Group #31807370 — subgroups

Subgroups — #31807370

Order 17792.

252 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×C₄×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#22580819	C₂	1	1	yes	maximal, characteristic
3	#22580715	C₂	1	4	yes	maximal
4	C₂×#125×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
7	C₄×C₁₃₉⋊C₈	—	2	2	no	—
8	C₁₃₉⋊#95	—	2	2	no	—
9	#125×C₁₃₉	—	2	2	no	—
10	#125×C₁₃₉	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₃₉	—	2	2	no	—
15	C₂×C₁₃₉⋊C₈	—	2	2	no	—
16	C₂×C₁₃₉⋊C₈	—	2	2	no	—
17	C₂×C₁₃₉⋊C₈	Q₈	1	4	yes	—
18	C₄⋊Dic₁₃₉	—	2	4	no	—
19	C₂²×C₄×C₁₃₉	—	2	2	no	—
20	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
21	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₃₉	—	2	8	no	—
23	C₄⋊C₄×C₁₃₉	—	2	2	no	—
24	C₄⋊C₄×C₁₃₉	C₂×C₄	1	4	yes	—
25	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
26	C₄²×C₁₃₉	D₄	1	2	yes	—
27	C₂×Dic₁₃₉	—	4	2	no	—
28	C₂×Dic₁₃₉	—	4	4	no	—
29	C₁₃₉⋊C₈	—	2	4	no	—
30	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₉	—	2	4	no	—
32	C₂×C₄×C₁₃₉	—	2	4	no	—
33	C₂×C₄×C₁₃₉	—	4	4	no	—
34	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₃₉	#41	1	2	yes	—
39	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₃₉	#124	1	1	yes	characteristic
41	C₂²×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₃₉	#135	1	4	yes	—
43	C₄×C₁₃₉	—	2	4	no	—
44	C₄×C₁₃₉	—	4	4	no	—
45	C₄×C₁₃₉	C₄×Q₈	1	2	yes	—
46	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
47	Dic₁₃₉	—	8	2	no	—
48	C₂×C₁₃₉	#391	1	1	yes	characteristic
49	C₂×C₁₃₉	#416	1	2	yes	—
50	C₂×C₁₃₉	#488	1	2	yes	—
51	C₂×C₁₃₉	#499	1	2	yes	—
52	C₁₃₉	#1991	1	1	yes	characteristic, complemented, Sylow
53	#1991	—	139	1	no	maximal, Sylow
54	C₂×#125	D₁₃₉	1	1	yes	characteristic
55	C₂×C₄×C₈	—	139	1	no	—
56	#382	—	139	1	no	—
57	#340	—	139	4	no	—
58	C₂²×C₈	—	139	2	no	—
59	#125	—	2	2	no	—
60	#125	Dic₁₃₉	1	4	yes	complemented
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	139	2	no	—
63	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
64	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
65	C₄×C₈	—	278	2	no	—
66	#95	—	278	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	278	2	no	—
69	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
70	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
71	C₂×C₈	—	278	2	no	—
72	C₂×C₈	—	278	2	no	—
73	C₂×C₈	—	139	4	no	—
74	C₄⋊C₄	—	2	8	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	278	4	no	—
77	C₄⋊C₄	C₂×Dic₁₃₉	1	4	yes	—
78	C₄²	C₂×Dic₁₃₉	1	2	yes	—
79	C₄²	C₂²⋊D₁₃₉	1	2	yes	—
80	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	556	2	no	—
85	C₂×C₄	—	556	4	no	—
86	C₂×C₄	C₂²⋊Dic₁₃₉	1	4	yes	—
87	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
89	C₂×C₄	Q₈×D₁₃₉	1	2	yes	—
90	C₂×C₄	#14555786	1	2	yes	—
91	C₈	—	278	4	no	—
92	C₂²	C₂×C₂²⋊Dic₁₃₉	1	1	yes	characteristic
93	C₂²	#17984831	1	1	yes	characteristic
94	C₂²	#17984840	1	1	yes	characteristic
95	C₂²	#17984862	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	4	4	no	—
98	C₄	—	1112	2	no	—
99	C₄	Q₈×Dic₁₃₉	1	2	yes	—
100	C₄	#17984818	1	2	yes	—
101	C₂	#22580860	1	1	yes	characteristic
102	C₂	#22580949	1	2	yes	—
103	C₂	#22581310	1	2	yes	—
104	C₂	#22581352	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic

#22580819

C₂

yes

maximal, characteristic

#22580715

C₂

yes

maximal

C₂×#125×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₄×C₁₃₉⋊C₈

—

C₁₃₉⋊#95

—

#125×C₁₃₉

—

#125×C₁₃₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

Q₈

yes

—

C₄⋊Dic₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

D₄

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₉

#124

yes

characteristic

C₂²×C₁₃₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃₉

#135

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×Q₈

yes

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

#391

yes

characteristic

C₂×C₁₃₉

#416

yes

—

C₂×C₁₃₉

#488

yes

—

C₂×C₁₃₉

#499

yes

—

C₁₃₉

#1991

yes

characteristic, complemented, Sylow

#1991

—

139

maximal, Sylow

C₂×#125

D₁₃₉

yes

characteristic

C₂×C₄×C₈

—

139

—

#382

—

139

—

#340

—

139

—

C₂²×C₈

—

139

—

#125

—

#125

Dic₁₃₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄×C₈

—

278

—

#95

—

278

—

C₂²×C₄

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

C₂×Dic₁₃₉

yes

—

C₄²

C₂×Dic₁₃₉

yes

—

C₄²

C₂²⋊D₁₃₉

yes

—

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

556

—

C₂×C₄

—

556

—

C₂×C₄

C₂²⋊Dic₁₃₉

yes

—

C₂×C₄

C₂²×Dic₁₃₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₂×C₄

Q₈×D₁₃₉

yes

—

C₂×C₄

#14555786

yes

—

C₈

—

278

—

C₂²

C₂×C₂²⋊Dic₁₃₉

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

1112

—

C₄

Q₈×Dic₁₃₉

yes

—

100

C₄

#17984818

yes

—

101

C₂

#22580860

yes

characteristic

102

C₂

#22580949

yes

—

103

C₂

#22581310

yes

—

104

C₂

#22581352

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#125×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#125	D₁₃₉
hypocenter	C₁₃₉	#1991
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×#125×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₁₃₉	C₄	semidirect
52. C₁₃₉	#1991	semidirect
60. #125	Dic₁₃₉	semidirect

H ≅ G/N

type

10. #125×C₁₃₉

C₄

semidirect

52. C₁₃₉

#1991

semidirect

60. #125

Dic₁₃₉

semidirect

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

subgroup	index
C₂×C₄×C₁₃₉⋊C₈	2
#22580819	2
#22580715	2
C₂×#125×C₁₃₉	2
#1991	139

subgroup

index

C₂×C₄×C₁₃₉⋊C₈

#22580819

#22580715

C₂×#125×C₁₃₉

#1991

139

Subgroups — #31807370

Conjugacy classes of subgroups

Complements

Maximal subgroups