Subgroups — C₅×#33652 #11956372

Order 1280.

568 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3061	C₂	1	1	yes	maximal, characteristic
2	C₅×#2913	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2237	C₂	1	2	yes	maximal, complemented
4	C₅×#2236	C₂	1	2	yes	maximal, complemented
5	C₅×#2234	C₂	1	2	yes	maximal, complemented
6	C₅×#2232	C₂	1	2	yes	maximal, complemented
7	C₅×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2139	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#2138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#2113	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×#135	C₂²	1	1	yes	characteristic
13	C₂×C₅×#131	C₂²	1	1	yes	characteristic
14	C₅×#537	—	2	1	no	—
15	C₅×#520	—	2	1	no	—
16	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
18	C₅×#468	—	2	2	no	—
19	C₅×#467	C₂²	1	2	yes	—
20	C₅×#466	C₂²	1	2	yes	—
21	C₅×#428	—	2	1	no	—
22	C₅×#428	—	2	2	no	—
23	C₅×#427	C₂²	1	1	yes	characteristic
24	C₅×#427	C₂²	1	2	yes	—
25	C₂×C₅×#107	C₂²	1	1	yes	characteristic
26	C₂×C₅×#107	C₂²	1	1	yes	characteristic
27	C₂×C₅×#107	C₂²	1	2	yes	—
28	C₅×#417	—	2	1	no	—
29	C₅×#367	C₂²	1	2	yes	—
30	C₅×#366	C₂²	1	2	yes	—
31	C₅×#366	C₂²	1	2	yes	—
32	C₅×#366	C₂²	1	2	yes	—
33	C₅×#358	C₂²	1	1	yes	characteristic
34	C₅×#358	C₂²	1	1	yes	characteristic
35	C₅×#358	C₂²	1	2	yes	—
36	C₅×#357	C₂²	1	1	yes	characteristic
37	C₅×#357	C₂²	1	1	yes	characteristic
38	C₅×#357	C₂²	1	2	yes	—
39	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
41	#33652	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₅×#41	—	2	1	no	—
43	C₂×C₅×#41	D₄	1	1	yes	characteristic
44	C₅×#135	—	2	2	no	—
45	C₅×#135	—	2	2	no	—
46	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₅×SD₁₆	—	2	1	no	—
48	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
49	C₅×#131	—	2	2	no	—
50	C₅×#131	—	2	2	no	—
51	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
52	C₂×C₅×M₁₆	D₄	1	2	yes	—
53	C₂²×C₅×C₈	C₂³	1	2	yes	—
54	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₅×C₂²⋊Q₈	—	2	2	no	—
56	C₅×C₂²⋊Q₈	—	4	1	no	—
57	C₅×#121	—	2	2	no	—
58	C₅×#121	—	4	1	no	—
59	C₅×#121	—	4	2	no	—
60	C₄×C₅×Q₈	—	4	1	no	—
61	C₄×C₅×D₄	—	4	2	no	—
62	C₄×C₅×D₄	—	4	1	no	—
63	C₅×#117	—	2	2	no	—
64	C₅×#117	—	2	1	no	—
65	C₂×C₅×C₄⋊C₄	—	2	1	no	—
66	C₂×C₅×C₄⋊C₄	—	2	2	no	—
67	C₂×C₅×C₄⋊C₄	—	2	1	no	—
68	C₅×M₃₂	—	2	2	no	—
69	C₂×C₅×C₁₆	—	2	2	no	—
70	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₅×C₁₆	C₂³	1	2	yes	—
73	C₅×#107	—	2	2	no	—
74	C₅×#107	—	2	1	no	—
75	C₅×#107	—	2	1	no	—
76	C₅×#107	C₂³	1	2	yes	—
77	C₅×#107	C₂³	1	1	yes	characteristic
78	C₅×#107	C₂³	1	2	yes	—
79	C₅×#107	C₂³	1	1	yes	characteristic
80	C₅×#106	—	2	2	no	—
81	C₅×#106	—	2	1	no	—
82	C₅×#103	—	2	2	no	—
83	C₅×#102	—	2	2	no	—
84	#3061	C₂×C₅	1	1	yes	characteristic
85	#2913	C₂×C₅	1	1	yes	characteristic, complemented
86	#2237	C₂×C₅	1	2	yes	complemented
87	#2236	C₂×C₅	1	2	yes	complemented
88	#2234	C₂×C₅	1	2	yes	complemented
89	#2232	C₂×C₅	1	2	yes	complemented
90	#2159	C₂×C₅	1	1	yes	characteristic, complemented
91	C₂×#366	C₂×C₅	1	1	yes	characteristic, complemented
92	#2139	C₂×C₅	1	1	yes	characteristic, complemented
93	#2138	C₂×C₅	1	1	yes	characteristic, complemented
94	#2113	C₂×C₅	1	1	yes	characteristic, complemented
95	C₅×#41	—	2	2	no	—
96	C₅×#41	—	2	2	no	—
97	C₅×#41	—	4	2	no	—
98	C₅×#41	—	4	2	no	—
99	C₂×C₅×Q₈	—	2	1	no	—
100	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
101	C₂×C₅×D₄	—	2	1	no	—
102	C₂×C₅×D₄	—	4	1	no	—
103	C₂×C₅×D₄	—	4	2	no	—
104	C₂×C₅×D₄	D₈	1	2	yes	—
105	C₂×C₅×D₄	D₈	1	1	yes	characteristic
106	C₂²×C₄×C₅	—	4	2	no	—
107	C₂²×C₄×C₅	—	4	1	no	—
108	C₂²×C₄×C₅	—	4	1	no	—
109	C₂²×C₄×C₅	—	4	1	no	—
110	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
112	C₅×Q₁₆	—	2	1	no	—
113	C₅×Q₁₆	—	2	1	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	C₅×M₁₆	—	2	4	no	—
119	C₂×C₅×C₈	—	2	2	no	—
120	C₂×C₅×C₈	—	2	2	no	—
121	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
122	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₅×C₈	#41	1	2	yes	—
124	C₂×C₅×C₈	#41	1	2	yes	—
125	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
126	C₅×C₄⋊C₄	—	2	1	no	—
127	C₅×C₄⋊C₄	—	2	2	no	—
128	C₅×C₄⋊C₄	—	2	2	no	—
129	C₅×C₄⋊C₄	—	2	1	no	—
130	C₅×C₄⋊C₄	—	4	2	no	—
131	C₅×C₄⋊C₄	—	4	1	no	—
132	C₅×C₄⋊C₄	—	4	1	no	—
133	C₅×C₄⋊C₄	—	4	2	no	—
134	C₅×C₂²⋊C₄	—	4	2	no	—
135	C₅×C₂²⋊C₄	—	4	1	no	—
136	C₅×C₂²⋊C₄	—	4	2	no	—
137	C₅×C₂²⋊C₄	—	4	2	no	—
138	C₄²×C₅	—	4	2	no	—
139	C₄²×C₅	—	4	1	no	—
140	C₅×C₁₆	—	2	1	no	—
141	C₅×C₁₆	—	2	1	no	—
142	C₅×C₁₆	—	2	2	no	—
143	C₂×#135	C₂²×C₅	1	1	yes	characteristic
144	C₂×#131	C₂²×C₅	1	1	yes	characteristic
145	#537	—	2	1	no	—
146	#520	—	2	1	no	—
147	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
148	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
149	#468	—	2	2	no	—
150	#467	C₂²×C₅	1	2	yes	—
151	#466	C₂²×C₅	1	2	yes	—
152	#428	—	2	1	no	—
153	#428	—	2	2	no	—
154	#427	C₂²×C₅	1	1	yes	characteristic
155	#427	C₂²×C₅	1	2	yes	—
156	C₂×#107	C₂²×C₅	1	1	yes	characteristic
157	C₂×#107	C₂²×C₅	1	2	yes	—
158	C₂×#107	C₂²×C₅	1	1	yes	characteristic
159	#417	—	2	1	no	—
160	#367	C₂²×C₅	1	2	yes	—
161	#366	C₂²×C₅	1	2	yes	—
162	#366	C₂²×C₅	1	2	yes	—
163	#366	C₂²×C₅	1	2	yes	—
164	#358	C₂²×C₅	1	1	yes	characteristic
165	#358	C₂²×C₅	1	1	yes	characteristic
166	#358	C₂²×C₅	1	2	yes	—
167	#357	C₂²×C₅	1	1	yes	characteristic
168	#357	C₂²×C₅	1	1	yes	characteristic
169	#357	C₂²×C₅	1	2	yes	—
170	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
171	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
172	C₂³×C₅	—	4	1	no	—
173	C₂³×C₅	C₂×D₈	1	2	yes	—
174	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
175	C₅×Q₈	—	2	2	no	—
176	C₅×Q₈	—	4	1	no	—
177	C₅×Q₈	—	4	1	no	—
178	C₅×D₄	—	2	4	no	—
179	C₅×D₄	—	2	2	no	—
180	C₅×D₄	—	4	1	no	—
181	C₅×D₄	—	4	1	no	—
182	C₅×D₄	—	8	2	no	—
183	C₅×D₄	—	16	2	no	—
184	C₂×C₄×C₅	—	2	2	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	4	1	no	—
187	C₂×C₄×C₅	—	4	2	no	—
188	C₂×C₄×C₅	—	4	1	no	—
189	C₂×C₄×C₅	—	4	2	no	—
190	C₂×C₄×C₅	—	4	1	no	—
191	C₂×C₄×C₅	—	8	2	no	—
192	C₂×C₄×C₅	—	16	1	no	—
193	C₂×C₄×C₅	—	16	2	no	—
194	C₂×C₄×C₅	—	16	1	no	—
195	C₂×C₄×C₅	#123	1	2	yes	—
196	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
197	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
198	C₂×C₄×C₅	#137	1	2	yes	—
199	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
200	C₅×C₈	—	2	2	no	—
201	C₅×C₈	#123	1	2	yes	—
202	C₅×C₈	C₂×#41	1	1	yes	characteristic
203	C₅×C₈	C₂×#41	1	1	yes	characteristic
204	C₂×#41	—	2	1	no	—
205	C₂×#41	C₅×D₄	1	1	yes	characteristic
206	#135	—	2	2	no	—
207	#135	—	2	2	no	—
208	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
209	C₂×SD₁₆	—	2	1	no	—
210	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
211	#131	—	2	2	no	—
212	#131	—	2	2	no	—
213	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
214	C₂×M₁₆	C₅×D₄	1	2	yes	—
215	C₂²×C₈	C₂³×C₅	1	2	yes	—
216	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
217	C₂²⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	4	1	no	—
219	#121	—	2	2	no	—
220	#121	—	4	1	no	—
221	#121	—	4	2	no	—
222	C₄×Q₈	—	4	1	no	—
223	C₄×D₄	—	4	2	no	—
224	C₄×D₄	—	4	1	no	—
225	#117	—	2	2	no	—
226	#117	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	M₃₂	—	2	2	no	—
231	C₂×C₁₆	—	2	2	no	—
232	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
233	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
234	C₂×C₁₆	C₂³×C₅	1	2	yes	—
235	#107	—	2	2	no	—
236	#107	—	2	1	no	—
237	#107	—	2	1	no	—
238	#107	C₂³×C₅	1	1	yes	characteristic
239	#107	C₂³×C₅	1	2	yes	—
240	#107	C₂³×C₅	1	1	yes	characteristic
241	#107	C₂³×C₅	1	2	yes	—
242	#106	—	2	2	no	—
243	#106	—	2	1	no	—
244	#103	—	2	2	no	—
245	#102	—	2	2	no	—
246	C₂²×C₅	—	2	1	no	—
247	C₂²×C₅	—	2	1	no	—
248	C₂²×C₅	—	2	2	no	—
249	C₂²×C₅	—	2	2	no	—
250	C₂²×C₅	—	2	2	no	—
251	C₂²×C₅	—	8	1	no	—
252	C₂²×C₅	—	8	1	no	—
253	C₂²×C₅	—	8	1	no	—
254	C₂²×C₅	#480	1	2	yes	—
255	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
256	C₄×C₅	—	2	2	no	—
257	C₄×C₅	—	8	1	no	—
258	C₄×C₅	—	8	2	no	—
259	C₄×C₅	—	8	1	no	—
260	C₄×C₅	—	8	2	no	—
261	C₄×C₅	—	8	1	no	—
262	C₄×C₅	#480	1	2	yes	—
263	C₄×C₅	C₂×#123	1	1	yes	characteristic
264	C₄×C₅	C₂×#137	1	1	yes	characteristic
265	#41	—	2	2	no	—
266	#41	—	2	2	no	—
267	#41	—	4	2	no	—
268	#41	—	4	2	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	C₅×D₈	1	2	yes	—
275	C₂×D₄	C₅×D₈	1	1	yes	characteristic
276	C₂²×C₄	—	4	2	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	4	1	no	—
279	C₂²×C₄	—	4	1	no	—
280	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
281	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
282	Q₁₆	—	2	1	no	—
283	Q₁₆	—	2	1	no	—
284	SD₁₆	—	4	2	no	—
285	D₈	—	2	1	no	—
286	D₈	—	2	1	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	4	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
292	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
293	C₂×C₈	C₅×#41	1	2	yes	—
294	C₂×C₈	C₅×#41	1	2	yes	—
295	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	4	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₂²⋊C₄	—	4	1	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₄²	—	4	2	no	—
309	C₄²	—	4	1	no	—
310	C₁₆	—	2	1	no	—
311	C₁₆	—	2	1	no	—
312	C₁₆	—	2	2	no	—
313	C₂×C₅	—	2	2	no	—
314	C₂×C₅	—	4	2	no	—
315	C₂×C₅	—	16	1	no	—
316	C₂×C₅	C₂×#480	1	1	yes	characteristic
317	C₂×C₅	#3415	1	1	yes	characteristic
318	C₂×C₅	#3417	1	1	yes	characteristic
319	C₂³	—	4	1	no	—
320	C₂³	C₂×C₅×D₈	1	2	yes	—
321	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
322	Q₈	—	2	2	no	—
323	Q₈	—	4	1	no	—
324	Q₈	—	4	1	no	—
325	D₄	—	2	4	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	1	no	—
328	D₄	—	4	1	no	—
329	D₄	—	8	2	no	—
330	D₄	—	16	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	—	16	1	no	—
340	C₂×C₄	—	16	2	no	—
341	C₂×C₄	—	16	1	no	—
342	C₂×C₄	C₅×#123	1	2	yes	—
343	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
344	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
345	C₂×C₄	C₅×#137	1	2	yes	—
346	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	C₅×#123	1	2	yes	—
349	C₈	C₂×C₅×#41	1	1	yes	characteristic
350	C₈	C₂×C₅×#41	1	1	yes	characteristic
351	C₅	#33652	1	1	yes	characteristic, direct product factor, Sylow
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	2	2	no	—
356	C₂²	—	2	2	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	1	no	—
359	C₂²	—	8	1	no	—
360	C₂²	C₅×#480	1	2	yes	—
361	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	2	no	—
365	C₄	—	8	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	8	1	no	—
368	C₄	C₅×#480	1	2	yes	—
369	C₄	C₂×C₅×#123	1	1	yes	characteristic
370	C₄	C₂×C₅×#137	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	4	2	no	—
373	C₂	—	16	1	no	—
374	C₂	C₂×C₅×#480	1	1	yes	characteristic
375	C₂	C₅×#3415	1	1	yes	characteristic
376	C₂	C₅×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₅×#3061	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41 ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2913	C₂	semidirect
3. C₅×#2237	C₂	semidirect
4. C₅×#2236	C₂	semidirect
5. C₅×#2234	C₂	semidirect
6. C₅×#2232	C₂	semidirect
7. C₅×#2159	C₂	semidirect
8. C₂×C₅×#366	C₂	semidirect
9. C₅×#2139	C₂	semidirect
10. C₅×#2138	C₂	semidirect
11. C₅×#2113	C₂	semidirect
41. #33652	C₅	direct (N × H)
85. #2913	C₂×C₅	semidirect
86. #2237	C₂×C₅	semidirect
87. #2236	C₂×C₅	semidirect
88. #2234	C₂×C₅	semidirect
89. #2232	C₂×C₅	semidirect
90. #2159	C₂×C₅	semidirect
91. C₂×#366	C₂×C₅	semidirect
92. #2139	C₂×C₅	semidirect
93. #2138	C₂×C₅	semidirect
94. #2113	C₂×C₅	semidirect
351. C₅	#33652	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×#3061	2
C₅×#2913	2
C₅×#2237	2
C₅×#2236	2
C₅×#2234	2
C₅×#2232	2
C₅×#2159	2
C₂×C₅×#366	2
C₅×#2139	2
C₅×#2138	2
C₅×#2113	2
#33652	5

Subgroups — C5×#33652 #11956372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33652 #11956372