Group #31806158 — subgroups

Subgroups — #31806158

Order 17792.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22580911	C₂	1	1	yes	maximal, characteristic
2	#22580909	C₂	1	2	yes	maximal
3	C₁₃₉⋊#338	C₂	1	1	yes	maximal, characteristic
4	#22580709	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	2	yes	—
7	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
8	#17984801	C₂²	1	1	yes	characteristic
9	#17984753	C₄	1	4	yes	complemented
10	#17984752	C₄	1	2	yes	complemented
11	C₄×C₁₃₉⋊C₈	C₄	1	4	yes	complemented
12	C₈×Dic₁₃₉	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
14	#117×C₁₃₉	C₂²	1	2	yes	—
15	#14555769	—	2	2	no	—
16	#14555769	—	2	2	no	—
17	C₂×C₁₃₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₁₃₉	—	2	1	no	—
23	C₂²⋊Dic₁₃₉	—	2	1	no	—
24	C₄×Dic₁₃₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₃₉	—	2	2	no	—
27	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₁₃₉	—	2	2	no	—
29	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
30	C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
31	C₂×Dic₁₃₉	—	2	2	no	—
32	C₁₃₉⋊C₈	—	2	4	no	—
33	C₁₃₉⋊C₈	C₄²	1	1	yes	characteristic
34	C₁₃₉⋊C₈	C₄²	1	1	yes	characteristic
35	C₁₃₉⋊C₈	C₄²	1	2	yes	—
36	C₂³×C₁₃₉	#41	1	1	yes	characteristic
37	C₂×C₄×C₁₃₉	—	2	4	no	—
38	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₉	#41	1	2	yes	—
41	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
42	C₈×C₁₃₉	—	2	2	no	—
43	C₂²×C₁₃₉	—	4	1	no	—
44	C₂²×C₁₃₉	#117	1	2	yes	—
45	C₂²×C₁₃₉	#117	1	1	yes	characteristic
46	C₄×C₁₃₉	—	4	4	no	—
47	C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₃₉	#117	1	1	yes	characteristic
49	C₄×C₁₃₉	#117	1	2	yes	—
50	Dic₁₃₉	—	4	2	no	—
51	C₂×C₁₃₉	—	2	1	no	—
52	C₂×C₁₃₉	—	2	2	no	—
53	C₂×C₁₃₉	#376	1	1	yes	characteristic
54	C₁₃₉	#1747	1	1	yes	characteristic, complemented, Sylow
55	#1747	—	139	1	no	maximal, Sylow
56	#405	—	139	1	no	—
57	#405	—	139	2	no	—
58	#338	—	139	1	no	—
59	H(3,ℤ/4)	—	139	2	no	—
60	H(3,ℤ/4)	D₁₃₉	1	1	yes	characteristic
61	C₂×M₁₆	—	139	2	no	—
62	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
63	C₂²×C₈	—	139	1	no	—
64	#117	—	139	1	no	—
65	#117	C₂×D₁₃₉	1	2	yes	—
66	#97	—	139	2	no	—
67	#97	—	139	4	no	—
68	C₄×C₈	—	139	2	no	—
69	C₄×C₈	—	139	4	no	—
70	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	278	2	no	—
73	M₁₆	—	278	2	no	—
74	C₂×C₈	—	139	4	no	—
75	C₂×C₈	—	139	1	no	—
76	C₂×C₈	—	139	2	no	—
77	C₂×C₈	—	139	1	no	—
78	C₂×C₈	—	139	2	no	—
79	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	278	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	278	1	no	—
84	C₄²	—	139	2	no	—
85	C₄²	C₄×D₁₃₉	1	4	yes	—
86	C₂³	#14555784	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	278	2	no	—
89	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
91	C₂×C₄	#14555784	1	2	yes	—
92	C₂×C₄	#14555786	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	278	4	no	—
95	C₈	—	139	1	no	—
96	C₈	—	139	1	no	—
97	C₈	—	139	2	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#17984798	1	2	yes	—
100	C₂²	#17984800	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	556	2	no	—
103	C₄	C₄²×D₁₃₉	1	1	yes	characteristic
104	C₄	#17984798	1	1	yes	characteristic
105	C₄	#17984800	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#22580798	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22580911

C₂

yes

maximal, characteristic

#22580909

C₂

yes

maximal

C₁₃₉⋊#338

C₂

yes

maximal, characteristic

#22580709

C₂

yes

maximal

H(3,ℤ/4)×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

#17984801

C₂²

yes

characteristic

#17984753

C₄

yes

complemented

#17984752

C₄

yes

complemented

C₄×C₁₃₉⋊C₈

C₄

yes

complemented

C₈×Dic₁₃₉

C₄

yes

complemented

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₂²

yes

—

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₃₉

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

C₄²

yes

characteristic

C₁₃₉⋊C₈

C₄²

yes

characteristic

C₁₃₉⋊C₈

C₄²

yes

—

C₂³×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#117

yes

—

C₂²×C₁₃₉

#117

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₄²

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#376

yes

characteristic

C₁₃₉

#1747

yes

characteristic, complemented, Sylow

#1747

—

139

maximal, Sylow

#405

—

139

—

#405

—

139

—

#338

—

139

—

H(3,ℤ/4)

—

139

—

H(3,ℤ/4)

D₁₃₉

yes

characteristic

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

—

139

—

#117

—

139

—

#117

C₂×D₁₃₉

yes

—

#97

—

139

—

#97

—

139

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

278

—

M₁₆

—

278

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₂²⋊C₄

—

C₂²⋊C₄

—

278

—

C₄²

—

139

—

C₄²

C₄×D₁₃₉

yes

—

C₂³

#14555784

yes

characteristic

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

C₂×C₄

C₂×C₄×D₁₃₉

yes

—

C₂×C₄

#14555784

yes

—

C₂×C₄

#14555786

yes

characteristic

C₈

—

C₈

—

278

—

C₈

—

139

—

C₈

—

139

—

C₈

—

139

—

C₂²

—

C₂²

#17984798

yes

—

100

C₂²

#17984800

yes

characteristic

101

C₄

—

102

C₄

—

556

—

103

C₄

C₄²×D₁₃₉

yes

characteristic

104

C₄

#17984798

yes

characteristic

105

C₄

#17984800

yes

—

106

C₂

—

107

C₂

—

108

C₂

#22580798

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₃₉
commutator	C₄×C₁₃₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	H(3,ℤ/4)×C₁₃₉	C₂
socle	C₂×C₁₃₉	#376
prime radical	C₂³×C₁₃₉	#41
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	H(3,ℤ/4)	D₁₃₉
hypocenter	C₁₃₉	#1747
LCS	G ▷ C₄×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₁₃₉

commutator

C₄×C₁₃₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

H(3,ℤ/4)×C₁₃₉

C₂

C₂×C₁₃₉

C₂³×C₁₃₉

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₄×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #17984753	C₄	semidirect
10. #17984752	C₄	semidirect
11. C₄×C₁₃₉⋊C₈	C₄	semidirect
12. C₈×Dic₁₃₉	C₄	semidirect
54. C₁₃₉	#1747	semidirect

H ≅ G/N

type

9. #17984753

C₄

semidirect

10. #17984752

C₄

semidirect

11. C₄×C₁₃₉⋊C₈

C₄

semidirect

12. C₈×Dic₁₃₉

C₄

semidirect

54. C₁₃₉

#1747

semidirect

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

subgroup	index
#22580911	2
#22580909	2
C₁₃₉⋊#338	2
#22580709	2
H(3,ℤ/4)×C₁₃₉	2
#1747	139

subgroup

index

C₁₃₉⋊#338

H(3,ℤ/4)×C₁₃₉

139

Subgroups — #31806158

Conjugacy classes of subgroups

Complements

Maximal subgroups