Group #31808003 — subgroups

Subgroups — #31808003

Order 17792.

184 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂²×C₁₃₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#22580908	C₂	1	1	yes	maximal, characteristic
3	#22580728	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₁₃₉	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₄	1	1	yes	characteristic
7	#17984801	C₄	1	1	yes	characteristic
8	C₂×C₁₃₉⋊C₁₆	C₄	1	4	yes	complemented
9	C₂×C₁₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#17984752	C₄	1	2	yes	—
12	C₈×Dic₁₃₉	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	M₃₂×C₁₃₉	C₄	1	4	yes	complemented
15	C₂×C₁₆×C₁₃₉	C₂²	1	2	yes	—
16	#14555769	C₈	1	4	yes	—
17	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
18	C₄⋊Dic₁₃₉	C₈	1	2	yes	—
19	C₂²⋊Dic₁₃₉	C₈	1	2	yes	—
20	C₄×Dic₁₃₉	C₂×C₄	1	2	yes	—
21	C₁₃₉⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₁₃₉⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
28	C₁₆×C₁₃₉	C₂×C₄	1	4	yes	—
29	C₂×Dic₁₃₉	C₂×C₈	1	4	yes	—
30	C₁₃₉⋊C₈	C₂×C₈	1	4	yes	—
31	C₂³×C₁₃₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₃₉	C₄²	1	2	yes	—
33	C₂×C₄×C₁₃₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₃₉	C₄²	1	2	yes	—
37	C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₁₃₉	—	2	2	no	—
40	C₂²×C₁₃₉	C₄×C₈	1	2	yes	—
41	C₂²×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₃₉	C₄×C₈	1	2	yes	—
43	C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
45	Dic₁₃₉	—	2	4	no	—
46	C₂×C₁₃₉	—	2	2	no	—
47	C₂×C₁₃₉	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂×C₁₃₉	#504	1	2	yes	—
49	C₁₃₉	#2108	1	1	yes	characteristic, complemented, Sylow
50	#2108	—	139	1	no	maximal, Sylow
51	C₂×M₃₂	D₁₃₉	1	1	yes	characteristic
52	C₂²×C₁₆	—	139	1	no	—
53	#405	—	139	1	no	—
54	#346	—	139	2	no	—
55	C₄×C₁₆	—	139	2	no	—
56	C₂×M₁₆	—	139	1	no	—
57	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
58	#117	—	139	1	no	—
59	M₃₂	Dic₁₃₉	1	4	yes	complemented
60	C₂×C₁₆	—	139	4	no	—
61	C₂×C₁₆	—	139	1	no	—
62	C₂×C₁₆	—	139	1	no	—
63	C₂×C₁₆	C₂×D₁₃₉	1	2	yes	—
64	#97	—	139	2	no	—
65	C₄×C₈	—	139	2	no	—
66	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
67	M₁₆	—	139	4	no	—
68	C₂×C₈	—	139	2	no	—
69	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
70	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
71	C₂×C₈	C₂×Dic₁₃₉	1	2	yes	—
72	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
73	C₄⋊C₄	—	139	2	no	—
74	C₂²⋊C₄	—	139	2	no	—
75	C₄²	—	139	2	no	—
76	C₁₆	—	139	2	no	—
77	C₁₆	—	139	2	no	—
78	C₁₆	C₂×Dic₁₃₉	1	4	yes	—
79	C₂³	C₈×D₁₃₉	1	1	yes	characteristic
80	C₂×C₄	—	139	4	no	—
81	C₂×C₄	C₄×Dic₁₃₉	1	2	yes	—
82	C₂×C₄	C₈×D₁₃₉	1	2	yes	—
83	C₂×C₄	C₈×D₁₃₉	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
85	C₈	—	139	4	no	—
86	C₈	C₄×Dic₁₃₉	1	2	yes	—
87	C₈	C₂×C₄×D₁₃₉	1	1	yes	characteristic
88	C₈	C₂²×Dic₁₃₉	1	1	yes	characteristic
89	C₂²	—	2	2	no	—
90	C₂²	C₈×Dic₁₃₉	1	2	yes	—
91	C₂²	C₂×C₈×D₁₃₉	1	1	yes	characteristic
92	C₄	—	278	4	no	—
93	C₄	C₈×Dic₁₃₉	1	2	yes	—
94	C₄	C₂×C₄×Dic₁₃₉	1	1	yes	characteristic
95	C₄	C₂×C₈×D₁₃₉	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₈×Dic₁₃₉	1	1	yes	characteristic
98	C₂	#22581371	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₃₉⋊C₁₆

C₂

yes

maximal, characteristic

#22580908

C₂

yes

maximal, characteristic

#22580728

C₂

yes

maximal, complemented

C₁₆×Dic₁₃₉

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₄

yes

characteristic

#17984801

C₄

yes

characteristic

C₂×C₁₃₉⋊C₁₆

C₄

yes

complemented

C₂×C₁₃₉⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₃₉⋊C₁₆

C₂²

yes

characteristic

#17984752

C₄

yes

—

C₈×Dic₁₃₉

C₂²

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

M₃₂×C₁₃₉

C₄

yes

complemented

C₂×C₁₆×C₁₃₉

C₂²

yes

—

#14555769

C₈

yes

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₃₉

C₈

yes

—

C₂²⋊Dic₁₃₉

C₈

yes

—

C₄×Dic₁₃₉

C₂×C₄

yes

—

C₁₃₉⋊C₁₆

C₂×C₄

yes

—

C₁₃₉⋊C₁₆

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₁₆×C₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

C₂×C₈

yes

—

C₁₃₉⋊C₈

C₂×C₈

yes

—

C₂³×C₁₃₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₉

C₄²

yes

—

C₂×C₄×C₁₃₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂×C₈

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₈×C₁₃₉

C₄²

yes

—

C₈×C₁₃₉

C₂²×C₄

yes

characteristic

C₈×C₁₃₉

C₂²×C₄

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₄×C₈

yes

—

C₂²×C₁₃₉

C₂²×C₈

yes

characteristic

C₄×C₁₃₉

C₄×C₈

yes

—

C₄×C₁₃₉

C₂×C₄²

yes

characteristic

C₄×C₁₃₉

C₂²×C₈

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₃₉

#504

yes

—

C₁₃₉

#2108

yes

characteristic, complemented, Sylow

#2108

—

139

maximal, Sylow

C₂×M₃₂

D₁₃₉

yes

characteristic

C₂²×C₁₆

—

139

—

#405

—

139

—

#346

—

139

—

C₄×C₁₆

—

139

—

C₂×M₁₆

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#117

—

139

—

M₃₂

Dic₁₃₉

yes

complemented

C₂×C₁₆

—

139

—

C₂×C₁₆

—

139

—

C₂×C₁₆

—

139

—

C₂×C₁₆

C₂×D₁₃₉

yes

—

#97

—

139

—

C₄×C₈

—

139

—

C₂²×C₄

C₄×D₁₃₉

yes

characteristic

M₁₆

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₂×C₈

C₄×D₁₃₉

yes

—

C₂×C₈

C₂×Dic₁₃₉

yes

—

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

C₄²

—

139

—

C₁₆

—

139

—

C₁₆

—

139

—

C₁₆

C₂×Dic₁₃₉

yes

—

C₂³

C₈×D₁₃₉

yes

characteristic

C₂×C₄

—

139

—

C₂×C₄

C₄×Dic₁₃₉

yes

—

C₂×C₄

C₈×D₁₃₉

yes

—

C₂×C₄

C₈×D₁₃₉

yes

characteristic

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

C₈

—

139

—

C₈

C₄×Dic₁₃₉

yes

—

C₈

C₂×C₄×D₁₃₉

yes

characteristic

C₈

C₂²×Dic₁₃₉

yes

characteristic

C₂²

—

C₂²

C₈×Dic₁₃₉

yes

—

C₂²

C₂×C₈×D₁₃₉

yes

characteristic

C₄

—

278

—

C₄

C₈×Dic₁₃₉

yes

—

C₄

C₂×C₄×Dic₁₃₉

yes

characteristic

C₄

C₂×C₈×D₁₃₉

yes

characteristic

C₂

—

C₂

C₂×C₈×Dic₁₃₉

yes

characteristic

C₂

#22581371

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₃₉
commutator	C₂×C₁₃₉	C₂×C₄×C₈
Fitting	C₂×M₃₂×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×C₈
prime radical	C₂³×C₁₃₉	C₂×C₈
cosocle	C₂×C₈×C₁₃₉	C₂³
hypercenter	C₂×M₃₂	D₁₃₉
hypocenter	C₁₃₉	#2108
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₃₉

commutator

C₂×C₁₃₉

C₂×C₄×C₈

Fitting

C₂×M₃₂×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×C₈

prime radical

C₂³×C₁₃₉

C₂×C₈

cosocle

C₂×C₈×C₁₃₉

C₂³

hypercenter

C₂×M₃₂

D₁₃₉

C₁₃₉

G ▷ C₂×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #22580728	C₂	semidirect
4. C₁₆×Dic₁₃₉	C₂	semidirect
8. C₂×C₁₃₉⋊C₁₆	C₄	semidirect
14. M₃₂×C₁₃₉	C₄	semidirect
49. C₁₃₉	#2108	semidirect
59. M₃₂	Dic₁₃₉	semidirect

H ≅ G/N

type

3. #22580728

C₂

semidirect

4. C₁₆×Dic₁₃₉

C₂

semidirect

8. C₂×C₁₃₉⋊C₁₆

C₄

semidirect

14. M₃₂×C₁₃₉

C₄

semidirect

49. C₁₃₉

#2108

semidirect

59. M₃₂

Dic₁₃₉

semidirect

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

subgroup	index
C₂²×C₁₃₉⋊C₁₆	2
#22580908	2
#22580728	2
C₁₆×Dic₁₃₉	2
C₂×M₃₂×C₁₃₉	2
#2108	139

subgroup

index

C₂²×C₁₃₉⋊C₁₆

#22580908

#22580728

C₁₆×Dic₁₃₉

C₂×M₃₂×C₁₃₉

#2108

139

Subgroups — #31808003

Conjugacy classes of subgroups

Complements

Maximal subgroups