Group #17990907 — subgroups

Subgroups — C₃₇×#2089 #17990907

Order 4736.

176 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₃₇×#428	C₂	1	2	yes	maximal
2	C₂×#107×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#424	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#343	C₂	1	2	yes	maximal
5	C₃₇×#341	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
8	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
9	#117×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
11	#107×C₃₇	—	2	2	no	—
12	#107×C₃₇	—	2	2	no	—
13	#106×C₃₇	—	2	2	no	—
14	C₄⋊C₈×C₃₇	—	2	2	no	—
15	C₂²×C₄×C₃₇	—	2	2	no	—
16	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
17	M₁₆×C₃₇	—	2	2	no	—
18	M₁₆×C₃₇	—	2	2	no	—
19	M₁₆×C₃₇	Q₈	1	4	yes	—
20	C₂×C₈×C₃₇	—	2	2	no	—
21	C₂×C₈×C₃₇	—	2	2	no	—
22	C₂×C₈×C₃₇	—	2	1	no	—
23	C₂×C₈×C₃₇	D₄	1	2	yes	—
24	C₄⋊C₄×C₃₇	—	2	2	no	—
25	C₄⋊C₄×C₃₇	—	2	2	no	—
26	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
28	C₂²⋊C₄×C₃₇	—	2	1	no	—
29	C₄²×C₃₇	—	2	1	no	—
30	C₂³×C₃₇	#41	1	1	yes	characteristic
31	C₂×C₄×C₃₇	—	2	1	no	—
32	C₂×C₄×C₃₇	—	2	1	no	—
33	C₂×C₄×C₃₇	—	4	2	no	—
34	C₂×C₄×C₃₇	—	8	2	no	—
35	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₃₇	#41	1	2	yes	—
39	C₈×C₃₇	—	2	4	no	—
40	C₈×C₃₇	—	2	2	no	—
41	C₈×C₃₇	—	4	1	no	—
42	C₂²×C₃₇	—	2	1	no	—
43	C₂²×C₃₇	—	2	1	no	—
44	C₂²×C₃₇	#121	1	1	yes	characteristic
45	C₂²×C₃₇	#125	1	2	yes	—
46	C₄×C₃₇	—	4	1	no	—
47	C₄×C₃₇	—	4	1	no	—
48	C₄×C₃₇	—	8	2	no	—
49	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₃₇	#123	1	1	yes	characteristic
51	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
52	#2089	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×C₃₇	—	2	2	no	—
54	C₂×C₃₇	#398	1	1	yes	characteristic
55	C₂×C₃₇	#472	1	1	yes	characteristic
56	C₂×C₃₇	#473	1	1	yes	characteristic
57	#428	C₂×C₃₇	1	2	yes	—
58	C₂×#107	C₂×C₃₇	1	1	yes	characteristic
59	#424	C₂×C₃₇	1	1	yes	characteristic
60	#343	C₂×C₃₇	1	2	yes	—
61	#341	C₂×C₃₇	1	1	yes	characteristic
62	C₃₇	#2089	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
64	C₂×M₁₆	C₂²×C₃₇	1	2	yes	—
65	C₂²×C₈	C₂²×C₃₇	1	1	yes	characteristic
66	#117	C₂²×C₃₇	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
68	#107	—	2	2	no	—
69	#107	—	2	2	no	—
70	#106	—	2	2	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	2	2	no	—
76	M₁₆	Q₈×C₃₇	1	4	yes	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	D₄×C₃₇	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
84	C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
85	C₂²⋊C₄	—	2	1	no	—
86	C₄²	—	2	1	no	—
87	C₂³	#41×C₃₇	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	8	2	no	—
92	C₂×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
93	C₂×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
94	C₂×C₄	C₂×Q₈×C₃₇	1	2	yes	—
95	C₂×C₄	#41×C₃₇	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	2	no	—
98	C₈	—	4	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#121×C₃₇	1	1	yes	characteristic
102	C₂²	#125×C₃₇	1	2	yes	—
103	C₄	—	4	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	8	2	no	—
106	C₄	C₂²⋊Q₈×C₃₇	1	2	yes	—
107	C₄	#123×C₃₇	1	1	yes	characteristic
108	C₄	C₄⋊Q₈×C₃₇	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	C₃₇×#398	1	1	yes	characteristic
111	C₂	C₃₇×#472	1	1	yes	characteristic
112	C₂	C₃₇×#473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₇×#428

C₂

yes

maximal

C₂×#107×C₃₇

C₂

yes

maximal, characteristic

C₃₇×#424

C₂

yes

maximal, characteristic

C₃₇×#343

C₂

yes

maximal

C₃₇×#341

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

#107×C₃₇

—

#107×C₃₇

—

#106×C₃₇

—

C₄⋊C₈×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

M₁₆×C₃₇

—

M₁₆×C₃₇

—

M₁₆×C₃₇

Q₈

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

D₄

yes

characteristic

C₄⋊C₄×C₃₇

D₄

yes

characteristic

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₂³×C₃₇

#41

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₈×C₃₇

—

C₈×C₃₇

—

C₈×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#121

yes

characteristic

C₂²×C₃₇

#125

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

#123

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

#2089

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×#107

C₂×C₃₇

yes

characteristic

#424

C₂×C₃₇

yes

characteristic

#343

C₂×C₃₇

yes

—

#341

C₂×C₃₇

yes

characteristic

C₃₇

#2089

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

C₂²×C₃₇

yes

characteristic

C₂×M₁₆

C₂²×C₃₇

yes

—

C₂²×C₈

C₂²×C₃₇

yes

characteristic

#117

C₂²×C₃₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₃₇

yes

—

#107

—

#107

—

#106

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂³×C₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

Q₈×C₃₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₃₇

yes

characteristic

C₄⋊C₄

D₄×C₃₇

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₂³

#41×C₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₃₇

yes

characteristic

C₂×C₄

C₂×D₄×C₃₇

yes

characteristic

C₂×C₄

C₂×Q₈×C₃₇

yes

—

C₂×C₄

#41×C₃₇

yes

—

C₈

—

C₈

—

C₈

—

C₂²

—

100

C₂²

—

101

C₂²

#121×C₃₇

yes

characteristic

102

C₂²

#125×C₃₇

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

C₂²⋊Q₈×C₃₇

yes

—

107

C₄

#123×C₃₇

yes

characteristic

108

C₄

C₄⋊Q₈×C₃₇

yes

characteristic

109

C₂

—

110

C₂

C₃₇×#398

yes

characteristic

111

C₂

C₃₇×#472

yes

characteristic

112

C₂

C₃₇×#473

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
prime radical	C₂³×C₃₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₇

#121

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₃₇

C₂³×C₃₇

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

N	H ≅ G/N	type
52. #2089	C₃₇	direct (N × H)
62. C₃₇	#2089	direct (N × H)

H ≅ G/N

type

52. #2089

C₃₇

direct (N × H)

62. C₃₇

#2089

direct (N × H)

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

subgroup	index
C₃₇×#428	2
C₂×#107×C₃₇	2
C₃₇×#424	2
C₃₇×#343	2
C₃₇×#341	2
#2089	37

subgroup

index

C₃₇×#428

C₂×#107×C₃₇

C₃₇×#424

C₃₇×#343

C₃₇×#341

#2089

Subgroups — C₃₇×#2089 #17990907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37×#2089 #17990907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#2089 #17990907