Group #20295768 — subgroups

Subgroups — #20295768

Order 6976.

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

normal subgroups only

This group has 260 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	#16812688	C₂	1	2	yes	maximal, complemented
2	#16812687	C₂	1	2	yes	maximal, complemented
3	#16812685	C₂	1	4	yes	maximal, complemented
4	#16812669	C₂	1	4	yes	maximal, complemented
5	#16812667	C₂	1	2	yes	maximal, complemented
6	#124×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₀₉	C₂²	1	2	yes	complemented
8	C₂×Dic₂₁₈	C₂²	1	2	yes	complemented
9	C₂×D₄₃₆	C₂²	1	2	yes	—
10	C₂×D₄₃₆	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₁₀₉	C₂²	1	4	yes	—
12	C₂²⋊Dic₁₀₉	C₂²	1	2	yes	complemented
13	#13182006	C₂²	1	4	yes	complemented
14	#13182006	C₂²	1	4	yes	complemented
15	#13182006	C₂²	1	4	yes	complemented
16	C₄×Dic₁₀₉	C₂²	1	2	yes	complemented
17	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
18	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
20	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
21	C₂³×C₁₀₉	C₂³	1	2	yes	—
22	C₂²×D₁₀₉	—	4	2	no	—
23	C₂²×D₁₀₉	—	4	4	no	—
24	C₂²×D₁₀₉	C₂³	1	4	yes	—
25	Q₈×C₁₀₉	—	4	1	no	—
26	D₄×C₁₀₉	—	4	1	no	—
27	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
28	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
30	C₂²⋊D₁₀₉	—	4	4	no	—
31	C₂×Dic₁₀₉	C₂³	1	4	yes	—
32	D₄₃₆	—	4	2	no	—
33	D₄₃₆	—	4	2	no	—
34	Dic₂₁₈	—	4	2	no	—
35	C₂²×C₁₀₉	—	2	4	no	—
36	C₂²×C₁₀₉	—	2	2	no	—
37	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₀₉	—	2	4	no	—
39	C₂×D₁₀₉	—	2	4	no	—
40	C₂×D₁₀₉	—	2	4	no	—
41	C₂×D₁₀₉	—	4	8	no	—
42	C₄×C₁₀₉	—	2	2	no	—
43	C₄×C₁₀₉	—	2	2	no	—
44	C₄×C₁₀₉	—	2	1	no	—
45	Dic₁₀₉	—	2	4	no	—
46	C₂×C₁₀₉	—	4	2	no	—
47	C₂×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
48	C₂×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
49	D₁₀₉	—	4	4	no	—
50	C₁₀₉	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
51	H(3,𝔽₄)	—	109	1	no	maximal, Sylow
52	#124	—	109	2	no	—
53	#124	—	109	4	no	—
54	#124	—	109	2	no	—
55	#124	D₁₀₉	1	1	yes	characteristic, complemented
56	#120	—	109	2	no	—
57	#120	—	109	4	no	—
58	C₂⁴	—	109	2	no	—
59	C₂×Q₈	—	109	2	no	—
60	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
61	C₂×D₄	—	109	4	no	—
62	C₂×D₄	—	109	2	no	—
63	C₂×D₄	—	109	2	no	—
64	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
65	C₂²⋊C₄	—	109	2	no	—
66	C₂²⋊C₄	—	109	4	no	—
67	C₂²⋊C₄	—	109	4	no	—
68	C₂²⋊C₄	—	109	4	no	—
69	C₂²⋊C₄	C₂×D₁₀₉	1	4	yes	complemented
70	C₄²	—	109	2	no	—
71	C₄²	C₂×D₁₀₉	1	1	yes	characteristic, complemented
72	C₂³	—	109	4	no	—
73	C₂³	—	436	2	no	—
74	C₂³	—	436	4	no	—
75	C₂³	C₂²×D₁₀₉	1	2	yes	—
76	Q₈	—	4	1	no	—
77	Q₈	—	436	2	no	—
78	D₄	—	4	1	no	—
79	D₄	—	436	2	no	—
80	D₄	—	436	2	no	—
81	D₄	—	436	4	no	—
82	C₂×C₄	—	109	4	no	—
83	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
84	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
85	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	4	no	—
88	C₂²	—	436	8	no	—
89	C₂²	—	218	4	no	—
90	C₂²	—	218	4	no	—
91	C₂²	—	218	4	no	—
92	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	218	4	no	—
97	C₂	—	4	2	no	—
98	C₂	—	436	4	no	—
99	C₂	#16812744	1	1	yes	characteristic
100	C₂	#16812745	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16812688

C₂

yes

maximal, complemented

#16812687

C₂

yes

maximal, complemented

#16812685

C₂

yes

maximal, complemented

#16812669

C₂

yes

maximal, complemented

#16812667

C₂

yes

maximal, complemented

#124×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₂²

yes

complemented

C₂×Dic₂₁₈

C₂²

yes

complemented

C₂×D₄₃₆

C₂²

yes

—

C₂×D₄₃₆

C₂²

yes

—

C₂×C₂²⋊D₁₀₉

C₂²

yes

—

C₂²⋊Dic₁₀₉

C₂²

yes

complemented

#13182006

C₂²

yes

complemented

#13182006

C₂²

yes

complemented

#13182006

C₂²

yes

complemented

C₄×Dic₁₀₉

C₂²

yes

complemented

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic, complemented

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₉

C₂²

yes

complemented

C₄²×C₁₀₉

C₂²

yes

characteristic, complemented

C₂³×C₁₀₉

C₂³

yes

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂³

yes

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

C₂³

yes

—

D₄₃₆

—

D₄₃₆

—

Dic₂₁₈

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₉

H(5,𝔽₂)

yes

—

D₁₀₉

—

C₁₀₉

H(3,𝔽₄)

yes

characteristic, complemented, Sylow

H(3,𝔽₄)

—

109

maximal, Sylow

#124

—

109

—

#124

—

109

—

#124

—

109

—

#124

D₁₀₉

yes

characteristic, complemented

#120

—

109

—

#120

—

109

—

C₂⁴

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic, complemented

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

C₂×D₁₀₉

yes

complemented

C₄²

—

109

—

C₄²

C₂×D₁₀₉

yes

characteristic, complemented

C₂³

—

109

—

C₂³

—

436

—

C₂³

—

436

—

C₂³

C₂²×D₁₀₉

yes

—

Q₈

—

Q₈

—

436

—

D₄

—

D₄

—

436

—

D₄

—

436

—

D₄

—

436

—

C₂×C₄

—

109

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂²

—

C₂²

—

C₂²

—

436

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂³×D₁₀₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

218

—

C₂

—

C₂

—

436

—

C₂

#16812744

yes

characteristic

100

C₂

#16812745

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂⁴
Fitting	#124×C₁₀₉	C₂
hypercenter	#124	D₁₀₉
hypocenter	C₁₀₉	H(3,𝔽₄)
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂⁴

Fitting

#124×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #16812688	C₂	semidirect
2. #16812687	C₂	semidirect
3. #16812685	C₂	semidirect
4. #16812669	C₂	semidirect
5. #16812667	C₂	semidirect
6. #124×C₁₀₉	C₂	semidirect
7. C₂³×D₁₀₉	C₂²	semidirect
8. C₂×Dic₂₁₈	C₂²	semidirect
12. C₂²⋊Dic₁₀₉	C₂²	semidirect
13. #13182006	C₂²	semidirect
14. #13182006	C₂²	semidirect
15. #13182006	C₂²	semidirect
16. C₄×Dic₁₀₉	C₂²	semidirect
17. C₂×Q₈×C₁₀₉	C₂²	semidirect
19. C₂²⋊C₄×C₁₀₉	C₂²	semidirect
20. C₄²×C₁₀₉	C₂²	semidirect
50. C₁₀₉	H(3,𝔽₄)	semidirect
55. #124	D₁₀₉	semidirect
60. C₂×Q₈	C₂×D₁₀₉	semidirect
69. C₂²⋊C₄	C₂×D₁₀₉	semidirect
71. C₄²	C₂×D₁₀₉	semidirect

H ≅ G/N

type

1. #16812688

C₂

semidirect

2. #16812687

C₂

semidirect

3. #16812685

C₂

semidirect

4. #16812669

C₂

semidirect

5. #16812667

C₂

semidirect

6. #124×C₁₀₉

C₂

semidirect

7. C₂³×D₁₀₉

C₂²

semidirect

8. C₂×Dic₂₁₈

C₂²

semidirect

12. C₂²⋊Dic₁₀₉

C₂²

semidirect

13. #13182006

C₂²

semidirect

14. #13182006

C₂²

semidirect

15. #13182006

C₂²

semidirect

16. C₄×Dic₁₀₉

C₂²

semidirect

17. C₂×Q₈×C₁₀₉

C₂²

semidirect

19. C₂²⋊C₄×C₁₀₉

C₂²

semidirect

20. C₄²×C₁₀₉

C₂²

semidirect

50. C₁₀₉

H(3,𝔽₄)

semidirect

55. #124

D₁₀₉

semidirect

60. C₂×Q₈

C₂×D₁₀₉

semidirect

69. C₂²⋊C₄

C₂×D₁₀₉

semidirect

71. C₄²

C₂×D₁₀₉

semidirect

Subgroups — #20295768

Conjugacy classes of subgroups

Complements