Subgroups — C₅×#68770 #14297810

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5522	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#825	C₂	1	1	yes	maximal, characteristic
3	C₅×#5118	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#805	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#339	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4532	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1806	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₃×C₅×#117	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×Dic₃⋊C₄	—	2	1	no	—
14	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₅×#208	C₂²	1	1	yes	characteristic
17	C₅×#893	C₄	1	1	yes	characteristic, complemented
18	C₅×#893	C₄	1	1	yes	characteristic, complemented
19	C₅×#887	C₄	1	1	yes	characteristic, complemented
20	C₅×#887	C₄	1	1	yes	characteristic, complemented
21	C₅×#67⋊C₄	—	2	1	no	—
22	C₅×#67⋊C₄	—	2	1	no	—
23	C₅×#872	—	2	1	no	—
24	C₄²×C₅×D₃	—	2	1	no	—
25	C₅×C₃⋊#95	—	2	1	no	—
26	C₅×#825	—	2	2	no	—
27	C₅×Dic₃⋊C₈	—	2	1	no	—
28	C₅×Dic₃⋊C₈	—	2	1	no	—
29	C₅×#805	—	2	2	no	—
30	#68770	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₄×C₅×#41	—	3	1	no	—
32	C₅×#423	—	3	1	no	—
33	C₂×C₅×#104	—	3	1	no	—
34	C₂×C₅×#104	—	3	1	no	—
35	C₄×C₅×C₄⋊C₄	—	3	1	no	—
36	C₅×#339	—	3	1	no	—
37	C₅×#339	D₃	1	1	yes	characteristic, complemented
38	C₂²×C₃×C₄×C₅	—	2	1	no	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₅×#67	—	2	1	no	—
41	C₂²×C₅×Dic₃	—	2	1	no	—
42	C₂²×C₅×Dic₃	D₄	1	1	yes	characteristic
43	C₂²×C₅×Dic₃	D₄	1	1	yes	characteristic
44	C₅×#235	—	2	1	no	—
45	C₅×#235	—	2	1	no	—
46	C₅×#235	D₄	1	2	yes	—
47	C₅×#235	Q₈	1	2	yes	—
48	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅×D₃	—	2	1	no	—
50	C₂×C₄×C₅×D₃	—	2	1	no	—
51	C₂×C₄×C₅×D₃	—	2	1	no	—
52	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
53	C₃×C₅×M₁₆	—	2	1	no	—
54	C₃×C₅×M₁₆	—	2	1	no	—
55	C₂×C₃×C₅×C₈	—	2	1	no	—
56	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
58	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
59	C₃×C₄²×C₅	—	2	1	no	—
60	C₃×C₄²×C₅	—	2	1	no	—
61	C₃×C₄²×C₅	—	2	1	no	—
62	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
63	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
64	C₅×Dic₃⋊C₄	—	2	4	no	—
65	C₅×Dic₃⋊C₄	—	2	1	no	—
66	C₅×Dic₃⋊C₄	—	2	1	no	—
67	C₄×C₅×Dic₃	—	2	1	no	—
68	C₄×C₅×Dic₃	—	2	1	no	—
69	C₄×C₅×Dic₃	—	2	1	no	—
70	C₄×C₅×Dic₃	—	2	1	no	—
71	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
74	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
75	C₅×#208	—	2	1	no	—
76	C₅×#208	—	2	1	no	—
77	C₂×C₅×C₃⋊C₈	—	2	1	no	—
78	#5522	C₂×C₅	1	1	yes	characteristic
79	C₂×#825	C₂×C₅	1	1	yes	characteristic
80	#5118	C₂×C₅	1	1	yes	characteristic, complemented
81	C₄×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
82	C₂×#805	C₂×C₅	1	1	yes	characteristic
83	C₃×#339	C₂×C₅	1	1	yes	characteristic, complemented
84	#4532	C₂×C₅	1	1	yes	characteristic, complemented
85	C₂×C₅×#41	—	3	1	no	—
86	C₂×C₅×M₁₆	—	3	1	no	—
87	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
88	C₄×C₅×Q₈	—	3	1	no	—
89	C₄×C₅×Q₈	—	3	1	no	—
90	C₄×C₅×D₄	—	3	1	no	—
91	C₄×C₅×D₄	—	3	1	no	—
92	C₄×C₅×D₄	—	6	1	no	—
93	C₄×C₅×D₄	—	6	1	no	—
94	C₅×#117	—	6	1	no	—
95	C₅×#117	C₂×D₃	1	1	yes	characteristic
96	C₂×C₅×C₄⋊C₄	—	6	1	no	—
97	C₂×C₄²×C₅	—	3	1	no	—
98	C₂×C₄²×C₅	—	3	1	no	—
99	C₂×C₄²×C₅	—	6	1	no	—
100	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
101	C₅×C₄⋊C₈	—	6	1	no	—
102	C₅×C₄⋊C₈	—	6	1	no	—
103	C₅×#104	—	6	2	no	—
104	C₅×#104	—	6	2	no	—
105	C₅×#95	—	6	1	no	—
106	#1806	—	3	1	no	Sylow
107	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₅×D₃	—	2	1	no	—
109	C₂×C₃×C₄×C₅	—	2	2	no	—
110	C₂×C₃×C₄×C₅	—	2	1	no	—
111	C₂×C₃×C₄×C₅	—	2	1	no	—
112	C₂×C₃×C₄×C₅	—	4	1	no	—
113	C₂×C₃×C₄×C₅	—	4	1	no	—
114	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
116	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
117	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
119	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
120	C₅×#67	—	2	2	no	—
121	C₅×#67	—	2	2	no	—
122	C₂×C₅×Dic₃	—	2	1	no	—
123	C₂×C₅×Dic₃	—	2	2	no	—
124	C₂×C₅×Dic₃	—	2	2	no	—
125	C₂×C₅×Dic₃	—	4	1	no	—
126	C₂×C₅×Dic₃	—	4	1	no	—
127	C₂×C₅×Dic₃	—	4	1	no	—
128	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
132	C₅×D₁₂	—	2	1	no	—
133	C₅×D₁₂	C₄⋊C₄	1	2	yes	complemented
134	C₄×C₅×D₃	—	2	2	no	—
135	C₄×C₅×D₃	—	2	2	no	—
136	C₄×C₅×D₃	—	4	2	no	—
137	C₄×C₅×D₃	—	4	2	no	—
138	C₅×Dic₆	—	2	1	no	—
139	C₅×Dic₆	C₄⋊C₄	1	2	yes	complemented
140	C₃×C₅×C₈	—	4	1	no	—
141	C₅×C₃⋊C₈	—	4	1	no	—
142	C₂×#235	C₂²×C₅	1	1	yes	characteristic
143	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
144	C₃×#117	C₂²×C₅	1	1	yes	characteristic
145	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
146	C₂×Dic₃⋊C₄	—	2	1	no	—
147	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
148	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
149	C₂×#208	C₂²×C₅	1	1	yes	characteristic
150	#893	C₄×C₅	1	1	yes	characteristic, complemented
151	#893	C₄×C₅	1	1	yes	characteristic, complemented
152	#887	C₄×C₅	1	1	yes	characteristic, complemented
153	#887	C₄×C₅	1	1	yes	characteristic, complemented
154	#67⋊C₄	—	2	1	no	—
155	#67⋊C₄	—	2	1	no	—
156	#872	—	2	1	no	—
157	C₄²×D₃	—	2	1	no	—
158	C₃⋊#95	—	2	1	no	—
159	#825	—	2	2	no	—
160	Dic₃⋊C₈	—	2	1	no	—
161	Dic₃⋊C₈	—	2	1	no	—
162	#805	—	2	2	no	—
163	C₅×#41	—	3	2	no	—
164	C₅×#41	—	3	2	no	—
165	C₅×#41	—	6	1	no	—
166	C₅×#41	—	6	1	no	—
167	C₂×C₅×Q₈	—	3	1	no	—
168	C₂×C₅×D₄	—	3	1	no	—
169	C₂×C₅×D₄	—	6	1	no	—
170	C₂²×C₄×C₅	—	2	1	no	—
171	C₂²×C₄×C₅	—	3	1	no	—
172	C₂²×C₄×C₅	—	3	1	no	—
173	C₂²×C₄×C₅	—	6	1	no	—
174	C₂²×C₄×C₅	—	6	1	no	—
175	C₂²×C₄×C₅	—	6	1	no	—
176	C₂²×C₄×C₅	—	6	1	no	—
177	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
178	C₅×M₁₆	—	2	1	no	—
179	C₅×M₁₆	—	2	1	no	—
180	C₅×M₁₆	—	6	1	no	—
181	C₅×M₁₆	—	6	1	no	—
182	C₂×C₅×C₈	—	2	1	no	—
183	C₂×C₅×C₈	—	6	1	no	—
184	C₅×C₄⋊C₄	—	6	1	no	—
185	C₅×C₄⋊C₄	—	6	1	no	—
186	C₅×C₄⋊C₄	—	6	4	no	—
187	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
188	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
189	C₅×C₂²⋊C₄	—	2	1	no	—
190	C₅×C₂²⋊C₄	—	6	1	no	—
191	C₅×C₂²⋊C₄	—	6	1	no	—
192	C₄²×C₅	—	2	1	no	—
193	C₄²×C₅	—	2	1	no	—
194	C₄²×C₅	—	2	1	no	—
195	C₄²×C₅	—	3	1	no	—
196	C₄²×C₅	—	3	1	no	—
197	C₄²×C₅	—	3	1	no	—
198	C₄²×C₅	—	3	1	no	—
199	C₄²×C₅	—	6	1	no	—
200	C₄²×C₅	—	6	1	no	—
201	C₄²×C₅	—	6	1	no	—
202	C₄²×C₅	—	6	1	no	—
203	C₄×#41	—	3	1	no	—
204	#423	—	3	1	no	—
205	C₂×#104	—	3	1	no	—
206	C₂×#104	—	3	1	no	—
207	C₄×C₄⋊C₄	—	3	1	no	—
208	#339	—	3	1	no	—
209	#339	C₅×D₃	1	1	yes	characteristic, complemented
210	C₂²×C₃×C₅	—	2	1	no	—
211	C₂²×C₃×C₅	—	2	1	no	—
212	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₃×C₅	#121	1	1	yes	characteristic
214	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
215	C₂×C₅×D₃	—	2	2	no	—
216	C₂×C₅×D₃	—	4	1	no	—
217	C₂×C₅×D₃	—	4	1	no	—
218	C₃×C₄×C₅	—	4	2	no	—
219	C₃×C₄×C₅	—	4	1	no	—
220	C₃×C₄×C₅	—	4	1	no	—
221	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
222	C₃×C₄×C₅	#117	1	1	yes	characteristic
223	C₃×C₄×C₅	#121	1	1	yes	characteristic
224	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
225	C₅×Dic₃	—	2	2	no	—
226	C₅×Dic₃	—	2	1	no	—
227	C₅×Dic₃	—	2	1	no	—
228	C₅×Dic₃	—	2	1	no	—
229	C₅×Dic₃	—	4	2	no	—
230	C₅×Dic₃	#104	1	2	yes	complemented
231	C₂²×C₃×C₄	—	2	1	no	—
232	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
233	C₂×#67	—	2	1	no	—
234	C₂²×Dic₃	—	2	1	no	—
235	C₂²×Dic₃	C₅×D₄	1	1	yes	characteristic
236	C₂²×Dic₃	C₅×D₄	1	1	yes	characteristic
237	#235	—	2	1	no	—
238	#235	—	2	1	no	—
239	#235	C₅×D₄	1	2	yes	—
240	#235	C₅×Q₈	1	2	yes	—
241	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
242	C₂×C₄×D₃	—	2	1	no	—
243	C₂×C₄×D₃	—	2	1	no	—
244	C₂×C₄×D₃	—	2	1	no	—
245	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
246	C₃×M₁₆	—	2	1	no	—
247	C₃×M₁₆	—	2	1	no	—
248	C₂×C₃×C₈	—	2	1	no	—
249	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
250	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
251	C₃×C₂²⋊C₄	—	2	1	no	—
252	C₃×C₄²	—	2	1	no	—
253	C₃×C₄²	—	2	1	no	—
254	C₃×C₄²	—	2	1	no	—
255	(C₂×D₃)⋊C₄	—	2	1	no	—
256	(C₂×D₃)⋊C₄	—	2	1	no	—
257	Dic₃⋊C₄	—	2	4	no	—
258	Dic₃⋊C₄	—	2	1	no	—
259	Dic₃⋊C₄	—	2	1	no	—
260	C₄×Dic₃	—	2	1	no	—
261	C₄×Dic₃	—	2	1	no	—
262	C₄×Dic₃	—	2	1	no	—
263	C₄×Dic₃	—	2	1	no	—
264	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
265	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
266	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
267	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
268	#208	—	2	1	no	—
269	#208	—	2	1	no	—
270	C₂×C₃⋊C₈	—	2	1	no	—
271	C₂³×C₅	—	6	1	no	—
272	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
273	C₅×Q₈	—	3	2	no	—
274	C₅×Q₈	—	6	1	no	—
275	C₅×D₄	—	3	2	no	—
276	C₅×D₄	—	6	2	no	—
277	C₅×D₄	—	6	2	no	—
278	C₅×D₄	—	6	1	no	—
279	C₂×C₄×C₅	—	2	2	no	—
280	C₂×C₄×C₅	—	2	1	no	—
281	C₂×C₄×C₅	—	2	1	no	—
282	C₂×C₄×C₅	—	3	1	no	—
283	C₂×C₄×C₅	—	3	1	no	—
284	C₂×C₄×C₅	—	3	1	no	—
285	C₂×C₄×C₅	—	3	1	no	—
286	C₂×C₄×C₅	—	4	1	no	—
287	C₂×C₄×C₅	—	4	1	no	—
288	C₂×C₄×C₅	—	6	2	no	—
289	C₂×C₄×C₅	—	6	1	no	—
290	C₂×C₄×C₅	—	6	2	no	—
291	C₂×C₄×C₅	—	6	2	no	—
292	C₂×C₄×C₅	—	6	2	no	—
293	C₂×C₄×C₅	—	12	2	no	—
294	C₂×C₄×C₅	—	12	2	no	—
295	C₂×C₄×C₅	—	12	1	no	—
296	C₂×C₄×C₅	—	12	1	no	—
297	C₂×C₄×C₅	—	12	1	no	—
298	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
299	C₂×C₄×C₅	#235	1	1	yes	characteristic
300	C₂×C₄×C₅	#235	1	1	yes	characteristic
301	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
302	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
303	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
304	C₅×C₈	—	4	1	no	—
305	C₅×C₈	—	12	1	no	—
306	C₂×#41	—	3	1	no	—
307	C₂×M₁₆	—	3	1	no	—
308	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
309	C₄×Q₈	—	3	1	no	—
310	C₄×Q₈	—	3	1	no	—
311	C₄×D₄	—	3	1	no	—
312	C₄×D₄	—	3	1	no	—
313	C₄×D₄	—	6	1	no	—
314	C₄×D₄	—	6	1	no	—
315	#117	—	6	1	no	—
316	#117	C₂×C₅×D₃	1	1	yes	characteristic
317	C₂×C₄⋊C₄	—	6	1	no	—
318	C₂×C₄²	—	3	1	no	—
319	C₂×C₄²	—	3	1	no	—
320	C₂×C₄²	—	6	1	no	—
321	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
322	C₄⋊C₈	—	6	1	no	—
323	C₄⋊C₈	—	6	1	no	—
324	#104	—	6	2	no	—
325	#104	—	6	2	no	—
326	#95	—	6	1	no	—
327	C₂×C₃×C₅	—	2	1	no	—
328	C₂×C₃×C₅	—	2	1	no	—
329	C₂×C₃×C₅	#380	1	1	yes	characteristic
330	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
331	C₂×C₃×C₅	#421	1	1	yes	characteristic
332	C₅×D₃	—	4	2	no	—
333	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
334	C₂²×D₃	—	2	1	no	—
335	C₂×C₃×C₄	—	2	2	no	—
336	C₂×C₃×C₄	—	2	1	no	—
337	C₂×C₃×C₄	—	2	1	no	—
338	C₂×C₃×C₄	—	4	1	no	—
339	C₂×C₃×C₄	—	4	1	no	—
340	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
341	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
342	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
343	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
344	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
345	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
346	#67	—	2	2	no	—
347	#67	—	2	2	no	—
348	C₂×Dic₃	—	2	1	no	—
349	C₂×Dic₃	—	2	2	no	—
350	C₂×Dic₃	—	2	2	no	—
351	C₂×Dic₃	—	4	1	no	—
352	C₂×Dic₃	—	4	1	no	—
353	C₂×Dic₃	—	4	1	no	—
354	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
355	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
356	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
357	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
358	D₁₂	—	2	1	no	—
359	D₁₂	C₅×C₄⋊C₄	1	2	yes	complemented
360	C₄×D₃	—	2	2	no	—
361	C₄×D₃	—	2	2	no	—
362	C₄×D₃	—	4	2	no	—
363	C₄×D₃	—	4	2	no	—
364	Dic₆	—	2	1	no	—
365	Dic₆	C₅×C₄⋊C₄	1	2	yes	complemented
366	C₃×C₈	—	4	1	no	—
367	C₃⋊C₈	—	4	1	no	—
368	C₂²×C₅	—	2	1	no	—
369	C₂²×C₅	—	2	1	no	—
370	C₂²×C₅	—	6	2	no	—
371	C₂²×C₅	—	12	1	no	—
372	C₂²×C₅	—	12	1	no	—
373	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
374	C₂²×C₅	#884	1	1	yes	characteristic
375	C₂²×C₅	#896	1	1	yes	characteristic
376	C₄×C₅	—	3	2	no	—
377	C₄×C₅	—	4	2	no	—
378	C₄×C₅	—	4	1	no	—
379	C₄×C₅	—	4	1	no	—
380	C₄×C₅	—	6	2	no	—
381	C₄×C₅	—	6	1	no	—
382	C₄×C₅	—	6	1	no	—
383	C₄×C₅	—	6	1	no	—
384	C₄×C₅	—	12	2	no	—
385	C₄×C₅	#872	1	1	yes	characteristic
386	C₄×C₅	#884	1	1	yes	characteristic
387	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
388	C₄×C₅	#896	1	1	yes	characteristic
389	#41	—	3	2	no	—
390	#41	—	3	2	no	—
391	#41	—	6	1	no	—
392	#41	—	6	1	no	—
393	C₂×Q₈	—	3	1	no	—
394	C₂×D₄	—	3	1	no	—
395	C₂×D₄	—	6	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	3	1	no	—
398	C₂²×C₄	—	3	1	no	—
399	C₂²×C₄	—	6	1	no	—
400	C₂²×C₄	—	6	1	no	—
401	C₂²×C₄	—	6	1	no	—
402	C₂²×C₄	—	6	1	no	—
403	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
404	M₁₆	—	2	1	no	—
405	M₁₆	—	2	1	no	—
406	M₁₆	—	6	1	no	—
407	M₁₆	—	6	1	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	6	1	no	—
410	C₄⋊C₄	—	6	1	no	—
411	C₄⋊C₄	—	6	1	no	—
412	C₄⋊C₄	—	6	4	no	—
413	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
414	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
415	C₂²⋊C₄	—	2	1	no	—
416	C₂²⋊C₄	—	6	1	no	—
417	C₂²⋊C₄	—	6	1	no	—
418	C₄²	—	2	1	no	—
419	C₄²	—	2	1	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	3	1	no	—
422	C₄²	—	3	1	no	—
423	C₄²	—	3	1	no	—
424	C₄²	—	3	1	no	—
425	C₄²	—	6	1	no	—
426	C₄²	—	6	1	no	—
427	C₄²	—	6	1	no	—
428	C₄²	—	6	1	no	—
429	C₃×C₅	#1806	1	1	yes	characteristic, complemented, Hall
430	C₂²×C₃	—	2	1	no	—
431	C₂²×C₃	—	2	1	no	—
432	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
433	C₂²×C₃	C₅×#121	1	1	yes	characteristic
434	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
435	C₂×D₃	—	2	2	no	—
436	C₂×D₃	—	4	1	no	—
437	C₂×D₃	—	4	1	no	—
438	C₃×C₄	—	4	2	no	—
439	C₃×C₄	—	4	1	no	—
440	C₃×C₄	—	4	1	no	—
441	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
442	C₃×C₄	C₅×#117	1	1	yes	characteristic
443	C₃×C₄	C₅×#121	1	1	yes	characteristic
444	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
445	Dic₃	—	2	2	no	—
446	Dic₃	—	2	1	no	—
447	Dic₃	—	2	1	no	—
448	Dic₃	—	2	1	no	—
449	Dic₃	—	4	2	no	—
450	Dic₃	C₅×#104	1	2	yes	complemented
451	C₂×C₅	—	2	1	no	—
452	C₂×C₅	—	2	1	no	—
453	C₂×C₅	—	12	2	no	—
454	C₂×C₅	#4667	1	1	yes	characteristic
455	C₂×C₅	D₃×#104	1	1	yes	characteristic
456	C₂×C₅	#4821	1	1	yes	characteristic
457	C₂³	—	6	1	no	—
458	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
459	Q₈	—	3	2	no	—
460	Q₈	—	6	1	no	—
461	D₄	—	3	2	no	—
462	D₄	—	6	2	no	—
463	D₄	—	6	2	no	—
464	D₄	—	6	1	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	3	1	no	—
469	C₂×C₄	—	3	1	no	—
470	C₂×C₄	—	3	1	no	—
471	C₂×C₄	—	3	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	6	2	no	—
475	C₂×C₄	—	6	1	no	—
476	C₂×C₄	—	6	2	no	—
477	C₂×C₄	—	6	2	no	—
478	C₂×C₄	—	6	2	no	—
479	C₂×C₄	—	12	2	no	—
480	C₂×C₄	—	12	2	no	—
481	C₂×C₄	—	12	1	no	—
482	C₂×C₄	—	12	1	no	—
483	C₂×C₄	—	12	1	no	—
484	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
485	C₂×C₄	C₅×#235	1	1	yes	characteristic
486	C₂×C₄	C₅×#235	1	1	yes	characteristic
487	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
488	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
489	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
490	C₈	—	4	1	no	—
491	C₈	—	12	1	no	—
492	C₂×C₃	—	2	1	no	—
493	C₂×C₃	—	2	1	no	—
494	C₂×C₃	C₅×#380	1	1	yes	characteristic
495	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
496	C₂×C₃	C₅×#421	1	1	yes	characteristic
497	D₃	—	4	2	no	—
498	C₅	#68770	1	1	yes	characteristic, direct product factor, Sylow
499	C₂²	—	2	1	no	—
500	C₂²	—	2	1	no	—
501	C₂²	—	6	2	no	—
502	C₂²	—	12	1	no	—
503	C₂²	—	12	1	no	—
504	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
505	C₂²	C₅×#884	1	1	yes	characteristic
506	C₂²	C₅×#896	1	1	yes	characteristic
507	C₄	—	3	2	no	—
508	C₄	—	4	2	no	—
509	C₄	—	4	1	no	—
510	C₄	—	4	1	no	—
511	C₄	—	6	2	no	—
512	C₄	—	6	1	no	—
513	C₄	—	6	1	no	—
514	C₄	—	6	1	no	—
515	C₄	—	12	2	no	—
516	C₄	C₅×#872	1	1	yes	characteristic
517	C₄	C₅×#884	1	1	yes	characteristic
518	C₄	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
519	C₄	C₅×#896	1	1	yes	characteristic
520	C₃	C₅×#1806	1	1	yes	characteristic, complemented, Sylow
521	C₂	—	2	1	no	—
522	C₂	—	2	1	no	—
523	C₂	—	12	2	no	—
524	C₂	C₅×#4667	1	1	yes	characteristic
525	C₂	C₅×D₃×#104	1	1	yes	characteristic
526	C₂	C₅×#4821	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#339	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₅×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#339	D₃
hypocenter	C₃	C₅×#1806
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#339

Complements

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

N	H ≅ G/N	type
3. C₅×#5118	C₂	semidirect
4. C₄×C₅×Dic₃⋊C₄	C₂	semidirect
6. C₃×C₅×#339	C₂	semidirect
7. C₅×#4532	C₂	semidirect
17. C₅×#893	C₄	semidirect
18. C₅×#893	C₄	semidirect
19. C₅×#887	C₄	semidirect
20. C₅×#887	C₄	semidirect
30. #68770	C₅	direct (N × H)
37. C₅×#339	D₃	semidirect
80. #5118	C₂×C₅	semidirect
81. C₄×Dic₃⋊C₄	C₂×C₅	semidirect
83. C₃×#339	C₂×C₅	semidirect
84. #4532	C₂×C₅	semidirect
133. C₅×D₁₂	C₄⋊C₄	semidirect
139. C₅×Dic₆	C₄⋊C₄	semidirect
150. #893	C₄×C₅	semidirect
151. #893	C₄×C₅	semidirect
152. #887	C₄×C₅	semidirect
153. #887	C₄×C₅	semidirect
209. #339	C₅×D₃	semidirect
230. C₅×Dic₃	#104	semidirect
359. D₁₂	C₅×C₄⋊C₄	semidirect
365. Dic₆	C₅×C₄⋊C₄	semidirect
429. C₃×C₅	#1806	semidirect
450. Dic₃	C₅×#104	semidirect
498. C₅	#68770	direct (N × H)
520. C₃	C₅×#1806	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5522	2
C₂×C₅×#825	2
C₅×#5118	2
C₄×C₅×Dic₃⋊C₄	2
C₂×C₅×#805	2
C₃×C₅×#339	2
C₅×#4532	2
C₅×#1806	3
#68770	5

Subgroups — C5×#68770 #14297810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68770 #14297810