Group #20295001 — subgroups

Subgroups — #100×D₁₀₉ #20295001

Order 6976.

186 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 53 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×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₁₀₉	C₂	1	2	yes	maximal, complemented
3	#16812616	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812615	C₂	1	2	yes	maximal, complemented
5	#100×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₀₉	C₄	1	2	yes	—
7	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊D₁₀₉	C₄	1	2	yes	—
9	D₄×D₁₀₉	—	2	2	no	—
10	D₄×D₁₀₉	—	2	2	no	—
11	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
12	#13182013	C₂²	1	2	yes	complemented
13	#13182012	—	2	2	no	—
14	C₈×D₁₀₉	—	2	2	no	—
15	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	M₁₆×C₁₀₉	C₂²	1	2	yes	complemented
17	C₂³×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂²×D₁₀₉	—	2	2	no	—
19	C₂²×D₁₀₉	—	2	2	no	—
20	C₂²×D₁₀₉	—	4	4	no	—
21	C₂²×D₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₀₉	C₂×C₄	1	2	yes	—
23	D₄×C₁₀₉	—	2	2	no	—
24	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₂²⋊D₁₀₉	—	2	4	no	—
26	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	D₄₃₆	—	2	2	no	—
28	C₄×D₁₀₉	D₄	1	2	yes	—
29	C₄×D₁₀₉	D₄	1	2	yes	—
30	C₈×C₁₀₉	—	2	2	no	—
31	#11724784	—	2	2	no	—
32	C₂²×C₁₀₉	—	2	2	no	—
33	C₂²×C₁₀₉	—	4	2	no	—
34	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×D₁₀₉	—	2	2	no	—
36	C₂×D₁₀₉	—	2	2	no	—
37	C₂×D₁₀₉	—	4	4	no	—
38	C₂×D₁₀₉	—	4	4	no	—
39	C₂×D₁₀₉	—	4	2	no	—
40	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×C₁₀₉	C₂×D₄	1	2	yes	—
43	Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
44	C₂×C₁₀₉	—	2	1	no	—
45	C₂×C₁₀₉	—	4	2	no	—
46	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	D₁₀₉	—	2	1	no	—
48	D₁₀₉	—	4	2	no	—
49	D₁₀₉	#100	1	2	yes	direct product factor
50	C₁₀₉	C₂×#100	1	1	yes	characteristic, complemented, Sylow
51	C₂×#100	—	109	1	no	maximal, Sylow
52	C₂²×D₄	—	109	1	no	—
53	C₂×M₁₆	—	109	2	no	—
54	#100	—	109	1	no	—
55	#100	—	109	2	no	—
56	#100	D₁₀₉	1	1	yes	characteristic, direct product factor
57	C₂⁴	—	109	2	no	—
58	C₂×D₄	—	109	2	no	—
59	C₂×D₄	—	109	1	no	—
60	C₂×D₄	—	218	2	no	—
61	C₂×D₄	—	218	2	no	—
62	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
63	C₂²×C₄	—	109	1	no	—
64	M₁₆	—	109	2	no	—
65	M₁₆	—	218	2	no	—
66	M₁₆	C₂×D₁₀₉	1	2	yes	complemented
67	C₂×C₈	—	218	2	no	—
68	C₂³	—	109	1	no	—
69	C₂³	—	109	2	no	—
70	C₂³	—	436	4	no	—
71	C₂³	—	218	2	no	—
72	C₂³	—	218	2	no	—
73	C₂³	C₄×D₁₀₉	1	2	yes	—
74	D₄	—	2	2	no	—
75	D₄	—	218	4	no	—
76	D₄	—	218	2	no	—
77	C₂×C₄	—	109	1	no	—
78	C₂×C₄	—	109	2	no	—
79	C₂×C₄	—	109	2	no	—
80	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	—	218	2	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	4	2	no	—
85	C₂²	—	109	1	no	—
86	C₂²	—	109	1	no	—
87	C₂²	—	436	2	no	—
88	C₂²	—	436	4	no	—
89	C₂²	—	436	4	no	—
90	C₂²	—	218	2	no	—
91	C₂²	—	218	2	no	—
92	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
93	C₄	—	109	2	no	—
94	C₄	D₄×D₁₀₉	1	2	yes	—
95	C₂	—	2	1	no	—
96	C₂	—	4	2	no	—
97	C₂	—	109	2	no	—
98	C₂	—	436	2	no	—
99	C₂	—	218	1	no	—
100	C₂	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×D₁₀₉

C₂

yes

maximal, characteristic

M₁₆×D₁₀₉

C₂

yes

maximal, complemented

#16812616

C₂

yes

maximal, characteristic, complemented

#16812615

C₂

yes

maximal, complemented

#100×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₄

yes

—

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

C₄

yes

—

D₄×D₁₀₉

—

D₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182013

C₂²

yes

complemented

#13182012

—

C₈×D₁₀₉

—

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

M₁₆×C₁₀₉

C₂²

yes

complemented

C₂³×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×C₄

yes

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

D₄₃₆

—

C₄×D₁₀₉

D₄

yes

—

C₄×D₁₀₉

D₄

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

—

Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

D₁₀₉

—

D₁₀₉

—

D₁₀₉

#100

yes

direct product factor

C₁₀₉

C₂×#100

yes

characteristic, complemented, Sylow

C₂×#100

—

109

maximal, Sylow

C₂²×D₄

—

109

—

C₂×M₁₆

—

109

—

#100

—

109

—

#100

—

109

—

#100

D₁₀₉

yes

characteristic, direct product factor

C₂⁴

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

C₂×D₁₀₉

yes

complemented

C₂×C₈

—

218

—

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

436

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₄×D₁₀₉

yes

—

D₄

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

218

—

C₂²

—

C₂²

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×C₄×D₁₀₉

yes

characteristic

C₄

—

109

—

C₄

D₄×D₁₀₉

yes

—

C₂

—

C₂

—

C₂

—

109

—

C₂

—

436

—

C₂

—

218

—

100

C₂

C₂²⋊C₄×D₁₀₉

yes

characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×D₁₀₉
commutator	C₂²×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#100×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×C₂²⋊C₄
hypercenter	#100	D₁₀₉
hypocenter	C₁₀₉	C₂×#100
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

subgroup

G/H

center

C₂

C₂²⋊C₄×D₁₀₉

commutator

C₂²×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#100×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₂×C₂²⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

N	H ≅ G/N	type
2. M₁₆×D₁₀₉	C₂	semidirect
3. #16812616	C₂	semidirect
4. #16812615	C₂	semidirect
5. #100×C₁₀₉	C₂	semidirect
12. #13182013	C₂²	semidirect
16. M₁₆×C₁₀₉	C₂²	semidirect
49. D₁₀₉	#100	direct (N × H)
50. C₁₀₉	C₂×#100	semidirect
56. #100	D₁₀₉	direct (N × H)
66. M₁₆	C₂×D₁₀₉	semidirect

H ≅ G/N

type

2. M₁₆×D₁₀₉

C₂

semidirect

3. #16812616

C₂

semidirect

4. #16812615

C₂

semidirect

5. #100×C₁₀₉

C₂

semidirect

12. #13182013

C₂²

semidirect

16. M₁₆×C₁₀₉

C₂²

semidirect

49. D₁₀₉

#100

direct (N × H)

50. C₁₀₉

C₂×#100

semidirect

56. #100

D₁₀₉

direct (N × H)

66. M₁₆

C₂×D₁₀₉

semidirect

Subgroups — #100×D₁₀₉ #20295001

Conjugacy classes of subgroups

Complements

Subgroups — #100×D109 #20295001

Conjugacy classes of subgroups

Complements

Subgroups — #100×D₁₀₉ #20295001