Group #18010522 — subgroups

Subgroups — #18010522

Order 4736.

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14558538	C₂	1	1	yes	maximal, characteristic
2	#14558537	C₂	1	1	yes	maximal, characteristic
3	#14558409	C₂	1	4	yes	maximal
4	#97×D₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596953	C₂²	1	2	yes	—
6	C₂×#10596953	C₂²	1	2	yes	—
7	#11924603	—	2	2	no	—
8	C₄⋊#64438	—	2	2	no	—
9	C₂×C₈×D₃₇	C₂²	1	2	yes	—
10	C₄²×D₃₇	C₂²	1	1	yes	characteristic
11	#11924447	C₄	1	1	yes	characteristic
12	#11924446	C₄	1	2	yes	—
13	#97×C₃₇	C₄	1	1	yes	characteristic
14	C₂×C₄×D₃₇	Q₈	1	2	yes	—
15	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
16	C₂×#64438	—	2	2	no	—
17	C₂×#64438	—	2	2	no	—
18	#10596953	—	2	4	no	—
19	#10596953	—	2	4	no	—
20	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
21	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₃₇	D₄	1	2	yes	—
24	C₂×#64436	C₂×C₄	1	2	yes	—
25	C₈×D₃₇	—	2	2	no	—
26	C₈×D₃₇	—	2	2	no	—
27	C₂²×D₃₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
32	C₄×D₃₇	—	4	2	no	—
33	C₄×D₃₇	#41	1	2	yes	—
34	C₄×D₃₇	#41	1	2	yes	—
35	#64438	—	4	2	no	—
36	#64438	—	4	2	no	—
37	C₈×C₃₇	—	2	2	no	—
38	#64436	—	2	2	no	—
39	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×D₃₇	—	2	2	no	—
41	C₂×D₃₇	#125	1	2	yes	—
42	C₄×C₃₇	—	2	2	no	—
43	C₄×C₃₇	#117	1	2	yes	—
44	Dic₃₇	—	2	2	no	—
45	Dic₃₇	#124	1	1	yes	characteristic
46	Dic₃₇	#124	1	1	yes	characteristic
47	#1838	—	37	1	no	maximal, Sylow
48	C₂×C₃₇	#382	1	1	yes	characteristic
49	C₂×C₃₇	#430	1	2	yes	—
50	D₃₇	—	2	2	no	—
51	#423	—	37	1	no	—
52	#423	—	37	1	no	—
53	C₂×#97	—	37	1	no	—
54	#341	—	37	4	no	—
55	C₃₇	#1838	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₁₆	—	37	2	no	—
57	C₂×M₁₆	—	37	2	no	—
58	C₂²×C₈	—	37	2	no	—
59	C₂×C₄²	—	37	1	no	—
60	C₄⋊C₈	—	74	2	no	—
61	C₄⋊C₈	—	74	2	no	—
62	#97	—	37	2	no	—
63	#97	—	37	1	no	—
64	#97	#3866	1	1	yes	characteristic
65	C₂²×C₄	—	37	1	no	—
66	C₂²×C₄	—	37	2	no	—
67	M₁₆	—	74	4	no	—
68	M₁₆	—	74	4	no	—
69	C₂×C₈	—	37	2	no	—
70	C₂×C₈	—	74	2	no	—
71	C₂×C₈	—	74	2	no	—
72	C₂×C₈	—	74	2	no	—
73	C₂×C₈	—	74	2	no	—
74	C₂×C₈	C₂×#3866	1	2	yes	—
75	C₄²	—	37	2	no	—
76	C₄²	—	37	1	no	—
77	C₄²	C₂×#3866	1	1	yes	characteristic
78	C₂³	—	37	1	no	—
79	C₂×C₄	—	37	1	no	—
80	C₂×C₄	—	37	2	no	—
81	C₂×C₄	—	37	2	no	—
82	C₂×C₄	—	37	2	no	—
83	C₂×C₄	—	148	2	no	—
84	C₂×C₄	C₄⋊#3866	1	2	yes	—
85	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	74	2	no	—
88	C₈	—	148	2	no	—
89	C₈	—	148	2	no	—
90	C₂²	—	37	2	no	—
91	C₂²	—	74	2	no	—
92	C₂²	C₂×C₄⋊#3866	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	37	1	no	—
95	C₄	—	37	1	no	—
96	C₄	—	74	2	no	—
97	C₄	#11924615	1	2	yes	—
98	C₂	—	74	2	no	—
99	C₂	#14558477	1	1	yes	characteristic
100	C₂	#14558548	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558538

C₂

yes

maximal, characteristic

#14558537

C₂

yes

maximal, characteristic

#14558409

C₂

yes

maximal

#97×D₃₇

C₂

yes

maximal, characteristic

C₂×#10596953

C₂²

yes

—

C₂×#10596953

C₂²

yes

—

#11924603

—

C₄⋊#64438

—

C₂×C₈×D₃₇

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

#11924447

C₄

yes

characteristic

#11924446

C₄

yes

—

#97×C₃₇

C₄

yes

characteristic

C₂×C₄×D₃₇

Q₈

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

—

—

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₂×#64436

C₂×C₄

yes

—

C₈×D₃₇

—

C₈×D₃₇

—

C₂²×D₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

#41

yes

—

C₄×D₃₇

#41

yes

—

#64438

—

#64438

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

#125

yes

—

C₄×C₃₇

—

C₄×C₃₇

#117

yes

—

Dic₃₇

—

Dic₃₇

yes

Dic₃₇

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

#423

—

#423

—

C₂×#97

—

#341

—

C₃₇

#1838

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

—

—

yes

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

C₄⋊#3866

yes

—

C₂×C₄

C₂²×#3866

yes

characteristic

C₈

—

C₈

—

C₈

—

148

—

C₈

—

148

—

C₂²

—

C₂²

—

C₂²

C₂×C₄⋊#3866

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#11924615

yes

—

C₂

—

C₂

#14558477

yes

characteristic

100

C₂

#14558548

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	#97×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₃₇	C₂×Q₈
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#97	#3866
hypocenter	C₃₇	#1838
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

#97×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₄⋊C₄

prime radical

C₂²×D₃₇

C₂×Q₈

cosocle

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
55. C₃₇	#1838	semidirect

H ≅ G/N

type

55. C₃₇

#1838

semidirect

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

subgroup	index
#14558538	2
#14558537	2
#14558409	2
#97×D₃₇	2
#1838	37

subgroup

index

#97×D₃₇

Subgroups — #18010522

Conjugacy classes of subgroups

Complements

Maximal subgroups