Subgroups — C₅×#21137 #11943857

Order 1280.

800 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₅×#2922	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#131	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1929	C₂	1	2	yes	maximal, complemented
5	C₅×#1922	C₂	1	2	yes	maximal, complemented
6	C₅×#1876	C₂	1	2	yes	maximal
7	C₅×#1866	C₂	1	2	yes	maximal, complemented
8	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₅×#104	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₅×#104	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1744	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	C₄×C₅×#41	C₂²	1	1	yes	characteristic
15	C₄×C₅×#41	C₂²	1	1	yes	characteristic
16	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
17	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
18	C₄×C₅×SD₁₆	C₄	1	1	yes	characteristic, complemented
19	C₄×C₅×SD₁₆	C₄	1	1	yes	characteristic, complemented
20	C₄×C₅×SD₁₆	C₄	1	1	yes	characteristic, complemented
21	C₄×C₅×SD₁₆	C₄	1	1	yes	characteristic, complemented
22	C₄×C₅×D₈	C₄	1	1	yes	characteristic, complemented
23	C₄×C₅×D₈	C₄	1	1	yes	characteristic, complemented
24	C₅×#433	—	2	2	no	—
25	C₅×#431	—	2	1	no	—
26	C₅×#431	—	2	1	no	—
27	C₂×C₅×#108	C₂²	1	2	yes	—
28	C₅×#427	C₂²	1	1	yes	characteristic
29	C₅×#424	C₂²	1	2	yes	—
30	C₂×C₅×#104	C₂²	1	2	yes	—
31	C₂×C₅×#104	C₂²	1	2	yes	—
32	C₅×#416	C₂²	1	1	yes	characteristic
33	C₅×#416	C₂²	1	1	yes	characteristic
34	C₅×#408	C₂²	1	2	yes	—
35	C₅×#405	—	2	2	no	—
36	C₅×#405	C₂²	1	2	yes	—
37	C₄×C₅×M₁₆	—	2	1	no	—
38	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
39	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₅×C₈	—	2	1	no	—
41	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
43	C₄³×C₅	C₂²	1	1	yes	characteristic
44	C₅×#341	C₂²	1	2	yes	—
45	C₅×#339	C₂²	1	2	yes	—
46	C₅×#339	C₂²	1	2	yes	—
47	C₅×#336	C₂²	1	2	yes	—
48	C₅×#336	C₂²	1	2	yes	—
49	#21137	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₅×#41	C₂³	1	1	yes	characteristic
51	C₂×C₅×#41	C₂³	1	1	yes	characteristic
52	C₅×#135	—	2	4	no	—
53	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
54	C₂×C₅×SD₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂×C₅×SD₁₆	C₂×C₄	1	1	yes	characteristic
56	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
57	C₅×#131	—	2	4	no	—
58	C₅×#131	—	2	4	no	—
59	C₂×C₅×M₁₆	—	2	2	no	—
60	C₂×C₅×M₁₆	C₂³	1	2	yes	—
61	C₂×C₅×M₁₆	C₂³	1	2	yes	—
62	C₂²×C₅×C₈	—	2	2	no	—
63	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
64	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₅×C₈	C₂³	1	2	yes	—
66	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₅×D₄	—	2	1	no	—
71	C₄×C₅×D₄	—	2	1	no	—
72	C₄×C₅×D₄	—	2	1	no	—
73	C₄×C₅×D₄	—	2	1	no	—
74	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
75	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
76	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
77	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
78	C₅×#117	—	2	1	no	—
79	C₅×#117	—	2	1	no	—
80	C₅×#117	C₂³	1	1	yes	characteristic
81	C₅×#117	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₅	—	2	1	no	—
83	C₂×C₄²×C₅	—	2	1	no	—
84	C₂×C₄²×C₅	—	2	1	no	—
85	C₂×C₄²×C₅	—	2	1	no	—
86	C₂×C₄²×C₅	C₂³	1	2	yes	—
87	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
88	C₅×#108	—	2	4	no	—
89	C₅×#107	C₂×C₄	1	1	yes	characteristic
90	C₅×#107	C₂×C₄	1	1	yes	characteristic
91	C₅×#106	C₂×C₄	1	1	yes	characteristic
92	C₅×#106	C₂×C₄	1	1	yes	characteristic
93	C₅×C₄⋊C₈	—	2	2	no	—
94	C₅×C₄⋊C₈	—	2	2	no	—
95	C₅×#104	—	2	4	no	—
96	C₅×#104	—	2	4	no	—
97	C₅×#103	C₂×C₄	1	1	yes	characteristic
98	C₅×#103	C₂×C₄	1	1	yes	characteristic
99	C₅×#103	C₂×C₄	1	1	yes	characteristic
100	C₅×#103	C₂×C₄	1	1	yes	characteristic
101	C₅×#102	C₂×C₄	1	1	yes	characteristic
102	C₅×#102	C₂×C₄	1	1	yes	characteristic
103	C₅×#102	C₂×C₄	1	1	yes	characteristic
104	C₅×#102	C₂×C₄	1	1	yes	characteristic
105	C₅×C₂²⋊C₈	—	2	2	no	—
106	C₅×C₂²⋊C₈	—	2	2	no	—
107	C₅×#97	—	2	2	no	—
108	C₅×#97	—	2	1	no	—
109	C₅×#97	—	2	1	no	—
110	C₄×C₅×C₈	—	2	2	no	—
111	C₄×C₅×C₈	—	2	1	no	—
112	C₄×C₅×C₈	—	2	1	no	—
113	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
114	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
115	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
116	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
117	C₄×C₅×C₈	D₄	1	2	yes	—
118	C₄×C₅×C₈	D₄	1	1	yes	characteristic
119	C₄×C₅×C₈	D₄	1	1	yes	characteristic
120	C₄×#135	C₂×C₅	1	1	yes	characteristic
121	#2922	C₂×C₅	1	1	yes	characteristic, complemented
122	C₄×#131	C₂×C₅	1	1	yes	characteristic, complemented
123	#1929	C₂×C₅	1	2	yes	complemented
124	#1922	C₂×C₅	1	2	yes	complemented
125	#1876	C₂×C₅	1	2	yes	—
126	#1866	C₂×C₅	1	2	yes	complemented
127	C₄×#108	C₂×C₅	1	1	yes	characteristic, complemented
128	C₄×#104	C₂×C₅	1	1	yes	characteristic, complemented
129	C₄×#104	C₂×C₅	1	1	yes	characteristic, complemented
130	#1744	C₂×C₅	1	1	yes	characteristic, complemented
131	C₅×#41	—	2	2	no	—
132	C₅×#41	—	2	2	no	—
133	C₅×#41	—	2	2	no	—
134	C₅×#41	#41	1	4	yes	—
135	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₅×D₄	—	2	1	no	—
138	C₂×C₅×D₄	—	2	1	no	—
139	C₂×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅	—	2	1	no	—
142	C₂²×C₄×C₅	—	2	2	no	—
143	C₂²×C₄×C₅	—	2	1	no	—
144	C₂²×C₄×C₅	—	2	1	no	—
145	C₂²×C₄×C₅	—	2	1	no	—
146	C₂²×C₄×C₅	—	2	1	no	—
147	C₂²×C₄×C₅	—	2	1	no	—
148	C₂²×C₄×C₅	#41	1	1	yes	characteristic
149	C₂²×C₄×C₅	#41	1	1	yes	characteristic
150	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
151	C₅×Q₁₆	—	2	2	no	—
152	C₅×SD₁₆	—	2	2	no	—
153	C₅×SD₁₆	—	2	2	no	—
154	C₅×D₈	—	2	2	no	—
155	C₅×M₁₆	—	2	8	no	—
156	C₅×M₁₆	—	2	2	no	—
157	C₅×M₁₆	—	2	2	no	—
158	C₅×M₁₆	—	2	2	no	—
159	C₅×M₁₆	#41	1	4	yes	—
160	C₂×C₅×C₈	—	2	8	no	—
161	C₂×C₅×C₈	—	2	2	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	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
172	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
173	C₅×C₄⋊C₄	—	2	1	no	—
174	C₅×C₄⋊C₄	—	2	1	no	—
175	C₅×C₄⋊C₄	—	2	1	no	—
176	C₅×C₄⋊C₄	—	2	1	no	—
177	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
178	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
179	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
180	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
181	C₅×C₂²⋊C₄	—	2	1	no	—
182	C₅×C₂²⋊C₄	—	2	1	no	—
183	C₅×C₂²⋊C₄	—	2	1	no	—
184	C₅×C₂²⋊C₄	—	2	1	no	—
185	C₅×C₂²⋊C₄	—	2	1	no	—
186	C₅×C₂²⋊C₄	—	2	1	no	—
187	C₄²×C₅	—	2	4	no	—
188	C₄²×C₅	—	2	1	no	—
189	C₄²×C₅	—	2	1	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	C₄²×C₅	—	2	2	no	—
195	C₄²×C₅	—	2	2	no	—
196	C₄²×C₅	—	4	2	no	—
197	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
198	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
199	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
201	C₂×#135	C₂²×C₅	1	1	yes	characteristic
202	C₂×#131	C₂²×C₅	1	2	yes	—
203	C₄×#41	C₂²×C₅	1	1	yes	characteristic
204	C₄×#41	C₂²×C₅	1	1	yes	characteristic
205	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic, complemented
206	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic, complemented
207	C₄×SD₁₆	C₄×C₅	1	1	yes	characteristic, complemented
208	C₄×SD₁₆	C₄×C₅	1	1	yes	characteristic, complemented
209	C₄×SD₁₆	C₄×C₅	1	1	yes	characteristic, complemented
210	C₄×SD₁₆	C₄×C₅	1	1	yes	characteristic, complemented
211	C₄×D₈	C₄×C₅	1	1	yes	characteristic, complemented
212	C₄×D₈	C₄×C₅	1	1	yes	characteristic, complemented
213	#433	—	2	2	no	—
214	#431	—	2	1	no	—
215	#431	—	2	1	no	—
216	C₂×#108	C₂²×C₅	1	2	yes	—
217	#427	C₂²×C₅	1	1	yes	characteristic
218	#424	C₂²×C₅	1	2	yes	—
219	C₂×#104	C₂²×C₅	1	2	yes	—
220	C₂×#104	C₂²×C₅	1	2	yes	—
221	#416	C₂²×C₅	1	1	yes	characteristic
222	#416	C₂²×C₅	1	1	yes	characteristic
223	#408	C₂²×C₅	1	2	yes	—
224	#405	—	2	2	no	—
225	#405	C₂²×C₅	1	2	yes	—
226	C₄×M₁₆	—	2	1	no	—
227	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
228	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
229	C₂×C₄×C₈	—	2	1	no	—
230	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
231	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
232	C₄³	C₂²×C₅	1	1	yes	characteristic
233	#341	C₂²×C₅	1	2	yes	—
234	#339	C₂²×C₅	1	2	yes	—
235	#339	C₂²×C₅	1	2	yes	—
236	#336	C₂²×C₅	1	2	yes	—
237	#336	C₂²×C₅	1	2	yes	—
238	C₂³×C₅	—	2	1	no	—
239	C₂³×C₅	—	2	1	no	—
240	C₂³×C₅	C₂×#41	1	1	yes	characteristic
241	C₅×Q₈	—	2	1	no	—
242	C₅×Q₈	—	2	1	no	—
243	C₅×Q₈	—	2	1	no	—
244	C₅×Q₈	#117	1	2	yes	—
245	C₅×D₄	—	2	4	no	—
246	C₅×D₄	—	2	1	no	—
247	C₅×D₄	—	2	1	no	—
248	C₅×D₄	—	2	1	no	—
249	C₅×D₄	—	4	2	no	—
250	C₅×D₄	#117	1	2	yes	—
251	C₂×C₄×C₅	—	2	1	no	—
252	C₂×C₄×C₅	—	2	4	no	—
253	C₂×C₄×C₅	—	2	2	no	—
254	C₂×C₄×C₅	—	2	2	no	—
255	C₂×C₄×C₅	—	2	4	no	—
256	C₂×C₄×C₅	—	2	1	no	—
257	C₂×C₄×C₅	—	2	1	no	—
258	C₂×C₄×C₅	—	2	1	no	—
259	C₂×C₄×C₅	—	2	1	no	—
260	C₂×C₄×C₅	—	2	1	no	—
261	C₂×C₄×C₅	—	4	2	no	—
262	C₂×C₄×C₅	—	4	2	no	—
263	C₂×C₄×C₅	—	4	2	no	—
264	C₂×C₄×C₅	—	4	2	no	—
265	C₂×C₄×C₅	—	4	2	no	—
266	C₂×C₄×C₅	—	4	2	no	—
267	C₂×C₄×C₅	—	4	2	no	—
268	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
269	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
270	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
271	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
272	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
273	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₅	C₂×#41	1	2	yes	—
275	C₂×C₄×C₅	C₂×#41	1	2	yes	—
276	C₅×C₈	—	2	1	no	—
277	C₅×C₈	—	2	1	no	—
278	C₅×C₈	—	2	1	no	—
279	C₅×C₈	—	2	1	no	—
280	C₅×C₈	—	2	2	no	—
281	C₅×C₈	—	2	4	no	—
282	C₅×C₈	—	2	2	no	—
283	C₅×C₈	—	4	2	no	—
284	C₂×#41	C₂³×C₅	1	1	yes	characteristic
285	C₂×#41	C₂³×C₅	1	1	yes	characteristic
286	#135	—	2	4	no	—
287	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
288	C₂×SD₁₆	C₂×C₄×C₅	1	1	yes	characteristic
289	C₂×SD₁₆	C₂×C₄×C₅	1	1	yes	characteristic
290	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
291	#131	—	2	4	no	—
292	#131	—	2	4	no	—
293	C₂×M₁₆	—	2	2	no	—
294	C₂×M₁₆	C₂³×C₅	1	2	yes	—
295	C₂×M₁₆	C₂³×C₅	1	2	yes	—
296	C₂²×C₈	—	2	2	no	—
297	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₅	1	2	yes	—
300	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
301	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
302	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
303	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
304	C₄×D₄	—	2	1	no	—
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	2	1	no	—
307	C₄×D₄	—	2	1	no	—
308	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
309	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
310	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
311	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
312	#117	—	2	1	no	—
313	#117	—	2	1	no	—
314	#117	C₂³×C₅	1	1	yes	characteristic
315	#117	C₂³×C₅	1	1	yes	characteristic
316	C₂×C₄²	—	2	1	no	—
317	C₂×C₄²	—	2	1	no	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	C₂³×C₅	1	2	yes	—
321	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
322	#108	—	2	4	no	—
323	#107	C₂×C₄×C₅	1	1	yes	characteristic
324	#107	C₂×C₄×C₅	1	1	yes	characteristic
325	#106	C₂×C₄×C₅	1	1	yes	characteristic
326	#106	C₂×C₄×C₅	1	1	yes	characteristic
327	C₄⋊C₈	—	2	2	no	—
328	C₄⋊C₈	—	2	2	no	—
329	#104	—	2	4	no	—
330	#104	—	2	4	no	—
331	#103	C₂×C₄×C₅	1	1	yes	characteristic
332	#103	C₂×C₄×C₅	1	1	yes	characteristic
333	#103	C₂×C₄×C₅	1	1	yes	characteristic
334	#103	C₂×C₄×C₅	1	1	yes	characteristic
335	#102	C₂×C₄×C₅	1	1	yes	characteristic
336	#102	C₂×C₄×C₅	1	1	yes	characteristic
337	#102	C₂×C₄×C₅	1	1	yes	characteristic
338	#102	C₂×C₄×C₅	1	1	yes	characteristic
339	C₂²⋊C₈	—	2	2	no	—
340	C₂²⋊C₈	—	2	2	no	—
341	#97	—	2	2	no	—
342	#97	—	2	1	no	—
343	#97	—	2	1	no	—
344	C₄×C₈	—	2	2	no	—
345	C₄×C₈	—	2	1	no	—
346	C₄×C₈	—	2	1	no	—
347	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
348	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
349	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
350	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
351	C₄×C₈	C₅×D₄	1	1	yes	characteristic
352	C₄×C₈	C₅×D₄	1	1	yes	characteristic
353	C₄×C₈	C₅×D₄	1	2	yes	—
354	C₂²×C₅	—	2	1	no	—
355	C₂²×C₅	—	2	1	no	—
356	C₂²×C₅	—	2	2	no	—
357	C₂²×C₅	—	4	1	no	—
358	C₂²×C₅	—	4	1	no	—
359	C₂²×C₅	—	4	1	no	—
360	C₂²×C₅	—	4	1	no	—
361	C₂²×C₅	—	4	1	no	—
362	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
363	C₂²×C₅	#529	1	2	yes	—
364	C₄×C₅	—	2	1	no	—
365	C₄×C₅	—	2	1	no	—
366	C₄×C₅	—	2	2	no	—
367	C₄×C₅	—	4	4	no	—
368	C₄×C₅	—	4	1	no	—
369	C₄×C₅	—	4	1	no	—
370	C₄×C₅	—	4	1	no	—
371	C₄×C₅	—	4	1	no	—
372	C₄×C₅	—	4	1	no	—
373	C₄×C₅	#443	1	2	yes	—
374	C₄×C₅	#444	1	2	yes	—
375	C₄×C₅	C₂×#117	1	1	yes	characteristic
376	C₄×C₅	C₄×#41	1	1	yes	characteristic
377	C₄×C₅	#525	1	2	yes	—
378	#41	—	2	2	no	—
379	#41	—	2	2	no	—
380	#41	—	2	2	no	—
381	#41	C₅×#41	1	4	yes	—
382	C₂×Q₈	C₂²×C₄×C₅	1	1	yes	characteristic
383	C₂×Q₈	C₂²×C₄×C₅	1	1	yes	characteristic
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	C₂²×C₄×C₅	1	1	yes	characteristic
387	C₂×D₄	C₂²×C₄×C₅	1	1	yes	characteristic
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	C₅×#41	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	2	no	—
400	SD₁₆	—	2	2	no	—
401	D₈	—	2	2	no	—
402	M₁₆	—	2	8	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	2	2	no	—
405	M₁₆	—	2	2	no	—
406	M₁₆	C₅×#41	1	4	yes	—
407	C₂×C₈	—	2	8	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
415	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
416	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
417	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
418	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
419	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
425	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
426	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
427	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
428	C₂²⋊C₄	—	2	1	no	—
429	C₂²⋊C₄	—	2	1	no	—
430	C₂²⋊C₄	—	2	1	no	—
431	C₂²⋊C₄	—	2	1	no	—
432	C₂²⋊C₄	—	2	1	no	—
433	C₂²⋊C₄	—	2	1	no	—
434	C₄²	—	2	4	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	1	no	—
437	C₄²	—	2	1	no	—
438	C₄²	—	2	1	no	—
439	C₄²	—	2	1	no	—
440	C₄²	—	2	1	no	—
441	C₄²	—	2	2	no	—
442	C₄²	—	2	2	no	—
443	C₄²	—	4	2	no	—
444	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
445	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
446	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
447	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
448	C₂×C₅	—	2	2	no	—
449	C₂×C₅	—	4	2	no	—
450	C₂×C₅	—	8	1	no	—
451	C₂×C₅	#2290	1	1	yes	characteristic
452	C₂×C₅	C₂×#443	1	1	yes	characteristic
453	C₂×C₅	C₂×#444	1	1	yes	characteristic
454	C₂³	—	2	1	no	—
455	C₂³	—	2	1	no	—
456	C₂³	C₂×C₅×#41	1	1	yes	characteristic
457	Q₈	—	2	1	no	—
458	Q₈	—	2	1	no	—
459	Q₈	—	2	1	no	—
460	Q₈	C₅×#117	1	2	yes	—
461	D₄	—	2	4	no	—
462	D₄	—	2	1	no	—
463	D₄	—	2	1	no	—
464	D₄	—	2	1	no	—
465	D₄	—	4	2	no	—
466	D₄	C₅×#117	1	2	yes	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	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	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
485	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
486	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
487	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
488	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
489	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
490	C₂×C₄	C₂×C₅×#41	1	2	yes	—
491	C₂×C₄	C₂×C₅×#41	1	2	yes	—
492	C₈	—	2	1	no	—
493	C₈	—	2	1	no	—
494	C₈	—	2	1	no	—
495	C₈	—	2	1	no	—
496	C₈	—	2	2	no	—
497	C₈	—	2	4	no	—
498	C₈	—	2	2	no	—
499	C₈	—	4	2	no	—
500	C₅	#21137	1	1	yes	characteristic, direct product factor, Sylow
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	4	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	4	1	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	1	no	—
509	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
510	C₂²	C₅×#529	1	2	yes	—
511	C₄	—	2	2	no	—
512	C₄	—	2	1	no	—
513	C₄	—	2	1	no	—
514	C₄	—	4	4	no	—
515	C₄	—	4	1	no	—
516	C₄	—	4	1	no	—
517	C₄	—	4	1	no	—
518	C₄	—	4	1	no	—
519	C₄	—	4	1	no	—
520	C₄	C₅×#443	1	2	yes	—
521	C₄	C₅×#444	1	2	yes	—
522	C₄	C₂×C₅×#117	1	1	yes	characteristic
523	C₄	C₄×C₅×#41	1	1	yes	characteristic
524	C₄	C₅×#525	1	2	yes	—
525	C₂	—	2	2	no	—
526	C₂	—	4	2	no	—
527	C₂	—	8	1	no	—
528	C₂	C₅×#2290	1	1	yes	characteristic
529	C₂	C₂×C₅×#443	1	1	yes	characteristic
530	C₂	C₂×C₅×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2922	C₂	semidirect
3. C₄×C₅×#131	C₂	semidirect
4. C₅×#1929	C₂	semidirect
5. C₅×#1922	C₂	semidirect
7. C₅×#1866	C₂	semidirect
8. C₄×C₅×#108	C₂	semidirect
9. C₄×C₅×#104	C₂	semidirect
10. C₄×C₅×#104	C₂	semidirect
11. C₅×#1744	C₂	semidirect
16. C₄×C₅×Q₁₆	C₄	semidirect
17. C₄×C₅×Q₁₆	C₄	semidirect
18. C₄×C₅×SD₁₆	C₄	semidirect
19. C₄×C₅×SD₁₆	C₄	semidirect
20. C₄×C₅×SD₁₆	C₄	semidirect
21. C₄×C₅×SD₁₆	C₄	semidirect
22. C₄×C₅×D₈	C₄	semidirect
23. C₄×C₅×D₈	C₄	semidirect
49. #21137	C₅	direct (N × H)
121. #2922	C₂×C₅	semidirect
122. C₄×#131	C₂×C₅	semidirect
123. #1929	C₂×C₅	semidirect
124. #1922	C₂×C₅	semidirect
126. #1866	C₂×C₅	semidirect
127. C₄×#108	C₂×C₅	semidirect
128. C₄×#104	C₂×C₅	semidirect
129. C₄×#104	C₂×C₅	semidirect
130. #1744	C₂×C₅	semidirect
205. C₄×Q₁₆	C₄×C₅	semidirect
206. C₄×Q₁₆	C₄×C₅	semidirect
207. C₄×SD₁₆	C₄×C₅	semidirect
208. C₄×SD₁₆	C₄×C₅	semidirect
209. C₄×SD₁₆	C₄×C₅	semidirect
210. C₄×SD₁₆	C₄×C₅	semidirect
211. C₄×D₈	C₄×C₅	semidirect
212. C₄×D₈	C₄×C₅	semidirect
500. C₅	#21137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#135	2
C₅×#2922	2
C₄×C₅×#131	2
C₅×#1929	2
C₅×#1922	2
C₅×#1876	2
C₅×#1866	2
C₄×C₅×#108	2
C₄×C₅×#104	2
C₅×#1744	2
#21137	5

Subgroups — C5×#21137 #11943857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21137 #11943857