Group #20295145 — subgroups

Subgroups — #20295145

Order 6976.

104 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16812707	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812706	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812634	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812627	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812610	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×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	—	2	1	no	—
14	#13182013	C₂²	1	1	yes	characteristic
15	#13182012	—	2	1	no	—
16	C₂×#11724784	—	2	1	no	—
17	C₂×#11724784	C₂²	1	1	yes	characteristic
18	C₂×#11724784	C₂²	1	1	yes	characteristic
19	C₈×D₁₀₉	—	2	1	no	—
20	#41×C₁₀₉	C₂²	1	1	yes	characteristic
21	M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄²×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	—	2	1	no	—
28	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂²⋊D₁₀₉	—	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	#11724784	—	2	1	no	—
35	#11724784	C₂×C₄	1	1	yes	characteristic
36	#11724784	C₂×C₄	1	1	yes	characteristic
37	#11724784	D₄	1	1	yes	characteristic
38	#11724784	D₄	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	—	4	1	no	—
44	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	Dic₁₀₉	—	2	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	—	109	1	no	—
56	#108	—	109	1	no	—
57	#104	—	109	1	no	—
58	#104	D₁₀₉	1	1	yes	characteristic, complemented
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	—	109	1	no	—
67	M₁₆	—	218	1	no	—
68	M₁₆	—	218	1	no	—
69	M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
70	C₂×C₈	—	109	1	no	—
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	218	1	no	—
73	C₂×C₈	—	218	1	no	—
74	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
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	—	2	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	—	109	1	no	—
89	C₈	—	109	1	no	—
90	C₈	—	218	1	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	218	1	no	—
93	C₂²	#13182028	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	4	1	no	—
96	C₄	—	218	1	no	—
97	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
98	C₄	D₄×D₁₀₉	1	1	yes	characteristic
99	C₂	—	2	1	no	—
100	C₂	—	4	1	no	—
101	C₂	—	436	1	no	—
102	C₂	#16812659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812719

C₂

yes

maximal, characteristic

#16812707

C₂

yes

maximal, characteristic, complemented

#16812706

C₂

yes

maximal, characteristic, complemented

#16812634

C₂

yes

maximal, characteristic, complemented

#16812627

C₂

yes

maximal, characteristic, complemented

#16812610

C₂

yes

maximal, characteristic, complemented

#104×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

—

#13182013

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#11724784

C₂²

yes

characteristic

C₈×D₁₀₉

—

#41×C₁₀₉

C₂²

yes

characteristic

M₁₆×C₁₀₉

C₂²

yes

characteristic

C₄²×C₁₀₉

C₂²

yes

characteristic

Q₈×C₁₀₉

C₂×C₄

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

D₄₃₆

C₂×C₄

yes

characteristic

C₄×D₁₀₉

—

Dic₂₁₈

C₂×C₄

yes

characteristic

C₈×C₁₀₉

—

#11724784

—

#11724784

C₂×C₄

yes

characteristic

#11724784

C₂×C₄

yes

characteristic

#11724784

D₄

yes

characteristic

#11724784

D₄

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#41

yes

characteristic

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

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

—

109

—

#108

—

109

—

#104

—

109

—

#104

D₁₀₉

yes

characteristic, complemented

#97

—

109

—

#41

—

109

—

#41

C₂×D₁₀₉

yes

characteristic

Q₁₆

—

109

—

SD₁₆

—

109

—

SD₁₆

—

109

—

D₈

—

109

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

C₂×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

C₂×D₁₀₉

yes

characteristic

Q₈

—

109

—

Q₈

C₄×D₁₀₉

yes

characteristic

D₄

—

D₄

—

109

—

D₄

—

218

—

D₄

C₄×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₂²

—

C₂²

—

218

—

C₂²

#13182028

yes

characteristic

C₄

—

C₄

—

C₄

—

218

—

C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₄

D₄×D₁₀₉

yes

characteristic

C₂

—

100

C₂

—

101

C₂

—

436

—

102

C₂

#16812659

yes

characteristic

	subgroup	G/H
center	C₄	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#104×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₄×D₄
hypercenter	#104	D₁₀₉
hypocenter	C₁₀₉	#444
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

subgroup

G/H

center

C₄

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#104×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₄×D₄

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

N	H ≅ G/N	type
2. #16812707	C₂	semidirect
3. #16812706	C₂	semidirect
4. #16812634	C₂	semidirect
5. #16812627	C₂	semidirect
6. #16812610	C₂	semidirect
7. #104×C₁₀₉	C₂	semidirect
9. #13182019	C₄	semidirect
10. #13182017	C₄	semidirect
11. #13182016	C₄	semidirect
12. #13182014	C₄	semidirect
51. C₁₀₉	#444	semidirect
58. #104	D₁₀₉	semidirect

H ≅ G/N

type

2. #16812707

C₂

semidirect

3. #16812706

C₂

semidirect

4. #16812634

C₂

semidirect

5. #16812627

C₂

semidirect

6. #16812610

C₂

semidirect

7. #104×C₁₀₉

C₂

semidirect

9. #13182019

C₄

semidirect

10. #13182017

C₄

semidirect

11. #13182016

C₄

semidirect

12. #13182014

C₄

semidirect

51. C₁₀₉

#444

semidirect

58. #104

D₁₀₉

semidirect

Subgroups — #20295145

Conjugacy classes of subgroups

Complements