Subgroups — C₅×#33631 #11956351

Order 1280.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#2948	C₂	1	1	yes	maximal, characteristic
3	C₅×#2228	C₂	1	2	yes	maximal, complemented
4	C₅×#2228	C₂	1	2	yes	maximal, complemented
5	C₅×#2226	C₂	1	2	yes	maximal, complemented
6	C₅×#2224	C₂	1	2	yes	maximal, complemented
7	C₅×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#2140	C₂	1	1	yes	maximal, characteristic
11	C₂×C₅×#357	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₅×#544	—	2	1	no	—
14	C₅×#518	—	2	1	no	—
15	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₅×#501	—	2	2	no	—
18	C₅×#500	C₂²	1	2	yes	complemented
19	C₅×#499	C₂²	1	2	yes	complemented
20	C₅×#442	—	2	2	no	—
21	C₄×C₅×D₈	C₂²	1	2	yes	—
22	C₄×C₅×D₈	C₂²	1	2	yes	—
23	C₅×#428	—	2	1	no	—
24	C₅×#427	C₂²	1	1	yes	characteristic
25	C₅×#427	C₂²	1	1	yes	characteristic
26	C₂×C₅×#107	C₂²	1	1	yes	characteristic
27	C₅×#418	—	2	1	no	—
28	C₅×#405	C₂²	1	1	yes	characteristic
29	C₅×#367	C₂²	1	1	yes	characteristic, complemented
30	C₅×#367	C₂²	1	1	yes	characteristic, complemented
31	C₅×#367	C₂²	1	2	yes	complemented
32	C₅×#366	C₂²	1	1	yes	characteristic, complemented
33	C₅×#366	C₂²	1	1	yes	characteristic, complemented
34	C₅×#366	C₂²	1	2	yes	complemented
35	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
36	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
37	C₅×#357	C₂²	1	2	yes	—
38	C₅×#357	C₂²	1	2	yes	—
39	C₅×#357	C₂²	1	2	yes	—
40	C₅×#357	C₂²	1	2	yes	—
41	#33631	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂²×C₅×D₄	—	2	1	no	—
43	C₂×C₅×D₈	—	2	1	no	—
44	C₂×C₅×D₈	—	2	2	no	—
45	C₂×C₅×D₈	Q₈	1	4	yes	complemented
46	C₂×C₅×D₈	C₂³	1	2	yes	—
47	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₅×C₄⋊Q₈	—	2	2	no	—
50	C₅×C₄⋊Q₈	—	4	1	no	—
51	C₅×#125	—	2	2	no	—
52	C₅×#125	—	4	1	no	—
53	C₅×C₂²⋊Q₈	—	4	2	no	—
54	C₄×C₅×D₄	—	2	2	no	—
55	C₄×C₅×D₄	—	2	2	no	—
56	C₄×C₅×D₄	—	4	1	no	—
57	C₄×C₅×D₄	—	4	1	no	—
58	C₅×#117	—	2	1	no	—
59	C₅×#117	—	2	1	no	—
60	C₅×#117	D₄	1	1	yes	characteristic
61	C₂×C₅×C₄⋊C₄	—	2	1	no	—
62	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
63	C₅×M₃₂	—	2	2	no	—
64	C₂×C₅×C₁₆	—	2	2	no	—
65	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₅×C₁₆	C₂³	1	2	yes	—
67	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
68	C₅×#107	—	2	1	no	—
69	C₅×#107	—	2	2	no	—
70	C₅×#107	C₂³	1	2	yes	—
71	C₅×#107	C₂³	1	1	yes	characteristic
72	C₅×#107	C₂³	1	1	yes	characteristic
73	C₅×#107	C₂³	1	1	yes	characteristic
74	C₅×#107	C₂³	1	1	yes	characteristic
75	C₅×#106	—	2	1	no	—
76	C₅×#106	—	2	1	no	—
77	C₅×#106	—	2	2	no	—
78	C₅×#102	—	2	2	no	—
79	C₅×#102	—	2	2	no	—
80	C₅×#97	D₄	1	2	yes	—
81	C₄×C₅×C₈	C₂³	1	2	yes	—
82	#3164	C₂×C₅	1	1	yes	characteristic, complemented
83	#2948	C₂×C₅	1	1	yes	characteristic
84	#2228	C₂×C₅	1	2	yes	complemented
85	#2228	C₂×C₅	1	2	yes	complemented
86	#2226	C₂×C₅	1	2	yes	complemented
87	#2224	C₂×C₅	1	2	yes	complemented
88	#2159	C₂×C₅	1	1	yes	characteristic, complemented
89	#2158	C₂×C₅	1	1	yes	characteristic, complemented
90	#2151	C₂×C₅	1	1	yes	characteristic, complemented
91	#2140	C₂×C₅	1	1	yes	characteristic
92	C₂×#357	C₂×C₅	1	1	yes	characteristic
93	C₂⁴×C₅	—	4	1	no	—
94	C₂×C₅×Q₈	—	4	2	no	—
95	C₂×C₅×D₄	—	2	4	no	—
96	C₂×C₅×D₄	—	2	2	no	—
97	C₂×C₅×D₄	—	4	1	no	—
98	C₂×C₅×D₄	—	4	2	no	—
99	C₂²×C₄×C₅	—	4	1	no	—
100	C₂²×C₄×C₅	—	4	2	no	—
101	C₂²×C₄×C₅	—	4	2	no	—
102	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
103	C₅×D₈	—	2	4	no	—
104	C₅×D₈	—	2	2	no	—
105	C₅×D₈	C₂×Q₈	1	4	yes	—
106	C₅×M₁₆	—	2	2	no	—
107	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
110	C₂×C₅×C₈	#41	1	2	yes	—
111	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
112	C₅×C₄⋊C₄	—	2	2	no	—
113	C₅×C₄⋊C₄	—	2	1	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₄⋊C₄	—	4	1	no	—
118	C₅×C₄⋊C₄	—	4	2	no	—
119	C₅×C₄⋊C₄	—	4	2	no	—
120	C₅×C₄⋊C₄	—	4	2	no	—
121	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
122	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
123	C₅×C₂²⋊C₄	—	4	1	no	—
124	C₅×C₂²⋊C₄	—	4	1	no	—
125	C₅×C₂²⋊C₄	—	4	2	no	—
126	C₅×C₂²⋊C₄	—	4	2	no	—
127	C₅×C₂²⋊C₄	D₈	1	2	yes	—
128	C₄²×C₅	—	4	1	no	—
129	C₄²×C₅	—	4	1	no	—
130	C₄²×C₅	C₂×D₄	1	2	yes	—
131	C₅×C₁₆	—	2	1	no	—
132	C₅×C₁₆	—	2	1	no	—
133	C₅×C₁₆	—	2	2	no	—
134	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
135	#544	—	2	1	no	—
136	#518	—	2	1	no	—
137	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
138	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
139	#501	—	2	2	no	—
140	#500	C₂²×C₅	1	2	yes	complemented
141	#499	C₂²×C₅	1	2	yes	complemented
142	#442	—	2	2	no	—
143	C₄×D₈	C₂²×C₅	1	2	yes	—
144	C₄×D₈	C₂²×C₅	1	2	yes	—
145	#428	—	2	1	no	—
146	#427	C₂²×C₅	1	1	yes	characteristic
147	#427	C₂²×C₅	1	1	yes	characteristic
148	C₂×#107	C₂²×C₅	1	1	yes	characteristic
149	#418	—	2	1	no	—
150	#405	C₂²×C₅	1	1	yes	characteristic
151	#367	C₂²×C₅	1	1	yes	characteristic, complemented
152	#367	C₂²×C₅	1	1	yes	characteristic, complemented
153	#367	C₂²×C₅	1	2	yes	complemented
154	#366	C₂²×C₅	1	1	yes	characteristic, complemented
155	#366	C₂²×C₅	1	1	yes	characteristic, complemented
156	#366	C₂²×C₅	1	2	yes	complemented
157	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
158	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
159	#357	C₂²×C₅	1	2	yes	—
160	#357	C₂²×C₅	1	2	yes	—
161	#357	C₂²×C₅	1	2	yes	—
162	#357	C₂²×C₅	1	2	yes	—
163	C₂³×C₅	—	4	2	no	—
164	C₂³×C₅	—	4	4	no	—
165	C₂³×C₅	—	16	1	no	—
166	C₂³×C₅	—	16	1	no	—
167	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
168	C₅×Q₈	—	16	2	no	—
169	C₅×D₄	—	2	4	no	—
170	C₅×D₄	—	4	2	no	—
171	C₅×D₄	—	4	4	no	—
172	C₂×C₄×C₅	—	4	1	no	—
173	C₂×C₄×C₅	—	4	1	no	—
174	C₂×C₄×C₅	—	4	1	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	—	16	2	no	—
178	C₂×C₄×C₅	—	16	2	no	—
179	C₂×C₄×C₅	—	16	1	no	—
180	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
181	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
182	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
183	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
184	C₂×C₄×C₅	#137	1	2	yes	—
185	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
186	C₅×C₈	—	2	2	no	—
187	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
188	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
189	C₅×C₈	C₂×#41	1	1	yes	characteristic
190	C₂²×D₄	—	2	1	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	2	no	—
193	C₂×D₈	C₅×Q₈	1	4	yes	complemented
194	C₂×D₈	C₂³×C₅	1	2	yes	—
195	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	4	1	no	—
199	#125	—	2	2	no	—
200	#125	—	4	1	no	—
201	C₂²⋊Q₈	—	4	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	—	2	2	no	—
204	C₄×D₄	—	4	1	no	—
205	C₄×D₄	—	4	1	no	—
206	#117	—	2	1	no	—
207	#117	—	2	1	no	—
208	#117	C₅×D₄	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₂²⋊C₄	—	4	2	no	—
211	M₃₂	—	2	2	no	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
214	C₂×C₁₆	C₂³×C₅	1	2	yes	—
215	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
216	#107	—	2	1	no	—
217	#107	—	2	2	no	—
218	#107	C₂³×C₅	1	2	yes	—
219	#107	C₂³×C₅	1	1	yes	characteristic
220	#107	C₂³×C₅	1	1	yes	characteristic
221	#107	C₂³×C₅	1	1	yes	characteristic
222	#107	C₂³×C₅	1	1	yes	characteristic
223	#106	—	2	1	no	—
224	#106	—	2	1	no	—
225	#106	—	2	2	no	—
226	#102	—	2	2	no	—
227	#102	—	2	2	no	—
228	#97	C₅×D₄	1	2	yes	—
229	C₄×C₈	C₂³×C₅	1	2	yes	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	2	1	no	—
232	C₂²×C₅	—	4	4	no	—
233	C₂²×C₅	—	8	2	no	—
234	C₂²×C₅	—	8	2	no	—
235	C₂²×C₅	—	16	4	no	—
236	C₂²×C₅	#474	1	2	yes	—
237	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
238	C₄×C₅	—	2	2	no	—
239	C₄×C₅	—	2	2	no	—
240	C₄×C₅	—	8	1	no	—
241	C₄×C₅	—	8	2	no	—
242	C₄×C₅	—	8	1	no	—
243	C₄×C₅	—	8	1	no	—
244	C₄×C₅	—	8	1	no	—
245	C₄×C₅	#474	1	2	yes	—
246	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
247	C₄×C₅	C₂×#137	1	1	yes	characteristic
248	C₂⁴	—	4	1	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×D₄	—	2	4	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
258	D₈	—	2	4	no	—
259	D₈	—	2	2	no	—
260	D₈	C₂×C₅×Q₈	1	4	yes	—
261	M₁₆	—	2	2	no	—
262	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
263	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
264	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
265	C₂×C₈	C₅×#41	1	2	yes	—
266	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
277	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
278	C₂²⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	C₅×D₈	1	2	yes	—
283	C₄²	—	4	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	C₂×C₅×D₄	1	2	yes	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	2	2	no	—
289	C₂×C₅	—	2	2	no	—
290	C₂×C₅	—	8	4	no	—
291	C₂×C₅	C₂×#474	1	1	yes	characteristic
292	C₂×C₅	#3415	1	1	yes	characteristic
293	C₂×C₅	#3416	1	1	yes	characteristic
294	C₂³	—	4	2	no	—
295	C₂³	—	4	4	no	—
296	C₂³	—	16	1	no	—
297	C₂³	—	16	1	no	—
298	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
299	Q₈	—	16	2	no	—
300	D₄	—	2	4	no	—
301	D₄	—	4	4	no	—
302	D₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	16	2	no	—
309	C₂×C₄	—	16	2	no	—
310	C₂×C₄	—	16	1	no	—
311	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
312	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
313	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
314	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
315	C₂×C₄	C₅×#137	1	2	yes	—
316	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
319	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
320	C₈	C₂×C₅×#41	1	1	yes	characteristic
321	C₅	#33631	1	1	yes	characteristic, direct product factor, Sylow
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	4	no	—
328	C₂²	C₅×#474	1	2	yes	—
329	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	C₅×#474	1	2	yes	—
338	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
339	C₄	C₂×C₅×#137	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	8	4	no	—
342	C₂	C₂×C₅×#474	1	1	yes	characteristic
343	C₂	C₅×#3415	1	1	yes	characteristic
344	C₂	C₅×#3416	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 ◁ C₅×#117 ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3164	C₂	semidirect
3. C₅×#2228	C₂	semidirect
4. C₅×#2228	C₂	semidirect
5. C₅×#2226	C₂	semidirect
6. C₅×#2224	C₂	semidirect
7. C₅×#2159	C₂	semidirect
8. C₅×#2158	C₂	semidirect
9. C₅×#2151	C₂	semidirect
18. C₅×#500	C₂²	semidirect
19. C₅×#499	C₂²	semidirect
29. C₅×#367	C₂²	semidirect
30. C₅×#367	C₂²	semidirect
31. C₅×#367	C₂²	semidirect
32. C₅×#366	C₂²	semidirect
33. C₅×#366	C₂²	semidirect
34. C₅×#366	C₂²	semidirect
35. C₅×C₄⋊C₁₆	C₂²	semidirect
36. C₅×C₄⋊C₁₆	C₂²	semidirect
41. #33631	C₅	direct (N × H)
45. C₂×C₅×D₈	Q₈	semidirect
82. #3164	C₂×C₅	semidirect
84. #2228	C₂×C₅	semidirect
85. #2228	C₂×C₅	semidirect
86. #2226	C₂×C₅	semidirect
87. #2224	C₂×C₅	semidirect
88. #2159	C₂×C₅	semidirect
89. #2158	C₂×C₅	semidirect
90. #2151	C₂×C₅	semidirect
140. #500	C₂²×C₅	semidirect
141. #499	C₂²×C₅	semidirect
151. #367	C₂²×C₅	semidirect
152. #367	C₂²×C₅	semidirect
153. #367	C₂²×C₅	semidirect
154. #366	C₂²×C₅	semidirect
155. #366	C₂²×C₅	semidirect
156. #366	C₂²×C₅	semidirect
157. C₄⋊C₁₆	C₂²×C₅	semidirect
158. C₄⋊C₁₆	C₂²×C₅	semidirect
193. C₂×D₈	C₅×Q₈	semidirect
321. C₅	#33631	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×#3164	2
C₅×#2948	2
C₅×#2228	2
C₅×#2226	2
C₅×#2224	2
C₅×#2159	2
C₅×#2158	2
C₅×#2151	2
C₅×#2140	2
C₂×C₅×#357	2
#33631	5

Subgroups — C5×#33631 #11956351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33631 #11956351