Subgroups — C₅×#23097 #11945817

Order 1280.

858 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#464	C₂	1	1	yes	maximal, characteristic
2	C₅×#2367	C₂	1	1	yes	maximal, characteristic
3	C₅×#2356	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1969	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1968	C₂	1	1	yes	maximal, characteristic
6	C₅×#1958	C₂	1	1	yes	maximal, characteristic
7	C₅×#1949	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1927	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1925	C₂	1	1	yes	maximal, characteristic
10	C₅×#1878	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1877	C₂	1	1	yes	maximal, characteristic
12	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₅×#103	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₅×#103	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₅×#102	C₂	1	1	yes	maximal, characteristic
16	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₅×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₅×#124	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
22	C₅×#464	C₄	1	8	yes	complemented
23	C₂×C₅×#107	C₂²	1	1	yes	characteristic
24	C₂×C₅×#106	C₂²	1	1	yes	characteristic
25	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₅×#103	C₂²	1	1	yes	characteristic
27	C₂×C₅×#103	C₂²	1	1	yes	characteristic
28	C₂×C₅×#103	C₂²	1	1	yes	characteristic
29	C₂×C₅×#103	C₂²	1	1	yes	characteristic
30	C₂×C₅×#103	C₂²	1	1	yes	characteristic
31	C₂×C₅×#102	C₂²	1	1	yes	characteristic
32	C₂×C₅×#102	C₂²	1	1	yes	characteristic
33	C₂×C₅×#102	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
37	C₅×#391	C₂²	1	1	yes	characteristic
38	C₅×#391	C₂²	1	1	yes	characteristic
39	C₅×#391	C₂²	1	1	yes	characteristic
40	C₅×#390	C₂²	1	1	yes	characteristic
41	C₅×#388	—	2	1	no	—
42	C₅×#388	—	2	1	no	—
43	C₅×#387	—	2	1	no	—
44	C₅×#387	—	2	1	no	—
45	C₅×#385	—	2	1	no	—
46	C₅×#383	—	2	1	no	—
47	C₅×#382	C₂²	1	1	yes	characteristic
48	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
49	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
50	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
51	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
52	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
53	C₅×#376	—	2	1	no	—
54	C₅×#340	C₂²	1	1	yes	characteristic
55	C₅×#340	C₂²	1	1	yes	characteristic
56	C₅×#340	C₂²	1	1	yes	characteristic
57	C₅×#340	C₂²	1	1	yes	characteristic
58	C₅×#336	C₂²	1	1	yes	characteristic
59	C₅×#336	C₂²	1	1	yes	characteristic
60	#23097	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
63	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₅	—	2	1	no	—
65	C₂×C₅×Q₁₆	—	4	1	no	—
66	C₂×C₅×Q₁₆	—	4	1	no	—
67	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	complemented
68	C₂×C₅×SD₁₆	—	4	1	no	—
69	C₂×C₅×SD₁₆	—	4	1	no	—
70	C₂×C₅×SD₁₆	C₂×C₄	1	4	yes	—
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×#124	—	2	2	no	—
76	C₅×#124	C₂×C₄	1	4	yes	—
77	C₄×C₅×Q₈	—	2	2	no	—
78	C₄×C₅×Q₈	—	2	2	no	—
79	C₄×C₅×Q₈	—	4	1	no	—
80	C₄×C₅×Q₈	C₂×C₄	1	4	yes	complemented
81	C₄×C₅×D₄	—	2	2	no	—
82	C₄×C₅×D₄	—	4	1	no	—
83	C₂×C₅×C₄⋊C₄	—	2	1	no	—
84	C₂×C₅×C₄⋊C₄	—	2	1	no	—
85	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
86	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
93	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
94	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
95	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
96	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
97	C₂×C₄²×C₅	—	2	1	no	—
98	C₂×C₄²×C₅	—	2	1	no	—
99	C₂×C₄²×C₅	—	2	1	no	—
100	C₂×C₄²×C₅	—	2	1	no	—
101	C₂×C₄²×C₅	—	2	1	no	—
102	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
105	C₅×#107	—	4	1	no	—
106	C₅×#106	—	4	1	no	—
107	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
108	C₅×#103	—	2	2	no	—
109	C₅×#103	—	2	2	no	—
110	C₅×#103	—	2	2	no	—
111	C₅×#103	—	4	1	no	—
112	C₅×#103	C₂×C₄	1	4	yes	complemented
113	C₅×#102	—	2	2	no	—
114	C₅×#102	—	4	1	no	—
115	C₅×#102	C₂×C₄	1	4	yes	—
116	C₄×C₅×C₈	—	2	2	no	—
117	C₄×C₅×C₈	—	4	1	no	—
118	C₄×C₅×C₈	—	4	1	no	—
119	C₅×#95	—	2	1	no	—
120	C₅×#95	—	2	1	no	—
121	C₅×#95	—	2	1	no	—
122	C₅×#95	—	2	1	no	—
123	C₅×#95	—	2	1	no	—
124	C₅×#95	—	2	1	no	—
125	C₅×#95	—	2	1	no	—
126	C₅×#95	—	2	1	no	—
127	C₅×#95	—	2	1	no	—
128	C₅×#95	—	2	1	no	—
129	C₅×#95	—	2	1	no	—
130	C₅×#95	D₄	1	1	yes	characteristic
131	C₅×#95	D₄	1	1	yes	characteristic
132	C₂×#464	C₂×C₅	1	1	yes	characteristic
133	#2367	C₂×C₅	1	1	yes	characteristic
134	#2356	C₂×C₅	1	1	yes	characteristic, complemented
135	#1969	C₂×C₅	1	1	yes	characteristic, complemented
136	#1968	C₂×C₅	1	1	yes	characteristic
137	#1958	C₂×C₅	1	1	yes	characteristic
138	#1949	C₂×C₅	1	1	yes	characteristic, complemented
139	#1927	C₂×C₅	1	1	yes	characteristic, complemented
140	#1925	C₂×C₅	1	1	yes	characteristic
141	#1878	C₂×C₅	1	1	yes	characteristic, complemented
142	#1877	C₂×C₅	1	1	yes	characteristic
143	C₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
144	C₄×#103	C₂×C₅	1	1	yes	characteristic, complemented
145	C₄×#103	C₂×C₅	1	1	yes	characteristic, complemented
146	C₄×#102	C₂×C₅	1	1	yes	characteristic
147	C₂⁴×C₅	—	2	1	no	—
148	C₂×C₅×Q₈	—	2	1	no	—
149	C₂×C₅×Q₈	—	2	1	no	—
150	C₂×C₅×Q₈	—	2	1	no	—
151	C₂×C₅×Q₈	—	2	1	no	—
152	C₂×C₅×Q₈	—	2	1	no	—
153	C₂×C₅×Q₈	—	2	1	no	—
154	C₂×C₅×Q₈	—	2	1	no	—
155	C₂×C₅×Q₈	—	2	1	no	—
156	C₂×C₅×Q₈	—	2	1	no	—
157	C₂×C₅×Q₈	—	2	1	no	—
158	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
159	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
160	C₂×C₅×D₄	—	2	1	no	—
161	C₂×C₅×D₄	—	2	1	no	—
162	C₂×C₅×D₄	—	2	1	no	—
163	C₂×C₅×D₄	—	2	1	no	—
164	C₂×C₅×D₄	—	2	1	no	—
165	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
166	C₂²×C₄×C₅	—	2	1	no	—
167	C₂²×C₄×C₅	—	2	1	no	—
168	C₂²×C₄×C₅	—	2	1	no	—
169	C₂²×C₄×C₅	—	2	1	no	—
170	C₂²×C₄×C₅	—	2	1	no	—
171	C₂²×C₄×C₅	—	2	1	no	—
172	C₂²×C₄×C₅	—	2	1	no	—
173	C₂²×C₄×C₅	—	4	2	no	—
174	C₂²×C₄×C₅	—	4	2	no	—
175	C₂²×C₄×C₅	—	8	1	no	—
176	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
183	C₅×Q₁₆	—	4	4	no	—
184	C₅×SD₁₆	—	4	4	no	—
185	C₂×C₅×C₈	—	2	1	no	—
186	C₂×C₅×C₈	—	2	1	no	—
187	C₂×C₅×C₈	—	2	1	no	—
188	C₂×C₅×C₈	—	2	1	no	—
189	C₂×C₅×C₈	—	2	1	no	—
190	C₂×C₅×C₈	—	2	1	no	—
191	C₂×C₅×C₈	—	2	1	no	—
192	C₂×C₅×C₈	—	2	1	no	—
193	C₂×C₅×C₈	—	2	1	no	—
194	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
195	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
196	C₂×C₅×C₈	#41	1	2	yes	—
197	C₅×C₄⋊C₄	—	2	4	no	—
198	C₅×C₄⋊C₄	—	2	2	no	—
199	C₅×C₄⋊C₄	—	2	2	no	—
200	C₅×C₄⋊C₄	—	2	1	no	—
201	C₅×C₄⋊C₄	—	2	1	no	—
202	C₅×C₄⋊C₄	—	2	1	no	—
203	C₅×C₄⋊C₄	—	2	1	no	—
204	C₅×C₄⋊C₄	—	2	1	no	—
205	C₅×C₄⋊C₄	—	2	1	no	—
206	C₅×C₄⋊C₄	—	2	1	no	—
207	C₅×C₄⋊C₄	—	2	1	no	—
208	C₅×C₄⋊C₄	—	4	2	no	—
209	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
210	C₅×C₄⋊C₄	#41	1	2	yes	—
211	C₅×C₂²⋊C₄	—	2	4	no	—
212	C₅×C₂²⋊C₄	—	2	4	no	—
213	C₅×C₂²⋊C₄	—	4	2	no	—
214	C₅×C₂²⋊C₄	—	4	2	no	—
215	C₅×C₂²⋊C₄	—	8	1	no	—
216	C₄²×C₅	—	2	4	no	—
217	C₄²×C₅	—	2	1	no	—
218	C₄²×C₅	—	2	1	no	—
219	C₄²×C₅	—	2	1	no	—
220	C₄²×C₅	—	2	1	no	—
221	C₄²×C₅	—	4	2	no	—
222	C₄²×C₅	—	4	2	no	—
223	C₄²×C₅	—	4	2	no	—
224	C₄²×C₅	—	8	1	no	—
225	C₄²×C₅	C₂²×C₄	1	2	yes	—
226	C₄²×C₅	#41	1	2	yes	—
227	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
228	C₂²×SD₁₆	C₂²×C₅	1	1	yes	characteristic
229	C₂×#124	C₂²×C₅	1	1	yes	characteristic
230	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
231	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
232	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
233	#464	C₄×C₅	1	8	yes	complemented
234	C₂×#107	C₂²×C₅	1	1	yes	characteristic
235	C₂×#106	C₂²×C₅	1	1	yes	characteristic
236	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
237	C₂×#103	C₂²×C₅	1	1	yes	characteristic
238	C₂×#103	C₂²×C₅	1	1	yes	characteristic
239	C₂×#103	C₂²×C₅	1	1	yes	characteristic
240	C₂×#103	C₂²×C₅	1	1	yes	characteristic
241	C₂×#103	C₂²×C₅	1	1	yes	characteristic
242	C₂×#102	C₂²×C₅	1	1	yes	characteristic
243	C₂×#102	C₂²×C₅	1	1	yes	characteristic
244	C₂×#102	C₂²×C₅	1	1	yes	characteristic
245	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
246	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
247	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
248	#391	C₂²×C₅	1	1	yes	characteristic
249	#391	C₂²×C₅	1	1	yes	characteristic
250	#391	C₂²×C₅	1	1	yes	characteristic
251	#390	C₂²×C₅	1	1	yes	characteristic
252	#388	—	2	1	no	—
253	#388	—	2	1	no	—
254	#387	—	2	1	no	—
255	#387	—	2	1	no	—
256	#385	—	2	1	no	—
257	#383	—	2	1	no	—
258	#382	C₂²×C₅	1	1	yes	characteristic
259	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
260	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
261	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
262	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
263	C₄×C₂²⋊C₄	—	2	1	no	—
264	#376	—	2	1	no	—
265	#340	C₂²×C₅	1	1	yes	characteristic
266	#340	C₂²×C₅	1	1	yes	characteristic
267	#340	C₂²×C₅	1	1	yes	characteristic
268	#340	C₂²×C₅	1	1	yes	characteristic
269	#336	C₂²×C₅	1	1	yes	characteristic
270	#336	C₂²×C₅	1	1	yes	characteristic
271	C₂³×C₅	—	2	2	no	—
272	C₂³×C₅	—	4	1	no	—
273	C₂³×C₅	—	4	1	no	—
274	C₂³×C₅	—	4	1	no	—
275	C₂³×C₅	—	4	1	no	—
276	C₂³×C₅	—	4	1	no	—
277	C₂³×C₅	—	4	1	no	—
278	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
279	C₅×Q₈	—	2	2	no	—
280	C₅×Q₈	—	2	2	no	—
281	C₅×Q₈	—	2	2	no	—
282	C₅×Q₈	—	2	2	no	—
283	C₅×Q₈	—	4	1	no	—
284	C₅×Q₈	—	4	1	no	—
285	C₅×Q₈	—	4	1	no	—
286	C₅×Q₈	—	4	1	no	—
287	C₅×D₄	—	2	2	no	—
288	C₅×D₄	—	2	2	no	—
289	C₅×D₄	—	4	1	no	—
290	C₅×D₄	—	4	1	no	—
291	C₂×C₄×C₅	—	2	1	no	—
292	C₂×C₄×C₅	—	2	1	no	—
293	C₂×C₄×C₅	—	2	1	no	—
294	C₂×C₄×C₅	—	2	1	no	—
295	C₂×C₄×C₅	—	2	1	no	—
296	C₂×C₄×C₅	—	2	1	no	—
297	C₂×C₄×C₅	—	2	1	no	—
298	C₂×C₄×C₅	—	2	1	no	—
299	C₂×C₄×C₅	—	2	2	no	—
300	C₂×C₄×C₅	—	2	2	no	—
301	C₂×C₄×C₅	—	2	2	no	—
302	C₂×C₄×C₅	—	2	1	no	—
303	C₂×C₄×C₅	—	2	1	no	—
304	C₂×C₄×C₅	—	2	2	no	—
305	C₂×C₄×C₅	—	2	1	no	—
306	C₂×C₄×C₅	—	2	1	no	—
307	C₂×C₄×C₅	—	2	1	no	—
308	C₂×C₄×C₅	—	2	1	no	—
309	C₂×C₄×C₅	—	4	1	no	—
310	C₂×C₄×C₅	—	4	1	no	—
311	C₂×C₄×C₅	—	4	1	no	—
312	C₂×C₄×C₅	—	4	1	no	—
313	C₂×C₄×C₅	—	4	1	no	—
314	C₂×C₄×C₅	—	4	1	no	—
315	C₂×C₄×C₅	—	4	1	no	—
316	C₂×C₄×C₅	—	4	1	no	—
317	C₂×C₄×C₅	—	4	1	no	—
318	C₂×C₄×C₅	—	4	1	no	—
319	C₂×C₄×C₅	—	4	1	no	—
320	C₂×C₄×C₅	—	4	1	no	—
321	C₂×C₄×C₅	—	4	1	no	—
322	C₂×C₄×C₅	—	4	1	no	—
323	C₂×C₄×C₅	—	4	1	no	—
324	C₂×C₄×C₅	—	4	1	no	—
325	C₂×C₄×C₅	—	4	1	no	—
326	C₂×C₄×C₅	—	8	4	no	—
327	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
328	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
329	C₂×C₄×C₅	#135	1	2	yes	—
330	C₂×C₄×C₅	#135	1	2	yes	—
331	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
332	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
333	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
334	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
335	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
336	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
337	C₅×C₈	—	2	2	no	—
338	C₅×C₈	—	2	2	no	—
339	C₅×C₈	—	2	2	no	—
340	C₅×C₈	—	4	1	no	—
341	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
342	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
343	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
344	C₂³×C₄	—	2	1	no	—
345	C₂×Q₁₆	—	4	1	no	—
346	C₂×Q₁₆	—	4	1	no	—
347	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	complemented
348	C₂×SD₁₆	—	4	1	no	—
349	C₂×SD₁₆	—	4	1	no	—
350	C₂×SD₁₆	C₂×C₄×C₅	1	4	yes	—
351	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
352	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
354	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
355	#124	—	2	2	no	—
356	#124	C₂×C₄×C₅	1	4	yes	—
357	C₄×Q₈	—	2	2	no	—
358	C₄×Q₈	—	2	2	no	—
359	C₄×Q₈	—	4	1	no	—
360	C₄×Q₈	C₂×C₄×C₅	1	4	yes	complemented
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	—	4	1	no	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
366	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
372	C₂×C₂²⋊C₄	—	2	1	no	—
373	C₂×C₂²⋊C₄	—	2	1	no	—
374	C₂×C₂²⋊C₄	—	2	1	no	—
375	C₂×C₂²⋊C₄	—	2	1	no	—
376	C₂×C₂²⋊C₄	—	2	1	no	—
377	C₂×C₄²	—	2	1	no	—
378	C₂×C₄²	—	2	1	no	—
379	C₂×C₄²	—	2	1	no	—
380	C₂×C₄²	—	2	1	no	—
381	C₂×C₄²	—	2	1	no	—
382	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
383	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
384	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
385	#107	—	4	1	no	—
386	#106	—	4	1	no	—
387	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	complemented
388	#103	—	2	2	no	—
389	#103	—	2	2	no	—
390	#103	—	2	2	no	—
391	#103	—	4	1	no	—
392	#103	C₂×C₄×C₅	1	4	yes	complemented
393	#102	—	2	2	no	—
394	#102	—	4	1	no	—
395	#102	C₂×C₄×C₅	1	4	yes	—
396	C₄×C₈	—	2	2	no	—
397	C₄×C₈	—	4	1	no	—
398	C₄×C₈	—	4	1	no	—
399	#95	—	2	1	no	—
400	#95	—	2	1	no	—
401	#95	—	2	1	no	—
402	#95	—	2	1	no	—
403	#95	—	2	1	no	—
404	#95	—	2	1	no	—
405	#95	—	2	1	no	—
406	#95	—	2	1	no	—
407	#95	—	2	1	no	—
408	#95	—	2	1	no	—
409	#95	—	2	1	no	—
410	#95	C₅×D₄	1	1	yes	characteristic
411	#95	C₅×D₄	1	1	yes	characteristic
412	C₂²×C₅	—	4	2	no	—
413	C₂²×C₅	—	4	2	no	—
414	C₂²×C₅	—	4	2	no	—
415	C₂²×C₅	—	8	1	no	—
416	C₂²×C₅	—	8	1	no	—
417	C₂²×C₅	—	8	1	no	—
418	C₂²×C₅	—	8	1	no	—
419	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
420	C₂²×C₅	#546	1	1	yes	characteristic
421	C₂²×C₅	#550	1	1	yes	characteristic
422	C₂²×C₅	C₂×#135	1	1	yes	characteristic
423	C₂²×C₅	C₂×#135	1	1	yes	characteristic
424	C₂²×C₅	#577	1	1	yes	characteristic
425	C₂²×C₅	#578	1	1	yes	characteristic
426	C₄×C₅	—	2	2	no	—
427	C₄×C₅	—	2	2	no	—
428	C₄×C₅	—	2	2	no	—
429	C₄×C₅	—	2	2	no	—
430	C₄×C₅	—	4	2	no	—
431	C₄×C₅	—	4	2	no	—
432	C₄×C₅	—	4	2	no	—
433	C₄×C₅	—	4	1	no	—
434	C₄×C₅	—	4	2	no	—
435	C₄×C₅	—	4	1	no	—
436	C₄×C₅	—	8	1	no	—
437	C₄×C₅	—	8	1	no	—
438	C₄×C₅	—	8	1	no	—
439	C₄×C₅	C₄×#41	1	1	yes	characteristic
440	C₄×C₅	#518	1	1	yes	characteristic
441	C₄×C₅	#553	1	1	yes	characteristic
442	C₄×C₅	#555	1	1	yes	characteristic
443	C₂⁴	—	2	1	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
455	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	—	8	1	no	—
472	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
473	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
474	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
475	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
476	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
477	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
479	Q₁₆	—	4	4	no	—
480	SD₁₆	—	4	4	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
491	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
492	C₂×C₈	C₅×#41	1	2	yes	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	4	2	no	—
505	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
506	C₄⋊C₄	C₅×#41	1	2	yes	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	4	2	no	—
510	C₂²⋊C₄	—	4	2	no	—
511	C₂²⋊C₄	—	8	1	no	—
512	C₄²	—	2	4	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	2	1	no	—
517	C₄²	—	4	2	no	—
518	C₄²	—	4	2	no	—
519	C₄²	—	4	2	no	—
520	C₄²	—	8	1	no	—
521	C₄²	C₂²×C₄×C₅	1	2	yes	—
522	C₄²	C₅×#41	1	2	yes	—
523	C₂×C₅	—	8	2	no	—
524	C₂×C₅	#2363	1	1	yes	characteristic
525	C₂×C₅	C₄×#135	1	1	yes	characteristic
526	C₂×C₅	#2947	1	1	yes	characteristic
527	C₂×C₅	#3296	1	1	yes	characteristic
528	C₂×C₅	#3300	1	1	yes	characteristic
529	C₂×C₅	#3393	1	1	yes	characteristic
530	C₂×C₅	#3394	1	1	yes	characteristic
531	C₂³	—	2	2	no	—
532	C₂³	—	4	1	no	—
533	C₂³	—	4	1	no	—
534	C₂³	—	4	1	no	—
535	C₂³	—	4	1	no	—
536	C₂³	—	4	1	no	—
537	C₂³	—	4	1	no	—
538	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
539	Q₈	—	2	2	no	—
540	Q₈	—	2	2	no	—
541	Q₈	—	2	2	no	—
542	Q₈	—	2	2	no	—
543	Q₈	—	4	1	no	—
544	Q₈	—	4	1	no	—
545	Q₈	—	4	1	no	—
546	Q₈	—	4	1	no	—
547	D₄	—	2	2	no	—
548	D₄	—	2	2	no	—
549	D₄	—	4	1	no	—
550	D₄	—	4	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	8	4	no	—
587	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
588	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
589	C₂×C₄	C₅×#135	1	2	yes	—
590	C₂×C₄	C₅×#135	1	2	yes	—
591	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
592	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
593	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
594	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
595	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
596	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
597	C₈	—	2	2	no	—
598	C₈	—	2	2	no	—
599	C₈	—	2	2	no	—
600	C₈	—	4	1	no	—
601	C₅	#23097	1	1	yes	characteristic, direct product factor, Sylow
602	C₂²	—	4	2	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	8	1	no	—
606	C₂²	—	8	1	no	—
607	C₂²	—	8	1	no	—
608	C₂²	—	8	1	no	—
609	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
610	C₂²	C₅×#546	1	1	yes	characteristic
611	C₂²	C₅×#550	1	1	yes	characteristic
612	C₂²	C₂×C₅×#135	1	1	yes	characteristic
613	C₂²	C₂×C₅×#135	1	1	yes	characteristic
614	C₂²	C₅×#577	1	1	yes	characteristic
615	C₂²	C₅×#578	1	1	yes	characteristic
616	C₄	—	2	2	no	—
617	C₄	—	2	2	no	—
618	C₄	—	2	2	no	—
619	C₄	—	2	2	no	—
620	C₄	—	4	1	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	1	no	—
623	C₄	—	4	2	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	2	no	—
626	C₄	—	8	1	no	—
627	C₄	—	8	1	no	—
628	C₄	—	8	1	no	—
629	C₄	C₄×C₅×#41	1	1	yes	characteristic
630	C₄	C₅×#518	1	1	yes	characteristic
631	C₄	C₅×#553	1	1	yes	characteristic
632	C₄	C₅×#555	1	1	yes	characteristic
633	C₂	—	8	2	no	—
634	C₂	C₅×#2363	1	1	yes	characteristic
635	C₂	C₄×C₅×#135	1	1	yes	characteristic
636	C₂	C₅×#2947	1	1	yes	characteristic
637	C₂	C₅×#3296	1	1	yes	characteristic
638	C₂	C₅×#3300	1	1	yes	characteristic
639	C₂	C₅×#3393	1	1	yes	characteristic
640	C₂	C₅×#3394	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₅×#2356	C₂	semidirect
4. C₅×#1969	C₂	semidirect
7. C₅×#1949	C₂	semidirect
8. C₅×#1927	C₂	semidirect
10. C₅×#1878	C₂	semidirect
12. C₅×C₈×C₄⋊C₄	C₂	semidirect
13. C₄×C₅×#103	C₂	semidirect
14. C₄×C₅×#103	C₂	semidirect
22. C₅×#464	C₄	semidirect
60. #23097	C₅	direct (N × H)
67. C₂×C₅×Q₁₆	C₂×C₄	semidirect
80. C₄×C₅×Q₈	C₂×C₄	semidirect
107. C₅×C₄⋊C₈	C₂×C₄	semidirect
112. C₅×#103	C₂×C₄	semidirect
134. #2356	C₂×C₅	semidirect
135. #1969	C₂×C₅	semidirect
138. #1949	C₂×C₅	semidirect
139. #1927	C₂×C₅	semidirect
141. #1878	C₂×C₅	semidirect
143. C₈×C₄⋊C₄	C₂×C₅	semidirect
144. C₄×#103	C₂×C₅	semidirect
145. C₄×#103	C₂×C₅	semidirect
233. #464	C₄×C₅	semidirect
347. C₂×Q₁₆	C₂×C₄×C₅	semidirect
360. C₄×Q₈	C₂×C₄×C₅	semidirect
387. C₄⋊C₈	C₂×C₄×C₅	semidirect
392. #103	C₂×C₄×C₅	semidirect
601. C₅	#23097	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#464	2
C₅×#2367	2
C₅×#2356	2
C₅×#1969	2
C₅×#1968	2
C₅×#1958	2
C₅×#1949	2
C₅×#1927	2
C₅×#1925	2
C₅×#1878	2
C₅×#1877	2
C₅×C₈×C₄⋊C₄	2
C₄×C₅×#103	2
C₄×C₅×#102	2
#23097	5

Subgroups — C5×#23097 #11945817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#23097 #11945817