Group #18010146 — subgroups

Subgroups — #18010146

Order 4736.

180 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 82 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	#14558552	C₂	1	2	yes	maximal
2	#14558396	C₂	1	2	yes	maximal, complemented
3	#14558395	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#64438	C₂²	1	4	yes	—
6	#11924593	—	2	2	no	—
7	C₄×#64438	—	2	2	no	—
8	C₂×C₈×D₃₇	C₄	1	1	yes	characteristic
9	C₂×C₈×D₃₇	C₄	1	1	yes	characteristic
10	C₄²×D₃₇	C₂²	1	1	yes	characteristic
11	C₄×#64436	C₄	1	1	yes	characteristic
12	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
13	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₃₇	C₄	1	1	yes	characteristic
15	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
17	C₂×C₄×D₃₇	Q₈	1	1	yes	characteristic
18	C₂×#64438	—	2	4	no	—
19	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₃₇	D₄	1	1	yes	characteristic
23	C₄×Dic₃₇	Q₈	1	1	yes	characteristic
24	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
25	C₂×#64436	C₂×C₄	1	1	yes	characteristic
26	C₂×#64436	C₂×C₄	1	1	yes	characteristic
27	C₈×D₃₇	—	2	2	no	—
28	C₈×D₃₇	C₈	1	4	yes	complemented
29	C₂²×D₃₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₃₇	C₂×Q₈	1	1	yes	characteristic
36	C₄×D₃₇	—	2	2	no	—
37	C₄×D₃₇	—	2	2	no	—
38	C₄×D₃₇	C₂×C₈	1	2	yes	—
39	C₄×D₃₇	M₁₆	1	2	yes	—
40	#64438	—	4	4	no	—
41	C₈×C₃₇	—	2	1	no	—
42	C₈×C₃₇	C₂×C₈	1	2	yes	—
43	#64436	—	2	1	no	—
44	#64436	C₂×C₈	1	2	yes	—
45	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×D₃₇	—	2	1	no	—
47	C₂×D₃₇	—	2	1	no	—
48	C₂×D₃₇	C₄⋊C₈	1	2	yes	—
49	C₄×C₃₇	—	2	1	no	—
50	C₄×C₃₇	#108	1	2	yes	—
51	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
52	C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
53	Dic₃₇	—	2	1	no	—
54	Dic₃₇	C₄⋊C₈	1	2	yes	—
55	Dic₃₇	#135	1	2	yes	—
56	#1564	—	37	1	no	maximal, Sylow
57	C₂×C₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₃₇	#427	1	1	yes	characteristic
59	C₂×C₃₇	C₂×#108	1	1	yes	characteristic
60	D₃₇	—	2	2	no	—
61	#431	—	37	2	no	—
62	C₂×C₄×C₈	—	37	1	no	—
63	#335	—	37	2	no	—
64	#334	—	37	2	no	—
65	C₃₇	#1564	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₈	—	37	1	no	—
67	C₂²×C₈	—	37	1	no	—
68	C₂×C₄²	—	37	1	no	—
69	C₄⋊C₈	—	37	4	no	—
70	C₂²⋊C₈	—	74	2	no	—
71	C₄×C₈	—	37	1	no	—
72	C₄×C₈	—	37	1	no	—
73	C₄×C₈	—	37	1	no	—
74	C₄×C₈	—	74	2	no	—
75	C₄×C₈	#3866	1	1	yes	characteristic
76	C₂²×C₄	—	37	1	no	—
77	C₂²×C₄	—	37	1	no	—
78	C₂²×C₄	—	37	1	no	—
79	C₂×C₈	—	37	1	no	—
80	C₂×C₈	—	37	4	no	—
81	C₂×C₈	—	37	1	no	—
82	C₂×C₈	—	74	2	no	—
83	C₂×C₈	—	74	4	no	—
84	C₂×C₈	C₂×#3866	1	1	yes	characteristic
85	C₂×C₈	C₂×#3866	1	1	yes	characteristic
86	C₄²	—	37	1	no	—
87	C₄²	—	37	1	no	—
88	C₄²	—	37	1	no	—
89	C₄²	C₂×#3866	1	1	yes	characteristic
90	C₂³	—	37	1	no	—
91	C₂×C₄	—	37	1	no	—
92	C₂×C₄	—	37	1	no	—
93	C₂×C₄	—	37	2	no	—
94	C₂×C₄	—	37	2	no	—
95	C₂×C₄	—	37	1	no	—
96	C₂×C₄	—	74	2	no	—
97	C₂×C₄	—	74	2	no	—
98	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
99	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
100	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	37	2	no	—
103	C₈	—	74	1	no	—
104	C₈	—	148	4	no	—
105	C₈	D₃₇⋊C₈	1	2	yes	—
106	C₂²	—	37	2	no	—
107	C₂²	—	74	1	no	—
108	C₂²	—	74	1	no	—
109	C₂²	C₂×C₄⋊#3866	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	37	2	no	—
112	C₄	—	37	2	no	—
113	C₄	—	74	1	no	—
114	C₄	#11924607	1	2	yes	—
115	C₄	C₂×D₃₇⋊C₈	1	1	yes	characteristic
116	C₄	D₃₇⋊M₁₆	1	1	yes	characteristic
117	C₂	—	74	2	no	—
118	C₂	D₃₇⋊(C₄⋊C₈)	1	1	yes	characteristic
119	C₂	#14558543	1	1	yes	characteristic
120	C₂	C₂×#11924607	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558552

C₂

yes

maximal

#14558396

C₂

yes

maximal, complemented

#14558395

C₂

yes

maximal, complemented

C₄×C₈×D₃₇

C₂

yes

maximal, characteristic

C₄⋊#64438

C₂²

yes

—

#11924593

—

C₄×#64438

—

C₂×C₈×D₃₇

C₄

yes

characteristic

C₂×C₈×D₃₇

C₄

yes

characteristic

C₄²×D₃₇

C₂²

yes

characteristic

C₄×#64436

C₄

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

C₄×C₈×C₃₇

C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

characteristic

C₂×C₄×D₃₇

Q₈

yes

characteristic

C₂×#64438

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

Q₈

yes

characteristic

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₈×D₃₇

—

C₈×D₃₇

C₈

yes

complemented

C₂²×D₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₄⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×Q₈

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂×C₈

yes

—

C₄×D₃₇

M₁₆

yes

—

#64438

—

C₈×C₃₇

—

C₈×C₃₇

C₂×C₈

yes

—

#64436

—

#64436

C₂×C₈

yes

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₄⋊C₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

#108

yes

—

C₄×C₃₇

C₂²×C₈

yes

characteristic

C₄×C₃₇

C₂×M₁₆

yes

characteristic

Dic₃₇

—

Dic₃₇

C₄⋊C₈

yes

—

Dic₃₇

#135

yes

—

#1564

—

maximal, Sylow

C₂×C₃₇

C₂×C₄⋊C₈

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

#431

—

C₂×C₄×C₈

—

#335

—

#334

—

C₃₇

#1564

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3866

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₂²×#3866

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

148

—

105

C₈

D₃₇⋊C₈

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₄⋊#3866

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#11924607

yes

—

115

C₄

C₂×D₃₇⋊C₈

yes

characteristic

116

C₄

D₃₇⋊M₁₆

yes

characteristic

117

C₂

—

118

C₂

D₃₇⋊(C₄⋊C₈)

yes

characteristic

119

C₂

#14558543

yes

characteristic

120

C₂

C₂×#11924607

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#3866
commutator	C₄×C₃₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄×C₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₃₇	C₄⋊C₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₄×C₈	#3866
hypocenter	C₃₇	#1564
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

C₄⋊#3866

commutator

C₄×C₃₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₄×C₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₄⋊C₄

prime radical

C₂²×D₃₇

C₄⋊C₄

cosocle

C₄×Dic₃₇

C₂³

C₄×C₈

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
2. #14558396	C₂	semidirect
3. #14558395	C₂	semidirect
28. C₈×D₃₇	C₈	semidirect
65. C₃₇	#1564	semidirect

H ≅ G/N

type

2. #14558396

C₂

semidirect

3. #14558395

C₂

semidirect

28. C₈×D₃₇

C₈

semidirect

65. C₃₇

#1564

semidirect

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

subgroup	index
#14558552	2
#14558396	2
#14558395	2
C₄×C₈×D₃₇	2
#1564	37

subgroup

index

C₄×C₈×D₃₇

Subgroups — #18010146

Conjugacy classes of subgroups

Complements

Maximal subgroups