Group #35216075 — subgroups

Subgroups — #35216075

Order 20864.

254 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 91 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×#19104668	C₂	1	2	yes	maximal
2	C₂×#19104658	C₂	1	2	yes	maximal
3	#23781571	C₂	1	1	yes	maximal, characteristic
4	#23781420	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₂₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
9	#19104668	—	2	4	no	—
10	#19104658	—	2	4	no	—
11	C₁₆₃⋊#95	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₆₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₂₆	—	2	2	no	—
16	C₂×Dic₃₂₆	—	2	1	no	—
17	C₂×Dic₃₂₆	—	2	2	no	—
18	C₂×Dic₃₂₆	Q₈	1	2	yes	—
19	C₂²×Dic₁₆₃	—	2	1	no	—
20	C₂²×Dic₁₆₃	—	2	1	no	—
21	C₂×C₁₆₃⋊C₈	—	2	2	no	—
22	C₂×C₁₆₃⋊C₈	—	2	2	no	—
23	C₂×C₁₆₃⋊C₈	D₄	1	4	yes	—
24	C₄⋊Dic₁₆₃	—	2	2	no	—
25	C₂²×C₄×C₁₆₃	—	2	2	no	—
26	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
27	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₆₃	—	2	2	no	—
29	C₄⋊C₄×C₁₆₃	—	2	2	no	—
30	C₄⋊C₄×C₁₆₃	—	4	4	no	—
31	C₄²×C₁₆₃	—	2	2	no	—
32	Dic₃₂₆	—	2	4	no	—
33	Dic₃₂₆	—	4	2	no	—
34	C₂×Dic₁₆₃	—	2	2	no	—
35	C₂×Dic₁₆₃	—	4	1	no	—
36	C₂×Dic₁₆₃	—	4	1	no	—
37	C₂×Dic₁₆₃	—	4	1	no	—
38	C₂×Dic₁₆₃	—	4	2	no	—
39	C₁₆₃⋊C₈	—	2	4	no	—
40	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₆₃	—	2	2	no	—
42	C₂×C₄×C₁₆₃	—	2	2	no	—
43	C₂×C₄×C₁₆₃	—	4	2	no	—
44	C₂×C₄×C₁₆₃	—	4	4	no	—
45	C₂×C₄×C₁₆₃	Q₁₆	1	4	yes	—
46	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₆₃	#41	1	2	yes	—
49	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
50	C₂²×C₁₆₃	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₆₃	#123	1	1	yes	characteristic
52	C₂²×C₁₆₃	#127	1	1	yes	characteristic
53	C₂²×C₁₆₃	C₂×Q₁₆	1	2	yes	—
54	C₂²×C₁₆₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₆₃	—	2	4	no	—
56	C₄×C₁₆₃	—	8	2	no	—
57	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₆₃	#123	1	2	yes	—
59	Dic₁₆₃	—	4	2	no	—
60	Dic₁₆₃	—	8	1	no	—
61	C₂×C₁₆₃	#399	1	1	yes	characteristic
62	C₂×C₁₆₃	#477	1	2	yes	—
63	C₂×C₁₆₃	#484	1	2	yes	—
64	C₂×C₁₆₃	#494	1	1	yes	characteristic
65	C₂×C₁₆₃	#496	1	1	yes	characteristic
66	C₁₆₃	#2095	1	1	yes	characteristic, complemented, Sylow
67	#2095	—	163	1	no	maximal, Sylow
68	C₂×#107	—	163	2	no	—
69	C₂×#103	—	163	2	no	—
70	#391	—	163	1	no	—
71	#384	D₁₆₃	1	1	yes	characteristic
72	#336	—	163	1	no	—
73	C₂²×Q₈	—	163	1	no	—
74	C₂²×C₈	—	163	2	no	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	—	163	1	no	—
77	C₂×C₄⋊C₄	C₂×D₁₆₃	1	2	yes	—
78	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
79	#107	—	326	4	no	—
80	#103	—	326	4	no	—
81	#95	—	326	2	no	—
82	C₂×Q₈	—	326	1	no	—
83	C₂×Q₈	—	326	2	no	—
84	C₂×Q₈	—	326	2	no	—
85	C₂×Q₈	—	163	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	326	1	no	—
88	C₂²×C₄	—	326	1	no	—
89	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
91	C₂×C₈	—	326	2	no	—
92	C₂×C₈	—	326	2	no	—
93	C₂×C₈	—	163	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	4	4	no	—
97	C₄⋊C₄	—	326	2	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
100	Q₈	—	326	4	no	—
101	Q₈	—	652	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	326	2	no	—
107	C₂×C₄	—	652	1	no	—
108	C₂×C₄	—	652	2	no	—
109	C₂×C₄	—	652	1	no	—
110	C₂×C₄	—	652	1	no	—
111	C₂×C₄	#14579689	1	4	yes	—
112	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
113	C₂×C₄	Q₈×D₁₆₃	1	1	yes	characteristic
114	C₂×C₄	#14579699	1	2	yes	—
115	C₂×C₄	#14579700	1	1	yes	characteristic
116	C₈	—	326	4	no	—
117	C₂²	#19104711	1	1	yes	characteristic
118	C₂²	#19104718	1	1	yes	characteristic
119	C₂²	#19104731	1	1	yes	characteristic
120	C₂²	C₂×#14579689	1	2	yes	—
121	C₂²	#19104761	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	8	2	no	—
124	C₄	—	1304	1	no	—
125	C₄	—	652	2	no	—
126	C₄	#19104713	1	2	yes	—
127	C₄	#19104720	1	2	yes	—
128	C₂	#23781600	1	1	yes	characteristic
129	C₂	#23781957	1	2	yes	—
130	C₂	#23782000	1	2	yes	—
131	C₂	#23782044	1	1	yes	characteristic
132	C₂	#23782052	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104668

C₂

yes

maximal

C₂×#19104658

C₂

yes

maximal

#23781571

C₂

yes

maximal, characteristic

#23781420

C₂

yes

maximal, characteristic

#384×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₂₆

C₂²

yes

characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

characteristic

#19104668

—

#19104658

—

C₁₆₃⋊#95

—

C₂×C₄⋊C₄×C₁₆₃

—

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

Q₈

yes

—

C₂²×Dic₁₆₃

—

C₂²×Dic₁₆₃

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

D₄

yes

—

C₄⋊Dic₁₆₃

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

—

Dic₃₂₆

—

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

Q₁₆

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

#41

yes

characteristic

C₂²×C₁₆₃

C₂²⋊Q₈

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

C₂×Q₁₆

yes

—

C₂²×C₁₆₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂²⋊Q₈

yes

—

C₄×C₁₆₃

#123

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

#399

yes

characteristic

C₂×C₁₆₃

#477

yes

—

C₂×C₁₆₃

#484

yes

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, complemented, Sylow

#2095

—

163

maximal, Sylow

C₂×#107

—

163

—

C₂×#103

—

163

—

#391

—

163

—

#384

D₁₆₃

yes

characteristic

#336

—

163

—

C₂²×Q₈

—

163

—

C₂²×C₈

—

163

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#107

—

326

—

#103

—

326

—

#95

—

326

—

C₂×Q₈

—

326

—

C₂×Q₈

—

326

—

C₂×Q₈

—

326

—

C₂×Q₈

—

163

—

C₂²×C₄

—

C₂²×C₄

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

163

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₄²

—

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

100

Q₈

—

326

—

101

Q₈

—

652

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

326

—

107

C₂×C₄

—

652

—

108

C₂×C₄

—

652

—

109

C₂×C₄

—

652

—

110

C₂×C₄

—

652

—

111

C₂×C₄

#14579689

yes

—

112

C₂×C₄

D₄×D₁₆₃

yes

—

113

C₂×C₄

Q₈×D₁₆₃

yes

characteristic

114

C₂×C₄

#14579699

yes

—

115

C₂×C₄

#14579700

yes

characteristic

116

C₈

—

326

—

117

C₂²

#19104711

yes

characteristic

118

C₂²

#19104718

yes

characteristic

119

C₂²

#19104731

yes

characteristic

120

C₂²

C₂×#14579689

yes

—

121

C₂²

#19104761

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

1304

—

125

C₄

—

652

—

126

C₄

#19104713

yes

—

127

C₄

#19104720

yes

—

128

C₂

#23781600

yes

characteristic

129

C₂

#23781957

yes

—

130

C₂

#23782000

yes

—

131

C₂

#23782044

yes

characteristic

132

C₂

#23782052

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₆₃
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#384×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂×D₄
hypercenter	#384	D₁₆₃
hypocenter	C₁₆₃	#2095
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₆₃

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#384×C₁₆₃

C₂

socle, prime radical

C₂³×C₁₆₃

C₂×D₄

D₁₆₃

C₁₆₃

G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
66. C₁₆₃	#2095	semidirect

H ≅ G/N

type

66. C₁₆₃

#2095

semidirect

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

subgroup	index
C₂×#19104668	2
C₂×#19104658	2
#23781571	2
#23781420	2
#384×C₁₆₃	2
#2095	163

subgroup

index

#384×C₁₆₃

163

Subgroups — #35216075

Conjugacy classes of subgroups

Complements

Maximal subgroups