Group #31805029 — subgroups

Subgroups — #31805029

Order 17792.

196 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 59 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	#125×D₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#22581008	C₂	1	2	yes	maximal
3	#22580695	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580694	C₂	1	2	yes	maximal, complemented
5	#330×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984834	C₂²	1	1	yes	characteristic
7	#17984833	C₄	1	2	yes	—
8	#17984832	—	2	2	no	—
9	C₄⋊C₄×D₁₃₉	—	2	2	no	—
10	C₄⋊C₄×D₁₃₉	C₄	1	2	yes	—
11	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
12	Dic₁₃₉⋊C₈	—	2	2	no	—
13	(C₂×D₁₃₉)⋊C₈	—	2	2	no	—
14	#17984753	C₂²	1	2	yes	—
15	#125×C₁₃₉	C₂²	1	1	yes	characteristic
16	#97×C₁₃₉	C₂²	1	2	yes	—
17	C₂×C₄×D₁₃₉	—	2	2	no	—
18	C₂×C₄×D₁₃₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₁₃₉	D₄	1	2	yes	—
20	C₂×C₁₃₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₃₉	—	2	2	no	—
22	C₄⋊Dic₁₃₉	C₂×C₄	1	2	yes	—
23	Dic₁₃₉⋊C₄	—	2	2	no	—
24	Dic₁₃₉⋊C₄	—	2	2	no	—
25	Dic₁₃₉⋊C₄	—	2	2	no	—
26	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₃₉	D₄	1	2	yes	—
28	C₂×C₈×C₁₃₉	—	2	2	no	—
29	C₄⋊C₄×C₁₃₉	—	2	2	no	—
30	C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
31	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
32	C₂²×D₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×Dic₁₃₉	—	2	2	no	—
34	C₂×Dic₁₃₉	C₂²⋊C₄	1	2	yes	—
35	C₂×Dic₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×D₁₃₉	—	2	2	no	—
37	C₄×D₁₃₉	—	4	2	no	—
38	C₄×D₁₃₉	—	4	4	no	—
39	C₁₃₉⋊C₈	—	4	2	no	—
40	C₂×C₄×C₁₃₉	—	2	2	no	—
41	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
43	C₈×C₁₃₉	—	4	2	no	—
44	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×D₁₃₉	—	2	2	no	—
46	C₂×D₁₃₉	#101	1	2	yes	—
47	C₄×C₁₃₉	—	2	1	no	—
48	C₄×C₁₃₉	—	2	2	no	—
49	C₄×C₁₃₉	—	4	2	no	—
50	Dic₁₃₉	—	2	1	no	—
51	Dic₁₃₉	—	2	2	no	—
52	Dic₁₃₉	—	4	2	no	—
53	C₂×C₁₃₉	C₂×#101	1	1	yes	characteristic
54	C₂×C₁₃₉	#421	1	2	yes	—
55	D₁₃₉	—	2	2	no	—
56	C₁₃₉	#1534	1	1	yes	characteristic, complemented, Sylow
57	#1534	—	139	1	no	maximal, Sylow
58	C₂×#125	—	139	1	no	—
59	#432	—	139	2	no	—
60	#330	—	139	1	no	—
61	#330	—	139	2	no	—
62	#330	D₁₃₉	1	1	yes	characteristic, complemented
63	#125	—	278	2	no	—
64	#125	—	139	2	no	—
65	#125	—	139	1	no	—
66	#125	C₂×D₁₃₉	1	1	yes	characteristic
67	C₂×C₄⋊C₄	—	278	2	no	—
68	C₂×C₄⋊C₄	—	139	2	no	—
69	C₂×C₄²	—	139	1	no	—
70	C₄⋊C₈	—	278	2	no	—
71	C₂²⋊C₈	—	278	2	no	—
72	#97	—	139	2	no	—
73	#97	C₂×D₁₃₉	1	2	yes	—
74	C₂²×C₄	—	278	2	no	—
75	C₂²×C₄	—	139	1	no	—
76	C₂²×C₄	—	139	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	278	2	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	278	2	no	—
81	C₄⋊C₄	—	278	2	no	—
82	C₄⋊C₄	—	278	2	no	—
83	C₄⋊C₄	—	278	2	no	—
84	C₄⋊C₄	—	139	2	no	—
85	C₄⋊C₄	C₄×D₁₃₉	1	2	yes	—
86	C₄²	—	139	1	no	—
87	C₄²	—	139	2	no	—
88	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
89	C₂³	—	139	1	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	278	2	no	—
92	C₂×C₄	—	278	2	no	—
93	C₂×C₄	—	556	2	no	—
94	C₂×C₄	—	556	4	no	—
95	C₂×C₄	—	139	1	no	—
96	C₂×C₄	—	139	2	no	—
97	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
98	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
99	C₈	—	4	2	no	—
100	C₈	—	556	2	no	—
101	C₂²	—	278	2	no	—
102	C₂²	—	139	2	no	—
103	C₂²	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	—	4	2	no	—
107	C₄	—	278	1	no	—
108	C₄	—	278	2	no	—
109	C₄	—	556	2	no	—
110	C₂	—	278	2	no	—
111	C₂	#101×D₁₃₉	1	1	yes	characteristic
112	C₂	#22580968	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#125×D₁₃₉

C₂

yes

maximal, characteristic

#22581008

C₂

yes

maximal

#22580695

C₂

yes

maximal, characteristic, complemented

#22580694

C₂

yes

maximal, complemented

#330×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#17984834

C₂²

yes

characteristic

#17984833

C₄

yes

—

#17984832

—

C₄⋊C₄×D₁₃₉

—

C₄⋊C₄×D₁₃₉

C₄

yes

—

C₄²×D₁₃₉

C₂²

yes

characteristic

Dic₁₃₉⋊C₈

—

(C₂×D₁₃₉)⋊C₈

—

#17984753

C₂²

yes

—

#125×C₁₃₉

C₂²

yes

characteristic

#97×C₁₃₉

C₂²

yes

—

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₉

D₄

yes

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂×C₄

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₄×Dic₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂²⋊C₄

yes

—

C₂×Dic₁₃₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

#101

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

C₂×#101

yes

characteristic

C₂×C₁₃₉

#421

yes

—

D₁₃₉

—

C₁₃₉

#1534

yes

characteristic, complemented, Sylow

#1534

—

139

maximal, Sylow

C₂×#125

—

139

—

#432

—

139

—

#330

—

139

—

#330

—

139

—

#330

D₁₃₉

yes

characteristic, complemented

#125

—

278

—

#125

—

139

—

#125

—

139

—

#125

C₂×D₁₃₉

yes

characteristic

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

—

139

—

C₄⋊C₈

—

278

—

C₂²⋊C₈

—

278

—

#97

—

139

—

#97

C₂×D₁₃₉

yes

—

C₂²×C₄

—

278

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂×C₈

—

C₂×C₈

—

278

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄⋊C₄

C₄×D₁₃₉

yes

—

C₄²

—

139

—

C₄²

—

139

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

—

139

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

556

—

C₂×C₄

—

556

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

C₂×C₄

D₄×D₁₃₉

yes

—

C₈

—

100

C₈

—

556

—

101

C₂²

—

278

—

102

C₂²

—

139

—

103

C₂²

C₂²⋊C₄×D₁₃₉

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

278

—

108

C₄

—

278

—

109

C₄

—

556

—

110

C₂

—

278

—

111

C₂

#101×D₁₃₉

yes

characteristic

112

C₂

#22580968

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#330×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₃₉	C₂²⋊C₄
cosocle	C₄²×C₁₃₉	C₂³
hypercenter	#330	D₁₃₉
hypocenter	C₁₃₉	#1534
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#330×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₃₉

C₂²⋊C₄

cosocle

C₄²×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
3. #22580695	C₂	semidirect
4. #22580694	C₂	semidirect
5. #330×C₁₃₉	C₂	semidirect
56. C₁₃₉	#1534	semidirect
62. #330	D₁₃₉	semidirect

H ≅ G/N

type

3. #22580695

C₂

semidirect

4. #22580694

C₂

semidirect

5. #330×C₁₃₉

C₂

semidirect

56. C₁₃₉

#1534

semidirect

62. #330

D₁₃₉

semidirect

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

subgroup	index
#125×D₁₃₉	2
#22581008	2
#22580695	2
#22580694	2
#330×C₁₃₉	2
#1534	139

subgroup

index

#125×D₁₃₉

#330×C₁₃₉

139

Subgroups — #31805029

Conjugacy classes of subgroups

Complements

Maximal subgroups