Group #35214652 — subgroups

Subgroups — #35214652

Order 20864.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄⋊(C₁₆₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₁₆₃	C₂	1	1	yes	maximal, characteristic
3	#23781419	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₆₃	C₄	1	2	yes	—
7	C₂×C₄×Dic₁₆₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₆₃	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₆₃	C₂²	1	2	yes	—
10	C₄⋊(C₁₆₃⋊C₈)	—	2	2	no	—
11	C₂²×C₈×C₁₆₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₆₃	—	2	2	no	—
14	C₂²×Dic₁₆₃	C₂×C₄	1	2	yes	—
15	C₂²×Dic₁₆₃	C₂×C₄	1	2	yes	—
16	C₂×C₁₆₃⋊C₈	—	2	2	no	—
17	C₂×C₁₆₃⋊C₈	—	2	4	no	—
18	C₄×Dic₁₆₃	—	2	4	no	—
19	C₄×Dic₁₆₃	—	2	2	no	—
20	C₄×Dic₁₆₃	—	2	2	no	—
21	C₄×Dic₁₆₃	—	2	2	no	—
22	C₄×Dic₁₆₃	C₈	1	4	yes	complemented
23	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₆₃	—	2	2	no	—
27	C₂×C₈×C₁₆₃	—	2	4	no	—
28	C₄²×C₁₆₃	D₄	1	2	yes	—
29	C₄²×C₁₆₃	Q₈	1	2	yes	—
30	C₂×Dic₁₆₃	—	2	2	no	—
31	C₂×Dic₁₆₃	—	2	2	no	—
32	C₂×Dic₁₆₃	—	2	2	no	—
33	C₂×Dic₁₆₃	—	2	2	no	—
34	C₂×Dic₁₆₃	C₂×C₈	1	4	yes	—
35	C₂×Dic₁₆₃	M₁₆	1	4	yes	—
36	C₁₆₃⋊C₈	—	4	2	no	—
37	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₆₃	—	2	2	no	—
39	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₆₃	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₁₆₃	M₁₆	1	2	yes	—
43	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₆₃	#41	1	2	yes	—
46	C₂×C₄×C₁₆₃	#41	1	2	yes	—
47	C₈×C₁₆₃	—	4	2	no	—
48	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₆₃	#117	1	2	yes	—
50	C₂²×C₁₆₃	C₂²×C₈	1	1	yes	characteristic
51	C₂²×C₁₆₃	C₂×M₁₆	1	2	yes	—
52	C₂²×C₁₆₃	C₂×M₁₆	1	1	yes	characteristic
53	C₄×C₁₆₃	—	2	2	no	—
54	C₄×C₁₆₃	C₄⋊C₈	1	4	yes	—
55	C₄×C₁₆₃	#124	1	1	yes	characteristic
56	C₄×C₁₆₃	#124	1	1	yes	characteristic
57	C₄×C₁₆₃	#125	1	2	yes	—
58	Dic₁₆₃	—	2	4	no	—
59	Dic₁₆₃	—	2	4	no	—
60	C₂×C₁₆₃	#382	1	1	yes	characteristic
61	C₂×C₁₆₃	C₂×C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₁₆₃	#423	1	1	yes	characteristic
63	C₂×C₁₆₃	#431	1	2	yes	—
64	C₂×C₁₆₃	#432	1	2	yes	—
65	C₁₆₃	#1831	1	1	yes	characteristic, complemented, Sylow
66	#1831	—	163	1	no	maximal, Sylow
67	C₂×C₄⋊C₈	—	163	1	no	—
68	C₂×C₄⋊C₈	D₁₆₃	1	1	yes	characteristic
69	C₄³	—	163	1	no	—
70	#336	—	163	4	no	—
71	C₂²×C₈	—	163	2	no	—
72	C₂²×C₈	C₂×D₁₆₃	1	2	yes	—
73	C₂×C₄²	—	163	2	no	—
74	C₂×C₄²	—	163	1	no	—
75	C₂×C₄²	—	163	2	no	—
76	C₂×C₄²	—	163	1	no	—
77	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	—	326	2	no	—
80	C₂²×C₄	—	163	2	no	—
81	C₂²×C₄	—	163	2	no	—
82	C₂²×C₄	C₄×D₁₆₃	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₁₆₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	326	2	no	—
88	C₂×C₈	—	326	4	no	—
89	C₄²	—	326	4	no	—
90	C₄²	—	326	2	no	—
91	C₄²	—	326	2	no	—
92	C₄²	—	326	2	no	—
93	C₄²	—	163	4	no	—
94	C₄²	C₂²⋊D₁₆₃	1	2	yes	—
95	C₄²	Dic₃₂₆	1	2	yes	—
96	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	326	2	no	—
99	C₂×C₄	—	326	2	no	—
100	C₂×C₄	—	326	2	no	—
101	C₂×C₄	—	326	2	no	—
102	C₂×C₄	—	163	4	no	—
103	C₂×C₄	—	163	4	no	—
104	C₂×C₄	Dic₁₆₃⋊C₄	1	2	yes	—
105	C₂×C₄	Dic₁₆₃⋊C₄	1	2	yes	—
106	C₂×C₄	C₈×D₁₆₃	1	2	yes	—
107	C₂×C₄	#14579682	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₃₂₆	1	1	yes	characteristic
110	C₂×C₄	#14579698	1	2	yes	—
111	C₂×C₄	#14579700	1	2	yes	—
112	C₈	—	4	2	no	—
113	C₈	—	652	2	no	—
114	C₂²	C₂×Dic₁₆₃⋊C₄	1	1	yes	characteristic
115	C₂²	#19104693	1	2	yes	—
116	C₂²	C₂×C₈×D₁₆₃	1	1	yes	characteristic
117	C₂²	M₁₆×D₁₆₃	1	2	yes	—
118	C₂²	C₂×#14579682	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	326	4	no	—
121	C₄	—	326	4	no	—
122	C₄	Dic₁₆₃⋊C₈	1	4	yes	—
123	C₄	#19104722	1	1	yes	characteristic
124	C₄	#19104724	1	1	yes	characteristic
125	C₄	#19104726	1	2	yes	—
126	C₂	#23781530	1	1	yes	characteristic
127	C₂	C₂×Dic₁₆₃⋊C₈	1	1	yes	characteristic
128	C₂	#23781688	1	1	yes	characteristic
129	C₂	#23781717	1	2	yes	—
130	C₂	#23781721	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₆₃⋊C₈)

C₂

yes

maximal, characteristic

C₄²×Dic₁₆₃

C₂

yes

maximal, characteristic

#23781419

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₆₃

C₄

yes

—

C₂×C₄×Dic₁₆₃

C₄

yes

characteristic

C₂×C₄×Dic₁₆₃

C₄

yes

characteristic

C₂×C₄×Dic₁₆₃

C₂²

yes

—

C₄⋊(C₁₆₃⋊C₈)

—

C₂²×C₈×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₃

—

C₂²×Dic₁₆₃

C₂×C₄

yes

—

C₂²×Dic₁₆₃

C₂×C₄

yes

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

C₈

yes

complemented

C₂²×C₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₄²×C₁₆₃

D₄

yes

—

C₄²×C₁₆₃

Q₈

yes

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

C₂×C₈

yes

—

C₂×Dic₁₆₃

M₁₆

yes

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂×C₈

yes

—

C₂×C₄×C₁₆₃

M₁₆

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₆₃

#117

yes

—

C₂²×C₁₆₃

C₂²×C₈

yes

characteristic

C₂²×C₁₆₃

C₂×M₁₆

yes

—

C₂²×C₁₆₃

C₂×M₁₆

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄⋊C₈

yes

—

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

#382

yes

characteristic

C₂×C₁₆₃

C₂×C₄⋊C₈

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

—

C₂×C₁₆₃

#432

yes

—

C₁₆₃

#1831

yes

characteristic, complemented, Sylow

#1831

—

163

maximal, Sylow

C₂×C₄⋊C₈

—

163

—

C₂×C₄⋊C₈

D₁₆₃

yes

characteristic

C₄³

—

163

—

#336

—

163

—

C₂²×C₈

—

163

—

C₂²×C₈

C₂×D₁₆₃

yes

—

C₂×C₄²

—

163

—

C₂×C₄²

—

163

—

C₂×C₄²

—

163

—

C₂×C₄²

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

326

—

C₂²×C₄

—

163

—

C₂²×C₄

—

163

—

C₂²×C₄

C₄×D₁₆₃

yes

characteristic

C₂²×C₄

C₄×D₁₆₃

yes

characteristic

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₄²

—

326

—

C₄²

—

326

—

C₄²

—

326

—

C₄²

—

326

—

C₄²

—

163

—

C₄²

C₂²⋊D₁₆₃

yes

—

C₄²

Dic₃₂₆

yes

—

C₂³

C₂×C₄×D₁₆₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

326

—

C₂×C₄

—

326

—

100

C₂×C₄

—

326

—

101

C₂×C₄

—

326

—

102

C₂×C₄

—

163

—

103

C₂×C₄

—

163

—

104

C₂×C₄

Dic₁₆₃⋊C₄

yes

—

105

C₂×C₄

Dic₁₆₃⋊C₄

yes

—

106

C₂×C₄

C₈×D₁₆₃

yes

—

107

C₂×C₄

#14579682

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

109

C₂×C₄

C₂×Dic₃₂₆

yes

characteristic

110

C₂×C₄

#14579698

yes

—

111

C₂×C₄

#14579700

yes

—

112

C₈

—

113

C₈

—

652

—

114

C₂²

C₂×Dic₁₆₃⋊C₄

yes

characteristic

115

C₂²

#19104693

yes

—

116

C₂²

C₂×C₈×D₁₆₃

yes

characteristic

117

C₂²

M₁₆×D₁₆₃

yes

—

118

C₂²

C₂×#14579682

yes

characteristic

119

C₄

—

120

C₄

—

326

—

121

C₄

—

326

—

122

C₄

Dic₁₆₃⋊C₈

yes

—

123

C₄

#19104722

yes

characteristic

124

C₄

#19104724

yes

characteristic

125

C₄

#19104726

yes

—

126

C₂

#23781530

yes

characteristic

127

C₂

C₂×Dic₁₆₃⋊C₈

yes

characteristic

128

C₂

#23781688

yes

characteristic

129

C₂

#23781717

yes

—

130

C₂

#23781721

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₆₃
commutator	C₂²×C₁₆₃	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₆₃
hypocenter	C₁₆₃	#1831
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₆₃

commutator

C₂²×C₁₆₃

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₁₆₃

C₂

socle, prime radical

C₂³×C₁₆₃

C₂²×C₄

cosocle

C₂²×C₄×C₁₆₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₆₃

C₁₆₃

G ▷ C₂²×C₁₆₃ ▷ C₁₆₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
22. C₄×Dic₁₆₃	C₈	semidirect
65. C₁₆₃	#1831	semidirect

H ≅ G/N

type

22. C₄×Dic₁₆₃

C₈

semidirect

65. C₁₆₃

#1831

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₆₃⋊C₈)	2
C₄²×Dic₁₆₃	2
#23781419	2
C₂×C₄⋊C₈×C₁₆₃	2
#1831	163

subgroup

index

C₂×C₄⋊(C₁₆₃⋊C₈)

C₄²×Dic₁₆₃

#23781419

C₂×C₄⋊C₈×C₁₆₃

#1831

163

Subgroups — #35214652

Conjugacy classes of subgroups

Complements

Maximal subgroups