Subgroups — C₅×#33669 #11956389

Order 1280.

650 conjugacy classes of subgroups.

The subgroup lattice has 352 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×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2233	C₂	1	2	yes	maximal, complemented
4	C₅×#2226	C₂	1	2	yes	maximal, complemented
5	C₅×#2206	C₂	1	2	yes	maximal, complemented
6	C₅×#2184	C₂	1	2	yes	maximal, complemented
7	C₅×#2177	C₂	1	4	yes	maximal, complemented
8	C₅×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₅×#532	—	2	1	no	—
11	C₂×C₄×C₅×D₄	—	2	1	no	—
12	C₂×C₅×D₁₆	C₂²	1	4	yes	—
13	C₅×#500	C₂²	1	1	yes	characteristic, complemented
14	C₅×#493	C₂²	1	1	yes	characteristic, complemented
15	C₅×#486	—	2	1	no	—
16	C₅×#466	—	2	1	no	—
17	C₅×#466	C₂²	1	2	yes	—
18	C₄×C₅×D₈	—	2	1	no	—
19	C₄×C₅×D₈	—	2	1	no	—
20	C₄×C₅×D₈	—	2	1	no	—
21	C₄×C₅×D₈	C₂²	1	2	yes	—
22	C₄×C₅×D₈	C₂²	1	2	yes	—
23	C₄×C₅×D₈	C₂²	1	2	yes	complemented
24	C₂×C₅×#107	C₂²	1	1	yes	characteristic
25	C₂×C₅×#102	—	2	1	no	—
26	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
27	C₅×#367	C₂²	1	4	yes	complemented
28	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
29	C₅×#357	C₂²	1	4	yes	—
30	C₅×#357	C₂²	1	4	yes	complemented
31	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₅×#346	C₂²	1	2	yes	complemented
33	#33669	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅×#41	—	4	1	no	—
35	C₂²×C₅×D₄	—	2	1	no	—
36	C₂³×C₄×C₅	—	4	1	no	—
37	C₂×C₅×D₈	—	2	4	no	—
38	C₂×C₅×D₈	—	2	1	no	—
39	C₂×C₅×D₈	—	2	1	no	—
40	C₂×C₅×D₈	C₂³	1	2	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×#124	—	4	1	no	—
47	C₅×#121	—	2	2	no	—
48	C₅×#121	—	4	1	no	—
49	C₄×C₅×D₄	—	2	2	no	—
50	C₄×C₅×D₄	—	2	2	no	—
51	C₄×C₅×D₄	—	2	2	no	—
52	C₄×C₅×D₄	—	4	1	no	—
53	C₄×C₅×D₄	—	4	1	no	—
54	C₄×C₅×D₄	—	4	1	no	—
55	C₂×C₅×C₄⋊C₄	—	2	1	no	—
56	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
57	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
58	C₅×D₁₆	—	2	8	no	—
59	C₂×C₅×C₁₆	C₂³	1	4	yes	—
60	C₅×#107	—	2	1	no	—
61	C₅×#107	—	2	1	no	—
62	C₅×#107	C₂³	1	2	yes	—
63	C₅×#107	C₂³	1	2	yes	—
64	C₅×#102	—	2	2	no	—
65	C₅×#102	—	2	2	no	—
66	C₅×#102	—	2	2	no	—
67	C₅×#102	—	2	2	no	—
68	C₄×C₅×C₈	D₄	1	1	yes	characteristic
69	C₄×C₅×C₈	D₄	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	#3169	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic, complemented
74	#2233	C₂×C₅	1	2	yes	complemented
75	#2226	C₂×C₅	1	2	yes	complemented
76	#2206	C₂×C₅	1	2	yes	complemented
77	#2184	C₂×C₅	1	2	yes	complemented
78	#2177	C₂×C₅	1	4	yes	complemented
79	#2163	C₂×C₅	1	1	yes	characteristic, complemented
80	C₂⁴×C₅	—	4	1	no	—
81	C₅×#41	—	8	4	no	—
82	C₂×C₅×Q₈	—	4	1	no	—
83	C₂×C₅×D₄	—	2	4	no	—
84	C₂×C₅×D₄	—	2	2	no	—
85	C₂×C₅×D₄	—	2	2	no	—
86	C₂×C₅×D₄	—	4	2	no	—
87	C₂×C₅×D₄	—	4	2	no	—
88	C₂×C₅×D₄	—	4	1	no	—
89	C₂×C₅×D₄	—	4	1	no	—
90	C₂×C₅×D₄	—	4	1	no	—
91	C₂²×C₄×C₅	—	4	1	no	—
92	C₂²×C₄×C₅	—	4	2	no	—
93	C₂²×C₄×C₅	—	4	2	no	—
94	C₂²×C₄×C₅	—	4	2	no	—
95	C₂²×C₄×C₅	—	8	4	no	—
96	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₅×D₈	—	2	4	no	—
100	C₅×D₈	—	2	2	no	—
101	C₅×D₈	—	2	4	no	—
102	C₅×D₈	—	4	2	no	—
103	C₅×D₈	#41	1	4	yes	complemented
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	4	1	no	—
106	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	—	4	1	no	—
113	C₅×C₄⋊C₄	—	4	1	no	—
114	C₅×C₂²⋊C₄	—	4	2	no	—
115	C₅×C₂²⋊C₄	—	4	2	no	—
116	C₅×C₂²⋊C₄	—	4	2	no	—
117	C₅×C₂²⋊C₄	—	4	2	no	—
118	C₄²×C₅	D₈	1	1	yes	characteristic
119	C₄²×C₅	D₈	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
122	C₅×C₁₆	—	2	4	no	—
123	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
124	#532	—	2	1	no	—
125	C₂×C₄×D₄	—	2	1	no	—
126	C₂×D₁₆	C₂²×C₅	1	4	yes	—
127	#500	C₂²×C₅	1	1	yes	characteristic, complemented
128	#493	C₂²×C₅	1	1	yes	characteristic, complemented
129	#486	—	2	1	no	—
130	#466	—	2	1	no	—
131	#466	C₂²×C₅	1	2	yes	—
132	C₄×D₈	—	2	1	no	—
133	C₄×D₈	—	2	1	no	—
134	C₄×D₈	—	2	1	no	—
135	C₄×D₈	C₂²×C₅	1	2	yes	—
136	C₄×D₈	C₂²×C₅	1	2	yes	—
137	C₄×D₈	C₂²×C₅	1	2	yes	complemented
138	C₂×#107	C₂²×C₅	1	1	yes	characteristic
139	C₂×#102	—	2	1	no	—
140	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
141	#367	C₂²×C₅	1	4	yes	complemented
142	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
143	#357	C₂²×C₅	1	4	yes	—
144	#357	C₂²×C₅	1	4	yes	complemented
145	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
146	#346	C₂²×C₅	1	2	yes	complemented
147	C₂³×C₅	—	4	4	no	—
148	C₂³×C₅	—	4	2	no	—
149	C₂³×C₅	—	4	2	no	—
150	C₂³×C₅	—	8	2	no	—
151	C₂³×C₅	—	8	2	no	—
152	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
153	C₅×Q₈	—	8	2	no	—
154	C₅×D₄	—	2	4	no	—
155	C₅×D₄	—	4	4	no	—
156	C₅×D₄	—	4	2	no	—
157	C₅×D₄	—	4	2	no	—
158	C₅×D₄	—	4	2	no	—
159	C₅×D₄	—	8	2	no	—
160	C₅×D₄	—	8	4	no	—
161	C₅×D₄	—	8	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	—	4	2	no	—
168	C₂×C₄×C₅	—	8	4	no	—
169	C₂×C₄×C₅	—	8	4	no	—
170	C₂×C₄×C₅	—	8	2	no	—
171	C₂×C₄×C₅	—	8	4	no	—
172	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
178	C₅×C₈	—	2	1	no	—
179	C₅×C₈	—	2	1	no	—
180	C₅×C₈	—	2	1	no	—
181	C₅×C₈	C₂×#41	1	2	yes	—
182	C₂×#41	—	4	1	no	—
183	C₂²×D₄	—	2	1	no	—
184	C₂³×C₄	—	4	1	no	—
185	C₂×D₈	—	2	4	no	—
186	C₂×D₈	—	2	1	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	C₂³×C₅	1	2	yes	—
189	C₂×D₈	C₂³×C₅	1	2	yes	—
190	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
192	C₄⋊Q₈	—	2	1	no	—
193	#127	—	2	1	no	—
194	#124	—	4	1	no	—
195	#121	—	2	2	no	—
196	#121	—	4	1	no	—
197	C₄×D₄	—	2	2	no	—
198	C₄×D₄	—	2	2	no	—
199	C₄×D₄	—	2	2	no	—
200	C₄×D₄	—	4	1	no	—
201	C₄×D₄	—	4	1	no	—
202	C₄×D₄	—	4	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₂²⋊C₄	—	4	1	no	—
205	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
206	D₁₆	—	2	8	no	—
207	C₂×C₁₆	C₂³×C₅	1	4	yes	—
208	#107	—	2	1	no	—
209	#107	—	2	1	no	—
210	#107	C₂³×C₅	1	2	yes	—
211	#107	C₂³×C₅	1	2	yes	—
212	#102	—	2	2	no	—
213	#102	—	2	2	no	—
214	#102	—	2	2	no	—
215	#102	—	2	2	no	—
216	C₄×C₈	C₅×D₄	1	1	yes	characteristic
217	C₄×C₈	C₅×D₄	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	4	4	no	—
223	C₂²×C₅	—	8	8	no	—
224	C₂²×C₅	—	8	2	no	—
225	C₂²×C₅	—	8	2	no	—
226	C₂²×C₅	—	8	2	no	—
227	C₂²×C₅	—	8	2	no	—
228	C₂²×C₅	—	8	2	no	—
229	C₂²×C₅	#509	1	2	yes	—
230	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	8	2	no	—
236	C₄×C₅	—	8	2	no	—
237	C₄×C₅	#509	1	2	yes	—
238	C₄×C₅	#525	1	1	yes	characteristic
239	C₄×C₅	#575	1	1	yes	characteristic
240	C₂⁴	—	4	1	no	—
241	#41	—	8	4	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	8	4	no	—
256	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
257	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
259	D₈	—	2	4	no	—
260	D₈	—	2	2	no	—
261	D₈	—	2	4	no	—
262	D₈	—	4	2	no	—
263	D₈	C₅×#41	1	4	yes	complemented
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
267	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
268	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
269	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₄²	C₅×D₈	1	1	yes	characteristic
279	C₄²	C₅×D₈	1	1	yes	characteristic
280	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
281	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
282	C₁₆	—	2	4	no	—
283	C₂×C₅	—	2	2	no	—
284	C₂×C₅	—	8	4	no	—
285	C₂×C₅	—	16	2	no	—
286	C₂×C₅	#3100	1	1	yes	characteristic
287	C₂×C₅	C₂×#509	1	1	yes	characteristic
288	C₂×C₅	C₂×#509	1	1	yes	characteristic
289	C₂³	—	4	4	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
295	Q₈	—	8	2	no	—
296	D₄	—	2	4	no	—
297	D₄	—	4	4	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	4	no	—
303	D₄	—	8	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	8	4	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	8	4	no	—
314	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
315	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
316	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
317	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
318	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	C₂×C₅×#41	1	2	yes	—
324	C₅	#33669	1	1	yes	characteristic, direct product factor, Sylow
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	8	8	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	C₅×#509	1	2	yes	—
335	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	C₅×#509	1	2	yes	—
343	C₄	C₅×#525	1	1	yes	characteristic
344	C₄	C₅×#575	1	1	yes	characteristic
345	C₂	—	2	2	no	—
346	C₂	—	8	4	no	—
347	C₂	—	16	2	no	—
348	C₂	C₅×#3100	1	1	yes	characteristic
349	C₂	C₂×C₅×#509	1	1	yes	characteristic
350	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×C₄×C₅×D₈	C₂	semidirect
3. C₅×#2233	C₂	semidirect
4. C₅×#2226	C₂	semidirect
5. C₅×#2206	C₂	semidirect
6. C₅×#2184	C₂	semidirect
7. C₅×#2177	C₂	semidirect
8. C₅×#2163	C₂	semidirect
13. C₅×#500	C₂²	semidirect
14. C₅×#493	C₂²	semidirect
23. C₄×C₅×D₈	C₂²	semidirect
27. C₅×#367	C₂²	semidirect
28. C₅×C₄⋊C₁₆	C₂²	semidirect
30. C₅×#357	C₂²	semidirect
32. C₅×#346	C₂²	semidirect
33. #33669	C₅	direct (N × H)
72. #3169	C₂×C₅	semidirect
73. C₂×C₄×D₈	C₂×C₅	semidirect
74. #2233	C₂×C₅	semidirect
75. #2226	C₂×C₅	semidirect
76. #2206	C₂×C₅	semidirect
77. #2184	C₂×C₅	semidirect
78. #2177	C₂×C₅	semidirect
79. #2163	C₂×C₅	semidirect
103. C₅×D₈	#41	semidirect
127. #500	C₂²×C₅	semidirect
128. #493	C₂²×C₅	semidirect
137. C₄×D₈	C₂²×C₅	semidirect
141. #367	C₂²×C₅	semidirect
142. C₄⋊C₁₆	C₂²×C₅	semidirect
144. #357	C₂²×C₅	semidirect
146. #346	C₂²×C₅	semidirect
263. D₈	C₅×#41	semidirect
324. C₅	#33669	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×C₄×C₅×D₈	2
C₅×#2233	2
C₅×#2226	2
C₅×#2206	2
C₅×#2184	2
C₅×#2177	2
C₅×#2163	2
#33669	5

Subgroups — C5×#33669 #11956389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33669 #11956389