Group #31808246 — subgroups

Subgroups — #31808246

Order 17792.

144 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22581370	C₂	1	2	yes	maximal, complemented
2	#22580774	C₂	1	1	yes	maximal, characteristic, complemented
3	#22580737	C₂	1	2	yes	maximal, complemented
4	#22580729	C₂	1	1	yes	maximal, characteristic, complemented
5	#443×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	#17984782	—	2	2	no	—
7	#17984782	C₂²	1	2	yes	—
8	C₂×C₁₃₉⋊C₁₆	—	2	2	no	—
9	C₂×C₁₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#135×C₁₃₉	C₄	1	1	yes	characteristic
12	#131×C₁₃₉	C₂²	1	2	yes	—
13	#108×C₁₃₉	C₄	1	1	yes	characteristic
14	#104×C₁₃₉	C₄	1	2	yes	—
15	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₁₃₉⋊C₁₆	—	2	2	no	—
17	C₁₃₉⋊C₁₆	—	2	1	no	—
18	C₁₃₉⋊C₁₆	D₄	1	2	yes	—
19	#41×C₁₃₉	C₂×C₄	1	2	yes	—
20	Q₁₆×C₁₃₉	C₈	1	1	yes	characteristic
21	SD₁₆×C₁₃₉	C₈	1	2	yes	—
22	D₈×C₁₃₉	C₈	1	1	yes	characteristic
23	M₁₆×C₁₃₉	—	2	2	no	—
24	M₁₆×C₁₃₉	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₃₉	—	2	2	no	—
26	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
28	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
29	Q₈×C₁₃₉	C₂×C₈	1	2	yes	—
30	D₄×C₁₃₉	—	2	2	no	—
31	D₄×C₁₃₉	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₁₃₉	—	2	2	no	—
33	C₂×C₄×C₁₃₉	—	2	1	no	—
34	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₃₉	—	2	2	no	—
36	C₈×C₁₃₉	C₂×C₈	1	1	yes	characteristic
37	C₈×C₁₃₉	C₂×C₈	1	1	yes	characteristic
38	C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₃₉	#41	1	1	yes	characteristic
40	C₂²×C₁₃₉	—	2	2	no	—
41	C₂²×C₁₃₉	#131	1	1	yes	characteristic
42	C₄×C₁₃₉	—	2	2	no	—
43	C₄×C₁₃₉	—	4	1	no	—
44	C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
45	C₄×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₁₃₉	—	2	1	no	—
47	C₂×C₁₃₉	—	4	2	no	—
48	C₂×C₁₃₉	C₈×D₄	1	1	yes	characteristic
49	C₁₃₉	#2171	1	1	yes	characteristic, complemented, Sylow
50	#2171	—	139	1	no	maximal, Sylow
51	#504	—	139	2	no	—
52	#443	D₁₃₉	1	1	yes	characteristic
53	#364	—	139	1	no	—
54	#350	—	139	2	no	—
55	#346	—	139	1	no	—
56	#135	Dic₁₃₉	1	1	yes	characteristic
57	#131	C₂×D₁₃₉	1	2	yes	—
58	M₃₂	—	278	2	no	—
59	M₃₂	—	139	2	no	—
60	C₂×C₁₆	—	278	2	no	—
61	C₂×C₁₆	—	139	1	no	—
62	C₂×C₁₆	—	139	1	no	—
63	#108	Dic₁₃₉	1	1	yes	characteristic
64	#104	Dic₁₃₉	1	2	yes	—
65	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
66	#41	C₂×Dic₁₃₉	1	2	yes	—
67	Q₁₆	C₁₃₉⋊C₈	1	1	yes	characteristic
68	SD₁₆	C₁₃₉⋊C₈	1	2	yes	—
69	D₈	C₁₃₉⋊C₈	1	1	yes	characteristic
70	M₁₆	—	2	2	no	—
71	M₁₆	C₂×Dic₁₃₉	1	2	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×Dic₁₃₉	1	1	yes	characteristic
74	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
75	C₄²	C₂×Dic₁₃₉	1	1	yes	characteristic
76	C₁₆	—	278	2	no	—
77	C₁₆	—	278	1	no	—
78	C₁₆	—	139	2	no	—
79	Q₈	C₂×C₁₃₉⋊C₈	1	2	yes	—
80	D₄	—	2	2	no	—
81	D₄	C₂×C₁₃₉⋊C₈	1	2	yes	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	C₂×C₁₃₉⋊C₈	1	1	yes	characteristic
87	C₈	C₂×C₁₃₉⋊C₈	1	1	yes	characteristic
88	C₈	D₄×D₁₃₉	1	1	yes	characteristic
89	C₈	#14555784	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	#17984849	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	4	1	no	—
94	C₄	D₄×Dic₁₃₉	1	1	yes	characteristic
95	C₄	C₂²×C₁₃₉⋊C₈	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	4	2	no	—
98	C₂	D₄×C₁₃₉⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581370

C₂

yes

maximal, complemented

#22580774

C₂

yes

maximal, characteristic, complemented

#22580737

C₂

yes

maximal, complemented

#22580729

C₂

yes

maximal, characteristic, complemented

#443×C₁₃₉

C₂

yes

maximal, characteristic

#17984782

—

#17984782

C₂²

yes

—

C₂×C₁₃₉⋊C₁₆

—

C₂×C₁₃₉⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₃₉⋊C₁₆

C₂²

yes

characteristic

#135×C₁₃₉

C₄

yes

characteristic

#131×C₁₃₉

C₂²

yes

—

#108×C₁₃₉

C₄

yes

characteristic

#104×C₁₃₉

C₄

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₁₃₉⋊C₁₆

—

C₁₃₉⋊C₁₆

—

C₁₃₉⋊C₁₆

D₄

yes

—

#41×C₁₃₉

C₂×C₄

yes

—

Q₁₆×C₁₃₉

C₈

yes

characteristic

SD₁₆×C₁₃₉

C₈

yes

—

D₈×C₁₃₉

C₈

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₄²×C₁₃₉

C₂×C₄

yes

characteristic

Q₈×C₁₃₉

C₂×C₈

yes

—

D₄×C₁₃₉

—

D₄×C₁₃₉

C₂×C₈

yes

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

C₂×C₈

yes

characteristic

C₈×C₁₃₉

C₂×C₈

yes

characteristic

C₈×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

#41

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#131

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×D₄

yes

characteristic

C₄×C₁₃₉

C₂²×C₈

yes

characteristic

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₈×D₄

yes

characteristic

C₁₃₉

#2171

yes

characteristic, complemented, Sylow

#2171

—

139

maximal, Sylow

#504

—

139

—

#443

D₁₃₉

yes

characteristic

#364

—

139

—

#350

—

139

—

#346

—

139

—

#135

Dic₁₃₉

yes

characteristic

#131

C₂×D₁₃₉

yes

—

M₃₂

—

278

—

M₃₂

—

139

—

C₂×C₁₆

—

278

—

C₂×C₁₆

—

139

—

C₂×C₁₆

—

139

—

#108

Dic₁₃₉

yes

characteristic

#104

Dic₁₃₉

yes

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

#41

C₂×Dic₁₃₉

yes

—

Q₁₆

C₁₃₉⋊C₈

yes

characteristic

SD₁₆

C₁₃₉⋊C₈

yes

—

D₈

C₁₃₉⋊C₈

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₁₃₉

yes

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₃₉

yes

characteristic

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

C₄²

C₂×Dic₁₃₉

yes

characteristic

C₁₆

—

278

—

C₁₆

—

278

—

C₁₆

—

139

—

Q₈

C₂×C₁₃₉⋊C₈

yes

—

D₄

—

D₄

C₂×C₁₃₉⋊C₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₁₃₉

yes

characteristic

C₈

—

C₈

C₂×C₁₃₉⋊C₈

yes

characteristic

C₈

C₂×C₁₃₉⋊C₈

yes

characteristic

C₈

D₄×D₁₃₉

yes

characteristic

C₈

#14555784

yes

characteristic

C₂²

—

C₂²

#17984849

yes

characteristic

C₄

—

C₄

—

C₄

D₄×Dic₁₃₉

yes

characteristic

C₄

C₂²×C₁₃₉⋊C₈

yes

characteristic

C₂

—

C₂

—

C₂

D₄×C₁₃₉⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₁₃₉
commutator	C₄×C₁₃₉	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₁₃₉
Fitting	#443×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₈×D₄
prime radical	#135×C₁₃₉	C₄
cosocle	C₂×C₈×C₁₃₉	C₂³
hypercenter	#443	D₁₃₉
hypocenter	C₁₃₉	#2171
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

subgroup

G/H

center

C₈

D₄×D₁₃₉

commutator

C₄×C₁₃₉

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₁₃₉

Fitting

#443×C₁₃₉

C₂

socle

C₂×C₁₃₉

C₈×D₄

prime radical

#135×C₁₃₉

C₄

cosocle

C₂×C₈×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

N	H ≅ G/N	type
1. #22581370	C₂	semidirect
2. #22580774	C₂	semidirect
3. #22580737	C₂	semidirect
4. #22580729	C₂	semidirect
49. C₁₃₉	#2171	semidirect

H ≅ G/N

type

1. #22581370

C₂

semidirect

2. #22580774

C₂

semidirect

3. #22580737

C₂

semidirect

4. #22580729

C₂

semidirect

49. C₁₃₉

#2171

semidirect

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

subgroup	index
#22581370	2
#22580774	2
#22580737	2
#22580729	2
#443×C₁₃₉	2
#2171	139

subgroup

index

#443×C₁₃₉

139

Subgroups — #31808246

Conjugacy classes of subgroups

Complements

Maximal subgroups