Subgroups — C₅×#82225 #14320814

Order 1920.

1304 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₅×#67	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#872	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4990	C₂	1	2	yes	maximal, complemented
6	C₅×#4976	C₂	1	2	yes	maximal, complemented
7	C₅×#4957	C₂	1	2	yes	maximal
8	C₅×#4945	C₂	1	2	yes	maximal, complemented
9	C₄×C₅×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2293	—	3	1	no	maximal, Hall
13	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂²×C₅×#67	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₃×C₅×#117	—	2	4	no	—
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₂×C₅×#217	C₂²	1	2	yes	—
22	C₂×C₅×#217	C₂²	1	1	yes	characteristic
23	C₄×C₅×#67	C₄	1	8	yes	complemented
24	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
32	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
33	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
35	C₅×#872	—	2	4	no	—
36	C₅×C₃⋊#95	C₂²	1	2	yes	—
37	C₅×C₃⋊#95	C₂²	1	2	yes	—
38	C₅×C₃⋊#95	C₂²	1	2	yes	—
39	#82225	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
40	C₂×C₄×C₅×D₄	—	3	1	no	—
41	C₂×C₅×#117	—	3	1	no	—
42	C₂×C₅×#117	D₃	1	1	yes	characteristic, complemented
43	C₅×#390	—	3	2	no	—
44	C₅×#389	—	3	2	no	—
45	C₅×#388	—	3	2	no	—
46	C₅×#388	—	3	2	no	—
47	C₄×C₅×C₄⋊C₄	—	3	1	no	—
48	C₄×C₅×C₂²⋊C₄	—	3	1	no	—
49	C₄×C₅×C₂²⋊C₄	—	3	1	no	—
50	C₄³×C₅	—	3	1	no	—
51	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
52	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄×C₅	—	2	2	no	—
54	C₂²×C₃×C₄×C₅	—	2	2	no	—
55	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
56	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
57	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
59	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂×C₅×#67	—	2	4	no	—
61	C₂×C₅×#67	C₂×C₄	1	4	yes	—
62	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
63	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
64	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
65	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
66	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
67	C₂×C₄×C₅×D₃	—	2	2	no	—
68	C₂×C₄×C₅×D₃	—	2	2	no	—
69	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	—
70	C₃×C₅×C₄⋊C₄	—	2	4	no	—
71	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
72	C₃×C₄²×C₅	—	2	2	no	—
73	C₃×C₄²×C₅	D₄	1	4	yes	complemented
74	C₅×#217	—	2	4	no	—
75	C₅×#217	C₂×C₄	1	4	yes	—
76	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
77	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
78	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
79	C₅×Dic₃⋊C₄	—	2	4	no	—
80	C₅×Dic₃⋊C₄	—	2	4	no	—
81	C₅×Dic₃⋊C₄	C₂×C₄	1	4	yes	—
82	C₄×C₅×Dic₃	—	2	2	no	—
83	C₄×C₅×Dic₃	—	2	2	no	—
84	C₄×C₅×Dic₃	—	2	4	no	—
85	C₄×C₅×Dic₃	—	2	4	no	—
86	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	—
87	C₂×C₃×#117	C₂×C₅	1	1	yes	characteristic, complemented
88	C₂×C₄×#67	C₂×C₅	1	1	yes	characteristic
89	C₂×#872	C₂×C₅	1	1	yes	characteristic, complemented
90	C₄×#217	C₂×C₅	1	1	yes	characteristic, complemented
91	#4990	C₂×C₅	1	2	yes	complemented
92	#4976	C₂×C₅	1	2	yes	complemented
93	#4957	C₂×C₅	1	2	yes	—
94	#4945	C₂×C₅	1	2	yes	complemented
95	C₄×(C₂×D₃)⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
96	C₄×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
97	C₄²×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
98	C₂²×C₅×D₄	—	3	1	no	—
99	C₂³×C₄×C₅	—	3	1	no	—
100	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
101	C₄×C₅×D₄	—	3	8	no	—
102	C₅×#117	—	2	4	no	—
103	C₅×#117	—	6	4	no	—
104	C₂×C₅×C₄⋊C₄	—	3	1	no	—
105	C₂×C₅×C₄⋊C₄	—	3	2	no	—
106	C₂×C₅×C₄⋊C₄	—	3	2	no	—
107	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
108	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
109	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
110	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
111	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
112	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
113	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
114	C₂×C₄²×C₅	—	3	1	no	—
115	C₂×C₄²×C₅	—	3	2	no	—
116	C₂×C₄²×C₅	—	3	2	no	—
117	C₂×C₄²×C₅	—	3	1	no	—
118	C₂×C₄²×C₅	—	3	1	no	—
119	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
120	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
121	C₅×#95	—	3	2	no	—
122	C₅×#95	—	3	2	no	—
123	C₅×#95	—	3	2	no	—
124	#2293	—	3	1	no	Sylow
125	C₂³×C₃×C₅	—	2	1	no	—
126	C₂³×C₃×C₅	—	2	2	no	—
127	C₂³×C₃×C₅	—	2	2	no	—
128	C₂³×C₃×C₅	—	2	1	no	—
129	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
130	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
131	C₂²×C₅×D₃	—	2	1	no	—
132	C₂²×C₅×D₃	—	2	2	no	—
133	C₂²×C₅×D₃	—	2	1	no	—
134	C₂²×C₅×D₃	—	2	2	no	—
135	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
136	C₂×C₃×C₄×C₅	—	2	1	no	—
137	C₂×C₃×C₄×C₅	—	2	2	no	—
138	C₂×C₃×C₄×C₅	—	2	2	no	—
139	C₂×C₃×C₄×C₅	—	2	1	no	—
140	C₂×C₃×C₄×C₅	—	2	8	no	—
141	C₂×C₃×C₄×C₅	—	2	2	no	—
142	C₂×C₃×C₄×C₅	—	2	2	no	—
143	C₂×C₃×C₄×C₅	—	2	2	no	—
144	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
145	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
146	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
147	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
148	C₂×C₃×C₄×C₅	#41	1	2	yes	—
149	C₅×#67	—	2	8	no	—
150	C₂×C₅×Dic₃	—	2	1	no	—
151	C₂×C₅×Dic₃	—	2	2	no	—
152	C₂×C₅×Dic₃	—	2	2	no	—
153	C₂×C₅×Dic₃	—	2	2	no	—
154	C₂×C₅×Dic₃	—	2	1	no	—
155	C₂×C₅×Dic₃	—	2	2	no	—
156	C₂×C₅×Dic₃	—	2	2	no	—
157	C₂×C₅×Dic₃	—	2	2	no	—
158	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
159	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
160	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
161	C₂×C₅×Dic₃	#41	1	4	yes	—
162	C₂×C₅×Dic₃	#41	1	4	yes	—
163	C₄×C₅×D₃	—	2	8	no	—
164	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
165	C₂²×#67	C₂²×C₅	1	1	yes	characteristic
166	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
167	C₃×#117	—	2	4	no	—
168	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
169	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
170	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
171	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
172	C₂×#217	C₂²×C₅	1	2	yes	—
173	C₂×#217	C₂²×C₅	1	1	yes	characteristic
174	C₄×#67	C₄×C₅	1	8	yes	complemented
175	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
176	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
177	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
178	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
179	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
180	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
181	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
182	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
183	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
184	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
185	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
186	#872	—	2	4	no	—
187	C₃⋊#95	C₂²×C₅	1	2	yes	—
188	C₃⋊#95	C₂²×C₅	1	2	yes	—
189	C₃⋊#95	C₂²×C₅	1	2	yes	—
190	C₂⁴×C₅	—	3	1	no	—
191	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
192	C₂×C₅×D₄	—	3	4	no	—
193	C₂×C₅×D₄	—	6	4	no	—
194	C₂²×C₄×C₅	—	2	2	no	—
195	C₂²×C₄×C₅	—	2	2	no	—
196	C₂²×C₄×C₅	—	3	1	no	—
197	C₂²×C₄×C₅	—	3	1	no	—
198	C₂²×C₄×C₅	—	3	2	no	—
199	C₂²×C₄×C₅	—	3	1	no	—
200	C₂²×C₄×C₅	—	3	4	no	—
201	C₂²×C₄×C₅	—	3	2	no	—
202	C₂²×C₄×C₅	—	6	2	no	—
203	C₂²×C₄×C₅	—	6	2	no	—
204	C₂²×C₄×C₅	C₄×D₃	1	4	yes	—
205	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
206	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
207	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
208	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
209	C₅×C₄⋊C₄	—	2	4	no	—
210	C₅×C₄⋊C₄	—	3	4	no	—
211	C₅×C₄⋊C₄	—	6	4	no	—
212	C₅×C₄⋊C₄	—	6	4	no	—
213	C₅×C₂²⋊C₄	—	2	4	no	—
214	C₅×C₂²⋊C₄	—	3	4	no	—
215	C₅×C₂²⋊C₄	—	3	4	no	—
216	C₅×C₂²⋊C₄	—	6	4	no	—
217	C₅×C₂²⋊C₄	—	6	4	no	—
218	C₅×C₂²⋊C₄	—	6	4	no	—
219	C₄²×C₅	—	2	2	no	—
220	C₄²×C₅	—	3	4	no	—
221	C₄²×C₅	—	6	4	no	—
222	C₄²×C₅	—	6	4	no	—
223	C₄²×C₅	—	6	2	no	—
224	C₄²×C₅	—	6	2	no	—
225	C₄²×C₅	#67	1	4	yes	complemented
226	C₂×C₄×D₄	—	3	1	no	—
227	C₂×#117	—	3	1	no	—
228	C₂×#117	C₅×D₃	1	1	yes	characteristic, complemented
229	#390	—	3	2	no	—
230	#389	—	3	2	no	—
231	#388	—	3	2	no	—
232	#388	—	3	2	no	—
233	C₄×C₄⋊C₄	—	3	1	no	—
234	C₄×C₂²⋊C₄	—	3	1	no	—
235	C₄×C₂²⋊C₄	—	3	1	no	—
236	C₄³	—	3	1	no	—
237	C₂²×C₃×C₅	—	2	2	no	—
238	C₂²×C₃×C₅	—	2	2	no	—
239	C₂²×C₃×C₅	—	2	2	no	—
240	C₂²×C₃×C₅	—	4	2	no	—
241	C₂²×C₃×C₅	—	4	2	no	—
242	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
243	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
244	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
245	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
246	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
247	C₂×C₅×D₃	—	2	2	no	—
248	C₂×C₅×D₃	—	2	2	no	—
249	C₂×C₅×D₃	—	2	2	no	—
250	C₂×C₅×D₃	—	4	2	no	—
251	C₂×C₅×D₃	—	4	2	no	—
252	C₃×C₄×C₅	—	2	2	no	—
253	C₃×C₄×C₅	—	2	4	no	—
254	C₃×C₄×C₅	—	4	2	no	—
255	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
256	C₅×Dic₃	—	2	2	no	—
257	C₅×Dic₃	—	2	2	no	—
258	C₅×Dic₃	—	2	4	no	—
259	C₅×Dic₃	—	4	2	no	—
260	C₅×Dic₃	#117	1	4	yes	complemented
261	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
262	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
263	C₂²×C₃×C₄	—	2	2	no	—
264	C₂²×C₃×C₄	—	2	2	no	—
265	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
266	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
267	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
268	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
269	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
270	C₂×#67	—	2	4	no	—
271	C₂×#67	C₂×C₄×C₅	1	4	yes	—
272	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
273	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
274	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
275	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
276	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
277	C₂×C₄×D₃	—	2	2	no	—
278	C₂×C₄×D₃	—	2	2	no	—
279	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	—
280	C₃×C₄⋊C₄	—	2	4	no	—
281	C₃×C₂²⋊C₄	—	2	4	no	—
282	C₃×C₄²	—	2	2	no	—
283	C₃×C₄²	C₅×D₄	1	4	yes	complemented
284	#217	—	2	4	no	—
285	#217	C₂×C₄×C₅	1	4	yes	—
286	(C₂×D₃)⋊C₄	—	2	4	no	—
287	(C₂×D₃)⋊C₄	—	2	4	no	—
288	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	4	yes	—
289	Dic₃⋊C₄	—	2	4	no	—
290	Dic₃⋊C₄	—	2	4	no	—
291	Dic₃⋊C₄	C₂×C₄×C₅	1	4	yes	—
292	C₄×Dic₃	—	2	2	no	—
293	C₄×Dic₃	—	2	2	no	—
294	C₄×Dic₃	—	2	4	no	—
295	C₄×Dic₃	—	2	4	no	—
296	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	—
297	C₂³×C₅	—	2	1	no	—
298	C₂³×C₅	—	2	2	no	—
299	C₂³×C₅	—	2	2	no	—
300	C₂³×C₅	—	2	1	no	—
301	C₂³×C₅	—	3	2	no	—
302	C₂³×C₅	—	6	1	no	—
303	C₂³×C₅	—	6	2	no	—
304	C₂³×C₅	—	6	2	no	—
305	C₂³×C₅	—	6	1	no	—
306	C₂³×C₅	C₂×C₄×D₃	1	2	yes	—
307	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
308	C₅×D₄	—	6	8	no	—
309	C₂×C₄×C₅	—	2	1	no	—
310	C₂×C₄×C₅	—	2	2	no	—
311	C₂×C₄×C₅	—	2	2	no	—
312	C₂×C₄×C₅	—	2	1	no	—
313	C₂×C₄×C₅	—	2	8	no	—
314	C₂×C₄×C₅	—	2	2	no	—
315	C₂×C₄×C₅	—	2	2	no	—
316	C₂×C₄×C₅	—	2	2	no	—
317	C₂×C₄×C₅	—	3	2	no	—
318	C₂×C₄×C₅	—	3	4	no	—
319	C₂×C₄×C₅	—	3	2	no	—
320	C₂×C₄×C₅	—	3	2	no	—
321	C₂×C₄×C₅	—	3	4	no	—
322	C₂×C₄×C₅	—	6	2	no	—
323	C₂×C₄×C₅	—	6	1	no	—
324	C₂×C₄×C₅	—	6	2	no	—
325	C₂×C₄×C₅	—	6	2	no	—
326	C₂×C₄×C₅	—	6	1	no	—
327	C₂×C₄×C₅	—	6	8	no	—
328	C₂×C₄×C₅	—	6	2	no	—
329	C₂×C₄×C₅	—	6	2	no	—
330	C₂×C₄×C₅	—	6	2	no	—
331	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
332	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
333	C₂×C₄×C₅	#235	1	2	yes	—
334	C₂×C₄×C₅	C₂×#67	1	4	yes	—
335	C₂×C₄×C₅	C₂×#67	1	2	yes	—
336	C₂²×D₄	—	3	1	no	—
337	C₂³×C₄	—	3	1	no	—
338	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
339	C₄×D₄	—	3	8	no	—
340	#117	—	2	4	no	—
341	#117	—	6	4	no	—
342	C₂×C₄⋊C₄	—	3	1	no	—
343	C₂×C₄⋊C₄	—	3	2	no	—
344	C₂×C₄⋊C₄	—	3	2	no	—
345	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
346	C₂×C₂²⋊C₄	—	3	1	no	—
347	C₂×C₂²⋊C₄	—	3	1	no	—
348	C₂×C₂²⋊C₄	—	3	2	no	—
349	C₂×C₂²⋊C₄	—	3	2	no	—
350	C₂×C₂²⋊C₄	—	3	2	no	—
351	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
352	C₂×C₄²	—	3	1	no	—
353	C₂×C₄²	—	3	1	no	—
354	C₂×C₄²	—	3	1	no	—
355	C₂×C₄²	—	3	2	no	—
356	C₂×C₄²	—	3	2	no	—
357	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
358	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
359	#95	—	3	2	no	—
360	#95	—	3	2	no	—
361	#95	—	3	2	no	—
362	C₂×C₃×C₅	—	4	2	no	—
363	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
364	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
365	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
366	C₂×C₃×C₅	#529	1	2	yes	—
367	C₂×C₃×C₅	#532	1	2	yes	—
368	C₅×D₃	—	4	2	no	—
369	C₂³×C₃	—	2	1	no	—
370	C₂³×C₃	—	2	2	no	—
371	C₂³×C₃	—	2	2	no	—
372	C₂³×C₃	—	2	1	no	—
373	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
374	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
375	C₂²×D₃	—	2	1	no	—
376	C₂²×D₃	—	2	2	no	—
377	C₂²×D₃	—	2	1	no	—
378	C₂²×D₃	—	2	2	no	—
379	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
380	C₂×C₃×C₄	—	2	1	no	—
381	C₂×C₃×C₄	—	2	2	no	—
382	C₂×C₃×C₄	—	2	2	no	—
383	C₂×C₃×C₄	—	2	1	no	—
384	C₂×C₃×C₄	—	2	8	no	—
385	C₂×C₃×C₄	—	2	2	no	—
386	C₂×C₃×C₄	—	2	2	no	—
387	C₂×C₃×C₄	—	2	2	no	—
388	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
389	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
390	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
391	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
392	C₂×C₃×C₄	C₅×#41	1	2	yes	—
393	#67	—	2	8	no	—
394	C₂×Dic₃	—	2	1	no	—
395	C₂×Dic₃	—	2	2	no	—
396	C₂×Dic₃	—	2	2	no	—
397	C₂×Dic₃	—	2	2	no	—
398	C₂×Dic₃	—	2	1	no	—
399	C₂×Dic₃	—	2	2	no	—
400	C₂×Dic₃	—	2	2	no	—
401	C₂×Dic₃	—	2	2	no	—
402	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
403	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
404	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
405	C₂×Dic₃	C₅×#41	1	4	yes	—
406	C₂×Dic₃	C₅×#41	1	4	yes	—
407	C₄×D₃	—	2	8	no	—
408	C₂²×C₅	—	2	2	no	—
409	C₂²×C₅	—	2	2	no	—
410	C₂²×C₅	—	2	2	no	—
411	C₂²×C₅	—	4	2	no	—
412	C₂²×C₅	—	4	2	no	—
413	C₂²×C₅	—	6	2	no	—
414	C₂²×C₅	—	6	2	no	—
415	C₂²×C₅	—	6	2	no	—
416	C₂²×C₅	—	12	2	no	—
417	C₂²×C₅	—	12	2	no	—
418	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
419	C₂²×C₅	C₂×#235	1	1	yes	characteristic
420	C₂²×C₅	D₃×#41	1	2	yes	—
421	C₂²×C₅	D₃×#41	1	2	yes	—
422	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
423	C₄×C₅	—	2	2	no	—
424	C₄×C₅	—	2	4	no	—
425	C₄×C₅	—	3	4	no	—
426	C₄×C₅	—	4	2	no	—
427	C₄×C₅	—	6	2	no	—
428	C₄×C₅	—	6	4	no	—
429	C₄×C₅	—	6	2	no	—
430	C₄×C₅	—	12	2	no	—
431	C₄×C₅	C₄×#67	1	4	yes	—
432	C₂⁴	—	3	1	no	—
433	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
434	C₂×D₄	—	3	4	no	—
435	C₂×D₄	—	6	4	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	3	1	no	—
439	C₂²×C₄	—	3	1	no	—
440	C₂²×C₄	—	3	2	no	—
441	C₂²×C₄	—	3	1	no	—
442	C₂²×C₄	—	3	4	no	—
443	C₂²×C₄	—	3	2	no	—
444	C₂²×C₄	—	6	2	no	—
445	C₂²×C₄	—	6	2	no	—
446	C₂²×C₄	C₄×C₅×D₃	1	4	yes	—
447	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
448	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
449	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
450	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	3	4	no	—
453	C₄⋊C₄	—	6	4	no	—
454	C₄⋊C₄	—	6	4	no	—
455	C₂²⋊C₄	—	2	4	no	—
456	C₂²⋊C₄	—	3	4	no	—
457	C₂²⋊C₄	—	3	4	no	—
458	C₂²⋊C₄	—	6	4	no	—
459	C₂²⋊C₄	—	6	4	no	—
460	C₂²⋊C₄	—	6	4	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	3	4	no	—
463	C₄²	—	6	4	no	—
464	C₄²	—	6	4	no	—
465	C₄²	—	6	2	no	—
466	C₄²	—	6	2	no	—
467	C₄²	C₅×#67	1	4	yes	complemented
468	C₃×C₅	#2293	1	1	yes	characteristic, complemented, Hall
469	C₂²×C₃	—	2	2	no	—
470	C₂²×C₃	—	2	2	no	—
471	C₂²×C₃	—	2	2	no	—
472	C₂²×C₃	—	4	2	no	—
473	C₂²×C₃	—	4	2	no	—
474	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
475	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
476	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
477	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
478	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
479	C₂×D₃	—	2	2	no	—
480	C₂×D₃	—	2	2	no	—
481	C₂×D₃	—	2	2	no	—
482	C₂×D₃	—	4	2	no	—
483	C₂×D₃	—	4	2	no	—
484	C₃×C₄	—	2	2	no	—
485	C₃×C₄	—	2	4	no	—
486	C₃×C₄	—	4	2	no	—
487	C₃×C₄	C₄×C₅×D₄	1	4	yes	—
488	Dic₃	—	2	2	no	—
489	Dic₃	—	2	2	no	—
490	Dic₃	—	2	4	no	—
491	Dic₃	—	4	2	no	—
492	Dic₃	C₅×#117	1	4	yes	complemented
493	C₂×C₅	—	4	2	no	—
494	C₂×C₅	—	12	2	no	—
495	C₂×C₅	D₃×#117	1	1	yes	characteristic
496	C₂×C₅	#5522	1	1	yes	characteristic
497	C₂×C₅	#5528	1	2	yes	—
498	C₂×C₅	C₂×C₄×#67	1	1	yes	characteristic
499	C₂×C₅	#5832	1	2	yes	—
500	C₂³	—	2	1	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	2	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	3	2	no	—
505	C₂³	—	6	1	no	—
506	C₂³	—	6	2	no	—
507	C₂³	—	6	2	no	—
508	C₂³	—	6	1	no	—
509	C₂³	C₂×C₄×C₅×D₃	1	2	yes	—
510	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
511	D₄	—	6	8	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	8	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	3	2	no	—
521	C₂×C₄	—	3	4	no	—
522	C₂×C₄	—	3	2	no	—
523	C₂×C₄	—	3	2	no	—
524	C₂×C₄	—	3	4	no	—
525	C₂×C₄	—	6	2	no	—
526	C₂×C₄	—	6	1	no	—
527	C₂×C₄	—	6	2	no	—
528	C₂×C₄	—	6	2	no	—
529	C₂×C₄	—	6	1	no	—
530	C₂×C₄	—	6	8	no	—
531	C₂×C₄	—	6	2	no	—
532	C₂×C₄	—	6	2	no	—
533	C₂×C₄	—	6	2	no	—
534	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
535	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
536	C₂×C₄	C₅×#235	1	2	yes	—
537	C₂×C₄	C₂×C₅×#67	1	4	yes	—
538	C₂×C₄	C₂×C₅×#67	1	2	yes	—
539	C₂×C₃	—	4	2	no	—
540	C₂×C₃	C₂×C₅×#117	1	1	yes	characteristic
541	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
542	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
543	C₂×C₃	C₅×#529	1	2	yes	—
544	C₂×C₃	C₅×#532	1	2	yes	—
545	D₃	—	4	2	no	—
546	C₅	#82225	1	1	yes	characteristic, direct product factor, Sylow
547	C₂²	—	2	2	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	2	2	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	6	2	no	—
553	C₂²	—	6	2	no	—
554	C₂²	—	6	2	no	—
555	C₂²	—	12	2	no	—
556	C₂²	—	12	2	no	—
557	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
558	C₂²	C₂×C₅×#235	1	1	yes	characteristic
559	C₂²	C₅×D₃×#41	1	2	yes	—
560	C₂²	C₅×D₃×#41	1	2	yes	—
561	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
562	C₄	—	2	2	no	—
563	C₄	—	2	4	no	—
564	C₄	—	3	4	no	—
565	C₄	—	4	2	no	—
566	C₄	—	6	2	no	—
567	C₄	—	6	4	no	—
568	C₄	—	6	2	no	—
569	C₄	—	12	2	no	—
570	C₄	C₄×C₅×#67	1	4	yes	—
571	C₃	C₅×#2293	1	1	yes	characteristic, complemented, Sylow
572	C₂	—	4	2	no	—
573	C₂	—	12	2	no	—
574	C₂	C₅×D₃×#117	1	1	yes	characteristic
575	C₂	C₅×#5522	1	1	yes	characteristic
576	C₂	C₅×#5528	1	2	yes	—
577	C₂	C₂×C₄×C₅×#67	1	1	yes	characteristic
578	C₂	C₅×#5832	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×#117	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂²×C₅×#67	C₂²
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₂×C₅×#117	D₃
hypocenter	C₃	C₅×#2293
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#117

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#117	C₂	semidirect
3. C₂×C₅×#872	C₂	semidirect
4. C₄×C₅×#217	C₂	semidirect
5. C₅×#4990	C₂	semidirect
6. C₅×#4976	C₂	semidirect
8. C₅×#4945	C₂	semidirect
9. C₄×C₅×(C₂×D₃)⋊C₄	C₂	semidirect
10. C₄×C₅×Dic₃⋊C₄	C₂	semidirect
11. C₄²×C₅×Dic₃	C₂	semidirect
23. C₄×C₅×#67	C₄	semidirect
39. #82225	C₅	direct (N × H)
42. C₂×C₅×#117	D₃	semidirect
73. C₃×C₄²×C₅	D₄	semidirect
87. C₂×C₃×#117	C₂×C₅	semidirect
89. C₂×#872	C₂×C₅	semidirect
90. C₄×#217	C₂×C₅	semidirect
91. #4990	C₂×C₅	semidirect
92. #4976	C₂×C₅	semidirect
94. #4945	C₂×C₅	semidirect
95. C₄×(C₂×D₃)⋊C₄	C₂×C₅	semidirect
96. C₄×Dic₃⋊C₄	C₂×C₅	semidirect
97. C₄²×Dic₃	C₂×C₅	semidirect
174. C₄×#67	C₄×C₅	semidirect
225. C₄²×C₅	#67	semidirect
228. C₂×#117	C₅×D₃	semidirect
260. C₅×Dic₃	#117	semidirect
283. C₃×C₄²	C₅×D₄	semidirect
467. C₄²	C₅×#67	semidirect
468. C₃×C₅	#2293	semidirect
492. Dic₃	C₅×#117	semidirect
546. C₅	#82225	direct (N × H)
571. C₃	C₅×#2293	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×#117	2
C₂×C₄×C₅×#67	2
C₂×C₅×#872	2
C₄×C₅×#217	2
C₅×#4990	2
C₅×#4976	2
C₅×#4957	2
C₅×#4945	2
C₄×C₅×(C₂×D₃)⋊C₄	2
C₄×C₅×Dic₃⋊C₄	2
C₄²×C₅×Dic₃	2
C₅×#2293	3
#82225	5

Subgroups — C5×#82225 #14320814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#82225 #14320814