Group #20295115 — subgroups

Subgroups — SD₁₆×Dic₁₀₉ #20295115

Order 6976.

122 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#16812631	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812623	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812620	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₉	—	2	1	no	—
9	C₂×#11724784	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₀₉	—	2	1	no	—
11	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₁₀₉	—	2	1	no	—
14	C₄×Dic₁₀₉	—	2	1	no	—
15	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	SD₁₆×C₁₀₉	C₄	1	4	yes	complemented
19	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂³×C₁₀₉	—	2	1	no	—
21	Q₈×C₁₀₉	—	2	1	no	—
22	Q₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
23	D₄×C₁₀₉	—	2	1	no	—
24	D₄×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₀₉	—	2	1	no	—
28	C₂×Dic₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	—	4	2	no	—
30	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
32	C₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
33	#11724784	—	2	1	no	—
34	C₂²×C₁₀₉	—	2	2	no	—
35	C₂²×C₁₀₉	—	4	1	no	—
36	C₂²×C₁₀₉	—	4	1	no	—
37	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₄×C₁₀₉	—	2	2	no	—
39	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₁₀₉	#41	1	1	yes	characteristic
41	Dic₁₀₉	—	2	1	no	—
42	Dic₁₀₉	—	4	1	no	—
43	Dic₁₀₉	—	4	1	no	—
44	Dic₁₀₉	SD₁₆	1	2	yes	direct product factor
45	C₂×C₁₀₉	—	4	2	no	—
46	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
47	C₂×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
48	C₂×C₁₀₉	#135	1	1	yes	characteristic
49	C₁₀₉	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
50	C₄×SD₁₆	—	109	1	no	maximal, Sylow
51	C₂×SD₁₆	D₁₀₉	1	1	yes	characteristic
52	C₄×Q₈	—	109	1	no	—
53	C₄×D₄	—	109	1	no	—
54	#106	—	109	1	no	—
55	#103	—	109	1	no	—
56	#102	—	109	1	no	—
57	C₄×C₈	—	109	1	no	—
58	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
59	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
60	C₂²×C₄	—	218	1	no	—
61	SD₁₆	Dic₁₀₉	1	4	yes	direct product factor
62	C₂×C₈	—	109	1	no	—
63	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
64	C₄⋊C₄	—	109	1	no	—
65	C₄⋊C₄	—	109	1	no	—
66	C₄⋊C₄	—	218	1	no	—
67	C₂²⋊C₄	—	218	1	no	—
68	C₄²	—	109	1	no	—
69	C₄²	—	218	1	no	—
70	C₂³	—	2	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	C₂×Dic₁₀₉	1	2	yes	complemented
73	D₄	—	2	1	no	—
74	D₄	C₂×Dic₁₀₉	1	2	yes	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	109	1	no	—
77	C₂×C₄	—	109	1	no	—
78	C₂×C₄	—	436	2	no	—
79	C₂×C₄	—	218	1	no	—
80	C₂×C₄	—	218	1	no	—
81	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
82	C₈	—	218	1	no	—
83	C₈	C₂×Dic₁₀₉	1	2	yes	complemented
84	C₂²	—	2	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	109	2	no	—
90	C₄	—	436	1	no	—
91	C₄	—	436	1	no	—
92	C₄	—	218	1	no	—
93	C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
94	C₄	#13182028	1	1	yes	characteristic
95	C₂	—	4	2	no	—
96	C₂	D₄×Dic₁₀₉	1	1	yes	characteristic
97	C₂	SD₁₆×D₁₀₉	1	1	yes	characteristic
98	C₂	#16812720	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₀₉

C₂

yes

maximal, characteristic

#16812631

C₂

yes

maximal, characteristic, complemented

#16812623

C₂

yes

maximal, characteristic, complemented

#16812620

C₂

yes

maximal, characteristic

C₈×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₉

—

C₂×#11724784

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

SD₁₆×C₁₀₉

C₄

yes

complemented

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×C₄

yes

complemented

D₄×C₁₀₉

—

D₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

C₂×Dic₁₀₉

D₄

yes

characteristic

C₈×C₁₀₉

C₂×C₄

yes

complemented

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

SD₁₆

yes

direct product factor

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×D₄

yes

characteristic

C₂×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×C₁₀₉

#135

yes

characteristic

C₁₀₉

C₄×SD₁₆

yes

characteristic, complemented, Sylow

C₄×SD₁₆

—

109

maximal, Sylow

C₂×SD₁₆

D₁₀₉

yes

characteristic

C₄×Q₈

—

109

—

C₄×D₄

—

109

—

#106

—

109

—

#103

—

109

—

#102

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

218

—

SD₁₆

Dic₁₀₉

yes

direct product factor

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

218

—

C₂³

—

Q₈

—

Q₈

C₂×Dic₁₀₉

yes

complemented

D₄

—

D₄

C₂×Dic₁₀₉

yes

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

218

—

C₈

C₂×Dic₁₀₉

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₀₉

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

436

—

C₄

—

436

—

C₄

—

218

—

C₄

C₂²×Dic₁₀₉

yes

characteristic

C₄

#13182028

yes

characteristic

C₂

—

C₂

D₄×Dic₁₀₉

yes

characteristic

C₂

SD₁₆×D₁₀₉

yes

characteristic

C₂

#16812720

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂×SD₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₀₉
hypocenter	C₁₀₉	C₄×SD₁₆
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

C₂×SD₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₀₉

hypocenter

C₁₀₉

C₄×SD₁₆

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. Q₈×Dic₁₀₉	C₂	semidirect
3. #16812631	C₂	semidirect
4. #16812623	C₂	semidirect
6. C₈×Dic₁₀₉	C₂	semidirect
18. SD₁₆×C₁₀₉	C₄	semidirect
22. Q₈×C₁₀₉	C₂×C₄	semidirect
32. C₈×C₁₀₉	C₂×C₄	semidirect
44. Dic₁₀₉	SD₁₆	direct (N × H)
49. C₁₀₉	C₄×SD₁₆	semidirect
61. SD₁₆	Dic₁₀₉	direct (N × H)
72. Q₈	C₂×Dic₁₀₉	semidirect
83. C₈	C₂×Dic₁₀₉	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₀₉

C₂

semidirect

3. #16812631

C₂

semidirect

4. #16812623

C₂

semidirect

6. C₈×Dic₁₀₉

C₂

semidirect

18. SD₁₆×C₁₀₉

C₄

semidirect

22. Q₈×C₁₀₉

C₂×C₄

semidirect

32. C₈×C₁₀₉

C₂×C₄

semidirect

44. Dic₁₀₉

SD₁₆

direct (N × H)

49. C₁₀₉

C₄×SD₁₆

semidirect

61. SD₁₆

Dic₁₀₉

direct (N × H)

72. Q₈

C₂×Dic₁₀₉

semidirect

83. C₈

C₂×Dic₁₀₉

semidirect

Subgroups — SD₁₆×Dic₁₀₉ #20295115

Conjugacy classes of subgroups

Complements

Subgroups — SD16×Dic109 #20295115

Conjugacy classes of subgroups

Complements

Subgroups — SD₁₆×Dic₁₀₉ #20295115