Group #17996758 — subgroups

Subgroups — #17996758

Order 4736.

124 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 49 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	#14557934	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557361	C₂	1	1	yes	maximal, characteristic
4	#14557358	C₂	1	1	yes	maximal, characteristic
5	#14557351	C₂	1	1	yes	maximal, characteristic
6	#14557337	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#358	C₂	1	1	yes	maximal, characteristic
8	#11924527	—	2	1	no	—
9	Q₈×Dic₃₇	—	2	1	no	—
10	C₂×#10596925	C₂²	1	1	yes	characteristic
11	#11924470	C₂²	1	1	yes	characteristic
12	#11924470	C₂²	1	1	yes	characteristic
13	#11924467	—	2	1	no	—
14	#11924460	—	2	1	no	—
15	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
18	#107×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₃₇	—	2	1	no	—
20	Q₁₆×C₃₇	—	2	1	no	—
21	Q₁₆×C₃₇	Q₈	1	2	yes	—
22	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₃₇	—	2	1	no	—
24	C₄⋊Dic₃₇	—	2	1	no	—
25	C₄⋊Dic₃₇	—	2	1	no	—
26	C₄⋊Dic₃₇	—	4	1	no	—
27	C₄⋊Dic₃₇	—	4	1	no	—
28	#10596936	—	4	1	no	—
29	C₄×Dic₃₇	—	4	1	no	—
30	C₄×Dic₃₇	D₄	1	1	yes	characteristic
31	C₂×#64436	D₄	1	1	yes	characteristic
32	C₁₆×C₃₇	—	2	1	no	—
33	#10596925	—	2	1	no	—
34	Q₈×C₃₇	—	2	2	no	—
35	Q₈×C₃₇	—	4	1	no	—
36	C₂×C₄×C₃₇	—	4	1	no	—
37	C₂×C₄×C₃₇	—	4	1	no	—
38	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₃₇	—	4	1	no	—
40	C₂×Dic₃₇	—	4	1	no	—
41	C₂×Dic₃₇	D₈	1	1	yes	characteristic
42	C₂×Dic₃₇	D₈	1	1	yes	characteristic
43	C₈×C₃₇	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₃₇	#41	1	1	yes	characteristic
45	#64436	—	2	1	no	—
46	C₂²×C₃₇	C₂×D₈	1	1	yes	characteristic
47	C₄×C₃₇	—	4	2	no	—
48	C₄×C₃₇	—	8	1	no	—
49	C₄×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₄×C₃₇	#137	1	1	yes	characteristic
51	Dic₃₇	—	2	1	no	—
52	Dic₃₇	—	2	1	no	—
53	Dic₃₇	—	8	1	no	—
54	Dic₃₇	—	8	1	no	—
55	#2229	—	37	1	no	maximal, Sylow
56	C₂×C₃₇	#474	1	1	yes	characteristic
57	C₂×C₃₇	#508	1	1	yes	characteristic
58	C₂×C₃₇	#510	1	1	yes	characteristic
59	#499	—	37	1	no	—
60	C₄×Q₁₆	—	37	1	no	—
61	#367	—	37	1	no	—
62	#366	—	37	1	no	—
63	C₄⋊C₁₆	—	37	1	no	—
64	#358	—	37	1	no	—
65	#358	D₃₇	1	1	yes	characteristic
66	C₃₇	#2229	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₁₆	C₂×D₃₇	1	1	yes	characteristic
68	#125	—	74	1	no	—
69	C₄×Q₈	—	74	1	no	—
70	C₂×C₁₆	—	37	1	no	—
71	C₂×C₁₆	C₂×D₃₇	1	1	yes	characteristic
72	#107	—	37	1	no	—
73	#107	—	37	1	no	—
74	#107	C₂×D₃₇	1	1	yes	characteristic
75	#106	—	74	1	no	—
76	#103	—	74	1	no	—
77	C₄×C₈	—	37	1	no	—
78	C₂×Q₈	—	2	1	no	—
79	Q₁₆	—	2	1	no	—
80	Q₁₆	Dic₇₄	1	2	yes	—
81	C₂×C₈	—	37	1	no	—
82	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	74	1	no	—
85	C₄⋊C₄	—	74	1	no	—
86	C₄⋊C₄	—	148	1	no	—
87	C₄⋊C₄	—	148	1	no	—
88	C₄⋊C₄	—	148	1	no	—
89	C₄²	—	37	1	no	—
90	C₄²	—	148	1	no	—
91	C₁₆	—	2	1	no	—
92	C₁₆	—	74	1	no	—
93	Q₈	—	2	2	no	—
94	Q₈	—	4	1	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	37	1	no	—
98	C₂×C₄	—	37	1	no	—
99	C₂×C₄	—	148	1	no	—
100	C₂×C₄	—	148	1	no	—
101	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
102	C₈	—	74	1	no	—
103	C₈	C₂×Dic₇₄	1	1	yes	characteristic
104	C₈	#10596964	1	1	yes	characteristic
105	C₂²	D₈×D₃₇	1	1	yes	characteristic
106	C₄	—	4	2	no	—
107	C₄	—	8	1	no	—
108	C₄	—	296	1	no	—
109	C₄	—	296	1	no	—
110	C₄	—	74	1	no	—
111	C₄	—	74	1	no	—
112	C₄	#11924513	1	1	yes	characteristic
113	C₄	#11924565	1	1	yes	characteristic
114	C₂	#14557811	1	1	yes	characteristic
115	C₂	#14557964	1	1	yes	characteristic
116	C₂	#14557975	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557934

C₂

yes

maximal, characteristic

Q₁₆×Dic₃₇

C₂

yes

maximal, characteristic

#14557361

C₂

yes

maximal, characteristic

#14557358

C₂

yes

maximal, characteristic

#14557351

C₂

yes

maximal, characteristic

#14557337

C₂

yes

maximal, characteristic

C₃₇×#358

C₂

yes

maximal, characteristic

#11924527

—

Q₈×Dic₃₇

—

C₂×#10596925

C₂²

yes

characteristic

#11924470

C₂²

yes

characteristic

#11924470

C₂²

yes

characteristic

#11924467

—

#11924460

—

C₈×Dic₃₇

C₂²

yes

characteristic

C₂×Q₁₆×C₃₇

C₂²

yes

characteristic

C₂×C₁₆×C₃₇

C₂²

yes

characteristic

#107×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

Q₁₆×C₃₇

—

Q₁₆×C₃₇

Q₈

yes

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

characteristic

C₂×#64436

D₄

yes

characteristic

C₁₆×C₃₇

—

#10596925

—

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

D₈

yes

characteristic

C₂×Dic₃₇

D₈

yes

characteristic

C₈×C₃₇

C₂×Q₈

yes

C₈×C₃₇

yes

—

C₂²×C₃₇

C₂×D₈

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

characteristic

C₄×C₃₇

#137

yes

characteristic

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₄×Q₁₆

—

#367

—

#366

—

C₄⋊C₁₆

—

#358

—

#358

D₃₇

yes

characteristic

C₃₇

#2229

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₃₇

yes

characteristic

#125

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃₇

yes

—

—

C₂×D₃₇

yes

characteristic

#106

—

#103

—

C₄×C₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

Dic₇₄

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

148

—

C₄⋊C₄

—

148

—

C₄⋊C₄

—

148

—

C₄²

—

C₄²

—

148

—

C₁₆

—

C₁₆

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

100

C₂×C₄

—

148

—

101

C₂×C₄

D₄×D₃₇

yes

characteristic

102

C₈

—

103

C₈

C₂×Dic₇₄

yes

characteristic

104

C₈

#10596964

yes

characteristic

105

C₂²

D₈×D₃₇

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

296

—

109

C₄

—

296

—

110

C₄

—

111

C₄

—

112

C₄

#11924513

yes

characteristic

113

C₄

#11924565

yes

characteristic

114

C₂

#14557811

yes

characteristic

115

C₂

#14557964

yes

characteristic

116

C₂

#14557975

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₃₇
commutator, cosocle	C₂×C₈×C₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₃₇
Fitting	C₃₇×#358	C₂
socle, prime radical	C₂²×C₃₇	C₂×D₈
hypercenter	#358	D₃₇
hypocenter	C₃₇	#2229
LCS	G ▷ C₂×C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₃₇

commutator, cosocle

C₂×C₈×C₃₇

C₂³

Frattini

C₂×C₈

C₂²×D₃₇

Fitting

C₃₇×#358

C₂

socle, prime radical

C₂²×C₃₇

C₂×D₈

D₃₇

C₃₇

G ▷ C₂×C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
66. C₃₇	#2229	semidirect

H ≅ G/N

type

66. C₃₇

#2229

semidirect

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

subgroup	index
#14557934	2
Q₁₆×Dic₃₇	2
#14557361	2
#14557358	2
#14557351	2
#14557337	2
C₃₇×#358	2
#2229	37

subgroup

index

Q₁₆×Dic₃₇

C₃₇×#358

Subgroups — #17996758

Conjugacy classes of subgroups

Complements

Maximal subgroups