Group #17994499 — subgroups

Subgroups — C₄×#11924471 #17994499

Order 4736.

228 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 121 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×#11924471	C₂	1	2	yes	maximal
2	C₄×#10596934	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#64436	C₂	1	1	yes	maximal, characteristic
4	#14557297	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	2	yes	—
7	C₂²×#64436	C₂²	1	1	yes	characteristic
8	C₂²×#64436	C₂²	1	1	yes	characteristic
9	#11924471	C₄	1	8	yes	direct product factor
10	#11924447	—	2	1	no	—
11	C₄×#64436	—	2	1	no	—
12	C₄×#64436	C₄	1	4	yes	—
13	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	#97×C₃₇	—	2	1	no	—
16	C₄×C₈×C₃₇	—	2	1	no	—
17	C₂²×C₄×C₃₇	Q₈	1	2	yes	—
18	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
19	M₁₆×C₃₇	—	2	2	no	—
20	M₁₆×C₃₇	C₂×C₄	1	4	yes	—
21	C₂×C₈×C₃₇	—	2	2	no	—
22	C₄²×C₃₇	C₂×C₄	1	2	yes	—
23	C₄²×C₃₇	D₄	1	1	yes	characteristic
24	C₄²×C₃₇	D₄	1	1	yes	characteristic
25	#10596934	—	2	2	no	—
26	#10596934	C₂×C₄	1	4	yes	—
27	C₂×#64436	—	2	2	no	—
28	C₂×#64436	—	2	2	no	—
29	C₂×#64436	C₂×C₄	1	2	yes	—
30	C₂×#64436	C₂×C₄	1	2	yes	—
31	C₂×#64436	C₂×C₄	1	4	yes	—
32	C₂³×C₃₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₃₇	—	2	4	no	—
34	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	#41	1	2	yes	—
40	C₈×C₃₇	—	2	4	no	—
41	#64436	—	2	2	no	—
42	#64436	—	2	4	no	—
43	#64436	C₄²	1	4	yes	—
44	C₂²×C₃₇	—	2	2	no	—
45	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
47	C₄×C₃₇	—	2	2	no	—
48	C₄×C₃₇	#108	1	4	yes	complemented
49	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₃₇	#117	1	1	yes	characteristic
51	C₄×C₃₇	C₄×D₄	1	2	yes	—
52	C₄×#108	—	37	1	no	maximal, Sylow
53	C₂×C₃₇	—	2	2	no	—
54	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₃₇	C₂×#108	1	2	yes	—
56	C₂×#108	—	37	2	no	—
57	C₄×M₁₆	—	37	1	no	—
58	C₄×M₁₆	D₃₇	1	1	yes	characteristic
59	C₂×C₄×C₈	—	37	1	no	—
60	#341	—	37	2	no	—
61	C₃₇	C₄×#108	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₁₆	—	37	2	no	—
63	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
64	C₂²×C₈	—	37	1	no	—
65	C₂²×C₈	—	37	1	no	—
66	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
67	#108	—	37	8	no	—
68	#97	—	2	1	no	—
69	#97	—	74	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	37	4	no	—
72	C₄×C₈	—	74	1	no	—
73	C₂²×C₄	Dic₇₄	1	2	yes	—
74	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	37	4	no	—
77	M₁₆	—	74	2	no	—
78	M₁₆	C₄×D₃₇	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	37	2	no	—
81	C₂×C₈	—	37	4	no	—
82	C₂×C₈	—	37	2	no	—
83	C₂×C₈	—	74	2	no	—
84	C₂×C₈	—	74	2	no	—
85	C₄²	C₄×D₃₇	1	2	yes	—
86	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
87	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
88	C₂³	C₂×Dic₇₄	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	#10596936	1	2	yes	—
91	C₂×C₄	#10596936	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
94	C₂×C₄	#10596962	1	2	yes	—
95	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
96	C₈	—	2	4	no	—
97	C₈	—	37	4	no	—
98	C₈	—	74	2	no	—
99	C₈	—	74	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×#10596936	1	1	yes	characteristic
102	C₂²	C₄×Dic₇₄	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	#11924471	1	4	yes	direct product factor
105	C₄	C₄²×D₃₇	1	1	yes	characteristic
106	C₄	#11924493	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₃₇	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×#10596936	1	1	yes	characteristic
110	C₂	C₂×#11924471	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924471

C₂

yes

maximal

C₄×#10596934

C₂

yes

maximal, characteristic

C₂×C₄×#64436

C₂

yes

maximal, characteristic

#14557297

C₂

yes

maximal

C₄×M₁₆×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

C₂²×#64436

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

characteristic

#11924471

C₄

yes

direct product factor

#11924447

—

C₄×#64436

—

C₄×#64436

C₄

yes

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

—

C₄×C₈×C₃₇

—

C₂²×C₄×C₃₇

Q₈

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

M₁₆×C₃₇

—

M₁₆×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

D₄

yes

characteristic

C₄²×C₃₇

D₄

yes

characteristic

#10596934

—

#10596934

C₂×C₄

yes

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

—

C₂³×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₈×C₃₇

—

#64436

—

#64436

—

#64436

C₄²

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

#108

yes

complemented

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

#117

yes

characteristic

C₄×C₃₇

C₄×D₄

yes

—

C₄×#108

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₇

C₂×#108

yes

—

C₂×#108

—

C₄×M₁₆

—

C₄×M₁₆

D₃₇

yes

characteristic

C₂×C₄×C₈

—

#341

—

C₃₇

C₄×#108

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₃₇

yes

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

Dic₇₄

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₄×D₃₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₃₇

yes

—

C₄²

C₂²⋊D₃₇

yes

characteristic

C₄²

C₂²⋊D₃₇

yes

characteristic

C₂³

C₂×Dic₇₄

yes

characteristic

C₂×C₄

—

C₂×C₄

#10596936

yes

—

C₂×C₄

#10596936

yes

—

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

C₂×C₄

C₂×C₄×D₃₇

yes

—

C₂×C₄

#10596962

yes

—

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

100

C₂²

—

101

C₂²

C₂×#10596936

yes

characteristic

102

C₂²

C₄×Dic₇₄

yes

—

103

C₄

—

104

C₄

#11924471

yes

direct product factor

105

C₄

C₄²×D₃₇

yes

characteristic

106

C₄

#11924493

yes

characteristic

107

C₄

C₄×C₂²⋊D₃₇

yes

—

108

C₂

—

109

C₂

C₄×#10596936

yes

characteristic

110

C₂

C₂×#11924471

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊D₃₇
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₄×M₁₆×C₃₇	C₂
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₄×M₁₆	D₃₇
hypocenter	C₃₇	C₄×#108
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²⋊D₃₇

commutator

C₄×C₃₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₄×M₁₆×C₃₇

C₂

socle

C₂²×C₃₇

C₂×C₄⋊C₄

prime radical

C₂³×C₃₇

C₂×Q₈

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₄×M₁₆

D₃₇

hypocenter

C₃₇

C₄×#108

LCS

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #11924471	C₄	direct (N × H)
48. C₄×C₃₇	#108	semidirect
61. C₃₇	C₄×#108	semidirect
104. C₄	#11924471	direct (N × H)

H ≅ G/N

type

9. #11924471

C₄

direct (N × H)

48. C₄×C₃₇

#108

semidirect

61. C₃₇

C₄×#108

semidirect

104. C₄

#11924471

direct (N × H)

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

subgroup	index
C₂×#11924471	2
C₄×#10596934	2
C₂×C₄×#64436	2
#14557297	2
C₄×M₁₆×C₃₇	2
C₄×#108	37

subgroup

index

C₂×C₄×#64436

C₄×M₁₆×C₃₇

Subgroups — C₄×#11924471 #17994499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#11924471 #17994499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11924471 #17994499