Group #18010312 — subgroups

Subgroups — #18010312

Order 4736.

272 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 98 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	#14558492	C₂	1	1	yes	maximal, characteristic
2	C₄²×#3866	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	C₄	1	4	yes	—
7	C₂×C₄×#3866	C₂²	1	2	yes	—
8	#11924591	C₄	1	4	yes	complemented
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	8	no	—
19	C₄×#3866	—	2	4	no	—
20	D₃₇⋊C₈	—	2	4	no	—
21	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
22	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
25	C₂×#64436	C₂×C₄	1	2	yes	—
26	C₈×D₃₇	—	2	4	no	—
27	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×#3866	—	2	4	no	—
29	C₂×#3866	—	2	4	no	—
30	C₂×#3866	M₁₆	1	8	yes	—
31	C₂×C₄×C₃₇	C₄²	1	2	yes	—
32	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₃₇	C₄²	1	2	yes	—
34	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×D₃₇	—	2	4	no	—
36	C₄×D₃₇	#41	1	4	yes	—
37	#64438	—	2	4	no	—
38	C₈×C₃₇	—	2	2	no	—
39	#64436	—	2	2	no	—
40	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
41	C₂×D₃₇	#117	1	2	yes	—
42	C₂×D₃₇	C₂×M₁₆	1	4	yes	—
43	#3866	—	2	8	no	—
44	C₄×C₃₇	—	2	2	no	—
45	C₄×C₃₇	#117	1	2	yes	—
46	Dic₃₇	—	2	2	no	—
47	Dic₃₇	#117	1	2	yes	—
48	#1745	—	37	1	no	maximal, Sylow
49	C₂×C₃₇	#376	1	1	yes	characteristic
50	C₂×C₃₇	C₄×M₁₆	1	2	yes	—
51	D₃₇	#432	1	4	yes	—
52	C₂×#97	—	37	1	no	—
53	C₂×#97	—	37	1	no	—
54	C₄³	—	37	1	no	—
55	#336	—	37	4	no	—
56	C₃₇	#1745	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₈	—	37	2	no	—
58	C₂²×C₈	—	37	2	no	—
59	C₂×C₄²	—	37	1	no	—
60	C₂×C₄²	—	37	2	no	—
61	C₂×C₄²	—	37	4	no	—
62	#97	—	37	4	no	—
63	#97	—	37	2	no	—
64	#97	—	37	1	no	—
65	#97	#3866	1	1	yes	characteristic, complemented
66	C₂²×C₄	—	37	1	no	—
67	C₂²×C₄	—	37	4	no	—
68	C₂²×C₄	—	37	2	no	—
69	C₂×C₈	—	37	4	no	—
70	C₂×C₈	—	37	2	no	—
71	C₂×C₈	—	74	4	no	—
72	C₂×C₈	—	74	4	no	—
73	C₂×C₈	C₂×#3866	1	2	yes	—
74	C₄²	—	37	2	no	—
75	C₄²	—	37	1	no	—
76	C₄²	—	74	4	no	—
77	C₄²	—	74	8	no	—
78	C₄²	C₂×#3866	1	1	yes	characteristic
79	C₂³	—	37	1	no	—
80	C₂×C₄	—	37	1	no	—
81	C₂×C₄	—	37	4	no	—
82	C₂×C₄	—	37	8	no	—
83	C₂×C₄	—	37	2	no	—
84	C₂×C₄	—	74	4	no	—
85	C₂×C₄	—	74	4	no	—
86	C₂×C₄	—	74	4	no	—
87	C₂×C₄	C₄×#3866	1	2	yes	—
88	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	74	4	no	—
91	C₈	—	74	2	no	—
92	C₂²	—	37	2	no	—
93	C₂²	—	37	4	no	—
94	C₂²	C₂×C₄×#3866	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	37	2	no	—
97	C₄	—	74	8	no	—
98	C₄	—	74	2	no	—
99	C₄	#11924615	1	2	yes	—
100	C₂	—	37	4	no	—
101	C₂	#14558469	1	1	yes	characteristic
102	C₂	M₁₆×#3866	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558492

C₂

yes

maximal, characteristic

C₄²×#3866

C₂

yes

maximal, characteristic

(C₂×#3866)⋊C₈

C₂

yes

maximal

#97×D₃₇

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

C₂²

yes

—

C₂×C₄×#3866

C₄

yes

—

C₂×C₄×#3866

C₂²

yes

—

#11924591

C₄

yes

complemented

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

—

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

—

—

M₁₆

yes

—

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

C₂×M₁₆

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

#117

yes

—

Dic₃₇

—

Dic₃₇

#117

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

C₄×M₁₆

yes

—

D₃₇

#432

yes

—

C₂×#97

—

C₂×#97

—

C₄³

—

#336

—

C₃₇

#1745

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

—

—

—

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

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

—

C₄

#11924615

yes

—

100

C₂

—

101

C₂

#14558469

yes

characteristic

102

C₂

M₁₆×#3866

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	#1745
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
8. #11924591	C₄	semidirect
11. #11924447	C₄	semidirect
12. #11924446	C₄	semidirect
13. #97×C₃₇	C₄	semidirect
56. C₃₇	#1745	semidirect
65. #97	#3866	semidirect

H ≅ G/N

type

8. #11924591

C₄

semidirect

11. #11924447

C₄

semidirect

12. #11924446

C₄

semidirect

13. #97×C₃₇

C₄

semidirect

56. C₃₇

#1745

semidirect

65. #97

#3866

semidirect

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

subgroup	index
#14558492	2
C₄²×#3866	2
(C₂×#3866)⋊C₈	2
#97×D₃₇	2
#1745	37

subgroup

index

#14558492

C₄²×#3866

(C₂×#3866)⋊C₈

#97×D₃₇

#1745

Subgroups — #18010312

Conjugacy classes of subgroups

Complements

Maximal subgroups