Subgroups — C₅×#19261 #11941981

Order 1280.

710 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅×#3037	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1682	C₂	1	4	yes	maximal, complemented
5	C₅×#1674	C₂	1	2	yes	maximal, complemented
6	C₅×#1668	C₂	1	2	yes	maximal, complemented
7	C₅×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1547	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#137	—	2	2	no	—
10	C₅×#532	C₂²	1	2	yes	—
11	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₅×#500	C₂²	1	1	yes	characteristic
13	C₅×#493	C₂²	1	1	yes	characteristic
14	C₅×#486	C₂²	1	2	yes	—
15	C₅×#466	—	2	1	no	—
16	C₅×#466	—	2	1	no	—
17	C₅×#464	—	2	4	no	—
18	C₅×#462	C₂²	1	4	yes	—
19	C₅×#460	C₂²	1	4	yes	—
20	C₅×C₈×Q₈	—	2	1	no	—
21	C₅×C₈×Q₈	—	2	1	no	—
22	C₅×C₈×Q₈	C₂²	1	2	yes	—
23	C₅×C₈×Q₈	C₂²	1	2	yes	—
24	C₄×C₅×D₈	—	2	1	no	—
25	C₄×C₅×D₈	—	2	1	no	—
26	C₅×#423	C₂²	1	2	yes	—
27	C₂×C₅×C₄⋊C₈	—	2	1	no	—
28	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₅×#417	—	2	2	no	—
30	C₂×C₄×C₅×C₈	—	2	1	no	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×#335	C₂²	1	2	yes	—
33	C₅×#335	C₂²	1	2	yes	—
34	C₅×#332	C₂²	1	4	yes	—
35	C₅×#331	C₂²	1	4	yes	—
36	#19261	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	2	no	—
38	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
39	C₅×#137	—	4	8	no	—
40	C₂×C₅×Q₁₆	—	2	4	no	—
41	C₂×C₅×SD₁₆	—	2	4	no	—
42	C₂×C₅×D₈	—	2	1	no	—
43	C₂×C₅×D₈	—	2	1	no	—
44	C₂×C₅×M₁₆	—	2	2	no	—
45	C₂²×C₅×C₈	—	2	1	no	—
46	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
48	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₅×#127	C₂³	1	2	yes	—
50	C₅×#124	—	2	2	no	—
51	C₅×#121	—	2	2	no	—
52	C₅×#121	—	2	2	no	—
53	C₄×C₅×Q₈	—	2	1	no	—
54	C₄×C₅×Q₈	—	2	1	no	—
55	C₄×C₅×Q₈	D₄	1	2	yes	—
56	C₄×C₅×Q₈	C₂³	1	2	yes	—
57	C₄×C₅×D₄	—	2	2	no	—
58	C₄×C₅×D₄	—	2	2	no	—
59	C₂×C₅×C₄⋊C₄	—	2	1	no	—
60	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₅	—	2	1	no	—
62	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
63	C₅×#107	—	2	1	no	—
64	C₅×#107	—	2	1	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	—	2	2	no	—
69	C₅×C₄⋊C₈	C₂³	1	2	yes	—
70	C₅×C₄⋊C₈	C₂³	1	4	yes	—
71	C₅×#103	—	2	4	no	—
72	C₅×#102	—	2	2	no	—
73	C₅×#102	—	2	2	no	—
74	C₅×#102	—	2	4	no	—
75	C₄×C₅×C₈	—	2	2	no	—
76	C₄×C₅×C₈	—	2	2	no	—
77	C₄×C₅×C₈	D₄	1	1	yes	characteristic
78	C₄×C₅×C₈	D₄	1	1	yes	characteristic
79	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
81	#3169	C₂×C₅	1	1	yes	characteristic
82	#3037	C₂×C₅	1	2	yes	complemented
83	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
84	#1682	C₂×C₅	1	4	yes	complemented
85	#1674	C₂×C₅	1	2	yes	complemented
86	#1668	C₂×C₅	1	2	yes	complemented
87	#1567	C₂×C₅	1	1	yes	characteristic, complemented
88	#1547	C₂×C₅	1	2	yes	complemented
89	C₅×#41	—	4	4	no	—
90	C₅×#41	—	4	4	no	—
91	C₂×C₅×Q₈	—	2	1	no	—
92	C₂×C₅×Q₈	—	2	2	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×Q₈	D₈	1	4	yes	complemented
95	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
96	C₂×C₅×D₄	—	2	2	no	—
97	C₂×C₅×D₄	—	2	2	no	—
98	C₂×C₅×D₄	—	2	2	no	—
99	C₂×C₅×D₄	—	4	2	no	—
100	C₂²×C₄×C₅	—	2	1	no	—
101	C₂²×C₄×C₅	—	2	1	no	—
102	C₂²×C₄×C₅	—	4	2	no	—
103	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅	#41	1	1	yes	characteristic
106	C₅×Q₁₆	—	4	8	no	—
107	C₅×SD₁₆	—	4	8	no	—
108	C₅×D₈	—	4	2	no	—
109	C₅×D₈	—	4	2	no	—
110	C₅×M₁₆	—	4	4	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	2	4	no	—
114	C₂×C₅×C₈	—	4	1	no	—
115	C₂×C₅×C₈	—	4	2	no	—
116	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	—	2	1	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	C₂×D₄	1	2	yes	—
126	C₅×C₂²⋊C₄	—	4	2	no	—
127	C₅×C₂²⋊C₄	—	4	2	no	—
128	C₄²×C₅	—	2	2	no	—
129	C₄²×C₅	—	2	2	no	—
130	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅	#41	1	1	yes	characteristic
133	C₄²×C₅	C₂⁴	1	1	yes	characteristic
134	C₂×#137	—	2	2	no	—
135	#532	C₂²×C₅	1	2	yes	—
136	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
137	#500	C₂²×C₅	1	1	yes	characteristic
138	#493	C₂²×C₅	1	1	yes	characteristic
139	#486	C₂²×C₅	1	2	yes	—
140	#466	—	2	1	no	—
141	#466	—	2	1	no	—
142	#464	—	2	4	no	—
143	#462	C₂²×C₅	1	4	yes	—
144	#460	C₂²×C₅	1	4	yes	—
145	C₈×Q₈	—	2	1	no	—
146	C₈×Q₈	—	2	1	no	—
147	C₈×Q₈	C₂²×C₅	1	2	yes	—
148	C₈×Q₈	C₂²×C₅	1	2	yes	—
149	C₄×D₈	—	2	1	no	—
150	C₄×D₈	—	2	1	no	—
151	#423	C₂²×C₅	1	2	yes	—
152	C₂×C₄⋊C₈	—	2	1	no	—
153	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
154	#417	—	2	2	no	—
155	C₂×C₄×C₈	—	2	1	no	—
156	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
157	#335	C₂²×C₅	1	2	yes	—
158	#335	C₂²×C₅	1	2	yes	—
159	#332	C₂²×C₅	1	4	yes	—
160	#331	C₂²×C₅	1	4	yes	—
161	C₂³×C₅	—	4	2	no	—
162	C₂³×C₅	#121	1	1	yes	characteristic
163	C₅×Q₈	—	2	2	no	—
164	C₅×Q₈	—	2	4	no	—
165	C₅×Q₈	—	4	2	no	—
166	C₅×Q₈	—	4	2	no	—
167	C₅×Q₈	C₂×D₈	1	4	yes	—
168	C₅×D₄	—	4	2	no	—
169	C₅×D₄	—	4	2	no	—
170	C₅×D₄	—	4	2	no	—
171	C₅×D₄	—	4	2	no	—
172	C₅×D₄	—	8	2	no	—
173	C₅×D₄	—	8	4	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	4	no	—
176	C₂×C₄×C₅	—	2	1	no	—
177	C₂×C₄×C₅	—	2	1	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	4	1	no	—
180	C₂×C₄×C₅	—	4	2	no	—
181	C₂×C₄×C₅	—	4	2	no	—
182	C₂×C₄×C₅	—	8	4	no	—
183	C₂×C₄×C₅	#121	1	1	yes	characteristic
184	C₂×C₄×C₅	#121	1	1	yes	characteristic
185	C₂×C₄×C₅	#121	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
187	C₂×C₄×C₅	#135	1	2	yes	—
188	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
191	C₅×C₈	—	2	1	no	—
192	C₅×C₈	—	2	1	no	—
193	C₅×C₈	—	2	1	no	—
194	C₅×C₈	—	4	4	no	—
195	C₅×C₈	—	4	2	no	—
196	C₅×C₈	#137	1	2	yes	complemented
197	C₂×#41	—	2	2	no	—
198	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
199	#137	—	4	8	no	—
200	C₂×Q₁₆	—	2	4	no	—
201	C₂×SD₁₆	—	2	4	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
207	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
208	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
209	#127	C₂³×C₅	1	2	yes	—
210	#124	—	2	2	no	—
211	#121	—	2	2	no	—
212	#121	—	2	2	no	—
213	C₄×Q₈	—	2	1	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	C₅×D₄	1	2	yes	—
216	C₄×Q₈	C₂³×C₅	1	2	yes	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
223	#107	—	2	1	no	—
224	#107	—	2	1	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	C₂³×C₅	1	2	yes	—
230	C₄⋊C₈	C₂³×C₅	1	4	yes	—
231	#103	—	2	4	no	—
232	#102	—	2	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	4	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	C₅×D₄	1	1	yes	characteristic
238	C₄×C₈	C₅×D₄	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
241	C₂²×C₅	—	2	1	no	—
242	C₂²×C₅	—	2	1	no	—
243	C₂²×C₅	—	8	2	no	—
244	C₂²×C₅	—	8	2	no	—
245	C₂²×C₅	—	8	2	no	—
246	C₂²×C₅	#466	1	2	yes	—
247	C₂²×C₅	C₂×#121	1	1	yes	characteristic
248	C₄×C₅	—	2	1	no	—
249	C₄×C₅	—	2	4	no	—
250	C₄×C₅	—	2	1	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	4	2	no	—
253	C₄×C₅	—	8	2	no	—
254	C₄×C₅	#466	1	2	yes	—
255	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
256	C₄×C₅	C₂×#135	1	1	yes	characteristic
257	C₄×C₅	C₂×#137	1	1	yes	characteristic
258	C₄×C₅	#575	1	1	yes	characteristic
259	#41	—	4	4	no	—
260	#41	—	4	4	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	C₅×D₈	1	4	yes	complemented
265	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
275	C₂²×C₄	C₅×#41	1	1	yes	characteristic
276	Q₁₆	—	4	8	no	—
277	SD₁₆	—	4	8	no	—
278	D₈	—	4	2	no	—
279	D₈	—	4	2	no	—
280	M₁₆	—	4	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
287	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
288	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
289	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
301	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
302	C₄²	C₅×#41	1	1	yes	characteristic
303	C₄²	C₂⁴×C₅	1	1	yes	characteristic
304	C₂×C₅	—	2	2	no	—
305	C₂×C₅	—	16	2	no	—
306	C₂×C₅	#3037	1	1	yes	characteristic
307	C₂×C₅	C₂×#466	1	1	yes	characteristic
308	C₂×C₅	#3067	1	1	yes	characteristic
309	C₂³	—	4	2	no	—
310	C₂³	C₅×#121	1	1	yes	characteristic
311	Q₈	—	2	4	no	—
312	Q₈	—	2	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	Q₈	C₂×C₅×D₈	1	4	yes	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	8	2	no	—
321	D₄	—	8	4	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	8	4	no	—
331	C₂×C₄	C₅×#121	1	1	yes	characteristic
332	C₂×C₄	C₅×#121	1	1	yes	characteristic
333	C₂×C₄	C₅×#121	1	1	yes	characteristic
334	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
335	C₂×C₄	C₅×#135	1	2	yes	—
336	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
337	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
338	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	4	4	no	—
343	C₈	—	4	2	no	—
344	C₈	C₅×#137	1	2	yes	complemented
345	C₅	#19261	1	1	yes	characteristic, direct product factor, Sylow
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	C₅×#466	1	2	yes	—
352	C₂²	C₂×C₅×#121	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	4	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	4	2	no	—
358	C₄	—	8	2	no	—
359	C₄	C₅×#466	1	2	yes	—
360	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
361	C₄	C₂×C₅×#135	1	1	yes	characteristic
362	C₄	C₂×C₅×#137	1	1	yes	characteristic
363	C₄	C₅×#575	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	16	2	no	—
366	C₂	C₅×#3037	1	1	yes	characteristic
367	C₂	C₂×C₅×#466	1	1	yes	characteristic
368	C₂	C₅×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#3169	C₂
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
2. C₅×#3037	C₂	semidirect
3. C₂×C₅×C₈×Q₈	C₂	semidirect
4. C₅×#1682	C₂	semidirect
5. C₅×#1674	C₂	semidirect
6. C₅×#1668	C₂	semidirect
7. C₅×#1567	C₂	semidirect
8. C₅×#1547	C₂	semidirect
36. #19261	C₅	direct (N × H)
82. #3037	C₂×C₅	semidirect
83. C₂×C₈×Q₈	C₂×C₅	semidirect
84. #1682	C₂×C₅	semidirect
85. #1674	C₂×C₅	semidirect
86. #1668	C₂×C₅	semidirect
87. #1567	C₂×C₅	semidirect
88. #1547	C₂×C₅	semidirect
94. C₂×C₅×Q₈	D₈	semidirect
196. C₅×C₈	#137	semidirect
264. C₂×Q₈	C₅×D₈	semidirect
344. C₈	C₅×#137	semidirect
345. C₅	#19261	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×#3169	2
C₅×#3037	2
C₂×C₅×C₈×Q₈	2
C₅×#1682	2
C₅×#1674	2
C₅×#1668	2
C₅×#1567	2
C₅×#1547	2
#19261	5

Subgroups — C5×#19261 #11941981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19261 #11941981