Group #17994646 — subgroups

Subgroups — #17994646

Order 4736.

252 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#11924472	C₂	1	2	yes	maximal, complemented
2	#14557517	C₂	1	2	yes	maximal
3	#14557291	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596934	C₂²	1	4	yes	—
6	#11924472	—	2	4	no	—
7	#11924455	—	2	2	no	—
8	#11924453	—	2	2	no	—
9	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
10	#131×C₃₇	—	2	4	no	—
11	C₂×M₁₆×C₃₇	C₄	1	2	yes	—
12	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₃₇	C₄	1	2	yes	—
14	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
15	#41×C₃₇	—	2	2	no	—
16	#41×C₃₇	—	2	2	no	—
17	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
18	C₂×D₄×C₃₇	D₄	1	1	yes	characteristic
19	C₂×D₄×C₃₇	Q₈	1	2	yes	—
20	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
22	M₁₆×C₃₇	—	2	2	no	—
23	M₁₆×C₃₇	—	2	4	no	—
24	C₂×C₈×C₃₇	—	2	4	no	—
25	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₃₇	D₄	1	4	yes	—
29	#10596934	—	2	4	no	—
30	#10596934	—	2	4	no	—
31	C₂×#64436	—	2	4	no	—
32	C₂³×C₃₇	C₄⋊C₄	1	2	yes	—
33	C₂³×C₃₇	C₂×Q₈	1	1	yes	characteristic
34	Q₈×C₃₇	—	2	2	no	—
35	D₄×C₃₇	—	2	4	no	—
36	D₄×C₃₇	—	2	2	no	—
37	C₂×C₄×C₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₃₇	#41	1	2	yes	—
44	C₈×C₃₇	—	2	2	no	—
45	C₈×C₃₇	C₂²⋊C₄	1	2	yes	—
46	C₈×C₃₇	C₂²⋊C₄	1	2	yes	—
47	#64436	—	4	4	no	—
48	C₂²×C₃₇	—	2	2	no	—
49	C₂²×C₃₇	—	4	1	no	—
50	C₂²×C₃₇	—	4	2	no	—
51	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₃₇	C₂²⋊Q₈	1	2	yes	—
53	C₄×C₃₇	—	2	2	no	—
54	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₃₇	#117	1	1	yes	characteristic
56	C₄×C₃₇	#121	1	2	yes	—
57	#1813	—	37	1	no	maximal, Sylow
58	C₂×C₃₇	—	2	1	no	—
59	C₂×C₃₇	—	2	2	no	—
60	C₂×C₃₇	—	4	2	no	—
61	C₂×C₃₇	#380	1	1	yes	characteristic
62	C₂×#131	D₃₇	1	1	yes	characteristic
63	C₂×#108	—	37	2	no	—
64	#413	—	37	2	no	—
65	#338	—	37	2	no	—
66	C₃₇	#1813	1	1	yes	characteristic, complemented, Sylow
67	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
68	#131	—	2	4	no	—
69	C₂×M₁₆	—	37	4	no	—
70	C₂×M₁₆	Dic₃₇	1	2	yes	—
71	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
72	C₂²×C₈	Dic₃₇	1	2	yes	—
73	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
74	#108	—	74	4	no	—
75	#101	—	74	2	no	—
76	#100	—	74	2	no	—
77	#41	—	2	2	no	—
78	#41	—	2	2	no	—
79	C₂×Q₈	D₁₄₈	1	1	yes	characteristic
80	C₂×D₄	Dic₇₄	1	2	yes	—
81	C₂×D₄	D₁₄₈	1	1	yes	characteristic
82	C₂²×C₄	C₂×Dic₃₇	1	2	yes	—
83	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	4	no	—
86	M₁₆	—	74	4	no	—
87	M₁₆	—	74	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	74	4	no	—
90	C₂×C₈	C₂×Dic₃₇	1	2	yes	—
91	C₂×C₈	C₂×Dic₃₇	1	1	yes	characteristic
92	C₂×C₈	C₂×Dic₃₇	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₃₇	1	4	yes	—
94	C₂³	C₄⋊Dic₃₇	1	2	yes	—
95	C₂³	C₂×Dic₇₄	1	1	yes	characteristic
96	Q₈	—	2	2	no	—
97	D₄	—	2	4	no	—
98	D₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄⋊Dic₃₇	1	2	yes	—
102	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
103	C₂×C₄	#10596962	1	2	yes	—
104	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
106	C₈	—	2	2	no	—
107	C₈	—	148	4	no	—
108	C₈	C₂²⋊Dic₃₇	1	2	yes	—
109	C₈	C₂²⋊Dic₃₇	1	2	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	C₂×C₄⋊Dic₃₇	1	1	yes	characteristic
114	C₂²	#11924515	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	C₂×C₂²⋊Dic₃₇	1	1	yes	characteristic
117	C₄	#11924495	1	1	yes	characteristic
118	C₄	#11924510	1	2	yes	—
119	C₂	—	2	1	no	—
120	C₂	—	2	2	no	—
121	C₂	—	4	2	no	—
122	C₂	#14557393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924472

C₂

yes

maximal, complemented

#14557517

C₂

yes

maximal

#14557291

C₂

yes

maximal, complemented

C₂×#131×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

#11924472

—

#11924455

—

#11924453

—

C₂×#41×C₃₇

C₂²

yes

characteristic

#131×C₃₇

—

C₂×M₁₆×C₃₇

C₄

yes

—

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₄

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

#41×C₃₇

—

#41×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

Q₈

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

D₄

yes

—

#10596934

—

#10596934

—

C₂×#64436

—

C₂³×C₃₇

C₄⋊C₄

yes

—

C₂³×C₃₇

C₂×Q₈

yes

characteristic

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₈×C₃₇

—

C₈×C₃₇

C₂²⋊C₄

yes

—

C₈×C₃₇

C₂²⋊C₄

yes

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₂²⋊C₄

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

—

#1813

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#380

yes

characteristic

C₂×#131

D₃₇

yes

—

—

—

C₃₇

#1813

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₃₇

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

Dic₃₇

yes

—

C₂×M₁₆

C₂×D₃₇

yes

characteristic

C₂²×C₈

Dic₃₇

yes

—

C₂²×C₈

C₂×D₃₇

yes

—

—

—

—

—

C₂×Q₈

D₁₄₈

yes

characteristic

C₂×D₄

Dic₇₄

yes

—

C₂×D₄

D₁₄₈

yes

characteristic

C₂²×C₄

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃₇

yes

—

C₂×C₈

C₂×Dic₃₇

yes

characteristic

C₂×C₈

C₂×Dic₃₇

yes

characteristic

C₂×C₈

C₂²⋊D₃₇

yes

—

C₂³

C₄⋊Dic₃₇

yes

—

C₂³

C₂×Dic₇₄

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₃₇

yes

—

102

C₂×C₄

C₂×D₁₄₈

yes

characteristic

103

C₂×C₄

#10596962

yes

—

104

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

106

C₈

—

107

C₈

—

148

—

108

C₈

C₂²⋊Dic₃₇

yes

—

109

C₈

C₂²⋊Dic₃₇

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₄⋊Dic₃₇

yes

characteristic

114

C₂²

#11924515

yes

—

115

C₄

—

116

C₄

C₂×C₂²⋊Dic₃₇

yes

characteristic

117

C₄

#11924495

yes

characteristic

118

C₄

#11924510

yes

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#14557393

yes

characteristic

	subgroup	G/H
center	C₄	#11924495
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×#131×C₃₇	C₂
socle	C₂×C₃₇	#380
prime radical	C₂×#41×C₃₇	C₂²
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×#131	D₃₇
hypocenter	C₃₇	#1813
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#11924495

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×#131×C₃₇

C₂

socle

C₂×C₃₇

#380

prime radical

C₂×#41×C₃₇

C₂²

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#11924472	C₂	semidirect
3. #14557291	C₂	semidirect
66. C₃₇	#1813	semidirect

H ≅ G/N

type

1. C₂×#11924472

C₂

semidirect

3. #14557291

C₂

semidirect

66. C₃₇

#1813

semidirect

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

subgroup	index
C₂×#11924472	2
#14557517	2
#14557291	2
C₂×#131×C₃₇	2
#1813	37

subgroup

index

C₂×#131×C₃₇

Subgroups — #17994646

Conjugacy classes of subgroups

Complements

Maximal subgroups