Finite Group Databasebeta

← C2×C5×#5835 #14509957

Subgroups — C2×C5×#5835 #14509957

Order 1920.

3024 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×C5×#1009C211yesmaximal, characteristic, complemented
3C5×#5835C2116yesmaximal, direct product factor
4C2×C5×#950C213yesmaximal, complemented
5C2×C5×#949C213yesmaximal, complemented
6C2×C5×#948C211yesmaximal, characteristic, complemented
7C2×C5×#932C213yesmaximal, complemented
8C22×C5×#213C213yesmaximal, complemented
9C2×C5×#576—31nomaximal, Hall
10C3×C5×H(5,𝔽2)—24no—
11C3×C5×H(5,𝔽2)C2218yescomplemented
12C2×C3×C5×#41—21no—
13C2×C3×C5×#41C2211yescharacteristic
14C2×C3×C5×#41C2213yescomplemented
15C22×C3×C5×D4—23no—
16C22×C3×C5×D4C2213yescomplemented
17C5×#1009—24no—
18C5×#1009C2218yescomplemented
19C2×C5×#239—21no—
20C2×C5×D3×D4—23no—
21C2×C5×#235C2213yescomplemented
22C22×C5×D12C2213yescomplemented
23C5×#950C22124yescomplemented
24C5×#949C22124yescomplemented
25C5×#948C2218yescomplemented
26C2×C5×#216C2211yescharacteristic, complemented
27C2×C5×#215C2213yescomplemented
28C2×C5×#214C2213yescomplemented
29C5×#932C22124yescomplemented
30C2×C5×#213C2213yescomplemented
31C2×C5×#213C2216yescomplemented
32C2×C5×#213C22124yescomplemented
33C2×C5×#208C2213yescomplemented
34C22×C5×C3⋊C8C2213yescomplemented
35C2×#5835C511yesmaximal, characteristic, direct product factor, Hall
36C2×C5×H(5,𝔽2)—31no—
37C2×C5×H(5,𝔽2)D311yescharacteristic, complemented
38C5×#576—316no—
39C2×C5×AGL(1,ℤ/8)—33no—
40C2×C5×AGL(1,ℤ/8)—33no—
41C2×C5×#135—33no—
42C22×C5×D8—33no—
43C2×C5×#131—31no—
44C24×C3×C5—23no—
45C23×C5×D3—23no—
46C3×C5×#41—28no—
47C3×C5×#41—26no—
48C3×C5×#41D414yes—
49C3×C5×#41C2314yes—
50C3×C5×#41C23112yescomplemented
51C2×C3×C5×Q8D411yescharacteristic
52C2×C3×C5×Q8C2311yescharacteristic
53C2×C3×C5×D4—23no—
54C2×C3×C5×D4—224no—
55C2×C3×C5×D4—23no—
56C2×C3×C5×D4—26no—
57C2×C3×C5×D4D413yes—
58C2×C3×C5×D4C2313yes—
59C2×C3×C5×D4C23112yescomplemented
60C22×C3×C4×C5—23no—
61C22×C3×C4×C5C2313yes—
62C2×C5×#67—23no—
63C5×#239—28no—
64C5×D3×D4—224no—
65C5×#235—26no—
66C5×#235C23112yescomplemented
67C2×C5×D12—23no—
68C2×C5×D12—26no—
69C2×C5×D12C2313yes—
70C2×C5×D12C23112yescomplemented
71C2×C4×C5×D3—23no—
72C2×C5×Dic6C2311yescharacteristic
73C5×#216C2314yescomplemented
74C5×#215C23112yescomplemented
75C5×#214C23112yescomplemented
76C5×#213C23112yescomplemented
77C5×#213C23124yescomplemented
78C5×#208C23112yescomplemented
79C2×C5×C3⋊C8C2311yescharacteristic
80C2×C5×C3⋊C8C2313yes—
81C2×C5×C3⋊C8C23112yescomplemented
82C2×C3×H(5,𝔽2)C2×C511yescharacteristic, complemented
83C2×#1009C2×C511yescharacteristic, complemented
84#5835C2×C5116yesdirect product factor
85C2×#950C2×C513yescomplemented
86C2×#949C2×C513yescomplemented
87C2×#948C2×C511yescharacteristic, complemented
88C2×#932C2×C513yescomplemented
89C22×#213C2×C513yescomplemented
90C5×H(5,𝔽2)—24no—
91C5×H(5,𝔽2)—38no—
92C5×H(5,𝔽2)—64no—
93C5×H(5,𝔽2)C2×D318yescomplemented
94C2×C5×#41—21no—
95C2×C5×#41—33no—
96C2×C5×#41—61no—
97C2×C5×#41C2×D311yescharacteristic
98C2×C5×#41C2×D313yescomplemented
99C22×C5×D4—23no—
100C22×C5×D4—33no—
101C22×C5×D4—63no—
102C22×C5×D4C2×D313yescomplemented
103C5×AGL(1,ℤ/8)—324no—
104C5×AGL(1,ℤ/8)—324no—
105C5×#135—324no—
106C2×C5×Q16—31no—
107C2×C5×SD16—33no—
108C2×C5×SD16—33no—
109C2×C5×D8—33no—
110C2×C5×D8—36no—
111C2×C5×D8—324no—
112C5×#131—38no—
113C2×C5×M16—33no—
114C22×C5×C8—33no—
115C2×#576—31noSylow
116C23×C3×C5—23no—
117C23×C3×C5—212no—
118C23×C3×C5—86no—
119C23×C3×C5C2×D413yes—
120C22×C5×D3—23no—
121C22×C5×D3—212no—
122C22×C5×D3—86no—
123C3×C5×Q8—21no—
124C3×C5×Q8C2×D414yes—
125C3×C5×Q8C2412yes—
126C3×C5×D4—212no—
127C3×C5×D4—23no—
128C3×C5×D4—212no—
129C3×C5×D4C2×D4112yes—
130C3×C5×D4C2416yes—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—212no—
133C2×C3×C4×C5C2×D413yes—
134C2×C3×C4×C5C2×D416yes—
135C2×C3×C4×C5C2411yescharacteristic
136C2×C3×C4×C5C2416yes—
137C5×#67—212no—
138C2×C5×Dic3—21no—
139C5×D12—23no—
140C5×D12—212no—
141C5×D12C2416yes—
142C4×C5×D3—212no—
143C5×Dic6—21no—
144C5×Dic6C2412yes—
145C5×C3⋊C8C2412yes—
146C5×C3⋊C8C2416yes—
147C3×H(5,𝔽2)—24no—
148C3×H(5,𝔽2)C22×C518yescomplemented
149C2×C3×#41—21no—
150C2×C3×#41C22×C511yescharacteristic
151C2×C3×#41C22×C513yescomplemented
152C22×C3×D4—23no—
153C22×C3×D4C22×C513yescomplemented
154#1009—24no—
155#1009C22×C518yescomplemented
156C2×#239—21no—
157C2×D3×D4—23no—
158C2×#235C22×C513yescomplemented
159C22×D12C22×C513yescomplemented
160#950C22×C5124yescomplemented
161#949C22×C5124yescomplemented
162#948C22×C518yescomplemented
163C2×#216C22×C511yescharacteristic, complemented
164C2×#215C22×C513yescomplemented
165C2×#214C22×C513yescomplemented
166#932C22×C5124yescomplemented
167C2×#213C22×C513yescomplemented
168C2×#213C22×C516yescomplemented
169C2×#213C22×C5124yescomplemented
170C2×#208C22×C513yescomplemented
171C22×C3⋊C8C22×C513yescomplemented
172C24×C5—23no—
173C24×C5—63no—
174C5×#41—28no—
175C5×#41—26no—
176C5×#41—312no—
177C5×#41—68no—
178C5×#41—66no—
179C5×#41#6714yes—
180C5×#41C22×D314yes—
181C5×#41C22×D3112yescomplemented
182C2×C5×Q8—31no—
183C2×C5×Q8#6711yescharacteristic
184C2×C5×Q8C22×D311yescharacteristic
185C2×C5×D4—23no—
186C2×C5×D4—224no—
187C2×C5×D4—23no—
188C2×C5×D4—26no—
189C2×C5×D4—33no—
190C2×C5×D4—312no—
191C2×C5×D4—63no—
192C2×C5×D4—624no—
193C2×C5×D4—63no—
194C2×C5×D4—66no—
195C2×C5×D4#6713yes—
196C2×C5×D4C22×D313yes—
197C2×C5×D4C22×D3112yescomplemented
198C22×C4×C5—23no—
199C22×C4×C5—63no—
200C22×C4×C5C22×D313yes—
201C5×Q16—34no—
202C5×SD16—312no—
203C5×SD16—312no—
204C5×D8—312no—
205C5×D8—324no—
206C5×M16—312no—
207C2×C5×C8—31no—
208C2×C5×C8—33no—
209C2×C5×C8—312no—
210C2×H(5,𝔽2)—31no—
211C2×H(5,𝔽2)C5×D311yescharacteristic, complemented
212#576—316no—
213C2×AGL(1,ℤ/8)—33no—
214C2×AGL(1,ℤ/8)—33no—
215C2×#135—33no—
216C22×D8—33no—
217C2×#131—31no—
218C22×C3×C5—26no—
219C22×C3×C5—26no—
220C22×C3×C5—46no—
221C22×C3×C5—812no—
222C22×C3×C5C22×D416yes—
223C22×C3×C5C22×D411yescharacteristic
224C2×C5×D3—26no—
225C2×C5×D3—46no—
226C2×C5×D3—812no—
227C3×C4×C5—22no—
228C3×C4×C5C22×D411yescharacteristic
229C3×C4×C5C22×D416yes—
230C3×C4×C5C2511yescharacteristic
231C5×Dic3—22no—
232C24×C3—23no—
233C23×D3—23no—
234C3×#41—28no—
235C3×#41—26no—
236C3×#41C5×D414yes—
237C3×#41C23×C514yes—
238C3×#41C23×C5112yescomplemented
239C2×C3×Q8C5×D411yescharacteristic
240C2×C3×Q8C23×C511yescharacteristic
241C2×C3×D4—23no—
242C2×C3×D4—224no—
243C2×C3×D4—23no—
244C2×C3×D4—26no—
245C2×C3×D4C5×D413yes—
246C2×C3×D4C23×C513yes—
247C2×C3×D4C23×C5112yescomplemented
248C22×C3×C4—23no—
249C22×C3×C4C23×C513yes—
250C2×#67—23no—
251#239—28no—
252D3×D4—224no—
253#235—26no—
254#235C23×C5112yescomplemented
255C2×D12—23no—
256C2×D12—26no—
257C2×D12C23×C513yes—
258C2×D12C23×C5112yescomplemented
259C2×C4×D3—23no—
260C2×Dic6C23×C511yescharacteristic
261#216C23×C514yescomplemented
262#215C23×C5112yescomplemented
263#214C23×C5112yescomplemented
264#213C23×C5112yescomplemented
265#213C23×C5124yescomplemented
266#208C23×C5112yescomplemented
267C2×C3⋊C8C23×C511yescharacteristic
268C2×C3⋊C8C23×C513yes—
269C2×C3⋊C8C23×C5112yescomplemented
270C23×C5—23no—
271C23×C5—212no—
272C23×C5—63no—
273C23×C5—612no—
274C23×C5—86no—
275C23×C5—246no—
276C23×C5C2×#6713yes—
277C5×Q8—21no—
278C5×Q8—32no—
279C5×Q8—61no—
280C5×Q8C2×#6714yes—
281C5×Q8C23×D312yes—
282C5×D4—212no—
283C5×D4—23no—
284C5×D4—212no—
285C5×D4—36no—
286C5×D4—612no—
287C5×D4—612no—
288C5×D4—63no—
289C5×D4C2×#67112yes—
290C5×D4C23×D316yes—
291C2×C4×C5—21no—
292C2×C4×C5—212no—
293C2×C4×C5—61no—
294C2×C4×C5—612no—
295C2×C4×C5C2×#6716yes—
296C2×C4×C5C2×#6713yes—
297C2×C4×C5C23×D316yes—
298C2×C4×C5C23×D311yescharacteristic
299C5×C8—32no—
300C5×C8—36no—
301H(5,𝔽2)—24no—
302H(5,𝔽2)—38no—
303H(5,𝔽2)—64no—
304H(5,𝔽2)C2×C5×D318yescomplemented
305C2×#41—21no—
306C2×#41—33no—
307C2×#41—61no—
308C2×#41C2×C5×D311yescharacteristic
309C2×#41C2×C5×D313yescomplemented
310C22×D4—23no—
311C22×D4—33no—
312C22×D4—63no—
313C22×D4C2×C5×D313yescomplemented
314AGL(1,ℤ/8)—324no—
315AGL(1,ℤ/8)—324no—
316#135—324no—
317C2×Q16—31no—
318C2×SD16—33no—
319C2×SD16—33no—
320C2×D8—33no—
321C2×D8—36no—
322C2×D8—324no—
323#131—38no—
324C2×M16—33no—
325C22×C8—33no—
326C2×C3×C5—26no—
327C2×C3×C5—46no—
328C2×C3×C5#57612yescomplemented
329C2×C3×C5C23×D411yescharacteristic
330C5×D3—46no—
331C23×C3—23no—
332C23×C3—212no—
333C23×C3—86no—
334C23×C3C2×C5×D413yes—
335C22×D3—23no—
336C22×D3—212no—
337C22×D3—86no—
338C3×Q8—21no—
339C3×Q8C2×C5×D414yes—
340C3×Q8C24×C512yes—
341C3×D4—212no—
342C3×D4—23no—
343C3×D4—212no—
344C3×D4C2×C5×D4112yes—
345C3×D4C24×C516yes—
346C2×C3×C4—21no—
347C2×C3×C4—212no—
348C2×C3×C4C2×C5×D413yes—
349C2×C3×C4C2×C5×D416yes—
350C2×C3×C4C24×C511yescharacteristic
351C2×C3×C4C24×C516yes—
352#67—212no—
353C2×Dic3—21no—
354D12—23no—
355D12—212no—
356D12C24×C516yes—
357C4×D3—212no—
358Dic6—21no—
359Dic6C24×C512yes—
360C3⋊C8C24×C512yes—
361C3⋊C8C24×C516yes—
362C22×C5—26no—
363C22×C5—26no—
364C22×C5—46no—
365C22×C5—66no—
366C22×C5—812no—
367C22×C5—126no—
368C22×C5—2412no—
369C22×C5C22×#6716yes—
370C22×C5C22×#6711yescharacteristic
371C4×C5—22no—
372C4×C5—62no—
373C4×C5C22×#6716yes—
374C4×C5C22×#6711yescharacteristic
375C4×C5C24×D311yescharacteristic
376C24—23no—
377C24—63no—
378#41—28no—
379#41—26no—
380#41—312no—
381#41—68no—
382#41—66no—
383#41C5×#6714yes—
384#41C22×C5×D314yes—
385#41C22×C5×D3112yescomplemented
386C2×Q8—31no—
387C2×Q8C5×#6711yescharacteristic
388C2×Q8C22×C5×D311yescharacteristic
389C2×D4—23no—
390C2×D4—224no—
391C2×D4—23no—
392C2×D4—26no—
393C2×D4—33no—
394C2×D4—312no—
395C2×D4—63no—
396C2×D4—624no—
397C2×D4—63no—
398C2×D4—66no—
399C2×D4C5×#6713yes—
400C2×D4C22×C5×D313yes—
401C2×D4C22×C5×D3112yescomplemented
402C22×C4—23no—
403C22×C4—63no—
404C22×C4C22×C5×D313yes—
405Q16—34no—
406SD16—312no—
407SD16—312no—
408D8—312no—
409D8—324no—
410M16—312no—
411C2×C8—31no—
412C2×C8—33no—
413C2×C8—312no—
414C3×C5C2×#57611yescharacteristic, complemented, Hall
415C22×C3—26no—
416C22×C3—26no—
417C22×C3—46no—
418C22×C3—812no—
419C22×C3C22×C5×D416yes—
420C22×C3C22×C5×D411yescharacteristic
421C2×D3—26no—
422C2×D3—46no—
423C2×D3—812no—
424C3×C4—22no—
425C3×C4C22×C5×D411yescharacteristic
426C3×C4C22×C5×D416yes—
427C3×C4C25×C511yescharacteristic
428Dic3—22no—
429C2×C5—26no—
430C2×C5—46no—
431C2×C5—126no—
432C2×C5#583512yesdirect product factor
433C2×C5C23×#6711yescharacteristic
434C23—23no—
435C23—212no—
436C23—63no—
437C23—612no—
438C23—86no—
439C23—246no—
440C23C2×C5×#6713yes—
441Q8—21no—
442Q8—32no—
443Q8—61no—
444Q8C2×C5×#6714yes—
445Q8C23×C5×D312yes—
446D4—212no—
447D4—23no—
448D4—212no—
449D4—36no—
450D4—612no—
451D4—612no—
452D4—63no—
453D4C2×C5×#67112yes—
454D4C23×C5×D316yes—
455C2×C4—21no—
456C2×C4—212no—
457C2×C4—61no—
458C2×C4—612no—
459C2×C4C2×C5×#6713yes—
460C2×C4C2×C5×#6716yes—
461C2×C4C23×C5×D311yescharacteristic
462C2×C4C23×C5×D316yes—
463C8—32no—
464C8—36no—
465C2×C3—26no—
466C2×C3—46no—
467C2×C3C5×#57612yescomplemented
468C2×C3C23×C5×D411yescharacteristic
469D3—46no—
470C5C2×#583511yescharacteristic, direct product factor, Sylow
471C22—26no—
472C22—26no—
473C22—46no—
474C22—66no—
475C22—812no—
476C22—126no—
477C22—2412no—
478C22C22×C5×#6716yes—
479C22C22×C5×#6711yescharacteristic
480C4—22no—
481C4—62no—
482C4C22×C5×#6716yes—
483C4C22×C5×#6711yescharacteristic
484C4C24×C5×D311yescharacteristic
485C3C2×C5×#57611yescharacteristic, complemented, Sylow
486C2—26no—
487C2—46no—
488C2—126no—
489C2C5×#583512yesdirect product factor
490C2C23×C5×#6711yescharacteristic
subgroupG/H
centerC22×C5C22×#67
commutator, cosocleC3×C4C25×C5
FrattiniC4C24×C5×D3
FittingC2×C3×C5×H(5,𝔽2)C2
socleC22×C3×C5C22×D4
hypercenterC2×C5×H(5,𝔽2)D3
hypocenterC3C2×C5×#576
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×H(5,𝔽2)C2semidirect
2. C2×C5×#1009C2semidirect
3. C5×#5835C2direct (N × H)
4. C2×C5×#950C2semidirect
5. C2×C5×#949C2semidirect
6. C2×C5×#948C2semidirect
7. C2×C5×#932C2semidirect
8. C22×C5×#213C2semidirect
11. C3×C5×H(5,𝔽2)C22semidirect
14. C2×C3×C5×#41C22semidirect
16. C22×C3×C5×D4C22semidirect
18. C5×#1009C22semidirect
21. C2×C5×#235C22semidirect
22. C22×C5×D12C22semidirect
23. C5×#950C22semidirect
24. C5×#949C22semidirect
25. C5×#948C22semidirect
26. C2×C5×#216C22semidirect
27. C2×C5×#215C22semidirect
28. C2×C5×#214C22semidirect
29. C5×#932C22semidirect
30. C2×C5×#213C22semidirect
31. C2×C5×#213C22semidirect
32. C2×C5×#213C22semidirect
33. C2×C5×#208C22semidirect
34. C22×C5×C3⋊C8C22semidirect
35. C2×#5835C5direct (N × H)
37. C2×C5×H(5,𝔽2)D3semidirect
50. C3×C5×#41C23semidirect
59. C2×C3×C5×D4C23semidirect
66. C5×#235C23semidirect
70. C2×C5×D12C23semidirect
73. C5×#216C23semidirect
74. C5×#215C23semidirect
75. C5×#214C23semidirect
76. C5×#213C23semidirect
77. C5×#213C23semidirect
78. C5×#208C23semidirect
81. C2×C5×C3⋊C8C23semidirect
82. C2×C3×H(5,𝔽2)C2×C5semidirect
83. C2×#1009C2×C5semidirect
84. #5835C2×C5direct (N × H)
85. C2×#950C2×C5semidirect
86. C2×#949C2×C5semidirect
87. C2×#948C2×C5semidirect
88. C2×#932C2×C5semidirect
89. C22×#213C2×C5semidirect
93. C5×H(5,𝔽2)C2×D3semidirect
98. C2×C5×#41C2×D3semidirect
102. C22×C5×D4C2×D3semidirect
148. C3×H(5,𝔽2)C22×C5semidirect
151. C2×C3×#41C22×C5semidirect
153. C22×C3×D4C22×C5semidirect
155. #1009C22×C5semidirect
158. C2×#235C22×C5semidirect
159. C22×D12C22×C5semidirect
160. #950C22×C5semidirect
161. #949C22×C5semidirect
162. #948C22×C5semidirect
163. C2×#216C22×C5semidirect
164. C2×#215C22×C5semidirect
165. C2×#214C22×C5semidirect
166. #932C22×C5semidirect
167. C2×#213C22×C5semidirect
168. C2×#213C22×C5semidirect
169. C2×#213C22×C5semidirect
170. C2×#208C22×C5semidirect
171. C22×C3⋊C8C22×C5semidirect
181. C5×#41C22×D3semidirect
197. C2×C5×D4C22×D3semidirect
211. C2×H(5,𝔽2)C5×D3semidirect
238. C3×#41C23×C5semidirect
247. C2×C3×D4C23×C5semidirect
254. #235C23×C5semidirect
258. C2×D12C23×C5semidirect
261. #216C23×C5semidirect
262. #215C23×C5semidirect
263. #214C23×C5semidirect
264. #213C23×C5semidirect
265. #213C23×C5semidirect
266. #208C23×C5semidirect
269. C2×C3⋊C8C23×C5semidirect
304. H(5,𝔽2)C2×C5×D3semidirect
309. C2×#41C2×C5×D3semidirect
313. C22×D4C2×C5×D3semidirect
328. C2×C3×C5#576semidirect
385. #41C22×C5×D3semidirect
401. C2×D4C22×C5×D3semidirect
414. C3×C5C2×#576semidirect
432. C2×C5#5835direct (N × H)
467. C2×C3C5×#576semidirect
470. C5C2×#5835direct (N × H)
485. C3C2×C5×#576semidirect
489. C2C5×#5835direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×H(5,𝔽2)2
C2×C5×#10092
C5×#58352
C2×C5×#9502
C2×C5×#9492
C2×C5×#9482
C2×C5×#9322
C22×C5×#2132
C2×C5×#5763
C2×#58355