Subgroups — C₅×#17993 #11940713

Order 1280.

576 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3168	C₂	1	1	yes	maximal, characteristic
2	C₅×#2993	C₂	1	2	yes	maximal, complemented
3	C₅×#1587	C₂	1	2	yes	maximal, complemented
4	C₅×#1586	C₂	1	2	yes	maximal, complemented
5	C₅×#1583	C₂	1	2	yes	maximal, complemented
6	C₅×#1582	C₂	1	2	yes	maximal, complemented
7	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1495	C₂	1	2	yes	maximal, complemented
9	C₅×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#529	C₂²	1	2	yes	—
11	C₅×#499	C₄	1	1	yes	characteristic
12	C₅×#495	C₄	1	1	yes	characteristic
13	C₅×#488	C₄	1	2	yes	—
14	C₅×#467	C₄	1	1	yes	characteristic
15	C₅×#466	C₄	1	1	yes	characteristic
16	C₅×#465	C₄	1	2	yes	—
17	C₅×#446	C₂²	1	2	yes	—
18	C₅×C₈×Q₈	C₂²	1	2	yes	—
19	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₅×SD₁₆	C₂²	1	2	yes	—
21	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
22	C₅×#436	—	2	2	no	—
23	C₅×#436	C₂²	1	2	yes	—
24	C₅×#435	—	2	2	no	—
25	C₅×C₈×D₄	C₂²	1	2	yes	—
26	C₅×#433	—	2	2	no	—
27	C₅×#431	C₂²	1	1	yes	characteristic
28	C₅×#431	C₂²	1	1	yes	characteristic
29	C₅×#424	—	2	2	no	—
30	C₅×#423	C₂²	1	2	yes	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×#334	C₂²	1	1	yes	characteristic
33	C₅×#334	C₂²	1	1	yes	characteristic
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×D₄⋊C₈	C₂²	1	2	yes	—
38	C₅×D₄⋊C₈	C₂²	1	2	yes	—
39	C₅×#322	C₂²	1	2	yes	—
40	C₅×#322	C₂²	1	2	yes	—
41	#17993	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅×SD₁₆	C₂×C₄	1	2	yes	—
44	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₅×M₁₆	—	2	2	no	—
46	C₂×C₅×M₁₆	—	2	2	no	—
47	C₂²×C₅×C₈	—	2	2	no	—
48	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₅×#126	—	2	2	no	—
51	C₅×#125	C₂×C₄	1	2	yes	—
52	C₅×#124	C₂×C₄	1	2	yes	—
53	C₅×#123	—	2	2	no	—
54	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
55	C₅×#121	C₂×C₄	1	2	yes	—
56	C₄×C₅×Q₈	C₂³	1	2	yes	—
57	C₄×C₅×D₄	—	2	2	no	—
58	C₄×C₅×D₄	C₂³	1	2	yes	—
59	C₅×#117	—	2	2	no	—
60	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
61	C₅×#107	C₂×C₄	1	1	yes	characteristic
62	C₅×#107	C₂×C₄	1	1	yes	characteristic
63	C₅×#106	C₂×C₄	1	2	yes	—
64	C₅×C₄⋊C₈	—	2	2	no	—
65	C₅×C₄⋊C₈	—	2	2	no	—
66	C₅×C₄⋊C₈	—	2	2	no	—
67	C₅×C₄⋊C₈	—	2	1	no	—
68	C₅×C₄⋊C₈	D₄	1	2	yes	—
69	C₅×C₄⋊C₈	C₂³	1	2	yes	—
70	C₅×C₄⋊C₈	C₂³	1	2	yes	—
71	C₅×#103	C₂×C₄	1	2	yes	—
72	C₅×#103	C₂×C₄	1	2	yes	—
73	C₅×#102	C₂×C₄	1	2	yes	—
74	C₅×#102	C₂×C₄	1	2	yes	—
75	C₅×C₂²⋊C₈	—	2	1	no	—
76	C₅×C₂²⋊C₈	—	2	2	no	—
77	C₅×C₂²⋊C₈	—	2	2	no	—
78	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
79	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
80	C₅×#97	—	2	2	no	—
81	C₄×C₅×C₈	—	2	2	no	—
82	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
83	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
84	C₄×C₅×C₈	C₂³	1	2	yes	—
85	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
87	C₄×C₅×C₈	C₂³	1	2	yes	—
88	#3168	C₂×C₅	1	1	yes	characteristic
89	#2993	C₂×C₅	1	2	yes	complemented
90	#1587	C₂×C₅	1	2	yes	complemented
91	#1586	C₂×C₅	1	2	yes	complemented
92	#1583	C₂×C₅	1	2	yes	complemented
93	#1582	C₂×C₅	1	2	yes	complemented
94	#1566	C₂×C₅	1	1	yes	characteristic, complemented
95	#1495	C₂×C₅	1	2	yes	complemented
96	#1452	C₂×C₅	1	1	yes	characteristic, complemented
97	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
98	C₂×C₅×D₄	—	2	2	no	—
99	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₅	—	2	2	no	—
101	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅	#41	1	1	yes	characteristic
104	C₅×Q₁₆	—	4	1	no	—
105	C₅×SD₁₆	—	4	2	no	—
106	C₅×D₈	—	4	1	no	—
107	C₅×M₁₆	—	8	2	no	—
108	C₅×M₁₆	—	8	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	4	1	no	—
113	C₂×C₅×C₈	—	4	1	no	—
114	C₂×C₅×C₈	—	8	2	no	—
115	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
120	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
121	C₅×C₄⋊C₄	—	2	2	no	—
122	C₅×C₄⋊C₄	—	2	2	no	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	—	2	2	no	—
125	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
126	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
127	C₅×C₂²⋊C₄	—	2	2	no	—
128	C₅×C₂²⋊C₄	—	2	2	no	—
129	C₅×C₂²⋊C₄	—	2	2	no	—
130	C₅×C₂²⋊C₄	—	2	2	no	—
131	C₅×C₂²⋊C₄	—	2	2	no	—
132	C₄²×C₅	—	2	2	no	—
133	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
134	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₅	#41	1	1	yes	characteristic
136	C₄²×C₅	C₂⁴	1	1	yes	characteristic
137	#529	C₂²×C₅	1	2	yes	—
138	#499	C₄×C₅	1	1	yes	characteristic
139	#495	C₄×C₅	1	1	yes	characteristic
140	#488	C₄×C₅	1	2	yes	—
141	#467	C₄×C₅	1	1	yes	characteristic
142	#466	C₄×C₅	1	1	yes	characteristic
143	#465	C₄×C₅	1	2	yes	—
144	#446	C₂²×C₅	1	2	yes	—
145	C₈×Q₈	C₂²×C₅	1	2	yes	—
146	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
147	C₄×SD₁₆	C₂²×C₅	1	2	yes	—
148	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
149	#436	—	2	2	no	—
150	#436	C₂²×C₅	1	2	yes	—
151	#435	—	2	2	no	—
152	C₈×D₄	C₂²×C₅	1	2	yes	—
153	#433	—	2	2	no	—
154	#431	C₂²×C₅	1	1	yes	characteristic
155	#431	C₂²×C₅	1	1	yes	characteristic
156	#424	—	2	2	no	—
157	#423	C₂²×C₅	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
159	#334	C₂²×C₅	1	1	yes	characteristic
160	#334	C₂²×C₅	1	1	yes	characteristic
161	#334	C₂²×C₅	1	2	yes	—
162	Q₈⋊C₈	C₂²×C₅	1	2	yes	—
163	Q₈⋊C₈	C₂²×C₅	1	2	yes	—
164	D₄⋊C₈	C₂²×C₅	1	2	yes	—
165	D₄⋊C₈	C₂²×C₅	1	2	yes	—
166	#322	C₂²×C₅	1	2	yes	—
167	#322	C₂²×C₅	1	2	yes	—
168	C₂³×C₅	—	2	2	no	—
169	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
170	C₅×Q₈	—	2	2	no	—
171	C₅×Q₈	—	2	2	no	—
172	C₅×D₄	—	2	2	no	—
173	C₅×D₄	—	2	2	no	—
174	C₅×D₄	—	8	2	no	—
175	C₂×C₄×C₅	—	2	1	no	—
176	C₂×C₄×C₅	—	2	1	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	4	1	no	—
182	C₂×C₄×C₅	—	4	2	no	—
183	C₂×C₄×C₅	—	4	2	no	—
184	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
190	C₅×C₈	—	2	1	no	—
191	C₅×C₈	—	2	1	no	—
192	C₅×C₈	—	2	1	no	—
193	C₅×C₈	—	2	1	no	—
194	C₅×C₈	—	2	2	no	—
195	C₅×C₈	—	2	2	no	—
196	C₅×C₈	—	4	2	no	—
197	C₅×C₈	—	4	2	no	—
198	C₅×C₈	—	4	2	no	—
199	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
200	C₂×SD₁₆	C₂×C₄×C₅	1	2	yes	—
201	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
206	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
207	#126	—	2	2	no	—
208	#125	C₂×C₄×C₅	1	2	yes	—
209	#124	C₂×C₄×C₅	1	2	yes	—
210	#123	—	2	2	no	—
211	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
212	#121	C₂×C₄×C₅	1	2	yes	—
213	C₄×Q₈	C₂³×C₅	1	2	yes	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	C₂³×C₅	1	2	yes	—
216	#117	—	2	2	no	—
217	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
218	#107	C₂×C₄×C₅	1	1	yes	characteristic
219	#107	C₂×C₄×C₅	1	1	yes	characteristic
220	#106	C₂×C₄×C₅	1	2	yes	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	C₅×D₄	1	2	yes	—
226	C₄⋊C₈	C₂³×C₅	1	2	yes	—
227	C₄⋊C₈	C₂³×C₅	1	2	yes	—
228	#103	C₂×C₄×C₅	1	2	yes	—
229	#103	C₂×C₄×C₅	1	2	yes	—
230	#102	C₂×C₄×C₅	1	2	yes	—
231	#102	C₂×C₄×C₅	1	2	yes	—
232	C₂²⋊C₈	—	2	1	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
236	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
237	#97	—	2	2	no	—
238	C₄×C₈	—	2	2	no	—
239	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
240	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₅	1	2	yes	—
242	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
244	C₄×C₈	C₂³×C₅	1	2	yes	—
245	C₂²×C₅	—	2	1	no	—
246	C₂²×C₅	—	2	1	no	—
247	C₂²×C₅	—	2	1	no	—
248	C₂²×C₅	—	4	2	no	—
249	C₂²×C₅	—	4	2	no	—
250	C₂²×C₅	—	4	2	no	—
251	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	1	no	—
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	2	1	no	—
256	C₄×C₅	—	4	2	no	—
257	C₄×C₅	—	4	2	no	—
258	C₄×C₅	—	4	2	no	—
259	C₄×C₅	#568	1	1	yes	characteristic
260	C₄×C₅	#568	1	1	yes	characteristic
261	C₄×C₅	#577	1	2	yes	—
262	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
268	C₂²×C₄	C₅×#41	1	1	yes	characteristic
269	Q₁₆	—	4	1	no	—
270	SD₁₆	—	4	2	no	—
271	D₈	—	4	1	no	—
272	M₁₆	—	8	2	no	—
273	M₁₆	—	8	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	8	2	no	—
280	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
281	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
282	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
283	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
284	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
285	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
291	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
292	C₂²⋊C₄	—	2	2	no	—
293	C₂²⋊C₄	—	2	2	no	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
299	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
300	C₄²	C₅×#41	1	1	yes	characteristic
301	C₄²	C₂⁴×C₅	1	1	yes	characteristic
302	C₂×C₅	—	4	1	no	—
303	C₂×C₅	—	8	2	no	—
304	C₂×C₅	#2932	1	1	yes	characteristic
305	C₂×C₅	#2952	1	1	yes	characteristic
306	C₂×C₅	#2957	1	1	yes	characteristic
307	C₂³	—	2	2	no	—
308	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
309	Q₈	—	2	2	no	—
310	Q₈	—	2	2	no	—
311	D₄	—	2	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	8	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
324	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
325	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
326	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
327	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
328	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	2	no	—
334	C₈	—	2	2	no	—
335	C₈	—	4	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₅	#17993	1	1	yes	characteristic, direct product factor, Sylow
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	C₅×#568	1	1	yes	characteristic
354	C₄	C₅×#568	1	1	yes	characteristic
355	C₄	C₅×#577	1	2	yes	—
356	C₂	—	4	1	no	—
357	C₂	—	8	2	no	—
358	C₂	C₅×#2932	1	1	yes	characteristic
359	C₂	C₅×#2952	1	1	yes	characteristic
360	C₂	C₅×#2957	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	C₅×#466	C₄
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
2. C₅×#2993	C₂	semidirect
3. C₅×#1587	C₂	semidirect
4. C₅×#1586	C₂	semidirect
5. C₅×#1583	C₂	semidirect
6. C₅×#1582	C₂	semidirect
7. C₅×#1566	C₂	semidirect
8. C₅×#1495	C₂	semidirect
9. C₅×#1452	C₂	semidirect
41. #17993	C₅	direct (N × H)
89. #2993	C₂×C₅	semidirect
90. #1587	C₂×C₅	semidirect
91. #1586	C₂×C₅	semidirect
92. #1583	C₂×C₅	semidirect
93. #1582	C₂×C₅	semidirect
94. #1566	C₂×C₅	semidirect
95. #1495	C₂×C₅	semidirect
96. #1452	C₂×C₅	semidirect
338. C₅	#17993	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×#3168	2
C₅×#2993	2
C₅×#1587	2
C₅×#1586	2
C₅×#1583	2
C₅×#1582	2
C₅×#1566	2
C₅×#1495	2
C₅×#1452	2
#17993	5

Subgroups — C5×#17993 #11940713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17993 #11940713