Group #35214572 — subgroups

Subgroups — #35214572

Order 20864.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#19104670	C₂	1	2	yes	maximal, complemented
3	#23781629	C₂	1	1	yes	maximal, characteristic
4	#23781429	C₂	1	2	yes	maximal, complemented
5	#407×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579683	C₂²	1	2	yes	—
7	C₂²×C₁₆₃⋊C₈	—	2	4	no	—
8	C₂²×C₁₆₃⋊C₈	C₄	1	4	yes	—
9	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
11	#19104670	—	2	4	no	—
12	C₂²⋊(C₁₆₃⋊C₈)	—	2	2	no	—
13	C₂³×C₄×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₆₃	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₆₃	—	2	2	no	—
16	#14579683	—	2	2	no	—
17	#14579683	—	2	2	no	—
18	C₂×C₁₆₃⋊C₈	—	2	8	no	—
19	C₂×C₁₆₃⋊C₈	—	2	2	no	—
20	C₂×C₁₆₃⋊C₈	—	2	2	no	—
21	C₂×C₁₆₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₁₆₃⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₁₆₃⋊C₈	D₄	1	4	yes	—
24	C₂⁴×C₁₆₃	Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₁₆₃	—	2	2	no	—
26	C₂²×C₄×C₁₆₃	—	2	2	no	—
27	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
29	C₂²×C₄×C₁₆₃	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₆₃	—	2	2	no	—
32	M₁₆×C₁₆₃	—	2	2	no	—
33	C₂×C₈×C₁₆₃	—	2	2	no	—
34	C₁₆₃⋊C₈	—	2	2	no	—
35	C₁₆₃⋊C₈	—	4	2	no	—
36	C₁₆₃⋊C₈	C₂²⋊C₄	1	4	yes	—
37	C₂³×C₁₆₃	—	2	4	no	—
38	C₂³×C₁₆₃	—	2	2	no	—
39	C₂³×C₁₆₃	C₄⋊C₄	1	2	yes	—
40	C₂³×C₁₆₃	C₂×Q₈	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	—	2	4	no	—
45	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₆₃	#41	1	2	yes	—
50	C₈×C₁₆₃	—	4	2	no	—
51	C₂²×C₁₆₃	—	2	8	no	—
52	C₂²×C₁₆₃	—	2	2	no	—
53	C₂²×C₁₆₃	—	2	2	no	—
54	C₂²×C₁₆₃	—	2	2	no	—
55	C₂²×C₁₆₃	#108	1	4	yes	complemented
56	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₆₃	—	2	2	no	—
59	C₄×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₆₃	#117	1	1	yes	characteristic
61	C₄×C₁₆₃	#121	1	2	yes	—
62	C₂×C₁₆₃	—	2	4	no	—
63	C₂×C₁₆₃	—	2	2	no	—
64	C₂×C₁₆₃	#380	1	1	yes	characteristic
65	C₂×C₁₆₃	C₂×#108	1	2	yes	—
66	C₁₆₃	#1810	1	1	yes	characteristic, complemented, Sylow
67	#1810	—	163	1	no	maximal, Sylow
68	C₂³×C₈	—	163	1	no	—
69	C₂×#108	—	163	2	no	—
70	#407	—	163	1	no	—
71	#407	D₁₆₃	1	1	yes	characteristic
72	#341	—	163	2	no	—
73	C₂³×C₄	C₂×D₁₆₃	1	1	yes	characteristic
74	C₂×M₁₆	—	163	2	no	—
75	C₂×M₁₆	C₂×D₁₆₃	1	2	yes	—
76	C₂²×C₈	—	326	4	no	—
77	C₂²×C₈	—	163	4	no	—
78	C₂²×C₈	—	163	1	no	—
79	C₂²×C₈	—	163	1	no	—
80	#108	—	326	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₂²⋊C₈	—	326	2	no	—
83	C₂⁴	Dic₃₂₆	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₄×D₁₆₃	1	2	yes	—
87	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
88	C₂²×C₄	Dic₃₂₆	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	326	2	no	—
93	M₁₆	—	326	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	326	2	no	—
96	C₂×C₈	—	326	8	no	—
97	C₂×C₈	—	326	2	no	—
98	C₂×C₈	—	163	2	no	—
99	C₂×C₈	—	163	2	no	—
100	C₂×C₈	—	163	4	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	2	2	no	—
103	C₂³	Dic₁₆₃⋊C₄	1	2	yes	—
104	C₂³	C₂×Dic₃₂₆	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	Dic₁₆₃⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
113	C₂×C₄	#14579698	1	2	yes	—
114	C₈	—	4	2	no	—
115	C₈	—	326	2	no	—
116	C₈	—	652	2	no	—
117	C₈	—	163	4	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#19104670	1	4	yes	complemented
123	C₂²	C₂×Dic₁₆₃⋊C₄	1	1	yes	characteristic
124	C₂²	#19104712	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	C₂²⋊C₄×D₁₆₃	1	1	yes	characteristic
127	C₄	#19104691	1	1	yes	characteristic
128	C₄	#19104707	1	2	yes	—
129	C₂	—	2	4	no	—
130	C₂	—	2	2	no	—
131	C₂	#23781523	1	1	yes	characteristic
132	C₂	C₂×#19104670	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₁₆₃⋊C₈

C₂

yes

maximal, characteristic

C₂×#19104670

C₂

yes

maximal, complemented

#23781629

C₂

yes

maximal, characteristic

#23781429

C₂

yes

maximal, complemented

#407×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

C₂²×C₁₆₃⋊C₈

—

C₂²×C₁₆₃⋊C₈

C₄

yes

—

C₂²×C₁₆₃⋊C₈

C₂²

yes

characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

characteristic

#19104670

—

C₂²⋊(C₁₆₃⋊C₈)

—

C₂³×C₄×C₁₆₃

C₂²

yes

characteristic

C₂×M₁₆×C₁₆₃

C₂²

yes

—

C₂²⋊C₈×C₁₆₃

—

#14579683

—

#14579683

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₆₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₆₃⋊C₈

D₄

yes

—

C₂⁴×C₁₆₃

Q₈

yes

characteristic

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

Q₈

yes

characteristic

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

M₁₆×C₁₆₃

—

M₁₆×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₆₃

—

C₂³×C₁₆₃

—

C₂³×C₁₆₃

C₄⋊C₄

yes

—

C₂³×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#108

yes

complemented

C₂²×C₁₆₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₆₃

C₂²⋊Q₈

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂×C₂²⋊C₄

yes

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#380

yes

characteristic

C₂×C₁₆₃

C₂×#108

yes

—

C₁₆₃

#1810

yes

characteristic, complemented, Sylow

#1810

—

163

maximal, Sylow

C₂³×C₈

—

163

—

C₂×#108

—

163

—

#407

—

163

—

#407

D₁₆₃

yes

characteristic

#341

—

163

—

C₂³×C₄

C₂×D₁₆₃

yes

characteristic

C₂×M₁₆

—

163

—

C₂×M₁₆

C₂×D₁₆₃

yes

—

C₂²×C₈

—

326

—

C₂²×C₈

—

163

—

C₂²×C₈

—

163

—

C₂²×C₈

—

163

—

#108

—

326

—

C₂²⋊C₈

—

C₂²⋊C₈

—

326

—

C₂⁴

Dic₃₂₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₆₃

yes

—

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

Dic₃₂₆

yes

characteristic

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

326

—

M₁₆

—

326

—

C₂×C₈

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

100

C₂×C₈

—

163

—

101

C₂³

—

102

C₂³

—

103

C₂³

Dic₁₆₃⋊C₄

yes

—

104

C₂³

C₂×Dic₃₂₆

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₁₆₃⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

111

C₂×C₄

D₄×D₁₆₃

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

113

C₂×C₄

#14579698

yes

—

114

C₈

—

115

C₈

—

326

—

116

C₈

—

652

—

117

C₈

—

163

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#19104670

yes

complemented

123

C₂²

C₂×Dic₁₆₃⋊C₄

yes

characteristic

124

C₂²

#19104712

yes

—

125

C₄

—

126

C₄

C₂²⋊C₄×D₁₆₃

yes

characteristic

127

C₄

#19104691

yes

characteristic

128

C₄

#19104707

yes

—

129

C₂

—

130

C₂

—

131

C₂

#23781523

yes

characteristic

132

C₂

C₂×#19104670

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₆₃
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#407×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₆₃	Q₈
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	#407	D₁₆₃
hypocenter	C₁₆₃	#1810
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₆₃

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#407×C₁₆₃

C₂

socle

C₂²×C₁₆₃

C₂×C₄⋊C₄

prime radical

C₂⁴×C₁₆₃

Q₈

cosocle

C₂²×C₄×C₁₆₃

C₂³

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#19104670	C₂	semidirect
4. #23781429	C₂	semidirect
55. C₂²×C₁₆₃	#108	semidirect
66. C₁₆₃	#1810	semidirect
122. C₂²	#19104670	semidirect

H ≅ G/N

type

2. C₂×#19104670

C₂

semidirect

4. #23781429

C₂

semidirect

55. C₂²×C₁₆₃

#108

semidirect

66. C₁₆₃

#1810

semidirect

122. C₂²

#19104670

semidirect

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

subgroup	index
C₂³×C₁₆₃⋊C₈	2
C₂×#19104670	2
#23781629	2
#23781429	2
#407×C₁₆₃	2
#1810	163

subgroup

index

C₂³×C₁₆₃⋊C₈

#407×C₁₆₃

163

Subgroups — #35214572

Conjugacy classes of subgroups

Complements

Maximal subgroups