Subgroups — C₅×#81576 #14458290

Order 1920.

988 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5726	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5717	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5087	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5072	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4775	C₂	1	2	yes	maximal, complemented
7	C₅×#4774	C₂	1	2	yes	maximal, complemented
8	C₅×#4773	C₂	1	2	yes	maximal, complemented
9	C₅×D₃×#102	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×#3294	—	3	1	no	maximal, Hall
13	C₂×C₅×#239	—	2	1	no	—
14	C₂×C₅×D₃×D₄	—	2	1	no	—
15	C₂×C₅×D₃×D₄	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₃×C₅×#127	C₂²	1	1	yes	characteristic
18	C₅×#946	—	2	1	no	—
19	C₅×#938	—	2	1	no	—
20	C₅×#938	C₂²	1	2	yes	—
21	C₅×#937	—	2	2	no	—
22	C₂×C₅×#213	C₂²	1	2	yes	complemented
23	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
24	C₂×C₅×D₃×C₈	C₂²	1	2	yes	complemented
25	C₅×#895	C₂²	1	2	yes	complemented
26	C₅×#894	—	2	2	no	—
27	C₅×D₃×C₄⋊C₄	C₂²	1	2	yes	complemented
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, complemented
31	C₃×C₅×#102	C₂²	1	2	yes	complemented
32	C₃×C₅×#102	C₂²	1	2	yes	complemented
33	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
34	C₅×#832	C₂²	1	2	yes	complemented
35	C₅×#821	C₂²	1	1	yes	characteristic
36	C₅×#821	C₂²	1	1	yes	characteristic, complemented
37	C₅×(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
38	C₅×#818	C₂²	1	1	yes	characteristic, complemented
39	C₅×#809	C₂²	1	2	yes	—
40	C₅×#807	C₂²	1	2	yes	complemented
41	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
42	#81576	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#547	—	3	1	no	—
44	C₅×D₄²	—	3	1	no	—
45	C₅×#486	D₃	1	1	yes	characteristic, complemented
46	C₅×#480	—	3	2	no	—
47	C₅×#474	—	3	1	no	—
48	C₅×#466	—	3	2	no	—
49	C₅×#459	—	3	1	no	—
50	C₅×#447	—	3	2	no	—
51	C₄×C₅×D₈	—	3	1	no	—
52	C₅×C₈×D₄	—	3	1	no	—
53	C₂×C₅×#102	—	3	2	no	—
54	C₂³×C₅×D₃	—	2	2	no	—
55	C₂×C₃×C₅×Q₈	—	2	1	no	—
56	C₂×C₃×C₅×D₄	—	2	1	no	—
57	C₂×C₃×C₅×D₄	—	2	1	no	—
58	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
59	C₂×C₅×#67	—	2	2	no	—
60	C₂×C₅×#67	—	2	2	no	—
61	C₅×#239	—	4	4	no	—
62	C₅×D₃×D₄	—	2	4	no	—
63	C₅×D₃×D₄	—	4	2	no	—
64	C₅×D₃×D₄	—	4	4	no	—
65	C₂×C₅×D₁₂	—	2	2	no	—
66	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
67	C₂×C₅×D₁₂	C₂³	1	1	yes	characteristic
68	C₂×C₅×D₁₂	C₂³	1	1	yes	characteristic
69	C₂×C₄×C₅×D₃	—	2	2	no	—
70	C₂×C₄×C₅×D₃	—	2	1	no	—
71	C₂×C₄×C₅×D₃	—	2	1	no	—
72	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
73	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₃×C₅×C₄⋊C₄	—	2	1	no	—
76	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
77	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
78	C₅×#217	—	2	2	no	—
79	C₅×#213	—	4	2	no	—
80	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
81	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
82	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
83	C₅×(C₂×D₃)⋊C₄	D₄	1	2	yes	—
84	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
85	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
86	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
87	C₅×Dic₃⋊C₄	—	2	2	no	—
88	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
89	C₅×D₂₄	—	2	2	no	—
90	C₅×D₃×C₈	—	4	2	no	—
91	#5726	C₂×C₅	1	1	yes	characteristic, complemented
92	#5717	C₂×C₅	1	1	yes	characteristic, complemented
93	C₃×#486	C₂×C₅	1	1	yes	characteristic, complemented
94	#5087	C₂×C₅	1	1	yes	characteristic, complemented
95	#5072	C₂×C₅	1	1	yes	characteristic, complemented
96	#4775	C₂×C₅	1	2	yes	complemented
97	#4774	C₂×C₅	1	2	yes	complemented
98	#4773	C₂×C₅	1	2	yes	complemented
99	D₃×#102	C₂×C₅	1	2	yes	complemented
100	C₄×D₂₄	C₂×C₅	1	1	yes	characteristic, complemented
101	C₈×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
102	C₂×C₅×#41	—	6	1	no	—
103	C₂²×C₅×D₄	—	3	2	no	—
104	C₂²×C₅×D₄	—	6	1	no	—
105	C₂×C₅×D₈	—	3	2	no	—
106	C₂×C₅×D₈	—	3	1	no	—
107	C₂²×C₅×C₈	—	3	2	no	—
108	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic, complemented
109	C₅×#127	C₂×D₃	1	1	yes	characteristic
110	C₅×#123	—	6	2	no	—
111	C₅×C₂²⋊Q₈	—	6	1	no	—
112	C₅×#121	—	3	2	no	—
113	C₅×#121	—	3	2	no	—
114	C₅×#121	—	6	1	no	—
115	C₅×#120	—	6	2	no	—
116	C₄×C₅×D₄	—	3	1	no	—
117	C₄×C₅×D₄	—	3	1	no	—
118	C₄×C₅×D₄	—	3	1	no	—
119	C₂×C₅×C₄⋊C₄	—	3	2	no	—
120	C₅×#107	—	3	1	no	—
121	C₅×#107	—	3	2	no	—
122	C₅×C₄⋊C₈	—	3	1	no	—
123	C₅×#102	—	3	2	no	—
124	C₅×#102	—	3	1	no	—
125	C₅×#102	—	3	1	no	—
126	C₅×#102	—	3	2	no	—
127	C₅×#102	C₂×D₃	1	2	yes	complemented
128	C₅×#102	C₂×D₃	1	2	yes	complemented
129	C₅×C₂²⋊C₈	—	3	2	no	—
130	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic, complemented
131	#3294	—	3	1	no	Sylow
132	C₂³×C₃×C₅	—	2	2	no	—
133	C₂²×C₅×D₃	—	2	1	no	—
134	C₂²×C₅×D₃	—	2	1	no	—
135	C₂²×C₅×D₃	—	4	4	no	—
136	C₂²×C₅×D₃	—	4	4	no	—
137	C₂²×C₅×D₃	—	8	2	no	—
138	C₂²×C₅×D₃	C₂×D₄	1	2	yes	—
139	C₃×C₅×Q₈	—	4	2	no	—
140	C₃×C₅×D₄	—	2	2	no	—
141	C₃×C₅×D₄	—	2	2	no	—
142	C₃×C₅×D₄	—	4	2	no	—
143	C₃×C₅×D₄	—	8	1	no	—
144	C₂×C₃×C₄×C₅	—	2	2	no	—
145	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
148	C₅×#67	—	4	4	no	—
149	C₅×#67	—	4	4	no	—
150	C₂×C₅×Dic₃	—	2	1	no	—
151	C₂×C₅×Dic₃	—	2	1	no	—
152	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
153	C₅×D₁₂	—	2	1	no	—
154	C₅×D₁₂	—	2	1	no	—
155	C₅×D₁₂	—	2	1	no	—
156	C₅×D₁₂	—	4	4	no	—
157	C₅×D₁₂	D₈	1	4	yes	complemented
158	C₅×D₁₂	#41	1	2	yes	complemented
159	C₄×C₅×D₃	—	2	2	no	—
160	C₄×C₅×D₃	—	2	2	no	—
161	C₄×C₅×D₃	—	4	4	no	—
162	C₄×C₅×D₃	—	4	2	no	—
163	C₄×C₅×D₃	—	4	2	no	—
164	C₃×C₅×C₈	—	2	1	no	—
165	C₃×C₅×C₈	—	2	1	no	—
166	C₅×C₃⋊C₈	—	2	2	no	—
167	C₂×#239	—	2	1	no	—
168	C₂×D₃×D₄	—	2	1	no	—
169	C₂×D₃×D₄	C₂²×C₅	1	2	yes	—
170	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
171	C₃×#127	C₂²×C₅	1	1	yes	characteristic
172	#946	—	2	1	no	—
173	#938	—	2	1	no	—
174	#938	C₂²×C₅	1	2	yes	—
175	#937	—	2	2	no	—
176	C₂×#213	C₂²×C₅	1	2	yes	complemented
177	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic
178	C₂×D₃×C₈	C₂²×C₅	1	2	yes	complemented
179	#895	C₂²×C₅	1	2	yes	complemented
180	#894	—	2	2	no	—
181	D₃×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
182	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
183	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
184	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic, complemented
185	C₃×#102	C₂²×C₅	1	2	yes	complemented
186	C₃×#102	C₂²×C₅	1	2	yes	complemented
187	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
188	#832	C₂²×C₅	1	2	yes	complemented
189	#821	C₂²×C₅	1	1	yes	characteristic
190	#821	C₂²×C₅	1	1	yes	characteristic, complemented
191	(C₂×D₃)⋊C₈	C₂²×C₅	1	2	yes	complemented
192	#818	C₂²×C₅	1	1	yes	characteristic, complemented
193	#809	C₂²×C₅	1	2	yes	—
194	#807	C₂²×C₅	1	2	yes	complemented
195	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic, complemented
196	C₂⁴×C₅	—	6	2	no	—
197	C₅×#41	—	12	4	no	—
198	C₂×C₅×Q₈	—	2	1	no	—
199	C₂×C₅×D₄	—	2	1	no	—
200	C₂×C₅×D₄	—	2	1	no	—
201	C₂×C₅×D₄	—	3	1	no	—
202	C₂×C₅×D₄	—	3	1	no	—
203	C₂×C₅×D₄	—	3	1	no	—
204	C₂×C₅×D₄	—	6	2	no	—
205	C₂×C₅×D₄	—	6	2	no	—
206	C₂×C₅×D₄	—	6	4	no	—
207	C₂×C₅×D₄	—	6	2	no	—
208	C₂×C₅×D₄	—	12	2	no	—
209	C₂×C₅×D₄	—	12	4	no	—
210	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
211	C₂²×C₄×C₅	—	3	2	no	—
212	C₂²×C₄×C₅	—	6	2	no	—
213	C₂²×C₄×C₅	—	6	1	no	—
214	C₂²×C₄×C₅	—	6	1	no	—
215	C₅×D₈	—	6	2	no	—
216	C₅×D₈	—	12	2	no	—
217	C₂×C₅×C₈	—	3	2	no	—
218	C₂×C₅×C₈	—	12	2	no	—
219	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
220	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
221	C₅×C₄⋊C₄	—	2	1	no	—
222	C₅×C₄⋊C₄	—	3	1	no	—
223	C₅×C₄⋊C₄	—	3	1	no	—
224	C₅×C₄⋊C₄	—	3	1	no	—
225	C₅×C₄⋊C₄	—	6	2	no	—
226	C₅×C₄⋊C₄	C₂²×D₃	1	2	yes	—
227	C₅×C₂²⋊C₄	—	3	2	no	—
228	C₅×C₂²⋊C₄	—	6	2	no	—
229	C₅×C₂²⋊C₄	—	6	2	no	—
230	C₅×C₂²⋊C₄	—	6	1	no	—
231	C₅×C₂²⋊C₄	—	6	1	no	—
232	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
233	#547	—	3	1	no	—
234	D₄²	—	3	1	no	—
235	#486	C₅×D₃	1	1	yes	characteristic, complemented
236	#480	—	3	2	no	—
237	#474	—	3	1	no	—
238	#466	—	3	2	no	—
239	#459	—	3	1	no	—
240	#447	—	3	2	no	—
241	C₄×D₈	—	3	1	no	—
242	C₈×D₄	—	3	1	no	—
243	C₂×#102	—	3	2	no	—
244	C₂²×C₃×C₅	—	4	2	no	—
245	C₂²×C₃×C₅	—	4	2	no	—
246	C₂²×C₃×C₅	—	4	2	no	—
247	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₅×D₃	—	2	2	no	—
249	C₂×C₅×D₃	—	2	2	no	—
250	C₂×C₅×D₃	—	2	2	no	—
251	C₂×C₅×D₃	—	4	1	no	—
252	C₂×C₅×D₃	—	4	1	no	—
253	C₂×C₅×D₃	—	4	1	no	—
254	C₂×C₅×D₃	—	4	1	no	—
255	C₂×C₅×D₃	—	4	1	no	—
256	C₂×C₅×D₃	—	8	8	no	—
257	C₂×C₅×D₃	C₂×D₈	1	4	yes	—
258	C₃×C₄×C₅	—	2	1	no	—
259	C₃×C₄×C₅	—	4	2	no	—
260	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
261	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
262	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
263	C₅×Dic₃	—	2	2	no	—
264	C₅×Dic₃	—	4	1	no	—
265	C₅×Dic₃	—	4	1	no	—
266	C₂³×D₃	—	2	2	no	—
267	C₂×C₃×Q₈	—	2	1	no	—
268	C₂×C₃×D₄	—	2	1	no	—
269	C₂×C₃×D₄	—	2	1	no	—
270	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
271	C₂×#67	—	2	2	no	—
272	C₂×#67	—	2	2	no	—
273	#239	—	4	4	no	—
274	D₃×D₄	—	2	4	no	—
275	D₃×D₄	—	4	2	no	—
276	D₃×D₄	—	4	4	no	—
277	C₂×D₁₂	—	2	2	no	—
278	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
279	C₂×D₁₂	C₂³×C₅	1	1	yes	characteristic
280	C₂×D₁₂	C₂³×C₅	1	1	yes	characteristic
281	C₂×C₄×D₃	—	2	2	no	—
282	C₂×C₄×D₃	—	2	1	no	—
283	C₂×C₄×D₃	—	2	1	no	—
284	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
285	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
286	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
287	C₃×C₄⋊C₄	—	2	1	no	—
288	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
289	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
290	#217	—	2	2	no	—
291	#213	—	4	2	no	—
292	(C₂×D₃)⋊C₄	—	2	1	no	—
293	(C₂×D₃)⋊C₄	—	2	1	no	—
294	(C₂×D₃)⋊C₄	—	2	2	no	—
295	(C₂×D₃)⋊C₄	C₅×D₄	1	2	yes	—
296	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
297	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
298	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
299	Dic₃⋊C₄	—	2	2	no	—
300	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
301	D₂₄	—	2	2	no	—
302	D₃×C₈	—	4	2	no	—
303	C₂³×C₅	—	2	2	no	—
304	C₂³×C₅	—	3	2	no	—
305	C₂³×C₅	—	6	1	no	—
306	C₂³×C₅	—	6	1	no	—
307	C₂³×C₅	—	12	4	no	—
308	C₂³×C₅	—	12	4	no	—
309	C₂³×C₅	—	24	2	no	—
310	C₅×Q₈	—	4	2	no	—
311	C₅×D₄	—	2	2	no	—
312	C₅×D₄	—	2	2	no	—
313	C₅×D₄	—	3	4	no	—
314	C₅×D₄	—	3	2	no	—
315	C₅×D₄	—	4	2	no	—
316	C₅×D₄	—	6	1	no	—
317	C₅×D₄	—	6	1	no	—
318	C₅×D₄	—	6	1	no	—
319	C₅×D₄	—	8	1	no	—
320	C₅×D₄	—	12	4	no	—
321	C₅×D₄	—	12	4	no	—
322	C₅×D₄	—	12	4	no	—
323	C₂×C₄×C₅	—	2	2	no	—
324	C₂×C₄×C₅	—	3	2	no	—
325	C₂×C₄×C₅	—	6	2	no	—
326	C₂×C₄×C₅	—	6	2	no	—
327	C₂×C₄×C₅	—	6	1	no	—
328	C₂×C₄×C₅	—	6	1	no	—
329	C₂×C₄×C₅	—	12	4	no	—
330	C₂×C₄×C₅	—	12	2	no	—
331	C₂×C₄×C₅	—	12	2	no	—
332	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
333	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
334	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
335	C₅×C₈	—	2	1	no	—
336	C₅×C₈	—	2	1	no	—
337	C₅×C₈	—	6	2	no	—
338	C₂×#41	—	6	1	no	—
339	C₂²×D₄	—	3	2	no	—
340	C₂²×D₄	—	6	1	no	—
341	C₂×D₈	—	3	2	no	—
342	C₂×D₈	—	3	1	no	—
343	C₂²×C₈	—	3	2	no	—
344	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
345	#127	C₂×C₅×D₃	1	1	yes	characteristic
346	#123	—	6	2	no	—
347	C₂²⋊Q₈	—	6	1	no	—
348	#121	—	3	2	no	—
349	#121	—	3	2	no	—
350	#121	—	6	1	no	—
351	#120	—	6	2	no	—
352	C₄×D₄	—	3	1	no	—
353	C₄×D₄	—	3	1	no	—
354	C₄×D₄	—	3	1	no	—
355	C₂×C₄⋊C₄	—	3	2	no	—
356	#107	—	3	1	no	—
357	#107	—	3	2	no	—
358	C₄⋊C₈	—	3	1	no	—
359	#102	—	3	2	no	—
360	#102	—	3	1	no	—
361	#102	—	3	1	no	—
362	#102	—	3	2	no	—
363	#102	C₂×C₅×D₃	1	2	yes	complemented
364	#102	C₂×C₅×D₃	1	2	yes	complemented
365	C₂²⋊C₈	—	3	2	no	—
366	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
367	C₂×C₃×C₅	—	8	2	no	—
368	C₂×C₃×C₅	#546	1	1	yes	characteristic
369	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
370	C₂×C₃×C₅	#577	1	1	yes	characteristic
371	C₅×D₃	—	2	4	no	—
372	C₅×D₃	—	4	2	no	—
373	C₅×D₃	—	8	1	no	—
374	C₂³×C₃	—	2	2	no	—
375	C₂²×D₃	—	2	1	no	—
376	C₂²×D₃	—	2	1	no	—
377	C₂²×D₃	—	4	4	no	—
378	C₂²×D₃	—	4	4	no	—
379	C₂²×D₃	—	8	2	no	—
380	C₂²×D₃	C₂×C₅×D₄	1	2	yes	—
381	C₃×Q₈	—	4	2	no	—
382	C₃×D₄	—	2	2	no	—
383	C₃×D₄	—	2	2	no	—
384	C₃×D₄	—	4	2	no	—
385	C₃×D₄	—	8	1	no	—
386	C₂×C₃×C₄	—	2	2	no	—
387	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
388	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
389	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
390	#67	—	4	4	no	—
391	#67	—	4	4	no	—
392	C₂×Dic₃	—	2	1	no	—
393	C₂×Dic₃	—	2	1	no	—
394	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
395	D₁₂	—	2	1	no	—
396	D₁₂	—	2	1	no	—
397	D₁₂	—	2	1	no	—
398	D₁₂	—	4	4	no	—
399	D₁₂	C₅×D₈	1	4	yes	complemented
400	D₁₂	C₅×#41	1	2	yes	complemented
401	C₄×D₃	—	2	2	no	—
402	C₄×D₃	—	2	2	no	—
403	C₄×D₃	—	4	4	no	—
404	C₄×D₃	—	4	2	no	—
405	C₄×D₃	—	4	2	no	—
406	C₃×C₈	—	2	1	no	—
407	C₃×C₈	—	2	1	no	—
408	C₃⋊C₈	—	2	2	no	—
409	C₂²×C₅	—	3	4	no	—
410	C₂²×C₅	—	4	2	no	—
411	C₂²×C₅	—	4	2	no	—
412	C₂²×C₅	—	4	2	no	—
413	C₂²×C₅	—	6	2	no	—
414	C₂²×C₅	—	6	2	no	—
415	C₂²×C₅	—	6	2	no	—
416	C₂²×C₅	—	12	1	no	—
417	C₂²×C₅	—	12	1	no	—
418	C₂²×C₅	—	12	1	no	—
419	C₂²×C₅	—	12	1	no	—
420	C₂²×C₅	—	12	1	no	—
421	C₂²×C₅	—	24	8	no	—
422	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
423	C₄×C₅	—	2	1	no	—
424	C₄×C₅	—	4	2	no	—
425	C₄×C₅	—	6	2	no	—
426	C₄×C₅	—	12	1	no	—
427	C₄×C₅	—	12	1	no	—
428	C₄×C₅	D₃×D₈	1	2	yes	—
429	C₄×C₅	D₃×#41	1	1	yes	characteristic
430	C₄×C₅	#1009	1	1	yes	characteristic
431	C₂⁴	—	6	2	no	—
432	#41	—	12	4	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	3	1	no	—
437	C₂×D₄	—	3	1	no	—
438	C₂×D₄	—	3	1	no	—
439	C₂×D₄	—	6	2	no	—
440	C₂×D₄	—	6	2	no	—
441	C₂×D₄	—	6	4	no	—
442	C₂×D₄	—	6	2	no	—
443	C₂×D₄	—	12	2	no	—
444	C₂×D₄	—	12	4	no	—
445	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
446	C₂²×C₄	—	3	2	no	—
447	C₂²×C₄	—	6	2	no	—
448	C₂²×C₄	—	6	1	no	—
449	C₂²×C₄	—	6	1	no	—
450	D₈	—	6	2	no	—
451	D₈	—	12	2	no	—
452	C₂×C₈	—	3	2	no	—
453	C₂×C₈	—	12	2	no	—
454	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
455	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	3	1	no	—
458	C₄⋊C₄	—	3	1	no	—
459	C₄⋊C₄	—	3	1	no	—
460	C₄⋊C₄	—	6	2	no	—
461	C₄⋊C₄	C₂²×C₅×D₃	1	2	yes	—
462	C₂²⋊C₄	—	3	2	no	—
463	C₂²⋊C₄	—	6	2	no	—
464	C₂²⋊C₄	—	6	2	no	—
465	C₂²⋊C₄	—	6	1	no	—
466	C₂²⋊C₄	—	6	1	no	—
467	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
468	C₃×C₅	#3294	1	1	yes	characteristic, complemented, Hall
469	C₂²×C₃	—	4	2	no	—
470	C₂²×C₃	—	4	2	no	—
471	C₂²×C₃	—	4	2	no	—
472	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
473	C₂×D₃	—	2	2	no	—
474	C₂×D₃	—	2	2	no	—
475	C₂×D₃	—	2	2	no	—
476	C₂×D₃	—	4	1	no	—
477	C₂×D₃	—	4	1	no	—
478	C₂×D₃	—	4	1	no	—
479	C₂×D₃	—	4	1	no	—
480	C₂×D₃	—	4	1	no	—
481	C₂×D₃	—	8	8	no	—
482	C₂×D₃	C₂×C₅×D₈	1	4	yes	—
483	C₃×C₄	—	2	1	no	—
484	C₃×C₄	—	4	2	no	—
485	C₃×C₄	C₂×C₅×D₈	1	2	yes	—
486	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
487	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
488	Dic₃	—	2	2	no	—
489	Dic₃	—	4	1	no	—
490	Dic₃	—	4	1	no	—
491	C₂×C₅	—	6	4	no	—
492	C₂×C₅	—	8	2	no	—
493	C₂×C₅	—	12	2	no	—
494	C₂×C₅	—	24	1	no	—
495	C₂×C₅	#5676	1	1	yes	characteristic
496	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
497	C₂×C₅	#5762	1	1	yes	characteristic
498	C₂³	—	2	2	no	—
499	C₂³	—	3	2	no	—
500	C₂³	—	6	1	no	—
501	C₂³	—	6	1	no	—
502	C₂³	—	12	4	no	—
503	C₂³	—	12	4	no	—
504	C₂³	—	24	2	no	—
505	Q₈	—	4	2	no	—
506	D₄	—	2	2	no	—
507	D₄	—	2	2	no	—
508	D₄	—	3	4	no	—
509	D₄	—	3	2	no	—
510	D₄	—	4	2	no	—
511	D₄	—	6	1	no	—
512	D₄	—	6	1	no	—
513	D₄	—	6	1	no	—
514	D₄	—	8	1	no	—
515	D₄	—	12	4	no	—
516	D₄	—	12	4	no	—
517	D₄	—	12	4	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	3	2	no	—
520	C₂×C₄	—	6	2	no	—
521	C₂×C₄	—	6	2	no	—
522	C₂×C₄	—	6	1	no	—
523	C₂×C₄	—	6	1	no	—
524	C₂×C₄	—	12	4	no	—
525	C₂×C₄	—	12	2	no	—
526	C₂×C₄	—	12	2	no	—
527	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
528	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
529	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	6	2	no	—
533	C₂×C₃	—	8	2	no	—
534	C₂×C₃	C₅×#546	1	1	yes	characteristic
535	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
536	C₂×C₃	C₅×#577	1	1	yes	characteristic
537	D₃	—	2	4	no	—
538	D₃	—	4	2	no	—
539	D₃	—	8	1	no	—
540	C₅	#81576	1	1	yes	characteristic, direct product factor, Sylow
541	C₂²	—	3	4	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	4	2	no	—
545	C₂²	—	6	2	no	—
546	C₂²	—	6	2	no	—
547	C₂²	—	6	2	no	—
548	C₂²	—	12	1	no	—
549	C₂²	—	12	1	no	—
550	C₂²	—	12	1	no	—
551	C₂²	—	12	1	no	—
552	C₂²	—	12	1	no	—
553	C₂²	—	24	8	no	—
554	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
555	C₄	—	2	1	no	—
556	C₄	—	4	2	no	—
557	C₄	—	6	2	no	—
558	C₄	—	12	1	no	—
559	C₄	—	12	1	no	—
560	C₄	C₅×D₃×D₈	1	2	yes	—
561	C₄	C₅×D₃×#41	1	1	yes	characteristic
562	C₄	C₅×#1009	1	1	yes	characteristic
563	C₃	C₅×#3294	1	1	yes	characteristic, complemented, Sylow
564	C₂	—	6	4	no	—
565	C₂	—	8	2	no	—
566	C₂	—	12	2	no	—
567	C₂	—	24	1	no	—
568	C₂	C₅×#5676	1	1	yes	characteristic
569	C₂	C₂×C₅×D₃×D₈	1	1	yes	characteristic
570	C₂	C₅×#5762	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×#486	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₅×#486	D₃
hypocenter	C₃	C₅×#3294
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#486

Complements

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

N	H ≅ G/N	type
1. C₅×#5726	C₂	semidirect
2. C₅×#5717	C₂	semidirect
3. C₃×C₅×#486	C₂	semidirect
4. C₅×#5087	C₂	semidirect
5. C₅×#5072	C₂	semidirect
6. C₅×#4775	C₂	semidirect
7. C₅×#4774	C₂	semidirect
8. C₅×#4773	C₂	semidirect
9. C₅×D₃×#102	C₂	semidirect
10. C₄×C₅×D₂₄	C₂	semidirect
11. C₅×C₈×D₁₂	C₂	semidirect
16. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
22. C₂×C₅×#213	C₂²	semidirect
24. C₂×C₅×D₃×C₈	C₂²	semidirect
25. C₅×#895	C₂²	semidirect
27. C₅×D₃×C₄⋊C₄	C₂²	semidirect
30. C₄×C₅×D₁₂	C₂²	semidirect
31. C₃×C₅×#102	C₂²	semidirect
32. C₃×C₅×#102	C₂²	semidirect
33. C₃×C₄×C₅×C₈	C₂²	semidirect
34. C₅×#832	C₂²	semidirect
36. C₅×#821	C₂²	semidirect
37. C₅×(C₂×D₃)⋊C₈	C₂²	semidirect
38. C₅×#818	C₂²	semidirect
40. C₅×#807	C₂²	semidirect
41. C₅×C₄⋊(C₃⋊C₈)	C₂²	semidirect
42. #81576	C₅	direct (N × H)
45. C₅×#486	D₃	semidirect
91. #5726	C₂×C₅	semidirect
92. #5717	C₂×C₅	semidirect
93. C₃×#486	C₂×C₅	semidirect
94. #5087	C₂×C₅	semidirect
95. #5072	C₂×C₅	semidirect
96. #4775	C₂×C₅	semidirect
97. #4774	C₂×C₅	semidirect
98. #4773	C₂×C₅	semidirect
99. D₃×#102	C₂×C₅	semidirect
100. C₄×D₂₄	C₂×C₅	semidirect
101. C₈×D₁₂	C₂×C₅	semidirect
108. C₅×C₄⋊Q₈	C₂×D₃	semidirect
127. C₅×#102	C₂×D₃	semidirect
128. C₅×#102	C₂×D₃	semidirect
130. C₄×C₅×C₈	C₂×D₃	semidirect
157. C₅×D₁₂	D₈	semidirect
158. C₅×D₁₂	#41	semidirect
170. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
176. C₂×#213	C₂²×C₅	semidirect
178. C₂×D₃×C₈	C₂²×C₅	semidirect
179. #895	C₂²×C₅	semidirect
181. D₃×C₄⋊C₄	C₂²×C₅	semidirect
184. C₄×D₁₂	C₂²×C₅	semidirect
185. C₃×#102	C₂²×C₅	semidirect
186. C₃×#102	C₂²×C₅	semidirect
187. C₃×C₄×C₈	C₂²×C₅	semidirect
188. #832	C₂²×C₅	semidirect
190. #821	C₂²×C₅	semidirect
191. (C₂×D₃)⋊C₈	C₂²×C₅	semidirect
192. #818	C₂²×C₅	semidirect
194. #807	C₂²×C₅	semidirect
195. C₄⋊(C₃⋊C₈)	C₂²×C₅	semidirect
235. #486	C₅×D₃	semidirect
344. C₄⋊Q₈	C₂×C₅×D₃	semidirect
363. #102	C₂×C₅×D₃	semidirect
364. #102	C₂×C₅×D₃	semidirect
366. C₄×C₈	C₂×C₅×D₃	semidirect
399. D₁₂	C₅×D₈	semidirect
400. D₁₂	C₅×#41	semidirect
468. C₃×C₅	#3294	semidirect
540. C₅	#81576	direct (N × H)
563. C₃	C₅×#3294	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5726	2
C₅×#5717	2
C₃×C₅×#486	2
C₅×#5087	2
C₅×#5072	2
C₅×#4775	2
C₅×#4774	2
C₅×#4773	2
C₅×D₃×#102	2
C₄×C₅×D₂₄	2
C₅×C₈×D₁₂	2
C₅×#3294	3
#81576	5

Subgroups — C5×#81576 #14458290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81576 #14458290