Subgroups — C₅×#74156 #14326776

Order 1920.

1008 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#872	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4990	C₂	1	2	yes	maximal
4	C₅×#4976	C₂	1	2	yes	maximal, complemented
5	C₅×#4932	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4669	C₂	1	2	yes	maximal
7	C₅×#4667	C₂	1	1	yes	maximal, characteristic
8	C₅×#4666	C₂	1	1	yes	maximal, characteristic
9	C₅×#4650	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4649	C₂	1	2	yes	maximal, complemented
11	C₅×#4648	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2346	—	3	1	no	maximal, Hall
13	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
17	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
18	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
21	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
22	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
25	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
26	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
27	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
28	C₅×#872	C₄	1	8	yes	complemented
29	C₃×C₅×#95	C₂²	1	1	yes	characteristic
30	C₃×C₅×#95	C₂²	1	2	yes	—
31	C₃×C₅×#95	C₂²	1	1	yes	characteristic
32	C₅×C₃⋊#95	C₂²	1	1	yes	characteristic
33	C₅×C₃⋊#95	C₂²	1	2	yes	—
34	C₅×C₃⋊#95	C₂²	1	2	yes	—
35	C₅×C₃⋊#95	C₂²	1	1	yes	characteristic
36	C₅×C₃⋊#95	C₂²	1	2	yes	—
37	#74156	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
38	C₂×C₅×#117	—	3	1	no	—
39	C₅×#389	—	3	2	no	—
40	C₅×#388	—	3	2	no	—
41	C₅×#388	—	3	2	no	—
42	C₅×#382	—	3	1	no	—
43	C₅×#382	D₃	1	1	yes	characteristic, complemented
44	C₅×#380	—	3	1	no	—
45	C₄×C₅×C₄⋊C₄	—	3	2	no	—
46	C₄×C₅×C₂²⋊C₄	—	3	1	no	—
47	C₅×#376	—	3	1	no	—
48	C₅×#376	—	3	1	no	—
49	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
50	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
56	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
59	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₅×D₃	—	2	4	no	—
61	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	—
62	C₃×C₅×C₄⋊C₄	—	4	2	no	—
63	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
64	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
65	C₅×(C₂×D₃)⋊C₄	—	4	2	no	—
66	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
67	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
68	C₅×Dic₃⋊C₄	—	2	4	no	—
69	C₅×Dic₃⋊C₄	—	4	2	no	—
70	C₅×Dic₃⋊C₄	C₂×C₄	1	4	yes	—
71	C₅×Dic₃⋊C₄	C₂×C₄	1	4	yes	—
72	C₄×C₅×Dic₃	—	2	4	no	—
73	C₄×C₅×Dic₃	—	2	4	no	—
74	C₄×C₅×Dic₃	—	2	4	no	—
75	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	—
76	C₂×#872	C₂×C₅	1	1	yes	characteristic
77	C₃×#382	C₂×C₅	1	1	yes	characteristic, complemented
78	#4990	C₂×C₅	1	2	yes	—
79	#4976	C₂×C₅	1	2	yes	complemented
80	#4932	C₂×C₅	1	1	yes	characteristic, complemented
81	#4669	C₂×C₅	1	2	yes	—
82	#4667	C₂×C₅	1	1	yes	characteristic
83	#4666	C₂×C₅	1	1	yes	characteristic
84	#4650	C₂×C₅	1	1	yes	characteristic, complemented
85	#4649	C₂×C₅	1	2	yes	complemented
86	#4648	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂³×C₄×C₅	—	3	1	no	—
88	C₅×#117	—	3	8	no	—
89	C₂×C₅×C₄⋊C₄	—	3	1	no	—
90	C₂×C₅×C₄⋊C₄	—	3	1	no	—
91	C₂×C₅×C₄⋊C₄	—	3	2	no	—
92	C₂×C₅×C₄⋊C₄	—	3	2	no	—
93	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
94	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
95	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
96	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
97	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
98	C₂×C₄²×C₅	—	3	1	no	—
99	C₂×C₄²×C₅	—	3	2	no	—
100	C₂×C₄²×C₅	—	3	2	no	—
101	C₂×C₄²×C₅	—	3	2	no	—
102	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
103	C₅×#95	—	3	1	no	—
104	C₅×#95	—	3	1	no	—
105	C₅×#95	—	3	2	no	—
106	C₅×#95	—	3	2	no	—
107	C₅×#95	—	3	2	no	—
108	C₅×#95	C₂×D₃	1	1	yes	characteristic
109	C₅×#95	C₂×D₃	1	1	yes	characteristic
110	C₅×#95	C₂×D₃	1	2	yes	—
111	#2346	—	3	1	no	Sylow
112	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
113	C₂²×C₅×D₃	—	2	2	no	—
114	C₂²×C₅×D₃	—	2	2	no	—
115	C₂²×C₅×D₃	—	2	1	no	—
116	C₂²×C₅×D₃	—	2	1	no	—
117	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
118	C₂×C₃×C₄×C₅	—	2	2	no	—
119	C₂×C₃×C₄×C₅	—	2	2	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	2	no	—
125	C₂×C₃×C₄×C₅	—	2	2	no	—
126	C₂×C₃×C₄×C₅	—	2	2	no	—
127	C₂×C₃×C₄×C₅	—	2	2	no	—
128	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
129	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
130	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
131	C₂×C₅×Dic₃	—	2	2	no	—
132	C₂×C₅×Dic₃	—	2	2	no	—
133	C₂×C₅×Dic₃	—	2	1	no	—
134	C₂×C₅×Dic₃	—	2	1	no	—
135	C₂×C₅×Dic₃	—	2	2	no	—
136	C₂×C₅×Dic₃	—	2	2	no	—
137	C₂×C₅×Dic₃	—	2	2	no	—
138	C₂×C₅×Dic₃	—	2	2	no	—
139	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
140	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
141	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
142	C₂×C₅×Dic₃	#41	1	4	yes	—
143	C₂×C₅×Dic₃	#41	1	4	yes	—
144	C₄×C₅×D₃	—	2	8	no	—
145	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
146	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
147	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
148	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
149	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
150	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
151	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
152	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
153	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
154	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
155	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
156	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
157	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
158	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
159	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
160	#872	C₄×C₅	1	8	yes	complemented
161	C₃×#95	C₂²×C₅	1	1	yes	characteristic
162	C₃×#95	C₂²×C₅	1	2	yes	—
163	C₃×#95	C₂²×C₅	1	1	yes	characteristic
164	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
165	C₃⋊#95	C₂²×C₅	1	2	yes	—
166	C₃⋊#95	C₂²×C₅	1	2	yes	—
167	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
168	C₃⋊#95	C₂²×C₅	1	2	yes	—
169	C₂⁴×C₅	—	3	1	no	—
170	C₂²×C₄×C₅	—	3	1	no	—
171	C₂²×C₄×C₅	—	3	1	no	—
172	C₂²×C₄×C₅	—	3	2	no	—
173	C₂²×C₄×C₅	—	3	1	no	—
174	C₂²×C₄×C₅	—	3	4	no	—
175	C₂²×C₄×C₅	—	3	2	no	—
176	C₂²×C₄×C₅	—	6	4	no	—
177	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
178	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
179	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
180	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
181	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
182	C₅×C₄⋊C₄	—	3	4	no	—
183	C₅×C₄⋊C₄	—	3	4	no	—
184	C₅×C₄⋊C₄	—	4	2	no	—
185	C₅×C₄⋊C₄	—	6	4	no	—
186	C₅×C₄⋊C₄	—	12	2	no	—
187	C₅×C₂²⋊C₄	—	3	4	no	—
188	C₅×C₂²⋊C₄	—	3	4	no	—
189	C₅×C₂²⋊C₄	—	6	4	no	—
190	C₅×C₂²⋊C₄	—	12	2	no	—
191	C₄²×C₅	—	3	4	no	—
192	C₄²×C₅	—	6	4	no	—
193	C₄²×C₅	—	6	4	no	—
194	C₄²×C₅	—	6	4	no	—
195	C₄²×C₅	C₄×D₃	1	4	yes	—
196	C₂×#117	—	3	1	no	—
197	#389	—	3	2	no	—
198	#388	—	3	2	no	—
199	#388	—	3	2	no	—
200	#382	—	3	1	no	—
201	#382	C₅×D₃	1	1	yes	characteristic, complemented
202	#380	—	3	1	no	—
203	C₄×C₄⋊C₄	—	3	2	no	—
204	C₄×C₂²⋊C₄	—	3	1	no	—
205	#376	—	3	1	no	—
206	#376	—	3	1	no	—
207	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
208	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
209	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
210	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
211	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
212	C₂×C₅×D₃	—	2	2	no	—
213	C₂×C₅×D₃	—	2	2	no	—
214	C₂×C₅×D₃	—	2	2	no	—
215	C₂×C₅×D₃	—	4	2	no	—
216	C₂×C₅×D₃	—	4	2	no	—
217	C₃×C₄×C₅	—	2	2	no	—
218	C₃×C₄×C₅	—	2	2	no	—
219	C₃×C₄×C₅	—	2	2	no	—
220	C₃×C₄×C₅	—	4	2	no	—
221	C₃×C₄×C₅	—	4	2	no	—
222	C₅×Dic₃	—	2	2	no	—
223	C₅×Dic₃	—	2	2	no	—
224	C₅×Dic₃	—	2	4	no	—
225	C₅×Dic₃	—	4	2	no	—
226	C₅×Dic₃	#117	1	4	yes	—
227	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
228	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
229	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
230	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
231	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
232	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
233	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
234	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
235	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
236	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
237	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
238	C₂×C₄×D₃	—	2	4	no	—
239	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	—
240	C₃×C₄⋊C₄	—	4	2	no	—
241	C₃×C₄²	C₂×C₄×C₅	1	4	yes	—
242	(C₂×D₃)⋊C₄	—	2	4	no	—
243	(C₂×D₃)⋊C₄	—	4	2	no	—
244	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	4	yes	—
245	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	4	yes	—
246	Dic₃⋊C₄	—	2	4	no	—
247	Dic₃⋊C₄	—	4	2	no	—
248	Dic₃⋊C₄	C₂×C₄×C₅	1	4	yes	—
249	Dic₃⋊C₄	C₂×C₄×C₅	1	4	yes	—
250	C₄×Dic₃	—	2	4	no	—
251	C₄×Dic₃	—	2	4	no	—
252	C₄×Dic₃	—	2	4	no	—
253	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	—
254	C₂³×C₅	—	3	2	no	—
255	C₂³×C₅	—	6	1	no	—
256	C₂³×C₅	—	6	1	no	—
257	C₂³×C₅	—	6	2	no	—
258	C₂³×C₅	—	6	2	no	—
259	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
260	C₂×C₄×C₅	—	2	1	no	—
261	C₂×C₄×C₅	—	2	1	no	—
262	C₂×C₄×C₅	—	2	2	no	—
263	C₂×C₄×C₅	—	2	2	no	—
264	C₂×C₄×C₅	—	2	2	no	—
265	C₂×C₄×C₅	—	2	2	no	—
266	C₂×C₄×C₅	—	2	2	no	—
267	C₂×C₄×C₅	—	2	2	no	—
268	C₂×C₄×C₅	—	2	2	no	—
269	C₂×C₄×C₅	—	2	2	no	—
270	C₂×C₄×C₅	—	3	2	no	—
271	C₂×C₄×C₅	—	3	4	no	—
272	C₂×C₄×C₅	—	3	2	no	—
273	C₂×C₄×C₅	—	3	2	no	—
274	C₂×C₄×C₅	—	3	4	no	—
275	C₂×C₄×C₅	—	6	8	no	—
276	C₂×C₄×C₅	—	6	1	no	—
277	C₂×C₄×C₅	—	6	2	no	—
278	C₂×C₄×C₅	—	6	1	no	—
279	C₂×C₄×C₅	—	6	2	no	—
280	C₂×C₄×C₅	—	6	2	no	—
281	C₂×C₄×C₅	—	6	2	no	—
282	C₂×C₄×C₅	—	6	2	no	—
283	C₂×C₄×C₅	—	6	2	no	—
284	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
285	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
286	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
287	C₂³×C₄	—	3	1	no	—
288	#117	—	3	8	no	—
289	C₂×C₄⋊C₄	—	3	1	no	—
290	C₂×C₄⋊C₄	—	3	1	no	—
291	C₂×C₄⋊C₄	—	3	2	no	—
292	C₂×C₄⋊C₄	—	3	2	no	—
293	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
294	C₂×C₂²⋊C₄	—	3	1	no	—
295	C₂×C₂²⋊C₄	—	3	1	no	—
296	C₂×C₂²⋊C₄	—	3	2	no	—
297	C₂×C₂²⋊C₄	—	3	2	no	—
298	C₂×C₄²	—	3	1	no	—
299	C₂×C₄²	—	3	2	no	—
300	C₂×C₄²	—	3	2	no	—
301	C₂×C₄²	—	3	2	no	—
302	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
303	#95	—	3	1	no	—
304	#95	—	3	1	no	—
305	#95	—	3	2	no	—
306	#95	—	3	2	no	—
307	#95	—	3	2	no	—
308	#95	C₂×C₅×D₃	1	1	yes	characteristic
309	#95	C₂×C₅×D₃	1	1	yes	characteristic
310	#95	C₂×C₅×D₃	1	2	yes	—
311	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
312	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
313	C₂×C₃×C₅	#520	1	1	yes	characteristic
314	C₂×C₃×C₅	#552	1	2	yes	—
315	C₂×C₃×C₅	#555	1	2	yes	—
316	C₅×D₃	—	4	2	no	—
317	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
318	C₂²×D₃	—	2	2	no	—
319	C₂²×D₃	—	2	2	no	—
320	C₂²×D₃	—	2	1	no	—
321	C₂²×D₃	—	2	1	no	—
322	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
323	C₂×C₃×C₄	—	2	2	no	—
324	C₂×C₃×C₄	—	2	2	no	—
325	C₂×C₃×C₄	—	2	1	no	—
326	C₂×C₃×C₄	—	2	1	no	—
327	C₂×C₃×C₄	—	2	2	no	—
328	C₂×C₃×C₄	—	2	2	no	—
329	C₂×C₃×C₄	—	2	2	no	—
330	C₂×C₃×C₄	—	2	2	no	—
331	C₂×C₃×C₄	—	2	2	no	—
332	C₂×C₃×C₄	—	2	2	no	—
333	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
334	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
335	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
336	C₂×Dic₃	—	2	2	no	—
337	C₂×Dic₃	—	2	2	no	—
338	C₂×Dic₃	—	2	1	no	—
339	C₂×Dic₃	—	2	1	no	—
340	C₂×Dic₃	—	2	2	no	—
341	C₂×Dic₃	—	2	2	no	—
342	C₂×Dic₃	—	2	2	no	—
343	C₂×Dic₃	—	2	2	no	—
344	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
345	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
346	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
347	C₂×Dic₃	C₅×#41	1	4	yes	—
348	C₂×Dic₃	C₅×#41	1	4	yes	—
349	C₄×D₃	—	2	8	no	—
350	C₂²×C₅	—	6	2	no	—
351	C₂²×C₅	—	6	2	no	—
352	C₂²×C₅	—	6	2	no	—
353	C₂²×C₅	—	12	2	no	—
354	C₂²×C₅	—	12	2	no	—
355	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
356	C₂²×C₅	#1004	1	1	yes	characteristic
357	C₂²×C₅	#1007	1	1	yes	characteristic
358	C₂²×C₅	D₃×#41	1	2	yes	—
359	C₂²×C₅	D₃×#41	1	2	yes	—
360	C₄×C₅	—	2	2	no	—
361	C₄×C₅	—	2	2	no	—
362	C₄×C₅	—	2	2	no	—
363	C₄×C₅	—	3	4	no	—
364	C₄×C₅	—	4	2	no	—
365	C₄×C₅	—	4	2	no	—
366	C₄×C₅	—	6	4	no	—
367	C₄×C₅	—	6	2	no	—
368	C₄×C₅	—	6	2	no	—
369	C₄×C₅	—	12	2	no	—
370	C₂⁴	—	3	1	no	—
371	C₂²×C₄	—	3	1	no	—
372	C₂²×C₄	—	3	2	no	—
373	C₂²×C₄	—	3	1	no	—
374	C₂²×C₄	—	3	1	no	—
375	C₂²×C₄	—	3	4	no	—
376	C₂²×C₄	—	3	2	no	—
377	C₂²×C₄	—	6	4	no	—
378	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
379	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
380	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
381	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
382	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
383	C₄⋊C₄	—	3	4	no	—
384	C₄⋊C₄	—	3	4	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	—	6	4	no	—
387	C₄⋊C₄	—	12	2	no	—
388	C₂²⋊C₄	—	3	4	no	—
389	C₂²⋊C₄	—	3	4	no	—
390	C₂²⋊C₄	—	6	4	no	—
391	C₂²⋊C₄	—	12	2	no	—
392	C₄²	—	3	4	no	—
393	C₄²	—	6	4	no	—
394	C₄²	—	6	4	no	—
395	C₄²	—	6	4	no	—
396	C₄²	C₄×C₅×D₃	1	4	yes	—
397	C₃×C₅	#2346	1	1	yes	characteristic, complemented, Hall
398	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
399	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
400	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
401	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
402	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
403	C₂×D₃	—	2	2	no	—
404	C₂×D₃	—	2	2	no	—
405	C₂×D₃	—	2	2	no	—
406	C₂×D₃	—	4	2	no	—
407	C₂×D₃	—	4	2	no	—
408	C₃×C₄	—	2	2	no	—
409	C₃×C₄	—	2	2	no	—
410	C₃×C₄	—	2	2	no	—
411	C₃×C₄	—	4	2	no	—
412	C₃×C₄	—	4	2	no	—
413	Dic₃	—	2	2	no	—
414	Dic₃	—	2	2	no	—
415	Dic₃	—	2	4	no	—
416	Dic₃	—	4	2	no	—
417	Dic₃	C₅×#117	1	4	yes	—
418	C₂×C₅	—	12	2	no	—
419	C₂×C₅	#5504	1	1	yes	characteristic
420	C₂×C₅	D₃×#117	1	1	yes	characteristic
421	C₂×C₅	#5522	1	1	yes	characteristic
422	C₂×C₅	#5670	1	2	yes	—
423	C₂×C₅	#5693	1	2	yes	—
424	C₂³	—	3	2	no	—
425	C₂³	—	6	1	no	—
426	C₂³	—	6	1	no	—
427	C₂³	—	6	2	no	—
428	C₂³	—	6	2	no	—
429	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	3	2	no	—
441	C₂×C₄	—	3	4	no	—
442	C₂×C₄	—	3	2	no	—
443	C₂×C₄	—	3	2	no	—
444	C₂×C₄	—	3	4	no	—
445	C₂×C₄	—	6	8	no	—
446	C₂×C₄	—	6	1	no	—
447	C₂×C₄	—	6	2	no	—
448	C₂×C₄	—	6	1	no	—
449	C₂×C₄	—	6	2	no	—
450	C₂×C₄	—	6	2	no	—
451	C₂×C₄	—	6	2	no	—
452	C₂×C₄	—	6	2	no	—
453	C₂×C₄	—	6	2	no	—
454	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
455	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
456	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
457	C₂×C₃	C₂×C₅×#117	1	1	yes	characteristic
458	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
459	C₂×C₃	C₅×#520	1	1	yes	characteristic
460	C₂×C₃	C₅×#552	1	2	yes	—
461	C₂×C₃	C₅×#555	1	2	yes	—
462	D₃	—	4	2	no	—
463	C₅	#74156	1	1	yes	characteristic, direct product factor, Sylow
464	C₂²	—	6	2	no	—
465	C₂²	—	6	2	no	—
466	C₂²	—	6	2	no	—
467	C₂²	—	12	2	no	—
468	C₂²	—	12	2	no	—
469	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
470	C₂²	C₅×#1004	1	1	yes	characteristic
471	C₂²	C₅×#1007	1	1	yes	characteristic
472	C₂²	C₅×D₃×#41	1	2	yes	—
473	C₂²	C₅×D₃×#41	1	2	yes	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	3	4	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	6	4	no	—
481	C₄	—	6	2	no	—
482	C₄	—	6	2	no	—
483	C₄	—	12	2	no	—
484	C₃	C₅×#2346	1	1	yes	characteristic, complemented, Sylow
485	C₂	—	12	2	no	—
486	C₂	C₅×#5504	1	1	yes	characteristic
487	C₂	C₅×D₃×#117	1	1	yes	characteristic
488	C₂	C₅×#5522	1	1	yes	characteristic
489	C₂	C₅×#5670	1	2	yes	—
490	C₂	C₅×#5693	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#382	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂³×C₅×D₃	C₂³
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#382	D₃
hypocenter	C₃	C₅×#2346
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#382

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#382	C₂	semidirect
4. C₅×#4976	C₂	semidirect
5. C₅×#4932	C₂	semidirect
9. C₅×#4650	C₂	semidirect
10. C₅×#4649	C₂	semidirect
11. C₅×#4648	C₂	semidirect
28. C₅×#872	C₄	semidirect
37. #74156	C₅	direct (N × H)
43. C₅×#382	D₃	semidirect
77. C₃×#382	C₂×C₅	semidirect
79. #4976	C₂×C₅	semidirect
80. #4932	C₂×C₅	semidirect
84. #4650	C₂×C₅	semidirect
85. #4649	C₂×C₅	semidirect
86. #4648	C₂×C₅	semidirect
160. #872	C₄×C₅	semidirect
201. #382	C₅×D₃	semidirect
397. C₃×C₅	#2346	semidirect
463. C₅	#74156	direct (N × H)
484. C₃	C₅×#2346	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#872	2
C₃×C₅×#382	2
C₅×#4990	2
C₅×#4976	2
C₅×#4932	2
C₅×#4669	2
C₅×#4667	2
C₅×#4666	2
C₅×#4650	2
C₅×#4649	2
C₅×#4648	2
C₅×#2346	3
#74156	5

Subgroups — C5×#74156 #14326776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#74156 #14326776