Subgroups — C₃×C₅×#2550 #14275584

Order 1920.

936 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#398	C₂	1	2	yes	maximal
4	C₃×C₅×#395	C₂	1	2	yes	maximal
5	C₃×C₅×#391	C₂	1	2	yes	maximal
6	C₃×C₅×#389	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#389	C₂	1	1	yes	maximal, characteristic
8	C₃×C₅×#389	C₂	1	2	yes	maximal
9	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic
10	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₅×#2550	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊Q₈	—	2	4	no	—
15	C₃×C₅×#125	—	2	4	no	—
16	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
30	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
31	C₃×C₅×#95	C₂²	1	1	yes	characteristic
32	C₃×C₅×#95	C₂²	1	1	yes	characteristic
33	C₃×C₅×#95	C₂²	1	2	yes	—
34	C₃×C₅×#95	C₂²	1	2	yes	—
35	C₃×C₅×#95	C₂²	1	2	yes	—
36	C₃×C₅×#95	C₂²	1	2	yes	—
37	C₃×C₅×#95	C₂²	1	2	yes	—
38	C₃×#2550	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
39	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
40	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic
41	C₅×#398	C₂×C₃	1	2	yes	—
42	C₅×#395	C₂×C₃	1	2	yes	—
43	C₅×#391	C₂×C₃	1	2	yes	—
44	C₅×#389	C₂×C₃	1	1	yes	characteristic
45	C₅×#389	C₂×C₃	1	1	yes	characteristic
46	C₅×#389	C₂×C₃	1	2	yes	—
47	C₅×#382	C₂×C₃	1	1	yes	characteristic
48	C₄×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
49	C₄×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
50	C₂×C₃×C₅×Q₈	—	2	4	no	—
51	C₂×C₃×C₅×Q₈	—	2	4	no	—
52	C₂×C₃×C₅×Q₈	—	4	2	no	—
53	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
56	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
57	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
58	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
60	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
64	C₃×C₅×C₄⋊C₄	—	2	4	no	—
65	C₃×C₅×C₄⋊C₄	—	2	4	no	—
66	C₃×C₅×C₄⋊C₄	—	2	2	no	—
67	C₃×C₅×C₄⋊C₄	—	2	4	no	—
68	C₃×C₅×C₄⋊C₄	—	2	2	no	—
69	C₃×C₅×C₄⋊C₄	—	2	2	no	—
70	C₃×C₅×C₄⋊C₄	—	2	4	no	—
71	C₃×C₅×C₄⋊C₄	—	2	2	no	—
72	C₃×C₅×C₄⋊C₄	—	4	2	no	—
73	C₃×C₅×C₄⋊C₄	D₄	1	4	yes	—
74	C₃×C₅×C₄⋊C₄	Q₈	1	4	yes	complemented
75	C₃×C₄²×C₅	—	2	2	no	—
76	C₃×C₄²×C₅	—	2	2	no	—
77	C₃×C₄²×C₅	—	4	2	no	—
78	C₃×C₄²×C₅	Q₈	1	4	yes	complemented
79	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
80	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic
81	C₃×#398	C₂×C₅	1	2	yes	—
82	C₃×#395	C₂×C₅	1	2	yes	—
83	C₃×#391	C₂×C₅	1	2	yes	—
84	C₃×#389	C₂×C₅	1	2	yes	—
85	C₃×#389	C₂×C₅	1	1	yes	characteristic
86	C₃×#389	C₂×C₅	1	1	yes	characteristic
87	C₃×#382	C₂×C₅	1	1	yes	characteristic
88	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
89	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
90	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
91	C₅×C₄⋊Q₈	—	2	4	no	—
92	C₅×#125	—	2	4	no	—
93	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
94	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
97	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
98	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
99	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
100	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
101	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
102	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
104	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
105	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
106	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
107	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
108	C₅×#95	C₂²×C₃	1	1	yes	characteristic
109	C₅×#95	C₂²×C₃	1	1	yes	characteristic
110	C₅×#95	C₂²×C₃	1	2	yes	—
111	C₅×#95	C₂²×C₃	1	2	yes	—
112	C₅×#95	C₂²×C₃	1	2	yes	—
113	C₅×#95	C₂²×C₃	1	2	yes	—
114	C₅×#95	C₂²×C₃	1	2	yes	—
115	#2550	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
116	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
117	C₃×C₅×Q₈	—	4	8	no	—
118	C₂×C₃×C₄×C₅	—	2	1	no	—
119	C₂×C₃×C₄×C₅	—	2	1	no	—
120	C₂×C₃×C₄×C₅	—	2	1	no	—
121	C₂×C₃×C₄×C₅	—	2	1	no	—
122	C₂×C₃×C₄×C₅	—	2	2	no	—
123	C₂×C₃×C₄×C₅	—	2	2	no	—
124	C₂×C₃×C₄×C₅	—	2	1	no	—
125	C₂×C₃×C₄×C₅	—	2	2	no	—
126	C₂×C₃×C₄×C₅	—	2	2	no	—
127	C₂×C₃×C₄×C₅	—	2	1	no	—
128	C₂×C₃×C₄×C₅	—	2	1	no	—
129	C₂×C₃×C₄×C₅	—	2	1	no	—
130	C₂×C₃×C₄×C₅	—	2	1	no	—
131	C₂×C₃×C₄×C₅	—	2	1	no	—
132	C₂×C₃×C₄×C₅	—	2	2	no	—
133	C₂×C₃×C₄×C₅	—	2	2	no	—
134	C₂×C₃×C₄×C₅	—	2	1	no	—
135	C₂×C₃×C₄×C₅	—	2	1	no	—
136	C₂×C₃×C₄×C₅	—	2	2	no	—
137	C₂×C₃×C₄×C₅	—	2	2	no	—
138	C₂×C₃×C₄×C₅	—	2	1	no	—
139	C₂×C₃×C₄×C₅	—	2	2	no	—
140	C₂×C₃×C₄×C₅	—	2	1	no	—
141	C₂×C₃×C₄×C₅	—	2	2	no	—
142	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
143	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
144	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
145	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
146	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
147	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
148	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
149	C₂×C₃×C₄×C₅	#41	1	2	yes	—
150	C₂×C₃×C₄×C₅	#41	1	2	yes	—
151	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
152	C₃×C₄⋊Q₈	—	2	4	no	—
153	C₃×#125	—	2	4	no	—
154	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
155	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
156	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
157	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
158	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
159	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
160	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
161	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
162	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
163	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
164	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
165	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
166	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
167	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
168	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
169	C₃×#95	C₂²×C₅	1	1	yes	characteristic
170	C₃×#95	C₂²×C₅	1	1	yes	characteristic
171	C₃×#95	C₂²×C₅	1	2	yes	—
172	C₃×#95	C₂²×C₅	1	2	yes	—
173	C₃×#95	C₂²×C₅	1	2	yes	—
174	C₃×#95	C₂²×C₅	1	2	yes	—
175	C₃×#95	C₂²×C₅	1	2	yes	—
176	C₂×C₅×Q₈	—	2	4	no	—
177	C₂×C₅×Q₈	—	2	4	no	—
178	C₂×C₅×Q₈	—	4	2	no	—
179	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
180	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
181	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
182	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
183	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
184	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
185	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
186	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
187	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
188	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
189	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
190	C₅×C₄⋊C₄	—	2	4	no	—
191	C₅×C₄⋊C₄	—	2	4	no	—
192	C₅×C₄⋊C₄	—	2	2	no	—
193	C₅×C₄⋊C₄	—	2	4	no	—
194	C₅×C₄⋊C₄	—	2	2	no	—
195	C₅×C₄⋊C₄	—	2	2	no	—
196	C₅×C₄⋊C₄	—	2	4	no	—
197	C₅×C₄⋊C₄	—	2	2	no	—
198	C₅×C₄⋊C₄	—	4	2	no	—
199	C₅×C₄⋊C₄	C₃×D₄	1	4	yes	—
200	C₅×C₄⋊C₄	C₃×Q₈	1	4	yes	complemented
201	C₄²×C₅	—	2	2	no	—
202	C₄²×C₅	—	2	2	no	—
203	C₄²×C₅	—	4	2	no	—
204	C₄²×C₅	C₃×Q₈	1	4	yes	complemented
205	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
206	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic
207	#398	C₂×C₃×C₅	1	2	yes	—
208	#395	C₂×C₃×C₅	1	2	yes	—
209	#391	C₂×C₃×C₅	1	2	yes	—
210	#389	C₂×C₃×C₅	1	2	yes	—
211	#389	C₂×C₃×C₅	1	1	yes	characteristic
212	#389	C₂×C₃×C₅	1	1	yes	characteristic
213	#382	C₂×C₃×C₅	1	1	yes	characteristic
214	C₄×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
215	C₄×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
216	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
217	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
218	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
219	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
220	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
221	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
223	C₃×C₄×C₅	—	2	2	no	—
224	C₃×C₄×C₅	—	2	2	no	—
225	C₃×C₄×C₅	—	2	4	no	—
226	C₃×C₄×C₅	—	2	2	no	—
227	C₃×C₄×C₅	—	2	2	no	—
228	C₃×C₄×C₅	—	2	4	no	—
229	C₃×C₄×C₅	—	4	1	no	—
230	C₃×C₄×C₅	—	4	2	no	—
231	C₃×C₄×C₅	—	4	1	no	—
232	C₃×C₄×C₅	—	4	2	no	—
233	C₃×C₄×C₅	C₄⋊Q₈	1	4	yes	—
234	C₂×C₃×Q₈	—	2	4	no	—
235	C₂×C₃×Q₈	—	2	4	no	—
236	C₂×C₃×Q₈	—	4	2	no	—
237	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
238	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
239	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
240	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
241	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
242	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
243	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
244	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
245	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
246	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
247	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
248	C₃×C₄⋊C₄	—	2	4	no	—
249	C₃×C₄⋊C₄	—	2	4	no	—
250	C₃×C₄⋊C₄	—	2	2	no	—
251	C₃×C₄⋊C₄	—	2	4	no	—
252	C₃×C₄⋊C₄	—	2	2	no	—
253	C₃×C₄⋊C₄	—	2	2	no	—
254	C₃×C₄⋊C₄	—	2	4	no	—
255	C₃×C₄⋊C₄	—	2	2	no	—
256	C₃×C₄⋊C₄	—	4	2	no	—
257	C₃×C₄⋊C₄	C₅×D₄	1	4	yes	—
258	C₃×C₄⋊C₄	C₅×Q₈	1	4	yes	complemented
259	C₃×C₄²	—	2	2	no	—
260	C₃×C₄²	—	2	2	no	—
261	C₃×C₄²	—	4	2	no	—
262	C₃×C₄²	C₅×Q₈	1	4	yes	complemented
263	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
264	C₅×Q₈	—	4	8	no	—
265	C₂×C₄×C₅	—	2	1	no	—
266	C₂×C₄×C₅	—	2	1	no	—
267	C₂×C₄×C₅	—	2	1	no	—
268	C₂×C₄×C₅	—	2	1	no	—
269	C₂×C₄×C₅	—	2	2	no	—
270	C₂×C₄×C₅	—	2	2	no	—
271	C₂×C₄×C₅	—	2	1	no	—
272	C₂×C₄×C₅	—	2	2	no	—
273	C₂×C₄×C₅	—	2	2	no	—
274	C₂×C₄×C₅	—	2	1	no	—
275	C₂×C₄×C₅	—	2	1	no	—
276	C₂×C₄×C₅	—	2	1	no	—
277	C₂×C₄×C₅	—	2	1	no	—
278	C₂×C₄×C₅	—	2	1	no	—
279	C₂×C₄×C₅	—	2	2	no	—
280	C₂×C₄×C₅	—	2	2	no	—
281	C₂×C₄×C₅	—	2	1	no	—
282	C₂×C₄×C₅	—	2	1	no	—
283	C₂×C₄×C₅	—	2	2	no	—
284	C₂×C₄×C₅	—	2	2	no	—
285	C₂×C₄×C₅	—	2	1	no	—
286	C₂×C₄×C₅	—	2	2	no	—
287	C₂×C₄×C₅	—	2	1	no	—
288	C₂×C₄×C₅	—	2	2	no	—
289	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
290	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
291	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
292	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
293	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
294	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
295	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
296	C₂×C₄×C₅	C₃×#41	1	2	yes	—
297	C₂×C₄×C₅	C₃×#41	1	2	yes	—
298	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
299	C₄⋊Q₈	—	2	4	no	—
300	#125	—	2	4	no	—
301	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
303	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
304	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
306	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
308	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
309	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
310	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
312	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
313	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
314	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
315	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
316	#95	C₂²×C₃×C₅	1	1	yes	characteristic
317	#95	C₂²×C₃×C₅	1	1	yes	characteristic
318	#95	C₂²×C₃×C₅	1	2	yes	—
319	#95	C₂²×C₃×C₅	1	2	yes	—
320	#95	C₂²×C₃×C₅	1	2	yes	—
321	#95	C₂²×C₃×C₅	1	2	yes	—
322	#95	C₂²×C₃×C₅	1	2	yes	—
323	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
324	C₂×C₃×C₅	#533	1	1	yes	characteristic
325	C₂×C₃×C₅	#544	1	1	yes	characteristic
326	C₂×C₃×C₅	#546	1	1	yes	characteristic
327	C₂×C₃×C₅	#548	1	1	yes	characteristic
328	C₂×C₃×C₅	#554	1	1	yes	characteristic
329	C₂×C₃×C₅	#557	1	1	yes	characteristic
330	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
331	C₃×Q₈	—	4	8	no	—
332	C₂×C₃×C₄	—	2	1	no	—
333	C₂×C₃×C₄	—	2	1	no	—
334	C₂×C₃×C₄	—	2	1	no	—
335	C₂×C₃×C₄	—	2	1	no	—
336	C₂×C₃×C₄	—	2	2	no	—
337	C₂×C₃×C₄	—	2	2	no	—
338	C₂×C₃×C₄	—	2	1	no	—
339	C₂×C₃×C₄	—	2	2	no	—
340	C₂×C₃×C₄	—	2	2	no	—
341	C₂×C₃×C₄	—	2	1	no	—
342	C₂×C₃×C₄	—	2	1	no	—
343	C₂×C₃×C₄	—	2	1	no	—
344	C₂×C₃×C₄	—	2	1	no	—
345	C₂×C₃×C₄	—	2	1	no	—
346	C₂×C₃×C₄	—	2	2	no	—
347	C₂×C₃×C₄	—	2	2	no	—
348	C₂×C₃×C₄	—	2	1	no	—
349	C₂×C₃×C₄	—	2	1	no	—
350	C₂×C₃×C₄	—	2	2	no	—
351	C₂×C₃×C₄	—	2	2	no	—
352	C₂×C₃×C₄	—	2	1	no	—
353	C₂×C₃×C₄	—	2	2	no	—
354	C₂×C₃×C₄	—	2	1	no	—
355	C₂×C₃×C₄	—	2	2	no	—
356	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
357	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
358	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
359	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
360	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
361	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
362	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
363	C₂×C₃×C₄	C₅×#41	1	2	yes	—
364	C₂×C₃×C₄	C₅×#41	1	2	yes	—
365	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
366	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
367	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
368	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
369	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
370	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
371	C₂²×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
372	C₄×C₅	—	2	2	no	—
373	C₄×C₅	—	2	2	no	—
374	C₄×C₅	—	2	4	no	—
375	C₄×C₅	—	2	2	no	—
376	C₄×C₅	—	2	2	no	—
377	C₄×C₅	—	2	4	no	—
378	C₄×C₅	—	4	1	no	—
379	C₄×C₅	—	4	2	no	—
380	C₄×C₅	—	4	1	no	—
381	C₄×C₅	—	4	2	no	—
382	C₄×C₅	C₃×C₄⋊Q₈	1	4	yes	—
383	C₂×Q₈	—	2	4	no	—
384	C₂×Q₈	—	2	4	no	—
385	C₂×Q₈	—	4	2	no	—
386	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
387	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
388	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
389	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
390	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
391	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
392	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
393	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
394	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
395	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
396	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	4	2	no	—
406	C₄⋊C₄	C₃×C₅×D₄	1	4	yes	—
407	C₄⋊C₄	C₃×C₅×Q₈	1	4	yes	complemented
408	C₄²	—	2	2	no	—
409	C₄²	—	2	2	no	—
410	C₄²	—	4	2	no	—
411	C₄²	C₃×C₅×Q₈	1	4	yes	complemented
412	C₃×C₅	#2550	1	1	yes	characteristic, direct product factor, Hall
413	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
414	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
415	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
416	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
417	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
418	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
419	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
420	C₃×C₄	—	2	2	no	—
421	C₃×C₄	—	2	2	no	—
422	C₃×C₄	—	2	4	no	—
423	C₃×C₄	—	2	2	no	—
424	C₃×C₄	—	2	2	no	—
425	C₃×C₄	—	2	4	no	—
426	C₃×C₄	—	4	1	no	—
427	C₃×C₄	—	4	2	no	—
428	C₃×C₄	—	4	1	no	—
429	C₃×C₄	—	4	2	no	—
430	C₃×C₄	C₅×C₄⋊Q₈	1	4	yes	—
431	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
432	C₂×C₅	C₃×#533	1	1	yes	characteristic
433	C₂×C₅	C₃×#544	1	1	yes	characteristic
434	C₂×C₅	C₃×#546	1	1	yes	characteristic
435	C₂×C₅	C₃×#548	1	1	yes	characteristic
436	C₂×C₅	C₃×#554	1	1	yes	characteristic
437	C₂×C₅	C₃×#557	1	1	yes	characteristic
438	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
439	Q₈	—	4	8	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
465	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
466	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
467	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
468	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
469	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
470	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
471	C₂×C₄	C₃×C₅×#41	1	2	yes	—
472	C₂×C₄	C₃×C₅×#41	1	2	yes	—
473	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
474	C₂×C₃	C₅×#533	1	1	yes	characteristic
475	C₂×C₃	C₅×#544	1	1	yes	characteristic
476	C₂×C₃	C₅×#546	1	1	yes	characteristic
477	C₂×C₃	C₅×#548	1	1	yes	characteristic
478	C₂×C₃	C₅×#554	1	1	yes	characteristic
479	C₂×C₃	C₅×#557	1	1	yes	characteristic
480	C₅	C₃×#2550	1	1	yes	characteristic, direct product factor, Sylow
481	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
482	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
483	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
484	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
485	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
486	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
487	C₂²	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
488	C₄	—	2	2	no	—
489	C₄	—	2	2	no	—
490	C₄	—	2	4	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	2	4	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	—	4	2	no	—
498	C₄	C₃×C₅×C₄⋊Q₈	1	4	yes	—
499	C₃	C₅×#2550	1	1	yes	characteristic, direct product factor, Sylow
500	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
501	C₂	C₃×C₅×#533	1	1	yes	characteristic
502	C₂	C₃×C₅×#544	1	1	yes	characteristic
503	C₂	C₃×C₅×#546	1	1	yes	characteristic
504	C₂	C₃×C₅×#548	1	1	yes	characteristic
505	C₂	C₃×C₅×#554	1	1	yes	characteristic
506	C₂	C₃×C₅×#557	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
12. C₅×#2550	C₃	direct (N × H)
38. C₃×#2550	C₅	direct (N × H)
74. C₃×C₅×C₄⋊C₄	Q₈	semidirect
78. C₃×C₄²×C₅	Q₈	semidirect
115. #2550	C₃×C₅	direct (N × H)
200. C₅×C₄⋊C₄	C₃×Q₈	semidirect
204. C₄²×C₅	C₃×Q₈	semidirect
258. C₃×C₄⋊C₄	C₅×Q₈	semidirect
262. C₃×C₄²	C₅×Q₈	semidirect
407. C₄⋊C₄	C₃×C₅×Q₈	semidirect
411. C₄²	C₃×C₅×Q₈	semidirect
412. C₃×C₅	#2550	direct (N × H)
480. C₅	C₃×#2550	direct (N × H)
499. C₃	C₅×#2550	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
C₂×C₃×C₅×#125	2
C₃×C₅×#398	2
C₃×C₅×#395	2
C₃×C₅×#391	2
C₃×C₅×#389	2
C₃×C₅×#382	2
C₃×C₄×C₅×C₄⋊C₄	2
C₅×#2550	3
C₃×#2550	5

Subgroups — C3×C5×#2550 #14275584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2550 #14275584