Group #28365203 — subgroups

Subgroups — #28365203

Order 13952.

126 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 49 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	#20295424	C₂	1	1	yes	maximal, characteristic, complemented
2	#20294850	C₂	1	2	yes	maximal, complemented
3	#20294847	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294836	C₂	1	2	yes	maximal
5	#443×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812699	—	2	1	no	—
7	#16812639	C₂²	1	2	yes	—
8	#16812631	—	2	1	no	—
9	#16812631	C₂²	1	2	yes	—
10	#135×C₁₀₉	C₂²	1	1	yes	characteristic
11	#131×C₁₀₉	—	2	1	no	—
12	#108×C₁₀₉	C₂²	1	1	yes	characteristic
13	#104×C₁₀₉	—	2	1	no	—
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₈	—	4	1	no	—
16	C₄⋊Dic₁₀₉	—	2	2	no	—
17	C₄⋊Dic₁₀₉	—	4	1	no	—
18	#13182004	—	2	2	no	—
19	#41×C₁₀₉	—	2	1	no	—
20	Q₁₆×C₁₀₉	Q₈	1	1	yes	characteristic, complemented
21	SD₁₆×C₁₀₉	—	2	1	no	—
22	D₈×C₁₀₉	Q₈	1	1	yes	characteristic, complemented
23	M₁₆×C₁₀₉	—	2	1	no	—
24	M₁₆×C₁₀₉	—	4	1	no	—
25	C₂×C₈×C₁₀₉	—	4	1	no	—
26	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
29	Q₈×C₁₀₉	—	2	1	no	—
30	D₄×C₁₀₉	—	2	1	no	—
31	D₄×C₁₀₉	—	4	1	no	—
32	C₂×C₄×C₁₀₉	—	2	1	no	—
33	C₂×C₄×C₁₀₉	—	4	1	no	—
34	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	—	4	2	no	—
36	Dic₂₁₈	—	8	2	no	—
37	C₈×C₁₀₉	—	4	1	no	—
38	C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
39	C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
40	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₁₀₉	#41	1	1	yes	characteristic
42	C₂²×C₁₀₉	—	4	1	no	—
43	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
44	C₄×C₁₀₉	—	2	2	no	—
45	C₄×C₁₀₉	—	4	1	no	—
46	C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
47	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
48	Dic₁₀₉	—	8	2	no	—
49	C₂×C₁₀₉	—	2	1	no	—
50	C₂×C₁₀₉	—	8	1	no	—
51	C₂×C₁₀₉	#476	1	1	yes	characteristic
52	#2230	—	109	1	no	maximal, Sylow
53	C₁₀₉	#2230	1	1	yes	characteristic, complemented, Sylow
54	#498	—	109	1	no	—
55	#443	D₁₀₉	1	1	yes	characteristic
56	#365	—	109	2	no	—
57	#364	—	109	1	no	—
58	#360	—	109	2	no	—
59	#135	C₂×D₁₀₉	1	1	yes	characteristic
60	#131	—	2	1	no	—
61	C₄⋊Q₈	—	218	1	no	—
62	M₃₂	—	109	2	no	—
63	#108	C₂×D₁₀₉	1	1	yes	characteristic
64	#106	—	109	2	no	—
65	#106	—	218	1	no	—
66	#104	—	2	1	no	—
67	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
68	#41	—	2	1	no	—
69	C₂×Q₈	—	436	1	no	—
70	Q₁₆	Dic₂₁₈	1	1	yes	characteristic, complemented
71	SD₁₆	—	2	1	no	—
72	D₈	Dic₂₁₈	1	1	yes	characteristic, complemented
73	M₁₆	—	2	1	no	—
74	M₁₆	—	4	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
77	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
78	C₄⋊C₄	—	436	1	no	—
79	C₄⋊C₄	—	218	2	no	—
80	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
81	C₁₆	—	218	2	no	—
82	Q₈	—	2	1	no	—
83	Q₈	—	872	2	no	—
84	D₄	—	2	1	no	—
85	D₄	—	4	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	436	2	no	—
89	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
90	C₈	—	4	1	no	—
91	C₈	#13182017	1	1	yes	characteristic
92	C₈	#13182017	1	1	yes	characteristic
93	C₈	C₂×Dic₂₁₈	1	1	yes	characteristic
94	C₈	#13182029	1	1	yes	characteristic
95	C₂²	—	4	1	no	—
96	C₂²	#16812724	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	4	1	no	—
99	C₄	—	872	2	no	—
100	C₄	#16812678	1	1	yes	characteristic
101	C₄	C₂×#13182017	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	8	1	no	—
104	C₂	#20295316	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295424

C₂

yes

maximal, characteristic, complemented

#20294850

C₂

yes

maximal, complemented

#20294847

C₂

yes

maximal, characteristic, complemented

#20294836

C₂

yes

maximal

#443×C₁₀₉

C₂

yes

maximal, characteristic

#16812699

—

#16812639

C₂²

yes

—

#16812631

—

#16812631

C₂²

yes

—

#135×C₁₀₉

C₂²

yes

characteristic

#131×C₁₀₉

—

#108×C₁₀₉

C₂²

yes

characteristic

#104×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

#13182004

—

#41×C₁₀₉

—

Q₁₆×C₁₀₉

Q₈

yes

characteristic, complemented

SD₁₆×C₁₀₉

—

D₈×C₁₀₉

Q₈

yes

characteristic, complemented

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

SD₁₆

yes

characteristic

C₈×C₁₀₉

SD₁₆

yes

characteristic

C₈×C₁₀₉

C₂×Q₈

yes

characteristic

C₈×C₁₀₉

#41

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#476

yes

characteristic

#2230

—

109

maximal, Sylow

C₁₀₉

#2230

yes

characteristic, complemented, Sylow

#498

—

109

—

#443

D₁₀₉

yes

characteristic

#365

—

109

—

#364

—

109

—

#360

—

109

—

#135

C₂×D₁₀₉

yes

characteristic

#131

—

C₄⋊Q₈

—

218

—

M₃₂

—

109

—

#108

C₂×D₁₀₉

yes

characteristic

#106

—

109

—

#106

—

218

—

#104

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

#41

—

C₂×Q₈

—

436

—

Q₁₆

Dic₂₁₈

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₂₁₈

yes

characteristic, complemented

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₁₆

—

218

—

Q₈

—

Q₈

—

872

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₈

—

C₈

yes

C₈

yes

C₈

C₂×Dic₂₁₈

yes

characteristic

C₈

#13182029

yes

characteristic

C₂²

—

C₂²

#16812724

yes

characteristic

C₄

—

C₄

—

C₄

—

872

—

100

C₄

#16812678

yes

characteristic

101

C₄

C₂×#13182017

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

#20295316

yes

characteristic

	subgroup	G/H
center	C₂	#20295316
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#443×C₁₀₉	C₂
socle	C₂×C₁₀₉	#476
prime radical	#135×C₁₀₉	C₂²
hypercenter	#443	D₁₀₉
hypocenter	C₁₀₉	#2230
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

subgroup

G/H

center

C₂

#20295316

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#443×C₁₀₉

C₂

socle

C₂×C₁₀₉

#476

prime radical

#135×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

N	H ≅ G/N	type
1. #20295424	C₂	semidirect
2. #20294850	C₂	semidirect
3. #20294847	C₂	semidirect
20. Q₁₆×C₁₀₉	Q₈	semidirect
22. D₈×C₁₀₉	Q₈	semidirect
53. C₁₀₉	#2230	semidirect
70. Q₁₆	Dic₂₁₈	semidirect
72. D₈	Dic₂₁₈	semidirect

H ≅ G/N

type

1. #20295424

C₂

semidirect

2. #20294850

C₂

semidirect

3. #20294847

C₂

semidirect

20. Q₁₆×C₁₀₉

Q₈

semidirect

22. D₈×C₁₀₉

Q₈

semidirect

53. C₁₀₉

#2230

semidirect

70. Q₁₆

Dic₂₁₈

semidirect

72. D₈

Dic₂₁₈

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295424	2
#20294850	2
#20294847	2
#20294836	2
#443×C₁₀₉	2
#2230	109

subgroup

index

#443×C₁₀₉

109

Subgroups — #28365203

Conjugacy classes of subgroups

Complements

Maximal subgroups