Subgroups — C₅×#33771 #11956491

Order 1280.

730 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×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2246	C₂	1	2	yes	maximal, complemented
4	C₅×#2240	C₂	1	2	yes	maximal, complemented
5	C₅×#2206	C₂	1	4	yes	maximal, complemented
6	C₅×#2184	C₂	1	4	yes	maximal, complemented
7	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×D₈	—	2	1	no	—
9	C₂²×C₅×D₈	C₂²	1	2	yes	—
10	C₅×#532	—	2	1	no	—
11	C₂×C₅×#127	—	2	1	no	—
12	C₂×C₅×D₁₆	C₂²	1	8	yes	complemented
13	C₅×#500	C₂²	1	1	yes	characteristic, complemented
14	C₅×#493	—	2	1	no	—
15	C₅×#493	—	2	1	no	—
16	C₅×#493	C₂²	1	2	yes	—
17	C₅×#493	C₂²	1	2	yes	—
18	C₅×#493	C₂²	1	1	yes	characteristic, complemented
19	C₅×#486	—	2	1	no	—
20	C₅×#466	—	2	1	no	—
21	C₅×#466	C₂²	1	2	yes	—
22	C₄×C₅×D₈	—	2	1	no	—
23	C₄×C₅×D₈	C₂²	1	2	yes	complemented
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
26	C₅×#357	C₂²	1	8	yes	complemented
27	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
29	#33771	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	4	1	no	—
31	C₂²×C₅×D₄	—	2	2	no	—
32	C₂²×C₅×D₄	—	4	1	no	—
33	C₂²×C₅×D₄	—	4	1	no	—
34	C₂×C₅×D₈	—	2	1	no	—
35	C₂×C₅×D₈	—	2	2	no	—
36	C₂×C₅×D₈	—	2	2	no	—
37	C₂×C₅×D₈	—	2	2	no	—
38	C₂×C₅×D₈	—	4	4	no	—
39	C₂×C₅×D₈	—	4	4	no	—
40	C₂×C₅×D₈	C₂³	1	4	yes	—
41	C₂×C₅×D₈	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₅×C₄⋊Q₈	—	2	1	no	—
45	C₅×#127	—	2	1	no	—
46	C₅×#127	—	2	2	no	—
47	C₅×#127	—	2	2	no	—
48	C₅×#127	—	4	1	no	—
49	C₅×#127	—	4	1	no	—
50	C₅×#124	—	4	1	no	—
51	C₅×#121	—	2	2	no	—
52	C₅×#121	—	4	1	no	—
53	C₄×C₅×D₄	—	2	2	no	—
54	C₄×C₅×D₄	—	4	1	no	—
55	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
56	C₅×D₁₆	—	4	8	no	—
57	C₂×C₅×C₁₆	C₂³	1	4	yes	—
58	C₅×#107	—	2	1	no	—
59	C₅×#107	C₂³	1	2	yes	—
60	C₅×#102	—	2	2	no	—
61	C₅×#102	—	2	2	no	—
62	C₄×C₅×C₈	D₄	1	1	yes	characteristic
63	C₄×C₅×C₈	D₄	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
66	#3169	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×#493	C₂×C₅	1	1	yes	characteristic, complemented
68	#2246	C₂×C₅	1	2	yes	complemented
69	#2240	C₂×C₅	1	2	yes	complemented
70	#2206	C₂×C₅	1	4	yes	complemented
71	#2184	C₂×C₅	1	4	yes	complemented
72	#2162	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂⁴×C₅	—	4	2	no	—
74	C₅×#41	—	8	4	no	—
75	C₂×C₅×Q₈	—	4	1	no	—
76	C₂×C₅×D₄	—	2	2	no	—
77	C₂×C₅×D₄	—	2	4	no	—
78	C₂×C₅×D₄	—	4	4	no	—
79	C₂×C₅×D₄	—	4	4	no	—
80	C₂×C₅×D₄	—	4	2	no	—
81	C₂×C₅×D₄	—	4	1	no	—
82	C₂×C₅×D₄	—	4	1	no	—
83	C₂×C₅×D₄	—	4	2	no	—
84	C₂×C₅×D₄	—	4	2	no	—
85	C₂×C₅×D₄	—	4	2	no	—
86	C₂×C₅×D₄	—	4	2	no	—
87	C₂×C₅×D₄	—	4	2	no	—
88	C₂×C₅×D₄	—	8	4	no	—
89	C₂×C₅×D₄	—	8	4	no	—
90	C₂²×C₄×C₅	—	4	2	no	—
91	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₅×D₈	—	2	8	no	—
95	C₅×D₈	—	2	4	no	—
96	C₅×D₈	—	4	4	no	—
97	C₅×D₈	—	4	4	no	—
98	C₅×D₈	—	4	2	no	—
99	C₅×D₈	—	4	4	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	4	1	no	—
108	C₅×C₂²⋊C₄	—	4	2	no	—
109	C₅×C₂²⋊C₄	—	4	2	no	—
110	C₄²×C₅	D₈	1	1	yes	characteristic
111	C₄²×C₅	D₈	1	1	yes	characteristic
112	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
114	C₅×C₁₆	—	2	4	no	—
115	C₂²×D₈	—	2	1	no	—
116	C₂²×D₈	C₂²×C₅	1	2	yes	—
117	#532	—	2	1	no	—
118	C₂×#127	—	2	1	no	—
119	C₂×D₁₆	C₂²×C₅	1	8	yes	complemented
120	#500	C₂²×C₅	1	1	yes	characteristic, complemented
121	#493	—	2	1	no	—
122	#493	—	2	1	no	—
123	#493	C₂²×C₅	1	2	yes	—
124	#493	C₂²×C₅	1	2	yes	—
125	#493	C₂²×C₅	1	1	yes	characteristic, complemented
126	#486	—	2	1	no	—
127	#466	—	2	1	no	—
128	#466	C₂²×C₅	1	2	yes	—
129	C₄×D₈	—	2	1	no	—
130	C₄×D₈	C₂²×C₅	1	2	yes	complemented
131	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
132	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
133	#357	C₂²×C₅	1	8	yes	complemented
134	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
135	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
136	C₂³×C₅	—	4	2	no	—
137	C₂³×C₅	—	4	4	no	—
138	C₂³×C₅	—	8	4	no	—
139	C₂³×C₅	—	8	4	no	—
140	C₂³×C₅	—	8	4	no	—
141	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
142	C₅×Q₈	—	8	2	no	—
143	C₅×D₄	—	4	4	no	—
144	C₅×D₄	—	4	2	no	—
145	C₅×D₄	—	4	4	no	—
146	C₅×D₄	—	4	2	no	—
147	C₅×D₄	—	8	4	no	—
148	C₅×D₄	—	8	4	no	—
149	C₅×D₄	—	8	4	no	—
150	C₅×D₄	—	8	4	no	—
151	C₅×D₄	—	8	4	no	—
152	C₅×D₄	—	8	2	no	—
153	C₅×D₄	—	8	2	no	—
154	C₅×D₄	—	8	4	no	—
155	C₂×C₄×C₅	—	2	1	no	—
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	—	8	4	no	—
160	C₂×C₄×C₅	D₁₆	1	4	yes	complemented
161	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
167	C₅×C₈	—	2	1	no	—
168	C₅×C₈	—	2	1	no	—
169	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
170	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
171	C₂×#41	—	4	1	no	—
172	C₂²×D₄	—	2	2	no	—
173	C₂²×D₄	—	4	1	no	—
174	C₂²×D₄	—	4	1	no	—
175	C₂×D₈	—	2	2	no	—
176	C₂×D₈	—	2	2	no	—
177	C₂×D₈	—	2	2	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	—	4	4	no	—
180	C₂×D₈	—	4	4	no	—
181	C₂×D₈	C₂³×C₅	1	4	yes	—
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	#127	—	2	2	no	—
187	#127	—	2	2	no	—
188	#127	—	2	1	no	—
189	#127	—	4	1	no	—
190	#127	—	4	1	no	—
191	#124	—	4	1	no	—
192	#121	—	2	2	no	—
193	#121	—	4	1	no	—
194	C₄×D₄	—	2	2	no	—
195	C₄×D₄	—	4	1	no	—
196	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
197	D₁₆	—	4	8	no	—
198	C₂×C₁₆	C₂³×C₅	1	4	yes	—
199	#107	—	2	1	no	—
200	#107	C₂³×C₅	1	2	yes	—
201	#102	—	2	2	no	—
202	#102	—	2	2	no	—
203	C₄×C₈	C₅×D₄	1	1	yes	characteristic
204	C₄×C₈	C₅×D₄	1	1	yes	characteristic
205	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	8	2	no	—
210	C₂²×C₅	—	8	4	no	—
211	C₂²×C₅	—	8	4	no	—
212	C₂²×C₅	—	8	2	no	—
213	C₂²×C₅	—	8	4	no	—
214	C₂²×C₅	—	8	2	no	—
215	C₂²×C₅	—	16	8	no	—
216	C₂²×C₅	C₂×D₁₆	1	2	yes	—
217	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	8	2	no	—
221	C₄×C₅	C₂×D₁₆	1	2	yes	—
222	C₄×C₅	C₂×D₁₆	1	2	yes	—
223	C₄×C₅	#509	1	2	yes	—
224	C₄×C₅	#535	1	1	yes	characteristic
225	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₂⁴	—	4	2	no	—
227	#41	—	8	4	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	8	4	no	—
242	C₂×D₄	—	8	4	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
245	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
247	D₈	—	2	4	no	—
248	D₈	—	2	8	no	—
249	D₈	—	4	4	no	—
250	D₈	—	4	4	no	—
251	D₈	—	4	4	no	—
252	D₈	—	4	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
256	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
257	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
258	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	C₅×D₈	1	1	yes	characteristic
264	C₄²	C₅×D₈	1	1	yes	characteristic
265	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
266	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
267	C₁₆	—	2	4	no	—
268	C₂×C₅	—	2	2	no	—
269	C₂×C₅	—	16	2	no	—
270	C₂×C₅	—	16	4	no	—
271	C₂×C₅	#3205	1	1	yes	characteristic
272	C₂×C₅	C₂²×D₁₆	1	1	yes	characteristic
273	C₂×C₅	C₂×#509	1	1	yes	characteristic
274	C₂³	—	4	2	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	8	4	no	—
277	C₂³	—	8	4	no	—
278	C₂³	—	8	4	no	—
279	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
280	Q₈	—	8	2	no	—
281	D₄	—	4	4	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	4	no	—
284	D₄	—	4	2	no	—
285	D₄	—	8	4	no	—
286	D₄	—	8	4	no	—
287	D₄	—	8	4	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	2	no	—
291	D₄	—	8	4	no	—
292	D₄	—	8	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₅×D₁₆	1	4	yes	complemented
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	C₂²×C₅×D₄	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
308	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
309	C₅	#33771	1	1	yes	characteristic, direct product factor, Sylow
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	4	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	4	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	16	8	no	—
319	C₂²	C₂×C₅×D₁₆	1	2	yes	—
320	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	C₂×C₅×D₁₆	1	2	yes	—
325	C₄	C₂×C₅×D₁₆	1	2	yes	—
326	C₄	C₅×#509	1	2	yes	—
327	C₄	C₅×#535	1	1	yes	characteristic
328	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	16	4	no	—
331	C₂	—	16	2	no	—
332	C₂	C₅×#3205	1	1	yes	characteristic
333	C₂	C₂²×C₅×D₁₆	1	1	yes	characteristic
334	C₂	C₂×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
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×#3169	C₂	semidirect
2. C₂×C₅×#493	C₂	semidirect
3. C₅×#2246	C₂	semidirect
4. C₅×#2240	C₂	semidirect
5. C₅×#2206	C₂	semidirect
6. C₅×#2184	C₂	semidirect
7. C₅×#2162	C₂	semidirect
12. C₂×C₅×D₁₆	C₂²	semidirect
13. C₅×#500	C₂²	semidirect
18. C₅×#493	C₂²	semidirect
23. C₄×C₅×D₈	C₂²	semidirect
25. C₅×C₄⋊C₁₆	C₂²	semidirect
26. C₅×#357	C₂²	semidirect
28. C₄×C₅×C₁₆	C₂²	semidirect
29. #33771	C₅	direct (N × H)
66. #3169	C₂×C₅	semidirect
67. C₂×#493	C₂×C₅	semidirect
68. #2246	C₂×C₅	semidirect
69. #2240	C₂×C₅	semidirect
70. #2206	C₂×C₅	semidirect
71. #2184	C₂×C₅	semidirect
72. #2162	C₂×C₅	semidirect
119. C₂×D₁₆	C₂²×C₅	semidirect
120. #500	C₂²×C₅	semidirect
125. #493	C₂²×C₅	semidirect
130. C₄×D₈	C₂²×C₅	semidirect
132. C₄⋊C₁₆	C₂²×C₅	semidirect
133. #357	C₂²×C₅	semidirect
135. C₄×C₁₆	C₂²×C₅	semidirect
160. C₂×C₄×C₅	D₁₆	semidirect
298. C₂×C₄	C₅×D₁₆	semidirect
309. C₅	#33771	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
C₂×C₅×#493	2
C₅×#2246	2
C₅×#2240	2
C₅×#2206	2
C₅×#2184	2
C₅×#2162	2
#33771	5

Subgroups — C5×#33771 #11956491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33771 #11956491