Finite Group Databasebeta

← C5×#36676 #11959396

Subgroups — C5×#36676 #11959396

Order 1280.

2006 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2842C211yesmaximal, characteristic, complemented
2C5×#2834C211yesmaximal, characteristic, complemented
3C5×#2827C216yesmaximal, complemented
4C5×#2826C213yesmaximal, complemented
5C5×#2825C212yesmaximal, complemented
6C5×#2812C213yesmaximal, complemented
7C5×#2401C212yesmaximal, complemented
8C5×#2397C211yesmaximal, characteristic, complemented
9C5×#2395C213yesmaximal, complemented
10C5×#2392C216yesmaximal, complemented
11C4×C5×#120C213yesmaximal, complemented
12C24×C4×C5C2211yescharacteristic
13C2×C5×#120C2213yes—
14C2×C4×C5×D4C2216yes—
15C2×C4×C5×D4C2213yes—
16C2×C4×C5×D4C2213yescomplemented
17C2×C5×#117C2213yescomplemented
18C5×#401C2212yescomplemented
19C5×#400C2216yescomplemented
20C5×#400C2213yescomplemented
21C5×#399C2213yes—
22C5×#399C2211yescharacteristic
23C5×#399C2213yescomplemented
24C5×#398C2213yescomplemented
25C5×#398C2216yescomplemented
26C5×#397C2216yes—
27C5×#397C2213yes—
28C5×#397C2216yescomplemented
29C5×#397C2216yescomplemented
30C5×#396C2212yes—
31C5×#396C2216yes—
32C5×#396C2216yescomplemented
33C5×#394C2216yes—
34C5×#394C2216yes—
35C5×#394C2216yes—
36C5×#394C2216yescomplemented
37C5×#392C2216yes—
38C5×#392C2213yes—
39C5×#392C2211yescharacteristic, complemented
40C5×#386C2216yes—
41C5×#386C2213yes—
42C5×#385C2213yes—
43C5×#385C2216yes—
44C5×#379C2213yes—
45C4×C5×C4⋊C4C2216yescomplemented
46C4×C5×C22⋊C4C2216yes—
47C4×C5×C22⋊C4C2213yes—
48C4×C5×C22⋊C4C2213yescomplemented
49C5×#376C2211yescharacteristic, complemented
50#36676C511yesmaximal, characteristic, direct product factor, Sylow
51C25×C5C2311yescharacteristic
52C22×C5×D4C2316yes—
53C22×C5×D4C2313yes—
54C22×C5×D4C2313yes—
55C23×C4×C5—46no—
56C23×C4×C5C2312yes—
57C23×C4×C5C2311yescharacteristic
58C23×C4×C5C2313yes—
59C23×C4×C5C2311yescharacteristic
60C23×C4×C5C2312yes—
61C5×#120—212no—
62C4×C5×D4—412no—
63C4×C5×D4—46no—
64C4×C5×D4—46no—
65C5×#117—46no—
66C2×C5×C4⋊C4C2316yes—
67C2×C5×C4⋊C4C2316yes—
68C2×C5×C4⋊C4C2316yes—
69C2×C5×C4⋊C4C2313yes—
70C2×C5×C4⋊C4C2313yes—
71C2×C5×C4⋊C4C2313yes—
72C2×C5×C4⋊C4C2313yes—
73C2×C5×C22⋊C4C2316yes—
74C2×C5×C22⋊C4C2313yes—
75C2×C5×C22⋊C4C2316yes—
76C2×C5×C22⋊C4C2313yes—
77C2×C5×C22⋊C4C2316yes—
78C2×C5×C22⋊C4C2313yes—
79C2×C5×C22⋊C4C2316yes—
80C2×C5×C22⋊C4C2313yes—
81C2×C5×C22⋊C4C2316yes—
82C2×C5×C22⋊C4C2316yes—
83C2×C5×C22⋊C4C2316yes—
84C2×C42×C5C2313yes—
85C2×C42×C5C2316yes—
86C5×#95C2313yes—
87C5×#95C2311yescharacteristic
88C5×#95C2316yes—
89C5×#95C2316yes—
90C5×#95C2316yes—
91C5×#95C2313yes—
92C5×#95C2313yes—
93C5×#95C2316yes—
94C5×#95C2316yes—
95#2842C2×C511yescharacteristic, complemented
96#2834C2×C511yescharacteristic, complemented
97#2827C2×C516yescomplemented
98#2826C2×C513yescomplemented
99#2825C2×C512yescomplemented
100#2812C2×C513yescomplemented
101#2401C2×C512yescomplemented
102#2397C2×C511yescharacteristic, complemented
103#2395C2×C513yescomplemented
104#2392C2×C516yescomplemented
105C4×#120C2×C513yescomplemented
106C24×C5—43no—
107C24×C5—43no—
108C24×C5#4114yes—
109C24×C5C2413yes—
110C24×C5C2412yes—
111C24×C5C2411yescharacteristic
112C2×C5×D4—212no—
113C2×C5×D4—26no—
114C2×C5×D4—43no—
115C2×C5×D4—43no—
116C2×C5×D4—46no—
117C2×C5×D4—46no—
118C2×C5×D4—46no—
119C2×C5×D4—43no—
120C22×C4×C5—26no—
121C22×C4×C5—212no—
122C22×C4×C5—212no—
123C22×C4×C5—26no—
124C22×C4×C5—43no—
125C22×C4×C5—46no—
126C22×C4×C5—416no—
127C22×C4×C5C2416yes—
128C22×C4×C5C2413yes—
129C22×C4×C5C2416yes—
130C22×C4×C5C2413yes—
131C22×C4×C5C2413yes—
132C22×C4×C5C2411yescharacteristic
133C22×C4×C5C2412yes—
134C22×C4×C5C2411yescharacteristic
135C5×C4⋊C4—46no—
136C5×C4⋊C4—46no—
137C5×C4⋊C4—43no—
138C5×C4⋊C4—43no—
139C5×C4⋊C4—43no—
140C5×C4⋊C4—43no—
141C5×C4⋊C4—46no—
142C5×C22⋊C4—212no—
143C5×C22⋊C4—26no—
144C5×C22⋊C4—43no—
145C5×C22⋊C4—46no—
146C5×C22⋊C4—46no—
147C5×C22⋊C4—46no—
148C5×C22⋊C4—46no—
149C5×C22⋊C4—46no—
150C5×C22⋊C4—46no—
151C5×C22⋊C4—43no—
152C5×C22⋊C4—43no—
153C42×C5—212no—
154C42×C5—26no—
155C24×C4C22×C511yescharacteristic
156C2×#120C22×C513yes—
157C2×C4×D4C22×C516yes—
158C2×C4×D4C22×C513yes—
159C2×C4×D4C22×C513yescomplemented
160C2×#117C22×C513yescomplemented
161#401C22×C512yescomplemented
162#400C22×C516yescomplemented
163#400C22×C513yescomplemented
164#399C22×C513yes—
165#399C22×C511yescharacteristic
166#399C22×C513yescomplemented
167#398C22×C513yescomplemented
168#398C22×C516yescomplemented
169#397C22×C516yes—
170#397C22×C513yes—
171#397C22×C516yescomplemented
172#397C22×C516yescomplemented
173#396C22×C512yes—
174#396C22×C516yes—
175#396C22×C516yescomplemented
176#394C22×C516yes—
177#394C22×C516yes—
178#394C22×C516yes—
179#394C22×C516yescomplemented
180#392C22×C516yes—
181#392C22×C513yes—
182#392C22×C511yescharacteristic, complemented
183#386C22×C516yes—
184#386C22×C513yes—
185#385C22×C513yes—
186#385C22×C516yes—
187#379C22×C513yes—
188C4×C4⋊C4C22×C516yescomplemented
189C4×C22⋊C4C22×C516yes—
190C4×C22⋊C4C22×C513yes—
191C4×C22⋊C4C22×C513yescomplemented
192#376C22×C511yescharacteristic, complemented
193C23×C5—26no—
194C23×C5—23no—
195C23×C5—23no—
196C23×C5—26no—
197C23×C5—23no—
198C23×C5—23no—
199C23×C5—26no—
200C23×C5—26no—
201C23×C5—23no—
202C23×C5—412no—
203C23×C5—44no—
204C23×C5—412no—
205C23×C5C2×#4114yes—
206C23×C5C2×#4112yes—
207C23×C5C2511yescharacteristic
208C5×D4—412no—
209C5×D4—424no—
210C5×D4—412no—
211C2×C4×C5—23no—
212C2×C4×C5—26no—
213C2×C4×C5—26no—
214C2×C4×C5—23no—
215C2×C4×C5—23no—
216C2×C4×C5—26no—
217C2×C4×C5—26no—
218C2×C4×C5—23no—
219C2×C4×C5—26no—
220C2×C4×C5—23no—
221C2×C4×C5—26no—
222C2×C4×C5—26no—
223C2×C4×C5—26no—
224C2×C4×C5—26no—
225C2×C4×C5—23no—
226C2×C4×C5—44no—
227C2×C4×C5—48no—
228C2×C4×C5—412no—
229C2×C4×C5—48no—
230C2×C4×C5—44no—
231C2×C4×C5H(5,𝔽2)16yes—
232C25C23×C511yescharacteristic
233C22×D4C23×C516yes—
234C22×D4C23×C513yes—
235C22×D4C23×C513yes—
236C23×C4—46no—
237C23×C4C23×C513yes—
238C23×C4C23×C512yes—
239C23×C4C23×C511yescharacteristic
240C23×C4C23×C511yescharacteristic
241C23×C4C23×C512yes—
242#120—212no—
243C4×D4—412no—
244C4×D4—46no—
245C4×D4—46no—
246#117—46no—
247C2×C4⋊C4C23×C513yes—
248C2×C4⋊C4C23×C513yes—
249C2×C4⋊C4C23×C516yes—
250C2×C4⋊C4C23×C513yes—
251C2×C4⋊C4C23×C513yes—
252C2×C4⋊C4C23×C516yes—
253C2×C4⋊C4C23×C516yes—
254C2×C22⋊C4C23×C516yes—
255C2×C22⋊C4C23×C513yes—
256C2×C22⋊C4C23×C516yes—
257C2×C22⋊C4C23×C513yes—
258C2×C22⋊C4C23×C516yes—
259C2×C22⋊C4C23×C513yes—
260C2×C22⋊C4C23×C516yes—
261C2×C22⋊C4C23×C516yes—
262C2×C22⋊C4C23×C516yes—
263C2×C22⋊C4C23×C513yes—
264C2×C22⋊C4C23×C516yes—
265C2×C42C23×C516yes—
266C2×C42C23×C513yes—
267#95C23×C513yes—
268#95C23×C511yescharacteristic
269#95C23×C516yes—
270#95C23×C516yes—
271#95C23×C516yes—
272#95C23×C513yes—
273#95C23×C513yes—
274#95C23×C516yes—
275#95C23×C516yes—
276C22×C5—26no—
277C22×C5—212no—
278C22×C5—416no—
279C22×C5—42no—
280C22×C5—41no—
281C22×C5—46no—
282C22×C5—43no—
283C22×C5—43no—
284C22×C5—46no—
285C22×C5—43no—
286C22×C5—46no—
287C22×C5—43no—
288C22×C5C22×#4111yescharacteristic
289C22×C5C2×H(5,𝔽2)13yes—
290C22×C5#58513yes—
291C4×C5—43no—
292C4×C5—46no—
293C4×C5—43no—
294C4×C5—42no—
295C4×C5—41no—
296C4×C5—46no—
297C4×C5—43no—
298C4×C5#56014yes—
299C24—43no—
300C24—43no—
301C24C5×#4114yes—
302C24C24×C513yes—
303C24C24×C511yescharacteristic
304C24C24×C512yes—
305C2×D4—212no—
306C2×D4—26no—
307C2×D4—43no—
308C2×D4—43no—
309C2×D4—46no—
310C2×D4—46no—
311C2×D4—46no—
312C2×D4—43no—
313C22×C4—26no—
314C22×C4—212no—
315C22×C4—212no—
316C22×C4—26no—
317C22×C4—43no—
318C22×C4—46no—
319C22×C4—416no—
320C22×C4C24×C513yes—
321C22×C4C24×C516yes—
322C22×C4C24×C513yes—
323C22×C4C24×C512yes—
324C22×C4C24×C511yescharacteristic
325C22×C4C24×C516yes—
326C22×C4C24×C513yes—
327C22×C4C24×C511yescharacteristic
328C4⋊C4—43no—
329C4⋊C4—43no—
330C4⋊C4—43no—
331C4⋊C4—43no—
332C4⋊C4—46no—
333C4⋊C4—46no—
334C4⋊C4—46no—
335C22⋊C4—212no—
336C22⋊C4—26no—
337C22⋊C4—46no—
338C22⋊C4—46no—
339C22⋊C4—46no—
340C22⋊C4—46no—
341C22⋊C4—43no—
342C22⋊C4—43no—
343C22⋊C4—43no—
344C22⋊C4—46no—
345C22⋊C4—46no—
346C42—212no—
347C42—26no—
348C2×C5—42no—
349C2×C5—44no—
350C2×C5—83no—
351C2×C5#345913yes—
352C2×C5C2×#56011yescharacteristic
353C2×C5#352913yes—
354C23—26no—
355C23—23no—
356C23—23no—
357C23—26no—
358C23—23no—
359C23—23no—
360C23—26no—
361C23—26no—
362C23—23no—
363C23—412no—
364C23—44no—
365C23—412no—
366C23C2×C5×#4112yes—
367C23C2×C5×#4114yes—
368C23C25×C511yescharacteristic
369D4—412no—
370D4—424no—
371D4—412no—
372C2×C4—23no—
373C2×C4—26no—
374C2×C4—26no—
375C2×C4—23no—
376C2×C4—23no—
377C2×C4—26no—
378C2×C4—26no—
379C2×C4—23no—
380C2×C4—26no—
381C2×C4—23no—
382C2×C4—26no—
383C2×C4—26no—
384C2×C4—26no—
385C2×C4—26no—
386C2×C4—23no—
387C2×C4—44no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4—412no—
392C2×C4C5×H(5,𝔽2)16yes—
393C5#3667611yescharacteristic, direct product factor, Sylow
394C22—26no—
395C22—212no—
396C22—416no—
397C22—42no—
398C22—41no—
399C22—46no—
400C22—43no—
401C22—43no—
402C22—46no—
403C22—43no—
404C22—46no—
405C22—43no—
406C22C22×C5×#4111yescharacteristic
407C22C2×C5×H(5,𝔽2)13yes—
408C22C5×#58513yes—
409C4—43no—
410C4—46no—
411C4—42no—
412C4—41no—
413C4—43no—
414C4—43no—
415C4—46no—
416C4C5×#56014yes—
417C2—42no—
418C2—44no—
419C2—83no—
420C2C5×#345913yes—
421C2C2×C5×#56011yescharacteristic
422C2C5×#352913yes—
subgroupG/H
centerC22×C4×C5C24
commutator, Frattini, cosocleC23C25×C5
socleC23×C5C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#2842C2semidirect
2. C5×#2834C2semidirect
3. C5×#2827C2semidirect
4. C5×#2826C2semidirect
5. C5×#2825C2semidirect
6. C5×#2812C2semidirect
7. C5×#2401C2semidirect
8. C5×#2397C2semidirect
9. C5×#2395C2semidirect
10. C5×#2392C2semidirect
11. C4×C5×#120C2semidirect
16. C2×C4×C5×D4C22semidirect
17. C2×C5×#117C22semidirect
18. C5×#401C22semidirect
19. C5×#400C22semidirect
20. C5×#400C22semidirect
23. C5×#399C22semidirect
24. C5×#398C22semidirect
25. C5×#398C22semidirect
28. C5×#397C22semidirect
29. C5×#397C22semidirect
32. C5×#396C22semidirect
36. C5×#394C22semidirect
39. C5×#392C22semidirect
45. C4×C5×C4⋊C4C22semidirect
48. C4×C5×C22⋊C4C22semidirect
49. C5×#376C22semidirect
50. #36676C5direct (N × H)
95. #2842C2×C5semidirect
96. #2834C2×C5semidirect
97. #2827C2×C5semidirect
98. #2826C2×C5semidirect
99. #2825C2×C5semidirect
100. #2812C2×C5semidirect
101. #2401C2×C5semidirect
102. #2397C2×C5semidirect
103. #2395C2×C5semidirect
104. #2392C2×C5semidirect
105. C4×#120C2×C5semidirect
159. C2×C4×D4C22×C5semidirect
160. C2×#117C22×C5semidirect
161. #401C22×C5semidirect
162. #400C22×C5semidirect
163. #400C22×C5semidirect
166. #399C22×C5semidirect
167. #398C22×C5semidirect
168. #398C22×C5semidirect
171. #397C22×C5semidirect
172. #397C22×C5semidirect
175. #396C22×C5semidirect
179. #394C22×C5semidirect
182. #392C22×C5semidirect
188. C4×C4⋊C4C22×C5semidirect
191. C4×C22⋊C4C22×C5semidirect
192. #376C22×C5semidirect
393. C5#36676direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C5×#28422
C5×#28342
C5×#28272
C5×#28262
C5×#28252
C5×#28122
C5×#24012
C5×#23972
C5×#23952
C5×#23922
C4×C5×#1202
#366765