Subgroups — C₅×#34363 #11957083

Order 1280.

2954 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#121	C₂	1	1	yes	maximal, characteristic
2	C₅×#2377	C₂	1	4	yes	maximal, complemented
3	C₅×#2363	C₂	1	4	yes	maximal, complemented
4	C₅×#2344	C₂	1	2	yes	maximal, complemented
5	C₅×#2342	C₂	1	4	yes	maximal, complemented
6	C₅×D₄×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₅×#2333	C₂	1	4	yes	maximal, complemented
8	C₅×#2324	C₂	1	4	yes	maximal, complemented
9	C₅×#2309	C₂	1	2	yes	maximal, complemented
10	C₅×#2308	C₂	1	2	yes	maximal, complemented
11	C₅×#2280	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2280	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂³×C₅×D₄	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₅×#121	C₄	1	16	yes	complemented
17	C₂×C₅×#121	C₂²	1	2	yes	—
18	C₂×C₅×#121	C₂²	1	2	yes	—
19	C₂×C₅×#121	C₂²	1	2	yes	—
20	C₂×C₅×#121	C₂²	1	2	yes	—
21	C₂×C₄×C₅×D₄	C₂²	1	4	yes	complemented
22	C₂×C₅×#117	C₂²	1	4	yes	—
23	C₂×C₅×#117	C₂²	1	2	yes	complemented
24	C₂×C₅×#117	C₂²	1	2	yes	complemented
25	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂²×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
27	C₂²×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₅×#390	C₂²	1	8	yes	complemented
29	C₅×#390	C₂²	1	8	yes	complemented
30	C₅×#390	C₂²	1	4	yes	complemented
31	C₅×#389	C₂²	1	4	yes	complemented
32	C₅×#388	C₂²	1	8	yes	complemented
33	C₅×#388	C₂²	1	8	yes	complemented
34	C₅×#387	C₂²	1	8	yes	complemented
35	C₅×#386	C₂²	1	8	yes	complemented
36	C₅×#386	C₂²	1	8	yes	complemented
37	C₅×#386	C₂²	1	8	yes	complemented
38	C₅×#385	C₂²	1	8	yes	complemented
39	C₅×#381	C₂²	1	4	yes	—
40	C₅×#381	C₂²	1	2	yes	complemented
41	C₅×#381	C₂²	1	2	yes	complemented
42	C₅×#380	C₂²	1	2	yes	—
43	C₅×#380	C₂²	1	2	yes	—
44	C₅×#380	C₂²	1	2	yes	complemented
45	C₅×#380	C₂²	1	2	yes	complemented
46	C₅×#379	C₂²	1	1	yes	characteristic
47	C₅×#379	C₂²	1	2	yes	—
48	C₅×#379	C₂²	1	1	yes	characteristic
49	C₅×#379	C₂²	1	4	yes	—
50	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	complemented
51	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
52	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
53	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
54	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
55	C₄×C₅×C₂²⋊C₄	C₂²	1	8	yes	complemented
56	#34363	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁵×C₅	C₂³	1	1	yes	characteristic
58	C₂⁵×C₅	C₂³	1	2	yes	—
59	C₂²×C₅×D₄	—	4	4	no	—
60	C₂²×C₅×D₄	—	4	8	no	—
61	C₂²×C₅×D₄	C₂×C₄	1	8	yes	—
62	C₂²×C₅×D₄	C₂×C₄	1	16	yes	complemented
63	C₂²×C₅×D₄	C₂³	1	2	yes	—
64	C₂²×C₅×D₄	C₂³	1	2	yes	—
65	C₂²×C₅×D₄	C₂³	1	4	yes	—
66	C₂²×C₅×D₄	C₂³	1	4	yes	—
67	C₂³×C₄×C₅	—	4	4	no	—
68	C₂³×C₄×C₅	C₂×C₄	1	8	yes	complemented
69	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₅	C₂³	1	2	yes	—
73	C₂³×C₄×C₅	C₂³	1	8	yes	—
74	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
77	C₅×#121	C₂×C₄	1	16	yes	complemented
78	C₅×#121	C₂×C₄	1	16	yes	complemented
79	C₅×#121	C₂×C₄	1	16	yes	complemented
80	C₅×#121	C₂×C₄	1	16	yes	complemented
81	C₄×C₅×D₄	—	4	8	no	—
82	C₅×#117	—	4	4	no	—
83	C₅×#117	—	4	8	no	—
84	C₅×#117	—	4	4	no	—
85	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
86	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
87	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
88	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
89	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
90	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
91	C₂×C₅×C₂²⋊C₄	—	4	8	no	—
92	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	16	yes	complemented
93	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
94	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
95	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
96	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
97	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
98	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
99	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
100	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
101	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
102	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
103	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
104	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
105	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
106	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
107	C₂×C₄²×C₅	C₂³	1	4	yes	—
108	C₂×C₄²×C₅	C₂³	1	4	yes	—
109	C₂×C₄²×C₅	C₂³	1	8	yes	—
110	C₅×#95	C₂³	1	4	yes	—
111	C₅×#95	C₂³	1	4	yes	—
112	C₅×#95	C₂³	1	4	yes	—
113	C₅×#95	C₂³	1	4	yes	—
114	C₅×#95	C₂³	1	8	yes	—
115	C₅×#95	C₂³	1	8	yes	—
116	C₂²×#121	C₂×C₅	1	1	yes	characteristic
117	#2377	C₂×C₅	1	4	yes	complemented
118	#2363	C₂×C₅	1	4	yes	complemented
119	#2344	C₂×C₅	1	2	yes	complemented
120	#2342	C₂×C₅	1	4	yes	complemented
121	D₄×C₂²⋊C₄	C₂×C₅	1	2	yes	complemented
122	#2333	C₂×C₅	1	4	yes	complemented
123	#2324	C₂×C₅	1	4	yes	complemented
124	#2309	C₂×C₅	1	2	yes	complemented
125	#2308	C₂×C₅	1	2	yes	complemented
126	#2280	C₂×C₅	1	1	yes	characteristic, complemented
127	#2280	C₂×C₅	1	1	yes	characteristic, complemented
128	C₂⁴×C₅	—	2	4	no	—
129	C₂⁴×C₅	—	2	8	no	—
130	C₂⁴×C₅	—	4	2	no	—
131	C₂⁴×C₅	—	4	2	no	—
132	C₂⁴×C₅	—	4	4	no	—
133	C₂⁴×C₅	—	4	4	no	—
134	C₂⁴×C₅	C₂²×C₄	1	8	yes	—
135	C₂⁴×C₅	C₂²×C₄	1	4	yes	—
136	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
137	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
138	C₂⁴×C₅	C₂⁴	1	4	yes	—
139	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
140	C₂×C₅×D₄	—	2	8	no	—
141	C₂×C₅×D₄	—	4	16	no	—
142	C₂×C₅×D₄	—	4	32	no	—
143	C₂×C₅×D₄	—	4	4	no	—
144	C₂×C₅×D₄	—	4	4	no	—
145	C₂×C₅×D₄	—	4	4	no	—
146	C₂×C₅×D₄	—	4	4	no	—
147	C₂×C₅×D₄	—	4	4	no	—
148	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
149	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
150	C₂×C₅×D₄	C₂²×C₄	1	16	yes	—
151	C₂×C₅×D₄	C₂²×C₄	1	16	yes	—
152	C₂²×C₄×C₅	—	2	8	no	—
153	C₂²×C₄×C₅	—	2	4	no	—
154	C₂²×C₄×C₅	—	2	4	no	—
155	C₂²×C₄×C₅	—	2	4	no	—
156	C₂²×C₄×C₅	—	4	16	no	—
157	C₂²×C₄×C₅	—	4	2	no	—
158	C₂²×C₄×C₅	—	4	2	no	—
159	C₂²×C₄×C₅	—	4	2	no	—
160	C₂²×C₄×C₅	—	4	8	no	—
161	C₂²×C₄×C₅	—	4	8	no	—
162	C₂²×C₄×C₅	—	4	8	no	—
163	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
164	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
165	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
166	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
167	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
169	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
172	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
178	C₅×C₄⋊C₄	—	2	8	no	—
179	C₅×C₄⋊C₄	—	4	4	no	—
180	C₅×C₄⋊C₄	—	4	8	no	—
181	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
182	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
183	C₅×C₂²⋊C₄	—	2	4	no	—
184	C₅×C₂²⋊C₄	—	2	8	no	—
185	C₅×C₂²⋊C₄	—	2	4	no	—
186	C₅×C₂²⋊C₄	—	4	8	no	—
187	C₅×C₂²⋊C₄	—	4	2	no	—
188	C₅×C₂²⋊C₄	—	4	8	no	—
189	C₅×C₂²⋊C₄	—	4	4	no	—
190	C₅×C₂²⋊C₄	—	4	2	no	—
191	C₅×C₂²⋊C₄	—	4	2	no	—
192	C₅×C₂²⋊C₄	—	4	2	no	—
193	C₅×C₂²⋊C₄	—	4	4	no	—
194	C₅×C₂²⋊C₄	—	4	8	no	—
195	C₅×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
196	C₅×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
197	C₄²×C₅	—	2	8	no	—
198	C₄²×C₅	—	4	4	no	—
199	C₄²×C₅	—	4	8	no	—
200	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic
201	C₂³×D₄	C₂²×C₅	1	2	yes	—
202	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
203	C₂×#121	C₄×C₅	1	16	yes	complemented
204	C₂×#121	C₂²×C₅	1	2	yes	—
205	C₂×#121	C₂²×C₅	1	2	yes	—
206	C₂×#121	C₂²×C₅	1	2	yes	—
207	C₂×#121	C₂²×C₅	1	2	yes	—
208	C₂×C₄×D₄	C₂²×C₅	1	4	yes	complemented
209	C₂×#117	C₂²×C₅	1	4	yes	—
210	C₂×#117	C₂²×C₅	1	2	yes	complemented
211	C₂×#117	C₂²×C₅	1	2	yes	complemented
212	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
213	C₂²×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
214	C₂²×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
215	#390	C₂²×C₅	1	8	yes	complemented
216	#390	C₂²×C₅	1	8	yes	complemented
217	#390	C₂²×C₅	1	4	yes	complemented
218	#389	C₂²×C₅	1	4	yes	complemented
219	#388	C₂²×C₅	1	8	yes	complemented
220	#388	C₂²×C₅	1	8	yes	complemented
221	#387	C₂²×C₅	1	8	yes	complemented
222	#386	C₂²×C₅	1	8	yes	complemented
223	#386	C₂²×C₅	1	8	yes	complemented
224	#386	C₂²×C₅	1	8	yes	complemented
225	#385	C₂²×C₅	1	8	yes	complemented
226	#381	C₂²×C₅	1	4	yes	—
227	#381	C₂²×C₅	1	2	yes	complemented
228	#381	C₂²×C₅	1	2	yes	complemented
229	#380	C₂²×C₅	1	2	yes	—
230	#380	C₂²×C₅	1	2	yes	—
231	#380	C₂²×C₅	1	2	yes	complemented
232	#380	C₂²×C₅	1	2	yes	complemented
233	#379	C₂²×C₅	1	1	yes	characteristic
234	#379	C₂²×C₅	1	2	yes	—
235	#379	C₂²×C₅	1	1	yes	characteristic
236	#379	C₂²×C₅	1	4	yes	—
237	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	complemented
238	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
239	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
240	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	complemented
241	C₄×C₂²⋊C₄	C₂²×C₅	1	8	yes	complemented
242	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	complemented
243	C₂³×C₅	—	2	2	no	—
244	C₂³×C₅	—	2	2	no	—
245	C₂³×C₅	—	2	8	no	—
246	C₂³×C₅	—	2	2	no	—
247	C₂³×C₅	—	2	2	no	—
248	C₂³×C₅	—	2	4	no	—
249	C₂³×C₅	—	2	8	no	—
250	C₂³×C₅	—	2	4	no	—
251	C₂³×C₅	—	2	2	no	—
252	C₂³×C₅	—	2	2	no	—
253	C₂³×C₅	—	2	2	no	—
254	C₂³×C₅	—	2	2	no	—
255	C₂³×C₅	—	2	8	no	—
256	C₂³×C₅	—	2	16	no	—
257	C₂³×C₅	—	4	4	no	—
258	C₂³×C₅	—	4	4	no	—
259	C₂³×C₅	—	4	8	no	—
260	C₂³×C₅	—	4	8	no	—
261	C₂³×C₅	—	4	16	no	—
262	C₂³×C₅	—	8	16	no	—
263	C₂³×C₅	C₂³×C₄	1	8	yes	—
264	C₂³×C₅	C₂³×C₄	1	2	yes	—
265	C₂³×C₅	C₂³×C₄	1	2	yes	—
266	C₂³×C₅	C₂³×C₄	1	2	yes	—
267	C₂³×C₅	H(5,𝔽₂)	1	4	yes	—
268	C₂³×C₅	C₂⁵	1	1	yes	characteristic
269	C₅×D₄	—	2	16	no	—
270	C₅×D₄	—	4	8	no	—
271	C₅×D₄	—	4	16	no	—
272	C₅×D₄	—	4	16	no	—
273	C₂×C₄×C₅	—	2	16	no	—
274	C₂×C₄×C₅	—	2	2	no	—
275	C₂×C₄×C₅	—	2	4	no	—
276	C₂×C₄×C₅	—	2	2	no	—
277	C₂×C₄×C₅	—	2	4	no	—
278	C₂×C₄×C₅	—	2	2	no	—
279	C₂×C₄×C₅	—	2	8	no	—
280	C₂×C₄×C₅	—	2	8	no	—
281	C₂×C₄×C₅	—	2	8	no	—
282	C₂×C₄×C₅	—	2	4	no	—
283	C₂×C₄×C₅	—	2	4	no	—
284	C₂×C₄×C₅	—	2	4	no	—
285	C₂×C₄×C₅	—	2	8	no	—
286	C₂×C₄×C₅	—	2	8	no	—
287	C₂×C₄×C₅	—	2	4	no	—
288	C₂×C₄×C₅	—	2	4	no	—
289	C₂×C₄×C₅	—	2	8	no	—
290	C₂×C₄×C₅	—	2	4	no	—
291	C₂×C₄×C₅	—	4	4	no	—
292	C₂×C₄×C₅	—	4	4	no	—
293	C₂×C₄×C₅	—	4	4	no	—
294	C₂×C₄×C₅	—	8	16	no	—
295	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
296	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
297	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
298	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
299	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
300	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
301	C₂⁵	C₂³×C₅	1	2	yes	—
302	C₂⁵	C₂³×C₅	1	1	yes	characteristic
303	C₂²×D₄	—	4	4	no	—
304	C₂²×D₄	—	4	8	no	—
305	C₂²×D₄	C₂×C₄×C₅	1	8	yes	—
306	C₂²×D₄	C₂×C₄×C₅	1	16	yes	complemented
307	C₂²×D₄	C₂³×C₅	1	2	yes	—
308	C₂²×D₄	C₂³×C₅	1	2	yes	—
309	C₂²×D₄	C₂³×C₅	1	4	yes	—
310	C₂²×D₄	C₂³×C₅	1	4	yes	—
311	C₂³×C₄	—	4	4	no	—
312	C₂³×C₄	C₂×C₄×C₅	1	8	yes	complemented
313	C₂³×C₄	C₂³×C₅	1	2	yes	—
314	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
315	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
316	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
317	C₂³×C₄	C₂³×C₅	1	8	yes	—
318	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
319	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
321	#121	C₂×C₄×C₅	1	16	yes	complemented
322	#121	C₂×C₄×C₅	1	16	yes	complemented
323	#121	C₂×C₄×C₅	1	16	yes	complemented
324	#121	C₂×C₄×C₅	1	16	yes	complemented
325	C₄×D₄	—	4	8	no	—
326	#117	—	4	4	no	—
327	#117	—	4	8	no	—
328	#117	—	4	4	no	—
329	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	complemented
330	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
331	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
332	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
333	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
335	C₂×C₂²⋊C₄	—	4	8	no	—
336	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	16	yes	complemented
337	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
338	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
339	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
340	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
341	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
342	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
343	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
344	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
345	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
351	C₂×C₄²	C₂³×C₅	1	4	yes	—
352	C₂×C₄²	C₂³×C₅	1	8	yes	—
353	C₂×C₄²	C₂³×C₅	1	4	yes	—
354	#95	C₂³×C₅	1	4	yes	—
355	#95	C₂³×C₅	1	4	yes	—
356	#95	C₂³×C₅	1	4	yes	—
357	#95	C₂³×C₅	1	4	yes	—
358	#95	C₂³×C₅	1	8	yes	—
359	#95	C₂³×C₅	1	8	yes	—
360	C₂²×C₅	—	2	2	no	—
361	C₂²×C₅	—	2	2	no	—
362	C₂²×C₅	—	2	8	no	—
363	C₂²×C₅	—	2	2	no	—
364	C₂²×C₅	—	2	2	no	—
365	C₂²×C₅	—	2	8	no	—
366	C₂²×C₅	—	2	2	no	—
367	C₂²×C₅	—	2	2	no	—
368	C₂²×C₅	—	2	8	no	—
369	C₂²×C₅	—	2	2	no	—
370	C₂²×C₅	—	2	2	no	—
371	C₂²×C₅	—	2	8	no	—
372	C₂²×C₅	—	4	16	no	—
373	C₂²×C₅	—	4	8	no	—
374	C₂²×C₅	—	4	4	no	—
375	C₂²×C₅	—	4	4	no	—
376	C₂²×C₅	—	4	8	no	—
377	C₂²×C₅	—	8	16	no	—
378	C₂²×C₅	#518	1	4	yes	—
379	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
380	C₂²×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
381	C₂²×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
382	C₂²×C₅	#585	1	2	yes	—
383	C₄×C₅	—	2	2	no	—
384	C₄×C₅	—	2	2	no	—
385	C₄×C₅	—	2	2	no	—
386	C₄×C₅	—	2	8	no	—
387	C₄×C₅	—	4	4	no	—
388	C₄×C₅	—	4	8	no	—
389	C₄×C₅	—	4	4	no	—
390	C₄×C₅	#518	1	4	yes	—
391	C₂⁴	—	2	4	no	—
392	C₂⁴	—	2	8	no	—
393	C₂⁴	—	4	2	no	—
394	C₂⁴	—	4	2	no	—
395	C₂⁴	—	4	4	no	—
396	C₂⁴	—	4	4	no	—
397	C₂⁴	C₂²×C₄×C₅	1	8	yes	—
398	C₂⁴	C₂²×C₄×C₅	1	4	yes	—
399	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
400	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
401	C₂⁴	C₂⁴×C₅	1	4	yes	—
402	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
403	C₂×D₄	—	2	8	no	—
404	C₂×D₄	—	4	16	no	—
405	C₂×D₄	—	4	32	no	—
406	C₂×D₄	—	4	4	no	—
407	C₂×D₄	—	4	4	no	—
408	C₂×D₄	—	4	4	no	—
409	C₂×D₄	—	4	4	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
412	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
413	C₂×D₄	C₂²×C₄×C₅	1	16	yes	—
414	C₂×D₄	C₂²×C₄×C₅	1	16	yes	—
415	C₂²×C₄	—	2	8	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	4	16	no	—
420	C₂²×C₄	—	4	2	no	—
421	C₂²×C₄	—	4	2	no	—
422	C₂²×C₄	—	4	2	no	—
423	C₂²×C₄	—	4	8	no	—
424	C₂²×C₄	—	4	8	no	—
425	C₂²×C₄	—	4	8	no	—
426	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
427	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
428	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
429	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
430	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
431	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
432	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
433	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
434	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
435	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
436	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
437	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
438	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
440	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	4	4	no	—
443	C₄⋊C₄	—	4	8	no	—
444	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
445	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	4	8	no	—
450	C₂²⋊C₄	—	4	2	no	—
451	C₂²⋊C₄	—	4	2	no	—
452	C₂²⋊C₄	—	4	4	no	—
453	C₂²⋊C₄	—	4	8	no	—
454	C₂²⋊C₄	—	4	2	no	—
455	C₂²⋊C₄	—	4	8	no	—
456	C₂²⋊C₄	—	4	4	no	—
457	C₂²⋊C₄	—	4	2	no	—
458	C₂²⋊C₄	C₂²×C₄×C₅	1	16	yes	—
459	C₂²⋊C₄	C₂²×C₄×C₅	1	16	yes	—
460	C₄²	—	2	8	no	—
461	C₄²	—	4	4	no	—
462	C₄²	—	4	8	no	—
463	C₂×C₅	—	2	4	no	—
464	C₂×C₅	—	4	2	no	—
465	C₂×C₅	—	4	2	no	—
466	C₂×C₅	—	4	8	no	—
467	C₂×C₅	C₂×#518	1	1	yes	characteristic
468	C₂×C₅	C₂×#518	1	1	yes	characteristic
469	C₂×C₅	#3428	1	1	yes	characteristic
470	C₂×C₅	#3542	1	4	yes	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	8	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	2	8	no	—
478	C₂³	—	2	4	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	8	no	—
484	C₂³	—	2	16	no	—
485	C₂³	—	4	4	no	—
486	C₂³	—	4	4	no	—
487	C₂³	—	4	8	no	—
488	C₂³	—	4	8	no	—
489	C₂³	—	4	16	no	—
490	C₂³	—	8	16	no	—
491	C₂³	C₂³×C₄×C₅	1	8	yes	—
492	C₂³	C₂³×C₄×C₅	1	2	yes	—
493	C₂³	C₂³×C₄×C₅	1	2	yes	—
494	C₂³	C₂³×C₄×C₅	1	2	yes	—
495	C₂³	C₅×H(5,𝔽₂)	1	4	yes	—
496	C₂³	C₂⁵×C₅	1	1	yes	characteristic
497	D₄	—	2	16	no	—
498	D₄	—	4	8	no	—
499	D₄	—	4	16	no	—
500	D₄	—	4	16	no	—
501	C₂×C₄	—	2	16	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	8	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	8	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	8	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	4	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	8	16	no	—
523	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
524	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
525	C₂×C₄	C₂³×C₄×C₅	1	8	yes	—
526	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
527	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
528	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
529	C₅	#34363	1	1	yes	characteristic, direct product factor, Sylow
530	C₂²	—	2	2	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	8	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	2	no	—
535	C₂²	—	2	8	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	8	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	8	no	—
542	C₂²	—	4	16	no	—
543	C₂²	—	4	8	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	8	no	—
547	C₂²	—	8	16	no	—
548	C₂²	C₅×#518	1	4	yes	—
549	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
550	C₂²	C₂×C₅×H(5,𝔽₂)	1	2	yes	—
551	C₂²	C₂×C₅×H(5,𝔽₂)	1	2	yes	—
552	C₂²	C₅×#585	1	2	yes	—
553	C₄	—	2	2	no	—
554	C₄	—	2	2	no	—
555	C₄	—	2	2	no	—
556	C₄	—	2	8	no	—
557	C₄	—	4	4	no	—
558	C₄	—	4	8	no	—
559	C₄	—	4	4	no	—
560	C₄	C₅×#518	1	4	yes	—
561	C₂	—	2	4	no	—
562	C₂	—	4	2	no	—
563	C₂	—	4	2	no	—
564	C₂	—	4	8	no	—
565	C₂	C₂×C₅×#518	1	1	yes	characteristic
566	C₂	C₂×C₅×#518	1	1	yes	characteristic
567	C₂	C₅×#3428	1	1	yes	characteristic
568	C₂	C₅×#3542	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅
Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂²×C₅×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2377	C₂	semidirect
3. C₅×#2363	C₂	semidirect
4. C₅×#2344	C₂	semidirect
5. C₅×#2342	C₂	semidirect
6. C₅×D₄×C₂²⋊C₄	C₂	semidirect
7. C₅×#2333	C₂	semidirect
8. C₅×#2324	C₂	semidirect
9. C₅×#2309	C₂	semidirect
10. C₅×#2308	C₂	semidirect
11. C₅×#2280	C₂	semidirect
12. C₅×#2280	C₂	semidirect
16. C₂×C₅×#121	C₄	semidirect
21. C₂×C₄×C₅×D₄	C₂²	semidirect
23. C₂×C₅×#117	C₂²	semidirect
24. C₂×C₅×#117	C₂²	semidirect
28. C₅×#390	C₂²	semidirect
29. C₅×#390	C₂²	semidirect
30. C₅×#390	C₂²	semidirect
31. C₅×#389	C₂²	semidirect
32. C₅×#388	C₂²	semidirect
33. C₅×#388	C₂²	semidirect
34. C₅×#387	C₂²	semidirect
35. C₅×#386	C₂²	semidirect
36. C₅×#386	C₂²	semidirect
37. C₅×#386	C₂²	semidirect
38. C₅×#385	C₂²	semidirect
40. C₅×#381	C₂²	semidirect
41. C₅×#381	C₂²	semidirect
44. C₅×#380	C₂²	semidirect
45. C₅×#380	C₂²	semidirect
50. C₄×C₅×C₄⋊C₄	C₂²	semidirect
53. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
54. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
55. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
56. #34363	C₅	direct (N × H)
62. C₂²×C₅×D₄	C₂×C₄	semidirect
68. C₂³×C₄×C₅	C₂×C₄	semidirect
77. C₅×#121	C₂×C₄	semidirect
78. C₅×#121	C₂×C₄	semidirect
79. C₅×#121	C₂×C₄	semidirect
80. C₅×#121	C₂×C₄	semidirect
85. C₂×C₅×C₄⋊C₄	C₂×C₄	semidirect
92. C₂×C₅×C₂²⋊C₄	C₂×C₄	semidirect
117. #2377	C₂×C₅	semidirect
118. #2363	C₂×C₅	semidirect
119. #2344	C₂×C₅	semidirect
120. #2342	C₂×C₅	semidirect
121. D₄×C₂²⋊C₄	C₂×C₅	semidirect
122. #2333	C₂×C₅	semidirect
123. #2324	C₂×C₅	semidirect
124. #2309	C₂×C₅	semidirect
125. #2308	C₂×C₅	semidirect
126. #2280	C₂×C₅	semidirect
127. #2280	C₂×C₅	semidirect
203. C₂×#121	C₄×C₅	semidirect
208. C₂×C₄×D₄	C₂²×C₅	semidirect
210. C₂×#117	C₂²×C₅	semidirect
211. C₂×#117	C₂²×C₅	semidirect
215. #390	C₂²×C₅	semidirect
216. #390	C₂²×C₅	semidirect
217. #390	C₂²×C₅	semidirect
218. #389	C₂²×C₅	semidirect
219. #388	C₂²×C₅	semidirect
220. #388	C₂²×C₅	semidirect
221. #387	C₂²×C₅	semidirect
222. #386	C₂²×C₅	semidirect
223. #386	C₂²×C₅	semidirect
224. #386	C₂²×C₅	semidirect
225. #385	C₂²×C₅	semidirect
227. #381	C₂²×C₅	semidirect
228. #381	C₂²×C₅	semidirect
231. #380	C₂²×C₅	semidirect
232. #380	C₂²×C₅	semidirect
237. C₄×C₄⋊C₄	C₂²×C₅	semidirect
240. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
241. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
242. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
306. C₂²×D₄	C₂×C₄×C₅	semidirect
312. C₂³×C₄	C₂×C₄×C₅	semidirect
321. #121	C₂×C₄×C₅	semidirect
322. #121	C₂×C₄×C₅	semidirect
323. #121	C₂×C₄×C₅	semidirect
324. #121	C₂×C₄×C₅	semidirect
329. C₂×C₄⋊C₄	C₂×C₄×C₅	semidirect
336. C₂×C₂²⋊C₄	C₂×C₄×C₅	semidirect
529. C₅	#34363	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#121	2
C₅×#2377	2
C₅×#2363	2
C₅×#2344	2
C₅×#2342	2
C₅×D₄×C₂²⋊C₄	2
C₅×#2333	2
C₅×#2324	2
C₅×#2309	2
C₅×#2308	2
C₅×#2280	2
#34363	5

Subgroups — C5×#34363 #11957083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#34363 #11957083