Subgroups — C₅×#80450 #14459130

Order 1920.

840 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5727	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5551	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#461	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5171	C₂	1	2	yes	maximal, complemented
5	C₅×#5170	C₂	1	2	yes	maximal, complemented
6	C₅×#5079	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₅×#214	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4851	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#4834	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4805	C₂	1	2	yes	maximal, complemented
11	C₅×D₃×#103	C₂	1	2	yes	maximal, complemented
12	C₅×#3298	—	3	1	no	maximal, Hall
13	C₂×C₅×D₃×Q₈	C₂²	1	2	yes	—
14	C₂×C₅×#235	—	2	1	no	—
15	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
18	C₅×#946	C₂²	1	2	yes	—
19	C₅×#938	C₂²	1	2	yes	—
20	C₂×C₅×#214	C₂²	1	1	yes	characteristic
21	C₅×#924	—	2	1	no	—
22	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
23	C₅×#896	—	2	2	no	—
24	C₅×#893	—	2	1	no	—
25	C₅×D₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₅×#888	—	2	1	no	—
27	C₅×#883	—	2	2	no	—
28	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
29	C₅×#869	C₂²	1	1	yes	characteristic
30	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₃×C₅×#103	C₂²	1	2	yes	—
33	C₅×#835	C₂²	1	2	yes	—
34	C₅×#835	C₂²	1	2	yes	—
35	C₅×#832	C₂²	1	1	yes	characteristic
36	C₅×(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
37	C₅×#817	C₂²	1	2	yes	—
38	C₅×#816	C₂²	1	2	yes	—
39	C₅×#810	C₂²	1	1	yes	characteristic
40	C₅×#808	C₂²	1	1	yes	characteristic
41	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
42	#80450	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×D₄×Q₈	—	3	1	no	—
44	C₅×#547	—	3	1	no	—
45	C₅×#487	—	3	1	no	—
46	C₅×#483	—	3	2	no	—
47	C₅×#475	—	3	1	no	—
48	C₅×#465	—	3	2	no	—
49	C₅×#461	D₃	1	1	yes	characteristic, complemented
50	C₅×#448	—	3	2	no	—
51	C₄×C₅×SD₁₆	—	3	1	no	—
52	C₅×C₈×D₄	—	3	1	no	—
53	C₂×C₅×#103	—	3	2	no	—
54	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
55	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₅×#67	—	2	2	no	—
58	C₅×D₃×Q₈	—	2	4	no	—
59	C₅×D₃×Q₈	—	4	2	no	—
60	C₅×#235	—	4	4	no	—
61	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
62	C₂×C₄×C₅×D₃	—	2	2	no	—
63	C₂×C₄×C₅×D₃	—	2	2	no	—
64	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
65	C₂×C₅×Dic₆	—	2	1	no	—
66	C₂×C₅×Dic₆	—	2	2	no	—
67	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
68	C₃×C₅×SD₁₆	—	4	2	no	—
69	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
70	C₃×C₅×C₄⋊C₄	—	2	1	no	—
71	C₃×C₅×C₄⋊C₄	—	2	1	no	—
72	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
73	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
74	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
75	C₅×#217	—	2	2	no	—
76	C₅×#214	—	2	2	no	—
77	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
78	C₅×(C₂×D₃)⋊C₄	D₄	1	2	yes	—
79	C₅×C₄⋊Dic₃	—	2	1	no	—
80	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
81	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
82	C₅×C₄⋊Dic₃	C₂³	1	2	yes	—
83	C₅×Dic₃⋊C₄	—	2	1	no	—
84	C₅×Dic₃⋊C₄	—	2	2	no	—
85	C₅×Dic₃⋊C₄	—	2	2	no	—
86	C₄×C₅×Dic₃	—	2	1	no	—
87	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
88	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
89	C₅×D₃×C₈	—	4	2	no	—
90	#5727	C₂×C₅	1	1	yes	characteristic, complemented
91	#5551	C₂×C₅	1	1	yes	characteristic
92	C₃×#461	C₂×C₅	1	1	yes	characteristic, complemented
93	#5171	C₂×C₅	1	2	yes	complemented
94	#5170	C₂×C₅	1	2	yes	complemented
95	#5079	C₂×C₅	1	1	yes	characteristic, complemented
96	C₄×#214	C₂×C₅	1	1	yes	characteristic, complemented
97	#4851	C₂×C₅	1	1	yes	characteristic, complemented
98	#4834	C₂×C₅	1	1	yes	characteristic, complemented
99	#4805	C₂×C₅	1	2	yes	complemented
100	D₃×#103	C₂×C₅	1	2	yes	complemented
101	C₂×C₅×#41	—	6	1	no	—
102	C₂²×C₅×Q₈	—	3	2	no	—
103	C₂×C₅×SD₁₆	—	3	1	no	—
104	C₂×C₅×SD₁₆	C₂×D₃	1	2	yes	—
105	C₂²×C₅×C₈	—	3	2	no	—
106	C₅×C₄⋊Q₈	—	3	1	no	—
107	C₅×C₄⋊Q₈	—	6	1	no	—
108	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
109	C₅×#123	—	6	2	no	—
110	C₅×C₂²⋊Q₈	—	3	2	no	—
111	C₅×C₂²⋊Q₈	—	6	2	no	—
112	C₅×C₂²⋊Q₈	—	6	1	no	—
113	C₅×#121	—	3	2	no	—
114	C₄×C₅×Q₈	—	3	1	no	—
115	C₄×C₅×D₄	—	3	1	no	—
116	C₄×C₅×D₄	—	6	1	no	—
117	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
118	C₂×C₅×C₄⋊C₄	—	3	2	no	—
119	C₅×#106	—	3	2	no	—
120	C₅×#106	—	3	1	no	—
121	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
122	C₅×#103	—	3	2	no	—
123	C₅×#103	—	3	2	no	—
124	C₅×#103	—	3	2	no	—
125	C₅×#103	—	3	1	no	—
126	C₅×#103	C₂×D₃	1	2	yes	—
127	C₅×#102	—	3	1	no	—
128	C₅×C₂²⋊C₈	—	3	2	no	—
129	C₄×C₅×C₈	—	3	1	no	—
130	#3298	—	3	1	no	Sylow
131	C₂³×C₃×C₅	—	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×D₄	—	2	1	no	—
136	C₃×C₅×D₄	—	2	1	no	—
137	C₂×C₃×C₄×C₅	—	2	1	no	—
138	C₂×C₃×C₄×C₅	—	2	2	no	—
139	C₂×C₃×C₄×C₅	—	4	2	no	—
140	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
143	C₅×#67	—	4	4	no	—
144	C₂×C₅×Dic₃	—	2	1	no	—
145	C₂×C₅×Dic₃	—	2	1	no	—
146	C₂×C₅×Dic₃	—	2	2	no	—
147	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
148	C₅×D₁₂	SD₁₆	1	4	yes	complemented
149	C₄×C₅×D₃	—	2	2	no	—
150	C₄×C₅×D₃	—	2	2	no	—
151	C₄×C₅×D₃	—	4	4	no	—
152	C₄×C₅×D₃	—	4	4	no	—
153	C₅×Dic₆	—	2	1	no	—
154	C₅×Dic₆	—	4	4	no	—
155	C₅×Dic₆	—	4	2	no	—
156	C₅×Dic₆	#41	1	2	yes	complemented
157	C₃×C₅×C₈	—	2	2	no	—
158	C₅×C₃⋊C₈	—	2	1	no	—
159	C₅×C₃⋊C₈	—	2	1	no	—
160	C₂×D₃×Q₈	C₂²×C₅	1	2	yes	—
161	C₂×#235	—	2	1	no	—
162	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	—
163	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
164	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
165	#946	C₂²×C₅	1	2	yes	—
166	#938	C₂²×C₅	1	2	yes	—
167	C₂×#214	C₂²×C₅	1	1	yes	characteristic
168	#924	—	2	1	no	—
169	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
170	#896	—	2	2	no	—
171	#893	—	2	1	no	—
172	D₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
173	#888	—	2	1	no	—
174	#883	—	2	2	no	—
175	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
176	#869	C₂²×C₅	1	1	yes	characteristic
177	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
178	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
179	C₃×#103	C₂²×C₅	1	2	yes	—
180	#835	C₂²×C₅	1	2	yes	—
181	#835	C₂²×C₅	1	2	yes	—
182	#832	C₂²×C₅	1	1	yes	characteristic
183	(C₂×D₃)⋊C₈	C₂²×C₅	1	2	yes	—
184	#817	C₂²×C₅	1	2	yes	—
185	#816	C₂²×C₅	1	2	yes	—
186	#810	C₂²×C₅	1	1	yes	characteristic
187	#808	C₂²×C₅	1	1	yes	characteristic
188	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
189	C₅×#41	—	12	4	no	—
190	C₂×C₅×Q₈	—	3	1	no	—
191	C₂×C₅×Q₈	—	6	2	no	—
192	C₂×C₅×Q₈	—	6	1	no	—
193	C₂×C₅×Q₈	—	6	4	no	—
194	C₂×C₅×Q₈	—	12	2	no	—
195	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
196	C₂×C₅×D₄	—	3	1	no	—
197	C₂×C₅×D₄	—	6	2	no	—
198	C₂×C₅×D₄	C₂²×D₃	1	1	yes	characteristic
199	C₂²×C₄×C₅	—	2	1	no	—
200	C₂²×C₄×C₅	—	3	2	no	—
201	C₂²×C₄×C₅	—	6	2	no	—
202	C₂²×C₄×C₅	—	6	2	no	—
203	C₅×SD₁₆	—	4	2	no	—
204	C₅×SD₁₆	—	6	2	no	—
205	C₂×C₅×C₈	—	3	1	no	—
206	C₂×C₅×C₈	—	3	1	no	—
207	C₂×C₅×C₈	—	12	2	no	—
208	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
209	C₅×C₄⋊C₄	—	2	1	no	—
210	C₅×C₄⋊C₄	—	2	1	no	—
211	C₅×C₄⋊C₄	—	3	1	no	—
212	C₅×C₄⋊C₄	—	3	2	no	—
213	C₅×C₄⋊C₄	—	3	1	no	—
214	C₅×C₄⋊C₄	—	6	2	no	—
215	C₅×C₄⋊C₄	—	6	2	no	—
216	C₅×C₄⋊C₄	—	6	1	no	—
217	C₅×C₄⋊C₄	—	6	1	no	—
218	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
219	C₅×C₂²⋊C₄	—	2	1	no	—
220	C₅×C₂²⋊C₄	—	3	2	no	—
221	C₅×C₂²⋊C₄	—	6	2	no	—
222	C₅×C₂²⋊C₄	—	6	2	no	—
223	C₄²×C₅	—	6	1	no	—
224	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
225	D₄×Q₈	—	3	1	no	—
226	#547	—	3	1	no	—
227	#487	—	3	1	no	—
228	#483	—	3	2	no	—
229	#475	—	3	1	no	—
230	#465	—	3	2	no	—
231	#461	C₅×D₃	1	1	yes	characteristic, complemented
232	#448	—	3	2	no	—
233	C₄×SD₁₆	—	3	1	no	—
234	C₈×D₄	—	3	1	no	—
235	C₂×#103	—	3	2	no	—
236	C₂²×C₃×C₅	—	4	1	no	—
237	C₂²×C₃×C₅	—	4	1	no	—
238	C₂²×C₃×C₅	—	4	1	no	—
239	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₅×D₃	—	2	2	no	—
241	C₂×C₅×D₃	—	2	2	no	—
242	C₂×C₅×D₃	C₂×SD₁₆	1	4	yes	—
243	C₃×C₄×C₅	—	2	1	no	—
244	C₃×C₄×C₅	—	4	2	no	—
245	C₃×C₄×C₅	—	4	1	no	—
246	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
247	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
248	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
249	C₅×Dic₃	—	2	2	no	—
250	C₅×Dic₃	—	2	2	no	—
251	C₅×Dic₃	—	4	2	no	—
252	C₅×Dic₃	—	4	1	no	—
253	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
254	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
255	C₂²×C₃×C₄	—	2	1	no	—
256	C₂×#67	—	2	2	no	—
257	D₃×Q₈	—	2	4	no	—
258	D₃×Q₈	—	4	2	no	—
259	#235	—	4	4	no	—
260	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
261	C₂×C₄×D₃	—	2	2	no	—
262	C₂×C₄×D₃	—	2	2	no	—
263	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
264	C₂×Dic₆	—	2	1	no	—
265	C₂×Dic₆	—	2	2	no	—
266	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
267	C₃×SD₁₆	—	4	2	no	—
268	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
269	C₃×C₄⋊C₄	—	2	1	no	—
270	C₃×C₄⋊C₄	—	2	1	no	—
271	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
272	C₃×C₂²⋊C₄	—	2	1	no	—
273	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
274	#217	—	2	2	no	—
275	#214	—	2	2	no	—
276	(C₂×D₃)⋊C₄	—	2	2	no	—
277	(C₂×D₃)⋊C₄	C₅×D₄	1	2	yes	—
278	C₄⋊Dic₃	—	2	1	no	—
279	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
280	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
281	C₄⋊Dic₃	C₂³×C₅	1	2	yes	—
282	Dic₃⋊C₄	—	2	1	no	—
283	Dic₃⋊C₄	—	2	2	no	—
284	Dic₃⋊C₄	—	2	2	no	—
285	C₄×Dic₃	—	2	1	no	—
286	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
287	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
288	D₃×C₈	—	4	2	no	—
289	C₂³×C₅	—	2	1	no	—
290	C₂³×C₅	—	3	2	no	—
291	C₅×Q₈	—	2	2	no	—
292	C₅×Q₈	—	2	2	no	—
293	C₅×Q₈	—	3	2	no	—
294	C₅×Q₈	—	6	1	no	—
295	C₅×Q₈	—	12	4	no	—
296	C₅×Q₈	—	12	2	no	—
297	C₅×D₄	—	2	1	no	—
298	C₅×D₄	—	2	1	no	—
299	C₅×D₄	—	3	4	no	—
300	C₅×D₄	—	12	4	no	—
301	C₂×C₄×C₅	—	2	1	no	—
302	C₂×C₄×C₅	—	2	2	no	—
303	C₂×C₄×C₅	—	3	2	no	—
304	C₂×C₄×C₅	—	4	2	no	—
305	C₂×C₄×C₅	—	6	1	no	—
306	C₂×C₄×C₅	—	6	2	no	—
307	C₂×C₄×C₅	—	6	2	no	—
308	C₂×C₄×C₅	—	6	1	no	—
309	C₂×C₄×C₅	—	6	2	no	—
310	C₂×C₄×C₅	—	12	4	no	—
311	C₂×C₄×C₅	—	12	4	no	—
312	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
313	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
314	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
315	C₅×C₈	—	2	2	no	—
316	C₅×C₈	—	6	1	no	—
317	C₅×C₈	—	6	1	no	—
318	C₂×#41	—	6	1	no	—
319	C₂²×Q₈	—	3	2	no	—
320	C₂×SD₁₆	—	3	1	no	—
321	C₂×SD₁₆	C₂×C₅×D₃	1	2	yes	—
322	C₂²×C₈	—	3	2	no	—
323	C₄⋊Q₈	—	3	1	no	—
324	C₄⋊Q₈	—	6	1	no	—
325	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
326	#123	—	6	2	no	—
327	C₂²⋊Q₈	—	3	2	no	—
328	C₂²⋊Q₈	—	6	2	no	—
329	C₂²⋊Q₈	—	6	1	no	—
330	#121	—	3	2	no	—
331	C₄×Q₈	—	3	1	no	—
332	C₄×D₄	—	3	1	no	—
333	C₄×D₄	—	6	1	no	—
334	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
335	C₂×C₄⋊C₄	—	3	2	no	—
336	#106	—	3	2	no	—
337	#106	—	3	1	no	—
338	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
339	#103	—	3	2	no	—
340	#103	—	3	2	no	—
341	#103	—	3	2	no	—
342	#103	—	3	1	no	—
343	#103	C₂×C₅×D₃	1	2	yes	—
344	#102	—	3	1	no	—
345	C₂²⋊C₈	—	3	2	no	—
346	C₄×C₈	—	3	1	no	—
347	C₂×C₃×C₅	—	8	1	no	—
348	C₂×C₃×C₅	#546	1	1	yes	characteristic
349	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
350	C₂×C₃×C₅	#578	1	1	yes	characteristic
351	C₅×D₃	—	2	4	no	—
352	C₂³×C₃	—	2	1	no	—
353	C₂²×D₃	C₂×C₅×D₄	1	2	yes	—
354	C₃×Q₈	—	2	2	no	—
355	C₃×Q₈	—	2	2	no	—
356	C₃×D₄	—	2	1	no	—
357	C₃×D₄	—	2	1	no	—
358	C₂×C₃×C₄	—	2	1	no	—
359	C₂×C₃×C₄	—	2	2	no	—
360	C₂×C₃×C₄	—	4	2	no	—
361	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
362	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
363	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
364	#67	—	4	4	no	—
365	C₂×Dic₃	—	2	1	no	—
366	C₂×Dic₃	—	2	1	no	—
367	C₂×Dic₃	—	2	2	no	—
368	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
369	D₁₂	C₅×SD₁₆	1	4	yes	complemented
370	C₄×D₃	—	2	2	no	—
371	C₄×D₃	—	2	2	no	—
372	C₄×D₃	—	4	4	no	—
373	C₄×D₃	—	4	4	no	—
374	Dic₆	—	2	1	no	—
375	Dic₆	—	4	4	no	—
376	Dic₆	—	4	2	no	—
377	Dic₆	C₅×#41	1	2	yes	complemented
378	C₃×C₈	—	2	2	no	—
379	C₃⋊C₈	—	2	1	no	—
380	C₃⋊C₈	—	2	1	no	—
381	C₂²×C₅	—	3	4	no	—
382	C₂²×C₅	—	4	1	no	—
383	C₂²×C₅	—	4	1	no	—
384	C₂²×C₅	—	4	1	no	—
385	C₂²×C₅	—	6	2	no	—
386	C₂²×C₅	—	6	2	no	—
387	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
388	C₄×C₅	—	2	1	no	—
389	C₄×C₅	—	4	1	no	—
390	C₄×C₅	—	4	2	no	—
391	C₄×C₅	—	6	2	no	—
392	C₄×C₅	—	6	2	no	—
393	C₄×C₅	—	12	2	no	—
394	C₄×C₅	—	12	1	no	—
395	C₄×C₅	D₃×SD₁₆	1	2	yes	—
396	C₄×C₅	#1004	1	1	yes	characteristic
397	C₄×C₅	D₃×#41	1	1	yes	characteristic
398	#41	—	12	4	no	—
399	C₂×Q₈	—	3	1	no	—
400	C₂×Q₈	—	6	2	no	—
401	C₂×Q₈	—	6	1	no	—
402	C₂×Q₈	—	6	4	no	—
403	C₂×Q₈	—	12	2	no	—
404	C₂×Q₈	C₂²×C₅×D₃	1	2	yes	—
405	C₂×D₄	—	3	1	no	—
406	C₂×D₄	—	6	2	no	—
407	C₂×D₄	C₂²×C₅×D₃	1	1	yes	characteristic
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	3	2	no	—
410	C₂²×C₄	—	6	2	no	—
411	C₂²×C₄	—	6	2	no	—
412	SD₁₆	—	4	2	no	—
413	SD₁₆	—	6	2	no	—
414	C₂×C₈	—	3	1	no	—
415	C₂×C₈	—	3	1	no	—
416	C₂×C₈	—	12	2	no	—
417	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	3	1	no	—
421	C₄⋊C₄	—	3	1	no	—
422	C₄⋊C₄	—	3	2	no	—
423	C₄⋊C₄	—	6	2	no	—
424	C₄⋊C₄	—	6	2	no	—
425	C₄⋊C₄	—	6	1	no	—
426	C₄⋊C₄	—	6	1	no	—
427	C₄⋊C₄	C₂²×C₅×D₃	1	1	yes	characteristic
428	C₂²⋊C₄	—	2	1	no	—
429	C₂²⋊C₄	—	3	2	no	—
430	C₂²⋊C₄	—	6	2	no	—
431	C₂²⋊C₄	—	6	2	no	—
432	C₄²	—	6	1	no	—
433	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
434	C₃×C₅	#3298	1	1	yes	characteristic, complemented, Hall
435	C₂²×C₃	—	4	1	no	—
436	C₂²×C₃	—	4	1	no	—
437	C₂²×C₃	—	4	1	no	—
438	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
439	C₂×D₃	—	2	2	no	—
440	C₂×D₃	—	2	2	no	—
441	C₂×D₃	C₂×C₅×SD₁₆	1	4	yes	—
442	C₃×C₄	—	2	1	no	—
443	C₃×C₄	—	4	2	no	—
444	C₃×C₄	—	4	1	no	—
445	C₃×C₄	C₂×C₅×SD₁₆	1	2	yes	—
446	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
447	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
448	Dic₃	—	2	2	no	—
449	Dic₃	—	2	2	no	—
450	Dic₃	—	4	2	no	—
451	Dic₃	—	4	1	no	—
452	C₂×C₅	—	6	4	no	—
453	C₂×C₅	—	8	1	no	—
454	C₂×C₅	#5612	1	1	yes	characteristic
455	C₂×C₅	C₂×D₃×SD₁₆	1	1	yes	characteristic
456	C₂×C₅	#5779	1	1	yes	characteristic
457	C₂³	—	2	1	no	—
458	C₂³	—	3	2	no	—
459	Q₈	—	2	2	no	—
460	Q₈	—	2	2	no	—
461	Q₈	—	3	2	no	—
462	Q₈	—	6	1	no	—
463	Q₈	—	12	4	no	—
464	Q₈	—	12	2	no	—
465	D₄	—	2	1	no	—
466	D₄	—	2	1	no	—
467	D₄	—	3	4	no	—
468	D₄	—	12	4	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	3	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	6	1	no	—
474	C₂×C₄	—	6	2	no	—
475	C₂×C₄	—	6	2	no	—
476	C₂×C₄	—	6	1	no	—
477	C₂×C₄	—	6	2	no	—
478	C₂×C₄	—	12	4	no	—
479	C₂×C₄	—	12	4	no	—
480	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
481	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
482	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
483	C₈	—	2	2	no	—
484	C₈	—	6	1	no	—
485	C₈	—	6	1	no	—
486	C₂×C₃	—	8	1	no	—
487	C₂×C₃	C₅×#546	1	1	yes	characteristic
488	C₂×C₃	C₂²×C₅×SD₁₆	1	1	yes	characteristic
489	C₂×C₃	C₅×#578	1	1	yes	characteristic
490	D₃	—	2	4	no	—
491	C₅	#80450	1	1	yes	characteristic, direct product factor, Sylow
492	C₂²	—	3	4	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	1	no	—
496	C₂²	—	6	2	no	—
497	C₂²	—	6	2	no	—
498	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
499	C₄	—	2	1	no	—
500	C₄	—	4	1	no	—
501	C₄	—	4	2	no	—
502	C₄	—	6	2	no	—
503	C₄	—	6	2	no	—
504	C₄	—	12	2	no	—
505	C₄	—	12	1	no	—
506	C₄	C₅×D₃×SD₁₆	1	2	yes	—
507	C₄	C₅×#1004	1	1	yes	characteristic
508	C₄	C₅×D₃×#41	1	1	yes	characteristic
509	C₃	C₅×#3298	1	1	yes	characteristic, complemented, Sylow
510	C₂	—	6	4	no	—
511	C₂	—	8	1	no	—
512	C₂	C₅×#5612	1	1	yes	characteristic
513	C₂	C₂×C₅×D₃×SD₁₆	1	1	yes	characteristic
514	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×#461	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₅×#5551	C₂
hypercenter	C₅×#461	D₃
hypocenter	C₃	C₅×#3298
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#461

Complements

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

N	H ≅ G/N	type
1. C₅×#5727	C₂	semidirect
3. C₃×C₅×#461	C₂	semidirect
4. C₅×#5171	C₂	semidirect
5. C₅×#5170	C₂	semidirect
6. C₅×#5079	C₂	semidirect
7. C₄×C₅×#214	C₂	semidirect
8. C₅×#4851	C₂	semidirect
9. C₅×#4834	C₂	semidirect
10. C₅×#4805	C₂	semidirect
11. C₅×D₃×#103	C₂	semidirect
42. #80450	C₅	direct (N × H)
49. C₅×#461	D₃	semidirect
90. #5727	C₂×C₅	semidirect
92. C₃×#461	C₂×C₅	semidirect
93. #5171	C₂×C₅	semidirect
94. #5170	C₂×C₅	semidirect
95. #5079	C₂×C₅	semidirect
96. C₄×#214	C₂×C₅	semidirect
97. #4851	C₂×C₅	semidirect
98. #4834	C₂×C₅	semidirect
99. #4805	C₂×C₅	semidirect
100. D₃×#103	C₂×C₅	semidirect
148. C₅×D₁₂	SD₁₆	semidirect
156. C₅×Dic₆	#41	semidirect
231. #461	C₅×D₃	semidirect
369. D₁₂	C₅×SD₁₆	semidirect
377. Dic₆	C₅×#41	semidirect
434. C₃×C₅	#3298	semidirect
491. C₅	#80450	direct (N × H)
509. 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×#5727	2
C₅×#5551	2
C₃×C₅×#461	2
C₅×#5171	2
C₅×#5170	2
C₅×#5079	2
C₄×C₅×#214	2
C₅×#4851	2
C₅×#4834	2
C₅×#4805	2
C₅×D₃×#103	2
C₅×#3298	3
#80450	5

Subgroups — C5×#80450 #14459130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#80450 #14459130