Group #31780118 — subgroups

Subgroups — #31780118

Order 17536.

172 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₁₃₇⋊#564	C₂	1	1	yes	maximal, characteristic
2	#22577812	C₂	1	2	yes	maximal
3	#22577773	C₂	1	1	yes	maximal, characteristic
4	#22577144	C₂	1	2	yes	maximal
5	#330×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#17984088	C₂²	1	1	yes	characteristic
7	#17984080	C₂²	1	2	yes	—
8	#17984079	—	2	2	no	—
9	#17984078	—	2	2	no	—
10	#17984078	—	2	2	no	—
11	#17984022	—	2	2	no	—
12	#17984020	—	2	2	no	—
13	#17984011	—	2	2	no	—
14	#17983999	C₂²	1	1	yes	characteristic
15	#125×C₁₃₇	C₂²	1	1	yes	characteristic
16	#97×C₁₃₇	C₂²	1	2	yes	—
17	C₂×Dic₂₇₄	—	2	1	no	—
18	C₂×#11764040	—	2	1	no	—
19	C₄⋊Dic₁₃₇	—	2	2	no	—
20	C₄⋊Dic₁₃₇	—	2	2	no	—
21	C₄⋊Dic₁₃₇	D₄	1	2	yes	—
22	C₄⋊Dic₁₃₇	D₄	1	2	yes	—
23	#14555489	—	2	2	no	—
24	#14555489	—	2	2	no	—
25	#14555489	—	2	2	no	—
26	#14555489	—	2	2	no	—
27	C₄×Dic₁₃₇	—	2	2	no	—
28	C₂×C₈×C₁₃₇	—	2	2	no	—
29	C₄⋊C₄×C₁₃₇	—	2	2	no	—
30	C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
31	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
32	Dic₂₇₄	—	4	2	no	—
33	C₂×Dic₁₃₇	—	2	2	no	—
34	C₂×Dic₁₃₇	—	2	2	no	—
35	C₂×Dic₁₃₇	—	4	1	no	—
36	#11764040	—	4	1	no	—
37	C₂×C₄×C₁₃₇	—	2	2	no	—
38	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
40	C₈×C₁₃₇	—	2	4	no	—
41	C₂²×C₁₃₇	#120	1	1	yes	characteristic
42	C₄×C₁₃₇	—	2	1	no	—
43	C₄×C₁₃₇	—	2	2	no	—
44	C₄×C₁₃₇	—	4	2	no	—
45	Dic₁₃₇	—	4	2	no	—
46	Dic₁₃₇	—	4	2	no	—
47	Dic₁₃₇	—	8	1	no	—
48	C₂×C₁₃₇	#454	1	1	yes	characteristic
49	C₂×C₁₃₇	#456	1	2	yes	—
50	C₁₃₇	#1663	1	1	yes	characteristic, complemented, Sylow
51	#1663	—	137	1	no	maximal, Sylow
52	#564	—	137	1	no	—
53	#501	—	137	2	no	—
54	#491	—	137	1	no	—
55	#330	—	137	2	no	—
56	#330	D₁₃₇	1	1	yes	characteristic
57	C₄⋊Q₈	—	137	1	no	—
58	#125	—	274	2	no	—
59	#125	—	274	2	no	—
60	#125	—	274	2	no	—
61	#125	—	137	2	no	—
62	#125	C₂×D₁₃₇	1	1	yes	characteristic
63	#107	—	274	2	no	—
64	#106	—	274	2	no	—
65	#103	—	274	2	no	—
66	#97	—	137	1	no	—
67	#97	C₂×D₁₃₇	1	2	yes	—
68	C₂×Q₈	—	274	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	274	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	274	2	no	—
73	C₄⋊C₄	—	274	2	no	—
74	C₄⋊C₄	—	274	2	no	—
75	C₄⋊C₄	—	274	2	no	—
76	C₄⋊C₄	—	274	2	no	—
77	C₄⋊C₄	—	274	2	no	—
78	C₄⋊C₄	—	137	2	no	—
79	C₄⋊C₄	—	137	2	no	—
80	C₄⋊C₄	D₅₄₈	1	2	yes	—
81	C₄²	—	274	2	no	—
82	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
83	Q₈	—	548	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	274	2	no	—
86	C₂×C₄	—	274	2	no	—
87	C₂×C₄	—	548	1	no	—
88	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
89	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	548	1	no	—
92	C₂²	#17984058	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	548	2	no	—
97	C₄	—	548	2	no	—
98	C₄	—	1096	1	no	—
99	C₂	#22577578	1	1	yes	characteristic
100	C₂	#22577587	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃₇⋊#564

C₂

yes

maximal, characteristic

#22577812

C₂

yes

maximal

#22577773

C₂

yes

maximal, characteristic

#22577144

C₂

yes

maximal

#330×C₁₃₇

C₂

yes

maximal, characteristic

#17984088

C₂²

yes

characteristic

#17984080

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

characteristic

#125×C₁₃₇

C₂²

yes

characteristic

#97×C₁₃₇

C₂²

yes

—

C₂×Dic₂₇₄

—

C₂×#11764040

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

D₄

yes

—

C₄⋊Dic₁₃₇

D₄

yes

—

—

—

—

—

C₄×Dic₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

#120

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

#454

yes

characteristic

C₂×C₁₃₇

#456

yes

—

C₁₃₇

#1663

yes

characteristic, complemented, Sylow

#1663

—

137

maximal, Sylow

#564

—

137

—

#501

—

137

—

#491

—

137

—

#330

—

137

—

#330

D₁₃₇

yes

characteristic

C₄⋊Q₈

—

137

—

#125

—

274

—

#125

—

274

—

#125

—

274

—

#125

—

137

—

#125

C₂×D₁₃₇

yes

characteristic

#107

—

274

—

#106

—

274

—

#103

—

274

—

#97

—

137

—

#97

C₂×D₁₃₇

yes

—

C₂×Q₈

—

274

—

C₂×C₈

—

C₂×C₈

—

274

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

D₅₄₈

yes

—

C₄²

—

274

—

C₄²

C₂²×D₁₃₇

yes

characteristic

Q₈

—

548

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

548

—

C₂×C₄

D₄×D₁₃₇

yes

—

C₂×C₄

C₂×D₅₄₈

yes

characteristic

C₈

—

C₈

—

548

—

C₂²

#17984058

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

548

—

C₄

—

548

—

C₄

—

1096

—

C₂

#22577578

yes

characteristic

100

C₂

#22577587

yes

—

	subgroup	G/H
center	C₂²	#17984058
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#330×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	#120
hypercenter	#330	D₁₃₇
hypocenter	C₁₃₇	#1663
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

#17984058

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#330×C₁₃₇

C₂

C₂²×C₁₃₇

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
50. C₁₃₇	#1663	semidirect

H ≅ G/N

type

50. C₁₃₇

#1663

semidirect

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

subgroup	index
C₁₃₇⋊#564	2
#22577812	2
#22577773	2
#22577144	2
#330×C₁₃₇	2
#1663	137

subgroup

index

C₁₃₇⋊#564

#330×C₁₃₇

137

Subgroups — #31780118

Conjugacy classes of subgroups

Complements

Maximal subgroups