Subgroups — C₃×C₅×#3018 #14276052

Order 1920.

1448 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#585	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#575	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#456	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#455	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#454	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#453	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×C₅×#104	C₂	1	2	yes	maximal, complemented
9	C₃×C₅×#413	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#3018	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₅×2₋¹⁺⁴	—	2	2	no	—
12	C₃×C₅×2₋¹⁺⁴	C₂²	1	2	yes	—
13	C₃×C₅×H(5,𝔽₂)	—	2	2	no	—
14	C₃×C₅×H(5,𝔽₂)	—	2	2	no	—
15	C₃×C₅×H(5,𝔽₂)	C₂²	1	2	yes	—
16	C₂×C₃×C₅×#41	—	2	4	no	—
17	C₂×C₃×C₅×#41	—	2	2	no	—
18	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
19	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic, complemented
21	C₃×C₅×#137	—	2	2	no	—
22	C₃×C₅×#137	C₂²	1	4	yes	complemented
23	C₃×C₅×AGL(1,ℤ/8)	—	2	2	no	—
24	C₃×C₅×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
25	C₃×C₅×#135	—	2	4	no	—
26	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	complemented
27	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
28	C₃×C₅×#127	C₂²	1	1	yes	characteristic, complemented
29	C₃×C₅×#124	C₂²	1	1	yes	characteristic, complemented
30	C₃×C₅×#124	C₂²	1	1	yes	characteristic, complemented
31	C₃×C₅×#121	—	2	2	no	—
32	C₃×C₄×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
34	C₃×C₅×#104	C₂²	1	4	yes	complemented
35	C₃×C₅×#104	C₂²	1	4	yes	complemented
36	C₃×C₅×#101	C₂²	1	2	yes	complemented
37	C₃×C₅×#100	C₂²	1	2	yes	complemented
38	C₃×#3018	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
39	C₅×#585	C₂×C₃	1	1	yes	characteristic, complemented
40	C₅×#575	C₂×C₃	1	2	yes	complemented
41	C₅×#532	C₂×C₃	1	1	yes	characteristic, complemented
42	C₅×#456	C₂×C₃	1	2	yes	complemented
43	C₅×#455	C₂×C₃	1	2	yes	complemented
44	C₅×#454	C₂×C₃	1	2	yes	complemented
45	C₅×#453	C₂×C₃	1	2	yes	complemented
46	C₂×C₅×#104	C₂×C₃	1	2	yes	complemented
47	C₅×#413	C₂×C₃	1	1	yes	characteristic, complemented
48	C₃×C₅×#41	—	2	4	no	—
49	C₃×C₅×#41	—	2	8	no	—
50	C₃×C₅×#41	—	2	4	no	—
51	C₃×C₅×#41	—	2	4	no	—
52	C₃×C₅×#41	—	2	4	no	—
53	C₃×C₅×#41	—	2	2	no	—
54	C₃×C₅×#41	—	2	2	no	—
55	C₃×C₅×#41	—	2	4	no	—
56	C₃×C₅×#41	—	2	2	no	—
57	C₃×C₅×#41	—	2	2	no	—
58	C₃×C₅×#41	—	2	2	no	—
59	C₃×C₅×#41	—	2	2	no	—
60	C₃×C₅×#41	D₄	1	4	yes	complemented
61	C₃×C₅×#41	C₂³	1	4	yes	—
62	C₂×C₃×C₅×Q₈	—	2	2	no	—
63	C₂×C₃×C₅×Q₈	—	2	4	no	—
64	C₂×C₃×C₅×Q₈	D₄	1	2	yes	complemented
65	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
67	C₂×C₃×C₅×D₄	—	2	8	no	—
68	C₂×C₃×C₅×D₄	—	2	2	no	—
69	C₂×C₃×C₅×D₄	—	2	4	no	—
70	C₂×C₃×C₅×D₄	—	2	2	no	—
71	C₂×C₃×C₅×D₄	—	2	4	no	—
72	C₂×C₃×C₅×D₄	—	2	2	no	—
73	C₂×C₃×C₅×D₄	—	2	1	no	—
74	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
75	C₂×C₃×C₅×D₄	D₄	1	2	yes	complemented
76	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
78	C₂²×C₃×C₄×C₅	—	2	4	no	—
79	C₂²×C₃×C₄×C₅	—	2	2	no	—
80	C₂²×C₃×C₄×C₅	—	2	1	no	—
81	C₂²×C₃×C₄×C₅	—	2	1	no	—
82	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
83	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
84	C₃×C₅×Q₁₆	—	2	4	no	—
85	C₃×C₅×SD₁₆	—	2	4	no	—
86	C₃×C₅×SD₁₆	—	2	4	no	—
87	C₃×C₅×D₈	—	2	4	no	—
88	C₃×C₅×M₁₆	—	2	2	no	—
89	C₃×C₅×M₁₆	C₂³	1	4	yes	—
90	C₂×C₃×C₅×C₈	—	2	2	no	—
91	C₃×C₅×C₄⋊C₄	—	2	2	no	—
92	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
93	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
94	C₃×C₄²×C₅	—	2	1	no	—
95	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
96	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
97	C₃×#585	C₂×C₅	1	1	yes	characteristic, complemented
98	C₃×#575	C₂×C₅	1	2	yes	complemented
99	C₃×#532	C₂×C₅	1	1	yes	characteristic, complemented
100	C₃×#456	C₂×C₅	1	2	yes	complemented
101	C₃×#455	C₂×C₅	1	2	yes	complemented
102	C₃×#454	C₂×C₅	1	2	yes	complemented
103	C₃×#453	C₂×C₅	1	2	yes	complemented
104	C₂×C₃×#104	C₂×C₅	1	2	yes	complemented
105	C₃×#413	C₂×C₅	1	1	yes	characteristic, complemented
106	C₅×2₋¹⁺⁴	—	2	2	no	—
107	C₅×2₋¹⁺⁴	C₂²×C₃	1	2	yes	—
108	C₅×H(5,𝔽₂)	—	2	2	no	—
109	C₅×H(5,𝔽₂)	—	2	2	no	—
110	C₅×H(5,𝔽₂)	C₂²×C₃	1	2	yes	—
111	C₂×C₅×#41	—	2	4	no	—
112	C₂×C₅×#41	—	2	2	no	—
113	C₂×C₅×#41	C₂²×C₃	1	2	yes	—
114	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
115	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic, complemented
116	C₅×#137	—	2	2	no	—
117	C₅×#137	C₂²×C₃	1	4	yes	complemented
118	C₅×AGL(1,ℤ/8)	—	2	2	no	—
119	C₅×AGL(1,ℤ/8)	C₂²×C₃	1	4	yes	complemented
120	C₅×#135	—	2	4	no	—
121	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	complemented
122	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
123	C₅×#127	C₂²×C₃	1	1	yes	characteristic, complemented
124	C₅×#124	C₂²×C₃	1	1	yes	characteristic, complemented
125	C₅×#124	C₂²×C₃	1	1	yes	characteristic, complemented
126	C₅×#121	—	2	2	no	—
127	C₄×C₅×D₄	—	2	2	no	—
128	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
129	C₅×#104	C₂²×C₃	1	4	yes	complemented
130	C₅×#104	C₂²×C₃	1	4	yes	complemented
131	C₅×#101	C₂²×C₃	1	2	yes	complemented
132	C₅×#100	C₂²×C₃	1	2	yes	complemented
133	#3018	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
134	C₂³×C₃×C₅	—	2	4	no	—
135	C₂³×C₃×C₅	—	2	2	no	—
136	C₂³×C₃×C₅	—	2	1	no	—
137	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
138	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
139	C₃×C₅×Q₈	—	2	4	no	—
140	C₃×C₅×Q₈	—	2	2	no	—
141	C₃×C₅×Q₈	—	2	2	no	—
142	C₃×C₅×Q₈	—	2	2	no	—
143	C₃×C₅×Q₈	C₂×D₄	1	4	yes	—
144	C₃×C₅×D₄	—	2	8	no	—
145	C₃×C₅×D₄	—	2	4	no	—
146	C₃×C₅×D₄	—	2	4	no	—
147	C₃×C₅×D₄	—	2	4	no	—
148	C₃×C₅×D₄	—	2	4	no	—
149	C₃×C₅×D₄	—	2	2	no	—
150	C₃×C₅×D₄	—	2	2	no	—
151	C₃×C₅×D₄	—	2	2	no	—
152	C₃×C₅×D₄	—	4	4	no	—
153	C₃×C₅×D₄	—	4	2	no	—
154	C₃×C₅×D₄	C₂×D₄	1	4	yes	—
155	C₂×C₃×C₄×C₅	—	2	4	no	—
156	C₂×C₃×C₄×C₅	—	2	8	no	—
157	C₂×C₃×C₄×C₅	—	2	2	no	—
158	C₂×C₃×C₄×C₅	—	2	1	no	—
159	C₂×C₃×C₄×C₅	—	2	2	no	—
160	C₂×C₃×C₄×C₅	—	2	4	no	—
161	C₂×C₃×C₄×C₅	—	2	2	no	—
162	C₂×C₃×C₄×C₅	—	2	4	no	—
163	C₂×C₃×C₄×C₅	—	2	1	no	—
164	C₂×C₃×C₄×C₅	—	2	1	no	—
165	C₂×C₃×C₄×C₅	—	4	2	no	—
166	C₂×C₃×C₄×C₅	—	4	1	no	—
167	C₂×C₃×C₄×C₅	—	4	1	no	—
168	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
169	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
171	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
172	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
173	C₃×C₅×C₈	—	2	4	no	—
174	C₃×2₋¹⁺⁴	—	2	2	no	—
175	C₃×2₋¹⁺⁴	C₂²×C₅	1	2	yes	—
176	C₃×H(5,𝔽₂)	—	2	2	no	—
177	C₃×H(5,𝔽₂)	—	2	2	no	—
178	C₃×H(5,𝔽₂)	C₂²×C₅	1	2	yes	—
179	C₂×C₃×#41	—	2	4	no	—
180	C₂×C₃×#41	—	2	2	no	—
181	C₂×C₃×#41	C₂²×C₅	1	2	yes	—
182	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
183	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic, complemented
184	C₃×#137	—	2	2	no	—
185	C₃×#137	C₂²×C₅	1	4	yes	complemented
186	C₃×AGL(1,ℤ/8)	—	2	2	no	—
187	C₃×AGL(1,ℤ/8)	C₂²×C₅	1	4	yes	complemented
188	C₃×#135	—	2	4	no	—
189	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	complemented
190	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
191	C₃×#127	C₂²×C₅	1	1	yes	characteristic, complemented
192	C₃×#124	C₂²×C₅	1	1	yes	characteristic, complemented
193	C₃×#124	C₂²×C₅	1	1	yes	characteristic, complemented
194	C₃×#121	—	2	2	no	—
195	C₃×C₄×D₄	—	2	2	no	—
196	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic, complemented
197	C₃×#104	C₂²×C₅	1	4	yes	complemented
198	C₃×#104	C₂²×C₅	1	4	yes	complemented
199	C₃×#101	C₂²×C₅	1	2	yes	complemented
200	C₃×#100	C₂²×C₅	1	2	yes	complemented
201	C₅×#41	—	2	4	no	—
202	C₅×#41	—	2	8	no	—
203	C₅×#41	—	2	4	no	—
204	C₅×#41	—	2	4	no	—
205	C₅×#41	—	2	4	no	—
206	C₅×#41	—	2	2	no	—
207	C₅×#41	—	2	2	no	—
208	C₅×#41	—	2	4	no	—
209	C₅×#41	—	2	2	no	—
210	C₅×#41	—	2	2	no	—
211	C₅×#41	—	2	2	no	—
212	C₅×#41	—	2	2	no	—
213	C₅×#41	C₃×D₄	1	4	yes	complemented
214	C₅×#41	C₂³×C₃	1	4	yes	—
215	C₂×C₅×Q₈	—	2	2	no	—
216	C₂×C₅×Q₈	—	2	4	no	—
217	C₂×C₅×Q₈	C₃×D₄	1	2	yes	complemented
218	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
219	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
220	C₂×C₅×D₄	—	2	8	no	—
221	C₂×C₅×D₄	—	2	2	no	—
222	C₂×C₅×D₄	—	2	4	no	—
223	C₂×C₅×D₄	—	2	2	no	—
224	C₂×C₅×D₄	—	2	4	no	—
225	C₂×C₅×D₄	—	2	2	no	—
226	C₂×C₅×D₄	—	2	1	no	—
227	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
228	C₂×C₅×D₄	C₃×D₄	1	2	yes	complemented
229	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
230	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
231	C₂²×C₄×C₅	—	2	4	no	—
232	C₂²×C₄×C₅	—	2	2	no	—
233	C₂²×C₄×C₅	—	2	1	no	—
234	C₂²×C₄×C₅	—	2	1	no	—
235	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
236	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
237	C₅×Q₁₆	—	2	4	no	—
238	C₅×SD₁₆	—	2	4	no	—
239	C₅×SD₁₆	—	2	4	no	—
240	C₅×D₈	—	2	4	no	—
241	C₅×M₁₆	—	2	2	no	—
242	C₅×M₁₆	C₂³×C₃	1	4	yes	—
243	C₂×C₅×C₈	—	2	2	no	—
244	C₅×C₄⋊C₄	—	2	2	no	—
245	C₅×C₂²⋊C₄	—	2	2	no	—
246	C₅×C₂²⋊C₄	—	2	2	no	—
247	C₄²×C₅	—	2	1	no	—
248	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
249	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
250	#585	C₂×C₃×C₅	1	1	yes	characteristic, complemented
251	#575	C₂×C₃×C₅	1	2	yes	complemented
252	#532	C₂×C₃×C₅	1	1	yes	characteristic, complemented
253	#456	C₂×C₃×C₅	1	2	yes	complemented
254	#455	C₂×C₃×C₅	1	2	yes	complemented
255	#454	C₂×C₃×C₅	1	2	yes	complemented
256	#453	C₂×C₃×C₅	1	2	yes	complemented
257	C₂×#104	C₂×C₃×C₅	1	2	yes	complemented
258	#413	C₂×C₃×C₅	1	1	yes	characteristic, complemented
259	C₂²×C₃×C₅	—	2	2	no	—
260	C₂²×C₃×C₅	—	2	4	no	—
261	C₂²×C₃×C₅	—	4	1	no	—
262	C₂²×C₃×C₅	—	4	2	no	—
263	C₂²×C₃×C₅	—	4	2	no	—
264	C₂²×C₃×C₅	—	4	1	no	—
265	C₂²×C₃×C₅	—	8	2	no	—
266	C₂²×C₃×C₅	—	8	4	no	—
267	C₂²×C₃×C₅	#120	1	2	yes	—
268	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
269	C₃×C₄×C₅	—	2	2	no	—
270	C₃×C₄×C₅	—	2	4	no	—
271	C₃×C₄×C₅	—	2	2	no	—
272	C₃×C₄×C₅	—	4	1	no	—
273	C₃×C₄×C₅	—	4	1	no	—
274	C₃×C₄×C₅	#120	1	1	yes	characteristic
275	C₃×C₄×C₅	#120	1	1	yes	characteristic
276	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
277	C₃×#41	—	2	4	no	—
278	C₃×#41	—	2	8	no	—
279	C₃×#41	—	2	4	no	—
280	C₃×#41	—	2	4	no	—
281	C₃×#41	—	2	4	no	—
282	C₃×#41	—	2	2	no	—
283	C₃×#41	—	2	2	no	—
284	C₃×#41	—	2	4	no	—
285	C₃×#41	—	2	2	no	—
286	C₃×#41	—	2	2	no	—
287	C₃×#41	—	2	2	no	—
288	C₃×#41	—	2	2	no	—
289	C₃×#41	C₅×D₄	1	4	yes	complemented
290	C₃×#41	C₂³×C₅	1	4	yes	—
291	C₂×C₃×Q₈	—	2	2	no	—
292	C₂×C₃×Q₈	—	2	4	no	—
293	C₂×C₃×Q₈	C₅×D₄	1	2	yes	complemented
294	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
295	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
296	C₂×C₃×D₄	—	2	8	no	—
297	C₂×C₃×D₄	—	2	2	no	—
298	C₂×C₃×D₄	—	2	4	no	—
299	C₂×C₃×D₄	—	2	2	no	—
300	C₂×C₃×D₄	—	2	4	no	—
301	C₂×C₃×D₄	—	2	2	no	—
302	C₂×C₃×D₄	—	2	1	no	—
303	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
304	C₂×C₃×D₄	C₅×D₄	1	2	yes	complemented
305	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
306	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
307	C₂²×C₃×C₄	—	2	4	no	—
308	C₂²×C₃×C₄	—	2	2	no	—
309	C₂²×C₃×C₄	—	2	1	no	—
310	C₂²×C₃×C₄	—	2	1	no	—
311	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
312	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
313	C₃×Q₁₆	—	2	4	no	—
314	C₃×SD₁₆	—	2	4	no	—
315	C₃×SD₁₆	—	2	4	no	—
316	C₃×D₈	—	2	4	no	—
317	C₃×M₁₆	—	2	2	no	—
318	C₃×M₁₆	C₂³×C₅	1	4	yes	—
319	C₂×C₃×C₈	—	2	2	no	—
320	C₃×C₄⋊C₄	—	2	2	no	—
321	C₃×C₂²⋊C₄	—	2	2	no	—
322	C₃×C₂²⋊C₄	—	2	2	no	—
323	C₃×C₄²	—	2	1	no	—
324	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
325	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
326	C₂³×C₅	—	2	4	no	—
327	C₂³×C₅	—	2	2	no	—
328	C₂³×C₅	—	2	1	no	—
329	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
330	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
331	C₅×Q₈	—	2	4	no	—
332	C₅×Q₈	—	2	2	no	—
333	C₅×Q₈	—	2	2	no	—
334	C₅×Q₈	—	2	2	no	—
335	C₅×Q₈	C₂×C₃×D₄	1	4	yes	—
336	C₅×D₄	—	2	8	no	—
337	C₅×D₄	—	2	4	no	—
338	C₅×D₄	—	2	4	no	—
339	C₅×D₄	—	2	4	no	—
340	C₅×D₄	—	2	4	no	—
341	C₅×D₄	—	2	2	no	—
342	C₅×D₄	—	2	2	no	—
343	C₅×D₄	—	2	2	no	—
344	C₅×D₄	—	4	4	no	—
345	C₅×D₄	—	4	2	no	—
346	C₅×D₄	C₂×C₃×D₄	1	4	yes	—
347	C₂×C₄×C₅	—	2	4	no	—
348	C₂×C₄×C₅	—	2	8	no	—
349	C₂×C₄×C₅	—	2	2	no	—
350	C₂×C₄×C₅	—	2	1	no	—
351	C₂×C₄×C₅	—	2	4	no	—
352	C₂×C₄×C₅	—	2	2	no	—
353	C₂×C₄×C₅	—	2	2	no	—
354	C₂×C₄×C₅	—	2	4	no	—
355	C₂×C₄×C₅	—	2	1	no	—
356	C₂×C₄×C₅	—	2	1	no	—
357	C₂×C₄×C₅	—	4	2	no	—
358	C₂×C₄×C₅	—	4	1	no	—
359	C₂×C₄×C₅	—	4	1	no	—
360	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
361	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
362	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
363	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
364	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
365	C₅×C₈	—	2	4	no	—
366	2₋¹⁺⁴	—	2	2	no	—
367	2₋¹⁺⁴	C₂²×C₃×C₅	1	2	yes	—
368	H(5,𝔽₂)	—	2	2	no	—
369	H(5,𝔽₂)	—	2	2	no	—
370	H(5,𝔽₂)	C₂²×C₃×C₅	1	2	yes	—
371	C₂×#41	—	2	4	no	—
372	C₂×#41	—	2	2	no	—
373	C₂×#41	C₂²×C₃×C₅	1	2	yes	—
374	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
375	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
376	#137	—	2	2	no	—
377	#137	C₂²×C₃×C₅	1	4	yes	complemented
378	AGL(1,ℤ/8)	—	2	2	no	—
379	AGL(1,ℤ/8)	C₂²×C₃×C₅	1	4	yes	complemented
380	#135	—	2	4	no	—
381	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	complemented
382	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
383	#127	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
384	#124	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
385	#124	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
386	#121	—	2	2	no	—
387	C₄×D₄	—	2	2	no	—
388	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
389	#104	C₂²×C₃×C₅	1	4	yes	complemented
390	#104	C₂²×C₃×C₅	1	4	yes	complemented
391	#101	C₂²×C₃×C₅	1	2	yes	complemented
392	#100	C₂²×C₃×C₅	1	2	yes	complemented
393	C₂×C₃×C₅	—	2	1	no	—
394	C₂×C₃×C₅	—	2	2	no	—
395	C₂×C₃×C₅	—	4	2	no	—
396	C₂×C₃×C₅	—	4	4	no	—
397	C₂×C₃×C₅	—	8	1	no	—
398	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
399	C₂³×C₃	—	2	4	no	—
400	C₂³×C₃	—	2	2	no	—
401	C₂³×C₃	—	2	1	no	—
402	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
403	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
404	C₃×Q₈	—	2	4	no	—
405	C₃×Q₈	—	2	2	no	—
406	C₃×Q₈	—	2	2	no	—
407	C₃×Q₈	—	2	2	no	—
408	C₃×Q₈	C₂×C₅×D₄	1	4	yes	—
409	C₃×D₄	—	2	8	no	—
410	C₃×D₄	—	2	4	no	—
411	C₃×D₄	—	2	4	no	—
412	C₃×D₄	—	2	4	no	—
413	C₃×D₄	—	2	4	no	—
414	C₃×D₄	—	2	2	no	—
415	C₃×D₄	—	2	2	no	—
416	C₃×D₄	—	2	2	no	—
417	C₃×D₄	—	4	4	no	—
418	C₃×D₄	—	4	2	no	—
419	C₃×D₄	C₂×C₅×D₄	1	4	yes	—
420	C₂×C₃×C₄	—	2	4	no	—
421	C₂×C₃×C₄	—	2	8	no	—
422	C₂×C₃×C₄	—	2	2	no	—
423	C₂×C₃×C₄	—	2	1	no	—
424	C₂×C₃×C₄	—	2	4	no	—
425	C₂×C₃×C₄	—	2	2	no	—
426	C₂×C₃×C₄	—	2	2	no	—
427	C₂×C₃×C₄	—	2	4	no	—
428	C₂×C₃×C₄	—	2	1	no	—
429	C₂×C₃×C₄	—	2	1	no	—
430	C₂×C₃×C₄	—	4	2	no	—
431	C₂×C₃×C₄	—	4	1	no	—
432	C₂×C₃×C₄	—	4	1	no	—
433	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
434	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
435	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
436	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
437	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
438	C₃×C₈	—	2	4	no	—
439	C₂²×C₅	—	2	2	no	—
440	C₂²×C₅	—	2	4	no	—
441	C₂²×C₅	—	4	1	no	—
442	C₂²×C₅	—	4	2	no	—
443	C₂²×C₅	—	4	2	no	—
444	C₂²×C₅	—	4	1	no	—
445	C₂²×C₅	—	8	2	no	—
446	C₂²×C₅	—	8	4	no	—
447	C₂²×C₅	C₃×#120	1	2	yes	—
448	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
449	C₄×C₅	—	2	2	no	—
450	C₄×C₅	—	2	4	no	—
451	C₄×C₅	—	2	2	no	—
452	C₄×C₅	—	4	1	no	—
453	C₄×C₅	—	4	1	no	—
454	C₄×C₅	C₃×#120	1	1	yes	characteristic
455	C₄×C₅	C₃×#120	1	1	yes	characteristic
456	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
457	#41	—	2	4	no	—
458	#41	—	2	8	no	—
459	#41	—	2	4	no	—
460	#41	—	2	4	no	—
461	#41	—	2	4	no	—
462	#41	—	2	2	no	—
463	#41	—	2	2	no	—
464	#41	—	2	4	no	—
465	#41	—	2	2	no	—
466	#41	—	2	2	no	—
467	#41	—	2	2	no	—
468	#41	—	2	2	no	—
469	#41	C₃×C₅×D₄	1	4	yes	complemented
470	#41	C₂³×C₃×C₅	1	4	yes	—
471	C₂×Q₈	—	2	2	no	—
472	C₂×Q₈	—	2	4	no	—
473	C₂×Q₈	C₃×C₅×D₄	1	2	yes	complemented
474	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
475	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
476	C₂×D₄	—	2	8	no	—
477	C₂×D₄	—	2	2	no	—
478	C₂×D₄	—	2	4	no	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	4	no	—
481	C₂×D₄	—	2	2	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	C₃×C₅×D₄	1	2	yes	—
484	C₂×D₄	C₃×C₅×D₄	1	2	yes	complemented
485	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
486	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
487	C₂²×C₄	—	2	4	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
492	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
493	Q₁₆	—	2	4	no	—
494	SD₁₆	—	2	4	no	—
495	SD₁₆	—	2	4	no	—
496	D₈	—	2	4	no	—
497	M₁₆	—	2	2	no	—
498	M₁₆	C₂³×C₃×C₅	1	4	yes	—
499	C₂×C₈	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₄²	—	2	1	no	—
504	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
505	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
506	C₃×C₅	#3018	1	1	yes	characteristic, direct product factor, Hall
507	C₂²×C₃	—	2	2	no	—
508	C₂²×C₃	—	2	4	no	—
509	C₂²×C₃	—	4	1	no	—
510	C₂²×C₃	—	4	2	no	—
511	C₂²×C₃	—	4	2	no	—
512	C₂²×C₃	—	4	1	no	—
513	C₂²×C₃	—	8	2	no	—
514	C₂²×C₃	—	8	4	no	—
515	C₂²×C₃	C₅×#120	1	2	yes	—
516	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
517	C₃×C₄	—	2	2	no	—
518	C₃×C₄	—	2	4	no	—
519	C₃×C₄	—	2	2	no	—
520	C₃×C₄	—	4	1	no	—
521	C₃×C₄	—	4	1	no	—
522	C₃×C₄	C₅×#120	1	1	yes	characteristic
523	C₃×C₄	C₅×#120	1	1	yes	characteristic
524	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
525	C₂×C₅	—	2	1	no	—
526	C₂×C₅	—	2	2	no	—
527	C₂×C₅	—	4	2	no	—
528	C₂×C₅	—	4	4	no	—
529	C₂×C₅	—	8	1	no	—
530	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
531	C₂³	—	2	4	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	1	no	—
534	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
535	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
536	Q₈	—	2	4	no	—
537	Q₈	—	2	2	no	—
538	Q₈	—	2	2	no	—
539	Q₈	—	2	2	no	—
540	Q₈	C₂×C₃×C₅×D₄	1	4	yes	—
541	D₄	—	2	8	no	—
542	D₄	—	2	4	no	—
543	D₄	—	2	4	no	—
544	D₄	—	2	4	no	—
545	D₄	—	2	4	no	—
546	D₄	—	2	2	no	—
547	D₄	—	2	2	no	—
548	D₄	—	2	2	no	—
549	D₄	—	4	4	no	—
550	D₄	—	4	2	no	—
551	D₄	C₂×C₃×C₅×D₄	1	4	yes	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	8	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
566	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
567	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
568	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
569	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
570	C₈	—	2	4	no	—
571	C₂×C₃	—	2	1	no	—
572	C₂×C₃	—	2	2	no	—
573	C₂×C₃	—	4	2	no	—
574	C₂×C₃	—	4	4	no	—
575	C₂×C₃	—	8	1	no	—
576	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
577	C₅	C₃×#3018	1	1	yes	characteristic, direct product factor, Sylow
578	C₂²	—	2	2	no	—
579	C₂²	—	2	4	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	8	2	no	—
585	C₂²	—	8	4	no	—
586	C₂²	C₃×C₅×#120	1	2	yes	—
587	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
588	C₄	—	2	2	no	—
589	C₄	—	2	4	no	—
590	C₄	—	2	2	no	—
591	C₄	—	4	1	no	—
592	C₄	—	4	1	no	—
593	C₄	C₃×C₅×#120	1	1	yes	characteristic
594	C₄	C₃×C₅×#120	1	1	yes	characteristic
595	C₄	C₂²×C₃×C₅×D₄	1	2	yes	—
596	C₃	C₅×#3018	1	1	yes	characteristic, direct product factor, Sylow
597	C₂	—	2	1	no	—
598	C₂	—	2	2	no	—
599	C₂	—	4	2	no	—
600	C₂	—	4	4	no	—
601	C₂	—	8	1	no	—
602	C₂	C₂×C₃×C₅×#120	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
socle	C₂×C₃×C₅	C₂×#120
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#585	C₂	semidirect
2. C₃×C₅×#575	C₂	semidirect
3. C₃×C₅×#532	C₂	semidirect
4. C₃×C₅×#456	C₂	semidirect
5. C₃×C₅×#455	C₂	semidirect
6. C₃×C₅×#454	C₂	semidirect
7. C₃×C₅×#453	C₂	semidirect
8. C₂×C₃×C₅×#104	C₂	semidirect
9. C₃×C₅×#413	C₂	semidirect
10. C₅×#3018	C₃	direct (N × H)
20. C₂×C₃×C₅×#41	C₂²	semidirect
22. C₃×C₅×#137	C₂²	semidirect
24. C₃×C₅×AGL(1,ℤ/8)	C₂²	semidirect
26. C₂×C₃×C₅×M₁₆	C₂²	semidirect
27. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
28. C₃×C₅×#127	C₂²	semidirect
29. C₃×C₅×#124	C₂²	semidirect
30. C₃×C₅×#124	C₂²	semidirect
33. C₂×C₃×C₄²×C₅	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×#3018	C₅	direct (N × H)
39. C₅×#585	C₂×C₃	semidirect
40. C₅×#575	C₂×C₃	semidirect
41. C₅×#532	C₂×C₃	semidirect
42. C₅×#456	C₂×C₃	semidirect
43. C₅×#455	C₂×C₃	semidirect
44. C₅×#454	C₂×C₃	semidirect
45. C₅×#453	C₂×C₃	semidirect
46. C₂×C₅×#104	C₂×C₃	semidirect
47. C₅×#413	C₂×C₃	semidirect
60. C₃×C₅×#41	D₄	semidirect
64. C₂×C₃×C₅×Q₈	D₄	semidirect
75. C₂×C₃×C₅×D₄	D₄	semidirect
97. C₃×#585	C₂×C₅	semidirect
98. C₃×#575	C₂×C₅	semidirect
99. C₃×#532	C₂×C₅	semidirect
100. C₃×#456	C₂×C₅	semidirect
101. C₃×#455	C₂×C₅	semidirect
102. C₃×#454	C₂×C₅	semidirect
103. C₃×#453	C₂×C₅	semidirect
104. C₂×C₃×#104	C₂×C₅	semidirect
105. C₃×#413	C₂×C₅	semidirect
115. C₂×C₅×#41	C₂²×C₃	semidirect
117. C₅×#137	C₂²×C₃	semidirect
119. C₅×AGL(1,ℤ/8)	C₂²×C₃	semidirect
121. C₂×C₅×M₁₆	C₂²×C₃	semidirect
122. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
123. C₅×#127	C₂²×C₃	semidirect
124. C₅×#124	C₂²×C₃	semidirect
125. C₅×#124	C₂²×C₃	semidirect
128. C₂×C₄²×C₅	C₂²×C₃	semidirect
129. C₅×#104	C₂²×C₃	semidirect
130. C₅×#104	C₂²×C₃	semidirect
131. C₅×#101	C₂²×C₃	semidirect
132. C₅×#100	C₂²×C₃	semidirect
133. #3018	C₃×C₅	direct (N × H)
183. C₂×C₃×#41	C₂²×C₅	semidirect
185. C₃×#137	C₂²×C₅	semidirect
187. C₃×AGL(1,ℤ/8)	C₂²×C₅	semidirect
189. C₂×C₃×M₁₆	C₂²×C₅	semidirect
190. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
191. C₃×#127	C₂²×C₅	semidirect
192. C₃×#124	C₂²×C₅	semidirect
193. C₃×#124	C₂²×C₅	semidirect
196. C₂×C₃×C₄²	C₂²×C₅	semidirect
197. C₃×#104	C₂²×C₅	semidirect
198. C₃×#104	C₂²×C₅	semidirect
199. C₃×#101	C₂²×C₅	semidirect
200. C₃×#100	C₂²×C₅	semidirect
213. C₅×#41	C₃×D₄	semidirect
217. C₂×C₅×Q₈	C₃×D₄	semidirect
228. C₂×C₅×D₄	C₃×D₄	semidirect
250. #585	C₂×C₃×C₅	semidirect
251. #575	C₂×C₃×C₅	semidirect
252. #532	C₂×C₃×C₅	semidirect
253. #456	C₂×C₃×C₅	semidirect
254. #455	C₂×C₃×C₅	semidirect
255. #454	C₂×C₃×C₅	semidirect
256. #453	C₂×C₃×C₅	semidirect
257. C₂×#104	C₂×C₃×C₅	semidirect
258. #413	C₂×C₃×C₅	semidirect
289. C₃×#41	C₅×D₄	semidirect
293. C₂×C₃×Q₈	C₅×D₄	semidirect
304. C₂×C₃×D₄	C₅×D₄	semidirect
375. C₂×#41	C₂²×C₃×C₅	semidirect
377. #137	C₂²×C₃×C₅	semidirect
379. AGL(1,ℤ/8)	C₂²×C₃×C₅	semidirect
381. C₂×M₁₆	C₂²×C₃×C₅	semidirect
382. C₄⋊Q₈	C₂²×C₃×C₅	semidirect
383. #127	C₂²×C₃×C₅	semidirect
384. #124	C₂²×C₃×C₅	semidirect
385. #124	C₂²×C₃×C₅	semidirect
388. C₂×C₄²	C₂²×C₃×C₅	semidirect
389. #104	C₂²×C₃×C₅	semidirect
390. #104	C₂²×C₃×C₅	semidirect
391. #101	C₂²×C₃×C₅	semidirect
392. #100	C₂²×C₃×C₅	semidirect
469. #41	C₃×C₅×D₄	semidirect
473. C₂×Q₈	C₃×C₅×D₄	semidirect
484. C₂×D₄	C₃×C₅×D₄	semidirect
506. C₃×C₅	#3018	direct (N × H)
577. C₅	C₃×#3018	direct (N × H)
596. C₃	C₅×#3018	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×#585	2
C₃×C₅×#575	2
C₃×C₅×#532	2
C₃×C₅×#456	2
C₃×C₅×#455	2
C₃×C₅×#454	2
C₃×C₅×#453	2
C₂×C₃×C₅×#104	2
C₃×C₅×#413	2
C₅×#3018	3
C₃×#3018	5

Subgroups — C3×C5×#3018 #14276052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3018 #14276052