Subgroups — C₅×#24989 #11947709

Order 1280.

620 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#473	C₂	1	1	yes	maximal, characteristic
2	C₅×#2950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2082	C₂	1	2	yes	maximal, complemented
4	C₅×#2049	C₂	1	2	yes	maximal
5	C₅×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1953	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1905	C₂	1	2	yes	maximal
8	C₅×#1833	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1830	C₂	1	2	yes	maximal, complemented
10	C₅×#1814	C₂	1	1	yes	maximal, characteristic
11	C₅×#1757	C₂	1	1	yes	maximal, characteristic
12	C₂×C₅×#137	C₂²	1	2	yes	—
13	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₅×#131	C₂²	1	1	yes	characteristic
15	C₅×#519	C₂²	1	2	yes	—
16	C₅×#473	—	2	2	no	—
17	C₅×#473	—	2	2	no	—
18	C₅×#441	—	2	2	no	—
19	C₄×C₅×Q₁₆	—	2	2	no	—
20	C₅×#430	C₂²	1	2	yes	—
21	C₂×C₅×#108	C₂²	1	1	yes	characteristic
22	C₂×C₅×#108	C₂²	1	1	yes	characteristic
23	C₅×#427	C₂²	1	1	yes	characteristic
24	C₅×#421	C₂²	1	2	yes	—
25	C₅×#419	C₂²	1	2	yes	—
26	C₅×#419	C₂²	1	2	yes	—
27	C₅×#413	C₂²	1	1	yes	characteristic
28	C₂×C₅×#101	C₂²	1	2	yes	—
29	C₂×C₅×#101	C₂²	1	2	yes	—
30	C₅×#410	C₂²	1	1	yes	characteristic
31	C₅×#405	C₂²	1	1	yes	characteristic
32	C₅×#405	C₂²	1	1	yes	characteristic
33	C₅×#344	C₂²	1	1	yes	characteristic
34	C₅×#344	C₂²	1	1	yes	characteristic
35	C₅×#344	C₂²	1	2	yes	—
36	C₅×#344	C₂²	1	2	yes	—
37	C₅×#344	C₂²	1	2	yes	—
38	C₅×H(3,ℤ/4)	C₂²	1	2	yes	—
39	#24989	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₅×#41	C₂³	1	1	yes	characteristic
41	C₂²×C₅×Q₈	C₂³	1	2	yes	—
42	C₅×#137	—	2	4	no	—
43	C₅×#137	—	4	2	no	—
44	C₂×C₅×Q₁₆	—	2	2	no	—
45	C₂×C₅×Q₁₆	—	2	1	no	—
46	C₂×C₅×Q₁₆	—	2	1	no	—
47	C₂×C₅×Q₁₆	—	2	1	no	—
48	C₂×C₅×Q₁₆	—	2	1	no	—
49	C₂×C₅×Q₁₆	D₄	1	2	yes	—
50	C₂×C₅×Q₁₆	D₄	1	2	yes	—
51	C₂×C₅×SD₁₆	—	2	2	no	—
52	C₅×#131	—	2	1	no	—
53	C₅×#131	—	2	1	no	—
54	C₅×#131	—	2	1	no	—
55	C₅×#131	—	2	1	no	—
56	C₂×C₅×M₁₆	—	2	1	no	—
57	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₅×M₁₆	C₂³	1	2	yes	—
60	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₅×M₁₆	C₂³	1	2	yes	—
62	C₂²×C₅×C₈	—	2	1	no	—
63	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
64	C₄×C₅×Q₈	—	2	4	no	—
65	C₄×C₅×Q₈	—	4	2	no	—
66	C₅×#117	—	2	4	no	—
67	C₅×#117	C₂³	1	1	yes	characteristic
68	C₅×#117	C₂³	1	2	yes	—
69	C₅×#117	C₂³	1	1	yes	characteristic
70	C₅×#108	—	2	1	no	—
71	C₅×#108	—	2	1	no	—
72	C₅×#108	—	2	1	no	—
73	C₅×#108	—	2	1	no	—
74	C₅×#108	—	4	2	no	—
75	C₅×#107	—	2	1	no	—
76	C₅×#106	—	2	1	no	—
77	C₅×#104	—	4	2	no	—
78	C₅×#103	—	2	4	no	—
79	C₅×#103	—	4	2	no	—
80	C₅×#101	—	2	4	no	—
81	C₅×#101	—	2	1	no	—
82	C₅×#101	—	4	2	no	—
83	C₅×#100	—	2	1	no	—
84	C₅×#99	—	2	2	no	—
85	C₅×#97	—	2	1	no	—
86	C₅×#97	—	2	1	no	—
87	C₄×C₅×C₈	—	2	1	no	—
88	C₄×C₅×C₈	—	2	1	no	—
89	C₂×#473	C₂×C₅	1	1	yes	characteristic
90	#2950	C₂×C₅	1	1	yes	characteristic, complemented
91	#2082	C₂×C₅	1	2	yes	complemented
92	#2049	C₂×C₅	1	2	yes	—
93	#1979	C₂×C₅	1	1	yes	characteristic, complemented
94	#1953	C₂×C₅	1	1	yes	characteristic, complemented
95	#1905	C₂×C₅	1	2	yes	—
96	#1833	C₂×C₅	1	1	yes	characteristic, complemented
97	#1830	C₂×C₅	1	2	yes	complemented
98	#1814	C₂×C₅	1	1	yes	characteristic
99	#1757	C₂×C₅	1	1	yes	characteristic
100	C₅×#41	—	2	1	no	—
101	C₅×#41	—	2	1	no	—
102	C₅×#41	—	2	1	no	—
103	C₅×#41	—	2	1	no	—
104	C₂×C₅×Q₈	—	2	2	no	—
105	C₂×C₅×Q₈	—	2	2	no	—
106	C₂×C₅×Q₈	—	2	2	no	—
107	C₂×C₅×Q₈	—	2	2	no	—
108	C₂×C₅×Q₈	—	2	2	no	—
109	C₂×C₅×Q₈	C₂×D₄	1	4	yes	—
110	C₂×C₅×Q₈	#41	1	1	yes	characteristic
111	C₂×C₅×D₄	—	2	1	no	—
112	C₂×C₅×D₄	#41	1	1	yes	characteristic
113	C₂²×C₄×C₅	—	2	2	no	—
114	C₂²×C₄×C₅	—	2	1	no	—
115	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
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	4	no	—
121	C₅×SD₁₆	—	4	4	no	—
122	C₅×M₁₆	—	2	1	no	—
123	C₅×M₁₆	—	2	2	no	—
124	C₅×M₁₆	—	2	1	no	—
125	C₅×M₁₆	—	2	1	no	—
126	C₅×M₁₆	—	2	1	no	—
127	C₅×M₁₆	—	2	2	no	—
128	C₅×M₁₆	—	2	2	no	—
129	C₅×M₁₆	—	2	1	no	—
130	C₅×M₁₆	—	2	2	no	—
131	C₅×M₁₆	—	2	1	no	—
132	C₅×M₁₆	—	4	1	no	—
133	C₅×M₁₆	—	4	1	no	—
134	C₂×C₅×C₈	—	2	1	no	—
135	C₂×C₅×C₈	—	2	1	no	—
136	C₂×C₅×C₈	—	2	2	no	—
137	C₂×C₅×C₈	—	2	1	no	—
138	C₂×C₅×C₈	—	2	2	no	—
139	C₂×C₅×C₈	—	4	1	no	—
140	C₂×C₅×C₈	—	4	1	no	—
141	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
142	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₅×C₈	#41	1	1	yes	characteristic
144	C₂×C₅×C₈	#41	1	1	yes	characteristic
145	C₂×C₅×C₈	#41	1	1	yes	characteristic
146	C₂×C₅×C₈	#41	1	1	yes	characteristic
147	C₅×C₄⋊C₄	—	2	1	no	—
148	C₅×C₄⋊C₄	—	2	2	no	—
149	C₅×C₄⋊C₄	—	2	1	no	—
150	C₅×C₄⋊C₄	—	4	4	no	—
151	C₅×C₂²⋊C₄	—	2	2	no	—
152	C₅×C₂²⋊C₄	—	2	1	no	—
153	C₅×C₂²⋊C₄	#41	1	2	yes	—
154	C₄²×C₅	—	2	1	no	—
155	C₄²×C₅	—	2	1	no	—
156	C₄²×C₅	—	2	2	no	—
157	C₄²×C₅	—	4	4	no	—
158	C₂×#137	C₂²×C₅	1	2	yes	—
159	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
160	C₂×#131	C₂²×C₅	1	1	yes	characteristic
161	#519	C₂²×C₅	1	2	yes	—
162	#473	—	2	2	no	—
163	#473	—	2	2	no	—
164	#441	—	2	2	no	—
165	C₄×Q₁₆	—	2	2	no	—
166	#430	C₂²×C₅	1	2	yes	—
167	C₂×#108	C₂²×C₅	1	1	yes	characteristic
168	C₂×#108	C₂²×C₅	1	1	yes	characteristic
169	#427	C₂²×C₅	1	1	yes	characteristic
170	#421	C₂²×C₅	1	2	yes	—
171	#419	C₂²×C₅	1	2	yes	—
172	#419	C₂²×C₅	1	2	yes	—
173	#413	C₂²×C₅	1	1	yes	characteristic
174	C₂×#101	C₂²×C₅	1	2	yes	—
175	C₂×#101	C₂²×C₅	1	2	yes	—
176	#410	C₂²×C₅	1	1	yes	characteristic
177	#405	C₂²×C₅	1	1	yes	characteristic
178	#405	C₂²×C₅	1	1	yes	characteristic
179	#344	C₂²×C₅	1	2	yes	—
180	#344	C₂²×C₅	1	2	yes	—
181	#344	C₂²×C₅	1	1	yes	characteristic
182	#344	C₂²×C₅	1	1	yes	characteristic
183	#344	C₂²×C₅	1	2	yes	—
184	H(3,ℤ/4)	C₂²×C₅	1	2	yes	—
185	C₂³×C₅	—	2	1	no	—
186	C₂³×C₅	C₂×#41	1	1	yes	characteristic
187	C₅×Q₈	—	2	4	no	—
188	C₅×Q₈	—	2	4	no	—
189	C₅×Q₈	—	2	1	no	—
190	C₅×Q₈	—	2	1	no	—
191	C₅×Q₈	—	4	2	no	—
192	C₅×Q₈	—	4	2	no	—
193	C₅×D₄	—	2	1	no	—
194	C₅×D₄	—	2	1	no	—
195	C₅×D₄	—	4	1	no	—
196	C₅×D₄	—	4	1	no	—
197	C₂×C₄×C₅	—	2	1	no	—
198	C₂×C₄×C₅	—	2	2	no	—
199	C₂×C₄×C₅	—	2	4	no	—
200	C₂×C₄×C₅	—	4	2	no	—
201	C₂×C₄×C₅	—	4	2	no	—
202	C₂×C₄×C₅	—	4	1	no	—
203	C₂×C₄×C₅	—	4	1	no	—
204	C₂×C₄×C₅	—	4	2	no	—
205	C₂×C₄×C₅	—	4	1	no	—
206	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
211	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
212	C₅×C₈	—	4	1	no	—
213	C₅×C₈	—	4	2	no	—
214	C₅×C₈	—	4	1	no	—
215	C₅×C₈	—	4	2	no	—
216	C₅×C₈	—	4	1	no	—
217	C₅×C₈	#123	1	1	yes	characteristic
218	C₅×C₈	#123	1	1	yes	characteristic
219	C₅×C₈	#123	1	1	yes	characteristic
220	C₅×C₈	#123	1	1	yes	characteristic
221	C₂×#41	C₂³×C₅	1	1	yes	characteristic
222	C₂²×Q₈	C₂³×C₅	1	2	yes	—
223	#137	—	2	4	no	—
224	#137	—	4	2	no	—
225	C₂×Q₁₆	—	2	2	no	—
226	C₂×Q₁₆	—	2	1	no	—
227	C₂×Q₁₆	—	2	1	no	—
228	C₂×Q₁₆	—	2	1	no	—
229	C₂×Q₁₆	—	2	1	no	—
230	C₂×Q₁₆	C₅×D₄	1	2	yes	—
231	C₂×Q₁₆	C₅×D₄	1	2	yes	—
232	C₂×SD₁₆	—	2	2	no	—
233	#131	—	2	1	no	—
234	#131	—	2	1	no	—
235	#131	—	2	1	no	—
236	#131	—	2	1	no	—
237	C₂×M₁₆	—	2	1	no	—
238	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
239	C₂×M₁₆	C₂³×C₅	1	2	yes	—
240	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
241	C₂×M₁₆	C₂³×C₅	1	2	yes	—
242	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
245	C₄×Q₈	—	2	4	no	—
246	C₄×Q₈	—	4	2	no	—
247	#117	—	2	4	no	—
248	#117	C₂³×C₅	1	2	yes	—
249	#117	C₂³×C₅	1	1	yes	characteristic
250	#117	C₂³×C₅	1	1	yes	characteristic
251	#108	—	2	1	no	—
252	#108	—	2	1	no	—
253	#108	—	2	1	no	—
254	#108	—	2	1	no	—
255	#108	—	4	2	no	—
256	#107	—	2	1	no	—
257	#106	—	2	1	no	—
258	#104	—	4	2	no	—
259	#103	—	2	4	no	—
260	#103	—	4	2	no	—
261	#101	—	2	4	no	—
262	#101	—	2	1	no	—
263	#101	—	4	2	no	—
264	#100	—	2	1	no	—
265	#99	—	2	2	no	—
266	#97	—	2	1	no	—
267	#97	—	2	1	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	2	1	no	—
270	C₂²×C₅	—	4	1	no	—
271	C₂²×C₅	—	4	1	no	—
272	C₂²×C₅	—	8	1	no	—
273	C₂²×C₅	#529	1	1	yes	characteristic
274	C₂²×C₅	#546	1	1	yes	characteristic
275	C₂²×C₅	#555	1	1	yes	characteristic
276	C₄×C₅	—	4	2	no	—
277	C₄×C₅	—	4	1	no	—
278	C₄×C₅	—	4	4	no	—
279	C₄×C₅	—	8	2	no	—
280	C₄×C₅	—	8	1	no	—
281	C₄×C₅	C₂×#123	1	1	yes	characteristic
282	C₄×C₅	#540	1	1	yes	characteristic
283	C₄×C₅	#546	1	1	yes	characteristic
284	C₄×C₅	#553	1	1	yes	characteristic
285	#41	—	2	1	no	—
286	#41	—	2	1	no	—
287	#41	—	2	1	no	—
288	#41	—	2	1	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	C₂×C₅×D₄	1	4	yes	—
295	C₂×Q₈	C₅×#41	1	1	yes	characteristic
296	C₂×D₄	—	2	1	no	—
297	C₂×D₄	C₅×#41	1	1	yes	characteristic
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
301	Q₁₆	—	2	2	no	—
302	Q₁₆	—	2	2	no	—
303	Q₁₆	—	2	2	no	—
304	Q₁₆	—	2	2	no	—
305	Q₁₆	—	4	4	no	—
306	SD₁₆	—	4	4	no	—
307	M₁₆	—	2	1	no	—
308	M₁₆	—	2	2	no	—
309	M₁₆	—	2	1	no	—
310	M₁₆	—	2	1	no	—
311	M₁₆	—	2	2	no	—
312	M₁₆	—	2	1	no	—
313	M₁₆	—	2	1	no	—
314	M₁₆	—	2	2	no	—
315	M₁₆	—	2	1	no	—
316	M₁₆	—	2	2	no	—
317	M₁₆	—	4	1	no	—
318	M₁₆	—	4	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	4	1	no	—
325	C₂×C₈	—	4	1	no	—
326	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
327	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
328	C₂×C₈	C₅×#41	1	1	yes	characteristic
329	C₂×C₈	C₅×#41	1	1	yes	characteristic
330	C₂×C₈	C₅×#41	1	1	yes	characteristic
331	C₂×C₈	C₅×#41	1	1	yes	characteristic
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	4	4	no	—
336	C₂²⋊C₄	—	2	1	no	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₂²⋊C₄	C₅×#41	1	2	yes	—
339	C₄²	—	2	1	no	—
340	C₄²	—	2	1	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	4	4	no	—
343	C₂×C₅	—	2	1	no	—
344	C₂×C₅	—	2	1	no	—
345	C₂×C₅	—	2	1	no	—
346	C₂×C₅	—	8	1	no	—
347	C₂×C₅	#2420	1	1	yes	characteristic
348	C₂³	—	2	1	no	—
349	C₂³	C₂×C₅×#41	1	1	yes	characteristic
350	Q₈	—	2	4	no	—
351	Q₈	—	2	4	no	—
352	Q₈	—	2	1	no	—
353	Q₈	—	2	1	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	4	2	no	—
356	D₄	—	2	1	no	—
357	D₄	—	2	1	no	—
358	D₄	—	4	1	no	—
359	D₄	—	4	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
370	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
371	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
372	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
373	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
374	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
375	C₈	—	4	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	4	2	no	—
378	C₈	—	4	1	no	—
379	C₈	—	4	1	no	—
380	C₈	C₅×#123	1	1	yes	characteristic
381	C₈	C₅×#123	1	1	yes	characteristic
382	C₈	C₅×#123	1	1	yes	characteristic
383	C₈	C₅×#123	1	1	yes	characteristic
384	C₅	#24989	1	1	yes	characteristic, direct product factor, Sylow
385	C₂²	—	4	1	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	8	1	no	—
388	C₂²	C₅×#529	1	1	yes	characteristic
389	C₂²	C₅×#546	1	1	yes	characteristic
390	C₂²	C₅×#555	1	1	yes	characteristic
391	C₄	—	4	2	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	4	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	1	no	—
396	C₄	C₂×C₅×#123	1	1	yes	characteristic
397	C₄	C₅×#540	1	1	yes	characteristic
398	C₄	C₅×#546	1	1	yes	characteristic
399	C₄	C₅×#553	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	8	1	no	—
404	C₂	C₅×#2420	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	#2420
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2950	C₂	semidirect
3. C₅×#2082	C₂	semidirect
5. C₅×#1979	C₂	semidirect
6. C₅×#1953	C₂	semidirect
8. C₅×#1833	C₂	semidirect
9. C₅×#1830	C₂	semidirect
39. #24989	C₅	direct (N × H)
90. #2950	C₂×C₅	semidirect
91. #2082	C₂×C₅	semidirect
93. #1979	C₂×C₅	semidirect
94. #1953	C₂×C₅	semidirect
96. #1833	C₂×C₅	semidirect
97. #1830	C₂×C₅	semidirect
384. C₅	#24989	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×#473	2
C₅×#2950	2
C₅×#2082	2
C₅×#2049	2
C₅×#1979	2
C₅×#1953	2
C₅×#1905	2
C₅×#1833	2
C₅×#1830	2
C₅×#1814	2
C₅×#1757	2
#24989	5

Subgroups — C5×#24989 #11947709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#24989 #11947709