Group #20295837 — subgroups

Subgroups — #20295837

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	#16812729	C₂	1	3	yes	maximal, complemented
3	#16812725	C₂	1	3	yes	maximal, complemented
4	#16812719	C₂	1	3	yes	maximal, complemented
5	C₂×#13182017	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	#13182029	C₂²	1	3	yes	complemented
9	#13182028	—	2	3	no	—
10	C₂×Dic₂₁₈	C₂²	1	3	yes	complemented
11	Q₈×D₁₀₉	—	2	1	no	—
12	#13182019	C₂²	1	3	yes	complemented
13	#13182017	C₂²	1	3	yes	complemented
14	#13182017	C₂²	1	6	yes	complemented
15	#13182016	C₂²	1	1	yes	characteristic, complemented
16	#13182014	C₂²	1	3	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	3	yes	—
21	#41×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₁₀₉	—	2	3	no	—
23	C₂×D₄×C₁₀₉	C₂²	1	3	yes	—
24	C₂³×C₁₀₉	—	2	3	no	—
25	Q₈×C₁₀₉	D₄	1	1	yes	characteristic
26	Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
27	D₄×C₁₀₉	—	2	3	no	—
28	D₄×C₁₀₉	—	2	3	no	—
29	D₄×C₁₀₉	D₄	1	3	yes	—
30	D₄×C₁₀₉	C₂³	1	3	yes	—
31	C₂×C₄×C₁₀₉	—	2	3	no	—
32	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
33	C₂²⋊D₁₀₉	—	2	3	no	—
34	C₂×Dic₁₀₉	—	2	3	no	—
35	D₄₃₆	C₂³	1	1	yes	characteristic
36	C₄×D₁₀₉	—	2	3	no	—
37	Dic₂₁₈	—	2	3	no	—
38	Dic₂₁₈	C₂³	1	3	yes	—
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	1	no	—
45	C₄×C₁₀₉	—	2	1	no	—
46	C₄×C₁₀₉	C₂×D₄	1	3	yes	—
47	C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
48	Dic₁₀₉	—	2	3	no	—
49	C₂×C₁₀₉	—	2	3	no	—
50	C₂×C₁₀₉	—	4	3	no	—
51	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
52	D₁₀₉	—	4	1	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)	D₁₀₉	1	1	yes	characteristic, complemented
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	—	109	1	no	—
62	#41	—	2	1	no	—
63	#41	—	109	3	no	—
64	#41	—	218	3	no	—
65	#41	C₂×D₁₀₉	1	3	yes	—
66	#41	C₂×D₁₀₉	1	1	yes	characteristic
67	C₂×Q₈	—	109	3	no	—
68	C₂×Q₈	—	218	1	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₁₀₉	1	3	yes	—
71	Q₁₆	—	109	3	no	—
72	SD₁₆	—	109	1	no	—
73	SD₁₆	—	109	3	no	—
74	SD₁₆	—	109	6	no	—
75	D₈	—	109	3	no	—
76	M₁₆	—	109	3	no	—
77	C₂×C₈	—	109	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	109	3	no	—
80	Q₈	—	218	3	no	—
81	Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
82	Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	109	1	no	—
86	D₄	—	218	3	no	—
87	D₄	C₂²⋊D₁₀₉	1	3	yes	—
88	D₄	C₂²×D₁₀₉	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	218	3	no	—
91	C₂×C₄	—	218	3	no	—
92	C₂×C₄	C₂²×D₁₀₉	1	3	yes	—
93	C₈	—	109	1	no	—
94	C₈	—	109	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	218	1	no	—
98	C₂²	C₂×C₂²⋊D₁₀₉	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	218	3	no	—
101	C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
102	C₄	C₂³×D₁₀₉	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	436	1	no	—
106	C₂	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812746

C₂

yes

maximal, characteristic, complemented

#16812729

C₂

yes

maximal, complemented

#16812725

C₂

yes

maximal, complemented

#16812719

C₂

yes

maximal, complemented

C₂×#13182017

C₂

yes

maximal, complemented

#16812706

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182029

C₂²

yes

complemented

#13182028

—

C₂×Dic₂₁₈

C₂²

yes

complemented

Q₈×D₁₀₉

—

#13182019

C₂²

yes

complemented

#13182017

C₂²

yes

complemented

#13182017

C₂²

yes

complemented

#13182016

C₂²

yes

characteristic, complemented

#13182014

C₂²

yes

complemented

#13182013

C₂²

yes

complemented

C₂×#11724784

C₂²

yes

complemented

#41×C₁₀₉

—

#41×C₁₀₉

C₂²

yes

—

#41×C₁₀₉

C₂²

yes

characteristic

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂²

yes

—

C₂³×C₁₀₉

—

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

—

C₂×Dic₁₀₉

—

D₄₃₆

C₂³

yes

characteristic

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

C₂³

yes

—

#11724784

C₂³

yes

characteristic

#11724784

C₂³

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

—

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

#577

yes

characteristic, complemented, Sylow

#577

—

109

maximal, Sylow

2₋¹⁺⁴

—

109

—

H(5,𝔽₂)

D₁₀₉

yes

characteristic, complemented

#137

—

109

—

AGL(1,ℤ/8)

—

109

—

#135

—

109

—

C₂×SD₁₆

—

109

—

#131

—

109

—

#41

—

#41

—

109

—

#41

—

218

—

#41

C₂×D₁₀₉

yes

—

#41

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×D₄

—

C₂×D₄

C₂×D₁₀₉

yes

—

Q₁₆

—

109

—

SD₁₆

—

109

—

SD₁₆

—

109

—

SD₁₆

—

109

—

D₈

—

109

—

M₁₆

—

109

—

C₂×C₈

—

109

—

C₂³

—

Q₈

—

109

—

Q₈

—

218

—

Q₈

C₂²⋊D₁₀₉

yes

characteristic

Q₈

C₂²×D₁₀₉

yes

characteristic

D₄

—

D₄

—

D₄

—

109

—

D₄

—

218

—

D₄

C₂²⋊D₁₀₉

yes

—

D₄

C₂²×D₁₀₉

yes

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₈

—

109

—

C₈

—

109

—

C₂²

—

C₂²

—

C₂²

—

218

—

C₂²

C₂×C₂²⋊D₁₀₉

yes

—

C₄

—

100

C₄

—

218

—

101

C₄

C₂×C₂²⋊D₁₀₉

yes

—

102

C₄

C₂³×D₁₀₉

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

436

—

106

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	#577
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. #16812746	C₂	semidirect
2. #16812729	C₂	semidirect
3. #16812725	C₂	semidirect
4. #16812719	C₂	semidirect
5. C₂×#13182017	C₂	semidirect
6. #16812706	C₂	semidirect
7. H(5,𝔽₂)×C₁₀₉	C₂	semidirect
8. #13182029	C₂²	semidirect
10. C₂×Dic₂₁₈	C₂²	semidirect
12. #13182019	C₂²	semidirect
13. #13182017	C₂²	semidirect
14. #13182017	C₂²	semidirect
15. #13182016	C₂²	semidirect
16. #13182014	C₂²	semidirect
17. #13182013	C₂²	semidirect
18. C₂×#11724784	C₂²	semidirect
53. C₁₀₉	#577	semidirect
56. H(5,𝔽₂)	D₁₀₉	semidirect

H ≅ G/N

type

1. #16812746

C₂

semidirect

2. #16812729

C₂

semidirect

3. #16812725

C₂

semidirect

4. #16812719

C₂

semidirect

5. C₂×#13182017

C₂

semidirect

6. #16812706

C₂

semidirect

7. H(5,𝔽₂)×C₁₀₉

C₂

semidirect

8. #13182029

C₂²

semidirect

10. C₂×Dic₂₁₈

C₂²

semidirect

12. #13182019

C₂²

semidirect

13. #13182017

C₂²

semidirect

14. #13182017

C₂²

semidirect

15. #13182016

C₂²

semidirect

16. #13182014

C₂²

semidirect

17. #13182013

C₂²

semidirect

18. C₂×#11724784

C₂²

semidirect

53. C₁₀₉

#577

semidirect

56. H(5,𝔽₂)

D₁₀₉

semidirect

Subgroups — #20295837

Conjugacy classes of subgroups

Complements