Group #31806540 — subgroups

Subgroups — #31806540

Order 17792.

188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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	#22580976	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#22580718	C₂	1	1	yes	maximal, characteristic
4	#22580717	C₂	1	2	yes	maximal
5	#341×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	2	yes	—
7	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
9	Dic₁₃₉⋊C₈	—	2	4	no	—
10	C₈×Dic₁₃₉	C₄	1	2	yes	—
11	C₈×Dic₁₃₉	C₄	1	2	yes	—
12	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₃₉	Q₈	1	2	yes	—
15	#14555769	—	2	2	no	—
16	#14555769	—	2	2	no	—
17	C₂×C₁₃₉⋊C₈	—	2	2	no	—
18	C₂×C₁₃₉⋊C₈	—	2	2	no	—
19	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₁₃₉	D₄	1	2	yes	—
23	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₃₉	—	2	2	no	—
25	M₁₆×C₁₃₉	—	2	2	no	—
26	C₂×C₈×C₁₃₉	—	2	2	no	—
27	C₂×C₈×C₁₃₉	—	2	2	no	—
28	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₃₉	—	2	4	no	—
30	C₂×Dic₁₃₉	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₁₃₉	C₄⋊C₄	1	2	yes	—
32	C₁₃₉⋊C₈	—	2	2	no	—
33	C₁₃₉⋊C₈	—	4	2	no	—
34	C₂³×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₉	#41	1	2	yes	—
38	C₂×C₄×C₁₃₉	#41	1	2	yes	—
39	C₈×C₁₃₉	—	2	2	no	—
40	C₈×C₁₃₉	—	4	2	no	—
41	C₂²×C₁₃₉	—	2	2	no	—
42	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₃₉	#125	1	2	yes	—
44	C₄×C₁₃₉	#117	1	1	yes	characteristic
45	C₄×C₁₃₉	#117	1	1	yes	characteristic
46	C₄×C₁₃₉	#124	1	2	yes	—
47	Dic₁₃₉	—	2	2	no	—
48	Dic₁₃₉	#108	1	4	yes	—
49	C₂×C₁₃₉	—	2	2	no	—
50	C₂×C₁₃₉	#382	1	1	yes	characteristic
51	C₂×C₁₃₉	C₂×#108	1	2	yes	—
52	C₁₃₉	#1837	1	1	yes	characteristic, complemented, Sylow
53	#1837	—	139	1	no	maximal, Sylow
54	#423	—	139	2	no	—
55	C₂×C₄×C₈	—	139	1	no	—
56	#341	—	139	2	no	—
57	#341	—	139	1	no	—
58	#341	D₁₃₉	1	1	yes	characteristic
59	C₂×M₁₆	—	139	2	no	—
60	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
61	C₂²×C₈	—	139	1	no	—
62	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
63	C₂×C₄²	—	139	1	no	—
64	C₄⋊C₈	—	278	4	no	—
65	C₄×C₈	—	139	2	no	—
66	C₄×C₈	—	139	2	no	—
67	C₂²×C₄	—	139	2	no	—
68	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	278	2	no	—
72	M₁₆	—	278	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	278	2	no	—
76	C₂×C₈	—	278	2	no	—
77	C₂×C₈	—	139	2	no	—
78	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
79	C₄²	—	139	1	no	—
80	C₄²	—	139	1	no	—
81	C₄²	—	139	2	no	—
82	C₂³	Q₈×D₁₃₉	1	1	yes	characteristic
83	C₂×C₄	—	278	4	no	—
84	C₂×C₄	—	139	2	no	—
85	C₂×C₄	—	139	2	no	—
86	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
87	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
88	C₂×C₄	#14555784	1	2	yes	—
89	C₂×C₄	#14555785	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	278	2	no	—
93	C₈	—	556	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
96	C₂²	#17984832	1	2	yes	—
97	C₄	—	278	2	no	—
98	C₄	—	139	4	no	—
99	C₄	#17984798	1	1	yes	characteristic
100	C₄	#17984799	1	1	yes	characteristic
101	C₄	#17984828	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#22580817	1	1	yes	characteristic
104	C₂	#108×D₁₃₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22580976

C₂

yes

maximal

C₂×C₈×Dic₁₃₉

C₂

yes

maximal, characteristic

#22580718

C₂

yes

maximal, characteristic

#22580717

C₂

yes

maximal

#341×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

Dic₁₃₉⋊C₈

—

C₈×Dic₁₃₉

C₄

yes

—

C₈×Dic₁₃₉

C₄

yes

—

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

Q₈

yes

—

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₄×Dic₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₄⋊C₄

yes

—

C₂×Dic₁₃₉

C₄⋊C₄

yes

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

#125

yes

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

—

Dic₁₃₉

—

Dic₁₃₉

#108

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#382

yes

characteristic

C₂×C₁₃₉

C₂×#108

yes

—

C₁₃₉

#1837

yes

characteristic, complemented, Sylow

#1837

—

139

maximal, Sylow

#423

—

139

—

C₂×C₄×C₈

—

139

—

#341

—

139

—

#341

—

139

—

#341

D₁₃₉

yes

characteristic

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂×C₄²

—

139

—

C₄⋊C₈

—

278

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

278

—

M₁₆

—

278

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

—

C₄²

—

139

—

C₄²

—

139

—

C₄²

—

139

—

C₂³

Q₈×D₁₃₉

yes

characteristic

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

C₂×C₄

D₄×D₁₃₉

yes

characteristic

C₂×C₄

#14555784

yes

—

C₂×C₄

#14555785

yes

—

C₈

—

C₈

—

C₈

—

278

—

C₈

—

556

—

C₂²

—

C₂²

C₄⋊C₄×D₁₃₉

yes

characteristic

C₂²

#17984832

yes

—

C₄

—

278

—

C₄

—

139

—

C₄

#17984798

yes

characteristic

100

C₄

#17984799

yes

characteristic

101

C₄

#17984828

yes

—

102

C₂

—

103

C₂

#22580817

yes

characteristic

104

C₂

#108×D₁₃₉

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#341×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#341	D₁₃₉
hypocenter	C₁₃₉	#1837
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#341×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₃₉

C₂×Q₈

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
52. C₁₃₉	#1837	semidirect

H ≅ G/N

type

52. C₁₃₉

#1837

semidirect

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

subgroup	index
#22580976	2
C₂×C₈×Dic₁₃₉	2
#22580718	2
#22580717	2
#341×C₁₃₉	2
#1837	139

subgroup

index

C₂×C₈×Dic₁₃₉

#341×C₁₃₉

139

Subgroups — #31806540

Conjugacy classes of subgroups

Complements

Maximal subgroups