Group #31820552 — subgroups

Subgroups — #31820552

Order 17792.

930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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	#22581707	C₂	1	10	yes	maximal, complemented
2	#22581701	C₂	1	5	yes	maximal, complemented
3	#22581509	C₂	1	5	yes	maximal, complemented
4	#22581488	C₂	1	10	yes	maximal, complemented
5	#564×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984841	C₂²	1	10	yes	—
7	#17984837	C₂²	1	20	yes	—
8	#17984834	C₂²	1	5	yes	—
9	#17984830	C₂²	1	10	yes	—
10	#17984827	C₂²	1	20	yes	—
11	#17984822	C₂²	1	10	yes	—
12	#17984820	C₂²	1	20	yes	—
13	#17984817	C₂²	1	10	yes	—
14	#17984807	C₂²	1	10	yes	—
15	#17984804	C₂²	1	10	yes	—
16	#17984800	C₂²	1	10	yes	—
17	C₄²×D₁₃₉	C₂²	1	5	yes	—
18	#125×C₁₃₉	C₂²	1	5	yes	—
19	#125×C₁₃₉	C₂²	1	10	yes	—
20	C₂×Dic₂₇₈	C₂³	1	10	yes	—
21	C₂×D₅₅₆	C₂³	1	10	yes	—
22	C₂×C₄×D₁₃₉	C₂³	1	5	yes	—
23	C₂×C₄×D₁₃₉	C₂³	1	10	yes	—
24	C₄⋊Dic₁₃₉	C₂³	1	10	yes	—
25	Dic₁₃₉⋊C₄	C₂³	1	20	yes	—
26	(C₂×D₁₃₉)⋊C₄	C₂³	1	20	yes	—
27	(C₂×D₁₃₉)⋊C₄	C₂³	1	20	yes	—
28	C₄×Dic₁₃₉	C₂³	1	10	yes	—
29	C₄×Dic₁₃₉	C₂³	1	5	yes	—
30	C₄⋊C₄×C₁₃₉	C₂³	1	10	yes	—
31	C₄⋊C₄×C₁₃₉	C₂³	1	10	yes	—
32	C₄⋊C₄×C₁₃₉	C₂³	1	10	yes	—
33	C₄²×C₁₃₉	C₂³	1	5	yes	—
34	C₂²×D₁₃₉	C₂⁴	1	5	yes	—
35	Dic₂₇₈	—	4	10	no	—
36	D₅₅₆	—	4	10	no	—
37	C₂×Dic₁₃₉	C₂⁴	1	10	yes	—
38	C₂×Dic₁₃₉	C₂⁴	1	1	yes	characteristic
39	C₄×D₁₃₉	—	2	10	no	—
40	C₄×D₁₃₉	—	4	10	no	—
41	C₂×C₄×C₁₃₉	C₂⁴	1	5	yes	—
42	C₂×C₄×C₁₃₉	C₂⁴	1	10	yes	—
43	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
44	C₂×D₁₃₉	—	2	5	no	—
45	C₂×D₁₃₉	—	2	10	no	—
46	C₄×C₁₃₉	—	2	5	no	—
47	C₄×C₁₃₉	—	2	10	no	—
48	Dic₁₃₉	—	2	10	no	—
49	Dic₁₃₉	2₋¹⁺⁴	1	2	yes	—
50	C₂×C₁₃₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
51	C₂×C₁₃₉	#585	1	2	yes	—
52	D₁₃₉	—	4	5	no	—
53	C₁₃₉	#3531	1	1	yes	characteristic, complemented, Sylow
54	#3531	—	139	1	no	maximal, Sylow
55	#564	D₁₃₉	1	1	yes	characteristic, complemented
56	#563	—	139	10	no	—
57	#562	—	139	5	no	—
58	#533	—	139	5	no	—
59	#529	—	139	10	no	—
60	C₄⋊Q₈	—	139	10	no	—
61	#126	—	139	20	no	—
62	#125	—	139	5	no	—
63	#125	C₂×D₁₃₉	1	10	yes	—
64	#125	C₂×D₁₃₉	1	5	yes	—
65	#124	—	139	10	no	—
66	#123	—	139	20	no	—
67	C₂²⋊Q₈	—	139	10	no	—
68	C₂²⋊Q₈	—	139	20	no	—
69	#121	—	139	10	no	—
70	C₄×Q₈	—	139	10	no	—
71	C₄×D₄	—	139	10	no	—
72	#117	—	139	10	no	—
73	C₂×C₄²	—	139	5	no	—
74	C₂×Q₈	—	139	10	no	—
75	C₂×D₄	—	139	10	no	—
76	C₂²×C₄	—	139	5	no	—
77	C₂²×C₄	—	139	10	no	—
78	C₄⋊C₄	—	139	20	no	—
79	C₄⋊C₄	—	139	10	no	—
80	C₄⋊C₄	C₂²×D₁₃₉	1	10	yes	—
81	C₄⋊C₄	C₂²×D₁₃₉	1	10	yes	—
82	C₄⋊C₄	C₂²×D₁₃₉	1	10	yes	—
83	C₂²⋊C₄	—	139	20	no	—
84	C₂²⋊C₄	—	139	20	no	—
85	C₄²	—	139	10	no	—
86	C₄²	—	139	5	no	—
87	C₄²	C₂²×D₁₃₉	1	5	yes	—
88	C₂³	—	139	5	no	—
89	Q₈	—	556	10	no	—
90	D₄	—	556	10	no	—
91	C₂×C₄	—	278	10	no	—
92	C₂×C₄	—	556	10	no	—
93	C₂×C₄	—	139	10	no	—
94	C₂×C₄	—	139	1	no	—
95	C₂×C₄	C₂³×D₁₃₉	1	5	yes	—
96	C₂×C₄	C₂³×D₁₃₉	1	10	yes	—
97	C₂²	—	278	5	no	—
98	C₂²	—	278	10	no	—
99	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	278	10	no	—
103	C₄	—	139	2	no	—
104	C₂	—	556	5	no	—
105	C₂	2₋¹⁺⁴×D₁₃₉	1	1	yes	characteristic
106	C₂	#22581790	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581707

C₂

yes

maximal, complemented

#22581701

C₂

yes

maximal, complemented

#22581509

C₂

yes

maximal, complemented

#22581488

C₂

yes

maximal, complemented

#564×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#17984841

C₂²

yes

—

#17984837

C₂²

yes

—

#17984834

C₂²

yes

—

#17984830

C₂²

yes

—

#17984827

C₂²

yes

—

#17984822

C₂²

yes

—

#17984820

C₂²

yes

—

#17984817

C₂²

yes

—

#17984807

C₂²

yes

—

#17984804

C₂²

yes

—

#17984800

C₂²

yes

—

C₄²×D₁₃₉

C₂²

yes

—

#125×C₁₃₉

C₂²

yes

—

#125×C₁₃₉

C₂²

yes

—

C₂×Dic₂₇₈

C₂³

yes

—

C₂×D₅₅₆

C₂³

yes

—

C₂×C₄×D₁₃₉

C₂³

yes

—

C₂×C₄×D₁₃₉

C₂³

yes

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

C₂³

yes

—

(C₂×D₁₃₉)⋊C₄

C₂³

yes

—

(C₂×D₁₃₉)⋊C₄

C₂³

yes

—

C₄×Dic₁₃₉

C₂³

yes

—

C₄×Dic₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

C₂³

yes

—

C₄²×C₁₃₉

C₂³

yes

—

C₂²×D₁₃₉

C₂⁴

yes

—

Dic₂₇₈

—

D₅₅₆

—

C₂×Dic₁₃₉

C₂⁴

yes

—

C₂×Dic₁₃₉

C₂⁴

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₂×C₄×C₁₃₉

C₂⁴

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

—

C₂²×C₁₃₉

C₂⁵

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

2₋¹⁺⁴

yes

—

C₂×C₁₃₉

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₁₃₉

#585

yes

—

D₁₃₉

—

C₁₃₉

#3531

yes

characteristic, complemented, Sylow

#3531

—

139

maximal, Sylow

#564

D₁₃₉

yes

characteristic, complemented

#563

—

139

—

#562

—

139

—

#533

—

139

—

#529

—

139

—

C₄⋊Q₈

—

139

—

#126

—

139

—

#125

—

139

—

#125

C₂×D₁₃₉

yes

—

#125

C₂×D₁₃₉

yes

—

#124

—

139

—

#123

—

139

—

C₂²⋊Q₈

—

139

—

C₂²⋊Q₈

—

139

—

#121

—

139

—

C₄×Q₈

—

139

—

C₄×D₄

—

139

—

#117

—

139

—

C₂×C₄²

—

139

—

C₂×Q₈

—

139

—

C₂×D₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

C₂²×D₁₃₉

yes

—

C₄⋊C₄

C₂²×D₁₃₉

yes

—

C₄⋊C₄

C₂²×D₁₃₉

yes

—

C₂²⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

C₄²

—

139

—

C₄²

—

139

—

C₄²

C₂²×D₁₃₉

yes

—

C₂³

—

139

—

Q₈

—

556

—

D₄

—

556

—

C₂×C₄

—

278

—

C₂×C₄

—

556

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₂³×D₁₃₉

yes

—

C₂×C₄

C₂³×D₁₃₉

yes

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

C₂⁴×D₁₃₉

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

278

—

103

C₄

—

139

—

104

C₂

—

556

—

105

C₂

2₋¹⁺⁴×D₁₃₉

yes

characteristic

106

C₂

#22581790

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₉
commutator, socle, cosocle	C₂²×C₁₃₉	C₂⁵
Fitting	#564×C₁₃₉	C₂
hypercenter	#564	D₁₃₉
hypocenter	C₁₃₉	#3531
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₃₉

commutator, socle, cosocle

C₂²×C₁₃₉

C₂⁵

Fitting

#564×C₁₃₉

C₂

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #22581707	C₂	semidirect
2. #22581701	C₂	semidirect
3. #22581509	C₂	semidirect
4. #22581488	C₂	semidirect
5. #564×C₁₃₉	C₂	semidirect
53. C₁₃₉	#3531	semidirect
55. #564	D₁₃₉	semidirect

H ≅ G/N

type

1. #22581707

C₂

semidirect

2. #22581701

C₂

semidirect

3. #22581509

C₂

semidirect

4. #22581488

C₂

semidirect

5. #564×C₁₃₉

C₂

semidirect

53. C₁₃₉

#3531

semidirect

55. #564

D₁₃₉

semidirect

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

subgroup	index
#22581707	2
#22581701	2
#22581509	2
#22581488	2
#564×C₁₃₉	2
#3531	139

subgroup

index

#564×C₁₃₉

139

Subgroups — #31820552

Conjugacy classes of subgroups

Complements

Maximal subgroups