Group #31805732 — subgroups

Subgroups — #31805732

Order 17792.

172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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	#22581363	C₂	1	2	yes	maximal
2	#22581323	C₂	1	1	yes	maximal, characteristic
3	#22580695	C₂	1	1	yes	maximal, characteristic
4	#22580695	C₂	1	2	yes	maximal
5	#564×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	#17984842	C₂²	1	1	yes	characteristic
7	#17984775	—	2	2	no	—
8	#17984773	—	2	2	no	—
9	#17984765	—	2	2	no	—
10	#17984753	C₂²	1	1	yes	characteristic
11	#17984753	C₂²	1	2	yes	—
12	#125×C₁₃₉	—	2	2	no	—
13	#125×C₁₃₉	—	2	2	no	—
14	#125×C₁₃₉	—	2	2	no	—
15	#125×C₁₃₉	C₂²	1	1	yes	characteristic
16	#125×C₁₃₉	C₂²	1	2	yes	—
17	C₂×Dic₂₇₈	—	2	1	no	—
18	C₂×C₁₃₉⋊C₈	—	2	2	no	—
19	C₂×C₁₃₉⋊C₈	—	2	1	no	—
20	C₄⋊Dic₁₃₉	—	2	2	no	—
21	C₄⋊C₄×C₁₃₉	—	2	2	no	—
22	C₄⋊C₄×C₁₃₉	—	2	2	no	—
23	C₄⋊C₄×C₁₃₉	—	2	2	no	—
24	C₄⋊C₄×C₁₃₉	—	2	2	no	—
25	C₄⋊C₄×C₁₃₉	—	2	2	no	—
26	C₄⋊C₄×C₁₃₉	—	2	2	no	—
27	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
28	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
30	C₄²×C₁₃₉	—	2	2	no	—
31	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
32	Dic₂₇₈	—	4	2	no	—
33	C₂×Dic₁₃₉	—	4	1	no	—
34	C₁₃₉⋊C₈	—	2	4	no	—
35	C₁₃₉⋊C₈	—	4	1	no	—
36	C₂×C₄×C₁₃₉	—	2	2	no	—
37	C₂×C₄×C₁₃₉	—	2	2	no	—
38	C₂×C₄×C₁₃₉	—	2	2	no	—
39	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₁₃₉	#120	1	1	yes	characteristic
42	C₄×C₁₃₉	—	2	1	no	—
43	C₄×C₁₃₉	—	2	2	no	—
44	C₄×C₁₃₉	—	4	2	no	—
45	C₄×C₁₃₉	—	4	2	no	—
46	C₄×C₁₃₉	—	4	2	no	—
47	Dic₁₃₉	—	8	1	no	—
48	C₂×C₁₃₉	#454	1	1	yes	characteristic
49	C₂×C₁₃₉	#456	1	2	yes	—
50	C₁₃₉	#1663	1	1	yes	characteristic, complemented, Sylow
51	#1663	—	139	1	no	maximal, Sylow
52	#564	D₁₃₉	1	1	yes	characteristic
53	#501	—	139	2	no	—
54	#491	—	139	1	no	—
55	#330	—	139	2	no	—
56	#330	—	139	1	no	—
57	C₄⋊Q₈	—	139	1	no	—
58	#125	—	2	2	no	—
59	#125	—	2	2	no	—
60	#125	—	2	2	no	—
61	#125	C₂×D₁₃₉	1	1	yes	characteristic
62	#125	C₂×D₁₃₉	1	2	yes	—
63	#107	—	278	2	no	—
64	#106	—	278	2	no	—
65	#103	—	278	2	no	—
66	#97	—	139	1	no	—
67	#97	—	139	2	no	—
68	C₂×Q₈	—	278	1	no	—
69	C₂×C₈	—	278	2	no	—
70	C₂×C₈	—	278	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	278	2	no	—
78	C₄⋊C₄	C₂²⋊D₁₃₉	1	2	yes	—
79	C₄⋊C₄	C₂²⋊D₁₃₉	1	2	yes	—
80	C₄⋊C₄	C₂²⋊D₁₃₉	1	2	yes	—
81	C₄²	—	2	2	no	—
82	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
83	Q₈	—	556	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	556	1	no	—
88	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
90	C₈	—	278	4	no	—
91	C₈	—	556	1	no	—
92	C₂²	#17984811	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	—	1112	1	no	—
99	C₂	#22581126	1	1	yes	characteristic
100	C₂	#22581136	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581363

C₂

yes

maximal

#22581323

C₂

yes

maximal, characteristic

#22580695

C₂

yes

maximal, characteristic

#22580695

C₂

yes

maximal

#564×C₁₃₉

C₂

yes

maximal, characteristic

#17984842

C₂²

yes

—

—

—

C₂²

yes

characteristic

#17984753

C₂²

yes

—

#125×C₁₃₉

—

#125×C₁₃₉

—

#125×C₁₃₉

—

#125×C₁₃₉

C₂²

yes

characteristic

#125×C₁₃₉

C₂²

yes

—

C₂×Dic₂₇₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂²×C₁₃₉

#120

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#454

yes

characteristic

C₂×C₁₃₉

#456

yes

—

C₁₃₉

#1663

yes

characteristic, complemented, Sylow

#1663

—

139

maximal, Sylow

#564

D₁₃₉

yes

characteristic

#501

—

139

—

#491

—

139

—

#330

—

139

—

#330

—

139

—

C₄⋊Q₈

—

139

—

—

—

—

C₂×D₁₃₉

yes

characteristic

#125

C₂×D₁₃₉

yes

—

#107

—

278

—

#106

—

278

—

#103

—

278

—

#97

—

139

—

#97

—

139

—

C₂×Q₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

C₂²⋊D₁₃₉

yes

—

C₄⋊C₄

C₂²⋊D₁₃₉

yes

—

C₄⋊C₄

C₂²⋊D₁₃₉

yes

—

C₄²

—

C₄²

C₂²×D₁₃₉

yes

characteristic

Q₈

—

556

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

556

—

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₈

—

278

—

C₈

—

556

—

C₂²

#17984811

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

1112

—

C₂

#22581126

yes

characteristic

100

C₂

#22581136

yes

—

	subgroup	G/H
center	C₂²	#17984811
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#564×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#120
hypercenter	#564	D₁₃₉
hypocenter	C₁₃₉	#1663
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #564

subgroup

G/H

center

C₂²

#17984811

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#564×C₁₃₉

C₂

C₂²×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #564

N	H ≅ G/N	type
50. C₁₃₉	#1663	semidirect

H ≅ G/N

type

50. C₁₃₉

#1663

semidirect

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

subgroup	index
#22581363	2
#22581323	2
#22580695	2
#564×C₁₃₉	2
#1663	139

subgroup

index

#564×C₁₃₉

139

Subgroups — #31805732

Conjugacy classes of subgroups

Complements

Maximal subgroups