Subgroups — C₃×C₅×#3255 #14276289

Order 1920.

684 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#533	C₂	1	2	yes	maximal
2	C₃×C₅×#475	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#470	C₂	1	4	yes	maximal
4	C₃×C₅×#462	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
6	C₅×#3255	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	—
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃×C₅×#125	—	2	2	no	—
12	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
13	C₃×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
14	C₃×C₄×C₅×Q₈	—	2	2	no	—
15	C₃×C₄×C₅×Q₈	C₂²	1	4	yes	—
16	C₃×C₅×#117	—	2	2	no	—
17	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₃×C₅×#106	C₂²	1	4	yes	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
20	C₃×C₅×#103	C₂²	1	8	yes	—
21	C₃×#3255	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#533	C₂×C₃	1	2	yes	—
23	C₅×#475	C₂×C₃	1	4	yes	complemented
24	C₅×#470	C₂×C₃	1	4	yes	—
25	C₅×#462	C₂×C₃	1	4	yes	complemented
26	C₅×#423	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
29	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₃×C₅×Q₁₆	—	4	4	no	—
32	C₃×C₅×M₁₆	—	4	2	no	—
33	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	4	no	—
38	C₃×C₅×C₄⋊C₄	—	2	4	no	—
39	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
40	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
41	C₃×C₄²×C₅	—	2	4	no	—
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	C₂³	1	2	yes	—
45	C₃×#533	C₂×C₅	1	2	yes	—
46	C₃×#475	C₂×C₅	1	4	yes	complemented
47	C₃×#470	C₂×C₅	1	4	yes	—
48	C₃×#462	C₂×C₅	1	4	yes	complemented
49	C₃×#423	C₂×C₅	1	1	yes	characteristic
50	C₂×C₅×Q₁₆	C₂²×C₃	1	4	yes	—
51	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
52	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
53	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
54	C₅×#125	—	2	2	no	—
55	C₅×C₂²⋊Q₈	—	2	2	no	—
56	C₅×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
57	C₄×C₅×Q₈	—	2	2	no	—
58	C₄×C₅×Q₈	C₂²×C₃	1	4	yes	—
59	C₅×#117	—	2	2	no	—
60	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
61	C₅×#106	C₂²×C₃	1	4	yes	—
62	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
63	C₅×#103	C₂²×C₃	1	8	yes	—
64	#3255	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
65	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
66	C₃×C₅×Q₈	—	2	4	no	—
67	C₃×C₅×Q₈	—	2	4	no	—
68	C₃×C₅×Q₈	—	4	4	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	—	2	1	no	—
71	C₂×C₃×C₄×C₅	—	2	1	no	—
72	C₂×C₃×C₄×C₅	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₃×C₅×C₈	—	2	4	no	—
79	C₂×C₃×Q₁₆	C₂²×C₅	1	4	yes	—
80	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
81	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
82	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
83	C₃×#125	—	2	2	no	—
84	C₃×C₂²⋊Q₈	—	2	2	no	—
85	C₃×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
86	C₃×C₄×Q₈	—	2	2	no	—
87	C₃×C₄×Q₈	C₂²×C₅	1	4	yes	—
88	C₃×#117	—	2	2	no	—
89	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
90	C₃×#106	C₂²×C₅	1	4	yes	—
91	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
92	C₃×#103	C₂²×C₅	1	8	yes	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
95	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
96	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
97	C₅×Q₁₆	—	4	4	no	—
98	C₅×M₁₆	—	4	2	no	—
99	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	4	no	—
104	C₅×C₄⋊C₄	—	2	4	no	—
105	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
106	C₅×C₂²⋊C₄	—	2	4	no	—
107	C₄²×C₅	—	2	4	no	—
108	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
109	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
110	C₄²×C₅	C₂³×C₃	1	2	yes	—
111	#533	C₂×C₃×C₅	1	2	yes	—
112	#475	C₂×C₃×C₅	1	4	yes	complemented
113	#470	C₂×C₃×C₅	1	4	yes	—
114	#462	C₂×C₃×C₅	1	4	yes	complemented
115	#423	C₂×C₃×C₅	1	1	yes	characteristic
116	C₂²×C₃×C₅	—	2	1	no	—
117	C₂²×C₃×C₅	—	2	2	no	—
118	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	—	2	2	no	—
121	C₃×C₄×C₅	—	4	4	no	—
122	C₃×C₄×C₅	—	4	4	no	—
123	C₃×C₄×C₅	#137	1	4	yes	—
124	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
125	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
126	C₂×C₃×Q₈	—	2	2	no	—
127	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
128	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
129	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
130	C₃×Q₁₆	—	4	4	no	—
131	C₃×M₁₆	—	4	2	no	—
132	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
133	C₃×C₄⋊C₄	—	2	2	no	—
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	—	2	4	no	—
137	C₃×C₄⋊C₄	—	2	4	no	—
138	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
139	C₃×C₂²⋊C₄	—	2	4	no	—
140	C₃×C₄²	—	2	4	no	—
141	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
142	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
143	C₃×C₄²	C₂³×C₅	1	2	yes	—
144	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
145	C₅×Q₈	—	2	4	no	—
146	C₅×Q₈	—	2	4	no	—
147	C₅×Q₈	—	4	4	no	—
148	C₂×C₄×C₅	—	2	4	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
155	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
156	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
157	C₅×C₈	—	2	4	no	—
158	C₂×Q₁₆	C₂²×C₃×C₅	1	4	yes	—
159	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
160	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
161	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
162	#125	—	2	2	no	—
163	C₂²⋊Q₈	—	2	2	no	—
164	C₂²⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
165	C₄×Q₈	—	2	2	no	—
166	C₄×Q₈	C₂²×C₃×C₅	1	4	yes	—
167	#117	—	2	2	no	—
168	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
169	#106	C₂²×C₃×C₅	1	4	yes	—
170	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
171	#103	C₂²×C₃×C₅	1	8	yes	—
172	C₂×C₃×C₅	—	4	1	no	—
173	C₂×C₃×C₅	#537	1	1	yes	characteristic
174	C₂×C₃×C₅	C₂×#137	1	2	yes	—
175	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
176	C₃×Q₈	—	2	4	no	—
177	C₃×Q₈	—	2	4	no	—
178	C₃×Q₈	—	4	4	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
186	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
187	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
188	C₃×C₈	—	2	4	no	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	4	4	no	—
195	C₄×C₅	—	4	4	no	—
196	C₄×C₅	C₃×#137	1	4	yes	—
197	C₄×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
198	C₄×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
201	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
202	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
203	Q₁₆	—	4	4	no	—
204	M₁₆	—	4	2	no	—
205	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₄²	—	2	4	no	—
214	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
215	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
216	C₄²	C₂³×C₃×C₅	1	2	yes	—
217	C₃×C₅	#3255	1	1	yes	characteristic, direct product factor, Hall
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	4	4	no	—
224	C₃×C₄	—	4	4	no	—
225	C₃×C₄	C₅×#137	1	4	yes	—
226	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
227	C₃×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
228	C₂×C₅	—	4	1	no	—
229	C₂×C₅	C₃×#537	1	1	yes	characteristic
230	C₂×C₅	C₂×C₃×#137	1	2	yes	—
231	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
232	Q₈	—	2	4	no	—
233	Q₈	—	2	4	no	—
234	Q₈	—	4	4	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
242	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
243	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
244	C₈	—	2	4	no	—
245	C₂×C₃	—	4	1	no	—
246	C₂×C₃	C₅×#537	1	1	yes	characteristic
247	C₂×C₃	C₂×C₅×#137	1	2	yes	—
248	C₅	C₃×#3255	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	4	4	no	—
255	C₄	—	4	4	no	—
256	C₄	C₃×C₅×#137	1	4	yes	—
257	C₄	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
258	C₄	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
259	C₃	C₅×#3255	1	1	yes	characteristic, direct product factor, Sylow
260	C₂	—	4	1	no	—
261	C₂	C₃×C₅×#537	1	1	yes	characteristic
262	C₂	C₂×C₃×C₅×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#475	C₂	semidirect
4. C₃×C₅×#462	C₂	semidirect
6. C₅×#3255	C₃	direct (N × H)
21. C₃×#3255	C₅	direct (N × H)
23. C₅×#475	C₂×C₃	semidirect
25. C₅×#462	C₂×C₃	semidirect
46. C₃×#475	C₂×C₅	semidirect
48. C₃×#462	C₂×C₅	semidirect
64. #3255	C₃×C₅	direct (N × H)
112. #475	C₂×C₃×C₅	semidirect
114. #462	C₂×C₃×C₅	semidirect
217. C₃×C₅	#3255	direct (N × H)
248. C₅	C₃×#3255	direct (N × H)
259. C₃	C₅×#3255	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×#533	2
C₃×C₅×#475	2
C₃×C₅×#470	2
C₃×C₅×#462	2
C₃×C₅×#423	2
C₅×#3255	3
C₃×#3255	5

Subgroups — C3×C5×#3255 #14276289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3255 #14276289