Group #37533759 — subgroups

Subgroups — #37533759

Order 23168.

292 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 117 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×#19153752	C₂	1	2	yes	maximal, complemented
2	#26041243	C₂	1	2	yes	maximal
3	#26041044	C₂	1	2	yes	maximal, complemented
4	C₂³×C₈×C₁₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668433	C₂²	1	4	yes	—
6	#19153752	—	2	4	no	—
7	C₂²⋊#13089678	—	2	4	no	—
8	C₂³×C₄×C₁₈₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₈₁	—	2	4	no	—
10	C₂²×C₈×C₁₈₁	C₄	1	4	yes	—
11	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
13	#15668433	—	2	4	no	—
14	#15668433	—	2	4	no	—
15	C₂×#13089678	—	2	4	no	—
16	C₂⁴×C₁₈₁	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₁₈₁	—	2	2	no	—
18	C₂²×C₄×C₁₈₁	—	2	2	no	—
19	C₂²×C₄×C₁₈₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₈₁	D₄	1	2	yes	—
21	C₂²×C₄×C₁₈₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₈₁	—	2	2	no	—
24	C₂×C₈×C₁₈₁	—	2	8	no	—
25	C₂×C₈×C₁₈₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₈₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₁₈₁	D₄	1	4	yes	—
28	#13089678	—	4	4	no	—
29	C₂³×C₁₈₁	—	2	4	no	—
30	C₂³×C₁₈₁	—	2	2	no	—
31	C₂³×C₁₈₁	C₄⋊C₄	1	2	yes	—
32	C₂³×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₈₁	—	2	2	no	—
34	C₂×C₄×C₁₈₁	—	2	2	no	—
35	C₂×C₄×C₁₈₁	—	2	2	no	—
36	C₂×C₄×C₁₈₁	—	2	4	no	—
37	C₂×C₄×C₁₈₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₈₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₈₁	#41	1	2	yes	—
42	C₈×C₁₈₁	—	2	2	no	—
43	C₈×C₁₈₁	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₁₈₁	—	2	8	no	—
45	C₂²×C₁₈₁	—	2	2	no	—
46	C₂²×C₁₈₁	—	2	2	no	—
47	C₂²×C₁₈₁	—	2	2	no	—
48	C₂²×C₁₈₁	#108	1	4	yes	complemented
49	C₂²×C₁₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
51	C₄×C₁₈₁	—	2	2	no	—
52	C₄×C₁₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₈₁	#117	1	1	yes	characteristic
54	C₄×C₁₈₁	#121	1	2	yes	—
55	C₂×C₁₈₁	—	2	4	no	—
56	C₂×C₁₈₁	—	2	2	no	—
57	C₂×C₁₈₁	#380	1	1	yes	characteristic
58	C₂×C₁₈₁	C₂×#108	1	2	yes	—
59	C₁₈₁	#1810	1	1	yes	characteristic, complemented, Sylow
60	#1810	—	181	1	no	maximal, Sylow
61	C₂³×C₈	D₁₈₁	1	1	yes	characteristic
62	C₂×#108	—	181	2	no	—
63	#407	—	181	2	no	—
64	#341	—	181	2	no	—
65	C₂³×C₄	C₂×D₁₈₁	1	1	yes	characteristic
66	C₂×M₁₆	—	181	4	no	—
67	C₂²×C₈	—	2	4	no	—
68	C₂²×C₈	Dic₁₈₁	1	4	yes	—
69	C₂²×C₈	C₂×D₁₈₁	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₁₈₁	1	1	yes	characteristic
71	#108	—	362	4	no	—
72	C₂²⋊C₈	—	362	4	no	—
73	C₂⁴	Dic₃₆₂	1	1	yes	characteristic
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂×Dic₁₈₁	1	2	yes	—
77	C₂²×C₄	D₇₂₄	1	2	yes	—
78	C₂²×C₄	Dic₃₆₂	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
80	M₁₆	—	362	4	no	—
81	M₁₆	—	362	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	8	no	—
84	C₂×C₈	—	362	4	no	—
85	C₂×C₈	C₂×Dic₁₈₁	1	2	yes	—
86	C₂×C₈	C₂×Dic₁₈₁	1	2	yes	—
87	C₂×C₈	C₂²⋊D₁₈₁	1	4	yes	—
88	C₂³	—	2	4	no	—
89	C₂³	—	2	2	no	—
90	C₂³	C₄⋊Dic₁₈₁	1	2	yes	—
91	C₂³	C₂×Dic₃₆₂	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	C₄⋊Dic₁₈₁	1	2	yes	—
97	C₂×C₄	C₂²×Dic₁₈₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	2	yes	—
99	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
100	C₂×C₄	#15668449	1	2	yes	—
101	C₈	—	2	2	no	—
102	C₈	—	724	4	no	—
103	C₈	C₂²⋊Dic₁₈₁	1	4	yes	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	#19153752	1	4	yes	complemented
109	C₂²	C₂×C₄⋊Dic₁₈₁	1	1	yes	characteristic
110	C₂²	#19153795	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	C₂×C₂²⋊Dic₁₈₁	1	1	yes	characteristic
113	C₄	#19153775	1	1	yes	characteristic
114	C₄	#19153790	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	—	2	2	no	—
117	C₂	#26041139	1	1	yes	characteristic
118	C₂	C₂×#19153752	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153752

C₂

yes

maximal, complemented

#26041243

C₂

yes

maximal

#26041044

C₂

yes

maximal, complemented

C₂³×C₈×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668433

C₂²

yes

—

#19153752

—

C₂²⋊#13089678

—

C₂³×C₄×C₁₈₁

C₂²

yes

characteristic

C₂²×C₈×C₁₈₁

—

C₂²×C₈×C₁₈₁

C₄

yes

—

C₂²×C₈×C₁₈₁

C₂²

yes

characteristic

C₂²×C₈×C₁₈₁

C₂²

yes

—

—

—

C₂⁴×C₁₈₁

Q₈

yes

characteristic

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

C₂×C₄

yes

—

C₂²×C₄×C₁₈₁

D₄

yes

—

C₂²×C₄×C₁₈₁

Q₈

yes

characteristic

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

C₂×C₄

yes

—

C₂×C₈×C₁₈₁

C₂×C₄

yes

—

C₂×C₈×C₁₈₁

D₄

yes

—

#13089678

—

C₂³×C₁₈₁

—

C₂³×C₁₈₁

—

C₂³×C₁₈₁

C₄⋊C₄

yes

—

C₂³×C₁₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

#41

yes

—

C₈×C₁₈₁

—

C₈×C₁₈₁

C₂²⋊C₄

yes

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

#108

yes

complemented

C₂²×C₁₈₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₈₁

C₂²⋊Q₈

yes

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂×C₂²⋊C₄

yes

C₄×C₁₈₁

yes

C₄×C₁₈₁

yes

—

C₂×C₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

#380

yes

characteristic

C₂×C₁₈₁

C₂×#108

yes

—

C₁₈₁

#1810

yes

characteristic, complemented, Sylow

#1810

—

181

maximal, Sylow

C₂³×C₈

D₁₈₁

yes

characteristic

C₂×#108

—

181

—

#407

—

181

—

#341

—

181

—

C₂³×C₄

C₂×D₁₈₁

yes

characteristic

C₂×M₁₆

—

181

—

C₂²×C₈

—

C₂²×C₈

Dic₁₈₁

yes

—

C₂²×C₈

C₂×D₁₈₁

yes

characteristic

C₂²×C₈

C₂×D₁₈₁

yes

characteristic

#108

—

362

—

C₂²⋊C₈

—

362

—

C₂⁴

Dic₃₆₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₈₁

yes

—

C₂²×C₄

D₇₂₄

yes

—

C₂²×C₄

Dic₃₆₂

yes

characteristic

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

M₁₆

—

362

—

M₁₆

—

362

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

362

—

C₂×C₈

C₂×Dic₁₈₁

yes

—

C₂×C₈

C₂×Dic₁₈₁

yes

—

C₂×C₈

C₂²⋊D₁₈₁

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₁₈₁

yes

—

C₂³

C₂×Dic₃₆₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₈₁

yes

—

C₂×C₄

C₂²×Dic₁₈₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

—

C₂×C₄

C₂×D₇₂₄

yes

characteristic

100

C₂×C₄

#15668449

yes

—

101

C₈

—

102

C₈

—

724

—

103

C₈

C₂²⋊Dic₁₈₁

yes

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#19153752

yes

complemented

109

C₂²

C₂×C₄⋊Dic₁₈₁

yes

characteristic

110

C₂²

#19153795

yes

—

111

C₄

—

112

C₄

C₂×C₂²⋊Dic₁₈₁

yes

characteristic

113

C₄

#19153775

yes

characteristic

114

C₄

#19153790

yes

—

115

C₂

—

116

C₂

—

117

C₂

#26041139

yes

characteristic

118

C₂

C₂×#19153752

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₂₄
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	C₂³×C₈×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₈₁	Q₈
cosocle	C₂²×C₄×C₁₈₁	C₂³
hypercenter	C₂³×C₈	D₁₈₁
hypocenter	C₁₈₁	#1810
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₇₂₄

commutator

C₂×C₄×C₁₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

C₂³×C₈×C₁₈₁

C₂

socle

C₂²×C₁₈₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₁₈₁

Q₈

cosocle

C₂²×C₄×C₁₈₁

C₂³

hypercenter

C₂³×C₈

D₁₈₁

hypocenter

C₁₈₁

#1810

LCS

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
1. C₂×#19153752	C₂	semidirect
3. #26041044	C₂	semidirect
48. C₂²×C₁₈₁	#108	semidirect
59. C₁₈₁	#1810	semidirect
108. C₂²	#19153752	semidirect

H ≅ G/N

type

1. C₂×#19153752

C₂

semidirect

3. #26041044

C₂

semidirect

48. C₂²×C₁₈₁

#108

semidirect

59. C₁₈₁

#1810

semidirect

108. C₂²

#19153752

semidirect

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

subgroup	index
C₂×#19153752	2
#26041243	2
#26041044	2
C₂³×C₈×C₁₈₁	2
#1810	181

subgroup

index

C₂³×C₈×C₁₈₁

181

Subgroups — #37533759

Conjugacy classes of subgroups

Complements

Maximal subgroups