Group #28363182 — subgroups

Subgroups — #28363182

Order 13952.

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

normal subgroups only

This group has 162 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	M₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#20294798	C₂	1	2	yes	maximal
3	#20294787	C₂	1	2	yes	maximal
4	#20294783	C₂	1	1	yes	maximal, characteristic
5	#338×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	1	yes	characteristic
7	#16812658	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812609	C₄	1	2	yes	complemented
10	C₈×Dic₁₀₉	C₄	1	2	yes	complemented
11	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
12	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₀₉	—	2	1	no	—
14	#13182013	—	2	2	no	—
15	C₂×#11724784	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₀₉	—	2	2	no	—
17	C₂²⋊Dic₁₀₉	—	2	2	no	—
18	C₄×Dic₁₀₉	—	2	1	no	—
19	C₄×Dic₁₀₉	—	2	2	no	—
20	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₀₉	—	2	2	no	—
24	M₁₆×C₁₀₉	—	2	2	no	—
25	M₁₆×C₁₀₉	—	2	2	no	—
26	C₂×C₈×C₁₀₉	—	2	2	no	—
27	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂³×C₁₀₉	#41	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	#41	1	2	yes	—
31	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	#41	1	2	yes	—
33	C₂×Dic₁₀₉	—	2	1	no	—
34	C₂×Dic₁₀₉	—	2	1	no	—
35	C₂×Dic₁₀₉	—	4	2	no	—
36	C₂×Dic₁₀₉	—	4	2	no	—
37	C₈×C₁₀₉	—	2	2	no	—
38	C₈×C₁₀₉	—	4	2	no	—
39	#11724784	—	2	2	no	—
40	C₂²×C₁₀₉	—	4	1	no	—
41	C₂²×C₁₀₉	#117	1	1	yes	characteristic
42	C₂²×C₁₀₉	#126	1	2	yes	—
43	C₄×C₁₀₉	#117	1	1	yes	characteristic
44	C₄×C₁₀₉	#117	1	1	yes	characteristic
45	C₄×C₁₀₉	#126	1	2	yes	—
46	Dic₁₀₉	—	2	2	no	—
47	Dic₁₀₉	—	4	1	no	—
48	Dic₁₀₉	—	8	2	no	—
49	C₂×C₁₀₉	—	2	1	no	—
50	C₂×C₁₀₉	—	2	2	no	—
51	C₂×C₁₀₉	#383	1	1	yes	characteristic
52	#1841	—	109	1	no	maximal, Sylow
53	C₁₀₉	#1841	1	1	yes	characteristic, complemented, Sylow
54	C₄×M₁₆	—	109	1	no	—
55	#344	—	109	2	no	—
56	#339	—	109	2	no	—
57	#338	D₁₀₉	1	1	yes	characteristic
58	H(3,ℤ/4)	—	109	1	no	—
59	C₂×M₁₆	—	109	1	no	—
60	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
61	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
62	#117	—	109	2	no	—
63	C₂×C₄²	—	109	1	no	—
64	#97	—	109	2	no	—
65	C₄×C₈	—	109	2	no	—
66	C₂²×C₄	—	218	1	no	—
67	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	218	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	109	2	no	—
74	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
75	C₄⋊C₄	—	218	2	no	—
76	C₂²⋊C₄	—	218	2	no	—
77	C₄²	—	109	1	no	—
78	C₄²	—	109	1	no	—
79	C₄²	—	218	2	no	—
80	C₄²	—	218	1	no	—
81	C₂³	#13182030	1	1	yes	characteristic
82	C₂×C₄	—	436	2	no	—
83	C₂×C₄	—	436	2	no	—
84	C₂×C₄	—	218	1	no	—
85	C₂×C₄	—	218	1	no	—
86	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
87	C₂×C₄	#13182028	1	2	yes	—
88	C₂×C₄	#13182028	1	1	yes	characteristic
89	C₂×C₄	#13182029	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	218	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#16812657	1	1	yes	characteristic
95	C₂²	#16812694	1	2	yes	—
96	C₄	—	872	2	no	—
97	C₄	—	436	1	no	—
98	C₄	—	218	2	no	—
99	C₄	#16812655	1	1	yes	characteristic
100	C₄	#16812656	1	1	yes	characteristic
101	C₄	#16812692	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#20294893	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294798

C₂

yes

maximal

#20294787

C₂

yes

maximal

#20294783

C₂

yes

maximal, characteristic

#338×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

characteristic

#16812658

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812609

C₄

yes

complemented

C₈×Dic₁₀₉

C₄

yes

complemented

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

#13182013

—

C₂×#11724784

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

#41

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#117

yes

characteristic

C₂²×C₁₀₉

#126

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#383

yes

characteristic

#1841

—

109

maximal, Sylow

C₁₀₉

#1841

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

109

—

#344

—

109

—

#339

—

109

—

#338

D₁₀₉

yes

characteristic

H(3,ℤ/4)

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

C₂×C₄²

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

—

218

—

C₂³

#13182030

yes

characteristic

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

#13182028

yes

—

C₂×C₄

#13182028

yes

characteristic

C₂×C₄

#13182029

yes

—

C₈

—

C₈

—

C₈

—

218

—

C₂²

—

C₂²

#16812657

yes

characteristic

C₂²

#16812694

yes

—

C₄

—

872

—

C₄

—

436

—

C₄

—

218

—

C₄

#16812655

yes

characteristic

100

C₄

#16812656

yes

characteristic

101

C₄

#16812692

yes

—

102

C₂

—

103

C₂

—

104

C₂

#20294893

yes

characteristic

	subgroup	G/H
center	C₄	#16812656
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#338×C₁₀₉	C₂
socle	C₂×C₁₀₉	#383
prime radical	C₂³×C₁₀₉	#41
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#338	D₁₀₉
hypocenter	C₁₀₉	#1841
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#16812656

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#338×C₁₀₉

C₂

C₂×C₁₀₉

C₂³×C₁₀₉

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
9. #16812609	C₄	semidirect
10. C₈×Dic₁₀₉	C₄	semidirect
53. C₁₀₉	#1841	semidirect

H ≅ G/N

type

9. #16812609

C₄

semidirect

10. C₈×Dic₁₀₉

C₄

semidirect

53. C₁₀₉

#1841

semidirect

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

subgroup	index
M₁₆×Dic₁₀₉	2
#20294798	2
#20294787	2
#20294783	2
#338×C₁₀₉	2
#1841	109

subgroup

index

M₁₆×Dic₁₀₉

#338×C₁₀₉

109

Subgroups — #28363182

Conjugacy classes of subgroups

Complements

Maximal subgroups