Subgroups — C₅×#23098 #11945818

Order 1280.

906 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#463	C₂	1	1	yes	maximal, characteristic
2	C₅×#2367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2357	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1970	C₂	1	1	yes	maximal, characteristic
5	C₅×#1968	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1958	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1949	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1928	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1926	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1880	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1879	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×#102	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₅×#102	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₅×SD₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₅×D₈	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×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
22	C₅×#463	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×#102	C₂²	1	1	yes	characteristic
30	C₂×C₅×#102	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×#390	C₂²	1	1	yes	characteristic
39	C₅×#390	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×#388	—	2	1	no	—
44	C₅×#388	—	2	1	no	—
45	C₅×#385	—	2	1	no	—
46	C₅×#382	C₂²	1	1	yes	characteristic
47	C₅×#381	—	2	1	no	—
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	C₄×C₅×C₂²⋊C₄	—	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	#23098	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
63	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₅	—	2	1	no	—
65	C₂³×C₄×C₅	—	2	1	no	—
66	C₂×C₅×SD₁₆	—	4	1	no	—
67	C₂×C₅×SD₁₆	—	4	1	no	—
68	C₂×C₅×SD₁₆	C₂×C₄	1	4	yes	complemented
69	C₂×C₅×D₈	—	4	1	no	—
70	C₂×C₅×D₈	—	4	1	no	—
71	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
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	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₅×#124	—	2	2	no	—
77	C₅×#124	C₂×C₄	1	4	yes	—
78	C₄×C₅×Q₈	—	2	2	no	—
79	C₄×C₅×Q₈	—	4	1	no	—
80	C₄×C₅×D₄	—	2	2	no	—
81	C₄×C₅×D₄	—	2	2	no	—
82	C₄×C₅×D₄	—	4	1	no	—
83	C₄×C₅×D₄	C₂×C₄	1	4	yes	complemented
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×C₅×C₂²⋊C₄	—	2	1	no	—
98	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
99	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
100	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
101	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
102	C₂×C₄²×C₅	—	2	1	no	—
103	C₂×C₄²×C₅	—	2	1	no	—
104	C₂×C₄²×C₅	—	2	1	no	—
105	C₂×C₄²×C₅	—	2	1	no	—
106	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
109	C₅×#107	—	4	1	no	—
110	C₅×#106	—	4	1	no	—
111	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
112	C₅×#103	—	2	2	no	—
113	C₅×#103	—	4	1	no	—
114	C₅×#103	C₂×C₄	1	4	yes	complemented
115	C₅×#102	—	2	2	no	—
116	C₅×#102	—	2	2	no	—
117	C₅×#102	—	2	2	no	—
118	C₅×#102	—	4	1	no	—
119	C₅×#102	C₂×C₄	1	4	yes	—
120	C₄×C₅×C₈	—	2	2	no	—
121	C₄×C₅×C₈	—	4	1	no	—
122	C₄×C₅×C₈	—	4	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×#463	C₂×C₅	1	1	yes	characteristic
133	#2367	C₂×C₅	1	1	yes	characteristic, complemented
134	#2357	C₂×C₅	1	1	yes	characteristic, complemented
135	#1970	C₂×C₅	1	1	yes	characteristic
136	#1968	C₂×C₅	1	1	yes	characteristic, complemented
137	#1958	C₂×C₅	1	1	yes	characteristic, complemented
138	#1949	C₂×C₅	1	1	yes	characteristic, complemented
139	#1928	C₂×C₅	1	1	yes	characteristic, complemented
140	#1926	C₂×C₅	1	1	yes	characteristic, complemented
141	#1880	C₂×C₅	1	1	yes	characteristic, complemented
142	#1879	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×#102	C₂×C₅	1	1	yes	characteristic, complemented
146	C₄×#102	C₂×C₅	1	1	yes	characteristic, complemented
147	C₂⁴×C₅	—	2	1	no	—
148	C₂⁴×C₅	—	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	C₂²×C₄	1	2	yes	—
155	C₂×C₅×D₄	—	2	1	no	—
156	C₂×C₅×D₄	—	2	1	no	—
157	C₂×C₅×D₄	—	2	1	no	—
158	C₂×C₅×D₄	—	2	1	no	—
159	C₂×C₅×D₄	—	2	1	no	—
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	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
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	4	no	—
171	C₂²×C₄×C₅	—	2	1	no	—
172	C₂²×C₄×C₅	—	2	1	no	—
173	C₂²×C₄×C₅	—	2	1	no	—
174	C₂²×C₄×C₅	—	4	2	no	—
175	C₂²×C₄×C₅	—	4	2	no	—
176	C₂²×C₄×C₅	—	8	1	no	—
177	C₂²×C₄×C₅	—	8	1	no	—
178	C₂²×C₄×C₅	—	8	1	no	—
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	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
186	C₅×SD₁₆	—	4	4	no	—
187	C₅×D₈	—	4	4	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	—	2	1	no	—
195	C₂×C₅×C₈	—	2	1	no	—
196	C₂×C₅×C₈	—	2	1	no	—
197	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
198	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
199	C₂×C₅×C₈	#41	1	2	yes	—
200	C₅×C₄⋊C₄	—	2	2	no	—
201	C₅×C₄⋊C₄	—	2	2	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	—	2	1	no	—
209	C₅×C₄⋊C₄	—	2	1	no	—
210	C₅×C₄⋊C₄	—	4	2	no	—
211	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
212	C₅×C₄⋊C₄	#41	1	2	yes	—
213	C₅×C₂²⋊C₄	—	2	4	no	—
214	C₅×C₂²⋊C₄	—	2	4	no	—
215	C₅×C₂²⋊C₄	—	2	4	no	—
216	C₅×C₂²⋊C₄	—	4	2	no	—
217	C₅×C₂²⋊C₄	—	4	2	no	—
218	C₅×C₂²⋊C₄	—	4	2	no	—
219	C₅×C₂²⋊C₄	—	4	2	no	—
220	C₅×C₂²⋊C₄	—	8	1	no	—
221	C₅×C₂²⋊C₄	—	8	1	no	—
222	C₅×C₂²⋊C₄	—	8	1	no	—
223	C₄²×C₅	—	2	1	no	—
224	C₄²×C₅	—	2	1	no	—
225	C₄²×C₅	—	2	1	no	—
226	C₄²×C₅	—	2	1	no	—
227	C₄²×C₅	—	4	2	no	—
228	C₄²×C₅	—	8	1	no	—
229	C₄²×C₅	—	8	1	no	—
230	C₄²×C₅	—	8	1	no	—
231	C₄²×C₅	C₂²×C₄	1	2	yes	—
232	C₄²×C₅	#41	1	2	yes	—
233	C₂²×SD₁₆	C₂²×C₅	1	1	yes	characteristic
234	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
235	C₂×#124	C₂²×C₅	1	1	yes	characteristic
236	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
237	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
238	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
239	#463	C₄×C₅	1	8	yes	complemented
240	C₂×#107	C₂²×C₅	1	1	yes	characteristic
241	C₂×#106	C₂²×C₅	1	1	yes	characteristic
242	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
243	C₂×#103	C₂²×C₅	1	1	yes	characteristic
244	C₂×#103	C₂²×C₅	1	1	yes	characteristic
245	C₂×#103	C₂²×C₅	1	1	yes	characteristic
246	C₂×#102	C₂²×C₅	1	1	yes	characteristic
247	C₂×#102	C₂²×C₅	1	1	yes	characteristic
248	C₂×#102	C₂²×C₅	1	1	yes	characteristic
249	C₂×#102	C₂²×C₅	1	1	yes	characteristic
250	C₂×#102	C₂²×C₅	1	1	yes	characteristic
251	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
252	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
253	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
254	#391	C₂²×C₅	1	1	yes	characteristic
255	#390	C₂²×C₅	1	1	yes	characteristic
256	#390	C₂²×C₅	1	1	yes	characteristic
257	#390	C₂²×C₅	1	1	yes	characteristic
258	#388	—	2	1	no	—
259	#388	—	2	1	no	—
260	#388	—	2	1	no	—
261	#388	—	2	1	no	—
262	#385	—	2	1	no	—
263	#382	C₂²×C₅	1	1	yes	characteristic
264	#381	—	2	1	no	—
265	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
266	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
267	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
268	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
269	C₄×C₂²⋊C₄	—	2	1	no	—
270	C₄×C₂²⋊C₄	—	2	1	no	—
271	#340	C₂²×C₅	1	1	yes	characteristic
272	#340	C₂²×C₅	1	1	yes	characteristic
273	#340	C₂²×C₅	1	1	yes	characteristic
274	#340	C₂²×C₅	1	1	yes	characteristic
275	#336	C₂²×C₅	1	1	yes	characteristic
276	#336	C₂²×C₅	1	1	yes	characteristic
277	C₂³×C₅	—	2	2	no	—
278	C₂³×C₅	—	2	2	no	—
279	C₂³×C₅	—	4	1	no	—
280	C₂³×C₅	—	4	1	no	—
281	C₂³×C₅	—	4	1	no	—
282	C₂³×C₅	—	4	1	no	—
283	C₂³×C₅	—	4	1	no	—
284	C₂³×C₅	—	4	1	no	—
285	C₂³×C₅	—	4	1	no	—
286	C₂³×C₅	—	4	1	no	—
287	C₂³×C₅	—	4	1	no	—
288	C₂³×C₅	—	4	1	no	—
289	C₂³×C₅	—	4	1	no	—
290	C₂³×C₅	—	4	1	no	—
291	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
292	C₅×Q₈	—	2	2	no	—
293	C₅×Q₈	—	2	2	no	—
294	C₅×Q₈	—	4	1	no	—
295	C₅×Q₈	—	4	1	no	—
296	C₅×D₄	—	2	2	no	—
297	C₅×D₄	—	2	2	no	—
298	C₅×D₄	—	2	2	no	—
299	C₅×D₄	—	2	2	no	—
300	C₅×D₄	—	4	1	no	—
301	C₅×D₄	—	4	1	no	—
302	C₅×D₄	—	4	1	no	—
303	C₅×D₄	—	4	1	no	—
304	C₂×C₄×C₅	—	2	1	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	—	2	1	no	—
310	C₂×C₄×C₅	—	2	1	no	—
311	C₂×C₄×C₅	—	2	1	no	—
312	C₂×C₄×C₅	—	2	2	no	—
313	C₂×C₄×C₅	—	2	2	no	—
314	C₂×C₄×C₅	—	2	1	no	—
315	C₂×C₄×C₅	—	2	1	no	—
316	C₂×C₄×C₅	—	2	2	no	—
317	C₂×C₄×C₅	—	2	1	no	—
318	C₂×C₄×C₅	—	2	1	no	—
319	C₂×C₄×C₅	—	2	1	no	—
320	C₂×C₄×C₅	—	2	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	—	4	1	no	—
327	C₂×C₄×C₅	—	4	1	no	—
328	C₂×C₄×C₅	—	4	1	no	—
329	C₂×C₄×C₅	—	4	1	no	—
330	C₂×C₄×C₅	—	4	1	no	—
331	C₂×C₄×C₅	—	4	1	no	—
332	C₂×C₄×C₅	—	4	1	no	—
333	C₂×C₄×C₅	—	4	1	no	—
334	C₂×C₄×C₅	—	4	1	no	—
335	C₂×C₄×C₅	—	4	1	no	—
336	C₂×C₄×C₅	—	8	4	no	—
337	C₂×C₄×C₅	—	8	4	no	—
338	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
339	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
340	C₂×C₄×C₅	#135	1	2	yes	—
341	C₂×C₄×C₅	#135	1	2	yes	—
342	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
343	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
344	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
345	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
346	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
347	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
348	C₅×C₈	—	2	2	no	—
349	C₅×C₈	—	2	2	no	—
350	C₅×C₈	—	2	2	no	—
351	C₅×C₈	—	4	1	no	—
352	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
353	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
354	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
355	C₂³×C₄	—	2	1	no	—
356	C₂³×C₄	—	2	1	no	—
357	C₂×SD₁₆	—	4	1	no	—
358	C₂×SD₁₆	—	4	1	no	—
359	C₂×SD₁₆	C₂×C₄×C₅	1	4	yes	complemented
360	C₂×D₈	—	4	1	no	—
361	C₂×D₈	—	4	1	no	—
362	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
363	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
365	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
367	#124	—	2	2	no	—
368	#124	C₂×C₄×C₅	1	4	yes	—
369	C₄×Q₈	—	2	2	no	—
370	C₄×Q₈	—	4	1	no	—
371	C₄×D₄	—	2	2	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	4	1	no	—
374	C₄×D₄	C₂×C₄×C₅	1	4	yes	complemented
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	1	no	—
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₂²⋊C₄	—	2	1	no	—
390	C₂×C₂²⋊C₄	—	2	1	no	—
391	C₂×C₂²⋊C₄	—	2	1	no	—
392	C₂×C₂²⋊C₄	—	2	1	no	—
393	C₂×C₄²	—	2	1	no	—
394	C₂×C₄²	—	2	1	no	—
395	C₂×C₄²	—	2	1	no	—
396	C₂×C₄²	—	2	1	no	—
397	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
400	#107	—	4	1	no	—
401	#106	—	4	1	no	—
402	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	complemented
403	#103	—	2	2	no	—
404	#103	—	4	1	no	—
405	#103	C₂×C₄×C₅	1	4	yes	complemented
406	#102	—	2	2	no	—
407	#102	—	2	2	no	—
408	#102	—	2	2	no	—
409	#102	—	4	1	no	—
410	#102	C₂×C₄×C₅	1	4	yes	—
411	C₄×C₈	—	2	2	no	—
412	C₄×C₈	—	4	1	no	—
413	C₄×C₈	—	4	1	no	—
414	#95	—	2	1	no	—
415	#95	—	2	1	no	—
416	#95	—	2	1	no	—
417	#95	—	2	1	no	—
418	#95	—	2	1	no	—
419	#95	—	2	1	no	—
420	#95	—	2	1	no	—
421	#95	C₅×D₄	1	1	yes	characteristic
422	#95	C₅×D₄	1	1	yes	characteristic
423	C₂²×C₅	—	4	2	no	—
424	C₂²×C₅	—	4	2	no	—
425	C₂²×C₅	—	4	2	no	—
426	C₂²×C₅	—	4	2	no	—
427	C₂²×C₅	—	4	2	no	—
428	C₂²×C₅	—	4	2	no	—
429	C₂²×C₅	—	8	1	no	—
430	C₂²×C₅	—	8	1	no	—
431	C₂²×C₅	—	8	1	no	—
432	C₂²×C₅	—	8	1	no	—
433	C₂²×C₅	—	8	1	no	—
434	C₂²×C₅	—	8	1	no	—
435	C₂²×C₅	—	8	1	no	—
436	C₂²×C₅	—	8	1	no	—
437	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
438	C₂²×C₅	#546	1	1	yes	characteristic
439	C₂²×C₅	#550	1	1	yes	characteristic
440	C₂²×C₅	C₂×#135	1	1	yes	characteristic
441	C₂²×C₅	C₂×#135	1	1	yes	characteristic
442	C₂²×C₅	#576	1	1	yes	characteristic
443	C₂²×C₅	#577	1	1	yes	characteristic
444	C₄×C₅	—	2	2	no	—
445	C₄×C₅	—	2	2	no	—
446	C₄×C₅	—	2	2	no	—
447	C₄×C₅	—	2	2	no	—
448	C₄×C₅	—	4	2	no	—
449	C₄×C₅	—	4	2	no	—
450	C₄×C₅	—	4	1	no	—
451	C₄×C₅	—	4	2	no	—
452	C₄×C₅	—	4	1	no	—
453	C₄×C₅	—	8	1	no	—
454	C₄×C₅	—	8	1	no	—
455	C₄×C₅	—	8	1	no	—
456	C₄×C₅	C₄×#41	1	1	yes	characteristic
457	C₄×C₅	#518	1	1	yes	characteristic
458	C₄×C₅	#553	1	1	yes	characteristic
459	C₄×C₅	#555	1	1	yes	characteristic
460	C₂⁴	—	2	1	no	—
461	C₂⁴	—	2	1	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	1	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	—	2	1	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
479	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	4	2	no	—
488	C₂²×C₄	—	4	2	no	—
489	C₂²×C₄	—	8	1	no	—
490	C₂²×C₄	—	8	1	no	—
491	C₂²×C₄	—	8	1	no	—
492	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
493	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
494	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
495	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
496	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
497	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
498	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
499	SD₁₆	—	4	4	no	—
500	D₈	—	4	4	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
511	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
512	C₂×C₈	C₅×#41	1	2	yes	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	4	2	no	—
524	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
525	C₄⋊C₄	C₅×#41	1	2	yes	—
526	C₂²⋊C₄	—	2	4	no	—
527	C₂²⋊C₄	—	2	4	no	—
528	C₂²⋊C₄	—	2	4	no	—
529	C₂²⋊C₄	—	4	2	no	—
530	C₂²⋊C₄	—	4	2	no	—
531	C₂²⋊C₄	—	4	2	no	—
532	C₂²⋊C₄	—	4	2	no	—
533	C₂²⋊C₄	—	8	1	no	—
534	C₂²⋊C₄	—	8	1	no	—
535	C₂²⋊C₄	—	8	1	no	—
536	C₄²	—	2	1	no	—
537	C₄²	—	2	1	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	4	2	no	—
541	C₄²	—	8	1	no	—
542	C₄²	—	8	1	no	—
543	C₄²	—	8	1	no	—
544	C₄²	C₂²×C₄×C₅	1	2	yes	—
545	C₄²	C₅×#41	1	2	yes	—
546	C₂×C₅	—	8	2	no	—
547	C₂×C₅	—	8	2	no	—
548	C₂×C₅	#2363	1	1	yes	characteristic
549	C₂×C₅	C₄×#135	1	1	yes	characteristic
550	C₂×C₅	#2946	1	1	yes	characteristic
551	C₂×C₅	#3297	1	1	yes	characteristic
552	C₂×C₅	#3301	1	1	yes	characteristic
553	C₂×C₅	#3394	1	1	yes	characteristic
554	C₂×C₅	#3396	1	1	yes	characteristic
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	4	1	no	—
558	C₂³	—	4	1	no	—
559	C₂³	—	4	1	no	—
560	C₂³	—	4	1	no	—
561	C₂³	—	4	1	no	—
562	C₂³	—	4	1	no	—
563	C₂³	—	4	1	no	—
564	C₂³	—	4	1	no	—
565	C₂³	—	4	1	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	4	1	no	—
568	C₂³	—	4	1	no	—
569	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
570	Q₈	—	2	2	no	—
571	Q₈	—	2	2	no	—
572	Q₈	—	4	1	no	—
573	Q₈	—	4	1	no	—
574	D₄	—	2	2	no	—
575	D₄	—	2	2	no	—
576	D₄	—	2	2	no	—
577	D₄	—	2	2	no	—
578	D₄	—	4	1	no	—
579	D₄	—	4	1	no	—
580	D₄	—	4	1	no	—
581	D₄	—	4	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	8	4	no	—
615	C₂×C₄	—	8	4	no	—
616	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
617	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
618	C₂×C₄	C₅×#135	1	2	yes	—
619	C₂×C₄	C₅×#135	1	2	yes	—
620	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
621	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
622	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
623	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
624	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
625	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
626	C₈	—	2	2	no	—
627	C₈	—	2	2	no	—
628	C₈	—	2	2	no	—
629	C₈	—	4	1	no	—
630	C₅	#23098	1	1	yes	characteristic, direct product factor, Sylow
631	C₂²	—	4	2	no	—
632	C₂²	—	4	2	no	—
633	C₂²	—	4	2	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	8	1	no	—
638	C₂²	—	8	1	no	—
639	C₂²	—	8	1	no	—
640	C₂²	—	8	1	no	—
641	C₂²	—	8	1	no	—
642	C₂²	—	8	1	no	—
643	C₂²	—	8	1	no	—
644	C₂²	—	8	1	no	—
645	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
646	C₂²	C₅×#546	1	1	yes	characteristic
647	C₂²	C₅×#550	1	1	yes	characteristic
648	C₂²	C₂×C₅×#135	1	1	yes	characteristic
649	C₂²	C₂×C₅×#135	1	1	yes	characteristic
650	C₂²	C₅×#576	1	1	yes	characteristic
651	C₂²	C₅×#577	1	1	yes	characteristic
652	C₄	—	2	2	no	—
653	C₄	—	2	2	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	2	no	—
656	C₄	—	4	1	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	1	no	—
659	C₄	—	4	2	no	—
660	C₄	—	4	2	no	—
661	C₄	—	8	1	no	—
662	C₄	—	8	1	no	—
663	C₄	—	8	1	no	—
664	C₄	C₄×C₅×#41	1	1	yes	characteristic
665	C₄	C₅×#518	1	1	yes	characteristic
666	C₄	C₅×#553	1	1	yes	characteristic
667	C₄	C₅×#555	1	1	yes	characteristic
668	C₂	—	8	2	no	—
669	C₂	—	8	2	no	—
670	C₂	C₅×#2363	1	1	yes	characteristic
671	C₂	C₄×C₅×#135	1	1	yes	characteristic
672	C₂	C₅×#2946	1	1	yes	characteristic
673	C₂	C₅×#3297	1	1	yes	characteristic
674	C₂	C₅×#3301	1	1	yes	characteristic
675	C₂	C₅×#3394	1	1	yes	characteristic
676	C₂	C₅×#3396	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×D₈	C₂²
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
2. C₅×#2367	C₂	semidirect
3. C₅×#2357	C₂	semidirect
5. C₅×#1968	C₂	semidirect
6. C₅×#1958	C₂	semidirect
7. C₅×#1949	C₂	semidirect
8. C₅×#1928	C₂	semidirect
9. C₅×#1926	C₂	semidirect
10. C₅×#1880	C₂	semidirect
12. C₅×C₈×C₄⋊C₄	C₂	semidirect
13. C₄×C₅×#103	C₂	semidirect
14. C₄×C₅×#102	C₂	semidirect
15. C₄×C₅×#102	C₂	semidirect
22. C₅×#463	C₄	semidirect
60. #23098	C₅	direct (N × H)
68. C₂×C₅×SD₁₆	C₂×C₄	semidirect
83. C₄×C₅×D₄	C₂×C₄	semidirect
111. C₅×C₄⋊C₈	C₂×C₄	semidirect
114. C₅×#103	C₂×C₄	semidirect
133. #2367	C₂×C₅	semidirect
134. #2357	C₂×C₅	semidirect
136. #1968	C₂×C₅	semidirect
137. #1958	C₂×C₅	semidirect
138. #1949	C₂×C₅	semidirect
139. #1928	C₂×C₅	semidirect
140. #1926	C₂×C₅	semidirect
141. #1880	C₂×C₅	semidirect
143. C₈×C₄⋊C₄	C₂×C₅	semidirect
144. C₄×#103	C₂×C₅	semidirect
145. C₄×#102	C₂×C₅	semidirect
146. C₄×#102	C₂×C₅	semidirect
239. #463	C₄×C₅	semidirect
359. C₂×SD₁₆	C₂×C₄×C₅	semidirect
374. C₄×D₄	C₂×C₄×C₅	semidirect
402. C₄⋊C₈	C₂×C₄×C₅	semidirect
405. #103	C₂×C₄×C₅	semidirect
630. C₅	#23098	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×#463	2
C₅×#2367	2
C₅×#2357	2
C₅×#1970	2
C₅×#1968	2
C₅×#1958	2
C₅×#1949	2
C₅×#1928	2
C₅×#1926	2
C₅×#1880	2
C₅×#1879	2
C₅×C₈×C₄⋊C₄	2
C₄×C₅×#103	2
C₄×C₅×#102	2
#23098	5

Subgroups — C5×#23098 #11945818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#23098 #11945818