Group #28379427 — subgroups

Subgroups — Q₃₂×#89137 #28379427

Order 13952.

150 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 50 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	Q₁₆×#89137	C₂	1	2	yes	maximal
2	#20295952	C₂	1	1	yes	maximal, characteristic
3	#20295939	C₂	1	2	yes	maximal
4	C₁₆×#89137	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×#89137	—	2	2	no	—
7	D₁₀₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16812768	C₂²	1	2	yes	—
9	#16812762	—	2	2	no	—
10	C₈×#89137	C₂²	1	1	yes	characteristic
11	Q₁₆×D₁₀₉	C₂²	1	2	yes	—
12	Dic₈₇₂	C₄	1	1	yes	characteristic, complemented
13	#16812645	C₄	1	2	yes	complemented
14	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
15	Q₃₂×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	D₁₀₉⋊C₈	D₄	1	1	yes	characteristic
17	C₄⋊#89137	—	2	2	no	—
18	C₄⋊#89137	—	4	2	no	—
19	C₄×#89137	—	4	2	no	—
20	C₄×#89137	D₄	1	1	yes	characteristic
21	#13182032	—	2	1	no	—
22	Q₈×D₁₀₉	—	2	2	no	—
23	Dic₄₃₆	C₂×C₄	1	2	yes	—
24	#13182019	—	2	2	no	—
25	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
26	#13182004	C₂×C₄	1	1	yes	characteristic
27	Q₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×#89137	—	4	2	no	—
30	C₂×#89137	D₈	1	1	yes	characteristic
31	C₂×#89137	D₈	1	1	yes	characteristic
32	Q₈×C₁₀₉	—	2	2	no	—
33	C₄×D₁₀₉	—	4	2	no	—
34	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
35	Dic₂₁₈	—	2	2	no	—
36	Dic₂₁₈	—	4	2	no	—
37	#11724786	—	2	1	no	—
38	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	#11724784	#41	1	1	yes	characteristic
40	C₂×D₁₀₉	C₂×D₈	1	1	yes	characteristic
41	#89137	—	2	1	no	—
42	#89137	—	8	2	no	—
43	#89137	Q₃₂	1	2	yes	direct product factor
44	C₄×C₁₀₉	—	4	2	no	—
45	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
46	Dic₁₀₉	—	4	2	no	—
47	Dic₁₀₉	#135	1	1	yes	characteristic
48	C₂×C₁₀₉	C₄×D₈	1	1	yes	characteristic
49	D₁₀₉	C₂×Q₃₂	1	1	yes	characteristic
50	D₁₀₉	#508	1	1	yes	characteristic
51	C₄×Q₃₂	—	109	1	no	maximal, Sylow
52	C₁₀₉	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₃₂	—	109	1	no	—
54	C₄×Q₁₆	—	109	2	no	—
55	#366	—	109	1	no	—
56	#358	—	109	2	no	—
57	C₄×C₁₆	—	109	1	no	—
58	C₂×Q₁₆	—	109	2	no	—
59	C₄×Q₈	—	218	2	no	—
60	Q₃₂	—	109	2	no	—
61	Q₃₂	—	109	1	no	—
62	Q₃₂	#89137	1	1	yes	characteristic, direct product factor
63	C₂×C₁₆	—	109	1	no	—
64	C₂×C₁₆	—	109	1	no	—
65	#107	—	109	2	no	—
66	#103	—	218	2	no	—
67	C₄×C₈	—	109	1	no	—
68	C₂×Q₈	—	218	2	no	—
69	Q₁₆	—	109	2	no	—
70	Q₁₆	—	218	2	no	—
71	Q₁₆	C₂×#89137	1	2	yes	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	—	109	1	no	—
74	C₄⋊C₄	—	436	2	no	—
75	C₄⋊C₄	—	218	2	no	—
76	C₄²	—	109	1	no	—
77	C₄²	—	436	2	no	—
78	C₁₆	—	109	1	no	—
79	C₁₆	—	218	1	no	—
80	C₁₆	C₂×#89137	1	1	yes	characteristic
81	Q₈	—	2	2	no	—
82	Q₈	—	436	2	no	—
83	Q₈	—	218	2	no	—
84	C₂×C₄	—	109	1	no	—
85	C₂×C₄	—	109	1	no	—
86	C₂×C₄	—	109	1	no	—
87	C₂×C₄	—	436	2	no	—
88	C₂×C₄	—	436	2	no	—
89	C₈	—	109	1	no	—
90	C₈	—	218	1	no	—
91	C₈	C₂²×#89137	1	1	yes	characteristic
92	C₂²	—	109	1	no	—
93	C₄	—	4	2	no	—
94	C₄	—	872	2	no	—
95	C₄	—	109	2	no	—
96	C₄	—	109	1	no	—
97	C₄	—	436	2	no	—
98	C₄	—	218	1	no	—
99	C₄	D₄×#89137	1	1	yes	characteristic
100	C₂	—	109	1	no	—
101	C₂	—	109	1	no	—
102	C₂	D₈×#89137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₁₆×#89137

C₂

yes

maximal

#20295952

C₂

yes

maximal, characteristic

#20295939

C₂

yes

maximal

C₁₆×#89137

C₂

yes

maximal, characteristic

Q₃₂×D₁₀₉

C₂

yes

maximal, characteristic

Q₈×#89137

—

D₁₀₉⋊C₁₆

C₂²

yes

characteristic

#16812768

C₂²

yes

—

#16812762

—

C₈×#89137

C₂²

yes

characteristic

Q₁₆×D₁₀₉

C₂²

yes

—

Dic₈₇₂

C₄

yes

characteristic, complemented

#16812645

C₄

yes

complemented

C₁₆×D₁₀₉

C₂²

yes

characteristic

Q₃₂×C₁₀₉

C₄

yes

characteristic, complemented

D₁₀₉⋊C₈

D₄

yes

—

—

—

D₄

yes

characteristic

#13182032

—

Q₈×D₁₀₉

—

Dic₄₃₆

C₂×C₄

yes

—

#13182019

—

C₈×D₁₀₉

C₂³

yes

characteristic

#13182004

C₂×C₄

yes

characteristic

Q₁₆×C₁₀₉

C₂×C₄

yes

—

C₁₆×C₁₀₉

C₂×C₄

yes

characteristic

C₂×#89137

—

C₂×#89137

D₈

yes

characteristic

C₂×#89137

D₈

yes

characteristic

Q₈×C₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×D₄

yes

characteristic

Dic₂₁₈

—

Dic₂₁₈

—

#11724786

—

C₈×C₁₀₉

C₂²×C₄

yes

yes

C₂×D₁₀₉

C₂×D₈

yes

—

—

Q₃₂

yes

direct product factor

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

#135

yes

characteristic

C₂×C₁₀₉

C₄×D₈

yes

characteristic

D₁₀₉

C₂×Q₃₂

yes

characteristic

D₁₀₉

#508

yes

characteristic

C₄×Q₃₂

—

109

maximal, Sylow

C₁₀₉

C₄×Q₃₂

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

109

—

C₄×Q₁₆

—

109

—

#366

—

109

—

#358

—

109

—

C₄×C₁₆

—

109

—

C₂×Q₁₆

—

109

—

C₄×Q₈

—

218

—

Q₃₂

—

109

—

Q₃₂

—

109

—

Q₃₂

#89137

yes

characteristic, direct product factor

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

#107

—

109

—

#103

—

218

—

C₄×C₈

—

109

—

C₂×Q₈

—

218

—

Q₁₆

—

109

—

Q₁₆

—

218

—

Q₁₆

C₂×#89137

yes

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

436

—

C₁₆

—

109

—

C₁₆

—

218

—

C₁₆

C₂×#89137

yes

characteristic

Q₈

—

Q₈

—

436

—

Q₈

—

218

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₈

—

109

—

C₈

—

218

—

C₈

C₂²×#89137

yes

characteristic

C₂²

—

109

—

C₄

—

C₄

—

872

—

C₄

—

109

—

C₄

—

109

—

C₄

—

436

—

C₄

—

218

—

C₄

D₄×#89137

yes

characteristic

100

C₂

—

109

—

101

C₂

—

109

—

102

C₂

D₈×#89137

yes

characteristic

	subgroup	G/H
center	C₂	D₈×#89137
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₈	C₂²×#89137
Fitting	Q₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₄×D₈
prime radical	C₂×D₁₀₉	C₂×D₈
cosocle	C₈×D₁₀₉	C₂³
hypercenter	Q₃₂	#89137
hypocenter	C₁₀₉	C₄×Q₃₂
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

subgroup

G/H

center

C₂

D₈×#89137

commutator

C₈×C₁₀₉

C₂²×C₄

Frattini

C₈

C₂²×#89137

Fitting

Q₃₂×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₄×D₈

prime radical

C₂×D₁₀₉

C₂×D₈

cosocle

C₈×D₁₀₉

C₂³

hypercenter

Q₃₂

#89137

hypocenter

C₁₀₉

C₄×Q₃₂

LCS

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

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

N	H ≅ G/N	type
12. Dic₈₇₂	C₄	semidirect
13. #16812645	C₄	semidirect
15. Q₃₂×C₁₀₉	C₄	semidirect
43. #89137	Q₃₂	direct (N × H)
52. C₁₀₉	C₄×Q₃₂	semidirect
62. Q₃₂	#89137	direct (N × H)

H ≅ G/N

type

12. Dic₈₇₂

C₄

semidirect

13. #16812645

C₄

semidirect

15. Q₃₂×C₁₀₉

C₄

semidirect

43. #89137

Q₃₂

direct (N × H)

52. C₁₀₉

C₄×Q₃₂

semidirect

62. Q₃₂

#89137

direct (N × H)

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

subgroup	index
Q₁₆×#89137	2
#20295952	2
#20295939	2
C₁₆×#89137	2
Q₃₂×D₁₀₉	2
C₄×Q₃₂	109

subgroup

index

Q₁₆×#89137

#20295952

#20295939

C₁₆×#89137

Q₃₂×D₁₀₉

C₄×Q₃₂

109

Subgroups — Q₃₂×#89137 #28379427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q32×#89137 #28379427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₃₂×#89137 #28379427