Subgroups — C₅×#82337 #14327675

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#926	C₂	1	2	yes	maximal, complemented
3	C₅×#5221	C₂	1	2	yes	maximal
4	C₅×#5212	C₂	1	2	yes	maximal, complemented
5	C₅×Dic₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₅×#4951	C₂	1	4	yes	maximal, complemented
7	C₅×#4935	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2354	—	3	1	no	maximal, Hall
10	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
12	C₃×C₄×C₅×D₄	C₄	1	8	yes	complemented
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₅×#217	C₂²	1	4	yes	—
17	C₂×C₅×#217	C₂²	1	4	yes	—
18	C₅×#926	—	2	8	no	—
19	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
21	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
22	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
23	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
25	C₂×C₄×C₅×Dic₃	C₂²	1	4	yes	—
26	C₅×C₃⋊#95	C₂²	1	2	yes	—
27	C₅×C₃⋊#95	C₂²	1	4	yes	—
28	#82337	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂×C₄×C₅×D₄	D₃	1	1	yes	characteristic
30	C₂×C₅×#117	—	3	2	no	—
31	C₅×#390	—	3	2	no	—
32	C₅×#388	—	3	4	no	—
33	C₅×#384	—	3	1	no	—
34	C₅×#380	—	3	2	no	—
35	C₄×C₅×C₄⋊C₄	—	3	2	no	—
36	C₄×C₅×C₄⋊C₄	—	3	1	no	—
37	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
38	C₂×C₃×C₅×D₄	—	2	2	no	—
39	C₂×C₃×C₅×D₄	—	2	2	no	—
40	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
41	C₂²×C₃×C₄×C₅	—	2	4	no	—
42	C₂²×C₃×C₄×C₅	—	2	4	no	—
43	C₂²×C₃×C₄×C₅	C₂×C₄	1	8	yes	—
44	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
45	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂²×C₅×Dic₃	C₂³	1	4	yes	—
49	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
50	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
51	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
52	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	8	yes	—
53	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
54	C₅×#217	—	2	8	no	—
55	C₅×#217	—	4	4	no	—
56	C₅×C₄⋊Dic₃	—	2	4	no	—
57	C₅×C₄⋊Dic₃	—	2	4	no	—
58	C₅×C₄⋊Dic₃	—	2	2	no	—
59	C₅×C₄⋊Dic₃	—	4	2	no	—
60	C₅×C₄⋊Dic₃	D₄	1	4	yes	complemented
61	C₄×C₅×Dic₃	—	2	4	no	—
62	C₄×C₅×Dic₃	—	2	8	no	—
63	C₂×C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic
64	C₂×#926	C₂×C₅	1	2	yes	complemented
65	#5221	C₂×C₅	1	2	yes	—
66	#5212	C₂×C₅	1	2	yes	complemented
67	Dic₃×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
68	#4951	C₂×C₅	1	4	yes	complemented
69	#4935	C₂×C₅	1	1	yes	characteristic, complemented
70	C₄×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
72	C₂³×C₄×C₅	C₂×D₃	1	2	yes	—
73	C₄×C₅×D₄	Dic₃	1	8	yes	complemented
74	C₅×#117	—	6	8	no	—
75	C₂×C₅×C₄⋊C₄	—	3	1	no	—
76	C₂×C₅×C₄⋊C₄	—	3	1	no	—
77	C₂×C₅×C₄⋊C₄	—	3	2	no	—
78	C₂×C₅×C₄⋊C₄	—	3	2	no	—
79	C₂×C₅×C₄⋊C₄	—	3	2	no	—
80	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
81	C₂×C₅×C₂²⋊C₄	—	3	4	no	—
82	C₂×C₅×C₂²⋊C₄	—	3	4	no	—
83	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
84	C₂×C₄²×C₅	—	3	2	no	—
85	C₂×C₄²×C₅	—	3	4	no	—
86	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
87	C₅×#95	—	3	4	no	—
88	C₅×#95	—	3	2	no	—
89	#2354	—	3	1	no	Sylow
90	C₂³×C₃×C₅	—	2	2	no	—
91	C₂³×C₃×C₅	—	2	2	no	—
92	C₂³×C₃×C₅	—	2	2	no	—
93	C₂³×C₃×C₅	—	2	2	no	—
94	C₂³×C₃×C₅	—	2	2	no	—
95	C₂³×C₃×C₅	—	2	2	no	—
96	C₂³×C₃×C₅	C₂²×C₄	1	4	yes	—
97	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
98	C₃×C₅×D₄	—	2	8	no	—
99	C₂×C₃×C₄×C₅	—	2	2	no	—
100	C₂×C₃×C₄×C₅	—	2	2	no	—
101	C₂×C₃×C₄×C₅	—	2	1	no	—
102	C₂×C₃×C₄×C₅	—	2	1	no	—
103	C₂×C₃×C₄×C₅	—	2	16	no	—
104	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
105	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
106	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
107	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
108	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
109	C₂×C₃×C₄×C₅	#41	1	2	yes	—
110	C₂×C₃×C₄×C₅	#41	1	2	yes	—
111	C₂×C₃×C₄×C₅	#41	1	2	yes	—
112	C₂×C₅×Dic₃	—	2	2	no	—
113	C₂×C₅×Dic₃	—	2	2	no	—
114	C₂×C₅×Dic₃	—	2	4	no	—
115	C₂×C₅×Dic₃	—	2	2	no	—
116	C₂×C₅×Dic₃	—	2	4	no	—
117	C₂×C₅×Dic₃	—	2	2	no	—
118	C₂×C₅×Dic₃	—	2	4	no	—
119	C₂×C₅×Dic₃	—	2	2	no	—
120	C₂×C₅×Dic₃	C₂×D₄	1	4	yes	—
121	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
122	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
123	C₃×C₄×D₄	C₄×C₅	1	8	yes	complemented
124	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
125	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
126	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
127	C₂×#217	C₂²×C₅	1	4	yes	—
128	C₂×#217	C₂²×C₅	1	4	yes	—
129	#926	—	2	8	no	—
130	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
131	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
132	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
133	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
134	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
135	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
136	C₂×C₄×Dic₃	C₂²×C₅	1	4	yes	—
137	C₃⋊#95	C₂²×C₅	1	2	yes	—
138	C₃⋊#95	C₂²×C₅	1	4	yes	—
139	C₂⁴×C₅	C₂²×D₃	1	2	yes	—
140	C₂×C₅×D₄	—	2	2	no	—
141	C₂×C₅×D₄	—	2	2	no	—
142	C₂×C₅×D₄	C₂×Dic₃	1	4	yes	—
143	C₂²×C₄×C₅	—	2	4	no	—
144	C₂²×C₄×C₅	—	2	4	no	—
145	C₂²×C₄×C₅	—	3	2	no	—
146	C₂²×C₄×C₅	—	3	4	no	—
147	C₂²×C₄×C₅	—	3	2	no	—
148	C₂²×C₄×C₅	C₂×Dic₃	1	8	yes	—
149	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
150	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
152	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
153	C₅×C₄⋊C₄	—	3	4	no	—
154	C₅×C₄⋊C₄	—	6	4	no	—
155	C₅×C₄⋊C₄	—	6	4	no	—
156	C₅×C₄⋊C₄	—	6	2	no	—
157	C₅×C₄⋊C₄	—	12	2	no	—
158	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
159	C₅×C₂²⋊C₄	—	6	8	no	—
160	C₅×C₂²⋊C₄	—	12	4	no	—
161	C₅×C₂²⋊C₄	C₂×Dic₃	1	8	yes	—
162	C₄²×C₅	—	6	8	no	—
163	C₄²×C₅	—	6	4	no	—
164	C₄²×C₅	C₂×Dic₃	1	4	yes	—
165	C₂×C₄×D₄	C₅×D₃	1	1	yes	characteristic
166	C₂×#117	—	3	2	no	—
167	#390	—	3	2	no	—
168	#388	—	3	4	no	—
169	#384	—	3	1	no	—
170	#380	—	3	2	no	—
171	C₄×C₄⋊C₄	—	3	2	no	—
172	C₄×C₄⋊C₄	—	3	1	no	—
173	C₂²×C₃×C₅	—	2	4	no	—
174	C₂²×C₃×C₅	—	2	4	no	—
175	C₂²×C₃×C₅	—	2	4	no	—
176	C₂²×C₃×C₅	—	4	2	no	—
177	C₂²×C₃×C₅	—	4	2	no	—
178	C₂²×C₃×C₅	—	4	2	no	—
179	C₂²×C₃×C₅	—	4	2	no	—
180	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
181	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
182	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
183	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
184	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
185	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
186	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
187	C₃×C₄×C₅	—	2	4	no	—
188	C₃×C₄×C₅	—	2	2	no	—
189	C₃×C₄×C₅	#117	1	4	yes	complemented
190	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
191	C₅×Dic₃	—	2	4	no	—
192	C₅×Dic₃	—	4	4	no	—
193	C₅×Dic₃	—	4	2	no	—
194	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
195	C₂×C₃×D₄	—	2	2	no	—
196	C₂×C₃×D₄	—	2	2	no	—
197	C₂×C₃×D₄	C₂×C₄×C₅	1	4	yes	—
198	C₂²×C₃×C₄	—	2	4	no	—
199	C₂²×C₃×C₄	—	2	4	no	—
200	C₂²×C₃×C₄	C₂×C₄×C₅	1	8	yes	—
201	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
202	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
203	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
204	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
205	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
206	C₂²×Dic₃	C₂³×C₅	1	4	yes	—
207	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
208	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
209	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	8	yes	—
210	C₃×C₄²	C₂×C₄×C₅	1	4	yes	—
211	#217	—	2	8	no	—
212	#217	—	4	4	no	—
213	C₄⋊Dic₃	—	2	4	no	—
214	C₄⋊Dic₃	—	2	4	no	—
215	C₄⋊Dic₃	—	2	2	no	—
216	C₄⋊Dic₃	—	4	2	no	—
217	C₄⋊Dic₃	C₅×D₄	1	4	yes	complemented
218	C₄×Dic₃	—	2	4	no	—
219	C₄×Dic₃	—	2	8	no	—
220	C₂³×C₅	—	2	2	no	—
221	C₂³×C₅	—	2	2	no	—
222	C₂³×C₅	—	2	2	no	—
223	C₂³×C₅	—	2	2	no	—
224	C₂³×C₅	—	2	2	no	—
225	C₂³×C₅	—	2	2	no	—
226	C₂³×C₅	C₂²×Dic₃	1	4	yes	—
227	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
228	C₅×D₄	—	2	8	no	—
229	C₂×C₄×C₅	—	2	2	no	—
230	C₂×C₄×C₅	—	2	2	no	—
231	C₂×C₄×C₅	—	2	1	no	—
232	C₂×C₄×C₅	—	2	1	no	—
233	C₂×C₄×C₅	—	2	16	no	—
234	C₂×C₄×C₅	—	3	4	no	—
235	C₂×C₄×C₅	—	6	2	no	—
236	C₂×C₄×C₅	—	6	2	no	—
237	C₂×C₄×C₅	—	6	2	no	—
238	C₂×C₄×C₅	—	6	4	no	—
239	C₂×C₄×C₅	—	6	4	no	—
240	C₂×C₄×C₅	—	6	2	no	—
241	C₂×C₄×C₅	—	6	2	no	—
242	C₂×C₄×C₅	—	6	4	no	—
243	C₂×C₄×C₅	#235	1	2	yes	—
244	C₂×C₄×C₅	#235	1	2	yes	—
245	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
246	C₂×C₄×C₅	#237	1	2	yes	—
247	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
248	C₂×C₄×C₅	C₂²×Dic₃	1	4	yes	—
249	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
250	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
251	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
252	C₂³×C₄	C₂×C₅×D₃	1	2	yes	—
253	C₄×D₄	C₅×Dic₃	1	8	yes	complemented
254	#117	—	6	8	no	—
255	C₂×C₄⋊C₄	—	3	1	no	—
256	C₂×C₄⋊C₄	—	3	2	no	—
257	C₂×C₄⋊C₄	—	3	1	no	—
258	C₂×C₄⋊C₄	—	3	2	no	—
259	C₂×C₄⋊C₄	—	3	2	no	—
260	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
261	C₂×C₂²⋊C₄	—	3	4	no	—
262	C₂×C₂²⋊C₄	—	3	4	no	—
263	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
264	C₂×C₄²	—	3	4	no	—
265	C₂×C₄²	—	3	2	no	—
266	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
267	#95	—	3	4	no	—
268	#95	—	3	2	no	—
269	C₂×C₃×C₅	—	4	4	no	—
270	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
271	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
272	C₂×C₃×C₅	#520	1	1	yes	characteristic
273	C₂×C₃×C₅	#547	1	1	yes	characteristic
274	C₂×C₃×C₅	#550	1	1	yes	characteristic
275	C₂×C₃×C₅	#555	1	1	yes	characteristic
276	C₂×C₃×C₅	#556	1	1	yes	characteristic
277	C₂³×C₃	—	2	2	no	—
278	C₂³×C₃	—	2	2	no	—
279	C₂³×C₃	—	2	2	no	—
280	C₂³×C₃	—	2	2	no	—
281	C₂³×C₃	—	2	2	no	—
282	C₂³×C₃	—	2	2	no	—
283	C₂³×C₃	C₂²×C₄×C₅	1	4	yes	—
284	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
285	C₃×D₄	—	2	8	no	—
286	C₂×C₃×C₄	—	2	2	no	—
287	C₂×C₃×C₄	—	2	2	no	—
288	C₂×C₃×C₄	—	2	1	no	—
289	C₂×C₃×C₄	—	2	1	no	—
290	C₂×C₃×C₄	—	2	16	no	—
291	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
292	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
293	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
294	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
295	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
296	C₂×C₃×C₄	C₅×#41	1	2	yes	—
297	C₂×C₃×C₄	C₅×#41	1	2	yes	—
298	C₂×C₃×C₄	C₅×#41	1	2	yes	—
299	C₂×Dic₃	—	2	2	no	—
300	C₂×Dic₃	—	2	2	no	—
301	C₂×Dic₃	—	2	4	no	—
302	C₂×Dic₃	—	2	2	no	—
303	C₂×Dic₃	—	2	4	no	—
304	C₂×Dic₃	—	2	2	no	—
305	C₂×Dic₃	—	2	4	no	—
306	C₂×Dic₃	—	2	2	no	—
307	C₂×Dic₃	C₂×C₅×D₄	1	4	yes	—
308	C₂²×C₅	—	2	4	no	—
309	C₂²×C₅	—	2	4	no	—
310	C₂²×C₅	—	2	4	no	—
311	C₂²×C₅	—	4	2	no	—
312	C₂²×C₅	—	4	2	no	—
313	C₂²×C₅	—	4	2	no	—
314	C₂²×C₅	—	4	2	no	—
315	C₂²×C₅	C₂×#235	1	1	yes	characteristic
316	C₂²×C₅	C₂×#235	1	1	yes	characteristic
317	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
318	C₂²×C₅	C₂×#237	1	1	yes	characteristic
319	C₂²×C₅	#1009	1	1	yes	characteristic
320	C₂²×C₅	#1010	1	1	yes	characteristic
321	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
322	C₄×C₅	—	2	4	no	—
323	C₄×C₅	—	2	2	no	—
324	C₄×C₅	—	6	4	no	—
325	C₄×C₅	—	12	4	no	—
326	C₄×C₅	—	12	2	no	—
327	C₄×C₅	#926	1	4	yes	complemented
328	C₄×C₅	D₄×Dic₃	1	4	yes	—
329	C₂⁴	C₂²×C₅×D₃	1	2	yes	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	C₂×C₅×Dic₃	1	4	yes	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	3	2	no	—
336	C₂²×C₄	—	3	2	no	—
337	C₂²×C₄	—	3	4	no	—
338	C₂²×C₄	C₂×C₅×Dic₃	1	8	yes	—
339	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
340	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
341	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
342	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
343	C₄⋊C₄	—	3	4	no	—
344	C₄⋊C₄	—	6	4	no	—
345	C₄⋊C₄	—	6	4	no	—
346	C₄⋊C₄	—	6	2	no	—
347	C₄⋊C₄	—	12	2	no	—
348	C₄⋊C₄	C₂×C₅×Dic₃	1	4	yes	—
349	C₂²⋊C₄	—	6	8	no	—
350	C₂²⋊C₄	—	12	4	no	—
351	C₂²⋊C₄	C₂×C₅×Dic₃	1	8	yes	—
352	C₄²	—	6	8	no	—
353	C₄²	—	6	4	no	—
354	C₄²	C₂×C₅×Dic₃	1	4	yes	—
355	C₃×C₅	#2354	1	1	yes	characteristic, complemented, Hall
356	C₂²×C₃	—	2	4	no	—
357	C₂²×C₃	—	2	4	no	—
358	C₂²×C₃	—	2	4	no	—
359	C₂²×C₃	—	4	2	no	—
360	C₂²×C₃	—	4	2	no	—
361	C₂²×C₃	—	4	2	no	—
362	C₂²×C₃	—	4	2	no	—
363	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
364	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
365	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
366	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
367	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
368	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
369	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
370	C₃×C₄	—	2	4	no	—
371	C₃×C₄	—	2	2	no	—
372	C₃×C₄	C₅×#117	1	4	yes	complemented
373	C₃×C₄	C₄×C₅×D₄	1	4	yes	—
374	Dic₃	—	2	4	no	—
375	Dic₃	—	4	4	no	—
376	Dic₃	—	4	2	no	—
377	C₂×C₅	—	4	4	no	—
378	C₂×C₅	#5542	1	1	yes	characteristic
379	C₂×C₅	#5551	1	1	yes	characteristic
380	C₂×C₅	#5553	1	1	yes	characteristic
381	C₂×C₅	#5563	1	1	yes	characteristic
382	C₂×C₅	C₂×#926	1	1	yes	characteristic
383	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
384	C₂×C₅	#5827	1	1	yes	characteristic
385	C₂³	—	2	2	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	2	2	no	—
391	C₂³	C₂²×C₅×Dic₃	1	4	yes	—
392	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
393	D₄	—	2	8	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	16	no	—
399	C₂×C₄	—	3	4	no	—
400	C₂×C₄	—	6	2	no	—
401	C₂×C₄	—	6	2	no	—
402	C₂×C₄	—	6	2	no	—
403	C₂×C₄	—	6	4	no	—
404	C₂×C₄	—	6	4	no	—
405	C₂×C₄	—	6	2	no	—
406	C₂×C₄	—	6	2	no	—
407	C₂×C₄	—	6	4	no	—
408	C₂×C₄	C₅×#235	1	2	yes	—
409	C₂×C₄	C₅×#235	1	2	yes	—
410	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
411	C₂×C₄	C₅×#237	1	2	yes	—
412	C₂×C₄	C₂²×C₅×Dic₃	1	2	yes	—
413	C₂×C₄	C₂²×C₅×Dic₃	1	4	yes	—
414	C₂×C₄	C₂²×C₅×Dic₃	1	2	yes	—
415	C₂×C₄	C₂²×C₅×Dic₃	1	2	yes	—
416	C₂×C₃	—	4	4	no	—
417	C₂×C₃	C₂×C₅×#117	1	1	yes	characteristic
418	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
419	C₂×C₃	C₅×#520	1	1	yes	characteristic
420	C₂×C₃	C₅×#547	1	1	yes	characteristic
421	C₂×C₃	C₅×#550	1	1	yes	characteristic
422	C₂×C₃	C₅×#555	1	1	yes	characteristic
423	C₂×C₃	C₅×#556	1	1	yes	characteristic
424	C₅	#82337	1	1	yes	characteristic, direct product factor, Sylow
425	C₂²	—	2	4	no	—
426	C₂²	—	2	4	no	—
427	C₂²	—	2	4	no	—
428	C₂²	—	4	2	no	—
429	C₂²	—	4	2	no	—
430	C₂²	—	4	2	no	—
431	C₂²	—	4	2	no	—
432	C₂²	C₂×C₅×#235	1	1	yes	characteristic
433	C₂²	C₂×C₅×#235	1	1	yes	characteristic
434	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
435	C₂²	C₂×C₅×#237	1	1	yes	characteristic
436	C₂²	C₅×#1009	1	1	yes	characteristic
437	C₂²	C₅×#1010	1	1	yes	characteristic
438	C₂²	C₂³×C₅×Dic₃	1	1	yes	characteristic
439	C₄	—	2	4	no	—
440	C₄	—	2	2	no	—
441	C₄	—	6	4	no	—
442	C₄	—	12	4	no	—
443	C₄	—	12	2	no	—
444	C₄	C₅×#926	1	4	yes	complemented
445	C₄	C₅×D₄×Dic₃	1	4	yes	—
446	C₃	C₅×#2354	1	1	yes	characteristic, complemented, Sylow
447	C₂	—	4	4	no	—
448	C₂	C₅×#5542	1	1	yes	characteristic
449	C₂	C₅×#5551	1	1	yes	characteristic
450	C₂	C₅×#5553	1	1	yes	characteristic
451	C₂	C₅×#5563	1	1	yes	characteristic
452	C₂	C₂×C₅×#926	1	1	yes	characteristic
453	C₂	C₂×C₅×D₄×Dic₃	1	1	yes	characteristic
454	C₂	C₅×#5827	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×D₄	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂²×C₃×C₅×D₄	C₂²
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₂×C₄×C₅×D₄	D₃
hypocenter	C₃	C₅×#2354
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄×C₅×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#926	C₂	semidirect
4. C₅×#5212	C₂	semidirect
5. C₅×Dic₃×C₄⋊C₄	C₂	semidirect
6. C₅×#4951	C₂	semidirect
7. C₅×#4935	C₂	semidirect
8. C₄×C₅×C₄⋊Dic₃	C₂	semidirect
12. C₃×C₄×C₅×D₄	C₄	semidirect
28. #82337	C₅	direct (N × H)
60. C₅×C₄⋊Dic₃	D₄	semidirect
64. C₂×#926	C₂×C₅	semidirect
66. #5212	C₂×C₅	semidirect
67. Dic₃×C₄⋊C₄	C₂×C₅	semidirect
68. #4951	C₂×C₅	semidirect
69. #4935	C₂×C₅	semidirect
70. C₄×C₄⋊Dic₃	C₂×C₅	semidirect
73. C₄×C₅×D₄	Dic₃	semidirect
123. C₃×C₄×D₄	C₄×C₅	semidirect
189. C₃×C₄×C₅	#117	semidirect
217. C₄⋊Dic₃	C₅×D₄	semidirect
253. C₄×D₄	C₅×Dic₃	semidirect
327. C₄×C₅	#926	semidirect
355. C₃×C₅	#2354	semidirect
372. C₃×C₄	C₅×#117	semidirect
424. C₅	#82337	direct (N × H)
444. C₄	C₅×#926	semidirect
446. C₃	C₅×#2354	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×D₄	2
C₂×C₅×#926	2
C₅×#5221	2
C₅×#5212	2
C₅×Dic₃×C₄⋊C₄	2
C₅×#4951	2
C₅×#4935	2
C₄×C₅×C₄⋊Dic₃	2
C₅×#2354	3
#82337	5

Subgroups — C5×#82337 #14327675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#82337 #14327675