Group #20295481 — subgroups

Subgroups — C₂×#16812657 #20295481

Order 6976.

330 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 167 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	C₂²×C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#16812657	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13182006	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
9	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₀₉	C₄	1	8	yes	complemented
11	C₂×C₄×D₁₀₉	C₂²	1	4	yes	—
12	C₄⋊Dic₁₀₉	C₂²	1	4	yes	complemented
13	#13182006	C₂²	1	8	yes	—
14	C₄×Dic₁₀₉	C₂²	1	8	yes	complemented
15	C₂²×C₄×C₁₀₉	C₂²	1	2	yes	—
16	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
18	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂²×D₁₀₉	—	2	2	no	—
20	C₂²×D₁₀₉	—	2	2	no	—
21	C₂²×D₁₀₉	C₂×C₄	1	4	yes	—
22	C₂²×D₁₀₉	C₂³	1	2	yes	—
23	C₂×C₄×C₁₀₉	—	2	4	no	—
24	C₂×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
25	C₂×C₄×C₁₀₉	C₂³	1	4	yes	—
26	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
27	C₂×Dic₁₀₉	—	2	4	no	—
28	C₂×Dic₁₀₉	C₂×C₄	1	4	yes	—
29	C₂×Dic₁₀₉	C₂³	1	4	yes	—
30	C₂×Dic₁₀₉	C₂³	1	2	yes	—
31	C₄×D₁₀₉	C₂×C₄	1	16	yes	complemented
32	C₂²×C₁₀₉	C₂²×C₄	1	2	yes	—
33	C₂²×C₁₀₉	#41	1	2	yes	—
34	C₂²×C₁₀₉	#41	1	2	yes	—
35	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
36	C₂×D₁₀₉	—	2	8	no	—
37	C₂×D₁₀₉	—	2	2	no	—
38	C₂×D₁₀₉	—	2	2	no	—
39	C₂×D₁₀₉	C₂²×C₄	1	4	yes	—
40	C₄×C₁₀₉	—	2	4	no	—
41	C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
42	Dic₁₀₉	—	2	4	no	—
43	Dic₁₀₉	C₂²×C₄	1	4	yes	—
44	C₂×C₁₀₉	#117	1	4	yes	complemented
45	C₂×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
46	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
47	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
48	D₁₀₉	—	2	4	no	—
49	C₁₀₉	C₂×#117	1	1	yes	characteristic, complemented, Sylow
50	C₂×#117	—	109	1	no	maximal, Sylow
51	C₂³×C₄	—	109	1	no	—
52	#117	—	109	8	no	—
53	C₂×C₄⋊C₄	—	109	1	no	—
54	C₂×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic, complemented
55	C₂×C₂²⋊C₄	—	109	2	no	—
56	C₂×C₄²	—	109	2	no	—
57	C₂⁴	—	109	1	no	—
58	C₂²×C₄	—	109	1	no	—
59	C₂²×C₄	—	109	8	no	—
60	C₂²×C₄	—	109	4	no	—
61	C₂²×C₄	—	109	2	no	—
62	C₂²×C₄	C₂×D₁₀₉	1	2	yes	—
63	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
64	C₄⋊C₄	—	109	4	no	—
65	C₄⋊C₄	C₂×D₁₀₉	1	4	yes	complemented
66	C₂²⋊C₄	—	109	8	no	—
67	C₄²	—	109	8	no	—
68	C₂³	—	109	4	no	—
69	C₂³	—	109	2	no	—
70	C₂³	—	218	2	no	—
71	C₂³	—	218	2	no	—
72	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	109	4	no	—
75	C₂×C₄	—	109	2	no	—
76	C₂×C₄	—	109	16	no	—
77	C₂×C₄	—	109	4	no	—
78	C₂×C₄	—	218	4	no	—
79	C₂×C₄	C₄×D₁₀₉	1	4	yes	—
80	C₂×C₄	C₂²×D₁₀₉	1	4	yes	—
81	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
82	C₂²	—	109	4	no	—
83	C₂²	—	218	8	no	—
84	C₂²	—	218	2	no	—
85	C₂²	—	218	2	no	—
86	C₂²	C₂×C₄×D₁₀₉	1	2	yes	—
87	C₂²	#13182028	1	2	yes	—
88	C₂²	#13182030	1	2	yes	—
89	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	109	4	no	—
92	C₄	—	218	4	no	—
93	C₄	C₂×C₄×D₁₀₉	1	4	yes	—
94	C₂	—	218	4	no	—
95	C₂	#16812657	1	4	yes	direct product factor
96	C₂	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
97	C₂	C₂×#13182028	1	1	yes	characteristic
98	C₂	C₂×#13182030	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₀₉

C₂

yes

maximal, characteristic

#16812657

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182006

C₂

yes

maximal

C₂×C₄×Dic₁₀₉

C₂

yes

maximal, complemented

C₂×C₄⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₄

yes

complemented

C₂×C₄×D₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

complemented

#13182006

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

complemented

C₂²×C₄×C₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

complemented

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₄

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₄×D₁₀₉

C₂×C₄

yes

complemented

C₂²×C₁₀₉

C₂²×C₄

yes

—

C₂²×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²×C₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

C₂²×C₄

yes

—

C₂×C₁₀₉

#117

yes

complemented

C₂×C₁₀₉

C₂³×C₄

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

characteristic

D₁₀₉

—

C₁₀₉

C₂×#117

yes

characteristic, complemented, Sylow

C₂×#117

—

109

maximal, Sylow

C₂³×C₄

—

109

—

#117

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

D₁₀₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂⁴

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

complemented

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₄×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×C₄×D₁₀₉

yes

—

C₂²

#13182028

yes

—

C₂²

#13182030

yes

—

C₂²

C₂³×D₁₀₉

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

218

—

C₄

C₂×C₄×D₁₀₉

yes

—

C₂

—

218

—

C₂

#16812657

yes

direct product factor

C₂

C₂²×C₄×D₁₀₉

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂³	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂³×C₄
Frattini	C₂²	C₂³×D₁₀₉
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	C₂×#117
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²×D₁₀₉

commutator

C₂×C₁₀₉

C₂³×C₄

Frattini

C₂²

C₂³×D₁₀₉

Fitting

C₂×C₄⋊C₄×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
2. #16812657	C₂	direct (N × H)
3. C₂×C₄⋊Dic₁₀₉	C₂	semidirect
5. C₂×C₄×Dic₁₀₉	C₂	semidirect
6. C₂×C₄⋊C₄×C₁₀₉	C₂	semidirect
10. C₂×C₄×D₁₀₉	C₄	semidirect
12. C₄⋊Dic₁₀₉	C₂²	semidirect
14. C₄×Dic₁₀₉	C₂²	semidirect
17. C₄⋊C₄×C₁₀₉	C₂²	semidirect
31. C₄×D₁₀₉	C₂×C₄	semidirect
44. C₂×C₁₀₉	#117	semidirect
49. C₁₀₉	C₂×#117	semidirect
54. C₂×C₄⋊C₄	D₁₀₉	semidirect
65. C₄⋊C₄	C₂×D₁₀₉	semidirect
95. C₂	#16812657	direct (N × H)

H ≅ G/N

type

2. #16812657

C₂

direct (N × H)

3. C₂×C₄⋊Dic₁₀₉

C₂

semidirect

5. C₂×C₄×Dic₁₀₉

C₂

semidirect

6. C₂×C₄⋊C₄×C₁₀₉

C₂

semidirect

10. C₂×C₄×D₁₀₉

C₄

semidirect

12. C₄⋊Dic₁₀₉

C₂²

semidirect

14. C₄×Dic₁₀₉

C₂²

semidirect

17. C₄⋊C₄×C₁₀₉

C₂²

semidirect

31. C₄×D₁₀₉

C₂×C₄

semidirect

44. C₂×C₁₀₉

#117

semidirect

49. C₁₀₉

C₂×#117

semidirect

54. C₂×C₄⋊C₄

D₁₀₉

semidirect

65. C₄⋊C₄

C₂×D₁₀₉

semidirect

95. C₂

#16812657

direct (N × H)

Subgroups — C₂×#16812657 #20295481

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16812657 #20295481

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16812657 #20295481