Group #31805707 — subgroups

Subgroups — #31805707

Order 17792.

182 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₁₃₉⋊#564	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581319	C₂	1	1	yes	maximal, characteristic
3	#22581318	C₂	1	2	yes	maximal
4	#22580694	C₂	1	2	yes	maximal, complemented
5	#330×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984839	C₂²	1	1	yes	characteristic
7	#17984834	C₂²	1	2	yes	—
8	#17984833	—	2	2	no	—
9	#17984832	—	2	2	no	—
10	#17984832	—	2	2	no	—
11	#17984764	—	2	2	no	—
12	#17984764	—	2	2	no	—
13	#17984762	—	2	2	no	—
14	#17984753	C₂²	1	1	yes	characteristic
15	#125×C₁₃₉	C₂²	1	1	yes	characteristic
16	#97×C₁₃₉	C₂²	1	2	yes	—
17	C₂×D₅₅₆	—	2	1	no	—
18	C₂×D₅₅₆	—	2	2	no	—
19	C₂×C₁₃₉⋊C₈	—	2	1	no	—
20	C₄⋊Dic₁₃₉	—	2	2	no	—
21	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
22	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
23	Dic₁₃₉⋊C₄	—	2	2	no	—
24	Dic₁₃₉⋊C₄	—	2	2	no	—
25	Dic₁₃₉⋊C₄	—	2	2	no	—
26	Dic₁₃₉⋊C₄	—	2	2	no	—
27	C₄×Dic₁₃₉	—	2	2	no	—
28	C₂×C₈×C₁₃₉	—	2	2	no	—
29	C₄⋊C₄×C₁₃₉	—	2	2	no	—
30	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
31	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
32	C₂²×D₁₃₉	—	4	1	no	—
33	D₅₅₆	—	4	4	no	—
34	D₅₅₆	—	4	2	no	—
35	C₂×Dic₁₃₉	—	2	2	no	—
36	C₂×Dic₁₃₉	—	2	2	no	—
37	C₁₃₉⋊C₈	—	4	1	no	—
38	C₂×C₄×C₁₃₉	—	2	2	no	—
39	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
41	C₈×C₁₃₉	—	4	2	no	—
42	C₂²×C₁₃₉	#120	1	1	yes	characteristic
43	C₂×D₁₃₉	—	8	2	no	—
44	C₂×D₁₃₉	—	8	1	no	—
45	C₄×C₁₃₉	—	2	2	no	—
46	C₄×C₁₃₉	—	2	1	no	—
47	C₄×C₁₃₉	—	4	2	no	—
48	Dic₁₃₉	—	4	2	no	—
49	Dic₁₃₉	—	4	2	no	—
50	C₂×C₁₃₉	#454	1	2	yes	—
51	C₂×C₁₃₉	#454	1	1	yes	characteristic
52	D₁₃₉	—	16	1	no	—
53	C₁₃₉	#1658	1	1	yes	characteristic, complemented, Sylow
54	#1658	—	139	1	no	maximal, Sylow
55	#564	—	139	1	no	—
56	#490	—	139	1	no	—
57	#490	—	139	2	no	—
58	#330	—	139	2	no	—
59	#330	D₁₃₉	1	1	yes	characteristic, complemented
60	#127	—	139	1	no	—
61	#125	—	278	2	no	—
62	#125	—	278	2	no	—
63	#125	—	278	2	no	—
64	#125	—	139	2	no	—
65	#125	C₂×D₁₃₉	1	1	yes	characteristic
66	#102	—	278	2	no	—
67	#102	—	278	2	no	—
68	#102	—	278	2	no	—
69	#97	—	139	1	no	—
70	#97	C₂×D₁₃₉	1	2	yes	—
71	C₂×D₄	—	278	2	no	—
72	C₂×D₄	—	278	1	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	278	1	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	278	2	no	—
77	C₄⋊C₄	—	278	2	no	—
78	C₄⋊C₄	—	278	2	no	—
79	C₄⋊C₄	—	278	2	no	—
80	C₄⋊C₄	—	278	2	no	—
81	C₄⋊C₄	—	139	2	no	—
82	C₄⋊C₄	—	139	2	no	—
83	C₄⋊C₄	D₅₅₆	1	2	yes	—
84	C₄²	—	278	2	no	—
85	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
86	C₂³	—	556	1	no	—
87	D₄	—	556	4	no	—
88	D₄	—	556	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	278	2	no	—
91	C₂×C₄	—	278	2	no	—
92	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
93	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
94	C₈	—	4	2	no	—
95	C₈	—	556	1	no	—
96	C₂²	—	1112	2	no	—
97	C₂²	—	1112	1	no	—
98	C₂²	#17984812	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	—	4	2	no	—
102	C₄	—	556	2	no	—
103	C₄	—	556	2	no	—
104	C₂	—	2224	1	no	—
105	C₂	#22581127	1	2	yes	—
106	C₂	#22581128	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃₉⋊#564

C₂

yes

maximal, characteristic, complemented

#22581319

C₂

yes

maximal, characteristic

#22581318

C₂

yes

maximal

#22580694

C₂

yes

maximal, complemented

#330×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#17984839

C₂²

yes

characteristic

#17984834

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

characteristic

#125×C₁₃₉

C₂²

yes

characteristic

#97×C₁₃₉

C₂²

yes

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

D₄

yes

—

C₄⋊Dic₁₃₉

D₄

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

—

D₅₅₆

—

D₅₅₆

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

#120

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#454

yes

—

C₂×C₁₃₉

#454

yes

characteristic

D₁₃₉

—

C₁₃₉

#1658

yes

characteristic, complemented, Sylow

#1658

—

139

maximal, Sylow

#564

—

139

—

#490

—

139

—

#490

—

139

—

#330

—

139

—

#330

D₁₃₉

yes

characteristic, complemented

#127

—

139

—

#125

—

278

—

#125

—

278

—

#125

—

278

—

#125

—

139

—

#125

C₂×D₁₃₉

yes

characteristic

#102

—

278

—

#102

—

278

—

#102

—

278

—

#97

—

139

—

#97

C₂×D₁₃₉

yes

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×C₈

—

C₂×C₈

—

278

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

D₅₅₆

yes

—

C₄²

—

278

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

—

556

—

D₄

—

556

—

D₄

—

556

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

D₄×D₁₃₉

yes

—

C₂×C₄

C₂×D₅₅₆

yes

characteristic

C₈

—

C₈

—

556

—

C₂²

—

1112

—

C₂²

—

1112

—

C₂²

#17984812

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

556

—

103

C₄

—

556

—

104

C₂

—

2224

—

105

C₂

#22581127

yes

—

106

C₂

#22581128

yes

characteristic

	subgroup	G/H
center	C₂²	#17984812
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#330×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#120
prime radical	#17984839	C₂²
hypercenter	#330	D₁₃₉
hypocenter	C₁₃₉	#1658
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

#17984812

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#330×C₁₃₉

C₂

C₂²×C₁₃₉

C₂²

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
1. C₁₃₉⋊#564	C₂	semidirect
4. #22580694	C₂	semidirect
5. #330×C₁₃₉	C₂	semidirect
53. C₁₃₉	#1658	semidirect
59. #330	D₁₃₉	semidirect

H ≅ G/N

type

1. C₁₃₉⋊#564

C₂

semidirect

4. #22580694

C₂

semidirect

5. #330×C₁₃₉

C₂

semidirect

53. C₁₃₉

#1658

semidirect

59. #330

D₁₃₉

semidirect

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

subgroup	index
C₁₃₉⋊#564	2
#22581319	2
#22581318	2
#22580694	2
#330×C₁₃₉	2
#1658	139

subgroup

index

C₁₃₉⋊#564

#330×C₁₃₉

139

Subgroups — #31805707

Conjugacy classes of subgroups

Complements

Maximal subgroups