Finite Group Databasebeta

← C2×C5×#3523 #11985507

Subgroups — C2×C5×#3523 #11985507

Order 1280.

7022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3523C2132yesmaximal, direct product factor
2C2×C5×#557C211yesmaximal, characteristic, complemented
3C2×C5×#557C213yesmaximal, complemented
4C2×C5×#556C213yesmaximal, complemented
5C2×C5×#552C216yesmaximal, complemented
6C2×C5×D4×Q8C212yesmaximal, complemented
7C2×C5×D4×Q8C211yesmaximal, characteristic, complemented
8C2×C5×#541C216yesmaximal, complemented
9C2×C5×#533C213yesmaximal, complemented
10C2×C5×#529C213yesmaximal
11C2×C5×#519C212yesmaximal, complemented
12C22×C4×C5×Q8C211yesmaximal, characteristic, complemented
13C23×C5×Q8C2211yescharacteristic
14C23×C5×Q8C2212yes—
15C5×#557C22116yescomplemented
16C5×#557C22148yescomplemented
17C5×#556C22148yescomplemented
18C5×#552C22196yescomplemented
19C5×D4×Q8C22132yescomplemented
20C5×D4×Q8C22116yescomplemented
21C5×#541C22196yescomplemented
22C5×#533C22148yescomplemented
23C2×C5×C4⋊Q8C2213yes—
24C2×C5×C4⋊Q8C2213yes—
25C2×C5×C4⋊Q8C2216yes—
26C5×#529C22148yes—
27C2×C5×#126C2216yes—
28C2×C5×#126C22112yes—
29C2×C5×#125C2213yes—
30C2×C5×#125C2216yes—
31C2×C5×#125C22112yes—
32C2×C5×#125C2216yes—
33C2×C5×#124C2213yes—
34C2×C5×#123C2216yes—
35C2×C5×C22⋊Q8C2213yes—
36C2×C5×C22⋊Q8C2216yes—
37C2×C5×C22⋊Q8C2216yes—
38C2×C5×C22⋊Q8C2216yes—
39C2×C5×#121C2211yescharacteristic
40C5×#519C22132yescomplemented
41C2×C4×C5×Q8C2211yescharacteristic
42C2×C4×C5×Q8C2213yes—
43C2×C4×C5×Q8C2211yescharacteristic
44C2×C4×C5×Q8C2213yes—
45C2×C4×C5×Q8C2216yes—
46C2×C4×C5×Q8C2212yes—
47C2×C4×C5×Q8C2216yes—
48C2×C4×C5×Q8C2213yes—
49C2×C4×C5×Q8C2216yes—
50C2×C4×C5×Q8C22116yescomplemented
51C2×C4×C5×D4C2216yes—
52C2×C4×C5×D4C2213yes—
53C2×C5×#117C2216yes—
54C2×C5×#117C2216yes—
55C2×C5×#117C2216yes—
56C22×C5×C4⋊C4C2213yes—
57C22×C42×C5C2213yes—
58C2×#3523C511yesmaximal, characteristic, direct product factor, Sylow
59C22×C5×Q8—28no—
60C22×C5×Q8—216no—
61C22×C5×Q8C2313yes—
62C22×C5×Q8C2313yes—
63C22×C5×Q8C2316yes—
64C22×C5×Q8C2318yes—
65C22×C5×Q8C23116yes—
66C22×C5×Q8C2311yescharacteristic
67C22×C5×D4C2312yes—
68C22×C5×D4C2311yescharacteristic
69C23×C4×C5C2311yescharacteristic
70C23×C4×C5C2313yes—
71C23×C4×C5C2313yes—
72C23×C4×C5C2316yes—
73C5×C4⋊Q8C23124yes—
74C5×C4⋊Q8C23124yes—
75C5×C4⋊Q8C23148yes—
76C5×#126C23148yes—
77C5×#126C23196yes—
78C5×#125C23124yes—
79C5×#125C23148yes—
80C5×#125C23196yes—
81C5×#125C23148yes—
82C5×#124C23124yes—
83C5×#123C23148yes—
84C5×C22⋊Q8C23124yes—
85C5×C22⋊Q8C23148yes—
86C5×C22⋊Q8C23148yes—
87C5×C22⋊Q8C23148yes—
88C5×#121C2318yes—
89C4×C5×Q8C2318yes—
90C4×C5×Q8C23124yes—
91C4×C5×Q8C2318yes—
92C4×C5×Q8C23124yes—
93C4×C5×Q8C23148yes—
94C4×C5×Q8C23116yes—
95C4×C5×Q8C23148yes—
96C4×C5×Q8C23124yes—
97C4×C5×Q8C23148yes—
98C4×C5×D4C23148yes—
99C4×C5×D4C23124yes—
100C5×#117C23148yes—
101C5×#117C23148yes—
102C5×#117C23148yes—
103C2×C5×C4⋊C4C2313yes—
104C2×C5×C4⋊C4C2313yes—
105C2×C5×C4⋊C4C2316yes—
106C2×C5×C4⋊C4C23124yes—
107C2×C5×C4⋊C4C2316yes—
108C2×C5×C4⋊C4C2316yes—
109C2×C5×C4⋊C4C2316yes—
110C2×C5×C4⋊C4C2316yes—
111C2×C5×C4⋊C4C2316yes—
112C2×C5×C4⋊C4C2313yes—
113C2×C5×C4⋊C4C2316yes—
114C2×C5×C4⋊C4C2311yescharacteristic
115C2×C5×C4⋊C4C23112yes—
116C2×C5×C4⋊C4C2316yes—
117C2×C5×C22⋊C4C2312yes—
118C2×C5×C22⋊C4C2316yes—
119C2×C5×C22⋊C4C2316yes—
120C2×C5×C22⋊C4C2316yes—
121C2×C5×C22⋊C4C2316yes—
122C2×C42×C5C2313yes—
123C2×C42×C5C2313yes—
124C2×C42×C5C2313yes—
125C2×C42×C5C2313yes—
126C2×C42×C5C23124yes—
127C2×C42×C5C23112yes—
128C2×C42×C5C2316yes—
129#3523C2×C5132yesdirect product factor
130C2×#557C2×C511yescharacteristic, complemented
131C2×#557C2×C513yescomplemented
132C2×#556C2×C513yescomplemented
133C2×#552C2×C516yescomplemented
134C2×D4×Q8C2×C512yescomplemented
135C2×D4×Q8C2×C511yescharacteristic, complemented
136C2×#541C2×C516yescomplemented
137C2×#533C2×C513yescomplemented
138C2×#529C2×C513yes—
139C2×#519C2×C512yescomplemented
140C22×C4×Q8C2×C511yescharacteristic, complemented
141C24×C5C2412yes—
142C24×C5C2411yescharacteristic
143C2×C5×Q8—232no—
144C2×C5×Q8—264no—
145C2×C5×Q8—212no—
146C2×C5×Q8—224no—
147C2×C5×Q8—212no—
148C2×C5×Q8#4118yes—
149C2×C5×Q8C24112yes—
150C2×C5×Q8C24112yes—
151C2×C5×Q8C24124yes—
152C2×C5×Q8C2414yes—
153C2×C5×D4—28no—
154C2×C5×D4—24no—
155C2×C5×D4C2418yes—
156C2×C5×D4C2414yes—
157C22×C4×C5—212no—
158C22×C4×C5—212no—
159C22×C4×C5—224no—
160C22×C4×C5—24no—
161C22×C4×C5C2414yes—
162C22×C4×C5C24112yes—
163C22×C4×C5C24112yes—
164C22×C4×C5C24124yes—
165C22×C4×C5C2411yescharacteristic
166C22×C4×C5C2413yes—
167C22×C4×C5C2416yes—
168C22×C4×C5C2412yes—
169C22×C4×C5C2413yes—
170C22×C4×C5C2413yes—
171C22×C4×C5C2416yes—
172C22×C4×C5C2413yes—
173C22×C4×C5C2411yescharacteristic
174C5×C4⋊C4C24112yes—
175C5×C4⋊C4C24112yes—
176C5×C4⋊C4C24124yes—
177C5×C4⋊C4C24124yes—
178C5×C4⋊C4C24124yes—
179C5×C4⋊C4C24124yes—
180C5×C4⋊C4C24124yes—
181C5×C4⋊C4C24124yes—
182C5×C4⋊C4C24112yes—
183C5×C4⋊C4C24124yes—
184C5×C4⋊C4C2414yes—
185C5×C4⋊C4C24148yes—
186C5×C4⋊C4C24124yes—
187C5×C22⋊C4C2418yes—
188C5×C22⋊C4C24124yes—
189C5×C22⋊C4C24124yes—
190C5×C22⋊C4C24124yes—
191C5×C22⋊C4C24124yes—
192C42×C5C24112yes—
193C42×C5C24112yes—
194C42×C5C24112yes—
195C42×C5C24112yes—
196C42×C5C24148yes—
197C42×C5C24124yes—
198C23×Q8C22×C511yescharacteristic
199C23×Q8C22×C512yes—
200#557C22×C5116yescomplemented
201#557C22×C5148yescomplemented
202#556C22×C5148yescomplemented
203#552C22×C5196yescomplemented
204D4×Q8C22×C5132yescomplemented
205D4×Q8C22×C5116yescomplemented
206#541C22×C5196yescomplemented
207#533C22×C5148yescomplemented
208C2×C4⋊Q8C22×C513yes—
209C2×C4⋊Q8C22×C513yes—
210C2×C4⋊Q8C22×C516yes—
211#529C22×C5148yes—
212C2×#126C22×C516yes—
213C2×#126C22×C5112yes—
214C2×#125C22×C513yes—
215C2×#125C22×C516yes—
216C2×#125C22×C5112yes—
217C2×#125C22×C516yes—
218C2×#124C22×C513yes—
219C2×#123C22×C516yes—
220C2×C22⋊Q8C22×C513yes—
221C2×C22⋊Q8C22×C516yes—
222C2×C22⋊Q8C22×C516yes—
223C2×C22⋊Q8C22×C516yes—
224C2×#121C22×C511yescharacteristic
225#519C22×C5132yescomplemented
226C2×C4×Q8C22×C511yescharacteristic
227C2×C4×Q8C22×C513yes—
228C2×C4×Q8C22×C511yescharacteristic
229C2×C4×Q8C22×C513yes—
230C2×C4×Q8C22×C516yes—
231C2×C4×Q8C22×C512yes—
232C2×C4×Q8C22×C516yes—
233C2×C4×Q8C22×C513yes—
234C2×C4×Q8C22×C516yes—
235C2×C4×Q8C22×C5116yescomplemented
236C2×C4×D4C22×C516yes—
237C2×C4×D4C22×C513yes—
238C2×#117C22×C516yes—
239C2×#117C22×C516yes—
240C2×#117C22×C516yes—
241C22×C4⋊C4C22×C513yes—
242C22×C42C22×C513yes—
243C23×C5—22no—
244C23×C5—22no—
245C23×C5—24no—
246C23×C5—24no—
247C23×C5—24no—
248C23×C52−1+414yes—
249C23×C5C2514yes—
250C23×C5C2512yes—
251C23×C5C2511yescharacteristic
252C5×Q8—224no—
253C5×Q8—248no—
254C5×Q8—224no—
255C5×Q8C2×#41116yes—
256C5×D4—216no—
257C5×D4—28no—
258C2×C4×C5—26no—
259C2×C4×C5—212no—
260C2×C4×C5—224no—
261C2×C4×C5—26no—
262C2×C4×C5—26no—
263C2×C4×C5—22no—
264C2×C4×C5—24no—
265C2×C4×C5—212no—
266C2×C4×C5—224no—
267C2×C4×C5—248no—
268C2×C4×C5—28no—
269C2×C4×C5C2×#41112yes—
270C2×C4×C52−1+414yes—
271C2×C4×C5C2512yes—
272C2×C4×C5C2516yes—
273C2×C4×C5C25112yes—
274C2×C4×C5C2514yes—
275C2×C4×C5C2516yes—
276C2×C4×C5C2516yes—
277C2×C4×C5C25112yes—
278C2×C4×C5C2516yes—
279C2×C4×C5C2512yes—
280C22×Q8—28no—
281C22×Q8—216no—
282C22×Q8C23×C513yes—
283C22×Q8C23×C513yes—
284C22×Q8C23×C516yes—
285C22×Q8C23×C518yes—
286C22×Q8C23×C5116yes—
287C22×Q8C23×C511yescharacteristic
288C22×D4C23×C512yes—
289C22×D4C23×C511yescharacteristic
290C23×C4C23×C513yes—
291C23×C4C23×C513yes—
292C23×C4C23×C516yes—
293C23×C4C23×C511yescharacteristic
294C4⋊Q8C23×C5124yes—
295C4⋊Q8C23×C5124yes—
296C4⋊Q8C23×C5148yes—
297#126C23×C5148yes—
298#126C23×C5196yes—
299#125C23×C5124yes—
300#125C23×C5148yes—
301#125C23×C5196yes—
302#125C23×C5148yes—
303#124C23×C5124yes—
304#123C23×C5148yes—
305C22⋊Q8C23×C5124yes—
306C22⋊Q8C23×C5148yes—
307C22⋊Q8C23×C5148yes—
308C22⋊Q8C23×C5148yes—
309#121C23×C518yes—
310C4×Q8C23×C518yes—
311C4×Q8C23×C5124yes—
312C4×Q8C23×C518yes—
313C4×Q8C23×C5124yes—
314C4×Q8C23×C5148yes—
315C4×Q8C23×C5116yes—
316C4×Q8C23×C5148yes—
317C4×Q8C23×C5124yes—
318C4×Q8C23×C5148yes—
319C4×D4C23×C5148yes—
320C4×D4C23×C5124yes—
321#117C23×C5148yes—
322#117C23×C5148yes—
323#117C23×C5148yes—
324C2×C4⋊C4C23×C513yes—
325C2×C4⋊C4C23×C513yes—
326C2×C4⋊C4C23×C516yes—
327C2×C4⋊C4C23×C5124yes—
328C2×C4⋊C4C23×C516yes—
329C2×C4⋊C4C23×C516yes—
330C2×C4⋊C4C23×C516yes—
331C2×C4⋊C4C23×C516yes—
332C2×C4⋊C4C23×C516yes—
333C2×C4⋊C4C23×C513yes—
334C2×C4⋊C4C23×C516yes—
335C2×C4⋊C4C23×C511yescharacteristic
336C2×C4⋊C4C23×C5112yes—
337C2×C4⋊C4C23×C516yes—
338C2×C22⋊C4C23×C512yes—
339C2×C22⋊C4C23×C516yes—
340C2×C22⋊C4C23×C516yes—
341C2×C22⋊C4C23×C516yes—
342C2×C22⋊C4C23×C516yes—
343C2×C42C23×C513yes—
344C2×C42C23×C5112yes—
345C2×C42C23×C516yes—
346C2×C42C23×C513yes—
347C2×C42C23×C513yes—
348C2×C42C23×C513yes—
349C2×C42C23×C5124yes—
350C22×C5—22no—
351C22×C5—22no—
352C22×C5—28no—
353C22×C5—24no—
354C22×C5—24no—
355C22×C5—24no—
356C22×C5—48no—
357C22×C5C22×#4112yes—
358C22×C5C2×2−1+414yes—
359C22×C5C2×2−1+412yes—
360C22×C5C2×2−1+412yes—
361C22×C5C2611yescharacteristic
362C4×C5—26no—
363C4×C5—212no—
364C4×C5—26no—
365C4×C5—26no—
366C4×C5—22no—
367C4×C5—24no—
368C4×C5—212no—
369C4×C5C22×#41112yes—
370C4×C5C2×2−1+414yes—
371C24C24×C512yes—
372C24C24×C511yescharacteristic
373C2×Q8—232no—
374C2×Q8—264no—
375C2×Q8—212no—
376C2×Q8—224no—
377C2×Q8—212no—
378C2×Q8C5×#4118yes—
379C2×Q8C24×C5112yes—
380C2×Q8C24×C5112yes—
381C2×Q8C24×C5124yes—
382C2×Q8C24×C514yes—
383C2×D4—28no—
384C2×D4—24no—
385C2×D4C24×C518yes—
386C2×D4C24×C514yes—
387C22×C4—212no—
388C22×C4—212no—
389C22×C4—224no—
390C22×C4—24no—
391C22×C4C24×C511yescharacteristic
392C22×C4C24×C513yes—
393C22×C4C24×C514yes—
394C22×C4C24×C516yes—
395C22×C4C24×C512yes—
396C22×C4C24×C513yes—
397C22×C4C24×C516yes—
398C22×C4C24×C513yes—
399C22×C4C24×C5112yes—
400C22×C4C24×C5112yes—
401C22×C4C24×C5124yes—
402C22×C4C24×C513yes—
403C22×C4C24×C511yescharacteristic
404C4⋊C4C24×C5112yes—
405C4⋊C4C24×C5112yes—
406C4⋊C4C24×C5124yes—
407C4⋊C4C24×C5124yes—
408C4⋊C4C24×C5124yes—
409C4⋊C4C24×C5124yes—
410C4⋊C4C24×C5124yes—
411C4⋊C4C24×C5124yes—
412C4⋊C4C24×C5112yes—
413C4⋊C4C24×C5124yes—
414C4⋊C4C24×C514yes—
415C4⋊C4C24×C5148yes—
416C4⋊C4C24×C5124yes—
417C22⋊C4C24×C518yes—
418C22⋊C4C24×C5124yes—
419C22⋊C4C24×C5124yes—
420C22⋊C4C24×C5124yes—
421C22⋊C4C24×C5124yes—
422C42C24×C5112yes—
423C42C24×C5112yes—
424C42C24×C5112yes—
425C42C24×C5112yes—
426C42C24×C5148yes—
427C42C24×C5124yes—
428C2×C5—24no—
429C2×C5—44no—
430C2×C5#352314yesdirect product factor
431C2×C5C23×#4111yescharacteristic
432C2×C5C22×2−1+411yescharacteristic
433C2×C5C22×2−1+411yescharacteristic
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23C5×2−1+414yes—
440C23C25×C514yes—
441C23C25×C512yes—
442C23C25×C511yescharacteristic
443Q8—224no—
444Q8—248no—
445Q8—224no—
446Q8C2×C5×#41116yes—
447D4—216no—
448D4—28no—
449C2×C4—26no—
450C2×C4—212no—
451C2×C4—224no—
452C2×C4—26no—
453C2×C4—26no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—212no—
457C2×C4—224no—
458C2×C4—248no—
459C2×C4—28no—
460C2×C4C2×C5×#41112yes—
461C2×C4C5×2−1+414yes—
462C2×C4C25×C512yes—
463C2×C4C25×C516yes—
464C2×C4C25×C5112yes—
465C2×C4C25×C514yes—
466C2×C4C25×C516yes—
467C2×C4C25×C516yes—
468C2×C4C25×C5112yes—
469C2×C4C25×C516yes—
470C2×C4C25×C512yes—
471C5C2×#352311yescharacteristic, direct product factor, Sylow
472C22—22no—
473C22—22no—
474C22—28no—
475C22—24no—
476C22—24no—
477C22—24no—
478C22—48no—
479C22C22×C5×#4112yes—
480C22C2×C5×2−1+414yes—
481C22C2×C5×2−1+412yes—
482C22C2×C5×2−1+412yes—
483C22C26×C511yescharacteristic
484C4—26no—
485C4—212no—
486C4—26no—
487C4—26no—
488C4—22no—
489C4—24no—
490C4—212no—
491C4C22×C5×#41112yes—
492C4C2×C5×2−1+414yes—
493C2—24no—
494C2—44no—
495C2C5×#352314yesdirect product factor
496C2C23×C5×#4111yescharacteristic
497C2C22×C5×2−1+411yescharacteristic
498C2C22×C5×2−1+411yescharacteristic
subgroupG/H
center, socleC23×C5C25
commutator, Frattini, cosocleC22C26×C5
prime radicalC2×C5×#121C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#3523C2direct (N × H)
2. C2×C5×#557C2semidirect
3. C2×C5×#557C2semidirect
4. C2×C5×#556C2semidirect
5. C2×C5×#552C2semidirect
6. C2×C5×D4×Q8C2semidirect
7. C2×C5×D4×Q8C2semidirect
8. C2×C5×#541C2semidirect
9. C2×C5×#533C2semidirect
11. C2×C5×#519C2semidirect
12. C22×C4×C5×Q8C2semidirect
15. C5×#557C22semidirect
16. C5×#557C22semidirect
17. C5×#556C22semidirect
18. C5×#552C22semidirect
19. C5×D4×Q8C22semidirect
20. C5×D4×Q8C22semidirect
21. C5×#541C22semidirect
22. C5×#533C22semidirect
40. C5×#519C22semidirect
50. C2×C4×C5×Q8C22semidirect
58. C2×#3523C5direct (N × H)
129. #3523C2×C5direct (N × H)
130. C2×#557C2×C5semidirect
131. C2×#557C2×C5semidirect
132. C2×#556C2×C5semidirect
133. C2×#552C2×C5semidirect
134. C2×D4×Q8C2×C5semidirect
135. C2×D4×Q8C2×C5semidirect
136. C2×#541C2×C5semidirect
137. C2×#533C2×C5semidirect
139. C2×#519C2×C5semidirect
140. C22×C4×Q8C2×C5semidirect
200. #557C22×C5semidirect
201. #557C22×C5semidirect
202. #556C22×C5semidirect
203. #552C22×C5semidirect
204. D4×Q8C22×C5semidirect
205. D4×Q8C22×C5semidirect
206. #541C22×C5semidirect
207. #533C22×C5semidirect
225. #519C22×C5semidirect
235. C2×C4×Q8C22×C5semidirect
430. C2×C5#3523direct (N × H)
471. C5C2×#3523direct (N × H)
495. C2C5×#3523direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35232
C2×C5×#5572
C2×C5×#5562
C2×C5×#5522
C2×C5×D4×Q82
C2×C5×#5412
C2×C5×#5332
C2×C5×#5292
C2×C5×#5192
C22×C4×C5×Q82
C2×#35235