Subgroups — C₃×C₅×#888 #13087900

Order 1440.

298 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Dic₆	C₂	1	2	yes	maximal
2	C₃²×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×Dic₃⋊C₄	C₂	1	2	yes	maximal
5	C₃×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×C₄⋊Q₈	—	3	1	no	maximal
7	C₅×#888	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
9	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₃×C₅×Dic₆	—	2	4	no	—
14	C₃×#888	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₃×C₅×Q₈	—	3	2	no	—
16	C₂×C₅×Dic₆	C₂×C₃	1	2	yes	—
17	C₃×C₅×C₄⋊C₄	—	2	1	no	—
18	C₃×C₅×C₄⋊C₄	—	3	2	no	—
19	C₃×C₅×C₄⋊C₄	—	3	1	no	—
20	C₃×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
22	C₃×C₄²×C₅	—	3	1	no	—
23	C₅×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
24	C₅×Dic₃⋊C₄	C₂×C₃	1	2	yes	—
25	C₄×C₅×Dic₃	C₂×C₃	1	1	yes	characteristic
26	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
27	C₃²×C₄×C₅	—	2	2	no	—
28	C₃²×C₄×C₅	Q₈	1	2	yes	complemented
29	C₃×C₅×Dic₃	—	2	2	no	—
30	C₃×C₅×Dic₃	D₄	1	2	yes	—
31	C₃×C₅×Dic₃	Q₈	1	2	yes	complemented
32	C₅×C₄⋊Q₈	—	3	1	no	Hall
33	C₂×C₃×Dic₆	C₂×C₅	1	2	yes	—
34	C₃²×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
35	C₃×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic
36	C₃×Dic₃⋊C₄	C₂×C₅	1	2	yes	—
37	C₃×C₄×Dic₃	C₂×C₅	1	1	yes	characteristic
38	C₃×C₅×Q₈	—	6	4	no	—
39	C₂×C₃×C₄×C₅	—	2	1	no	—
40	C₂×C₃×C₄×C₅	—	2	2	no	—
41	C₂×C₃×C₄×C₅	—	3	1	no	—
42	C₂×C₃×C₄×C₅	—	3	1	no	—
43	C₂×C₃×C₄×C₅	—	3	2	no	—
44	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	—
45	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₂²×C₃	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
49	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
51	C₅×Dic₆	—	2	4	no	—
52	C₃×C₄⋊Q₈	—	3	1	no	—
53	#888	C₃×C₅	1	1	yes	characteristic, direct product factor
54	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₅×Q₈	—	3	2	no	—
58	C₅×C₄⋊C₄	—	3	2	no	—
59	C₅×C₄⋊C₄	—	3	1	no	—
60	C₅×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
61	C₄²×C₅	—	3	1	no	—
62	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	—
63	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
65	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
66	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
67	C₃×Dic₆	—	2	4	no	—
68	C₂²×C₃×C₅	—	2	1	no	—
69	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
70	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
71	C₃×C₄×C₅	—	2	2	no	—
72	C₃×C₄×C₅	—	2	2	no	—
73	C₃×C₄×C₅	—	2	2	no	—
74	C₃×C₄×C₅	—	3	2	no	—
75	C₃×C₄×C₅	—	3	2	no	—
76	C₃×C₄×C₅	—	4	2	no	—
77	C₃×C₄×C₅	—	6	2	no	—
78	C₃×C₄×C₅	Dic₆	1	2	yes	complemented
79	C₃×C₄×C₅	C₃×Q₈	1	2	yes	complemented
80	C₅×Dic₃	—	2	2	no	—
81	C₅×Dic₃	C₃×D₄	1	2	yes	—
82	C₅×Dic₃	C₃×Q₈	1	2	yes	complemented
83	C₂×C₃×Q₈	—	3	2	no	—
84	C₂×Dic₆	C₂×C₃×C₅	1	2	yes	—
85	C₃×C₄⋊C₄	—	2	1	no	—
86	C₃×C₄⋊C₄	—	3	2	no	—
87	C₃×C₄⋊C₄	—	3	1	no	—
88	C₃×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic
89	C₃×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
90	C₃×C₄²	—	3	1	no	—
91	C₄⋊Dic₃	C₂×C₃×C₅	1	1	yes	characteristic
92	Dic₃⋊C₄	C₂×C₃×C₅	1	2	yes	—
93	C₄×Dic₃	C₂×C₃×C₅	1	1	yes	characteristic
94	C₃²×C₅	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
95	C₅×Q₈	—	6	4	no	—
96	C₂×C₄×C₅	—	3	1	no	—
97	C₂×C₄×C₅	—	3	1	no	—
98	C₂×C₄×C₅	—	3	2	no	—
99	C₂×C₄×C₅	C₂×C₃×D₃	1	2	yes	—
100	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
101	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
102	C₃²×C₄	—	2	2	no	—
103	C₃²×C₄	C₅×Q₈	1	2	yes	complemented
104	C₃×Dic₃	—	2	2	no	—
105	C₃×Dic₃	C₅×D₄	1	2	yes	—
106	C₃×Dic₃	C₅×Q₈	1	2	yes	complemented
107	C₄⋊Q₈	—	3	1	no	Sylow
108	C₂×C₃×C₅	—	2	1	no	—
109	C₂×C₃×C₅	—	2	1	no	—
110	C₂×C₃×C₅	—	2	1	no	—
111	C₂×C₃×C₅	C₂×Dic₆	1	1	yes	characteristic
112	C₂×C₃×C₅	D₃×D₄	1	1	yes	characteristic
113	C₂×C₃×C₅	D₃×Q₈	1	1	yes	characteristic
114	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
115	C₂×C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
116	C₂×C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
117	C₃×Q₈	—	6	4	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	3	1	no	—
121	C₂×C₃×C₄	—	3	1	no	—
122	C₂×C₃×C₄	—	3	2	no	—
123	C₂×C₃×C₄	C₂×C₅×D₃	1	2	yes	—
124	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂²×C₃×C₅	1	2	yes	—
126	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
127	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
128	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
129	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
130	Dic₆	—	2	4	no	—
131	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
132	C₄×C₅	—	2	2	no	—
133	C₄×C₅	—	3	2	no	—
134	C₄×C₅	—	3	2	no	—
135	C₄×C₅	—	6	2	no	—
136	C₄×C₅	C₃×Dic₆	1	2	yes	complemented
137	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
138	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
139	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
140	C₂×Q₈	—	3	2	no	—
141	C₄⋊C₄	—	3	2	no	—
142	C₄⋊C₄	—	3	1	no	—
143	C₄⋊C₄	C₃×C₅×D₃	1	1	yes	characteristic
144	C₄²	—	3	1	no	—
145	C₃×C₅	—	2	1	no	—
146	C₃×C₅	#888	1	1	yes	characteristic, direct product factor
147	C₃×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic, complemented
148	C₂²×C₃	—	2	1	no	—
149	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
150	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
151	C₃×C₄	—	2	2	no	—
152	C₃×C₄	—	2	2	no	—
153	C₃×C₄	—	2	2	no	—
154	C₃×C₄	—	3	2	no	—
155	C₃×C₄	—	3	2	no	—
156	C₃×C₄	—	4	2	no	—
157	C₃×C₄	—	6	2	no	—
158	C₃×C₄	C₅×Dic₆	1	2	yes	complemented
159	C₃×C₄	C₃×C₅×Q₈	1	2	yes	complemented
160	Dic₃	—	2	2	no	—
161	Dic₃	C₃×C₅×D₄	1	2	yes	—
162	Dic₃	C₃×C₅×Q₈	1	2	yes	complemented
163	C₂×C₅	C₂×C₃×Dic₆	1	1	yes	characteristic
164	C₂×C₅	C₃×D₃×D₄	1	1	yes	characteristic
165	C₂×C₅	C₃×D₃×Q₈	1	1	yes	characteristic
166	C₃²	C₅×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
167	Q₈	—	6	4	no	—
168	C₂×C₄	—	3	1	no	—
169	C₂×C₄	—	3	1	no	—
170	C₂×C₄	—	3	2	no	—
171	C₂×C₄	C₂×C₃×C₅×D₃	1	2	yes	—
172	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
173	C₂×C₃	—	2	1	no	—
174	C₂×C₃	—	2	1	no	—
175	C₂×C₃	—	2	1	no	—
176	C₂×C₃	C₂×C₅×Dic₆	1	1	yes	characteristic
177	C₂×C₃	C₅×D₃×D₄	1	1	yes	characteristic
178	C₂×C₃	C₅×D₃×Q₈	1	1	yes	characteristic
179	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
180	C₂×C₃	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
181	C₂×C₃	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
182	C₅	C₃×#888	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	3	2	no	—
186	C₄	—	3	2	no	—
187	C₄	—	6	2	no	—
188	C₄	C₃×C₅×Dic₆	1	2	yes	complemented
189	C₃	—	2	1	no	—
190	C₃	C₅×#888	1	1	yes	characteristic, direct product factor
191	C₃	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic, complemented
192	C₂	C₂×C₃×C₅×Dic₆	1	1	yes	characteristic
193	C₂	C₃×C₅×D₃×D₄	1	1	yes	characteristic
194	C₂	C₃×C₅×D₃×Q₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₅	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₃×C₅
Frattini	C₂²	C₂²×C₃×C₅×D₃
Fitting	C₃²×C₅×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₃²×C₅	C₂³
hypercenter	C₃×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₃×C₅×C₄⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₅×#888	C₃	direct (N × H)
14. C₃×#888	C₅	direct (N × H)
28. C₃²×C₄×C₅	Q₈	semidirect
31. C₃×C₅×Dic₃	Q₈	semidirect
53. #888	C₃×C₅	direct (N × H)
78. C₃×C₄×C₅	Dic₆	semidirect
79. C₃×C₄×C₅	C₃×Q₈	semidirect
82. C₅×Dic₃	C₃×Q₈	semidirect
94. C₃²×C₅	C₄⋊Q₈	semidirect
103. C₃²×C₄	C₅×Q₈	semidirect
106. C₃×Dic₃	C₅×Q₈	semidirect
136. C₄×C₅	C₃×Dic₆	semidirect
146. C₃×C₅	#888	direct (N × H)
147. C₃×C₅	C₃×C₄⋊Q₈	semidirect
158. C₃×C₄	C₅×Dic₆	semidirect
159. C₃×C₄	C₃×C₅×Q₈	semidirect
162. Dic₃	C₃×C₅×Q₈	semidirect
166. C₃²	C₅×C₄⋊Q₈	semidirect
182. C₅	C₃×#888	direct (N × H)
188. C₄	C₃×C₅×Dic₆	semidirect
190. C₃	C₅×#888	direct (N × H)
191. C₃	C₃×C₅×C₄⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×Dic₆	2
C₃²×C₅×C₄⋊C₄	2
C₃×C₅×C₄⋊Dic₃	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₄×C₅×Dic₃	2
C₃×C₅×C₄⋊Q₈	3
C₅×#888	3
C₃×#888	5

Subgroups — C3×C5×#888 #13087900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#888 #13087900