Group #17996503 — subgroups

Subgroups — #17996503

Order 4736.

126 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₈×Dic₇₄	C₂	1	1	yes	maximal, characteristic
2	#14557351	C₂	1	1	yes	maximal, characteristic
3	#14557351	C₂	1	1	yes	maximal, characteristic
4	#14557309	C₂	1	1	yes	maximal, characteristic
5	#14557308	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic
7	C₃₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₇₄	C₄	1	1	yes	characteristic
9	C₂×#10596925	C₂²	1	1	yes	characteristic
10	C₂×#10596925	C₂²	1	1	yes	characteristic
11	C₄⋊#64436	C₄	1	1	yes	characteristic
12	#11924465	C₄	1	1	yes	characteristic
13	#11924465	C₄	1	1	yes	characteristic
14	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
15	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₇₄	C₈	1	1	yes	characteristic
20	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
22	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄⋊Dic₃₇	C₈	1	1	yes	characteristic
24	#10596936	C₈	1	1	yes	characteristic
25	#10596936	C₈	1	1	yes	characteristic
26	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×#64436	C₂×C₄	1	1	yes	characteristic
29	C₂×#64436	C₂×C₄	1	1	yes	characteristic
30	C₁₆×C₃₇	—	2	1	no	—
31	C₁₆×C₃₇	—	2	1	no	—
32	#10596925	—	2	1	no	—
33	#10596925	Q₈	1	2	yes	complemented
34	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₃₇	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₃₇	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₃₇	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₃₇	C₂×C₈	1	1	yes	characteristic
41	Dic₇₄	—	2	2	no	—
42	C₈×C₃₇	—	2	1	no	—
43	C₈×C₃₇	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₃₇	#41	1	1	yes	characteristic
45	#64436	—	2	1	no	—
46	#64436	—	2	1	no	—
47	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₃₇	—	2	1	no	—
49	C₄×C₃₇	—	2	1	no	—
50	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₃₇	#131	1	1	yes	characteristic
52	Dic₃₇	—	2	1	no	—
53	Dic₃₇	—	2	1	no	—
54	Dic₃₇	—	2	1	no	—
55	Dic₃₇	M₃₂	1	2	yes	—
56	#2183	—	37	1	no	maximal, Sylow
57	C₂×C₃₇	C₈×Q₈	1	1	yes	characteristic
58	C₂×C₃₇	C₂×M₃₂	1	1	yes	characteristic
59	C₂×C₃₇	#504	1	1	yes	characteristic
60	C₈×Q₈	—	37	1	no	—
61	C₄⋊C₁₆	—	37	1	no	—
62	C₄⋊C₁₆	—	37	1	no	—
63	C₄⋊C₁₆	D₃₇	1	1	yes	characteristic
64	#346	—	37	1	no	—
65	#346	—	37	1	no	—
66	C₄×C₁₆	—	37	1	no	—
67	C₃₇	#2183	1	1	yes	characteristic, complemented, Sylow
68	C₄×Q₈	—	37	1	no	—
69	C₂×C₁₆	—	37	1	no	—
70	C₂×C₁₆	—	37	1	no	—
71	C₂×C₁₆	C₂×D₃₇	1	1	yes	characteristic
72	C₂×C₁₆	C₂×D₃₇	1	1	yes	characteristic
73	C₄⋊C₈	—	37	1	no	—
74	C₄⋊C₈	—	37	1	no	—
75	C₄⋊C₈	—	37	1	no	—
76	C₄×C₈	—	37	1	no	—
77	C₄×C₈	—	37	1	no	—
78	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
79	C₂×Q₈	—	37	1	no	—
80	C₂×C₈	—	37	1	no	—
81	C₂×C₈	—	37	1	no	—
82	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
83	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
84	C₄⋊C₄	—	37	1	no	—
85	C₄⋊C₄	—	37	1	no	—
86	C₄⋊C₄	—	37	1	no	—
87	C₄²	—	37	1	no	—
88	C₄²	—	37	1	no	—
89	C₄²	C₄×D₃₇	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	—	2	1	no	—
92	C₁₆	—	37	2	no	—
93	C₁₆	—	74	1	no	—
94	Q₈	—	74	2	no	—
95	C₂×C₄	—	37	1	no	—
96	C₂×C₄	—	37	1	no	—
97	C₂×C₄	—	37	1	no	—
98	C₂×C₄	—	37	1	no	—
99	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
100	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	74	1	no	—
104	C₈	—	74	1	no	—
105	C₈	#10596964	1	1	yes	characteristic
106	C₈	Q₈×D₃₇	1	1	yes	characteristic
107	C₂²	C₂×C₈×D₃₇	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	37	2	no	—
111	C₄	—	74	1	no	—
112	C₄	—	74	1	no	—
113	C₄	—	74	1	no	—
114	C₄	#11924501	1	1	yes	characteristic
115	C₄	#11924544	1	1	yes	characteristic
116	C₂	#14557657	1	1	yes	characteristic
117	C₂	M₃₂×D₃₇	1	1	yes	characteristic
118	C₂	#14557949	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₇₄

C₂

yes

maximal, characteristic

#14557351

C₂

yes

maximal, characteristic

#14557351

C₂

yes

maximal, characteristic

#14557309

C₂

yes

maximal, characteristic

#14557308

C₂

yes

maximal, characteristic

C₁₆×Dic₃₇

C₂

yes

maximal, characteristic

C₃₇×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₄×Dic₇₄

C₄

yes

characteristic

C₂×#10596925

C₂²

yes

characteristic

C₂×#10596925

C₂²

yes

characteristic

C₄⋊#64436

C₄

yes

characteristic

#11924465

C₄

yes

characteristic

#11924465

C₄

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

C₂×C₁₆×C₃₇

C₂²

yes

characteristic

C₂×C₁₆×C₃₇

C₂²

yes

characteristic

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Dic₇₄

C₈

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₃₇

C₈

yes

characteristic

#10596936

C₈

yes

characteristic

#10596936

C₈

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₁₆×C₃₇

—

C₁₆×C₃₇

—

#10596925

—

#10596925

Q₈

yes

complemented

C₂×C₄×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂×C₈

yes

characteristic

C₂×Dic₃₇

C₂×C₈

yes

characteristic

C₂×Dic₃₇

C₂×C₈

yes

characteristic

C₂×Dic₃₇

C₂×C₈

yes

characteristic

Dic₇₄

—

C₈×C₃₇

—

C₈×C₃₇

C₂×Q₈

yes

C₈×C₃₇

yes

—

—

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×Q₈

yes

characteristic

C₄×C₃₇

#131

yes

characteristic

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

M₃₂

yes

—

#2183

—

maximal, Sylow

C₂×C₃₇

C₈×Q₈

yes

characteristic

C₂×C₃₇

C₂×M₃₂

yes

characteristic

C₂×C₃₇

#504

yes

characteristic

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₃₇

yes

characteristic

#346

—

#346

—

C₄×C₁₆

—

C₃₇

#2183

yes

characteristic, complemented, Sylow

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃₇

yes

characteristic

C₂×C₁₆

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₇

yes

characteristic

C₂×C₈

C₂²×D₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄×D₃₇

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₃₇

yes

characteristic

100

C₂×C₄

C₈×D₃₇

yes

characteristic

101

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

#10596964

yes

characteristic

106

C₈

Q₈×D₃₇

yes

characteristic

107

C₂²

C₂×C₈×D₃₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#11924501

yes

characteristic

115

C₄

#11924544

yes

characteristic

116

C₂

#14557657

yes

characteristic

117

C₂

M₃₂×D₃₇

yes

characteristic

118

C₂

#14557949

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃₇
commutator, socle, prime radical	C₂²×C₃₇	C₂²×C₈
Fitting	C₃₇×C₄⋊C₁₆	C₂
cosocle	C₂×C₈×C₃₇	C₂³
hypercenter	C₄⋊C₁₆	D₃₇
hypocenter	C₃₇	#2183
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₃₇

commutator, socle, prime radical

C₂²×C₃₇

C₂²×C₈

Fitting

C₃₇×C₄⋊C₁₆

C₂

cosocle

C₂×C₈×C₃₇

C₂³

hypercenter

C₄⋊C₁₆

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
33. #10596925	Q₈	semidirect
67. C₃₇	#2183	semidirect

H ≅ G/N

type

33. #10596925

Q₈

semidirect

67. C₃₇

#2183

semidirect

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

subgroup	index
C₈×Dic₇₄	2
#14557351	2
#14557309	2
#14557308	2
C₁₆×Dic₃₇	2
C₃₇×C₄⋊C₁₆	2
#2183	37

subgroup

index

C₈×Dic₇₄

#14557351

#14557309

#14557308

C₁₆×Dic₃₇

C₃₇×C₄⋊C₁₆

#2183

Subgroups — #17996503

Conjugacy classes of subgroups

Complements

Maximal subgroups