Group #31806554 — subgroups

Subgroups — #31806554

Order 17792.

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

normal subgroups only

This group has 162 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	M₁₆×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#22580725	C₂	1	2	yes	maximal
3	#22580714	C₂	1	2	yes	maximal
4	#22580710	C₂	1	1	yes	maximal, characteristic
5	#338×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	1	yes	characteristic
7	#17984801	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
9	#17984752	C₄	1	2	yes	complemented
10	C₈×Dic₁₃₉	C₄	1	2	yes	complemented
11	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
12	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₃₉	—	2	1	no	—
14	#14555769	—	2	2	no	—
15	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₃₉	—	2	2	no	—
17	C₂²⋊Dic₁₃₉	—	2	2	no	—
18	C₄×Dic₁₃₉	—	2	1	no	—
19	C₄×Dic₁₃₉	—	2	2	no	—
20	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₃₉	—	2	2	no	—
24	M₁₆×C₁₃₉	—	2	2	no	—
25	M₁₆×C₁₃₉	—	2	2	no	—
26	C₂×C₈×C₁₃₉	—	2	2	no	—
27	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
28	C₂×Dic₁₃₉	—	2	1	no	—
29	C₂×Dic₁₃₉	—	2	1	no	—
30	C₂×Dic₁₃₉	—	4	2	no	—
31	C₂×Dic₁₃₉	—	4	2	no	—
32	C₁₃₉⋊C₈	—	2	2	no	—
33	C₂³×C₁₃₉	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₉	#41	1	2	yes	—
36	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
37	C₂×C₄×C₁₃₉	#41	1	2	yes	—
38	C₈×C₁₃₉	—	2	2	no	—
39	C₈×C₁₃₉	—	4	2	no	—
40	C₂²×C₁₃₉	—	4	1	no	—
41	C₂²×C₁₃₉	#117	1	1	yes	characteristic
42	C₂²×C₁₃₉	#126	1	2	yes	—
43	C₄×C₁₃₉	#117	1	1	yes	characteristic
44	C₄×C₁₃₉	#117	1	1	yes	characteristic
45	C₄×C₁₃₉	#126	1	2	yes	—
46	Dic₁₃₉	—	2	2	no	—
47	Dic₁₃₉	—	4	1	no	—
48	Dic₁₃₉	—	8	2	no	—
49	C₂×C₁₃₉	—	2	1	no	—
50	C₂×C₁₃₉	—	2	2	no	—
51	C₂×C₁₃₉	#383	1	1	yes	characteristic
52	C₁₃₉	#1841	1	1	yes	characteristic, complemented, Sylow
53	#1841	—	139	1	no	maximal, Sylow
54	C₄×M₁₆	—	139	1	no	—
55	#344	—	139	2	no	—
56	#339	—	139	2	no	—
57	#338	D₁₃₉	1	1	yes	characteristic
58	H(3,ℤ/4)	—	139	1	no	—
59	C₂×M₁₆	—	139	1	no	—
60	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
61	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
62	#117	—	139	2	no	—
63	C₂×C₄²	—	139	1	no	—
64	#97	—	139	2	no	—
65	C₄×C₈	—	139	2	no	—
66	C₂²×C₄	—	278	1	no	—
67	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	278	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	139	2	no	—
74	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
75	C₄⋊C₄	—	278	2	no	—
76	C₂²⋊C₄	—	278	2	no	—
77	C₄²	—	278	1	no	—
78	C₄²	—	278	2	no	—
79	C₄²	—	139	1	no	—
80	C₄²	—	139	1	no	—
81	C₂³	#14555786	1	1	yes	characteristic
82	C₂×C₄	—	278	1	no	—
83	C₂×C₄	—	278	1	no	—
84	C₂×C₄	—	556	2	no	—
85	C₂×C₄	—	556	2	no	—
86	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
87	C₂×C₄	#14555784	1	2	yes	—
88	C₂×C₄	#14555784	1	1	yes	characteristic
89	C₂×C₄	#14555785	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	278	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#17984800	1	1	yes	characteristic
95	C₂²	#17984837	1	2	yes	—
96	C₄	—	278	2	no	—
97	C₄	—	556	1	no	—
98	C₄	—	1112	2	no	—
99	C₄	#17984798	1	1	yes	characteristic
100	C₄	#17984799	1	1	yes	characteristic
101	C₄	#17984835	1	2	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#22580820	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₁₃₉

C₂

yes

maximal, characteristic

#22580725

C₂

yes

maximal

#22580714

C₂

yes

maximal

#22580710

C₂

yes

maximal, characteristic

#338×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

characteristic

#17984801

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

#17984752

C₄

yes

complemented

C₈×Dic₁₃₉

C₄

yes

complemented

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

—

#14555769

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#117

yes

characteristic

C₂²×C₁₃₉

#126

yes

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#383

yes

characteristic

C₁₃₉

#1841

yes

characteristic, complemented, Sylow

#1841

—

139

maximal, Sylow

C₄×M₁₆

—

139

—

#344

—

139

—

#339

—

139

—

#338

D₁₃₉

yes

characteristic

H(3,ℤ/4)

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

#117

—

139

—

C₂×C₄²

—

139

—

#97

—

139

—

C₄×C₈

—

139

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

278

—

C₂×C₈

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

—

C₄⋊C₄

—

278

—

C₂²⋊C₄

—

278

—

C₄²

—

278

—

C₄²

—

278

—

C₄²

—

139

—

C₄²

—

139

—

C₂³

#14555786

yes

characteristic

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

556

—

C₂×C₄

—

556

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

C₂×C₄

#14555784

yes

—

C₂×C₄

#14555784

yes

characteristic

C₂×C₄

#14555785

yes

—

C₈

—

C₈

—

C₈

—

278

—

C₂²

—

C₂²

#17984800

yes

characteristic

C₂²

#17984837

yes

—

C₄

—

278

—

C₄

—

556

—

C₄

—

1112

—

C₄

#17984798

yes

characteristic

100

C₄

#17984799

yes

characteristic

101

C₄

#17984835

yes

—

102

C₂

—

103

C₂

—

104

C₂

#22580820

yes

characteristic

	subgroup	G/H
center	C₄	#17984799
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#338×C₁₃₉	C₂
socle	C₂×C₁₃₉	#383
prime radical	C₂³×C₁₃₉	#41
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#338	D₁₃₉
hypocenter	C₁₃₉	#1841
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#17984799

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#338×C₁₃₉

C₂

C₂×C₁₃₉

C₂³×C₁₃₉

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
9. #17984752	C₄	semidirect
10. C₈×Dic₁₃₉	C₄	semidirect
52. C₁₃₉	#1841	semidirect

H ≅ G/N

type

9. #17984752

C₄

semidirect

10. C₈×Dic₁₃₉

C₄

semidirect

52. C₁₃₉

#1841

semidirect

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

subgroup	index
M₁₆×Dic₁₃₉	2
#22580725	2
#22580714	2
#22580710	2
#338×C₁₃₉	2
#1841	139

subgroup

index

M₁₆×Dic₁₃₉

#338×C₁₃₉

139

Subgroups — #31806554

Conjugacy classes of subgroups

Complements

Maximal subgroups