Group #17994729 — subgroups

Subgroups — #17994729

Order 4736.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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×#11924447	C₂	1	1	yes	maximal, characteristic
2	#14557403	C₂	1	1	yes	maximal, characteristic
3	#14557295	C₂	1	4	yes	maximal
4	C₃₇×#382	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64436	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₃₇	—	2	2	no	—
7	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
8	#11924447	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
11	#95×C₃₇	—	2	2	no	—
12	C₂²×C₄×C₃₇	—	2	2	no	—
13	C₂²×C₄×C₃₇	D₄	1	2	yes	—
14	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
15	C₂²×Dic₃₇	—	2	2	no	—
16	C₄⋊C₄×C₃₇	—	2	4	no	—
17	C₄²×C₃₇	C₂×C₄	1	2	yes	—
18	C₄²×C₃₇	Q₈	1	2	yes	—
19	C₄⋊Dic₃₇	—	2	4	no	—
20	C₄⋊Dic₃₇	—	4	4	no	—
21	C₂×#64436	—	2	2	no	—
22	C₂×#64436	—	2	2	no	—
23	C₂×#64436	C₂×C₄	1	4	yes	—
24	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₃₇	—	2	2	no	—
26	C₂×C₄×C₃₇	—	2	2	no	—
27	C₂×C₄×C₃₇	—	4	2	no	—
28	C₂×C₄×C₃₇	—	4	2	no	—
29	C₂×C₄×C₃₇	—	4	2	no	—
30	C₂×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₃₇	#41	1	2	yes	—
34	C₂×C₄×C₃₇	#41	1	2	yes	—
35	C₂×Dic₃₇	—	4	2	no	—
36	C₂×Dic₃₇	—	4	4	no	—
37	#64436	—	2	4	no	—
38	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₃₇	#124	1	1	yes	characteristic
40	C₂²×C₃₇	#124	1	1	yes	characteristic
41	C₂²×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₃₇	#137	1	2	yes	—
43	C₄×C₃₇	—	2	4	no	—
44	C₄×C₃₇	—	8	2	no	—
45	C₄×C₃₇	#117	1	2	yes	—
46	C₄×C₃₇	#125	1	2	yes	—
47	Dic₃₇	—	8	2	no	—
48	#1836	—	37	1	no	maximal, Sylow
49	C₂×C₃₇	#382	1	1	yes	characteristic
50	C₂×C₃₇	#428	1	2	yes	—
51	C₂×C₃₇	#489	1	2	yes	—
52	C₂×C₃₇	#490	1	1	yes	characteristic
53	C₂×C₃₇	#491	1	1	yes	characteristic
54	C₂×#97	—	37	1	no	—
55	#384	—	37	1	no	—
56	#382	D₃₇	1	1	yes	characteristic
57	#340	—	37	4	no	—
58	C₃₇	#1836	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	—	37	2	no	—
60	C₂×C₄⋊C₄	—	37	2	no	—
61	C₂×C₄⋊C₄	—	74	2	no	—
62	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
63	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
64	#97	—	37	4	no	—
65	#95	—	2	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	74	2	no	—
68	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
69	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
70	C₂×C₈	—	37	4	no	—
71	C₂×C₈	—	74	2	no	—
72	C₂×C₈	—	74	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	74	4	no	—
75	C₄⋊C₄	—	148	4	no	—
76	C₄²	Dic₇₄	1	2	yes	—
77	C₄²	C₄×D₃₇	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	148	2	no	—
85	C₂×C₄	—	148	4	no	—
86	C₂×C₄	#10596936	1	4	yes	—
87	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
88	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
89	C₂×C₄	#10596964	1	2	yes	—
90	C₂×C₄	#10596966	1	2	yes	—
91	C₈	—	74	4	no	—
92	C₂²	C₂×#10596936	1	1	yes	characteristic
93	C₂²	#11924523	1	1	yes	characteristic
94	C₂²	#11924525	1	1	yes	characteristic
95	C₂²	#11924561	1	2	yes	—
96	C₂²	#11924566	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	296	2	no	—
100	C₄	#11924494	1	2	yes	—
101	C₄	#11924527	1	2	yes	—
102	C₂	#14557398	1	1	yes	characteristic
103	C₂	#14557576	1	2	yes	—
104	C₂	#14557892	1	2	yes	—
105	C₂	#14557899	1	1	yes	characteristic
106	C₂	#14557903	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924447

C₂

yes

maximal, characteristic

#14557403

C₂

yes

maximal, characteristic

#14557295

C₂

yes

maximal

C₃₇×#382

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

#11924447

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#95×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

Q₈

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

C₂×C₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

—

C₂²×C₃₇

#137

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#117

yes

—

C₄×C₃₇

#125

yes

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#489

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×#97

—

#384

—

#382

D₃₇

yes

characteristic

#340

—

C₃₇

#1836

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#97

—

#95

—

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

—

148

—

C₄²

Dic₇₄

yes

—

C₄²

C₄×D₃₇

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

#10596936

yes

—

C₂×C₄

C₂×Dic₇₄

yes

characteristic

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

C₂×C₄

#10596964

yes

—

C₂×C₄

#10596966

yes

—

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#11924566

yes

—

C₄

—

C₄

—

C₄

—

296

—

100

C₄

#11924494

yes

—

101

C₄

#11924527

yes

—

102

C₂

#14557398

yes

characteristic

103

C₂

#14557576

yes

—

104

C₂

#14557892

yes

—

105

C₂

#14557899

yes

characteristic

106

C₂

#14557903

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#382	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#382	D₃₇
hypocenter	C₃₇	#1836
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#382

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 ◁ #382

N	H ≅ G/N	type
8. #11924447	C₄	semidirect
58. C₃₇	#1836	semidirect

H ≅ G/N

type

8. #11924447

C₄

semidirect

58. C₃₇

#1836

semidirect

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

subgroup	index
C₂×#11924447	2
#14557403	2
#14557295	2
C₃₇×#382	2
#1836	37

subgroup

index

C₃₇×#382

Subgroups — #17994729

Conjugacy classes of subgroups

Complements

Maximal subgroups