Subgroups — C₃×C₅×#872 #13087883

Order 1440.

334 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal
4	C₃×C₅×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#117	—	3	1	no	maximal
7	C₅×#872	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×D₃	C₄	1	4	yes	complemented
14	C₃×#872	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₃×C₄×C₅	—	3	1	no	—
16	C₂×C₄×C₅×D₃	C₂×C₃	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	—	3	2	no	—
18	C₃×C₅×C₂²⋊C₄	—	3	2	no	—
19	C₃×C₄²×C₅	—	2	1	no	—
20	C₃×C₄²×C₅	—	3	1	no	—
21	C₃×C₄²×C₅	D₃	1	1	yes	characteristic, complemented
22	C₃×C₄²×C₅	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×(C₂×D₃)⋊C₄	C₂×C₃	1	2	yes	—
24	C₅×Dic₃⋊C₄	C₂×C₃	1	2	yes	complemented
25	C₄×C₅×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₅×D₃	—	2	2	no	—
28	C₂×C₃×C₅×D₃	C₂×C₄	1	2	yes	—
29	C₃²×C₄×C₅	—	2	2	no	—
30	C₃²×C₄×C₅	C₂×C₄	1	2	yes	—
31	C₃×C₅×Dic₃	—	2	2	no	—
32	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	—
33	C₅×#117	—	3	1	no	Hall
34	C₂×C₃×C₄×D₃	C₂×C₅	1	1	yes	characteristic
35	C₃²×C₄²	C₂×C₅	1	1	yes	characteristic, complemented
36	C₃×(C₂×D₃)⋊C₄	C₂×C₅	1	2	yes	—
37	C₃×Dic₃⋊C₄	C₂×C₅	1	2	yes	complemented
38	C₃×C₄×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
39	C₂³×C₃×C₅	—	3	1	no	—
40	C₂²×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	—	2	2	no	—
42	C₂×C₃×C₄×C₅	—	2	1	no	—
43	C₂×C₃×C₄×C₅	—	3	1	no	—
44	C₂×C₃×C₄×C₅	—	3	4	no	—
45	C₂×C₃×C₄×C₅	—	3	2	no	—
46	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂²×C₃	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
51	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
52	C₄×C₅×D₃	C₃×C₄	1	4	yes	complemented
53	C₃×#117	—	3	1	no	—
54	#872	C₃×C₅	1	1	yes	characteristic, direct product factor
55	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃²×C₅	#41	1	2	yes	—
57	C₃×C₅×D₃	—	2	2	no	—
58	C₂²×C₄×C₅	—	3	1	no	—
59	C₅×C₄⋊C₄	—	3	2	no	—
60	C₅×C₂²⋊C₄	—	3	2	no	—
61	C₄²×C₅	—	3	1	no	—
62	C₄²×C₅	C₃×D₃	1	1	yes	characteristic, complemented
63	C₂²×C₃×D₃	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	—
65	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
66	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
67	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
68	C₃×C₄×D₃	C₄×C₅	1	4	yes	complemented
69	C₂²×C₃×C₅	—	2	1	no	—
70	C₂²×C₃×C₅	—	3	2	no	—
71	C₂²×C₃×C₅	—	6	2	no	—
72	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
73	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
74	C₂×C₅×D₃	—	2	2	no	—
75	C₂×C₅×D₃	C₂×C₃×C₄	1	2	yes	—
76	C₃×C₄×C₅	—	2	4	no	—
77	C₃×C₄×C₅	—	2	2	no	—
78	C₃×C₄×C₅	—	2	2	no	—
79	C₃×C₄×C₅	—	2	2	no	—
80	C₃×C₄×C₅	—	3	2	no	—
81	C₃×C₄×C₅	—	6	2	no	—
82	C₃×C₄×C₅	C₄×D₃	1	2	yes	—
83	C₃×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
84	C₅×Dic₃	—	2	2	no	—
85	C₅×Dic₃	C₂×C₃×C₄	1	2	yes	—
86	C₂²×C₃×C₄	—	3	1	no	—
87	C₂×C₄×D₃	C₂×C₃×C₅	1	1	yes	characteristic
88	C₃×C₄⋊C₄	—	3	2	no	—
89	C₃×C₂²⋊C₄	—	3	2	no	—
90	C₃×C₄²	—	2	1	no	—
91	C₃×C₄²	—	3	1	no	—
92	C₃×C₄²	C₅×D₃	1	1	yes	characteristic, complemented
93	C₃×C₄²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
94	(C₂×D₃)⋊C₄	C₂×C₃×C₅	1	2	yes	—
95	Dic₃⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
96	C₄×Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
97	C₃²×C₅	#117	1	1	yes	characteristic, complemented, Hall
98	C₂³×C₅	—	3	1	no	—
99	C₂×C₄×C₅	—	3	1	no	—
100	C₂×C₄×C₅	—	3	4	no	—
101	C₂×C₄×C₅	—	3	2	no	—
102	C₂×C₄×C₅	C₂×C₃×D₃	1	2	yes	—
103	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
104	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
105	C₂×C₃×D₃	—	2	2	no	—
106	C₂×C₃×D₃	C₂×C₄×C₅	1	2	yes	—
107	C₃²×C₄	—	2	2	no	—
108	C₃²×C₄	C₂×C₄×C₅	1	2	yes	—
109	C₃×Dic₃	—	2	2	no	—
110	C₃×Dic₃	C₂×C₄×C₅	1	2	yes	—
111	#117	—	3	1	no	Sylow
112	C₂×C₃×C₅	—	2	1	no	—
113	C₂×C₃×C₅	—	2	2	no	—
114	C₂×C₃×C₅	—	6	2	no	—
115	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
116	C₂×C₃×C₅	#235	1	2	yes	—
117	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
118	C₂×C₃×C₅	C₃×#41	1	2	yes	—
119	C₅×D₃	—	2	2	no	—
120	C₂³×C₃	—	3	1	no	—
121	C₂²×D₃	C₂²×C₃×C₅	1	1	yes	characteristic
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	3	1	no	—
125	C₂×C₃×C₄	—	3	4	no	—
126	C₂×C₃×C₄	—	3	2	no	—
127	C₂×C₃×C₄	C₂×C₅×D₃	1	2	yes	—
128	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂²×C₃×C₅	1	2	yes	—
130	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
131	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
132	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
133	C₄×D₃	C₃×C₄×C₅	1	4	yes	complemented
134	C₂²×C₅	—	3	2	no	—
135	C₂²×C₅	—	6	2	no	—
136	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
137	C₄×C₅	—	2	2	no	—
138	C₄×C₅	—	3	2	no	—
139	C₄×C₅	—	6	2	no	—
140	C₄×C₅	C₃×C₄×D₃	1	2	yes	—
141	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
142	C₂×C₃²	C₅×#41	1	2	yes	—
143	C₃×D₃	—	2	2	no	—
144	C₂²×C₄	—	3	1	no	—
145	C₄⋊C₄	—	3	2	no	—
146	C₂²⋊C₄	—	3	2	no	—
147	C₄²	—	3	1	no	—
148	C₄²	C₃×C₅×D₃	1	1	yes	characteristic, complemented
149	C₃×C₅	—	2	1	no	—
150	C₃×C₅	#872	1	1	yes	characteristic, direct product factor
151	C₃×C₅	C₃×#117	1	1	yes	characteristic, complemented
152	C₂²×C₃	—	2	1	no	—
153	C₂²×C₃	—	3	2	no	—
154	C₂²×C₃	—	6	2	no	—
155	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
156	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
157	C₂×D₃	—	2	2	no	—
158	C₂×D₃	C₂×C₃×C₄×C₅	1	2	yes	—
159	C₃×C₄	—	2	4	no	—
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	—	2	2	no	—
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	—	3	2	no	—
164	C₃×C₄	—	6	2	no	—
165	C₃×C₄	C₄×C₅×D₃	1	2	yes	—
166	C₃×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
167	Dic₃	—	2	2	no	—
168	Dic₃	C₂×C₃×C₄×C₅	1	2	yes	—
169	C₂×C₅	—	6	2	no	—
170	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
171	C₂×C₅	C₃×#235	1	2	yes	—
172	C₃²	C₅×#117	1	1	yes	characteristic, complemented, Sylow
173	C₂³	—	3	1	no	—
174	C₂×C₄	—	3	1	no	—
175	C₂×C₄	—	3	4	no	—
176	C₂×C₄	—	3	2	no	—
177	C₂×C₄	C₂×C₃×C₅×D₃	1	2	yes	—
178	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
179	C₂×C₃	—	2	1	no	—
180	C₂×C₃	—	2	2	no	—
181	C₂×C₃	—	6	2	no	—
182	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
183	C₂×C₃	C₅×#235	1	2	yes	—
184	C₂×C₃	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
185	C₂×C₃	C₃×C₅×#41	1	2	yes	—
186	D₃	—	2	2	no	—
187	C₅	C₃×#872	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²	—	3	2	no	—
189	C₂²	—	6	2	no	—
190	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	3	2	no	—
193	C₄	—	6	2	no	—
194	C₄	C₃×C₄×C₅×D₃	1	2	yes	—
195	C₃	—	2	1	no	—
196	C₃	C₅×#872	1	1	yes	characteristic, direct product factor
197	C₃	C₃×C₅×#117	1	1	yes	characteristic, complemented
198	C₂	—	6	2	no	—
199	C₂	C₂×C₃×C₄×C₅×D₃	1	1	yes	characteristic
200	C₂	C₃×C₅×#235	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₃
commutator	C₂×C₃	C₂²×C₃×C₄×C₅
Frattini	C₂²	C₂²×C₃×C₅×D₃
Fitting	C₃²×C₄²×C₅	C₂
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂²×C₃×C₅×D₃	C₂²
cosocle	C₂²×C₃	C₂³×C₃×C₅
hypercenter	C₃×C₄²×C₅	D₃
hypocenter	C₃	C₃×C₅×#117
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₄²×C₅

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄²×C₅	C₂	semidirect
4. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
5. C₃×C₄×C₅×Dic₃	C₂	semidirect
7. C₅×#872	C₃	direct (N × H)
13. C₃×C₄×C₅×D₃	C₄	semidirect
14. C₃×#872	C₅	direct (N × H)
21. C₃×C₄²×C₅	D₃	semidirect
22. C₃×C₄²×C₅	C₂×C₃	semidirect
24. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
25. C₄×C₅×Dic₃	C₂×C₃	semidirect
35. C₃²×C₄²	C₂×C₅	semidirect
37. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
38. C₃×C₄×Dic₃	C₂×C₅	semidirect
52. C₄×C₅×D₃	C₃×C₄	semidirect
54. #872	C₃×C₅	direct (N × H)
62. C₄²×C₅	C₃×D₃	semidirect
68. C₃×C₄×D₃	C₄×C₅	semidirect
92. C₃×C₄²	C₅×D₃	semidirect
93. C₃×C₄²	C₂×C₃×C₅	semidirect
95. Dic₃⋊C₄	C₂×C₃×C₅	semidirect
96. C₄×Dic₃	C₂×C₃×C₅	semidirect
97. C₃²×C₅	#117	semidirect
133. C₄×D₃	C₃×C₄×C₅	semidirect
148. C₄²	C₃×C₅×D₃	semidirect
150. C₃×C₅	#872	direct (N × H)
151. C₃×C₅	C₃×#117	semidirect
172. C₃²	C₅×#117	semidirect
187. C₅	C₃×#872	direct (N × H)
196. C₃	C₅×#872	direct (N × H)
197. C₃	C₃×C₅×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×D₃	2
C₃²×C₄²×C₅	2
C₃×C₅×(C₂×D₃)⋊C₄	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₄×C₅×Dic₃	2
C₃×C₅×#117	3
C₅×#872	3
C₃×#872	5

Subgroups — C3×C5×#872 #13087883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#872 #13087883