Subgroups — C₅×#23027 #11945747

Order 1280.

1050 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2363	C₂	1	2	yes	maximal, complemented
5	C₅×#1970	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1942	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1926	C₂	1	2	yes	maximal, complemented
8	C₅×#1875	C₂	1	2	yes	maximal, complemented
9	C₅×#1864	C₂	1	2	yes	maximal, complemented
10	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#121	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
16	C₂×C₅×#117	C₂²	1	2	yes	—
17	C₅×#466	C₄	1	8	yes	complemented
18	C₄×C₅×D₈	—	2	4	no	—
19	C₅×#427	—	2	4	no	—
20	C₂×C₅×#107	C₂²	1	1	yes	characteristic
21	C₂×C₅×#107	C₂²	1	1	yes	characteristic
22	C₂×C₅×#107	C₂²	1	1	yes	characteristic
23	C₂×C₅×#106	C₂²	1	2	yes	—
24	C₂×C₅×#102	C₂²	1	2	yes	—
25	C₂×C₅×#102	C₂²	1	2	yes	—
26	C₂×C₅×#102	C₂²	1	2	yes	—
27	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
30	C₅×#390	—	2	2	no	—
31	C₅×#390	C₂²	1	2	yes	—
32	C₅×#389	C₂²	1	2	yes	—
33	C₅×#388	—	2	2	no	—
34	C₅×#386	—	2	2	no	—
35	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
36	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
37	C₄×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
38	C₅×#340	C₂²	1	2	yes	—
39	C₅×#340	C₂²	1	2	yes	—
40	C₅×#336	C₂²	1	1	yes	characteristic
41	C₅×#336	C₂²	1	1	yes	characteristic
42	#23027	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₅×D₄	—	2	2	no	—
44	C₂²×C₅×D₄	C₂³	1	2	yes	—
45	C₂³×C₄×C₅	—	2	2	no	—
46	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₅×D₈	—	2	2	no	—
48	C₂×C₅×D₈	—	2	2	no	—
49	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
50	C₂²×C₅×C₈	—	2	2	no	—
51	C₂²×C₅×C₈	—	2	2	no	—
52	C₂²×C₅×C₈	C₂×C₄	1	4	yes	complemented
53	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
54	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
57	C₅×#121	—	2	4	no	—
58	C₅×#121	C₂×C₄	1	8	yes	complemented
59	C₄×C₅×D₄	—	2	4	no	—
60	C₄×C₅×D₄	—	4	2	no	—
61	C₅×#117	—	2	4	no	—
62	C₅×#117	—	4	2	no	—
63	C₂×C₅×C₄⋊C₄	—	2	2	no	—
64	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
65	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
66	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
67	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
68	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
69	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
70	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
71	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
72	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
73	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₅	—	2	2	no	—
75	C₂×C₄²×C₅	—	2	2	no	—
76	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
78	C₅×#107	—	2	2	no	—
79	C₅×#107	C₂×C₄	1	4	yes	complemented
80	C₅×#107	D₄	1	4	yes	complemented
81	C₅×#106	—	2	4	no	—
82	C₅×#102	—	2	4	no	—
83	C₅×#102	—	4	2	no	—
84	C₅×#102	C₂×C₄	1	8	yes	complemented
85	C₅×C₂²⋊C₈	—	2	2	no	—
86	C₄×C₅×C₈	—	2	2	no	—
87	C₄×C₅×C₈	—	2	2	no	—
88	C₅×#95	—	2	2	no	—
89	C₅×#95	—	2	2	no	—
90	C₅×#95	D₄	1	1	yes	characteristic
91	C₅×#95	D₄	1	1	yes	characteristic
92	C₂×#466	C₂×C₅	1	1	yes	characteristic
93	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic, complemented
94	C₂×#427	C₂×C₅	1	1	yes	characteristic, complemented
95	#2363	C₂×C₅	1	2	yes	complemented
96	#1970	C₂×C₅	1	1	yes	characteristic, complemented
97	#1942	C₂×C₅	1	1	yes	characteristic, complemented
98	#1926	C₂×C₅	1	2	yes	complemented
99	#1875	C₂×C₅	1	2	yes	complemented
100	#1864	C₂×C₅	1	2	yes	complemented
101	C₄×#107	C₂×C₅	1	1	yes	characteristic, complemented
102	C₈×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
103	C₂⁴×C₅	—	2	2	no	—
104	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅×D₄	—	2	8	no	—
106	C₂×C₅×D₄	—	2	2	no	—
107	C₂×C₅×D₄	—	2	2	no	—
108	C₂×C₅×D₄	—	2	2	no	—
109	C₂×C₅×D₄	—	2	2	no	—
110	C₂×C₅×D₄	—	2	2	no	—
111	C₂×C₅×D₄	—	8	2	no	—
112	C₂×C₅×D₄	—	8	2	no	—
113	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
114	C₂²×C₄×C₅	—	2	1	no	—
115	C₂²×C₄×C₅	—	2	1	no	—
116	C₂²×C₄×C₅	—	2	1	no	—
117	C₂²×C₄×C₅	—	2	1	no	—
118	C₂²×C₄×C₅	—	2	2	no	—
119	C₂²×C₄×C₅	—	2	2	no	—
120	C₂²×C₄×C₅	—	2	2	no	—
121	C₂²×C₄×C₅	—	4	4	no	—
122	C₂²×C₄×C₅	—	4	4	no	—
123	C₂²×C₄×C₅	—	8	2	no	—
124	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₅	#41	1	2	yes	—
131	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
132	C₅×D₈	—	2	8	no	—
133	C₂×C₅×C₈	—	2	1	no	—
134	C₂×C₅×C₈	—	2	1	no	—
135	C₂×C₅×C₈	—	2	8	no	—
136	C₂×C₅×C₈	—	2	1	no	—
137	C₂×C₅×C₈	—	2	1	no	—
138	C₂×C₅×C₈	—	2	1	no	—
139	C₂×C₅×C₈	—	2	1	no	—
140	C₂×C₅×C₈	—	2	1	no	—
141	C₂×C₅×C₈	—	2	1	no	—
142	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
143	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
144	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
145	C₂×C₅×C₈	#41	1	2	yes	—
146	C₅×C₄⋊C₄	—	2	2	no	—
147	C₅×C₄⋊C₄	—	2	2	no	—
148	C₅×C₄⋊C₄	—	2	2	no	—
149	C₅×C₄⋊C₄	—	2	2	no	—
150	C₅×C₄⋊C₄	—	8	2	no	—
151	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
152	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
153	C₅×C₂²⋊C₄	—	2	8	no	—
154	C₅×C₂²⋊C₄	—	2	2	no	—
155	C₅×C₂²⋊C₄	—	2	2	no	—
156	C₅×C₂²⋊C₄	—	4	4	no	—
157	C₅×C₂²⋊C₄	—	4	4	no	—
158	C₅×C₂²⋊C₄	—	4	4	no	—
159	C₅×C₂²⋊C₄	—	8	2	no	—
160	C₄²×C₅	—	2	1	no	—
161	C₄²×C₅	—	2	1	no	—
162	C₄²×C₅	—	2	1	no	—
163	C₄²×C₅	—	2	1	no	—
164	C₄²×C₅	—	4	4	no	—
165	C₄²×C₅	—	8	2	no	—
166	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
167	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
168	C₂×#121	C₂²×C₅	1	2	yes	—
169	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
170	C₂×#117	C₂²×C₅	1	2	yes	—
171	#466	C₄×C₅	1	8	yes	complemented
172	C₄×D₈	—	2	4	no	—
173	#427	—	2	4	no	—
174	C₂×#107	C₂²×C₅	1	1	yes	characteristic
175	C₂×#107	C₂²×C₅	1	1	yes	characteristic
176	C₂×#107	C₂²×C₅	1	1	yes	characteristic
177	C₂×#106	C₂²×C₅	1	2	yes	—
178	C₂×#102	C₂²×C₅	1	2	yes	—
179	C₂×#102	C₂²×C₅	1	2	yes	—
180	C₂×#102	C₂²×C₅	1	2	yes	—
181	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
184	#390	—	2	2	no	—
185	#390	C₂²×C₅	1	2	yes	—
186	#389	C₂²×C₅	1	2	yes	—
187	#388	—	2	2	no	—
188	#386	—	2	2	no	—
189	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
190	C₄×C₂²⋊C₄	—	2	2	no	—
191	C₄×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
192	#340	C₂²×C₅	1	2	yes	—
193	#340	C₂²×C₅	1	2	yes	—
194	#336	C₂²×C₅	1	1	yes	characteristic
195	#336	C₂²×C₅	1	1	yes	characteristic
196	C₂³×C₅	—	2	1	no	—
197	C₂³×C₅	—	2	1	no	—
198	C₂³×C₅	—	2	1	no	—
199	C₂³×C₅	—	2	1	no	—
200	C₂³×C₅	—	2	1	no	—
201	C₂³×C₅	—	2	1	no	—
202	C₂³×C₅	—	2	4	no	—
203	C₂³×C₅	—	4	2	no	—
204	C₂³×C₅	—	4	2	no	—
205	C₂³×C₅	—	4	2	no	—
206	C₂³×C₅	—	4	2	no	—
207	C₂³×C₅	—	4	2	no	—
208	C₂³×C₅	—	4	2	no	—
209	C₂³×C₅	C₄×D₄	1	2	yes	—
210	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
211	C₅×D₄	—	2	4	no	—
212	C₅×D₄	—	2	4	no	—
213	C₅×D₄	—	4	2	no	—
214	C₅×D₄	—	4	2	no	—
215	C₅×D₄	—	8	8	no	—
216	C₂×C₄×C₅	—	2	1	no	—
217	C₂×C₄×C₅	—	2	1	no	—
218	C₂×C₄×C₅	—	2	1	no	—
219	C₂×C₄×C₅	—	2	1	no	—
220	C₂×C₄×C₅	—	2	1	no	—
221	C₂×C₄×C₅	—	2	1	no	—
222	C₂×C₄×C₅	—	2	2	no	—
223	C₂×C₄×C₅	—	2	2	no	—
224	C₂×C₄×C₅	—	2	2	no	—
225	C₂×C₄×C₅	—	2	2	no	—
226	C₂×C₄×C₅	—	2	4	no	—
227	C₂×C₄×C₅	—	2	1	no	—
228	C₂×C₄×C₅	—	2	4	no	—
229	C₂×C₄×C₅	—	2	1	no	—
230	C₂×C₄×C₅	—	2	1	no	—
231	C₂×C₄×C₅	—	2	1	no	—
232	C₂×C₄×C₅	—	2	1	no	—
233	C₂×C₄×C₅	—	2	1	no	—
234	C₂×C₄×C₅	—	4	2	no	—
235	C₂×C₄×C₅	—	4	2	no	—
236	C₂×C₄×C₅	—	4	2	no	—
237	C₂×C₄×C₅	—	4	2	no	—
238	C₂×C₄×C₅	—	4	2	no	—
239	C₂×C₄×C₅	—	4	2	no	—
240	C₂×C₄×C₅	—	4	2	no	—
241	C₂×C₄×C₅	—	8	8	no	—
242	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
243	C₂×C₄×C₅	#135	1	2	yes	—
244	C₂×C₄×C₅	#135	1	2	yes	—
245	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
246	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
250	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
251	C₅×C₈	—	2	2	no	—
252	C₅×C₈	—	4	1	no	—
253	C₅×C₈	—	4	1	no	—
254	C₅×C₈	C₄×D₄	1	4	yes	—
255	C₂²×D₄	—	2	2	no	—
256	C₂²×D₄	C₂³×C₅	1	2	yes	—
257	C₂³×C₄	—	2	2	no	—
258	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
259	C₂×D₈	—	2	2	no	—
260	C₂×D₈	—	2	2	no	—
261	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
262	C₂²×C₈	—	2	2	no	—
263	C₂²×C₈	—	2	2	no	—
264	C₂²×C₈	C₂×C₄×C₅	1	4	yes	complemented
265	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
269	#121	—	2	4	no	—
270	#121	C₂×C₄×C₅	1	8	yes	complemented
271	C₄×D₄	—	2	4	no	—
272	C₄×D₄	—	4	2	no	—
273	#117	—	2	4	no	—
274	#117	—	4	2	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
277	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
278	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
279	C₂×C₂²⋊C₄	—	2	2	no	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₂²⋊C₄	—	2	2	no	—
284	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
285	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
286	C₂×C₄²	—	2	2	no	—
287	C₂×C₄²	—	2	2	no	—
288	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
290	#107	—	2	2	no	—
291	#107	C₂×C₄×C₅	1	4	yes	complemented
292	#107	C₅×D₄	1	4	yes	complemented
293	#106	—	2	4	no	—
294	#102	—	2	4	no	—
295	#102	—	4	2	no	—
296	#102	C₂×C₄×C₅	1	8	yes	complemented
297	C₂²⋊C₈	—	2	2	no	—
298	C₄×C₈	—	2	2	no	—
299	C₄×C₈	—	2	2	no	—
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	#95	C₅×D₄	1	1	yes	characteristic
303	#95	C₅×D₄	1	1	yes	characteristic
304	C₂²×C₅	—	2	2	no	—
305	C₂²×C₅	—	2	2	no	—
306	C₂²×C₅	—	2	2	no	—
307	C₂²×C₅	—	4	1	no	—
308	C₂²×C₅	—	4	1	no	—
309	C₂²×C₅	—	4	1	no	—
310	C₂²×C₅	—	4	1	no	—
311	C₂²×C₅	—	4	4	no	—
312	C₂²×C₅	—	4	4	no	—
313	C₂²×C₅	—	4	4	no	—
314	C₂²×C₅	—	8	2	no	—
315	C₂²×C₅	—	8	2	no	—
316	C₂²×C₅	—	8	2	no	—
317	C₂²×C₅	—	8	2	no	—
318	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
319	C₂²×C₅	D₄²	1	1	yes	characteristic
320	C₂²×C₅	#547	1	1	yes	characteristic
321	C₂²×C₅	C₂×#135	1	1	yes	characteristic
322	C₂²×C₅	C₂×#135	1	1	yes	characteristic
323	C₂²×C₅	#576	1	1	yes	characteristic
324	C₂²×C₅	#576	1	1	yes	characteristic
325	C₄×C₅	—	2	2	no	—
326	C₄×C₅	—	2	2	no	—
327	C₄×C₅	—	2	2	no	—
328	C₄×C₅	—	4	1	no	—
329	C₄×C₅	—	4	4	no	—
330	C₄×C₅	—	4	4	no	—
331	C₄×C₅	—	4	1	no	—
332	C₄×C₅	—	8	2	no	—
333	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
334	C₄×C₅	#520	1	1	yes	characteristic
335	C₄×C₅	#548	1	1	yes	characteristic
336	C₄×C₅	#553	1	1	yes	characteristic
337	C₂⁴	—	2	2	no	—
338	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
339	C₂×D₄	—	2	8	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	8	2	no	—
346	C₂×D₄	—	8	2	no	—
347	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	4	4	no	—
356	C₂²×C₄	—	4	4	no	—
357	C₂²×C₄	—	8	2	no	—
358	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
359	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
360	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
361	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
362	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
363	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
364	C₂²×C₄	C₅×#41	1	2	yes	—
365	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
366	D₈	—	2	8	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	8	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
377	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
378	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
379	C₂×C₈	C₅×#41	1	2	yes	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	8	2	no	—
385	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
386	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
387	C₂²⋊C₄	—	2	8	no	—
388	C₂²⋊C₄	—	2	2	no	—
389	C₂²⋊C₄	—	2	2	no	—
390	C₂²⋊C₄	—	4	4	no	—
391	C₂²⋊C₄	—	4	4	no	—
392	C₂²⋊C₄	—	4	4	no	—
393	C₂²⋊C₄	—	8	2	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	2	1	no	—
398	C₄²	—	4	4	no	—
399	C₄²	—	8	2	no	—
400	C₂×C₅	—	4	2	no	—
401	C₂×C₅	—	8	4	no	—
402	C₂×C₅	#2361	1	1	yes	characteristic
403	C₂×C₅	C₄×#135	1	1	yes	characteristic
404	C₂×C₅	#2948	1	1	yes	characteristic
405	C₂×C₅	#3280	1	1	yes	characteristic
406	C₂×C₅	#3287	1	1	yes	characteristic
407	C₂×C₅	#3339	1	1	yes	characteristic
408	C₂×C₅	#3341	1	1	yes	characteristic
409	C₂³	—	2	1	no	—
410	C₂³	—	2	1	no	—
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	2	4	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	C₄×C₅×D₄	1	2	yes	—
423	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
424	D₄	—	2	4	no	—
425	D₄	—	2	4	no	—
426	D₄	—	4	2	no	—
427	D₄	—	4	2	no	—
428	D₄	—	8	8	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	8	8	no	—
455	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
456	C₂×C₄	C₅×#135	1	2	yes	—
457	C₂×C₄	C₅×#135	1	2	yes	—
458	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
459	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
460	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
461	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
462	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
463	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
464	C₈	—	2	2	no	—
465	C₈	—	4	1	no	—
466	C₈	—	4	1	no	—
467	C₈	C₄×C₅×D₄	1	4	yes	—
468	C₅	#23027	1	1	yes	characteristic, direct product factor, Sylow
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	8	2	no	—
480	C₂²	—	8	2	no	—
481	C₂²	—	8	2	no	—
482	C₂²	—	8	2	no	—
483	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
484	C₂²	C₅×D₄²	1	1	yes	characteristic
485	C₂²	C₅×#547	1	1	yes	characteristic
486	C₂²	C₂×C₅×#135	1	1	yes	characteristic
487	C₂²	C₂×C₅×#135	1	1	yes	characteristic
488	C₂²	C₅×#576	1	1	yes	characteristic
489	C₂²	C₅×#576	1	1	yes	characteristic
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	4	4	no	—
494	C₄	—	4	1	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	4	no	—
497	C₄	—	8	2	no	—
498	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
499	C₄	C₅×#520	1	1	yes	characteristic
500	C₄	C₅×#548	1	1	yes	characteristic
501	C₄	C₅×#553	1	1	yes	characteristic
502	C₂	—	4	2	no	—
503	C₂	—	8	4	no	—
504	C₂	C₅×#2361	1	1	yes	characteristic
505	C₂	C₄×C₅×#135	1	1	yes	characteristic
506	C₂	C₅×#2948	1	1	yes	characteristic
507	C₂	C₅×#3280	1	1	yes	characteristic
508	C₂	C₅×#3287	1	1	yes	characteristic
509	C₂	C₅×#3339	1	1	yes	characteristic
510	C₂	C₅×#3341	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	C₂×C₅×#466	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄×C₅×C₂²⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×D₈	C₂	semidirect
3. C₂×C₅×#427	C₂	semidirect
4. C₅×#2363	C₂	semidirect
5. C₅×#1970	C₂	semidirect
6. C₅×#1942	C₂	semidirect
7. C₅×#1926	C₂	semidirect
8. C₅×#1875	C₂	semidirect
9. C₅×#1864	C₂	semidirect
10. C₄×C₅×#107	C₂	semidirect
11. C₅×C₈×C₂²⋊C₄	C₂	semidirect
17. C₅×#466	C₄	semidirect
42. #23027	C₅	direct (N × H)
52. C₂²×C₅×C₈	C₂×C₄	semidirect
58. C₅×#121	C₂×C₄	semidirect
79. C₅×#107	C₂×C₄	semidirect
80. C₅×#107	D₄	semidirect
84. C₅×#102	C₂×C₄	semidirect
93. C₂×C₄×D₈	C₂×C₅	semidirect
94. C₂×#427	C₂×C₅	semidirect
95. #2363	C₂×C₅	semidirect
96. #1970	C₂×C₅	semidirect
97. #1942	C₂×C₅	semidirect
98. #1926	C₂×C₅	semidirect
99. #1875	C₂×C₅	semidirect
100. #1864	C₂×C₅	semidirect
101. C₄×#107	C₂×C₅	semidirect
102. C₈×C₂²⋊C₄	C₂×C₅	semidirect
171. #466	C₄×C₅	semidirect
264. C₂²×C₈	C₂×C₄×C₅	semidirect
270. #121	C₂×C₄×C₅	semidirect
291. #107	C₂×C₄×C₅	semidirect
292. #107	C₅×D₄	semidirect
296. #102	C₂×C₄×C₅	semidirect
468. C₅	#23027	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×#466	2
C₂×C₄×C₅×D₈	2
C₂×C₅×#427	2
C₅×#2363	2
C₅×#1970	2
C₅×#1942	2
C₅×#1926	2
C₅×#1875	2
C₅×#1864	2
C₄×C₅×#107	2
C₅×C₈×C₂²⋊C₄	2
#23027	5

Subgroups — C5×#23027 #11945747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#23027 #11945747