Subgroups — C₅×#80049 #14459100

Order 1920.

840 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×#67	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5611	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#448	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#835	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5171	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#5170	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#5162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×Dic₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#5041	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4805	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×D₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#4792	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#4791	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#67⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#4721	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#3298	—	3	1	no	maximal, Hall
17	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×D₃×Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₅×#237	—	2	1	no	—
20	C₂×C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic
22	C₃×C₅×#121	C₂²	1	1	yes	characteristic
23	C₅×#946	—	2	1	no	—
24	C₅×#946	C₂²	1	1	yes	characteristic
25	C₅×Q₈×Dic₃	C₂²	1	1	yes	characteristic
26	C₅×#944	—	2	1	no	—
27	C₅×#944	C₂²	1	1	yes	characteristic
28	C₅×#938	C₂²	1	1	yes	characteristic
29	C₅×#935	—	2	1	no	—
30	C₅×D₄×Dic₃	C₂²	1	1	yes	characteristic
31	C₂×C₅×#214	C₂²	1	1	yes	characteristic
32	C₄×C₅×#67	—	2	1	no	—
33	C₄×C₅×#67	C₂²	1	1	yes	characteristic
34	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
35	C₅×#924	—	2	1	no	—
36	C₅×#924	C₂²	1	1	yes	characteristic
37	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
38	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
39	C₅×D₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
40	C₅×#888	C₂²	1	1	yes	characteristic
41	C₅×#883	—	2	1	no	—
42	C₅×#878	—	2	1	no	—
43	C₃×C₅×#103	C₂²	1	1	yes	characteristic
44	C₃×C₅×#103	C₂²	1	1	yes	characteristic
45	C₃×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
46	C₅×#835	C₂²	1	1	yes	characteristic
47	C₅×#835	C₂²	1	1	yes	characteristic
48	C₅×#835	C₂²	1	1	yes	characteristic
49	C₅×#835	C₂²	1	1	yes	characteristic
50	C₅×#832	C₂²	1	1	yes	characteristic
51	C₅×(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
52	C₅×#817	C₂²	1	1	yes	characteristic
53	C₅×#817	C₂²	1	1	yes	characteristic
54	C₅×#816	C₂²	1	1	yes	characteristic
55	C₅×#816	C₂²	1	1	yes	characteristic
56	C₅×Dic₃⋊C₈	C₂²	1	1	yes	characteristic
57	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
58	C₅×#810	C₂²	1	1	yes	characteristic
59	C₅×#808	C₂²	1	1	yes	characteristic
60	#80049	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
61	C₅×D₄×Q₈	—	3	1	no	—
62	C₅×#547	—	3	1	no	—
63	C₅×#487	—	3	1	no	—
64	C₅×#483	—	3	1	no	—
65	C₅×#483	—	3	1	no	—
66	C₅×#475	—	3	1	no	—
67	C₅×#465	—	3	1	no	—
68	C₅×#465	—	3	1	no	—
69	C₅×#461	—	3	1	no	—
70	C₅×#448	—	3	1	no	—
71	C₅×#448	D₃	1	1	yes	characteristic, complemented
72	C₄×C₅×SD₁₆	—	3	1	no	—
73	C₅×C₈×D₄	—	3	1	no	—
74	C₂×C₅×#103	—	3	1	no	—
75	C₂×C₅×#103	—	3	1	no	—
76	C₂×C₃×C₅×Q₈	—	2	1	no	—
77	C₂×C₃×C₅×Q₈	—	2	2	no	—
78	C₂×C₃×C₅×Q₈	—	4	1	no	—
79	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
80	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
81	C₂×C₃×C₅×D₄	—	2	1	no	—
82	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
83	C₂²×C₃×C₄×C₅	—	2	1	no	—
84	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
85	C₂×C₅×#67	—	2	1	no	—
86	C₂×C₅×#67	D₄	1	1	yes	characteristic
87	C₂²×C₅×Dic₃	—	2	1	no	—
88	C₂²×C₅×Dic₃	—	2	1	no	—
89	C₅×D₃×Q₈	—	2	2	no	—
90	C₅×D₃×Q₈	—	4	1	no	—
91	C₅×#237	—	4	4	no	—
92	C₂×C₄×C₅×D₃	—	2	1	no	—
93	C₂×C₄×C₅×D₃	—	2	1	no	—
94	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
95	C₂×C₅×Dic₆	—	2	1	no	—
96	C₂×C₅×Dic₆	—	2	1	no	—
97	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
98	C₃×C₅×SD₁₆	—	2	2	no	—
99	C₃×C₅×SD₁₆	—	4	1	no	—
100	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
101	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
102	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
103	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
104	C₅×#217	—	2	1	no	—
105	C₅×#217	—	2	1	no	—
106	C₅×#217	—	2	1	no	—
107	C₅×#217	D₄	1	1	yes	characteristic
108	C₅×#214	—	4	1	no	—
109	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
110	C₅×(C₂×D₃)⋊C₄	D₄	1	1	yes	characteristic
111	C₅×C₄⋊Dic₃	—	2	1	no	—
112	C₅×C₄⋊Dic₃	—	2	1	no	—
113	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
114	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
115	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
116	C₅×Dic₃⋊C₄	—	2	1	no	—
117	C₅×Dic₃⋊C₄	—	2	1	no	—
118	C₅×Dic₃⋊C₄	—	2	1	no	—
119	C₅×Dic₃⋊C₄	—	2	1	no	—
120	C₅×Dic₃⋊C₄	—	2	1	no	—
121	C₅×Dic₃⋊C₄	—	2	1	no	—
122	C₅×Dic₃⋊C₄	D₄	1	1	yes	characteristic
123	C₄×C₅×Dic₃	—	2	1	no	—
124	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
125	C₂×C₅×C₃⋊C₈	—	4	1	no	—
126	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
127	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
128	C₅×D₃×C₈	—	4	1	no	—
129	Q₈×#67	C₂×C₅	1	1	yes	characteristic, complemented
130	#5611	C₂×C₅	1	1	yes	characteristic
131	C₃×#448	C₂×C₅	1	1	yes	characteristic, complemented
132	C₂×#835	C₂×C₅	1	1	yes	characteristic, complemented
133	#5171	C₂×C₅	1	1	yes	characteristic, complemented
134	#5170	C₂×C₅	1	1	yes	characteristic, complemented
135	#5162	C₂×C₅	1	1	yes	characteristic, complemented
136	Dic₃×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
137	#5041	C₂×C₅	1	1	yes	characteristic, complemented
138	#4805	C₂×C₅	1	1	yes	characteristic, complemented
139	D₃×#103	C₂×C₅	1	1	yes	characteristic, complemented
140	#4792	C₂×C₅	1	1	yes	characteristic, complemented
141	#4791	C₂×C₅	1	1	yes	characteristic, complemented
142	#67⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
143	#4721	C₂×C₅	1	1	yes	characteristic, complemented
144	C₂×C₅×#41	—	6	1	no	—
145	C₂²×C₅×Q₈	—	3	1	no	—
146	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
147	C₂×C₅×SD₁₆	—	3	1	no	—
148	C₂×C₅×SD₁₆	C₂×D₃	1	1	yes	characteristic
149	C₂×C₅×SD₁₆	C₂×D₃	1	1	yes	characteristic
150	C₂²×C₅×C₈	—	3	1	no	—
151	C₂²×C₅×C₈	—	3	1	no	—
152	C₅×C₄⋊Q₈	—	3	1	no	—
153	C₅×C₄⋊Q₈	—	3	1	no	—
154	C₅×C₄⋊Q₈	—	6	1	no	—
155	C₅×#123	—	6	1	no	—
156	C₅×#123	—	6	1	no	—
157	C₅×C₂²⋊Q₈	—	3	1	no	—
158	C₅×C₂²⋊Q₈	—	3	1	no	—
159	C₅×C₂²⋊Q₈	—	6	1	no	—
160	C₅×C₂²⋊Q₈	—	6	1	no	—
161	C₅×C₂²⋊Q₈	—	6	1	no	—
162	C₅×#121	—	3	1	no	—
163	C₅×#121	C₂×D₃	1	1	yes	characteristic
164	C₄×C₅×Q₈	—	3	1	no	—
165	C₄×C₅×D₄	—	3	1	no	—
166	C₄×C₅×D₄	—	3	1	no	—
167	C₄×C₅×D₄	—	6	1	no	—
168	C₂×C₅×C₄⋊C₄	—	3	1	no	—
169	C₂×C₅×C₄⋊C₄	—	3	1	no	—
170	C₅×#106	—	3	1	no	—
171	C₅×#106	—	3	1	no	—
172	C₅×#106	—	3	1	no	—
173	C₅×C₄⋊C₈	—	3	1	no	—
174	C₅×#103	—	3	1	no	—
175	C₅×#103	—	3	1	no	—
176	C₅×#103	—	3	1	no	—
177	C₅×#103	—	3	1	no	—
178	C₅×#103	—	3	1	no	—
179	C₅×#103	—	3	1	no	—
180	C₅×#103	—	3	1	no	—
181	C₅×#103	C₂×D₃	1	1	yes	characteristic
182	C₅×#103	C₂×D₃	1	1	yes	characteristic
183	C₅×#102	—	3	1	no	—
184	C₅×C₂²⋊C₈	—	3	1	no	—
185	C₅×C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
186	C₄×C₅×C₈	—	3	1	no	—
187	#3298	—	3	1	no	Sylow
188	C₂³×C₃×C₅	—	2	1	no	—
189	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
191	C₃×C₅×Q₈	—	2	1	no	—
192	C₃×C₅×Q₈	—	2	1	no	—
193	C₃×C₅×Q₈	—	2	1	no	—
194	C₃×C₅×Q₈	—	4	2	no	—
195	C₃×C₅×Q₈	#41	1	2	yes	complemented
196	C₃×C₅×D₄	—	2	1	no	—
197	C₃×C₅×D₄	—	2	1	no	—
198	C₃×C₅×D₄	—	4	2	no	—
199	C₂×C₃×C₄×C₅	—	2	1	no	—
200	C₂×C₃×C₄×C₅	—	2	1	no	—
201	C₂×C₃×C₄×C₅	—	2	1	no	—
202	C₂×C₃×C₄×C₅	—	2	1	no	—
203	C₂×C₃×C₄×C₅	—	2	1	no	—
204	C₂×C₃×C₄×C₅	—	4	2	no	—
205	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
206	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
207	C₅×#67	—	4	2	no	—
208	C₅×#67	SD₁₆	1	4	yes	complemented
209	C₂×C₅×Dic₃	—	2	1	no	—
210	C₂×C₅×Dic₃	—	2	1	no	—
211	C₂×C₅×Dic₃	—	2	1	no	—
212	C₂×C₅×Dic₃	—	2	1	no	—
213	C₂×C₅×Dic₃	—	4	2	no	—
214	C₂×C₅×Dic₃	—	4	2	no	—
215	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
216	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
217	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
218	C₄×C₅×D₃	—	2	1	no	—
219	C₄×C₅×D₃	—	2	1	no	—
220	C₄×C₅×D₃	—	4	2	no	—
221	C₄×C₅×D₃	—	4	2	no	—
222	C₅×Dic₆	—	2	1	no	—
223	C₅×Dic₆	—	2	1	no	—
224	C₅×Dic₆	—	4	2	no	—
225	C₅×Dic₆	—	4	2	no	—
226	C₃×C₅×C₈	—	2	1	no	—
227	C₃×C₅×C₈	—	2	1	no	—
228	C₅×C₃⋊C₈	—	2	1	no	—
229	C₅×C₃⋊C₈	—	2	1	no	—
230	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
231	C₂×D₃×Q₈	C₂²×C₅	1	1	yes	characteristic
232	C₂×#237	—	2	1	no	—
233	C₂×C₃×SD₁₆	C₂²×C₅	1	1	yes	characteristic
234	C₂×C₃×SD₁₆	C₂²×C₅	1	1	yes	characteristic
235	C₃×#121	C₂²×C₅	1	1	yes	characteristic
236	#946	—	2	1	no	—
237	#946	C₂²×C₅	1	1	yes	characteristic
238	Q₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
239	#944	—	2	1	no	—
240	#944	C₂²×C₅	1	1	yes	characteristic
241	#938	C₂²×C₅	1	1	yes	characteristic
242	#935	—	2	1	no	—
243	D₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
244	C₂×#214	C₂²×C₅	1	1	yes	characteristic
245	C₄×#67	—	2	1	no	—
246	C₄×#67	C₂²×C₅	1	1	yes	characteristic
247	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
248	#924	—	2	1	no	—
249	#924	C₂²×C₅	1	1	yes	characteristic
250	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
251	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
252	D₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
253	#888	C₂²×C₅	1	1	yes	characteristic
254	#883	—	2	1	no	—
255	#878	—	2	1	no	—
256	C₃×#103	C₂²×C₅	1	1	yes	characteristic
257	C₃×#103	C₂²×C₅	1	1	yes	characteristic
258	C₃×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
259	#835	C₂²×C₅	1	1	yes	characteristic
260	#835	C₂²×C₅	1	1	yes	characteristic
261	#835	C₂²×C₅	1	1	yes	characteristic
262	#835	C₂²×C₅	1	1	yes	characteristic
263	#832	C₂²×C₅	1	1	yes	characteristic
264	(C₂×D₃)⋊C₈	C₂²×C₅	1	1	yes	characteristic
265	#817	C₂²×C₅	1	1	yes	characteristic
266	#817	C₂²×C₅	1	1	yes	characteristic
267	#816	C₂²×C₅	1	1	yes	characteristic
268	#816	C₂²×C₅	1	1	yes	characteristic
269	Dic₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
270	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
271	#810	C₂²×C₅	1	1	yes	characteristic
272	#808	C₂²×C₅	1	1	yes	characteristic
273	C₅×#41	—	12	4	no	—
274	C₂×C₅×Q₈	—	2	1	no	—
275	C₂×C₅×Q₈	—	2	2	no	—
276	C₂×C₅×Q₈	—	3	1	no	—
277	C₂×C₅×Q₈	—	4	1	no	—
278	C₂×C₅×Q₈	—	6	1	no	—
279	C₂×C₅×Q₈	—	6	1	no	—
280	C₂×C₅×Q₈	—	6	2	no	—
281	C₂×C₅×Q₈	—	12	1	no	—
282	C₂×C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
283	C₂×C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
284	C₂×C₅×D₄	—	2	1	no	—
285	C₂×C₅×D₄	—	3	1	no	—
286	C₂×C₅×D₄	—	6	1	no	—
287	C₂×C₅×D₄	C₂²×D₃	1	1	yes	characteristic
288	C₂²×C₄×C₅	—	2	1	no	—
289	C₂²×C₄×C₅	—	3	1	no	—
290	C₂²×C₄×C₅	—	6	1	no	—
291	C₂²×C₄×C₅	—	6	1	no	—
292	C₂²×C₄×C₅	—	6	1	no	—
293	C₂²×C₄×C₅	—	6	1	no	—
294	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
295	C₅×SD₁₆	—	2	2	no	—
296	C₅×SD₁₆	—	4	1	no	—
297	C₅×SD₁₆	—	12	1	no	—
298	C₂×C₅×C₈	—	3	1	no	—
299	C₂×C₅×C₈	—	3	1	no	—
300	C₂×C₅×C₈	—	12	1	no	—
301	C₂×C₅×C₈	—	12	1	no	—
302	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
303	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
304	C₅×C₄⋊C₄	—	3	1	no	—
305	C₅×C₄⋊C₄	—	3	1	no	—
306	C₅×C₄⋊C₄	—	3	1	no	—
307	C₅×C₄⋊C₄	—	3	1	no	—
308	C₅×C₄⋊C₄	—	6	1	no	—
309	C₅×C₄⋊C₄	—	6	1	no	—
310	C₅×C₄⋊C₄	—	6	1	no	—
311	C₅×C₄⋊C₄	—	6	1	no	—
312	C₅×C₄⋊C₄	—	6	1	no	—
313	C₅×C₄⋊C₄	—	6	1	no	—
314	C₅×C₄⋊C₄	—	6	1	no	—
315	C₅×C₄⋊C₄	—	6	1	no	—
316	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
317	C₅×C₂²⋊C₄	—	2	1	no	—
318	C₅×C₂²⋊C₄	—	3	1	no	—
319	C₅×C₂²⋊C₄	—	3	1	no	—
320	C₅×C₂²⋊C₄	—	6	1	no	—
321	C₅×C₂²⋊C₄	—	6	1	no	—
322	C₅×C₂²⋊C₄	—	6	1	no	—
323	C₅×C₂²⋊C₄	—	6	1	no	—
324	C₄²×C₅	—	3	1	no	—
325	C₄²×C₅	—	6	1	no	—
326	D₄×Q₈	—	3	1	no	—
327	#547	—	3	1	no	—
328	#487	—	3	1	no	—
329	#483	—	3	1	no	—
330	#483	—	3	1	no	—
331	#475	—	3	1	no	—
332	#465	—	3	1	no	—
333	#465	—	3	1	no	—
334	#461	—	3	1	no	—
335	#448	—	3	1	no	—
336	#448	C₅×D₃	1	1	yes	characteristic, complemented
337	C₄×SD₁₆	—	3	1	no	—
338	C₈×D₄	—	3	1	no	—
339	C₂×#103	—	3	1	no	—
340	C₂×#103	—	3	1	no	—
341	C₂²×C₃×C₅	—	2	1	no	—
342	C₂²×C₃×C₅	—	2	1	no	—
343	C₂²×C₃×C₅	—	4	1	no	—
344	C₂²×C₃×C₅	—	4	1	no	—
345	C₂²×C₃×C₅	—	4	1	no	—
346	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
347	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
348	C₂×C₅×D₃	—	2	1	no	—
349	C₂×C₅×D₃	—	2	1	no	—
350	C₂×C₅×D₃	C₂×SD₁₆	1	2	yes	—
351	C₃×C₄×C₅	—	2	2	no	—
352	C₃×C₄×C₅	—	2	1	no	—
353	C₃×C₄×C₅	—	4	1	no	—
354	C₃×C₄×C₅	—	4	1	no	—
355	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
356	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
357	C₅×Dic₃	—	2	1	no	—
358	C₅×Dic₃	—	2	1	no	—
359	C₅×Dic₃	—	4	1	no	—
360	C₅×Dic₃	—	4	1	no	—
361	C₅×Dic₃	—	4	1	no	—
362	C₅×Dic₃	—	4	1	no	—
363	C₅×Dic₃	C₂×SD₁₆	1	2	yes	—
364	C₂×C₃×Q₈	—	2	1	no	—
365	C₂×C₃×Q₈	—	2	2	no	—
366	C₂×C₃×Q₈	—	4	1	no	—
367	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
368	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
369	C₂×C₃×D₄	—	2	1	no	—
370	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
371	C₂²×C₃×C₄	—	2	1	no	—
372	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
373	C₂×#67	—	2	1	no	—
374	C₂×#67	C₅×D₄	1	1	yes	characteristic
375	C₂²×Dic₃	—	2	1	no	—
376	C₂²×Dic₃	—	2	1	no	—
377	D₃×Q₈	—	2	2	no	—
378	D₃×Q₈	—	4	1	no	—
379	#237	—	4	4	no	—
380	C₂×C₄×D₃	—	2	1	no	—
381	C₂×C₄×D₃	—	2	1	no	—
382	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
383	C₂×Dic₆	—	2	1	no	—
384	C₂×Dic₆	—	2	1	no	—
385	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
386	C₃×SD₁₆	—	2	2	no	—
387	C₃×SD₁₆	—	4	1	no	—
388	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
389	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
390	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
391	C₃×C₂²⋊C₄	—	2	1	no	—
392	#217	—	2	1	no	—
393	#217	—	2	1	no	—
394	#217	—	2	1	no	—
395	#217	C₅×D₄	1	1	yes	characteristic
396	#214	—	4	1	no	—
397	(C₂×D₃)⋊C₄	—	2	1	no	—
398	(C₂×D₃)⋊C₄	C₅×D₄	1	1	yes	characteristic
399	C₄⋊Dic₃	—	2	1	no	—
400	C₄⋊Dic₃	—	2	1	no	—
401	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
402	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
403	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
404	Dic₃⋊C₄	—	2	1	no	—
405	Dic₃⋊C₄	—	2	1	no	—
406	Dic₃⋊C₄	—	2	1	no	—
407	Dic₃⋊C₄	—	2	1	no	—
408	Dic₃⋊C₄	—	2	1	no	—
409	Dic₃⋊C₄	—	2	1	no	—
410	Dic₃⋊C₄	C₅×D₄	1	1	yes	characteristic
411	C₄×Dic₃	—	2	1	no	—
412	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
413	C₂×C₃⋊C₈	—	4	1	no	—
414	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
415	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
416	D₃×C₈	—	4	1	no	—
417	C₂³×C₅	—	2	1	no	—
418	C₂³×C₅	—	3	1	no	—
419	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
420	C₅×Q₈	—	2	1	no	—
421	C₅×Q₈	—	2	1	no	—
422	C₅×Q₈	—	2	1	no	—
423	C₅×Q₈	—	4	2	no	—
424	C₅×Q₈	—	6	1	no	—
425	C₅×Q₈	—	6	1	no	—
426	C₅×Q₈	—	12	2	no	—
427	C₅×Q₈	—	12	2	no	—
428	C₅×Q₈	#237	1	2	yes	complemented
429	C₅×D₄	—	2	1	no	—
430	C₅×D₄	—	2	1	no	—
431	C₅×D₄	—	3	4	no	—
432	C₅×D₄	—	4	2	no	—
433	C₅×D₄	—	12	2	no	—
434	C₂×C₄×C₅	—	2	1	no	—
435	C₂×C₄×C₅	—	2	1	no	—
436	C₂×C₄×C₅	—	2	1	no	—
437	C₂×C₄×C₅	—	2	1	no	—
438	C₂×C₄×C₅	—	2	1	no	—
439	C₂×C₄×C₅	—	3	1	no	—
440	C₂×C₄×C₅	—	3	1	no	—
441	C₂×C₄×C₅	—	3	1	no	—
442	C₂×C₄×C₅	—	4	2	no	—
443	C₂×C₄×C₅	—	6	1	no	—
444	C₂×C₄×C₅	—	6	1	no	—
445	C₂×C₄×C₅	—	6	1	no	—
446	C₂×C₄×C₅	—	6	1	no	—
447	C₂×C₄×C₅	—	6	1	no	—
448	C₂×C₄×C₅	—	6	1	no	—
449	C₂×C₄×C₅	—	12	2	no	—
450	C₂×C₄×C₅	—	12	2	no	—
451	C₂×C₄×C₅	—	12	2	no	—
452	C₂×C₄×C₅	—	12	2	no	—
453	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
454	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
455	C₅×C₈	—	2	1	no	—
456	C₅×C₈	—	2	1	no	—
457	C₅×C₈	—	6	1	no	—
458	C₅×C₈	—	6	1	no	—
459	C₂×#41	—	6	1	no	—
460	C₂²×Q₈	—	3	1	no	—
461	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
462	C₂×SD₁₆	—	3	1	no	—
463	C₂×SD₁₆	C₂×C₅×D₃	1	1	yes	characteristic
464	C₂×SD₁₆	C₂×C₅×D₃	1	1	yes	characteristic
465	C₂²×C₈	—	3	1	no	—
466	C₂²×C₈	—	3	1	no	—
467	C₄⋊Q₈	—	3	1	no	—
468	C₄⋊Q₈	—	3	1	no	—
469	C₄⋊Q₈	—	6	1	no	—
470	#123	—	6	1	no	—
471	#123	—	6	1	no	—
472	C₂²⋊Q₈	—	3	1	no	—
473	C₂²⋊Q₈	—	3	1	no	—
474	C₂²⋊Q₈	—	6	1	no	—
475	C₂²⋊Q₈	—	6	1	no	—
476	C₂²⋊Q₈	—	6	1	no	—
477	#121	—	3	1	no	—
478	#121	C₂×C₅×D₃	1	1	yes	characteristic
479	C₄×Q₈	—	3	1	no	—
480	C₄×D₄	—	3	1	no	—
481	C₄×D₄	—	3	1	no	—
482	C₄×D₄	—	6	1	no	—
483	C₂×C₄⋊C₄	—	3	1	no	—
484	C₂×C₄⋊C₄	—	3	1	no	—
485	#106	—	3	1	no	—
486	#106	—	3	1	no	—
487	#106	—	3	1	no	—
488	C₄⋊C₈	—	3	1	no	—
489	#103	—	3	1	no	—
490	#103	—	3	1	no	—
491	#103	—	3	1	no	—
492	#103	—	3	1	no	—
493	#103	—	3	1	no	—
494	#103	—	3	1	no	—
495	#103	—	3	1	no	—
496	#103	C₂×C₅×D₃	1	1	yes	characteristic
497	#103	C₂×C₅×D₃	1	1	yes	characteristic
498	#102	—	3	1	no	—
499	C₂²⋊C₈	—	3	1	no	—
500	C₂²⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
501	C₄×C₈	—	3	1	no	—
502	C₂×C₃×C₅	—	2	2	no	—
503	C₂×C₃×C₅	—	8	1	no	—
504	C₂×C₃×C₅	#546	1	1	yes	characteristic
505	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
506	C₂×C₃×C₅	#578	1	1	yes	characteristic
507	C₅×D₃	—	2	2	no	—
508	C₂³×C₃	—	2	1	no	—
509	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
510	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
511	C₃×Q₈	—	2	1	no	—
512	C₃×Q₈	—	2	1	no	—
513	C₃×Q₈	—	2	1	no	—
514	C₃×Q₈	—	4	2	no	—
515	C₃×Q₈	C₅×#41	1	2	yes	complemented
516	C₃×D₄	—	2	1	no	—
517	C₃×D₄	—	2	1	no	—
518	C₃×D₄	—	4	2	no	—
519	C₂×C₃×C₄	—	2	1	no	—
520	C₂×C₃×C₄	—	2	1	no	—
521	C₂×C₃×C₄	—	2	1	no	—
522	C₂×C₃×C₄	—	2	1	no	—
523	C₂×C₃×C₄	—	2	1	no	—
524	C₂×C₃×C₄	—	4	2	no	—
525	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
526	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
527	#67	—	4	2	no	—
528	#67	C₅×SD₁₆	1	4	yes	complemented
529	C₂×Dic₃	—	2	1	no	—
530	C₂×Dic₃	—	2	1	no	—
531	C₂×Dic₃	—	2	1	no	—
532	C₂×Dic₃	—	2	1	no	—
533	C₂×Dic₃	—	4	2	no	—
534	C₂×Dic₃	—	4	2	no	—
535	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
536	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
537	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
538	C₄×D₃	—	2	1	no	—
539	C₄×D₃	—	2	1	no	—
540	C₄×D₃	—	4	2	no	—
541	C₄×D₃	—	4	2	no	—
542	Dic₆	—	2	1	no	—
543	Dic₆	—	2	1	no	—
544	Dic₆	—	4	2	no	—
545	Dic₆	—	4	2	no	—
546	C₃×C₈	—	2	1	no	—
547	C₃×C₈	—	2	1	no	—
548	C₃⋊C₈	—	2	1	no	—
549	C₃⋊C₈	—	2	1	no	—
550	C₂²×C₅	—	2	1	no	—
551	C₂²×C₅	—	2	1	no	—
552	C₂²×C₅	—	3	2	no	—
553	C₂²×C₅	—	4	1	no	—
554	C₂²×C₅	—	4	1	no	—
555	C₂²×C₅	—	4	1	no	—
556	C₂²×C₅	—	6	1	no	—
557	C₂²×C₅	—	6	1	no	—
558	C₂²×C₅	D₃×SD₁₆	1	2	yes	—
559	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
560	C₄×C₅	—	2	1	no	—
561	C₄×C₅	—	2	2	no	—
562	C₄×C₅	—	3	2	no	—
563	C₄×C₅	—	4	1	no	—
564	C₄×C₅	—	4	1	no	—
565	C₄×C₅	—	6	1	no	—
566	C₄×C₅	—	6	1	no	—
567	C₄×C₅	—	12	1	no	—
568	C₄×C₅	—	12	1	no	—
569	C₄×C₅	—	12	1	no	—
570	C₄×C₅	—	12	1	no	—
571	C₄×C₅	C₂×#237	1	1	yes	characteristic
572	C₄×C₅	#1004	1	1	yes	characteristic
573	#41	—	12	4	no	—
574	C₂×Q₈	—	2	1	no	—
575	C₂×Q₈	—	2	2	no	—
576	C₂×Q₈	—	3	1	no	—
577	C₂×Q₈	—	4	1	no	—
578	C₂×Q₈	—	6	1	no	—
579	C₂×Q₈	—	6	1	no	—
580	C₂×Q₈	—	6	2	no	—
581	C₂×Q₈	—	12	1	no	—
582	C₂×Q₈	C₂²×C₅×D₃	1	1	yes	characteristic
583	C₂×Q₈	C₂²×C₅×D₃	1	1	yes	characteristic
584	C₂×D₄	—	2	1	no	—
585	C₂×D₄	—	3	1	no	—
586	C₂×D₄	—	6	1	no	—
587	C₂×D₄	C₂²×C₅×D₃	1	1	yes	characteristic
588	C₂²×C₄	—	2	1	no	—
589	C₂²×C₄	—	3	1	no	—
590	C₂²×C₄	—	6	1	no	—
591	C₂²×C₄	—	6	1	no	—
592	C₂²×C₄	—	6	1	no	—
593	C₂²×C₄	—	6	1	no	—
594	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
595	SD₁₆	—	2	2	no	—
596	SD₁₆	—	4	1	no	—
597	SD₁₆	—	12	1	no	—
598	C₂×C₈	—	3	1	no	—
599	C₂×C₈	—	3	1	no	—
600	C₂×C₈	—	12	1	no	—
601	C₂×C₈	—	12	1	no	—
602	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
603	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
604	C₄⋊C₄	—	3	1	no	—
605	C₄⋊C₄	—	3	1	no	—
606	C₄⋊C₄	—	3	1	no	—
607	C₄⋊C₄	—	3	1	no	—
608	C₄⋊C₄	—	6	1	no	—
609	C₄⋊C₄	—	6	1	no	—
610	C₄⋊C₄	—	6	1	no	—
611	C₄⋊C₄	—	6	1	no	—
612	C₄⋊C₄	—	6	1	no	—
613	C₄⋊C₄	—	6	1	no	—
614	C₄⋊C₄	—	6	1	no	—
615	C₄⋊C₄	—	6	1	no	—
616	C₄⋊C₄	C₂²×C₅×D₃	1	1	yes	characteristic
617	C₂²⋊C₄	—	2	1	no	—
618	C₂²⋊C₄	—	3	1	no	—
619	C₂²⋊C₄	—	3	1	no	—
620	C₂²⋊C₄	—	6	1	no	—
621	C₂²⋊C₄	—	6	1	no	—
622	C₂²⋊C₄	—	6	1	no	—
623	C₂²⋊C₄	—	6	1	no	—
624	C₄²	—	3	1	no	—
625	C₄²	—	6	1	no	—
626	C₃×C₅	#3298	1	1	yes	characteristic, complemented, Hall
627	C₂²×C₃	—	2	1	no	—
628	C₂²×C₃	—	2	1	no	—
629	C₂²×C₃	—	4	1	no	—
630	C₂²×C₃	—	4	1	no	—
631	C₂²×C₃	—	4	1	no	—
632	C₂²×C₃	C₂×C₅×SD₁₆	1	2	yes	—
633	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
634	C₂×D₃	—	2	1	no	—
635	C₂×D₃	—	2	1	no	—
636	C₂×D₃	C₂×C₅×SD₁₆	1	2	yes	—
637	C₃×C₄	—	2	2	no	—
638	C₃×C₄	—	2	1	no	—
639	C₃×C₄	—	4	1	no	—
640	C₃×C₄	—	4	1	no	—
641	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
642	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
643	Dic₃	—	2	1	no	—
644	Dic₃	—	2	1	no	—
645	Dic₃	—	4	1	no	—
646	Dic₃	—	4	1	no	—
647	Dic₃	—	4	1	no	—
648	Dic₃	—	4	1	no	—
649	Dic₃	C₂×C₅×SD₁₆	1	2	yes	—
650	C₂×C₅	—	2	2	no	—
651	C₂×C₅	—	6	2	no	—
652	C₂×C₅	—	8	1	no	—
653	C₂×C₅	#5591	1	1	yes	characteristic
654	C₂×C₅	C₂×D₃×SD₁₆	1	1	yes	characteristic
655	C₂×C₅	#5779	1	1	yes	characteristic
656	C₂³	—	2	1	no	—
657	C₂³	—	3	1	no	—
658	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
659	Q₈	—	2	1	no	—
660	Q₈	—	2	1	no	—
661	Q₈	—	2	1	no	—
662	Q₈	—	4	2	no	—
663	Q₈	—	6	1	no	—
664	Q₈	—	6	1	no	—
665	Q₈	—	12	2	no	—
666	Q₈	—	12	2	no	—
667	Q₈	C₅×#237	1	2	yes	complemented
668	D₄	—	2	1	no	—
669	D₄	—	2	1	no	—
670	D₄	—	3	4	no	—
671	D₄	—	4	2	no	—
672	D₄	—	12	2	no	—
673	C₂×C₄	—	2	1	no	—
674	C₂×C₄	—	2	1	no	—
675	C₂×C₄	—	2	1	no	—
676	C₂×C₄	—	2	1	no	—
677	C₂×C₄	—	2	1	no	—
678	C₂×C₄	—	3	1	no	—
679	C₂×C₄	—	3	1	no	—
680	C₂×C₄	—	3	1	no	—
681	C₂×C₄	—	4	2	no	—
682	C₂×C₄	—	6	1	no	—
683	C₂×C₄	—	6	1	no	—
684	C₂×C₄	—	6	1	no	—
685	C₂×C₄	—	6	1	no	—
686	C₂×C₄	—	6	1	no	—
687	C₂×C₄	—	6	1	no	—
688	C₂×C₄	—	12	2	no	—
689	C₂×C₄	—	12	2	no	—
690	C₂×C₄	—	12	2	no	—
691	C₂×C₄	—	12	2	no	—
692	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
693	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
694	C₈	—	2	1	no	—
695	C₈	—	2	1	no	—
696	C₈	—	6	1	no	—
697	C₈	—	6	1	no	—
698	C₂×C₃	—	2	2	no	—
699	C₂×C₃	—	8	1	no	—
700	C₂×C₃	C₅×#546	1	1	yes	characteristic
701	C₂×C₃	C₂²×C₅×SD₁₆	1	1	yes	characteristic
702	C₂×C₃	C₅×#578	1	1	yes	characteristic
703	D₃	—	2	2	no	—
704	C₅	#80049	1	1	yes	characteristic, direct product factor, Sylow
705	C₂²	—	2	1	no	—
706	C₂²	—	2	1	no	—
707	C₂²	—	3	2	no	—
708	C₂²	—	4	1	no	—
709	C₂²	—	4	1	no	—
710	C₂²	—	4	1	no	—
711	C₂²	—	6	1	no	—
712	C₂²	—	6	1	no	—
713	C₂²	C₅×D₃×SD₁₆	1	2	yes	—
714	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
715	C₄	—	2	1	no	—
716	C₄	—	2	2	no	—
717	C₄	—	3	2	no	—
718	C₄	—	4	1	no	—
719	C₄	—	4	1	no	—
720	C₄	—	6	1	no	—
721	C₄	—	6	1	no	—
722	C₄	—	12	1	no	—
723	C₄	—	12	1	no	—
724	C₄	—	12	1	no	—
725	C₄	—	12	1	no	—
726	C₄	C₂×C₅×#237	1	1	yes	characteristic
727	C₄	C₅×#1004	1	1	yes	characteristic
728	C₃	C₅×#3298	1	1	yes	characteristic, complemented, Sylow
729	C₂	—	2	2	no	—
730	C₂	—	6	2	no	—
731	C₂	—	8	1	no	—
732	C₂	C₅×#5591	1	1	yes	characteristic
733	C₂	C₂×C₅×D₃×SD₁₆	1	1	yes	characteristic
734	C₂	C₅×#5779	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×#448	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₅×#5611	C₂
hypercenter	C₅×#448	D₃
hypocenter	C₃	C₅×#3298
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#448

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×#67	C₂	semidirect
3. C₃×C₅×#448	C₂	semidirect
4. C₂×C₅×#835	C₂	semidirect
5. C₅×#5171	C₂	semidirect
6. C₅×#5170	C₂	semidirect
7. C₅×#5162	C₂	semidirect
8. C₅×Dic₃×SD₁₆	C₂	semidirect
9. C₅×#5041	C₂	semidirect
10. C₅×#4805	C₂	semidirect
11. C₅×D₃×#103	C₂	semidirect
12. C₅×#4792	C₂	semidirect
13. C₅×#4791	C₂	semidirect
14. C₅×#67⋊C₈	C₂	semidirect
15. C₅×#4721	C₂	semidirect
60. #80049	C₅	direct (N × H)
71. C₅×#448	D₃	semidirect
129. Q₈×#67	C₂×C₅	semidirect
131. C₃×#448	C₂×C₅	semidirect
132. C₂×#835	C₂×C₅	semidirect
133. #5171	C₂×C₅	semidirect
134. #5170	C₂×C₅	semidirect
135. #5162	C₂×C₅	semidirect
136. Dic₃×SD₁₆	C₂×C₅	semidirect
137. #5041	C₂×C₅	semidirect
138. #4805	C₂×C₅	semidirect
139. D₃×#103	C₂×C₅	semidirect
140. #4792	C₂×C₅	semidirect
141. #4791	C₂×C₅	semidirect
142. #67⋊C₈	C₂×C₅	semidirect
143. #4721	C₂×C₅	semidirect
195. C₃×C₅×Q₈	#41	semidirect
208. C₅×#67	SD₁₆	semidirect
336. #448	C₅×D₃	semidirect
428. C₅×Q₈	#237	semidirect
515. C₃×Q₈	C₅×#41	semidirect
528. #67	C₅×SD₁₆	semidirect
626. C₃×C₅	#3298	semidirect
667. Q₈	C₅×#237	semidirect
704. C₅	#80049	direct (N × H)
728. C₃	C₅×#3298	semidirect

Maximal subgroups

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

subgroup	index
C₅×Q₈×#67	2
C₅×#5611	2
C₃×C₅×#448	2
C₂×C₅×#835	2
C₅×#5171	2
C₅×#5170	2
C₅×#5162	2
C₅×Dic₃×SD₁₆	2
C₅×#5041	2
C₅×#4805	2
C₅×D₃×#103	2
C₅×#4792	2
C₅×#4791	2
C₅×#67⋊C₈	2
C₅×#4721	2
C₅×#3298	3
#80049	5

Subgroups — C5×#80049 #14459100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#80049 #14459100