Group #17994677 — subgroups

Subgroups — #17994677

Order 4736.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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×#11924495	C₂	1	1	yes	maximal, characteristic
2	#14557497	C₂	1	1	yes	maximal, characteristic
3	#14557301	C₂	1	4	yes	maximal, complemented
4	C₃₇×#407	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596934	C₂²	1	2	yes	—
6	#11924495	—	2	2	no	—
7	#11924495	—	2	2	no	—
8	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₃₇	C₄	1	2	yes	—
10	C₂×C₄×Dic₃₇	C₄	1	2	yes	—
11	C₂²⋊#64436	—	2	2	no	—
12	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
14	C₂²⋊C₈×C₃₇	—	2	2	no	—
15	C₂⁴×C₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₃₇	—	2	4	no	—
17	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₇	D₄	1	2	yes	—
19	C₂²×C₄×C₃₇	D₄	1	2	yes	—
20	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
21	C₂²×Dic₃₇	C₂×C₄	1	4	yes	—
22	M₁₆×C₃₇	—	2	4	no	—
23	C₂×C₈×C₃₇	—	2	2	no	—
24	C₂²⋊Dic₃₇	—	2	4	no	—
25	C₄⋊Dic₃₇	—	2	2	no	—
26	C₄⋊Dic₃₇	—	2	2	no	—
27	C₄×Dic₃₇	—	2	4	no	—
28	#10596934	—	2	4	no	—
29	C₂×#64436	—	2	2	no	—
30	C₂³×C₃₇	—	2	2	no	—
31	C₂³×C₃₇	—	4	2	no	—
32	C₂³×C₃₇	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₃₇	—	2	2	no	—
35	C₂×C₄×C₃₇	—	2	4	no	—
36	C₂×C₄×C₃₇	—	2	4	no	—
37	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₃₇	#41	1	4	yes	—
41	C₂×Dic₃₇	—	2	4	no	—
42	C₂×Dic₃₇	—	4	4	no	—
43	C₈×C₃₇	—	4	2	no	—
44	#64436	—	4	2	no	—
45	C₂²×C₃₇	—	2	2	no	—
46	C₂²×C₃₇	—	2	2	no	—
47	C₂²×C₃₇	—	2	4	no	—
48	C₂²×C₃₇	—	4	4	no	—
49	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₃₇	#117	1	2	yes	—
51	C₄×C₃₇	—	2	2	no	—
52	C₄×C₃₇	#123	1	2	yes	—
53	C₄×C₃₇	#123	1	2	yes	—
54	Dic₃₇	—	4	4	no	—
55	#1822	—	37	1	no	maximal, Sylow
56	C₂×C₃₇	—	2	2	no	—
57	C₂×C₃₇	—	4	2	no	—
58	C₂×C₃₇	#381	1	1	yes	characteristic
59	C₂×C₃₇	#413	1	2	yes	—
60	C₂×#117	—	37	1	no	—
61	#407	—	37	1	no	—
62	#407	D₃₇	1	1	yes	characteristic
63	#343	—	37	4	no	—
64	C₃₇	#1822	1	1	yes	characteristic, complemented, Sylow
65	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
66	C₂×M₁₆	—	37	2	no	—
67	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
68	#117	—	74	2	no	—
69	#117	—	74	2	no	—
70	C₂×C₄⋊C₄	—	37	2	no	—
71	C₂×C₂²⋊C₄	—	37	2	no	—
72	C₂×C₄²	—	37	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	74	2	no	—
75	C₂⁴	C₄×D₃₇	1	1	yes	characteristic
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	37	4	no	—
78	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
79	C₂²×C₄	D₁₄₈	1	2	yes	—
80	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	74	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	74	2	no	—
86	C₄⋊C₄	—	74	2	no	—
87	C₄⋊C₄	—	74	2	no	—
88	C₂²⋊C₄	—	74	4	no	—
89	C₄²	—	74	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	#10596938	1	2	yes	—
93	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	74	4	no	—
98	C₂×C₄	—	148	4	no	—
99	C₂×C₄	#10596938	1	2	yes	—
100	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
101	C₂×C₄	#10596964	1	4	yes	—
102	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	148	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	4	4	no	—
109	C₂²	C₂×#10596938	1	1	yes	characteristic
110	C₂²	#11924492	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	—	148	4	no	—
113	C₄	#11924517	1	2	yes	—
114	C₄	#11924520	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	4	2	no	—
117	C₂	#14557394	1	1	yes	characteristic
118	C₂	#14557515	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924495

C₂

yes

maximal, characteristic

#14557497

C₂

yes

maximal, characteristic

#14557301

C₂

yes

maximal, complemented

C₃₇×#407

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

#11924495

—

#11924495

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₂²⋊Dic₃₇

C₄

yes

—

C₂×C₄×Dic₃₇

C₄

yes

—

C₂²⋊#64436

—

C₂³×C₄×C₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂²⋊C₈×C₃₇

—

C₂⁴×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

#10596934

—

C₂×#64436

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂²⋊C₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

#117

yes

—

C₄×C₃₇

—

C₄×C₃₇

#123

yes

—

C₄×C₃₇

#123

yes

—

Dic₃₇

—

#1822

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#381

yes

characteristic

C₂×C₃₇

#413

yes

—

C₂×#117

—

#407

—

#407

D₃₇

yes

characteristic

#343

—

C₃₇

#1822

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₃₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂⁴

C₄×D₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₇

yes

characteristic

C₂²×C₄

D₁₄₈

yes

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

#10596938

yes

—

C₂³

C₂×C₄×D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

#10596938

yes

—

100

C₂×C₄

C₂×D₁₄₈

yes

characteristic

101

C₂×C₄

#10596964

yes

—

102

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

103

C₈

—

104

C₈

—

148

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×#10596938

yes

characteristic

110

C₂²

#11924492

yes

—

111

C₄

—

112

C₄

—

148

—

113

C₄

#11924517

yes

—

114

C₄

#11924520

yes

—

115

C₂

—

116

C₂

—

117

C₂

#14557394

yes

characteristic

118

C₂

#14557515

yes

—

	subgroup	G/H
center	C₂²	C₂×#10596938
commutator	C₂³×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#407	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃₇	C₂×C₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#407	D₃₇
hypocenter	C₃₇	#1822
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×#10596938

commutator

C₂³×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#407

C₂

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₃₇

C₂×C₄

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
3. #14557301	C₂	semidirect
64. C₃₇	#1822	semidirect

H ≅ G/N

type

3. #14557301

C₂

semidirect

64. C₃₇

#1822

semidirect

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

subgroup	index
C₂×#11924495	2
#14557497	2
#14557301	2
C₃₇×#407	2
#1822	37

subgroup

index

C₃₇×#407

Subgroups — #17994677

Conjugacy classes of subgroups

Complements

Maximal subgroups