Group #37549524 — subgroups

Subgroups — #37549524

Order 23168.

232 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 90 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×#19153886	C₂	1	1	yes	maximal, characteristic
2	#26042244	C₂	1	2	yes	maximal
3	C₂×C₈×#10624382	C₂	1	1	yes	maximal, characteristic
4	#26042149	C₂	1	2	yes	maximal
5	#26041314	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668459	C₂²	1	2	yes	—
7	C₂×D₁₈₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10624382	C₂²	1	1	yes	characteristic
9	#19153886	C₄	1	4	yes	complemented
10	#19153871	—	2	2	no	—
11	C₄×#13089685	—	2	2	no	—
12	C₈×#10624382	C₄	1	4	yes	complemented
13	C₂×C₈×D₁₈₁	C₂²	1	1	yes	characteristic
14	#19153774	C₂²	1	2	yes	—
15	#19153749	C₄	1	2	yes	complemented
16	#19153748	C₄	1	1	yes	characteristic, complemented
17	#106×C₁₈₁	C₄	1	1	yes	characteristic, complemented
18	C₂²×#10624382	—	2	1	no	—
19	#15668459	—	2	2	no	—
20	#15668459	C₂×C₄	1	4	yes	—
21	D₁₈₁⋊C₈	C₂×C₄	1	2	yes	—
22	D₁₈₁⋊C₈	Q₈	1	2	yes	—
23	C₂×#13089685	—	2	2	no	—
24	C₂×#13089685	D₄	1	1	yes	characteristic
25	C₂×#13089685	D₄	1	1	yes	characteristic
26	C₄×#10624382	—	2	1	no	—
27	C₄×#10624382	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
29	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
30	C₈×D₁₈₁	C₂×C₄	1	2	yes	—
31	C₈×D₁₈₁	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₁₈₁	C₂×C₄	1	2	yes	—
33	#15668426	—	2	2	no	—
34	C₄×Dic₁₈₁	—	2	2	no	—
35	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₈₁	C₂×C₄	1	2	yes	—
37	C₂×#10624382	—	2	4	no	—
38	C₂×#10624382	—	2	2	no	—
39	#13089685	—	2	4	no	—
40	#13089685	C₄⋊C₄	1	2	yes	—
41	#13089685	C₄⋊C₄	1	2	yes	—
42	C₂²×D₁₈₁	#41	1	1	yes	characteristic
43	C₂×Dic₁₈₁	—	2	2	no	—
44	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
45	C₄×D₁₈₁	C₂²×C₄	1	1	yes	characteristic
46	C₄×D₁₈₁	C₂²×C₄	1	1	yes	characteristic
47	C₄×D₁₈₁	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₁₈₁	#41	1	1	yes	characteristic
49	#13089678	C₄²	1	2	yes	—
50	C₂×C₄×C₁₈₁	—	2	2	no	—
51	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₁₈₁	C₄²	1	2	yes	—
53	C₂²×C₁₈₁	C₄×D₄	1	1	yes	characteristic
54	C₂×D₁₈₁	—	2	2	no	—
55	C₂×D₁₈₁	#117	1	1	yes	characteristic
56	C₂×D₁₈₁	C₄×Q₈	1	1	yes	characteristic
57	#10624382	—	2	4	no	—
58	C₄×C₁₈₁	—	4	2	no	—
59	C₄×C₁₈₁	C₂×C₄²	1	1	yes	characteristic
60	C₄×C₁₈₁	C₄×Q₈	1	1	yes	characteristic
61	Dic₁₈₁	—	4	2	no	—
62	Dic₁₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	Dic₁₈₁	C₄×D₄	1	1	yes	characteristic
64	C₂×C₁₈₁	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₁₈₁	#443	1	2	yes	—
66	D₁₈₁	—	2	1	no	—
67	D₁₈₁	—	2	1	no	—
68	C₁₈₁	#1772	1	1	yes	characteristic, complemented, Sylow
69	#1772	—	181	1	no	maximal, Sylow
70	C₂×#108	—	181	1	no	—
71	#427	—	181	1	no	—
72	#405	—	181	2	no	—
73	C₂×C₄×C₈	—	181	1	no	—
74	#339	—	181	2	no	—
75	C₂×M₁₆	—	181	2	no	—
76	C₂²×C₈	—	181	1	no	—
77	C₂²×C₈	—	181	1	no	—
78	#117	—	181	2	no	—
79	C₂×C₄²	—	181	1	no	—
80	#108	—	181	4	no	—
81	#107	—	181	2	no	—
82	#106	—	181	1	no	—
83	#106	#10624382	1	1	yes	characteristic, complemented
84	#97	—	362	2	no	—
85	C₄×C₈	—	362	2	no	—
86	C₄×C₈	—	181	4	no	—
87	C₂²×C₄	—	362	1	no	—
88	C₂²×C₄	—	181	1	no	—
89	M₁₆	—	362	2	no	—
90	M₁₆	—	181	4	no	—
91	C₂×C₈	—	362	2	no	—
92	C₂×C₈	—	181	1	no	—
93	C₂×C₈	—	181	2	no	—
94	C₂×C₈	—	181	1	no	—
95	C₂×C₈	—	181	2	no	—
96	C₂×C₈	—	181	1	no	—
97	C₂×C₈	—	181	2	no	—
98	C₂×C₈	—	181	2	no	—
99	C₂×C₈	C₂×#10624382	1	1	yes	characteristic
100	C₄⋊C₄	—	181	2	no	—
101	C₄⋊C₄	C₂×#10624382	1	2	yes	—
102	C₂²⋊C₄	—	362	2	no	—
103	C₄²	—	362	2	no	—
104	C₄²	—	362	1	no	—
105	C₄²	—	181	2	no	—
106	C₂³	—	181	1	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	362	4	no	—
109	C₂×C₄	—	362	2	no	—
110	C₂×C₄	—	362	2	no	—
111	C₂×C₄	—	181	1	no	—
112	C₂×C₄	—	181	1	no	—
113	C₂×C₄	—	181	1	no	—
114	C₂×C₄	—	181	1	no	—
115	C₂×C₄	—	181	1	no	—
116	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
117	C₈	—	362	4	no	—
118	C₈	—	181	2	no	—
119	C₈	—	181	2	no	—
120	C₈	—	181	2	no	—
121	C₈	C₄×#10624382	1	2	yes	—
122	C₂²	—	362	2	no	—
123	C₂²	—	181	1	no	—
124	C₂²	—	181	1	no	—
125	C₂²	D₄×#10624382	1	1	yes	characteristic
126	C₄	—	4	2	no	—
127	C₄	—	362	4	no	—
128	C₄	—	181	1	no	—
129	C₄	—	181	1	no	—
130	C₄	—	724	2	no	—
131	C₄	C₂×C₄×#10624382	1	1	yes	characteristic
132	C₄	Q₈×#10624382	1	1	yes	characteristic
133	C₂	—	362	1	no	—
134	C₂	—	362	1	no	—
135	C₂	C₄⋊C₄×#10624382	1	1	yes	characteristic
136	C₂	#26042322	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153886

C₂

yes

maximal, characteristic

#26042244

C₂

yes

maximal

C₂×C₈×#10624382

C₂

yes

maximal, characteristic

#26042149

C₂

yes

maximal

#26041314

C₂

yes

maximal, characteristic

C₂×#15668459

C₂²

yes

—

C₂×D₁₈₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×#10624382

C₂²

yes

characteristic

#19153886

C₄

yes

—

—

C₄

yes

complemented

C₂×C₈×D₁₈₁

C₂²

yes

characteristic

#19153774

C₂²

yes

—

#19153749

C₄

yes

complemented

#19153748

C₄

yes

characteristic, complemented

#106×C₁₈₁

C₄

yes

characteristic, complemented

C₂²×#10624382

—

#15668459

—

#15668459

C₂×C₄

yes

—

D₁₈₁⋊C₈

C₂×C₄

yes

—

D₁₈₁⋊C₈

Q₈

yes

—

C₂×#13089685

—

C₂×#13089685

D₄

yes

characteristic

C₂×#13089685

D₄

yes

characteristic

C₄×#10624382

—

C₄×#10624382

C₂×C₄

yes

—

C₂×C₄×D₁₈₁

C₂³

yes

characteristic

C₂×#13089678

C₂×C₄

yes

characteristic

C₈×D₁₈₁

C₂×C₄

yes

—

C₈×D₁₈₁

C₂×C₄

yes

—

C₄⋊Dic₁₈₁

C₂×C₄

yes

—

#15668426

—

C₄×Dic₁₈₁

—

C₂×C₈×C₁₈₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₈₁

C₂×C₄

yes

—

—

—

—

C₄⋊C₄

yes

—

#13089685

C₄⋊C₄

yes

—

C₂²×D₁₈₁

#41

yes

characteristic

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

C₂×D₄

yes

characteristic

C₄×D₁₈₁

C₂²×C₄

yes

characteristic

C₄×D₁₈₁

C₂²×C₄

yes

characteristic

C₄×D₁₈₁

C₂×Q₈

yes

C₄×D₁₈₁

yes

C₄²

yes

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₈×C₁₈₁

C₄²

yes

—

C₂²×C₁₈₁

C₄×D₄

yes

characteristic

C₂×D₁₈₁

—

C₂×D₁₈₁

#117

yes

characteristic

C₂×D₁₈₁

C₄×Q₈

yes

characteristic

#10624382

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂×C₄²

yes

characteristic

C₄×C₁₈₁

C₄×Q₈

yes

characteristic

Dic₁₈₁

—

Dic₁₈₁

C₂×C₄⋊C₄

yes

characteristic

Dic₁₈₁

C₄×D₄

yes

characteristic

C₂×C₁₈₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₈₁

#443

yes

—

D₁₈₁

—

D₁₈₁

—

C₁₈₁

#1772

yes

characteristic, complemented, Sylow

#1772

—

181

maximal, Sylow

C₂×#108

—

181

—

#427

—

181

—

#405

—

181

—

C₂×C₄×C₈

—

181

—

#339

—

181

—

C₂×M₁₆

—

181

—

C₂²×C₈

—

181

—

C₂²×C₈

—

181

—

#117

—

181

—

C₂×C₄²

—

181

—

#108

—

181

—

#107

—

181

—

#106

—

181

—

#106

#10624382

yes

characteristic, complemented

#97

—

362

—

C₄×C₈

—

362

—

C₄×C₈

—

181

—

C₂²×C₄

—

362

—

C₂²×C₄

—

181

—

M₁₆

—

362

—

M₁₆

—

181

—

C₂×C₈

—

362

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

C₂×#10624382

yes

characteristic

100

C₄⋊C₄

—

181

—

101

C₄⋊C₄

C₂×#10624382

yes

—

102

C₂²⋊C₄

—

362

—

103

C₄²

—

362

—

104

C₄²

—

362

—

105

C₄²

—

181

—

106

C₂³

—

181

—

107

C₂×C₄

—

108

C₂×C₄

—

362

—

109

C₂×C₄

—

362

—

110

C₂×C₄

—

362

—

111

C₂×C₄

—

181

—

112

C₂×C₄

—

181

—

113

C₂×C₄

—

181

—

114

C₂×C₄

—

181

—

115

C₂×C₄

—

181

—

116

C₂×C₄

C₂²×#10624382

yes

characteristic

117

C₈

—

362

—

118

C₈

—

181

—

119

C₈

—

181

—

120

C₈

—

181

—

121

C₈

C₄×#10624382

yes

—

122

C₂²

—

362

—

123

C₂²

—

181

—

124

C₂²

—

181

—

125

C₂²

D₄×#10624382

yes

characteristic

126

C₄

—

127

C₄

—

362

—

128

C₄

—

181

—

129

C₄

—

181

—

130

C₄

—

724

—

131

C₄

C₂×C₄×#10624382

yes

characteristic

132

C₄

Q₈×#10624382

yes

characteristic

133

C₂

—

362

—

134

C₂

—

362

—

135

C₂

C₄⋊C₄×#10624382

yes

characteristic

136

C₂

#26042322

yes

—

	subgroup	G/H
center	C₂²	D₄×#10624382
commutator	C₄×C₁₈₁	C₂×C₄²
Frattini	C₂×C₄	C₂²×#10624382
Fitting	#106×C₁₈₁	C₄
socle	C₂²×C₁₈₁	C₄×D₄
prime radical	C₂²×D₁₈₁	#41
cosocle	C₂×C₄×D₁₈₁	C₂³
hypercenter	#106	#10624382
hypocenter	C₁₈₁	#1772
LCS	G ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×#10624382

commutator

C₄×C₁₈₁

C₂×C₄²

Frattini

C₂×C₄

C₂²×#10624382

Fitting

#106×C₁₈₁

C₄

socle

C₂²×C₁₈₁

C₄×D₄

prime radical

C₂²×D₁₈₁

#41

cosocle

C₂×C₄×D₁₈₁

C₂³

C₁₈₁

G ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
9. #19153886	C₄	semidirect
12. C₈×#10624382	C₄	semidirect
15. #19153749	C₄	semidirect
16. #19153748	C₄	semidirect
17. #106×C₁₈₁	C₄	semidirect
68. C₁₈₁	#1772	semidirect
83. #106	#10624382	semidirect

H ≅ G/N

type

9. #19153886

C₄

semidirect

12. C₈×#10624382

C₄

semidirect

15. #19153749

C₄

semidirect

16. #19153748

C₄

semidirect

17. #106×C₁₈₁

C₄

semidirect

68. C₁₈₁

#1772

semidirect

83. #106

#10624382

semidirect

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

subgroup	index
C₂×#19153886	2
#26042244	2
C₂×C₈×#10624382	2
#26042149	2
#26041314	2
#1772	181

subgroup

index

181

Subgroups — #37549524

Conjugacy classes of subgroups

Complements

Maximal subgroups