Group #17993737 — subgroups

Subgroups — #17993737

Order 4736.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14558263	C₂	1	1	yes	maximal, characteristic
2	#14557912	C₂	1	1	yes	maximal, characteristic
3	#14557829	C₂	1	2	yes	maximal
4	#14557263	C₂	1	2	yes	maximal
5	C₃₇×#333	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596942	—	2	2	no	—
7	#11924536	C₂²	1	1	yes	characteristic
8	#11924535	—	2	2	no	—
9	#11924534	—	2	2	no	—
10	C₄×Dic₇₄	C₂²	1	2	yes	—
11	#11924501	—	2	2	no	—
12	#11924470	—	2	2	no	—
13	#11924461	—	2	2	no	—
14	C₄×#64436	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×Q₈×C₃₇	D₄	1	2	yes	—
18	C₂×Dic₇₄	—	2	1	no	—
19	C₂×Dic₇₄	—	2	2	no	—
20	C₂×Dic₇₄	D₄	1	2	yes	—
21	C₂×C₈×C₃₇	—	2	2	no	—
22	C₄⋊C₄×C₃₇	—	2	2	no	—
23	C₄²×C₃₇	C₂³	1	1	yes	characteristic
24	#10596942	—	4	4	no	—
25	C₄⋊Dic₃₇	—	2	2	no	—
26	C₄⋊Dic₃₇	D₄	1	2	yes	—
27	#10596936	—	2	4	no	—
28	#10596936	—	2	2	no	—
29	C₄×Dic₃₇	—	2	2	no	—
30	C₄×Dic₃₇	—	2	2	no	—
31	C₂×#64436	—	2	1	no	—
32	Q₈×C₃₇	—	4	2	no	—
33	C₂×C₄×C₃₇	—	2	2	no	—
34	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
36	C₂×Dic₃₇	—	2	2	no	—
37	C₂×Dic₃₇	—	2	2	no	—
38	C₂×Dic₃₇	—	4	1	no	—
39	Dic₇₄	—	2	2	no	—
40	Dic₇₄	—	4	2	no	—
41	Dic₇₄	—	8	2	no	—
42	Dic₇₄	Q₁₆	1	4	yes	complemented
43	C₈×C₃₇	—	4	2	no	—
44	#64436	—	2	2	no	—
45	C₂²×C₃₇	#120	1	1	yes	characteristic
46	C₄×C₃₇	—	2	1	no	—
47	C₄×C₃₇	—	4	2	no	—
48	C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
49	C₄×C₃₇	#137	1	2	yes	—
50	Dic₃₇	—	2	4	no	—
51	Dic₃₇	—	4	2	no	—
52	Dic₃₇	—	8	1	no	—
53	#1633	—	37	1	no	maximal, Sylow
54	C₂×C₃₇	#451	1	2	yes	—
55	C₂×C₃₇	#453	1	1	yes	characteristic
56	Q₈²	—	37	1	no	—
57	#494	—	37	1	no	—
58	#477	—	37	2	no	—
59	#333	D₃₇	1	1	yes	characteristic
60	Q₈⋊C₈	—	37	2	no	—
61	C₃₇	#1633	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₁₆	—	74	2	no	—
63	C₄⋊Q₈	—	37	1	no	—
64	C₄⋊Q₈	—	74	2	no	—
65	C₄⋊Q₈	—	74	2	no	—
66	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
67	C₄×Q₈	—	37	2	no	—
68	C₄×Q₈	—	74	2	no	—
69	#107	—	74	2	no	—
70	C₄⋊C₈	C₂×D₃₇	1	2	yes	—
71	#103	—	74	2	no	—
72	C₄×C₈	—	37	1	no	—
73	C₂×Q₈	—	37	2	no	—
74	C₂×Q₈	—	74	2	no	—
75	C₂×Q₈	—	74	1	no	—
76	C₂×Q₈	D₁₄₈	1	2	yes	—
77	Q₁₆	—	148	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	74	1	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	37	2	no	—
82	C₄⋊C₄	—	74	4	no	—
83	C₄⋊C₄	—	74	2	no	—
84	C₄⋊C₄	—	74	2	no	—
85	C₄²	—	74	2	no	—
86	C₄²	—	74	2	no	—
87	C₄²	C₂²×D₃₇	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	37	4	no	—
90	Q₈	—	296	2	no	—
91	Q₈	—	74	2	no	—
92	Q₈	—	148	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	74	2	no	—
95	C₂×C₄	—	74	2	no	—
96	C₂×C₄	—	148	1	no	—
97	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
98	C₂×C₄	D₄×D₃₇	1	2	yes	—
99	C₈	—	4	2	no	—
100	C₈	—	74	2	no	—
101	C₂²	#11924506	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	296	1	no	—
105	C₄	—	74	4	no	—
106	C₄	—	148	2	no	—
107	C₄	Q₁₆×D₃₇	1	2	yes	—
108	C₄	#11924565	1	2	yes	—
109	C₂	#14557690	1	2	yes	—
110	C₂	#14557703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558263

C₂

yes

maximal, characteristic

#14557912

C₂

yes

maximal, characteristic

#14557829

C₂

yes

maximal

#14557263

C₂

yes

maximal

C₃₇×#333

C₂

yes

maximal, characteristic

C₂×#10596942

—

#11924536

C₂²

yes

characteristic

#11924535

—

#11924534

—

C₄×Dic₇₄

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₂²

yes

—

C₂×Q₈×C₃₇

D₄

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

D₄

yes

—

C₂×C₈×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596942

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

D₄

yes

—

#10596936

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₂×#64436

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

Q₁₆

yes

complemented

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

#120

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×Q₁₆

yes

—

C₄×C₃₇

#137

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

#1633

—

maximal, Sylow

C₂×C₃₇

#451

yes

—

C₂×C₃₇

#453

yes

characteristic

Q₈²

—

#494

—

#477

—

#333

D₃₇

yes

characteristic

Q₈⋊C₈

—

C₃₇

#1633

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#107

—

C₄⋊C₈

C₂×D₃₇

yes

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₄₈

yes

—

Q₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

296

—

Q₈

—

Q₈

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

C₂×D₁₄₈

yes

characteristic

C₂×C₄

D₄×D₃₇

yes

—

C₈

—

100

C₈

—

101

C₂²

#11924506

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

296

—

105

C₄

—

106

C₄

—

148

—

107

C₄

Q₁₆×D₃₇

yes

—

108

C₄

#11924565

yes

—

109

C₂

#14557690

yes

—

110

C₂

#14557703

yes

characteristic

	subgroup	G/H
center	C₂²	#11924506
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#333	C₂
socle, prime radical	C₂²×C₃₇	#120
hypercenter	#333	D₃₇
hypocenter	C₃₇	#1633
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#11924506

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#333

C₂

C₂²×C₃₇

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
42. Dic₇₄	Q₁₆	semidirect
61. C₃₇	#1633	semidirect

H ≅ G/N

type

42. Dic₇₄

Q₁₆

semidirect

61. C₃₇

#1633

semidirect

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

subgroup	index
#14558263	2
#14557912	2
#14557829	2
#14557263	2
C₃₇×#333	2
#1633	37

subgroup

index

C₃₇×#333

Subgroups — #17993737

Conjugacy classes of subgroups

Complements

Maximal subgroups