Group #35215956 — subgroups

Subgroups — #35215956

Order 20864.

222 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 79 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×#19104660	C₂	1	2	yes	maximal
2	#23781571	C₂	1	1	yes	maximal, characteristic
3	#23781531	C₂	1	1	yes	maximal, characteristic
4	#23781428	C₂	1	2	yes	maximal
5	#336×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₂₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
9	#19104660	—	2	4	no	—
10	C₁₆₃⋊#95	—	2	2	no	—
11	C₁₆₃⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₆₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₂₆	—	2	1	no	—
15	C₂×Dic₃₂₆	—	2	2	no	—
16	C₂×Dic₃₂₆	—	2	2	no	—
17	C₂×Dic₃₂₆	D₄	1	2	yes	—
18	C₂²×Dic₁₆₃	—	2	1	no	—
19	C₂²×Dic₁₆₃	—	2	2	no	—
20	C₂²×Dic₁₆₃	—	2	1	no	—
21	C₄⋊Dic₁₆₃	—	2	2	no	—
22	C₄⋊Dic₁₆₃	—	2	2	no	—
23	C₄⋊Dic₁₆₃	—	2	2	no	—
24	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
25	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₆₃	—	2	2	no	—
27	C₂×C₈×C₁₆₃	—	2	2	no	—
28	C₂×C₈×C₁₆₃	—	2	2	no	—
29	C₄²×C₁₆₃	—	2	2	no	—
30	Dic₃₂₆	—	2	4	no	—
31	Dic₃₂₆	—	4	2	no	—
32	C₂×Dic₁₆₃	—	2	2	no	—
33	C₂×Dic₁₆₃	—	4	1	no	—
34	C₂×Dic₁₆₃	—	4	1	no	—
35	C₂×Dic₁₆₃	—	4	2	no	—
36	C₂×Dic₁₆₃	—	4	1	no	—
37	C₂×Dic₁₆₃	—	4	2	no	—
38	C₂×Dic₁₆₃	—	4	2	no	—
39	C₂×Dic₁₆₃	—	4	2	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	—	2	2	no	—
44	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₆₃	#41	1	2	yes	—
46	C₂×C₄×C₁₆₃	#41	1	2	yes	—
47	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
48	C₈×C₁₆₃	—	4	2	no	—
49	C₂²×C₁₆₃	#121	1	1	yes	characteristic
50	C₂²×C₁₆₃	#123	1	1	yes	characteristic
51	C₂²×C₁₆₃	#124	1	1	yes	characteristic
52	C₂²×C₁₆₃	#135	1	2	yes	—
53	C₂²×C₁₆₃	#137	1	2	yes	—
54	C₄×C₁₆₃	—	4	2	no	—
55	C₄×C₁₆₃	#123	1	2	yes	—
56	C₄×C₁₆₃	#126	1	2	yes	—
57	Dic₁₆₃	—	4	2	no	—
58	Dic₁₆₃	—	8	2	no	—
59	Dic₁₆₃	—	8	1	no	—
60	C₂×C₁₆₃	#397	1	1	yes	characteristic
61	C₂×C₁₆₃	#464	1	2	yes	—
62	C₂×C₁₆₃	#485	1	2	yes	—
63	C₂×C₁₆₃	#488	1	1	yes	characteristic
64	C₂×C₁₆₃	#491	1	1	yes	characteristic
65	C₁₆₃	#2074	1	1	yes	characteristic, complemented, Sylow
66	#2074	—	163	1	no	maximal, Sylow
67	C₂×#103	—	163	2	no	—
68	#391	—	163	1	no	—
69	#382	—	163	1	no	—
70	#340	—	163	2	no	—
71	#336	D₁₆₃	1	1	yes	characteristic
72	C₂²×Q₈	—	163	1	no	—
73	C₂²×C₈	C₂×D₁₆₃	1	2	yes	—
74	C₂×C₄⋊C₄	—	163	1	no	—
75	C₂×C₄⋊C₄	—	163	2	no	—
76	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
77	#103	—	326	4	no	—
78	#95	—	326	2	no	—
79	#95	—	326	2	no	—
80	C₂×Q₈	—	326	1	no	—
81	C₂×Q₈	—	326	2	no	—
82	C₂×Q₈	—	326	2	no	—
83	C₂×Q₈	—	163	2	no	—
84	C₂²×C₄	—	326	1	no	—
85	C₂²×C₄	—	326	2	no	—
86	C₂²×C₄	—	326	1	no	—
87	C₂²×C₄	D₆₅₂	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	326	2	no	—
93	C₄⋊C₄	—	326	2	no	—
94	C₄⋊C₄	—	326	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×D₆₅₂	1	1	yes	characteristic
97	Q₈	—	326	4	no	—
98	Q₈	—	652	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	326	2	no	—
103	C₂×C₄	—	652	2	no	—
104	C₂×C₄	—	652	2	no	—
105	C₂×C₄	—	652	1	no	—
106	C₂×C₄	—	652	2	no	—
107	C₂×C₄	—	652	2	no	—
108	C₂×C₄	—	652	1	no	—
109	C₂×C₄	—	652	1	no	—
110	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
111	C₂×C₄	#14579698	1	2	yes	—
112	C₂×C₄	#14579699	1	2	yes	—
113	C₂×C₄	#14579700	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₂²	#19104710	1	1	yes	characteristic
116	C₂²	#19104719	1	1	yes	characteristic
117	C₂²	#19104723	1	1	yes	characteristic
118	C₂²	#19104757	1	2	yes	—
119	C₂²	#19104768	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	1304	2	no	—
122	C₄	—	1304	1	no	—
123	C₄	—	652	2	no	—
124	C₄	#19104717	1	2	yes	—
125	C₄	#19104730	1	2	yes	—
126	C₂	#23781592	1	1	yes	characteristic
127	C₂	#23781888	1	2	yes	—
128	C₂	#23782007	1	2	yes	—
129	C₂	#23782020	1	1	yes	characteristic
130	C₂	#23782034	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104660

C₂

yes

maximal

#23781571

C₂

yes

maximal, characteristic

#23781531

C₂

yes

maximal, characteristic

#23781428

C₂

yes

maximal

#336×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₂₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

characteristic

#19104660

—

C₁₆₃⋊#95

—

C₁₆₃⋊#95

—

C₂²×C₈×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

D₄

yes

—

C₂²×Dic₁₆₃

—

C₂²×Dic₁₆₃

—

C₂²×Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₄²×C₁₆₃

—

Dic₃₂₆

—

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

#41

yes

characteristic

C₈×C₁₆₃

—

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

—

C₂²×C₁₆₃

#137

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#123

yes

—

C₄×C₁₆₃

#126

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

#397

yes

characteristic

C₂×C₁₆₃

#464

yes

—

C₂×C₁₆₃

#485

yes

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, complemented, Sylow

#2074

—

163

maximal, Sylow

C₂×#103

—

163

—

#391

—

163

—

#382

—

163

—

#340

—

163

—

#336

D₁₆₃

yes

characteristic

C₂²×Q₈

—

163

—

C₂²×C₈

C₂×D₁₆₃

yes

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#103

—

326

—

#95

—

326

—

#95

—

326

—

C₂×Q₈

—

326

—

C₂×Q₈

—

326

—

C₂×Q₈

—

326

—

C₂×Q₈

—

163

—

C₂²×C₄

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

D₆₅₂

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄²

—

C₂³

C₂×D₆₅₂

yes

characteristic

Q₈

—

326

—

Q₈

—

652

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

326

—

103

C₂×C₄

—

652

—

104

C₂×C₄

—

652

—

105

C₂×C₄

—

652

—

106

C₂×C₄

—

652

—

107

C₂×C₄

—

652

—

108

C₂×C₄

—

652

—

109

C₂×C₄

—

652

—

110

C₂×C₄

D₄×D₁₆₃

yes

characteristic

111

C₂×C₄

#14579698

yes

—

112

C₂×C₄

#14579699

yes

—

113

C₂×C₄

#14579700

yes

characteristic

114

C₈

—

115

C₂²

#19104710

yes

characteristic

116

C₂²

#19104719

yes

characteristic

117

C₂²

#19104723

yes

characteristic

118

C₂²

#19104757

yes

—

119

C₂²

#19104768

yes

—

120

C₄

—

121

C₄

—

1304

—

122

C₄

—

1304

—

123

C₄

—

652

—

124

C₄

#19104717

yes

—

125

C₄

#19104730

yes

—

126

C₂

#23781592

yes

characteristic

127

C₂

#23781888

yes

—

128

C₂

#23782007

yes

—

129

C₂

#23782020

yes

characteristic

130

C₂

#23782034

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₅₂
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#336×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂×D₄
hypercenter	#336	D₁₆₃
hypocenter	C₁₆₃	#2074
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₆₅₂

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#336×C₁₆₃

C₂

socle, prime radical

C₂³×C₁₆₃

C₂×D₄

D₁₆₃

C₁₆₃

G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
65. C₁₆₃	#2074	semidirect

H ≅ G/N

type

65. C₁₆₃

#2074

semidirect

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

subgroup	index
C₂×#19104660	2
#23781571	2
#23781531	2
#23781428	2
#336×C₁₆₃	2
#2074	163

subgroup

index

#336×C₁₆₃

163

Subgroups — #35215956

Conjugacy classes of subgroups

Complements

Maximal subgroups