Group #37534470 — subgroups

Subgroups — #37534470

Order 23168.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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	1	yes	maximal, characteristic
2	#26041306	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₈₁	C₂	1	1	yes	maximal, characteristic
4	#26041038	C₂	1	2	yes	maximal
5	#427×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668433	C₂²	1	2	yes	—
7	C₂²×#13089678	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
9	#19153752	—	2	2	no	—
10	C₄⋊#13089678	—	2	4	no	—
11	C₈×Dic₁₈₁	—	2	2	no	—
12	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
13	#117×C₁₈₁	C₂²	1	2	yes	—
14	#107×C₁₈₁	C₄	1	2	yes	complemented
15	#106×C₁₈₁	C₄	1	2	yes	complemented
16	C₂²×Dic₁₈₁	—	2	1	no	—
17	#15668433	—	2	2	no	—
18	#15668433	—	2	2	no	—
19	C₂×#13089678	—	2	1	no	—
20	C₂×#13089678	—	2	2	no	—
21	C₂×#13089678	—	2	1	no	—
22	C₂×#13089678	D₄	1	2	yes	—
23	C₄×Dic₁₈₁	—	2	1	no	—
24	C₄×Dic₁₈₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₈₁	D₄	1	2	yes	—
29	C₂×C₈×C₁₈₁	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₈₁	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₈₁	—	2	2	no	—
32	C₄²×C₁₈₁	—	2	2	no	—
33	C₂×Dic₁₈₁	—	2	4	no	—
34	C₂×Dic₁₈₁	—	2	2	no	—
35	#13089678	—	2	2	no	—
36	#13089678	—	4	2	no	—
37	C₂³×C₁₈₁	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₈₁	—	2	4	no	—
39	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₈₁	#41	1	1	yes	characteristic
45	C₈×C₁₈₁	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₈₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₈₁	—	2	2	no	—
48	C₂²×C₁₈₁	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₈₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₈₁	#125	1	1	yes	characteristic
51	C₄×C₁₈₁	—	4	4	no	—
52	C₄×C₁₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₈₁	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₈₁	#121	1	1	yes	characteristic
55	C₄×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₈₁	—	2	4	no	—
57	C₂×C₁₈₁	—	2	1	no	—
58	C₂×C₁₈₁	—	2	1	no	—
59	C₂×C₁₈₁	#389	1	1	yes	characteristic
60	C₂×C₁₈₁	#443	1	2	yes	—
61	C₁₈₁	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	181	1	no	maximal, Sylow
63	C₂×#108	—	181	1	no	—
64	#427	D₁₈₁	1	1	yes	characteristic
65	#424	—	181	2	no	—
66	C₂×C₄×C₈	—	181	1	no	—
67	#339	—	181	2	no	—
68	C₂×M₁₆	—	181	2	no	—
69	C₂²×C₈	—	181	1	no	—
70	C₂²×C₈	C₂×D₁₈₁	1	1	yes	characteristic
71	#117	C₂×D₁₈₁	1	2	yes	—
72	C₂×C₄²	—	181	1	no	—
73	#108	—	362	2	no	—
74	#107	Dic₁₈₁	1	2	yes	complemented
75	#106	Dic₁₈₁	1	2	yes	complemented
76	C₄⋊C₈	—	362	4	no	—
77	C₄×C₈	—	362	2	no	—
78	C₂²×C₄	—	362	1	no	—
79	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
80	M₁₆	—	362	2	no	—
81	M₁₆	—	362	2	no	—
82	C₂×C₈	—	362	1	no	—
83	C₂×C₈	—	362	1	no	—
84	C₂×C₈	—	362	2	no	—
85	C₂×C₈	—	181	2	no	—
86	C₂×C₈	C₂×Dic₁₈₁	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₁₈₁	1	1	yes	characteristic
88	C₂×C₈	D₇₂₄	1	2	yes	—
89	C₂×C₈	Dic₃₆₂	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₁₈₁	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	362	1	no	—
94	C₄²	—	181	2	no	—
95	C₂³	#15668448	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	362	4	no	—
98	C₂×C₄	—	362	2	no	—
99	C₂×C₄	C₂²×Dic₁₈₁	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
101	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₈₁	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₃₆₂	1	1	yes	characteristic
104	C₂×C₄	#15668450	1	1	yes	characteristic
105	C₈	—	362	2	no	—
106	C₈	—	724	2	no	—
107	C₈	C₄⋊Dic₁₈₁	1	2	yes	—
108	C₈	C₄⋊Dic₁₈₁	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₁₈₁	1	1	yes	characteristic
111	C₂²	#19153796	1	1	yes	characteristic
112	C₂²	#19153807	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	362	4	no	—
115	C₄	C₂×C₄⋊Dic₁₈₁	1	1	yes	characteristic
116	C₄	Q₈×Dic₁₈₁	1	1	yes	characteristic
117	C₄	#19153791	1	1	yes	characteristic
118	C₄	#19153815	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#26041178	1	1	yes	characteristic
122	C₂	#26041396	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153752

C₂

yes

maximal, characteristic

#26041306

C₂

yes

maximal

C₂×C₈×Dic₁₈₁

C₂

yes

maximal, characteristic

#26041038

C₂

yes

maximal

#427×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668433

C₂²

yes

—

C₂²×#13089678

C₂²

yes

characteristic

C₂×C₄×Dic₁₈₁

C₂²

yes

characteristic

#19153752

—

C₄⋊#13089678

—

C₈×Dic₁₈₁

—

C₂²×C₈×C₁₈₁

C₂²

yes

characteristic

#117×C₁₈₁

C₂²

yes

—

#107×C₁₈₁

C₄

yes

complemented

#106×C₁₈₁

C₄

yes

complemented

C₂²×Dic₁₈₁

—

—

—

—

—

—

D₄

yes

—

C₄×Dic₁₈₁

—

C₄×Dic₁₈₁

Q₈

yes

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₂×C₈×C₁₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₈₁

D₄

yes

—

C₂×C₈×C₁₈₁

Q₈

yes

—

C₄⋊C₄×C₁₈₁

C₂×C₄

yes

—

C₂²⋊C₄×C₁₈₁

—

C₄²×C₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

—

#13089678

—

C₂³×C₁₈₁

#41

yes

characteristic

C₂×C₄×C₁₈₁

—

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

characteristic

C₂×C₄×C₁₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₈₁

#41

yes

characteristic

C₈×C₁₈₁

C₄⋊C₄

yes

—

C₈×C₁₈₁

C₄⋊C₄

yes

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

C₄×D₄

yes

characteristic

C₂²×C₁₈₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₈₁

#125

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₈₁

C₄×Q₈

yes

characteristic

C₄×C₁₈₁

#121

yes

characteristic

C₄×C₁₈₁

C₄⋊Q₈

yes

characteristic

Dic₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

#389

yes

characteristic

C₂×C₁₈₁

#443

yes

—

C₁₈₁

#1945

yes

characteristic, complemented, Sylow

#1945

—

181

maximal, Sylow

C₂×#108

—

181

—

#427

D₁₈₁

yes

characteristic

#424

—

181

—

C₂×C₄×C₈

—

181

—

#339

—

181

—

C₂×M₁₆

—

181

—

C₂²×C₈

—

181

—

C₂²×C₈

C₂×D₁₈₁

yes

characteristic

#117

C₂×D₁₈₁

yes

—

C₂×C₄²

—

181

—

#108

—

362

—

#107

Dic₁₈₁

yes

complemented

#106

Dic₁₈₁

yes

complemented

C₄⋊C₈

—

362

—

C₄×C₈

—

362

—

C₂²×C₄

—

362

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

M₁₆

—

362

—

M₁₆

—

362

—

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₂×C₈

—

181

—

C₂×C₈

C₂×Dic₁₈₁

yes

characteristic

C₂×C₈

C₂×Dic₁₈₁

yes

characteristic

C₂×C₈

D₇₂₄

yes

—

C₂×C₈

Dic₃₆₂

yes

—

C₄⋊C₄

C₂×Dic₁₈₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

362

—

C₄²

—

181

—

C₂³

#15668448

yes

characteristic

C₂×C₄

—

C₂×C₄

—

362

—

C₂×C₄

—

362

—

C₂×C₄

C₂²×Dic₁₈₁

yes

characteristic

100

C₂×C₄

D₄×D₁₈₁

yes

characteristic

101

C₂×C₄

C₂×D₇₂₄

yes

characteristic

102

C₂×C₄

Q₈×D₁₈₁

yes

characteristic

103

C₂×C₄

C₂×Dic₃₆₂

yes

characteristic

104

C₂×C₄

#15668450

yes

characteristic

105

C₈

—

362

—

106

C₈

—

724

—

107

C₈

C₄⋊Dic₁₈₁

yes

—

108

C₈

C₄⋊Dic₁₈₁

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₁₈₁

yes

characteristic

111

C₂²

#19153796

yes

characteristic

112

C₂²

#19153807

yes

characteristic

113

C₄

—

114

C₄

—

362

—

115

C₄

C₂×C₄⋊Dic₁₈₁

yes

characteristic

116

C₄

Q₈×Dic₁₈₁

yes

characteristic

117

C₄

#19153791

yes

characteristic

118

C₄

#19153815

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#26041178

yes

characteristic

122

C₂

#26041396

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₈₁
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	#427×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₄×D₄
prime radical	C₂³×C₁₈₁	#41
cosocle	C₂²×C₄×C₁₈₁	C₂³
hypercenter	#427	D₁₈₁
hypocenter	C₁₈₁	#1945
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₈₁

commutator

C₂×C₄×C₁₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

#427×C₁₈₁

C₂

socle

C₂²×C₁₈₁

C₄×D₄

prime radical

C₂³×C₁₈₁

#41

cosocle

C₂²×C₄×C₁₈₁

C₂³

D₁₈₁

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₁₈₁	C₄	semidirect
15. #106×C₁₈₁	C₄	semidirect
61. C₁₈₁	#1945	semidirect
74. #107	Dic₁₈₁	semidirect
75. #106	Dic₁₈₁	semidirect

H ≅ G/N

type

14. #107×C₁₈₁

C₄

semidirect

15. #106×C₁₈₁

C₄

semidirect

61. C₁₈₁

#1945

semidirect

74. #107

Dic₁₈₁

semidirect

75. #106

Dic₁₈₁

semidirect

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

subgroup	index
C₂×#19153752	2
#26041306	2
C₂×C₈×Dic₁₈₁	2
#26041038	2
#427×C₁₈₁	2
#1945	181

subgroup

index

C₂×C₈×Dic₁₈₁

#427×C₁₈₁

181

Subgroups — #37534470

Conjugacy classes of subgroups

Complements

Maximal subgroups