Group #35225040 — subgroups

Subgroups — C₂×#23782067 #35225040

Order 20864.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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×#19104727	C₂	1	2	yes	maximal
2	#23782067	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19104669	C₂	1	2	yes	maximal
4	C₂×#19104667	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	#19104727	—	2	4	no	—
7	#19104727	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₁₆₃	C₂²	1	4	yes	—
9	C₂×Dic₁₆₃⋊C₄	—	2	4	no	—
10	#19104669	C₂²	1	8	yes	—
11	#19104667	C₂²	1	8	yes	—
12	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₆₃	C₂²	1	4	yes	—
16	C₂²×Dic₁₆₃	—	2	4	no	—
17	C₄⋊Dic₁₆₃	—	2	4	no	—
18	C₄⋊Dic₁₆₃	C₂³	1	8	yes	—
19	Dic₁₆₃⋊C₄	—	2	16	no	—
20	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
21	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₆₃	Q₈	1	4	yes	—
23	C₂×C₈×C₁₆₃	Q₈	1	4	yes	—
24	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
25	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
26	C₄²×C₁₆₃	D₄	1	2	yes	—
27	C₄²×C₁₆₃	C₂³	1	2	yes	—
28	C₂×Dic₁₆₃	—	2	8	no	—
29	C₂×Dic₁₆₃	—	4	8	no	—
30	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₆₃	—	2	4	no	—
32	C₂×C₄×C₁₆₃	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
37	C₈×C₁₆₃	C₂×Q₈	1	4	yes	—
38	C₈×C₁₆₃	C₂×Q₈	1	4	yes	—
39	C₂²×C₁₆₃	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₁₆₃	#135	1	4	yes	—
41	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₁₆₃	—	2	4	no	—
43	C₄×C₁₆₃	C₄⋊Q₈	1	2	yes	—
44	C₄×C₁₆₃	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₁₆₃	C₂²×Q₈	1	1	yes	characteristic
46	Dic₁₆₃	—	4	8	no	—
47	C₂×C₁₆₃	#499	1	4	yes	complemented
48	C₂×C₁₆₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₁₆₃	C₂×#135	1	2	yes	—
50	C₁₆₃	C₂×#499	1	1	yes	characteristic, complemented, Sylow
51	C₂×#499	—	163	1	no	maximal, Sylow
52	C₂×#125	—	163	2	no	—
53	#499	—	163	8	no	—
54	C₂×#107	—	163	2	no	—
55	C₂×#106	—	163	2	no	—
56	C₂×C₄×C₈	D₁₆₃	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
59	#125	—	326	4	no	—
60	#125	—	163	8	no	—
61	C₂×C₄⋊C₄	—	326	4	no	—
62	C₂×C₄⋊C₄	—	163	4	no	—
63	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
64	#107	—	163	8	no	—
65	#106	—	163	8	no	—
66	C₄×C₈	C₂×D₁₆₃	1	4	yes	—
67	C₂²×C₄	—	326	4	no	—
68	C₂²×C₄	D₆₅₂	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
70	C₂×C₈	Dic₃₂₆	1	4	yes	—
71	C₂×C₈	Dic₃₂₆	1	4	yes	—
72	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
73	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
74	C₄⋊C₄	—	326	16	no	—
75	C₄⋊C₄	—	326	4	no	—
76	C₄⋊C₄	—	163	8	no	—
77	C₄²	D₆₅₂	1	2	yes	—
78	C₄²	C₂²×D₁₆₃	1	2	yes	—
79	C₂³	C₂×D₆₅₂	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	326	8	no	—
82	C₂×C₄	—	652	8	no	—
83	C₂×C₄	C₂×D₆₅₂	1	4	yes	—
84	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₃₂₆	1	2	yes	—
86	C₂×C₄	C₂×Dic₃₂₆	1	2	yes	—
87	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
88	C₈	C₂×Dic₃₂₆	1	4	yes	—
89	C₈	C₂×Dic₃₂₆	1	4	yes	—
90	C₂²	#19104735	1	2	yes	—
91	C₂²	#19104757	1	4	yes	—
92	C₂²	C₂²×D₆₅₂	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	652	8	no	—
95	C₄	#19104735	1	2	yes	—
96	C₄	C₂²×Dic₃₂₆	1	1	yes	characteristic
97	C₄	C₂²×Dic₃₂₆	1	1	yes	characteristic
98	C₂	#23782067	1	4	yes	direct product factor
99	C₂	C₂×#19104735	1	1	yes	characteristic
100	C₂	C₂×#19104757	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104727

C₂

yes

maximal

#23782067

C₂

yes

maximal, direct product factor

C₂×#19104669

C₂

yes

maximal

C₂×#19104667

C₂

yes

maximal

C₂×C₄×C₈×C₁₆₃

C₂

yes

maximal, characteristic

#19104727

—

#19104727

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

C₂×Dic₁₆₃⋊C₄

—

#19104669

C₂²

yes

—

#19104667

C₂²

yes

—

C₂²×C₈×C₁₆₃

C₂²

yes

characteristic

C₂²×C₈×C₁₆₃

C₂²

yes

characteristic

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₄×C₈×C₁₆₃

C₂²

yes

—

C₂²×Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

C₂³

yes

—

Dic₁₆₃⋊C₄

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₈×C₁₆₃

Q₈

yes

—

C₂×C₈×C₁₆₃

Q₈

yes

—

C₂×C₈×C₁₆₃

C₂³

yes

—

C₂×C₈×C₁₆₃

C₂³

yes

—

C₄²×C₁₆₃

D₄

yes

—

C₄²×C₁₆₃

C₂³

yes

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

C₂⁴

yes

characteristic

C₈×C₁₆₃

C₂×Q₈

yes

—

C₈×C₁₆₃

C₂×Q₈

yes

—

C₂²×C₁₆₃

C₄⋊Q₈

yes

—

C₂²×C₁₆₃

#135

yes

—

C₂²×C₁₆₃

C₂²×D₄

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄⋊Q₈

yes

—

C₄×C₁₆₃

C₂²×Q₈

yes

characteristic

C₄×C₁₆₃

C₂²×Q₈

yes

characteristic

Dic₁₆₃

—

C₂×C₁₆₃

#499

yes

complemented

C₂×C₁₆₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₆₃

C₂×#135

yes

—

C₁₆₃

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

163

maximal, Sylow

C₂×#125

—

163

—

#499

—

163

—

C₂×#107

—

163

—

C₂×#106

—

163

—

C₂×C₄×C₈

D₁₆₃

yes

characteristic

C₂²×C₈

C₂×D₁₆₃

yes

characteristic

C₂²×C₈

C₂×D₁₆₃

yes

characteristic

#125

—

326

—

#125

—

163

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#107

—

163

—

#106

—

163

—

C₄×C₈

C₂×D₁₆₃

yes

—

C₂²×C₄

—

326

—

C₂²×C₄

D₆₅₂

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

Dic₃₂₆

yes

—

C₂×C₈

Dic₃₂₆

yes

—

C₂×C₈

C₂²×D₁₆₃

yes

—

C₂×C₈

C₂²×D₁₆₃

yes

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

163

—

C₄²

D₆₅₂

yes

—

C₄²

C₂²×D₁₆₃

yes

—

C₂³

C₂×D₆₅₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

326

—

C₂×C₄

—

652

—

C₂×C₄

C₂×D₆₅₂

yes

—

C₂×C₄

C₂×D₆₅₂

yes

characteristic

C₂×C₄

C₂×Dic₃₂₆

yes

—

C₂×C₄

C₂×Dic₃₂₆

yes

—

C₂×C₄

C₂³×D₁₆₃

yes

characteristic

C₈

C₂×Dic₃₂₆

yes

—

C₈

C₂×Dic₃₂₆

yes

—

C₂²

#19104735

yes

—

C₂²

#19104757

yes

—

C₂²

C₂²×D₆₅₂

yes

characteristic

C₄

—

C₄

—

652

—

C₄

#19104735

yes

—

C₄

C₂²×Dic₃₂₆

yes

characteristic

C₄

C₂²×Dic₃₂₆

yes

characteristic

C₂

#23782067

yes

direct product factor

C₂

C₂×#19104735

yes

characteristic

100

C₂

C₂×#19104757

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₅₂
commutator, cosocle	C₂×C₄×C₁₆₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	C₂×C₄×C₈×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₆₃
hypocenter	C₁₆₃	C₂×#499
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₆₅₂

commutator, cosocle

C₂×C₄×C₁₆₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₆₃

Fitting

C₂×C₄×C₈×C₁₆₃

C₂

socle, prime radical

C₂³×C₁₆₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₆₃

hypocenter

C₁₆₃

C₂×#499

LCS

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #23782067	C₂	direct (N × H)
47. C₂×C₁₆₃	#499	semidirect
50. C₁₆₃	C₂×#499	semidirect
98. C₂	#23782067	direct (N × H)

H ≅ G/N

type

2. #23782067

C₂

direct (N × H)

47. C₂×C₁₆₃

#499

semidirect

50. C₁₆₃

C₂×#499

semidirect

98. C₂

#23782067

direct (N × H)

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

subgroup	index
C₂×#19104727	2
#23782067	2
C₂×#19104669	2
C₂×#19104667	2
C₂×C₄×C₈×C₁₆₃	2
C₂×#499	163

subgroup

index

C₂×C₄×C₈×C₁₆₃

163

Subgroups — C₂×#23782067 #35225040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23782067 #35225040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23782067 #35225040