Group #17993371 — subgroups

Subgroups — #17993371

Order 4736.

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

normal subgroups only

This group has 168 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	C₂×C₄⋊#64436	C₂	1	1	yes	maximal, characteristic
3	#14557286	C₂	1	4	yes	maximal, complemented
4	C₃₇×#431	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596934	—	2	1	no	—
6	C₂²×#64436	—	2	1	no	—
7	C₄⋊#64436	—	2	1	no	—
8	C₄⋊#64436	—	2	1	no	—
9	C₄⋊#64436	C₂²	1	2	yes	—
10	C₄⋊#64436	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₃₇	C₄	1	2	yes	—
13	C₂²⋊C₈×C₃₇	C₄	1	2	yes	—
14	C₄×C₈×C₃₇	C₂²	1	2	yes	—
15	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	Q₈	1	1	yes	characteristic
18	C₂×C₈×C₃₇	C₂×C₄	1	4	yes	—
19	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₃₇	D₄	1	1	yes	characteristic
21	C₄²×C₃₇	Q₈	1	1	yes	characteristic
22	C₄²×C₃₇	C₂³	1	1	yes	characteristic
23	#10596934	—	4	2	no	—
24	C₂×#64436	—	2	2	no	—
25	C₂×#64436	—	2	2	no	—
26	C₂×#64436	—	4	2	no	—
27	C₂³×C₃₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₃₇	—	2	2	no	—
29	C₂×C₄×C₃₇	—	4	1	no	—
30	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₃₇	D₈	1	2	yes	—
34	C₂×C₄×C₃₇	Q₁₆	1	2	yes	—
35	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₃₇	—	2	4	no	—
39	#64436	—	4	2	no	—
40	#64436	—	4	2	no	—
41	C₂²×C₃₇	—	2	1	no	—
42	C₂²×C₃₇	—	2	1	no	—
43	C₂²×C₃₇	#107	1	2	yes	—
44	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₃₇	—	2	1	no	—
46	C₄×C₃₇	—	2	1	no	—
47	C₄×C₃₇	—	2	1	no	—
48	C₄×C₃₇	#107	1	2	yes	—
49	C₄×C₃₇	#131	1	2	yes	—
50	C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
51	C₄×C₃₇	C₂×Q₁₆	1	1	yes	characteristic
52	#1571	—	37	1	no	maximal, Sylow
53	C₂×C₃₇	—	2	2	no	—
54	C₂×C₃₇	#424	1	1	yes	characteristic
55	C₂×C₃₇	C₂×#107	1	1	yes	characteristic
56	C₂×C₃₇	#430	1	1	yes	characteristic
57	#431	D₃₇	1	1	yes	characteristic
58	#423	—	37	1	no	—
59	C₂×C₄⋊C₈	—	37	1	no	—
60	#335	—	37	4	no	—
61	C₃₇	#1571	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₁₆	—	74	1	no	—
63	C₂²×C₈	—	74	1	no	—
64	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
65	C₄⋊C₈	—	37	2	no	—
66	C₄⋊C₈	—	37	2	no	—
67	C₄⋊C₈	—	74	1	no	—
68	C₄⋊C₈	—	74	1	no	—
69	C₄⋊C₈	Dic₃₇	1	2	yes	—
70	C₂²⋊C₈	Dic₃₇	1	2	yes	—
71	C₄×C₈	C₂×D₃₇	1	2	yes	—
72	C₂²×C₄	Dic₇₄	1	1	yes	characteristic
73	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₃₇	1	1	yes	characteristic
75	M₁₆	—	148	2	no	—
76	C₂×C₈	—	74	2	no	—
77	C₂×C₈	—	74	2	no	—
78	C₂×C₈	—	148	2	no	—
79	C₂×C₈	C₂×Dic₃₇	1	4	yes	—
80	C₄²	Dic₇₄	1	1	yes	characteristic
81	C₄²	D₁₄₈	1	1	yes	characteristic
82	C₄²	C₂×Dic₃₇	1	1	yes	characteristic
83	C₄²	C₂²×D₃₇	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₃₇	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	D₂₉₆	1	2	yes	—
88	C₂×C₄	Dic₁₄₈	1	2	yes	—
89	C₂×C₄	C₄⋊Dic₃₇	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₃₇	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₃₇	1	1	yes	characteristic
92	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
93	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
94	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	148	2	no	—
97	C₈	—	148	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#11924470	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₃₇	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#11924470	1	2	yes	—
106	C₄	#11924543	1	2	yes	—
107	C₄	C₂×D₂₉₆	1	1	yes	characteristic
108	C₄	C₂×Dic₁₄₈	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#14557560	1	1	yes	characteristic
111	C₂	C₂×#11924470	1	1	yes	characteristic
112	C₂	#14557582	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557557

C₂

yes

maximal, characteristic

C₂×C₄⋊#64436

C₂

yes

maximal, characteristic

#14557286

C₂

yes

maximal, complemented

C₃₇×#431

C₂

yes

maximal, characteristic

C₂×#10596934

—

C₂²×#64436

—

C₄⋊#64436

—

C₄⋊#64436

—

C₄⋊#64436

C₂²

yes

—

C₄⋊#64436

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₄

yes

—

C₂²⋊C₈×C₃₇

C₄

yes

—

C₄×C₈×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

Q₈

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

D₄

yes

characteristic

C₄²×C₃₇

Q₈

yes

characteristic

C₄²×C₃₇

C₂³

yes

—

—

—

—

C₂³×C₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

D₈

yes

—

C₂×C₄×C₃₇

Q₁₆

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₈×C₃₇

—

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#107

yes

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#107

yes

—

C₄×C₃₇

#131

yes

—

C₄×C₃₇

C₂×D₈

yes

characteristic

C₄×C₃₇

C₂×Q₁₆

yes

characteristic

#1571

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

#335

—

C₃₇

#1571

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₃₇

yes

—

C₂²⋊C₈

Dic₃₇

yes

—

C₄×C₈

C₂×D₃₇

yes

—

C₂²×C₄

Dic₇₄

yes

characteristic

C₂²×C₄

D₁₄₈

yes

characteristic

C₂²×C₄

C₂×Dic₃₇

yes

characteristic

M₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

148

—

C₂×C₈

C₂×Dic₃₇

yes

—

C₄²

Dic₇₄

yes

characteristic

C₄²

D₁₄₈

yes

characteristic

C₄²

C₂×Dic₃₇

yes

characteristic

C₄²

C₂²×D₃₇

yes

characteristic

C₂³

C₄⋊Dic₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₂₉₆

yes

—

C₂×C₄

Dic₁₄₈

yes

—

C₂×C₄

C₄⋊Dic₃₇

yes

characteristic

C₂×C₄

C₄⋊Dic₃₇

yes

characteristic

C₂×C₄

C₄⋊Dic₃₇

yes

characteristic

C₂×C₄

C₂×Dic₇₄

yes

characteristic

C₂×C₄

C₂×D₁₄₈

yes

characteristic

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

C₈

—

C₈

—

148

—

C₈

—

148

—

C₂²

—

C₂²

—

100

C₂²

#11924470

yes

—

101

C₂²

C₂×C₄⋊Dic₃₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#11924470

yes

—

106

C₄

#11924543

yes

—

107

C₄

C₂×D₂₉₆

yes

characteristic

108

C₄

C₂×Dic₁₄₈

yes

characteristic

109

C₂

—

110

C₂

#14557560

yes

characteristic

111

C₂

C₂×#11924470

yes

characteristic

112

C₂

#14557582

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₄₈
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#431	C₂
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₃₇	C₄⋊C₄
cosocle	C₄²×C₃₇	C₂³
hypercenter	#431	D₃₇
hypocenter	C₃₇	#1571
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₁₄₈

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#431

C₂

socle

C₂²×C₃₇

C₂×C₄⋊C₄

prime radical

C₂³×C₃₇

C₄⋊C₄

cosocle

C₄²×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #14557286	C₂	semidirect
61. C₃₇	#1571	semidirect

H ≅ G/N

type

3. #14557286

C₂

semidirect

61. C₃₇

#1571

semidirect

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

subgroup	index
#14557557	2
C₂×C₄⋊#64436	2
#14557286	2
C₃₇×#431	2
#1571	37

subgroup

index

#14557557

C₂×C₄⋊#64436

#14557286

C₃₇×#431

#1571

Subgroups — #17993371

Conjugacy classes of subgroups

Complements

Maximal subgroups