Group #20295530 — subgroups

Subgroups — C₂×#16812683 #20295530

Order 6976.

342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 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×C₂²⋊D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812683	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊Dic₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₂×#13182006	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊D₁₀₉	—	2	4	no	—
9	C₂×C₂²⋊D₁₀₉	C₂²	1	4	yes	—
10	C₂²×Dic₁₀₉	—	2	4	no	—
11	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
12	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₀₉	C₂²	1	4	yes	complemented
14	C₄⋊Dic₁₀₉	C₂²	1	8	yes	complemented
15	#13182006	C₂²	1	8	yes	complemented
16	C₂⁴×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
19	C₂³×C₁₀₉	—	2	4	no	—
20	C₂³×C₁₀₉	D₄	1	4	yes	—
21	C₂³×C₁₀₉	C₂³	1	2	yes	—
22	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
23	C₂²×D₁₀₉	—	2	2	no	—
24	C₂²×D₁₀₉	—	2	4	no	—
25	C₂²×D₁₀₉	C₂³	1	2	yes	—
26	C₂×C₄×C₁₀₉	—	2	4	no	—
27	C₂×C₄×C₁₀₉	C₂³	1	4	yes	—
28	C₂²⋊D₁₀₉	—	2	8	no	—
29	C₂×Dic₁₀₉	—	2	4	no	—
30	C₂×Dic₁₀₉	—	2	8	no	—
31	C₂×Dic₁₀₉	—	2	2	no	—
32	C₂×Dic₁₀₉	C₂³	1	4	yes	—
33	C₂×Dic₁₀₉	C₂³	1	2	yes	—
34	C₂²×C₁₀₉	—	2	8	no	—
35	C₂²×C₁₀₉	—	2	4	no	—
36	C₂²×C₁₀₉	C₂×D₄	1	4	yes	—
37	C₂²×C₁₀₉	C₂×D₄	1	2	yes	—
38	C₂²×C₁₀₉	#41	1	4	yes	—
39	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
40	C₂×D₁₀₉	—	2	2	no	—
41	C₂×D₁₀₉	—	2	4	no	—
42	C₂×D₁₀₉	—	4	4	no	—
43	C₄×C₁₀₉	—	2	4	no	—
44	Dic₁₀₉	—	2	4	no	—
45	Dic₁₀₉	—	2	2	no	—
46	C₂×C₁₀₉	—	2	4	no	—
47	C₂×C₁₀₉	#123	1	4	yes	complemented
48	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
49	C₂×C₁₀₉	C₂×#41	1	2	yes	—
50	D₁₀₉	—	4	2	no	—
51	C₁₀₉	C₂×#123	1	1	yes	characteristic, complemented, Sylow
52	C₂×#123	—	109	1	no	maximal, Sylow
53	C₂²×D₄	—	109	1	no	—
54	C₂³×C₄	—	109	1	no	—
55	#123	—	109	8	no	—
56	C₂×C₄⋊C₄	—	109	2	no	—
57	C₂×C₂²⋊C₄	—	109	2	no	—
58	C₂×C₂²⋊C₄	D₁₀₉	1	1	yes	characteristic, complemented
59	C₂⁴	—	109	1	no	—
60	C₂⁴	C₂×D₁₀₉	1	1	yes	characteristic
61	C₂×D₄	—	109	4	no	—
62	C₂×D₄	—	218	4	no	—
63	C₂²×C₄	—	109	4	no	—
64	C₂²×C₄	—	109	2	no	—
65	C₂²×C₄	—	109	1	no	—
66	C₂²×C₄	—	218	4	no	—
67	C₂²×C₄	C₂×D₁₀₉	1	2	yes	—
68	C₄⋊C₄	—	109	8	no	—
69	C₂²⋊C₄	—	109	8	no	—
70	C₂²⋊C₄	C₂×D₁₀₉	1	4	yes	complemented
71	C₂³	—	2	4	no	—
72	C₂³	—	109	2	no	—
73	C₂³	—	218	2	no	—
74	C₂³	—	218	4	no	—
75	C₂³	D₄₃₆	1	4	yes	—
76	C₂³	C₂²×D₁₀₉	1	2	yes	—
77	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
78	D₄	—	218	8	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	—	109	4	no	—
81	C₂×C₄	—	109	2	no	—
82	C₂×C₄	—	218	8	no	—
83	C₂×C₄	—	218	4	no	—
84	C₂×C₄	—	218	2	no	—
85	C₂×C₄	C₂²×D₁₀₉	1	4	yes	—
86	C₂²	—	2	8	no	—
87	C₂²	—	2	4	no	—
88	C₂²	—	436	4	no	—
89	C₂²	—	218	2	no	—
90	C₂²	—	218	4	no	—
91	C₂²	C₂×D₄₃₆	1	4	yes	—
92	C₂²	C₂×D₄₃₆	1	2	yes	—
93	C₂²	#13182028	1	4	yes	—
94	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
95	C₄	—	2	4	no	—
96	C₄	—	218	4	no	—
97	C₄	—	218	2	no	—
98	C₂	—	2	4	no	—
99	C₂	—	436	2	no	—
100	C₂	#16812683	1	4	yes	direct product factor
101	C₂	C₂²×D₄₃₆	1	1	yes	characteristic
102	C₂	C₂×#13182028	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊D₁₀₉

C₂

yes

maximal, characteristic

C₂³×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812683

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, complemented

C₂×#13182006

C₂

yes

maximal, complemented

C₂×C₂²⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

—

C₂×C₂²⋊D₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

complemented

C₄⋊Dic₁₀₉

C₂²

yes

complemented

#13182006

C₂²

yes

complemented

C₂⁴×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

—

C₂²⋊C₄×C₁₀₉

C₂²

yes

complemented

C₂³×C₁₀₉

—

C₂³×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#123

yes

complemented

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

—

D₁₀₉

—

C₁₀₉

C₂×#123

yes

characteristic, complemented, Sylow

C₂×#123

—

109

maximal, Sylow

C₂²×D₄

—

109

—

C₂³×C₄

—

109

—

#123

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

D₁₀₉

yes

characteristic, complemented

C₂⁴

—

109

—

C₂⁴

C₂×D₁₀₉

yes

characteristic

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

C₂×D₁₀₉

yes

complemented

C₂³

—

C₂³

—

109

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

D₄₃₆

yes

—

C₂³

C₂²×D₁₀₉

yes

—

C₂³

C₂²×D₁₀₉

yes

characteristic

D₄

—

218

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂²

—

C₂²

—

C₂²

—

436

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×D₄₃₆

yes

—

C₂²

C₂×D₄₃₆

yes

—

C₂²

#13182028

yes

—

C₂²

C₂³×D₁₀₉

yes

characteristic

C₄

—

C₄

—

218

—

C₄

—

218

—

C₂

—

C₂

—

436

—

100

C₂

#16812683

yes

direct product factor

101

C₂

C₂²×D₄₃₆

yes

characteristic

102

C₂

C₂×#13182028

yes

—

	subgroup	G/H
center	C₂³	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂⁴
Frattini	C₂²	C₂³×D₁₀₉
Fitting	C₂×C₂²⋊C₄×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	C₂×C₂²⋊C₄	D₁₀₉
hypocenter	C₁₀₉	C₂×#123
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center

C₂³

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂⁴

Frattini

C₂²

C₂³×D₁₀₉

Fitting

C₂×C₂²⋊C₄×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂³

hypercenter

C₂×C₂²⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂³×Dic₁₀₉	C₂	semidirect
3. #16812683	C₂	direct (N × H)
4. C₂×C₄⋊Dic₁₀₉	C₂	semidirect
5. C₂×#13182006	C₂	semidirect
6. C₂×C₂²⋊C₄×C₁₀₉	C₂	semidirect
13. C₂²×Dic₁₀₉	C₂²	semidirect
14. C₄⋊Dic₁₀₉	C₂²	semidirect
15. #13182006	C₂²	semidirect
18. C₂²⋊C₄×C₁₀₉	C₂²	semidirect
47. C₂×C₁₀₉	#123	semidirect
51. C₁₀₉	C₂×#123	semidirect
58. C₂×C₂²⋊C₄	D₁₀₉	semidirect
70. C₂²⋊C₄	C₂×D₁₀₉	semidirect
100. C₂	#16812683	direct (N × H)

H ≅ G/N

type

2. C₂³×Dic₁₀₉

C₂

semidirect

3. #16812683

C₂

direct (N × H)

4. C₂×C₄⋊Dic₁₀₉

C₂

semidirect

5. C₂×#13182006

C₂

semidirect

6. C₂×C₂²⋊C₄×C₁₀₉

C₂

semidirect

13. C₂²×Dic₁₀₉

C₂²

semidirect

14. C₄⋊Dic₁₀₉

C₂²

semidirect

15. #13182006

C₂²

semidirect

18. C₂²⋊C₄×C₁₀₉

C₂²

semidirect

47. C₂×C₁₀₉

#123

semidirect

51. C₁₀₉

C₂×#123

semidirect

58. C₂×C₂²⋊C₄

D₁₀₉

semidirect

70. C₂²⋊C₄

C₂×D₁₀₉

semidirect

100. C₂

#16812683

direct (N × H)

Subgroups — C₂×#16812683 #20295530

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16812683 #20295530

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16812683 #20295530