Subgroups — C₅×#9818 #11932538

Order 1280.

536 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
2	C₅×#1810	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1752	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1507	C₂	1	2	yes	maximal, complemented
5	C₅×#1471	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×D₈	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#121	C₂²	1	2	yes	—
9	C₅×#466	—	2	1	no	—
10	C₅×#466	—	2	1	no	—
11	C₅×#466	—	2	1	no	—
12	C₅×#466	—	2	1	no	—
13	C₂×C₅×#108	—	2	1	no	—
14	C₂×C₅×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₅×#102	C₄	1	2	yes	complemented
16	C₅×#407	C₂²	1	2	yes	—
17	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×#97	—	2	1	no	—
19	C₄×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
20	C₅×#341	—	2	1	no	—
21	C₅×#336	—	2	1	no	—
22	C₅×#327	—	2	4	no	—
23	C₅×#323	—	2	4	no	—
24	#9818	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×D₄	—	2	2	no	—
26	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂×C₅×D₈	—	2	1	no	—
30	C₂×C₅×D₈	—	4	2	no	—
31	C₂×C₅×M₁₆	—	2	2	no	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₂²×C₅×C₈	—	4	2	no	—
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₅×#121	—	2	2	no	—
41	C₅×#121	—	2	2	no	—
42	C₅×#121	—	2	2	no	—
43	C₅×#121	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
46	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
47	C₂×C₄²×C₅	—	2	1	no	—
48	C₅×#108	—	4	2	no	—
49	C₅×#107	—	2	1	no	—
50	C₅×#107	—	2	1	no	—
51	C₅×#102	—	2	2	no	—
52	C₅×#102	—	2	2	no	—
53	C₅×C₂²⋊C₈	—	2	1	no	—
54	C₅×C₂²⋊C₈	—	2	2	no	—
55	C₅×C₂²⋊C₈	—	2	2	no	—
56	C₅×C₂²⋊C₈	D₄	1	2	yes	—
57	C₅×#97	—	4	2	no	—
58	C₅×#95	D₄	1	2	yes	—
59	C₂×#466	C₂×C₅	1	1	yes	characteristic
60	#1810	C₂×C₅	1	1	yes	characteristic, complemented
61	#1752	C₂×C₅	1	1	yes	characteristic, complemented
62	#1507	C₂×C₅	1	2	yes	complemented
63	#1471	C₂×C₅	1	2	yes	complemented
64	C₂⁴×C₅	—	2	2	no	—
65	C₂⁴×C₅	#41	1	1	yes	characteristic
66	C₂×C₅×D₄	—	2	2	no	—
67	C₂×C₅×D₄	—	2	2	no	—
68	C₂×C₅×D₄	—	4	2	no	—
69	C₂×C₅×D₄	—	4	2	no	—
70	C₂×C₅×D₄	—	4	2	no	—
71	C₂×C₅×D₄	—	4	2	no	—
72	C₂×C₅×D₄	—	8	4	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	—	4	1	no	—
76	C₂²×C₄×C₅	—	4	1	no	—
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	#41	1	1	yes	characteristic
83	C₅×D₈	—	4	2	no	—
84	C₅×D₈	—	4	2	no	—
85	C₅×M₁₆	—	4	2	no	—
86	C₅×M₁₆	—	4	2	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	4	1	no	—
92	C₂×C₅×C₈	—	4	1	no	—
93	C₂×C₅×C₈	—	4	1	no	—
94	C₂×C₅×C₈	—	4	1	no	—
95	C₂×C₅×C₈	—	4	1	no	—
96	C₂×C₅×C₈	—	4	2	no	—
97	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₂²⋊C₄	—	4	2	no	—
104	C₅×C₂²⋊C₄	—	4	2	no	—
105	C₅×C₂²⋊C₄	—	4	2	no	—
106	C₄²×C₅	—	4	1	no	—
107	C₄²×C₅	—	4	1	no	—
108	C₂²×D₈	C₄×C₅	1	1	yes	characteristic, complemented
109	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂×#121	C₂²×C₅	1	2	yes	—
111	#466	—	2	1	no	—
112	#466	—	2	1	no	—
113	#466	—	2	1	no	—
114	#466	—	2	1	no	—
115	C₂×#108	—	2	1	no	—
116	C₂×#107	C₄×C₅	1	1	yes	characteristic, complemented
117	C₂×#102	C₄×C₅	1	2	yes	complemented
118	#407	C₂²×C₅	1	2	yes	—
119	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
120	C₂×#97	—	2	1	no	—
121	C₄×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
122	#341	—	2	1	no	—
123	#336	—	2	1	no	—
124	#327	—	2	4	no	—
125	#323	—	2	4	no	—
126	C₂³×C₅	—	2	2	no	—
127	C₂³×C₅	—	2	2	no	—
128	C₂³×C₅	—	4	1	no	—
129	C₂³×C₅	—	4	2	no	—
130	C₂³×C₅	—	4	4	no	—
131	C₂³×C₅	—	8	2	no	—
132	C₂³×C₅	—	8	2	no	—
133	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
134	C₂³×C₅	#121	1	1	yes	characteristic
135	C₂³×C₅	#124	1	1	yes	characteristic
136	C₅×D₄	—	4	2	no	—
137	C₅×D₄	—	4	2	no	—
138	C₅×D₄	—	8	2	no	—
139	C₅×D₄	—	8	4	no	—
140	C₅×D₄	—	8	4	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	4	1	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	—	4	1	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	—	8	2	no	—
155	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅	#121	1	1	yes	characteristic
158	C₂×C₄×C₅	#127	1	1	yes	characteristic
159	C₅×C₈	—	2	1	no	—
160	C₅×C₈	—	2	1	no	—
161	C₅×C₈	—	2	1	no	—
162	C₅×C₈	—	4	2	no	—
163	C₅×C₈	—	8	2	no	—
164	C₅×C₈	#99	1	2	yes	—
165	C₂²×D₄	—	2	2	no	—
166	C₂²×D₄	C₂×C₄×C₅	1	2	yes	—
167	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
168	C₂×D₈	—	2	1	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	4	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	4	2	no	—
174	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
175	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
176	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
177	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
178	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
179	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
180	#121	—	2	2	no	—
181	#121	—	2	2	no	—
182	#121	—	2	2	no	—
183	#121	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	—	2	1	no	—
188	#108	—	4	2	no	—
189	#107	—	2	1	no	—
190	#107	—	2	1	no	—
191	#102	—	2	2	no	—
192	#102	—	2	2	no	—
193	C₂²⋊C₈	—	2	1	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	C₅×D₄	1	2	yes	—
197	#97	—	4	2	no	—
198	#95	C₅×D₄	1	2	yes	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	—	2	1	no	—
206	C₂²×C₅	—	4	4	no	—
207	C₂²×C₅	—	8	2	no	—
208	C₂²×C₅	—	8	2	no	—
209	C₂²×C₅	—	8	2	no	—
210	C₂²×C₅	—	16	4	no	—
211	C₂²×C₅	#390	1	1	yes	characteristic
212	C₂²×C₅	#444	1	2	yes	—
213	C₂²×C₅	#472	1	2	yes	—
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	—	8	2	no	—
218	C₄×C₅	—	8	2	no	—
219	C₄×C₅	C₂×#99	1	1	yes	characteristic
220	C₄×C₅	#410	1	1	yes	characteristic
221	C₂⁴	—	2	2	no	—
222	C₂⁴	C₅×#41	1	1	yes	characteristic
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	8	4	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂²×C₄	C₅×#41	1	1	yes	characteristic
240	D₈	—	4	2	no	—
241	D₈	—	4	2	no	—
242	M₁₆	—	4	2	no	—
243	M₁₆	—	4	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	—	4	1	no	—
264	C₄²	—	4	1	no	—
265	C₂×C₅	—	2	2	no	—
266	C₂×C₅	—	2	2	no	—
267	C₂×C₅	—	4	1	no	—
268	C₂×C₅	—	16	2	no	—
269	C₂×C₅	#1964	1	1	yes	characteristic
270	C₂×C₅	#1975	1	1	yes	characteristic
271	C₂×C₅	#1978	1	1	yes	characteristic
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	4	1	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	8	2	no	—
278	C₂³	—	8	2	no	—
279	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
280	C₂³	C₅×#121	1	1	yes	characteristic
281	C₂³	C₅×#124	1	1	yes	characteristic
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	2	no	—
285	D₄	—	8	4	no	—
286	D₄	—	8	4	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
302	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
303	C₂×C₄	C₅×#121	1	1	yes	characteristic
304	C₂×C₄	C₅×#127	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	4	2	no	—
309	C₈	—	8	2	no	—
310	C₈	C₅×#99	1	2	yes	—
311	C₅	#9818	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	4	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	16	4	no	—
324	C₂²	C₅×#390	1	1	yes	characteristic
325	C₂²	C₅×#444	1	2	yes	—
326	C₂²	C₅×#472	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	C₂×C₅×#99	1	1	yes	characteristic
333	C₄	C₅×#410	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	2	2	no	—
336	C₂	—	4	1	no	—
337	C₂	—	16	2	no	—
338	C₂	C₅×#1964	1	1	yes	characteristic
339	C₂	C₅×#1975	1	1	yes	characteristic
340	C₂	C₅×#1978	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#390
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂×C₅×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1810	C₂	semidirect
3. C₅×#1752	C₂	semidirect
4. C₅×#1507	C₂	semidirect
5. C₅×#1471	C₂	semidirect
6. C₂²×C₅×D₈	C₄	semidirect
14. C₂×C₅×#107	C₄	semidirect
15. C₂×C₅×#102	C₄	semidirect
24. #9818	C₅	direct (N × H)
60. #1810	C₂×C₅	semidirect
61. #1752	C₂×C₅	semidirect
62. #1507	C₂×C₅	semidirect
63. #1471	C₂×C₅	semidirect
108. C₂²×D₈	C₄×C₅	semidirect
116. C₂×#107	C₄×C₅	semidirect
117. C₂×#102	C₄×C₅	semidirect
311. C₅	#9818	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×#466	2
C₅×#1810	2
C₅×#1752	2
C₅×#1507	2
C₅×#1471	2
#9818	5

Subgroups — C5×#9818 #11932538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9818 #11932538