Group #20295266 — subgroups

Subgroups — #20295266

Order 6976.

114 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 41 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	C₂×#13182019	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#16812679	C₂	1	1	yes	maximal, characteristic
4	#16812675	C₂	1	1	yes	maximal, characteristic
5	#16812632	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812624	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
10	#13182019	—	2	2	no	—
11	C₂×#11724784	C₂²	1	1	yes	characteristic
12	C₈×D₁₀₉	—	2	2	no	—
13	C₄⋊Dic₁₀₉	—	2	1	no	—
14	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
15	#13182008	—	2	1	no	—
16	#13182006	—	2	1	no	—
17	#13182006	—	2	1	no	—
18	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
22	Q₈×C₁₀₉	—	2	1	no	—
23	Q₈×C₁₀₉	—	2	1	no	—
24	C₂×C₄×C₁₀₉	—	2	1	no	—
25	C₂×C₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₀₉	—	2	1	no	—
28	C₂×Dic₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
30	C₄×D₁₀₉	—	2	1	no	—
31	C₄×D₁₀₉	—	2	1	no	—
32	Dic₂₁₈	—	2	1	no	—
33	Dic₂₁₈	—	2	1	no	—
34	C₈×C₁₀₉	—	2	1	no	—
35	#11724784	D₄	1	2	yes	—
36	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×D₁₀₉	—	2	1	no	—
38	C₂×D₁₀₉	—	2	1	no	—
39	C₂×D₁₀₉	Q₁₆	1	2	yes	complemented
40	C₄×C₁₀₉	—	4	1	no	—
41	C₄×C₁₀₉	—	4	1	no	—
42	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₄×C₁₀₉	#41	1	1	yes	characteristic
44	Dic₁₀₉	—	2	1	no	—
45	Dic₁₀₉	—	4	1	no	—
46	Dic₁₀₉	—	4	1	no	—
47	C₂×C₁₀₉	#121	1	1	yes	characteristic
48	C₂×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₁₀₉	#135	1	1	yes	characteristic
50	D₁₀₉	—	2	2	no	—
51	C₁₀₉	#467	1	1	yes	characteristic, complemented, Sylow
52	#467	—	109	1	no	maximal, Sylow
53	C₂×Q₁₆	—	109	1	no	—
54	C₂²×C₈	—	109	1	no	—
55	C₂²⋊Q₈	—	109	1	no	—
56	C₂²⋊Q₈	—	109	1	no	—
57	#107	—	109	1	no	—
58	#103	—	109	1	no	—
59	#103	D₁₀₉	1	1	yes	characteristic, complemented
60	C₂×Q₈	—	109	1	no	—
61	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
62	C₂²×C₄	—	109	1	no	—
63	Q₁₆	—	218	2	no	—
64	C₂×C₈	—	109	1	no	—
65	C₂×C₈	—	218	2	no	—
66	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
67	C₄⋊C₄	—	109	1	no	—
68	C₄⋊C₄	—	218	1	no	—
69	C₄⋊C₄	—	218	1	no	—
70	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
71	C₂²⋊C₄	—	218	1	no	—
72	C₂²⋊C₄	—	218	1	no	—
73	C₂³	—	109	1	no	—
74	Q₈	—	2	1	no	—
75	Q₈	—	2	1	no	—
76	Q₈	—	218	1	no	—
77	Q₈	—	218	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	109	1	no	—
81	C₂×C₄	—	218	1	no	—
82	C₂×C₄	—	218	1	no	—
83	C₂×C₄	—	218	1	no	—
84	C₂×C₄	—	218	1	no	—
85	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	109	2	no	—
88	C₂²	—	109	2	no	—
89	C₂²	—	218	1	no	—
90	C₂²	—	218	1	no	—
91	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
92	C₄	—	4	1	no	—
93	C₄	—	4	1	no	—
94	C₄	—	436	1	no	—
95	C₄	—	436	1	no	—
96	C₄	—	218	1	no	—
97	C₄	D₄×D₁₀₉	1	1	yes	characteristic
98	C₄	#13182029	1	1	yes	characteristic
99	C₂	—	218	2	no	—
100	C₂	#16812672	1	1	yes	characteristic
101	C₂	Q₁₆×D₁₀₉	1	1	yes	characteristic
102	C₂	#16812720	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182019

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#16812679

C₂

yes

maximal, characteristic

#16812675

C₂

yes

maximal, characteristic

#16812632

C₂

yes

maximal, characteristic, complemented

#16812624

C₂

yes

maximal, characteristic, complemented

#103×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₈

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182019

—

C₂×#11724784

C₂²

yes

characteristic

C₈×D₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂²

yes

—

—

—

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×D₁₀₉

D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

D₄

yes

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

Q₁₆

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#121

yes

characteristic

C₂×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₂×C₁₀₉

#135

yes

characteristic

D₁₀₉

—

C₁₀₉

#467

yes

characteristic, complemented, Sylow

#467

—

109

maximal, Sylow

C₂×Q₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

109

—

#107

—

109

—

#103

—

109

—

#103

D₁₀₉

yes

characteristic, complemented

C₂×Q₈

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

Q₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂³

—

109

—

Q₈

—

Q₈

—

Q₈

—

218

—

Q₈

—

218

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

109

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

D₄×D₁₀₉

yes

characteristic

C₄

—

C₄

—

C₄

—

436

—

C₄

—

436

—

C₄

—

218

—

C₄

D₄×D₁₀₉

yes

characteristic

C₄

#13182029

yes

characteristic

C₂

—

218

—

100

C₂

#16812672

yes

characteristic

101

C₂

Q₁₆×D₁₀₉

yes

characteristic

102

C₂

#16812720

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#103×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#103	D₁₀₉
hypocenter	C₁₀₉	#467
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#103×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
1. C₂×#13182019	C₂	semidirect
5. #16812632	C₂	semidirect
6. #16812624	C₂	semidirect
7. #103×C₁₀₉	C₂	semidirect
39. C₂×D₁₀₉	Q₁₆	semidirect
51. C₁₀₉	#467	semidirect
59. #103	D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#13182019

C₂

semidirect

5. #16812632

C₂

semidirect

6. #16812624

C₂

semidirect

7. #103×C₁₀₉

C₂

semidirect

39. C₂×D₁₀₉

Q₁₆

semidirect

51. C₁₀₉

#467

semidirect

59. #103

D₁₀₉

semidirect

Subgroups — #20295266

Conjugacy classes of subgroups

Complements