Group #20295767 — subgroups

Subgroups — #20295767

Order 6976.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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	#16812696	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812693	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812684	C₂	1	3	yes	maximal, complemented
4	#16812674	C₂	1	3	yes	maximal, complemented
5	#16812672	C₂	1	3	yes	maximal, complemented
6	#16812669	C₂	1	3	yes	maximal, complemented
7	#126×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×D₄₃₆	C₂²	1	3	yes	—
10	C₂×D₄₃₆	C₂²	1	3	yes	—
11	C₂×D₄₃₆	C₂²	1	3	yes	complemented
12	C₂×C₂²⋊D₁₀₉	C₂²	1	3	yes	—
13	C₂×C₄×D₁₀₉	C₂²	1	3	yes	complemented
14	#13182008	C₂²	1	3	yes	complemented
15	#13182006	C₂²	1	3	yes	—
16	#13182006	C₂²	1	3	yes	—
17	#13182006	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₁₀₉	C₂²	1	3	yes	—
20	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
21	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂²×D₁₀₉	—	4	3	no	—
23	C₂²×D₁₀₉	C₂³	1	1	yes	characteristic
24	C₂²×D₁₀₉	C₂³	1	1	yes	characteristic
25	C₂²×D₁₀₉	C₂³	1	3	yes	—
26	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
27	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
28	C₂²⋊D₁₀₉	—	4	3	no	—
29	C₂×Dic₁₀₉	C₂³	1	3	yes	—
30	D₄₃₆	—	4	3	no	—
31	D₄₃₆	—	4	3	no	—
32	D₄₃₆	—	4	3	no	—
33	C₄×D₁₀₉	—	4	3	no	—
34	C₂²×C₁₀₉	—	2	3	no	—
35	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
36	C₂×D₁₀₉	—	2	3	no	—
37	C₂×D₁₀₉	—	2	3	no	—
38	C₂×D₁₀₉	—	2	3	no	—
39	C₂×D₁₀₉	—	2	3	no	—
40	C₂×D₁₀₉	—	2	3	no	—
41	C₂×D₁₀₉	—	4	4	no	—
42	C₄×C₁₀₉	—	2	3	no	—
43	C₄×C₁₀₉	—	2	3	no	—
44	Dic₁₀₉	—	2	3	no	—
45	C₂×C₁₀₉	—	4	1	no	—
46	C₂×C₁₀₉	H(5,𝔽₂)	1	3	yes	—
47	D₁₀₉	—	4	3	no	—
48	D₁₀₉	—	4	1	no	—
49	D₁₀₉	—	4	1	no	—
50	C₁₀₉	#560	1	1	yes	characteristic, complemented, Sylow
51	#560	—	109	1	no	maximal, Sylow
52	#127	—	109	1	no	—
53	#126	—	109	1	no	—
54	#126	D₁₀₉	1	1	yes	characteristic, complemented
55	#123	—	109	3	no	—
56	#121	—	109	3	no	—
57	#121	—	109	3	no	—
58	#120	—	109	3	no	—
59	C₂⁴	—	109	1	no	—
60	C₂×D₄	—	109	3	no	—
61	C₂×D₄	—	109	3	no	—
62	C₂×D₄	—	109	3	no	—
63	C₂×D₄	—	109	3	no	—
64	C₂²×C₄	—	109	3	no	—
65	C₄⋊C₄	—	109	3	no	—
66	C₄⋊C₄	C₂×D₁₀₉	1	3	yes	complemented
67	C₂²⋊C₄	—	109	3	no	—
68	C₂²⋊C₄	—	109	3	no	—
69	C₂²⋊C₄	—	109	3	no	—
70	C₂²⋊C₄	C₂×D₁₀₉	1	3	yes	—
71	C₄²	C₂×D₁₀₉	1	1	yes	characteristic, complemented
72	C₂³	—	109	1	no	—
73	C₂³	—	109	1	no	—
74	C₂³	—	109	3	no	—
75	C₂³	—	436	3	no	—
76	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
77	D₄	—	436	3	no	—
78	D₄	—	436	3	no	—
79	D₄	—	436	3	no	—
80	D₄	—	436	3	no	—
81	C₂×C₄	—	109	3	no	—
82	C₂×C₄	—	436	3	no	—
83	C₂×C₄	C₂²×D₁₀₉	1	3	yes	—
84	C₂×C₄	C₂²×D₁₀₉	1	3	yes	—
85	C₂²	—	2	3	no	—
86	C₂²	—	436	4	no	—
87	C₂²	—	218	3	no	—
88	C₂²	—	218	3	no	—
89	C₂²	—	218	3	no	—
90	C₂²	—	218	3	no	—
91	C₂²	—	218	3	no	—
92	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
93	C₄	—	2	3	no	—
94	C₄	—	2	3	no	—
95	C₄	—	218	3	no	—
96	C₂	—	4	1	no	—
97	C₂	—	436	1	no	—
98	C₂	—	436	1	no	—
99	C₂	—	436	3	no	—
100	C₂	#16812745	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16812696

C₂

yes

maximal, characteristic, complemented

#16812693

C₂

yes

maximal, characteristic, complemented

#16812684

C₂

yes

maximal, complemented

#16812674

C₂

yes

maximal, complemented

#16812672

C₂

yes

maximal, complemented

#16812669

C₂

yes

maximal, complemented

#126×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₂²

yes

characteristic

C₂×D₄₃₆

C₂²

yes

—

C₂×D₄₃₆

C₂²

yes

—

C₂×D₄₃₆

C₂²

yes

complemented

C₂×C₂²⋊D₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

complemented

#13182008

C₂²

yes

complemented

#13182006

C₂²

yes

—

#13182006

C₂²

yes

—

#13182006

C₂²

yes

complemented

C₄⋊C₄×C₁₀₉

C₂²

yes

complemented

C₂²⋊C₄×C₁₀₉

C₂²

yes

—

C₄²×C₁₀₉

C₂²

yes

characteristic, complemented

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

C₂³

yes

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

H(5,𝔽₂)

yes

—

D₁₀₉

—

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#560

yes

characteristic, complemented, Sylow

#560

—

109

maximal, Sylow

#127

—

109

—

#126

—

109

—

#126

D₁₀₉

yes

characteristic, complemented

#123

—

109

—

#121

—

109

—

#121

—

109

—

#120

—

109

—

C₂⁴

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

complemented

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

C₂×D₁₀₉

yes

—

C₄²

C₂×D₁₀₉

yes

characteristic, complemented

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

436

—

C₂³

C₂²×D₁₀₉

yes

characteristic

D₄

—

436

—

D₄

—

436

—

D₄

—

436

—

D₄

—

436

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂²

—

C₂²

—

436

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂³×D₁₀₉

yes

characteristic

C₄

—

C₄

—

C₄

—

218

—

C₂

—

C₂

—

436

—

C₂

—

436

—

C₂

—

436

—

100

C₂

#16812745

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂⁴
Fitting	#126×C₁₀₉	C₂
hypercenter	#126	D₁₀₉
hypocenter	C₁₀₉	#560
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #126

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂⁴

Fitting

#126×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ #126

N	H ≅ G/N	type
1. #16812696	C₂	semidirect
2. #16812693	C₂	semidirect
3. #16812684	C₂	semidirect
4. #16812674	C₂	semidirect
5. #16812672	C₂	semidirect
6. #16812669	C₂	semidirect
7. #126×C₁₀₉	C₂	semidirect
11. C₂×D₄₃₆	C₂²	semidirect
13. C₂×C₄×D₁₀₉	C₂²	semidirect
14. #13182008	C₂²	semidirect
17. #13182006	C₂²	semidirect
18. C₄⋊C₄×C₁₀₉	C₂²	semidirect
20. C₄²×C₁₀₉	C₂²	semidirect
50. C₁₀₉	#560	semidirect
54. #126	D₁₀₉	semidirect
66. C₄⋊C₄	C₂×D₁₀₉	semidirect
71. C₄²	C₂×D₁₀₉	semidirect

H ≅ G/N

type

1. #16812696

C₂

semidirect

2. #16812693

C₂

semidirect

3. #16812684

C₂

semidirect

4. #16812674

C₂

semidirect

5. #16812672

C₂

semidirect

6. #16812669

C₂

semidirect

7. #126×C₁₀₉

C₂

semidirect

11. C₂×D₄₃₆

C₂²

semidirect

13. C₂×C₄×D₁₀₉

C₂²

semidirect

14. #13182008

C₂²

semidirect

17. #13182006

C₂²

semidirect

18. C₄⋊C₄×C₁₀₉

C₂²

semidirect

20. C₄²×C₁₀₉

C₂²

semidirect

50. C₁₀₉

#560

semidirect

54. #126

D₁₀₉

semidirect

66. C₄⋊C₄

C₂×D₁₀₉

semidirect

71. C₄²

C₂×D₁₀₉

semidirect

Subgroups — #20295767

Conjugacy classes of subgroups

Complements