Subgroups — C₃×C₅×#877 #13087888

Order 1440.

338 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#217	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₅×Dic₃	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#117	—	3	1	no	maximal
7	C₅×#877	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
10	C₂×C₃×C₅×Dic₃	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
14	C₃×#877	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₃×C₄×C₅	—	3	1	no	—
16	C₂²×C₅×Dic₃	C₂×C₃	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	—	3	2	no	—
18	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
19	C₃×C₅×C₂²⋊C₄	—	3	1	no	—
20	C₃×C₅×C₂²⋊C₄	D₃	1	1	yes	characteristic
21	C₃×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
22	C₃×C₄²×C₅	—	3	2	no	—
23	C₅×#217	C₂×C₃	1	1	yes	characteristic
24	C₅×Dic₃⋊C₄	C₂×C₃	1	2	yes	complemented
25	C₄×C₅×Dic₃	C₂×C₃	1	2	yes	complemented
26	C₂²×C₃²×C₅	—	2	2	no	—
27	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
29	C₃²×C₄×C₅	—	2	2	no	—
30	C₃×C₅×Dic₃	—	2	2	no	—
31	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	—
32	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	—
33	C₅×#117	—	3	1	no	Hall
34	C₂²×C₃×Dic₃	C₂×C₅	1	1	yes	characteristic
35	C₃²×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
36	C₃×#217	C₂×C₅	1	1	yes	characteristic
37	C₃×Dic₃⋊C₄	C₂×C₅	1	2	yes	complemented
38	C₃×C₄×Dic₃	C₂×C₅	1	2	yes	complemented
39	C₂³×C₃×C₅	—	2	1	no	—
40	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
41	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	—	2	2	no	—
43	C₂×C₃×C₄×C₅	—	3	4	no	—
44	C₂×C₃×C₄×C₅	—	3	1	no	—
45	C₂×C₃×C₄×C₅	—	3	1	no	—
46	C₂×C₃×C₄×C₅	—	3	2	no	—
47	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂²×C₃	1	2	yes	—
49	C₂×C₅×Dic₃	C₃×C₄	1	4	yes	complemented
50	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
51	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
53	C₃×#117	—	3	1	no	—
54	#877	C₃×C₅	1	1	yes	characteristic, direct product factor
55	C₂×C₃²×C₅	—	2	2	no	—
56	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃²×C₅	#41	1	2	yes	—
58	C₂²×C₄×C₅	—	3	1	no	—
59	C₅×C₄⋊C₄	—	3	2	no	—
60	C₅×C₂²⋊C₄	—	3	1	no	—
61	C₅×C₂²⋊C₄	C₃×D₃	1	1	yes	characteristic
62	C₄²×C₅	—	3	2	no	—
63	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	—
65	C₂×C₃×Dic₃	C₄×C₅	1	4	yes	complemented
66	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
67	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
68	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
69	C₂²×C₃×C₅	—	2	2	no	—
70	C₂²×C₃×C₅	—	2	2	no	—
71	C₂²×C₃×C₅	—	2	2	no	—
72	C₂²×C₃×C₅	—	2	1	no	—
73	C₂²×C₃×C₅	—	4	2	no	—
74	C₂²×C₃×C₅	C₄×D₃	1	2	yes	—
75	C₂²×C₃×C₅	C₂×C₃×C₄	1	2	yes	—
76	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
77	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
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	—	3	2	no	—
82	C₃×C₄×C₅	—	4	2	no	—
83	C₃×C₄×C₅	—	6	2	no	—
84	C₅×Dic₃	—	2	2	no	—
85	C₅×Dic₃	C₂×C₃×C₄	1	2	yes	—
86	C₅×Dic₃	C₂×C₃×C₄	1	2	yes	—
87	C₂²×C₃×C₄	—	3	1	no	—
88	C₂²×Dic₃	C₂×C₃×C₅	1	1	yes	characteristic
89	C₃×C₄⋊C₄	—	3	2	no	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₂²⋊C₄	—	3	1	no	—
92	C₃×C₂²⋊C₄	C₅×D₃	1	1	yes	characteristic
93	C₃×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
94	C₃×C₄²	—	3	2	no	—
95	#217	C₂×C₃×C₅	1	1	yes	characteristic
96	Dic₃⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
97	C₄×Dic₃	C₂×C₃×C₅	1	2	yes	complemented
98	C₃²×C₅	#117	1	1	yes	characteristic, complemented, Hall
99	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
100	C₂×C₄×C₅	—	3	4	no	—
101	C₂×C₄×C₅	—	3	1	no	—
102	C₂×C₄×C₅	—	3	1	no	—
103	C₂×C₄×C₅	—	3	2	no	—
104	C₂×C₄×C₅	C₂×C₃×D₃	1	2	yes	—
105	C₂²×C₃²	—	2	2	no	—
106	C₂²×C₃²	C₂×C₄×C₅	1	2	yes	—
107	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
108	C₃²×C₄	—	2	2	no	—
109	C₃×Dic₃	—	2	2	no	—
110	C₃×Dic₃	C₂×C₄×C₅	1	2	yes	—
111	C₃×Dic₃	C₂×C₄×C₅	1	2	yes	—
112	#117	—	3	1	no	Sylow
113	C₂×C₃×C₅	—	2	2	no	—
114	C₂×C₃×C₅	—	2	2	no	—
115	C₂×C₃×C₅	—	2	1	no	—
116	C₂×C₃×C₅	—	2	2	no	—
117	C₂×C₃×C₅	—	4	2	no	—
118	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
119	C₂×C₃×C₅	#237	1	2	yes	—
120	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
121	C₂×C₃×C₅	C₃×#41	1	2	yes	—
122	C₂³×C₃	—	2	1	no	—
123	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
124	C₂³×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	3	4	no	—
127	C₂×C₃×C₄	—	3	1	no	—
128	C₂×C₃×C₄	—	3	1	no	—
129	C₂×C₃×C₄	—	3	2	no	—
130	C₂×C₃×C₄	C₂×C₅×D₃	1	2	yes	—
131	C₂×C₃×C₄	C₂²×C₃×C₅	1	2	yes	—
132	C₂×Dic₃	C₃×C₄×C₅	1	4	yes	complemented
133	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
134	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
135	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	C₃×C₄×D₃	1	2	yes	—
138	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
139	C₄×C₅	—	2	2	no	—
140	C₄×C₅	—	3	2	no	—
141	C₄×C₅	—	3	2	no	—
142	C₄×C₅	—	6	2	no	—
143	C₂×C₃²	—	2	2	no	—
144	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
145	C₂×C₃²	C₅×#41	1	2	yes	—
146	C₂²×C₄	—	3	1	no	—
147	C₄⋊C₄	—	3	2	no	—
148	C₂²⋊C₄	—	3	1	no	—
149	C₂²⋊C₄	C₃×C₅×D₃	1	1	yes	characteristic
150	C₄²	—	3	2	no	—
151	C₃×C₅	—	2	1	no	—
152	C₃×C₅	#877	1	1	yes	characteristic, direct product factor
153	C₃×C₅	C₃×#117	1	1	yes	characteristic, complemented
154	C₂²×C₃	—	2	2	no	—
155	C₂²×C₃	—	2	2	no	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	—	4	2	no	—
159	C₂²×C₃	C₄×C₅×D₃	1	2	yes	—
160	C₂²×C₃	C₂×C₃×C₄×C₅	1	2	yes	—
161	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
162	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	3	2	no	—
166	C₃×C₄	—	3	2	no	—
167	C₃×C₄	—	4	2	no	—
168	C₃×C₄	—	6	2	no	—
169	Dic₃	—	2	2	no	—
170	Dic₃	C₂×C₃×C₄×C₅	1	2	yes	—
171	Dic₃	C₂×C₃×C₄×C₅	1	2	yes	—
172	C₂×C₅	—	2	2	no	—
173	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
174	C₂×C₅	C₃×#237	1	2	yes	—
175	C₃²	C₅×#117	1	1	yes	characteristic, complemented, Sylow
176	C₂³	C₂×C₃×C₅×D₃	1	1	yes	characteristic
177	C₂×C₄	—	3	4	no	—
178	C₂×C₄	—	3	1	no	—
179	C₂×C₄	—	3	1	no	—
180	C₂×C₄	—	3	2	no	—
181	C₂×C₄	C₂×C₃×C₅×D₃	1	2	yes	—
182	C₂×C₃	—	2	2	no	—
183	C₂×C₃	—	2	2	no	—
184	C₂×C₃	—	2	1	no	—
185	C₂×C₃	—	2	2	no	—
186	C₂×C₃	—	4	2	no	—
187	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
188	C₂×C₃	C₅×#237	1	2	yes	—
189	C₂×C₃	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
190	C₂×C₃	C₃×C₅×#41	1	2	yes	—
191	C₅	C₃×#877	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	—	2	2	no	—
193	C₂²	C₃×C₄×C₅×D₃	1	2	yes	—
194	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	3	2	no	—
197	C₄	—	3	2	no	—
198	C₄	—	6	2	no	—
199	C₃	—	2	1	no	—
200	C₃	C₅×#877	1	1	yes	characteristic, direct product factor
201	C₃	C₃×C₅×#117	1	1	yes	characteristic, complemented
202	C₂	—	2	2	no	—
203	C₂	C₂×C₃×C₄×C₅×D₃	1	1	yes	characteristic
204	C₂	C₃×C₅×#237	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
5. C₃×C₄×C₅×Dic₃	C₂	semidirect
7. C₅×#877	C₃	direct (N × H)
10. C₂×C₃×C₅×Dic₃	C₄	semidirect
14. C₃×#877	C₅	direct (N × H)
24. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
25. C₄×C₅×Dic₃	C₂×C₃	semidirect
37. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
38. C₃×C₄×Dic₃	C₂×C₅	semidirect
49. C₂×C₅×Dic₃	C₃×C₄	semidirect
54. #877	C₃×C₅	direct (N × H)
65. C₂×C₃×Dic₃	C₄×C₅	semidirect
96. Dic₃⋊C₄	C₂×C₃×C₅	semidirect
97. C₄×Dic₃	C₂×C₃×C₅	semidirect
98. C₃²×C₅	#117	semidirect
132. C₂×Dic₃	C₃×C₄×C₅	semidirect
152. C₃×C₅	#877	direct (N × H)
153. C₃×C₅	C₃×#117	semidirect
175. C₃²	C₅×#117	semidirect
191. C₅	C₃×#877	direct (N × H)
200. C₃	C₅×#877	direct (N × H)
201. 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
C₂²×C₃×C₅×Dic₃	2
C₃²×C₅×C₂²⋊C₄	2
C₃×C₅×#217	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₄×C₅×Dic₃	2
C₃×C₅×#117	3
C₅×#877	3
C₃×#877	5

Subgroups — C3×C5×#877 #13087888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#877 #13087888