Subgroups — C₅×#81615 #14460105

Order 1920.

796 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5734	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5726	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5084	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5083	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4814	C₂	1	2	yes	maximal, complemented
7	C₅×#4810	C₂	1	2	yes	maximal, complemented
8	C₅×#4807	C₂	1	2	yes	maximal, complemented
9	C₅×#4804	C₂	1	2	yes	maximal, complemented
10	C₄×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#3303	—	3	1	no	maximal, Hall
13	C₂×C₅×#239	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₅×#947	—	2	2	no	—
17	C₅×#946	C₂²	1	2	yes	—
18	C₂×C₅×#215	C₂²	1	2	yes	—
19	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
20	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
21	C₅×#895	—	2	1	no	—
22	C₅×#895	—	2	1	no	—
23	C₅×#895	C₂²	1	2	yes	—
24	C₅×#894	—	2	2	no	—
25	C₅×#892	—	2	1	no	—
26	C₅×#892	C₂²	1	2	yes	—
27	C₅×D₃×C₄⋊C₄	—	2	1	no	—
28	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
29	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
30	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
31	C₃×C₅×#103	C₂²	1	2	yes	—
32	C₃×C₅×#103	C₂²	1	2	yes	—
33	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
34	C₅×#835	C₂²	1	2	yes	—
35	C₅×#821	C₂²	1	1	yes	characteristic
36	C₅×#821	C₂²	1	1	yes	characteristic
37	C₅×(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
38	C₅×#818	C₂²	1	1	yes	characteristic
39	C₅×#809	C₂²	1	2	yes	—
40	C₅×#808	C₂²	1	2	yes	—
41	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#81615	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#555	—	3	1	no	—
44	C₅×#547	—	3	1	no	—
45	C₅×#487	D₃	1	1	yes	characteristic, complemented
46	C₅×#485	—	3	2	no	—
47	C₅×#476	—	3	1	no	—
48	C₅×#465	—	3	2	no	—
49	C₅×#461	—	3	1	no	—
50	C₅×#449	—	3	2	no	—
51	C₄×C₅×D₈	—	3	1	no	—
52	C₅×C₈×D₄	—	3	1	no	—
53	C₅×#416	—	3	2	no	—
54	C₂×C₃×C₅×Q₈	—	2	1	no	—
55	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
56	C₅×#239	—	2	4	no	—
57	C₅×#239	—	4	2	no	—
58	C₂×C₅×D₁₂	—	2	2	no	—
59	C₂×C₅×D₁₂	—	2	2	no	—
60	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
61	C₂×C₅×D₁₂	C₂³	1	1	yes	characteristic
62	C₂×C₅×D₁₂	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₅×D₃	—	2	1	no	—
64	C₂×C₄×C₅×D₃	—	2	1	no	—
65	C₂×C₄×C₅×D₃	—	2	2	no	—
66	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
67	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
69	C₃×C₅×C₄⋊C₄	—	2	1	no	—
70	C₃×C₅×C₄⋊C₄	—	2	1	no	—
71	C₃×C₅×C₄⋊C₄	—	2	1	no	—
72	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
73	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
74	C₅×#215	—	4	2	no	—
75	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
76	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
77	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
78	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
79	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
80	C₅×(C₂×D₃)⋊C₄	D₄	1	2	yes	—
81	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
82	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
83	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
84	C₅×Dic₃⋊C₄	—	2	2	no	—
85	C₄×C₅×Dic₃	—	2	2	no	—
86	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
87	C₅×D₂₄	—	2	2	no	—
88	C₅×D₃×C₈	—	4	2	no	—
89	#5734	C₂×C₅	1	1	yes	characteristic, complemented
90	#5726	C₂×C₅	1	1	yes	characteristic, complemented
91	C₃×#487	C₂×C₅	1	1	yes	characteristic, complemented
92	#5084	C₂×C₅	1	1	yes	characteristic, complemented
93	#5083	C₂×C₅	1	1	yes	characteristic, complemented
94	#4814	C₂×C₅	1	2	yes	complemented
95	#4810	C₂×C₅	1	2	yes	complemented
96	#4807	C₂×C₅	1	2	yes	complemented
97	#4804	C₂×C₅	1	2	yes	complemented
98	C₄×D₂₄	C₂×C₅	1	1	yes	characteristic, complemented
99	C₈×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
100	C₂×C₅×#41	—	3	2	no	—
101	C₂×C₅×SD₁₆	—	3	2	no	—
102	C₂×C₅×D₈	—	3	1	no	—
103	C₂²×C₅×C₈	—	3	2	no	—
104	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
105	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
106	C₅×#124	—	6	2	no	—
107	C₅×#123	—	6	2	no	—
108	C₅×C₂²⋊Q₈	—	3	2	no	—
109	C₅×#121	—	3	2	no	—
110	C₅×#121	—	6	1	no	—
111	C₅×#121	—	6	1	no	—
112	C₄×C₅×D₄	—	3	1	no	—
113	C₄×C₅×D₄	—	3	1	no	—
114	C₄×C₅×D₄	—	3	1	no	—
115	C₅×#117	—	3	2	no	—
116	C₅×#117	—	6	1	no	—
117	C₂×C₅×C₄⋊C₄	—	6	1	no	—
118	C₅×#107	—	3	1	no	—
119	C₅×#106	—	3	2	no	—
120	C₅×C₄⋊C₈	—	3	1	no	—
121	C₅×#103	—	3	2	no	—
122	C₅×#103	C₂×D₃	1	2	yes	—
123	C₅×#103	C₂×D₃	1	2	yes	—
124	C₅×#102	—	3	1	no	—
125	C₅×#102	—	3	1	no	—
126	C₅×#102	—	3	2	no	—
127	C₅×C₂²⋊C₈	—	3	2	no	—
128	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
129	#3303	—	3	1	no	Sylow
130	C₂²×C₅×D₃	—	2	1	no	—
131	C₂²×C₅×D₃	—	2	1	no	—
132	C₂²×C₅×D₃	C₂×D₄	1	2	yes	—
133	C₃×C₅×Q₈	—	2	2	no	—
134	C₃×C₅×Q₈	—	2	2	no	—
135	C₃×C₅×Q₈	—	8	1	no	—
136	C₂×C₃×C₄×C₅	—	2	2	no	—
137	C₂×C₃×C₄×C₅	—	2	2	no	—
138	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
141	C₂×C₅×Dic₃	—	2	1	no	—
142	C₂×C₅×Dic₃	—	2	1	no	—
143	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
144	C₅×D₁₂	—	2	2	no	—
145	C₅×D₁₂	—	2	1	no	—
146	C₅×D₁₂	—	2	1	no	—
147	C₅×D₁₂	—	2	1	no	—
148	C₅×D₁₂	—	4	4	no	—
149	C₅×D₁₂	—	8	2	no	—
150	C₅×D₁₂	#41	1	2	yes	—
151	C₄×C₅×D₃	—	2	2	no	—
152	C₄×C₅×D₃	—	2	2	no	—
153	C₄×C₅×D₃	—	4	4	no	—
154	C₄×C₅×D₃	—	4	2	no	—
155	C₄×C₅×D₃	—	4	2	no	—
156	C₃×C₅×C₈	—	2	1	no	—
157	C₃×C₅×C₈	—	2	1	no	—
158	C₅×C₃⋊C₈	—	2	2	no	—
159	C₂×#239	C₂²×C₅	1	2	yes	—
160	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
161	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
162	#947	—	2	2	no	—
163	#946	C₂²×C₅	1	2	yes	—
164	C₂×#215	C₂²×C₅	1	2	yes	—
165	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic
166	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
167	#895	—	2	1	no	—
168	#895	—	2	1	no	—
169	#895	C₂²×C₅	1	2	yes	—
170	#894	—	2	2	no	—
171	#892	—	2	1	no	—
172	#892	C₂²×C₅	1	2	yes	—
173	D₃×C₄⋊C₄	—	2	1	no	—
174	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
175	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
176	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
177	C₃×#103	C₂²×C₅	1	2	yes	—
178	C₃×#103	C₂²×C₅	1	2	yes	—
179	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
180	#835	C₂²×C₅	1	2	yes	—
181	#821	C₂²×C₅	1	1	yes	characteristic
182	#821	C₂²×C₅	1	1	yes	characteristic
183	(C₂×D₃)⋊C₈	C₂²×C₅	1	2	yes	—
184	#818	C₂²×C₅	1	1	yes	characteristic
185	#809	C₂²×C₅	1	2	yes	—
186	#808	C₂²×C₅	1	2	yes	—
187	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
188	C₅×#41	—	6	4	no	—
189	C₅×#41	—	12	2	no	—
190	C₂×C₅×Q₈	—	2	1	no	—
191	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
192	C₂×C₅×D₄	—	3	1	no	—
193	C₂×C₅×D₄	—	3	1	no	—
194	C₂×C₅×D₄	—	3	1	no	—
195	C₂×C₅×D₄	—	6	2	no	—
196	C₂×C₅×D₄	—	6	2	no	—
197	C₂²×C₄×C₅	—	3	2	no	—
198	C₂²×C₄×C₅	—	6	2	no	—
199	C₂²×C₄×C₅	—	6	1	no	—
200	C₂²×C₄×C₅	—	6	1	no	—
201	C₅×SD₁₆	—	12	2	no	—
202	C₅×D₈	—	6	2	no	—
203	C₂×C₅×C₈	—	3	2	no	—
204	C₂×C₅×C₈	—	12	2	no	—
205	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
206	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
207	C₅×C₄⋊C₄	—	2	1	no	—
208	C₅×C₄⋊C₄	—	2	1	no	—
209	C₅×C₄⋊C₄	—	2	1	no	—
210	C₅×C₄⋊C₄	—	3	1	no	—
211	C₅×C₄⋊C₄	—	3	1	no	—
212	C₅×C₄⋊C₄	—	3	1	no	—
213	C₅×C₄⋊C₄	—	6	2	no	—
214	C₅×C₄⋊C₄	C₂²×D₃	1	2	yes	—
215	C₅×C₂²⋊C₄	—	3	2	no	—
216	C₅×C₂²⋊C₄	—	6	2	no	—
217	C₅×C₂²⋊C₄	—	6	2	no	—
218	C₅×C₂²⋊C₄	—	6	2	no	—
219	C₅×C₂²⋊C₄	—	6	1	no	—
220	C₅×C₂²⋊C₄	—	6	1	no	—
221	C₄²×C₅	—	6	2	no	—
222	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
223	#555	—	3	1	no	—
224	#547	—	3	1	no	—
225	#487	C₅×D₃	1	1	yes	characteristic, complemented
226	#485	—	3	2	no	—
227	#476	—	3	1	no	—
228	#465	—	3	2	no	—
229	#461	—	3	1	no	—
230	#449	—	3	2	no	—
231	C₄×D₈	—	3	1	no	—
232	C₈×D₄	—	3	1	no	—
233	#416	—	3	2	no	—
234	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₅×D₃	—	2	2	no	—
236	C₂×C₅×D₃	—	2	2	no	—
237	C₂×C₅×D₃	—	2	2	no	—
238	C₂×C₅×D₃	—	2	2	no	—
239	C₂×C₅×D₃	—	4	1	no	—
240	C₂×C₅×D₃	—	4	1	no	—
241	C₂×C₅×D₃	—	4	1	no	—
242	C₂×C₅×D₃	—	4	1	no	—
243	C₂×C₅×D₃	—	4	1	no	—
244	C₃×C₄×C₅	—	2	1	no	—
245	C₃×C₄×C₅	—	4	2	no	—
246	C₃×C₄×C₅	—	4	2	no	—
247	C₃×C₄×C₅	#135	1	2	yes	—
248	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
249	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
250	C₅×Dic₃	—	2	2	no	—
251	C₅×Dic₃	—	4	1	no	—
252	C₅×Dic₃	—	4	1	no	—
253	C₂×C₃×Q₈	—	2	1	no	—
254	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
255	#239	—	2	4	no	—
256	#239	—	4	2	no	—
257	C₂×D₁₂	—	2	2	no	—
258	C₂×D₁₂	—	2	2	no	—
259	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
260	C₂×D₁₂	C₂³×C₅	1	1	yes	characteristic
261	C₂×D₁₂	C₂³×C₅	1	1	yes	characteristic
262	C₂×C₄×D₃	—	2	1	no	—
263	C₂×C₄×D₃	—	2	1	no	—
264	C₂×C₄×D₃	—	2	2	no	—
265	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
266	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
267	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
268	C₃×C₄⋊C₄	—	2	1	no	—
269	C₃×C₄⋊C₄	—	2	1	no	—
270	C₃×C₄⋊C₄	—	2	1	no	—
271	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
272	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
273	#215	—	4	2	no	—
274	(C₂×D₃)⋊C₄	—	2	2	no	—
275	(C₂×D₃)⋊C₄	—	2	2	no	—
276	(C₂×D₃)⋊C₄	—	2	1	no	—
277	(C₂×D₃)⋊C₄	—	2	1	no	—
278	(C₂×D₃)⋊C₄	—	2	2	no	—
279	(C₂×D₃)⋊C₄	C₅×D₄	1	2	yes	—
280	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
281	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
282	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
283	Dic₃⋊C₄	—	2	2	no	—
284	C₄×Dic₃	—	2	2	no	—
285	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
286	D₂₄	—	2	2	no	—
287	D₃×C₈	—	4	2	no	—
288	C₂³×C₅	—	3	2	no	—
289	C₂³×C₅	—	6	1	no	—
290	C₂³×C₅	—	6	1	no	—
291	C₅×Q₈	—	2	2	no	—
292	C₅×Q₈	—	2	2	no	—
293	C₅×Q₈	—	8	1	no	—
294	C₅×D₄	—	3	2	no	—
295	C₅×D₄	—	6	2	no	—
296	C₅×D₄	—	6	1	no	—
297	C₅×D₄	—	6	1	no	—
298	C₅×D₄	—	6	1	no	—
299	C₅×D₄	—	12	4	no	—
300	C₅×D₄	—	24	2	no	—
301	C₂×C₄×C₅	—	2	2	no	—
302	C₂×C₄×C₅	—	2	2	no	—
303	C₂×C₄×C₅	—	3	2	no	—
304	C₂×C₄×C₅	—	6	2	no	—
305	C₂×C₄×C₅	—	6	2	no	—
306	C₂×C₄×C₅	—	6	1	no	—
307	C₂×C₄×C₅	—	6	1	no	—
308	C₂×C₄×C₅	—	12	4	no	—
309	C₂×C₄×C₅	—	12	2	no	—
310	C₂×C₄×C₅	—	12	2	no	—
311	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
312	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
313	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
314	C₅×C₈	—	2	1	no	—
315	C₅×C₈	—	2	1	no	—
316	C₅×C₈	—	6	2	no	—
317	C₂×#41	—	3	2	no	—
318	C₂×SD₁₆	—	3	2	no	—
319	C₂×D₈	—	3	1	no	—
320	C₂²×C₈	—	3	2	no	—
321	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
322	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
323	#124	—	6	2	no	—
324	#123	—	6	2	no	—
325	C₂²⋊Q₈	—	3	2	no	—
326	#121	—	3	2	no	—
327	#121	—	6	1	no	—
328	#121	—	6	1	no	—
329	C₄×D₄	—	3	1	no	—
330	C₄×D₄	—	3	1	no	—
331	C₄×D₄	—	3	1	no	—
332	#117	—	3	2	no	—
333	#117	—	6	1	no	—
334	C₂×C₄⋊C₄	—	6	1	no	—
335	#107	—	3	1	no	—
336	#106	—	3	2	no	—
337	C₄⋊C₈	—	3	1	no	—
338	#103	—	3	2	no	—
339	#103	C₂×C₅×D₃	1	2	yes	—
340	#103	C₂×C₅×D₃	1	2	yes	—
341	#102	—	3	1	no	—
342	#102	—	3	1	no	—
343	#102	—	3	2	no	—
344	C₂²⋊C₈	—	3	2	no	—
345	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
346	C₂×C₃×C₅	#546	1	1	yes	characteristic
347	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
348	C₂×C₃×C₅	#577	1	1	yes	characteristic
349	C₅×D₃	—	4	2	no	—
350	C₅×D₃	—	4	2	no	—
351	C₅×D₃	—	8	1	no	—
352	C₂²×D₃	—	2	1	no	—
353	C₂²×D₃	—	2	1	no	—
354	C₂²×D₃	C₂×C₅×D₄	1	2	yes	—
355	C₃×Q₈	—	2	2	no	—
356	C₃×Q₈	—	2	2	no	—
357	C₃×Q₈	—	8	1	no	—
358	C₂×C₃×C₄	—	2	2	no	—
359	C₂×C₃×C₄	—	2	2	no	—
360	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
361	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
362	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
363	C₂×Dic₃	—	2	1	no	—
364	C₂×Dic₃	—	2	1	no	—
365	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
366	D₁₂	—	2	2	no	—
367	D₁₂	—	2	1	no	—
368	D₁₂	—	2	1	no	—
369	D₁₂	—	2	1	no	—
370	D₁₂	—	4	4	no	—
371	D₁₂	—	8	2	no	—
372	D₁₂	C₅×#41	1	2	yes	—
373	C₄×D₃	—	2	2	no	—
374	C₄×D₃	—	2	2	no	—
375	C₄×D₃	—	4	4	no	—
376	C₄×D₃	—	4	2	no	—
377	C₄×D₃	—	4	2	no	—
378	C₃×C₈	—	2	1	no	—
379	C₃×C₈	—	2	1	no	—
380	C₃⋊C₈	—	2	2	no	—
381	C₂²×C₅	—	6	2	no	—
382	C₂²×C₅	—	6	2	no	—
383	C₂²×C₅	—	6	2	no	—
384	C₂²×C₅	—	6	2	no	—
385	C₂²×C₅	—	12	1	no	—
386	C₂²×C₅	—	12	1	no	—
387	C₂²×C₅	—	12	1	no	—
388	C₂²×C₅	—	12	1	no	—
389	C₂²×C₅	—	12	1	no	—
390	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
391	C₄×C₅	—	2	1	no	—
392	C₄×C₅	—	4	2	no	—
393	C₄×C₅	—	4	2	no	—
394	C₄×C₅	—	6	2	no	—
395	C₄×C₅	—	12	1	no	—
396	C₄×C₅	—	12	1	no	—
397	C₄×C₅	#919	1	2	yes	—
398	C₄×C₅	D₃×#41	1	1	yes	characteristic
399	C₄×C₅	#1009	1	1	yes	characteristic
400	#41	—	6	4	no	—
401	#41	—	12	2	no	—
402	C₂×Q₈	—	2	1	no	—
403	C₂×Q₈	C₂²×C₅×D₃	1	2	yes	—
404	C₂×D₄	—	3	1	no	—
405	C₂×D₄	—	3	1	no	—
406	C₂×D₄	—	3	1	no	—
407	C₂×D₄	—	6	2	no	—
408	C₂×D₄	—	6	2	no	—
409	C₂²×C₄	—	3	2	no	—
410	C₂²×C₄	—	6	2	no	—
411	C₂²×C₄	—	6	1	no	—
412	C₂²×C₄	—	6	1	no	—
413	SD₁₆	—	12	2	no	—
414	D₈	—	6	2	no	—
415	C₂×C₈	—	3	2	no	—
416	C₂×C₈	—	12	2	no	—
417	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
418	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	3	1	no	—
423	C₄⋊C₄	—	3	1	no	—
424	C₄⋊C₄	—	3	1	no	—
425	C₄⋊C₄	—	6	2	no	—
426	C₄⋊C₄	C₂²×C₅×D₃	1	2	yes	—
427	C₂²⋊C₄	—	3	2	no	—
428	C₂²⋊C₄	—	6	2	no	—
429	C₂²⋊C₄	—	6	2	no	—
430	C₂²⋊C₄	—	6	2	no	—
431	C₂²⋊C₄	—	6	1	no	—
432	C₂²⋊C₄	—	6	1	no	—
433	C₄²	—	6	2	no	—
434	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
435	C₃×C₅	#3303	1	1	yes	characteristic, complemented, Hall
436	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
437	C₂×D₃	—	2	2	no	—
438	C₂×D₃	—	2	2	no	—
439	C₂×D₃	—	2	2	no	—
440	C₂×D₃	—	2	2	no	—
441	C₂×D₃	—	4	1	no	—
442	C₂×D₃	—	4	1	no	—
443	C₂×D₃	—	4	1	no	—
444	C₂×D₃	—	4	1	no	—
445	C₂×D₃	—	4	1	no	—
446	C₃×C₄	—	2	1	no	—
447	C₃×C₄	—	4	2	no	—
448	C₃×C₄	—	4	2	no	—
449	C₃×C₄	C₅×#135	1	2	yes	—
450	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
451	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
452	Dic₃	—	2	2	no	—
453	Dic₃	—	4	1	no	—
454	Dic₃	—	4	1	no	—
455	C₂×C₅	—	12	2	no	—
456	C₂×C₅	—	12	2	no	—
457	C₂×C₅	—	24	1	no	—
458	C₂×C₅	#5676	1	1	yes	characteristic
459	C₂×C₅	#5762	1	1	yes	characteristic
460	C₂×C₅	C₂×#919	1	1	yes	characteristic
461	C₂³	—	3	2	no	—
462	C₂³	—	6	1	no	—
463	C₂³	—	6	1	no	—
464	Q₈	—	2	2	no	—
465	Q₈	—	2	2	no	—
466	Q₈	—	8	1	no	—
467	D₄	—	3	2	no	—
468	D₄	—	6	2	no	—
469	D₄	—	6	1	no	—
470	D₄	—	6	1	no	—
471	D₄	—	6	1	no	—
472	D₄	—	12	4	no	—
473	D₄	—	24	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	3	2	no	—
477	C₂×C₄	—	6	2	no	—
478	C₂×C₄	—	6	2	no	—
479	C₂×C₄	—	6	1	no	—
480	C₂×C₄	—	6	1	no	—
481	C₂×C₄	—	12	4	no	—
482	C₂×C₄	—	12	2	no	—
483	C₂×C₄	—	12	2	no	—
484	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
485	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
486	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
487	C₈	—	2	1	no	—
488	C₈	—	2	1	no	—
489	C₈	—	6	2	no	—
490	C₂×C₃	C₅×#546	1	1	yes	characteristic
491	C₂×C₃	C₂×C₅×#135	1	1	yes	characteristic
492	C₂×C₃	C₅×#577	1	1	yes	characteristic
493	D₃	—	4	2	no	—
494	D₃	—	4	2	no	—
495	D₃	—	8	1	no	—
496	C₅	#81615	1	1	yes	characteristic, direct product factor, Sylow
497	C₂²	—	6	2	no	—
498	C₂²	—	6	2	no	—
499	C₂²	—	6	2	no	—
500	C₂²	—	6	2	no	—
501	C₂²	—	12	1	no	—
502	C₂²	—	12	1	no	—
503	C₂²	—	12	1	no	—
504	C₂²	—	12	1	no	—
505	C₂²	—	12	1	no	—
506	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
507	C₄	—	2	1	no	—
508	C₄	—	4	2	no	—
509	C₄	—	4	2	no	—
510	C₄	—	6	2	no	—
511	C₄	—	12	1	no	—
512	C₄	—	12	1	no	—
513	C₄	C₅×#919	1	2	yes	—
514	C₄	C₅×D₃×#41	1	1	yes	characteristic
515	C₄	C₅×#1009	1	1	yes	characteristic
516	C₃	C₅×#3303	1	1	yes	characteristic, complemented, Sylow
517	C₂	—	12	2	no	—
518	C₂	—	12	2	no	—
519	C₂	—	24	1	no	—
520	C₂	C₅×#5676	1	1	yes	characteristic
521	C₂	C₅×#5762	1	1	yes	characteristic
522	C₂	C₂×C₅×#919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#487	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₅×#487	D₃
hypocenter	C₃	C₅×#3303
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#487

Complements

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

N	H ≅ G/N	type
1. C₅×#5734	C₂	semidirect
2. C₅×#5726	C₂	semidirect
3. C₃×C₅×#487	C₂	semidirect
4. C₅×#5084	C₂	semidirect
5. C₅×#5083	C₂	semidirect
6. C₅×#4814	C₂	semidirect
7. C₅×#4810	C₂	semidirect
8. C₅×#4807	C₂	semidirect
9. C₅×#4804	C₂	semidirect
10. C₄×C₅×D₂₄	C₂	semidirect
11. C₅×C₈×D₁₂	C₂	semidirect
42. #81615	C₅	direct (N × H)
45. C₅×#487	D₃	semidirect
89. #5734	C₂×C₅	semidirect
90. #5726	C₂×C₅	semidirect
91. C₃×#487	C₂×C₅	semidirect
92. #5084	C₂×C₅	semidirect
93. #5083	C₂×C₅	semidirect
94. #4814	C₂×C₅	semidirect
95. #4810	C₂×C₅	semidirect
96. #4807	C₂×C₅	semidirect
97. #4804	C₂×C₅	semidirect
98. C₄×D₂₄	C₂×C₅	semidirect
99. C₈×D₁₂	C₂×C₅	semidirect
225. #487	C₅×D₃	semidirect
435. C₃×C₅	#3303	semidirect
496. C₅	#81615	direct (N × H)
516. C₃	C₅×#3303	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5734	2
C₅×#5726	2
C₃×C₅×#487	2
C₅×#5084	2
C₅×#5083	2
C₅×#4814	2
C₅×#4810	2
C₅×#4807	2
C₅×#4804	2
C₄×C₅×D₂₄	2
C₅×C₈×D₁₂	2
C₅×#3303	3
#81615	5

Subgroups — C5×#81615 #14460105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81615 #14460105