Group #46812396 — subgroups

Subgroups — #46812396

Order 34432.

176 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 67 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	#31770645	C₂	1	2	yes	maximal
2	#31770631	C₂	1	1	yes	maximal, characteristic
3	#31770374	C₂	1	2	yes	maximal
4	#31770370	C₂	1	1	yes	maximal, characteristic
5	C₂×#106×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16864386	C₂²	1	1	yes	characteristic
7	C₂×#16864386	C₂²	1	2	yes	—
8	#22575412	C₂²	1	1	yes	characteristic
9	#22575386	—	2	2	no	—
10	#22575384	—	2	2	no	—
11	#22575382	—	2	2	no	—
12	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
14	#106×C₂₆₉	—	2	2	no	—
15	#16864386	—	2	2	no	—
16	#16864386	—	2	2	no	—
17	#16864386	Q₈	1	4	yes	—
18	C₂×#14524130	—	2	2	no	—
19	C₂×#14524130	—	2	1	no	—
20	C₄⋊Dic₂₆₉	D₄	1	2	yes	—
21	C₂²⋊Dic₂₆₉	—	2	1	no	—
22	C₄×Dic₂₆₉	—	2	1	no	—
23	C₂²×C₄×C₂₆₉	—	2	2	no	—
24	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₂₆₉	—	2	2	no	—
26	C₂×C₈×C₂₆₉	D₄	1	2	yes	—
27	C₄⋊C₄×C₂₆₉	—	2	2	no	—
28	C₄⋊C₄×C₂₆₉	—	2	2	no	—
29	C₂×Dic₂₆₉	—	2	2	no	—
30	#14524130	—	2	4	no	—
31	#14524130	—	4	1	no	—
32	C₂³×C₂₆₉	#41	1	1	yes	characteristic
33	C₂×C₄×C₂₆₉	—	4	2	no	—
34	C₂×C₄×C₂₆₉	—	8	2	no	—
35	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₂₆₉	#41	1	2	yes	—
39	C₈×C₂₆₉	—	2	2	no	—
40	C₂²×C₂₆₉	—	2	2	no	—
41	C₂²×C₂₆₉	#121	1	1	yes	characteristic
42	C₂²×C₂₆₉	#125	1	2	yes	—
43	C₄×C₂₆₉	—	8	2	no	—
44	C₄×C₂₆₉	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₂₆₉	#123	1	1	yes	characteristic
46	C₄×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₂₆₉	—	4	2	no	—
48	C₂×C₂₆₉	—	2	2	no	—
49	C₂×C₂₆₉	#398	1	1	yes	characteristic
50	C₂×C₂₆₉	#471	1	2	yes	—
51	C₂₆₉	#2090	1	1	yes	characteristic, complemented, Sylow
52	#2090	—	269	1	no	maximal, Sylow
53	#428	—	269	2	no	—
54	C₂×#106	D₂₆₉	1	1	yes	characteristic
55	#424	—	269	1	no	—
56	#343	—	269	2	no	—
57	#341	—	269	1	no	—
58	C₂×M₁₆	—	269	2	no	—
59	C₂×M₁₆	—	269	1	no	—
60	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
61	#117	—	269	1	no	—
62	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
63	#107	—	538	2	no	—
64	#106	—	2	2	no	—
65	#106	—	538	2	no	—
66	C₄⋊C₈	—	538	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
69	M₁₆	—	269	4	no	—
70	M₁₆	—	538	2	no	—
71	M₁₆	—	538	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	538	2	no	—
74	C₂×C₈	—	538	1	no	—
75	C₂×C₈	C₂²⋊D₂₆₉	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	269	2	no	—
79	C₂²⋊C₄	—	538	1	no	—
80	C₄²	—	538	1	no	—
81	C₂³	#16864401	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	8	2	no	—
84	C₂×C₄	—	538	2	no	—
85	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
86	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₂₆₉	1	2	yes	—
88	C₂×C₄	#16864402	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	538	4	no	—
91	C₈	—	1076	1	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#22575424	1	1	yes	characteristic
94	C₂²	#22575443	1	2	yes	—
95	C₄	—	8	2	no	—
96	C₄	—	1076	2	no	—
97	C₄	#22575431	1	2	yes	—
98	C₄	#22575436	1	1	yes	characteristic
99	C₄	#22575451	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#31770536	1	1	yes	characteristic
102	C₂	#31770865	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31770645

C₂

yes

maximal

#31770631

C₂

yes

maximal, characteristic

#31770374

C₂

yes

maximal

#31770370

C₂

yes

maximal, characteristic

C₂×#106×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864386

C₂²

yes

characteristic

C₂×#16864386

C₂²

yes

—

#22575412

C₂²

yes

—

—

—

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

—

#106×C₂₆₉

—

#16864386

—

#16864386

—

#16864386

Q₈

yes

—

C₂×#14524130

—

C₂×#14524130

—

C₄⋊Dic₂₆₉

D₄

yes

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

D₄

yes

—

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

#14524130

—

C₂³×C₂₆₉

#41

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

—

C₂×C₄×C₂₆₉

#41

yes

—

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

#121

yes

characteristic

C₂²×C₂₆₉

#125

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂²⋊Q₈

yes

—

C₄×C₂₆₉

#123

yes

characteristic

C₄×C₂₆₉

C₄⋊Q₈

yes

characteristic

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#398

yes

characteristic

C₂×C₂₆₉

#471

yes

—

C₂₆₉

#2090

yes

characteristic, complemented, Sylow

#2090

—

269

maximal, Sylow

#428

—

269

—

C₂×#106

D₂₆₉

yes

characteristic

#424

—

269

—

#343

—

269

—

#341

—

269

—

C₂×M₁₆

—

269

—

C₂×M₁₆

—

269

—

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

#117

—

269

—

C₂×C₄⋊C₄

C₂×D₂₆₉

yes

—

#107

—

538

—

#106

—

#106

—

538

—

C₄⋊C₈

—

538

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

M₁₆

—

269

—

M₁₆

—

538

—

M₁₆

—

538

—

C₂×C₈

—

C₂×C₈

—

538

—

C₂×C₈

—

538

—

C₂×C₈

C₂²⋊D₂₆₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

269

—

C₂²⋊C₄

—

538

—

C₄²

—

538

—

C₂³

#16864401

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

538

—

C₂×C₄

D₄×D₂₆₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

characteristic

C₂×C₄

Q₈×D₂₆₉

yes

—

C₂×C₄

#16864402

yes

—

C₈

—

C₈

—

538

—

C₈

—

1076

—

C₂²

—

C₂²

#22575424

yes

characteristic

C₂²

#22575443

yes

—

C₄

—

C₄

—

1076

—

C₄

#22575431

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#31770536

yes

characteristic

102

C₂

#31770865

yes

—

	subgroup	G/H
center	C₂²	#22575424
commutator, cosocle	C₂²×C₄×C₂₆₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	C₂×#106×C₂₆₉	C₂
socle	C₂²×C₂₆₉	#121
prime radical	C₂³×C₂₆₉	#41
hypercenter	C₂×#106	D₂₆₉
hypocenter	C₂₆₉	#2090
LCS	G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

#22575424

commutator, cosocle

C₂²×C₄×C₂₆₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

C₂×#106×C₂₆₉

C₂

C₂²×C₂₆₉

C₂³×C₂₆₉

D₂₆₉

C₂₆₉

G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
51. C₂₆₉	#2090	semidirect

H ≅ G/N

type

51. C₂₆₉

#2090

semidirect

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

subgroup	index
#31770645	2
#31770631	2
#31770374	2
#31770370	2
C₂×#106×C₂₆₉	2
#2090	269

subgroup

index

C₂×#106×C₂₆₉

269

Subgroups — #46812396

Conjugacy classes of subgroups

Complements

Maximal subgroups