Group #35215477 — subgroups

Subgroups — #35215477

Order 20864.

280 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂×#19104646	C₂	1	1	yes	maximal, characteristic
3	#23781420	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₁₆₃⋊C₈)	—	2	2	no	—
8	#19104646	—	2	2	no	—
9	C₂²×Q₈×C₁₆₃	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₁₆₃	—	2	2	no	—
11	C₄×Q₈×C₁₆₃	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₆₃	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₆₃	C₄	1	2	yes	—
14	C₂×C₄²×C₁₆₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆₃⋊C₈	—	2	2	no	—
17	C₂×C₁₆₃⋊C₈	—	2	2	no	—
18	C₂×C₁₆₃⋊C₈	—	2	2	no	—
19	C₂×C₁₆₃⋊C₈	—	2	4	no	—
20	C₂×C₁₆₃⋊C₈	Q₈	1	4	yes	complemented
21	C₂×Q₈×C₁₆₃	—	2	2	no	—
22	C₂×Q₈×C₁₆₃	—	2	4	no	—
23	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₆₃	—	2	4	no	—
29	C₄⋊C₄×C₁₆₃	—	2	2	no	—
30	C₄²×C₁₆₃	—	2	2	no	—
31	C₄²×C₁₆₃	—	2	2	no	—
32	C₄²×C₁₆₃	D₄	1	2	yes	—
33	C₄²×C₁₆₃	D₄	1	2	yes	—
34	C₁₆₃⋊C₈	—	2	4	no	—
35	C₁₆₃⋊C₈	—	4	2	no	—
36	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₁₆₃	—	2	8	no	—
38	C₂×C₄×C₁₆₃	—	2	2	no	—
39	C₂×C₄×C₁₆₃	—	2	2	no	—
40	C₂×C₄×C₁₆₃	—	2	2	no	—
41	C₂×C₄×C₁₆₃	—	2	2	no	—
42	C₂×C₄×C₁₆₃	—	2	2	no	—
43	C₂×C₄×C₁₆₃	—	2	4	no	—
44	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₆₃	M₁₆	1	4	yes	—
47	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₆₃	#41	1	2	yes	—
51	C₂²×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₆₃	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₆₃	C₂×M₁₆	1	2	yes	—
54	C₂²×C₁₆₃	#131	1	2	yes	—
55	C₄×C₁₆₃	—	2	2	no	—
56	C₄×C₁₆₃	—	2	4	no	—
57	C₄×C₁₆₃	—	2	2	no	—
58	C₄×C₁₆₃	—	4	2	no	—
59	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₁₆₃	#124	1	1	yes	characteristic
61	C₄×C₁₆₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₁₆₃	#391	1	1	yes	characteristic
63	C₂×C₁₆₃	#407	1	1	yes	characteristic
64	C₂×C₁₆₃	#408	1	1	yes	characteristic
65	C₂×C₁₆₃	#446	1	4	yes	—
66	C₁₆₃	#1985	1	1	yes	characteristic, complemented, Sylow
67	#1985	—	163	1	no	maximal, Sylow
68	C₂×C₄×Q₈	D₁₆₃	1	1	yes	characteristic
69	C₂×C₄⋊C₈	—	163	1	no	—
70	C₂×#97	—	163	1	no	—
71	#336	—	163	4	no	—
72	C₂²×Q₈	Dic₁₆₃	1	1	yes	characteristic
73	C₂²×C₈	—	163	2	no	—
74	C₂²×C₈	—	163	2	no	—
75	C₄×Q₈	—	2	2	no	—
76	C₄×Q₈	—	2	2	no	—
77	C₂×C₄⋊C₄	Dic₁₆₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	Dic₁₆₃	1	2	yes	—
79	C₂×C₄²	C₂×D₁₆₃	1	2	yes	—
80	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
81	C₄⋊C₈	—	326	2	no	—
82	#97	—	326	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂²×C₄	C₂×Dic₁₆₃	1	2	yes	—
86	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₆₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
90	C₂×C₈	—	326	2	no	—
91	C₂×C₈	—	326	2	no	—
92	C₂×C₈	—	326	2	no	—
93	C₂×C₈	—	326	4	no	—
94	C₂×C₈	—	163	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	2	2	no	—
99	C₄²	C₂²⋊D₁₆₃	1	2	yes	—
100	C₄²	C₂²⋊D₁₆₃	1	2	yes	—
101	C₂³	C₂²×Dic₁₆₃	1	1	yes	characteristic
102	Q₈	—	2	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
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	C₂²⋊Dic₁₆₃	1	2	yes	—
110	C₂×C₄	C₂²⋊Dic₁₆₃	1	2	yes	—
111	C₂×C₄	#14579683	1	4	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₁₆₃	1	2	yes	—
115	C₂×C₄	#14579700	1	2	yes	—
116	C₈	—	326	4	no	—
117	C₈	—	652	2	no	—
118	C₂²	C₂×C₂²⋊Dic₁₆₃	1	1	yes	characteristic
119	C₂²	Q₈×Dic₁₆₃	1	2	yes	—
120	C₂²	C₂×#14579683	1	2	yes	—
121	C₂²	#19104742	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	4	no	—
125	C₄	—	4	2	no	—
126	C₄	#19104711	1	2	yes	—
127	C₄	#19104724	1	1	yes	characteristic
128	C₄	#19104733	1	1	yes	characteristic
129	C₂	#23781572	1	1	yes	characteristic
130	C₂	#23781629	1	1	yes	characteristic
131	C₂	#23781632	1	1	yes	characteristic
132	C₂	#23781796	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₆₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×#19104646

C₂

yes

maximal, characteristic

#23781420

C₂

yes

maximal

C₂×C₄×Q₈×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

—

C₂²×C₁₆₃⋊C₈

C₂²

yes

—

C₄⋊(C₁₆₃⋊C₈)

—

#19104646

—

C₂²×Q₈×C₁₆₃

C₄

yes

characteristic

C₄×Q₈×C₁₆₃

—

C₄×Q₈×C₁₆₃

—

C₂×C₄⋊C₄×C₁₆₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₁₆₃

C₄

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

Q₈

yes

complemented

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

—

C₄²×C₁₆₃

—

C₄²×C₁₆₃

D₄

yes

—

C₄²×C₁₆₃

D₄

yes

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂²×C₄

yes

characteristic

Q₈×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₃

M₁₆

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₂²×C₁₆₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₆₃

C₄×Q₈

yes

—

C₂²×C₁₆₃

C₂×M₁₆

yes

—

C₂²×C₁₆₃

#131

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂²⋊Q₈

yes

—

C₄×C₁₆₃

#124

yes

characteristic

C₄×C₁₆₃

C₄⋊Q₈

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

—

C₁₆₃

#1985

yes

characteristic, complemented, Sylow

#1985

—

163

maximal, Sylow

C₂×C₄×Q₈

D₁₆₃

yes

characteristic

C₂×C₄⋊C₈

—

163

—

C₂×#97

—

163

—

#336

—

163

—

C₂²×Q₈

Dic₁₆₃

yes

characteristic

C₂²×C₈

—

163

—

C₂²×C₈

—

163

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₁₆₃

yes

characteristic

C₂×C₄⋊C₄

Dic₁₆₃

yes

—

C₂×C₄²

C₂×D₁₆₃

yes

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

C₄⋊C₈

—

326

—

#97

—

326

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₁₆₃

yes

—

C₂²×C₄

C₂×Dic₁₆₃

yes

characteristic

C₂²×C₄

C₂×Dic₁₆₃

yes

characteristic

C₂²×C₄

C₂×Dic₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

163

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₁₆₃

yes

—

100

C₄²

C₂²⋊D₁₆₃

yes

—

101

C₂³

C₂²×Dic₁₆₃

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊Dic₁₆₃

yes

—

110

C₂×C₄

C₂²⋊Dic₁₆₃

yes

—

111

C₂×C₄

#14579683

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

114

C₂×C₄

Q₈×D₁₆₃

yes

—

115

C₂×C₄

#14579700

yes

—

116

C₈

—

326

—

117

C₈

—

652

—

118

C₂²

C₂×C₂²⋊Dic₁₆₃

yes

characteristic

119

C₂²

Q₈×Dic₁₆₃

yes

—

120

C₂²

C₂×#14579683

yes

—

121

C₂²

#19104742

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#19104711

yes

—

127

C₄

#19104724

yes

characteristic

128

C₄

#19104733

yes

characteristic

129

C₂

#23781572

yes

characteristic

130

C₂

#23781629

yes

characteristic

131

C₂

#23781632

yes

characteristic

132

C₂

#23781796

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₆₃
commutator, socle, prime radical	C₂³×C₁₆₃	C₂²×C₄
Fitting	C₂×C₄×Q₈×C₁₆₃	C₂
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₂×C₄×Q₈	D₁₆₃
hypocenter	C₁₆₃	#1985
LCS	G ▷ C₂³×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₆₃

commutator, socle, prime radical

C₂³×C₁₆₃

C₂²×C₄

Fitting

C₂×C₄×Q₈×C₁₆₃

C₂

cosocle

C₂²×C₄×C₁₆₃

C₂³

hypercenter

C₂×C₄×Q₈

D₁₆₃

C₁₆₃

G ▷ C₂³×C₁₆₃ ▷ C₁₆₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
20. C₂×C₁₆₃⋊C₈	Q₈	semidirect
66. C₁₆₃	#1985	semidirect

H ≅ G/N

type

20. C₂×C₁₆₃⋊C₈

Q₈

semidirect

66. C₁₆₃

#1985

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
C₂×#19104646	2
#23781420	2
C₂×C₄×Q₈×C₁₆₃	2
#1985	163

subgroup

index

C₂×C₄⋊(C₁₆₃⋊C₈)

C₂×#19104646

#23781420

C₂×C₄×Q₈×C₁₆₃

#1985

163

Subgroups — #35215477

Conjugacy classes of subgroups

Complements

Maximal subgroups