Finite Group Databasebeta

← C2×C5×#3518 #11985511

Subgroups — C2×C5×#3518 #11985511

Order 1280.

7118 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3518C2132yesmaximal, direct product factor
2C2×C5×Q82C211yesmaximal, characteristic, complemented
3C2×C5×#556C216yesmaximal, complemented
4C2×C5×#556C213yesmaximal, complemented
5C2×C5×#555C213yesmaximal, complemented
6C2×C5×D4×Q8C211yesmaximal, characteristic, complemented
7C2×C5×#548C212yesmaximal, complemented
8C2×C5×#542C216yesmaximal, complemented
9C2×C5×#533C213yesmaximal, complemented
10C2×C5×#529C213yesmaximal, complemented
11C2×C5×#519C212yesmaximal, complemented
12C2×C4×C5×#41C211yesmaximal, characteristic, complemented
13C22×C5×#41C2211yescharacteristic
14C23×C5×Q8C2212yes—
15C5×Q82C22116yescomplemented
16C5×#556C22196yescomplemented
17C5×#556C22148yescomplemented
18C5×#555C22148yescomplemented
19C5×D4×Q8C22116yescomplemented
20C5×#548C22132yescomplemented
21C5×#542C22196yescomplemented
22C5×#533C22148yescomplemented
23C2×C5×C4⋊Q8C2213yes—
24C2×C5×C4⋊Q8C2213yes—
25C2×C5×C4⋊Q8C2216yes—
26C5×#529C22148yescomplemented
27C2×C5×#126C2216yes—
28C2×C5×#126C22112yes—
29C2×C5×#125C2213yes—
30C2×C5×#124C2211yescharacteristic
31C2×C5×#124C2216yes—
32C2×C5×#124C22112yes—
33C2×C5×#123C2216yes—
34C2×C5×C22⋊Q8C2213yes—
35C2×C5×C22⋊Q8C2216yes—
36C2×C5×C22⋊Q8C2216yes—
37C2×C5×C22⋊Q8C2216yes—
38C2×C5×#121C2213yes—
39C2×C5×#121C2216yes—
40C5×#519C22132yescomplemented
41C4×C5×#41C22116yescomplemented
42C2×C4×C5×Q8C2211yescharacteristic
43C2×C4×C5×Q8C2211yescharacteristic
44C2×C4×C5×Q8C2216yes—
45C2×C4×C5×Q8C2212yes—
46C2×C4×C5×Q8C2216yes—
47C2×C4×C5×Q8C2213yes—
48C2×C4×C5×Q8C2216yes—
49C2×C4×C5×D4C2213yes—
50C2×C4×C5×D4C2213yes—
51C2×C4×C5×D4C2216yes—
52C2×C4×C5×D4C2213yes—
53C2×C5×#117C2213yes—
54C2×C5×#117C2216yes—
55C2×C5×#117C2216yes—
56C2×C5×#117C2216yes—
57C22×C42×C5C2213yes—
58C2×#3518C511yesmaximal, characteristic, direct product factor, Sylow
59C2×C5×#41—28no—
60C2×C5×#41C2318yes—
61C22×C5×Q8—216no—
62C22×C5×Q8C2311yescharacteristic
63C22×C5×Q8C2316yes—
64C22×C5×Q8C23116yes—
65C22×C5×Q8C2316yes—
66C22×C5×Q8C2311yescharacteristic
67C22×C5×D4C2311yescharacteristic
68C22×C5×D4C2313yes—
69C22×C5×D4C2316yes—
70C23×C4×C5C2313yes—
71C23×C4×C5C2313yes—
72C23×C4×C5C2316yes—
73C23×C4×C5C2313yes—
74C5×C4⋊Q8C23124yes—
75C5×C4⋊Q8C23124yes—
76C5×C4⋊Q8C23148yes—
77C5×#126C23148yes—
78C5×#126C23196yes—
79C5×#125C23124yes—
80C5×#124C2318yes—
81C5×#124C23148yes—
82C5×#124C23196yes—
83C5×#123C23148yes—
84C5×C22⋊Q8C23124yes—
85C5×C22⋊Q8C23148yes—
86C5×C22⋊Q8C23148yes—
87C5×C22⋊Q8C23148yes—
88C5×#121C23124yes—
89C5×#121C23148yes—
90C4×C5×Q8C2318yes—
91C4×C5×Q8C2318yes—
92C4×C5×Q8C23148yes—
93C4×C5×Q8C23116yes—
94C4×C5×Q8C23148yes—
95C4×C5×Q8C23124yes—
96C4×C5×Q8C23148yes—
97C4×C5×D4C23124yes—
98C4×C5×D4C23124yes—
99C4×C5×D4C23148yes—
100C4×C5×D4C23124yes—
101C5×#117C23124yes—
102C5×#117C23148yes—
103C5×#117C23148yes—
104C5×#117C23148yes—
105C2×C5×C4⋊C4C2313yes—
106C2×C5×C4⋊C4C2313yes—
107C2×C5×C4⋊C4C2313yes—
108C2×C5×C4⋊C4C2313yes—
109C2×C5×C4⋊C4C2316yes—
110C2×C5×C4⋊C4C2316yes—
111C2×C5×C4⋊C4C2316yes—
112C2×C5×C4⋊C4C2316yes—
113C2×C5×C4⋊C4C2316yes—
114C2×C5×C4⋊C4C2316yes—
115C2×C5×C22⋊C4C2316yes—
116C2×C5×C22⋊C4C2312yes—
117C2×C5×C22⋊C4C2316yes—
118C2×C5×C22⋊C4C23112yes—
119C2×C5×C22⋊C4C2312yes—
120C2×C5×C22⋊C4C2316yes—
121C2×C5×C22⋊C4C2316yes—
122C2×C42×C5C2311yescharacteristic
123C2×C42×C5C2313yes—
124C2×C42×C5C2313yes—
125C2×C42×C5C2313yes—
126C2×C42×C5C23124yes—
127C2×C42×C5C23112yes—
128C2×C42×C5C2316yes—
129#3518C2×C5132yesdirect product factor
130C2×Q82C2×C511yescharacteristic, complemented
131C2×#556C2×C516yescomplemented
132C2×#556C2×C513yescomplemented
133C2×#555C2×C513yescomplemented
134C2×D4×Q8C2×C511yescharacteristic, complemented
135C2×#548C2×C512yescomplemented
136C2×#542C2×C516yescomplemented
137C2×#533C2×C513yescomplemented
138C2×#529C2×C513yescomplemented
139C2×#519C2×C512yescomplemented
140C2×C4×#41C2×C511yescharacteristic, complemented
141C24×C5C2412yes—
142C24×C5C2413yes—
143C5×#41—232no—
144C2×C5×Q8—264no—
145C2×C5×Q8—224no—
146C2×C5×Q8—24no—
147C2×C5×Q8—412no—
148C2×C5×Q8#4118yes—
149C2×C5×Q8C2414yes—
150C2×C5×Q8C24124yes—
151C2×C5×Q8C24124yes—
152C2×C5×Q8C2414yes—
153C2×C5×D4—212no—
154C2×C5×D4—24no—
155C2×C5×D4—412no—
156C2×C5×D4C2414yes—
157C2×C5×D4C24112yes—
158C2×C5×D4C24124yes—
159C22×C4×C5—224no—
160C22×C4×C5—212no—
161C22×C4×C5—212no—
162C22×C4×C5—46no—
163C22×C4×C5C24112yes—
164C22×C4×C5C24112yes—
165C22×C4×C5C24124yes—
166C22×C4×C5C2413yes—
167C22×C4×C5C2411yescharacteristic
168C22×C4×C5C2411yescharacteristic
169C22×C4×C5C2412yes—
170C22×C4×C5C2416yes—
171C22×C4×C5C2416yes—
172C22×C4×C5C2413yes—
173C22×C4×C5C24112yes—
174C22×C4×C5C2413yes—
175C22×C4×C5C2411yescharacteristic
176C5×C4⋊C4C24112yes—
177C5×C4⋊C4C24112yes—
178C5×C4⋊C4C24112yes—
179C5×C4⋊C4C24112yes—
180C5×C4⋊C4C24124yes—
181C5×C4⋊C4C24124yes—
182C5×C4⋊C4C24124yes—
183C5×C4⋊C4C24124yes—
184C5×C4⋊C4C24124yes—
185C5×C4⋊C4C24124yes—
186C5×C22⋊C4C24124yes—
187C5×C22⋊C4C2418yes—
188C5×C22⋊C4C24124yes—
189C5×C22⋊C4C24148yes—
190C5×C22⋊C4C2418yes—
191C5×C22⋊C4C24124yes—
192C5×C22⋊C4C24124yes—
193C42×C5C2414yes—
194C42×C5C24112yes—
195C42×C5C24112yes—
196C42×C5C24112yes—
197C42×C5C24148yes—
198C42×C5C24124yes—
199C22×#41C22×C511yescharacteristic
200C23×Q8C22×C512yes—
201Q82C22×C5116yescomplemented
202#556C22×C5196yescomplemented
203#556C22×C5148yescomplemented
204#555C22×C5148yescomplemented
205D4×Q8C22×C5116yescomplemented
206#548C22×C5132yescomplemented
207#542C22×C5196yescomplemented
208#533C22×C5148yescomplemented
209C2×C4⋊Q8C22×C513yes—
210C2×C4⋊Q8C22×C513yes—
211C2×C4⋊Q8C22×C516yes—
212#529C22×C5148yescomplemented
213C2×#126C22×C516yes—
214C2×#126C22×C5112yes—
215C2×#125C22×C513yes—
216C2×#124C22×C511yescharacteristic
217C2×#124C22×C516yes—
218C2×#124C22×C5112yes—
219C2×#123C22×C516yes—
220C2×C22⋊Q8C22×C513yes—
221C2×C22⋊Q8C22×C516yes—
222C2×C22⋊Q8C22×C516yes—
223C2×C22⋊Q8C22×C516yes—
224C2×#121C22×C513yes—
225C2×#121C22×C516yes—
226#519C22×C5132yescomplemented
227C4×#41C22×C5116yescomplemented
228C2×C4×Q8C22×C511yescharacteristic
229C2×C4×Q8C22×C511yescharacteristic
230C2×C4×Q8C22×C516yes—
231C2×C4×Q8C22×C512yes—
232C2×C4×Q8C22×C516yes—
233C2×C4×Q8C22×C513yes—
234C2×C4×Q8C22×C516yes—
235C2×C4×D4C22×C513yes—
236C2×C4×D4C22×C513yes—
237C2×C4×D4C22×C516yes—
238C2×C4×D4C22×C513yes—
239C2×#117C22×C513yes—
240C2×#117C22×C516yes—
241C2×#117C22×C516yes—
242C2×#117C22×C516yes—
243C22×C42C22×C513yes—
244C23×C5—24no—
245C23×C5—26no—
246C23×C5—26no—
247C23×C5—24no—
248C23×C5—26no—
249C23×C5—24no—
250C23×C5C2514yes—
251C23×C5C2516yes—
252C23×C5C2511yescharacteristic
253C5×Q8—248no—
254C5×Q8—28no—
255C5×Q8—424no—
256C5×Q8C2×#41116yes—
257C5×D4—224no—
258C5×D4—28no—
259C5×D4—424no—
260C2×C4×C5—22no—
261C2×C4×C5—212no—
262C2×C4×C5—26no—
263C2×C4×C5—24no—
264C2×C4×C5—22no—
265C2×C4×C5—26no—
266C2×C4×C5—212no—
267C2×C4×C5—248no—
268C2×C4×C5—224no—
269C2×C4×C5—224no—
270C2×C4×C5—412no—
271C2×C4×C5C2×#41112yes—
272C2×C4×C52−1+414yes—
273C2×C4×C5C2516yes—
274C2×C4×C5C2512yes—
275C2×C4×C5C2512yes—
276C2×C4×C5C2514yes—
277C2×C4×C5C25112yes—
278C2×C4×C5C25112yes—
279C2×C4×C5C2516yes—
280C2×C4×C5C2516yes—
281C2×C4×C5C2512yes—
282C2×#41—28no—
283C2×#41C23×C518yes—
284C22×Q8—216no—
285C22×Q8C23×C511yescharacteristic
286C22×Q8C23×C516yes—
287C22×Q8C23×C5116yes—
288C22×Q8C23×C516yes—
289C22×Q8C23×C511yescharacteristic
290C22×D4C23×C511yescharacteristic
291C22×D4C23×C513yes—
292C22×D4C23×C516yes—
293C23×C4C23×C516yes—
294C23×C4C23×C513yes—
295C23×C4C23×C513yes—
296C23×C4C23×C513yes—
297C4⋊Q8C23×C5124yes—
298C4⋊Q8C23×C5124yes—
299C4⋊Q8C23×C5148yes—
300#126C23×C5148yes—
301#126C23×C5196yes—
302#125C23×C5124yes—
303#124C23×C518yes—
304#124C23×C5148yes—
305#124C23×C5196yes—
306#123C23×C5148yes—
307C22⋊Q8C23×C5124yes—
308C22⋊Q8C23×C5148yes—
309C22⋊Q8C23×C5148yes—
310C22⋊Q8C23×C5148yes—
311#121C23×C5124yes—
312#121C23×C5148yes—
313C4×Q8C23×C518yes—
314C4×Q8C23×C518yes—
315C4×Q8C23×C5148yes—
316C4×Q8C23×C5116yes—
317C4×Q8C23×C5148yes—
318C4×Q8C23×C5124yes—
319C4×Q8C23×C5148yes—
320C4×D4C23×C5124yes—
321C4×D4C23×C5124yes—
322C4×D4C23×C5148yes—
323C4×D4C23×C5124yes—
324#117C23×C5124yes—
325#117C23×C5148yes—
326#117C23×C5148yes—
327#117C23×C5148yes—
328C2×C4⋊C4C23×C513yes—
329C2×C4⋊C4C23×C513yes—
330C2×C4⋊C4C23×C513yes—
331C2×C4⋊C4C23×C513yes—
332C2×C4⋊C4C23×C516yes—
333C2×C4⋊C4C23×C516yes—
334C2×C4⋊C4C23×C516yes—
335C2×C4⋊C4C23×C516yes—
336C2×C4⋊C4C23×C516yes—
337C2×C4⋊C4C23×C516yes—
338C2×C22⋊C4C23×C516yes—
339C2×C22⋊C4C23×C512yes—
340C2×C22⋊C4C23×C516yes—
341C2×C22⋊C4C23×C5112yes—
342C2×C22⋊C4C23×C512yes—
343C2×C22⋊C4C23×C516yes—
344C2×C22⋊C4C23×C516yes—
345C2×C42C23×C513yes—
346C2×C42C23×C5112yes—
347C2×C42C23×C513yes—
348C2×C42C23×C516yes—
349C2×C42C23×C513yes—
350C2×C42C23×C5124yes—
351C2×C42C23×C511yescharacteristic
352C22×C5—24no—
353C22×C5—26no—
354C22×C5—26no—
355C22×C5—24no—
356C22×C5—26no—
357C22×C5—24no—
358C22×C5—48no—
359C22×C5—412no—
360C22×C5C22×#4112yes—
361C22×C5C2×2−1+412yes—
362C22×C5#58512yes—
363C22×C5C2611yescharacteristic
364C4×C5—22no—
365C4×C5—212no—
366C4×C5—26no—
367C4×C5—24no—
368C4×C5—22no—
369C4×C5—26no—
370C4×C5—212no—
371C4×C5C22×#41112yes—
372C4×C5C2×2−1+414yes—
373C24C24×C512yes—
374C24C24×C513yes—
375#41—232no—
376C2×Q8—264no—
377C2×Q8—224no—
378C2×Q8—24no—
379C2×Q8—412no—
380C2×Q8C5×#4118yes—
381C2×Q8C24×C514yes—
382C2×Q8C24×C5124yes—
383C2×Q8C24×C5124yes—
384C2×Q8C24×C514yes—
385C2×D4—212no—
386C2×D4—24no—
387C2×D4—412no—
388C2×D4C24×C514yes—
389C2×D4C24×C5112yes—
390C2×D4C24×C5124yes—
391C22×C4—224no—
392C22×C4—212no—
393C22×C4—212no—
394C22×C4—46no—
395C22×C4C24×C513yes—
396C22×C4C24×C511yescharacteristic
397C22×C4C24×C511yescharacteristic
398C22×C4C24×C512yes—
399C22×C4C24×C516yes—
400C22×C4C24×C516yes—
401C22×C4C24×C513yes—
402C22×C4C24×C5112yes—
403C22×C4C24×C5112yes—
404C22×C4C24×C5124yes—
405C22×C4C24×C5112yes—
406C22×C4C24×C513yes—
407C22×C4C24×C511yescharacteristic
408C4⋊C4C24×C5112yes—
409C4⋊C4C24×C5112yes—
410C4⋊C4C24×C5112yes—
411C4⋊C4C24×C5112yes—
412C4⋊C4C24×C5124yes—
413C4⋊C4C24×C5124yes—
414C4⋊C4C24×C5124yes—
415C4⋊C4C24×C5124yes—
416C4⋊C4C24×C5124yes—
417C4⋊C4C24×C5124yes—
418C22⋊C4C24×C5124yes—
419C22⋊C4C24×C518yes—
420C22⋊C4C24×C5124yes—
421C22⋊C4C24×C5148yes—
422C22⋊C4C24×C518yes—
423C22⋊C4C24×C5124yes—
424C22⋊C4C24×C5124yes—
425C42C24×C514yes—
426C42C24×C5112yes—
427C42C24×C5112yes—
428C42C24×C5112yes—
429C42C24×C5148yes—
430C42C24×C5124yes—
431C2×C5—44no—
432C2×C5—46no—
433C2×C5#351814yesdirect product factor
434C2×C5C23×#4111yescharacteristic
435C2×C5C22×2−1+411yescharacteristic
436C2×C5C2×#58511yescharacteristic
437C23—24no—
438C23—26no—
439C23—26no—
440C23—24no—
441C23—26no—
442C23—24no—
443C23C25×C514yes—
444C23C25×C516yes—
445C23C25×C511yescharacteristic
446Q8—248no—
447Q8—28no—
448Q8—424no—
449Q8C2×C5×#41116yes—
450D4—224no—
451D4—28no—
452D4—424no—
453C2×C4—22no—
454C2×C4—212no—
455C2×C4—26no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—26no—
459C2×C4—212no—
460C2×C4—248no—
461C2×C4—224no—
462C2×C4—224no—
463C2×C4—412no—
464C2×C4C2×C5×#41112yes—
465C2×C4C5×2−1+414yes—
466C2×C4C25×C516yes—
467C2×C4C25×C512yes—
468C2×C4C25×C512yes—
469C2×C4C25×C514yes—
470C2×C4C25×C5112yes—
471C2×C4C25×C5112yes—
472C2×C4C25×C516yes—
473C2×C4C25×C516yes—
474C2×C4C25×C512yes—
475C5C2×#351811yescharacteristic, direct product factor, Sylow
476C22—24no—
477C22—26no—
478C22—26no—
479C22—24no—
480C22—26no—
481C22—24no—
482C22—48no—
483C22—412no—
484C22C22×C5×#4112yes—
485C22C2×C5×2−1+412yes—
486C22C5×#58512yes—
487C22C26×C511yescharacteristic
488C4—22no—
489C4—212no—
490C4—26no—
491C4—24no—
492C4—22no—
493C4—26no—
494C4—212no—
495C4C22×C5×#41112yes—
496C4C2×C5×2−1+414yes—
497C2—44no—
498C2—46no—
499C2C5×#351814yesdirect product factor
500C2C23×C5×#4111yescharacteristic
501C2C22×C5×2−1+411yescharacteristic
502C2C2×C5×#58511yescharacteristic
subgroupG/H
center, socleC23×C5C25
commutator, Frattini, cosocleC22C26×C5
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#3518C2direct (N × H)
2. C2×C5×Q82C2semidirect
3. C2×C5×#556C2semidirect
4. C2×C5×#556C2semidirect
5. C2×C5×#555C2semidirect
6. C2×C5×D4×Q8C2semidirect
7. C2×C5×#548C2semidirect
8. C2×C5×#542C2semidirect
9. C2×C5×#533C2semidirect
10. C2×C5×#529C2semidirect
11. C2×C5×#519C2semidirect
12. C2×C4×C5×#41C2semidirect
15. C5×Q82C22semidirect
16. C5×#556C22semidirect
17. C5×#556C22semidirect
18. C5×#555C22semidirect
19. C5×D4×Q8C22semidirect
20. C5×#548C22semidirect
21. C5×#542C22semidirect
22. C5×#533C22semidirect
26. C5×#529C22semidirect
40. C5×#519C22semidirect
41. C4×C5×#41C22semidirect
58. C2×#3518C5direct (N × H)
129. #3518C2×C5direct (N × H)
130. C2×Q82C2×C5semidirect
131. C2×#556C2×C5semidirect
132. C2×#556C2×C5semidirect
133. C2×#555C2×C5semidirect
134. C2×D4×Q8C2×C5semidirect
135. C2×#548C2×C5semidirect
136. C2×#542C2×C5semidirect
137. C2×#533C2×C5semidirect
138. C2×#529C2×C5semidirect
139. C2×#519C2×C5semidirect
140. C2×C4×#41C2×C5semidirect
201. Q82C22×C5semidirect
202. #556C22×C5semidirect
203. #556C22×C5semidirect
204. #555C22×C5semidirect
205. D4×Q8C22×C5semidirect
206. #548C22×C5semidirect
207. #542C22×C5semidirect
208. #533C22×C5semidirect
212. #529C22×C5semidirect
226. #519C22×C5semidirect
227. C4×#41C22×C5semidirect
433. C2×C5#3518direct (N × H)
475. C5C2×#3518direct (N × H)
499. C2C5×#3518direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35182
C2×C5×Q822
C2×C5×#5562
C2×C5×#5552
C2×C5×D4×Q82
C2×C5×#5482
C2×C5×#5422
C2×C5×#5332
C2×C5×#5292
C2×C5×#5192
C2×C4×C5×#412
C2×#35185