Group #35223942 — subgroups

Subgroups — #35223942

Order 20864.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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	#23782236	C₂	1	3	yes	maximal, complemented
2	#23781855	C₂	1	8	yes	maximal, complemented
3	#23781639	C₂	1	3	yes	maximal, complemented
4	#585×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₂₆	C₂²	1	1	yes	characteristic, complemented
6	#19104733	—	2	3	no	—
7	#19104722	—	2	3	no	—
8	#19104716	C₂²	1	12	yes	complemented
9	#19104714	—	2	6	no	—
10	#19104694	C₂²	1	3	yes	complemented
11	#19104650	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₁₆₃	—	2	3	no	—
13	H(5,𝔽₂)×C₁₆₃	—	2	3	no	—
14	H(5,𝔽₂)×C₁₆₃	C₂²	1	4	yes	—
15	C₂×#41×C₁₆₃	—	2	6	no	—
16	C₂×#41×C₁₆₃	C₂²	1	3	yes	—
17	C₂×Dic₃₂₆	—	2	3	no	—
18	C₂×Dic₃₂₆	—	2	3	no	—
19	C₂²×Dic₁₆₃	C₂³	1	2	yes	—
20	C₄⋊Dic₁₆₃	—	2	3	no	—
21	Dic₁₆₃⋊C₄	—	2	12	no	—
22	C₂²⋊Dic₁₆₃	—	2	12	no	—
23	C₂²⋊Dic₁₆₃	C₂³	1	6	yes	—
24	C₄×Dic₁₆₃	—	2	3	no	—
25	#41×C₁₆₃	—	2	4	no	—
26	#41×C₁₆₃	—	2	12	no	—
27	#41×C₁₆₃	—	2	6	no	—
28	#41×C₁₆₃	—	2	12	no	—
29	#41×C₁₆₃	—	2	6	no	—
30	C₂×Q₈×C₁₆₃	—	2	6	no	—
31	C₂×Q₈×C₁₆₃	D₄	1	3	yes	—
32	C₂×D₄×C₁₆₃	—	2	6	no	—
33	C₂×D₄×C₁₆₃	—	2	12	no	—
34	C₂×D₄×C₁₆₃	D₄	1	3	yes	—
35	C₂×D₄×C₁₆₃	C₂³	1	6	yes	—
36	C₂²×C₄×C₁₆₃	—	2	4	no	—
37	C₂²×C₄×C₁₆₃	D₄	1	6	yes	—
38	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
39	Dic₃₂₆	—	4	1	no	—
40	Dic₃₂₆	—	4	3	no	—
41	C₂×Dic₁₆₃	—	2	6	no	—
42	C₂×Dic₁₆₃	—	2	6	no	—
43	C₂³×C₁₆₃	—	2	4	no	—
44	C₂³×C₁₆₃	C₂×D₄	1	6	yes	—
45	C₂³×C₁₆₃	C₂⁴	1	1	yes	characteristic
46	Q₈×C₁₆₃	—	2	4	no	—
47	Q₈×C₁₆₃	—	2	6	no	—
48	D₄×C₁₆₃	—	2	12	no	—
49	D₄×C₁₆₃	—	2	12	no	—
50	D₄×C₁₆₃	—	2	6	no	—
51	C₂×C₄×C₁₆₃	—	2	12	no	—
52	C₂×C₄×C₁₆₃	—	2	6	no	—
53	C₂×C₄×C₁₆₃	—	2	6	no	—
54	C₂×C₄×C₁₆₃	C₂×D₄	1	6	yes	—
55	C₂×C₄×C₁₆₃	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₁₆₃	C₂×D₄	1	3	yes	—
57	C₂²×C₁₆₃	—	2	6	no	—
58	C₂²×C₁₆₃	—	4	1	no	—
59	C₂²×C₁₆₃	—	4	6	no	—
60	C₂²×C₁₆₃	—	8	4	no	—
61	C₂²×C₁₆₃	C₂²×D₄	1	3	yes	—
62	C₄×C₁₆₃	—	2	6	no	—
63	C₄×C₁₆₃	#120	1	1	yes	characteristic
64	C₄×C₁₆₃	#120	1	3	yes	—
65	Dic₁₆₃	—	4	6	no	—
66	Dic₁₆₃	—	4	2	no	—
67	C₂×C₁₆₃	—	2	3	no	—
68	C₂×C₁₆₃	—	4	6	no	—
69	C₂×C₁₆₃	C₂×#120	1	1	yes	characteristic
70	C₁₆₃	#3028	1	1	yes	characteristic, complemented, Sylow
71	#3028	—	163	1	no	maximal, Sylow
72	#585	D₁₆₃	1	1	yes	characteristic
73	#536	—	163	3	no	—
74	#458	—	163	8	no	—
75	#410	—	163	3	no	—
76	2₋¹⁺⁴	—	2	3	no	—
77	H(5,𝔽₂)	—	2	3	no	—
78	H(5,𝔽₂)	C₂×D₁₆₃	1	4	yes	—
79	C₂×#41	—	2	6	no	—
80	C₂×#41	C₂×D₁₆₃	1	3	yes	—
81	C₂²×Q₈	—	163	1	no	—
82	C₄⋊Q₈	—	326	3	no	—
83	#124	—	326	3	no	—
84	#123	—	163	12	no	—
85	C₂²⋊Q₈	—	326	6	no	—
86	#117	—	163	3	no	—
87	#99	—	163	12	no	—
88	#41	—	2	4	no	—
89	#41	—	2	12	no	—
90	#41	—	2	6	no	—
91	#41	—	2	12	no	—
92	#41	—	2	6	no	—
93	C₂×Q₈	—	2	6	no	—
94	C₂×Q₈	—	326	3	no	—
95	C₂×Q₈	—	326	3	no	—
96	C₂×Q₈	C₂²⋊D₁₆₃	1	3	yes	—
97	C₂×D₄	—	2	6	no	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	C₂²⋊D₁₆₃	1	3	yes	—
100	C₂×D₄	C₂²×D₁₆₃	1	6	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	163	2	no	—
103	C₂²×C₄	C₂²⋊D₁₆₃	1	6	yes	—
104	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
105	C₄⋊C₄	—	326	3	no	—
106	C₄⋊C₄	—	326	12	no	—
107	C₂²⋊C₄	—	326	12	no	—
108	C₂²⋊C₄	—	163	6	no	—
109	C₄²	—	326	3	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂×C₂²⋊D₁₆₃	1	6	yes	—
112	C₂³	C₂³×D₁₆₃	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	6	no	—
115	Q₈	—	652	1	no	—
116	Q₈	—	652	3	no	—
117	D₄	—	2	12	no	—
118	D₄	—	2	12	no	—
119	D₄	—	2	6	no	—
120	C₂×C₄	—	2	12	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	326	6	no	—
124	C₂×C₄	—	326	6	no	—
125	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	3	yes	—
127	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	3	yes	—
128	C₂²	—	2	6	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	8	4	no	—
132	C₂²	C₂²×C₂²⋊D₁₆₃	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	652	6	no	—
135	C₄	—	652	2	no	—
136	C₄	#19104704	1	1	yes	characteristic
137	C₄	#19104704	1	3	yes	—
138	C₂	—	2	3	no	—
139	C₂	—	4	6	no	—
140	C₂	C₂×#19104704	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23782236

C₂

yes

maximal, complemented

#23781855

C₂

yes

maximal, complemented

#23781639

C₂

yes

maximal, complemented

#585×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₂₆

C₂²

yes

characteristic, complemented

#19104733

—

#19104722

—

#19104716

C₂²

yes

complemented

#19104714

—

#19104694

C₂²

yes

complemented

#19104650

C₂²

yes

complemented

2₋¹⁺⁴×C₁₆₃

—

H(5,𝔽₂)×C₁₆₃

—

H(5,𝔽₂)×C₁₆₃

C₂²

yes

—

C₂×#41×C₁₆₃

—

C₂×#41×C₁₆₃

C₂²

yes

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂²×Dic₁₆₃

C₂³

yes

—

C₄⋊Dic₁₆₃

—

Dic₁₆₃⋊C₄

—

C₂²⋊Dic₁₆₃

—

C₂²⋊Dic₁₆₃

C₂³

yes

—

C₄×Dic₁₆₃

—

#41×C₁₆₃

—

#41×C₁₆₃

—

#41×C₁₆₃

—

#41×C₁₆₃

—

#41×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

D₄

yes

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

D₄

yes

—

C₂×D₄×C₁₆₃

C₂³

yes

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

Dic₃₂₆

—

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂³×C₁₆₃

—

C₂³×C₁₆₃

C₂×D₄

yes

—

C₂³×C₁₆₃

C₂⁴

yes

characteristic

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂²×D₄

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#120

yes

characteristic

C₄×C₁₆₃

#120

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

C₂×#120

yes

characteristic

C₁₆₃

#3028

yes

characteristic, complemented, Sylow

#3028

—

163

maximal, Sylow

#585

D₁₆₃

yes

characteristic

#536

—

163

—

#458

—

163

—

#410

—

163

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₁₆₃

yes

—

C₂×#41

—

C₂×#41

C₂×D₁₆₃

yes

—

C₂²×Q₈

—

163

—

C₄⋊Q₈

—

326

—

#124

—

326

—

#123

—

163

—

C₂²⋊Q₈

—

326

—

#117

—

163

—

#99

—

163

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

326

—

C₂×Q₈

—

326

—

C₂×Q₈

C₂²⋊D₁₆₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₆₃

yes

—

100

C₂×D₄

C₂²×D₁₆₃

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

163

—

103

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

104

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

105

C₄⋊C₄

—

326

—

106

C₄⋊C₄

—

326

—

107

C₂²⋊C₄

—

326

—

108

C₂²⋊C₄

—

163

—

109

C₄²

—

326

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₁₆₃

yes

—

112

C₂³

C₂³×D₁₆₃

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

652

—

116

Q₈

—

652

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

326

—

124

C₂×C₄

—

326

—

125

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

126

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

127

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×C₂²⋊D₁₆₃

yes

—

133

C₄

—

134

C₄

—

652

—

135

C₄

—

652

—

136

C₄

#19104704

yes

characteristic

137

C₄

#19104704

yes

—

138

C₂

—

139

C₂

—

140

C₂

C₂×#19104704

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#19104704
commutator, cosocle	C₂³×C₁₆₃	C₂⁴
Frattini	C₂³	C₂³×D₁₆₃
Fitting, prime radical	#585×C₁₆₃	C₂
socle	C₂×C₁₆₃	C₂×#120
hypercenter	#585	D₁₆₃
hypocenter	C₁₆₃	#3028
LCS	G ▷ C₂³×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#19104704

commutator, cosocle

C₂³×C₁₆₃

C₂⁴

Frattini

C₂³

C₂³×D₁₆₃

Fitting, prime radical

#585×C₁₆₃

C₂

C₂×C₁₆₃

D₁₆₃

C₁₆₃

G ▷ C₂³×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #23782236	C₂	semidirect
2. #23781855	C₂	semidirect
3. #23781639	C₂	semidirect
5. C₂²×Dic₃₂₆	C₂²	semidirect
8. #19104716	C₂²	semidirect
10. #19104694	C₂²	semidirect
11. #19104650	C₂²	semidirect
70. C₁₆₃	#3028	semidirect

H ≅ G/N

type

1. #23782236

C₂

semidirect

2. #23781855

C₂

semidirect

3. #23781639

C₂

semidirect

5. C₂²×Dic₃₂₆

C₂²

semidirect

8. #19104716

C₂²

semidirect

10. #19104694

C₂²

semidirect

11. #19104650

C₂²

semidirect

70. C₁₆₃

#3028

semidirect

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

subgroup	index
#23782236	2
#23781855	2
#23781639	2
#585×C₁₆₃	2
#3028	163

subgroup

index

#585×C₁₆₃

163

Subgroups — #35223942

Conjugacy classes of subgroups

Complements

Maximal subgroups