Group #28362661 — subgroups

Subgroups — #28362661

Order 13952.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20295403	C₂	1	1	yes	maximal, characteristic
2	#20295400	C₂	1	2	yes	maximal, complemented
3	#20295374	C₂	1	2	yes	maximal, complemented
4	#20294747	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13182016	—	2	4	no	—
7	C₂×D₈₇₂	—	2	2	no	—
8	#16812696	C₂²	1	1	yes	characteristic
9	#16812696	C₂²	1	1	yes	characteristic
10	#16812619	—	2	4	no	—
11	C₄×#11724784	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
13	#106×C₁₀₉	—	2	2	no	—
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×D₄₃₆	—	2	2	no	—
16	C₂×D₄₃₆	—	2	1	no	—
17	C₂×D₄₃₆	—	2	1	no	—
18	C₂×D₄₃₆	—	2	2	no	—
19	#13182016	—	4	8	no	—
20	D₈₇₂	—	4	4	no	—
21	C₂×#11724784	D₄	1	4	yes	—
22	C₂×Q₈×C₁₀₉	—	2	2	no	—
23	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₀₉	—	2	2	no	—
25	C₄⋊C₄×C₁₀₉	—	2	2	no	—
26	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂²×D₁₀₉	—	4	1	no	—
28	C₂²×D₁₀₉	—	4	1	no	—
29	Q₈×C₁₀₉	—	4	4	no	—
30	C₂×C₄×C₁₀₉	—	4	2	no	—
31	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	D₄₃₆	—	4	2	no	—
34	D₄₃₆	—	4	2	no	—
35	D₄₃₆	—	4	4	no	—
36	D₄₃₆	—	4	4	no	—
37	C₈×C₁₀₉	SD₁₆	1	4	yes	complemented
38	#11724784	—	2	4	no	—
39	C₂²×C₁₀₉	#127	1	1	yes	characteristic
40	C₂×D₁₀₉	—	8	1	no	—
41	C₂×D₁₀₉	—	8	1	no	—
42	C₂×D₁₀₉	—	8	2	no	—
43	C₂×D₁₀₉	—	8	2	no	—
44	C₄×C₁₀₉	—	8	2	no	—
45	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
46	C₄×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
47	C₂×C₁₀₉	#492	1	1	yes	characteristic
48	C₂×C₁₀₉	#496	1	2	yes	—
49	D₁₀₉	—	16	1	no	—
50	D₁₀₉	—	16	1	no	—
51	#1715	—	109	1	no	maximal, Sylow
52	C₁₀₉	#1715	1	1	yes	characteristic, complemented, Sylow
53	#498	D₁₀₉	1	1	yes	characteristic, complemented
54	#493	—	109	1	no	—
55	#492	—	109	2	no	—
56	#486	—	109	2	no	—
57	#322	—	109	1	no	—
58	C₂×SD₁₆	—	218	4	no	—
59	C₂×D₈	—	218	2	no	—
60	C₄⋊Q₈	C₂×D₁₀₉	1	2	yes	—
61	#127	—	109	1	no	—
62	#127	—	109	1	no	—
63	#106	—	2	2	no	—
64	#102	—	218	4	no	—
65	C₄×C₈	—	109	2	no	—
66	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	218	2	no	—
69	C₂×D₄	—	218	2	no	—
70	C₂×D₄	—	218	1	no	—
71	C₂×D₄	—	218	1	no	—
72	SD₁₆	—	436	8	no	—
73	D₈	—	436	4	no	—
74	C₂×C₈	—	109	4	no	—
75	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
79	C₂³	—	436	1	no	—
80	C₂³	—	436	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	436	2	no	—
83	D₄	—	436	2	no	—
84	D₄	—	436	4	no	—
85	D₄	—	436	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
89	C₈	—	218	4	no	—
90	C₈	#13182016	1	4	yes	complemented
91	C₂²	—	872	1	no	—
92	C₂²	—	872	1	no	—
93	C₂²	—	872	2	no	—
94	C₂²	—	872	2	no	—
95	C₂²	#16812695	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#13182016	1	2	yes	—
98	C₄	#16812726	1	4	yes	—
99	C₂	—	1744	1	no	—
100	C₂	—	1744	1	no	—
101	C₂	#20295400	1	1	yes	characteristic
102	C₂	#20295412	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295403

C₂

yes

maximal, characteristic

#20295400

C₂

yes

maximal, complemented

#20295374

C₂

yes

maximal, complemented

#20294747

C₂

yes

maximal, characteristic, complemented

#498×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182016

—

C₂×D₈₇₂

—

#16812696

C₂²

yes

characteristic

#16812696

C₂²

yes

characteristic

#16812619

—

C₄×#11724784

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

#13182016

—

D₈₇₂

—

C₂×#11724784

D₄

yes

—

C₂×Q₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₈×C₁₀₉

SD₁₆

yes

complemented

#11724784

—

C₂²×C₁₀₉

#127

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂×C₁₀₉

#492

yes

characteristic

C₂×C₁₀₉

#496

yes

—

D₁₀₉

—

D₁₀₉

—

#1715

—

109

maximal, Sylow

C₁₀₉

#1715

yes

characteristic, complemented, Sylow

#498

D₁₀₉

yes

characteristic, complemented

#493

—

109

—

#492

—

109

—

#486

—

109

—

#322

—

109

—

C₂×SD₁₆

—

218

—

C₂×D₈

—

218

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

#127

—

109

—

#127

—

109

—

#106

—

#102

—

218

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

SD₁₆

—

436

—

D₈

—

436

—

C₂×C₈

—

109

—

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

436

—

C₂³

—

436

—

Q₈

—

D₄

—

436

—

D₄

—

436

—

D₄

—

436

—

D₄

—

436

—

C₂×C₄

—

C₂×C₄

D₄×D₁₀₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₈

—

218

—

C₈

#13182016

yes

complemented

C₂²

—

872

—

C₂²

—

872

—

C₂²

—

872

—

C₂²

—

872

—

C₂²

#16812695

yes

characteristic

C₄

—

C₄

C₂×#13182016

yes

—

C₄

#16812726

yes

—

C₂

—

1744

—

100

C₂

—

1744

—

101

C₂

#20295400

yes

characteristic

102

C₂

#20295412

yes

—

	subgroup	G/H
center	C₂²	#16812695
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#498×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#127
prime radical	#20295403	C₂
hypercenter	#498	D₁₀₉
hypocenter	C₁₀₉	#1715
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#16812695

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#498×C₁₀₉

C₂

C₂²×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #20295400	C₂	semidirect
3. #20295374	C₂	semidirect
4. #20294747	C₂	semidirect
5. #498×C₁₀₉	C₂	semidirect
37. C₈×C₁₀₉	SD₁₆	semidirect
52. C₁₀₉	#1715	semidirect
53. #498	D₁₀₉	semidirect
90. C₈	#13182016	semidirect

H ≅ G/N

type

2. #20295400

C₂

semidirect

3. #20295374

C₂

semidirect

4. #20294747

C₂

semidirect

5. #498×C₁₀₉

C₂

semidirect

37. C₈×C₁₀₉

SD₁₆

semidirect

52. C₁₀₉

#1715

semidirect

53. #498

D₁₀₉

semidirect

90. C₈

#13182016

semidirect

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

subgroup	index
#20295403	2
#20295400	2
#20295374	2
#20294747	2
#498×C₁₀₉	2
#1715	109

subgroup

index

#498×C₁₀₉

109

Subgroups — #28362661

Conjugacy classes of subgroups

Complements

Maximal subgroups