Group #35214705 — subgroups

Subgroups — #35214705

Order 20864.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#19104668	C₂	1	2	yes	maximal
2	C₂×#19104666	C₂	1	2	yes	maximal
3	C₂×#19104646	C₂	1	1	yes	maximal, characteristic
4	#23781535	C₂	1	1	yes	maximal, characteristic
5	#382×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₆₃	—	2	2	no	—
8	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
9	#19104668	—	2	4	no	—
10	#19104666	—	2	4	no	—
11	#19104646	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
15	#95×C₁₆₃	—	2	2	no	—
16	C₂²×Dic₁₆₃	—	2	2	no	—
17	C₂×C₁₆₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₁₆₃⋊C₈	D₄	1	4	yes	—
19	C₂×C₁₆₃⋊C₈	Q₈	1	4	yes	—
20	C₄⋊Dic₁₆₃	—	2	2	no	—
21	C₄⋊Dic₁₆₃	—	2	2	no	—
22	C₄⋊Dic₁₆₃	—	4	4	no	—
23	C₂²×C₄×C₁₆₃	—	2	1	no	—
24	C₂²×C₄×C₁₆₃	—	2	1	no	—
25	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
26	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₆₃	—	2	2	no	—
28	C₄⋊C₄×C₁₆₃	—	2	2	no	—
29	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
30	C₄²×C₁₆₃	Q₈	1	2	yes	—
31	C₂×Dic₁₆₃	—	4	2	no	—
32	C₂×Dic₁₆₃	—	4	4	no	—
33	C₁₆₃⋊C₈	C₄⋊C₄	1	8	yes	complemented
34	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₆₃	—	2	2	no	—
36	C₂×C₄×C₁₆₃	—	2	2	no	—
37	C₂×C₄×C₁₆₃	—	4	1	no	—
38	C₂×C₄×C₁₆₃	—	4	1	no	—
39	C₂×C₄×C₁₆₃	—	4	2	no	—
40	C₂×C₄×C₁₆₃	—	4	2	no	—
41	C₂×C₄×C₁₆₃	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₆₃	#127	1	1	yes	characteristic
48	C₂²×C₁₆₃	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₁₆₃	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₁₆₃	#137	1	2	yes	—
51	C₄×C₁₆₃	—	2	4	no	—
52	C₄×C₁₆₃	—	8	1	no	—
53	C₄×C₁₆₃	—	8	1	no	—
54	C₄×C₁₆₃	C₂×C₄⋊C₄	1	2	yes	—
55	C₄×C₁₆₃	C₄⋊Q₈	1	2	yes	—
56	Dic₁₆₃	—	8	2	no	—
57	C₂×C₁₆₃	#384	1	1	yes	characteristic
58	C₂×C₁₆₃	#428	1	2	yes	—
59	C₂×C₁₆₃	#496	1	1	yes	characteristic
60	C₂×C₁₆₃	#497	1	1	yes	characteristic
61	C₂×C₁₆₃	#501	1	2	yes	—
62	C₁₆₃	#1847	1	1	yes	characteristic, complemented, Sylow
63	#1847	—	163	1	no	maximal, Sylow
64	C₂×#107	—	163	2	no	—
65	C₂×#106	—	163	2	no	—
66	C₂×#97	—	163	1	no	—
67	#384	—	163	1	no	—
68	#382	D₁₆₃	1	1	yes	characteristic
69	C₂²×C₈	—	163	2	no	—
70	C₂×C₄⋊C₄	—	326	2	no	—
71	C₂×C₄⋊C₄	—	163	2	no	—
72	C₂×C₄⋊C₄	C₂×D₁₆₃	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂×D₁₆₃	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
75	#107	—	326	4	no	—
76	#106	—	326	4	no	—
77	#97	—	163	4	no	—
78	#95	—	2	2	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	326	2	no	—
82	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
84	C₂×C₈	—	163	4	no	—
85	C₂×C₈	—	163	4	no	—
86	C₂×C₈	—	163	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	326	2	no	—
90	C₄⋊C₄	—	326	2	no	—
91	C₄⋊C₄	—	652	4	no	—
92	C₄²	C₄×D₁₆₃	1	2	yes	—
93	C₄²	Dic₃₂₆	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	652	2	no	—
102	C₂×C₄	—	652	4	no	—
103	C₂×C₄	Dic₁₆₃⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
105	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
106	C₂×C₄	Q₈×D₁₆₃	1	2	yes	—
107	C₂×C₄	C₂×Dic₃₂₆	1	1	yes	characteristic
108	C₈	—	163	8	no	—
109	C₂²	C₂×Dic₁₆₃⋊C₄	1	1	yes	characteristic
110	C₂²	#19104731	1	1	yes	characteristic
111	C₂²	#19104733	1	1	yes	characteristic
112	C₂²	#19104760	1	2	yes	—
113	C₂²	#19104765	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	1304	2	no	—
118	C₄	C₄⋊C₄×D₁₆₃	1	2	yes	—
119	C₄	#19104734	1	2	yes	—
120	C₂	#23781534	1	1	yes	characteristic
121	C₂	#23781708	1	2	yes	—
122	C₂	#23782055	1	1	yes	characteristic
123	C₂	#23782060	1	1	yes	characteristic
124	C₂	#23782075	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104668

C₂

yes

maximal

C₂×#19104666

C₂

yes

maximal

C₂×#19104646

C₂

yes

maximal, characteristic

#23781535

C₂

yes

maximal, characteristic

#382×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

#19104668

—

#19104666

—

#19104646

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

characteristic

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

#95×C₁₆₃

—

C₂²×Dic₁₆₃

—

C₂×C₁₆₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₆₃⋊C₈

D₄

yes

—

C₂×C₁₆₃⋊C₈

Q₈

yes

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂×C₄

yes

—

C₄²×C₁₆₃

Q₈

yes

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

C₄⋊C₄

yes

complemented

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂²×C₁₆₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₆₃

#127

yes

characteristic

C₂²×C₁₆₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₆₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₆₃

#137

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂×C₄⋊C₄

yes

—

C₄×C₁₆₃

C₄⋊Q₈

yes

—

Dic₁₆₃

—

C₂×C₁₆₃

#384

yes

characteristic

C₂×C₁₆₃

#428

yes

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

—

C₁₆₃

#1847

yes

characteristic, complemented, Sylow

#1847

—

163

maximal, Sylow

C₂×#107

—

163

—

C₂×#106

—

163

—

C₂×#97

—

163

—

#384

—

163

—

#382

D₁₆₃

yes

characteristic

C₂²×C₈

—

163

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

characteristic

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#107

—

326

—

#106

—

326

—

#97

—

163

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

326

—

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

163

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

652

—

C₄²

C₄×D₁₆₃

yes

—

C₄²

Dic₃₂₆

yes

—

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

652

—

102

C₂×C₄

—

652

—

103

C₂×C₄

Dic₁₆₃⋊C₄

yes

—

104

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

105

C₂×C₄

D₄×D₁₆₃

yes

—

106

C₂×C₄

Q₈×D₁₆₃

yes

—

107

C₂×C₄

C₂×Dic₃₂₆

yes

characteristic

108

C₈

—

163

—

109

C₂²

C₂×Dic₁₆₃⋊C₄

yes

characteristic

110

C₂²

#19104731

yes

characteristic

111

C₂²

#19104733

yes

characteristic

112

C₂²

#19104760

yes

—

113

C₂²

#19104765

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

1304

—

118

C₄

C₄⋊C₄×D₁₆₃

yes

—

119

C₄

#19104734

yes

—

120

C₂

#23781534

yes

characteristic

121

C₂

#23781708

yes

—

122

C₂

#23782055

yes

characteristic

123

C₂

#23782060

yes

characteristic

124

C₂

#23782075

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₆₃
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#382×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂×D₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	#382	D₁₆₃
hypocenter	C₁₆₃	#1847
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₆₃

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#382×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 ◁ #382

N	H ≅ G/N	type
11. #19104646	C₄	semidirect
33. C₁₆₃⋊C₈	C₄⋊C₄	semidirect
62. C₁₆₃	#1847	semidirect

H ≅ G/N

type

11. #19104646

C₄

semidirect

33. C₁₆₃⋊C₈

C₄⋊C₄

semidirect

62. C₁₆₃

#1847

semidirect

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

subgroup	index
C₂×#19104668	2
C₂×#19104666	2
C₂×#19104646	2
#23781535	2
#382×C₁₆₃	2
#1847	163

subgroup

index

#382×C₁₆₃

163

Subgroups — #35214705

Conjugacy classes of subgroups

Complements

Maximal subgroups