Group #46811021 — subgroups

Subgroups — #46811021

Order 34432.

188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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	#31770628	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
3	#31770370	C₂	1	1	yes	maximal, characteristic
4	#31770369	C₂	1	2	yes	maximal
5	#341×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16864386	C₂²	1	2	yes	—
7	C₂²×#14524130	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
9	#22575382	—	2	4	no	—
10	C₈×Dic₂₆₉	C₄	1	2	yes	—
11	C₈×Dic₂₆₉	C₄	1	2	yes	—
12	C₂×M₁₆×C₂₆₉	C₂²	1	2	yes	—
13	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₆₉	Q₈	1	2	yes	—
15	#16864386	—	2	2	no	—
16	#16864386	—	2	2	no	—
17	C₂×#14524130	—	2	2	no	—
18	C₂×#14524130	—	2	2	no	—
19	C₂×#14524130	C₂×C₄	1	2	yes	—
20	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₂₆₉	D₄	1	2	yes	—
23	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
24	M₁₆×C₂₆₉	—	2	2	no	—
25	M₁₆×C₂₆₉	—	2	2	no	—
26	C₂×C₈×C₂₆₉	—	2	2	no	—
27	C₂×C₈×C₂₆₉	—	2	2	no	—
28	C₂×C₈×C₂₆₉	C₂×C₄	1	2	yes	—
29	C₂×Dic₂₆₉	—	2	4	no	—
30	C₂×Dic₂₆₉	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₂₆₉	C₄⋊C₄	1	2	yes	—
32	#14524130	—	2	2	no	—
33	#14524130	—	4	2	no	—
34	C₂³×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₆₉	#41	1	2	yes	—
38	C₂×C₄×C₂₆₉	#41	1	2	yes	—
39	C₈×C₂₆₉	—	2	2	no	—
40	C₈×C₂₆₉	—	4	2	no	—
41	C₂²×C₂₆₉	—	2	2	no	—
42	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₂₆₉	#125	1	2	yes	—
44	C₄×C₂₆₉	#117	1	1	yes	characteristic
45	C₄×C₂₆₉	#117	1	1	yes	characteristic
46	C₄×C₂₆₉	#124	1	2	yes	—
47	Dic₂₆₉	—	2	2	no	—
48	Dic₂₆₉	#108	1	4	yes	—
49	C₂×C₂₆₉	—	2	2	no	—
50	C₂×C₂₆₉	#382	1	1	yes	characteristic
51	C₂×C₂₆₉	C₂×#108	1	2	yes	—
52	C₂₆₉	#1837	1	1	yes	characteristic, complemented, Sylow
53	#1837	—	269	1	no	maximal, Sylow
54	#423	—	269	2	no	—
55	C₂×C₄×C₈	—	269	1	no	—
56	#341	—	269	2	no	—
57	#341	—	269	1	no	—
58	#341	D₂₆₉	1	1	yes	characteristic
59	C₂×M₁₆	—	269	2	no	—
60	C₂×M₁₆	C₂×D₂₆₉	1	2	yes	—
61	C₂²×C₈	—	269	1	no	—
62	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
63	C₂×C₄²	—	269	1	no	—
64	C₄⋊C₈	—	538	4	no	—
65	C₄×C₈	—	269	2	no	—
66	C₄×C₈	—	269	2	no	—
67	C₂²×C₄	—	269	2	no	—
68	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	538	2	no	—
72	M₁₆	—	538	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	269	2	no	—
76	C₂×C₈	—	538	2	no	—
77	C₂×C₈	—	538	2	no	—
78	C₂×C₈	C₄×D₂₆₉	1	2	yes	—
79	C₄²	—	269	1	no	—
80	C₄²	—	269	1	no	—
81	C₄²	—	269	2	no	—
82	C₂³	Q₈×D₂₆₉	1	1	yes	characteristic
83	C₂×C₄	—	269	2	no	—
84	C₂×C₄	—	269	2	no	—
85	C₂×C₄	—	538	4	no	—
86	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
87	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
88	C₂×C₄	#16864401	1	2	yes	—
89	C₂×C₄	#16864402	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	538	2	no	—
93	C₈	—	1076	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₄×D₂₆₉	1	1	yes	characteristic
96	C₂²	#22575443	1	2	yes	—
97	C₄	—	269	4	no	—
98	C₄	—	538	2	no	—
99	C₄	#22575409	1	1	yes	characteristic
100	C₄	#22575410	1	1	yes	characteristic
101	C₄	#22575439	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#31770469	1	1	yes	characteristic
104	C₂	#108×D₂₆₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31770628

C₂

yes

maximal

C₂×C₈×Dic₂₆₉

C₂

yes

maximal, characteristic

#31770370

C₂

yes

maximal, characteristic

#31770369

C₂

yes

maximal

#341×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864386

C₂²

yes

—

C₂²×#14524130

C₂²

yes

characteristic

C₂×C₄×Dic₂₆₉

C₂²

yes

characteristic

#22575382

—

C₈×Dic₂₆₉

C₄

yes

—

C₈×Dic₂₆₉

C₄

yes

—

C₂×M₁₆×C₂₆₉

C₂²

yes

—

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

C₂²×Dic₂₆₉

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

C₄×Dic₂₆₉

C₂×C₄

yes

characteristic

C₄×Dic₂₆₉

C₂×C₄

yes

characteristic

C₄×Dic₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

M₁₆×C₂₆₉

—

M₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂×C₄

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

C₄⋊C₄

yes

—

C₂×Dic₂₆₉

C₄⋊C₄

yes

—

#14524130

—

#14524130

—

C₂³×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

#41

yes

—

C₂×C₄×C₂₆₉

#41

yes

—

C₈×C₂₆₉

—

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₆₉

#125

yes

—

C₄×C₂₆₉

yes

C₄×C₂₆₉

yes

C₄×C₂₆₉

yes

—

Dic₂₆₉

—

Dic₂₆₉

#108

yes

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#382

yes

characteristic

C₂×C₂₆₉

C₂×#108

yes

—

C₂₆₉

#1837

yes

characteristic, complemented, Sylow

#1837

—

269

maximal, Sylow

#423

—

269

—

C₂×C₄×C₈

—

269

—

#341

—

269

—

#341

—

269

—

#341

D₂₆₉

yes

characteristic

C₂×M₁₆

—

269

—

C₂×M₁₆

C₂×D₂₆₉

yes

—

C₂²×C₈

—

269

—

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

C₂×C₄²

—

269

—

C₄⋊C₈

—

538

—

C₄×C₈

—

269

—

C₄×C₈

—

269

—

C₂²×C₄

—

269

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

538

—

M₁₆

—

538

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

269

—

C₂×C₈

—

538

—

C₂×C₈

—

538

—

C₂×C₈

C₄×D₂₆₉

yes

—

C₄²

—

269

—

C₄²

—

269

—

C₄²

—

269

—

C₂³

Q₈×D₂₆₉

yes

characteristic

C₂×C₄

—

269

—

C₂×C₄

—

269

—

C₂×C₄

—

538

—

C₂×C₄

C₂×C₄×D₂₆₉

yes

characteristic

C₂×C₄

D₄×D₂₆₉

yes

characteristic

C₂×C₄

#16864401

yes

—

C₂×C₄

#16864402

yes

—

C₈

—

C₈

—

C₈

—

538

—

C₈

—

1076

—

C₂²

—

C₂²

C₄⋊C₄×D₂₆₉

yes

characteristic

C₂²

#22575443

yes

—

C₄

—

269

—

C₄

—

538

—

C₄

#22575409

yes

characteristic

100

C₄

#22575410

yes

characteristic

101

C₄

#22575439

yes

—

102

C₂

—

103

C₂

#31770469

yes

characteristic

104

C₂

#108×D₂₆₉

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₆₉
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	#341×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×C₄⋊C₄
prime radical	C₂³×C₂₆₉	C₂×Q₈
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	#341	D₂₆₉
hypocenter	C₂₆₉	#1837
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₂₆₉

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

#341×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂×C₄⋊C₄

prime radical

C₂³×C₂₆₉

C₂×Q₈

cosocle

C₂²×C₄×C₂₆₉

C₂³

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
52. C₂₆₉	#1837	semidirect

H ≅ G/N

type

52. C₂₆₉

#1837

semidirect

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

subgroup	index
#31770628	2
C₂×C₈×Dic₂₆₉	2
#31770370	2
#31770369	2
#341×C₂₆₉	2
#1837	269

subgroup

index

C₂×C₈×Dic₂₆₉

#341×C₂₆₉

269

Subgroups — #46811021

Conjugacy classes of subgroups

Complements

Maximal subgroups