Subgroups — C₅×#73416 #14326413

Order 1920.

1064 conjugacy classes of subgroups.

The subgroup lattice has 496 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×#376	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal
4	C₅×#4936	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4933	C₂	1	2	yes	maximal, complemented
6	C₅×#4668	C₂	1	2	yes	maximal
7	C₅×#4664	C₂	1	1	yes	maximal, characteristic
8	C₅×D₃×#95	C₂	1	1	yes	maximal, characteristic
9	C₅×#4655	C₂	1	2	yes	maximal, complemented
10	C₅×#4651	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#4649	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2343	—	3	1	no	maximal, Hall
13	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
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×C₄⋊Dic₃	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	1	yes	characteristic
26	C₅×#872	C₄	1	8	yes	complemented
27	C₃×C₅×#95	C₂²	1	1	yes	characteristic
28	C₃×C₅×#95	C₂²	1	1	yes	characteristic
29	C₃×C₅×#95	C₂²	1	2	yes	—
30	C₅×C₃⋊#95	C₂²	1	1	yes	characteristic
31	C₅×C₃⋊#95	C₂²	1	2	yes	—
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	2	yes	—
36	C₅×C₃⋊#95	C₂²	1	2	yes	—
37	#73416	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
38	C₂×C₅×#117	—	3	1	no	—
39	C₅×#387	—	3	2	no	—
40	C₅×#387	—	3	2	no	—
41	C₅×#385	—	3	2	no	—
42	C₅×#383	—	3	1	no	—
43	C₅×#382	—	3	1	no	—
44	C₅×#381	—	3	1	no	—
45	C₄×C₅×C₄⋊C₄	—	3	1	no	—
46	C₄×C₅×C₂²⋊C₄	—	3	2	no	—
47	C₅×#376	D₃	1	1	yes	characteristic, complemented
48	C₂×C₅×#95	—	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	1	yes	characteristic
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	2	yes	—
55	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
56	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
57	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₅×D₃	—	2	4	no	—
61	C₂×C₄×C₅×D₃	—	2	4	no	—
62	C₂×C₄×C₅×D₃	—	2	4	no	—
63	C₂×C₄×C₅×D₃	—	4	2	no	—
64	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	—
65	C₃×C₄²×C₅	—	2	4	no	—
66	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
67	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
68	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
69	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
70	C₅×C₄⋊Dic₃	—	4	2	no	—
71	C₅×Dic₃⋊C₄	—	2	4	no	—
72	C₅×Dic₃⋊C₄	C₂×C₄	1	4	yes	complemented
73	C₅×Dic₃⋊C₄	C₂×C₄	1	4	yes	complemented
74	C₄×C₅×Dic₃	—	4	2	no	—
75	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	complemented
76	C₂×#872	C₂×C₅	1	1	yes	characteristic
77	C₃×#376	C₂×C₅	1	1	yes	characteristic, complemented
78	C₄×(C₂×D₃)⋊C₄	C₂×C₅	1	2	yes	—
79	#4936	C₂×C₅	1	1	yes	characteristic, complemented
80	#4933	C₂×C₅	1	2	yes	complemented
81	#4668	C₂×C₅	1	2	yes	—
82	#4664	C₂×C₅	1	1	yes	characteristic
83	D₃×#95	C₂×C₅	1	1	yes	characteristic
84	#4655	C₂×C₅	1	2	yes	complemented
85	#4651	C₂×C₅	1	1	yes	characteristic, complemented
86	#4649	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂³×C₄×C₅	—	3	1	no	—
88	C₂³×C₄×C₅	—	3	2	no	—
89	C₅×#117	—	3	8	no	—
90	C₂×C₅×C₄⋊C₄	—	3	1	no	—
91	C₂×C₅×C₄⋊C₄	—	3	1	no	—
92	C₂×C₅×C₄⋊C₄	—	3	2	no	—
93	C₂×C₅×C₄⋊C₄	—	3	2	no	—
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×C₄²×C₅	—	3	1	no	—
98	C₂×C₄²×C₅	—	3	2	no	—
99	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
100	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
101	C₅×#95	—	3	1	no	—
102	C₅×#95	—	3	2	no	—
103	C₅×#95	—	3	2	no	—
104	C₅×#95	—	3	2	no	—
105	C₅×#95	—	3	1	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	#2343	—	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	1	no	—
115	C₂²×C₅×D₃	—	2	1	no	—
116	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
117	C₂²×C₅×D₃	#41	1	4	yes	—
118	C₂×C₃×C₄×C₅	—	2	2	no	—
119	C₂×C₃×C₄×C₅	—	2	2	no	—
120	C₂×C₃×C₄×C₅	—	2	2	no	—
121	C₂×C₃×C₄×C₅	—	2	2	no	—
122	C₂×C₃×C₄×C₅	—	2	1	no	—
123	C₂×C₃×C₄×C₅	—	2	1	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	C₂²×C₄	1	2	yes	—
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	#41	1	4	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	—	2	2	no	—
140	C₂×C₅×Dic₃	—	2	2	no	—
141	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
142	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
143	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
144	C₄×C₅×D₃	—	2	8	no	—
145	C₄×C₅×D₃	—	4	8	no	—
146	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
147	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
148	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
149	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
150	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
151	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
152	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
153	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
154	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
155	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
156	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
157	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
158	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
159	#872	C₄×C₅	1	8	yes	complemented
160	C₃×#95	C₂²×C₅	1	1	yes	characteristic
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	2	yes	—
165	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
166	C₃⋊#95	C₂²×C₅	1	2	yes	—
167	C₃⋊#95	C₂²×C₅	1	2	yes	—
168	C₃⋊#95	C₂²×C₅	1	2	yes	—
169	C₃⋊#95	C₂²×C₅	1	2	yes	—
170	C₂⁴×C₅	—	3	1	no	—
171	C₂²×C₄×C₅	—	3	1	no	—
172	C₂²×C₄×C₅	—	3	1	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	—	3	2	no	—
177	C₂²×C₄×C₅	—	6	4	no	—
178	C₂²×C₄×C₅	—	6	4	no	—
179	C₂²×C₄×C₅	—	6	4	no	—
180	C₂²×C₄×C₅	—	12	2	no	—
181	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
182	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
183	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
184	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
185	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
186	C₅×C₄⋊C₄	—	3	4	no	—
187	C₅×C₄⋊C₄	—	3	4	no	—
188	C₅×C₄⋊C₄	—	6	4	no	—
189	C₅×C₄⋊C₄	—	12	2	no	—
190	C₅×C₂²⋊C₄	—	3	4	no	—
191	C₅×C₂²⋊C₄	—	3	4	no	—
192	C₅×C₂²⋊C₄	—	6	4	no	—
193	C₄²×C₅	—	2	4	no	—
194	C₄²×C₅	—	3	4	no	—
195	C₄²×C₅	—	12	2	no	—
196	C₄²×C₅	C₄×D₃	1	4	yes	complemented
197	C₂×#117	—	3	1	no	—
198	#387	—	3	2	no	—
199	#387	—	3	2	no	—
200	#385	—	3	2	no	—
201	#383	—	3	1	no	—
202	#382	—	3	1	no	—
203	#381	—	3	1	no	—
204	C₄×C₄⋊C₄	—	3	1	no	—
205	C₄×C₂²⋊C₄	—	3	2	no	—
206	#376	C₅×D₃	1	1	yes	characteristic, complemented
207	C₂×#95	—	3	1	no	—
208	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
209	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
210	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
211	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
213	C₂×C₅×D₃	—	2	2	no	—
214	C₂×C₅×D₃	—	2	2	no	—
215	C₂×C₅×D₃	—	2	4	no	—
216	C₂×C₅×D₃	—	2	4	no	—
217	C₂×C₅×D₃	#117	1	4	yes	—
218	C₃×C₄×C₅	—	2	2	no	—
219	C₃×C₄×C₅	—	2	2	no	—
220	C₃×C₄×C₅	—	2	2	no	—
221	C₃×C₄×C₅	—	2	4	no	—
222	C₃×C₄×C₅	—	4	2	no	—
223	C₅×Dic₃	—	2	2	no	—
224	C₅×Dic₃	—	2	2	no	—
225	C₅×Dic₃	—	2	2	no	—
226	C₅×Dic₃	—	4	2	no	—
227	C₅×Dic₃	—	4	2	no	—
228	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
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×C₃×C₄	C₂³×C₅	1	2	yes	—
234	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
235	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
236	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
237	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
238	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
239	C₂×C₄×D₃	—	2	4	no	—
240	C₂×C₄×D₃	—	2	4	no	—
241	C₂×C₄×D₃	—	2	4	no	—
242	C₂×C₄×D₃	—	4	2	no	—
243	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	—
244	C₃×C₄²	—	2	4	no	—
245	C₃×C₄²	C₂×C₄×C₅	1	4	yes	complemented
246	(C₂×D₃)⋊C₄	—	2	4	no	—
247	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	4	yes	—
248	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	4	yes	—
249	C₄⋊Dic₃	—	4	2	no	—
250	Dic₃⋊C₄	—	2	4	no	—
251	Dic₃⋊C₄	C₂×C₄×C₅	1	4	yes	complemented
252	Dic₃⋊C₄	C₂×C₄×C₅	1	4	yes	complemented
253	C₄×Dic₃	—	4	2	no	—
254	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	complemented
255	C₂³×C₅	—	3	2	no	—
256	C₂³×C₅	—	3	4	no	—
257	C₂³×C₅	—	6	2	no	—
258	C₂³×C₅	—	6	1	no	—
259	C₂³×C₅	—	6	1	no	—
260	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
261	C₂×C₄×C₅	—	2	2	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	1	no	—
266	C₂×C₄×C₅	—	2	1	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	2	no	—
272	C₂×C₄×C₅	—	3	2	no	—
273	C₂×C₄×C₅	—	6	8	no	—
274	C₂×C₄×C₅	—	6	2	no	—
275	C₂×C₄×C₅	—	6	2	no	—
276	C₂×C₄×C₅	—	6	1	no	—
277	C₂×C₄×C₅	—	6	1	no	—
278	C₂×C₄×C₅	—	6	2	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	—	12	8	no	—
285	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
286	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
287	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
288	C₂×C₄×C₅	#235	1	4	yes	—
289	C₂³×C₄	—	3	1	no	—
290	C₂³×C₄	—	3	2	no	—
291	#117	—	3	8	no	—
292	C₂×C₄⋊C₄	—	3	1	no	—
293	C₂×C₄⋊C₄	—	3	1	no	—
294	C₂×C₄⋊C₄	—	3	2	no	—
295	C₂×C₄⋊C₄	—	3	2	no	—
296	C₂×C₂²⋊C₄	—	3	1	no	—
297	C₂×C₂²⋊C₄	—	3	1	no	—
298	C₂×C₂²⋊C₄	—	3	2	no	—
299	C₂×C₄²	—	3	1	no	—
300	C₂×C₄²	—	3	2	no	—
301	C₂×C₄²	C₂×C₅×D₃	1	2	yes	—
302	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
303	#95	—	3	1	no	—
304	#95	—	3	2	no	—
305	#95	—	3	2	no	—
306	#95	—	3	2	no	—
307	#95	—	3	1	no	—
308	#95	—	3	2	no	—
309	#95	—	3	2	no	—
310	#95	C₂×C₅×D₃	1	1	yes	characteristic
311	#95	C₂×C₅×D₃	1	1	yes	characteristic
312	#95	C₂×C₅×D₃	1	2	yes	—
313	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
314	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
315	C₂×C₃×C₅	#520	1	1	yes	characteristic
316	C₂×C₃×C₅	#551	1	2	yes	—
317	C₂×C₃×C₅	#552	1	2	yes	—
318	C₅×D₃	—	2	4	no	—
319	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
320	C₂²×D₃	—	2	2	no	—
321	C₂²×D₃	—	2	1	no	—
322	C₂²×D₃	—	2	1	no	—
323	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
324	C₂²×D₃	C₅×#41	1	4	yes	—
325	C₂×C₃×C₄	—	2	2	no	—
326	C₂×C₃×C₄	—	2	2	no	—
327	C₂×C₃×C₄	—	2	2	no	—
328	C₂×C₃×C₄	—	2	2	no	—
329	C₂×C₃×C₄	—	2	1	no	—
330	C₂×C₃×C₄	—	2	1	no	—
331	C₂×C₃×C₄	—	2	2	no	—
332	C₂×C₃×C₄	—	2	2	no	—
333	C₂×C₃×C₄	—	2	2	no	—
334	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
335	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
336	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
337	C₂×C₃×C₄	C₅×#41	1	4	yes	—
338	C₂×Dic₃	—	2	2	no	—
339	C₂×Dic₃	—	2	2	no	—
340	C₂×Dic₃	—	2	1	no	—
341	C₂×Dic₃	—	2	1	no	—
342	C₂×Dic₃	—	2	2	no	—
343	C₂×Dic₃	—	2	2	no	—
344	C₂×Dic₃	—	2	2	no	—
345	C₂×Dic₃	—	2	2	no	—
346	C₂×Dic₃	—	2	2	no	—
347	C₂×Dic₃	—	2	2	no	—
348	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
349	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
350	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
351	C₄×D₃	—	2	8	no	—
352	C₄×D₃	—	4	8	no	—
353	C₂²×C₅	—	3	4	no	—
354	C₂²×C₅	—	6	2	no	—
355	C₂²×C₅	—	6	2	no	—
356	C₂²×C₅	—	6	4	no	—
357	C₂²×C₅	—	6	4	no	—
358	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
359	C₂²×C₅	C₂×#235	1	2	yes	—
360	C₂²×C₅	D₃×#41	1	2	yes	—
361	C₂²×C₅	#1009	1	1	yes	characteristic
362	C₂²×C₅	#1010	1	1	yes	characteristic
363	C₄×C₅	—	2	2	no	—
364	C₄×C₅	—	2	2	no	—
365	C₄×C₅	—	2	2	no	—
366	C₄×C₅	—	2	4	no	—
367	C₄×C₅	—	4	2	no	—
368	C₄×C₅	—	6	2	no	—
369	C₄×C₅	—	6	2	no	—
370	C₄×C₅	—	6	2	no	—
371	C₄×C₅	—	12	2	no	—
372	C₄×C₅	—	12	2	no	—
373	C₂⁴	—	3	1	no	—
374	C₂²×C₄	—	3	1	no	—
375	C₂²×C₄	—	3	1	no	—
376	C₂²×C₄	—	3	1	no	—
377	C₂²×C₄	—	3	4	no	—
378	C₂²×C₄	—	3	2	no	—
379	C₂²×C₄	—	3	2	no	—
380	C₂²×C₄	—	6	4	no	—
381	C₂²×C₄	—	6	4	no	—
382	C₂²×C₄	—	6	4	no	—
383	C₂²×C₄	—	12	2	no	—
384	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
385	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
386	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
387	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
388	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
389	C₄⋊C₄	—	3	4	no	—
390	C₄⋊C₄	—	3	4	no	—
391	C₄⋊C₄	—	6	4	no	—
392	C₄⋊C₄	—	12	2	no	—
393	C₂²⋊C₄	—	3	4	no	—
394	C₂²⋊C₄	—	3	4	no	—
395	C₂²⋊C₄	—	6	4	no	—
396	C₄²	—	2	4	no	—
397	C₄²	—	3	4	no	—
398	C₄²	—	12	2	no	—
399	C₄²	C₄×C₅×D₃	1	4	yes	complemented
400	C₃×C₅	#2343	1	1	yes	characteristic, complemented, Hall
401	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
402	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
403	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
404	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
405	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
406	C₂×D₃	—	2	2	no	—
407	C₂×D₃	—	2	2	no	—
408	C₂×D₃	—	2	4	no	—
409	C₂×D₃	—	2	4	no	—
410	C₂×D₃	C₅×#117	1	4	yes	—
411	C₃×C₄	—	2	2	no	—
412	C₃×C₄	—	2	2	no	—
413	C₃×C₄	—	2	2	no	—
414	C₃×C₄	—	2	4	no	—
415	C₃×C₄	—	4	2	no	—
416	Dic₃	—	2	2	no	—
417	Dic₃	—	2	2	no	—
418	Dic₃	—	2	2	no	—
419	Dic₃	—	4	2	no	—
420	Dic₃	—	4	2	no	—
421	C₂×C₅	—	6	4	no	—
422	C₂×C₅	C₄×#235	1	1	yes	characteristic
423	C₂×C₅	D₃×#117	1	1	yes	characteristic
424	C₂×C₅	#5523	1	1	yes	characteristic
425	C₂×C₅	#5527	1	2	yes	—
426	C₂×C₅	#5529	1	2	yes	—
427	C₂³	—	3	2	no	—
428	C₂³	—	3	4	no	—
429	C₂³	—	6	2	no	—
430	C₂³	—	6	1	no	—
431	C₂³	—	6	1	no	—
432	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
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	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	3	2	no	—
443	C₂×C₄	—	3	2	no	—
444	C₂×C₄	—	3	2	no	—
445	C₂×C₄	—	6	8	no	—
446	C₂×C₄	—	6	2	no	—
447	C₂×C₄	—	6	2	no	—
448	C₂×C₄	—	6	1	no	—
449	C₂×C₄	—	6	1	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	—	6	2	no	—
455	C₂×C₄	—	6	2	no	—
456	C₂×C₄	—	12	8	no	—
457	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
458	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
459	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
460	C₂×C₄	C₅×#235	1	4	yes	—
461	C₂×C₃	C₂×C₅×#117	1	1	yes	characteristic
462	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
463	C₂×C₃	C₅×#520	1	1	yes	characteristic
464	C₂×C₃	C₅×#551	1	2	yes	—
465	C₂×C₃	C₅×#552	1	2	yes	—
466	D₃	—	2	4	no	—
467	C₅	#73416	1	1	yes	characteristic, direct product factor, Sylow
468	C₂²	—	3	4	no	—
469	C₂²	—	6	2	no	—
470	C₂²	—	6	2	no	—
471	C₂²	—	6	4	no	—
472	C₂²	—	6	4	no	—
473	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
474	C₂²	C₂×C₅×#235	1	2	yes	—
475	C₂²	C₅×D₃×#41	1	2	yes	—
476	C₂²	C₅×#1009	1	1	yes	characteristic
477	C₂²	C₅×#1010	1	1	yes	characteristic
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	4	no	—
482	C₄	—	4	2	no	—
483	C₄	—	6	2	no	—
484	C₄	—	6	2	no	—
485	C₄	—	6	2	no	—
486	C₄	—	12	2	no	—
487	C₄	—	12	2	no	—
488	C₃	C₅×#2343	1	1	yes	characteristic, complemented, Sylow
489	C₂	—	6	4	no	—
490	C₂	C₄×C₅×#235	1	1	yes	characteristic
491	C₂	C₅×D₃×#117	1	1	yes	characteristic
492	C₂	C₅×#5523	1	1	yes	characteristic
493	C₂	C₅×#5527	1	2	yes	—
494	C₂	C₅×#5529	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×#376	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂³×C₅×D₃	C₂³
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#376	D₃
hypocenter	C₃	C₅×#2343
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#376	C₂	semidirect
4. C₅×#4936	C₂	semidirect
5. C₅×#4933	C₂	semidirect
9. C₅×#4655	C₂	semidirect
10. C₅×#4651	C₂	semidirect
11. C₅×#4649	C₂	semidirect
26. C₅×#872	C₄	semidirect
37. #73416	C₅	direct (N × H)
47. C₅×#376	D₃	semidirect
66. C₃×C₄²×C₅	C₂×C₄	semidirect
72. C₅×Dic₃⋊C₄	C₂×C₄	semidirect
73. C₅×Dic₃⋊C₄	C₂×C₄	semidirect
75. C₄×C₅×Dic₃	C₂×C₄	semidirect
77. C₃×#376	C₂×C₅	semidirect
79. #4936	C₂×C₅	semidirect
80. #4933	C₂×C₅	semidirect
84. #4655	C₂×C₅	semidirect
85. #4651	C₂×C₅	semidirect
86. #4649	C₂×C₅	semidirect
159. #872	C₄×C₅	semidirect
196. C₄²×C₅	C₄×D₃	semidirect
206. #376	C₅×D₃	semidirect
245. C₃×C₄²	C₂×C₄×C₅	semidirect
251. Dic₃⋊C₄	C₂×C₄×C₅	semidirect
252. Dic₃⋊C₄	C₂×C₄×C₅	semidirect
254. C₄×Dic₃	C₂×C₄×C₅	semidirect
399. C₄²	C₄×C₅×D₃	semidirect
400. C₃×C₅	#2343	semidirect
467. C₅	#73416	direct (N × H)
488. C₃	C₅×#2343	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×#376	2
C₄×C₅×(C₂×D₃)⋊C₄	2
C₅×#4936	2
C₅×#4933	2
C₅×#4668	2
C₅×#4664	2
C₅×D₃×#95	2
C₅×#4655	2
C₅×#4651	2
C₅×#4649	2
C₅×#2343	3
#73416	5

Subgroups — C5×#73416 #14326413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#73416 #14326413