Subgroups — C₅×#19426 #11942146

Order 1280.

650 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3229	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3077	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#461	C₂	1	1	yes	maximal, characteristic
4	C₅×#2975	C₂	1	1	yes	maximal, characteristic
5	C₅×#1699	C₂	1	4	yes	maximal, complemented
6	C₅×#1691	C₂	1	4	yes	maximal, complemented
7	C₅×#1571	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1546	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₅×SD₁₆	—	2	1	no	—
11	C₅×#533	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₅×#498	C₂²	1	2	yes	complemented
15	C₅×#487	C₂²	1	2	yes	complemented
16	C₅×#483	—	2	2	no	—
17	C₅×#478	—	2	2	no	—
18	C₅×#476	C₂²	1	2	yes	—
19	C₅×#475	—	2	2	no	—
20	C₅×#474	C₂²	1	2	yes	—
21	C₅×#461	—	2	2	no	—
22	C₅×#461	C₂²	1	4	yes	—
23	C₅×#436	—	2	2	no	—
24	C₅×#436	C₂²	1	4	yes	—
25	C₅×#431	C₂²	1	1	yes	characteristic
26	C₅×#431	C₂²	1	1	yes	characteristic
27	C₅×#428	—	2	1	no	—
28	C₅×#423	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₅×#418	—	2	1	no	—
31	C₂×C₅×#103	—	2	1	no	—
32	C₅×#407	—	2	2	no	—
33	C₅×#335	C₂²	1	4	yes	complemented
34	C₅×#333	C₂²	1	2	yes	complemented
35	C₅×#332	C₂²	1	2	yes	complemented
36	C₅×#332	C₂²	1	4	yes	complemented
37	#19426	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₅×Q₈	—	2	1	no	—
39	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₅	—	2	1	no	—
41	C₂×C₅×SD₁₆	—	2	4	no	—
42	C₂×C₅×SD₁₆	—	4	4	no	—
43	C₂×C₅×M₁₆	—	2	1	no	—
44	C₂×C₅×M₁₆	—	2	4	no	—
45	C₂²×C₅×C₈	—	2	1	no	—
46	C₅×C₄⋊Q₈	—	2	1	no	—
47	C₅×C₄⋊Q₈	—	2	1	no	—
48	C₅×C₄⋊Q₈	—	2	1	no	—
49	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
51	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
52	C₅×#125	—	2	1	no	—
53	C₅×C₂²⋊Q₈	—	2	1	no	—
54	C₅×C₂²⋊Q₈	—	2	1	no	—
55	C₄×C₅×Q₈	—	2	1	no	—
56	C₄×C₅×Q₈	—	2	1	no	—
57	C₄×C₅×D₄	—	2	1	no	—
58	C₄×C₅×D₄	—	2	1	no	—
59	C₄×C₅×D₄	D₄	1	2	yes	—
60	C₄×C₅×D₄	C₂³	1	2	yes	—
61	C₅×#117	—	2	1	no	—
62	C₂×C₅×C₄⋊C₄	—	2	1	no	—
63	C₂×C₅×C₄⋊C₄	—	2	1	no	—
64	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
65	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
66	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
67	C₅×#107	—	2	2	no	—
68	C₅×#106	—	2	4	no	—
69	C₅×#106	—	2	2	no	—
70	C₅×C₄⋊C₈	—	2	1	no	—
71	C₅×C₄⋊C₈	—	2	1	no	—
72	C₅×C₄⋊C₈	D₄	1	2	yes	—
73	C₅×C₄⋊C₈	C₂³	1	2	yes	—
74	C₅×C₄⋊C₈	C₂³	1	2	yes	—
75	C₅×C₄⋊C₈	C₂³	1	2	yes	—
76	C₅×#103	—	2	4	no	—
77	C₅×#103	—	2	4	no	—
78	C₅×#102	—	2	2	no	—
79	C₅×#102	—	2	2	no	—
80	C₅×C₂²⋊C₈	—	2	1	no	—
81	C₅×C₂²⋊C₈	—	2	4	no	—
82	C₅×C₂²⋊C₈	D₄	1	2	yes	—
83	C₄×C₅×C₈	—	2	1	no	—
84	C₄×C₅×C₈	C₂³	1	2	yes	—
85	#3229	C₂×C₅	1	1	yes	characteristic, complemented
86	#3077	C₂×C₅	1	1	yes	characteristic
87	C₂×#461	C₂×C₅	1	1	yes	characteristic
88	#2975	C₂×C₅	1	1	yes	characteristic
89	#1699	C₂×C₅	1	4	yes	complemented
90	#1691	C₂×C₅	1	4	yes	complemented
91	#1571	C₂×C₅	1	1	yes	characteristic, complemented
92	#1546	C₂×C₅	1	1	yes	characteristic, complemented
93	#1542	C₂×C₅	1	1	yes	characteristic, complemented
94	C₂⁴×C₅	—	2	1	no	—
95	C₂×C₅×Q₈	—	2	1	no	—
96	C₂×C₅×Q₈	—	2	2	no	—
97	C₂×C₅×Q₈	—	2	1	no	—
98	C₂×C₅×Q₈	—	4	1	no	—
99	C₂×C₅×Q₈	—	4	2	no	—
100	C₂×C₅×Q₈	—	4	2	no	—
101	C₂×C₅×D₄	—	2	1	no	—
102	C₂×C₅×D₄	—	2	2	no	—
103	C₂×C₅×D₄	—	2	2	no	—
104	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₅	—	2	1	no	—
106	C₂²×C₄×C₅	—	2	2	no	—
107	C₂²×C₄×C₅	—	2	2	no	—
108	C₂²×C₄×C₅	—	4	1	no	—
109	C₂²×C₄×C₅	—	8	2	no	—
110	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅	#41	1	1	yes	characteristic
113	C₅×SD₁₆	—	4	8	no	—
114	C₅×M₁₆	—	4	2	no	—
115	C₅×M₁₆	—	8	4	no	—
116	C₂×C₅×C₈	—	2	2	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	—	2	2	no	—
119	C₂×C₅×C₈	—	4	2	no	—
120	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	—	2	2	no	—
123	C₅×C₄⋊C₄	—	2	1	no	—
124	C₅×C₄⋊C₄	—	2	1	no	—
125	C₅×C₄⋊C₄	—	2	1	no	—
126	C₅×C₄⋊C₄	—	2	2	no	—
127	C₅×C₄⋊C₄	—	2	1	no	—
128	C₅×C₄⋊C₄	—	4	1	no	—
129	C₅×C₄⋊C₄	—	4	1	no	—
130	C₅×C₄⋊C₄	—	4	1	no	—
131	C₅×C₄⋊C₄	—	4	1	no	—
132	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
133	C₅×C₂²⋊C₄	—	2	2	no	—
134	C₅×C₂²⋊C₄	—	2	2	no	—
135	C₅×C₂²⋊C₄	—	4	1	no	—
136	C₄²×C₅	—	4	1	no	—
137	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₅	#41	1	1	yes	characteristic
140	C₄²×C₅	C₂⁴	1	1	yes	characteristic
141	C₂²×SD₁₆	—	2	1	no	—
142	#533	C₂²×C₅	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
144	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
145	#498	C₂²×C₅	1	2	yes	complemented
146	#487	C₂²×C₅	1	2	yes	complemented
147	#483	—	2	2	no	—
148	#478	—	2	2	no	—
149	#476	C₂²×C₅	1	2	yes	—
150	#475	—	2	2	no	—
151	#474	C₂²×C₅	1	2	yes	—
152	#461	—	2	2	no	—
153	#461	C₂²×C₅	1	4	yes	—
154	#436	—	2	2	no	—
155	#436	C₂²×C₅	1	4	yes	—
156	#431	C₂²×C₅	1	1	yes	characteristic
157	#431	C₂²×C₅	1	1	yes	characteristic
158	#428	—	2	1	no	—
159	#423	C₂²×C₅	1	1	yes	characteristic
160	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
161	#418	—	2	1	no	—
162	C₂×#103	—	2	1	no	—
163	#407	—	2	2	no	—
164	#335	C₂²×C₅	1	4	yes	complemented
165	#333	C₂²×C₅	1	2	yes	complemented
166	#332	C₂²×C₅	1	2	yes	complemented
167	#332	C₂²×C₅	1	4	yes	complemented
168	C₂³×C₅	—	2	2	no	—
169	C₂³×C₅	—	4	2	no	—
170	C₂³×C₅	—	8	1	no	—
171	C₂³×C₅	—	8	1	no	—
172	C₂³×C₅	#121	1	1	yes	characteristic
173	C₅×Q₈	—	4	2	no	—
174	C₅×Q₈	—	4	2	no	—
175	C₅×Q₈	—	4	2	no	—
176	C₅×Q₈	—	8	1	no	—
177	C₅×Q₈	—	8	2	no	—
178	C₅×D₄	—	2	2	no	—
179	C₅×D₄	—	2	2	no	—
180	C₅×D₄	—	4	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	4	1	no	—
184	C₂×C₄×C₅	—	4	4	no	—
185	C₂×C₄×C₅	—	4	1	no	—
186	C₂×C₄×C₅	—	4	1	no	—
187	C₂×C₄×C₅	—	4	2	no	—
188	C₂×C₄×C₅	—	8	2	no	—
189	C₂×C₄×C₅	—	8	1	no	—
190	C₂×C₄×C₅	—	8	2	no	—
191	C₂×C₄×C₅	#121	1	1	yes	characteristic
192	C₂×C₄×C₅	#121	1	1	yes	characteristic
193	C₂×C₄×C₅	#121	1	1	yes	characteristic
194	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
195	C₂×C₄×C₅	#137	1	2	yes	—
196	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
199	C₅×C₈	—	2	4	no	—
200	C₅×C₈	—	4	2	no	—
201	C₅×C₈	—	4	2	no	—
202	C₅×C₈	—	4	2	no	—
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
205	C₂³×C₄	—	2	1	no	—
206	C₂×SD₁₆	—	2	4	no	—
207	C₂×SD₁₆	—	4	4	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	2	4	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
215	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
216	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
217	#125	—	2	1	no	—
218	C₂²⋊Q₈	—	2	1	no	—
219	C₂²⋊Q₈	—	2	1	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×D₄	—	2	1	no	—
223	C₄×D₄	—	2	1	no	—
224	C₄×D₄	C₅×D₄	1	2	yes	—
225	C₄×D₄	C₂³×C₅	1	2	yes	—
226	#117	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
230	C₂×C₂²⋊C₄	—	2	1	no	—
231	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
232	#107	—	2	2	no	—
233	#106	—	2	4	no	—
234	#106	—	2	2	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	C₅×D₄	1	2	yes	—
238	C₄⋊C₈	C₂³×C₅	1	2	yes	—
239	C₄⋊C₈	C₂³×C₅	1	2	yes	—
240	C₄⋊C₈	C₂³×C₅	1	2	yes	—
241	#103	—	2	4	no	—
242	#103	—	2	4	no	—
243	#102	—	2	2	no	—
244	#102	—	2	2	no	—
245	C₂²⋊C₈	—	2	1	no	—
246	C₂²⋊C₈	—	2	4	no	—
247	C₂²⋊C₈	C₅×D₄	1	2	yes	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	C₂³×C₅	1	2	yes	—
250	C₂²×C₅	—	2	1	no	—
251	C₂²×C₅	—	2	1	no	—
252	C₂²×C₅	—	4	2	no	—
253	C₂²×C₅	—	4	2	no	—
254	C₂²×C₅	—	4	2	no	—
255	C₂²×C₅	—	8	4	no	—
256	C₂²×C₅	#468	1	2	yes	—
257	C₂²×C₅	C₂×#121	1	1	yes	characteristic
258	C₄×C₅	—	2	1	no	—
259	C₄×C₅	—	2	1	no	—
260	C₄×C₅	—	2	1	no	—
261	C₄×C₅	—	4	2	no	—
262	C₄×C₅	—	8	1	no	—
263	C₄×C₅	—	8	2	no	—
264	C₄×C₅	—	8	1	no	—
265	C₄×C₅	#468	1	2	yes	—
266	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
267	C₄×C₅	C₂×#137	1	1	yes	characteristic
268	C₄×C₅	#577	1	2	yes	—
269	C₂⁴	—	2	1	no	—
270	C₂×Q₈	—	2	1	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×Q₈	—	4	2	no	—
275	C₂×Q₈	—	4	2	no	—
276	C₂×D₄	—	2	1	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	8	2	no	—
285	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
286	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
287	C₂²×C₄	C₅×#41	1	1	yes	characteristic
288	SD₁₆	—	4	8	no	—
289	M₁₆	—	4	2	no	—
290	M₁₆	—	8	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	2	no	—
295	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
308	C₂²⋊C₄	—	2	2	no	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	4	1	no	—
311	C₄²	—	4	1	no	—
312	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
313	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
314	C₄²	C₅×#41	1	1	yes	characteristic
315	C₄²	C₂⁴×C₅	1	1	yes	characteristic
316	C₂×C₅	—	2	2	no	—
317	C₂×C₅	—	8	2	no	—
318	C₂×C₅	#3042	1	1	yes	characteristic
319	C₂×C₅	C₂×#468	1	1	yes	characteristic
320	C₂×C₅	#3069	1	1	yes	characteristic
321	C₂³	—	2	2	no	—
322	C₂³	—	4	2	no	—
323	C₂³	—	8	1	no	—
324	C₂³	—	8	1	no	—
325	C₂³	C₅×#121	1	1	yes	characteristic
326	Q₈	—	4	2	no	—
327	Q₈	—	4	2	no	—
328	Q₈	—	4	2	no	—
329	Q₈	—	8	1	no	—
330	Q₈	—	8	2	no	—
331	D₄	—	2	2	no	—
332	D₄	—	2	2	no	—
333	D₄	—	4	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	8	2	no	—
342	C₂×C₄	—	8	1	no	—
343	C₂×C₄	—	8	2	no	—
344	C₂×C₄	C₅×#121	1	1	yes	characteristic
345	C₂×C₄	C₅×#121	1	1	yes	characteristic
346	C₂×C₄	C₅×#121	1	1	yes	characteristic
347	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
348	C₂×C₄	C₅×#137	1	2	yes	—
349	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
350	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
351	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
352	C₈	—	2	4	no	—
353	C₈	—	4	2	no	—
354	C₈	—	4	2	no	—
355	C₈	—	4	2	no	—
356	C₅	#19426	1	1	yes	characteristic, direct product factor, Sylow
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	8	4	no	—
363	C₂²	C₅×#468	1	2	yes	—
364	C₂²	C₂×C₅×#121	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	8	1	no	—
370	C₄	—	8	2	no	—
371	C₄	—	8	1	no	—
372	C₄	C₅×#468	1	2	yes	—
373	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
374	C₄	C₂×C₅×#137	1	1	yes	characteristic
375	C₄	C₅×#577	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	8	2	no	—
378	C₂	C₅×#3042	1	1	yes	characteristic
379	C₂	C₂×C₅×#468	1	1	yes	characteristic
380	C₂	C₅×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3229	C₂	semidirect
5. C₅×#1699	C₂	semidirect
6. C₅×#1691	C₂	semidirect
7. C₅×#1571	C₂	semidirect
8. C₅×#1546	C₂	semidirect
9. C₅×#1542	C₂	semidirect
14. C₅×#498	C₂²	semidirect
15. C₅×#487	C₂²	semidirect
33. C₅×#335	C₂²	semidirect
34. C₅×#333	C₂²	semidirect
35. C₅×#332	C₂²	semidirect
36. C₅×#332	C₂²	semidirect
37. #19426	C₅	direct (N × H)
85. #3229	C₂×C₅	semidirect
89. #1699	C₂×C₅	semidirect
90. #1691	C₂×C₅	semidirect
91. #1571	C₂×C₅	semidirect
92. #1546	C₂×C₅	semidirect
93. #1542	C₂×C₅	semidirect
145. #498	C₂²×C₅	semidirect
146. #487	C₂²×C₅	semidirect
164. #335	C₂²×C₅	semidirect
165. #333	C₂²×C₅	semidirect
166. #332	C₂²×C₅	semidirect
167. #332	C₂²×C₅	semidirect
356. C₅	#19426	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×#3229	2
C₅×#3077	2
C₂×C₅×#461	2
C₅×#2975	2
C₅×#1699	2
C₅×#1691	2
C₅×#1571	2
C₅×#1546	2
C₅×#1542	2
#19426	5

Subgroups — C5×#19426 #11942146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19426 #11942146