Group #37534714 — subgroups

Subgroups — #37534714

Order 23168.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#19153808	C₂	1	1	yes	maximal, characteristic
2	#26041303	C₂	1	1	yes	maximal, characteristic
3	#26041047	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₁₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668433	C₂²	1	2	yes	—
6	#19153808	—	2	1	no	—
7	#19153808	—	2	1	no	—
8	#19153808	—	2	2	no	—
9	C₂×C₄⋊Dic₁₈₁	C₂²	1	2	yes	—
10	C₂×#15668428	C₄	1	4	yes	—
11	C₄⋊#13089678	—	2	2	no	—
12	C₂×M₁₆×C₁₈₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
14	#97×C₁₈₁	—	2	1	no	—
15	C₄×C₈×C₁₈₁	—	2	1	no	—
16	C₂²×Dic₁₈₁	C₂×C₄	1	4	yes	—
17	#15668433	—	2	4	no	—
18	C₂×#13089678	—	2	2	no	—
19	C₄⋊Dic₁₈₁	—	2	2	no	—
20	C₄⋊Dic₁₈₁	—	2	2	no	—
21	#15668428	—	2	4	no	—
22	#15668428	—	2	4	no	—
23	C₂²×C₄×C₁₈₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₈₁	—	2	2	no	—
26	M₁₆×C₁₈₁	Q₈	1	4	yes	—
27	C₂×C₈×C₁₈₁	—	2	2	no	—
28	C₄²×C₁₈₁	D₄	1	1	yes	characteristic
29	C₄²×C₁₈₁	D₄	1	1	yes	characteristic
30	C₄²×C₁₈₁	D₄	1	2	yes	—
31	C₂×Dic₁₈₁	—	2	4	no	—
32	C₂×Dic₁₈₁	—	4	4	no	—
33	#13089678	—	4	2	no	—
34	C₂³×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₈₁	—	4	2	no	—
36	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₈₁	#41	1	2	yes	—
42	C₈×C₁₈₁	—	2	4	no	—
43	C₂²×C₁₈₁	—	2	2	no	—
44	C₂²×C₁₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₈₁	C₄×Q₈	1	2	yes	—
46	C₄×C₁₈₁	—	2	2	no	—
47	C₄×C₁₈₁	—	2	2	no	—
48	C₄×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₁₈₁	#124	1	1	yes	characteristic
50	C₄×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₁₈₁	—	4	4	no	—
52	C₂×C₁₈₁	—	2	2	no	—
53	C₂×C₁₈₁	#391	1	1	yes	characteristic
54	C₂×C₁₈₁	#413	1	2	yes	—
55	C₁₈₁	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	181	1	no	maximal, Sylow
57	C₂×#125	—	181	1	no	—
58	#423	—	181	1	no	—
59	C₄×M₁₆	D₁₈₁	1	1	yes	characteristic
60	#343	—	181	4	no	—
61	C₂×M₁₆	—	181	2	no	—
62	C₂×M₁₆	C₂×D₁₈₁	1	2	yes	—
63	#125	—	362	1	no	—
64	#125	—	362	1	no	—
65	#125	—	362	2	no	—
66	C₂×C₄⋊C₄	—	181	4	no	—
67	C₂×C₄⋊C₄	—	181	2	no	—
68	C₂×C₄²	C₂×D₁₈₁	1	1	yes	characteristic
69	C₄⋊C₈	—	362	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	—	181	4	no	—
73	C₂²×C₄	C₄×D₁₈₁	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	362	4	no	—
77	M₁₆	Dic₃₆₂	1	4	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	362	2	no	—
80	C₄⋊C₄	—	362	4	no	—
81	C₄⋊C₄	—	362	4	no	—
82	C₄⋊C₄	—	362	2	no	—
83	C₄⋊C₄	—	362	2	no	—
84	C₄²	C₂²⋊D₁₈₁	1	1	yes	characteristic
85	C₄²	C₂²⋊D₁₈₁	1	1	yes	characteristic
86	C₄²	D₇₂₄	1	2	yes	—
87	C₂³	C₂×C₄×D₁₈₁	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	362	4	no	—
90	C₂×C₄	—	724	4	no	—
91	C₂×C₄	#15668426	1	2	yes	—
92	C₂×C₄	#15668426	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
94	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
95	C₂×C₄	C₂×Dic₃₆₂	1	2	yes	—
96	C₂×C₄	#15668449	1	2	yes	—
97	C₈	—	2	4	no	—
98	C₈	—	724	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×#15668426	1	1	yes	characteristic
101	C₂²	C₄×Dic₃₆₂	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	724	4	no	—
105	C₄	#19153795	1	2	yes	—
106	C₄	#19153804	1	1	yes	characteristic
107	C₄	#19153816	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#26041185	1	1	yes	characteristic
110	C₂	#26041261	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153808

C₂

yes

maximal, characteristic

#26041303

C₂

yes

maximal, characteristic

#26041047

C₂

yes

maximal

C₄×M₁₆×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668433

C₂²

yes

—

#19153808

—

#19153808

—

#19153808

—

C₂×C₄⋊Dic₁₈₁

C₂²

yes

—

C₂×#15668428

C₄

yes

—

C₄⋊#13089678

—

C₂×M₁₆×C₁₈₁

C₂²

yes

—

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

#97×C₁₈₁

—

C₄×C₈×C₁₈₁

—

C₂²×Dic₁₈₁

C₂×C₄

yes

—

#15668433

—

C₂×#13089678

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

#15668428

—

#15668428

—

C₂²×C₄×C₁₈₁

C₂×C₄

yes

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

M₁₆×C₁₈₁

—

M₁₆×C₁₈₁

Q₈

yes

—

C₂×C₈×C₁₈₁

—

C₄²×C₁₈₁

D₄

yes

characteristic

C₄²×C₁₈₁

D₄

yes

characteristic

C₄²×C₁₈₁

D₄

yes

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

—

C₂³×C₁₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

—

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

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₈₁

C₄×Q₈

yes

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂²⋊Q₈

yes

—

C₄×C₁₈₁

#124

yes

characteristic

C₄×C₁₈₁

C₄⋊Q₈

yes

characteristic

Dic₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

#391

yes

characteristic

C₂×C₁₈₁

#413

yes

—

C₁₈₁

#1998

yes

characteristic, complemented, Sylow

#1998

—

181

maximal, Sylow

C₂×#125

—

181

—

#423

—

181

—

C₄×M₁₆

D₁₈₁

yes

characteristic

#343

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

C₂×D₁₈₁

yes

—

#125

—

362

—

#125

—

362

—

#125

—

362

—

C₂×C₄⋊C₄

—

181

—

C₂×C₄⋊C₄

—

181

—

C₂×C₄²

C₂×D₁₈₁

yes

characteristic

C₄⋊C₈

—

362

—

#97

—

C₄×C₈

—

C₂²×C₄

—

181

—

C₂²×C₄

C₄×D₁₈₁

yes

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

M₁₆

—

M₁₆

—

362

—

M₁₆

Dic₃₆₂

yes

—

C₂×C₈

—

C₂×C₈

—

362

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

362

—

C₄²

C₂²⋊D₁₈₁

yes

characteristic

C₄²

C₂²⋊D₁₈₁

yes

characteristic

C₄²

D₇₂₄

yes

—

C₂³

C₂×C₄×D₁₈₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

362

—

C₂×C₄

—

724

—

C₂×C₄

#15668426

yes

—

C₂×C₄

#15668426

yes

—

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

characteristic

C₂×C₄

C₂×D₇₂₄

yes

characteristic

C₂×C₄

C₂×Dic₃₆₂

yes

—

C₂×C₄

#15668449

yes

—

C₈

—

C₈

—

724

—

C₂²

—

100

C₂²

C₂×#15668426

yes

characteristic

101

C₂²

C₄×Dic₃₆₂

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

724

—

105

C₄

#19153795

yes

—

106

C₄

#19153804

yes

characteristic

107

C₄

#19153816

yes

characteristic

108

C₂

—

109

C₂

#26041185

yes

characteristic

110

C₂

#26041261

yes

—

	subgroup	G/H
center	C₂²	C₂×#15668426
commutator, prime radical	C₂³×C₁₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	C₄×M₁₆×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₈₁	C₂³
hypercenter	C₄×M₁₆	D₁₈₁
hypocenter	C₁₈₁	#1998
LCS	G ▷ C₂³×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#15668426

commutator, prime radical

C₂³×C₁₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

C₄×M₁₆×C₁₈₁

C₂

socle

C₂²×C₁₈₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₈₁

C₂³

hypercenter

C₄×M₁₆

D₁₈₁

hypocenter

C₁₈₁

#1998

LCS

G ▷ C₂³×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
55. C₁₈₁	#1998	semidirect

H ≅ G/N

type

55. C₁₈₁

#1998

semidirect

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

subgroup	index
C₂×#19153808	2
#26041303	2
#26041047	2
C₄×M₁₆×C₁₈₁	2
#1998	181

subgroup

index

C₄×M₁₆×C₁₈₁

181

Subgroups — #37534714

Conjugacy classes of subgroups

Complements

Maximal subgroups