Group #18010316 — subgroups

Subgroups — #18010316

Order 4736.

240 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₄×D₃₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#14558469	C₂	1	1	yes	maximal, characteristic
3	(C₂×#3866)⋊C₈	C₂	1	4	yes	maximal
4	#97×D₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×D₃₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×#3866	C₂²	1	2	yes	—
7	C₄×#64438	C₄	1	4	yes	complemented
8	(C₂×#3866)⋊C₄	C₄	1	4	yes	—
9	C₂×C₈×D₃₇	C₂²	1	2	yes	—
10	C₄²×D₃₇	C₂²	1	1	yes	characteristic
11	#11924447	C₄	1	1	yes	characteristic, complemented
12	#11924446	C₄	1	2	yes	complemented
13	#97×C₃₇	C₄	1	1	yes	characteristic, complemented
14	C₂²×#3866	C₂×C₄	1	4	yes	—
15	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
17	C₂×#64438	C₂×C₄	1	4	yes	—
18	C₄×#3866	—	2	4	no	—
19	D₃₇⋊C₈	—	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	C₂×C₄	1	2	yes	—
24	C₂×#64436	C₂×C₄	1	2	yes	—
25	C₈×D₃₇	—	2	4	no	—
26	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×#3866	—	2	4	no	—
28	C₂×#3866	—	2	8	no	—
29	C₂×C₄×C₃₇	C₄²	1	2	yes	—
30	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₃₇	C₄²	1	2	yes	—
32	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×D₃₇	—	2	4	no	—
34	C₄×D₃₇	#41	1	4	yes	—
35	#64438	—	2	4	no	—
36	C₈×C₃₇	—	2	2	no	—
37	#64436	—	2	2	no	—
38	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
39	C₂×D₃₇	#117	1	2	yes	—
40	C₂×D₃₇	#131	1	4	yes	—
41	#3866	—	4	4	no	—
42	C₄×C₃₇	—	2	2	no	—
43	C₄×C₃₇	#117	1	2	yes	—
44	Dic₃₇	—	2	2	no	—
45	Dic₃₇	#117	1	2	yes	—
46	#1746	—	37	1	no	maximal, Sylow
47	C₂×C₃₇	#376	1	1	yes	characteristic
48	C₂×C₃₇	#405	1	2	yes	—
49	D₃₇	#433	1	4	yes	—
50	C₂×#97	—	37	1	no	—
51	C₂×C₄×C₈	—	37	1	no	—
52	#376	—	37	1	no	—
53	#336	—	37	4	no	—
54	C₃₇	#1746	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₈	—	37	2	no	—
56	C₂²×C₈	—	37	2	no	—
57	C₂×C₄²	—	37	1	no	—
58	C₂×C₄²	—	37	2	no	—
59	#97	—	37	2	no	—
60	#97	—	37	1	no	—
61	#97	#3866	1	1	yes	characteristic, complemented
62	C₄×C₈	—	37	4	no	—
63	#95	—	37	4	no	—
64	C₂²×C₄	—	37	1	no	—
65	C₂²×C₄	—	37	2	no	—
66	C₂²×C₄	—	37	4	no	—
67	C₂×C₈	—	37	4	no	—
68	C₂×C₈	—	37	2	no	—
69	C₂×C₈	—	74	4	no	—
70	C₂×C₈	—	74	4	no	—
71	C₂×C₈	C₂×#3866	1	2	yes	—
72	C₄²	—	37	2	no	—
73	C₄²	—	37	1	no	—
74	C₄²	—	74	4	no	—
75	C₄²	C₂×#3866	1	1	yes	characteristic
76	C₂³	—	37	1	no	—
77	C₂×C₄	—	37	1	no	—
78	C₂×C₄	—	37	4	no	—
79	C₂×C₄	—	37	2	no	—
80	C₂×C₄	—	74	4	no	—
81	C₂×C₄	—	74	4	no	—
82	C₂×C₄	—	74	8	no	—
83	C₂×C₄	C₄×#3866	1	2	yes	—
84	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	—	74	4	no	—
87	C₈	—	74	2	no	—
88	C₂²	—	37	2	no	—
89	C₂²	—	37	4	no	—
90	C₂²	C₂×C₄×#3866	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	37	2	no	—
93	C₄	—	74	2	no	—
94	C₄	—	148	4	no	—
95	C₄	#11924615	1	2	yes	—
96	C₂	—	37	4	no	—
97	C₂	#14558469	1	1	yes	characteristic
98	C₂	#14558499	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₃₇⋊C₈

C₂

yes

maximal, characteristic

#14558469

C₂

yes

maximal, characteristic

(C₂×#3866)⋊C₈

C₂

yes

maximal

#97×D₃₇

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

C₂²

yes

—

C₂×C₄×#3866

C₂²

yes

—

C₄×#64438

C₄

yes

complemented

(C₂×#3866)⋊C₄

C₄

yes

—

C₂×C₈×D₃₇

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

#11924447

C₄

yes

characteristic, complemented

#11924446

C₄

yes

complemented

#97×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×#64438

C₂×C₄

yes

—

C₄×#3866

—

D₃₇⋊C₈

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

—

C₈×D₃₇

—

C₂²×D₃₇

C₂²×C₄

yes

characteristic

C₂×#3866

—

C₂×#3866

—

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₄²

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

#41

yes

—

#64438

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₄²

yes

characteristic

C₂×D₃₇

#117

yes

—

C₂×D₃₇

#131

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

#117

yes

—

Dic₃₇

—

Dic₃₇

#117

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

#433

yes

—

C₂×#97

—

C₂×C₄×C₈

—

#376

—

#336

—

C₃₇

#1746

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

—

—

yes

characteristic, complemented

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×#3866

yes

—

C₂×C₄

C₂²×#3866

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

C₂×C₄×#3866

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

148

—

C₄

#11924615

yes

—

C₂

—

C₂

#14558469

yes

characteristic

C₂

#14558499

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3866
commutator, socle	C₂²×C₃₇	C₂×C₄²
Fitting	#97×C₃₇	C₄
prime radical	C₂²×D₃₇	C₂²×C₄
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#97	#3866
hypocenter	C₃₇	#1746
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#3866

commutator, socle

C₂²×C₃₇

C₂×C₄²

Fitting

#97×C₃₇

C₄

prime radical

C₂²×D₃₇

C₂²×C₄

cosocle

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
7. C₄×#64438	C₄	semidirect
11. #11924447	C₄	semidirect
12. #11924446	C₄	semidirect
13. #97×C₃₇	C₄	semidirect
54. C₃₇	#1746	semidirect
61. #97	#3866	semidirect

H ≅ G/N

type

7. C₄×#64438

C₄

semidirect

11. #11924447

C₄

semidirect

12. #11924446

C₄

semidirect

13. #97×C₃₇

C₄

semidirect

54. C₃₇

#1746

semidirect

61. #97

#3866

semidirect

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

subgroup	index
C₄×D₃₇⋊C₈	2
#14558469	2
(C₂×#3866)⋊C₈	2
#97×D₃₇	2
#1746	37

subgroup

index

C₄×D₃₇⋊C₈

#14558469

(C₂×#3866)⋊C₈

#97×D₃₇

#1746

Subgroups — #18010316

Conjugacy classes of subgroups

Complements

Maximal subgroups