Subgroups — C₃×C₄²×#168 #14294186

Order 1920.

816 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×#168	C₂	1	6	yes	maximal
3	C₄²×#168	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₄²×C₅	C₄	1	4	yes	—
5	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
6	C₂²×C₃×#168	C₂²	1	4	yes	—
7	C₃×C₄×#168	C₄	1	24	yes	direct product factor
8	C₃×C₄²×C₈	—	5	1	no	maximal, Hall
9	C₄³×C₅	C₂×C₃	1	1	yes	characteristic
10	C₂×C₄×#168	C₂×C₃	1	6	yes	—
11	C₂²×C₃×C₄×C₅	C₂×C₄	1	6	yes	—
12	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
13	C₃×C₄²×C₅	C₈	1	16	yes	complemented
14	C₃×C₄²×C₅	C₂×C₄	1	12	yes	—
15	C₂×C₃×#168	C₂×C₄	1	24	yes	—
16	C₂×C₃×C₄×C₈	—	5	6	no	—
17	C₃×C₄³	D₅	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₃×C₄	1	4	yes	—
19	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
20	C₂²×#168	C₂²×C₃	1	4	yes	—
21	C₄×#168	C₃×C₄	1	24	yes	direct product factor
22	C₄²×C₈	—	5	1	no	Sylow
23	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₅	C₄²	1	12	yes	—
25	C₂×C₃×C₄×C₅	C₂×C₈	1	24	yes	—
26	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
27	C₃×#168	C₄²	1	16	yes	direct product factor
28	C₂²×C₃×C₈	—	5	4	no	—
29	C₂×C₃×C₄²	Dic₅	1	4	yes	—
30	C₂×C₃×C₄²	C₂×D₅	1	3	yes	—
31	C₃×C₄×C₈	—	5	24	no	—
32	C₂²×C₄×C₅	C₂×C₃×C₄	1	6	yes	—
33	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
34	C₄²×C₅	C₃×C₈	1	16	yes	complemented
35	C₄²×C₅	C₂×C₃×C₄	1	12	yes	—
36	C₂×#168	C₂×C₃×C₄	1	24	yes	—
37	C₂×C₄×C₈	—	5	6	no	—
38	C₄³	C₃×D₅	1	1	yes	characteristic
39	C₂²×C₃×C₅	C₂×C₄²	1	3	yes	—
40	C₂²×C₃×C₅	C₂²×C₈	1	4	yes	—
41	C₃×C₄×C₅	C₄×C₈	1	24	yes	complemented
42	C₃×C₄×C₅	C₂×C₄²	1	4	yes	—
43	C₂²×C₃×C₄	C₂×Dic₅	1	6	yes	—
44	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
45	C₂×C₃×C₈	—	5	24	no	—
46	C₃×C₄²	#168	1	16	yes	direct product factor
47	C₃×C₄²	C₄×D₅	1	12	yes	—
48	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅	C₃×C₄²	1	12	yes	—
50	C₂×C₄×C₅	C₂×C₃×C₈	1	24	yes	—
51	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
52	#168	C₃×C₄²	1	16	yes	direct product factor
53	C₂²×C₈	—	5	4	no	—
54	C₂×C₄²	C₃×Dic₅	1	4	yes	—
55	C₂×C₄²	C₂×C₃×D₅	1	3	yes	—
56	C₄×C₈	—	5	24	no	—
57	C₂×C₃×C₅	C₄³	1	1	yes	characteristic
58	C₂×C₃×C₅	C₂×C₄×C₈	1	6	yes	—
59	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂×#168	1	24	yes	—
61	C₂×C₃×C₄	C₄×Dic₅	1	12	yes	—
62	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
63	C₃×C₈	—	5	16	no	—
64	C₂²×C₅	C₂×C₃×C₄²	1	3	yes	—
65	C₂²×C₅	C₂²×C₃×C₈	1	4	yes	—
66	C₄×C₅	C₃×C₄×C₈	1	24	yes	complemented
67	C₄×C₅	C₂×C₃×C₄²	1	4	yes	—
68	C₂²×C₄	C₂×C₃×Dic₅	1	6	yes	—
69	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
70	C₂×C₈	—	5	24	no	—
71	C₄²	C₃×#168	1	16	yes	direct product factor
72	C₄²	C₃×C₄×D₅	1	12	yes	—
73	C₃×C₅	C₄²×C₈	1	1	yes	characteristic, complemented, Hall
74	C₂²×C₃	C₂²×#168	1	4	yes	—
75	C₂²×C₃	C₂×C₄×Dic₅	1	3	yes	—
76	C₃×C₄	C₄×#168	1	24	yes	direct product factor
77	C₃×C₄	C₄²×D₅	1	4	yes	—
78	C₂×C₅	C₃×C₄³	1	1	yes	characteristic
79	C₂×C₅	C₂×C₃×C₄×C₈	1	6	yes	—
80	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
81	C₂×C₄	C₂×C₃×#168	1	24	yes	—
82	C₂×C₄	C₃×C₄×Dic₅	1	12	yes	—
83	C₂×C₄	C₂×C₃×C₄×D₅	1	6	yes	—
84	C₈	—	5	16	no	—
85	C₂×C₃	C₂×C₄×#168	1	6	yes	—
86	C₂×C₃	C₄²×Dic₅	1	1	yes	characteristic
87	C₅	C₃×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
88	C₂²	C₂²×C₃×#168	1	4	yes	—
89	C₂²	C₂×C₃×C₄×Dic₅	1	3	yes	—
90	C₄	C₃×C₄×#168	1	24	yes	direct product factor
91	C₄	C₃×C₄²×D₅	1	4	yes	—
92	C₃	C₄²×#168	1	1	yes	characteristic, direct product factor, Sylow
93	C₂	C₂×C₃×C₄×#168	1	6	yes	—
94	C₂	C₃×C₄²×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃×C₄³	D₅
commutator, hypocenter	C₅	C₃×C₄²×C₈
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₄³×C₅	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₄³

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₄²×#168	C₃	direct (N × H)
7. C₃×C₄×#168	C₄	direct (N × H)
13. C₃×C₄²×C₅	C₈	semidirect
21. C₄×#168	C₃×C₄	direct (N × H)
27. C₃×#168	C₄²	direct (N × H)
34. C₄²×C₅	C₃×C₈	semidirect
41. C₃×C₄×C₅	C₄×C₈	semidirect
46. C₃×C₄²	#168	direct (N × H)
52. #168	C₃×C₄²	direct (N × H)
66. C₄×C₅	C₃×C₄×C₈	semidirect
71. C₄²	C₃×#168	direct (N × H)
73. C₃×C₅	C₄²×C₈	semidirect
76. C₃×C₄	C₄×#168	direct (N × H)
87. C₅	C₃×C₄²×C₈	semidirect
90. C₄	C₃×C₄×#168	direct (N × H)
92. C₃	C₄²×#168	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄³×C₅	2
C₂×C₃×C₄×#168	2
C₄²×#168	3
C₃×C₄²×C₈	5

Subgroups — C3×C42×#168 #14294186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×#168 #14294186