Group #37534509 — subgroups

Subgroups — #37534509

Order 23168.

184 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 85 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	#26041327	C₂	1	2	yes	maximal
2	C₂×#19153752	C₂	1	1	yes	maximal, characteristic
3	#26041234	C₂	1	1	yes	maximal, characteristic
4	#26041050	C₂	1	2	yes	maximal
5	C₂×#108×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668433	C₂²	1	2	yes	—
7	C₂×#15668433	C₂²	1	1	yes	characteristic
8	#19153775	C₂²	1	1	yes	characteristic
9	#19153752	C₄	1	4	yes	complemented
10	#19153751	—	2	4	no	—
11	#19153726	—	2	1	no	—
12	C₈×Dic₁₈₁	—	2	1	no	—
13	C₂×M₁₆×C₁₈₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
15	#108×C₁₈₁	—	2	1	no	—
16	#108×C₁₈₁	—	2	1	no	—
17	#15668433	—	2	1	no	—
18	#15668433	—	2	2	no	—
19	#15668433	C₂×C₄	1	4	yes	—
20	#15668433	D₄	1	2	yes	—
21	C₂×#13089678	—	2	1	no	—
22	C₂×#13089678	—	2	2	no	—
23	C₄⋊Dic₁₈₁	—	2	1	no	—
24	C₂²⋊Dic₁₈₁	Q₈	1	2	yes	—
25	C₄×Dic₁₈₁	—	2	1	no	—
26	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₈₁	—	2	2	no	—
28	M₁₆×C₁₈₁	—	2	2	no	—
29	C₂×C₈×C₁₈₁	—	2	2	no	—
30	C₂×C₈×C₁₈₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₈₁	Q₈	1	2	yes	—
34	C₂×Dic₁₈₁	—	2	2	no	—
35	#13089678	—	2	4	no	—
36	#13089678	—	2	2	no	—
37	C₂³×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₈₁	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₈₁	#41	1	1	yes	characteristic
44	C₈×C₁₈₁	—	4	2	no	—
45	C₈×C₁₈₁	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₈₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₈₁	—	4	1	no	—
48	C₂²×C₁₈₁	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₁₈₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₈₁	C₄×D₄	1	1	yes	characteristic
53	C₄×C₁₈₁	#121	1	1	yes	characteristic
54	C₄×C₁₈₁	#125	1	1	yes	characteristic
55	Dic₁₈₁	—	4	2	no	—
56	C₂×C₁₈₁	—	2	1	no	—
57	C₂×C₁₈₁	—	2	1	no	—
58	C₂×C₁₈₁	—	2	1	no	—
59	C₂×C₁₈₁	#389	1	1	yes	characteristic
60	C₁₈₁	#1953	1	1	yes	characteristic, complemented, Sylow
61	#1953	—	181	1	no	maximal, Sylow
62	#430	—	181	2	no	—
63	C₂×#108	—	181	1	no	—
64	C₂×#108	D₁₈₁	1	1	yes	characteristic
65	#405	—	181	1	no	—
66	#344	—	181	2	no	—
67	C₂×M₁₆	—	181	2	no	—
68	C₂×M₁₆	—	181	1	no	—
69	C₂×M₁₆	C₂×D₁₈₁	1	2	yes	—
70	C₂²×C₈	C₂×D₁₈₁	1	1	yes	characteristic
71	#117	—	181	1	no	—
72	#108	—	2	1	no	—
73	#108	—	2	1	no	—
74	#108	—	362	4	no	—
75	#108	—	181	4	no	—
76	#97	—	362	1	no	—
77	C₄×C₈	—	362	1	no	—
78	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	362	1	no	—
82	M₁₆	—	362	2	no	—
83	M₁₆	—	181	4	no	—
84	M₁₆	—	181	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	362	2	no	—
87	C₂×C₈	—	362	1	no	—
88	C₂×C₈	C₄×D₁₈₁	1	2	yes	—
89	C₂×C₈	C₂²⋊D₁₈₁	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₁₈₁	1	1	yes	characteristic
91	C₂×C₈	Dic₃₆₂	1	2	yes	—
92	C₄⋊C₄	—	362	1	no	—
93	C₂²⋊C₄	—	181	2	no	—
94	C₄²	—	362	1	no	—
95	C₂³	Q₈×D₁₈₁	1	1	yes	characteristic
96	C₂×C₄	—	362	2	no	—
97	C₂×C₄	C₂×C₄×D₁₈₁	1	1	yes	characteristic
98	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₃₆₂	1	1	yes	characteristic
101	C₂×C₄	#15668448	1	1	yes	characteristic
102	C₂×C₄	#15668450	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	362	4	no	—
105	C₈	—	362	2	no	—
106	C₈	#15668428	1	2	yes	—
107	C₈	#15668428	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#19153781	1	1	yes	characteristic
110	C₂²	#19153792	1	1	yes	characteristic
111	C₂²	#19153815	1	1	yes	characteristic
112	C₄	—	724	2	no	—
113	C₄	C₂×#15668428	1	1	yes	characteristic
114	C₄	#19153778	1	1	yes	characteristic
115	C₄	#19153787	1	1	yes	characteristic
116	C₄	#19153807	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#26041175	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26041327

C₂

yes

maximal

C₂×#19153752

C₂

yes

maximal, characteristic

#26041234

C₂

yes

maximal, characteristic

#26041050

C₂

yes

maximal

C₂×#108×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668433

C₂²

yes

—

C₂×#15668433

C₂²

yes

characteristic

#19153775

C₂²

yes

characteristic

#19153752

C₄

yes

complemented

#19153751

—

#19153726

—

C₈×Dic₁₈₁

—

C₂×M₁₆×C₁₈₁

C₂²

yes

—

C₂²×C₈×C₁₈₁

C₂²

yes

characteristic

#108×C₁₈₁

—

#108×C₁₈₁

—

#15668433

—

#15668433

—

#15668433

C₂×C₄

yes

—

#15668433

D₄

yes

—

C₂×#13089678

—

C₂×#13089678

—

C₄⋊Dic₁₈₁

—

C₂²⋊Dic₁₈₁

Q₈

yes

—

C₄×Dic₁₈₁

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

M₁₆×C₁₈₁

—

M₁₆×C₁₈₁

—

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

C₂×C₄

yes

—

C₂×C₈×C₁₈₁

D₄

yes

characteristic

C₂×C₈×C₁₈₁

D₄

yes

characteristic

C₂×C₈×C₁₈₁

Q₈

yes

—

C₂×Dic₁₈₁

—

#13089678

—

#13089678

—

C₂³×C₁₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×Q₈

yes

C₂×C₄×C₁₈₁

yes

C₂×C₄×C₁₈₁

yes

C₈×C₁₈₁

—

C₈×C₁₈₁

C₄⋊C₄

yes

—

C₈×C₁₈₁

C₄⋊C₄

yes

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

C₄×Q₈

yes

characteristic

C₂²×C₁₈₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₈₁

C₄⋊Q₈

yes

characteristic

C₄×C₁₈₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₈₁

C₄×D₄

yes

C₄×C₁₈₁

yes

C₄×C₁₈₁

yes

Dic₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

#389

yes

characteristic

C₁₈₁

#1953

yes

characteristic, complemented, Sylow

#1953

—

181

maximal, Sylow

#430

—

181

—

C₂×#108

—

181

—

C₂×#108

D₁₈₁

yes

characteristic

#405

—

181

—

#344

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

C₂×D₁₈₁

yes

—

C₂²×C₈

C₂×D₁₈₁

yes

characteristic

#117

—

181

—

#108

—

#108

—

#108

—

362

—

#108

—

181

—

#97

—

362

—

C₄×C₈

—

362

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

362

—

M₁₆

—

362

—

M₁₆

—

181

—

M₁₆

—

181

—

C₂×C₈

—

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₂×C₈

C₄×D₁₈₁

yes

—

C₂×C₈

C₂²⋊D₁₈₁

yes

characteristic

C₂×C₈

C₂²⋊D₁₈₁

yes

characteristic

C₂×C₈

Dic₃₆₂

yes

—

C₄⋊C₄

—

362

—

C₂²⋊C₄

—

181

—

C₄²

—

362

—

C₂³

Q₈×D₁₈₁

yes

characteristic

C₂×C₄

—

362

—

C₂×C₄

C₂×C₄×D₁₈₁

yes

characteristic

C₂×C₄

D₄×D₁₈₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

characteristic

100

C₂×C₄

C₂×Dic₃₆₂

yes

characteristic

101

C₂×C₄

#15668448

yes

characteristic

102

C₂×C₄

#15668450

yes

characteristic

103

C₈

—

104

C₈

—

362

—

105

C₈

—

362

—

106

C₈

#15668428

yes

—

107

C₈

#15668428

yes

—

108

C₂²

—

109

C₂²

#19153781

yes

characteristic

110

C₂²

#19153792

yes

characteristic

111

C₂²

#19153815

yes

characteristic

112

C₄

—

724

—

113

C₄

C₂×#15668428

yes

characteristic

114

C₄

#19153778

yes

characteristic

115

C₄

#19153787

yes

characteristic

116

C₄

#19153807

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#26041175

yes

characteristic

	subgroup	G/H
center	C₄	#19153787
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	C₂×#108×C₁₈₁	C₂
socle	C₂×C₁₈₁	#389
prime radical	C₂³×C₁₈₁	C₂×Q₈
cosocle	C₂²×C₄×C₁₈₁	C₂³
hypercenter	C₂×#108	D₁₈₁
hypocenter	C₁₈₁	#1953
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#19153787

commutator

C₂×C₄×C₁₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

C₂×#108×C₁₈₁

C₂

socle

C₂×C₁₈₁

#389

prime radical

C₂³×C₁₈₁

C₂×Q₈

cosocle

C₂²×C₄×C₁₈₁

C₂³

D₁₈₁

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #19153752	C₄	semidirect
60. C₁₈₁	#1953	semidirect

H ≅ G/N

type

9. #19153752

C₄

semidirect

60. C₁₈₁

#1953

semidirect

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

subgroup	index
#26041327	2
C₂×#19153752	2
#26041234	2
#26041050	2
C₂×#108×C₁₈₁	2
#1953	181

subgroup

index

C₂×#108×C₁₈₁

181

Subgroups — #37534509

Conjugacy classes of subgroups

Complements

Maximal subgroups