Group #17991804 — subgroups

Subgroups — C₃₇×#2986 #17991804

Order 4736.

330 conjugacy classes of subgroups; only the 256 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 256 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×#533	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#446	C₂	1	4	yes	maximal, complemented
3	C₈×Q₈×C₃₇	C₂	1	4	yes	maximal, complemented
4	C₃₇×#433	C₂	1	4	yes	maximal
5	C₃₇×#431	C₂	1	2	yes	maximal
6	C₄⋊Q₈×C₃₇	C₄	1	2	yes	—
7	#125×C₃₇	C₄	1	2	yes	—
8	C₂²⋊Q₈×C₃₇	C₄	1	4	yes	—
9	C₄×Q₈×C₃₇	C₂²	1	4	yes	—
10	#117×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
13	C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
14	C₄⋊C₈×C₃₇	C₂²	1	8	yes	—
15	C₂²⋊C₈×C₃₇	C₂²	1	4	yes	—
16	#97×C₃₇	C₂²	1	4	yes	—
17	C₄×C₈×C₃₇	C₂²	1	4	yes	—
18	C₄×C₈×C₃₇	C₂²	1	4	yes	—
19	C₂×Q₈×C₃₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₃₇	C₂³	1	4	yes	—
23	C₂×C₈×C₃₇	C₂³	1	4	yes	—
24	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
25	C₄⋊C₄×C₃₇	C₂×C₄	1	8	yes	—
26	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
27	C₂²⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
28	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₃₇	C₂³	1	2	yes	—
31	C₄²×C₃₇	C₂³	1	4	yes	—
32	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
33	Q₈×C₃₇	—	2	8	no	—
34	C₂×C₄×C₃₇	—	2	4	no	—
35	C₂×C₄×C₃₇	—	2	2	no	—
36	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₃₇	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
41	C₈×C₃₇	—	2	4	no	—
42	C₈×C₃₇	—	2	4	no	—
43	C₂²×C₃₇	—	2	1	no	—
44	C₂²×C₃₇	—	2	2	no	—
45	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
46	C₄×C₃₇	—	2	1	no	—
47	C₄×C₃₇	—	2	2	no	—
48	C₄×C₃₇	—	2	8	no	—
49	C₄×C₃₇	#131	1	4	yes	—
50	C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
51	#2986	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×C₃₇	—	4	1	no	—
53	C₂×C₃₇	#519	1	1	yes	characteristic
54	C₂×C₃₇	C₂×#131	1	2	yes	—
55	#533	C₂×C₃₇	1	1	yes	characteristic
56	#446	C₂×C₃₇	1	4	yes	complemented
57	C₈×Q₈	C₂×C₃₇	1	4	yes	complemented
58	#433	C₂×C₃₇	1	4	yes	—
59	#431	C₂×C₃₇	1	2	yes	—
60	C₃₇	#2986	1	1	yes	characteristic, direct product factor, Sylow
61	C₄⋊Q₈	C₄×C₃₇	1	2	yes	—
62	#125	C₄×C₃₇	1	2	yes	—
63	C₂²⋊Q₈	C₄×C₃₇	1	4	yes	—
64	C₄×Q₈	C₂²×C₃₇	1	4	yes	—
65	#117	C₂²×C₃₇	1	2	yes	—
66	C₂×C₄²	C₂²×C₃₇	1	1	yes	characteristic
67	C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
68	C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
69	C₄⋊C₈	C₂²×C₃₇	1	8	yes	—
70	C₂²⋊C₈	C₂²×C₃₇	1	4	yes	—
71	#97	C₂²×C₃₇	1	4	yes	—
72	C₄×C₈	C₂²×C₃₇	1	4	yes	—
73	C₄×C₈	C₂²×C₃₇	1	4	yes	—
74	C₂×Q₈	C₂×C₄×C₃₇	1	4	yes	—
75	C₂²×C₄	C₂×C₄×C₃₇	1	2	yes	—
76	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
77	C₂×C₈	C₂³×C₃₇	1	4	yes	—
78	C₂×C₈	C₂³×C₃₇	1	4	yes	—
79	C₄⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
80	C₄⋊C₄	C₂×C₄×C₃₇	1	8	yes	—
81	C₄⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
82	C₂²⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
83	C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
84	C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
85	C₄²	C₂³×C₃₇	1	2	yes	—
86	C₄²	C₂³×C₃₇	1	4	yes	—
87	C₂³	C₂²×C₄×C₃₇	1	1	yes	characteristic
88	Q₈	—	2	8	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
92	C₂×C₄	C₂²×C₄×C₃₇	1	2	yes	—
93	C₂×C₄	C₂²×C₄×C₃₇	1	8	yes	—
94	C₂×C₄	C₂²×C₄×C₃₇	1	2	yes	—
95	C₂×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
96	C₈	—	2	4	no	—
97	C₈	—	2	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂³×C₄×C₃₇	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	2	no	—
103	C₄	—	2	8	no	—
104	C₄	#131×C₃₇	1	4	yes	—
105	C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
106	C₂	—	4	1	no	—
107	C₂	C₃₇×#519	1	1	yes	characteristic
108	C₂	C₂×#131×C₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₇×#533

C₂

yes

maximal, characteristic

C₃₇×#446

C₂

yes

maximal, complemented

C₈×Q₈×C₃₇

C₂

yes

maximal, complemented

C₃₇×#433

C₂

yes

maximal

C₃₇×#431

C₂

yes

maximal

C₄⋊Q₈×C₃₇

C₄

yes

—

#125×C₃₇

C₄

yes

—

C₂²⋊Q₈×C₃₇

C₄

yes

—

C₄×Q₈×C₃₇

C₂²

yes

—

#117×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₂²

yes

—

C₄⋊C₈×C₃₇

C₂²

yes

—

C₄⋊C₈×C₃₇

C₂²

yes

—

C₂²⋊C₈×C₃₇

C₂²

yes

—

#97×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂³

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₈×C₃₇

—

C₈×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#131

yes

—

C₄×C₃₇

2₋¹⁺⁴

yes

—

#2986

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃₇

—

C₂×C₃₇

#519

yes

characteristic

C₂×C₃₇

C₂×#131

yes

—

#533

C₂×C₃₇

yes

characteristic

#446

C₂×C₃₇

yes

complemented

C₈×Q₈

C₂×C₃₇

yes

complemented

#433

C₂×C₃₇

yes

—

#431

C₂×C₃₇

yes

—

C₃₇

#2986

yes

characteristic, direct product factor, Sylow

C₄⋊Q₈

C₄×C₃₇

yes

—

#125

C₄×C₃₇

yes

—

C₂²⋊Q₈

C₄×C₃₇

yes

—

C₄×Q₈

C₂²×C₃₇

yes

—

#117

C₂²×C₃₇

yes

—

C₂×C₄²

C₂²×C₃₇

yes

characteristic

C₄⋊C₈

C₂²×C₃₇

yes

—

C₄⋊C₈

C₂²×C₃₇

yes

—

C₄⋊C₈

C₂²×C₃₇

yes

—

C₂²⋊C₈

C₂²×C₃₇

yes

—

#97

C₂²×C₃₇

yes

—

C₄×C₈

C₂²×C₃₇

yes

—

C₄×C₈

C₂²×C₃₇

yes

—

C₂×Q₈

C₂×C₄×C₃₇

yes

—

C₂²×C₄

C₂×C₄×C₃₇

yes

—

C₂²×C₄

C₂³×C₃₇

yes

characteristic

C₂×C₈

C₂³×C₃₇

yes

—

C₂×C₈

C₂³×C₃₇

yes

—

C₄⋊C₄

C₂×C₄×C₃₇

yes

—

C₄⋊C₄

C₂×C₄×C₃₇

yes

—

C₄⋊C₄

C₂×C₄×C₃₇

yes

—

C₂²⋊C₄

C₂×C₄×C₃₇

yes

—

C₄²

C₂×C₄×C₃₇

yes

characteristic

C₄²

C₂×C₄×C₃₇

yes

characteristic

C₄²

C₂³×C₃₇

yes

—

C₄²

C₂³×C₃₇

yes

—

C₂³

C₂²×C₄×C₃₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₃₇

yes

characteristic

C₂×C₄

C₂²×C₄×C₃₇

yes

—

C₂×C₄

C₂²×C₄×C₃₇

yes

—

C₂×C₄

C₂²×C₄×C₃₇

yes

—

C₂×C₄

C₂⁴×C₃₇

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

100

C₂²

C₂³×C₄×C₃₇

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#131×C₃₇

yes

—

105

C₄

2₋¹⁺⁴×C₃₇

yes

—

106

C₂

—

107

C₂

C₃₇×#519

yes

characteristic

108

C₂

C₂×#131×C₃₇

yes

—

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₂³×C₄
prime radical	C₂³×C₃₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₃₇

C₂⁴

commutator

C₂²

C₂³×C₄×C₃₇

Frattini, cosocle

C₂×C₄

C₂⁴×C₃₇

socle

C₂²×C₃₇

C₂³×C₄

prime radical

C₂³×C₃₇

C₂²×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₃₇ ◁ G

N	H ≅ G/N	type
2. C₃₇×#446	C₂	semidirect
3. C₈×Q₈×C₃₇	C₂	semidirect
51. #2986	C₃₇	direct (N × H)
56. #446	C₂×C₃₇	semidirect
57. C₈×Q₈	C₂×C₃₇	semidirect
60. C₃₇	#2986	direct (N × H)

H ≅ G/N

type

2. C₃₇×#446

C₂

semidirect

3. C₈×Q₈×C₃₇

C₂

semidirect

51. #2986

C₃₇

direct (N × H)

56. #446

C₂×C₃₇

semidirect

57. C₈×Q₈

C₂×C₃₇

semidirect

60. C₃₇

#2986

direct (N × H)

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

subgroup	index
C₃₇×#533	2
C₃₇×#446	2
C₈×Q₈×C₃₇	2
C₃₇×#433	2
C₃₇×#431	2
#2986	37

subgroup

index

C₃₇×#533

C₃₇×#446

C₈×Q₈×C₃₇

C₃₇×#433

C₃₇×#431

#2986

Subgroups — C₃₇×#2986 #17991804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37×#2986 #17991804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#2986 #17991804