Group #37533769 — subgroups

Subgroups — #37533769

Order 23168.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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×#19153823	C₂	1	1	yes	maximal, characteristic
2	C₂×#19153751	C₂	1	2	yes	maximal, complemented
3	#26041263	C₂	1	1	yes	maximal, characteristic
4	#26041037	C₂	1	2	yes	maximal, complemented
5	#413×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	#19153823	—	2	4	no	—
7	C₂×#15668433	C₄	1	2	yes	—
8	C₂×#15668433	C₂²	1	2	yes	—
9	C₂×#15668433	C₂²	1	1	yes	characteristic
10	C₂²×#13089678	C₄	1	2	yes	—
11	C₂²×#13089678	C₂²	1	1	yes	characteristic
12	#19153751	—	2	4	no	—
13	#19153735	—	2	1	no	—
14	#19153733	—	2	1	no	—
15	C₂×#41×C₁₈₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₈₁	C₂²	1	2	yes	—
17	#101×C₁₈₁	—	2	1	no	—
18	#100×C₁₈₁	—	2	1	no	—
19	#15668433	—	2	4	no	—
20	#15668433	—	2	2	no	—
21	#15668433	—	2	2	no	—
22	#15668433	—	2	2	no	—
23	C₂×#13089678	—	2	4	no	—
24	C₂×#13089678	—	2	2	no	—
25	C₂×#13089678	C₂×C₄	1	2	yes	—
26	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
27	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
28	C₂×#13089678	D₄	1	4	yes	—
29	#41×C₁₈₁	—	2	2	no	—
30	#41×C₁₈₁	—	2	2	no	—
31	C₂×Q₈×C₁₈₁	D₄	1	1	yes	characteristic
32	C₂×D₄×C₁₈₁	D₄	1	1	yes	characteristic
33	C₂×D₄×C₁₈₁	Q₈	1	2	yes	—
34	C₂²×C₄×C₁₈₁	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₈₁	—	2	2	no	—
37	M₁₆×C₁₈₁	—	2	2	no	—
38	C₂×C₈×C₁₈₁	—	2	2	no	—
39	#13089678	—	2	2	no	—
40	#13089678	—	4	2	no	—
41	#13089678	C₂²⋊C₄	1	2	yes	—
42	#13089678	C₂²⋊C₄	1	2	yes	—
43	C₂³×C₁₈₁	C₄⋊C₄	1	2	yes	—
44	C₂³×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
45	Q₈×C₁₈₁	—	2	2	no	—
46	D₄×C₁₈₁	—	2	4	no	—
47	D₄×C₁₈₁	—	2	2	no	—
48	C₂×C₄×C₁₈₁	—	2	2	no	—
49	C₂×C₄×C₁₈₁	—	2	2	no	—
50	C₂×C₄×C₁₈₁	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₈₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₈₁	#41	1	2	yes	—
55	C₈×C₁₈₁	—	4	2	no	—
56	C₂²×C₁₈₁	—	2	2	no	—
57	C₂²×C₁₈₁	—	4	1	no	—
58	C₂²×C₁₈₁	—	4	2	no	—
59	C₂²×C₁₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₁₈₁	—	2	2	no	—
62	C₄×C₁₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₁₈₁	#117	1	1	yes	characteristic
64	C₄×C₁₈₁	#121	1	2	yes	—
65	C₂×C₁₈₁	—	2	1	no	—
66	C₂×C₁₈₁	—	2	2	no	—
67	C₂×C₁₈₁	—	4	2	no	—
68	C₂×C₁₈₁	#380	1	1	yes	characteristic
69	C₁₈₁	#1813	1	1	yes	characteristic, complemented, Sylow
70	#1813	—	181	1	no	maximal, Sylow
71	C₂×#131	—	181	1	no	—
72	C₂×#108	—	181	2	no	—
73	#413	—	181	1	no	—
74	#413	D₁₈₁	1	1	yes	characteristic
75	#338	—	181	2	no	—
76	C₂×#41	C₂×D₁₈₁	1	1	yes	characteristic
77	#131	—	362	4	no	—
78	C₂×M₁₆	—	181	2	no	—
79	C₂×M₁₆	—	181	2	no	—
80	C₂×M₁₆	—	181	1	no	—
81	C₂×M₁₆	C₂×D₁₈₁	1	2	yes	—
82	C₂²×C₈	—	181	2	no	—
83	C₂²×C₈	—	181	1	no	—
84	#108	—	362	4	no	—
85	#101	—	2	1	no	—
86	#101	—	362	1	no	—
87	#100	—	2	1	no	—
88	#100	—	362	1	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	C₂×Q₈	C₂²⋊D₁₈₁	1	1	yes	characteristic
92	C₂×D₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
93	C₂×D₄	Dic₃₆₂	1	2	yes	—
94	C₂²×C₄	C₄×D₁₈₁	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	362	2	no	—
99	M₁₆	—	362	4	no	—
100	M₁₆	—	362	2	no	—
101	M₁₆	—	362	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	362	4	no	—
104	C₂×C₈	—	362	2	no	—
105	C₂×C₈	—	181	2	no	—
106	C₂×C₈	—	181	1	no	—
107	C₂×C₈	—	181	1	no	—
108	C₂×C₈	—	181	4	no	—
109	C₂³	#15668428	1	2	yes	—
110	C₂³	C₂×Dic₃₆₂	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	D₄	—	2	4	no	—
113	D₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	#15668428	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₈₁	1	1	yes	characteristic
118	C₂×C₄	D₄×D₁₈₁	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
120	C₂×C₄	#15668448	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₈	—	362	2	no	—
123	C₈	—	181	2	no	—
124	C₈	—	181	2	no	—
125	C₈	—	724	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	4	1	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂×#15668428	1	1	yes	characteristic
130	C₂²	#19153793	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	C₂²⋊C₄×D₁₈₁	1	1	yes	characteristic
133	C₄	#19153772	1	1	yes	characteristic
134	C₄	#19153788	1	2	yes	—
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	—	4	2	no	—
138	C₂	#26041137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153823

C₂

yes

maximal, characteristic

C₂×#19153751

C₂

yes

maximal, complemented

#26041263

C₂

yes

maximal, characteristic

#26041037

C₂

yes

maximal, complemented

#413×C₁₈₁

C₂

yes

maximal, characteristic

#19153823

—

C₂×#15668433

C₄

yes

—

C₂×#15668433

C₂²

yes

—

C₂×#15668433

C₂²

yes

characteristic

C₂²×#13089678

C₄

yes

—

C₂²×#13089678

C₂²

yes

—

—

—

C₂×#41×C₁₈₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₈₁

C₂²

yes

—

#101×C₁₈₁

—

#100×C₁₈₁

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#13089678

C₂×C₄

yes

characteristic

C₂×#13089678

C₂×C₄

yes

characteristic

C₂×#13089678

D₄

yes

—

#41×C₁₈₁

—

#41×C₁₈₁

—

C₂×Q₈×C₁₈₁

D₄

yes

characteristic

C₂×D₄×C₁₈₁

D₄

yes

characteristic

C₂×D₄×C₁₈₁

Q₈

yes

—

C₂²×C₄×C₁₈₁

C₂×C₄

yes

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

M₁₆×C₁₈₁

—

M₁₆×C₁₈₁

—

C₂×C₈×C₁₈₁

—

#13089678

—

#13089678

—

#13089678

C₂²⋊C₄

yes

—

#13089678

C₂²⋊C₄

yes

—

C₂³×C₁₈₁

C₄⋊C₄

yes

—

C₂³×C₁₈₁

C₂×Q₈

yes

characteristic

Q₈×C₁₈₁

—

D₄×C₁₈₁

—

D₄×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

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

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

—

C₂×C₁₈₁

#380

yes

characteristic

C₁₈₁

#1813

yes

characteristic, complemented, Sylow

#1813

—

181

maximal, Sylow

C₂×#131

—

181

—

C₂×#108

—

181

—

#413

—

181

—

#413

D₁₈₁

yes

characteristic

#338

—

181

—

C₂×#41

C₂×D₁₈₁

yes

characteristic

#131

—

362

—

C₂×M₁₆

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

C₂×D₁₈₁

yes

—

C₂²×C₈

—

181

—

C₂²×C₈

—

181

—

#108

—

362

—

#101

—

#101

—

362

—

#100

—

#100

—

362

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₁₈₁

yes

characteristic

C₂×D₄

C₂²⋊D₁₈₁

yes

characteristic

C₂×D₄

Dic₃₆₂

yes

—

C₂²×C₄

C₄×D₁₈₁

yes

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

362

—

M₁₆

—

362

—

100

M₁₆

—

362

—

101

M₁₆

—

362

—

102

C₂×C₈

—

103

C₂×C₈

—

362

—

104

C₂×C₈

—

362

—

105

C₂×C₈

—

181

—

106

C₂×C₈

—

181

—

107

C₂×C₈

—

181

—

108

C₂×C₈

—

181

—

109

C₂³

#15668428

yes

—

110

C₂³

C₂×Dic₃₆₂

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#15668428

yes

—

117

C₂×C₄

C₂×C₄×D₁₈₁

yes

characteristic

118

C₂×C₄

D₄×D₁₈₁

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

characteristic

120

C₂×C₄

#15668448

yes

—

121

C₈

—

122

C₈

—

362

—

123

C₈

—

181

—

124

C₈

—

181

—

125

C₈

—

724

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×#15668428

yes

characteristic

130

C₂²

#19153793

yes

—

131

C₄

—

132

C₄

C₂²⋊C₄×D₁₈₁

yes

characteristic

133

C₄

#19153772

yes

characteristic

134

C₄

#19153788

yes

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#26041137

yes

characteristic

	subgroup	G/H
center	C₄	#19153772
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	#413×C₁₈₁	C₂
socle	C₂×C₁₈₁	#380
prime radical	C₂×#41×C₁₈₁	C₂²
cosocle	C₂²×C₄×C₁₈₁	C₂³
hypercenter	#413	D₁₈₁
hypocenter	C₁₈₁	#1813
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#19153772

commutator

C₂×C₄×C₁₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

#413×C₁₈₁

C₂

socle

C₂×C₁₈₁

#380

prime radical

C₂×#41×C₁₈₁

C₂²

cosocle

C₂²×C₄×C₁₈₁

C₂³

D₁₈₁

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. C₂×#19153751	C₂	semidirect
4. #26041037	C₂	semidirect
69. C₁₈₁	#1813	semidirect

H ≅ G/N

type

2. C₂×#19153751

C₂

semidirect

4. #26041037

C₂

semidirect

69. C₁₈₁

#1813

semidirect

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

subgroup	index
C₂×#19153823	2
C₂×#19153751	2
#26041263	2
#26041037	2
#413×C₁₈₁	2
#1813	181

subgroup

index

#413×C₁₈₁

181

Subgroups — #37533769

Conjugacy classes of subgroups

Complements

Maximal subgroups