Subgroups — C₅×#23018 #11945738

Order 1280.

892 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2933	C₂	1	2	yes	maximal, complemented
4	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1974	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1945	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1929	C₂	1	2	yes	maximal, complemented
8	C₅×#1871	C₂	1	2	yes	maximal, complemented
9	C₅×#1866	C₂	1	2	yes	maximal, complemented
10	C₅×#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×#135	C₂²	1	1	yes	characteristic
13	C₂×C₅×#131	C₂²	1	2	yes	—
14	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
15	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#525	C₂²	1	2	yes	—
17	C₅×#467	C₄	1	1	yes	characteristic, complemented
18	C₅×#467	C₄	1	1	yes	characteristic, complemented
19	C₅×#466	C₄	1	1	yes	characteristic, complemented
20	C₅×#466	C₄	1	1	yes	characteristic, complemented
21	C₅×#465	C₄	1	2	yes	complemented
22	C₅×#465	C₄	1	2	yes	complemented
23	C₅×#443	—	2	4	no	—
24	C₅×#436	—	2	2	no	—
25	C₅×#435	—	2	2	no	—
26	C₅×#435	—	2	2	no	—
27	C₅×C₈×D₄	—	2	2	no	—
28	C₂×C₅×#108	—	2	4	no	—
29	C₂×C₅×#108	C₂²	1	2	yes	—
30	C₂×C₅×#108	C₂²	1	1	yes	characteristic
31	C₂×C₅×#108	C₂²	1	1	yes	characteristic
32	C₅×#427	C₂²	1	1	yes	characteristic
33	C₅×#424	C₂²	1	2	yes	—
34	C₂×C₅×#104	C₂²	1	2	yes	—
35	C₂×C₅×#104	C₂²	1	2	yes	—
36	C₅×#416	C₂²	1	2	yes	—
37	C₅×#408	C₂²	1	2	yes	—
38	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₅×#405	C₂²	1	2	yes	—
40	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
42	C₄×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
43	C₅×#339	C₂²	1	2	yes	—
44	C₅×#339	C₂²	1	2	yes	—
45	C₅×#336	C₂²	1	1	yes	characteristic
46	C₅×#336	C₂²	1	1	yes	characteristic
47	#23018	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₅×#41	C₂³	1	2	yes	—
49	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₅×#135	—	2	2	no	—
51	C₅×#135	—	2	2	no	—
52	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
53	C₂×C₅×SD₁₆	C₂×C₄	1	2	yes	—
54	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
55	C₅×#131	—	2	4	no	—
56	C₅×#131	—	4	2	no	—
57	C₂×C₅×M₁₆	—	2	2	no	—
58	C₂×C₅×M₁₆	—	2	2	no	—
59	C₂×C₅×M₁₆	—	2	4	no	—
60	C₂×C₅×M₁₆	—	4	2	no	—
61	C₂×C₅×M₁₆	C₂³	1	2	yes	—
62	C₂×C₅×M₁₆	C₂³	1	2	yes	—
63	C₂²×C₅×C₈	—	2	2	no	—
64	C₂²×C₅×C₈	—	2	2	no	—
65	C₂²×C₅×C₈	—	2	2	no	—
66	C₂²×C₅×C₈	—	2	2	no	—
67	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₅×#123	—	2	2	no	—
76	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
77	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
78	C₅×#121	C₂×C₄	1	2	yes	—
79	C₅×#121	C₂×C₄	1	2	yes	—
80	C₅×#120	—	2	2	no	—
81	C₄×C₅×D₄	—	2	2	no	—
82	C₄×C₅×D₄	—	2	2	no	—
83	C₅×#117	C₂³	1	2	yes	—
84	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
85	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
87	C₅×#108	—	2	4	no	—
88	C₅×#108	—	2	2	no	—
89	C₅×#108	D₄	1	4	yes	complemented
90	C₅×#107	C₂×C₄	1	1	yes	characteristic
91	C₅×#107	C₂×C₄	1	1	yes	characteristic
92	C₅×#106	C₂×C₄	1	2	yes	—
93	C₅×C₄⋊C₈	—	2	2	no	—
94	C₅×C₄⋊C₈	—	2	2	no	—
95	C₅×#104	—	2	4	no	—
96	C₅×#104	—	4	2	no	—
97	C₅×#103	C₂×C₄	1	2	yes	—
98	C₅×#103	C₂×C₄	1	2	yes	—
99	C₅×#102	C₂×C₄	1	2	yes	—
100	C₅×#102	C₂×C₄	1	2	yes	—
101	C₅×C₂²⋊C₈	—	2	2	no	—
102	C₅×C₂²⋊C₈	—	2	2	no	—
103	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
104	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
105	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
106	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
107	C₅×#97	—	2	2	no	—
108	C₄×C₅×C₈	—	2	2	no	—
109	C₄×C₅×C₈	—	2	2	no	—
110	C₄×C₅×C₈	—	2	2	no	—
111	C₅×#95	—	2	1	no	—
112	#3050	C₂×C₅	1	1	yes	characteristic
113	C₂×#443	C₂×C₅	1	1	yes	characteristic, complemented
114	#2933	C₂×C₅	1	2	yes	complemented
115	C₂²×#108	C₂×C₅	1	1	yes	characteristic, complemented
116	#1974	C₂×C₅	1	1	yes	characteristic, complemented
117	#1945	C₂×C₅	1	1	yes	characteristic, complemented
118	#1929	C₂×C₅	1	2	yes	complemented
119	#1871	C₂×C₅	1	2	yes	complemented
120	#1866	C₂×C₅	1	2	yes	complemented
121	#1772	C₂×C₅	1	1	yes	characteristic, complemented
122	C₈×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
123	C₂⁴×C₅	#41	1	1	yes	characteristic
124	C₅×#41	—	2	2	no	—
125	C₅×#41	—	2	2	no	—
126	C₅×#41	—	2	2	no	—
127	C₅×#41	—	2	2	no	—
128	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
129	C₂×C₅×D₄	—	2	2	no	—
130	C₂×C₅×D₄	—	2	2	no	—
131	C₂×C₅×D₄	—	2	2	no	—
132	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₅	—	2	1	no	—
134	C₂²×C₄×C₅	—	2	1	no	—
135	C₂²×C₄×C₅	—	2	1	no	—
136	C₂²×C₄×C₅	—	2	1	no	—
137	C₂²×C₄×C₅	—	2	1	no	—
138	C₂²×C₄×C₅	—	2	1	no	—
139	C₂²×C₄×C₅	—	2	2	no	—
140	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₅	#41	1	1	yes	characteristic
145	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
146	C₅×Q₁₆	—	2	2	no	—
147	C₅×SD₁₆	—	2	4	no	—
148	C₅×D₈	—	2	2	no	—
149	C₅×M₁₆	—	2	2	no	—
150	C₅×M₁₆	—	2	2	no	—
151	C₅×M₁₆	—	2	2	no	—
152	C₅×M₁₆	—	4	4	no	—
153	C₅×M₁₆	—	4	4	no	—
154	C₅×M₁₆	C₂×D₄	1	4	yes	—
155	C₂×C₅×C₈	—	2	1	no	—
156	C₂×C₅×C₈	—	2	1	no	—
157	C₂×C₅×C₈	—	2	2	no	—
158	C₂×C₅×C₈	—	2	1	no	—
159	C₂×C₅×C₈	—	2	1	no	—
160	C₂×C₅×C₈	—	2	1	no	—
161	C₂×C₅×C₈	—	2	1	no	—
162	C₂×C₅×C₈	—	2	2	no	—
163	C₂×C₅×C₈	—	2	2	no	—
164	C₂×C₅×C₈	—	2	2	no	—
165	C₂×C₅×C₈	—	2	2	no	—
166	C₂×C₅×C₈	—	2	2	no	—
167	C₂×C₅×C₈	—	4	4	no	—
168	C₂×C₅×C₈	—	4	4	no	—
169	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
174	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
175	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
176	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
177	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
178	C₂×C₅×C₈	#41	1	2	yes	—
179	C₅×C₄⋊C₄	—	2	2	no	—
180	C₅×C₄⋊C₄	—	2	2	no	—
181	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
182	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
183	C₅×C₂²⋊C₄	—	2	4	no	—
184	C₅×C₂²⋊C₄	—	2	2	no	—
185	C₅×C₂²⋊C₄	—	2	2	no	—
186	C₅×C₂²⋊C₄	—	2	2	no	—
187	C₅×C₂²⋊C₄	—	2	2	no	—
188	C₅×C₂²⋊C₄	—	2	2	no	—
189	C₄²×C₅	—	2	2	no	—
190	C₄²×C₅	—	2	1	no	—
191	C₄²×C₅	—	2	1	no	—
192	C₄²×C₅	—	2	1	no	—
193	C₄²×C₅	—	2	1	no	—
194	C₂×#135	C₂²×C₅	1	1	yes	characteristic
195	C₂×#131	C₂²×C₅	1	2	yes	—
196	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
197	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
198	#525	C₂²×C₅	1	2	yes	—
199	#467	C₄×C₅	1	1	yes	characteristic, complemented
200	#467	C₄×C₅	1	1	yes	characteristic, complemented
201	#466	C₄×C₅	1	1	yes	characteristic, complemented
202	#466	C₄×C₅	1	1	yes	characteristic, complemented
203	#465	C₄×C₅	1	2	yes	complemented
204	#465	C₄×C₅	1	2	yes	complemented
205	#443	—	2	4	no	—
206	#436	—	2	2	no	—
207	#435	—	2	2	no	—
208	#435	—	2	2	no	—
209	C₈×D₄	—	2	2	no	—
210	C₂×#108	—	2	4	no	—
211	C₂×#108	C₂²×C₅	1	2	yes	—
212	C₂×#108	C₂²×C₅	1	1	yes	characteristic
213	C₂×#108	C₂²×C₅	1	1	yes	characteristic
214	#427	C₂²×C₅	1	1	yes	characteristic
215	#424	C₂²×C₅	1	2	yes	—
216	C₂×#104	C₂²×C₅	1	2	yes	—
217	C₂×#104	C₂²×C₅	1	2	yes	—
218	#416	C₂²×C₅	1	2	yes	—
219	#408	C₂²×C₅	1	2	yes	—
220	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
221	#405	C₂²×C₅	1	2	yes	—
222	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
223	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
224	C₄×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
225	#339	C₂²×C₅	1	2	yes	—
226	#339	C₂²×C₅	1	2	yes	—
227	#336	C₂²×C₅	1	1	yes	characteristic
228	#336	C₂²×C₅	1	1	yes	characteristic
229	C₂³×C₅	—	2	1	no	—
230	C₂³×C₅	—	2	1	no	—
231	C₂³×C₅	—	2	2	no	—
232	C₂³×C₅	—	2	2	no	—
233	C₂³×C₅	—	4	1	no	—
234	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
235	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
236	C₂³×C₅	C₂×#41	1	1	yes	characteristic
237	C₅×Q₈	—	2	2	no	—
238	C₅×Q₈	—	2	2	no	—
239	C₅×D₄	—	2	2	no	—
240	C₅×D₄	—	2	2	no	—
241	C₅×D₄	—	4	2	no	—
242	C₅×D₄	—	4	2	no	—
243	C₅×D₄	—	4	4	no	—
244	C₅×D₄	—	8	2	no	—
245	C₂×C₄×C₅	—	2	1	no	—
246	C₂×C₄×C₅	—	2	1	no	—
247	C₂×C₄×C₅	—	2	1	no	—
248	C₂×C₄×C₅	—	2	1	no	—
249	C₂×C₄×C₅	—	2	1	no	—
250	C₂×C₄×C₅	—	2	1	no	—
251	C₂×C₄×C₅	—	2	1	no	—
252	C₂×C₄×C₅	—	2	1	no	—
253	C₂×C₄×C₅	—	2	1	no	—
254	C₂×C₄×C₅	—	2	1	no	—
255	C₂×C₄×C₅	—	2	2	no	—
256	C₂×C₄×C₅	—	2	2	no	—
257	C₂×C₄×C₅	—	2	2	no	—
258	C₂×C₄×C₅	—	2	2	no	—
259	C₂×C₄×C₅	—	2	2	no	—
260	C₂×C₄×C₅	—	4	1	no	—
261	C₂×C₄×C₅	—	4	1	no	—
262	C₂×C₄×C₅	—	4	1	no	—
263	C₂×C₄×C₅	—	4	1	no	—
264	C₂×C₄×C₅	—	4	2	no	—
265	C₂×C₄×C₅	—	4	2	no	—
266	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
267	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
268	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
269	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
272	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
273	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
274	C₅×C₈	—	2	1	no	—
275	C₅×C₈	—	2	1	no	—
276	C₅×C₈	—	2	1	no	—
277	C₅×C₈	—	2	1	no	—
278	C₅×C₈	—	2	4	no	—
279	C₅×C₈	—	2	1	no	—
280	C₅×C₈	—	2	1	no	—
281	C₅×C₈	—	4	2	no	—
282	C₅×C₈	C₄×D₄	1	2	yes	—
283	C₅×C₈	C₄×D₄	1	2	yes	—
284	C₂×#41	C₂³×C₅	1	2	yes	—
285	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
286	#135	—	2	2	no	—
287	#135	—	2	2	no	—
288	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
289	C₂×SD₁₆	C₂×C₄×C₅	1	2	yes	—
290	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
291	#131	—	2	4	no	—
292	#131	—	4	2	no	—
293	C₂×M₁₆	—	2	2	no	—
294	C₂×M₁₆	—	2	2	no	—
295	C₂×M₁₆	—	2	4	no	—
296	C₂×M₁₆	—	4	2	no	—
297	C₂×M₁₆	C₂³×C₅	1	2	yes	—
298	C₂×M₁₆	C₂³×C₅	1	2	yes	—
299	C₂²×C₈	—	2	2	no	—
300	C₂²×C₈	—	2	2	no	—
301	C₂²×C₈	—	2	2	no	—
302	C₂²×C₈	—	2	2	no	—
303	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
304	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
305	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
306	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
307	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
308	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
311	#123	—	2	2	no	—
312	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
313	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
314	#121	C₂×C₄×C₅	1	2	yes	—
315	#121	C₂×C₄×C₅	1	2	yes	—
316	#120	—	2	2	no	—
317	C₄×D₄	—	2	2	no	—
318	C₄×D₄	—	2	2	no	—
319	#117	C₂³×C₅	1	2	yes	—
320	C₂×C₂²⋊C₄	—	2	1	no	—
321	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
323	#108	—	2	4	no	—
324	#108	—	2	2	no	—
325	#108	C₅×D₄	1	4	yes	complemented
326	#107	C₂×C₄×C₅	1	1	yes	characteristic
327	#107	C₂×C₄×C₅	1	1	yes	characteristic
328	#106	C₂×C₄×C₅	1	2	yes	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	2	2	no	—
331	#104	—	2	4	no	—
332	#104	—	4	2	no	—
333	#103	C₂×C₄×C₅	1	2	yes	—
334	#103	C₂×C₄×C₅	1	2	yes	—
335	#102	C₂×C₄×C₅	1	2	yes	—
336	#102	C₂×C₄×C₅	1	2	yes	—
337	C₂²⋊C₈	—	2	2	no	—
338	C₂²⋊C₈	—	2	2	no	—
339	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
340	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
341	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
342	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
343	#97	—	2	2	no	—
344	C₄×C₈	—	2	2	no	—
345	C₄×C₈	—	2	2	no	—
346	C₄×C₈	—	2	2	no	—
347	#95	—	2	1	no	—
348	C₂²×C₅	—	2	1	no	—
349	C₂²×C₅	—	2	1	no	—
350	C₂²×C₅	—	2	1	no	—
351	C₂²×C₅	—	2	1	no	—
352	C₂²×C₅	—	2	1	no	—
353	C₂²×C₅	—	2	1	no	—
354	C₂²×C₅	—	2	1	no	—
355	C₂²×C₅	—	4	2	no	—
356	C₂²×C₅	—	4	2	no	—
357	C₂²×C₅	—	4	2	no	—
358	C₂²×C₅	—	4	2	no	—
359	C₂²×C₅	—	4	2	no	—
360	C₂²×C₅	#443	1	2	yes	complemented
361	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
362	C₂²×C₅	#548	1	1	yes	characteristic
363	C₂²×C₅	#553	1	1	yes	characteristic
364	C₄×C₅	—	2	1	no	—
365	C₄×C₅	—	2	1	no	—
366	C₄×C₅	—	4	2	no	—
367	C₄×C₅	—	4	2	no	—
368	C₄×C₅	—	4	2	no	—
369	C₄×C₅	—	4	2	no	—
370	C₄×C₅	—	4	2	no	—
371	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
372	C₄×C₅	#520	1	1	yes	characteristic
373	C₄×C₅	D₄²	1	1	yes	characteristic
374	C₄×C₅	#547	1	1	yes	characteristic
375	C₂⁴	C₅×#41	1	1	yes	characteristic
376	#41	—	2	2	no	—
377	#41	—	2	2	no	—
378	#41	—	2	2	no	—
379	#41	—	2	2	no	—
380	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
393	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
394	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
395	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
396	C₂²×C₄	C₅×#41	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
398	Q₁₆	—	2	2	no	—
399	SD₁₆	—	2	4	no	—
400	D₈	—	2	2	no	—
401	M₁₆	—	2	2	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	4	4	no	—
405	M₁₆	—	4	4	no	—
406	M₁₆	C₂×C₅×D₄	1	4	yes	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	4	4	no	—
420	C₂×C₈	—	4	4	no	—
421	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
422	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
423	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
424	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
425	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
426	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
427	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
428	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
429	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
430	C₂×C₈	C₅×#41	1	2	yes	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
434	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	2	2	no	—
440	C₂²⋊C₄	—	2	2	no	—
441	C₄²	—	2	2	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	2	1	no	—
445	C₄²	—	2	1	no	—
446	C₂×C₅	—	2	1	no	—
447	C₂×C₅	—	2	1	no	—
448	C₂×C₅	—	2	2	no	—
449	C₂×C₅	—	4	1	no	—
450	C₂×C₅	—	8	2	no	—
451	C₂×C₅	#2361	1	1	yes	characteristic
452	C₂×C₅	C₂×#443	1	1	yes	characteristic
453	C₂×C₅	#2957	1	1	yes	characteristic
454	C₂³	—	2	2	no	—
455	C₂³	—	2	1	no	—
456	C₂³	—	2	1	no	—
457	C₂³	—	2	2	no	—
458	C₂³	—	4	1	no	—
459	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
460	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
461	C₂³	C₂×C₅×#41	1	1	yes	characteristic
462	Q₈	—	2	2	no	—
463	Q₈	—	2	2	no	—
464	D₄	—	2	2	no	—
465	D₄	—	2	2	no	—
466	D₄	—	4	4	no	—
467	D₄	—	4	2	no	—
468	D₄	—	4	2	no	—
469	D₄	—	8	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
492	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
493	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
494	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
495	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
496	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
497	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
498	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
499	C₈	—	2	1	no	—
500	C₈	—	2	1	no	—
501	C₈	—	2	1	no	—
502	C₈	—	2	1	no	—
503	C₈	—	2	4	no	—
504	C₈	—	2	1	no	—
505	C₈	—	2	1	no	—
506	C₈	—	4	2	no	—
507	C₈	C₄×C₅×D₄	1	2	yes	—
508	C₈	C₄×C₅×D₄	1	2	yes	—
509	C₅	#23018	1	1	yes	characteristic, direct product factor, Sylow
510	C₂²	—	2	1	no	—
511	C₂²	—	2	1	no	—
512	C₂²	—	2	1	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	1	no	—
516	C₂²	—	2	1	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	2	no	—
522	C₂²	C₅×#443	1	2	yes	complemented
523	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
524	C₂²	C₅×#548	1	1	yes	characteristic
525	C₂²	C₅×#553	1	1	yes	characteristic
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	2	no	—
532	C₄	—	4	2	no	—
533	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
534	C₄	C₅×#520	1	1	yes	characteristic
535	C₄	C₅×D₄²	1	1	yes	characteristic
536	C₄	C₅×#547	1	1	yes	characteristic
537	C₂	—	2	1	no	—
538	C₂	—	2	1	no	—
539	C₂	—	2	2	no	—
540	C₂	—	4	1	no	—
541	C₂	—	8	2	no	—
542	C₂	C₅×#2361	1	1	yes	characteristic
543	C₂	C₂×C₅×#443	1	1	yes	characteristic
544	C₂	C₅×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₅×#3050	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₂²⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#443	C₂	semidirect
3. C₅×#2933	C₂	semidirect
4. C₂²×C₅×#108	C₂	semidirect
5. C₅×#1974	C₂	semidirect
6. C₅×#1945	C₂	semidirect
7. C₅×#1929	C₂	semidirect
8. C₅×#1871	C₂	semidirect
9. C₅×#1866	C₂	semidirect
10. C₅×#1772	C₂	semidirect
11. C₅×C₈×C₂²⋊C₄	C₂	semidirect
17. C₅×#467	C₄	semidirect
18. C₅×#467	C₄	semidirect
19. C₅×#466	C₄	semidirect
20. C₅×#466	C₄	semidirect
21. C₅×#465	C₄	semidirect
22. C₅×#465	C₄	semidirect
47. #23018	C₅	direct (N × H)
89. C₅×#108	D₄	semidirect
113. C₂×#443	C₂×C₅	semidirect
114. #2933	C₂×C₅	semidirect
115. C₂²×#108	C₂×C₅	semidirect
116. #1974	C₂×C₅	semidirect
117. #1945	C₂×C₅	semidirect
118. #1929	C₂×C₅	semidirect
119. #1871	C₂×C₅	semidirect
120. #1866	C₂×C₅	semidirect
121. #1772	C₂×C₅	semidirect
122. C₈×C₂²⋊C₄	C₂×C₅	semidirect
199. #467	C₄×C₅	semidirect
200. #467	C₄×C₅	semidirect
201. #466	C₄×C₅	semidirect
202. #466	C₄×C₅	semidirect
203. #465	C₄×C₅	semidirect
204. #465	C₄×C₅	semidirect
325. #108	C₅×D₄	semidirect
360. C₂²×C₅	#443	semidirect
509. C₅	#23018	direct (N × H)
522. C₂²	C₅×#443	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3050	2
C₂×C₅×#443	2
C₅×#2933	2
C₂²×C₅×#108	2
C₅×#1974	2
C₅×#1945	2
C₅×#1929	2
C₅×#1871	2
C₅×#1866	2
C₅×#1772	2
C₅×C₈×C₂²⋊C₄	2
#23018	5

Subgroups — C5×#23018 #11945738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#23018 #11945738