Subgroups — C₅×#33627 #11956347

Order 1280.

598 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#2228	C₂	1	4	yes	maximal, complemented
4	C₅×#2226	C₂	1	2	yes	maximal, complemented
5	C₅×#2224	C₂	1	2	yes	maximal, complemented
6	C₅×#2159	C₂	1	2	yes	maximal, complemented
7	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2140	C₂	1	2	yes	maximal
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₅×#533	—	2	1	no	—
11	C₂×C₄×C₅×D₄	—	2	1	no	—
12	C₂×C₅×M₃₂	C₂²	1	2	yes	—
13	C₅×#500	C₂²	1	1	yes	characteristic, complemented
14	C₅×#500	C₂²	1	1	yes	characteristic, complemented
15	C₅×#499	—	2	1	no	—
16	C₅×#499	C₂²	1	2	yes	complemented
17	C₅×#498	—	2	1	no	—
18	C₄×C₅×D₈	—	2	1	no	—
19	C₄×C₅×D₈	—	2	1	no	—
20	C₄×C₅×D₈	C₂²	1	2	yes	—
21	C₄×C₅×D₈	C₂²	1	2	yes	—
22	C₅×#427	—	2	1	no	—
23	C₅×#427	C₂²	1	2	yes	—
24	C₂×C₅×#107	C₂²	1	1	yes	characteristic
25	C₂×C₅×#102	—	2	1	no	—
26	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
27	C₅×#367	C₂²	1	4	yes	complemented
28	C₅×#366	C₂²	1	4	yes	complemented
29	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
30	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
31	C₅×#357	C₂²	1	4	yes	—
32	C₅×#357	C₂²	1	4	yes	—
33	#33627	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₅×D₄	—	2	1	no	—
35	C₂³×C₄×C₅	—	4	1	no	—
36	C₂×C₅×D₈	—	2	1	no	—
37	C₂×C₅×D₈	—	2	2	no	—
38	C₂×C₅×D₈	Q₈	1	4	yes	complemented
39	C₂×C₅×D₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊Q₈	—	2	1	no	—
43	C₅×C₄⋊Q₈	—	2	1	no	—
44	C₅×#125	—	2	2	no	—
45	C₅×C₂²⋊Q₈	—	4	1	no	—
46	C₅×C₂²⋊Q₈	—	4	1	no	—
47	C₄×C₅×Q₈	—	4	1	no	—
48	C₄×C₅×Q₈	—	4	1	no	—
49	C₄×C₅×D₄	—	2	2	no	—
50	C₄×C₅×D₄	—	2	2	no	—
51	C₄×C₅×D₄	—	4	1	no	—
52	C₄×C₅×D₄	—	4	1	no	—
53	C₅×#117	—	2	2	no	—
54	C₂×C₅×C₄⋊C₄	—	2	1	no	—
55	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
56	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
57	C₅×M₃₂	—	2	4	no	—
58	C₂×C₅×C₁₆	C₂³	1	4	yes	—
59	C₅×#107	—	2	1	no	—
60	C₅×#107	—	2	1	no	—
61	C₅×#107	—	2	1	no	—
62	C₅×#107	C₂³	1	2	yes	—
63	C₅×#107	C₂³	1	2	yes	—
64	C₅×#107	C₂³	1	2	yes	—
65	C₅×#106	—	2	2	no	—
66	C₅×#106	—	2	2	no	—
67	C₅×#102	—	2	2	no	—
68	C₅×#102	—	2	2	no	—
69	C₄×C₅×C₈	D₄	1	1	yes	characteristic
70	C₄×C₅×C₈	D₄	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	#3160	C₂×C₅	1	1	yes	characteristic, complemented
74	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
75	#2228	C₂×C₅	1	4	yes	complemented
76	#2226	C₂×C₅	1	2	yes	complemented
77	#2224	C₂×C₅	1	2	yes	complemented
78	#2159	C₂×C₅	1	2	yes	complemented
79	#2150	C₂×C₅	1	1	yes	characteristic, complemented
80	#2140	C₂×C₅	1	2	yes	—
81	C₂⁴×C₅	—	4	1	no	—
82	C₂×C₅×Q₈	—	4	1	no	—
83	C₂×C₅×Q₈	—	4	1	no	—
84	C₂×C₅×D₄	—	2	4	no	—
85	C₂×C₅×D₄	—	2	2	no	—
86	C₂×C₅×D₄	—	4	1	no	—
87	C₂×C₅×D₄	—	4	2	no	—
88	C₂²×C₄×C₅	—	4	1	no	—
89	C₂²×C₄×C₅	—	4	2	no	—
90	C₂²×C₄×C₅	—	4	2	no	—
91	C₂²×C₄×C₅	—	8	4	no	—
92	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₅×D₈	—	2	4	no	—
96	C₅×D₈	—	2	2	no	—
97	C₅×D₈	C₂×Q₈	1	4	yes	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
103	C₂×C₅×C₈	#41	1	2	yes	—
104	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	4	1	no	—
109	C₅×C₄⋊C₄	—	4	2	no	—
110	C₅×C₄⋊C₄	—	4	2	no	—
111	C₅×C₄⋊C₄	—	4	1	no	—
112	C₅×C₄⋊C₄	—	4	1	no	—
113	C₅×C₂²⋊C₄	—	4	2	no	—
114	C₅×C₂²⋊C₄	—	4	2	no	—
115	C₅×C₂²⋊C₄	—	4	2	no	—
116	C₄²×C₅	—	4	2	no	—
117	C₄²×C₅	D₈	1	1	yes	characteristic
118	C₄²×C₅	D₈	1	1	yes	characteristic
119	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
121	C₅×C₁₆	—	2	4	no	—
122	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
123	#533	—	2	1	no	—
124	C₂×C₄×D₄	—	2	1	no	—
125	C₂×M₃₂	C₂²×C₅	1	2	yes	—
126	#500	C₂²×C₅	1	1	yes	characteristic, complemented
127	#500	C₂²×C₅	1	1	yes	characteristic, complemented
128	#499	—	2	1	no	—
129	#499	C₂²×C₅	1	2	yes	complemented
130	#498	—	2	1	no	—
131	C₄×D₈	—	2	1	no	—
132	C₄×D₈	—	2	1	no	—
133	C₄×D₈	C₂²×C₅	1	2	yes	—
134	C₄×D₈	C₂²×C₅	1	2	yes	—
135	#427	—	2	1	no	—
136	#427	C₂²×C₅	1	2	yes	—
137	C₂×#107	C₂²×C₅	1	1	yes	characteristic
138	C₂×#102	—	2	1	no	—
139	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
140	#367	C₂²×C₅	1	4	yes	complemented
141	#366	C₂²×C₅	1	4	yes	complemented
142	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
143	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
144	#357	C₂²×C₅	1	4	yes	—
145	#357	C₂²×C₅	1	4	yes	—
146	C₂³×C₅	—	4	2	no	—
147	C₂³×C₅	—	4	4	no	—
148	C₂³×C₅	—	8	2	no	—
149	C₂³×C₅	—	8	2	no	—
150	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
151	C₅×Q₈	—	8	2	no	—
152	C₅×Q₈	—	8	2	no	—
153	C₅×D₄	—	2	4	no	—
154	C₅×D₄	—	4	2	no	—
155	C₅×D₄	—	4	4	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	4	1	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	—	4	2	no	—
160	C₂×C₄×C₅	—	4	2	no	—
161	C₂×C₄×C₅	—	8	4	no	—
162	C₂×C₄×C₅	—	8	2	no	—
163	C₂×C₄×C₅	—	8	4	no	—
164	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
165	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
170	C₂×C₄×C₅	#137	1	2	yes	—
171	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
172	C₅×C₈	—	2	1	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
175	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
176	C₅×C₈	C₂×#41	1	1	yes	characteristic
177	C₂²×D₄	—	2	1	no	—
178	C₂³×C₄	—	4	1	no	—
179	C₂×D₈	—	2	1	no	—
180	C₂×D₈	—	2	2	no	—
181	C₂×D₈	C₅×Q₈	1	4	yes	complemented
182	C₂×D₈	C₂³×C₅	1	2	yes	—
183	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	#125	—	2	2	no	—
188	C₂²⋊Q₈	—	4	1	no	—
189	C₂²⋊Q₈	—	4	1	no	—
190	C₄×Q₈	—	4	1	no	—
191	C₄×Q₈	—	4	1	no	—
192	C₄×D₄	—	2	2	no	—
193	C₄×D₄	—	2	2	no	—
194	C₄×D₄	—	4	1	no	—
195	C₄×D₄	—	4	1	no	—
196	#117	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₂²⋊C₄	—	4	1	no	—
199	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
200	M₃₂	—	2	4	no	—
201	C₂×C₁₆	C₂³×C₅	1	4	yes	—
202	#107	—	2	1	no	—
203	#107	—	2	1	no	—
204	#107	—	2	1	no	—
205	#107	C₂³×C₅	1	2	yes	—
206	#107	C₂³×C₅	1	2	yes	—
207	#107	C₂³×C₅	1	2	yes	—
208	#106	—	2	2	no	—
209	#106	—	2	2	no	—
210	#102	—	2	2	no	—
211	#102	—	2	2	no	—
212	C₄×C₈	C₅×D₄	1	1	yes	characteristic
213	C₄×C₈	C₅×D₄	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
215	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	4	4	no	—
219	C₂²×C₅	—	8	8	no	—
220	C₂²×C₅	—	8	2	no	—
221	C₂²×C₅	—	8	2	no	—
222	C₂²×C₅	#474	1	2	yes	—
223	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	8	2	no	—
228	C₄×C₅	—	8	2	no	—
229	C₄×C₅	—	8	2	no	—
230	C₄×C₅	#474	1	2	yes	—
231	C₄×C₅	#509	1	2	yes	—
232	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
233	C₄×C₅	C₂×#137	1	1	yes	characteristic
234	C₂⁴	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	8	4	no	—
245	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
246	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
248	D₈	—	2	4	no	—
249	D₈	—	2	2	no	—
250	D₈	C₂×C₅×Q₈	1	4	yes	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
253	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
254	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
255	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
256	C₂×C₈	C₅×#41	1	2	yes	—
257	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	C₅×D₈	1	1	yes	characteristic
271	C₄²	C₅×D₈	1	1	yes	characteristic
272	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
273	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
274	C₁₆	—	2	4	no	—
275	C₂×C₅	—	2	2	no	—
276	C₂×C₅	—	8	4	no	—
277	C₂×C₅	C₂×#474	1	1	yes	characteristic
278	C₂×C₅	C₂×#509	1	1	yes	characteristic
279	C₂×C₅	#3414	1	1	yes	characteristic
280	C₂³	—	4	4	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	8	2	no	—
283	C₂³	—	8	2	no	—
284	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
285	Q₈	—	8	2	no	—
286	Q₈	—	8	2	no	—
287	D₄	—	2	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
299	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
300	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
301	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
304	C₂×C₄	C₅×#137	1	2	yes	—
305	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
309	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
310	C₈	C₂×C₅×#41	1	1	yes	characteristic
311	C₅	#33627	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	8	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	C₅×#474	1	2	yes	—
319	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₅×#474	1	2	yes	—
327	C₄	C₅×#509	1	2	yes	—
328	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
329	C₄	C₂×C₅×#137	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	8	4	no	—
332	C₂	C₂×C₅×#474	1	1	yes	characteristic
333	C₂	C₂×C₅×#509	1	1	yes	characteristic
334	C₂	C₅×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3160	C₂	semidirect
3. C₅×#2228	C₂	semidirect
4. C₅×#2226	C₂	semidirect
5. C₅×#2224	C₂	semidirect
6. C₅×#2159	C₂	semidirect
7. C₅×#2150	C₂	semidirect
13. C₅×#500	C₂²	semidirect
14. C₅×#500	C₂²	semidirect
16. C₅×#499	C₂²	semidirect
27. C₅×#367	C₂²	semidirect
28. C₅×#366	C₂²	semidirect
29. C₅×C₄⋊C₁₆	C₂²	semidirect
30. C₅×C₄⋊C₁₆	C₂²	semidirect
33. #33627	C₅	direct (N × H)
38. C₂×C₅×D₈	Q₈	semidirect
73. #3160	C₂×C₅	semidirect
75. #2228	C₂×C₅	semidirect
76. #2226	C₂×C₅	semidirect
77. #2224	C₂×C₅	semidirect
78. #2159	C₂×C₅	semidirect
79. #2150	C₂×C₅	semidirect
126. #500	C₂²×C₅	semidirect
127. #500	C₂²×C₅	semidirect
129. #499	C₂²×C₅	semidirect
140. #367	C₂²×C₅	semidirect
141. #366	C₂²×C₅	semidirect
142. C₄⋊C₁₆	C₂²×C₅	semidirect
143. C₄⋊C₁₆	C₂²×C₅	semidirect
181. C₂×D₈	C₅×Q₈	semidirect
311. C₅	#33627	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×#3160	2
C₂×C₄×C₅×D₈	2
C₅×#2228	2
C₅×#2226	2
C₅×#2224	2
C₅×#2159	2
C₅×#2150	2
C₅×#2140	2
#33627	5

Subgroups — C5×#33627 #11956347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33627 #11956347