Group #37535167 — subgroups

Subgroups — #37535167

Order 23168.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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	#26041278	C₂	1	2	yes	maximal
2	#26041245	C₂	1	1	yes	maximal, characteristic
3	#26041048	C₂	1	2	yes	maximal, complemented
4	#26041044	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#106×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668449	C₂²	1	1	yes	characteristic
7	C₂×#15668433	C₂²	1	1	yes	characteristic
8	C₂×#15668433	C₂²	1	2	yes	—
9	#19153739	—	2	2	no	—
10	#19153736	—	2	2	no	—
11	#19153728	—	2	2	no	—
12	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	2	yes	—
14	#106×C₁₈₁	—	2	2	no	—
15	#15668449	—	2	2	no	—
16	#15668449	—	2	2	no	—
17	C₂×Dic₃₆₂	D₄	1	1	yes	characteristic
18	C₂×D₇₂₄	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₈₁	—	2	1	no	—
20	C₂×C₄×D₁₈₁	—	2	1	no	—
21	#15668433	—	2	2	no	—
22	#15668433	—	2	2	no	—
23	#15668433	—	2	2	no	—
24	C₂×#13089678	—	2	1	no	—
25	C₂×#13089678	—	2	2	no	—
26	C₂²×C₄×C₁₈₁	—	2	2	no	—
27	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₈₁	—	2	2	no	—
29	C₂×C₈×C₁₈₁	D₄	1	2	yes	—
30	C₄⋊C₄×C₁₈₁	—	2	2	no	—
31	C₄⋊C₄×C₁₈₁	—	2	2	no	—
32	C₂²×D₁₈₁	—	2	1	no	—
33	Dic₃₆₂	—	2	2	no	—
34	D₇₂₄	—	2	2	no	—
35	C₂²⋊D₁₈₁	—	4	2	no	—
36	C₂×Dic₁₈₁	—	2	1	no	—
37	C₄×D₁₈₁	—	4	1	no	—
38	C₄×D₁₈₁	—	4	1	no	—
39	#13089678	—	4	2	no	—
40	#13089678	—	4	1	no	—
41	C₂³×C₁₈₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₈₁	—	4	2	no	—
43	C₂×C₄×C₁₈₁	—	8	2	no	—
44	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₈₁	#41	1	2	yes	—
47	C₂×C₄×C₁₈₁	#41	1	2	yes	—
48	C₈×C₁₈₁	—	2	2	no	—
49	C₂²×C₁₈₁	—	2	2	no	—
50	C₂²×C₁₈₁	#121	1	1	yes	characteristic
51	C₂²×C₁₈₁	#126	1	2	yes	—
52	C₂×D₁₈₁	—	4	1	no	—
53	C₂×D₁₈₁	—	4	2	no	—
54	C₄×C₁₈₁	—	8	2	no	—
55	C₄×C₁₈₁	#123	1	1	yes	characteristic
56	C₄×C₁₈₁	#123	1	2	yes	—
57	C₄×C₁₈₁	#124	1	1	yes	characteristic
58	Dic₁₈₁	—	4	1	no	—
59	C₂×C₁₈₁	—	2	2	no	—
60	C₂×C₁₈₁	#397	1	1	yes	characteristic
61	C₂×C₁₈₁	#471	1	2	yes	—
62	D₁₈₁	—	8	1	no	—
63	C₁₈₁	#2079	1	1	yes	characteristic, complemented, Sylow
64	#2079	—	181	1	no	maximal, Sylow
65	C₂×#106	D₁₈₁	1	1	yes	characteristic, complemented
66	#417	—	181	2	no	—
67	#408	—	181	1	no	—
68	#343	—	181	2	no	—
69	#341	—	181	1	no	—
70	C₂×#41	—	181	1	no	—
71	C₂×M₁₆	—	181	2	no	—
72	C₂×M₁₆	—	181	1	no	—
73	C₂²×C₈	C₂×D₁₈₁	1	1	yes	characteristic
74	C₂×C₄⋊C₄	C₂×D₁₈₁	1	2	yes	—
75	#106	—	2	2	no	—
76	#103	—	362	2	no	—
77	#102	—	362	2	no	—
78	C₂²⋊C₈	—	362	2	no	—
79	#41	—	362	2	no	—
80	#41	—	362	2	no	—
81	C₂×Q₈	—	181	1	no	—
82	C₂×D₄	—	362	1	no	—
83	C₂×D₄	—	181	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	362	1	no	—
86	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
87	M₁₆	—	362	2	no	—
88	M₁₆	—	362	2	no	—
89	M₁₆	—	362	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	362	2	no	—
92	C₂×C₈	—	362	1	no	—
93	C₂×C₈	C₂²⋊D₁₈₁	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₂³	—	362	1	no	—
97	C₂³	#15668448	1	1	yes	characteristic
98	Q₈	—	362	2	no	—
99	D₄	—	362	2	no	—
100	D₄	—	724	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	—	362	1	no	—
104	C₂×C₄	—	724	1	no	—
105	C₂×C₄	—	724	1	no	—
106	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
108	C₂×C₄	#15668449	1	2	yes	—
109	C₂×C₄	#15668450	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	724	2	no	—
112	C₈	—	724	1	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	724	1	no	—
115	C₂²	—	724	2	no	—
116	C₂²	#19153787	1	1	yes	characteristic
117	C₂²	#19153811	1	2	yes	—
118	C₄	—	8	2	no	—
119	C₄	—	724	1	no	—
120	C₄	#19153799	1	1	yes	characteristic
121	C₄	#19153801	1	2	yes	—
122	C₄	#19153805	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	1448	1	no	—
125	C₂	#26041205	1	1	yes	characteristic
126	C₂	#26041539	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26041278

C₂

yes

maximal

#26041245

C₂

yes

maximal, characteristic

#26041048

C₂

yes

maximal, complemented

#26041044

C₂

yes

maximal, characteristic, complemented

C₂×#106×C₁₈₁

C₂

yes

maximal, characteristic, complemented

C₂×#15668449

C₂²

yes

characteristic

C₂×#15668433

C₂²

yes

characteristic

C₂×#15668433

C₂²

yes

—

#19153739

—

#19153736

—

#19153728

—

C₂²×C₈×C₁₈₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

—

#106×C₁₈₁

—

#15668449

—

#15668449

—

C₂×Dic₃₆₂

D₄

yes

characteristic

C₂×D₇₂₄

D₄

yes

characteristic

C₂×C₂²⋊D₁₈₁

—

C₂×C₄×D₁₈₁

—

—

—

—

—

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

D₄

yes

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₂²×D₁₈₁

—

Dic₃₆₂

—

D₇₂₄

—

C₂²⋊D₁₈₁

—

C₂×Dic₁₈₁

—

C₄×D₁₈₁

—

C₄×D₁₈₁

—

#13089678

—

#13089678

—

C₂³×C₁₈₁

#41

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

#41

yes

—

C₂×C₄×C₁₈₁

#41

yes

—

C₈×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

#121

yes

characteristic

C₂²×C₁₈₁

#126

yes

—

C₂×D₁₈₁

—

C₂×D₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

#123

yes

characteristic

C₄×C₁₈₁

#123

yes

—

C₄×C₁₈₁

#124

yes

characteristic

Dic₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

#397

yes

characteristic

C₂×C₁₈₁

#471

yes

—

D₁₈₁

—

C₁₈₁

#2079

yes

characteristic, complemented, Sylow

#2079

—

181

maximal, Sylow

C₂×#106

D₁₈₁

yes

characteristic, complemented

#417

—

181

—

#408

—

181

—

#343

—

181

—

#341

—

181

—

C₂×#41

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

—

181

—

C₂²×C₈

C₂×D₁₈₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₈₁

yes

—

#106

—

#103

—

362

—

#102

—

362

—

C₂²⋊C₈

—

362

—

#41

—

362

—

#41

—

362

—

C₂×Q₈

—

181

—

C₂×D₄

—

362

—

C₂×D₄

—

181

—

C₂²×C₄

—

C₂²×C₄

—

362

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

M₁₆

—

362

—

M₁₆

—

362

—

M₁₆

—

362

—

C₂×C₈

—

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₂×C₈

C₂²⋊D₁₈₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

362

—

C₂³

#15668448

yes

characteristic

Q₈

—

362

—

D₄

—

362

—

100

D₄

—

724

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

362

—

104

C₂×C₄

—

724

—

105

C₂×C₄

—

724

—

106

C₂×C₄

D₄×D₁₈₁

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

characteristic

108

C₂×C₄

#15668449

yes

—

109

C₂×C₄

#15668450

yes

—

110

C₈

—

111

C₈

—

724

—

112

C₈

—

724

—

113

C₂²

—

114

C₂²

—

724

—

115

C₂²

—

724

—

116

C₂²

#19153787

yes

characteristic

117

C₂²

#19153811

yes

—

118

C₄

—

119

C₄

—

724

—

120

C₄

#19153799

yes

characteristic

121

C₄

#19153801

yes

—

122

C₄

#19153805

yes

characteristic

123

C₂

—

124

C₂

—

1448

—

125

C₂

#26041205

yes

characteristic

126

C₂

#26041539

yes

—

	subgroup	G/H
center	C₂²	#19153787
commutator, cosocle	C₂²×C₄×C₁₈₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	C₂×#106×C₁₈₁	C₂
socle	C₂²×C₁₈₁	#121
prime radical	C₂×#15668449	C₂²
hypercenter	C₂×#106	D₁₈₁
hypocenter	C₁₈₁	#2079
LCS	G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

#19153787

commutator, cosocle

C₂²×C₄×C₁₈₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

C₂×#106×C₁₈₁

C₂

C₂²×C₁₈₁

C₂²

D₁₈₁

C₁₈₁

G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #26041048	C₂	semidirect
4. #26041044	C₂	semidirect
5. C₂×#106×C₁₈₁	C₂	semidirect
63. C₁₈₁	#2079	semidirect
65. C₂×#106	D₁₈₁	semidirect

H ≅ G/N

type

3. #26041048

C₂

semidirect

4. #26041044

C₂

semidirect

5. C₂×#106×C₁₈₁

C₂

semidirect

63. C₁₈₁

#2079

semidirect

65. C₂×#106

D₁₈₁

semidirect

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

subgroup	index
#26041278	2
#26041245	2
#26041048	2
#26041044	2
C₂×#106×C₁₈₁	2
#2079	181

subgroup

index

C₂×#106×C₁₈₁

181

Subgroups — #37535167

Conjugacy classes of subgroups

Complements

Maximal subgroups