Group #22628495 — subgroups

Subgroups — C₃₇×#7242 #22628495

Order 9472.

190 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 94 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×C₃₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#1568	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₃₇	C₄	1	2	yes	—
4	C₃₇×#432	—	2	2	no	—
5	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
6	C₃₇×C₄⋊C₁₆	—	2	2	no	—
7	C₃₇×#335	—	2	2	no	—
8	C₃₇×#334	C₄	1	4	yes	—
9	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₈×C₃₇	Q₈	1	1	yes	characteristic
11	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
12	C₂×C₁₆×C₃₇	—	2	2	no	—
13	C₂×C₁₆×C₃₇	—	4	2	no	—
14	C₄⋊C₈×C₃₇	—	2	4	no	—
15	C₂²⋊C₈×C₃₇	—	4	2	no	—
16	#97×C₃₇	—	4	2	no	—
17	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₃₇	D₄	1	2	yes	—
19	C₂²×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₇	D₈	1	1	yes	characteristic
21	C₂²×C₄×C₃₇	Q₁₆	1	1	yes	characteristic
22	C₂×C₈×C₃₇	—	2	1	no	—
23	C₂×C₈×C₃₇	—	2	1	no	—
24	C₂×C₈×C₃₇	—	4	1	no	—
25	C₂×C₈×C₃₇	—	4	4	no	—
26	C₂×C₈×C₃₇	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₃₇	C₄⋊C₄	1	2	yes	—
29	C₄²×C₃₇	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₃₇	SD₁₆	1	2	yes	—
31	C₁₆×C₃₇	—	4	2	no	—
32	C₂³×C₃₇	#107	1	1	yes	characteristic
33	C₂×C₄×C₃₇	—	2	2	no	—
34	C₂×C₄×C₃₇	—	2	1	no	—
35	C₂×C₄×C₃₇	—	2	1	no	—
36	C₂×C₄×C₃₇	—	4	1	no	—
37	C₂×C₄×C₃₇	#95	1	1	yes	characteristic
38	C₂×C₄×C₃₇	#102	1	2	yes	—
39	C₂×C₄×C₃₇	#103	1	2	yes	—
40	C₂×C₄×C₃₇	#106	1	1	yes	characteristic
41	C₈×C₃₇	—	2	2	no	—
42	C₈×C₃₇	—	2	1	no	—
43	C₈×C₃₇	—	8	4	no	—
44	C₈×C₃₇	#100	1	1	yes	characteristic
45	C₈×C₃₇	#101	1	1	yes	characteristic
46	#7242	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂²×C₃₇	—	2	1	no	—
48	C₂²×C₃₇	—	2	1	no	—
49	C₂²×C₃₇	#340	1	1	yes	characteristic
50	C₂²×C₃₇	#368	1	2	yes	—
51	C₄×C₃₇	—	2	1	no	—
52	C₄×C₃₇	—	2	2	no	—
53	C₄×C₃₇	—	2	2	no	—
54	C₄×C₃₇	#338	1	1	yes	characteristic
55	C₄×C₃₇	#343	1	1	yes	characteristic
56	C₂×C₄⋊C₁₆	C₂×C₃₇	1	1	yes	characteristic
57	#1568	C₂×C₃₇	1	2	yes	—
58	C₂×C₃₇	—	2	2	no	—
59	C₂×C₃₇	#1292	1	1	yes	characteristic
60	C₂×C₃₇	#1381	1	1	yes	characteristic
61	C₂×C₃₇	#1384	1	1	yes	characteristic
62	C₂²×C₁₆	C₄×C₃₇	1	2	yes	—
63	#432	—	2	2	no	—
64	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
65	C₄⋊C₁₆	—	2	2	no	—
66	#335	—	2	2	no	—
67	#334	C₄×C₃₇	1	4	yes	—
68	C₃₇	#7242	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
70	C₂²×C₈	Q₈×C₃₇	1	1	yes	characteristic
71	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	C₄⋊C₈	—	2	4	no	—
75	C₂²⋊C₈	—	4	2	no	—
76	#97	—	4	2	no	—
77	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
78	C₄×C₈	D₄×C₃₇	1	2	yes	—
79	C₂²×C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
80	C₂²×C₄	D₈×C₃₇	1	1	yes	characteristic
81	C₂²×C₄	Q₁₆×C₃₇	1	1	yes	characteristic
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	4	no	—
86	C₂×C₈	C₄²×C₃₇	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
88	C₂×C₈	C₄⋊C₄×C₃₇	1	2	yes	—
89	C₄²	C₂²⋊C₄×C₃₇	1	2	yes	—
90	C₄²	SD₁₆×C₃₇	1	2	yes	—
91	C₁₆	—	4	2	no	—
92	C₂³	#107×C₃₇	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	#95×C₃₇	1	1	yes	characteristic
98	C₂×C₄	#102×C₃₇	1	2	yes	—
99	C₂×C₄	#103×C₃₇	1	2	yes	—
100	C₂×C₄	#106×C₃₇	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	2	1	no	—
103	C₈	—	8	4	no	—
104	C₈	#100×C₃₇	1	1	yes	characteristic
105	C₈	#101×C₃₇	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	C₃₇×#340	1	1	yes	characteristic
109	C₂²	C₃₇×#368	1	2	yes	—
110	C₄	—	2	2	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	C₃₇×#338	1	1	yes	characteristic
114	C₄	C₃₇×#343	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	C₃₇×#1292	1	1	yes	characteristic
117	C₂	C₃₇×#1381	1	1	yes	characteristic
118	C₂	C₃₇×#1384	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₇×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₃₇×#1568

C₂

yes

maximal

C₂²×C₁₆×C₃₇

C₄

yes

—

C₃₇×#432

—

C₂×C₄×C₈×C₃₇

C₂²

yes

characteristic

C₃₇×C₄⋊C₁₆

—

C₃₇×#335

—

C₃₇×#334

C₄

yes

—

C₂²×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₃₇

Q₈

yes

characteristic

C₂×C₄²×C₃₇

D₄

yes

characteristic

C₂×C₁₆×C₃₇

—

C₂×C₁₆×C₃₇

—

C₄⋊C₈×C₃₇

—

C₂²⋊C₈×C₃₇

—

#97×C₃₇

—

C₄×C₈×C₃₇

C₂×C₄

yes

—

C₄×C₈×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₃₇

D₈

yes

characteristic

C₂²×C₄×C₃₇

Q₁₆

yes

characteristic

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₄²

yes

characteristic

C₂×C₈×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₃₇

C₄⋊C₄

yes

—

C₄²×C₃₇

C₂²⋊C₄

yes

—

C₄²×C₃₇

SD₁₆

yes

—

C₁₆×C₃₇

—

C₂³×C₃₇

#107

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

#95

yes

characteristic

C₂×C₄×C₃₇

#102

yes

—

C₂×C₄×C₃₇

#103

yes

—

C₂×C₄×C₃₇

#106

yes

characteristic

C₈×C₃₇

—

C₈×C₃₇

—

C₈×C₃₇

—

C₈×C₃₇

yes

C₈×C₃₇

yes

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#340

yes

characteristic

C₂²×C₃₇

#368

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

yes

C₄×C₃₇

yes

C₂×C₄⋊C₁₆

C₂×C₃₇

yes

characteristic

#1568

C₂×C₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂²×C₁₆

C₄×C₃₇

yes

—

#432

—

C₂×C₄×C₈

C₂²×C₃₇

yes

characteristic

C₄⋊C₁₆

—

#335

—

#334

C₄×C₃₇

yes

—

C₃₇

#7242

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂×C₄×C₃₇

yes

characteristic

C₂²×C₈

Q₈×C₃₇

yes

characteristic

C₂×C₄²

D₄×C₃₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×C₄×C₃₇

yes

—

C₄×C₈

D₄×C₃₇

yes

—

C₂²×C₄

C₄⋊C₄×C₃₇

yes

characteristic

C₂²×C₄

D₈×C₃₇

yes

characteristic

C₂²×C₄

Q₁₆×C₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₃₇

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₃₇

yes

characteristic

C₂×C₈

C₄⋊C₄×C₃₇

yes

—

C₄²

C₂²⋊C₄×C₃₇

yes

—

C₄²

SD₁₆×C₃₇

yes

—

C₁₆

—

C₂³

#107×C₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95×C₃₇

yes

characteristic

C₂×C₄

#102×C₃₇

yes

—

C₂×C₄

#103×C₃₇

yes

—

100

C₂×C₄

#106×C₃₇

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

#100×C₃₇

yes

characteristic

105

C₈

#101×C₃₇

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

C₃₇×#340

yes

characteristic

109

C₂²

C₃₇×#368

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₃₇×#338

yes

characteristic

114

C₄

C₃₇×#343

yes

characteristic

115

C₂

—

116

C₂

C₃₇×#1292

yes

characteristic

117

C₂

C₃₇×#1381

yes

characteristic

118

C₂

C₃₇×#1384

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	#340
commutator	C₂×C₈	C₄²×C₃₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃₇
prime radical	C₂³×C₃₇	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ C₂×C₄×C₈×C₃₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₇

#340

commutator

C₂×C₈

C₄²×C₃₇

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₃₇

prime radical

C₂³×C₃₇

#107

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ C₂×C₄×C₈×C₃₇ ◁ G

N	H ≅ G/N	type
46. #7242	C₃₇	direct (N × H)
68. C₃₇	#7242	direct (N × H)

H ≅ G/N

type

46. #7242

C₃₇

direct (N × H)

68. C₃₇

#7242

direct (N × H)

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

subgroup	index
C₂×C₃₇×C₄⋊C₁₆	2
C₃₇×#1568	2
#7242	37

subgroup

index

C₂×C₃₇×C₄⋊C₁₆

C₃₇×#1568

#7242

Subgroups — C₃₇×#7242 #22628495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37×#7242 #22628495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#7242 #22628495