Group #20295829 — subgroups

Subgroups — #20295829

Order 6976.

268 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 107 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	#16812745	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812725	C₂	1	3	yes	maximal, complemented
3	#16812724	C₂	1	3	yes	maximal, complemented
4	#16812719	C₂	1	3	yes	maximal, complemented
5	C₂×#13182014	C₂	1	3	yes	maximal, complemented
6	#16812706	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13182030	—	2	1	no	—
9	#13182029	C₂²	1	3	yes	complemented
10	C₂×D₄₃₆	C₂²	1	3	yes	complemented
11	D₄×D₁₀₉	—	2	3	no	—
12	#13182019	C₂²	1	1	yes	characteristic, complemented
13	#13182017	C₂²	1	3	yes	complemented
14	#13182016	C₂²	1	3	yes	complemented
15	#13182014	C₂²	1	3	yes	complemented
16	#13182014	C₂²	1	6	yes	complemented
17	#13182013	C₂²	1	3	yes	complemented
18	C₂×#11724784	C₂²	1	3	yes	complemented
19	#41×C₁₀₉	—	2	1	no	—
20	#41×C₁₀₉	C₂²	1	1	yes	characteristic
21	#41×C₁₀₉	C₂²	1	3	yes	complemented
22	C₂×D₄×C₁₀₉	—	2	3	no	—
23	C₂×D₄×C₁₀₉	C₂²	1	3	yes	complemented
24	C₂³×C₁₀₉	—	2	3	no	—
25	C₂²×D₁₀₉	—	2	3	no	—
26	Q₈×C₁₀₉	D₄	1	1	yes	characteristic
27	Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
28	D₄×C₁₀₉	—	2	3	no	—
29	D₄×C₁₀₉	—	2	3	no	—
30	D₄×C₁₀₉	D₄	1	3	yes	—
31	D₄×C₁₀₉	C₂³	1	3	yes	—
32	C₂×C₄×C₁₀₉	—	2	3	no	—
33	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
34	C₂²⋊D₁₀₉	—	2	3	no	—
35	D₄₃₆	—	2	3	no	—
36	D₄₃₆	C₂³	1	3	yes	—
37	C₄×D₁₀₉	—	2	3	no	—
38	Dic₂₁₈	C₂³	1	1	yes	characteristic
39	#11724784	C₂³	1	1	yes	characteristic
40	#11724784	C₂³	1	3	yes	—
41	C₂²×C₁₀₉	—	2	3	no	—
42	C₂²×C₁₀₉	—	8	3	no	—
43	C₂²×C₁₀₉	C₂×D₄	1	3	yes	—
44	C₂×D₁₀₉	—	2	3	no	—
45	C₂×D₁₀₉	—	8	3	no	—
46	C₄×C₁₀₉	—	2	1	no	—
47	C₄×C₁₀₉	C₂×D₄	1	3	yes	—
48	C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
49	Dic₁₀₉	—	2	1	no	—
50	C₂×C₁₀₉	—	2	3	no	—
51	C₂×C₁₀₉	—	4	3	no	—
52	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
53	D₁₀₉	—	4	3	no	—
54	C₁₀₉	#576	1	1	yes	characteristic, complemented, Sylow
55	#576	—	109	1	no	maximal, Sylow
56	H(5,𝔽₂)	—	109	1	no	—
57	H(5,𝔽₂)	D₁₀₉	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	109	3	no	—
59	AGL(1,ℤ/8)	—	109	3	no	—
60	#135	—	109	3	no	—
61	C₂×D₈	—	109	3	no	—
62	#131	—	109	1	no	—
63	#41	—	2	1	no	—
64	#41	—	109	3	no	—
65	#41	—	218	1	no	—
66	#41	C₂×D₁₀₉	1	1	yes	characteristic
67	#41	C₂×D₁₀₉	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	109	3	no	—
70	C₂×D₄	—	218	3	no	—
71	C₂×D₄	C₂×D₁₀₉	1	3	yes	complemented
72	Q₁₆	—	109	1	no	—
73	SD₁₆	—	109	3	no	—
74	SD₁₆	—	109	3	no	—
75	D₈	—	109	3	no	—
76	D₈	—	109	6	no	—
77	M₁₆	—	109	3	no	—
78	C₂×C₈	—	109	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	218	3	no	—
81	Q₈	—	109	1	no	—
82	Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
83	Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	109	3	no	—
87	D₄	—	218	3	no	—
88	D₄	—	218	3	no	—
89	D₄	C₂²⋊D₁₀₉	1	3	yes	—
90	D₄	C₂²×D₁₀₉	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	218	3	no	—
93	C₂×C₄	C₂²×D₁₀₉	1	3	yes	—
94	C₈	—	109	1	no	—
95	C₈	—	109	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	872	3	no	—
99	C₂²	—	218	3	no	—
100	C₂²	C₂×C₂²⋊D₁₀₉	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	218	1	no	—
103	C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
104	C₄	C₂³×D₁₀₉	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	436	3	no	—
108	C₂	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812745

C₂

yes

maximal, characteristic, complemented

#16812725

C₂

yes

maximal, complemented

#16812724

C₂

yes

maximal, complemented

#16812719

C₂

yes

maximal, complemented

C₂×#13182014

C₂

yes

maximal, complemented

#16812706

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182030

—

#13182029

C₂²

yes

complemented

C₂×D₄₃₆

C₂²

yes

complemented

D₄×D₁₀₉

—

#13182019

C₂²

yes

characteristic, complemented

#13182017

C₂²

yes

complemented

#13182016

C₂²

yes

complemented

#13182014

C₂²

yes

complemented

#13182014

C₂²

yes

complemented

#13182013

C₂²

yes

complemented

C₂×#11724784

C₂²

yes

complemented

#41×C₁₀₉

—

#41×C₁₀₉

C₂²

yes

characteristic

#41×C₁₀₉

C₂²

yes

complemented

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂²

yes

complemented

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

D₄

yes

characteristic

Q₈×C₁₀₉

C₂³

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

D₄

yes

—

D₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂²⋊D₁₀₉

—

D₄₃₆

—

D₄₃₆

C₂³

yes

—

C₄×D₁₀₉

—

Dic₂₁₈

C₂³

yes

characteristic

#11724784

C₂³

yes

characteristic

#11724784

C₂³

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

—

C₄×C₁₀₉

C₂⁴

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

D₁₀₉

—

C₁₀₉

#576

yes

characteristic, complemented, Sylow

#576

—

109

maximal, Sylow

H(5,𝔽₂)

—

109

—

H(5,𝔽₂)

D₁₀₉

yes

characteristic, complemented

AGL(1,ℤ/8)

—

109

—

AGL(1,ℤ/8)

—

109

—

#135

—

109

—

C₂×D₈

—

109

—

#131

—

109

—

#41

—

#41

—

109

—

#41

—

218

—

#41

C₂×D₁₀₉

yes

characteristic

#41

C₂×D₁₀₉

yes

complemented

C₂×D₄

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂×D₄

C₂×D₁₀₉

yes

complemented

Q₁₆

—

109

—

SD₁₆

—

109

—

SD₁₆

—

109

—

D₈

—

109

—

D₈

—

109

—

M₁₆

—

109

—

C₂×C₈

—

109

—

C₂³

—

C₂³

—

218

—

Q₈

—

109

—

Q₈

C₂²⋊D₁₀₉

yes

characteristic

Q₈

C₂²×D₁₀₉

yes

characteristic

D₄

—

D₄

—

D₄

—

109

—

D₄

—

218

—

D₄

—

218

—

D₄

C₂²⋊D₁₀₉

yes

—

D₄

C₂²×D₁₀₉

yes

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₈

—

109

—

C₈

—

109

—

C₂²

—

C₂²

—

C₂²

—

872

—

C₂²

—

218

—

100

C₂²

C₂×C₂²⋊D₁₀₉

yes

—

101

C₄

—

102

C₄

—

218

—

103

C₄

C₂×C₂²⋊D₁₀₉

yes

—

104

C₄

C₂³×D₁₀₉

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

436

—

108

C₂

C₂²×C₂²⋊D₁₀₉

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂⁴
Frattini	C₄	C₂³×D₁₀₉
Fitting	H(5,𝔽₂)×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₁₀₉
hypocenter	C₁₀₉	#576
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₀₉

commutator

C₄×C₁₀₉

C₂⁴

Frattini

C₄

C₂³×D₁₀₉

Fitting

H(5,𝔽₂)×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₂²×D₄

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #16812745	C₂	semidirect
2. #16812725	C₂	semidirect
3. #16812724	C₂	semidirect
4. #16812719	C₂	semidirect
5. C₂×#13182014	C₂	semidirect
6. #16812706	C₂	semidirect
7. H(5,𝔽₂)×C₁₀₉	C₂	semidirect
9. #13182029	C₂²	semidirect
10. C₂×D₄₃₆	C₂²	semidirect
12. #13182019	C₂²	semidirect
13. #13182017	C₂²	semidirect
14. #13182016	C₂²	semidirect
15. #13182014	C₂²	semidirect
16. #13182014	C₂²	semidirect
17. #13182013	C₂²	semidirect
18. C₂×#11724784	C₂²	semidirect
21. #41×C₁₀₉	C₂²	semidirect
23. C₂×D₄×C₁₀₉	C₂²	semidirect
54. C₁₀₉	#576	semidirect
57. H(5,𝔽₂)	D₁₀₉	semidirect
67. #41	C₂×D₁₀₉	semidirect
71. C₂×D₄	C₂×D₁₀₉	semidirect

H ≅ G/N

type

1. #16812745

C₂

semidirect

2. #16812725

C₂

semidirect

3. #16812724

C₂

semidirect

4. #16812719

C₂

semidirect

5. C₂×#13182014

C₂

semidirect

6. #16812706

C₂

semidirect

7. H(5,𝔽₂)×C₁₀₉

C₂

semidirect

9. #13182029

C₂²

semidirect

10. C₂×D₄₃₆

C₂²

semidirect

12. #13182019

C₂²

semidirect

13. #13182017

C₂²

semidirect

14. #13182016

C₂²

semidirect

15. #13182014

C₂²

semidirect

16. #13182014

C₂²

semidirect

17. #13182013

C₂²

semidirect

18. C₂×#11724784

C₂²

semidirect

21. #41×C₁₀₉

C₂²

semidirect

23. C₂×D₄×C₁₀₉

C₂²

semidirect

54. C₁₀₉

#576

semidirect

57. H(5,𝔽₂)

D₁₀₉

semidirect

67. #41

C₂×D₁₀₉

semidirect

71. C₂×D₄

C₂×D₁₀₉

semidirect

Subgroups — #20295829

Conjugacy classes of subgroups

Complements