Group #18008744 — subgroups

Subgroups — #18008744

Order 4736.

930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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	#14558287	C₂	1	10	yes	maximal, complemented
2	#14558281	C₂	1	5	yes	maximal, complemented
3	#14558089	C₂	1	5	yes	maximal, complemented
4	#14558068	C₂	1	10	yes	maximal, complemented
5	C₃₇×#564	C₂	1	1	yes	maximal, characteristic, complemented
6	#11924535	C₂²	1	10	yes	—
7	#11924531	C₂²	1	20	yes	—
8	#11924528	C₂²	1	5	yes	—
9	#11924524	C₂²	1	10	yes	—
10	#11924521	C₂²	1	20	yes	—
11	#11924516	C₂²	1	10	yes	—
12	#11924514	C₂²	1	20	yes	—
13	#11924511	C₂²	1	10	yes	—
14	#11924501	C₂²	1	10	yes	—
15	#11924498	C₂²	1	10	yes	—
16	#11924494	C₂²	1	10	yes	—
17	C₄²×D₃₇	C₂²	1	5	yes	—
18	#125×C₃₇	C₂²	1	5	yes	—
19	#125×C₃₇	C₂²	1	10	yes	—
20	C₂×D₁₄₈	C₂³	1	10	yes	—
21	C₂×C₄×D₃₇	C₂³	1	5	yes	—
22	C₂×C₄×D₃₇	C₂³	1	10	yes	—
23	C₂×Dic₇₄	C₂³	1	10	yes	—
24	C₄⋊C₄×C₃₇	C₂³	1	10	yes	—
25	C₄⋊C₄×C₃₇	C₂³	1	10	yes	—
26	C₄⋊C₄×C₃₇	C₂³	1	10	yes	—
27	C₄²×C₃₇	C₂³	1	5	yes	—
28	#10596938	C₂³	1	20	yes	—
29	#10596938	C₂³	1	20	yes	—
30	C₄⋊Dic₃₇	C₂³	1	10	yes	—
31	#10596936	C₂³	1	20	yes	—
32	C₄×Dic₃₇	C₂³	1	10	yes	—
33	C₄×Dic₃₇	C₂³	1	5	yes	—
34	C₂²×D₃₇	C₂⁴	1	5	yes	—
35	C₂×C₄×C₃₇	C₂⁴	1	10	yes	—
36	C₂×C₄×C₃₇	C₂⁴	1	5	yes	—
37	C₂×Dic₃₇	C₂⁴	1	10	yes	—
38	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
39	D₁₄₈	—	4	10	no	—
40	C₄×D₃₇	—	2	10	no	—
41	C₄×D₃₇	—	4	10	no	—
42	Dic₇₄	—	4	10	no	—
43	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
44	C₂×D₃₇	—	2	5	no	—
45	C₂×D₃₇	—	2	10	no	—
46	C₄×C₃₇	—	2	5	no	—
47	C₄×C₃₇	—	2	10	no	—
48	Dic₃₇	—	2	10	no	—
49	Dic₃₇	2₋¹⁺⁴	1	2	yes	—
50	#3531	—	37	1	no	maximal, Sylow
51	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
52	C₂×C₃₇	#585	1	2	yes	—
53	D₃₇	—	4	5	no	—
54	#564	D₃₇	1	1	yes	characteristic, complemented
55	#563	—	37	10	no	—
56	#562	—	37	5	no	—
57	#533	—	37	5	no	—
58	#529	—	37	10	no	—
59	C₃₇	#3531	1	1	yes	characteristic, complemented, Sylow
60	C₄⋊Q₈	—	37	10	no	—
61	#126	—	37	20	no	—
62	#125	—	37	5	no	—
63	#125	C₂×D₃₇	1	10	yes	—
64	#125	C₂×D₃₇	1	5	yes	—
65	#124	—	37	10	no	—
66	#123	—	37	20	no	—
67	C₂²⋊Q₈	—	37	10	no	—
68	C₂²⋊Q₈	—	37	20	no	—
69	#121	—	37	10	no	—
70	C₄×Q₈	—	37	10	no	—
71	C₄×D₄	—	37	10	no	—
72	#117	—	37	10	no	—
73	C₂×C₄²	—	37	5	no	—
74	C₂×Q₈	—	37	10	no	—
75	C₂×D₄	—	37	10	no	—
76	C₂²×C₄	—	37	5	no	—
77	C₂²×C₄	—	37	10	no	—
78	C₄⋊C₄	—	37	20	no	—
79	C₄⋊C₄	—	37	10	no	—
80	C₄⋊C₄	C₂²×D₃₇	1	10	yes	—
81	C₄⋊C₄	C₂²×D₃₇	1	10	yes	—
82	C₄⋊C₄	C₂²×D₃₇	1	10	yes	—
83	C₂²⋊C₄	—	37	20	no	—
84	C₂²⋊C₄	—	37	20	no	—
85	C₄²	—	37	10	no	—
86	C₄²	—	37	5	no	—
87	C₄²	C₂²×D₃₇	1	5	yes	—
88	C₂³	—	37	5	no	—
89	Q₈	—	148	10	no	—
90	D₄	—	148	10	no	—
91	C₂×C₄	—	37	1	no	—
92	C₂×C₄	—	37	10	no	—
93	C₂×C₄	—	74	10	no	—
94	C₂×C₄	—	148	10	no	—
95	C₂×C₄	C₂³×D₃₇	1	5	yes	—
96	C₂×C₄	C₂³×D₃₇	1	10	yes	—
97	C₂²	—	74	5	no	—
98	C₂²	—	74	10	no	—
99	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	37	2	no	—
103	C₄	—	74	10	no	—
104	C₂	—	148	5	no	—
105	C₂	2₋¹⁺⁴×D₃₇	1	1	yes	characteristic
106	C₂	#14558370	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558287

C₂

yes

maximal, complemented

#14558281

C₂

yes

maximal, complemented

#14558089

C₂

yes

maximal, complemented

#14558068

C₂

yes

maximal, complemented

C₃₇×#564

C₂

yes

maximal, characteristic, complemented

#11924535

C₂²

yes

—

#11924531

C₂²

yes

—

#11924528

C₂²

yes

—

#11924524

C₂²

yes

—

#11924521

C₂²

yes

—

#11924516

C₂²

yes

—

#11924514

C₂²

yes

—

#11924511

C₂²

yes

—

#11924501

C₂²

yes

—

#11924498

C₂²

yes

—

#11924494

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

—

#125×C₃₇

C₂²

yes

—

#125×C₃₇

C₂²

yes

—

C₂×D₁₄₈

C₂³

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×Dic₇₄

C₂³

yes

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂³

yes

—

#10596938

C₂³

yes

—

#10596938

C₂³

yes

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

C₂³

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂²×D₃₇

C₂⁴

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

—

C₂×Dic₃₇

C₂⁴

yes

—

C₂×Dic₃₇

C₂⁴

yes

characteristic

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₂²×C₃₇

C₂⁵

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

2₋¹⁺⁴

yes

—

#3531

—

maximal, Sylow

C₂×C₃₇

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₃₇

#585

yes

—

D₃₇

—

#564

D₃₇

yes

characteristic, complemented

—

—

—

—

C₃₇

#3531

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#126

—

#125

—

#125

C₂×D₃₇

yes

—

#125

C₂×D₃₇

yes

—

#124

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₃₇

yes

—

C₄⋊C₄

C₂²×D₃₇

yes

—

C₄⋊C₄

C₂²×D₃₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃₇

yes

—

C₂³

—

Q₈

—

148

—

D₄

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

C₂³×D₃₇

yes

—

C₂×C₄

C₂³×D₃₇

yes

—

C₂²

—

C₂²

—

C₂²

C₂⁴×D₃₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₂

—

148

—

105

C₂

2₋¹⁺⁴×D₃₇

yes

characteristic

106

C₂

#14558370

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁵
Fitting	C₃₇×#564	C₂
hypercenter	#564	D₃₇
hypocenter	C₃₇	#3531
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₃₇

commutator, socle, cosocle

C₂²×C₃₇

C₂⁵

Fitting

C₃₇×#564

C₂

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #14558287	C₂	semidirect
2. #14558281	C₂	semidirect
3. #14558089	C₂	semidirect
4. #14558068	C₂	semidirect
5. C₃₇×#564	C₂	semidirect
54. #564	D₃₇	semidirect
59. C₃₇	#3531	semidirect

H ≅ G/N

type

1. #14558287

C₂

semidirect

2. #14558281

C₂

semidirect

3. #14558089

C₂

semidirect

4. #14558068

C₂

semidirect

5. C₃₇×#564

C₂

semidirect

54. #564

D₃₇

semidirect

59. C₃₇

#3531

semidirect

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

subgroup	index
#14558287	2
#14558281	2
#14558089	2
#14558068	2
C₃₇×#564	2
#3531	37

subgroup

index

C₃₇×#564

Subgroups — #18008744

Conjugacy classes of subgroups

Complements

Maximal subgroups