Subgroups — C₅×#33629 #11956349

Order 1280.

574 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic
3	C₅×#2227	C₂	1	2	yes	maximal, complemented
4	C₅×#2226	C₂	1	2	yes	maximal, complemented
5	C₅×#2225	C₂	1	2	yes	maximal, complemented
6	C₅×#2224	C₂	1	2	yes	maximal, complemented
7	C₅×#2159	C₂	1	2	yes	maximal, complemented
8	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2139	C₂	1	2	yes	maximal
10	C₂×C₅×#135	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
12	C₄×C₅×#41	—	2	1	no	—
13	C₂×C₅×M₃₂	C₂²	1	2	yes	—
14	C₅×#500	—	2	1	no	—
15	C₅×#500	—	2	1	no	—
16	C₅×#500	C₂²	1	2	yes	—
17	C₅×#500	C₂²	1	2	yes	—
18	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
19	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₅×SD₁₆	—	2	1	no	—
21	C₄×C₅×SD₁₆	—	2	1	no	—
22	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
23	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
24	C₅×#427	C₂²	1	1	yes	characteristic
25	C₂×C₅×#107	—	2	1	no	—
26	C₂×C₅×#107	C₂²	1	2	yes	—
27	C₅×#416	—	2	1	no	—
28	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	C₅×#367	C₂²	1	4	yes	—
30	C₅×#366	C₂²	1	4	yes	—
31	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₅×#358	C₂²	1	4	yes	—
34	C₅×#357	C₂²	1	4	yes	—
35	#33629	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅×#41	—	2	1	no	—
37	C₂²×C₅×Q₈	—	4	1	no	—
38	C₅×#135	—	2	2	no	—
39	C₅×#135	Q₈	1	4	yes	complemented
40	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₅×SD₁₆	—	2	1	no	—
42	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₅×C₄⋊Q₈	—	2	2	no	—
46	C₅×C₄⋊Q₈	—	2	2	no	—
47	C₅×C₄⋊Q₈	—	4	1	no	—
48	C₅×C₄⋊Q₈	—	4	1	no	—
49	C₄×C₅×Q₈	—	2	1	no	—
50	C₄×C₅×Q₈	—	2	1	no	—
51	C₄×C₅×D₄	—	2	1	no	—
52	C₄×C₅×D₄	—	2	1	no	—
53	C₄×C₅×D₄	—	4	1	no	—
54	C₄×C₅×D₄	—	4	1	no	—
55	C₅×#117	—	2	1	no	—
56	C₅×#117	—	4	1	no	—
57	C₂×C₅×C₄⋊C₄	—	2	2	no	—
58	C₂×C₅×C₄⋊C₄	—	4	1	no	—
59	C₂×C₄²×C₅	—	4	1	no	—
60	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
61	C₅×M₃₂	—	2	4	no	—
62	C₂×C₅×C₁₆	C₂³	1	4	yes	—
63	C₅×#107	—	2	2	no	—
64	C₅×#107	—	2	2	no	—
65	C₅×#107	—	2	2	no	—
66	C₅×#107	C₂³	1	4	yes	—
67	C₅×#107	C₂³	1	1	yes	characteristic
68	C₅×#107	C₂³	1	1	yes	characteristic
69	C₅×#106	—	2	1	no	—
70	C₅×#103	—	2	1	no	—
71	C₅×#103	—	2	1	no	—
72	C₅×#102	—	2	1	no	—
73	C₅×#102	—	2	1	no	—
74	C₄×C₅×C₈	D₄	1	1	yes	characteristic
75	C₄×C₅×C₈	D₄	1	1	yes	characteristic
76	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₂×#500	C₂×C₅	1	1	yes	characteristic, complemented
79	C₄×#135	C₂×C₅	1	1	yes	characteristic
80	#2227	C₂×C₅	1	2	yes	complemented
81	#2226	C₂×C₅	1	2	yes	complemented
82	#2225	C₂×C₅	1	2	yes	complemented
83	#2224	C₂×C₅	1	2	yes	complemented
84	#2159	C₂×C₅	1	2	yes	complemented
85	#2150	C₂×C₅	1	1	yes	characteristic, complemented
86	#2139	C₂×C₅	1	2	yes	—
87	C₅×#41	—	2	4	no	—
88	C₅×#41	—	4	2	no	—
89	C₂×C₅×Q₈	—	2	1	no	—
90	C₂×C₅×Q₈	—	4	2	no	—
91	C₂×C₅×Q₈	—	4	2	no	—
92	C₂×C₅×Q₈	—	8	4	no	—
93	C₂×C₅×D₄	—	2	1	no	—
94	C₂×C₅×D₄	—	4	1	no	—
95	C₂²×C₄×C₅	—	4	1	no	—
96	C₂²×C₄×C₅	—	4	2	no	—
97	C₂²×C₄×C₅	—	4	1	no	—
98	C₂²×C₄×C₅	—	4	1	no	—
99	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
100	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
102	C₅×Q₁₆	—	2	1	no	—
103	C₅×Q₁₆	C₂×Q₈	1	2	yes	complemented
104	C₅×SD₁₆	—	2	4	no	—
105	C₅×D₈	—	2	1	no	—
106	C₅×D₈	C₂×Q₈	1	2	yes	complemented
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
112	C₂×C₅×C₈	#41	1	2	yes	—
113	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
114	C₅×C₄⋊C₄	—	2	4	no	—
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₄⋊C₄	—	4	2	no	—
118	C₅×C₄⋊C₄	—	4	2	no	—
119	C₅×C₄⋊C₄	—	4	2	no	—
120	C₅×C₄⋊C₄	—	4	1	no	—
121	C₅×C₄⋊C₄	—	4	1	no	—
122	C₅×C₂²⋊C₄	—	4	1	no	—
123	C₅×C₂²⋊C₄	—	4	1	no	—
124	C₅×C₂²⋊C₄	—	4	1	no	—
125	C₄²×C₅	—	4	1	no	—
126	C₄²×C₅	—	4	1	no	—
127	C₄²×C₅	—	4	1	no	—
128	C₄²×C₅	D₈	1	1	yes	characteristic
129	C₄²×C₅	D₈	1	1	yes	characteristic
130	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
132	C₅×C₁₆	—	2	4	no	—
133	C₂×#135	C₂²×C₅	1	1	yes	characteristic
134	C₂×C₄⋊Q₈	—	2	1	no	—
135	C₄×#41	—	2	1	no	—
136	C₂×M₃₂	C₂²×C₅	1	2	yes	—
137	#500	—	2	1	no	—
138	#500	—	2	1	no	—
139	#500	C₂²×C₅	1	2	yes	—
140	#500	C₂²×C₅	1	2	yes	—
141	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
142	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
143	C₄×SD₁₆	—	2	1	no	—
144	C₄×SD₁₆	—	2	1	no	—
145	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
146	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
147	#427	C₂²×C₅	1	1	yes	characteristic
148	C₂×#107	—	2	1	no	—
149	C₂×#107	C₂²×C₅	1	2	yes	—
150	#416	—	2	1	no	—
151	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
152	#367	C₂²×C₅	1	4	yes	—
153	#366	C₂²×C₅	1	4	yes	—
154	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
155	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
156	#358	C₂²×C₅	1	4	yes	—
157	#357	C₂²×C₅	1	4	yes	—
158	C₂³×C₅	—	4	1	no	—
159	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
160	C₅×Q₈	—	2	2	no	—
161	C₅×Q₈	—	4	1	no	—
162	C₅×Q₈	—	8	4	no	—
163	C₅×Q₈	—	8	4	no	—
164	C₅×D₄	—	2	2	no	—
165	C₅×D₄	—	4	4	no	—
166	C₅×D₄	—	4	1	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	—	4	4	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	—	4	1	no	—
173	C₂×C₄×C₅	—	4	4	no	—
174	C₂×C₄×C₅	—	8	4	no	—
175	C₂×C₄×C₅	—	8	2	no	—
176	C₂×C₄×C₅	—	8	2	no	—
177	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
178	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
183	C₂×C₄×C₅	#137	1	2	yes	—
184	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
185	C₅×C₈	—	2	1	no	—
186	C₅×C₈	—	2	1	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	C₂×#41	—	2	1	no	—
191	C₂²×Q₈	—	4	1	no	—
192	#135	—	2	2	no	—
193	#135	C₅×Q₈	1	4	yes	complemented
194	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
195	C₂×SD₁₆	—	2	1	no	—
196	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
197	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	4	1	no	—
202	C₄⋊Q₈	—	4	1	no	—
203	C₄×Q₈	—	2	1	no	—
204	C₄×Q₈	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	—	2	1	no	—
207	C₄×D₄	—	4	1	no	—
208	C₄×D₄	—	4	1	no	—
209	#117	—	2	1	no	—
210	#117	—	4	1	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	4	1	no	—
213	C₂×C₄²	—	4	1	no	—
214	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
215	M₃₂	—	2	4	no	—
216	C₂×C₁₆	C₂³×C₅	1	4	yes	—
217	#107	—	2	2	no	—
218	#107	—	2	2	no	—
219	#107	—	2	2	no	—
220	#107	C₂³×C₅	1	4	yes	—
221	#107	C₂³×C₅	1	1	yes	characteristic
222	#107	C₂³×C₅	1	1	yes	characteristic
223	#106	—	2	1	no	—
224	#103	—	2	1	no	—
225	#103	—	2	1	no	—
226	#102	—	2	1	no	—
227	#102	—	2	1	no	—
228	C₄×C₈	C₅×D₄	1	1	yes	characteristic
229	C₄×C₈	C₅×D₄	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	4	2	no	—
235	C₂²×C₅	—	8	1	no	—
236	C₂²×C₅	—	8	1	no	—
237	C₂²×C₅	#474	1	2	yes	—
238	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	4	2	no	—
242	C₄×C₅	—	8	1	no	—
243	C₄×C₅	—	8	4	no	—
244	C₄×C₅	—	8	1	no	—
245	C₄×C₅	#474	1	2	yes	—
246	C₄×C₅	#509	1	2	yes	—
247	C₄×C₅	#510	1	2	yes	—
248	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
249	C₄×C₅	C₂×#137	1	1	yes	characteristic
250	#41	—	2	4	no	—
251	#41	—	4	2	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×Q₈	—	8	4	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
263	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
265	Q₁₆	—	2	1	no	—
266	Q₁₆	C₂×C₅×Q₈	1	2	yes	complemented
267	SD₁₆	—	2	4	no	—
268	D₈	—	2	1	no	—
269	D₈	C₂×C₅×Q₈	1	2	yes	complemented
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
272	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
273	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
274	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
275	C₂×C₈	C₅×#41	1	2	yes	—
276	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	4	1	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	4	1	no	—
291	C₄²	C₅×D₈	1	1	yes	characteristic
292	C₄²	C₅×D₈	1	1	yes	characteristic
293	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
294	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
295	C₁₆	—	2	4	no	—
296	C₂×C₅	—	2	2	no	—
297	C₂×C₅	—	8	2	no	—
298	C₂×C₅	C₂×#474	1	1	yes	characteristic
299	C₂×C₅	C₂×#509	1	1	yes	characteristic
300	C₂×C₅	C₂×#510	1	1	yes	characteristic
301	C₂³	—	4	1	no	—
302	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
303	Q₈	—	2	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	8	4	no	—
306	Q₈	—	8	4	no	—
307	D₄	—	2	2	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
321	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
322	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
323	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
324	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
325	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
326	C₂×C₄	C₅×#137	1	2	yes	—
327	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
331	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
332	C₈	C₂×C₅×#41	1	1	yes	characteristic
333	C₅	#33629	1	1	yes	characteristic, direct product factor, Sylow
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₅×#474	1	2	yes	—
340	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	4	2	no	—
344	C₄	—	8	4	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	C₅×#474	1	2	yes	—
348	C₄	C₅×#509	1	2	yes	—
349	C₄	C₅×#510	1	2	yes	—
350	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
351	C₄	C₂×C₅×#137	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	8	2	no	—
354	C₂	C₂×C₅×#474	1	1	yes	characteristic
355	C₂	C₂×C₅×#509	1	1	yes	characteristic
356	C₂	C₂×C₅×#510	1	1	yes	characteristic

	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 ◁ 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. C₂×C₅×#500	C₂	semidirect
3. C₅×#2227	C₂	semidirect
4. C₅×#2226	C₂	semidirect
5. C₅×#2225	C₂	semidirect
6. C₅×#2224	C₂	semidirect
7. C₅×#2159	C₂	semidirect
8. C₅×#2150	C₂	semidirect
35. #33629	C₅	direct (N × H)
39. C₅×#135	Q₈	semidirect
78. C₂×#500	C₂×C₅	semidirect
80. #2227	C₂×C₅	semidirect
81. #2226	C₂×C₅	semidirect
82. #2225	C₂×C₅	semidirect
83. #2224	C₂×C₅	semidirect
84. #2159	C₂×C₅	semidirect
85. #2150	C₂×C₅	semidirect
103. C₅×Q₁₆	C₂×Q₈	semidirect
106. C₅×D₈	C₂×Q₈	semidirect
193. #135	C₅×Q₈	semidirect
266. Q₁₆	C₂×C₅×Q₈	semidirect
269. D₈	C₂×C₅×Q₈	semidirect
333. C₅	#33629	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×#500	2
C₄×C₅×#135	2
C₅×#2227	2
C₅×#2226	2
C₅×#2225	2
C₅×#2224	2
C₅×#2159	2
C₅×#2150	2
C₅×#2139	2
#33629	5

Subgroups — C5×#33629 #11956349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33629 #11956349