Subgroups — C₅×#17953 #11940673

Order 1280.

558 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3104	C₂	1	1	yes	maximal, characteristic
2	C₅×#2985	C₂	1	1	yes	maximal, characteristic
3	C₅×#2983	C₂	1	1	yes	maximal, characteristic
4	C₅×#1593	C₂	1	2	yes	maximal, complemented
5	C₅×#1591	C₂	1	2	yes	maximal, complemented
6	C₅×#1587	C₂	1	2	yes	maximal, complemented
7	C₅×#1584	C₂	1	2	yes	maximal, complemented
8	C₅×#1572	C₂	1	1	yes	maximal, characteristic
9	C₅×#1491	C₂	1	1	yes	maximal, characteristic
10	C₅×#1485	C₂	1	1	yes	maximal, characteristic
11	C₅×#1453	C₂	1	1	yes	maximal, characteristic
12	C₅×#533	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₅×#484	C₄	1	2	yes	—
15	C₅×#477	C₄	1	2	yes	—
16	C₅×#462	C₄	1	2	yes	—
17	C₅×#451	C₄	1	2	yes	—
18	C₅×#446	—	2	2	no	—
19	C₅×#446	—	2	2	no	—
20	C₅×#446	C₂²	1	2	yes	—
21	C₅×#446	C₂²	1	2	yes	—
22	C₅×#446	C₂²	1	2	yes	—
23	C₅×C₈×Q₈	C₂²	1	2	yes	—
24	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
25	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
26	C₅×#433	—	2	2	no	—
27	C₅×#432	—	2	2	no	—
28	C₅×#432	C₂²	1	1	yes	characteristic
29	C₅×#431	C₂²	1	1	yes	characteristic
30	C₅×#431	C₂²	1	1	yes	characteristic
31	C₅×#431	C₂²	1	1	yes	characteristic
32	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
33	C₅×#335	C₂²	1	2	yes	—
34	C₅×#334	C₂²	1	2	yes	—
35	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
39	C₅×#322	C₂²	1	1	yes	characteristic
40	C₅×#322	C₂²	1	1	yes	characteristic
41	C₅×#322	C₂²	1	1	yes	characteristic
42	C₅×C₈²	C₂²	1	1	yes	characteristic
43	#17953	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₅×M₁₆	—	2	1	no	—
48	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
49	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
50	C₅×#125	—	2	1	no	—
51	C₅×C₂²⋊Q₈	—	2	1	no	—
52	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
53	C₄×C₅×Q₈	—	2	1	no	—
54	C₄×C₅×Q₈	—	2	1	no	—
55	C₄×C₅×Q₈	—	2	1	no	—
56	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
57	C₄×C₅×Q₈	C₂³	1	2	yes	—
58	C₄×C₅×Q₈	C₂³	1	2	yes	—
59	C₅×#117	—	2	1	no	—
60	C₂×C₅×C₄⋊C₄	—	2	1	no	—
61	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅	—	2	1	no	—
63	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
64	C₅×#107	C₂×C₄	1	2	yes	—
65	C₅×#107	C₂×C₄	1	2	yes	—
66	C₅×C₄⋊C₈	—	2	2	no	—
67	C₅×C₄⋊C₈	—	2	2	no	—
68	C₅×C₄⋊C₈	—	2	2	no	—
69	C₅×C₄⋊C₈	—	2	2	no	—
70	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
71	C₅×C₄⋊C₈	D₄	1	2	yes	—
72	C₅×C₄⋊C₈	C₂³	1	2	yes	—
73	C₅×C₄⋊C₈	C₂³	1	2	yes	—
74	C₅×#103	C₂×C₄	1	2	yes	—
75	C₅×#103	C₂×C₄	1	2	yes	—
76	C₅×#103	C₂×C₄	1	2	yes	—
77	C₅×#103	C₂×C₄	1	2	yes	—
78	C₅×C₂²⋊C₈	—	2	1	no	—
79	C₅×C₂²⋊C₈	—	2	1	no	—
80	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
81	C₅×C₂²⋊C₈	D₄	1	2	yes	—
82	C₅×#97	—	2	2	no	—
83	C₅×#97	—	2	2	no	—
84	C₅×#97	—	2	2	no	—
85	C₅×#97	—	2	1	no	—
86	C₅×#97	—	2	1	no	—
87	C₄×C₅×C₈	—	2	2	no	—
88	C₄×C₅×C₈	—	2	1	no	—
89	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
90	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
91	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
92	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
94	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
95	#3104	C₂×C₅	1	1	yes	characteristic
96	#2985	C₂×C₅	1	1	yes	characteristic
97	#2983	C₂×C₅	1	1	yes	characteristic
98	#1593	C₂×C₅	1	2	yes	complemented
99	#1591	C₂×C₅	1	2	yes	complemented
100	#1587	C₂×C₅	1	2	yes	complemented
101	#1584	C₂×C₅	1	2	yes	complemented
102	#1572	C₂×C₅	1	1	yes	characteristic
103	#1491	C₂×C₅	1	1	yes	characteristic
104	#1485	C₂×C₅	1	1	yes	characteristic
105	#1453	C₂×C₅	1	1	yes	characteristic
106	C₂×C₅×Q₈	—	2	1	no	—
107	C₂×C₅×Q₈	—	2	1	no	—
108	C₂×C₅×Q₈	—	4	2	no	—
109	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
110	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₅	—	2	1	no	—
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅	#41	1	1	yes	characteristic
116	C₅×Q₁₆	—	4	2	no	—
117	C₅×Q₁₆	—	4	2	no	—
118	C₅×M₁₆	—	4	2	no	—
119	C₂×C₅×C₈	—	2	1	no	—
120	C₂×C₅×C₈	—	2	2	no	—
121	C₂×C₅×C₈	—	2	2	no	—
122	C₂×C₅×C₈	—	2	1	no	—
123	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
124	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
125	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
126	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
127	C₅×C₄⋊C₄	—	2	2	no	—
128	C₅×C₄⋊C₄	—	2	1	no	—
129	C₅×C₄⋊C₄	—	2	1	no	—
130	C₅×C₄⋊C₄	—	2	1	no	—
131	C₅×C₄⋊C₄	—	2	1	no	—
132	C₅×C₄⋊C₄	—	2	2	no	—
133	C₅×C₄⋊C₄	—	2	2	no	—
134	C₅×C₄⋊C₄	—	2	2	no	—
135	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
136	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
137	C₅×C₂²⋊C₄	—	2	2	no	—
138	C₄²×C₅	—	2	2	no	—
139	C₄²×C₅	—	2	2	no	—
140	C₄²×C₅	—	2	2	no	—
141	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
142	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
143	C₄²×C₅	#41	1	1	yes	characteristic
144	C₄²×C₅	C₂⁴	1	1	yes	characteristic
145	#533	C₂²×C₅	1	1	yes	characteristic
146	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
147	#484	C₄×C₅	1	2	yes	—
148	#477	C₄×C₅	1	2	yes	—
149	#462	C₄×C₅	1	2	yes	—
150	#451	C₄×C₅	1	2	yes	—
151	#446	—	2	2	no	—
152	#446	—	2	2	no	—
153	#446	C₂²×C₅	1	2	yes	—
154	#446	C₂²×C₅	1	2	yes	—
155	#446	C₂²×C₅	1	2	yes	—
156	C₈×Q₈	C₂²×C₅	1	2	yes	—
157	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
158	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
159	#433	—	2	2	no	—
160	#432	—	2	2	no	—
161	#432	C₂²×C₅	1	1	yes	characteristic
162	#431	C₂²×C₅	1	1	yes	characteristic
163	#431	C₂²×C₅	1	1	yes	characteristic
164	#431	C₂²×C₅	1	1	yes	characteristic
165	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
166	#335	C₂²×C₅	1	2	yes	—
167	#334	C₂²×C₅	1	2	yes	—
168	Q₈⋊C₈	C₂²×C₅	1	2	yes	—
169	Q₈⋊C₈	C₂²×C₅	1	2	yes	—
170	Q₈⋊C₈	C₂²×C₅	1	2	yes	—
171	Q₈⋊C₈	C₂²×C₅	1	2	yes	—
172	#322	C₂²×C₅	1	1	yes	characteristic
173	#322	C₂²×C₅	1	1	yes	characteristic
174	#322	C₂²×C₅	1	1	yes	characteristic
175	C₈²	C₂²×C₅	1	1	yes	characteristic
176	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
177	C₅×Q₈	—	2	2	no	—
178	C₅×Q₈	—	2	2	no	—
179	C₅×Q₈	—	2	2	no	—
180	C₅×Q₈	—	2	2	no	—
181	C₅×Q₈	—	4	2	no	—
182	C₅×Q₈	—	8	1	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	—	2	2	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	4	1	no	—
190	C₂×C₄×C₅	—	4	2	no	—
191	C₂×C₄×C₅	—	8	1	no	—
192	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
198	C₅×C₈	—	2	2	no	—
199	C₅×C₈	—	2	2	no	—
200	C₅×C₈	—	2	2	no	—
201	C₅×C₈	—	2	2	no	—
202	C₅×C₈	—	2	2	no	—
203	C₅×C₈	—	4	1	no	—
204	C₅×C₈	—	4	2	no	—
205	C₅×C₈	—	4	2	no	—
206	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
207	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
208	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
209	C₂×M₁₆	—	2	1	no	—
210	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
211	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
212	#125	—	2	1	no	—
213	C₂²⋊Q₈	—	2	1	no	—
214	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	—	2	1	no	—
218	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
219	C₄×Q₈	C₂³×C₅	1	2	yes	—
220	C₄×Q₈	C₂³×C₅	1	2	yes	—
221	#117	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
224	C₂×C₄²	—	2	1	no	—
225	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
226	#107	C₂×C₄×C₅	1	2	yes	—
227	#107	C₂×C₄×C₅	1	2	yes	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
233	C₄⋊C₈	C₅×D₄	1	2	yes	—
234	C₄⋊C₈	C₂³×C₅	1	2	yes	—
235	C₄⋊C₈	C₂³×C₅	1	2	yes	—
236	#103	C₂×C₄×C₅	1	2	yes	—
237	#103	C₂×C₄×C₅	1	2	yes	—
238	#103	C₂×C₄×C₅	1	2	yes	—
239	#103	C₂×C₄×C₅	1	2	yes	—
240	C₂²⋊C₈	—	2	1	no	—
241	C₂²⋊C₈	—	2	1	no	—
242	C₂²⋊C₈	C₂×C₄×C₅	1	2	yes	—
243	C₂²⋊C₈	C₅×D₄	1	2	yes	—
244	#97	—	2	2	no	—
245	#97	—	2	2	no	—
246	#97	—	2	2	no	—
247	#97	—	2	1	no	—
248	#97	—	2	1	no	—
249	C₄×C₈	—	2	2	no	—
250	C₄×C₈	—	2	1	no	—
251	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
252	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
254	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
257	C₂²×C₅	—	2	1	no	—
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	—	2	1	no	—
260	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
261	C₄×C₅	—	2	1	no	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	2	1	no	—
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	—	4	2	no	—
266	C₄×C₅	—	4	2	no	—
267	C₄×C₅	—	4	2	no	—
268	C₄×C₅	—	4	2	no	—
269	C₄×C₅	#568	1	2	yes	—
270	C₄×C₅	#578	1	2	yes	—
271	C₂×Q₈	—	2	1	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	4	2	no	—
274	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
275	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	2	1	no	—
278	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
280	C₂²×C₄	C₅×#41	1	1	yes	characteristic
281	Q₁₆	—	4	2	no	—
282	Q₁₆	—	4	2	no	—
283	M₁₆	—	4	2	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
289	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
290	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
291	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
301	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
307	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
308	C₄²	C₅×#41	1	1	yes	characteristic
309	C₄²	C₂⁴×C₅	1	1	yes	characteristic
310	C₂×C₅	—	4	1	no	—
311	C₂×C₅	#2930	1	1	yes	characteristic
312	C₂×C₅	#2950	1	1	yes	characteristic
313	C₂×C₅	#2956	1	1	yes	characteristic
314	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
315	Q₈	—	2	2	no	—
316	Q₈	—	2	2	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	4	2	no	—
320	Q₈	—	8	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
331	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
332	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
333	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
335	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
336	C₈	—	2	2	no	—
337	C₈	—	2	2	no	—
338	C₈	—	2	2	no	—
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	4	1	no	—
342	C₈	—	4	2	no	—
343	C₈	—	4	2	no	—
344	C₅	#17953	1	1	yes	characteristic, direct product factor, Sylow
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	C₅×#568	1	2	yes	—
358	C₄	C₅×#578	1	2	yes	—
359	C₂	—	4	1	no	—
360	C₂	C₅×#2930	1	1	yes	characteristic
361	C₂	C₅×#2950	1	1	yes	characteristic
362	C₂	C₅×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#431 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1593	C₂	semidirect
5. C₅×#1591	C₂	semidirect
6. C₅×#1587	C₂	semidirect
7. C₅×#1584	C₂	semidirect
43. #17953	C₅	direct (N × H)
98. #1593	C₂×C₅	semidirect
99. #1591	C₂×C₅	semidirect
100. #1587	C₂×C₅	semidirect
101. #1584	C₂×C₅	semidirect
344. C₅	#17953	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×#3104	2
C₅×#2985	2
C₅×#2983	2
C₅×#1593	2
C₅×#1591	2
C₅×#1587	2
C₅×#1584	2
C₅×#1572	2
C₅×#1491	2
C₅×#1485	2
C₅×#1453	2
#17953	5

Subgroups — C5×#17953 #11940673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17953 #11940673