Group #46811049 — subgroups

Subgroups — #46811049

Order 34432.

248 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22575387	C₂	1	2	yes	maximal
2	C₂×#22575385	C₂	1	2	yes	maximal
3	#31770471	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₂₆₉	C₂²	1	4	yes	—
6	#22575387	—	2	4	no	—
7	#22575385	—	2	4	no	—
8	#22575360	—	2	4	no	—
9	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₂₆₉	C₄	1	4	yes	complemented
13	C₂²×Dic₂₆₉	—	2	4	no	—
14	C₄⋊Dic₂₆₉	—	2	4	no	—
15	C₄⋊Dic₂₆₉	—	2	4	no	—
16	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
17	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₂₆₉	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₂₆₉	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₂₆₉	D₄	1	2	yes	—
21	C₂×C₈×C₂₆₉	D₄	1	2	yes	—
22	C₂×C₈×C₂₆₉	Q₈	1	2	yes	—
23	C₂×C₈×C₂₆₉	Q₈	1	2	yes	—
24	C₄²×C₂₆₉	C₂×C₄	1	2	yes	—
25	C₄²×C₂₆₉	Q₈	1	2	yes	—
26	C₂×Dic₂₆₉	—	4	4	no	—
27	C₂×Dic₂₆₉	—	4	8	no	—
28	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₆₉	—	2	4	no	—
30	C₂×C₄×C₂₆₉	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
37	C₈×C₂₆₉	C₄⋊C₄	1	4	yes	—
38	C₈×C₂₆₉	C₄⋊C₄	1	4	yes	—
39	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₂₆₉	#127	1	1	yes	characteristic
41	C₂²×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₂₆₉	#135	1	4	yes	—
43	C₄×C₂₆₉	—	2	4	no	—
44	C₄×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
48	Dic₂₆₉	—	8	4	no	—
49	C₂×C₂₆₉	#384	1	1	yes	characteristic
50	C₂×C₂₆₉	#427	1	2	yes	—
51	C₂×C₂₆₉	#495	1	2	yes	—
52	C₂×C₂₆₉	#499	1	2	yes	—
53	C₂₆₉	#1846	1	1	yes	characteristic, complemented, Sylow
54	#1846	—	269	1	no	maximal, Sylow
55	C₂×#107	—	269	2	no	—
56	C₂×#106	—	269	2	no	—
57	C₂×C₄×C₈	D₂₆₉	1	1	yes	characteristic
58	#382	—	269	2	no	—
59	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	269	4	no	—
62	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
63	#107	—	538	4	no	—
64	#106	—	538	4	no	—
65	C₄×C₈	Dic₂₆₉	1	4	yes	complemented
66	#95	—	538	4	no	—
67	C₂²×C₄	—	538	4	no	—
68	C₂²×C₄	D₁₀₇₆	1	2	yes	—
69	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
70	C₂×C₈	C₂×Dic₂₆₉	1	2	yes	—
71	C₂×C₈	C₂×Dic₂₆₉	1	2	yes	—
72	C₂×C₈	D₁₀₇₆	1	2	yes	—
73	C₂×C₈	D₁₀₇₆	1	2	yes	—
74	C₂×C₈	Dic₅₃₈	1	2	yes	—
75	C₂×C₈	Dic₅₃₈	1	2	yes	—
76	C₄⋊C₄	—	538	4	no	—
77	C₄⋊C₄	—	538	4	no	—
78	C₄²	C₂×Dic₂₆₉	1	2	yes	—
79	C₄²	Dic₅₃₈	1	2	yes	—
80	C₂³	C₂×D₁₀₇₆	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	1076	4	no	—
83	C₂×C₄	—	1076	8	no	—
84	C₂×C₄	C₄⋊Dic₂₆₉	1	4	yes	—
85	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
87	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
89	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
90	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
91	C₈	C₄⋊Dic₂₆₉	1	4	yes	—
92	C₈	C₄⋊Dic₂₆₉	1	4	yes	—
93	C₂²	C₂×C₄⋊Dic₂₆₉	1	1	yes	characteristic
94	C₂²	#22575450	1	1	yes	characteristic
95	C₂²	#22575453	1	1	yes	characteristic
96	C₂²	#22575475	1	4	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	2152	4	no	—
99	C₄	C₂×C₄⋊Dic₂₆₉	1	1	yes	characteristic
100	C₄	C₂×C₄⋊Dic₂₆₉	1	1	yes	characteristic
101	C₄	#22575453	1	1	yes	characteristic
102	C₄	#22575453	1	1	yes	characteristic
103	C₂	#31770475	1	1	yes	characteristic
104	C₂	#31770643	1	2	yes	—
105	C₂	#31770989	1	2	yes	—
106	C₂	#31771007	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575387

C₂

yes

maximal

C₂×#22575385

C₂

yes

maximal

#31770471

C₂

yes

maximal

C₂×C₄×C₈×C₂₆₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

#22575387

—

#22575385

—

#22575360

—

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₄×C₈×C₂₆₉

C₄

yes

complemented

C₂²×Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₂²×C₄×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

C₂×C₄

yes

—

C₂×C₈×C₂₆₉

C₂×C₄

yes

—

C₂×C₈×C₂₆₉

D₄

yes

—

C₂×C₈×C₂₆₉

D₄

yes

—

C₂×C₈×C₂₆₉

Q₈

yes

—

C₂×C₈×C₂₆₉

Q₈

yes

—

C₄²×C₂₆₉

C₂×C₄

yes

—

C₄²×C₂₆₉

Q₈

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₈×C₂₆₉

C₄⋊C₄

yes

—

C₈×C₂₆₉

C₄⋊C₄

yes

—

C₂²×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₆₉

#127

yes

characteristic

C₂²×C₂₆₉

C₄⋊Q₈

yes

characteristic

C₂²×C₂₆₉

#135

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₆₉

C₄⋊Q₈

yes

characteristic

C₄×C₂₆₉

C₄⋊Q₈

yes

characteristic

Dic₂₆₉

—

C₂×C₂₆₉

#384

yes

characteristic

C₂×C₂₆₉

#427

yes

—

C₂×C₂₆₉

#495

yes

—

C₂×C₂₆₉

#499

yes

—

C₂₆₉

#1846

yes

characteristic, complemented, Sylow

#1846

—

269

maximal, Sylow

C₂×#107

—

269

—

C₂×#106

—

269

—

C₂×C₄×C₈

D₂₆₉

yes

characteristic

#382

—

269

—

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

C₂×C₄⋊C₄

—

269

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

#107

—

538

—

#106

—

538

—

C₄×C₈

Dic₂₆₉

yes

complemented

#95

—

538

—

C₂²×C₄

—

538

—

C₂²×C₄

D₁₀₇₆

yes

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂×C₈

C₂×Dic₂₆₉

yes

—

C₂×C₈

C₂×Dic₂₆₉

yes

—

C₂×C₈

D₁₀₇₆

yes

—

C₂×C₈

D₁₀₇₆

yes

—

C₂×C₈

Dic₅₃₈

yes

—

C₂×C₈

Dic₅₃₈

yes

—

C₄⋊C₄

—

538

—

C₄⋊C₄

—

538

—

C₄²

C₂×Dic₂₆₉

yes

—

C₄²

Dic₅₃₈

yes

—

C₂³

C₂×D₁₀₇₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

1076

—

C₂×C₄

—

1076

—

C₂×C₄

C₄⋊Dic₂₆₉

yes

—

C₂×C₄

C₂²×Dic₂₆₉

yes

characteristic

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

C₈

C₄⋊Dic₂₆₉

yes

—

C₈

C₄⋊Dic₂₆₉

yes

—

C₂²

C₂×C₄⋊Dic₂₆₉

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

2152

—

C₄

C₂×C₄⋊Dic₂₆₉

yes

characteristic

100

C₄

C₂×C₄⋊Dic₂₆₉

yes

characteristic

101

C₄

#22575453

yes

characteristic

102

C₄

#22575453

yes

characteristic

103

C₂

#31770475

yes

characteristic

104

C₂

#31770643

yes

—

105

C₂

#31770989

yes

—

106

C₂

#31771007

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₀₇₆
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	C₂×C₄×C₈×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂×D₄
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	C₂×C₄×C₈	D₂₆₉
hypocenter	C₂₆₉	#1846
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₀₇₆

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

C₂×C₄×C₈×C₂₆₉

C₂

socle, prime radical

C₂³×C₂₆₉

C₂×D₄

cosocle

C₂²×C₄×C₂₆₉

C₂³

hypercenter

C₂×C₄×C₈

D₂₆₉

hypocenter

C₂₆₉

#1846

LCS

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
12. C₄×C₈×C₂₆₉	C₄	semidirect
53. C₂₆₉	#1846	semidirect
65. C₄×C₈	Dic₂₆₉	semidirect

H ≅ G/N

type

12. C₄×C₈×C₂₆₉

C₄

semidirect

53. C₂₆₉

#1846

semidirect

65. C₄×C₈

Dic₂₆₉

semidirect

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

subgroup	index
C₂×#22575387	2
C₂×#22575385	2
#31770471	2
C₂×C₄×C₈×C₂₆₉	2
#1846	269

subgroup

index

C₂×C₄×C₈×C₂₆₉

269

Subgroups — #46811049

Conjugacy classes of subgroups

Complements

Maximal subgroups