Subgroups — C₅×#23646 #11946366

Order 1280.

906 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#2896	C₂	1	2	yes	maximal
3	C₂×C₅×#408	C₂	1	1	yes	maximal, characteristic
4	C₅×#2079	C₂	1	4	yes	maximal, complemented
5	C₅×#2078	C₂	1	4	yes	maximal, complemented
6	C₅×#1742	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#341	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×#117	C₂²	1	2	yes	—
13	C₅×#427	—	2	2	no	—
14	C₅×#427	—	2	2	no	—
15	C₂×C₅×#107	C₂²	1	2	yes	complemented
16	C₂×C₅×#106	C₂²	1	2	yes	complemented
17	C₅×#419	—	2	2	no	—
18	C₅×#418	—	2	2	no	—
19	C₅×#417	C₂²	1	8	yes	—
20	C₅×#416	—	2	4	no	—
21	C₅×#408	—	2	2	no	—
22	C₅×#408	C₂²	1	2	yes	—
23	C₅×#408	C₂²	1	2	yes	—
24	C₂×C₅×C₂²⋊C₈	—	2	2	no	—
25	C₅×#343	C₂²	1	8	yes	complemented
26	C₅×#341	C₂²	1	2	yes	complemented
27	C₅×#341	C₂²	1	2	yes	complemented
28	#23646	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂×C₅×#41	—	2	1	no	—
30	C₂×C₅×#41	—	2	1	no	—
31	C₂×C₅×#41	—	2	4	no	—
32	C₂×C₅×#41	—	2	4	no	—
33	C₂×C₅×#41	D₄	1	2	yes	—
34	C₂×C₅×#41	C₂³	1	2	yes	—
35	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
36	C₂²×C₅×D₄	—	2	1	no	—
37	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₅	—	2	1	no	—
39	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₅×M₁₆	—	2	1	no	—
41	C₂×C₅×M₁₆	—	2	2	no	—
42	C₂×C₅×M₁₆	—	2	4	no	—
43	C₂×C₅×M₁₆	—	2	4	no	—
44	C₂×C₅×M₁₆	—	2	4	no	—
45	C₂×C₅×M₁₆	C₂³	1	2	yes	—
46	C₂×C₅×M₁₆	C₂³	1	4	yes	—
47	C₂²×C₅×C₈	—	2	4	no	—
48	C₂²×C₅×C₈	—	2	1	no	—
49	C₂²×C₅×C₈	—	2	2	no	—
50	C₂²×C₅×C₈	D₄	1	2	yes	—
51	C₂²×C₅×C₈	D₄	1	2	yes	—
52	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
53	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
54	C₅×#117	—	2	2	no	—
55	C₅×#117	—	2	2	no	—
56	C₅×#117	—	2	2	no	—
57	C₅×#117	—	2	2	no	—
58	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
59	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
60	C₂×C₄²×C₅	—	2	2	no	—
61	C₅×#107	—	2	4	no	—
62	C₅×#106	—	2	4	no	—
63	C₅×#103	—	2	8	no	—
64	C₅×#102	—	2	8	no	—
65	C₅×C₂²⋊C₈	—	2	4	no	—
66	C₅×C₂²⋊C₈	—	2	4	no	—
67	C₂×#427	C₂×C₅	1	1	yes	characteristic, complemented
68	#2896	C₂×C₅	1	2	yes	—
69	C₂×#408	C₂×C₅	1	1	yes	characteristic
70	#2079	C₂×C₅	1	4	yes	complemented
71	#2078	C₂×C₅	1	4	yes	complemented
72	#1742	C₂×C₅	1	2	yes	complemented
73	C₂×#341	C₂×C₅	1	1	yes	characteristic, complemented
74	C₂⁴×C₅	—	2	1	no	—
75	C₂⁴×C₅	#41	1	1	yes	characteristic
76	C₅×#41	—	2	8	no	—
77	C₅×#41	—	2	4	no	—
78	C₅×#41	—	2	4	no	—
79	C₅×#41	—	4	2	no	—
80	C₅×#41	—	4	2	no	—
81	C₅×#41	—	4	4	no	—
82	C₂×C₅×Q₈	—	2	1	no	—
83	C₂×C₅×Q₈	—	2	4	no	—
84	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
85	C₂×C₅×D₄	—	2	2	no	—
86	C₂×C₅×D₄	—	2	2	no	—
87	C₂×C₅×D₄	—	2	1	no	—
88	C₂×C₅×D₄	—	2	4	no	—
89	C₂×C₅×D₄	—	4	4	no	—
90	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₅	—	2	1	no	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	—	2	4	no	—
95	C₂²×C₄×C₅	—	4	2	no	—
96	C₂²×C₄×C₅	—	4	2	no	—
97	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	#41	1	1	yes	characteristic
102	C₂²×C₄×C₅	#41	1	4	yes	—
103	C₂²×C₄×C₅	#41	1	4	yes	—
104	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
105	C₅×M₁₆	—	2	4	no	—
106	C₅×M₁₆	—	2	4	no	—
107	C₅×M₁₆	—	2	4	no	—
108	C₅×M₁₆	—	4	4	no	—
109	C₅×M₁₆	—	4	2	no	—
110	C₂×C₅×C₈	—	2	4	no	—
111	C₂×C₅×C₈	—	2	4	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	—	2	4	no	—
116	C₂×C₅×C₈	—	4	4	no	—
117	C₂×C₅×C₈	—	4	2	no	—
118	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
119	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
120	C₅×C₄⋊C₄	—	2	4	no	—
121	C₅×C₄⋊C₄	—	2	4	no	—
122	C₅×C₂²⋊C₄	—	4	2	no	—
123	C₅×C₂²⋊C₄	—	4	2	no	—
124	C₄²×C₅	—	4	2	no	—
125	C₄²×C₅	—	4	2	no	—
126	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
127	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
128	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
129	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
130	C₂×#117	C₂²×C₅	1	2	yes	—
131	#427	—	2	2	no	—
132	#427	—	2	2	no	—
133	C₂×#107	C₂²×C₅	1	2	yes	complemented
134	C₂×#106	C₂²×C₅	1	2	yes	complemented
135	#419	—	2	2	no	—
136	#418	—	2	2	no	—
137	#417	C₂²×C₅	1	8	yes	—
138	#416	—	2	4	no	—
139	#408	—	2	2	no	—
140	#408	C₂²×C₅	1	2	yes	—
141	#408	C₂²×C₅	1	2	yes	—
142	C₂×C₂²⋊C₈	—	2	2	no	—
143	#343	C₂²×C₅	1	8	yes	complemented
144	#341	C₂²×C₅	1	2	yes	complemented
145	#341	C₂²×C₅	1	2	yes	complemented
146	C₂³×C₅	—	2	2	no	—
147	C₂³×C₅	—	4	2	no	—
148	C₂³×C₅	—	4	4	no	—
149	C₂³×C₅	—	4	2	no	—
150	C₂³×C₅	#121	1	1	yes	characteristic
151	C₂³×C₅	#121	1	1	yes	characteristic
152	C₂³×C₅	#126	1	4	yes	—
153	C₂³×C₅	C₂×#41	1	1	yes	characteristic
154	C₅×Q₈	—	2	4	no	—
155	C₅×Q₈	—	4	2	no	—
156	C₅×D₄	—	2	4	no	—
157	C₅×D₄	—	4	4	no	—
158	C₅×D₄	—	4	4	no	—
159	C₅×D₄	—	4	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	4	2	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	—	4	4	no	—
165	C₂×C₄×C₅	—	4	4	no	—
166	C₂×C₄×C₅	—	8	4	no	—
167	C₂×C₄×C₅	#121	1	1	yes	characteristic
168	C₂×C₄×C₅	#121	1	1	yes	characteristic
169	C₂×C₄×C₅	#123	1	2	yes	—
170	C₂×C₄×C₅	#123	1	2	yes	—
171	C₂×C₄×C₅	#123	1	4	yes	—
172	C₂×C₄×C₅	#123	1	4	yes	—
173	C₂×C₄×C₅	#124	1	2	yes	—
174	C₂×C₄×C₅	#124	1	2	yes	—
175	C₂×C₄×C₅	#126	1	4	yes	—
176	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×#41	1	2	yes	—
179	C₂×C₄×C₅	C₂×#41	1	2	yes	—
180	C₅×C₈	—	2	2	no	—
181	C₅×C₈	—	2	2	no	—
182	C₅×C₈	—	4	4	no	—
183	C₅×C₈	—	4	2	no	—
184	C₂×#41	—	2	1	no	—
185	C₂×#41	—	2	1	no	—
186	C₂×#41	—	2	4	no	—
187	C₂×#41	—	2	4	no	—
188	C₂×#41	C₅×D₄	1	2	yes	—
189	C₂×#41	C₂³×C₅	1	2	yes	—
190	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
193	C₂³×C₄	—	2	1	no	—
194	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	C₂³×C₅	1	4	yes	—
201	C₂×M₁₆	C₂³×C₅	1	2	yes	—
202	C₂²×C₈	—	2	4	no	—
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	C₅×D₄	1	2	yes	—
206	C₂²×C₈	C₅×D₄	1	2	yes	—
207	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
209	#117	—	2	2	no	—
210	#117	—	2	2	no	—
211	#117	—	2	2	no	—
212	#117	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	#107	—	2	4	no	—
217	#106	—	2	4	no	—
218	#103	—	2	8	no	—
219	#102	—	2	8	no	—
220	C₂²⋊C₈	—	2	4	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	—	2	2	no	—
224	C₂²×C₅	—	2	2	no	—
225	C₂²×C₅	—	4	4	no	—
226	C₂²×C₅	—	4	2	no	—
227	C₂²×C₅	—	4	4	no	—
228	C₂²×C₅	—	8	4	no	—
229	C₂²×C₅	#397	1	2	yes	—
230	C₂²×C₅	#397	1	2	yes	—
231	C₂²×C₅	C₂×#121	1	1	yes	characteristic
232	C₂²×C₅	C₂×#126	1	2	yes	—
233	C₄×C₅	—	4	2	no	—
234	C₄×C₅	—	8	4	no	—
235	C₄×C₅	#397	1	2	yes	—
236	C₄×C₅	#397	1	2	yes	—
237	C₄×C₅	C₂×#123	1	1	yes	characteristic
238	C₄×C₅	C₂×#123	1	2	yes	—
239	C₄×C₅	C₂×#124	1	1	yes	characteristic
240	C₂⁴	—	2	1	no	—
241	C₂⁴	C₅×#41	1	1	yes	characteristic
242	#41	—	2	8	no	—
243	#41	—	2	4	no	—
244	#41	—	2	4	no	—
245	#41	—	4	2	no	—
246	#41	—	4	2	no	—
247	#41	—	4	4	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	4	4	no	—
256	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	2	no	—
263	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
267	C₂²×C₄	C₅×#41	1	1	yes	characteristic
268	C₂²×C₄	C₅×#41	1	4	yes	—
269	C₂²×C₄	C₅×#41	1	4	yes	—
270	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
271	M₁₆	—	2	4	no	—
272	M₁₆	—	2	4	no	—
273	M₁₆	—	2	4	no	—
274	M₁₆	—	4	4	no	—
275	M₁₆	—	4	2	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	4	4	no	—
283	C₂×C₈	—	4	2	no	—
284	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
285	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	—	4	2	no	—
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	—	2	2	no	—
294	C₂×C₅	—	2	2	no	—
295	C₂×C₅	—	8	2	no	—
296	C₂×C₅	C₂×#397	1	1	yes	characteristic
297	C₂×C₅	#3067	1	2	yes	—
298	C₂³	—	2	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	4	2	no	—
302	C₂³	C₅×#121	1	1	yes	characteristic
303	C₂³	C₅×#121	1	1	yes	characteristic
304	C₂³	C₅×#126	1	4	yes	—
305	C₂³	C₂×C₅×#41	1	1	yes	characteristic
306	Q₈	—	2	4	no	—
307	Q₈	—	4	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	8	4	no	—
319	C₂×C₄	C₅×#121	1	1	yes	characteristic
320	C₂×C₄	C₅×#121	1	1	yes	characteristic
321	C₂×C₄	C₅×#123	1	2	yes	—
322	C₂×C₄	C₅×#123	1	2	yes	—
323	C₂×C₄	C₅×#123	1	4	yes	—
324	C₂×C₄	C₅×#123	1	4	yes	—
325	C₂×C₄	C₅×#124	1	2	yes	—
326	C₂×C₄	C₅×#124	1	2	yes	—
327	C₂×C₄	C₅×#126	1	4	yes	—
328	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
330	C₂×C₄	C₂×C₅×#41	1	2	yes	—
331	C₂×C₄	C₂×C₅×#41	1	2	yes	—
332	C₈	—	2	2	no	—
333	C₈	—	2	2	no	—
334	C₈	—	4	4	no	—
335	C₈	—	4	2	no	—
336	C₅	#23646	1	1	yes	characteristic, direct product factor, Sylow
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	4	no	—
344	C₂²	C₅×#397	1	2	yes	—
345	C₂²	C₅×#397	1	2	yes	—
346	C₂²	C₂×C₅×#121	1	1	yes	characteristic
347	C₂²	C₂×C₅×#126	1	2	yes	—
348	C₄	—	4	2	no	—
349	C₄	—	8	4	no	—
350	C₄	C₅×#397	1	2	yes	—
351	C₄	C₅×#397	1	2	yes	—
352	C₄	C₂×C₅×#123	1	1	yes	characteristic
353	C₄	C₂×C₅×#123	1	2	yes	—
354	C₄	C₂×C₅×#124	1	1	yes	characteristic
355	C₂	—	2	2	no	—
356	C₂	—	2	2	no	—
357	C₂	—	2	2	no	—
358	C₂	—	8	2	no	—
359	C₂	C₂×C₅×#397	1	1	yes	characteristic
360	C₂	C₅×#3067	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×#121
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#427	C₂	semidirect
4. C₅×#2079	C₂	semidirect
5. C₅×#2078	C₂	semidirect
6. C₅×#1742	C₂	semidirect
7. C₂×C₅×#341	C₂	semidirect
15. C₂×C₅×#107	C₂²	semidirect
16. C₂×C₅×#106	C₂²	semidirect
25. C₅×#343	C₂²	semidirect
26. C₅×#341	C₂²	semidirect
27. C₅×#341	C₂²	semidirect
28. #23646	C₅	direct (N × H)
67. C₂×#427	C₂×C₅	semidirect
70. #2079	C₂×C₅	semidirect
71. #2078	C₂×C₅	semidirect
72. #1742	C₂×C₅	semidirect
73. C₂×#341	C₂×C₅	semidirect
133. C₂×#107	C₂²×C₅	semidirect
134. C₂×#106	C₂²×C₅	semidirect
143. #343	C₂²×C₅	semidirect
144. #341	C₂²×C₅	semidirect
145. #341	C₂²×C₅	semidirect
336. C₅	#23646	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×#427	2
C₅×#2896	2
C₂×C₅×#408	2
C₅×#2079	2
C₅×#2078	2
C₅×#1742	2
C₂×C₅×#341	2
#23646	5

Subgroups — C5×#23646 #11946366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#23646 #11946366