Group #35214200 — subgroups

Subgroups — #35214200

Order 20864.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#23782042	C₂	1	1	yes	maximal, characteristic, complemented
2	#23782041	C₂	1	2	yes	maximal, complemented
3	#23782013	C₂	1	2	yes	maximal, complemented
4	#23781386	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₃₀₄	—	2	2	no	—
7	C₂×#14579684	—	2	4	no	—
8	#19104732	C₂²	1	2	yes	—
9	#19104655	—	2	4	no	—
10	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₆₃	—	2	2	no	—
12	C₄⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
13	#127×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₂×D₆₅₂	—	2	2	no	—
16	C₂×D₆₅₂	—	2	2	no	—
17	C₂×D₆₅₂	—	2	2	no	—
18	D₁₃₀₄	—	4	4	no	—
19	#14579684	—	4	8	no	—
20	C₂×C₁₆₃⋊C₈	D₄	1	4	yes	—
21	C₂×Q₈×C₁₆₃	—	2	1	no	—
22	C₂×D₄×C₁₆₃	—	2	1	no	—
23	C₂×D₄×C₁₆₃	—	2	2	no	—
24	SD₁₆×C₁₆₃	—	4	4	no	—
25	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₆₃	—	2	2	no	—
27	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
28	C₂²×D₁₆₃	—	4	2	no	—
29	D₆₅₂	—	4	4	no	—
30	D₆₅₂	—	4	4	no	—
31	D₆₅₂	—	4	4	no	—
32	C₁₆₃⋊C₈	—	2	4	no	—
33	C₂³×C₁₆₃	—	4	1	no	—
34	Q₈×C₁₆₃	—	4	2	no	—
35	D₄×C₁₆₃	—	4	2	no	—
36	D₄×C₁₆₃	—	4	4	no	—
37	C₂×C₄×C₁₆₃	—	4	1	no	—
38	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
40	C₈×C₁₆₃	D₈	1	4	yes	complemented
41	C₂²×C₁₆₃	—	8	1	no	—
42	C₂²×C₁₆₃	—	8	2	no	—
43	C₂²×C₁₆₃	#127	1	1	yes	characteristic
44	C₂×D₁₆₃	—	8	2	no	—
45	C₂×D₁₆₃	—	8	2	no	—
46	C₂×D₁₆₃	—	8	2	no	—
47	C₄×C₁₆₃	—	8	1	no	—
48	C₄×C₁₆₃	C₂×D₈	1	2	yes	—
49	C₄×C₁₆₃	AGL(1,ℤ/8)	1	4	yes	—
50	C₂×C₁₆₃	—	16	1	no	—
51	C₂×C₁₆₃	#493	1	1	yes	characteristic
52	C₂×C₁₆₃	#496	1	2	yes	—
53	D₁₆₃	—	16	2	no	—
54	C₁₆₃	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	163	1	no	maximal, Sylow
56	#493	—	163	2	no	—
57	#493	—	163	1	no	—
58	#492	D₁₆₃	1	1	yes	characteristic, complemented
59	#486	—	163	2	no	—
60	#322	—	163	1	no	—
61	C₂×SD₁₆	—	2	2	no	—
62	C₂×D₈	—	326	4	no	—
63	C₂×D₈	—	326	2	no	—
64	C₄⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
65	#127	—	163	2	no	—
66	#127	C₂×D₁₆₃	1	1	yes	characteristic
67	#102	—	326	4	no	—
68	C₄×C₈	—	163	2	no	—
69	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
70	C₂×Q₈	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	326	2	no	—
74	C₂×D₄	—	326	2	no	—
75	C₂×D₄	—	326	2	no	—
76	SD₁₆	—	4	4	no	—
77	D₈	—	652	8	no	—
78	D₈	—	652	4	no	—
79	C₂×C₈	—	163	4	no	—
80	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
83	C₂³	—	4	1	no	—
84	C₂³	—	652	2	no	—
85	Q₈	—	4	2	no	—
86	D₄	—	4	2	no	—
87	D₄	—	4	4	no	—
88	D₄	—	652	4	no	—
89	D₄	—	652	4	no	—
90	D₄	—	652	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
94	C₈	—	326	4	no	—
95	C₈	#14579684	1	4	yes	complemented
96	C₂²	—	8	1	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	1304	2	no	—
99	C₂²	—	1304	2	no	—
100	C₂²	—	1304	2	no	—
101	C₂²	#19104731	1	1	yes	characteristic
102	C₄	—	8	1	no	—
103	C₄	C₂×#14579684	1	2	yes	—
104	C₄	#19104762	1	4	yes	—
105	C₂	—	16	1	no	—
106	C₂	—	2608	2	no	—
107	C₂	#23782041	1	1	yes	characteristic
108	C₂	#23782051	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23782042

C₂

yes

maximal, characteristic, complemented

#23782041

C₂

yes

maximal, complemented

#23782013

C₂

yes

maximal, complemented

#23781386

C₂

yes

maximal, characteristic, complemented

#492×C₁₆₃

C₂

yes

maximal, characteristic, complemented

C₂×D₁₃₀₄

—

C₂×#14579684

—

#19104732

C₂²

yes

—

#19104655

—

C₄×C₁₆₃⋊C₈

C₂²

yes

—

C₂×SD₁₆×C₁₆₃

—

C₄⋊Q₈×C₁₆₃

C₂²

yes

characteristic

#127×C₁₆₃

C₂²

yes

characteristic

C₄×C₈×C₁₆₃

C₂²

yes

characteristic

C₂×D₆₅₂

—

C₂×D₆₅₂

—

C₂×D₆₅₂

—

D₁₃₀₄

—

#14579684

—

C₂×C₁₆₃⋊C₈

D₄

yes

—

C₂×Q₈×C₁₆₃

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

—

SD₁₆×C₁₆₃

—

C₂×C₈×C₁₆₃

D₄

yes

—

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂³

yes

characteristic

C₂²×D₁₆₃

—

D₆₅₂

—

D₆₅₂

—

D₆₅₂

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

—

Q₈×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₈×C₁₆₃

D₈

yes

complemented

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#127

yes

characteristic

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂×D₈

yes

—

C₄×C₁₆₃

AGL(1,ℤ/8)

yes

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#493

yes

characteristic

C₂×C₁₆₃

#496

yes

—

D₁₆₃

—

C₁₆₃

#1721

yes

characteristic, complemented, Sylow

#1721

—

163

maximal, Sylow

#493

—

163

—

#493

—

163

—

#492

D₁₆₃

yes

characteristic, complemented

#486

—

163

—

#322

—

163

—

C₂×SD₁₆

—

C₂×D₈

—

326

—

C₂×D₈

—

326

—

C₄⋊Q₈

C₂×D₁₆₃

yes

characteristic

#127

—

163

—

#127

C₂×D₁₆₃

yes

characteristic

#102

—

326

—

C₄×C₈

—

163

—

C₄×C₈

C₂×D₁₆₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

326

—

SD₁₆

—

D₈

—

652

—

D₈

—

652

—

C₂×C₈

—

163

—

C₂×C₈

C₂²⋊D₁₆₃

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₆₃

yes

characteristic

C₂³

—

C₂³

—

652

—

Q₈

—

D₄

—

D₄

—

D₄

—

652

—

D₄

—

652

—

D₄

—

652

—

C₂×C₄

—

C₂×C₄

D₄×D₁₆₃

yes

—

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

C₈

—

326

—

C₈

#14579684

yes

complemented

C₂²

—

C₂²

—

C₂²

—

1304

—

C₂²

—

1304

—

100

C₂²

—

1304

—

101

C₂²

#19104731

yes

characteristic

102

C₄

—

103

C₄

C₂×#14579684

yes

—

104

C₄

#19104762

yes

—

105

C₂

—

106

C₂

—

2608

—

107

C₂

#23782041

yes

characteristic

108

C₂

#23782051

yes

—

	subgroup	G/H
center	C₂²	#19104731
commutator, cosocle	C₄²×C₁₆₃	C₂³
Frattini	C₄²	C₂²×D₁₆₃
Fitting	#492×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#127
hypercenter	#492	D₁₆₃
hypocenter	C₁₆₃	#1721
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#19104731

commutator, cosocle

C₄²×C₁₆₃

C₂³

Frattini

C₄²

C₂²×D₁₆₃

Fitting

#492×C₁₆₃

C₂

C₂²×C₁₆₃

D₁₆₃

C₁₆₃

G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #23782042	C₂	semidirect
2. #23782041	C₂	semidirect
3. #23782013	C₂	semidirect
4. #23781386	C₂	semidirect
5. #492×C₁₆₃	C₂	semidirect
40. C₈×C₁₆₃	D₈	semidirect
54. C₁₆₃	#1721	semidirect
58. #492	D₁₆₃	semidirect
95. C₈	#14579684	semidirect

H ≅ G/N

type

1. #23782042

C₂

semidirect

2. #23782041

C₂

semidirect

3. #23782013

C₂

semidirect

4. #23781386

C₂

semidirect

5. #492×C₁₆₃

C₂

semidirect

40. C₈×C₁₆₃

D₈

semidirect

54. C₁₆₃

#1721

semidirect

58. #492

D₁₆₃

semidirect

95. C₈

#14579684

semidirect

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

subgroup	index
#23782042	2
#23782041	2
#23782013	2
#23781386	2
#492×C₁₆₃	2
#1721	163

subgroup

index

#492×C₁₆₃

163

Subgroups — #35214200

Conjugacy classes of subgroups

Complements

Maximal subgroups