Subgroups — C₅×#73646 #14327987

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#893	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4988	C₂	1	1	yes	maximal, characteristic
4	C₅×#4977	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×Dic₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4940	C₂	1	2	yes	maximal, complemented
7	C₅×#4939	C₂	1	1	yes	maximal, characteristic
8	C₄×C₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₄×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4669	C₂	1	2	yes	maximal, complemented
11	C₅×#4664	C₂	1	2	yes	maximal, complemented
12	C₅×#2357	—	3	1	no	maximal, Hall
13	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
19	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
20	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
21	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
22	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
29	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
30	C₅×#893	C₄	1	8	yes	complemented
31	C₃×C₅×#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	1	yes	characteristic
37	#73646	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
38	C₂×C₄×C₅×D₄	—	3	1	no	—
39	C₅×#390	—	3	1	no	—
40	C₅×#390	—	3	1	no	—
41	C₅×#388	—	3	2	no	—
42	C₅×#388	—	3	2	no	—
43	C₅×#382	—	3	1	no	—
44	C₅×#381	—	3	2	no	—
45	C₄×C₅×C₄⋊C₄	—	3	1	no	—
46	C₄×C₅×C₄⋊C₄	—	3	1	no	—
47	C₄×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
48	C₄×C₅×C₂²⋊C₄	—	3	2	no	—
49	C₂³×C₅×D₃	C₂³	1	2	yes	—
50	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
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	1	yes	characteristic
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×C₅×D₁₂	—	2	2	no	—
61	C₂×C₅×D₁₂	—	2	2	no	—
62	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
63	C₂×C₄×C₅×D₃	—	4	2	no	—
64	C₂×C₄×C₅×D₃	—	4	2	no	—
65	C₂×C₄×C₅×D₃	C₂×C₄	1	8	yes	—
66	C₃×C₅×C₄⋊C₄	—	2	2	no	—
67	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
68	C₃×C₄²×C₅	—	2	2	no	—
69	C₃×C₄²×C₅	—	4	2	no	—
70	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
71	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
72	C₅×(C₂×D₃)⋊C₄	—	4	2	no	—
73	C₅×(C₂×D₃)⋊C₄	—	4	2	no	—
74	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	—
75	C₅×C₄⋊Dic₃	—	2	2	no	—
76	C₅×Dic₃⋊C₄	—	2	2	no	—
77	C₅×Dic₃⋊C₄	—	2	2	no	—
78	C₄×C₅×Dic₃	—	2	2	no	—
79	C₄×C₅×Dic₃	—	4	2	no	—
80	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	—
81	C₂×#893	C₂×C₅	1	1	yes	characteristic
82	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
83	#4988	C₂×C₅	1	1	yes	characteristic
84	#4977	C₂×C₅	1	1	yes	characteristic, complemented
85	Dic₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
86	#4940	C₂×C₅	1	2	yes	complemented
87	#4939	C₂×C₅	1	1	yes	characteristic
88	C₄×(C₂×D₃)⋊C₄	C₂×C₅	1	2	yes	complemented
89	C₄×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
90	#4669	C₂×C₅	1	2	yes	complemented
91	#4664	C₂×C₅	1	2	yes	complemented
92	C₂²×C₅×D₄	—	3	1	no	—
93	C₂³×C₄×C₅	—	3	2	no	—
94	C₄×C₅×D₄	—	3	8	no	—
95	C₂×C₅×C₄⋊C₄	—	3	1	no	—
96	C₂×C₅×C₄⋊C₄	—	3	1	no	—
97	C₂×C₅×C₄⋊C₄	—	3	1	no	—
98	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
99	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
100	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
101	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
102	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
103	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
104	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
105	C₂×C₄²×C₅	—	3	1	no	—
106	C₂×C₄²×C₅	—	3	1	no	—
107	C₂×C₄²×C₅	—	3	2	no	—
108	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
109	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
110	C₅×#95	—	3	1	no	—
111	C₅×#95	—	3	2	no	—
112	C₅×#95	—	3	1	no	—
113	C₅×#95	—	3	2	no	—
114	C₅×#95	—	3	2	no	—
115	C₅×#95	C₂×D₃	1	2	yes	—
116	#2357	—	3	1	no	Sylow
117	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
118	C₂²×C₅×D₃	—	2	2	no	—
119	C₂²×C₅×D₃	—	2	2	no	—
120	C₂²×C₅×D₃	—	2	2	no	—
121	C₂²×C₅×D₃	—	2	2	no	—
122	C₂²×C₅×D₃	—	2	2	no	—
123	C₂²×C₅×D₃	—	2	2	no	—
124	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
125	C₂×C₃×C₄×C₅	—	2	1	no	—
126	C₂×C₃×C₄×C₅	—	2	2	no	—
127	C₂×C₃×C₄×C₅	—	2	1	no	—
128	C₂×C₃×C₄×C₅	—	2	2	no	—
129	C₂×C₃×C₄×C₅	—	2	1	no	—
130	C₂×C₃×C₄×C₅	—	2	2	no	—
131	C₂×C₃×C₄×C₅	—	2	1	no	—
132	C₂×C₃×C₄×C₅	—	2	2	no	—
133	C₂×C₃×C₄×C₅	—	2	1	no	—
134	C₂×C₃×C₄×C₅	—	2	2	no	—
135	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
136	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
137	C₂×C₃×C₄×C₅	#41	1	2	yes	—
138	C₂×C₃×C₄×C₅	#41	1	2	yes	—
139	C₂×C₃×C₄×C₅	#41	1	2	yes	—
140	C₂×C₅×Dic₃	—	2	1	no	—
141	C₂×C₅×Dic₃	—	2	1	no	—
142	C₂×C₅×Dic₃	—	2	1	no	—
143	C₂×C₅×Dic₃	—	2	1	no	—
144	C₂×C₅×Dic₃	—	2	1	no	—
145	C₂×C₅×Dic₃	—	2	1	no	—
146	C₂×C₅×Dic₃	—	2	1	no	—
147	C₂×C₅×Dic₃	—	2	1	no	—
148	C₂×C₅×Dic₃	—	2	2	no	—
149	C₂×C₅×Dic₃	—	2	2	no	—
150	C₂×C₅×Dic₃	—	2	2	no	—
151	C₂×C₅×Dic₃	—	2	1	no	—
152	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
153	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
154	C₂×C₅×Dic₃	#41	1	2	yes	—
155	C₅×D₁₂	—	2	8	no	—
156	C₄×C₅×D₃	—	4	8	no	—
157	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
158	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
159	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
160	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
161	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
162	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
163	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
164	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
165	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
166	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
167	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
168	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
169	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
170	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
171	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
172	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
173	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
174	#893	C₄×C₅	1	8	yes	complemented
175	C₃×#95	C₂²×C₅	1	2	yes	—
176	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
177	C₃⋊#95	C₂²×C₅	1	2	yes	—
178	C₃⋊#95	C₂²×C₅	1	2	yes	—
179	C₃⋊#95	C₂²×C₅	1	2	yes	—
180	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
181	C₂⁴×C₅	—	3	2	no	—
182	C₂×C₅×D₄	—	3	4	no	—
183	C₂×C₅×D₄	—	6	2	no	—
184	C₂×C₅×D₄	—	6	2	no	—
185	C₂²×C₄×C₅	—	3	1	no	—
186	C₂²×C₄×C₅	—	3	1	no	—
187	C₂²×C₄×C₅	—	3	1	no	—
188	C₂²×C₄×C₅	—	3	8	no	—
189	C₂²×C₄×C₅	—	3	1	no	—
190	C₂²×C₄×C₅	—	3	2	no	—
191	C₂²×C₄×C₅	—	12	2	no	—
192	C₂²×C₄×C₅	—	12	2	no	—
193	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
194	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
195	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
196	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
197	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
198	C₅×C₄⋊C₄	—	2	2	no	—
199	C₅×C₄⋊C₄	—	6	2	no	—
200	C₅×C₄⋊C₄	—	6	2	no	—
201	C₅×C₄⋊C₄	—	6	2	no	—
202	C₅×C₄⋊C₄	C₄×D₃	1	4	yes	—
203	C₅×C₂²⋊C₄	—	3	8	no	—
204	C₅×C₂²⋊C₄	—	6	4	no	—
205	C₅×C₂²⋊C₄	—	6	4	no	—
206	C₅×C₂²⋊C₄	—	12	2	no	—
207	C₅×C₂²⋊C₄	—	12	2	no	—
208	C₄²×C₅	—	2	2	no	—
209	C₄²×C₅	—	3	4	no	—
210	C₄²×C₅	—	4	2	no	—
211	C₄²×C₅	—	6	2	no	—
212	C₄²×C₅	—	12	2	no	—
213	C₂×C₄×D₄	—	3	1	no	—
214	#390	—	3	1	no	—
215	#390	—	3	1	no	—
216	#388	—	3	2	no	—
217	#388	—	3	2	no	—
218	#382	—	3	1	no	—
219	#381	—	3	2	no	—
220	C₄×C₄⋊C₄	—	3	1	no	—
221	C₄×C₄⋊C₄	—	3	1	no	—
222	C₄×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
223	C₄×C₂²⋊C₄	—	3	2	no	—
224	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
225	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
226	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
227	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
228	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
229	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
230	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
231	C₂×C₅×D₃	—	2	4	no	—
232	C₂×C₅×D₃	—	2	4	no	—
233	C₂×C₅×D₃	—	2	4	no	—
234	C₂×C₅×D₃	—	4	2	no	—
235	C₂×C₅×D₃	—	4	2	no	—
236	C₂×C₅×D₃	—	4	2	no	—
237	C₂×C₅×D₃	—	4	2	no	—
238	C₃×C₄×C₅	—	2	4	no	—
239	C₃×C₄×C₅	—	2	2	no	—
240	C₃×C₄×C₅	—	4	1	no	—
241	C₃×C₄×C₅	—	4	2	no	—
242	C₃×C₄×C₅	#117	1	4	yes	—
243	C₅×Dic₃	—	2	2	no	—
244	C₅×Dic₃	—	2	2	no	—
245	C₅×Dic₃	—	2	2	no	—
246	C₅×Dic₃	—	4	1	no	—
247	C₅×Dic₃	—	4	2	no	—
248	C₂³×D₃	C₂³×C₅	1	2	yes	—
249	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
250	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
251	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
252	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
253	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
254	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
255	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
256	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
257	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
258	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
259	C₂×D₁₂	—	2	2	no	—
260	C₂×D₁₂	—	2	2	no	—
261	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
262	C₂×C₄×D₃	—	4	2	no	—
263	C₂×C₄×D₃	—	4	2	no	—
264	C₂×C₄×D₃	C₂×C₄×C₅	1	8	yes	—
265	C₃×C₄⋊C₄	—	2	2	no	—
266	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
267	C₃×C₄²	—	2	2	no	—
268	C₃×C₄²	—	4	2	no	—
269	(C₂×D₃)⋊C₄	—	2	4	no	—
270	(C₂×D₃)⋊C₄	—	2	4	no	—
271	(C₂×D₃)⋊C₄	—	4	2	no	—
272	(C₂×D₃)⋊C₄	—	4	2	no	—
273	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	8	yes	—
274	C₄⋊Dic₃	—	2	2	no	—
275	Dic₃⋊C₄	—	2	2	no	—
276	Dic₃⋊C₄	—	2	2	no	—
277	C₄×Dic₃	—	2	2	no	—
278	C₄×Dic₃	—	4	2	no	—
279	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	—
280	C₂³×C₅	—	3	4	no	—
281	C₂³×C₅	—	6	2	no	—
282	C₂³×C₅	—	6	2	no	—
283	C₂³×C₅	—	6	2	no	—
284	C₂³×C₅	—	6	2	no	—
285	C₂³×C₅	—	6	2	no	—
286	C₂³×C₅	—	6	2	no	—
287	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
288	C₅×D₄	—	6	8	no	—
289	C₂×C₄×C₅	—	2	1	no	—
290	C₂×C₄×C₅	—	2	1	no	—
291	C₂×C₄×C₅	—	2	2	no	—
292	C₂×C₄×C₅	—	2	2	no	—
293	C₂×C₄×C₅	—	2	1	no	—
294	C₂×C₄×C₅	—	2	2	no	—
295	C₂×C₄×C₅	—	2	2	no	—
296	C₂×C₄×C₅	—	2	2	no	—
297	C₂×C₄×C₅	—	2	1	no	—
298	C₂×C₄×C₅	—	2	1	no	—
299	C₂×C₄×C₅	—	3	2	no	—
300	C₂×C₄×C₅	—	3	2	no	—
301	C₂×C₄×C₅	—	3	2	no	—
302	C₂×C₄×C₅	—	6	1	no	—
303	C₂×C₄×C₅	—	6	1	no	—
304	C₂×C₄×C₅	—	6	1	no	—
305	C₂×C₄×C₅	—	6	1	no	—
306	C₂×C₄×C₅	—	6	1	no	—
307	C₂×C₄×C₅	—	6	1	no	—
308	C₂×C₄×C₅	—	6	1	no	—
309	C₂×C₄×C₅	—	6	2	no	—
310	C₂×C₄×C₅	—	6	1	no	—
311	C₂×C₄×C₅	—	6	2	no	—
312	C₂×C₄×C₅	—	6	1	no	—
313	C₂×C₄×C₅	—	6	2	no	—
314	C₂×C₄×C₅	—	12	8	no	—
315	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
316	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
317	C₂×C₄×C₅	#235	1	2	yes	—
318	C₂×C₄×C₅	#235	1	2	yes	—
319	C₂×C₄×C₅	#239	1	2	yes	—
320	C₂²×D₄	—	3	1	no	—
321	C₂³×C₄	—	3	2	no	—
322	C₄×D₄	—	3	8	no	—
323	C₂×C₄⋊C₄	—	3	1	no	—
324	C₂×C₄⋊C₄	—	3	1	no	—
325	C₂×C₄⋊C₄	—	3	1	no	—
326	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	—	3	2	no	—
329	C₂×C₂²⋊C₄	—	3	2	no	—
330	C₂×C₂²⋊C₄	—	3	2	no	—
331	C₂×C₂²⋊C₄	—	3	2	no	—
332	C₂×C₂²⋊C₄	—	3	2	no	—
333	C₂×C₄²	—	3	1	no	—
334	C₂×C₄²	—	3	1	no	—
335	C₂×C₄²	—	3	2	no	—
336	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
337	C₂×C₄²	C₂×C₅×D₃	1	2	yes	—
338	#95	—	3	1	no	—
339	#95	—	3	2	no	—
340	#95	—	3	1	no	—
341	#95	—	3	2	no	—
342	#95	—	3	2	no	—
343	#95	C₂×C₅×D₃	1	2	yes	—
344	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
345	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
346	C₂×C₃×C₅	#518	1	1	yes	characteristic
347	C₂×C₃×C₅	#553	1	1	yes	characteristic
348	C₂×C₃×C₅	#553	1	1	yes	characteristic
349	C₂×C₃×C₅	#559	1	1	yes	characteristic
350	C₂×C₃×C₅	#559	1	1	yes	characteristic
351	C₅×D₃	—	4	4	no	—
352	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
353	C₂²×D₃	—	2	2	no	—
354	C₂²×D₃	—	2	2	no	—
355	C₂²×D₃	—	2	2	no	—
356	C₂²×D₃	—	2	2	no	—
357	C₂²×D₃	—	2	2	no	—
358	C₂²×D₃	—	2	2	no	—
359	C₂²×D₃	C₂²×C₄×C₅	1	4	yes	—
360	C₂×C₃×C₄	—	2	1	no	—
361	C₂×C₃×C₄	—	2	2	no	—
362	C₂×C₃×C₄	—	2	1	no	—
363	C₂×C₃×C₄	—	2	2	no	—
364	C₂×C₃×C₄	—	2	1	no	—
365	C₂×C₃×C₄	—	2	2	no	—
366	C₂×C₃×C₄	—	2	1	no	—
367	C₂×C₃×C₄	—	2	2	no	—
368	C₂×C₃×C₄	—	2	1	no	—
369	C₂×C₃×C₄	—	2	2	no	—
370	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
371	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
372	C₂×C₃×C₄	C₅×#41	1	2	yes	—
373	C₂×C₃×C₄	C₅×#41	1	2	yes	—
374	C₂×C₃×C₄	C₅×#41	1	2	yes	—
375	C₂×Dic₃	—	2	1	no	—
376	C₂×Dic₃	—	2	1	no	—
377	C₂×Dic₃	—	2	1	no	—
378	C₂×Dic₃	—	2	1	no	—
379	C₂×Dic₃	—	2	1	no	—
380	C₂×Dic₃	—	2	1	no	—
381	C₂×Dic₃	—	2	1	no	—
382	C₂×Dic₃	—	2	1	no	—
383	C₂×Dic₃	—	2	2	no	—
384	C₂×Dic₃	—	2	2	no	—
385	C₂×Dic₃	—	2	2	no	—
386	C₂×Dic₃	—	2	1	no	—
387	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
388	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
389	C₂×Dic₃	C₅×#41	1	2	yes	—
390	D₁₂	—	2	8	no	—
391	C₄×D₃	—	4	8	no	—
392	C₂²×C₅	—	6	4	no	—
393	C₂²×C₅	—	6	4	no	—
394	C₂²×C₅	—	6	4	no	—
395	C₂²×C₅	—	12	2	no	—
396	C₂²×C₅	—	12	2	no	—
397	C₂²×C₅	—	12	2	no	—
398	C₂²×C₅	—	12	2	no	—
399	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
400	C₂²×C₅	C₂×#235	1	1	yes	characteristic
401	C₂²×C₅	C₂×#235	1	1	yes	characteristic
402	C₂²×C₅	#1004	1	1	yes	characteristic
403	C₂²×C₅	C₂×#239	1	1	yes	characteristic
404	C₂²×C₅	D₃×#41	1	1	yes	characteristic
405	C₂²×C₅	#1009	1	1	yes	characteristic
406	C₄×C₅	—	2	2	no	—
407	C₄×C₅	—	2	4	no	—
408	C₄×C₅	—	4	1	no	—
409	C₄×C₅	—	4	2	no	—
410	C₄×C₅	—	6	2	no	—
411	C₄×C₅	—	6	2	no	—
412	C₄×C₅	—	6	2	no	—
413	C₄×C₅	—	12	1	no	—
414	C₄×C₅	—	12	2	no	—
415	C₄×C₅	#872	1	4	yes	—
416	C₂⁴	—	3	2	no	—
417	C₂×D₄	—	3	4	no	—
418	C₂×D₄	—	6	2	no	—
419	C₂×D₄	—	6	2	no	—
420	C₂²×C₄	—	3	1	no	—
421	C₂²×C₄	—	3	1	no	—
422	C₂²×C₄	—	3	1	no	—
423	C₂²×C₄	—	3	8	no	—
424	C₂²×C₄	—	3	1	no	—
425	C₂²×C₄	—	3	2	no	—
426	C₂²×C₄	—	12	2	no	—
427	C₂²×C₄	—	12	2	no	—
428	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
429	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
430	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
431	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
432	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	6	2	no	—
435	C₄⋊C₄	—	6	2	no	—
436	C₄⋊C₄	—	6	2	no	—
437	C₄⋊C₄	C₄×C₅×D₃	1	4	yes	—
438	C₂²⋊C₄	—	3	8	no	—
439	C₂²⋊C₄	—	6	4	no	—
440	C₂²⋊C₄	—	6	4	no	—
441	C₂²⋊C₄	—	12	2	no	—
442	C₂²⋊C₄	—	12	2	no	—
443	C₄²	—	2	2	no	—
444	C₄²	—	3	4	no	—
445	C₄²	—	4	2	no	—
446	C₄²	—	6	2	no	—
447	C₄²	—	12	2	no	—
448	C₃×C₅	#2357	1	1	yes	characteristic, complemented, Hall
449	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
450	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
451	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
452	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
453	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
454	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
455	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
456	C₂×D₃	—	2	4	no	—
457	C₂×D₃	—	2	4	no	—
458	C₂×D₃	—	2	4	no	—
459	C₂×D₃	—	4	2	no	—
460	C₂×D₃	—	4	2	no	—
461	C₂×D₃	—	4	2	no	—
462	C₂×D₃	—	4	2	no	—
463	C₃×C₄	—	2	4	no	—
464	C₃×C₄	—	2	2	no	—
465	C₃×C₄	—	4	1	no	—
466	C₃×C₄	—	4	2	no	—
467	C₃×C₄	C₅×#117	1	4	yes	—
468	Dic₃	—	2	2	no	—
469	Dic₃	—	2	2	no	—
470	Dic₃	—	2	2	no	—
471	Dic₃	—	4	1	no	—
472	Dic₃	—	4	2	no	—
473	C₂×C₅	—	12	4	no	—
474	C₂×C₅	C₂×#872	1	1	yes	characteristic
475	C₂×C₅	#5514	1	1	yes	characteristic
476	C₂×C₅	#5530	1	1	yes	characteristic
477	C₂×C₅	#5545	1	1	yes	characteristic
478	C₂×C₅	#5559	1	1	yes	characteristic
479	C₂×C₅	C₄×#239	1	1	yes	characteristic
480	C₂×C₅	#5585	1	1	yes	characteristic
481	C₂³	—	3	4	no	—
482	C₂³	—	6	2	no	—
483	C₂³	—	6	2	no	—
484	C₂³	—	6	2	no	—
485	C₂³	—	6	2	no	—
486	C₂³	—	6	2	no	—
487	C₂³	—	6	2	no	—
488	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
489	D₄	—	6	8	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	3	2	no	—
501	C₂×C₄	—	3	2	no	—
502	C₂×C₄	—	3	2	no	—
503	C₂×C₄	—	6	1	no	—
504	C₂×C₄	—	6	1	no	—
505	C₂×C₄	—	6	1	no	—
506	C₂×C₄	—	6	1	no	—
507	C₂×C₄	—	6	1	no	—
508	C₂×C₄	—	6	1	no	—
509	C₂×C₄	—	6	1	no	—
510	C₂×C₄	—	6	2	no	—
511	C₂×C₄	—	6	1	no	—
512	C₂×C₄	—	6	2	no	—
513	C₂×C₄	—	6	1	no	—
514	C₂×C₄	—	6	2	no	—
515	C₂×C₄	—	12	8	no	—
516	C₂×C₄	C₂×C₄×C₅×D₃	1	4	yes	—
517	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
518	C₂×C₄	C₅×#235	1	2	yes	—
519	C₂×C₄	C₅×#235	1	2	yes	—
520	C₂×C₄	C₅×#239	1	2	yes	—
521	C₂×C₃	C₂×C₅×#117	1	1	yes	characteristic
522	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
523	C₂×C₃	C₅×#518	1	1	yes	characteristic
524	C₂×C₃	C₅×#553	1	1	yes	characteristic
525	C₂×C₃	C₅×#553	1	1	yes	characteristic
526	C₂×C₃	C₅×#559	1	1	yes	characteristic
527	C₂×C₃	C₅×#559	1	1	yes	characteristic
528	D₃	—	4	4	no	—
529	C₅	#73646	1	1	yes	characteristic, direct product factor, Sylow
530	C₂²	—	6	4	no	—
531	C₂²	—	6	4	no	—
532	C₂²	—	6	4	no	—
533	C₂²	—	12	2	no	—
534	C₂²	—	12	2	no	—
535	C₂²	—	12	2	no	—
536	C₂²	—	12	2	no	—
537	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
538	C₂²	C₂×C₅×#235	1	1	yes	characteristic
539	C₂²	C₂×C₅×#235	1	1	yes	characteristic
540	C₂²	C₅×#1004	1	1	yes	characteristic
541	C₂²	C₂×C₅×#239	1	1	yes	characteristic
542	C₂²	C₅×D₃×#41	1	1	yes	characteristic
543	C₂²	C₅×#1009	1	1	yes	characteristic
544	C₄	—	2	2	no	—
545	C₄	—	2	4	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	2	no	—
548	C₄	—	6	2	no	—
549	C₄	—	6	2	no	—
550	C₄	—	6	2	no	—
551	C₄	—	12	1	no	—
552	C₄	—	12	2	no	—
553	C₄	C₅×#872	1	4	yes	—
554	C₃	C₅×#2357	1	1	yes	characteristic, complemented, Sylow
555	C₂	—	12	4	no	—
556	C₂	C₂×C₅×#872	1	1	yes	characteristic
557	C₂	C₅×#5514	1	1	yes	characteristic
558	C₂	C₅×#5530	1	1	yes	characteristic
559	C₂	C₅×#5545	1	1	yes	characteristic
560	C₂	C₅×#5559	1	1	yes	characteristic
561	C₂	C₄×C₅×#239	1	1	yes	characteristic
562	C₂	C₅×#5585	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₄×C₅×C₄⋊C₄	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂²×C₅×D₁₂	C₂²
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₄×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₅×#2357
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₄×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×C₄⋊C₄	C₂	semidirect
4. C₅×#4977	C₂	semidirect
5. C₅×Dic₃×C₄⋊C₄	C₂	semidirect
6. C₅×#4940	C₂	semidirect
8. C₄×C₅×(C₂×D₃)⋊C₄	C₂	semidirect
9. C₄×C₅×Dic₃⋊C₄	C₂	semidirect
10. C₅×#4669	C₂	semidirect
11. C₅×#4664	C₂	semidirect
30. C₅×#893	C₄	semidirect
37. #73646	C₅	direct (N × H)
47. C₄×C₅×C₄⋊C₄	D₃	semidirect
82. C₃×C₄×C₄⋊C₄	C₂×C₅	semidirect
84. #4977	C₂×C₅	semidirect
85. Dic₃×C₄⋊C₄	C₂×C₅	semidirect
86. #4940	C₂×C₅	semidirect
88. C₄×(C₂×D₃)⋊C₄	C₂×C₅	semidirect
89. C₄×Dic₃⋊C₄	C₂×C₅	semidirect
90. #4669	C₂×C₅	semidirect
91. #4664	C₂×C₅	semidirect
174. #893	C₄×C₅	semidirect
222. C₄×C₄⋊C₄	C₅×D₃	semidirect
448. C₃×C₅	#2357	semidirect
529. C₅	#73646	direct (N × H)
554. C₃	C₅×#2357	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#893	2
C₃×C₄×C₅×C₄⋊C₄	2
C₅×#4988	2
C₅×#4977	2
C₅×Dic₃×C₄⋊C₄	2
C₅×#4940	2
C₅×#4939	2
C₄×C₅×(C₂×D₃)⋊C₄	2
C₄×C₅×Dic₃⋊C₄	2
C₅×#4669	2
C₅×#4664	2
C₅×#2357	3
#73646	5

Subgroups — C5×#73646 #14327987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#73646 #14327987