Group #20295487 — subgroups

Subgroups — C₂×D₄×Dic₁₀₉ #20295487

Order 6976.

426 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 215 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	C₂³×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₁₀₉	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊Dic₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₉	—	2	8	no	—
8	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
10	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₀₉	C₂²	1	8	yes	complemented
12	C₄⋊Dic₁₀₉	C₂²	1	4	yes	complemented
13	C₂²⋊Dic₁₀₉	C₂²	1	8	yes	complemented
14	C₄×Dic₁₀₉	C₂²	1	4	yes	complemented
15	C₂⁴×C₁₀₉	C₂²	1	2	yes	—
16	C₂×D₄×C₁₀₉	C₄	1	8	yes	complemented
17	C₂×D₄×C₁₀₉	C₂²	1	4	yes	—
18	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂³×C₁₀₉	—	2	4	no	—
20	C₂³×C₁₀₉	—	2	4	no	—
21	C₂³×C₁₀₉	C₂×C₄	1	8	yes	complemented
22	C₂³×C₁₀₉	C₂³	1	4	yes	—
23	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
24	D₄×C₁₀₉	C₂×C₄	1	16	yes	complemented
25	C₂×C₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
26	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
27	C₂×Dic₁₀₉	—	2	4	no	—
28	C₂×Dic₁₀₉	—	2	2	no	—
29	C₂×Dic₁₀₉	—	2	16	no	—
30	C₂×Dic₁₀₉	D₄	1	4	yes	direct product factor
31	C₂×Dic₁₀₉	C₂³	1	2	yes	—
32	C₂×Dic₁₀₉	C₂³	1	4	yes	—
33	C₂×Dic₁₀₉	C₂³	1	2	yes	—
34	C₂²×C₁₀₉	—	2	16	no	—
35	C₂²×C₁₀₉	—	2	4	no	—
36	C₂²×C₁₀₉	—	2	4	no	—
37	C₂²×C₁₀₉	C₂²×C₄	1	2	yes	—
38	C₂²×C₁₀₉	C₂²×C₄	1	8	yes	complemented
39	C₂²×C₁₀₉	C₂×D₄	1	2	yes	—
40	C₂²×C₁₀₉	#41	1	2	yes	—
41	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
42	C₄×C₁₀₉	C₂²×C₄	1	4	yes	complemented
43	Dic₁₀₉	—	2	4	no	—
44	Dic₁₀₉	—	2	2	no	—
45	Dic₁₀₉	C₂×D₄	1	4	yes	direct product factor
46	C₂×C₁₀₉	—	2	8	no	—
47	C₂×C₁₀₉	C₄×D₄	1	4	yes	complemented
48	C₂×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
49	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
51	C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₄×D₄	—	109	1	no	maximal, Sylow
53	C₂²×D₄	D₁₀₉	1	1	yes	characteristic
54	C₂³×C₄	—	109	2	no	—
55	C₄×D₄	—	109	8	no	—
56	C₂×C₄⋊C₄	—	109	1	no	—
57	C₂×C₂²⋊C₄	—	109	2	no	—
58	C₂×C₄²	—	109	1	no	—
59	C₂⁴	C₂×D₁₀₉	1	2	yes	—
60	C₂×D₄	Dic₁₀₉	1	8	yes	direct product factor
61	C₂×D₄	C₂×D₁₀₉	1	4	yes	—
62	C₂²×C₄	—	109	1	no	—
63	C₂²×C₄	—	109	8	no	—
64	C₂²×C₄	—	109	1	no	—
65	C₂²×C₄	—	109	2	no	—
66	C₂²×C₄	—	218	8	no	—
67	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
68	C₄⋊C₄	—	109	4	no	—
69	C₂²⋊C₄	—	109	8	no	—
70	C₄²	—	109	4	no	—
71	C₂³	—	2	4	no	—
72	C₂³	—	2	4	no	—
73	C₂³	C₂×Dic₁₀₉	1	8	yes	complemented
74	C₂³	C₂²×D₁₀₉	1	4	yes	—
75	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
76	D₄	C₂×Dic₁₀₉	1	16	yes	direct product factor
77	C₂×C₄	—	109	4	no	—
78	C₂×C₄	—	109	2	no	—
79	C₂×C₄	—	109	2	no	—
80	C₂×C₄	—	109	4	no	—
81	C₂×C₄	—	218	16	no	—
82	C₂×C₄	—	218	4	no	—
83	C₂×C₄	—	218	2	no	—
84	C₂×C₄	C₂×Dic₁₀₉	1	4	yes	complemented
85	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
86	C₂²	—	2	16	no	—
87	C₂²	—	2	4	no	—
88	C₂²	—	2	4	no	—
89	C₂²	C₂²×Dic₁₀₉	1	2	yes	—
90	C₂²	C₂²×Dic₁₀₉	1	8	yes	complemented
91	C₂²	D₄×D₁₀₉	1	2	yes	—
92	C₂²	#13182028	1	2	yes	—
93	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
94	C₄	—	109	4	no	—
95	C₄	—	218	4	no	—
96	C₄	—	218	2	no	—
97	C₄	C₂²×Dic₁₀₉	1	4	yes	complemented
98	C₂	—	2	8	no	—
99	C₂	D₄×Dic₁₀₉	1	4	yes	direct product factor
100	C₂	C₂³×Dic₁₀₉	1	1	yes	characteristic
101	C₂	C₂×D₄×D₁₀₉	1	1	yes	characteristic
102	C₂	C₂×#13182028	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₁₀₉

C₂

yes

maximal, complemented

D₄×Dic₁₀₉

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₀₉

C₂

yes

maximal, complemented

C₂×C₄×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

complemented

C₄⋊Dic₁₀₉

C₂²

yes

complemented

C₂²⋊Dic₁₀₉

C₂²

yes

complemented

C₄×Dic₁₀₉

C₂²

yes

complemented

C₂⁴×C₁₀₉

C₂²

yes

—

C₂×D₄×C₁₀₉

C₄

yes

complemented

C₂×D₄×C₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×C₄

yes

complemented

C₂³×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂³

yes

characteristic

D₄×C₁₀₉

C₂×C₄

yes

complemented

C₂×C₄×C₁₀₉

C₂×C₄

yes

complemented

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

direct product factor

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×C₄

yes

—

C₂²×C₁₀₉

C₂²×C₄

yes

complemented

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₄×C₁₀₉

C₂²×C₄

yes

complemented

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×D₄

yes

direct product factor

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×D₄

yes

complemented

C₂×C₁₀₉

C₂³×C₄

yes

characteristic

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

characteristic

C₁₀₉

C₂×C₄×D₄

yes

characteristic, complemented, Sylow

C₂×C₄×D₄

—

109

maximal, Sylow

C₂²×D₄

D₁₀₉

yes

characteristic

C₂³×C₄

—

109

—

C₄×D₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂⁴

C₂×D₁₀₉

yes

—

C₂×D₄

Dic₁₀₉

yes

direct product factor

C₂×D₄

C₂×D₁₀₉

yes

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₁₀₉

yes

complemented

C₂³

C₂²×D₁₀₉

yes

—

C₂³

C₂²×D₁₀₉

yes

characteristic

D₄

C₂×Dic₁₀₉

yes

direct product factor

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×Dic₁₀₉

yes

complemented

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×Dic₁₀₉

yes

—

C₂²

C₂²×Dic₁₀₉

yes

complemented

C₂²

D₄×D₁₀₉

yes

—

C₂²

#13182028

yes

—

C₂²

C₂³×D₁₀₉

yes

characteristic

C₄

—

109

—

C₄

—

218

—

C₄

—

218

—

C₄

C₂²×Dic₁₀₉

yes

complemented

C₂

—

C₂

D₄×Dic₁₀₉

yes

direct product factor

100

C₂

C₂³×Dic₁₀₉

yes

characteristic

101

C₂

C₂×D₄×D₁₀₉

yes

characteristic

102

C₂

C₂×#13182028

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂³×C₄
Frattini	C₂²	C₂³×D₁₀₉
Fitting	C₂²×D₄×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	C₂²×D₄	D₁₀₉
hypocenter	C₁₀₉	C₂×C₄×D₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center

C₂³

C₂²×D₁₀₉

commutator

C₂×C₁₀₉

C₂³×C₄

Frattini

C₂²

C₂³×D₁₀₉

Fitting

C₂²×D₄×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂³

hypercenter

C₂²×D₄

D₁₀₉

hypocenter

C₁₀₉

C₂×C₄×D₄

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂³×Dic₁₀₉	C₂	semidirect
2. D₄×Dic₁₀₉	C₂	direct (N × H)
3. C₂×C₄⋊Dic₁₀₉	C₂	semidirect
4. C₂×C₂²⋊Dic₁₀₉	C₂	semidirect
5. C₂×C₄×Dic₁₀₉	C₂	semidirect
11. C₂²×Dic₁₀₉	C₂²	semidirect
12. C₄⋊Dic₁₀₉	C₂²	semidirect
13. C₂²⋊Dic₁₀₉	C₂²	semidirect
14. C₄×Dic₁₀₉	C₂²	semidirect
16. C₂×D₄×C₁₀₉	C₄	semidirect
21. C₂³×C₁₀₉	C₂×C₄	semidirect
24. D₄×C₁₀₉	C₂×C₄	semidirect
25. C₂×C₄×C₁₀₉	C₂×C₄	semidirect
30. C₂×Dic₁₀₉	D₄	direct (N × H)
38. C₂²×C₁₀₉	C₂²×C₄	semidirect
42. C₄×C₁₀₉	C₂²×C₄	semidirect
45. Dic₁₀₉	C₂×D₄	direct (N × H)
47. C₂×C₁₀₉	C₄×D₄	semidirect
51. C₁₀₉	C₂×C₄×D₄	semidirect
60. C₂×D₄	Dic₁₀₉	direct (N × H)
73. C₂³	C₂×Dic₁₀₉	semidirect
76. D₄	C₂×Dic₁₀₉	direct (N × H)
84. C₂×C₄	C₂×Dic₁₀₉	semidirect
90. C₂²	C₂²×Dic₁₀₉	semidirect
97. C₄	C₂²×Dic₁₀₉	semidirect
99. C₂	D₄×Dic₁₀₉	direct (N × H)

H ≅ G/N

type

1. C₂³×Dic₁₀₉

C₂

semidirect

2. D₄×Dic₁₀₉

C₂

direct (N × H)

3. C₂×C₄⋊Dic₁₀₉

C₂

semidirect

4. C₂×C₂²⋊Dic₁₀₉

C₂

semidirect

5. C₂×C₄×Dic₁₀₉

C₂

semidirect

11. C₂²×Dic₁₀₉

C₂²

semidirect

12. C₄⋊Dic₁₀₉

C₂²

semidirect

13. C₂²⋊Dic₁₀₉

C₂²

semidirect

14. C₄×Dic₁₀₉

C₂²

semidirect

16. C₂×D₄×C₁₀₉

C₄

semidirect

21. C₂³×C₁₀₉

C₂×C₄

semidirect

24. D₄×C₁₀₉

C₂×C₄

semidirect

25. C₂×C₄×C₁₀₉

C₂×C₄

semidirect

30. C₂×Dic₁₀₉

D₄

direct (N × H)

38. C₂²×C₁₀₉

C₂²×C₄

semidirect

42. C₄×C₁₀₉

C₂²×C₄

semidirect

45. Dic₁₀₉

C₂×D₄

direct (N × H)

47. C₂×C₁₀₉

C₄×D₄

semidirect

51. C₁₀₉

C₂×C₄×D₄

semidirect

60. C₂×D₄

Dic₁₀₉

direct (N × H)

73. C₂³

C₂×Dic₁₀₉

semidirect

76. D₄

C₂×Dic₁₀₉

direct (N × H)

84. C₂×C₄

C₂×Dic₁₀₉

semidirect

90. C₂²

C₂²×Dic₁₀₉

semidirect

97. C₄

C₂²×Dic₁₀₉

semidirect

99. C₂

D₄×Dic₁₀₉

direct (N × H)

Subgroups — C₂×D₄×Dic₁₀₉ #20295487

Conjugacy classes of subgroups

Complements

Subgroups — C2×D4×Dic109 #20295487

Conjugacy classes of subgroups

Complements

Subgroups — C₂×D₄×Dic₁₀₉ #20295487