Group #35223466 — subgroups

Subgroups — #35223466

Order 20864.

364 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 189 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	#23782222	C₂	1	1	yes	maximal, characteristic
2	#23781792	C₂	1	4	yes	maximal, complemented
3	#23781789	C₂	1	4	yes	maximal, complemented
4	#23781693	C₂	1	2	yes	maximal
5	#23781623	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#423×C₁₆₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
10	#19104735	C₄	1	2	yes	—
11	#19104727	C₄	1	2	yes	—
12	#19104714	C₄	1	4	yes	—
13	C₄×Dic₃₂₆	C₂²	1	4	yes	—
14	#19104694	C₂²	1	2	yes	—
15	Dic₁₆₃⋊C₈	C₂²	1	8	yes	—
16	#19104645	C₂²	1	4	yes	—
17	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	—
18	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	—
19	C₈×Dic₁₆₃	C₂²	1	4	yes	—
20	C₂×M₁₆×C₁₆₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₆₃	C₂²	1	4	yes	—
23	C₂×Dic₃₂₆	C₂×C₄	1	4	yes	—
24	C₂×C₁₆₃⋊C₈	—	2	2	no	—
25	C₂×C₁₆₃⋊C₈	Q₈	1	4	yes	complemented
26	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
27	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
28	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₆₃	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₁₆₃	C₂×C₄	1	4	yes	—
31	Dic₁₆₃⋊C₄	C₂×C₄	1	8	yes	—
32	C₂²⋊Dic₁₆₃	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₆₃	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₆₃	—	2	4	no	—
37	C₂×C₈×C₁₆₃	C₂³	1	4	yes	—
38	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₁₆₃	C₂³	1	2	yes	—
41	Dic₃₂₆	—	2	8	no	—
42	C₂×Dic₁₆₃	C₂²×C₄	1	8	yes	—
43	C₁₆₃⋊C₈	—	2	2	no	—
44	C₁₆₃⋊C₈	C₂×Q₈	1	2	yes	—
45	C₁₆₃⋊C₈	C₂×Q₈	1	2	yes	—
46	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₆₃	—	2	4	no	—
48	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₁₆₃	#41	1	2	yes	—
53	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₆₃	—	2	4	no	—
55	C₂²×C₁₆₃	—	2	2	no	—
56	C₂²×C₁₆₃	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₆₃	—	2	2	no	—
59	C₄×C₁₆₃	C₄×Q₈	1	2	yes	—
60	C₄×C₁₆₃	#131	1	4	yes	—
61	C₄×C₁₆₃	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
63	Dic₁₆₃	—	2	8	no	—
64	C₂×C₁₆₃	—	2	2	no	—
65	C₂×C₁₆₃	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₆₃	C₂×#131	1	2	yes	—
67	C₁₆₃	#2960	1	1	yes	characteristic, complemented, Sylow
68	#2960	—	163	1	no	maximal, Sylow
69	#533	—	163	1	no	—
70	#446	—	163	4	no	—
71	C₈×Q₈	—	163	4	no	—
72	#424	—	163	2	no	—
73	#423	D₁₆₃	1	1	yes	characteristic
74	#405	—	163	2	no	—
75	C₂×C₄×C₈	—	163	1	no	—
76	C₂×M₁₆	C₂×D₁₆₃	1	2	yes	—
77	C₂²×C₈	—	163	1	no	—
78	C₂²×C₈	—	163	1	no	—
79	C₄⋊Q₈	—	163	2	no	—
80	#125	—	163	2	no	—
81	C₂²⋊Q₈	—	163	4	no	—
82	C₄×Q₈	—	163	4	no	—
83	#117	—	163	2	no	—
84	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
85	C₄⋊C₈	—	163	8	no	—
86	C₄⋊C₈	C₂×D₁₆₃	1	4	yes	—
87	#97	—	163	4	no	—
88	C₄×C₈	—	163	4	no	—
89	C₄×C₈	—	163	2	no	—
90	C₄×C₈	—	163	2	no	—
91	C₂×Q₈	—	163	4	no	—
92	C₂²×C₄	C₄×D₁₆₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
94	M₁₆	—	2	4	no	—
95	C₂×C₈	—	326	2	no	—
96	C₂×C₈	—	163	2	no	—
97	C₂×C₈	—	163	4	no	—
98	C₂×C₈	—	163	1	no	—
99	C₂×C₈	—	163	1	no	—
100	C₂×C₈	C₂²×D₁₆₃	1	4	yes	—
101	C₄⋊C₄	—	163	4	no	—
102	C₄⋊C₄	—	163	8	no	—
103	C₄⋊C₄	—	163	4	no	—
104	C₂²⋊C₄	—	163	4	no	—
105	C₄²	—	163	4	no	—
106	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
107	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
108	C₄²	C₂²×D₁₆₃	1	2	yes	—
109	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
110	Q₈	—	326	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	163	8	no	—
113	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
116	C₂×C₄	Q₈×D₁₆₃	1	2	yes	—
117	C₂×C₄	#14579698	1	2	yes	—
118	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	326	2	no	—
121	C₈	—	163	2	no	—
122	C₈	—	163	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#19104700	1	2	yes	—
125	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	326	8	no	—
128	C₄	#19104700	1	2	yes	—
129	C₄	#19104743	1	4	yes	—
130	C₄	C₂×Q₈×D₁₆₃	1	1	yes	characteristic
131	C₄	C₂×#14579698	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#19104700	1	1	yes	characteristic
134	C₂	C₂×#19104743	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23782222

C₂

yes

maximal, characteristic

#23781792

C₂

yes

maximal, complemented

#23781789

C₂

yes

maximal, complemented

#23781693

C₂

yes

maximal

#23781623

C₂

yes

maximal

C₂×C₄×C₁₆₃⋊C₈

C₂

yes

maximal, characteristic

#423×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

characteristic

#19104735

C₄

yes

—

#19104727

C₄

yes

—

#19104714

C₄

yes

—

C₄×Dic₃₂₆

C₂²

yes

—

#19104694

C₂²

yes

—

Dic₁₆₃⋊C₈

C₂²

yes

—

#19104645

C₂²

yes

—

C₄×C₁₆₃⋊C₈

C₂²

yes

—

C₄×C₁₆₃⋊C₈

C₂²

yes

—

C₈×Dic₁₆₃

C₂²

yes

—

C₂×M₁₆×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₃

C₂²

yes

—

C₂×Dic₃₂₆

C₂×C₄

yes

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

Q₈

yes

complemented

C₂×C₁₆₃⋊C₈

C₂³

yes

—

C₂×C₁₆₃⋊C₈

C₂³

yes

characteristic

C₂×C₁₆₃⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₆₃

C₂×C₄

yes

—

C₄⋊Dic₁₆₃

C₂×C₄

yes

—

Dic₁₆₃⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₁₆₃

C₂×C₄

yes

—

C₄×Dic₁₆₃

C₂³

yes

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

M₁₆×C₁₆₃

—

C₂×C₈×C₁₆₃

C₂³

yes

—

C₄²×C₁₆₃

C₂×C₄

yes

characteristic

C₄²×C₁₆₃

C₂×C₄

yes

characteristic

C₄²×C₁₆₃

C₂³

yes

—

Dic₃₂₆

—

C₂×Dic₁₆₃

C₂²×C₄

yes

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

C₂×Q₈

yes

—

C₁₆₃⋊C₈

C₂×Q₈

yes

—

C₂³×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂²×C₄

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

C₂⁴

yes

characteristic

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₄×Q₈

yes

—

C₂²×C₁₆₃

C₂³×C₄

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄×Q₈

yes

—

C₄×C₁₆₃

#131

yes

—

C₄×C₁₆₃

C₂²×Q₈

yes

characteristic

C₄×C₁₆₃

C₂×#41

yes

characteristic

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₆₃

C₂×#131

yes

—

C₁₆₃

#2960

yes

characteristic, complemented, Sylow

#2960

—

163

maximal, Sylow

#533

—

163

—

#446

—

163

—

C₈×Q₈

—

163

—

#424

—

163

—

#423

D₁₆₃

yes

characteristic

#405

—

163

—

C₂×C₄×C₈

—

163

—

C₂×M₁₆

C₂×D₁₆₃

yes

—

C₂²×C₈

—

163

—

C₂²×C₈

—

163

—

C₄⋊Q₈

—

163

—

#125

—

163

—

C₂²⋊Q₈

—

163

—

C₄×Q₈

—

163

—

#117

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

C₄⋊C₈

—

163

—

C₄⋊C₈

C₂×D₁₆₃

yes

—

#97

—

163

—

C₄×C₈

—

163

—

C₄×C₈

—

163

—

C₄×C₈

—

163

—

C₂×Q₈

—

163

—

C₂²×C₄

C₄×D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

C₂×C₈

—

326

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

100

C₂×C₈

C₂²×D₁₆₃

yes

—

101

C₄⋊C₄

—

163

—

102

C₄⋊C₄

—

163

—

103

C₄⋊C₄

—

163

—

104

C₂²⋊C₄

—

163

—

105

C₄²

—

163

—

106

C₄²

C₄×D₁₆₃

yes

characteristic

107

C₄²

C₄×D₁₆₃

yes

characteristic

108

C₄²

C₂²×D₁₆₃

yes

—

109

C₂³

C₂×C₄×D₁₆₃

yes

characteristic

110

Q₈

—

326

—

111

C₂×C₄

—

112

C₂×C₄

—

163

—

113

C₂×C₄

C₂×C₄×D₁₆₃

yes

—

114

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

115

C₂×C₄

C₂×C₄×D₁₆₃

yes

—

116

C₂×C₄

Q₈×D₁₆₃

yes

—

117

C₂×C₄

#14579698

yes

—

118

C₂×C₄

C₂³×D₁₆₃

yes

characteristic

119

C₈

—

120

C₈

—

326

—

121

C₈

—

163

—

122

C₈

—

163

—

123

C₂²

—

124

C₂²

#19104700

yes

—

125

C₂²

C₂²×C₄×D₁₆₃

yes

characteristic

126

C₄

—

127

C₄

—

326

—

128

C₄

#19104700

yes

—

129

C₄

#19104743

yes

—

130

C₄

C₂×Q₈×D₁₆₃

yes

characteristic

131

C₄

C₂×#14579698

yes

characteristic

132

C₂

—

133

C₂

C₂×#19104700

yes

characteristic

134

C₂

C₂×#19104743

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Fitting	#423×C₁₆₃	C₂
prime radical	C₂³×C₁₆₃	C₂²×C₄
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#423	D₁₆₃
hypocenter	C₁₆₃	#2960
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₆₃

commutator, socle

C₂²×C₁₆₃

C₂³×C₄

Fitting

#423×C₁₆₃

C₂

prime radical

C₂³×C₁₆₃

C₂²×C₄

cosocle

C₂×C₄×C₁₆₃

C₂⁴

D₁₆₃

C₁₆₃

G ▷ C₂²×C₁₆₃ ▷ C₁₆₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #23781792	C₂	semidirect
3. #23781789	C₂	semidirect
25. C₂×C₁₆₃⋊C₈	Q₈	semidirect
67. C₁₆₃	#2960	semidirect

H ≅ G/N

type

2. #23781792

C₂

semidirect

3. #23781789

C₂

semidirect

25. C₂×C₁₆₃⋊C₈

Q₈

semidirect

67. C₁₆₃

#2960

semidirect

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

subgroup	index
#23782222	2
#23781792	2
#23781789	2
#23781693	2
#23781623	2
C₂×C₄×C₁₆₃⋊C₈	2
#423×C₁₆₃	2
#2960	163

subgroup

index

C₂×C₄×C₁₆₃⋊C₈

#423×C₁₆₃

#2960

163

Subgroups — #35223466

Conjugacy classes of subgroups

Complements

Maximal subgroups