Group #18010189 — subgroups

Subgroups — #18010189

Order 4736.

160 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 72 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	Q₈×#64438	C₂	1	2	yes	maximal
2	#14558394	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#64438	C₂	1	2	yes	maximal
4	#14558374	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#64438	—	2	2	no	—
7	C₄⋊#64438	C₂²	1	2	yes	—
8	C₄×#64438	—	2	2	no	—
9	C₄×#64438	C₂²	1	1	yes	characteristic
10	C₄×#64438	C₂²	1	1	yes	characteristic
11	Q₈×Dic₃₇	C₂²	1	2	yes	—
12	#11924470	C₄	1	1	yes	characteristic
13	#11924460	C₄	1	2	yes	—
14	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₃₇	C₄	1	1	yes	characteristic
16	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
17	C₂×#64438	—	2	2	no	—
18	C₂×#64438	—	2	1	no	—
19	C₂×#64438	D₄	1	2	yes	—
20	Q₁₆×C₃₇	C₈	1	4	yes	complemented
21	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₃₇	—	2	2	no	—
23	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₃₇	—	2	2	no	—
25	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
26	C₂×#64436	C₂×C₄	1	1	yes	characteristic
27	Q₈×C₃₇	—	2	2	no	—
28	Q₈×C₃₇	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₃₇	—	2	2	no	—
30	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₃₇	—	2	2	no	—
32	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₃₇	#41	1	1	yes	characteristic
34	#64438	—	2	2	no	—
35	#64438	—	4	1	no	—
36	#64438	—	4	2	no	—
37	C₈×C₃₇	C₂×C₈	1	2	yes	—
38	#64436	—	2	1	no	—
39	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
40	C₄×C₃₇	—	2	4	no	—
41	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₃₇	#131	1	1	yes	characteristic
43	Dic₃₇	—	2	1	no	—
44	Dic₃₇	—	4	2	no	—
45	Dic₃₇	#137	1	2	yes	—
46	#1579	—	37	1	no	maximal, Sylow
47	C₂×C₃₇	C₈×D₄	1	1	yes	characteristic
48	C₂×C₃₇	#441	1	1	yes	characteristic
49	C₂×C₃₇	#444	1	1	yes	characteristic
50	C₈×Q₈	—	37	2	no	—
51	C₄×Q₁₆	—	37	1	no	—
52	#334	—	37	1	no	—
53	Q₈⋊C₈	—	37	2	no	—
54	#322	—	37	1	no	—
55	C₃₇	#1579	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₁₆	#3866	1	1	yes	characteristic
57	C₄×Q₈	—	37	2	no	—
58	#107	—	37	1	no	—
59	C₄⋊C₈	—	37	2	no	—
60	C₄⋊C₈	—	74	2	no	—
61	#103	—	37	2	no	—
62	C₄×C₈	—	37	1	no	—
63	C₄×C₈	—	37	1	no	—
64	C₄×C₈	—	37	1	no	—
65	C₄×C₈	—	74	2	no	—
66	C₂×Q₈	C₂×#3866	1	2	yes	—
67	Q₁₆	#64438	1	4	yes	complemented
68	C₂×C₈	—	37	2	no	—
69	C₂×C₈	—	37	1	no	—
70	C₂×C₈	—	74	1	no	—
71	C₂×C₈	—	74	2	no	—
72	C₂×C₈	C₂×#3866	1	1	yes	characteristic
73	C₄⋊C₄	—	37	2	no	—
74	C₄⋊C₄	—	74	2	no	—
75	C₄²	—	37	1	no	—
76	C₄²	—	74	2	no	—
77	Q₈	—	2	2	no	—
78	Q₈	C₂×#64438	1	4	yes	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	37	1	no	—
81	C₂×C₄	—	37	1	no	—
82	C₂×C₄	—	74	2	no	—
83	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
84	C₈	—	74	2	no	—
85	C₈	—	74	1	no	—
86	C₈	—	148	2	no	—
87	C₈	—	148	1	no	—
88	C₈	C₂×#64438	1	2	yes	—
89	C₂²	D₄×#3866	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	37	2	no	—
92	C₄	—	74	1	no	—
93	C₄	—	148	2	no	—
94	C₄	C₂²×#64438	1	1	yes	characteristic
95	C₄	#11924623	1	1	yes	characteristic
96	C₂	D₄×#64438	1	1	yes	characteristic
97	C₂	#14558573	1	1	yes	characteristic
98	C₂	#14558579	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#64438

C₂

yes

maximal

#14558394

C₂

yes

maximal, characteristic

Q₈⋊#64438

C₂

yes

maximal

#14558374

C₂

yes

maximal, characteristic

Q₁₆×Dic₃₇

C₂

yes

maximal, characteristic

C₄⋊#64438

—

C₄⋊#64438

C₂²

yes

—

C₄×#64438

—

C₄×#64438

C₂²

yes

characteristic

C₄×#64438

C₂²

yes

characteristic

Q₈×Dic₃₇

C₂²

yes

—

#11924470

C₄

yes

characteristic

#11924460

C₄

yes

—

C₈×Dic₃₇

C₂²

yes

characteristic

C₂×Q₁₆×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂×#64438

—

C₂×#64438

—

C₂×#64438

D₄

yes

—

Q₁₆×C₃₇

C₈

yes

complemented

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

C₂×C₈

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

C₂×Dic₃₇

yes

—

—

—

C₈×C₃₇

C₂×C₈

yes

—

#64436

—

C₂²×C₃₇

C₄×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₂²×C₈

yes

characteristic

C₄×C₃₇

#131

yes

characteristic

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#137

yes

—

#1579

—

maximal, Sylow

C₂×C₃₇

C₈×D₄

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₈×Q₈

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

—

C₃₇

#1579

yes

characteristic, complemented, Sylow

C₂×Q₁₆

#3866

yes

characteristic

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

C₂×#3866

yes

—

Q₁₆

#64438

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

Q₈

—

Q₈

C₂×#64438

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×#3866

yes

characteristic

C₈

—

C₈

—

C₈

—

148

—

C₈

—

148

—

C₈

C₂×#64438

yes

—

C₂²

D₄×#3866

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

148

—

C₄

C₂²×#64438

yes

C₄

yes

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	D₄×#3866
commutator	C₄×C₃₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₂×Q₁₆×C₃₇	C₄
socle, prime radical	C₂²×C₃₇	C₄×D₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₂×Q₁₆	#3866
hypocenter	C₃₇	#1579
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#3866

commutator

C₄×C₃₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₂×Q₁₆×C₃₇

C₄

socle, prime radical

C₂²×C₃₇

C₄×D₄

cosocle

C₄×Dic₃₇

C₂³

C₂×Q₁₆

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
20. Q₁₆×C₃₇	C₈	semidirect
55. C₃₇	#1579	semidirect
67. Q₁₆	#64438	semidirect

H ≅ G/N

type

20. Q₁₆×C₃₇

C₈

semidirect

55. C₃₇

#1579

semidirect

67. Q₁₆

#64438

semidirect

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

subgroup	index
Q₈×#64438	2
#14558394	2
Q₈⋊#64438	2
#14558374	2
Q₁₆×Dic₃₇	2
#1579	37

subgroup

index

Q₁₆×Dic₃₇

Subgroups — #18010189

Conjugacy classes of subgroups

Complements

Maximal subgroups