Group #28363181 — subgroups

Subgroups — #28363181

Order 13952.

162 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 73 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	#20294797	C₂	1	2	yes	maximal
2	#20294786	C₂	1	2	yes	maximal
3	#20294784	C₂	1	1	yes	maximal, characteristic
4	#20294783	C₂	1	1	yes	maximal, characteristic
5	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	#16812658	C₂²	1	2	yes	—
8	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
11	#97×C₁₀₉	C₄	1	2	yes	complemented
12	C₄×C₈×C₁₀₉	C₄	1	2	yes	complemented
13	#13182013	—	2	2	no	—
14	#13182013	—	2	2	no	—
15	C₂×#11724784	—	2	2	no	—
16	C₄⋊Dic₁₀₉	—	2	2	no	—
17	C₂²⋊Dic₁₀₉	—	2	2	no	—
18	C₄×Dic₁₀₉	—	2	2	no	—
19	C₂²×C₄×C₁₀₉	—	2	1	no	—
20	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
21	M₁₆×C₁₀₉	—	2	2	no	—
22	M₁₆×C₁₀₉	—	2	2	no	—
23	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₄²×C₁₀₉	—	2	1	no	—
26	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₁₀₉	#41	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₁₀₉	—	4	2	no	—
32	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	#41	1	2	yes	—
34	C₂×C₄×C₁₀₉	#41	1	2	yes	—
35	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
36	C₂×Dic₁₀₉	—	4	2	no	—
37	C₈×C₁₀₉	—	2	2	no	—
38	C₈×C₁₀₉	—	2	2	no	—
39	#11724784	—	4	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	—	2	2	no	—
44	C₄×C₁₀₉	—	4	1	no	—
45	C₄×C₁₀₉	#117	1	1	yes	characteristic
46	C₄×C₁₀₉	#117	1	1	yes	characteristic
47	C₄×C₁₀₉	#126	1	2	yes	—
48	Dic₁₀₉	—	8	2	no	—
49	C₂×C₁₀₉	—	2	2	no	—
50	C₂×C₁₀₉	—	2	1	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	D₁₀₉	1	1	yes	characteristic
55	#344	—	109	2	no	—
56	#339	—	109	2	no	—
57	#338	—	109	1	no	—
58	H(3,ℤ/4)	—	109	1	no	—
59	C₂×M₁₆	—	109	2	no	—
60	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
61	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
62	#117	—	109	2	no	—
63	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
64	#97	Dic₁₀₉	1	2	yes	complemented
65	C₄×C₈	Dic₁₀₉	1	2	yes	complemented
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	218	2	no	—
71	M₁₆	—	218	2	no	—
72	C₂×C₈	—	218	2	no	—
73	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
74	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
75	C₄⋊C₄	—	218	2	no	—
76	C₂²⋊C₄	—	218	2	no	—
77	C₄²	—	2	1	no	—
78	C₄²	—	218	2	no	—
79	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
80	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
81	C₂³	#13182029	1	1	yes	characteristic
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	436	2	no	—
86	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
87	C₂×C₄	#13182029	1	2	yes	—
88	C₂×C₄	#13182029	1	2	yes	—
89	C₂×C₄	#13182029	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	2	2	no	—
92	C₈	—	436	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#16812658	1	1	yes	characteristic
95	C₂²	#16812693	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	—	4	1	no	—
98	C₄	—	872	2	no	—
99	C₄	#16812658	1	1	yes	characteristic
100	C₄	#16812658	1	1	yes	characteristic
101	C₄	#16812693	1	2	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#20294894	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20294797

C₂

yes

maximal

#20294786

C₂

yes

maximal

#20294784

C₂

yes

maximal, characteristic

#20294783

C₂

yes

maximal, characteristic

C₄×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

#16812658

C₂²

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

C₄

yes

complemented

C₄×C₈×C₁₀₉

C₄

yes

—

—

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

#41

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#117

yes

characteristic

C₂²×C₁₀₉

#126

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#383

yes

characteristic

#1841

—

109

maximal, Sylow

C₁₀₉

#1841

yes

characteristic, complemented, Sylow

C₄×M₁₆

D₁₀₉

yes

characteristic

#344

—

109

—

#339

—

109

—

#338

—

109

—

H(3,ℤ/4)

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#97

Dic₁₀₉

yes

complemented

C₄×C₈

Dic₁₀₉

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

C₄²

—

218

—

C₄²

C₂×Dic₁₀₉

yes

characteristic

C₄²

C₂×Dic₁₀₉

yes

characteristic

C₂³

#13182029

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

C₂×C₄

#13182029

yes

—

C₂×C₄

#13182029

yes

—

C₂×C₄

#13182029

yes

characteristic

C₈

—

C₈

—

C₈

—

436

—

C₂²

—

C₂²

#16812658

yes

characteristic

C₂²

#16812693

yes

—

C₄

—

C₄

—

C₄

—

872

—

C₄

#16812658

yes

characteristic

100

C₄

#16812658

yes

characteristic

101

C₄

#16812693

yes

—

102

C₂

—

103

C₂

—

104

C₂

#20294894

yes

characteristic

	subgroup	G/H
center	C₄	#16812658
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₄×M₁₆×C₁₀₉	C₂
socle	C₂×C₁₀₉	#383
prime radical	C₂³×C₁₀₉	#41
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₄×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#1841
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₄×M₁₆

subgroup

G/H

center

C₄

#16812658

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₄×M₁₆×C₁₀₉

C₂

C₂×C₁₀₉

C₂³×C₁₀₉

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₄×M₁₆

D₁₀₉

hypocenter

C₁₀₉

#1841

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₄×M₁₆

N	H ≅ G/N	type
11. #97×C₁₀₉	C₄	semidirect
12. C₄×C₈×C₁₀₉	C₄	semidirect
53. C₁₀₉	#1841	semidirect
64. #97	Dic₁₀₉	semidirect
65. C₄×C₈	Dic₁₀₉	semidirect

H ≅ G/N

type

11. #97×C₁₀₉

C₄

semidirect

12. C₄×C₈×C₁₀₉

C₄

semidirect

53. C₁₀₉

#1841

semidirect

64. #97

Dic₁₀₉

semidirect

65. C₄×C₈

Dic₁₀₉

semidirect

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

subgroup	index
#20294797	2
#20294786	2
#20294784	2
#20294783	2
C₄×M₁₆×C₁₀₉	2
#1841	109

subgroup

index

C₄×M₁₆×C₁₀₉

109

Subgroups — #28363181

Conjugacy classes of subgroups

Complements

Maximal subgroups