Subgroups — C₅×#19330 #11942050

Order 1280.

616 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3235	C₂	1	2	yes	maximal
2	C₅×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2931	C₂	1	1	yes	maximal, characteristic
4	C₅×#1703	C₂	1	4	yes	maximal, complemented
5	C₅×#1675	C₂	1	2	yes	maximal, complemented
6	C₅×#1669	C₂	1	2	yes	maximal, complemented
7	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1559	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#131	—	2	1	no	—
10	C₅×#533	C₂²	1	2	yes	—
11	C₅×#532	C₂²	1	1	yes	characteristic
12	C₅×#501	—	2	2	no	—
13	C₅×#500	C₂²	1	1	yes	characteristic
14	C₅×#494	C₂²	1	1	yes	characteristic
15	C₅×#490	—	2	2	no	—
16	C₅×#487	C₂²	1	2	yes	—
17	C₅×#482	—	2	4	no	—
18	C₅×#476	C₂²	1	4	yes	—
19	C₅×#474	C₂²	1	4	yes	—
20	C₅×#467	—	2	1	no	—
21	C₅×#467	—	2	1	no	—
22	C₄×C₅×Q₁₆	—	2	1	no	—
23	C₄×C₅×Q₁₆	—	2	1	no	—
24	C₅×#436	—	2	1	no	—
25	C₅×#436	C₂²	1	2	yes	—
26	C₅×C₈×D₄	—	2	1	no	—
27	C₅×C₈×D₄	C₂²	1	2	yes	—
28	C₅×#432	C₂²	1	2	yes	—
29	C₅×#423	C₂²	1	1	yes	characteristic
30	C₅×#408	—	2	1	no	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×#335	C₂²	1	2	yes	—
33	C₅×#334	C₂²	1	2	yes	—
34	C₅×#333	C₂²	1	4	yes	—
35	C₅×#332	C₂²	1	4	yes	—
36	#19330	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	1	no	—
38	C₂×C₅×Q₁₆	—	2	1	no	—
39	C₂×C₅×Q₁₆	—	2	1	no	—
40	C₅×#131	—	4	4	no	—
41	C₂×C₅×M₁₆	—	2	1	no	—
42	C₂×C₅×M₁₆	—	2	2	no	—
43	C₂²×C₅×C₈	—	2	2	no	—
44	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
46	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₅×#127	D₄	1	1	yes	characteristic
50	C₅×#125	—	2	2	no	—
51	C₅×#124	—	2	1	no	—
52	C₅×C₂²⋊Q₈	—	2	2	no	—
53	C₅×C₂²⋊Q₈	—	2	2	no	—
54	C₅×#121	—	2	1	no	—
55	C₅×#121	D₄	1	2	yes	—
56	C₄×C₅×Q₈	—	2	2	no	—
57	C₄×C₅×Q₈	—	2	2	no	—
58	C₄×C₅×D₄	—	2	1	no	—
59	C₄×C₅×D₄	C₂³	1	2	yes	—
60	C₅×#117	—	2	2	no	—
61	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
62	C₅×#107	—	2	1	no	—
63	C₅×#107	—	2	1	no	—
64	C₅×#107	—	2	4	no	—
65	C₅×#106	—	2	4	no	—
66	C₅×C₄⋊C₈	—	2	1	no	—
67	C₅×C₄⋊C₈	C₂³	1	2	yes	—
68	C₅×C₄⋊C₈	C₂³	1	4	yes	—
69	C₅×#103	—	2	2	no	—
70	C₅×#103	—	2	2	no	—
71	C₅×#102	—	2	4	no	—
72	C₅×#102	—	2	4	no	—
73	C₅×C₂²⋊C₈	—	2	2	no	—
74	C₅×C₂²⋊C₈	—	2	2	no	—
75	C₅×C₂²⋊C₈	—	2	2	no	—
76	C₅×#97	—	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	#3235	C₂×C₅	1	2	yes	—
82	#3170	C₂×C₅	1	1	yes	characteristic, complemented
83	#2931	C₂×C₅	1	1	yes	characteristic
84	#1703	C₂×C₅	1	4	yes	complemented
85	#1675	C₂×C₅	1	2	yes	complemented
86	#1669	C₂×C₅	1	2	yes	complemented
87	#1568	C₂×C₅	1	1	yes	characteristic, complemented
88	#1559	C₂×C₅	1	2	yes	complemented
89	C₅×#41	—	4	2	no	—
90	C₅×#41	—	4	2	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×D₄	—	2	2	no	—
95	C₂×C₅×D₄	—	2	1	no	—
96	C₂×C₅×D₄	—	2	1	no	—
97	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₅	—	2	2	no	—
99	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	#41	1	1	yes	characteristic
102	C₅×Q₁₆	—	4	2	no	—
103	C₅×Q₁₆	—	4	2	no	—
104	C₅×M₁₆	—	4	4	no	—
105	C₅×M₁₆	—	4	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	4	no	—
109	C₂×C₅×C₈	—	4	1	no	—
110	C₂×C₅×C₈	—	4	4	no	—
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	2	2	no	—
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	—	4	2	no	—
121	C₅×C₄⋊C₄	—	4	2	no	—
122	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
123	C₅×C₂²⋊C₄	—	2	2	no	—
124	C₅×C₂²⋊C₄	—	2	2	no	—
125	C₅×C₂²⋊C₄	—	4	2	no	—
126	C₄²×C₅	—	4	2	no	—
127	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₅	#41	1	1	yes	characteristic
130	C₄²×C₅	C₂⁴	1	1	yes	characteristic
131	C₂×#131	—	2	1	no	—
132	#533	C₂²×C₅	1	2	yes	—
133	#532	C₂²×C₅	1	1	yes	characteristic
134	#501	—	2	2	no	—
135	#500	C₂²×C₅	1	1	yes	characteristic
136	#494	C₂²×C₅	1	1	yes	characteristic
137	#490	—	2	2	no	—
138	#487	C₂²×C₅	1	2	yes	—
139	#482	—	2	4	no	—
140	#476	C₂²×C₅	1	4	yes	—
141	#474	C₂²×C₅	1	4	yes	—
142	#467	—	2	1	no	—
143	#467	—	2	1	no	—
144	C₄×Q₁₆	—	2	1	no	—
145	C₄×Q₁₆	—	2	1	no	—
146	#436	—	2	1	no	—
147	#436	C₂²×C₅	1	2	yes	—
148	C₈×D₄	—	2	1	no	—
149	C₈×D₄	C₂²×C₅	1	2	yes	—
150	#432	C₂²×C₅	1	2	yes	—
151	#423	C₂²×C₅	1	1	yes	characteristic
152	#408	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
154	#335	C₂²×C₅	1	2	yes	—
155	#334	C₂²×C₅	1	2	yes	—
156	#333	C₂²×C₅	1	4	yes	—
157	#332	C₂²×C₅	1	4	yes	—
158	C₂³×C₅	—	2	2	no	—
159	C₂³×C₅	#121	1	1	yes	characteristic
160	C₅×Q₈	—	4	2	no	—
161	C₅×Q₈	—	4	2	no	—
162	C₅×Q₈	—	4	2	no	—
163	C₅×Q₈	—	8	2	no	—
164	C₅×D₄	—	2	4	no	—
165	C₅×D₄	—	4	2	no	—
166	C₅×D₄	—	4	2	no	—
167	C₅×D₄	—	4	4	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	4	1	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	—	4	2	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	#121	1	1	yes	characteristic
178	C₂×C₄×C₅	#121	1	1	yes	characteristic
179	C₂×C₄×C₅	#121	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
183	C₅×C₈	—	2	1	no	—
184	C₅×C₈	—	2	1	no	—
185	C₅×C₈	—	2	1	no	—
186	C₅×C₈	—	4	2	no	—
187	C₅×C₈	—	4	4	no	—
188	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
189	C₂×#41	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	#131	—	4	4	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
197	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
198	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
199	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
200	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
201	#127	C₅×D₄	1	1	yes	characteristic
202	#125	—	2	2	no	—
203	#124	—	2	1	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	#121	—	2	1	no	—
207	#121	C₅×D₄	1	2	yes	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	C₂³×C₅	1	2	yes	—
212	#117	—	2	2	no	—
213	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
214	#107	—	2	1	no	—
215	#107	—	2	1	no	—
216	#107	—	2	4	no	—
217	#106	—	2	4	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	C₂³×C₅	1	2	yes	—
220	C₄⋊C₈	C₂³×C₅	1	4	yes	—
221	#103	—	2	2	no	—
222	#103	—	2	2	no	—
223	#102	—	2	4	no	—
224	#102	—	2	4	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	#97	—	2	2	no	—
229	C₄×C₈	C₅×D₄	1	1	yes	characteristic
230	C₄×C₈	C₅×D₄	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	2	1	no	—
235	C₂²×C₅	—	2	1	no	—
236	C₂²×C₅	—	4	2	no	—
237	C₂²×C₅	—	4	2	no	—
238	C₂²×C₅	—	4	2	no	—
239	C₂²×C₅	C₂×#121	1	1	yes	characteristic
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	4	2	no	—
245	C₄×C₅	—	8	2	no	—
246	C₄×C₅	—	8	2	no	—
247	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₄×C₅	#575	1	1	yes	characteristic
249	C₄×C₅	#578	1	2	yes	—
250	#41	—	4	2	no	—
251	#41	—	4	2	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
262	C₂²×C₄	C₅×#41	1	1	yes	characteristic
263	Q₁₆	—	4	2	no	—
264	Q₁₆	—	4	2	no	—
265	M₁₆	—	4	4	no	—
266	M₁₆	—	4	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	4	no	—
272	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
273	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
274	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
275	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
284	C₂²⋊C₄	—	2	2	no	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₄²	—	4	2	no	—
288	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
289	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
290	C₄²	C₅×#41	1	1	yes	characteristic
291	C₄²	C₂⁴×C₅	1	1	yes	characteristic
292	C₂×C₅	—	4	1	no	—
293	C₂×C₅	—	8	2	no	—
294	C₂×C₅	#3036	1	1	yes	characteristic
295	C₂×C₅	#3059	1	1	yes	characteristic
296	C₂×C₅	#3067	1	1	yes	characteristic
297	C₂³	—	2	2	no	—
298	C₂³	C₅×#121	1	1	yes	characteristic
299	Q₈	—	4	2	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	8	2	no	—
303	D₄	—	2	4	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	4	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	C₅×#121	1	1	yes	characteristic
317	C₂×C₄	C₅×#121	1	1	yes	characteristic
318	C₂×C₄	C₅×#121	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
321	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	4	2	no	—
326	C₈	—	4	4	no	—
327	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
328	C₅	#19330	1	1	yes	characteristic, direct product factor, Sylow
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	C₂×C₅×#121	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	2	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
344	C₄	C₅×#575	1	1	yes	characteristic
345	C₄	C₅×#578	1	2	yes	—
346	C₂	—	4	1	no	—
347	C₂	—	8	2	no	—
348	C₂	C₅×#3036	1	1	yes	characteristic
349	C₂	C₅×#3059	1	1	yes	characteristic
350	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×#532	C₂²
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×#3170	C₂	semidirect
4. C₅×#1703	C₂	semidirect
5. C₅×#1675	C₂	semidirect
6. C₅×#1669	C₂	semidirect
7. C₅×#1568	C₂	semidirect
8. C₅×#1559	C₂	semidirect
36. #19330	C₅	direct (N × H)
82. #3170	C₂×C₅	semidirect
84. #1703	C₂×C₅	semidirect
85. #1675	C₂×C₅	semidirect
86. #1669	C₂×C₅	semidirect
87. #1568	C₂×C₅	semidirect
88. #1559	C₂×C₅	semidirect
328. C₅	#19330	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×#3235	2
C₅×#3170	2
C₅×#2931	2
C₅×#1703	2
C₅×#1675	2
C₅×#1669	2
C₅×#1568	2
C₅×#1559	2
#19330	5

Subgroups — C5×#19330 #11942050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19330 #11942050