Subgroups — C₃×C₅×#2253 #14275287

Order 1920.

232 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#499	C₂	1	2	yes	maximal
2	C₃×C₅×#367	C₂	1	2	yes	maximal
3	C₃×C₅×#366	C₂	1	2	yes	maximal
4	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₅×#2253	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₅×#125	—	2	2	no	—
7	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	C₃×C₅×#107	C₂²	1	4	yes	—
10	C₃×C₅×#106	—	2	2	no	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×#2253	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₅×#499	C₂×C₃	1	2	yes	—
14	C₅×#367	C₂×C₃	1	2	yes	—
15	C₅×#366	C₂×C₃	1	2	yes	—
16	C₄×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₄	—	2	4	no	—
20	C₃×C₅×C₄⋊C₄	—	4	4	no	—
21	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
22	C₃×C₅×C₁₆	Q₈	1	2	yes	—
23	C₃×C₅×C₁₆	Q₈	1	2	yes	—
24	C₃×#499	C₂×C₅	1	2	yes	—
25	C₃×#367	C₂×C₅	1	2	yes	—
26	C₃×#366	C₂×C₅	1	2	yes	—
27	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
28	C₅×#125	—	2	2	no	—
29	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
30	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
31	C₅×#107	C₂²×C₃	1	4	yes	—
32	C₅×#106	—	2	2	no	—
33	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
34	#2253	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₃×C₄×C₅	—	4	4	no	—
36	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
37	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
38	C₃×C₅×C₈	Q₁₆	1	2	yes	—
39	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
40	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
41	C₃×#125	—	2	2	no	—
42	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
43	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
44	C₃×#107	C₂²×C₅	1	4	yes	—
45	C₃×#106	—	2	2	no	—
46	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
47	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
49	C₅×C₄⋊C₄	—	2	4	no	—
50	C₅×C₄⋊C₄	—	4	4	no	—
51	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
52	C₅×C₁₆	C₃×Q₈	1	2	yes	—
53	C₅×C₁₆	C₃×Q₈	1	2	yes	—
54	#499	C₂×C₃×C₅	1	2	yes	—
55	#367	C₂×C₃×C₅	1	2	yes	—
56	#366	C₂×C₃×C₅	1	2	yes	—
57	C₄×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
58	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
59	C₃×C₄×C₅	—	2	2	no	—
60	C₃×C₄×C₅	—	8	4	no	—
61	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
62	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
65	C₃×C₄⋊C₄	—	2	4	no	—
66	C₃×C₄⋊C₄	—	4	4	no	—
67	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
68	C₃×C₁₆	C₅×Q₈	1	2	yes	—
69	C₃×C₁₆	C₅×Q₈	1	2	yes	—
70	C₂×C₄×C₅	—	4	4	no	—
71	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
72	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
73	C₅×C₈	C₃×Q₁₆	1	2	yes	—
74	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
75	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
76	#125	—	2	2	no	—
77	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
78	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
79	#107	C₂²×C₃×C₅	1	4	yes	—
80	#106	—	2	2	no	—
81	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
82	C₂×C₃×C₅	#500	1	1	yes	characteristic
83	C₂×C₃×C₅	#508	1	2	yes	—
84	C₂×C₃×C₄	—	4	4	no	—
85	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
86	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
87	C₃×C₈	C₅×Q₁₆	1	2	yes	—
88	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
89	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
90	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
91	C₄×C₅	—	2	2	no	—
92	C₄×C₅	—	8	4	no	—
93	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
94	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
95	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
96	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
97	C₄⋊C₄	—	2	4	no	—
98	C₄⋊C₄	—	4	4	no	—
99	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
100	C₁₆	C₃×C₅×Q₈	1	2	yes	—
101	C₁₆	C₃×C₅×Q₈	1	2	yes	—
102	C₃×C₅	#2253	1	1	yes	characteristic, direct product factor, Hall
103	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
104	C₃×C₄	—	2	2	no	—
105	C₃×C₄	—	8	4	no	—
106	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
107	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
108	C₂×C₅	C₃×#500	1	1	yes	characteristic
109	C₂×C₅	C₃×#508	1	2	yes	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
112	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
113	C₈	C₃×C₅×Q₁₆	1	2	yes	—
114	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
115	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
116	C₂×C₃	C₅×#500	1	1	yes	characteristic
117	C₂×C₃	C₅×#508	1	2	yes	—
118	C₅	C₃×#2253	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	8	4	no	—
122	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
123	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
124	C₃	C₅×#2253	1	1	yes	characteristic, direct product factor, Sylow
125	C₂	C₃×C₅×#500	1	1	yes	characteristic
126	C₂	C₃×C₅×#508	1	2	yes	—

	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
5. C₅×#2253	C₃	direct (N × H)
12. C₃×#2253	C₅	direct (N × H)
34. #2253	C₃×C₅	direct (N × H)
102. C₃×C₅	#2253	direct (N × H)
118. C₅	C₃×#2253	direct (N × H)
124. C₃	C₅×#2253	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×#499	2
C₃×C₅×#367	2
C₃×C₅×#366	2
C₃×C₄×C₅×C₁₆	2
C₅×#2253	3
C₃×#2253	5

Subgroups — C3×C5×#2253 #14275287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2253 #14275287