Subgroups — C₅×#33588 #11956308

Order 1280.

680 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#510	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3156	C₂	1	1	yes	maximal, characteristic
4	C₅×#2950	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2211	C₂	1	4	yes	maximal, complemented
6	C₅×#2209	C₂	1	4	yes	maximal, complemented
7	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2138	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×#137	—	2	1	no	—
11	C₂×C₅×#135	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
14	C₅×#536	—	2	1	no	—
15	C₅×#519	—	2	1	no	—
16	C₅×#510	—	2	4	no	—
17	C₂×C₅×Q₃₂	—	2	4	no	—
18	C₂×C₅×Q₃₂	C₂²	1	4	yes	—
19	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
20	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
21	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
22	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
23	C₅×#497	—	2	2	no	—
24	C₅×#495	C₂²	1	2	yes	—
25	C₅×#494	C₂²	1	2	yes	—
26	C₅×#441	—	2	2	no	—
27	C₄×C₅×Q₁₆	C₂²	1	4	yes	—
28	C₅×#427	C₂²	1	1	yes	characteristic
29	C₅×#419	—	2	1	no	—
30	C₅×#405	C₂²	1	1	yes	characteristic
31	C₅×C₄⋊C₁₆	C₂²	1	4	yes	—
32	C₅×#358	C₂²	1	2	yes	—
33	C₅×#358	C₂²	1	4	yes	—
34	C₅×#357	C₂²	1	2	yes	—
35	#33588	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅×#41	—	2	1	no	—
37	C₂²×C₅×Q₈	—	2	1	no	—
38	C₂²×C₅×Q₈	—	2	1	no	—
39	C₅×#137	—	4	4	no	—
40	C₅×#135	—	2	2	no	—
41	C₅×#135	—	2	2	no	—
42	C₂×C₅×Q₁₆	—	2	1	no	—
43	C₂×C₅×Q₁₆	—	2	2	no	—
44	C₂×C₅×Q₁₆	—	2	2	no	—
45	C₂×C₅×Q₁₆	—	2	1	no	—
46	C₂×C₅×Q₁₆	—	2	2	no	—
47	C₂×C₅×Q₁₆	—	2	2	no	—
48	C₂×C₅×Q₁₆	D₄	1	4	yes	—
49	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
50	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
51	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
52	C₂×C₅×SD₁₆	—	2	2	no	—
53	C₂×C₅×SD₁₆	—	2	1	no	—
54	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
55	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
56	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
57	C₅×C₄⋊Q₈	—	2	2	no	—
58	C₅×#124	—	2	2	no	—
59	C₅×#123	—	4	2	no	—
60	C₅×C₂²⋊Q₈	—	4	2	no	—
61	C₄×C₅×Q₈	—	2	4	no	—
62	C₄×C₅×Q₈	—	4	2	no	—
63	C₅×#117	—	2	1	no	—
64	C₅×#117	—	4	2	no	—
65	C₅×#117	D₄	1	1	yes	characteristic
66	C₅×Q₃₂	—	2	8	no	—
67	C₅×Q₃₂	—	2	4	no	—
68	C₅×SD₃₂	—	2	4	no	—
69	C₅×M₃₂	—	2	2	no	—
70	C₂×C₅×C₁₆	—	2	2	no	—
71	C₂×C₅×C₁₆	C₂³	1	2	yes	—
72	C₂×C₅×C₁₆	C₂³	1	2	yes	—
73	C₅×#107	C₂³	1	2	yes	—
74	C₅×#106	—	2	1	no	—
75	C₅×#103	—	2	4	no	—
76	C₅×#97	D₄	1	2	yes	—
77	C₄×C₅×C₈	C₂³	1	2	yes	—
78	C₂×#510	C₂×C₅	1	1	yes	characteristic
79	C₂²×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
80	#3156	C₂×C₅	1	1	yes	characteristic
81	#2950	C₂×C₅	1	1	yes	characteristic, complemented
82	#2211	C₂×C₅	1	4	yes	complemented
83	#2209	C₂×C₅	1	4	yes	complemented
84	#2151	C₂×C₅	1	1	yes	characteristic, complemented
85	#2141	C₂×C₅	1	1	yes	characteristic, complemented
86	#2138	C₂×C₅	1	1	yes	characteristic
87	C₅×#41	—	4	2	no	—
88	C₅×#41	—	4	2	no	—
89	C₂×C₅×Q₈	—	2	4	no	—
90	C₂×C₅×Q₈	—	2	2	no	—
91	C₂×C₅×Q₈	—	2	2	no	—
92	C₂×C₅×Q₈	—	2	1	no	—
93	C₂×C₅×Q₈	—	4	1	no	—
94	C₂×C₅×Q₈	—	4	1	no	—
95	C₂×C₅×Q₈	—	4	2	no	—
96	C₂×C₅×Q₈	—	4	2	no	—
97	C₂×C₅×Q₈	—	4	2	no	—
98	C₂×C₅×D₄	—	2	1	no	—
99	C₂×C₅×D₄	—	4	1	no	—
100	C₂²×C₄×C₅	—	4	1	no	—
101	C₂²×C₄×C₅	—	4	1	no	—
102	C₂²×C₄×C₅	—	4	1	no	—
103	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
104	C₅×Q₁₆	—	2	4	no	—
105	C₅×Q₁₆	—	2	2	no	—
106	C₅×Q₁₆	—	2	1	no	—
107	C₅×Q₁₆	—	2	2	no	—
108	C₅×Q₁₆	—	2	2	no	—
109	C₅×Q₁₆	—	2	1	no	—
110	C₅×Q₁₆	—	4	4	no	—
111	C₅×Q₁₆	—	4	2	no	—
112	C₅×Q₁₆	C₂×D₄	1	4	yes	—
113	C₅×SD₁₆	—	4	4	no	—
114	C₅×SD₁₆	—	4	2	no	—
115	C₅×D₈	—	2	1	no	—
116	C₅×D₈	—	2	1	no	—
117	C₅×M₁₆	—	2	2	no	—
118	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
119	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
121	C₂×C₅×C₈	#41	1	2	yes	—
122	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	—	4	4	no	—
125	C₅×C₄⋊C₄	—	4	4	no	—
126	C₅×C₄⋊C₄	D₈	1	2	yes	—
127	C₅×C₂²⋊C₄	—	4	1	no	—
128	C₅×C₂²⋊C₄	—	4	4	no	—
129	C₅×C₂²⋊C₄	D₈	1	2	yes	—
130	C₄²×C₅	—	4	1	no	—
131	C₄²×C₅	—	4	4	no	—
132	C₄²×C₅	C₂×D₄	1	2	yes	—
133	C₅×C₁₆	—	2	2	no	—
134	C₅×C₁₆	—	2	2	no	—
135	C₂×#137	—	2	1	no	—
136	C₂×#135	C₂²×C₅	1	1	yes	characteristic
137	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
138	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
139	#536	—	2	1	no	—
140	#519	—	2	1	no	—
141	#510	—	2	4	no	—
142	C₂×Q₃₂	—	2	4	no	—
143	C₂×Q₃₂	C₂²×C₅	1	4	yes	—
144	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
145	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
146	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
147	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
148	#497	—	2	2	no	—
149	#495	C₂²×C₅	1	2	yes	—
150	#494	C₂²×C₅	1	2	yes	—
151	#441	—	2	2	no	—
152	C₄×Q₁₆	C₂²×C₅	1	4	yes	—
153	#427	C₂²×C₅	1	1	yes	characteristic
154	#419	—	2	1	no	—
155	#405	C₂²×C₅	1	1	yes	characteristic
156	C₄⋊C₁₆	C₂²×C₅	1	4	yes	—
157	#358	C₂²×C₅	1	2	yes	—
158	#358	C₂²×C₅	1	4	yes	—
159	#357	C₂²×C₅	1	2	yes	—
160	C₂³×C₅	—	4	1	no	—
161	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
162	C₅×Q₈	—	2	4	no	—
163	C₅×Q₈	—	4	2	no	—
164	C₅×Q₈	—	4	4	no	—
165	C₅×Q₈	—	4	1	no	—
166	C₅×Q₈	—	4	2	no	—
167	C₅×Q₈	—	4	2	no	—
168	C₅×Q₈	—	4	1	no	—
169	C₅×Q₈	—	8	2	no	—
170	C₅×D₄	—	4	1	no	—
171	C₅×D₄	—	4	1	no	—
172	C₅×D₄	—	8	2	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	4	4	no	—
175	C₂×C₄×C₅	—	4	2	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	—	8	2	no	—
179	C₂×C₄×C₅	—	8	2	no	—
180	C₂×C₄×C₅	#121	1	2	yes	—
181	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
182	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
183	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
184	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
185	C₅×C₈	—	2	2	no	—
186	C₅×C₈	#121	1	2	yes	—
187	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
188	C₅×C₈	C₂×#41	1	1	yes	characteristic
189	C₂×#41	—	2	1	no	—
190	C₂²×Q₈	—	2	1	no	—
191	C₂²×Q₈	—	2	1	no	—
192	#137	—	4	4	no	—
193	#135	—	2	2	no	—
194	#135	—	2	2	no	—
195	C₂×Q₁₆	—	2	1	no	—
196	C₂×Q₁₆	—	2	2	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	—	2	2	no	—
199	C₂×Q₁₆	—	2	2	no	—
200	C₂×Q₁₆	—	2	2	no	—
201	C₂×Q₁₆	C₅×D₄	1	4	yes	—
202	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
203	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
204	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
205	C₂×SD₁₆	—	2	1	no	—
206	C₂×SD₁₆	—	2	2	no	—
207	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
208	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	2	no	—
211	#124	—	2	2	no	—
212	#123	—	4	2	no	—
213	C₂²⋊Q₈	—	4	2	no	—
214	C₄×Q₈	—	2	4	no	—
215	C₄×Q₈	—	4	2	no	—
216	#117	—	2	1	no	—
217	#117	—	4	2	no	—
218	#117	C₅×D₄	1	1	yes	characteristic
219	Q₃₂	—	2	8	no	—
220	Q₃₂	—	2	4	no	—
221	SD₃₂	—	2	4	no	—
222	M₃₂	—	2	2	no	—
223	C₂×C₁₆	—	2	2	no	—
224	C₂×C₁₆	C₂³×C₅	1	2	yes	—
225	C₂×C₁₆	C₂³×C₅	1	2	yes	—
226	#107	C₂³×C₅	1	2	yes	—
227	#106	—	2	1	no	—
228	#103	—	2	4	no	—
229	#97	C₅×D₄	1	2	yes	—
230	C₄×C₈	C₂³×C₅	1	2	yes	—
231	C₂²×C₅	—	2	2	no	—
232	C₂²×C₅	—	8	2	no	—
233	C₂²×C₅	—	8	1	no	—
234	C₂²×C₅	#459	1	2	yes	—
235	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
236	C₄×C₅	—	2	4	no	—
237	C₄×C₅	—	4	4	no	—
238	C₄×C₅	—	8	2	no	—
239	C₄×C₅	—	8	1	no	—
240	C₄×C₅	—	8	2	no	—
241	C₄×C₅	#459	1	2	yes	—
242	C₄×C₅	C₂×#121	1	1	yes	characteristic
243	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
244	#41	—	4	2	no	—
245	#41	—	4	2	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
261	Q₁₆	—	2	4	no	—
262	Q₁₆	—	2	2	no	—
263	Q₁₆	—	2	1	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	2	2	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	4	4	no	—
268	Q₁₆	—	4	2	no	—
269	Q₁₆	C₂×C₅×D₄	1	4	yes	—
270	SD₁₆	—	4	4	no	—
271	SD₁₆	—	4	2	no	—
272	D₈	—	2	1	no	—
273	D₈	—	2	1	no	—
274	M₁₆	—	2	2	no	—
275	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
276	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
277	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
278	C₂×C₈	C₅×#41	1	2	yes	—
279	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₄⋊C₄	—	4	4	no	—
283	C₄⋊C₄	C₅×D₈	1	2	yes	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	C₅×D₈	1	2	yes	—
287	C₄²	—	4	1	no	—
288	C₄²	—	4	4	no	—
289	C₄²	C₂×C₅×D₄	1	2	yes	—
290	C₁₆	—	2	2	no	—
291	C₁₆	—	2	2	no	—
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	—	16	1	no	—
294	C₂×C₅	C₂×#459	1	1	yes	characteristic
295	C₂×C₅	#3417	1	2	yes	—
296	C₂³	—	4	1	no	—
297	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
298	Q₈	—	2	4	no	—
299	Q₈	—	4	4	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	4	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	8	2	no	—
306	D₄	—	4	1	no	—
307	D₄	—	4	1	no	—
308	D₄	—	8	2	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	C₅×#121	1	2	yes	—
317	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
318	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
319	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
320	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	C₅×#121	1	2	yes	—
323	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
324	C₈	C₂×C₅×#41	1	1	yes	characteristic
325	C₅	#33588	1	1	yes	characteristic, direct product factor, Sylow
326	C₂²	—	2	2	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	C₅×#459	1	2	yes	—
330	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
331	C₄	—	2	4	no	—
332	C₄	—	4	4	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	C₅×#459	1	2	yes	—
337	C₄	C₂×C₅×#121	1	1	yes	characteristic
338	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	16	1	no	—
341	C₂	C₂×C₅×#459	1	1	yes	characteristic
342	C₂	C₅×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#135	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
2. C₂²×C₅×Q₃₂	C₂	semidirect
4. C₅×#2950	C₂	semidirect
5. C₅×#2211	C₂	semidirect
6. C₅×#2209	C₂	semidirect
7. C₅×#2151	C₂	semidirect
8. C₅×#2141	C₂	semidirect
35. #33588	C₅	direct (N × H)
79. C₂²×Q₃₂	C₂×C₅	semidirect
81. #2950	C₂×C₅	semidirect
82. #2211	C₂×C₅	semidirect
83. #2209	C₂×C₅	semidirect
84. #2151	C₂×C₅	semidirect
85. #2141	C₂×C₅	semidirect
325. C₅	#33588	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#510	2
C₂²×C₅×Q₃₂	2
C₅×#3156	2
C₅×#2950	2
C₅×#2211	2
C₅×#2209	2
C₅×#2151	2
C₅×#2141	2
C₅×#2138	2
#33588	5

Subgroups — C5×#33588 #11956308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33588 #11956308