Group #37535297 — subgroups

Subgroups — #37535297

Order 23168.

158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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	#26041328	C₂	1	1	yes	maximal, characteristic
2	#26041302	C₂	1	1	yes	maximal, characteristic
3	#26041049	C₂	1	2	yes	maximal
4	#26041038	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668433	C₂²	1	2	yes	—
7	C₂×#15668433	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
9	#19153752	—	2	2	no	—
10	#19153745	—	2	2	no	—
11	C₂×M₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
12	#117×C₁₈₁	C₂²	1	2	yes	—
13	C₂²×Dic₁₈₁	—	2	1	no	—
14	#15668433	—	2	2	no	—
15	#15668433	—	2	2	no	—
16	#15668433	—	2	2	no	—
17	C₂×#13089678	—	2	1	no	—
18	C₂×#13089678	—	2	2	no	—
19	C₄×Dic₁₈₁	—	2	1	no	—
20	C₄×Dic₁₈₁	Q₈	1	1	yes	characteristic
21	C₄×Dic₁₈₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₈₁	—	2	2	no	—
24	C₂×C₈×C₁₈₁	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₈₁	—	2	2	no	—
26	C₂²⋊C₄×C₁₈₁	—	2	2	no	—
27	C₄²×C₁₈₁	—	2	2	no	—
28	C₂×Dic₁₈₁	—	2	1	no	—
29	C₂×Dic₁₈₁	—	2	1	no	—
30	C₂×Dic₁₈₁	—	4	2	no	—
31	#13089678	—	4	1	no	—
32	#13089678	—	4	2	no	—
33	C₂³×C₁₈₁	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₈₁	—	4	2	no	—
35	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₈₁	#41	1	2	yes	—
38	C₂×C₄×C₁₈₁	#41	1	2	yes	—
39	C₈×C₁₈₁	—	2	2	no	—
40	C₂²×C₁₈₁	—	4	1	no	—
41	C₂²×C₁₈₁	C₂²⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₈₁	#126	1	2	yes	—
43	C₄×C₁₈₁	—	8	2	no	—
44	C₄×C₁₈₁	#123	1	1	yes	characteristic
45	C₄×C₁₈₁	#124	1	1	yes	characteristic
46	C₄×C₁₈₁	#125	1	2	yes	—
47	Dic₁₈₁	—	2	2	no	—
48	Dic₁₈₁	—	4	1	no	—
49	C₂×C₁₈₁	—	2	1	no	—
50	C₂×C₁₈₁	—	2	2	no	—
51	C₂×C₁₈₁	#400	1	1	yes	characteristic
52	C₁₈₁	#2102	1	1	yes	characteristic, complemented, Sylow
53	#2102	—	181	1	no	maximal, Sylow
54	#430	—	181	1	no	—
55	#423	—	181	1	no	—
56	#344	—	181	2	no	—
57	#339	—	181	2	no	—
58	H(3,ℤ/4)	D₁₈₁	1	1	yes	characteristic
59	C₂×M₁₆	—	181	2	no	—
60	C₂×M₁₆	—	181	1	no	—
61	C₂×M₁₆	C₂×D₁₈₁	1	1	yes	characteristic
62	#117	C₂×D₁₈₁	1	2	yes	—
63	C₂×C₄²	—	181	1	no	—
64	#108	—	362	2	no	—
65	C₄⋊C₈	—	362	2	no	—
66	C₂²×C₄	—	362	1	no	—
67	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	362	2	no	—
70	M₁₆	—	362	2	no	—
71	M₁₆	—	362	2	no	—
72	C₂×C₈	—	362	1	no	—
73	C₂×C₈	—	362	2	no	—
74	C₂×C₈	D₇₂₄	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	362	1	no	—
79	C₄²	—	181	1	no	—
80	C₄²	—	181	1	no	—
81	C₂³	#15668448	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	362	1	no	—
84	C₂×C₄	—	362	1	no	—
85	C₂×C₄	—	724	2	no	—
86	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₈₁	1	1	yes	characteristic
88	C₂×C₄	#15668448	1	2	yes	—
89	C₂×C₄	#15668449	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	724	1	no	—
92	C₈	—	724	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#19153796	1	1	yes	characteristic
95	C₂²	#19153809	1	2	yes	—
96	C₄	—	8	2	no	—
97	C₄	—	362	2	no	—
98	C₄	—	724	1	no	—
99	C₄	#19153800	1	1	yes	characteristic
100	C₄	#19153804	1	1	yes	characteristic
101	C₄	#19153806	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#26041219	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26041328

C₂

yes

maximal, characteristic

#26041302

C₂

yes

maximal, characteristic

#26041049

C₂

yes

maximal

#26041038

C₂

yes

maximal

H(3,ℤ/4)×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668433

C₂²

yes

—

C₂×#15668433

C₂²

yes

characteristic

C₂×C₄×Dic₁₈₁

C₂²

yes

characteristic

#19153752

—

#19153745

—

C₂×M₁₆×C₁₈₁

C₂²

yes

characteristic

#117×C₁₈₁

C₂²

yes

—

C₂²×Dic₁₈₁

—

—

—

—

—

—

C₄×Dic₁₈₁

—

C₄×Dic₁₈₁

Q₈

yes

characteristic

C₄×Dic₁₈₁

Q₈

yes

characteristic

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

M₁₆×C₁₈₁

—

C₂×C₈×C₁₈₁

D₄

yes

—

C₄⋊C₄×C₁₈₁

—

C₂²⋊C₄×C₁₈₁

—

C₄²×C₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

—

#13089678

—

C₂³×C₁₈₁

#41

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₈₁

#41

yes

—

C₂×C₄×C₁₈₁

#41

yes

—

C₈×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₈₁

#126

yes

—

C₄×C₁₈₁

—

C₄×C₁₈₁

yes

C₄×C₁₈₁

yes

C₄×C₁₈₁

yes

—

Dic₁₈₁

—

Dic₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

#400

yes

characteristic

C₁₈₁

#2102

yes

characteristic, complemented, Sylow

#2102

—

181

maximal, Sylow

#430

—

181

—

#423

—

181

—

#344

—

181

—

#339

—

181

—

H(3,ℤ/4)

D₁₈₁

yes

characteristic

C₂×M₁₆

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

C₂×D₁₈₁

yes

characteristic

#117

C₂×D₁₈₁

yes

—

C₂×C₄²

—

181

—

#108

—

362

—

C₄⋊C₈

—

362

—

C₂²×C₄

—

362

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

M₁₆

—

M₁₆

—

362

—

M₁₆

—

362

—

M₁₆

—

362

—

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₂×C₈

D₇₂₄

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

362

—

C₄²

—

181

—

C₄²

—

181

—

C₂³

#15668448

yes

characteristic

C₂×C₄

—

C₂×C₄

—

362

—

C₂×C₄

—

362

—

C₂×C₄

—

724

—

C₂×C₄

C₂×D₇₂₄

yes

characteristic

C₂×C₄

Q₈×D₁₈₁

yes

characteristic

C₂×C₄

#15668448

yes

—

C₂×C₄

#15668449

yes

—

C₈

—

C₈

—

724

—

C₈

—

724

—

C₂²

—

C₂²

#19153796

yes

characteristic

C₂²

#19153809

yes

—

C₄

—

C₄

—

362

—

C₄

—

724

—

C₄

#19153800

yes

characteristic

100

C₄

#19153804

yes

characteristic

101

C₄

#19153806

yes

—

102

C₂

—

103

C₂

—

104

C₂

#26041219

yes

characteristic

	subgroup	G/H
center	C₄	#19153804
commutator, cosocle	C₂²×C₄×C₁₈₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	H(3,ℤ/4)×C₁₈₁	C₂
socle	C₂×C₁₈₁	#400
prime radical	C₂³×C₁₈₁	#41
hypercenter	H(3,ℤ/4)	D₁₈₁
hypocenter	C₁₈₁	#2102
LCS	G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#19153804

commutator, cosocle

C₂²×C₄×C₁₈₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

H(3,ℤ/4)×C₁₈₁

C₂

C₂×C₁₈₁

C₂³×C₁₈₁

D₁₈₁

C₁₈₁

G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
52. C₁₈₁	#2102	semidirect

H ≅ G/N

type

52. C₁₈₁

#2102

semidirect

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

subgroup	index
#26041328	2
#26041302	2
#26041049	2
#26041038	2
H(3,ℤ/4)×C₁₈₁	2
#2102	181

subgroup

index

H(3,ℤ/4)×C₁₈₁

181

Subgroups — #37535297

Conjugacy classes of subgroups

Complements

Maximal subgroups