Subgroups — C₅×#3557 #10619342

Order 640.

930 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#563	C₂	1	4	yes	maximal, complemented
2	C₅×#559	C₂	1	2	yes	maximal, complemented
3	C₅×#557	C₂	1	2	yes	maximal, complemented
4	C₅×#556	C₂	1	2	yes	maximal, complemented
5	C₅×#554	C₂	1	2	yes	maximal, complemented
6	C₅×#552	C₂	1	4	yes	maximal, complemented
7	C₅×#542	C₂	1	2	yes	maximal, complemented
8	C₅×#542	C₂	1	4	yes	maximal, complemented
9	C₅×#541	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#540	C₂	1	1	yes	maximal, characteristic
11	C₅×#539	C₂	1	4	yes	maximal, complemented
12	C₅×#533	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#529	C₂	1	2	yes	maximal, complemented
14	C₅×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₅×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₅×C₄⋊Q₈	C₂²	1	4	yes	—
18	C₅×#127	C₂²	1	1	yes	characteristic
19	C₅×#126	C₂²	1	4	yes	—
20	C₅×#126	C₂²	1	4	yes	—
21	C₅×#126	C₂²	1	4	yes	—
22	C₅×#126	C₂²	1	4	yes	—
23	C₅×#126	C₂²	1	4	yes	—
24	C₅×#125	C₂²	1	2	yes	—
25	C₅×#125	C₂²	1	2	yes	—
26	C₅×#125	C₂²	1	4	yes	—
27	C₅×#125	C₂²	1	4	yes	—
28	C₅×#125	C₂²	1	1	yes	characteristic
29	C₅×#125	C₂²	1	4	yes	—
30	C₅×#125	C₂²	1	1	yes	characteristic
31	C₅×#124	C₂²	1	2	yes	—
32	C₅×#124	C₂²	1	4	yes	—
33	C₅×#124	C₂²	1	2	yes	—
34	C₅×#124	C₂²	1	4	yes	—
35	C₅×#123	C₂²	1	4	yes	—
36	C₅×#123	C₂²	1	4	yes	—
37	C₅×#123	C₂²	1	4	yes	—
38	C₅×#123	C₂²	1	4	yes	—
39	C₅×#123	C₂²	1	4	yes	—
40	C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
41	C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
42	C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
43	C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
44	C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
45	C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
46	C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
47	C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
48	C₅×#121	C₂²	1	2	yes	—
49	C₅×#121	C₂²	1	4	yes	—
50	C₄×C₅×Q₈	C₂²	1	2	yes	—
51	C₄×C₅×Q₈	C₂²	1	2	yes	—
52	C₄×C₅×Q₈	C₂²	1	2	yes	—
53	C₄×C₅×Q₈	C₂²	1	2	yes	—
54	C₄×C₅×Q₈	C₂²	1	4	yes	—
55	C₄×C₅×Q₈	C₂²	1	2	yes	—
56	C₄×C₅×D₄	C₂²	1	4	yes	—
57	C₄×C₅×D₄	C₂²	1	2	yes	—
58	C₄×C₅×D₄	C₂²	1	4	yes	—
59	C₄×C₅×D₄	C₂²	1	2	yes	—
60	C₄×C₅×D₄	C₂²	1	2	yes	—
61	C₅×#117	C₂²	1	4	yes	—
62	C₅×#117	C₂²	1	1	yes	characteristic
63	C₅×#117	C₂²	1	1	yes	characteristic
64	C₅×#117	C₂²	1	4	yes	—
65	C₂×C₅×C₄⋊C₄	C₂²	1	4	yes	—
66	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
67	#3557	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₅×Q₈	C₂³	1	2	yes	—
69	C₂×C₅×Q₈	C₂³	1	2	yes	—
70	C₂×C₅×Q₈	C₂³	1	4	yes	—
71	C₂×C₅×Q₈	C₂³	1	2	yes	—
72	C₂×C₅×D₄	C₂³	1	2	yes	—
73	C₂×C₅×D₄	C₂³	1	4	yes	—
74	C₂×C₅×D₄	C₂³	1	4	yes	—
75	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂³	1	2	yes	—
77	C₂²×C₄×C₅	C₂³	1	4	yes	—
78	C₂²×C₄×C₅	C₂³	1	4	yes	—
79	C₂²×C₄×C₅	C₂³	1	4	yes	—
80	C₅×C₄⋊C₄	C₂³	1	2	yes	—
81	C₅×C₄⋊C₄	C₂³	1	4	yes	—
82	C₅×C₄⋊C₄	C₂³	1	4	yes	—
83	C₅×C₄⋊C₄	C₂³	1	4	yes	—
84	C₅×C₄⋊C₄	C₂³	1	2	yes	—
85	C₅×C₄⋊C₄	C₂³	1	4	yes	—
86	C₅×C₄⋊C₄	C₂³	1	2	yes	—
87	C₅×C₄⋊C₄	C₂³	1	4	yes	—
88	C₅×C₄⋊C₄	C₂³	1	2	yes	—
89	C₅×C₄⋊C₄	C₂³	1	2	yes	—
90	C₅×C₄⋊C₄	C₂³	1	2	yes	—
91	C₅×C₄⋊C₄	C₂³	1	2	yes	—
92	C₅×C₄⋊C₄	C₂³	1	4	yes	—
93	C₅×C₄⋊C₄	C₂³	1	2	yes	—
94	C₅×C₄⋊C₄	C₂³	1	4	yes	—
95	C₅×C₄⋊C₄	C₂³	1	2	yes	—
96	C₅×C₄⋊C₄	C₂³	1	4	yes	—
97	C₅×C₄⋊C₄	C₂³	1	2	yes	—
98	C₅×C₄⋊C₄	C₂³	1	4	yes	—
99	C₅×C₄⋊C₄	C₂³	1	4	yes	—
100	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
101	C₅×C₂²⋊C₄	C₂³	1	2	yes	—
102	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
103	C₅×C₂²⋊C₄	C₂³	1	1	yes	characteristic
104	C₅×C₂²⋊C₄	C₂³	1	1	yes	characteristic
105	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
106	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
107	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
108	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
109	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
110	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
111	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
112	C₄²×C₅	C₂³	1	1	yes	characteristic
113	C₄²×C₅	C₂³	1	2	yes	—
114	C₄²×C₅	C₂³	1	1	yes	characteristic
115	C₄²×C₅	C₂³	1	4	yes	—
116	C₄²×C₅	C₂³	1	1	yes	characteristic
117	C₄²×C₅	C₂³	1	2	yes	—
118	C₄²×C₅	C₂³	1	1	yes	characteristic
119	C₄²×C₅	C₂³	1	4	yes	—
120	C₄²×C₅	C₂³	1	4	yes	—
121	#563	C₂×C₅	1	4	yes	complemented
122	#559	C₂×C₅	1	2	yes	complemented
123	#557	C₂×C₅	1	2	yes	complemented
124	#556	C₂×C₅	1	2	yes	complemented
125	#554	C₂×C₅	1	2	yes	complemented
126	#552	C₂×C₅	1	4	yes	complemented
127	#542	C₂×C₅	1	2	yes	complemented
128	#542	C₂×C₅	1	4	yes	complemented
129	#541	C₂×C₅	1	1	yes	characteristic, complemented
130	#540	C₂×C₅	1	1	yes	characteristic
131	#539	C₂×C₅	1	4	yes	complemented
132	#533	C₂×C₅	1	1	yes	characteristic, complemented
133	#529	C₂×C₅	1	2	yes	complemented
134	C₂³×C₅	C₂⁴	1	1	yes	characteristic
135	C₂³×C₅	C₂⁴	1	4	yes	—
136	C₅×Q₈	—	2	4	no	—
137	C₅×Q₈	—	4	2	no	—
138	C₅×Q₈	—	4	4	no	—
139	C₅×Q₈	—	4	2	no	—
140	C₅×D₄	—	2	4	no	—
141	C₅×D₄	—	4	4	no	—
142	C₅×D₄	—	4	4	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	4	4	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	4	4	no	—
147	C₂×C₄×C₅	—	4	4	no	—
148	C₂×C₄×C₅	C₂⁴	1	2	yes	—
149	C₂×C₄×C₅	C₂⁴	1	4	yes	—
150	C₂×C₄×C₅	C₂⁴	1	2	yes	—
151	C₂×C₄×C₅	C₂⁴	1	2	yes	—
152	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂⁴	1	2	yes	—
154	C₂×C₄×C₅	C₂⁴	1	4	yes	—
155	C₂×C₄×C₅	C₂⁴	1	2	yes	—
156	C₂×C₄×C₅	C₂⁴	1	2	yes	—
157	C₂×C₄×C₅	C₂⁴	1	4	yes	—
158	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
159	C₄⋊Q₈	C₂²×C₅	1	2	yes	—
160	C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
161	C₄⋊Q₈	C₂²×C₅	1	2	yes	—
162	C₄⋊Q₈	C₂²×C₅	1	4	yes	—
163	#127	C₂²×C₅	1	1	yes	characteristic
164	#126	C₂²×C₅	1	4	yes	—
165	#126	C₂²×C₅	1	4	yes	—
166	#126	C₂²×C₅	1	4	yes	—
167	#126	C₂²×C₅	1	4	yes	—
168	#126	C₂²×C₅	1	4	yes	—
169	#125	C₂²×C₅	1	2	yes	—
170	#125	C₂²×C₅	1	2	yes	—
171	#125	C₂²×C₅	1	4	yes	—
172	#125	C₂²×C₅	1	4	yes	—
173	#125	C₂²×C₅	1	1	yes	characteristic
174	#125	C₂²×C₅	1	4	yes	—
175	#125	C₂²×C₅	1	1	yes	characteristic
176	#124	C₂²×C₅	1	2	yes	—
177	#124	C₂²×C₅	1	4	yes	—
178	#124	C₂²×C₅	1	2	yes	—
179	#124	C₂²×C₅	1	4	yes	—
180	#123	C₂²×C₅	1	4	yes	—
181	#123	C₂²×C₅	1	4	yes	—
182	#123	C₂²×C₅	1	4	yes	—
183	#123	C₂²×C₅	1	4	yes	—
184	#123	C₂²×C₅	1	4	yes	—
185	C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
186	C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
187	C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
188	C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
189	C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
190	C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
191	C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
192	C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
193	#121	C₂²×C₅	1	2	yes	—
194	#121	C₂²×C₅	1	4	yes	—
195	C₄×Q₈	C₂²×C₅	1	2	yes	—
196	C₄×Q₈	C₂²×C₅	1	2	yes	—
197	C₄×Q₈	C₂²×C₅	1	2	yes	—
198	C₄×Q₈	C₂²×C₅	1	2	yes	—
199	C₄×Q₈	C₂²×C₅	1	4	yes	—
200	C₄×Q₈	C₂²×C₅	1	2	yes	—
201	C₄×D₄	C₂²×C₅	1	4	yes	—
202	C₄×D₄	C₂²×C₅	1	2	yes	—
203	C₄×D₄	C₂²×C₅	1	4	yes	—
204	C₄×D₄	C₂²×C₅	1	2	yes	—
205	C₄×D₄	C₂²×C₅	1	2	yes	—
206	#117	C₂²×C₅	1	4	yes	—
207	#117	C₂²×C₅	1	1	yes	characteristic
208	#117	C₂²×C₅	1	1	yes	characteristic
209	#117	C₂²×C₅	1	4	yes	—
210	C₂×C₄⋊C₄	C₂²×C₅	1	4	yes	—
211	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	2	4	no	—
214	C₂²×C₅	—	2	4	no	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	2	4	no	—
217	C₂²×C₅	C₂⁵	1	1	yes	characteristic
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	4	no	—
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	—	2	4	no	—
222	C₄×C₅	—	2	2	no	—
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	—	2	4	no	—
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	2₋¹⁺⁴	1	2	yes	—
229	C₂×Q₈	C₂³×C₅	1	2	yes	—
230	C₂×Q₈	C₂³×C₅	1	2	yes	—
231	C₂×Q₈	C₂³×C₅	1	4	yes	—
232	C₂×Q₈	C₂³×C₅	1	2	yes	—
233	C₂×D₄	C₂³×C₅	1	2	yes	—
234	C₂×D₄	C₂³×C₅	1	4	yes	—
235	C₂×D₄	C₂³×C₅	1	4	yes	—
236	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
237	C₂²×C₄	C₂³×C₅	1	2	yes	—
238	C₂²×C₄	C₂³×C₅	1	4	yes	—
239	C₂²×C₄	C₂³×C₅	1	4	yes	—
240	C₂²×C₄	C₂³×C₅	1	4	yes	—
241	C₄⋊C₄	C₂³×C₅	1	2	yes	—
242	C₄⋊C₄	C₂³×C₅	1	4	yes	—
243	C₄⋊C₄	C₂³×C₅	1	4	yes	—
244	C₄⋊C₄	C₂³×C₅	1	4	yes	—
245	C₄⋊C₄	C₂³×C₅	1	2	yes	—
246	C₄⋊C₄	C₂³×C₅	1	4	yes	—
247	C₄⋊C₄	C₂³×C₅	1	2	yes	—
248	C₄⋊C₄	C₂³×C₅	1	4	yes	—
249	C₄⋊C₄	C₂³×C₅	1	2	yes	—
250	C₄⋊C₄	C₂³×C₅	1	2	yes	—
251	C₄⋊C₄	C₂³×C₅	1	2	yes	—
252	C₄⋊C₄	C₂³×C₅	1	2	yes	—
253	C₄⋊C₄	C₂³×C₅	1	4	yes	—
254	C₄⋊C₄	C₂³×C₅	1	2	yes	—
255	C₄⋊C₄	C₂³×C₅	1	4	yes	—
256	C₄⋊C₄	C₂³×C₅	1	2	yes	—
257	C₄⋊C₄	C₂³×C₅	1	4	yes	—
258	C₄⋊C₄	C₂³×C₅	1	2	yes	—
259	C₄⋊C₄	C₂³×C₅	1	4	yes	—
260	C₄⋊C₄	C₂³×C₅	1	4	yes	—
261	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
262	C₂²⋊C₄	C₂³×C₅	1	2	yes	—
263	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
264	C₂²⋊C₄	C₂³×C₅	1	1	yes	characteristic
265	C₂²⋊C₄	C₂³×C₅	1	1	yes	characteristic
266	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
267	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
268	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
269	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
270	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
271	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
272	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
273	C₄²	C₂³×C₅	1	1	yes	characteristic
274	C₄²	C₂³×C₅	1	2	yes	—
275	C₄²	C₂³×C₅	1	1	yes	characteristic
276	C₄²	C₂³×C₅	1	4	yes	—
277	C₄²	C₂³×C₅	1	1	yes	characteristic
278	C₄²	C₂³×C₅	1	2	yes	—
279	C₄²	C₂³×C₅	1	1	yes	characteristic
280	C₄²	C₂³×C₅	1	4	yes	—
281	C₄²	C₂³×C₅	1	4	yes	—
282	C₂×C₅	—	4	1	no	—
283	C₂×C₅	—	4	4	no	—
284	C₂×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
285	C₂×C₅	#585	1	2	yes	—
286	C₂³	C₂⁴×C₅	1	1	yes	characteristic
287	C₂³	C₂⁴×C₅	1	4	yes	—
288	Q₈	—	2	4	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	4	no	—
291	Q₈	—	4	2	no	—
292	D₄	—	2	4	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	C₂⁴×C₅	1	2	yes	—
301	C₂×C₄	C₂⁴×C₅	1	4	yes	—
302	C₂×C₄	C₂⁴×C₅	1	2	yes	—
303	C₂×C₄	C₂⁴×C₅	1	2	yes	—
304	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
305	C₂×C₄	C₂⁴×C₅	1	2	yes	—
306	C₂×C₄	C₂⁴×C₅	1	4	yes	—
307	C₂×C₄	C₂⁴×C₅	1	2	yes	—
308	C₂×C₄	C₂⁴×C₅	1	2	yes	—
309	C₂×C₄	C₂⁴×C₅	1	4	yes	—
310	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
311	C₅	#3557	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²	—	2	1	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	4	no	—
317	C₂²	C₂⁵×C₅	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	4	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	4	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	4	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	C₅×2₋¹⁺⁴	1	2	yes	—
329	C₂	—	4	1	no	—
330	C₂	—	4	4	no	—
331	C₂	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic
332	C₂	C₅×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅
prime radical	C₅×#540	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#563	C₂	semidirect
2. C₅×#559	C₂	semidirect
3. C₅×#557	C₂	semidirect
4. C₅×#556	C₂	semidirect
5. C₅×#554	C₂	semidirect
6. C₅×#552	C₂	semidirect
7. C₅×#542	C₂	semidirect
8. C₅×#542	C₂	semidirect
9. C₅×#541	C₂	semidirect
11. C₅×#539	C₂	semidirect
12. C₅×#533	C₂	semidirect
13. C₅×#529	C₂	semidirect
67. #3557	C₅	direct (N × H)
121. #563	C₂×C₅	semidirect
122. #559	C₂×C₅	semidirect
123. #557	C₂×C₅	semidirect
124. #556	C₂×C₅	semidirect
125. #554	C₂×C₅	semidirect
126. #552	C₂×C₅	semidirect
127. #542	C₂×C₅	semidirect
128. #542	C₂×C₅	semidirect
129. #541	C₂×C₅	semidirect
131. #539	C₂×C₅	semidirect
132. #533	C₂×C₅	semidirect
133. #529	C₂×C₅	semidirect
311. C₅	#3557	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#563	2
C₅×#559	2
C₅×#557	2
C₅×#556	2
C₅×#554	2
C₅×#552	2
C₅×#542	2
C₅×#541	2
C₅×#540	2
C₅×#539	2
C₅×#533	2
C₅×#529	2
#3557	5

Subgroups — C5×#3557 #10619342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3557 #10619342