Group #37535232 — subgroups

Subgroups — #37535232

Order 23168.

176 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 67 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	#26041319	C₂	1	2	yes	maximal
2	#26041305	C₂	1	1	yes	maximal, characteristic
3	#26041048	C₂	1	2	yes	maximal
4	#26041044	C₂	1	1	yes	maximal, characteristic
5	C₂×#106×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668433	C₂²	1	1	yes	characteristic
7	C₂×#15668433	C₂²	1	2	yes	—
8	#19153775	C₂²	1	1	yes	characteristic
9	#19153749	—	2	2	no	—
10	#19153747	—	2	2	no	—
11	#19153745	—	2	2	no	—
12	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	2	yes	—
14	#106×C₁₈₁	—	2	2	no	—
15	#15668433	—	2	2	no	—
16	#15668433	—	2	2	no	—
17	#15668433	Q₈	1	4	yes	—
18	C₂×#13089678	—	2	2	no	—
19	C₂×#13089678	—	2	1	no	—
20	C₄⋊Dic₁₈₁	D₄	1	2	yes	—
21	C₂²⋊Dic₁₈₁	—	2	1	no	—
22	C₄×Dic₁₈₁	—	2	1	no	—
23	C₂²×C₄×C₁₈₁	—	2	2	no	—
24	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₈₁	—	2	2	no	—
26	C₂×C₈×C₁₈₁	D₄	1	2	yes	—
27	C₄⋊C₄×C₁₈₁	—	2	2	no	—
28	C₄⋊C₄×C₁₈₁	—	2	2	no	—
29	C₂×Dic₁₈₁	—	2	2	no	—
30	#13089678	—	2	4	no	—
31	#13089678	—	4	1	no	—
32	C₂³×C₁₈₁	#41	1	1	yes	characteristic
33	C₂×C₄×C₁₈₁	—	4	2	no	—
34	C₂×C₄×C₁₈₁	—	8	2	no	—
35	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₈₁	#41	1	2	yes	—
39	C₈×C₁₈₁	—	2	2	no	—
40	C₂²×C₁₈₁	—	2	2	no	—
41	C₂²×C₁₈₁	#121	1	1	yes	characteristic
42	C₂²×C₁₈₁	#125	1	2	yes	—
43	C₄×C₁₈₁	—	8	2	no	—
44	C₄×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₁₈₁	#123	1	1	yes	characteristic
46	C₄×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₁₈₁	—	4	2	no	—
48	C₂×C₁₈₁	—	2	2	no	—
49	C₂×C₁₈₁	#398	1	1	yes	characteristic
50	C₂×C₁₈₁	#471	1	2	yes	—
51	C₁₈₁	#2090	1	1	yes	characteristic, complemented, Sylow
52	#2090	—	181	1	no	maximal, Sylow
53	#428	—	181	2	no	—
54	C₂×#106	D₁₈₁	1	1	yes	characteristic
55	#424	—	181	1	no	—
56	#343	—	181	2	no	—
57	#341	—	181	1	no	—
58	C₂×M₁₆	—	181	1	no	—
59	C₂×M₁₆	—	181	2	no	—
60	C₂²×C₈	C₂×D₁₈₁	1	1	yes	characteristic
61	#117	—	181	1	no	—
62	C₂×C₄⋊C₄	C₂×D₁₈₁	1	2	yes	—
63	#107	—	362	2	no	—
64	#106	—	2	2	no	—
65	#106	—	362	2	no	—
66	C₄⋊C₈	—	362	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
69	M₁₆	—	362	2	no	—
70	M₁₆	—	362	2	no	—
71	M₁₆	—	181	4	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	362	2	no	—
74	C₂×C₈	—	362	1	no	—
75	C₂×C₈	C₂²⋊D₁₈₁	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	181	2	no	—
79	C₂²⋊C₄	—	362	1	no	—
80	C₄²	—	362	1	no	—
81	C₂³	#15668448	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	8	2	no	—
84	C₂×C₄	—	362	2	no	—
85	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
86	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₈₁	1	2	yes	—
88	C₂×C₄	#15668449	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	362	4	no	—
91	C₈	—	724	1	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#19153787	1	1	yes	characteristic
94	C₂²	#19153806	1	2	yes	—
95	C₄	—	8	2	no	—
96	C₄	—	724	2	no	—
97	C₄	#19153794	1	2	yes	—
98	C₄	#19153799	1	1	yes	characteristic
99	C₄	#19153814	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#26041210	1	1	yes	characteristic
102	C₂	#26041539	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26041319

C₂

yes

maximal

#26041305

C₂

yes

maximal, characteristic

#26041048

C₂

yes

maximal

#26041044

C₂

yes

maximal, characteristic

C₂×#106×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668433

C₂²

yes

characteristic

C₂×#15668433

C₂²

yes

—

#19153775

C₂²

yes

—

—

—

C₂²×C₈×C₁₈₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

—

#106×C₁₈₁

—

#15668433

—

#15668433

—

#15668433

Q₈

yes

—

C₂×#13089678

—

C₂×#13089678

—

C₄⋊Dic₁₈₁

D₄

yes

—

C₂²⋊Dic₁₈₁

—

C₄×Dic₁₈₁

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

D₄

yes

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

—

#13089678

—

C₂³×C₁₈₁

#41

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

#41

yes

—

C₈×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

#121

yes

characteristic

C₂²×C₁₈₁

#125

yes

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂²⋊Q₈

yes

—

C₄×C₁₈₁

#123

yes

characteristic

C₄×C₁₈₁

C₄⋊Q₈

yes

characteristic

Dic₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

#398

yes

characteristic

C₂×C₁₈₁

#471

yes

—

C₁₈₁

#2090

yes

characteristic, complemented, Sylow

#2090

—

181

maximal, Sylow

#428

—

181

—

C₂×#106

D₁₈₁

yes

characteristic

#424

—

181

—

#343

—

181

—

#341

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

—

181

—

C₂²×C₈

C₂×D₁₈₁

yes

characteristic

#117

—

181

—

C₂×C₄⋊C₄

C₂×D₁₈₁

yes

—

#107

—

362

—

#106

—

#106

—

362

—

C₄⋊C₈

—

362

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

M₁₆

—

362

—

M₁₆

—

362

—

M₁₆

—

181

—

C₂×C₈

—

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₂×C₈

C₂²⋊D₁₈₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

181

—

C₂²⋊C₄

—

362

—

C₄²

—

362

—

C₂³

#15668448

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

362

—

C₂×C₄

D₄×D₁₈₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

characteristic

C₂×C₄

Q₈×D₁₈₁

yes

—

C₂×C₄

#15668449

yes

—

C₈

—

C₈

—

362

—

C₈

—

724

—

C₂²

—

C₂²

#19153787

yes

characteristic

C₂²

#19153806

yes

—

C₄

—

C₄

—

724

—

C₄

#19153794

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#26041210

yes

characteristic

102

C₂

#26041539

yes

—

	subgroup	G/H
center	C₂²	#19153787
commutator, cosocle	C₂²×C₄×C₁₈₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	C₂×#106×C₁₈₁	C₂
socle	C₂²×C₁₈₁	#121
prime radical	C₂³×C₁₈₁	#41
hypercenter	C₂×#106	D₁₈₁
hypocenter	C₁₈₁	#2090
LCS	G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

#19153787

commutator, cosocle

C₂²×C₄×C₁₈₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

C₂×#106×C₁₈₁

C₂

C₂²×C₁₈₁

C₂³×C₁₈₁

D₁₈₁

C₁₈₁

G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
51. C₁₈₁	#2090	semidirect

H ≅ G/N

type

51. C₁₈₁

#2090

semidirect

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

subgroup	index
#26041319	2
#26041305	2
#26041048	2
#26041044	2
C₂×#106×C₁₈₁	2
#2090	181

subgroup

index

C₂×#106×C₁₈₁

181

Subgroups — #37535232

Conjugacy classes of subgroups

Complements

Maximal subgroups