Group #35214663 — subgroups

Subgroups — #35214663

Order 20864.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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	#23781535	C₂	1	1	yes	maximal, characteristic
3	#23781427	C₂	1	4	yes	maximal
4	#384×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₆₃	—	2	1	no	—
7	C₂×C₄⋊Dic₁₆₃	—	2	1	no	—
8	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
9	C₄×C₁₆₃⋊C₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₆₃	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₆₃	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₆₃	—	2	2	no	—
15	C₂×C₁₆₃⋊C₈	—	2	2	no	—
16	C₂×C₁₆₃⋊C₈	—	2	2	no	—
17	C₂×C₁₆₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₁₆₃	—	2	4	no	—
19	C₄⋊Dic₁₆₃	—	4	2	no	—
20	C₄⋊Dic₁₆₃	—	4	2	no	—
21	C₂²×C₄×C₁₆₃	—	2	2	no	—
22	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₆₃	—	2	4	no	—
26	C₄⋊C₄×C₁₆₃	—	4	2	no	—
27	C₄⋊C₄×C₁₆₃	—	4	2	no	—
28	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
29	C₄²×C₁₆₃	Q₈	1	2	yes	—
30	C₂×Dic₁₆₃	—	4	2	no	—
31	C₂×Dic₁₆₃	—	4	2	no	—
32	C₂×Dic₁₆₃	—	4	2	no	—
33	C₁₆₃⋊C₈	—	2	4	no	—
34	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₆₃	—	4	2	no	—
36	C₂×C₄×C₁₆₃	—	4	2	no	—
37	C₂×C₄×C₁₆₃	—	4	2	no	—
38	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₆₃	D₈	1	2	yes	—
41	C₂×C₄×C₁₆₃	SD₁₆	1	2	yes	—
42	C₂×C₄×C₁₆₃	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₆₃	Q₁₆	1	2	yes	—
44	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₆₃	#41	1	2	yes	—
47	C₂×C₄×C₁₆₃	#41	1	2	yes	—
48	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₆₃	#124	1	1	yes	characteristic
50	C₂²×C₁₆₃	#124	1	1	yes	characteristic
51	C₂²×C₁₆₃	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₁₆₃	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₆₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₁₆₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₁₆₃	—	8	2	no	—
56	C₄×C₁₆₃	#106	1	4	yes	—
57	C₄×C₁₆₃	#107	1	4	yes	—
58	C₄×C₁₆₃	#117	1	2	yes	—
59	C₄×C₁₆₃	#125	1	2	yes	—
60	Dic₁₆₃	—	8	2	no	—
61	C₂×C₁₆₃	#382	1	1	yes	characteristic
62	C₂×C₁₆₃	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₆₃	C₂×#107	1	1	yes	characteristic
64	C₂×C₁₆₃	#486	1	1	yes	characteristic
65	C₂×C₁₆₃	#486	1	1	yes	characteristic
66	C₂×C₁₆₃	#487	1	1	yes	characteristic
67	C₂×C₁₆₃	#487	1	1	yes	characteristic
68	C₁₆₃	#1834	1	1	yes	characteristic, complemented, Sylow
69	#1834	—	163	1	no	maximal, Sylow
70	C₂×C₄×C₈	—	163	1	no	—
71	#384	—	163	1	no	—
72	#384	D₁₆₃	1	1	yes	characteristic
73	#340	—	163	4	no	—
74	C₂²×C₈	—	163	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	326	1	no	—
78	C₂×C₄⋊C₄	—	326	1	no	—
79	C₂×C₄⋊C₄	—	163	2	no	—
80	C₂×C₄⋊C₄	C₂×D₁₆₃	1	2	yes	—
81	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
82	C₄×C₈	—	163	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	326	2	no	—
85	C₂²×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
88	C₂×C₈	—	326	2	no	—
89	C₂×C₈	—	326	2	no	—
90	C₂×C₈	—	163	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	326	4	no	—
95	C₄⋊C₄	—	652	2	no	—
96	C₄⋊C₄	—	652	2	no	—
97	C₄²	C₄×D₁₆₃	1	2	yes	—
98	C₄²	Dic₃₂₆	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	652	2	no	—
104	C₂×C₄	—	652	2	no	—
105	C₂×C₄	—	652	2	no	—
106	C₂×C₄	Dic₁₆₃⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₁₆₃⋊C₄	1	2	yes	—
108	C₂×C₄	#14579684	1	2	yes	—
109	C₂×C₄	#14579686	1	2	yes	—
110	C₂×C₄	#14579687	1	2	yes	—
111	C₂×C₄	#14579689	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₃₂₆	1	1	yes	characteristic
114	C₂×C₄	#14579698	1	2	yes	—
115	C₂×C₄	#14579700	1	2	yes	—
116	C₈	—	326	4	no	—
117	C₂²	C₂×Dic₁₆₃⋊C₄	1	1	yes	characteristic
118	C₂²	#19104722	1	1	yes	characteristic
119	C₂²	#19104724	1	1	yes	characteristic
120	C₂²	C₂×#14579684	1	1	yes	characteristic
121	C₂²	C₂×#14579686	1	1	yes	characteristic
122	C₂²	C₂×#14579687	1	1	yes	characteristic
123	C₂²	C₂×#14579689	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	1304	2	no	—
126	C₄	#19104666	1	4	yes	—
127	C₄	#19104668	1	4	yes	—
128	C₄	#19104693	1	2	yes	—
129	C₄	#19104726	1	2	yes	—
130	C₂	#23781530	1	1	yes	characteristic
131	C₂	C₂×#19104666	1	1	yes	characteristic
132	C₂	C₂×#19104668	1	1	yes	characteristic
133	C₂	#23782011	1	1	yes	characteristic
134	C₂	#23782013	1	1	yes	characteristic
135	C₂	#23782015	1	1	yes	characteristic
136	C₂	#23782017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₆₃⋊C₈

C₂

yes

maximal, characteristic

#23781535

C₂

yes

maximal, characteristic

#23781427

C₂

yes

maximal

#384×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

—

C₂×C₄⋊Dic₁₆₃

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

C₄×C₁₆₃⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₆₃

—

C₂×C₄⋊C₄×C₁₆₃

—

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₂²×Dic₁₆₃

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

characteristic

C₂²×C₄×C₁₆₃

D₄

yes

characteristic

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂×C₄

yes

—

C₄²×C₁₆₃

Q₈

yes

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₃

D₈

yes

—

C₂×C₄×C₁₆₃

SD₁₆

yes

—

C₂×C₄×C₁₆₃

SD₁₆

yes

—

C₂×C₄×C₁₆₃

Q₁₆

yes

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₂²×C₁₆₃

C₂×C₄⋊C₄

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

C₂×D₈

yes

characteristic

C₂²×C₁₆₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₆₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₆₃

C₂×Q₁₆

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

#106

yes

—

C₄×C₁₆₃

#107

yes

—

C₄×C₁₆₃

#117

yes

—

C₄×C₁₆₃

#125

yes

—

Dic₁₆₃

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, complemented, Sylow

#1834

—

163

maximal, Sylow

C₂×C₄×C₈

—

163

—

#384

—

163

—

#384

D₁₆₃

yes

characteristic

#340

—

163

—

C₂²×C₈

—

163

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

C₄×C₈

—

163

—

C₂²×C₄

—

C₂²×C₄

—

326

—

C₂²×C₄

C₂²⋊D₁₆₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₆₃

yes

characteristic

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

163

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

652

—

C₄⋊C₄

—

652

—

C₄²

C₄×D₁₆₃

yes

—

C₄²

Dic₃₂₆

yes

—

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

652

—

104

C₂×C₄

—

652

—

105

C₂×C₄

—

652

—

106

C₂×C₄

Dic₁₆₃⋊C₄

yes

—

107

C₂×C₄

Dic₁₆₃⋊C₄

yes

—

108

C₂×C₄

#14579684

yes

—

109

C₂×C₄

#14579686

yes

—

110

C₂×C₄

#14579687

yes

—

111

C₂×C₄

#14579689

yes

—

112

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

113

C₂×C₄

C₂×Dic₃₂₆

yes

characteristic

114

C₂×C₄

#14579698

yes

—

115

C₂×C₄

#14579700

yes

—

116

C₈

—

326

—

117

C₂²

C₂×Dic₁₆₃⋊C₄

yes

characteristic

118

C₂²

#19104722

yes

characteristic

119

C₂²

#19104724

yes

characteristic

120

C₂²

C₂×#14579684

yes

characteristic

121

C₂²

C₂×#14579686

yes

characteristic

122

C₂²

C₂×#14579687

yes

characteristic

123

C₂²

C₂×#14579689

yes

characteristic

124

C₄

—

125

C₄

—

1304

—

126

C₄

#19104666

yes

—

127

C₄

#19104668

yes

—

128

C₄

#19104693

yes

—

129

C₄

#19104726

yes

—

130

C₂

#23781530

yes

characteristic

131

C₂

C₂×#19104666

yes

characteristic

132

C₂

C₂×#19104668

yes

characteristic

133

C₂

#23782011

yes

characteristic

134

C₂

#23782013

yes

characteristic

135

C₂

#23782015

yes

characteristic

136

C₂

#23782017

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₆₃
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#384×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂×D₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	#384	D₁₆₃
hypocenter	C₁₆₃	#1834
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₆₃

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#384×C₁₆₃

C₂

socle, prime radical

C₂³×C₁₆₃

C₂×D₄

cosocle

C₂²×C₄×C₁₆₃

C₂³

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×C₁₆₃⋊C₈	C₄	semidirect
68. C₁₆₃	#1834	semidirect

H ≅ G/N

type

9. C₄×C₁₆₃⋊C₈

C₄

semidirect

68. C₁₆₃

#1834

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
#23781535	2
#23781427	2
#384×C₁₆₃	2
#1834	163

subgroup

index

C₂×C₄×C₁₆₃⋊C₈

#23781535

#23781427

#384×C₁₆₃

#1834

163

Subgroups — #35214663

Conjugacy classes of subgroups

Complements

Maximal subgroups