Group #17995590 — subgroups

Subgroups — #17995590

Order 4736.

228 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 97 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	#14557557	C₂	1	1	yes	maximal, characteristic
2	C₄×#10596934	C₂	1	1	yes	maximal, characteristic
3	#14557302	C₂	1	4	yes	maximal
4	C₂×#125×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596934	C₂²	1	2	yes	—
6	C₂×#10596934	C₂²	1	2	yes	—
7	C₄⋊#64436	—	2	2	no	—
8	#11924447	—	2	1	no	—
9	C₄×#64436	—	2	1	no	—
10	#125×C₃₇	—	2	1	no	—
11	#125×C₃₇	—	2	1	no	—
12	#125×C₃₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₃₇	C₄	1	4	yes	—
14	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₃₇	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₃₇	—	2	4	no	—
20	C₄⋊C₄×C₃₇	—	2	4	no	—
21	C₄⋊C₄×C₃₇	—	2	2	no	—
22	C₄⋊C₄×C₃₇	—	2	2	no	—
23	C₄²×C₃₇	D₄	1	1	yes	characteristic
24	C₄²×C₃₇	D₄	1	1	yes	characteristic
25	C₄²×C₃₇	D₄	1	2	yes	—
26	#10596934	—	2	2	no	—
27	#10596934	—	2	4	no	—
28	#10596934	Q₈	1	4	yes	—
29	C₂×#64436	—	2	2	no	—
30	C₂×#64436	—	2	2	no	—
31	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	—	2	4	no	—
33	C₂×C₄×C₃₇	—	4	2	no	—
34	C₂×C₄×C₃₇	—	4	4	no	—
35	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₃₇	#41	1	2	yes	—
41	#64436	—	2	4	no	—
42	#64436	—	4	2	no	—
43	C₂²×C₃₇	—	2	2	no	—
44	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
46	C₄×C₃₇	—	2	2	no	—
47	C₄×C₃₇	—	2	2	no	—
48	C₄×C₃₇	—	4	4	no	—
49	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₃₇	#124	1	1	yes	characteristic
51	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
52	#1998	—	37	1	no	maximal, Sylow
53	C₂×C₃₇	—	2	2	no	—
54	C₂×C₃₇	#391	1	1	yes	characteristic
55	C₂×C₃₇	#413	1	2	yes	—
56	C₂×#125	D₃₇	1	1	yes	characteristic
57	#423	—	37	1	no	—
58	C₄×M₁₆	—	37	1	no	—
59	#343	—	37	4	no	—
60	C₃₇	#1998	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	37	2	no	—
62	C₂×M₁₆	—	37	2	no	—
63	#125	—	2	1	no	—
64	#125	—	2	1	no	—
65	#125	—	2	2	no	—
66	C₂×C₄⋊C₄	Dic₃₇	1	4	yes	—
67	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
68	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
69	C₄⋊C₈	—	74	2	no	—
70	#97	—	74	1	no	—
71	C₄×C₈	—	74	1	no	—
72	C₂²×C₄	C₂×Dic₃₇	1	4	yes	—
73	C₂²×C₄	C₂×Dic₃₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
75	M₁₆	—	37	4	no	—
76	M₁₆	—	74	2	no	—
77	M₁₆	—	74	4	no	—
78	C₂×C₈	—	74	2	no	—
79	C₂×C₈	—	74	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
85	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
86	C₄²	C₂²⋊D₃₇	1	2	yes	—
87	C₂³	C₂²×Dic₃₇	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	C₂²⋊Dic₃₇	1	2	yes	—
92	C₂×C₄	C₂²⋊Dic₃₇	1	2	yes	—
93	C₂×C₄	Q₈×D₃₇	1	2	yes	—
94	C₂×C₄	#10596966	1	2	yes	—
95	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
97	C₈	—	74	4	no	—
98	C₈	—	148	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×C₂²⋊Dic₃₇	1	1	yes	characteristic
101	C₂²	Q₈×Dic₃₇	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	#11924512	1	2	yes	—
106	C₄	#11924525	1	1	yes	characteristic
107	C₄	#11924534	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#14557440	1	1	yes	characteristic
110	C₂	#14557517	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14557557

C₂

yes

maximal, characteristic

C₄×#10596934

C₂

yes

maximal, characteristic

#14557302

C₂

yes

maximal

C₂×#125×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

C₂×#10596934

C₂²

yes

—

C₄⋊#64436

—

#11924447

—

C₄×#64436

—

#125×C₃₇

—

#125×C₃₇

—

#125×C₃₇

—

C₂×C₄⋊C₄×C₃₇

C₄

yes

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

D₄

yes

characteristic

C₄²×C₃₇

D₄

yes

characteristic

C₄²×C₃₇

D₄

yes

—

#10596934

—

#10596934

—

#10596934

Q₈

yes

—

C₂×#64436

—

C₂×#64436

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

#41

yes

—

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

#124

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

#1998

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#391

yes

characteristic

C₂×C₃₇

#413

yes

—

C₂×#125

D₃₇

yes

characteristic

#423

—

C₄×M₁₆

—

#343

—

C₃₇

#1998

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

Dic₃₇

yes

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂²×C₄

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₃₇

yes

characteristic

C₄²

C₂²⋊D₃₇

yes

characteristic

C₄²

C₂²⋊D₃₇

yes

—

C₂³

C₂²×Dic₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₃₇

yes

—

C₂×C₄

C₂²⋊Dic₃₇

yes

—

C₂×C₄

Q₈×D₃₇

yes

—

C₂×C₄

#10596966

yes

—

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

C₈

—

C₈

—

148

—

C₂²

—

100

C₂²

C₂×C₂²⋊Dic₃₇

yes

characteristic

101

C₂²

Q₈×Dic₃₇

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#11924512

yes

—

106

C₄

#11924525

yes

characteristic

107

C₄

#11924534

yes

characteristic

108

C₂

—

109

C₂

#14557440

yes

characteristic

110

C₂

#14557517

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₃₇
commutator, prime radical	C₂³×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×#125×C₃₇	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×#125	D₃₇
hypocenter	C₃₇	#1998
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₃₇

commutator, prime radical

C₂³×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×#125×C₃₇

C₂

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
60. C₃₇	#1998	semidirect

H ≅ G/N

type

60. C₃₇

#1998

semidirect

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

subgroup	index
#14557557	2
C₄×#10596934	2
#14557302	2
C₂×#125×C₃₇	2
#1998	37

subgroup

index

C₂×#125×C₃₇

Subgroups — #17995590

Conjugacy classes of subgroups

Complements

Maximal subgroups