Group #31808007 — subgroups

Subgroups — #31808007

Order 17792.

160 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 85 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	M₃₂×D₁₃₉	C₂	1	2	yes	maximal
2	#22580910	C₂	1	1	yes	maximal, characteristic
3	#22580731	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580730	C₂	1	2	yes	maximal, complemented
5	#347×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555768	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
8	#17984799	C₄	1	1	yes	characteristic
9	#17984782	C₂²	1	2	yes	—
10	#17984781	C₄	1	4	yes	complemented
11	C₁₆×D₁₃₉	C₄	1	4	yes	complemented
12	#17984752	C₂²	1	1	yes	characteristic
13	C₄×C₁₃₉⋊C₈	C₄	1	1	yes	characteristic
14	C₈×Dic₁₃₉	C₄	1	1	yes	characteristic
15	M₃₂×C₁₃₉	C₂²	1	2	yes	—
16	#97×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₁₃₉	C₂×C₄	1	1	yes	characteristic
18	#14555768	—	2	2	no	—
19	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₈×D₁₃₉	C₂×C₄	1	2	yes	—
22	C₈×D₁₃₉	C₂×C₄	1	2	yes	—
23	Dic₁₃₉⋊C₄	—	2	1	no	—
24	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
25	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
26	C₁₃₉⋊C₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
29	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
30	C₁₆×C₁₃₉	C₂×C₄	1	4	yes	—
31	C₂²×D₁₃₉	M₁₆	1	1	yes	characteristic
32	C₂×Dic₁₃₉	—	2	1	no	—
33	C₂×Dic₁₃₉	M₁₆	1	1	yes	characteristic
34	C₄×D₁₃₉	C₄²	1	2	yes	—
35	C₄×D₁₃₉	M₁₆	1	2	yes	—
36	C₁₃₉⋊C₈	—	2	1	no	—
37	C₁₃₉⋊C₈	C₄²	1	1	yes	characteristic
38	C₁₃₉⋊C₈	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₁₃₉	—	2	1	no	—
40	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
41	C₈×C₁₃₉	—	2	1	no	—
42	C₈×C₁₃₉	C₂²×C₄	1	2	yes	—
43	C₂²×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
44	C₂×D₁₃₉	—	4	1	no	—
45	C₂×D₁₃₉	#97	1	2	yes	—
46	C₄×C₁₃₉	—	4	1	no	—
47	C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
49	Dic₁₃₉	—	4	1	no	—
50	Dic₁₃₉	#97	1	2	yes	—
51	C₂×C₁₃₉	—	2	1	no	—
52	C₂×C₁₃₉	C₂×#97	1	1	yes	characteristic
53	D₁₃₉	—	2	2	no	—
54	C₁₃₉	#2110	1	1	yes	characteristic, complemented, Sylow
55	#2110	—	139	1	no	maximal, Sylow
56	C₂×M₃₂	—	139	2	no	—
57	#405	—	139	1	no	—
58	#347	—	139	2	no	—
59	#347	—	139	1	no	—
60	#347	D₁₃₉	1	1	yes	characteristic, complemented
61	C₂×M₁₆	—	139	1	no	—
62	C₂²×C₈	—	139	1	no	—
63	#117	—	139	1	no	—
64	M₃₂	—	139	2	no	—
65	M₃₂	—	139	4	no	—
66	M₃₂	C₂×D₁₃₉	1	2	yes	—
67	C₂×C₁₆	—	139	4	no	—
68	#97	—	139	1	no	—
69	#97	C₂×D₁₃₉	1	1	yes	characteristic
70	C₄×C₈	—	139	1	no	—
71	C₄×C₈	—	139	1	no	—
72	C₂²×C₄	—	139	1	no	—
73	M₁₆	—	278	2	no	—
74	C₂×C₈	—	139	1	no	—
75	C₂×C₈	—	139	1	no	—
76	C₂×C₈	—	139	2	no	—
77	C₂×C₈	—	139	2	no	—
78	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
79	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
80	C₄⋊C₄	—	278	1	no	—
81	C₂²⋊C₄	—	278	1	no	—
82	C₄²	—	139	1	no	—
83	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
84	C₁₆	—	139	4	no	—
85	C₁₆	C₄×D₁₃₉	1	4	yes	—
86	C₂³	—	139	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	278	1	no	—
89	C₂×C₄	—	139	1	no	—
90	C₂×C₄	—	139	2	no	—
91	C₂×C₄	—	139	2	no	—
92	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	278	1	no	—
95	C₈	—	139	1	no	—
96	C₈	—	139	1	no	—
97	C₈	C₂×C₄×D₁₃₉	1	2	yes	—
98	C₂²	—	556	1	no	—
99	C₂²	—	139	2	no	—
100	C₂²	M₁₆×D₁₃₉	1	1	yes	characteristic
101	C₄	—	4	1	no	—
102	C₄	—	556	1	no	—
103	C₄	—	139	2	no	—
104	C₄	C₄²×D₁₃₉	1	1	yes	characteristic
105	C₄	M₁₆×D₁₃₉	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	278	2	no	—
108	C₂	#97×D₁₃₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×D₁₃₉

C₂

yes

maximal

#22580910

C₂

yes

maximal, characteristic

#22580731

C₂

yes

maximal, characteristic, complemented

#22580730

C₂

yes

maximal, complemented

#347×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555768

C₄

yes

characteristic

C₂×C₈×D₁₃₉

C₂²

yes

characteristic

#17984799

C₄

yes

characteristic

#17984782

C₂²

yes

—

#17984781

C₄

yes

complemented

C₁₆×D₁₃₉

C₄

yes

complemented

#17984752

C₂²

yes

characteristic

C₄×C₁₃₉⋊C₈

C₄

yes

characteristic

C₈×Dic₁₃₉

C₄

yes

characteristic

M₃₂×C₁₃₉

C₂²

yes

—

#97×C₁₃₉

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

C₂×C₄

yes

characteristic

#14555768

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₈×D₁₃₉

C₂×C₄

yes

—

C₈×D₁₃₉

C₂×C₄

yes

—

Dic₁₃₉⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₁₃₉⋊C₁₆

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₄²×C₁₃₉

C₂×C₄

yes

characteristic

C₁₆×C₁₃₉

C₂×C₄

yes

—

C₂²×D₁₃₉

M₁₆

yes

characteristic

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

M₁₆

yes

characteristic

C₄×D₁₃₉

C₄²

yes

—

C₄×D₁₃₉

M₁₆

yes

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

C₄²

yes

characteristic

C₁₃₉⋊C₈

C₄²

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

C₂²×C₄

yes

—

C₂²×C₁₃₉

C₂×M₁₆

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

#97

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₄²

yes

characteristic

C₄×C₁₃₉

C₂×M₁₆

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

#97

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×#97

yes

characteristic

D₁₃₉

—

C₁₃₉

#2110

yes

characteristic, complemented, Sylow

#2110

—

139

maximal, Sylow

C₂×M₃₂

—

139

—

#405

—

139

—

#347

—

139

—

#347

—

139

—

#347

D₁₃₉

yes

characteristic, complemented

C₂×M₁₆

—

139

—

C₂²×C₈

—

139

—

#117

—

139

—

M₃₂

—

139

—

M₃₂

—

139

—

M₃₂

C₂×D₁₃₉

yes

—

C₂×C₁₆

—

139

—

#97

—

139

—

#97

C₂×D₁₃₉

yes

characteristic

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₂²×C₄

—

139

—

M₁₆

—

278

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₂²⋊C₄

—

278

—

C₄²

—

139

—

C₄²

C₄×D₁₃₉

yes

characteristic

C₁₆

—

139

—

C₁₆

C₄×D₁₃₉

yes

—

C₂³

—

139

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

C₈

—

C₈

—

278

—

C₈

—

139

—

C₈

—

139

—

C₈

C₂×C₄×D₁₃₉

yes

—

C₂²

—

556

—

C₂²

—

139

—

100

C₂²

M₁₆×D₁₃₉

yes

characteristic

101

C₄

—

102

C₄

—

556

—

103

C₄

—

139

—

104

C₄

C₄²×D₁₃₉

yes

characteristic

105

C₄

M₁₆×D₁₃₉

yes

characteristic

106

C₂

—

107

C₂

—

278

—

108

C₂

#97×D₁₃₉

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₃₉
commutator	C₄×C₁₃₉	C₂×C₄²
Frattini	C₂×C₈	C₂²×D₁₃₉
Fitting	#347×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₂×#97
prime radical	C₂²×D₁₃₉	M₁₆
cosocle	C₂×C₈×C₁₃₉	C₂³
hypercenter	#347	D₁₃₉
hypocenter	C₁₃₉	#2110
LCS	G ▷ C₄×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ #347

subgroup

G/H

center

C₄

C₄²×D₁₃₉

commutator

C₄×C₁₃₉

C₂×C₄²

Frattini

C₂×C₈

C₂²×D₁₃₉

Fitting

#347×C₁₃₉

C₂

socle

C₂×C₁₃₉

C₂×#97

prime radical

C₂²×D₁₃₉

M₁₆

cosocle

C₂×C₈×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₄×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₄ ◁ #347

N	H ≅ G/N	type
3. #22580731	C₂	semidirect
4. #22580730	C₂	semidirect
5. #347×C₁₃₉	C₂	semidirect
10. #17984781	C₄	semidirect
11. C₁₆×D₁₃₉	C₄	semidirect
54. C₁₃₉	#2110	semidirect
60. #347	D₁₃₉	semidirect

H ≅ G/N

type

3. #22580731

C₂

semidirect

4. #22580730

C₂

semidirect

5. #347×C₁₃₉

C₂

semidirect

10. #17984781

C₄

semidirect

11. C₁₆×D₁₃₉

C₄

semidirect

54. C₁₃₉

#2110

semidirect

60. #347

D₁₃₉

semidirect

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

subgroup	index
M₃₂×D₁₃₉	2
#22580910	2
#22580731	2
#22580730	2
#347×C₁₃₉	2
#2110	139

subgroup

index

M₃₂×D₁₃₉

#347×C₁₃₉

139

Subgroups — #31808007

Conjugacy classes of subgroups

Complements

Maximal subgroups