characteristic

124

C₈

—

122

—

125

C₈

#11758224

yes

characteristic

126

C₈

C₂×C₂²⋊D₆₁

yes

characteristic

127

C₂²

D₈×D₆₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

122

—

132

C₄

—

488

—

133

C₄

—

488

—

134

C₄

#14519749

yes

characteristic

135

C₄

#14519801

yes

characteristic

136

C₂

#16839955

yes

characteristic

137

C₂

Q₃₂×D₆₁

yes

characteristic

138

C₂

#16840204

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₁
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₂×Q₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₂×Q₃₂	D₆₁
hypocenter	C₆₁	#2209
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₆₁

commutator

C₂×C₈×C₆₁

C₂³

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₂×Q₃₂×C₆₁

C₂

socle

C₂²×C₆₁

C₂×D₈

hypercenter

C₂×Q₃₂

D₆₁

hypocenter

C₆₁

#2209

LCS

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
65. Dic₆₁	Q₃₂	semidirect
77. C₆₁	#2209	semidirect

H ≅ G/N

type

65. Dic₆₁

Q₃₂

semidirect

77. C₆₁

#2209

semidirect

Subgroups — #21447867

Conjugacy classes of subgroups

Complements