Group #31806159 — subgroups

Subgroups — #31806159

Order 17792.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	#22580908	C₂	1	2	yes	maximal
2	C₁₃₉⋊#338	C₂	1	1	yes	maximal, characteristic
3	#22580710	C₂	1	1	yes	maximal, characteristic
4	#22580709	C₂	1	2	yes	maximal
5	#405×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	2	yes	—
7	#17984801	C₂²	1	2	yes	—
8	#17984752	C₄	1	4	yes	complemented
9	C₈×Dic₁₃₉	C₄	1	4	yes	complemented
10	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
12	#117×C₁₃₉	C₂²	1	1	yes	characteristic
13	#97×C₁₃₉	C₄	1	2	yes	complemented
14	C₄×C₈×C₁₃₉	C₄	1	2	yes	complemented
15	#14555769	—	2	2	no	—
16	#14555769	—	2	2	no	—
17	C₂×C₁₃₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₁₃₉	—	2	2	no	—
19	C₂²⋊Dic₁₃₉	—	2	2	no	—
20	C₄×Dic₁₃₉	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₃₉	—	2	2	no	—
23	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₁₃₉	—	2	1	no	—
29	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
30	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
31	C₂×Dic₁₃₉	—	2	4	no	—
32	C₁₃₉⋊C₈	—	2	4	no	—
33	C₂³×C₁₃₉	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	—	2	2	no	—
35	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₃₉	#41	1	2	yes	—
39	C₈×C₁₃₉	—	2	2	no	—
40	C₈×C₁₃₉	C₄²	1	2	yes	—
41	C₈×C₁₃₉	C₄²	1	1	yes	characteristic
42	C₈×C₁₃₉	C₄²	1	1	yes	characteristic
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	2	no	—
47	C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₃₉	#117	1	2	yes	—
49	C₄×C₁₃₉	#117	1	1	yes	characteristic
50	Dic₁₃₉	—	4	4	no	—
51	C₂×C₁₃₉	—	2	2	no	—
52	C₂×C₁₃₉	—	2	1	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	2	no	—
57	#405	D₁₃₉	1	1	yes	characteristic
58	#338	—	139	1	no	—
59	H(3,ℤ/4)	—	139	1	no	—
60	H(3,ℤ/4)	—	139	2	no	—
61	C₂×M₁₆	—	139	2	no	—
62	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
64	#117	—	139	2	no	—
65	#117	C₂×D₁₃₉	1	1	yes	characteristic
66	#97	—	139	4	no	—
67	#97	Dic₁₃₉	1	2	yes	complemented
68	C₄×C₈	—	139	4	no	—
69	C₄×C₈	Dic₁₃₉	1	2	yes	complemented
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	C₄×D₁₃₉	1	2	yes	—
76	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
77	C₂×C₈	C₂×Dic₁₃₉	1	2	yes	—
78	C₂×C₈	C₂×Dic₁₃₉	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₁₃₉	1	1	yes	characteristic
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	278	2	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	—	278	2	no	—
84	C₄²	—	139	4	no	—
85	C₄²	C₂×Dic₁₃₉	1	2	yes	—
86	C₂³	#14555785	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	278	4	no	—
89	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
90	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
91	C₂×C₄	#14555785	1	1	yes	characteristic
92	C₂×C₄	#14555785	1	2	yes	—
93	C₈	—	2	2	no	—
94	C₈	—	278	4	no	—
95	C₈	C₄×Dic₁₃₉	1	1	yes	characteristic
96	C₈	C₄×Dic₁₃₉	1	2	yes	—
97	C₈	C₄×Dic₁₃₉	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#17984799	1	2	yes	—
100	C₂²	#17984801	1	1	yes	characteristic
101	C₄	—	4	2	no	—
102	C₄	—	556	4	no	—
103	C₄	C₂×C₄×Dic₁₃₉	1	1	yes	characteristic
104	C₄	#17984799	1	2	yes	—
105	C₄	#17984801	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	2	2	no	—
108	C₂	#22580799	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22580908

C₂

yes

maximal

C₁₃₉⋊#338

C₂

yes

maximal, characteristic

#22580710

C₂

yes

maximal, characteristic

#22580709

C₂

yes

maximal

#405×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

—

#17984801

C₂²

yes

—

#17984752

C₄

yes

complemented

C₈×Dic₁₃₉

C₄

yes

complemented

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₂²

yes

characteristic

#97×C₁₃₉

C₄

yes

complemented

C₄×C₈×C₁₃₉

C₄

yes

complemented

#14555769

—

#14555769

—

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

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

—

C₈×C₁₃₉

—

C₈×C₁₃₉

C₄²

yes

—

C₈×C₁₃₉

C₄²

yes

characteristic

C₈×C₁₃₉

C₄²

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#117

yes

—

C₂²×C₁₃₉

#117

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₄²

yes

characteristic

C₄×C₁₃₉

#117

yes

—

C₄×C₁₃₉

#117

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#376

yes

characteristic

C₁₃₉

#1747

yes

characteristic, complemented, Sylow

#1747

—

139

maximal, Sylow

#405

—

139

—

#405

D₁₃₉

yes

characteristic

#338

—

139

—

H(3,ℤ/4)

—

139

—

H(3,ℤ/4)

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#117

—

139

—

#117

C₂×D₁₃₉

yes

characteristic

#97

—

139

—

#97

Dic₁₃₉

yes

complemented

C₄×C₈

—

139

—

C₄×C₈

Dic₁₃₉

yes

complemented

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

278

—

M₁₆

—

278

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

—

C₂×C₈

C₄×D₁₃₉

yes

—

C₂×C₈

C₂×Dic₁₃₉

yes

—

C₂×C₈

C₂×Dic₁₃₉

yes

characteristic

C₂×C₈

C₂×Dic₁₃₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₂²⋊C₄

—

C₂²⋊C₄

—

278

—

C₄²

—

139

—

C₄²

C₂×Dic₁₃₉

yes

—

C₂³

#14555785

yes

characteristic

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

—

C₂×C₄

C₂²×Dic₁₃₉

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

278

—

C₈

C₄×Dic₁₃₉

yes

characteristic

C₈

C₄×Dic₁₃₉

yes

—

C₈

C₄×Dic₁₃₉

yes

characteristic

C₂²

—

C₂²

#17984799

yes

—

100

C₂²

#17984801

yes

characteristic

101

C₄

—

102

C₄

—

556

—

103

C₄

C₂×C₄×Dic₁₃₉

yes

characteristic

104

C₄

#17984799

yes

—

105

C₄

#17984801

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#22580799

yes

characteristic

	subgroup	G/H
center	C₈	C₄×Dic₁₃₉
commutator	C₄×C₁₃₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#405×C₁₃₉	C₂
socle	C₂×C₁₃₉	#376
prime radical	C₂³×C₁₃₉	#41
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#405	D₁₃₉
hypocenter	C₁₃₉	#1747
LCS	G ▷ C₄×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₈ ◁ #405

subgroup

G/H

center

C₈

C₄×Dic₁₃₉

commutator

C₄×C₁₃₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#405×C₁₃₉

C₂

C₂×C₁₃₉

C₂³×C₁₃₉

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₄×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₈ ◁ #405

N	H ≅ G/N	type
8. #17984752	C₄	semidirect
9. C₈×Dic₁₃₉	C₄	semidirect
13. #97×C₁₃₉	C₄	semidirect
14. C₄×C₈×C₁₃₉	C₄	semidirect
54. C₁₃₉	#1747	semidirect
67. #97	Dic₁₃₉	semidirect
69. C₄×C₈	Dic₁₃₉	semidirect

H ≅ G/N

type

8. #17984752

C₄

semidirect

9. C₈×Dic₁₃₉

C₄

semidirect

13. #97×C₁₃₉

C₄

semidirect

14. C₄×C₈×C₁₃₉

C₄

semidirect

54. C₁₃₉

#1747

semidirect

67. #97

Dic₁₃₉

semidirect

69. C₄×C₈

Dic₁₃₉

semidirect

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

subgroup	index
#22580908	2
C₁₃₉⋊#338	2
#22580710	2
#22580709	2
#405×C₁₃₉	2
#1747	139

subgroup

index

C₁₃₉⋊#338

#405×C₁₃₉

139

Subgroups — #31806159

Conjugacy classes of subgroups

Complements

Maximal subgroups