Subgroups — C₃×C₅×#2252 #14275286

Order 1920.

256 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#500	C₂	1	2	yes	maximal
2	C₃×C₅×#366	C₂	1	4	yes	maximal
3	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×#2252	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
6	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
7	C₃×C₅×#107	—	2	2	no	—
8	C₃×C₅×#107	C₂²	1	4	yes	—
9	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₃×#2252	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₅×#500	C₂×C₃	1	2	yes	—
12	C₅×#366	C₂×C₃	1	4	yes	—
13	C₄×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic
14	C₂×C₃×C₅×Q₈	—	4	2	no	—
15	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
16	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	—	2	4	no	—
18	C₃×C₅×C₄⋊C₄	—	4	2	no	—
19	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
20	C₃×C₅×C₁₆	Q₈	1	4	yes	complemented
21	C₃×#500	C₂×C₅	1	2	yes	—
22	C₃×#366	C₂×C₅	1	4	yes	—
23	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
24	C₅×C₄⋊Q₈	—	2	2	no	—
25	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
26	C₅×#107	—	2	2	no	—
27	C₅×#107	C₂²×C₃	1	4	yes	—
28	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
29	#2252	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
30	C₃×C₅×Q₈	—	8	4	no	—
31	C₂×C₃×C₄×C₅	—	4	4	no	—
32	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
33	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
34	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
35	C₃×C₅×C₈	Q₁₆	1	2	yes	—
36	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
37	C₃×C₄⋊Q₈	—	2	2	no	—
38	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
39	C₃×#107	—	2	2	no	—
40	C₃×#107	C₂²×C₅	1	4	yes	—
41	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
42	C₂×C₅×Q₈	—	4	2	no	—
43	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
45	C₅×C₄⋊C₄	—	2	4	no	—
46	C₅×C₄⋊C₄	—	4	2	no	—
47	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
48	C₅×C₁₆	C₃×Q₈	1	4	yes	complemented
49	#500	C₂×C₃×C₅	1	2	yes	—
50	#366	C₂×C₃×C₅	1	4	yes	—
51	C₄×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
52	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
53	C₃×C₄×C₅	—	8	4	no	—
54	C₃×C₄×C₅	D₁₆	1	2	yes	—
55	C₃×C₄×C₅	Q₃₂	1	2	yes	—
56	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
57	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₃×Q₈	—	4	2	no	—
59	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
60	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
61	C₃×C₄⋊C₄	—	2	4	no	—
62	C₃×C₄⋊C₄	—	4	2	no	—
63	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
64	C₃×C₁₆	C₅×Q₈	1	4	yes	complemented
65	C₅×Q₈	—	8	4	no	—
66	C₂×C₄×C₅	—	4	4	no	—
67	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
70	C₅×C₈	C₃×Q₁₆	1	2	yes	—
71	C₅×C₈	C₂×C₃×Q₈	1	2	yes	—
72	C₄⋊Q₈	—	2	2	no	—
73	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
74	#107	—	2	2	no	—
75	#107	C₂²×C₃×C₅	1	4	yes	—
76	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
77	C₂×C₃×C₅	#500	1	1	yes	characteristic
78	C₂×C₃×C₅	C₂×D₁₆	1	1	yes	characteristic
79	C₂×C₃×C₅	C₂×Q₃₂	1	1	yes	characteristic
80	C₃×Q₈	—	8	4	no	—
81	C₂×C₃×C₄	—	4	4	no	—
82	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
83	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
85	C₃×C₈	C₅×Q₁₆	1	2	yes	—
86	C₃×C₈	C₂×C₅×Q₈	1	2	yes	—
87	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
88	C₄×C₅	—	8	4	no	—
89	C₄×C₅	C₃×D₁₆	1	2	yes	—
90	C₄×C₅	C₃×Q₃₂	1	2	yes	—
91	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
92	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
93	C₂×Q₈	—	4	2	no	—
94	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
99	C₁₆	C₃×C₅×Q₈	1	4	yes	complemented
100	C₃×C₅	#2252	1	1	yes	characteristic, direct product factor, Hall
101	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
102	C₃×C₄	—	8	4	no	—
103	C₃×C₄	C₅×D₁₆	1	2	yes	—
104	C₃×C₄	C₅×Q₃₂	1	2	yes	—
105	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
106	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
107	C₂×C₅	C₃×#500	1	1	yes	characteristic
108	C₂×C₅	C₂×C₃×D₁₆	1	1	yes	characteristic
109	C₂×C₅	C₂×C₃×Q₃₂	1	1	yes	characteristic
110	Q₈	—	8	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
113	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
114	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
115	C₈	C₃×C₅×Q₁₆	1	2	yes	—
116	C₈	C₂×C₃×C₅×Q₈	1	2	yes	—
117	C₂×C₃	C₅×#500	1	1	yes	characteristic
118	C₂×C₃	C₂×C₅×D₁₆	1	1	yes	characteristic
119	C₂×C₃	C₂×C₅×Q₃₂	1	1	yes	characteristic
120	C₅	C₃×#2252	1	1	yes	characteristic, direct product factor, Sylow
121	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
122	C₄	—	8	4	no	—
123	C₄	C₃×C₅×D₁₆	1	2	yes	—
124	C₄	C₃×C₅×Q₃₂	1	2	yes	—
125	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
126	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
127	C₃	C₅×#2252	1	1	yes	characteristic, direct product factor, Sylow
128	C₂	C₃×C₅×#500	1	1	yes	characteristic
129	C₂	C₂×C₃×C₅×D₁₆	1	1	yes	characteristic
130	C₂	C₂×C₃×C₅×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#2252	C₃	direct (N × H)
10. C₃×#2252	C₅	direct (N × H)
20. C₃×C₅×C₁₆	Q₈	semidirect
29. #2252	C₃×C₅	direct (N × H)
48. C₅×C₁₆	C₃×Q₈	semidirect
64. C₃×C₁₆	C₅×Q₈	semidirect
99. C₁₆	C₃×C₅×Q₈	semidirect
100. C₃×C₅	#2252	direct (N × H)
120. C₅	C₃×#2252	direct (N × H)
127. C₃	C₅×#2252	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×C₅×#500	2
C₃×C₅×#366	2
C₃×C₄×C₅×C₁₆	2
C₅×#2252	3
C₃×#2252	5

Subgroups — C3×C5×#2252 #14275286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2252 #14275286