Group #35215878 — subgroups

Subgroups — #35215878

Order 20864.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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	#23781708	C₂	1	2	yes	maximal
2	#23781534	C₂	1	1	yes	maximal, characteristic
3	#23781434	C₂	1	1	yes	maximal, characteristic
4	#23781424	C₂	1	2	yes	maximal
5	#544×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579683	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
8	C₂×Dic₁₆₃⋊C₄	—	2	2	no	—
9	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
10	#19104668	—	2	2	no	—
11	#19104666	—	2	2	no	—
12	C₄⋊Q₈×C₁₆₃	—	2	2	no	—
13	#125×C₁₆₃	—	2	2	no	—
14	C₂²⋊Q₈×C₁₆₃	—	2	2	no	—
15	#117×C₁₆₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₆₃	—	2	1	no	—
18	C₂²×Dic₁₆₃	—	2	1	no	—
19	#14579683	—	2	2	no	—
20	#14579683	Q₈	1	4	yes	complemented
21	C₂×C₁₆₃⋊C₈	—	2	2	no	—
22	C₄⋊Dic₁₆₃	—	2	2	no	—
23	Dic₁₆₃⋊C₄	—	4	4	no	—
24	C₄×Dic₁₆₃	—	2	1	no	—
25	C₄×Dic₁₆₃	Q₈	1	2	yes	complemented
26	C₂×Q₈×C₁₆₃	—	2	2	no	—
27	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
28	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₆₃	—	2	2	no	—
30	C₄⋊C₄×C₁₆₃	—	2	4	no	—
31	C₄⋊C₄×C₁₆₃	—	2	2	no	—
32	C₄⋊C₄×C₁₆₃	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₆₃	D₄	1	2	yes	—
34	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
35	C₄²×C₁₆₃	—	2	2	no	—
36	C₂×Dic₁₆₃	—	2	2	no	—
37	C₂×Dic₁₆₃	—	2	2	no	—
38	C₂×Dic₁₆₃	—	2	2	no	—
39	C₂×Dic₁₆₃	—	4	2	no	—
40	C₂×Dic₁₆₃	—	4	1	no	—
41	C₁₆₃⋊C₈	—	2	4	no	—
42	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₁₆₃	—	8	2	no	—
44	C₂×C₄×C₁₆₃	—	2	2	no	—
45	C₂×C₄×C₁₆₃	—	2	2	no	—
46	C₂×C₄×C₁₆₃	—	2	2	no	—
47	C₂×C₄×C₁₆₃	—	4	2	no	—
48	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
52	C₂²×C₁₆₃	—	2	1	no	—
53	C₂²×C₁₆₃	—	2	1	no	—
54	C₂²×C₁₆₃	#120	1	1	yes	characteristic
55	C₂²×C₁₆₃	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₁₆₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₄×C₁₆₃	—	4	2	no	—
58	C₄×C₁₆₃	—	4	2	no	—
59	C₄×C₁₆₃	—	4	2	no	—
60	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₁₆₃	C₄⋊Q₈	1	2	yes	—
62	Dic₁₆₃	—	2	4	no	—
63	Dic₁₆₃	—	8	1	no	—
64	C₂×C₁₆₃	—	2	1	no	—
65	C₂×C₁₆₃	—	2	1	no	—
66	C₂×C₁₆₃	#395	1	1	yes	characteristic
67	C₂×C₁₆₃	#453	1	1	yes	characteristic
68	C₂×C₁₆₃	#456	1	1	yes	characteristic
69	C₁₆₃	#2060	1	1	yes	characteristic, complemented, Sylow
70	#2060	—	163	1	no	maximal, Sylow
71	#544	D₁₆₃	1	1	yes	characteristic
72	#428	—	163	2	no	—
73	#384	—	163	1	no	—
74	#343	—	163	1	no	—
75	#339	—	163	2	no	—
76	C₂×M₁₆	—	163	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	#125	—	2	2	no	—
79	C₂²⋊Q₈	—	2	2	no	—
80	#117	C₂×D₁₆₃	1	2	yes	—
81	C₂×C₄⋊C₄	—	326	2	no	—
82	C₂×C₄⋊C₄	—	163	1	no	—
83	C₂×C₄⋊C₄	C₂×D₁₆₃	1	1	yes	characteristic
84	C₂×C₄²	—	163	1	no	—
85	#107	—	326	2	no	—
86	#106	—	326	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂²×C₄	—	326	1	no	—
89	C₂²×C₄	—	326	1	no	—
90	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
92	M₁₆	—	326	2	no	—
93	M₁₆	—	163	4	no	—
94	C₂×C₈	—	326	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	326	2	no	—
99	C₄⋊C₄	—	652	4	no	—
100	C₄⋊C₄	C₂²⋊D₁₆₃	1	2	yes	—
101	C₄⋊C₄	C₂²⋊D₁₆₃	1	2	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	326	1	no	—
105	C₄²	—	163	2	no	—
106	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
107	Q₈	—	8	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	326	2	no	—
113	C₂×C₄	—	326	2	no	—
114	C₂×C₄	—	326	2	no	—
115	C₂×C₄	—	652	2	no	—
116	C₂×C₄	—	652	1	no	—
117	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
118	C₂×C₄	Q₈×D₁₆₃	1	2	yes	—
119	C₂×C₄	Q₈×D₁₆₃	1	1	yes	characteristic
120	C₂×C₄	#14579700	1	1	yes	characteristic
121	C₈	—	326	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#19104704	1	1	yes	characteristic
125	C₂²	#19104711	1	1	yes	characteristic
126	C₂²	#19104733	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	1304	1	no	—
131	C₄	—	326	4	no	—
132	C₄	#19104711	1	2	yes	—
133	C₄	#19104733	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#23781584	1	1	yes	characteristic
137	C₂	#23781833	1	1	yes	characteristic
138	C₂	#23781848	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23781708

C₂

yes

maximal

#23781534

C₂

yes

maximal, characteristic

#23781434

C₂

yes

maximal, characteristic

#23781424

C₂

yes

maximal

#544×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

characteristic

C₂×Dic₁₆₃⋊C₄

—

C₂×C₄×Dic₁₆₃

C₂²

yes

characteristic

#19104668

—

#19104666

—

C₄⋊Q₈×C₁₆₃

—

#125×C₁₆₃

—

C₂²⋊Q₈×C₁₆₃

—

#117×C₁₆₃

C₂²

yes

—

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

characteristic

C₂²×Dic₁₆₃

—

C₂²×Dic₁₆₃

—

#14579683

—

#14579683

Q₈

yes

complemented

C₂×C₁₆₃⋊C₈

—

C₄⋊Dic₁₆₃

—

Dic₁₆₃⋊C₄

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

Q₈

yes

complemented

C₂×Q₈×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

D₄

yes

—

C₄⋊C₄×C₁₆₃

D₄

yes

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

Q₈×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

characteristic

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#120

yes

characteristic

C₂²×C₁₆₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₆₃

C₄⋊Q₈

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂²⋊Q₈

yes

—

C₄×C₁₆₃

C₄⋊Q₈

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, complemented, Sylow

#2060

—

163

maximal, Sylow

#544

D₁₆₃

yes

characteristic

#428

—

163

—

#384

—

163

—

#343

—

163

—

#339

—

163

—

C₂×M₁₆

—

163

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₁₆₃

yes

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

characteristic

C₂×C₄²

—

163

—

#107

—

326

—

#106

—

326

—

C₂×Q₈

—

C₂²×C₄

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

326

—

M₁₆

—

163

—

C₂×C₈

—

326

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

652

—

100

C₄⋊C₄

C₂²⋊D₁₆₃

yes

—

101

C₄⋊C₄

C₂²⋊D₁₆₃

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

326

—

105

C₄²

—

163

—

106

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

326

—

113

C₂×C₄

—

326

—

114

C₂×C₄

—

326

—

115

C₂×C₄

—

652

—

116

C₂×C₄

—

652

—

117

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

118

C₂×C₄

Q₈×D₁₆₃

yes

—

119

C₂×C₄

Q₈×D₁₆₃

yes

characteristic

120

C₂×C₄

#14579700

yes

characteristic

121

C₈

—

326

—

122

C₂²

—

123

C₂²

—

124

C₂²

#19104704

yes

characteristic

125

C₂²

#19104711

yes

characteristic

126

C₂²

#19104733

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

1304

—

131

C₄

—

326

—

132

C₄

#19104711

yes

—

133

C₄

#19104733

yes

—

134

C₂

—

135

C₂

—

136

C₂

#23781584

yes

characteristic

137

C₂

#23781833

yes

characteristic

138

C₂

#23781848

yes

characteristic

	subgroup	G/H
center	C₂²	#19104704
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#544×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#120
prime radical	C₂³×C₁₆₃	C₂×D₄
hypercenter	#544	D₁₆₃
hypocenter	C₁₆₃	#2060
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#19104704

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#544×C₁₆₃

C₂

socle

C₂²×C₁₆₃

#120

prime radical

C₂³×C₁₆₃

C₂×D₄

D₁₆₃

C₁₆₃

G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
20. #14579683	Q₈	semidirect
25. C₄×Dic₁₆₃	Q₈	semidirect
69. C₁₆₃	#2060	semidirect

H ≅ G/N

type

20. #14579683

Q₈

semidirect

25. C₄×Dic₁₆₃

Q₈

semidirect

69. C₁₆₃

#2060

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23781708	2
#23781534	2
#23781434	2
#23781424	2
#544×C₁₆₃	2
#2060	163

subgroup

index

#544×C₁₆₃

163

Subgroups — #35215878

Conjugacy classes of subgroups

Complements

Maximal subgroups