Group #35215887 — subgroups

Subgroups — #35215887

Order 20864.

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

normal subgroups only

This group has 230 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	#23782296	C₂	1	1	yes	maximal, characteristic
2	#23781707	C₂	1	2	yes	maximal
3	#23781530	C₂	1	1	yes	maximal, characteristic
4	#23781426	C₂	1	2	yes	maximal
5	#343×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	#19104734	—	2	2	no	—
7	#19104726	—	2	2	no	—
8	#19104714	—	2	2	no	—
9	#19104694	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
12	#19104669	—	2	2	no	—
13	#19104667	—	2	2	no	—
14	C₁₆₃⋊#95	—	2	2	no	—
15	C₂×M₁₆×C₁₆₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₂₆	—	2	2	no	—
18	C₂²×Dic₁₆₃	—	2	1	no	—
19	C₂²×Dic₁₆₃	—	2	1	no	—
20	C₄⋊Dic₁₆₃	—	2	2	no	—
21	C₄⋊Dic₁₆₃	—	2	2	no	—
22	C₄⋊Dic₁₆₃	D₄	1	2	yes	—
23	C₄⋊Dic₁₆₃	D₄	1	2	yes	—
24	Dic₁₆₃⋊C₄	—	2	4	no	—
25	C₂²⋊Dic₁₆₃	—	2	2	no	—
26	C₄×Dic₁₆₃	—	2	2	no	—
27	C₄×Dic₁₆₃	—	2	1	no	—
28	C₄×Dic₁₆₃	Q₈	1	2	yes	complemented
29	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
30	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₆₃	—	2	2	no	—
32	M₁₆×C₁₆₃	Q₈	1	4	yes	complemented
33	C₂×C₈×C₁₆₃	—	2	2	no	—
34	C₄⋊C₄×C₁₆₃	—	2	2	no	—
35	Dic₃₂₆	—	8	2	no	—
36	C₂×Dic₁₆₃	—	2	2	no	—
37	C₂×Dic₁₆₃	—	2	2	no	—
38	C₂×Dic₁₆₃	—	2	2	no	—
39	C₂×Dic₁₆₃	—	4	1	no	—
40	C₂×Dic₁₆₃	—	4	1	no	—
41	C₂×Dic₁₆₃	—	4	2	no	—
42	C₂×Dic₁₆₃	—	4	1	no	—
43	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₆₃	—	2	2	no	—
45	C₂×C₄×C₁₆₃	—	4	2	no	—
46	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
50	C₈×C₁₆₃	—	2	4	no	—
51	C₂²×C₁₆₃	—	2	1	no	—
52	C₂²×C₁₆₃	—	2	1	no	—
53	C₂²×C₁₆₃	#120	1	1	yes	characteristic
54	C₂²×C₁₆₃	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂²×C₁₆₃	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₁₆₃	—	4	2	no	—
57	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₆₃	C₄⋊Q₈	1	2	yes	—
59	Dic₁₆₃	—	4	2	no	—
60	Dic₁₆₃	—	4	2	no	—
61	Dic₁₆₃	—	4	2	no	—
62	Dic₁₆₃	—	8	1	no	—
63	C₂×C₁₆₃	—	2	1	no	—
64	C₂×C₁₆₃	—	2	1	no	—
65	C₂×C₁₆₃	#395	1	1	yes	characteristic
66	C₂×C₁₆₃	#454	1	1	yes	characteristic
67	C₂×C₁₆₃	#455	1	1	yes	characteristic
68	C₁₆₃	#2061	1	1	yes	characteristic, complemented, Sylow
69	#2061	—	163	1	no	maximal, Sylow
70	#544	—	163	1	no	—
71	#428	—	163	2	no	—
72	#382	—	163	1	no	—
73	#343	D₁₆₃	1	1	yes	characteristic
74	#339	—	163	2	no	—
75	C₂×M₁₆	C₂×D₁₆₃	1	2	yes	—
76	C₄⋊Q₈	—	326	2	no	—
77	#125	—	326	2	no	—
78	C₂²⋊Q₈	—	326	2	no	—
79	#117	—	163	2	no	—
80	C₂×C₄⋊C₄	—	163	1	no	—
81	C₂×C₄⋊C₄	C₂×D₁₆₃	1	1	yes	characteristic
82	C₂×C₄²	—	163	1	no	—
83	#107	—	326	2	no	—
84	#106	—	326	2	no	—
85	#95	—	326	2	no	—
86	C₂×Q₈	—	326	2	no	—
87	C₂²×C₄	—	326	1	no	—
88	C₂²×C₄	—	326	1	no	—
89	C₂²×C₄	D₆₅₂	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
91	M₁₆	—	2	2	no	—
92	M₁₆	Dic₃₂₆	1	4	yes	complemented
93	C₂×C₈	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	326	2	no	—
96	C₄⋊C₄	—	326	4	no	—
97	C₄⋊C₄	—	326	2	no	—
98	C₄⋊C₄	—	163	2	no	—
99	C₄⋊C₄	—	163	2	no	—
100	C₂²⋊C₄	—	326	2	no	—
101	C₄²	—	326	2	no	—
102	C₄²	—	326	1	no	—
103	C₄²	—	163	2	no	—
104	C₂³	C₂×D₆₅₂	1	1	yes	characteristic
105	Q₈	—	1304	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	326	2	no	—
109	C₂×C₄	—	326	2	no	—
110	C₂×C₄	—	326	2	no	—
111	C₂×C₄	—	652	2	no	—
112	C₂×C₄	—	652	1	no	—
113	C₂×C₄	—	652	1	no	—
114	C₂×C₄	—	652	1	no	—
115	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
116	C₂×C₄	Q₈×D₁₆₃	1	1	yes	characteristic
117	C₂×C₄	C₂×Dic₃₂₆	1	2	yes	—
118	C₂×C₄	#14579698	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#19104705	1	1	yes	characteristic
123	C₂²	#19104715	1	1	yes	characteristic
124	C₂²	#19104735	1	1	yes	characteristic
125	C₄	—	4	2	no	—
126	C₄	—	1304	1	no	—
127	C₄	—	652	2	no	—
128	C₄	—	652	2	no	—
129	C₄	—	652	2	no	—
130	C₄	#19104712	1	2	yes	—
131	C₄	#19104734	1	2	yes	—
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#23781586	1	1	yes	characteristic
135	C₂	#23781840	1	1	yes	characteristic
136	C₂	#23781845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23782296

C₂

yes

maximal, characteristic

#23781707

C₂

yes

maximal

#23781530

C₂

yes

maximal, characteristic

#23781426

C₂

yes

maximal

#343×C₁₆₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₆₃

C₂²

yes

characteristic

#19104669

—

#19104667

—

C₁₆₃⋊#95

—

C₂×M₁₆×C₁₆₃

C₂²

yes

—

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

characteristic

C₂×Dic₃₂₆

—

C₂²×Dic₁₆₃

—

C₂²×Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

D₄

yes

—

C₄⋊Dic₁₆₃

D₄

yes

—

Dic₁₆₃⋊C₄

—

C₂²⋊Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

Q₈

yes

complemented

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

M₁₆×C₁₆₃

—

M₁₆×C₁₆₃

Q₈

yes

complemented

C₂×C₈×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

Dic₃₂₆

—

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×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

#41

yes

characteristic

C₈×C₁₆₃

—

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

C₂²⋊Q₈

yes

—

C₄×C₁₆₃

C₄⋊Q₈

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, complemented, Sylow

#2061

—

163

maximal, Sylow

#544

—

163

—

#428

—

163

—

#382

—

163

—

#343

D₁₆₃

yes

characteristic

#339

—

163

—

C₂×M₁₆

C₂×D₁₆₃

yes

—

C₄⋊Q₈

—

326

—

#125

—

326

—

C₂²⋊Q₈

—

326

—

#117

—

163

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

characteristic

C₂×C₄²

—

163

—

#107

—

326

—

#106

—

326

—

#95

—

326

—

C₂×Q₈

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

D₆₅₂

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

M₁₆

Dic₃₂₆

yes

complemented

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

163

—

C₄⋊C₄

—

163

—

100

C₂²⋊C₄

—

326

—

101

C₄²

—

326

—

102

C₄²

—

326

—

103

C₄²

—

163

—

104

C₂³

C₂×D₆₅₂

yes

characteristic

105

Q₈

—

1304

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

326

—

109

C₂×C₄

—

326

—

110

C₂×C₄

—

326

—

111

C₂×C₄

—

652

—

112

C₂×C₄

—

652

—

113

C₂×C₄

—

652

—

114

C₂×C₄

—

652

—

115

C₂×C₄

D₄×D₁₆₃

yes

—

116

C₂×C₄

Q₈×D₁₆₃

yes

characteristic

117

C₂×C₄

C₂×Dic₃₂₆

yes

—

118

C₂×C₄

#14579698

yes

characteristic

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

#19104705

yes

characteristic

123

C₂²

#19104715

yes

characteristic

124

C₂²

#19104735

yes

characteristic

125

C₄

—

126

C₄

—

1304

—

127

C₄

—

652

—

128

C₄

—

652

—

129

C₄

—

652

—

130

C₄

#19104712

yes

—

131

C₄

#19104734

yes

—

132

C₂

—

133

C₂

—

134

C₂

#23781586

yes

characteristic

135

C₂

#23781840

yes

characteristic

136

C₂

#23781845

yes

characteristic

	subgroup	G/H
center	C₂²	#19104705
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#343×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#120
prime radical	C₂³×C₁₆₃	C₂×D₄
hypercenter	#343	D₁₆₃
hypocenter	C₁₆₃	#2061
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#19104705

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#343×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 ◁ #343

N	H ≅ G/N	type
28. C₄×Dic₁₆₃	Q₈	semidirect
32. M₁₆×C₁₆₃	Q₈	semidirect
68. C₁₆₃	#2061	semidirect
92. M₁₆	Dic₃₂₆	semidirect

H ≅ G/N

type

28. C₄×Dic₁₆₃

Q₈

semidirect

32. M₁₆×C₁₆₃

Q₈

semidirect

68. C₁₆₃

#2061

semidirect

92. M₁₆

Dic₃₂₆

semidirect

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

subgroup	index
#23782296	2
#23781707	2
#23781530	2
#23781426	2
#343×C₁₆₃	2
#2061	163

subgroup

index

#343×C₁₆₃

163

Subgroups — #35215887

Conjugacy classes of subgroups

Complements

Maximal subgroups