Group #28365322 — subgroups

Subgroups — M₆₄×D₁₀₉ #28365322

Order 13952.

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

normal subgroups only

This group has 112 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	C₂×C₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#20294860	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294859	C₂	1	2	yes	maximal, complemented
4	C₃₂×D₁₀₉	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
7	C₂×#13182004	C₄	1	1	yes	characteristic
8	C₁₆×D₁₀₉	C₄	1	1	yes	characteristic
9	C₁₆×D₁₀₉	C₄	1	1	yes	characteristic
10	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
11	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
12	#16812605	C₂²	1	2	yes	complemented
13	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₀₉	C₂²	1	2	yes	complemented
15	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
16	C₂×#11724784	C₈	1	1	yes	characteristic
17	C₈×D₁₀₉	C₈	1	1	yes	characteristic
18	C₈×D₁₀₉	C₈	1	1	yes	characteristic
19	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
21	#13182004	C₂×C₄	1	1	yes	characteristic
22	#13182004	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂²×D₁₀₉	C₁₆	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
28	C₂×Dic₁₀₉	C₁₆	1	1	yes	characteristic
29	C₄×D₁₀₉	C₁₆	1	1	yes	characteristic
30	C₄×D₁₀₉	C₁₆	1	1	yes	characteristic
31	C₄×D₁₀₉	C₂×C₈	1	1	yes	characteristic
32	C₄×D₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
34	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	#11724784	C₂×C₈	1	1	yes	characteristic
36	#11724784	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
38	C₂×D₁₀₉	—	2	2	no	—
39	C₂×D₁₀₉	C₂×C₁₆	1	1	yes	characteristic
40	C₂×D₁₀₉	C₂×C₁₆	1	1	yes	characteristic
41	C₄×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
42	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
43	Dic₁₀₉	C₂×C₁₆	1	1	yes	characteristic
44	Dic₁₀₉	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₁₀₉	—	2	1	no	—
46	C₂×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
47	D₁₀₉	—	2	1	no	—
48	D₁₀₉	M₆₄	1	2	yes	direct product factor
49	C₂×M₆₄	—	109	1	no	maximal, Sylow
50	C₁₀₉	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₁₆	—	109	1	no	—
52	M₆₄	—	109	1	no	—
53	M₆₄	—	109	2	no	—
54	M₆₄	D₁₀₉	1	1	yes	characteristic, direct product factor
55	C₂×C₃₂	—	109	2	no	—
56	C₂²×C₈	—	109	1	no	—
57	C₂×C₁₆	—	109	1	no	—
58	C₂×C₁₆	—	109	1	no	—
59	C₂×C₁₆	—	109	1	no	—
60	C₂×C₁₆	—	109	1	no	—
61	C₂×C₁₆	—	109	1	no	—
62	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
63	C₃₂	—	109	2	no	—
64	C₃₂	C₂×D₁₀₉	1	2	yes	complemented
65	C₂²×C₄	—	109	1	no	—
66	C₂×C₈	—	109	1	no	—
67	C₂×C₈	—	109	1	no	—
68	C₂×C₈	—	109	1	no	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	109	1	no	—
71	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
72	C₁₆	—	109	1	no	—
73	C₁₆	—	109	1	no	—
74	C₁₆	C₄×D₁₀₉	1	1	yes	characteristic
75	C₁₆	C₂²×D₁₀₉	1	1	yes	characteristic
76	C₂³	—	109	1	no	—
77	C₂×C₄	—	109	1	no	—
78	C₂×C₄	—	109	1	no	—
79	C₂×C₄	—	109	1	no	—
80	C₂×C₄	—	109	1	no	—
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
83	C₈	—	109	1	no	—
84	C₈	—	109	1	no	—
85	C₈	C₈×D₁₀₉	1	1	yes	characteristic
86	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
87	C₂²	—	109	1	no	—
88	C₂²	—	109	1	no	—
89	C₂²	—	218	2	no	—
90	C₂²	C₁₆×D₁₀₉	1	1	yes	characteristic
91	C₄	—	109	1	no	—
92	C₄	—	109	1	no	—
93	C₄	C₁₆×D₁₀₉	1	1	yes	characteristic
94	C₄	C₂×C₈×D₁₀₉	1	1	yes	characteristic
95	C₂	—	2	1	no	—
96	C₂	—	109	2	no	—
97	C₂	—	218	1	no	—
98	C₂	C₂×C₁₆×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×D₁₀₉

C₂

yes

maximal, characteristic

#20294860

C₂

yes

maximal, characteristic, complemented

#20294859

C₂

yes

maximal, complemented

C₃₂×D₁₀₉

C₂

yes

maximal, complemented

M₆₄×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×#13182004

C₄

yes

characteristic

C₁₆×D₁₀₉

C₄

yes

characteristic

C₁₆×D₁₀₉

C₄

yes

characteristic

C₁₆×D₁₀₉

C₂²

yes

characteristic

C₁₆×D₁₀₉

C₂²

yes

characteristic

#16812605

C₂²

yes

complemented

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₃₂×C₁₀₉

C₂²

yes

complemented

C₂×C₄×D₁₀₉

C₈

yes

characteristic

C₂×#11724784

C₈

yes

characteristic

C₈×D₁₀₉

C₈

yes

characteristic

C₈×D₁₀₉

C₈

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

#13182004

C₂×C₄

yes

characteristic

#13182004

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₁₆

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₁₆

yes

characteristic

C₄×D₁₀₉

C₁₆

yes

characteristic

C₄×D₁₀₉

C₁₆

yes

characteristic

C₄×D₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

C₂×C₈

yes

characteristic

C₈×C₁₀₉

C₂×C₈

yes

characteristic

C₈×C₁₀₉

C₂²×C₄

yes

characteristic

#11724784

C₂×C₈

yes

characteristic

#11724784

C₂×C₈

yes

characteristic

C₂²×C₁₀₉

C₂×C₁₆

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×C₁₆

yes

characteristic

C₂×D₁₀₉

C₂×C₁₆

yes

characteristic

C₄×C₁₀₉

C₂×C₁₆

yes

characteristic

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

Dic₁₀₉

C₂×C₁₆

yes

characteristic

Dic₁₀₉

C₂×C₁₆

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²×C₁₆

yes

characteristic

D₁₀₉

—

D₁₀₉

M₆₄

yes

direct product factor

C₂×M₆₄

—

109

maximal, Sylow

C₁₀₉

C₂×M₆₄

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

109

—

M₆₄

—

109

—

M₆₄

—

109

—

M₆₄

D₁₀₉

yes

characteristic, direct product factor

C₂×C₃₂

—

109

—

C₂²×C₈

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

C₃₂

—

109

—

C₃₂

C₂×D₁₀₉

yes

complemented

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₁₆

—

109

—

C₁₆

—

109

—

C₁₆

C₄×D₁₀₉

yes

characteristic

C₁₆

C₂²×D₁₀₉

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

C₈×D₁₀₉

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

C₈×D₁₀₉

yes

characteristic

C₈

C₂×C₄×D₁₀₉

yes

characteristic

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

C₁₆×D₁₀₉

yes

characteristic

C₄

—

109

—

C₄

—

109

—

C₄

C₁₆×D₁₀₉

yes

characteristic

C₄

C₂×C₈×D₁₀₉

yes

characteristic

C₂

—

C₂

—

109

—

C₂

—

218

—

C₂

C₂×C₁₆×D₁₀₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₁₀₉
commutator, socle	C₂×C₁₀₉	C₂²×C₁₆
Fitting	M₆₄×C₁₀₉	C₂
prime radical	C₂²×D₁₀₉	C₁₆
cosocle	C₁₆×C₁₀₉	C₂³
hypercenter	M₆₄	D₁₀₉
hypocenter	C₁₀₉	C₂×M₆₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₁₆ ◁ M₆₄

subgroup

G/H

center, Frattini

C₁₆

C₂²×D₁₀₉

commutator, socle

C₂×C₁₀₉

C₂²×C₁₆

Fitting

M₆₄×C₁₀₉

C₂

prime radical

C₂²×D₁₀₉

C₁₆

cosocle

C₁₆×C₁₀₉

C₂³

hypercenter

M₆₄

D₁₀₉

hypocenter

C₁₀₉

C₂×M₆₄

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₁₆ ◁ M₆₄

N	H ≅ G/N	type
2. #20294860	C₂	semidirect
3. #20294859	C₂	semidirect
4. C₃₂×D₁₀₉	C₂	semidirect
5. M₆₄×C₁₀₉	C₂	semidirect
12. #16812605	C₂²	semidirect
14. C₃₂×C₁₀₉	C₂²	semidirect
48. D₁₀₉	M₆₄	direct (N × H)
50. C₁₀₉	C₂×M₆₄	semidirect
54. M₆₄	D₁₀₉	direct (N × H)
64. C₃₂	C₂×D₁₀₉	semidirect

H ≅ G/N

type

2. #20294860

C₂

semidirect

3. #20294859

C₂

semidirect

4. C₃₂×D₁₀₉

C₂

semidirect

5. M₆₄×C₁₀₉

C₂

semidirect

12. #16812605

C₂²

semidirect

14. C₃₂×C₁₀₉

C₂²

semidirect

48. D₁₀₉

M₆₄

direct (N × H)

50. C₁₀₉

C₂×M₆₄

semidirect

54. M₆₄

D₁₀₉

direct (N × H)

64. C₃₂

C₂×D₁₀₉

semidirect

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

subgroup	index
C₂×C₁₆×D₁₀₉	2
#20294860	2
#20294859	2
C₃₂×D₁₀₉	2
M₆₄×C₁₀₉	2
C₂×M₆₄	109

subgroup

index

C₂×C₁₆×D₁₀₉

#20294860

#20294859

C₃₂×D₁₀₉

M₆₄×C₁₀₉

C₂×M₆₄

109

Subgroups — M₆₄×D₁₀₉ #28365322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M64×D109 #28365322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₆₄×D₁₀₉ #28365322