Group #17995564 — subgroups

Subgroups — #17995564

Order 4736.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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×#11924528	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924447	C₂	1	1	yes	maximal, characteristic
3	#14557295	C₂	1	4	yes	maximal
4	C₃₇×#384	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64436	C₂²	1	2	yes	—
6	#11924528	—	2	2	no	—
7	#11924528	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
9	C₂×#10596936	—	2	2	no	—
10	#11924447	—	2	2	no	—
11	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₃₇	—	2	2	no	—
15	C₂²×C₄×C₃₇	D₄	1	2	yes	—
16	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
17	C₂²×Dic₃₇	—	2	2	no	—
18	C₄⋊C₄×C₃₇	—	2	4	no	—
19	C₄⋊C₄×C₃₇	—	4	4	no	—
20	C₄²×C₃₇	C₂×C₄	1	2	yes	—
21	C₄²×C₃₇	D₄	1	2	yes	—
22	C₄⋊Dic₃₇	—	2	2	no	—
23	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
24	#10596936	—	2	8	no	—
25	C₂×#64436	—	2	2	no	—
26	C₂×#64436	—	2	2	no	—
27	C₂×#64436	Q₈	1	4	yes	—
28	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₇	—	2	2	no	—
30	C₂×C₄×C₃₇	—	2	2	no	—
31	C₂×C₄×C₃₇	—	4	2	no	—
32	C₂×C₄×C₃₇	—	4	4	no	—
33	C₂×C₄×C₃₇	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₃₇	#41	1	2	yes	—
38	C₂×Dic₃₇	—	2	4	no	—
39	C₂×Dic₃₇	—	4	2	no	—
40	C₂×Dic₃₇	—	4	2	no	—
41	#64436	—	2	4	no	—
42	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₃₇	#124	1	1	yes	characteristic
44	C₂²×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₃₇	#137	1	2	yes	—
47	C₄×C₃₇	—	2	4	no	—
48	C₄×C₃₇	—	8	2	no	—
49	C₄×C₃₇	C₄×Q₈	1	2	yes	—
50	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
51	Dic₃₇	—	4	4	no	—
52	#1992	—	37	1	no	maximal, Sylow
53	C₂×C₃₇	#391	1	1	yes	characteristic
54	C₂×C₃₇	#417	1	2	yes	—
55	C₂×C₃₇	#490	1	1	yes	characteristic
56	C₂×C₃₇	#491	1	1	yes	characteristic
57	C₂×C₃₇	#501	1	2	yes	—
58	C₂×#125	—	37	1	no	—
59	C₂×#97	—	37	1	no	—
60	#384	D₃₇	1	1	yes	characteristic
61	#340	—	37	4	no	—
62	C₃₇	#1992	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	—	37	2	no	—
64	#125	—	37	4	no	—
65	#125	—	74	2	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	37	2	no	—
68	C₂×C₄⋊C₄	—	74	2	no	—
69	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
70	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
71	#97	—	74	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	74	2	no	—
74	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
75	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
76	C₂×C₈	—	37	4	no	—
77	C₂×C₈	—	74	2	no	—
78	C₂×C₈	—	74	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	4	4	no	—
81	C₄⋊C₄	—	37	4	no	—
82	C₄⋊C₄	—	74	8	no	—
83	C₄⋊C₄	—	74	2	no	—
84	C₄²	C₄×D₃₇	1	2	yes	—
85	C₄²	D₁₄₈	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	—	74	4	no	—
92	C₂×C₄	—	148	2	no	—
93	C₂×C₄	—	148	2	no	—
94	C₂×C₄	#10596938	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
96	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
97	C₂×C₄	#10596964	1	2	yes	—
98	C₂×C₄	Q₈×D₃₇	1	2	yes	—
99	C₈	—	74	4	no	—
100	C₂²	C₂×#10596938	1	1	yes	characteristic
101	C₂²	#11924523	1	1	yes	characteristic
102	C₂²	#11924534	1	1	yes	characteristic
103	C₂²	#11924562	1	2	yes	—
104	C₂²	#11924567	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	8	2	no	—
107	C₄	—	148	4	no	—
108	C₄	#11924501	1	2	yes	—
109	C₄	#11924516	1	2	yes	—
110	C₂	#14557438	1	1	yes	characteristic
111	C₂	#14557532	1	2	yes	—
112	C₂	#14557897	1	1	yes	characteristic
113	C₂	#14557901	1	1	yes	characteristic
114	C₂	#14557939	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924528

C₂

yes

maximal, characteristic

C₂×#11924447

C₂

yes

maximal, characteristic

#14557295

C₂

yes

maximal

C₃₇×#384

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

#11924528

—

#11924528

C₄

yes

complemented

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

#11924447

—

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

D₄

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

—

—

—

Q₈

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

#124

yes

characteristic

C₂²×C₃₇

C₄⋊Q₈

yes

characteristic

C₂²×C₃₇

AGL(1,ℤ/8)

yes

—

C₂²×C₃₇

#137

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×Q₈

yes

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×#125

—

C₂×#97

—

#384

D₃₇

yes

characteristic

#340

—

C₃₇

#1992

yes

characteristic, complemented, Sylow

C₂²×C₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#97

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₃₇

yes

—

C₄²

D₁₄₈

yes

—

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

#10596938

yes

—

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

C₂×C₄

C₂×D₁₄₈

yes

characteristic

C₂×C₄

#10596964

yes

—

C₂×C₄

Q₈×D₃₇

yes

—

C₈

—

100

C₂²

C₂×#10596938

yes

characteristic

101

C₂²

#11924523

yes

characteristic

102

C₂²

#11924534

yes

characteristic

103

C₂²

#11924562

yes

—

104

C₂²

#11924567

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

148

—

108

C₄

#11924501

yes

—

109

C₄

#11924516

yes

—

110

C₂

#14557438

yes

characteristic

111

C₂

#14557532

yes

—

112

C₂

#14557897

yes

characteristic

113

C₂

#14557901

yes

characteristic

114

C₂

#14557939

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#384	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#384	D₃₇
hypocenter	C₃₇	#1992
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#384

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #11924528	C₄	semidirect
62. C₃₇	#1992	semidirect

H ≅ G/N

type

7. #11924528

C₄

semidirect

62. C₃₇

#1992

semidirect

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

subgroup	index
C₂×#11924528	2
C₂×#11924447	2
#14557295	2
C₃₇×#384	2
#1992	37

subgroup

index

C₃₇×#384

Subgroups — #17995564

Conjugacy classes of subgroups

Complements

Maximal subgroups