Subgroups — C₅×#33789 #11956509

Order 1280.

562 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3156	C₂	1	1	yes	maximal, characteristic
3	C₅×#2251	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2247	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2245	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2242	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2211	C₂	1	2	yes	maximal, complemented
8	C₅×#2209	C₂	1	2	yes	maximal, complemented
9	C₅×#2189	C₂	1	2	yes	maximal
10	C₅×#2188	C₂	1	2	yes	maximal, complemented
11	C₅×#2164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×#137	—	2	1	no	—
13	C₂×C₅×#135	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
15	C₅×#541	—	2	1	no	—
16	C₅×#536	—	2	1	no	—
17	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
18	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
19	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
20	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
21	C₅×#499	C₂²	1	1	yes	characteristic
22	C₅×#497	—	2	1	no	—
23	C₅×#497	—	2	1	no	—
24	C₅×#495	C₂²	1	1	yes	characteristic
25	C₅×#495	C₂²	1	1	yes	characteristic
26	C₅×#494	C₂²	1	1	yes	characteristic
27	C₅×#494	C₂²	1	1	yes	characteristic
28	C₅×#494	C₂²	1	1	yes	characteristic
29	C₅×#491	—	2	1	no	—
30	C₅×#470	—	2	1	no	—
31	C₅×#467	C₂²	1	2	yes	—
32	C₅×#441	—	2	1	no	—
33	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
34	C₅×#405	C₂²	1	1	yes	characteristic
35	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
36	C₅×#358	C₂²	1	2	yes	—
37	C₅×#358	C₂²	1	2	yes	—
38	C₅×#358	C₂²	1	2	yes	—
39	C₅×#357	C₂²	1	2	yes	—
40	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
41	C₅×#346	C₂²	1	1	yes	characteristic
42	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
43	#33789	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₅×#41	—	2	1	no	—
45	C₂²×C₅×Q₈	—	2	1	no	—
46	C₅×#137	—	4	4	no	—
47	C₅×#135	—	4	2	no	—
48	C₂×C₅×Q₁₆	—	2	1	no	—
49	C₂×C₅×Q₁₆	—	2	1	no	—
50	C₂×C₅×Q₁₆	—	2	1	no	—
51	C₂×C₅×Q₁₆	—	2	1	no	—
52	C₂×C₅×Q₁₆	—	4	2	no	—
53	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
54	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
57	C₂×C₅×SD₁₆	—	2	1	no	—
58	C₂×C₅×SD₁₆	—	2	1	no	—
59	C₂×C₅×SD₁₆	—	2	1	no	—
60	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
61	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
62	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
63	C₅×C₄⋊Q₈	—	2	1	no	—
64	C₅×C₄⋊Q₈	—	2	1	no	—
65	C₅×C₄⋊Q₈	—	2	1	no	—
66	C₅×#126	—	4	2	no	—
67	C₅×#125	—	2	1	no	—
68	C₅×#125	—	4	2	no	—
69	C₅×#124	—	2	1	no	—
70	C₅×#124	—	2	1	no	—
71	C₅×#123	—	4	2	no	—
72	C₅×C₂²⋊Q₈	—	2	2	no	—
73	C₅×C₂²⋊Q₈	—	4	2	no	—
74	C₄×C₅×Q₈	—	2	2	no	—
75	C₅×#117	D₄	1	1	yes	characteristic
76	C₅×Q₃₂	—	4	2	no	—
77	C₅×Q₃₂	—	4	2	no	—
78	C₅×Q₃₂	—	4	2	no	—
79	C₅×SD₃₂	—	4	2	no	—
80	C₂×C₅×C₁₆	C₂³	1	2	yes	—
81	C₂×C₅×C₁₆	C₂³	1	2	yes	—
82	C₅×#107	C₂³	1	2	yes	—
83	C₅×#106	—	2	1	no	—
84	C₅×#103	—	2	2	no	—
85	C₅×#103	—	2	2	no	—
86	C₅×#97	D₄	1	1	yes	characteristic
87	C₅×#97	D₄	1	1	yes	characteristic
88	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
89	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
90	#3177	C₂×C₅	1	1	yes	characteristic, complemented
91	#3156	C₂×C₅	1	1	yes	characteristic
92	#2251	C₂×C₅	1	1	yes	characteristic, complemented
93	#2247	C₂×C₅	1	1	yes	characteristic, complemented
94	#2245	C₂×C₅	1	1	yes	characteristic, complemented
95	#2242	C₂×C₅	1	1	yes	characteristic, complemented
96	#2211	C₂×C₅	1	2	yes	complemented
97	#2209	C₂×C₅	1	2	yes	complemented
98	#2189	C₂×C₅	1	2	yes	—
99	#2188	C₂×C₅	1	2	yes	complemented
100	#2164	C₂×C₅	1	1	yes	characteristic, complemented
101	C₅×#41	—	4	2	no	—
102	C₅×#41	—	4	2	no	—
103	C₂×C₅×Q₈	—	2	2	no	—
104	C₂×C₅×Q₈	—	2	1	no	—
105	C₂×C₅×Q₈	—	2	1	no	—
106	C₂×C₅×Q₈	—	2	1	no	—
107	C₂×C₅×Q₈	—	4	2	no	—
108	C₂×C₅×Q₈	—	4	2	no	—
109	C₂×C₅×Q₈	—	4	1	no	—
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₅×D₄	—	4	1	no	—
112	C₂²×C₄×C₅	—	4	1	no	—
113	C₂²×C₄×C₅	—	4	1	no	—
114	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
115	C₅×Q₁₆	—	2	2	no	—
116	C₅×Q₁₆	—	2	2	no	—
117	C₅×Q₁₆	—	2	2	no	—
118	C₅×Q₁₆	—	2	2	no	—
119	C₅×Q₁₆	—	2	2	no	—
120	C₅×Q₁₆	—	4	2	no	—
121	C₅×Q₁₆	—	4	2	no	—
122	C₅×Q₁₆	—	4	2	no	—
123	C₅×Q₁₆	—	4	2	no	—
124	C₅×SD₁₆	—	4	2	no	—
125	C₅×SD₁₆	—	4	2	no	—
126	C₅×SD₁₆	—	4	2	no	—
127	C₅×D₈	—	2	2	no	—
128	C₅×M₁₆	—	2	2	no	—
129	C₂×C₅×C₈	—	2	2	no	—
130	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
134	C₅×C₄⋊C₄	—	2	2	no	—
135	C₅×C₄⋊C₄	—	4	2	no	—
136	C₅×C₄⋊C₄	—	4	2	no	—
137	C₅×C₄⋊C₄	—	4	2	no	—
138	C₅×C₄⋊C₄	—	4	2	no	—
139	C₅×C₄⋊C₄	—	4	2	no	—
140	C₅×C₄⋊C₄	—	4	2	no	—
141	C₅×C₄⋊C₄	D₈	1	2	yes	—
142	C₅×C₂²⋊C₄	—	4	2	no	—
143	C₅×C₂²⋊C₄	—	4	2	no	—
144	C₅×C₂²⋊C₄	—	4	2	no	—
145	C₅×C₂²⋊C₄	D₈	1	2	yes	—
146	C₄²×C₅	—	4	2	no	—
147	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
149	C₅×C₁₆	—	2	2	no	—
150	C₅×C₁₆	—	2	2	no	—
151	C₂×#137	—	2	1	no	—
152	C₂×#135	C₂²×C₅	1	1	yes	characteristic
153	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
154	#541	—	2	1	no	—
155	#536	—	2	1	no	—
156	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
157	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
158	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
159	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
160	#499	C₂²×C₅	1	1	yes	characteristic
161	#497	—	2	1	no	—
162	#497	—	2	1	no	—
163	#495	C₂²×C₅	1	1	yes	characteristic
164	#495	C₂²×C₅	1	1	yes	characteristic
165	#494	C₂²×C₅	1	1	yes	characteristic
166	#494	C₂²×C₅	1	1	yes	characteristic
167	#494	C₂²×C₅	1	1	yes	characteristic
168	#491	—	2	1	no	—
169	#470	—	2	1	no	—
170	#467	C₂²×C₅	1	2	yes	—
171	#441	—	2	1	no	—
172	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
173	#405	C₂²×C₅	1	1	yes	characteristic
174	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
175	#358	C₂²×C₅	1	2	yes	—
176	#358	C₂²×C₅	1	2	yes	—
177	#358	C₂²×C₅	1	2	yes	—
178	#357	C₂²×C₅	1	2	yes	—
179	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
180	#346	C₂²×C₅	1	1	yes	characteristic
181	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
182	C₂³×C₅	—	4	1	no	—
183	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
184	C₅×Q₈	—	4	1	no	—
185	C₅×Q₈	—	4	1	no	—
186	C₅×Q₈	—	4	1	no	—
187	C₅×Q₈	—	4	2	no	—
188	C₅×Q₈	—	4	1	no	—
189	C₅×Q₈	—	4	1	no	—
190	C₅×Q₈	—	4	1	no	—
191	C₅×Q₈	—	4	2	no	—
192	C₅×Q₈	—	8	2	no	—
193	C₅×D₄	—	4	1	no	—
194	C₅×D₄	—	4	1	no	—
195	C₅×D₄	—	8	2	no	—
196	C₂×C₄×C₅	—	2	1	no	—
197	C₂×C₄×C₅	—	2	1	no	—
198	C₂×C₄×C₅	—	4	2	no	—
199	C₂×C₄×C₅	—	4	2	no	—
200	C₂×C₄×C₅	—	4	1	no	—
201	C₂×C₄×C₅	—	4	1	no	—
202	C₂×C₄×C₅	—	4	1	no	—
203	C₂×C₄×C₅	—	8	2	no	—
204	C₂×C₄×C₅	—	8	2	no	—
205	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
206	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
207	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
208	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
209	C₅×C₈	—	2	1	no	—
210	C₅×C₈	—	2	1	no	—
211	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
212	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
213	C₂×#41	—	2	1	no	—
214	C₂²×Q₈	—	2	1	no	—
215	#137	—	4	4	no	—
216	#135	—	4	2	no	—
217	C₂×Q₁₆	—	2	1	no	—
218	C₂×Q₁₆	—	2	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	—	4	2	no	—
222	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
223	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
224	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
225	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
226	C₂×SD₁₆	—	2	1	no	—
227	C₂×SD₁₆	—	2	1	no	—
228	C₂×SD₁₆	—	2	1	no	—
229	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
230	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
231	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	#126	—	4	2	no	—
236	#125	—	2	1	no	—
237	#125	—	4	2	no	—
238	#124	—	2	1	no	—
239	#124	—	2	1	no	—
240	#123	—	4	2	no	—
241	C₂²⋊Q₈	—	2	2	no	—
242	C₂²⋊Q₈	—	4	2	no	—
243	C₄×Q₈	—	2	2	no	—
244	#117	C₅×D₄	1	1	yes	characteristic
245	Q₃₂	—	4	2	no	—
246	Q₃₂	—	4	2	no	—
247	Q₃₂	—	4	2	no	—
248	SD₃₂	—	4	2	no	—
249	C₂×C₁₆	C₂³×C₅	1	2	yes	—
250	C₂×C₁₆	C₂³×C₅	1	2	yes	—
251	#107	C₂³×C₅	1	2	yes	—
252	#106	—	2	1	no	—
253	#103	—	2	2	no	—
254	#103	—	2	2	no	—
255	#97	C₅×D₄	1	1	yes	characteristic
256	#97	C₅×D₄	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
258	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
259	C₂²×C₅	—	2	2	no	—
260	C₂²×C₅	—	2	1	no	—
261	C₂²×C₅	—	8	2	no	—
262	C₂²×C₅	—	8	1	no	—
263	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
264	C₄×C₅	—	2	2	no	—
265	C₄×C₅	—	2	2	no	—
266	C₄×C₅	—	2	1	no	—
267	C₄×C₅	—	8	2	no	—
268	C₄×C₅	—	8	1	no	—
269	C₄×C₅	—	8	1	no	—
270	C₄×C₅	—	8	1	no	—
271	C₄×C₅	—	8	2	no	—
272	C₄×C₅	#535	1	1	yes	characteristic
273	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
274	#41	—	4	2	no	—
275	#41	—	4	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	4	2	no	—
281	C₂×Q₈	—	4	2	no	—
282	C₂×Q₈	—	4	1	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	2	2	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	4	2	no	—
294	Q₁₆	—	4	2	no	—
295	Q₁₆	—	4	2	no	—
296	Q₁₆	—	4	2	no	—
297	SD₁₆	—	4	2	no	—
298	SD₁₆	—	4	2	no	—
299	SD₁₆	—	4	2	no	—
300	D₈	—	2	2	no	—
301	M₁₆	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
304	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
305	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
306	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	4	2	no	—
309	C₄⋊C₄	—	4	2	no	—
310	C₄⋊C₄	—	4	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	4	2	no	—
314	C₄⋊C₄	C₅×D₈	1	2	yes	—
315	C₂²⋊C₄	—	4	2	no	—
316	C₂²⋊C₄	—	4	2	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	C₅×D₈	1	2	yes	—
319	C₄²	—	4	2	no	—
320	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
321	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
322	C₁₆	—	2	2	no	—
323	C₁₆	—	2	2	no	—
324	C₂×C₅	—	4	1	no	—
325	C₂×C₅	—	16	1	no	—
326	C₂×C₅	#3205	1	1	yes	characteristic
327	C₂×C₅	#3417	1	2	yes	—
328	C₂³	—	4	1	no	—
329	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
330	Q₈	—	4	1	no	—
331	Q₈	—	4	1	no	—
332	Q₈	—	4	1	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	1	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	4	1	no	—
337	Q₈	—	4	2	no	—
338	Q₈	—	8	2	no	—
339	D₄	—	4	1	no	—
340	D₄	—	4	1	no	—
341	D₄	—	8	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	—	8	2	no	—
351	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
352	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
353	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
354	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
358	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
359	C₅	#33789	1	1	yes	characteristic, direct product factor, Sylow
360	C₂²	—	2	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	8	2	no	—
363	C₂²	—	8	1	no	—
364	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	1	no	—
368	C₄	—	8	2	no	—
369	C₄	—	8	1	no	—
370	C₄	—	8	1	no	—
371	C₄	—	8	1	no	—
372	C₄	—	8	2	no	—
373	C₄	C₅×#535	1	1	yes	characteristic
374	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
375	C₂	—	4	1	no	—
376	C₂	—	16	1	no	—
377	C₂	C₅×#3205	1	1	yes	characteristic
378	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
1. C₅×#3177	C₂	semidirect
3. C₅×#2251	C₂	semidirect
4. C₅×#2247	C₂	semidirect
5. C₅×#2245	C₂	semidirect
6. C₅×#2242	C₂	semidirect
7. C₅×#2211	C₂	semidirect
8. C₅×#2209	C₂	semidirect
10. C₅×#2188	C₂	semidirect
11. C₅×#2164	C₂	semidirect
43. #33789	C₅	direct (N × H)
90. #3177	C₂×C₅	semidirect
92. #2251	C₂×C₅	semidirect
93. #2247	C₂×C₅	semidirect
94. #2245	C₂×C₅	semidirect
95. #2242	C₂×C₅	semidirect
96. #2211	C₂×C₅	semidirect
97. #2209	C₂×C₅	semidirect
99. #2188	C₂×C₅	semidirect
100. #2164	C₂×C₅	semidirect
359. C₅	#33789	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×#3177	2
C₅×#3156	2
C₅×#2251	2
C₅×#2247	2
C₅×#2245	2
C₅×#2242	2
C₅×#2211	2
C₅×#2209	2
C₅×#2189	2
C₅×#2188	2
C₅×#2164	2
#33789	5

Subgroups — C5×#33789 #11956509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33789 #11956509