Subgroups — C₅×#19205 #11941925

Order 1280.

678 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×#3246	C₂	1	2	yes	maximal, complemented
2	C₅×#3147	C₂	1	1	yes	maximal, characteristic
3	C₅×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1690	C₂	1	2	yes	maximal, complemented
5	C₅×#1688	C₂	1	2	yes	maximal, complemented
6	C₅×#1666	C₂	1	4	yes	maximal, complemented
7	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1547	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#135	—	2	1	no	—
10	C₂×C₅×#135	—	2	1	no	—
11	C₅×#533	C₂²	1	1	yes	characteristic
12	C₅×#532	C₂²	1	2	yes	—
13	C₅×#495	—	2	1	no	—
14	C₅×#495	—	2	1	no	—
15	C₅×#494	C₂²	1	1	yes	characteristic
16	C₅×#493	C₂²	1	1	yes	characteristic
17	C₅×#492	C₂²	1	2	yes	—
18	C₅×#476	—	2	2	no	—
19	C₅×#475	C₂²	1	4	yes	—
20	C₅×#465	—	2	4	no	—
21	C₅×#461	—	2	2	no	—
22	C₅×#460	C₂²	1	4	yes	—
23	C₅×#446	—	2	1	no	—
24	C₅×#446	C₂²	1	2	yes	—
25	C₅×C₈×Q₈	—	2	1	no	—
26	C₅×C₈×Q₈	C₂²	1	2	yes	—
27	C₅×#424	—	2	1	no	—
28	C₅×#423	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
30	C₅×#405	—	2	1	no	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×#334	C₂²	1	2	yes	—
33	C₅×#334	C₂²	1	2	yes	—
34	C₅×#332	C₂²	1	4	yes	—
35	C₅×#331	C₂²	1	4	yes	—
36	#19205	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	2	no	—
38	C₅×#135	—	4	4	no	—
39	C₅×#135	—	4	4	no	—
40	C₂×C₅×Q₁₆	—	2	1	no	—
41	C₂×C₅×Q₁₆	—	2	1	no	—
42	C₂×C₅×SD₁₆	—	2	4	no	—
43	C₂×C₅×SD₁₆	—	2	2	no	—
44	C₂×C₅×SD₁₆	—	2	2	no	—
45	C₂×C₅×D₈	—	2	1	no	—
46	C₂×C₅×D₈	—	2	1	no	—
47	C₂×C₅×M₁₆	—	2	1	no	—
48	C₂²×C₅×C₈	—	2	2	no	—
49	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
51	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
52	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
53	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₅×#127	C₂³	1	2	yes	—
55	C₅×#125	—	2	1	no	—
56	C₅×#124	—	2	2	no	—
57	C₅×C₂²⋊Q₈	—	2	1	no	—
58	C₅×C₂²⋊Q₈	D₄	1	2	yes	—
59	C₅×#121	—	2	2	no	—
60	C₅×#121	—	2	2	no	—
61	C₄×C₅×Q₈	—	2	1	no	—
62	C₄×C₅×Q₈	C₂³	1	2	yes	—
63	C₄×C₅×D₄	—	2	2	no	—
64	C₄×C₅×D₄	—	2	2	no	—
65	C₅×#117	—	2	1	no	—
66	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
67	C₅×#106	—	2	4	no	—
68	C₅×C₄⋊C₈	—	2	2	no	—
69	C₅×C₄⋊C₈	—	2	2	no	—
70	C₅×C₄⋊C₈	—	2	2	no	—
71	C₅×C₄⋊C₈	—	2	1	no	—
72	C₅×C₄⋊C₈	C₂³	1	2	yes	—
73	C₅×C₄⋊C₈	C₂³	1	4	yes	—
74	C₅×#103	—	2	4	no	—
75	C₅×#102	—	2	4	no	—
76	C₅×#97	—	2	2	no	—
77	C₄×C₅×C₈	—	2	2	no	—
78	C₄×C₅×C₈	D₄	1	1	yes	characteristic
79	C₄×C₅×C₈	D₄	1	1	yes	characteristic
80	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
82	#3246	C₂×C₅	1	2	yes	complemented
83	#3147	C₂×C₅	1	1	yes	characteristic
84	#2960	C₂×C₅	1	1	yes	characteristic, complemented
85	#1690	C₂×C₅	1	2	yes	complemented
86	#1688	C₂×C₅	1	2	yes	complemented
87	#1666	C₂×C₅	1	4	yes	complemented
88	C₂×#334	C₂×C₅	1	1	yes	characteristic, complemented
89	#1547	C₂×C₅	1	2	yes	complemented
90	C₅×#41	—	4	4	no	—
91	C₅×#41	—	4	4	no	—
92	C₂×C₅×Q₈	—	2	1	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
95	C₂×C₅×D₄	—	2	2	no	—
96	C₂×C₅×D₄	—	2	2	no	—
97	C₂×C₅×D₄	—	2	2	no	—
98	C₂×C₅×D₄	—	4	2	no	—
99	C₂²×C₄×C₅	—	4	2	no	—
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	#41	1	1	yes	characteristic
103	C₅×Q₁₆	—	4	2	no	—
104	C₅×Q₁₆	—	4	2	no	—
105	C₅×SD₁₆	—	4	4	no	—
106	C₅×SD₁₆	—	4	4	no	—
107	C₅×SD₁₆	—	8	4	no	—
108	C₅×D₈	—	4	2	no	—
109	C₅×D₈	—	4	2	no	—
110	C₅×M₁₆	—	4	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	2	4	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	8	2	no	—
115	C₂×C₅×C₈	SD₁₆	1	4	yes	complemented
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	1	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	—	2	1	no	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	—	2	2	no	—
125	C₅×C₄⋊C₄	—	2	2	no	—
126	C₅×C₄⋊C₄	—	2	2	no	—
127	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
128	C₅×C₂²⋊C₄	—	2	2	no	—
129	C₅×C₂²⋊C₄	—	4	2	no	—
130	C₅×C₂²⋊C₄	—	4	2	no	—
131	C₄²×C₅	—	2	2	no	—
132	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₅	#41	1	1	yes	characteristic
135	C₄²×C₅	C₂⁴	1	1	yes	characteristic
136	C₂×#135	—	2	1	no	—
137	C₂×#135	—	2	1	no	—
138	#533	C₂²×C₅	1	1	yes	characteristic
139	#532	C₂²×C₅	1	2	yes	—
140	#495	—	2	1	no	—
141	#495	—	2	1	no	—
142	#494	C₂²×C₅	1	1	yes	characteristic
143	#493	C₂²×C₅	1	1	yes	characteristic
144	#492	C₂²×C₅	1	2	yes	—
145	#476	—	2	2	no	—
146	#475	C₂²×C₅	1	4	yes	—
147	#465	—	2	4	no	—
148	#461	—	2	2	no	—
149	#460	C₂²×C₅	1	4	yes	—
150	#446	—	2	1	no	—
151	#446	C₂²×C₅	1	2	yes	—
152	C₈×Q₈	—	2	1	no	—
153	C₈×Q₈	C₂²×C₅	1	2	yes	—
154	#424	—	2	1	no	—
155	#423	C₂²×C₅	1	1	yes	characteristic
156	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
157	#405	—	2	1	no	—
158	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
159	#334	C₂²×C₅	1	2	yes	—
160	#334	C₂²×C₅	1	2	yes	—
161	#332	C₂²×C₅	1	4	yes	—
162	#331	C₂²×C₅	1	4	yes	—
163	C₂³×C₅	—	4	2	no	—
164	C₂³×C₅	#121	1	1	yes	characteristic
165	C₅×Q₈	—	2	4	no	—
166	C₅×Q₈	—	4	2	no	—
167	C₅×Q₈	—	4	2	no	—
168	C₅×Q₈	—	4	2	no	—
169	C₅×D₄	—	4	2	no	—
170	C₅×D₄	—	4	2	no	—
171	C₅×D₄	—	4	2	no	—
172	C₅×D₄	—	4	2	no	—
173	C₅×D₄	—	8	2	no	—
174	C₅×D₄	—	8	4	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	4	1	no	—
179	C₂×C₄×C₅	—	4	2	no	—
180	C₂×C₄×C₅	—	8	4	no	—
181	C₂×C₄×C₅	#121	1	1	yes	characteristic
182	C₂×C₄×C₅	#121	1	1	yes	characteristic
183	C₂×C₄×C₅	#121	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
185	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
186	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
189	C₅×C₈	—	2	1	no	—
190	C₅×C₈	—	2	1	no	—
191	C₅×C₈	—	4	2	no	—
192	C₅×C₈	—	4	4	no	—
193	C₅×C₈	C₂×SD₁₆	1	2	yes	—
194	C₅×C₈	C₂×SD₁₆	1	2	yes	—
195	C₂×#41	—	2	2	no	—
196	#135	—	4	4	no	—
197	#135	—	4	4	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×SD₁₆	—	2	4	no	—
201	C₂×SD₁₆	—	2	2	no	—
202	C₂×SD₁₆	—	2	2	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
208	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
209	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
210	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
211	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
212	#127	C₂³×C₅	1	2	yes	—
213	#125	—	2	1	no	—
214	#124	—	2	2	no	—
215	C₂²⋊Q₈	—	2	1	no	—
216	C₂²⋊Q₈	C₅×D₄	1	2	yes	—
217	#121	—	2	2	no	—
218	#121	—	2	2	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	C₂³×C₅	1	2	yes	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	—	2	2	no	—
223	#117	—	2	1	no	—
224	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
225	#106	—	2	4	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	C₂³×C₅	1	2	yes	—
231	C₄⋊C₈	C₂³×C₅	1	4	yes	—
232	#103	—	2	4	no	—
233	#102	—	2	4	no	—
234	#97	—	2	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	C₅×D₄	1	1	yes	characteristic
237	C₄×C₈	C₅×D₄	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
240	C₂²×C₅	—	2	1	no	—
241	C₂²×C₅	—	2	1	no	—
242	C₂²×C₅	—	8	2	no	—
243	C₂²×C₅	—	8	2	no	—
244	C₂²×C₅	—	8	2	no	—
245	C₂²×C₅	#460	1	2	yes	—
246	C₂²×C₅	C₂×#121	1	1	yes	characteristic
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	2	1	no	—
249	C₄×C₅	—	2	1	no	—
250	C₄×C₅	—	4	2	no	—
251	C₄×C₅	—	4	2	no	—
252	C₄×C₅	—	8	2	no	—
253	C₄×C₅	#460	1	2	yes	—
254	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
255	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
256	C₄×C₅	#577	1	2	yes	—
257	#41	—	4	4	no	—
258	#41	—	4	4	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂²×C₄	—	4	2	no	—
267	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
269	C₂²×C₄	C₅×#41	1	1	yes	characteristic
270	Q₁₆	—	4	2	no	—
271	Q₁₆	—	4	2	no	—
272	SD₁₆	—	4	4	no	—
273	SD₁₆	—	4	4	no	—
274	SD₁₆	—	8	4	no	—
275	D₈	—	4	2	no	—
276	D₈	—	4	2	no	—
277	M₁₆	—	4	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	8	2	no	—
282	C₂×C₈	C₅×SD₁₆	1	4	yes	complemented
283	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
284	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
285	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
286	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
300	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
301	C₄²	C₅×#41	1	1	yes	characteristic
302	C₄²	C₂⁴×C₅	1	1	yes	characteristic
303	C₂×C₅	—	2	2	no	—
304	C₂×C₅	—	16	2	no	—
305	C₂×C₅	C₂×#460	1	1	yes	characteristic
306	C₂×C₅	#3060	1	1	yes	characteristic
307	C₂×C₅	#3067	1	1	yes	characteristic
308	C₂³	—	4	2	no	—
309	C₂³	C₅×#121	1	1	yes	characteristic
310	Q₈	—	2	4	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	C₅×#121	1	1	yes	characteristic
327	C₂×C₄	C₅×#121	1	1	yes	characteristic
328	C₂×C₄	C₅×#121	1	1	yes	characteristic
329	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
330	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
331	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
333	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	4	no	—
338	C₈	C₂×C₅×SD₁₆	1	2	yes	—
339	C₈	C₂×C₅×SD₁₆	1	2	yes	—
340	C₅	#19205	1	1	yes	characteristic, direct product factor, Sylow
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	C₅×#460	1	2	yes	—
347	C₂²	C₂×C₅×#121	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	2	no	—
354	C₄	C₅×#460	1	2	yes	—
355	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
356	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
357	C₄	C₅×#577	1	2	yes	—
358	C₂	—	2	2	no	—
359	C₂	—	16	2	no	—
360	C₂	C₂×C₅×#460	1	1	yes	characteristic
361	C₂	C₅×#3060	1	1	yes	characteristic
362	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×#3147	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
1. C₅×#3246	C₂	semidirect
3. C₅×#2960	C₂	semidirect
4. C₅×#1690	C₂	semidirect
5. C₅×#1688	C₂	semidirect
6. C₅×#1666	C₂	semidirect
7. C₂×C₅×#334	C₂	semidirect
8. C₅×#1547	C₂	semidirect
36. #19205	C₅	direct (N × H)
82. #3246	C₂×C₅	semidirect
84. #2960	C₂×C₅	semidirect
85. #1690	C₂×C₅	semidirect
86. #1688	C₂×C₅	semidirect
87. #1666	C₂×C₅	semidirect
88. C₂×#334	C₂×C₅	semidirect
89. #1547	C₂×C₅	semidirect
115. C₂×C₅×C₈	SD₁₆	semidirect
282. C₂×C₈	C₅×SD₁₆	semidirect
340. C₅	#19205	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×#3246	2
C₅×#3147	2
C₅×#2960	2
C₅×#1690	2
C₅×#1688	2
C₅×#1666	2
C₂×C₅×#334	2
C₅×#1547	2
#19205	5

Subgroups — C5×#19205 #11941925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19205 #11941925