Subgroups — C₅×#34280 #11957000

Order 1280.

2486 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#540	C₂	1	1	yes	maximal, characteristic
2	C₅×#2374	C₂	1	4	yes	maximal, complemented
3	C₅×#2364	C₂	1	4	yes	maximal, complemented
4	C₅×#2363	C₂	1	4	yes	maximal, complemented
5	C₅×#2359	C₂	1	4	yes	maximal, complemented
6	C₅×#2330	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2328	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2324	C₂	1	2	yes	maximal, complemented
9	C₅×#2314	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₅×#121	C₂	1	2	yes	maximal, complemented
13	C₅×#2293	C₂	1	2	yes	maximal, complemented
14	C₅×#2275	C₂	1	2	yes	maximal, complemented
15	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#540	C₄	1	16	yes	complemented
17	C₂×C₅×#127	C₂²	1	1	yes	characteristic
18	C₂×C₅×#125	C₂²	1	1	yes	characteristic
19	C₂×C₅×#123	C₂²	1	4	yes	—
20	C₂×C₅×#121	C₂²	1	2	yes	—
21	C₂×C₅×#121	C₂²	1	4	yes	—
22	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
23	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
24	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
25	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
26	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
27	C₂×C₅×#117	C₂²	1	1	yes	characteristic
28	C₂×C₅×#117	C₂²	1	4	yes	—
29	C₂×C₅×#117	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
31	C₅×#390	C₂²	1	2	yes	—
32	C₅×#390	C₂²	1	2	yes	—
33	C₅×#390	C₂²	1	4	yes	—
34	C₅×#390	C₂²	1	4	yes	—
35	C₅×#389	C₂²	1	2	yes	—
36	C₅×#389	C₂²	1	2	yes	—
37	C₅×#389	C₂²	1	4	yes	—
38	C₅×#388	C₂²	1	4	yes	—
39	C₅×#388	C₂²	1	4	yes	—
40	C₅×#388	C₂²	1	4	yes	—
41	C₅×#388	C₂²	1	4	yes	—
42	C₅×#388	C₂²	1	4	yes	—
43	C₅×#387	C₂²	1	4	yes	—
44	C₅×#387	C₂²	1	4	yes	—
45	C₅×#387	C₂²	1	4	yes	—
46	C₅×#386	C₂²	1	4	yes	—
47	C₅×#386	C₂²	1	4	yes	—
48	C₅×#386	C₂²	1	4	yes	—
49	C₅×#386	C₂²	1	4	yes	—
50	C₅×#385	C₂²	1	4	yes	—
51	C₅×#384	C₂²	1	1	yes	characteristic
52	C₅×#382	C₂²	1	1	yes	characteristic
53	C₅×#380	C₂²	1	2	yes	—
54	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
55	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
56	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
57	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
58	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
59	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
60	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
61	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
62	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
63	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
64	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
65	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
66	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
67	C₅×#376	C₂²	1	1	yes	characteristic
68	C₅×#376	C₂²	1	1	yes	characteristic
69	C₂×C₅×#95	C₂²	1	4	yes	—
70	C₄³×C₅	C₂²	1	1	yes	characteristic
71	C₄³×C₅	C₂²	1	1	yes	characteristic
72	#34280	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂²×C₅×D₄	C₂³	1	2	yes	—
74	C₂²×C₅×D₄	C₂³	1	4	yes	—
75	C₂²×C₅×D₄	C₂³	1	4	yes	—
76	C₂³×C₄×C₅	C₂³	1	2	yes	—
77	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₅	C₂³	1	4	yes	—
79	C₂³×C₄×C₅	C₂³	1	4	yes	—
80	C₂³×C₄×C₅	C₂³	1	4	yes	—
81	C₅×#127	C₂×C₄	1	8	yes	complemented
82	C₅×#125	C₂×C₄	1	8	yes	complemented
83	C₅×#123	C₂×C₄	1	32	yes	complemented
84	C₅×#121	C₂×C₄	1	32	yes	complemented
85	C₅×#121	C₂×C₄	1	16	yes	complemented
86	C₄×C₅×D₄	—	2	8	no	—
87	C₄×C₅×D₄	—	2	8	no	—
88	C₄×C₅×D₄	—	4	8	no	—
89	C₄×C₅×D₄	—	4	8	no	—
90	C₄×C₅×D₄	C₂×C₄	1	16	yes	complemented
91	C₅×#117	—	2	4	no	—
92	C₅×#117	—	4	8	no	—
93	C₅×#117	C₂×C₄	1	8	yes	complemented
94	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
95	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
96	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
97	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
98	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
99	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
100	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
101	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
102	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
103	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
104	C₂×C₅×C₂²⋊C₄	C₂³	1	1	yes	characteristic
105	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
106	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
107	C₂×C₅×C₂²⋊C₄	C₂³	1	1	yes	characteristic
108	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
109	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
110	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
111	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
112	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
113	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
114	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
115	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
116	C₂×C₄²×C₅	—	2	4	no	—
117	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₅	C₂³	1	2	yes	—
120	C₂×C₄²×C₅	C₂³	1	2	yes	—
121	C₂×C₄²×C₅	C₂³	1	2	yes	—
122	C₂×C₄²×C₅	C₂³	1	2	yes	—
123	C₂×C₄²×C₅	C₂³	1	2	yes	—
124	C₂×C₄²×C₅	C₂³	1	4	yes	—
125	C₂×C₄²×C₅	C₂³	1	4	yes	—
126	C₂×C₄²×C₅	C₂³	1	4	yes	—
127	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
131	C₅×#95	C₂³	1	2	yes	—
132	C₅×#95	C₂³	1	4	yes	—
133	C₅×#95	C₂³	1	4	yes	—
134	C₅×#95	C₂³	1	4	yes	—
135	C₅×#95	C₂³	1	2	yes	—
136	C₅×#95	C₂³	1	2	yes	—
137	C₅×#95	C₂³	1	2	yes	—
138	C₅×#95	C₂³	1	4	yes	—
139	C₅×#95	C₂³	1	4	yes	—
140	C₅×#95	C₂³	1	4	yes	—
141	C₅×#95	C₂³	1	4	yes	—
142	C₂×#540	C₂×C₅	1	1	yes	characteristic
143	#2374	C₂×C₅	1	4	yes	complemented
144	#2364	C₂×C₅	1	4	yes	complemented
145	#2363	C₂×C₅	1	4	yes	complemented
146	#2359	C₂×C₅	1	4	yes	complemented
147	#2330	C₂×C₅	1	1	yes	characteristic, complemented
148	#2328	C₂×C₅	1	1	yes	characteristic, complemented
149	#2324	C₂×C₅	1	2	yes	complemented
150	#2314	C₂×C₅	1	1	yes	characteristic, complemented
151	C₄×#127	C₂×C₅	1	1	yes	characteristic, complemented
152	C₄×#125	C₂×C₅	1	1	yes	characteristic, complemented
153	C₄×#121	C₂×C₅	1	2	yes	complemented
154	#2293	C₂×C₅	1	2	yes	complemented
155	#2275	C₂×C₅	1	2	yes	complemented
156	C₄×#117	C₂×C₅	1	1	yes	characteristic, complemented
157	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
158	C₂⁴×C₅	C₂⁴	1	4	yes	—
159	C₂×C₅×D₄	—	2	4	no	—
160	C₂×C₅×D₄	—	2	4	no	—
161	C₂×C₅×D₄	—	4	4	no	—
162	C₂×C₅×D₄	—	4	4	no	—
163	C₂×C₅×D₄	—	4	4	no	—
164	C₂×C₅×D₄	—	4	4	no	—
165	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
166	C₂×C₅×D₄	C₂²×C₄	1	16	yes	—
167	C₂×C₅×D₄	C₂²×C₄	1	16	yes	—
168	C₂²×C₄×C₅	—	2	2	no	—
169	C₂²×C₄×C₅	—	2	2	no	—
170	C₂²×C₄×C₅	—	2	4	no	—
171	C₂²×C₄×C₅	—	2	4	no	—
172	C₂²×C₄×C₅	—	2	8	no	—
173	C₂²×C₄×C₅	—	2	8	no	—
174	C₂²×C₄×C₅	—	4	4	no	—
175	C₂²×C₄×C₅	—	4	2	no	—
176	C₂²×C₄×C₅	—	4	4	no	—
177	C₂²×C₄×C₅	—	4	4	no	—
178	C₂²×C₄×C₅	—	4	4	no	—
179	C₂²×C₄×C₅	—	4	4	no	—
180	C₂²×C₄×C₅	—	4	4	no	—
181	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
182	C₂²×C₄×C₅	C₂²×C₄	1	16	yes	—
183	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
187	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
189	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
191	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
194	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
196	C₅×C₄⋊C₄	—	2	2	no	—
197	C₅×C₄⋊C₄	—	2	2	no	—
198	C₅×C₄⋊C₄	—	2	4	no	—
199	C₅×C₄⋊C₄	—	2	4	no	—
200	C₅×C₄⋊C₄	—	2	8	no	—
201	C₅×C₄⋊C₄	—	4	4	no	—
202	C₅×C₄⋊C₄	—	4	4	no	—
203	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
204	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
205	C₅×C₄⋊C₄	C₂²×C₄	1	16	yes	—
206	C₅×C₂²⋊C₄	—	2	2	no	—
207	C₅×C₂²⋊C₄	—	2	2	no	—
208	C₅×C₂²⋊C₄	—	2	8	no	—
209	C₅×C₂²⋊C₄	—	2	8	no	—
210	C₅×C₂²⋊C₄	—	2	8	no	—
211	C₅×C₂²⋊C₄	—	2	8	no	—
212	C₅×C₂²⋊C₄	—	4	4	no	—
213	C₅×C₂²⋊C₄	—	4	4	no	—
214	C₅×C₂²⋊C₄	—	4	4	no	—
215	C₅×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
216	C₅×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
217	C₅×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
218	C₄²×C₅	—	2	4	no	—
219	C₄²×C₅	—	2	2	no	—
220	C₄²×C₅	—	2	2	no	—
221	C₄²×C₅	—	2	4	no	—
222	C₄²×C₅	—	2	4	no	—
223	C₄²×C₅	—	4	2	no	—
224	C₄²×C₅	—	4	4	no	—
225	C₄²×C₅	—	4	2	no	—
226	C₄²×C₅	—	4	4	no	—
227	C₄²×C₅	—	4	4	no	—
228	C₄²×C₅	C₂²×C₄	1	4	yes	—
229	C₄²×C₅	C₂²×C₄	1	4	yes	—
230	C₄²×C₅	#41	1	4	yes	—
231	C₄²×C₅	#41	1	4	yes	—
232	#540	C₄×C₅	1	16	yes	complemented
233	C₂×#127	C₂²×C₅	1	1	yes	characteristic
234	C₂×#125	C₂²×C₅	1	1	yes	characteristic
235	C₂×#123	C₂²×C₅	1	4	yes	—
236	C₂×#121	C₂²×C₅	1	2	yes	—
237	C₂×#121	C₂²×C₅	1	4	yes	—
238	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
239	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
240	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
241	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
242	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
243	C₂×#117	C₂²×C₅	1	1	yes	characteristic
244	C₂×#117	C₂²×C₅	1	4	yes	—
245	C₂×#117	C₂²×C₅	1	1	yes	characteristic
246	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
247	#390	C₂²×C₅	1	4	yes	—
248	#390	C₂²×C₅	1	2	yes	—
249	#390	C₂²×C₅	1	2	yes	—
250	#390	C₂²×C₅	1	4	yes	—
251	#389	C₂²×C₅	1	2	yes	—
252	#389	C₂²×C₅	1	2	yes	—
253	#389	C₂²×C₅	1	4	yes	—
254	#388	C₂²×C₅	1	4	yes	—
255	#388	C₂²×C₅	1	4	yes	—
256	#388	C₂²×C₅	1	4	yes	—
257	#388	C₂²×C₅	1	4	yes	—
258	#388	C₂²×C₅	1	4	yes	—
259	#387	C₂²×C₅	1	4	yes	—
260	#387	C₂²×C₅	1	4	yes	—
261	#387	C₂²×C₅	1	4	yes	—
262	#386	C₂²×C₅	1	4	yes	—
263	#386	C₂²×C₅	1	4	yes	—
264	#386	C₂²×C₅	1	4	yes	—
265	#386	C₂²×C₅	1	4	yes	—
266	#385	C₂²×C₅	1	4	yes	—
267	#384	C₂²×C₅	1	1	yes	characteristic
268	#382	C₂²×C₅	1	1	yes	characteristic
269	#380	C₂²×C₅	1	2	yes	—
270	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
271	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
272	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
273	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
274	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
275	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
276	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
277	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
278	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
279	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
280	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
281	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
282	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
283	#376	C₂²×C₅	1	1	yes	characteristic
284	#376	C₂²×C₅	1	1	yes	characteristic
285	C₂×#95	C₂²×C₅	1	4	yes	—
286	C₄³	C₂²×C₅	1	1	yes	characteristic
287	C₄³	C₂²×C₅	1	1	yes	characteristic
288	C₂³×C₅	—	2	1	no	—
289	C₂³×C₅	—	2	4	no	—
290	C₂³×C₅	—	2	4	no	—
291	C₂³×C₅	—	2	1	no	—
292	C₂³×C₅	—	2	4	no	—
293	C₂³×C₅	—	2	4	no	—
294	C₂³×C₅	—	2	2	no	—
295	C₂³×C₅	—	2	4	no	—
296	C₂³×C₅	—	2	4	no	—
297	C₂³×C₅	—	2	2	no	—
298	C₂³×C₅	C₂³×C₄	1	2	yes	—
299	C₂³×C₅	C₂³×C₄	1	8	yes	—
300	C₂³×C₅	C₂⁵	1	1	yes	characteristic
301	C₅×D₄	—	2	16	no	—
302	C₅×D₄	—	4	16	no	—
303	C₅×D₄	—	4	16	no	—
304	C₂×C₄×C₅	—	2	1	no	—
305	C₂×C₄×C₅	—	2	2	no	—
306	C₂×C₄×C₅	—	2	1	no	—
307	C₂×C₄×C₅	—	2	2	no	—
308	C₂×C₄×C₅	—	2	2	no	—
309	C₂×C₄×C₅	—	2	2	no	—
310	C₂×C₄×C₅	—	2	2	no	—
311	C₂×C₄×C₅	—	2	4	no	—
312	C₂×C₄×C₅	—	2	2	no	—
313	C₂×C₄×C₅	—	2	2	no	—
314	C₂×C₄×C₅	—	2	2	no	—
315	C₂×C₄×C₅	—	2	4	no	—
316	C₂×C₄×C₅	—	2	2	no	—
317	C₂×C₄×C₅	—	2	2	no	—
318	C₂×C₄×C₅	—	2	8	no	—
319	C₂×C₄×C₅	—	2	1	no	—
320	C₂×C₄×C₅	—	2	1	no	—
321	C₂×C₄×C₅	—	2	4	no	—
322	C₂×C₄×C₅	—	2	4	no	—
323	C₂×C₄×C₅	—	2	2	no	—
324	C₂×C₄×C₅	—	2	4	no	—
325	C₂×C₄×C₅	—	2	4	no	—
326	C₂×C₄×C₅	—	2	4	no	—
327	C₂×C₄×C₅	—	2	2	no	—
328	C₂×C₄×C₅	—	2	4	no	—
329	C₂×C₄×C₅	—	4	16	no	—
330	C₂×C₄×C₅	—	4	8	no	—
331	C₂×C₄×C₅	—	8	8	no	—
332	C₂×C₄×C₅	—	8	8	no	—
333	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
334	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
335	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
336	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
337	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
338	C₂×C₄×C₅	C₂×#41	1	4	yes	—
339	C₂×C₄×C₅	C₂×#41	1	2	yes	—
340	C₂×C₄×C₅	C₂×#41	1	2	yes	—
341	C₂×C₄×C₅	C₂×#41	1	2	yes	—
342	C₂×C₄×C₅	C₂×#41	1	2	yes	—
343	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
344	C₂²×D₄	C₂³×C₅	1	2	yes	—
345	C₂²×D₄	C₂³×C₅	1	4	yes	—
346	C₂²×D₄	C₂³×C₅	1	4	yes	—
347	C₂³×C₄	C₂³×C₅	1	2	yes	—
348	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
349	C₂³×C₄	C₂³×C₅	1	4	yes	—
350	C₂³×C₄	C₂³×C₅	1	4	yes	—
351	C₂³×C₄	C₂³×C₅	1	4	yes	—
352	#127	C₂×C₄×C₅	1	8	yes	complemented
353	#125	C₂×C₄×C₅	1	8	yes	complemented
354	#123	C₂×C₄×C₅	1	32	yes	complemented
355	#121	C₂×C₄×C₅	1	32	yes	complemented
356	#121	C₂×C₄×C₅	1	16	yes	complemented
357	C₄×D₄	—	2	8	no	—
358	C₄×D₄	—	2	8	no	—
359	C₄×D₄	—	4	8	no	—
360	C₄×D₄	—	4	8	no	—
361	C₄×D₄	C₂×C₄×C₅	1	16	yes	complemented
362	#117	—	2	4	no	—
363	#117	—	4	8	no	—
364	#117	C₂×C₄×C₅	1	8	yes	complemented
365	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
366	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
372	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
374	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₅	1	1	yes	characteristic
377	C₂×C₂²⋊C₄	C₂³×C₅	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
387	C₂×C₄²	—	2	4	no	—
388	C₂×C₄²	C₂³×C₅	1	2	yes	—
389	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₅	1	2	yes	—
391	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₅	1	2	yes	—
393	C₂×C₄²	C₂³×C₅	1	4	yes	—
394	C₂×C₄²	C₂³×C₅	1	4	yes	—
395	C₂×C₄²	C₂³×C₅	1	2	yes	—
396	C₂×C₄²	C₂³×C₅	1	2	yes	—
397	C₂×C₄²	C₂³×C₅	1	4	yes	—
398	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
400	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
402	#95	C₂³×C₅	1	2	yes	—
403	#95	C₂³×C₅	1	4	yes	—
404	#95	C₂³×C₅	1	4	yes	—
405	#95	C₂³×C₅	1	4	yes	—
406	#95	C₂³×C₅	1	2	yes	—
407	#95	C₂³×C₅	1	2	yes	—
408	#95	C₂³×C₅	1	2	yes	—
409	#95	C₂³×C₅	1	4	yes	—
410	#95	C₂³×C₅	1	4	yes	—
411	#95	C₂³×C₅	1	4	yes	—
412	#95	C₂³×C₅	1	4	yes	—
413	C₂²×C₅	—	2	2	no	—
414	C₂²×C₅	—	2	8	no	—
415	C₂²×C₅	—	2	8	no	—
416	C₂²×C₅	—	2	8	no	—
417	C₂²×C₅	—	2	4	no	—
418	C₂²×C₅	—	4	2	no	—
419	C₂²×C₅	—	4	4	no	—
420	C₂²×C₅	—	4	4	no	—
421	C₂²×C₅	—	4	4	no	—
422	C₂²×C₅	—	4	1	no	—
423	C₂²×C₅	—	4	1	no	—
424	C₂²×C₅	—	4	4	no	—
425	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
426	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
427	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
428	C₂²×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
429	C₂²×C₅	#585	1	2	yes	—
430	C₄×C₅	—	2	2	no	—
431	C₄×C₅	—	2	4	no	—
432	C₄×C₅	—	2	8	no	—
433	C₄×C₅	—	2	4	no	—
434	C₄×C₅	—	2	4	no	—
435	C₄×C₅	—	2	4	no	—
436	C₄×C₅	—	2	2	no	—
437	C₄×C₅	—	2	2	no	—
438	C₄×C₅	—	4	4	no	—
439	C₄×C₅	—	4	1	no	—
440	C₄×C₅	—	4	4	no	—
441	C₄×C₅	—	4	1	no	—
442	C₄×C₅	C₄×#41	1	4	yes	—
443	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
444	C₂⁴	C₂⁴×C₅	1	4	yes	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	4	4	no	—
448	C₂×D₄	—	4	4	no	—
449	C₂×D₄	—	4	4	no	—
450	C₂×D₄	—	4	4	no	—
451	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
452	C₂×D₄	C₂²×C₄×C₅	1	16	yes	—
453	C₂×D₄	C₂²×C₄×C₅	1	16	yes	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	8	no	—
459	C₂²×C₄	—	2	8	no	—
460	C₂²×C₄	—	4	4	no	—
461	C₂²×C₄	—	4	2	no	—
462	C₂²×C₄	—	4	4	no	—
463	C₂²×C₄	—	4	4	no	—
464	C₂²×C₄	—	4	4	no	—
465	C₂²×C₄	—	4	4	no	—
466	C₂²×C₄	—	4	4	no	—
467	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
468	C₂²×C₄	C₂²×C₄×C₅	1	16	yes	—
469	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
471	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
473	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
475	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
476	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
478	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
479	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
481	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	8	no	—
487	C₄⋊C₄	—	4	4	no	—
488	C₄⋊C₄	—	4	4	no	—
489	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
490	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
491	C₄⋊C₄	C₂²×C₄×C₅	1	16	yes	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	8	no	—
495	C₂²⋊C₄	—	2	8	no	—
496	C₂²⋊C₄	—	2	8	no	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	—	4	4	no	—
499	C₂²⋊C₄	—	4	4	no	—
500	C₂²⋊C₄	—	4	4	no	—
501	C₂²⋊C₄	C₂²×C₄×C₅	1	8	yes	—
502	C₂²⋊C₄	C₂²×C₄×C₅	1	16	yes	—
503	C₂²⋊C₄	C₂²×C₄×C₅	1	16	yes	—
504	C₄²	—	2	4	no	—
505	C₄²	—	2	2	no	—
506	C₄²	—	2	2	no	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	4	no	—
509	C₄²	—	4	4	no	—
510	C₄²	—	4	4	no	—
511	C₄²	—	4	2	no	—
512	C₄²	—	4	4	no	—
513	C₄²	—	4	2	no	—
514	C₄²	C₂²×C₄×C₅	1	4	yes	—
515	C₄²	C₂²×C₄×C₅	1	4	yes	—
516	C₄²	C₅×#41	1	4	yes	—
517	C₄²	C₅×#41	1	4	yes	—
518	C₂×C₅	—	4	2	no	—
519	C₂×C₅	—	4	8	no	—
520	C₂×C₅	C₂×C₄×#41	1	1	yes	characteristic
521	C₂×C₅	C₄×H(5,𝔽₂)	1	2	yes	—
522	C₂×C₅	#3494	1	2	yes	—
523	C₂×C₅	#3508	1	1	yes	characteristic
524	C₂×C₅	#3512	1	1	yes	characteristic
525	C₂³	—	2	1	no	—
526	C₂³	—	2	4	no	—
527	C₂³	—	2	4	no	—
528	C₂³	—	2	1	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	2	4	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	4	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	2	no	—
535	C₂³	C₂³×C₄×C₅	1	2	yes	—
536	C₂³	C₂³×C₄×C₅	1	8	yes	—
537	C₂³	C₂⁵×C₅	1	1	yes	characteristic
538	D₄	—	2	16	no	—
539	D₄	—	4	16	no	—
540	D₄	—	4	16	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	8	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	4	16	no	—
567	C₂×C₄	—	4	8	no	—
568	C₂×C₄	—	8	8	no	—
569	C₂×C₄	—	8	8	no	—
570	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
571	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
572	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
573	C₂×C₄	C₂³×C₄×C₅	1	8	yes	—
574	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
575	C₂×C₄	C₂×C₅×#41	1	4	yes	—
576	C₂×C₄	C₂×C₅×#41	1	2	yes	—
577	C₂×C₄	C₂×C₅×#41	1	2	yes	—
578	C₂×C₄	C₂×C₅×#41	1	2	yes	—
579	C₂×C₄	C₂×C₅×#41	1	2	yes	—
580	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
581	C₅	#34280	1	1	yes	characteristic, direct product factor, Sylow
582	C₂²	—	2	2	no	—
583	C₂²	—	2	8	no	—
584	C₂²	—	2	8	no	—
585	C₂²	—	2	8	no	—
586	C₂²	—	2	4	no	—
587	C₂²	—	4	2	no	—
588	C₂²	—	4	4	no	—
589	C₂²	—	4	4	no	—
590	C₂²	—	4	4	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	4	1	no	—
593	C₂²	—	4	4	no	—
594	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
595	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
596	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
597	C₂²	C₂×C₅×H(5,𝔽₂)	1	2	yes	—
598	C₂²	C₅×#585	1	2	yes	—
599	C₄	—	2	2	no	—
600	C₄	—	2	4	no	—
601	C₄	—	2	8	no	—
602	C₄	—	2	4	no	—
603	C₄	—	2	4	no	—
604	C₄	—	2	4	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	2	no	—
607	C₄	—	4	4	no	—
608	C₄	—	4	4	no	—
609	C₄	—	4	1	no	—
610	C₄	—	4	1	no	—
611	C₄	C₄×C₅×#41	1	4	yes	—
612	C₂	—	4	2	no	—
613	C₂	—	4	8	no	—
614	C₂	C₂×C₄×C₅×#41	1	1	yes	characteristic
615	C₂	C₄×C₅×H(5,𝔽₂)	1	2	yes	—
616	C₂	C₅×#3494	1	2	yes	—
617	C₂	C₅×#3508	1	1	yes	characteristic
618	C₂	C₅×#3512	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅
Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂×C₅×#540	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×#2374	C₂	semidirect
3. C₅×#2364	C₂	semidirect
4. C₅×#2363	C₂	semidirect
5. C₅×#2359	C₂	semidirect
6. C₅×#2330	C₂	semidirect
7. C₅×#2328	C₂	semidirect
8. C₅×#2324	C₂	semidirect
9. C₅×#2314	C₂	semidirect
10. C₄×C₅×#127	C₂	semidirect
11. C₄×C₅×#125	C₂	semidirect
12. C₄×C₅×#121	C₂	semidirect
13. C₅×#2293	C₂	semidirect
14. C₅×#2275	C₂	semidirect
15. C₄×C₅×#117	C₂	semidirect
16. C₅×#540	C₄	semidirect
72. #34280	C₅	direct (N × H)
81. C₅×#127	C₂×C₄	semidirect
82. C₅×#125	C₂×C₄	semidirect
83. C₅×#123	C₂×C₄	semidirect
84. C₅×#121	C₂×C₄	semidirect
85. C₅×#121	C₂×C₄	semidirect
90. C₄×C₅×D₄	C₂×C₄	semidirect
93. C₅×#117	C₂×C₄	semidirect
143. #2374	C₂×C₅	semidirect
144. #2364	C₂×C₅	semidirect
145. #2363	C₂×C₅	semidirect
146. #2359	C₂×C₅	semidirect
147. #2330	C₂×C₅	semidirect
148. #2328	C₂×C₅	semidirect
149. #2324	C₂×C₅	semidirect
150. #2314	C₂×C₅	semidirect
151. C₄×#127	C₂×C₅	semidirect
152. C₄×#125	C₂×C₅	semidirect
153. C₄×#121	C₂×C₅	semidirect
154. #2293	C₂×C₅	semidirect
155. #2275	C₂×C₅	semidirect
156. C₄×#117	C₂×C₅	semidirect
232. #540	C₄×C₅	semidirect
352. #127	C₂×C₄×C₅	semidirect
353. #125	C₂×C₄×C₅	semidirect
354. #123	C₂×C₄×C₅	semidirect
355. #121	C₂×C₄×C₅	semidirect
356. #121	C₂×C₄×C₅	semidirect
361. C₄×D₄	C₂×C₄×C₅	semidirect
364. #117	C₂×C₄×C₅	semidirect
581. C₅	#34280	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×#540	2
C₅×#2374	2
C₅×#2364	2
C₅×#2363	2
C₅×#2359	2
C₅×#2330	2
C₅×#2328	2
C₅×#2324	2
C₅×#2314	2
C₄×C₅×#127	2
C₄×C₅×#125	2
C₄×C₅×#121	2
C₅×#2293	2
C₅×#2275	2
C₄×C₅×#117	2
#34280	5

Subgroups — C5×#34280 #11957000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#34280 #11957000