Group #20295146 — subgroups

Subgroups — #20295146

Order 6976.

104 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16812719	C₂	1	1	yes	maximal, characteristic
2	#16812705	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812634	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812626	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812625	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812609	C₂	1	1	yes	maximal, characteristic, complemented
7	#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13182029	C₂²	1	1	yes	characteristic
9	#13182019	C₄	1	1	yes	characteristic, complemented
10	#13182017	C₄	1	1	yes	characteristic, complemented
11	#13182016	C₄	1	1	yes	characteristic, complemented
12	#13182014	C₄	1	1	yes	characteristic, complemented
13	#13182013	C₂²	1	1	yes	characteristic
14	#13182012	—	2	1	no	—
15	C₂×#11724784	C₂²	1	1	yes	characteristic
16	C₈×D₁₀₉	—	2	1	no	—
17	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	#41×C₁₀₉	C₂²	1	1	yes	characteristic
19	M₁₆×C₁₀₉	—	2	1	no	—
20	M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	—	2	1	no	—
22	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	D₄×C₁₀₉	—	2	1	no	—
25	D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂²⋊D₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	—	2	1	no	—
30	D₄₃₆	C₂×C₄	1	1	yes	characteristic
31	C₄×D₁₀₉	—	2	1	no	—
32	Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
33	C₈×C₁₀₉	—	2	1	no	—
34	C₈×C₁₀₉	D₄	1	1	yes	characteristic
35	C₈×C₁₀₉	D₄	1	1	yes	characteristic
36	#11724784	—	2	1	no	—
37	#11724784	C₂×C₄	1	1	yes	characteristic
38	#11724784	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₁₀₉	—	2	1	no	—
40	C₂²×C₁₀₉	#41	1	1	yes	characteristic
41	C₂×D₁₀₉	—	2	1	no	—
42	C₄×C₁₀₉	—	2	1	no	—
43	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	Dic₁₀₉	—	2	1	no	—
46	Dic₁₀₉	—	4	1	no	—
47	C₂×C₁₀₉	—	2	1	no	—
48	C₂×C₁₀₉	—	4	1	no	—
49	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
50	D₁₀₉	—	4	1	no	—
51	C₁₀₉	#444	1	1	yes	characteristic, complemented, Sylow
52	#444	—	109	1	no	maximal, Sylow
53	#135	—	109	1	no	—
54	#131	—	109	1	no	—
55	#131	D₁₀₉	1	1	yes	characteristic, complemented
56	#108	—	109	1	no	—
57	#104	—	109	1	no	—
58	#104	—	109	1	no	—
59	#97	—	109	1	no	—
60	#41	—	109	1	no	—
61	#41	C₂×D₁₀₉	1	1	yes	characteristic
62	Q₁₆	—	109	1	no	—
63	SD₁₆	—	109	1	no	—
64	SD₁₆	—	109	1	no	—
65	D₈	—	109	1	no	—
66	M₁₆	—	2	1	no	—
67	M₁₆	—	109	1	no	—
68	M₁₆	—	218	1	no	—
69	M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	218	1	no	—
73	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
74	C₄²	—	109	1	no	—
75	Q₈	—	109	1	no	—
76	Q₈	C₄×D₁₀₉	1	1	yes	characteristic
77	D₄	—	2	1	no	—
78	D₄	—	109	1	no	—
79	D₄	—	218	1	no	—
80	D₄	C₄×D₁₀₉	1	1	yes	characteristic
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	218	1	no	—
83	C₂×C₄	—	218	1	no	—
84	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
85	C₈	—	2	1	no	—
86	C₈	—	109	1	no	—
87	C₈	—	109	1	no	—
88	C₈	—	218	1	no	—
89	C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
90	C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
91	C₂²	—	2	1	no	—
92	C₂²	—	218	1	no	—
93	C₂²	#13182029	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	436	1	no	—
96	C₄	—	218	1	no	—
97	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
98	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
99	C₂	—	2	1	no	—
100	C₂	—	4	1	no	—
101	C₂	—	436	1	no	—
102	C₂	C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812719

C₂

yes

maximal, characteristic

#16812705

C₂

yes

maximal, characteristic, complemented

#16812634

C₂

yes

maximal, characteristic, complemented

#16812626

C₂

yes

maximal, characteristic, complemented

#16812625

C₂

yes

maximal, characteristic, complemented

#16812609

C₂

yes

maximal, characteristic, complemented

#131×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182029

C₂²

yes

characteristic

#13182019

C₄

yes

characteristic, complemented

#13182017

C₄

yes

characteristic, complemented

#13182016

C₄

yes

characteristic, complemented

#13182014

C₄

yes

characteristic, complemented

#13182013

C₂²

yes

characteristic

#13182012

—

C₂×#11724784

C₂²

yes

characteristic

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂²

yes

characteristic

#41×C₁₀₉

C₂²

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

Q₈×C₁₀₉

C₂×C₄

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

C₂×C₄

yes

characteristic

C₄×D₁₀₉

—

Dic₂₁₈

C₂×C₄

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

D₄

yes

characteristic

C₈×C₁₀₉

D₄

yes

characteristic

#11724784

—

#11724784

C₂×C₄

yes

characteristic

#11724784

C₂×C₄

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#41

yes

characteristic

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×D₄

yes

characteristic

D₁₀₉

—

C₁₀₉

#444

yes

characteristic, complemented, Sylow

#444

—

109

maximal, Sylow

#135

—

109

—

#131

—

109

—

#131

D₁₀₉

yes

characteristic, complemented

#108

—

109

—

#104

—

109

—

#104

—

109

—

#97

—

109

—

#41

—

109

—

#41

C₂×D₁₀₉

yes

characteristic

Q₁₆

—

109

—

SD₁₆

—

109

—

SD₁₆

—

109

—

D₈

—

109

—

M₁₆

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

C₂×D₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄²

—

109

—

Q₈

—

109

—

Q₈

C₄×D₁₀₉

yes

characteristic

D₄

—

D₄

—

109

—

D₄

—

218

—

D₄

C₄×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₈

C₂²⋊D₁₀₉

yes

characteristic

C₈

C₂²⋊D₁₀₉

yes

characteristic

C₂²

—

C₂²

—

218

—

C₂²

#13182029

yes

characteristic

C₄

—

C₄

—

436

—

C₄

—

218

—

C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂

—

100

C₂

—

101

C₂

—

436

—

102

C₂

C₄×C₂²⋊D₁₀₉

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#131×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₄×D₄
hypercenter	#131	D₁₀₉
hypocenter	C₁₀₉	#444
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #131

subgroup

G/H

center

C₄

C₂×C₂²⋊D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#131×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₄×D₄

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #131

N	H ≅ G/N	type
2. #16812705	C₂	semidirect
3. #16812634	C₂	semidirect
4. #16812626	C₂	semidirect
5. #16812625	C₂	semidirect
6. #16812609	C₂	semidirect
7. #131×C₁₀₉	C₂	semidirect
9. #13182019	C₄	semidirect
10. #13182017	C₄	semidirect
11. #13182016	C₄	semidirect
12. #13182014	C₄	semidirect
51. C₁₀₉	#444	semidirect
55. #131	D₁₀₉	semidirect

H ≅ G/N

type

2. #16812705

C₂

semidirect

3. #16812634

C₂

semidirect

4. #16812626

C₂

semidirect

5. #16812625

C₂

semidirect

6. #16812609

C₂

semidirect

7. #131×C₁₀₉

C₂

semidirect

9. #13182019

C₄

semidirect

10. #13182017

C₄

semidirect

11. #13182016

C₄

semidirect

12. #13182014

C₄

semidirect

51. C₁₀₉

#444

semidirect

55. #131

D₁₀₉

semidirect

Subgroups — #20295146

Conjugacy classes of subgroups

Complements