Subgroups — C₅×#33866 #11956586

Order 1280.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3338	C₂	1	1	yes	maximal, characteristic
2	C₅×D₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2236	C₂	1	2	yes	maximal, complemented
5	C₅×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2216	C₂	1	2	yes	maximal, complemented
7	C₅×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2188	C₂	1	2	yes	maximal, complemented
9	C₄×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
11	C₂×C₅×#358	C₂	1	2	yes	maximal, complemented
12	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
13	C₅×#554	—	2	1	no	—
14	C₅×D₄×Q₈	—	2	1	no	—
15	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
17	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
18	C₅×#500	C₂²	1	1	yes	characteristic
19	C₅×#494	C₂²	1	1	yes	characteristic
20	C₅×#484	—	2	2	no	—
21	C₅×#477	—	2	1	no	—
22	C₅×#467	C₂²	1	2	yes	—
23	C₅×#467	C₂²	1	2	yes	—
24	C₅×#462	—	2	1	no	—
25	C₅×#451	—	2	2	no	—
26	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
28	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
29	C₂×C₅×#107	C₂²	1	2	yes	—
30	C₅×#366	C₂²	1	1	yes	characteristic
31	C₅×#366	C₂²	1	2	yes	—
32	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₅×#358	C₂²	1	1	yes	characteristic
34	C₅×#358	C₂²	1	2	yes	—
35	C₅×#358	C₂²	1	2	yes	—
36	C₅×#358	C₂²	1	2	yes	—
37	C₅×#358	C₂²	1	1	yes	characteristic
38	C₅×#358	C₂²	1	2	yes	—
39	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
41	#33866	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂²×C₅×Q₈	—	2	2	no	—
43	C₂×C₅×Q₁₆	—	2	4	no	—
44	C₂×C₅×Q₁₆	—	2	2	no	—
45	C₂×C₅×Q₁₆	—	2	1	no	—
46	C₂×C₅×Q₁₆	—	4	2	no	—
47	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
48	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₂³	1	2	yes	—
51	C₅×C₄⋊Q₈	—	2	1	no	—
52	C₅×C₄⋊Q₈	—	2	1	no	—
53	C₅×C₄⋊Q₈	—	4	1	no	—
54	C₅×#125	—	4	1	no	—
55	C₅×C₂²⋊Q₈	—	2	2	no	—
56	C₅×C₂²⋊Q₈	—	2	2	no	—
57	C₅×C₂²⋊Q₈	—	4	2	no	—
58	C₅×C₂²⋊Q₈	—	4	2	no	—
59	C₄×C₅×Q₈	—	2	1	no	—
60	C₄×C₅×Q₈	—	2	1	no	—
61	C₄×C₅×D₄	—	4	1	no	—
62	C₄×C₅×D₄	—	4	1	no	—
63	C₄×C₅×D₄	D₄	1	1	yes	characteristic
64	C₂×C₅×C₄⋊C₄	—	2	2	no	—
65	C₅×Q₃₂	—	2	2	no	—
66	C₅×Q₃₂	—	4	2	no	—
67	C₂×C₅×C₁₆	—	4	2	no	—
68	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₅×C₁₆	C₂³	1	2	yes	—
71	C₅×#107	—	2	2	no	—
72	C₅×#107	—	2	1	no	—
73	C₅×#107	C₂³	1	1	yes	characteristic
74	C₅×#107	C₂³	1	2	yes	—
75	C₅×#107	C₂³	1	1	yes	characteristic
76	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
77	C₅×#103	—	2	1	no	—
78	C₅×#103	—	2	1	no	—
79	C₅×#103	—	2	2	no	—
80	C₅×#103	—	2	2	no	—
81	C₅×C₂²⋊C₈	D₄	1	2	yes	—
82	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
83	#3338	C₂×C₅	1	1	yes	characteristic
84	D₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
85	#2241	C₂×C₅	1	1	yes	characteristic, complemented
86	#2236	C₂×C₅	1	2	yes	complemented
87	#2227	C₂×C₅	1	1	yes	characteristic, complemented
88	#2216	C₂×C₅	1	2	yes	complemented
89	#2209	C₂×C₅	1	1	yes	characteristic, complemented
90	#2188	C₂×C₅	1	2	yes	complemented
91	C₄×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
92	D₄×C₁₆	C₂×C₅	1	1	yes	characteristic
93	C₂×#358	C₂×C₅	1	2	yes	complemented
94	C₂×C₅×Q₈	—	2	1	no	—
95	C₂×C₅×Q₈	—	2	1	no	—
96	C₂×C₅×Q₈	—	2	2	no	—
97	C₂×C₅×Q₈	—	4	4	no	—
98	C₂×C₅×Q₈	—	4	2	no	—
99	C₂×C₅×Q₈	—	4	1	no	—
100	C₂×C₅×Q₈	—	8	2	no	—
101	C₂×C₅×D₄	D₈	1	1	yes	characteristic
102	C₂²×C₄×C₅	—	4	2	no	—
103	C₂²×C₄×C₅	—	4	2	no	—
104	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
105	C₅×Q₁₆	—	2	2	no	—
106	C₅×Q₁₆	—	2	1	no	—
107	C₅×Q₁₆	—	2	1	no	—
108	C₅×Q₁₆	—	2	2	no	—
109	C₅×Q₁₆	—	2	1	no	—
110	C₅×Q₁₆	—	4	4	no	—
111	C₅×Q₁₆	—	8	1	no	—
112	C₅×Q₁₆	#41	1	2	yes	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
118	C₅×C₄⋊C₄	—	2	1	no	—
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	—	2	1	no	—
121	C₅×C₄⋊C₄	—	4	2	no	—
122	C₅×C₄⋊C₄	—	4	1	no	—
123	C₅×C₄⋊C₄	—	4	1	no	—
124	C₅×C₄⋊C₄	—	4	1	no	—
125	C₅×C₄⋊C₄	—	4	1	no	—
126	C₅×C₄⋊C₄	—	4	2	no	—
127	C₅×C₄⋊C₄	—	4	2	no	—
128	C₅×C₄⋊C₄	—	4	1	no	—
129	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
130	C₅×C₂²⋊C₄	—	4	2	no	—
131	C₅×C₂²⋊C₄	—	4	2	no	—
132	C₅×C₂²⋊C₄	D₈	1	2	yes	—
133	C₄²×C₅	—	4	1	no	—
134	C₄²×C₅	—	4	1	no	—
135	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
136	C₅×C₁₆	—	2	1	no	—
137	C₅×C₁₆	—	2	1	no	—
138	C₅×C₁₆	—	2	2	no	—
139	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
140	#554	—	2	1	no	—
141	D₄×Q₈	—	2	1	no	—
142	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic
143	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
144	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
145	#500	C₂²×C₅	1	1	yes	characteristic
146	#494	C₂²×C₅	1	1	yes	characteristic
147	#484	—	2	2	no	—
148	#477	—	2	1	no	—
149	#467	C₂²×C₅	1	2	yes	—
150	#467	C₂²×C₅	1	2	yes	—
151	#462	—	2	1	no	—
152	#451	—	2	2	no	—
153	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
154	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
155	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
156	C₂×#107	C₂²×C₅	1	2	yes	—
157	#366	C₂²×C₅	1	1	yes	characteristic
158	#366	C₂²×C₅	1	2	yes	—
159	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
160	#358	C₂²×C₅	1	1	yes	characteristic
161	#358	C₂²×C₅	1	2	yes	—
162	#358	C₂²×C₅	1	2	yes	—
163	#358	C₂²×C₅	1	2	yes	—
164	#358	C₂²×C₅	1	1	yes	characteristic
165	#358	C₂²×C₅	1	2	yes	—
166	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
167	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
168	C₂³×C₅	C₂×D₈	1	2	yes	—
169	C₅×Q₈	—	2	2	no	—
170	C₅×Q₈	—	4	1	no	—
171	C₅×Q₈	—	4	2	no	—
172	C₅×Q₈	—	4	1	no	—
173	C₅×Q₈	—	4	2	no	—
174	C₅×Q₈	—	4	1	no	—
175	C₅×Q₈	—	8	4	no	—
176	C₅×Q₈	—	16	1	no	—
177	C₅×D₄	Q₃₂	1	4	yes	complemented
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	4	1	no	—
181	C₂×C₄×C₅	—	4	1	no	—
182	C₂×C₄×C₅	—	4	1	no	—
183	C₂×C₄×C₅	—	4	1	no	—
184	C₂×C₄×C₅	—	4	2	no	—
185	C₂×C₄×C₅	—	4	2	no	—
186	C₂×C₄×C₅	—	8	4	no	—
187	C₂×C₄×C₅	—	16	2	no	—
188	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
190	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
192	C₅×C₈	—	2	2	no	—
193	C₅×C₈	—	2	1	no	—
194	C₅×C₈	C₂×#41	1	1	yes	characteristic
195	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
196	C₂²×Q₈	—	2	2	no	—
197	C₂×Q₁₆	—	2	4	no	—
198	C₂×Q₁₆	—	2	2	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	4	2	no	—
201	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
202	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
203	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₅	1	2	yes	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	4	1	no	—
208	#125	—	4	1	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	4	2	no	—
212	C₂²⋊Q₈	—	4	2	no	—
213	C₄×Q₈	—	2	1	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×D₄	—	4	1	no	—
216	C₄×D₄	—	4	1	no	—
217	C₄×D₄	C₅×D₄	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	2	2	no	—
219	Q₃₂	—	2	2	no	—
220	Q₃₂	—	4	2	no	—
221	C₂×C₁₆	—	4	2	no	—
222	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
223	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
224	C₂×C₁₆	C₂³×C₅	1	2	yes	—
225	#107	—	2	2	no	—
226	#107	—	2	1	no	—
227	#107	C₂³×C₅	1	1	yes	characteristic
228	#107	C₂³×C₅	1	2	yes	—
229	#107	C₂³×C₅	1	1	yes	characteristic
230	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
231	#103	—	2	1	no	—
232	#103	—	2	1	no	—
233	#103	—	2	2	no	—
234	#103	—	2	2	no	—
235	C₂²⋊C₈	C₅×D₄	1	2	yes	—
236	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
237	C₂²×C₅	—	2	2	no	—
238	C₂²×C₅	—	2	2	no	—
239	C₂²×C₅	C₂×Q₃₂	1	4	yes	—
240	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	2	no	—
243	C₄×C₅	—	4	2	no	—
244	C₄×C₅	—	8	1	no	—
245	C₄×C₅	—	8	1	no	—
246	C₄×C₅	—	8	2	no	—
247	C₄×C₅	—	8	2	no	—
248	C₄×C₅	—	8	1	no	—
249	C₄×C₅	C₂×Q₃₂	1	2	yes	—
250	C₄×C₅	#546	1	1	yes	characteristic
251	C₄×C₅	#577	1	1	yes	characteristic
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	4	4	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	8	2	no	—
259	C₂×D₄	C₅×D₈	1	1	yes	characteristic
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
263	Q₁₆	—	2	1	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	4	4	no	—
269	Q₁₆	—	8	1	no	—
270	Q₁₆	C₅×#41	1	2	yes	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
274	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
275	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	C₅×D₈	1	2	yes	—
291	C₄²	—	4	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
294	C₁₆	—	2	1	no	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	2	no	—
297	C₂×C₅	—	2	4	no	—
298	C₂×C₅	#3294	1	1	yes	characteristic
299	C₂×C₅	C₂²×Q₃₂	1	1	yes	characteristic
300	C₂×C₅	#3416	1	1	yes	characteristic
301	C₂³	C₂×C₅×D₈	1	2	yes	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	4	1	no	—
308	Q₈	—	8	4	no	—
309	Q₈	—	16	1	no	—
310	D₄	C₅×Q₃₂	1	4	yes	complemented
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	—	16	2	no	—
321	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
322	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
323	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	1	no	—
327	C₈	C₂×C₅×#41	1	1	yes	characteristic
328	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
329	C₅	#33866	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	C₂×C₅×Q₃₂	1	4	yes	—
333	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	1	no	—
336	C₄	—	4	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	1	no	—
339	C₄	—	8	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	1	no	—
342	C₄	C₂×C₅×Q₃₂	1	2	yes	—
343	C₄	C₅×#546	1	1	yes	characteristic
344	C₄	C₅×#577	1	1	yes	characteristic
345	C₂	—	2	4	no	—
346	C₂	C₅×#3294	1	1	yes	characteristic
347	C₂	C₂²×C₅×Q₃₂	1	1	yes	characteristic
348	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	C₂×C₅×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄ ◁ C₅×C₈×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2241	C₂	semidirect
4. C₅×#2236	C₂	semidirect
5. C₅×#2227	C₂	semidirect
6. C₅×#2216	C₂	semidirect
7. C₅×#2209	C₂	semidirect
8. C₅×#2188	C₂	semidirect
9. C₄×C₅×Q₃₂	C₂	semidirect
11. C₂×C₅×#358	C₂	semidirect
41. #33866	C₅	direct (N × H)
85. #2241	C₂×C₅	semidirect
86. #2236	C₂×C₅	semidirect
87. #2227	C₂×C₅	semidirect
88. #2216	C₂×C₅	semidirect
89. #2209	C₂×C₅	semidirect
90. #2188	C₂×C₅	semidirect
91. C₄×Q₃₂	C₂×C₅	semidirect
93. C₂×#358	C₂×C₅	semidirect
177. C₅×D₄	Q₃₂	semidirect
310. D₄	C₅×Q₃₂	semidirect
329. C₅	#33866	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×#3338	2
C₅×D₄×Q₁₆	2
C₅×#2241	2
C₅×#2236	2
C₅×#2227	2
C₅×#2216	2
C₅×#2209	2
C₅×#2188	2
C₄×C₅×Q₃₂	2
C₅×D₄×C₁₆	2
C₂×C₅×#358	2
#33866	5

Subgroups — C5×#33866 #11956586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33866 #11956586