Subgroups — C₃×C₅×#3288 #14276322

Order 1920.

928 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#577	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#576	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#496	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#472	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#471	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#455	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#454	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#453	C₂	1	2	yes	maximal, complemented
9	C₃×C₅×#444	C₂	1	2	yes	maximal, complemented
10	C₅×#3288	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₅×2₋¹⁺⁴	C₂²	1	1	yes	characteristic, complemented
12	C₃×C₅×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
13	C₃×C₅×H(5,𝔽₂)	C₂²	1	2	yes	complemented
14	C₃×C₅×#137	—	2	2	no	—
15	C₃×C₅×#137	C₂²	1	2	yes	complemented
16	C₃×C₅×AGL(1,ℤ/8)	—	2	2	no	—
17	C₃×C₅×AGL(1,ℤ/8)	—	2	2	no	—
18	C₃×C₅×AGL(1,ℤ/8)	—	2	2	no	—
19	C₃×C₅×AGL(1,ℤ/8)	C₂²	1	2	yes	—
20	C₃×C₅×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
21	C₃×C₅×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
22	C₃×C₅×#135	—	2	2	no	—
23	C₃×C₅×#135	—	2	2	no	—
24	C₃×C₅×#135	C₂²	1	2	yes	—
25	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
26	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	complemented
27	C₂×C₃×C₅×D₈	—	2	2	no	—
28	C₂×C₃×C₅×D₈	C₂²	1	2	yes	complemented
29	C₃×C₅×#131	C₂²	1	2	yes	complemented
30	C₃×C₅×#131	C₂²	1	2	yes	complemented
31	C₃×C₅×#127	C₂²	1	1	yes	characteristic, complemented
32	C₃×C₅×#124	C₂²	1	1	yes	characteristic, complemented
33	C₃×C₅×#108	C₂²	1	2	yes	complemented
34	C₃×C₅×#104	C₂²	1	2	yes	complemented
35	C₃×C₅×#104	C₂²	1	2	yes	complemented
36	C₃×C₅×#101	C₂²	1	1	yes	characteristic, complemented
37	C₃×C₅×#100	C₂²	1	2	yes	complemented
38	C₃×C₅×#100	C₂²	1	1	yes	characteristic, complemented
39	C₃×C₅×#97	C₂²	1	1	yes	characteristic, complemented
40	C₃×#3288	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
41	C₅×#577	C₂×C₃	1	2	yes	complemented
42	C₅×#576	C₂×C₃	1	2	yes	complemented
43	C₅×#496	C₂×C₃	1	1	yes	characteristic, complemented
44	C₅×#472	C₂×C₃	1	2	yes	complemented
45	C₅×#471	C₂×C₃	1	2	yes	complemented
46	C₅×#455	C₂×C₃	1	1	yes	characteristic, complemented
47	C₅×#454	C₂×C₃	1	1	yes	characteristic, complemented
48	C₅×#453	C₂×C₃	1	2	yes	complemented
49	C₅×#444	C₂×C₃	1	2	yes	complemented
50	C₃×C₅×#41	—	2	2	no	—
51	C₃×C₅×#41	—	2	2	no	—
52	C₃×C₅×#41	—	2	2	no	—
53	C₃×C₅×#41	—	2	2	no	—
54	C₃×C₅×#41	—	4	1	no	—
55	C₃×C₅×#41	C₂³	1	2	yes	—
56	C₃×C₅×#41	C₂³	1	2	yes	—
57	C₂×C₃×C₅×Q₈	—	2	2	no	—
58	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₅×D₄	—	2	2	no	—
60	C₂×C₃×C₅×D₄	—	2	2	no	—
61	C₂×C₃×C₅×D₄	—	2	2	no	—
62	C₂×C₃×C₅×D₄	—	4	1	no	—
63	C₂×C₃×C₅×D₄	—	4	1	no	—
64	C₂×C₃×C₅×D₄	—	4	2	no	—
65	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
67	C₃×C₅×Q₁₆	—	2	2	no	—
68	C₃×C₅×Q₁₆	D₄	1	2	yes	complemented
69	C₃×C₅×SD₁₆	—	2	2	no	—
70	C₃×C₅×SD₁₆	—	2	2	no	—
71	C₃×C₅×SD₁₆	—	2	2	no	—
72	C₃×C₅×SD₁₆	—	2	2	no	—
73	C₃×C₅×SD₁₆	—	2	2	no	—
74	C₃×C₅×SD₁₆	—	2	2	no	—
75	C₃×C₅×SD₁₆	D₄	1	2	yes	complemented
76	C₃×C₅×SD₁₆	D₄	1	2	yes	complemented
77	C₃×C₅×D₈	—	2	2	no	—
78	C₃×C₅×D₈	—	2	2	no	—
79	C₃×C₅×D₈	—	2	2	no	—
80	C₃×C₅×D₈	—	2	2	no	—
81	C₃×C₅×D₈	—	2	2	no	—
82	C₃×C₅×D₈	D₄	1	2	yes	complemented
83	C₃×C₅×M₁₆	—	2	2	no	—
84	C₃×C₅×M₁₆	—	2	2	no	—
85	C₃×C₅×M₁₆	C₂³	1	2	yes	—
86	C₃×C₅×M₁₆	C₂³	1	2	yes	—
87	C₂×C₃×C₅×C₈	—	2	2	no	—
88	C₂×C₃×C₅×C₈	—	2	2	no	—
89	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
90	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
91	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
92	C₃×#577	C₂×C₅	1	2	yes	complemented
93	C₃×#576	C₂×C₅	1	2	yes	complemented
94	C₃×#496	C₂×C₅	1	1	yes	characteristic, complemented
95	C₃×#472	C₂×C₅	1	2	yes	complemented
96	C₃×#471	C₂×C₅	1	2	yes	complemented
97	C₃×#455	C₂×C₅	1	1	yes	characteristic, complemented
98	C₃×#454	C₂×C₅	1	1	yes	characteristic, complemented
99	C₃×#453	C₂×C₅	1	2	yes	complemented
100	C₃×#444	C₂×C₅	1	2	yes	complemented
101	C₅×2₋¹⁺⁴	C₂²×C₃	1	1	yes	characteristic, complemented
102	C₅×H(5,𝔽₂)	C₂²×C₃	1	1	yes	characteristic
103	C₅×H(5,𝔽₂)	C₂²×C₃	1	2	yes	complemented
104	C₅×#137	—	2	2	no	—
105	C₅×#137	C₂²×C₃	1	2	yes	complemented
106	C₅×AGL(1,ℤ/8)	—	2	2	no	—
107	C₅×AGL(1,ℤ/8)	—	2	2	no	—
108	C₅×AGL(1,ℤ/8)	—	2	2	no	—
109	C₅×AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	—
110	C₅×AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	complemented
111	C₅×AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	complemented
112	C₅×#135	—	2	2	no	—
113	C₅×#135	—	2	2	no	—
114	C₅×#135	C₂²×C₃	1	2	yes	—
115	C₂×C₅×SD₁₆	—	2	2	no	—
116	C₂×C₅×SD₁₆	C₂²×C₃	1	2	yes	complemented
117	C₂×C₅×D₈	—	2	2	no	—
118	C₂×C₅×D₈	C₂²×C₃	1	2	yes	complemented
119	C₅×#131	C₂²×C₃	1	2	yes	complemented
120	C₅×#131	C₂²×C₃	1	2	yes	complemented
121	C₅×#127	C₂²×C₃	1	1	yes	characteristic, complemented
122	C₅×#124	C₂²×C₃	1	1	yes	characteristic, complemented
123	C₅×#108	C₂²×C₃	1	2	yes	complemented
124	C₅×#104	C₂²×C₃	1	2	yes	complemented
125	C₅×#104	C₂²×C₃	1	2	yes	complemented
126	C₅×#101	C₂²×C₃	1	1	yes	characteristic, complemented
127	C₅×#100	C₂²×C₃	1	2	yes	complemented
128	C₅×#100	C₂²×C₃	1	1	yes	characteristic, complemented
129	C₅×#97	C₂²×C₃	1	1	yes	characteristic, complemented
130	#3288	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×C₃×C₅	—	2	1	no	—
132	C₂³×C₃×C₅	—	2	2	no	—
133	C₂³×C₃×C₅	—	4	2	no	—
134	C₂³×C₃×C₅	—	4	1	no	—
135	C₃×C₅×Q₈	—	2	2	no	—
136	C₃×C₅×Q₈	—	4	1	no	—
137	C₃×C₅×Q₈	C₂×D₄	1	2	yes	—
138	C₃×C₅×Q₈	C₂×D₄	1	2	yes	—
139	C₃×C₅×D₄	—	2	2	no	—
140	C₃×C₅×D₄	—	2	2	no	—
141	C₃×C₅×D₄	—	2	2	no	—
142	C₃×C₅×D₄	—	2	2	no	—
143	C₃×C₅×D₄	—	2	2	no	—
144	C₃×C₅×D₄	—	4	2	no	—
145	C₃×C₅×D₄	—	4	2	no	—
146	C₃×C₅×D₄	—	4	2	no	—
147	C₃×C₅×D₄	—	4	1	no	—
148	C₃×C₅×D₄	—	8	2	no	—
149	C₃×C₅×D₄	C₂×D₄	1	2	yes	—
150	C₃×C₅×D₄	C₂×D₄	1	2	yes	—
151	C₂×C₃×C₄×C₅	—	2	2	no	—
152	C₂×C₃×C₄×C₅	—	2	2	no	—
153	C₂×C₃×C₄×C₅	—	2	1	no	—
154	C₂×C₃×C₄×C₅	—	2	1	no	—
155	C₂×C₃×C₄×C₅	—	4	1	no	—
156	C₂×C₃×C₄×C₅	—	4	2	no	—
157	C₂×C₃×C₄×C₅	—	4	2	no	—
158	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
159	C₃×C₅×C₈	—	2	2	no	—
160	C₃×C₅×C₈	—	2	2	no	—
161	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
162	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
163	C₃×2₋¹⁺⁴	C₂²×C₅	1	1	yes	characteristic, complemented
164	C₃×H(5,𝔽₂)	C₂²×C₅	1	1	yes	characteristic
165	C₃×H(5,𝔽₂)	C₂²×C₅	1	2	yes	complemented
166	C₃×#137	—	2	2	no	—
167	C₃×#137	C₂²×C₅	1	2	yes	complemented
168	C₃×AGL(1,ℤ/8)	—	2	2	no	—
169	C₃×AGL(1,ℤ/8)	—	2	2	no	—
170	C₃×AGL(1,ℤ/8)	—	2	2	no	—
171	C₃×AGL(1,ℤ/8)	C₂²×C₅	1	2	yes	—
172	C₃×AGL(1,ℤ/8)	C₂²×C₅	1	2	yes	complemented
173	C₃×AGL(1,ℤ/8)	C₂²×C₅	1	2	yes	complemented
174	C₃×#135	—	2	2	no	—
175	C₃×#135	—	2	2	no	—
176	C₃×#135	C₂²×C₅	1	2	yes	—
177	C₂×C₃×SD₁₆	—	2	2	no	—
178	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	complemented
179	C₂×C₃×D₈	—	2	2	no	—
180	C₂×C₃×D₈	C₂²×C₅	1	2	yes	complemented
181	C₃×#131	C₂²×C₅	1	2	yes	complemented
182	C₃×#131	C₂²×C₅	1	2	yes	complemented
183	C₃×#127	C₂²×C₅	1	1	yes	characteristic, complemented
184	C₃×#124	C₂²×C₅	1	1	yes	characteristic, complemented
185	C₃×#108	C₂²×C₅	1	2	yes	complemented
186	C₃×#104	C₂²×C₅	1	2	yes	complemented
187	C₃×#104	C₂²×C₅	1	2	yes	complemented
188	C₃×#101	C₂²×C₅	1	1	yes	characteristic, complemented
189	C₃×#100	C₂²×C₅	1	2	yes	complemented
190	C₃×#100	C₂²×C₅	1	1	yes	characteristic, complemented
191	C₃×#97	C₂²×C₅	1	1	yes	characteristic, complemented
192	C₅×#41	—	2	2	no	—
193	C₅×#41	—	2	2	no	—
194	C₅×#41	—	2	2	no	—
195	C₅×#41	—	2	2	no	—
196	C₅×#41	—	4	1	no	—
197	C₅×#41	C₂³×C₃	1	2	yes	—
198	C₅×#41	C₂³×C₃	1	2	yes	—
199	C₂×C₅×Q₈	—	2	2	no	—
200	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
201	C₂×C₅×D₄	—	2	2	no	—
202	C₂×C₅×D₄	—	2	2	no	—
203	C₂×C₅×D₄	—	2	2	no	—
204	C₂×C₅×D₄	—	4	1	no	—
205	C₂×C₅×D₄	—	4	1	no	—
206	C₂×C₅×D₄	—	4	2	no	—
207	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
208	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
209	C₅×Q₁₆	—	2	2	no	—
210	C₅×Q₁₆	C₃×D₄	1	2	yes	complemented
211	C₅×SD₁₆	—	2	2	no	—
212	C₅×SD₁₆	—	2	2	no	—
213	C₅×SD₁₆	—	2	2	no	—
214	C₅×SD₁₆	—	2	2	no	—
215	C₅×SD₁₆	—	2	2	no	—
216	C₅×SD₁₆	—	2	2	no	—
217	C₅×SD₁₆	C₃×D₄	1	2	yes	complemented
218	C₅×SD₁₆	C₃×D₄	1	2	yes	complemented
219	C₅×D₈	—	2	2	no	—
220	C₅×D₈	—	2	2	no	—
221	C₅×D₈	—	2	2	no	—
222	C₅×D₈	—	2	2	no	—
223	C₅×D₈	—	2	2	no	—
224	C₅×D₈	C₃×D₄	1	2	yes	complemented
225	C₅×M₁₆	—	2	2	no	—
226	C₅×M₁₆	—	2	2	no	—
227	C₅×M₁₆	C₂³×C₃	1	2	yes	—
228	C₅×M₁₆	C₂³×C₃	1	2	yes	—
229	C₂×C₅×C₈	—	2	2	no	—
230	C₂×C₅×C₈	—	2	2	no	—
231	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
232	C₅×C₂²⋊C₄	—	4	1	no	—
233	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
234	#577	C₂×C₃×C₅	1	2	yes	complemented
235	#576	C₂×C₃×C₅	1	2	yes	complemented
236	#496	C₂×C₃×C₅	1	1	yes	characteristic, complemented
237	#472	C₂×C₃×C₅	1	2	yes	complemented
238	#471	C₂×C₃×C₅	1	2	yes	complemented
239	#455	C₂×C₃×C₅	1	1	yes	characteristic, complemented
240	#454	C₂×C₃×C₅	1	1	yes	characteristic, complemented
241	#453	C₂×C₃×C₅	1	2	yes	complemented
242	#444	C₂×C₃×C₅	1	2	yes	complemented
243	C₂²×C₃×C₅	—	2	2	no	—
244	C₂²×C₃×C₅	—	2	2	no	—
245	C₂²×C₃×C₅	—	4	1	no	—
246	C₂²×C₃×C₅	—	4	2	no	—
247	C₂²×C₃×C₅	—	8	2	no	—
248	C₂²×C₃×C₅	—	8	1	no	—
249	C₂²×C₃×C₅	—	16	2	no	—
250	C₂²×C₃×C₅	—	16	1	no	—
251	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
252	C₃×C₄×C₅	—	2	2	no	—
253	C₃×C₄×C₅	—	2	2	no	—
254	C₃×C₄×C₅	—	4	1	no	—
255	C₃×C₄×C₅	—	4	1	no	—
256	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
257	C₃×#41	—	2	2	no	—
258	C₃×#41	—	2	2	no	—
259	C₃×#41	—	2	2	no	—
260	C₃×#41	—	2	2	no	—
261	C₃×#41	—	4	1	no	—
262	C₃×#41	C₂³×C₅	1	2	yes	—
263	C₃×#41	C₂³×C₅	1	2	yes	—
264	C₂×C₃×Q₈	—	2	2	no	—
265	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
266	C₂×C₃×D₄	—	2	2	no	—
267	C₂×C₃×D₄	—	2	2	no	—
268	C₂×C₃×D₄	—	2	2	no	—
269	C₂×C₃×D₄	—	4	1	no	—
270	C₂×C₃×D₄	—	4	1	no	—
271	C₂×C₃×D₄	—	4	2	no	—
272	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
273	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
274	C₃×Q₁₆	—	2	2	no	—
275	C₃×Q₁₆	C₅×D₄	1	2	yes	complemented
276	C₃×SD₁₆	—	2	2	no	—
277	C₃×SD₁₆	—	2	2	no	—
278	C₃×SD₁₆	—	2	2	no	—
279	C₃×SD₁₆	—	2	2	no	—
280	C₃×SD₁₆	—	2	2	no	—
281	C₃×SD₁₆	—	2	2	no	—
282	C₃×SD₁₆	C₅×D₄	1	2	yes	complemented
283	C₃×SD₁₆	C₅×D₄	1	2	yes	complemented
284	C₃×D₈	—	2	2	no	—
285	C₃×D₈	—	2	2	no	—
286	C₃×D₈	—	2	2	no	—
287	C₃×D₈	—	2	2	no	—
288	C₃×D₈	—	2	2	no	—
289	C₃×D₈	C₅×D₄	1	2	yes	complemented
290	C₃×M₁₆	—	2	2	no	—
291	C₃×M₁₆	—	2	2	no	—
292	C₃×M₁₆	C₂³×C₅	1	2	yes	—
293	C₃×M₁₆	C₂³×C₅	1	2	yes	—
294	C₂×C₃×C₈	—	2	2	no	—
295	C₂×C₃×C₈	—	2	2	no	—
296	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
297	C₃×C₂²⋊C₄	—	4	1	no	—
298	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
299	C₂³×C₅	—	2	1	no	—
300	C₂³×C₅	—	2	2	no	—
301	C₂³×C₅	—	4	2	no	—
302	C₂³×C₅	—	4	1	no	—
303	C₅×Q₈	—	2	2	no	—
304	C₅×Q₈	—	4	1	no	—
305	C₅×Q₈	C₂×C₃×D₄	1	2	yes	—
306	C₅×Q₈	C₂×C₃×D₄	1	2	yes	—
307	C₅×D₄	—	2	2	no	—
308	C₅×D₄	—	2	2	no	—
309	C₅×D₄	—	2	2	no	—
310	C₅×D₄	—	2	2	no	—
311	C₅×D₄	—	2	2	no	—
312	C₅×D₄	—	4	2	no	—
313	C₅×D₄	—	4	2	no	—
314	C₅×D₄	—	4	2	no	—
315	C₅×D₄	—	4	1	no	—
316	C₅×D₄	—	8	2	no	—
317	C₅×D₄	C₂×C₃×D₄	1	2	yes	—
318	C₅×D₄	C₂×C₃×D₄	1	2	yes	—
319	C₂×C₄×C₅	—	2	2	no	—
320	C₂×C₄×C₅	—	2	2	no	—
321	C₂×C₄×C₅	—	2	1	no	—
322	C₂×C₄×C₅	—	2	1	no	—
323	C₂×C₄×C₅	—	4	1	no	—
324	C₂×C₄×C₅	—	4	2	no	—
325	C₂×C₄×C₅	—	4	2	no	—
326	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
327	C₅×C₈	—	2	2	no	—
328	C₅×C₈	—	2	2	no	—
329	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
330	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
331	2₋¹⁺⁴	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
332	H(5,𝔽₂)	C₂²×C₃×C₅	1	1	yes	characteristic
333	H(5,𝔽₂)	C₂²×C₃×C₅	1	2	yes	complemented
334	#137	—	2	2	no	—
335	#137	C₂²×C₃×C₅	1	2	yes	complemented
336	AGL(1,ℤ/8)	—	2	2	no	—
337	AGL(1,ℤ/8)	—	2	2	no	—
338	AGL(1,ℤ/8)	—	2	2	no	—
339	AGL(1,ℤ/8)	C₂²×C₃×C₅	1	2	yes	—
340	AGL(1,ℤ/8)	C₂²×C₃×C₅	1	2	yes	complemented
341	AGL(1,ℤ/8)	C₂²×C₃×C₅	1	2	yes	complemented
342	#135	—	2	2	no	—
343	#135	—	2	2	no	—
344	#135	C₂²×C₃×C₅	1	2	yes	—
345	C₂×SD₁₆	—	2	2	no	—
346	C₂×SD₁₆	C₂²×C₃×C₅	1	2	yes	complemented
347	C₂×D₈	—	2	2	no	—
348	C₂×D₈	C₂²×C₃×C₅	1	2	yes	complemented
349	#131	C₂²×C₃×C₅	1	2	yes	complemented
350	#131	C₂²×C₃×C₅	1	2	yes	complemented
351	#127	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
352	#124	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
353	#108	C₂²×C₃×C₅	1	2	yes	complemented
354	#104	C₂²×C₃×C₅	1	2	yes	complemented
355	#104	C₂²×C₃×C₅	1	2	yes	complemented
356	#101	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
357	#100	C₂²×C₃×C₅	1	2	yes	complemented
358	#100	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
359	#97	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
360	C₂×C₃×C₅	—	2	1	no	—
361	C₂×C₃×C₅	—	4	2	no	—
362	C₂×C₃×C₅	—	4	2	no	—
363	C₂×C₃×C₅	—	8	2	no	—
364	C₂×C₃×C₅	—	8	1	no	—
365	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
366	C₂³×C₃	—	2	1	no	—
367	C₂³×C₃	—	2	2	no	—
368	C₂³×C₃	—	4	2	no	—
369	C₂³×C₃	—	4	1	no	—
370	C₃×Q₈	—	2	2	no	—
371	C₃×Q₈	—	4	1	no	—
372	C₃×Q₈	C₂×C₅×D₄	1	2	yes	—
373	C₃×Q₈	C₂×C₅×D₄	1	2	yes	—
374	C₃×D₄	—	2	2	no	—
375	C₃×D₄	—	2	2	no	—
376	C₃×D₄	—	2	2	no	—
377	C₃×D₄	—	2	2	no	—
378	C₃×D₄	—	2	2	no	—
379	C₃×D₄	—	4	2	no	—
380	C₃×D₄	—	4	2	no	—
381	C₃×D₄	—	4	2	no	—
382	C₃×D₄	—	4	1	no	—
383	C₃×D₄	—	8	2	no	—
384	C₃×D₄	C₂×C₅×D₄	1	2	yes	—
385	C₃×D₄	C₂×C₅×D₄	1	2	yes	—
386	C₂×C₃×C₄	—	2	2	no	—
387	C₂×C₃×C₄	—	2	2	no	—
388	C₂×C₃×C₄	—	2	1	no	—
389	C₂×C₃×C₄	—	2	1	no	—
390	C₂×C₃×C₄	—	4	1	no	—
391	C₂×C₃×C₄	—	4	2	no	—
392	C₂×C₃×C₄	—	4	2	no	—
393	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
394	C₃×C₈	—	2	2	no	—
395	C₃×C₈	—	2	2	no	—
396	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
397	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
398	C₂²×C₅	—	2	2	no	—
399	C₂²×C₅	—	2	2	no	—
400	C₂²×C₅	—	4	1	no	—
401	C₂²×C₅	—	4	2	no	—
402	C₂²×C₅	—	8	2	no	—
403	C₂²×C₅	—	8	1	no	—
404	C₂²×C₅	—	16	1	no	—
405	C₂²×C₅	—	16	2	no	—
406	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
407	C₄×C₅	—	2	2	no	—
408	C₄×C₅	—	2	2	no	—
409	C₄×C₅	—	4	1	no	—
410	C₄×C₅	—	4	1	no	—
411	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
412	#41	—	2	2	no	—
413	#41	—	2	2	no	—
414	#41	—	2	2	no	—
415	#41	—	2	2	no	—
416	#41	—	4	1	no	—
417	#41	C₂³×C₃×C₅	1	2	yes	—
418	#41	C₂³×C₃×C₅	1	2	yes	—
419	C₂×Q₈	—	2	2	no	—
420	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	4	1	no	—
425	C₂×D₄	—	4	1	no	—
426	C₂×D₄	—	4	2	no	—
427	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
428	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
429	Q₁₆	—	2	2	no	—
430	Q₁₆	C₃×C₅×D₄	1	2	yes	complemented
431	SD₁₆	—	2	2	no	—
432	SD₁₆	—	2	2	no	—
433	SD₁₆	—	2	2	no	—
434	SD₁₆	—	2	2	no	—
435	SD₁₆	—	2	2	no	—
436	SD₁₆	—	2	2	no	—
437	SD₁₆	C₃×C₅×D₄	1	2	yes	complemented
438	SD₁₆	C₃×C₅×D₄	1	2	yes	complemented
439	D₈	—	2	2	no	—
440	D₈	—	2	2	no	—
441	D₈	—	2	2	no	—
442	D₈	—	2	2	no	—
443	D₈	—	2	2	no	—
444	D₈	C₃×C₅×D₄	1	2	yes	complemented
445	M₁₆	—	2	2	no	—
446	M₁₆	—	2	2	no	—
447	M₁₆	C₂³×C₃×C₅	1	2	yes	—
448	M₁₆	C₂³×C₃×C₅	1	2	yes	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
452	C₂²⋊C₄	—	4	1	no	—
453	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
454	C₃×C₅	#3288	1	1	yes	characteristic, direct product factor, Hall
455	C₂²×C₃	—	2	2	no	—
456	C₂²×C₃	—	2	2	no	—
457	C₂²×C₃	—	4	1	no	—
458	C₂²×C₃	—	4	2	no	—
459	C₂²×C₃	—	8	2	no	—
460	C₂²×C₃	—	8	1	no	—
461	C₂²×C₃	—	16	1	no	—
462	C₂²×C₃	—	16	2	no	—
463	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
464	C₃×C₄	—	2	2	no	—
465	C₃×C₄	—	2	2	no	—
466	C₃×C₄	—	4	1	no	—
467	C₃×C₄	—	4	1	no	—
468	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
469	C₂×C₅	—	2	1	no	—
470	C₂×C₅	—	4	2	no	—
471	C₂×C₅	—	4	2	no	—
472	C₂×C₅	—	8	2	no	—
473	C₂×C₅	—	8	1	no	—
474	C₂×C₅	C₃×D₄²	1	1	yes	characteristic
475	C₂³	—	2	1	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	4	2	no	—
478	C₂³	—	4	1	no	—
479	Q₈	—	2	2	no	—
480	Q₈	—	4	1	no	—
481	Q₈	C₂×C₃×C₅×D₄	1	2	yes	—
482	Q₈	C₂×C₃×C₅×D₄	1	2	yes	—
483	D₄	—	2	2	no	—
484	D₄	—	2	2	no	—
485	D₄	—	2	2	no	—
486	D₄	—	2	2	no	—
487	D₄	—	2	2	no	—
488	D₄	—	4	2	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	2	no	—
491	D₄	—	4	1	no	—
492	D₄	—	8	2	no	—
493	D₄	C₂×C₃×C₅×D₄	1	2	yes	—
494	D₄	C₂×C₃×C₅×D₄	1	2	yes	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
503	C₈	—	2	2	no	—
504	C₈	—	2	2	no	—
505	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
506	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
507	C₂×C₃	—	2	1	no	—
508	C₂×C₃	—	4	2	no	—
509	C₂×C₃	—	4	2	no	—
510	C₂×C₃	—	8	2	no	—
511	C₂×C₃	—	8	1	no	—
512	C₂×C₃	C₅×D₄²	1	1	yes	characteristic
513	C₅	C₃×#3288	1	1	yes	characteristic, direct product factor, Sylow
514	C₂²	—	2	2	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	8	2	no	—
519	C₂²	—	8	1	no	—
520	C₂²	—	16	1	no	—
521	C₂²	—	16	2	no	—
522	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	4	1	no	—
526	C₄	—	4	1	no	—
527	C₄	C₂²×C₃×C₅×D₄	1	2	yes	—
528	C₃	C₅×#3288	1	1	yes	characteristic, direct product factor, Sylow
529	C₂	—	2	1	no	—
530	C₂	—	4	2	no	—
531	C₂	—	4	2	no	—
532	C₂	—	8	2	no	—
533	C₂	—	8	1	no	—
534	C₂	C₃×C₅×D₄²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#577	C₂	semidirect
2. C₃×C₅×#576	C₂	semidirect
3. C₃×C₅×#496	C₂	semidirect
4. C₃×C₅×#472	C₂	semidirect
5. C₃×C₅×#471	C₂	semidirect
6. C₃×C₅×#455	C₂	semidirect
7. C₃×C₅×#454	C₂	semidirect
8. C₃×C₅×#453	C₂	semidirect
9. C₃×C₅×#444	C₂	semidirect
10. C₅×#3288	C₃	direct (N × H)
11. C₃×C₅×2₋¹⁺⁴	C₂²	semidirect
13. C₃×C₅×H(5,𝔽₂)	C₂²	semidirect
15. C₃×C₅×#137	C₂²	semidirect
20. C₃×C₅×AGL(1,ℤ/8)	C₂²	semidirect
21. C₃×C₅×AGL(1,ℤ/8)	C₂²	semidirect
26. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
28. C₂×C₃×C₅×D₈	C₂²	semidirect
29. C₃×C₅×#131	C₂²	semidirect
30. C₃×C₅×#131	C₂²	semidirect
31. C₃×C₅×#127	C₂²	semidirect
32. C₃×C₅×#124	C₂²	semidirect
33. C₃×C₅×#108	C₂²	semidirect
34. C₃×C₅×#104	C₂²	semidirect
35. C₃×C₅×#104	C₂²	semidirect
36. C₃×C₅×#101	C₂²	semidirect
37. C₃×C₅×#100	C₂²	semidirect
38. C₃×C₅×#100	C₂²	semidirect
39. C₃×C₅×#97	C₂²	semidirect
40. C₃×#3288	C₅	direct (N × H)
41. C₅×#577	C₂×C₃	semidirect
42. C₅×#576	C₂×C₃	semidirect
43. C₅×#496	C₂×C₃	semidirect
44. C₅×#472	C₂×C₃	semidirect
45. C₅×#471	C₂×C₃	semidirect
46. C₅×#455	C₂×C₃	semidirect
47. C₅×#454	C₂×C₃	semidirect
48. C₅×#453	C₂×C₃	semidirect
49. C₅×#444	C₂×C₃	semidirect
68. C₃×C₅×Q₁₆	D₄	semidirect
75. C₃×C₅×SD₁₆	D₄	semidirect
76. C₃×C₅×SD₁₆	D₄	semidirect
82. C₃×C₅×D₈	D₄	semidirect
92. C₃×#577	C₂×C₅	semidirect
93. C₃×#576	C₂×C₅	semidirect
94. C₃×#496	C₂×C₅	semidirect
95. C₃×#472	C₂×C₅	semidirect
96. C₃×#471	C₂×C₅	semidirect
97. C₃×#455	C₂×C₅	semidirect
98. C₃×#454	C₂×C₅	semidirect
99. C₃×#453	C₂×C₅	semidirect
100. C₃×#444	C₂×C₅	semidirect
101. C₅×2₋¹⁺⁴	C₂²×C₃	semidirect
103. C₅×H(5,𝔽₂)	C₂²×C₃	semidirect
105. C₅×#137	C₂²×C₃	semidirect
110. C₅×AGL(1,ℤ/8)	C₂²×C₃	semidirect
111. C₅×AGL(1,ℤ/8)	C₂²×C₃	semidirect
116. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
118. C₂×C₅×D₈	C₂²×C₃	semidirect
119. C₅×#131	C₂²×C₃	semidirect
120. C₅×#131	C₂²×C₃	semidirect
121. C₅×#127	C₂²×C₃	semidirect
122. C₅×#124	C₂²×C₃	semidirect
123. C₅×#108	C₂²×C₃	semidirect
124. C₅×#104	C₂²×C₃	semidirect
125. C₅×#104	C₂²×C₃	semidirect
126. C₅×#101	C₂²×C₃	semidirect
127. C₅×#100	C₂²×C₃	semidirect
128. C₅×#100	C₂²×C₃	semidirect
129. C₅×#97	C₂²×C₃	semidirect
130. #3288	C₃×C₅	direct (N × H)
163. C₃×2₋¹⁺⁴	C₂²×C₅	semidirect
165. C₃×H(5,𝔽₂)	C₂²×C₅	semidirect
167. C₃×#137	C₂²×C₅	semidirect
172. C₃×AGL(1,ℤ/8)	C₂²×C₅	semidirect
173. C₃×AGL(1,ℤ/8)	C₂²×C₅	semidirect
178. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
180. C₂×C₃×D₈	C₂²×C₅	semidirect
181. C₃×#131	C₂²×C₅	semidirect
182. C₃×#131	C₂²×C₅	semidirect
183. C₃×#127	C₂²×C₅	semidirect
184. C₃×#124	C₂²×C₅	semidirect
185. C₃×#108	C₂²×C₅	semidirect
186. C₃×#104	C₂²×C₅	semidirect
187. C₃×#104	C₂²×C₅	semidirect
188. C₃×#101	C₂²×C₅	semidirect
189. C₃×#100	C₂²×C₅	semidirect
190. C₃×#100	C₂²×C₅	semidirect
191. C₃×#97	C₂²×C₅	semidirect
210. C₅×Q₁₆	C₃×D₄	semidirect
217. C₅×SD₁₆	C₃×D₄	semidirect
218. C₅×SD₁₆	C₃×D₄	semidirect
224. C₅×D₈	C₃×D₄	semidirect
234. #577	C₂×C₃×C₅	semidirect
235. #576	C₂×C₃×C₅	semidirect
236. #496	C₂×C₃×C₅	semidirect
237. #472	C₂×C₃×C₅	semidirect
238. #471	C₂×C₃×C₅	semidirect
239. #455	C₂×C₃×C₅	semidirect
240. #454	C₂×C₃×C₅	semidirect
241. #453	C₂×C₃×C₅	semidirect
242. #444	C₂×C₃×C₅	semidirect
275. C₃×Q₁₆	C₅×D₄	semidirect
282. C₃×SD₁₆	C₅×D₄	semidirect
283. C₃×SD₁₆	C₅×D₄	semidirect
289. C₃×D₈	C₅×D₄	semidirect
331. 2₋¹⁺⁴	C₂²×C₃×C₅	semidirect
333. H(5,𝔽₂)	C₂²×C₃×C₅	semidirect
335. #137	C₂²×C₃×C₅	semidirect
340. AGL(1,ℤ/8)	C₂²×C₃×C₅	semidirect
341. AGL(1,ℤ/8)	C₂²×C₃×C₅	semidirect
346. C₂×SD₁₆	C₂²×C₃×C₅	semidirect
348. C₂×D₈	C₂²×C₃×C₅	semidirect
349. #131	C₂²×C₃×C₅	semidirect
350. #131	C₂²×C₃×C₅	semidirect
351. #127	C₂²×C₃×C₅	semidirect
352. #124	C₂²×C₃×C₅	semidirect
353. #108	C₂²×C₃×C₅	semidirect
354. #104	C₂²×C₃×C₅	semidirect
355. #104	C₂²×C₃×C₅	semidirect
356. #101	C₂²×C₃×C₅	semidirect
357. #100	C₂²×C₃×C₅	semidirect
358. #100	C₂²×C₃×C₅	semidirect
359. #97	C₂²×C₃×C₅	semidirect
430. Q₁₆	C₃×C₅×D₄	semidirect
437. SD₁₆	C₃×C₅×D₄	semidirect
438. SD₁₆	C₃×C₅×D₄	semidirect
444. D₈	C₃×C₅×D₄	semidirect
454. C₃×C₅	#3288	direct (N × H)
513. C₅	C₃×#3288	direct (N × H)
528. C₃	C₅×#3288	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×#577	2
C₃×C₅×#576	2
C₃×C₅×#496	2
C₃×C₅×#472	2
C₃×C₅×#471	2
C₃×C₅×#455	2
C₃×C₅×#454	2
C₃×C₅×#453	2
C₃×C₅×#444	2
C₅×#3288	3
C₃×#3288	5

Subgroups — C3×C5×#3288 #14276322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3288 #14276322