Group #31808209 — subgroups

Subgroups — #31808209

Order 17792.

138 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22580907	C₂	1	1	yes	maximal, characteristic
2	Dic₁₃₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	(C₂×D₁₃₉)⋊C₁₆	C₂	1	2	yes	maximal
4	#22580729	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₁₆×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	C₂×C₁₃₉⋊C₁₆	C₂²	1	2	yes	—
10	#17984753	C₄	1	1	yes	characteristic
11	#17984752	C₄	1	1	yes	characteristic
12	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₃₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₃₉	C₂×C₄	1	1	yes	characteristic
16	#14555768	—	2	2	no	—
17	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₈×D₁₃₉	—	2	2	no	—
20	Dic₁₃₉⋊C₄	C₈	1	2	yes	—
21	(C₂×D₁₃₉)⋊C₄	C₈	1	2	yes	—
22	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
23	C₁₃₉⋊C₁₆	—	2	2	no	—
24	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
26	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₁₃₉	—	2	2	no	—
28	C₂²×D₁₃₉	C₂×C₈	1	1	yes	characteristic
29	C₂×Dic₁₃₉	C₂×C₈	1	2	yes	—
30	C₂×Dic₁₃₉	C₂×C₈	1	1	yes	characteristic
31	C₄×D₁₃₉	—	2	1	no	—
32	C₄×D₁₃₉	—	2	1	no	—
33	C₁₃₉⋊C₈	—	2	1	no	—
34	C₁₃₉⋊C₈	—	2	1	no	—
35	C₂×C₄×C₁₃₉	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₁₃₉	M₁₆	1	2	yes	—
38	C₈×C₁₃₉	#41	1	2	yes	—
39	C₂²×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
40	C₂×D₁₃₉	—	2	1	no	—
41	C₂×D₁₃₉	—	2	2	no	—
42	C₄×C₁₃₉	—	2	2	no	—
43	C₄×C₁₃₉	#117	1	1	yes	characteristic
44	C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
45	Dic₁₃₉	—	2	1	no	—
46	Dic₁₃₉	—	2	2	no	—
47	C₂×C₁₃₉	#431	1	1	yes	characteristic
48	C₂×C₁₃₉	#504	1	2	yes	—
49	D₁₃₉	—	4	1	no	—
50	C₁₃₉	#2164	1	1	yes	characteristic, complemented, Sylow
51	#2164	—	139	1	no	maximal, Sylow
52	#405	—	139	1	no	—
53	C₄⋊C₁₆	—	139	2	no	—
54	C₂²⋊C₁₆	—	139	2	no	—
55	#346	—	139	1	no	—
56	C₄×C₁₆	D₁₃₉	1	1	yes	characteristic, complemented
57	C₂×M₁₆	—	139	1	no	—
58	C₂²×C₈	—	139	1	no	—
59	#117	—	139	1	no	—
60	C₂×C₁₆	—	139	2	no	—
61	C₂×C₁₆	C₂×D₁₃₉	1	2	yes	—
62	#97	—	139	1	no	—
63	#97	—	139	1	no	—
64	C₄×C₈	—	139	1	no	—
65	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
66	C₂²×C₄	—	139	1	no	—
67	M₁₆	—	278	2	no	—
68	C₂×C₈	—	278	2	no	—
69	C₂×C₈	—	139	1	no	—
70	C₂×C₈	—	139	1	no	—
71	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
73	C₄⋊C₄	—	139	2	no	—
74	C₂²⋊C₄	—	139	2	no	—
75	C₄²	—	139	1	no	—
76	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	278	2	no	—
79	C₂³	—	139	1	no	—
80	C₂×C₄	—	278	1	no	—
81	C₂×C₄	—	278	1	no	—
82	C₂×C₄	—	139	2	no	—
83	C₂×C₄	—	139	1	no	—
84	C₂×C₄	C₈×D₁₃₉	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
86	C₈	—	278	1	no	—
87	C₈	—	278	1	no	—
88	C₈	#14555768	1	2	yes	—
89	C₈	#14555785	1	2	yes	—
90	C₂²	—	278	1	no	—
91	C₂²	—	278	2	no	—
92	C₂²	C₂×C₈×D₁₃₉	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	278	1	no	—
95	C₄	—	278	2	no	—
96	C₄	#17984799	1	1	yes	characteristic
97	C₄	C₂×#14555768	1	1	yes	characteristic
98	C₂	—	556	1	no	—
99	C₂	#22581004	1	1	yes	characteristic
100	C₂	#22581369	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22580907

C₂

yes

maximal, characteristic

Dic₁₃₉⋊C₁₆

C₂

yes

maximal, complemented

(C₂×D₁₃₉)⋊C₁₆

C₂

yes

maximal

#22580729

C₂

yes

maximal, characteristic, complemented

C₄×C₁₆×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555768

C₄

yes

characteristic

C₂×C₈×D₁₃₉

C₂²

yes

characteristic

#17984799

C₄

yes

characteristic

C₂×C₁₃₉⋊C₁₆

C₂²

yes

—

#17984753

C₄

yes

characteristic

#17984752

C₄

yes

characteristic

C₈×Dic₁₃₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₃₉

C₂²

yes

—

C₄×C₈×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

—

Dic₁₃₉⋊C₄

C₈

yes

—

(C₂×D₁₃₉)⋊C₄

C₈

yes

—

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₁₃₉⋊C₁₆

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₄²×C₁₃₉

C₂×C₄

yes

characteristic

C₁₆×C₁₃₉

—

C₂²×D₁₃₉

C₂×C₈

yes

characteristic

C₂×Dic₁₃₉

C₂×C₈

yes

—

C₂×Dic₁₃₉

C₂×C₈

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂×C₄×C₁₃₉

C₂×C₈

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₈×C₁₃₉

M₁₆

yes

—

C₈×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂²×C₈

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#117

yes

characteristic

C₄×C₁₃₉

C₂×M₁₆

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#431

yes

characteristic

C₂×C₁₃₉

#504

yes

—

D₁₃₉

—

C₁₃₉

#2164

yes

characteristic, complemented, Sylow

#2164

—

139

maximal, Sylow

#405

—

139

—

C₄⋊C₁₆

—

139

—

C₂²⋊C₁₆

—

139

—

#346

—

139

—

C₄×C₁₆

D₁₃₉

yes

characteristic, complemented

C₂×M₁₆

—

139

—

C₂²×C₈

—

139

—

#117

—

139

—

C₂×C₁₆

—

139

—

C₂×C₁₆

C₂×D₁₃₉

yes

—

#97

—

139

—

#97

—

139

—

C₄×C₈

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

—

139

—

M₁₆

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

C₄²

—

139

—

C₄²

C₄×D₁₃₉

yes

characteristic

C₁₆

—

C₁₆

—

278

—

C₂³

—

139

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₈×D₁₃₉

yes

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

C₈

—

278

—

C₈

—

278

—

C₈

#14555768

yes

—

C₈

#14555785

yes

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

C₂×C₈×D₁₃₉

yes

characteristic

C₄

—

C₄

—

278

—

C₄

—

278

—

C₄

yes

C₄

yes

C₂

—

556

—

C₂

#22581004

yes

characteristic

100

C₂

#22581369

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂²×C₈
Fitting	C₄×C₁₆×C₁₃₉	C₂
prime radical	C₂²×D₁₃₉	C₂×C₈
cosocle	C₂×C₈×C₁₃₉	C₂³
hypercenter	C₄×C₁₆	D₁₃₉
hypocenter	C₁₃₉	#2164
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂²×C₈

Fitting

C₄×C₁₆×C₁₃₉

C₂

prime radical

C₂²×D₁₃₉

C₂×C₈

cosocle

C₂×C₈×C₁₃₉

C₂³

hypercenter

C₄×C₁₆

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. Dic₁₃₉⋊C₁₆	C₂	semidirect
4. #22580729	C₂	semidirect
5. C₄×C₁₆×C₁₃₉	C₂	semidirect
50. C₁₃₉	#2164	semidirect
56. C₄×C₁₆	D₁₃₉	semidirect

H ≅ G/N

type

2. Dic₁₃₉⋊C₁₆

C₂

semidirect

4. #22580729

C₂

semidirect

5. C₄×C₁₆×C₁₃₉

C₂

semidirect

50. C₁₃₉

#2164

semidirect

56. C₄×C₁₆

D₁₃₉

semidirect

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

subgroup	index
#22580907	2
Dic₁₃₉⋊C₁₆	2
(C₂×D₁₃₉)⋊C₁₆	2
#22580729	2
C₄×C₁₆×C₁₃₉	2
#2164	139

subgroup

index

#22580907

Dic₁₃₉⋊C₁₆

(C₂×D₁₃₉)⋊C₁₆

#22580729

C₄×C₁₆×C₁₃₉

#2164

139

Subgroups — #31808209

Conjugacy classes of subgroups

Complements

Maximal subgroups