Group #20295791 — subgroups

Subgroups — #20295791

Order 6976.

258 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 147 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	#41×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#16812707	C₂	1	3	yes	maximal, complemented
3	#16812706	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812705	C₂	1	3	yes	maximal, complemented
5	C₂×#13182012	C₂	1	3	yes	maximal, complemented
6	M₁₆×D₁₀₉	C₂	1	3	yes	maximal, complemented
7	#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13182030	C₄	1	1	yes	characteristic
9	#13182029	C₂²	1	3	yes	—
10	#13182028	C₄	1	3	yes	—
11	Q₈×D₁₀₉	C₄	1	1	yes	characteristic
12	D₄×D₁₀₉	C₄	1	3	yes	—
13	C₂×C₄×D₁₀₉	C₂²	1	3	yes	—
14	#13182013	C₂²	1	3	yes	complemented
15	#13182012	C₂²	1	3	yes	—
16	#13182012	C₂²	1	1	yes	characteristic
17	#13182012	C₂²	1	3	yes	complemented
18	#13182012	C₂²	1	3	yes	complemented
19	C₂×#11724784	C₂²	1	3	yes	complemented
20	C₈×D₁₀₉	C₂²	1	6	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	C₂×C₄	1	3	yes	—
25	Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	D₄×C₁₀₉	C₂×C₄	1	3	yes	—
27	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
28	C₂²⋊D₁₀₉	C₂×C₄	1	6	yes	—
29	C₂×Dic₁₀₉	C₂×C₄	1	3	yes	—
30	D₄₃₆	C₂×C₄	1	3	yes	—
31	C₄×D₁₀₉	C₂×C₄	1	3	yes	—
32	C₄×D₁₀₉	C₂×C₄	1	3	yes	—
33	C₄×D₁₀₉	C₂³	1	3	yes	—
34	C₄×D₁₀₉	C₂³	1	1	yes	characteristic
35	Dic₂₁₈	C₂×C₄	1	3	yes	—
36	C₈×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₈×C₁₀₉	C₂³	1	3	yes	—
38	#11724784	C₂³	1	3	yes	—
39	#11724784	C₂³	1	1	yes	characteristic
40	C₂²×C₁₀₉	C₂²×C₄	1	3	yes	—
41	C₂×D₁₀₉	—	4	3	no	—
42	C₂×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×D₁₀₉	C₂²×C₄	1	3	yes	—
44	C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
45	C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
46	Dic₁₀₉	C₂²×C₄	1	3	yes	—
47	Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₁₀₉	—	2	3	no	—
49	C₂×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
50	D₁₀₉	—	2	1	no	—
51	D₁₀₉	—	2	3	no	—
52	C₁₀₉	#568	1	1	yes	characteristic, complemented, Sylow
53	#568	—	109	1	no	maximal, Sylow
54	C₂×#41	—	109	1	no	—
55	#131	—	109	1	no	—
56	#131	—	109	3	no	—
57	#131	—	109	3	no	—
58	#131	D₁₀₉	1	1	yes	characteristic, complemented
59	C₂×M₁₆	—	109	3	no	—
60	C₂×M₁₆	—	109	3	no	—
61	#41	—	109	1	no	—
62	#41	—	109	3	no	—
63	#41	—	109	3	no	—
64	#41	C₂×D₁₀₉	1	1	yes	characteristic
65	C₂×Q₈	—	109	1	no	—
66	C₂×D₄	—	109	3	no	—
67	C₂²×C₄	—	109	3	no	—
68	M₁₆	—	109	3	no	—
69	M₁₆	—	109	3	no	—
70	M₁₆	—	109	1	no	—
71	M₁₆	—	109	3	no	—
72	M₁₆	—	109	3	no	—
73	M₁₆	C₂×D₁₀₉	1	3	yes	complemented
74	C₂×C₈	—	109	3	no	—
75	C₂×C₈	—	109	6	no	—
76	C₂×C₈	C₂×D₁₀₉	1	3	yes	complemented
77	C₂³	—	109	3	no	—
78	Q₈	—	109	3	no	—
79	Q₈	C₄×D₁₀₉	1	1	yes	characteristic
80	D₄	—	109	6	no	—
81	D₄	—	109	3	no	—
82	D₄	C₄×D₁₀₉	1	3	yes	—
83	C₂×C₄	—	109	3	no	—
84	C₂×C₄	—	109	3	no	—
85	C₂×C₄	—	109	3	no	—
86	C₂×C₄	—	109	3	no	—
87	C₂×C₄	—	109	1	no	—
88	C₂×C₄	C₂²×D₁₀₉	1	3	yes	—
89	C₈	—	109	1	no	—
90	C₈	—	109	3	no	—
91	C₈	C₂²×D₁₀₉	1	3	yes	—
92	C₈	C₂²×D₁₀₉	1	1	yes	characteristic
93	C₂²	—	109	1	no	—
94	C₂²	—	109	3	no	—
95	C₂²	—	436	3	no	—
96	C₂²	C₂×C₄×D₁₀₉	1	3	yes	—
97	C₄	—	109	1	no	—
98	C₄	—	109	3	no	—
99	C₄	C₂×C₄×D₁₀₉	1	3	yes	—
100	C₄	C₂³×D₁₀₉	1	1	yes	characteristic
101	C₂	—	2	3	no	—
102	C₂	—	218	1	no	—
103	C₂	—	218	3	no	—
104	C₂	C₂²×C₄×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×D₁₀₉

C₂

yes

maximal, characteristic

#16812707

C₂

yes

maximal, complemented

#16812706

C₂

yes

maximal, characteristic, complemented

#16812705

C₂

yes

maximal, complemented

C₂×#13182012

C₂

yes

maximal, complemented

M₁₆×D₁₀₉

C₂

yes

maximal, complemented

#131×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182030

C₄

yes

characteristic

#13182029

C₂²

yes

—

#13182028

C₄

yes

—

Q₈×D₁₀₉

C₄

yes

characteristic

D₄×D₁₀₉

C₄

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

—

#13182013

C₂²

yes

complemented

#13182012

C₂²

yes

—

#13182012

C₂²

yes

characteristic

#13182012

C₂²

yes

complemented

#13182012

C₂²

yes

complemented

C₂×#11724784

C₂²

yes

complemented

C₈×D₁₀₉

C₂²

yes

complemented

#41×C₁₀₉

C₂²

yes

characteristic

M₁₆×C₁₀₉

C₂²

yes

complemented

C₂×C₈×C₁₀₉

C₂²

yes

complemented

C₂²×D₁₀₉

C₂×C₄

yes

—

Q₈×C₁₀₉

C₂×C₄

yes

characteristic

D₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂²⋊D₁₀₉

C₂×C₄

yes

—

C₂×Dic₁₀₉

C₂×C₄

yes

—

D₄₃₆

C₂×C₄

yes

—

C₄×D₁₀₉

C₂×C₄

yes

—

C₄×D₁₀₉

C₂×C₄

yes

—

C₄×D₁₀₉

C₂³

yes

—

C₄×D₁₀₉

C₂³

yes

characteristic

Dic₂₁₈

C₂×C₄

yes

—

C₈×C₁₀₉

C₂³

yes

characteristic

C₈×C₁₀₉

C₂³

yes

—

#11724784

C₂³

yes

—

#11724784

C₂³

yes

characteristic

C₂²×C₁₀₉

C₂²×C₄

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²×C₄

yes

characteristic

C₂×D₁₀₉

C₂²×C₄

yes

—

C₄×C₁₀₉

C₂²×C₄

yes

—

C₄×C₁₀₉

C₂⁴

yes

characteristic

Dic₁₀₉

C₂²×C₄

yes

—

Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂³×C₄

yes

characteristic

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#568

yes

characteristic, complemented, Sylow

#568

—

109

maximal, Sylow

C₂×#41

—

109

—

#131

—

109

—

#131

—

109

—

#131

—

109

—

#131

D₁₀₉

yes

characteristic, complemented

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

#41

—

109

—

#41

—

109

—

#41

—

109

—

#41

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

109

—

M₁₆

—

109

—

M₁₆

—

109

—

M₁₆

—

109

—

M₁₆

—

109

—

M₁₆

C₂×D₁₀₉

yes

complemented

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

complemented

C₂³

—

109

—

Q₈

—

109

—

Q₈

C₄×D₁₀₉

yes

characteristic

D₄

—

109

—

D₄

—

109

—

D₄

C₄×D₁₀₉

yes

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₈

—

109

—

C₈

—

109

—

C₈

C₂²×D₁₀₉

yes

—

C₈

C₂²×D₁₀₉

yes

characteristic

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

436

—

C₂²

C₂×C₄×D₁₀₉

yes

—

C₄

—

109

—

C₄

—

109

—

C₄

C₂×C₄×D₁₀₉

yes

—

100

C₄

C₂³×D₁₀₉

yes

characteristic

101

C₂

—

102

C₂

—

218

—

103

C₂

—

218

—

104

C₂

C₂²×C₄×D₁₀₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×D₁₀₉
commutator, socle	C₂×C₁₀₉	C₂³×C₄
Fitting	#131×C₁₀₉	C₂
hypercenter	#131	D₁₀₉
hypocenter	C₁₀₉	#568
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ #131

subgroup

G/H

center, Frattini

C₄

C₂³×D₁₀₉

commutator, socle

C₂×C₁₀₉

C₂³×C₄

Fitting

#131×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₄ ◁ #131

N	H ≅ G/N	type
2. #16812707	C₂	semidirect
3. #16812706	C₂	semidirect
4. #16812705	C₂	semidirect
5. C₂×#13182012	C₂	semidirect
6. M₁₆×D₁₀₉	C₂	semidirect
7. #131×C₁₀₉	C₂	semidirect
14. #13182013	C₂²	semidirect
17. #13182012	C₂²	semidirect
18. #13182012	C₂²	semidirect
19. C₂×#11724784	C₂²	semidirect
20. C₈×D₁₀₉	C₂²	semidirect
22. M₁₆×C₁₀₉	C₂²	semidirect
23. C₂×C₈×C₁₀₉	C₂²	semidirect
52. C₁₀₉	#568	semidirect
58. #131	D₁₀₉	semidirect
73. M₁₆	C₂×D₁₀₉	semidirect
76. C₂×C₈	C₂×D₁₀₉	semidirect

H ≅ G/N

type

2. #16812707

C₂

semidirect

3. #16812706

C₂

semidirect

4. #16812705

C₂

semidirect

5. C₂×#13182012

C₂

semidirect

6. M₁₆×D₁₀₉

C₂

semidirect

7. #131×C₁₀₉

C₂

semidirect

14. #13182013

C₂²

semidirect

17. #13182012

C₂²

semidirect

18. #13182012

C₂²

semidirect

19. C₂×#11724784

C₂²

semidirect

20. C₈×D₁₀₉

C₂²

semidirect

22. M₁₆×C₁₀₉

C₂²

semidirect

23. C₂×C₈×C₁₀₉

C₂²

semidirect

52. C₁₀₉

#568

semidirect

58. #131

D₁₀₉

semidirect

73. M₁₆

C₂×D₁₀₉

semidirect

76. C₂×C₈

C₂×D₁₀₉

semidirect

Subgroups — #20295791

Conjugacy classes of subgroups

Complements