Group #20295838 — subgroups

Subgroups — #20295838

Order 6976.

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

normal subgroups only

This group has 258 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	#16812746	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812745	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812732	C₂	1	3	yes	maximal, complemented
4	#16812727	C₂	1	3	yes	maximal, complemented
5	#16812721	C₂	1	3	yes	maximal, complemented
6	C₂×#13182018	C₂	1	3	yes	maximal, complemented
7	#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13182030	—	2	1	no	—
9	#13182029	C₂²	1	3	yes	—
10	#13182029	C₂²	1	3	yes	complemented
11	#13182028	—	2	3	no	—
12	C₂×Dic₂₁₈	C₂²	1	3	yes	complemented
13	Q₈×D₁₀₉	—	2	1	no	—
14	C₂×D₄₃₆	C₂²	1	3	yes	—
15	D₄×D₁₀₉	—	2	3	no	—
16	Dic₄₃₆	C₂²	1	3	yes	complemented
17	#13182018	C₂²	1	3	yes	complemented
18	#13182018	C₂²	1	6	yes	complemented
19	#13182018	C₂²	1	1	yes	characteristic, complemented
20	D₈₇₂	C₂²	1	3	yes	complemented
21	#41×C₁₀₉	C₂²	1	1	yes	characteristic
22	M₁₆×C₁₀₉	C₂²	1	3	yes	complemented
23	C₂×C₈×C₁₀₉	C₂²	1	3	yes	complemented
24	C₂²×D₁₀₉	—	2	3	no	—
25	Q₈×C₁₀₉	D₄	1	1	yes	characteristic
26	D₄×C₁₀₉	D₄	1	3	yes	—
27	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
28	C₂²⋊D₁₀₉	—	2	3	no	—
29	C₂²⋊D₁₀₉	—	2	3	no	—
30	C₂×Dic₁₀₉	—	2	3	no	—
31	D₄₃₆	—	2	3	no	—
32	D₄₃₆	C₂³	1	1	yes	characteristic
33	D₄₃₆	C₂³	1	3	yes	—
34	C₄×D₁₀₉	—	2	3	no	—
35	C₄×D₁₀₉	—	2	3	no	—
36	Dic₂₁₈	—	2	3	no	—
37	Dic₂₁₈	C₂³	1	3	yes	—
38	Dic₂₁₈	C₂³	1	1	yes	characteristic
39	C₈×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₈×C₁₀₉	C₂³	1	3	yes	—
41	C₂²×C₁₀₉	C₂×D₄	1	3	yes	—
42	C₂×D₁₀₉	—	2	1	no	—
43	C₂×D₁₀₉	—	2	3	no	—
44	C₂×D₁₀₉	—	8	3	no	—
45	C₄×C₁₀₉	C₂×D₄	1	3	yes	—
46	C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
47	Dic₁₀₉	—	2	3	no	—
48	Dic₁₀₉	—	2	1	no	—
49	C₂×C₁₀₉	—	2	3	no	—
50	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
51	D₁₀₉	—	4	1	no	—
52	D₁₀₉	—	4	3	no	—
53	C₁₀₉	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	109	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	109	1	no	—
56	H(5,𝔽₂)	—	109	1	no	—
57	#137	—	109	3	no	—
58	AGL(1,ℤ/8)	—	109	3	no	—
59	#135	—	109	3	no	—
60	C₂×SD₁₆	—	109	3	no	—
61	#131	D₁₀₉	1	1	yes	characteristic, complemented
62	#41	—	109	3	no	—
63	#41	—	109	3	no	—
64	#41	—	218	1	no	—
65	#41	—	218	3	no	—
66	#41	C₂×D₁₀₉	1	1	yes	characteristic
67	C₂×Q₈	—	109	3	no	—
68	C₂×Q₈	—	218	1	no	—
69	C₂×D₄	—	109	3	no	—
70	C₂×D₄	—	218	3	no	—
71	Q₁₆	—	109	3	no	—
72	SD₁₆	—	109	3	no	—
73	SD₁₆	—	109	6	no	—
74	SD₁₆	—	109	1	no	—
75	D₈	—	109	3	no	—
76	M₁₆	C₂×D₁₀₉	1	3	yes	complemented
77	C₂×C₈	C₂×D₁₀₉	1	3	yes	complemented
78	C₂³	—	218	3	no	—
79	Q₈	—	109	3	no	—
80	Q₈	—	109	1	no	—
81	Q₈	—	218	3	no	—
82	Q₈	D₄₃₆	1	1	yes	characteristic
83	D₄	—	109	1	no	—
84	D₄	—	109	3	no	—
85	D₄	—	218	3	no	—
86	D₄	—	218	3	no	—
87	D₄	—	218	3	no	—
88	D₄	D₄₃₆	1	3	yes	—
89	C₂×C₄	—	218	3	no	—
90	C₂×C₄	—	218	3	no	—
91	C₂×C₄	—	218	3	no	—
92	C₂×C₄	C₂²×D₁₀₉	1	3	yes	—
93	C₈	C₂²×D₁₀₉	1	1	yes	characteristic
94	C₈	C₂²×D₁₀₉	1	3	yes	—
95	C₂²	—	872	3	no	—
96	C₂²	—	218	1	no	—
97	C₂²	—	218	3	no	—
98	C₂²	C₂×D₄₃₆	1	3	yes	—
99	C₄	—	218	1	no	—
100	C₄	—	218	3	no	—
101	C₄	C₂×D₄₃₆	1	3	yes	—
102	C₄	C₂³×D₁₀₉	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	436	3	no	—
105	C₂	—	436	1	no	—
106	C₂	C₂²×D₄₃₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812746

C₂

yes

maximal, characteristic, complemented

#16812745

C₂

yes

maximal, characteristic, complemented

#16812732

C₂

yes

maximal, complemented

#16812727

C₂

yes

maximal, complemented

#16812721

C₂

yes

maximal, complemented

C₂×#13182018

C₂

yes

maximal, complemented

#131×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182030

—

#13182029

C₂²

yes

—

#13182029

C₂²

yes

complemented

#13182028

—

C₂×Dic₂₁₈

C₂²

yes

complemented

Q₈×D₁₀₉

—

C₂×D₄₃₆

C₂²

yes

—

D₄×D₁₀₉

—

Dic₄₃₆

C₂²

yes

complemented

#13182018

C₂²

yes

complemented

#13182018

C₂²

yes

complemented

#13182018

C₂²

yes

characteristic, complemented

D₈₇₂

C₂²

yes

complemented

#41×C₁₀₉

C₂²

yes

characteristic

M₁₆×C₁₀₉

C₂²

yes

complemented

C₂×C₈×C₁₀₉

C₂²

yes

complemented

C₂²×D₁₀₉

—

Q₈×C₁₀₉

D₄

yes

characteristic

D₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂²⋊D₁₀₉

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

C₂³

yes

characteristic

D₄₃₆

C₂³

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

C₂³

yes

—

Dic₂₁₈

C₂³

yes

characteristic

C₈×C₁₀₉

C₂³

yes

characteristic

C₈×C₁₀₉

C₂³

yes

—

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

—

C₄×C₁₀₉

C₂⁴

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#577

yes

characteristic, complemented, Sylow

#577

—

109

maximal, Sylow

2₋¹⁺⁴

—

109

—

H(5,𝔽₂)

—

109

—

#137

—

109

—

AGL(1,ℤ/8)

—

109

—

#135

—

109

—

C₂×SD₁₆

—

109

—

#131

D₁₀₉

yes

characteristic, complemented

#41

—

109

—

#41

—

109

—

#41

—

218

—

#41

—

218

—

#41

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

Q₁₆

—

109

—

SD₁₆

—

109

—

SD₁₆

—

109

—

SD₁₆

—

109

—

D₈

—

109

—

M₁₆

C₂×D₁₀₉

yes

complemented

C₂×C₈

C₂×D₁₀₉

yes

complemented

C₂³

—

218

—

Q₈

—

109

—

Q₈

—

109

—

Q₈

—

218

—

Q₈

D₄₃₆

yes

characteristic

D₄

—

109

—

D₄

—

109

—

D₄

—

218

—

D₄

—

218

—

D₄

—

218

—

D₄

D₄₃₆

yes

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₈

C₂²×D₁₀₉

yes

characteristic

C₈

C₂²×D₁₀₉

yes

—

C₂²

—

872

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×D₄₃₆

yes

—

C₄

—

218

—

100

C₄

—

218

—

101

C₄

C₂×D₄₃₆

yes

—

102

C₄

C₂³×D₁₀₉

yes

characteristic

103

C₂

—

104

C₂

—

436

—

105

C₂

—

436

—

106

C₂

C₂²×D₄₃₆

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₄₃₆
commutator	C₄×C₁₀₉	C₂⁴
Frattini	C₄	C₂³×D₁₀₉
Fitting	#131×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂²×D₄
hypercenter	#131	D₁₀₉
hypocenter	C₁₀₉	#577
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₄₃₆

commutator

C₄×C₁₀₉

C₂⁴

Frattini

C₄

C₂³×D₁₀₉

Fitting

#131×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₂²×D₄

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #16812746	C₂	semidirect
2. #16812745	C₂	semidirect
3. #16812732	C₂	semidirect
4. #16812727	C₂	semidirect
5. #16812721	C₂	semidirect
6. C₂×#13182018	C₂	semidirect
7. #131×C₁₀₉	C₂	semidirect
10. #13182029	C₂²	semidirect
12. C₂×Dic₂₁₈	C₂²	semidirect
16. Dic₄₃₆	C₂²	semidirect
17. #13182018	C₂²	semidirect
18. #13182018	C₂²	semidirect
19. #13182018	C₂²	semidirect
20. D₈₇₂	C₂²	semidirect
22. M₁₆×C₁₀₉	C₂²	semidirect
23. C₂×C₈×C₁₀₉	C₂²	semidirect
53. C₁₀₉	#577	semidirect
61. #131	D₁₀₉	semidirect
76. M₁₆	C₂×D₁₀₉	semidirect
77. C₂×C₈	C₂×D₁₀₉	semidirect

H ≅ G/N

type

1. #16812746

C₂

semidirect

2. #16812745

C₂

semidirect

3. #16812732

C₂

semidirect

4. #16812727

C₂

semidirect

5. #16812721

C₂

semidirect

6. C₂×#13182018

C₂

semidirect

7. #131×C₁₀₉

C₂

semidirect

10. #13182029

C₂²

semidirect

12. C₂×Dic₂₁₈

C₂²

semidirect

16. Dic₄₃₆

C₂²

semidirect

17. #13182018

C₂²

semidirect

18. #13182018

C₂²

semidirect

19. #13182018

C₂²

semidirect

20. D₈₇₂

C₂²

semidirect

22. M₁₆×C₁₀₉

C₂²

semidirect

23. C₂×C₈×C₁₀₉

C₂²

semidirect

53. C₁₀₉

#577

semidirect

61. #131

D₁₀₉

semidirect

76. M₁₆

C₂×D₁₀₉

semidirect

77. C₂×C₈

C₂×D₁₀₉

semidirect

Subgroups — #20295838

Conjugacy classes of subgroups

Complements