Group #28365290 — subgroups

Subgroups — #28365290

Order 13952.

184 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 53 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×Dic₈₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812643	C₂	1	2	yes	maximal, complemented
3	#20295409	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₃₆	C₂²	1	2	yes	—
7	C₂×#13182017	—	2	2	no	—
8	#16812686	—	2	2	no	—
9	Dic₈₇₂	—	2	2	no	—
10	#16812643	—	2	4	no	—
11	C₂×#13182004	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
14	D₁₆×C₁₀₉	—	2	2	no	—
15	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₈	—	2	2	no	—
17	Dic₄₃₆	—	2	2	no	—
18	Dic₄₃₆	—	2	2	no	—
19	#13182017	—	4	4	no	—
20	C₂×#11724784	D₄	1	1	yes	characteristic
21	C₂²⋊Dic₁₀₉	—	4	4	no	—
22	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
23	#13182004	D₄	1	2	yes	—
24	C₂×D₄×C₁₀₉	—	2	2	no	—
25	D₈×C₁₀₉	—	2	2	no	—
26	D₈×C₁₀₉	—	2	2	no	—
27	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₁₆×C₁₀₉	D₄	1	2	yes	—
29	C₂³×C₁₀₉	—	4	2	no	—
30	D₄×C₁₀₉	—	4	2	no	—
31	D₄×C₁₀₉	—	4	2	no	—
32	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₁₀₉	—	4	2	no	—
34	C₂×Dic₁₀₉	D₈	1	2	yes	—
35	Dic₂₁₈	—	4	2	no	—
36	Dic₂₁₈	—	4	2	no	—
37	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	#11724784	D₈	1	2	yes	—
40	C₂²×C₁₀₉	—	8	2	no	—
41	C₂²×C₁₀₉	—	8	4	no	—
42	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
43	C₄×C₁₀₉	#127	1	1	yes	characteristic
44	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
45	Dic₁₀₉	—	2	2	no	—
46	Dic₁₀₉	—	8	2	no	—
47	C₂×C₁₀₉	—	16	2	no	—
48	C₂×C₁₀₉	#493	1	1	yes	characteristic
49	C₂×C₁₀₉	#508	1	2	yes	—
50	#2249	—	109	1	no	maximal, Sylow
51	C₁₀₉	#2249	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₃₂	—	109	1	no	—
53	C₂×SD₃₂	—	109	2	no	—
54	C₂×D₁₆	D₁₀₉	1	1	yes	characteristic
55	#495	—	109	2	no	—
56	C₄×C₁₆	—	109	1	no	—
57	C₂×Q₁₆	—	109	2	no	—
58	C₂×SD₁₆	—	218	2	no	—
59	C₂×D₈	C₂×D₁₀₉	1	2	yes	—
60	#124	—	218	2	no	—
61	Q₃₂	—	218	2	no	—
62	SD₃₂	—	218	4	no	—
63	D₁₆	—	2	2	no	—
64	C₂×C₁₆	—	109	1	no	—
65	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
66	C₄×C₈	—	109	1	no	—
67	C₂×Q₈	—	218	2	no	—
68	C₂×D₄	—	2	2	no	—
69	Q₁₆	—	218	2	no	—
70	Q₁₆	—	218	2	no	—
71	SD₁₆	—	436	4	no	—
72	D₈	—	2	2	no	—
73	D₈	—	2	2	no	—
74	C₂×C₈	—	109	1	no	—
75	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
76	C₂²⋊C₄	—	436	4	no	—
77	C₄²	—	109	1	no	—
78	C₁₆	—	109	2	no	—
79	C₁₆	C₂²⋊D₁₀₉	1	2	yes	—
80	C₂³	—	4	2	no	—
81	Q₈	—	436	2	no	—
82	Q₈	—	436	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	2	no	—
85	C₂×C₄	—	109	2	no	—
86	C₂×C₄	—	436	2	no	—
87	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
88	C₈	—	109	2	no	—
89	C₈	D₄×D₁₀₉	1	1	yes	characteristic
90	C₈	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
91	C₂²	—	8	4	no	—
92	C₂²	—	8	2	no	—
93	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
94	C₄	—	872	2	no	—
95	C₄	—	218	2	no	—
96	C₄	#16812695	1	1	yes	characteristic
97	C₄	D₈×D₁₀₉	1	1	yes	characteristic
98	C₂	—	16	2	no	—
99	C₂	#20295401	1	1	yes	characteristic
100	C₂	#20295455	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₈₇₂

C₂

yes

maximal, characteristic, complemented

C₂×#16812643

C₂

yes

maximal, complemented

#20295409

C₂

yes

maximal, complemented

C₁₆×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₄₃₆

C₂²

yes

—

C₂×#13182017

—

#16812686

—

Dic₈₇₂

—

#16812643

—

C₂×#13182004

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×D₈×C₁₀₉

C₂²

yes

—

D₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

Dic₄₃₆

—

Dic₄₃₆

—

#13182017

—

C₂×#11724784

D₄

yes

characteristic

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

#13182004

D₄

yes

—

C₂×D₄×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₁₆×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₈

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

C₂×D₄

yes

characteristic

#11724784

D₈

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₄×C₁₀₉

#127

yes

characteristic

C₄×C₁₀₉

C₂×D₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#493

yes

characteristic

C₂×C₁₀₉

#508

yes

—

#2249

—

109

maximal, Sylow

C₁₀₉

#2249

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

109

—

C₂×SD₃₂

—

109

—

C₂×D₁₆

D₁₀₉

yes

characteristic

#495

—

109

—

C₄×C₁₆

—

109

—

C₂×Q₁₆

—

109

—

C₂×SD₁₆

—

218

—

C₂×D₈

C₂×D₁₀₉

yes

—

#124

—

218

—

Q₃₂

—

218

—

SD₃₂

—

218

—

D₁₆

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

C₄×C₈

—

109

—

C₂×Q₈

—

218

—

C₂×D₄

—

Q₁₆

—

218

—

Q₁₆

—

218

—

SD₁₆

—

436

—

D₈

—

D₈

—

C₂×C₈

—

109

—

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

436

—

C₄²

—

109

—

C₁₆

—

109

—

C₁₆

C₂²⋊D₁₀₉

yes

—

C₂³

—

Q₈

—

436

—

Q₈

—

436

—

D₄

—

D₄

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

D₄×D₁₀₉

yes

characteristic

C₈

—

109

—

C₈

D₄×D₁₀₉

yes

characteristic

C₈

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂²

—

C₂²

—

C₂²

D₈×D₁₀₉

yes

characteristic

C₄

—

872

—

C₄

—

218

—

C₄

#16812695

yes

characteristic

C₄

D₈×D₁₀₉

yes

characteristic

C₂

—

C₂

#20295401

yes

characteristic

100

C₂

#20295455

yes

—

	subgroup	G/H
center	C₂²	D₈×D₁₀₉
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting, prime radical	C₂×D₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₈
hypercenter	C₂×D₁₆	D₁₀₉
hypocenter	C₁₀₉	#2249
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

subgroup

G/H

center

C₂²

D₈×D₁₀₉

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting, prime radical

C₂×D₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₈

hypercenter

C₂×D₁₆

D₁₀₉

hypocenter

C₁₀₉

#2249

LCS

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. C₂×Dic₈₇₂	C₂	semidirect
2. C₂×#16812643	C₂	semidirect
3. #20295409	C₂	semidirect
4. C₁₆×Dic₁₀₉	C₂	semidirect
51. C₁₀₉	#2249	semidirect

H ≅ G/N

type

1. C₂×Dic₈₇₂

C₂

semidirect

2. C₂×#16812643

C₂

semidirect

3. #20295409

C₂

semidirect

4. C₁₆×Dic₁₀₉

C₂

semidirect

51. C₁₀₉

#2249

semidirect

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

subgroup	index
C₂×Dic₈₇₂	2
C₂×#16812643	2
#20295409	2
C₁₆×Dic₁₀₉	2
C₂×D₁₆×C₁₀₉	2
#2249	109

subgroup

index

C₂×Dic₈₇₂

C₂×#16812643

#20295409

C₁₆×Dic₁₀₉

C₂×D₁₆×C₁₀₉

#2249

109

Subgroups — #28365290

Conjugacy classes of subgroups

Complements

Maximal subgroups