Finite Group Databasebeta

← C2×C5×#5837 #14510024

Subgroups — C2×C5×#5837 #14510024

Order 1920.

2920 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×2−1+4C211yesmaximal, characteristic, complemented
2C2×C5×#1009C211yesmaximal, characteristic, complemented
3C5×#5837C2116yesmaximal, direct product factor
4C2×C5×#950C213yesmaximal, complemented
5C2×C5×#949C213yesmaximal, complemented
6C2×C5×#948C211yesmaximal, characteristic, complemented
7C2×C5×#942C213yesmaximal, complemented
8C22×C5×#215C213yesmaximal, complemented
9C2×C5×#577—31nomaximal, Hall
10C3×C5×2−1+4—24no—
11C3×C5×2−1+4C2218yescomplemented
12C2×C3×C5×#41—23no—
13C2×C3×C5×#41C2211yescharacteristic
14C2×C3×C5×#41C2213yescomplemented
15C22×C3×C5×Q8—21no—
16C22×C3×C5×Q8C2213yescomplemented
17C5×#1009—24no—
18C5×#1009C2218yescomplemented
19C2×C5×#239—21no—
20C2×C5×D3×D4—23no—
21C2×C5×#235C2213yes—
22C22×C5×D12C2213yes—
23C5×#950C22124yescomplemented
24C5×#949C22124yescomplemented
25C5×#948C2218yescomplemented
26C2×C5×#216C2213yescomplemented
27C5×#942C22124yescomplemented
28C2×C5×#215C2213yescomplemented
29C2×C5×#215C2216yescomplemented
30C2×C5×#215C22124yescomplemented
31C2×C5×#214C2211yescharacteristic, complemented
32C2×C5×#213C2213yescomplemented
33C2×C5×#208C2213yescomplemented
34C22×C5×C3⋊C8C2213yescomplemented
35C2×#5837C511yesmaximal, characteristic, direct product factor, Hall
36C2×C5×2−1+4D311yescharacteristic, complemented
37C2×C5×H(5,𝔽2)—31no—
38C5×#577—316no—
39C2×C5×#137—33no—
40C2×C5×AGL(1,ℤ/8)—33no—
41C2×C5×#135—33no—
42C22×C5×SD16—33no—
43C2×C5×#131—31no—
44C23×C5×D3—23no—
45C3×C5×#41—224no—
46C3×C5×#41—26no—
47C3×C5×#41D414yes—
48C3×C5×#41C2314yes—
49C3×C5×#41C23112yescomplemented
50C2×C3×C5×Q8—28no—
51C2×C3×C5×Q8—23no—
52C2×C3×C5×Q8—26no—
53C2×C3×C5×Q8D411yescharacteristic
54C2×C3×C5×Q8C2313yes—
55C2×C3×C5×Q8C23112yescomplemented
56C2×C3×C5×D4—23no—
57C2×C3×C5×D4D413yes—
58C2×C3×C5×D4C2311yescharacteristic
59C22×C3×C4×C5—23no—
60C22×C3×C4×C5—23no—
61C22×C3×C4×C5C2313yes—
62C2×C5×#67—23no—
63C5×#239—28no—
64C5×D3×D4—224no—
65C5×#235—26no—
66C5×#235C23112yes—
67C2×C5×D12—23no—
68C2×C5×D12—26no—
69C2×C5×D12C2313yes—
70C2×C5×D12C23112yes—
71C2×C4×C5×D3—23no—
72C2×C5×Dic6C2311yescharacteristic
73C5×#216C23112yescomplemented
74C5×#215C23112yescomplemented
75C5×#215C23124yescomplemented
76C5×#214C2314yescomplemented
77C5×#213C23112yescomplemented
78C5×#208C23112yescomplemented
79C2×C5×C3⋊C8C2311yescharacteristic
80C2×C5×C3⋊C8C2313yes—
81C2×C5×C3⋊C8C23112yescomplemented
82C2×C3×2−1+4C2×C511yescharacteristic, complemented
83C2×#1009C2×C511yescharacteristic, complemented
84#5837C2×C5116yesdirect product factor
85C2×#950C2×C513yescomplemented
86C2×#949C2×C513yescomplemented
87C2×#948C2×C511yescharacteristic, complemented
88C2×#942C2×C513yescomplemented
89C22×#215C2×C513yescomplemented
90C5×2−1+4—24no—
91C5×2−1+4C2×D318yescomplemented
92C5×H(5,𝔽2)—38no—
93C5×H(5,𝔽2)—64no—
94C2×C5×#41—23no—
95C2×C5×#41—33no—
96C2×C5×#41—61no—
97C2×C5×#41C2×D311yescharacteristic
98C2×C5×#41C2×D313yescomplemented
99C22×C5×Q8—21no—
100C22×C5×Q8C2×D313yescomplemented
101C22×C5×D4—33no—
102C22×C5×D4—63no—
103C5×#137—324no—
104C5×AGL(1,ℤ/8)—324no—
105C5×#135—324no—
106C2×C5×Q16—33no—
107C2×C5×SD16—33no—
108C2×C5×SD16—36no—
109C2×C5×SD16—324no—
110C2×C5×SD16—31no—
111C2×C5×D8—33no—
112C5×#131—38no—
113C2×C5×M16—33no—
114C22×C5×C8—33no—
115C2×#577—31noSylow
116C23×C3×C5—21no—
117C23×C3×C5C2×D413yes—
118C22×C5×D3—23no—
119C22×C5×D3—212no—
120C22×C5×D3—86no—
121C3×C5×Q8—23no—
122C3×C5×Q8—212no—
123C3×C5×Q8C2×D414yes—
124C3×C5×Q8C2416yes—
125C3×C5×D4—212no—
126C3×C5×D4—21no—
127C3×C5×D4C2×D4112yes—
128C3×C5×D4C2412yes—
129C2×C3×C4×C5—23no—
130C2×C3×C4×C5—212no—
131C2×C3×C4×C5—212no—
132C2×C3×C4×C5C2×D413yes—
133C2×C3×C4×C5C2×D416yes—
134C2×C3×C4×C5C2411yescharacteristic
135C2×C3×C4×C5C2416yes—
136C5×#67—212no—
137C2×C5×Dic3—21no—
138C5×D12—23no—
139C5×D12—212no—
140C5×D12C2416yes—
141C4×C5×D3—212no—
142C5×Dic6—21no—
143C5×Dic6C2412yes—
144C5×C3⋊C8C2412yes—
145C5×C3⋊C8C2416yes—
146C3×2−1+4—24no—
147C3×2−1+4C22×C518yescomplemented
148C2×C3×#41—23no—
149C2×C3×#41C22×C511yescharacteristic
150C2×C3×#41C22×C513yescomplemented
151C22×C3×Q8—21no—
152C22×C3×Q8C22×C513yescomplemented
153#1009—24no—
154#1009C22×C518yescomplemented
155C2×#239—21no—
156C2×D3×D4—23no—
157C2×#235C22×C513yes—
158C22×D12C22×C513yes—
159#950C22×C5124yescomplemented
160#949C22×C5124yescomplemented
161#948C22×C518yescomplemented
162C2×#216C22×C513yescomplemented
163#942C22×C5124yescomplemented
164C2×#215C22×C513yescomplemented
165C2×#215C22×C516yescomplemented
166C2×#215C22×C5124yescomplemented
167C2×#214C22×C511yescharacteristic, complemented
168C2×#213C22×C513yescomplemented
169C2×#208C22×C513yescomplemented
170C22×C3⋊C8C22×C513yescomplemented
171C24×C5—63no—
172C5×#41—224no—
173C5×#41—26no—
174C5×#41—312no—
175C5×#41—68no—
176C5×#41—66no—
177C5×#41#6714yes—
178C5×#41C22×D314yes—
179C5×#41C22×D3112yescomplemented
180C2×C5×Q8—28no—
181C2×C5×Q8—23no—
182C2×C5×Q8—26no—
183C2×C5×Q8—31no—
184C2×C5×Q8#6711yescharacteristic
185C2×C5×Q8C22×D313yes—
186C2×C5×Q8C22×D3112yescomplemented
187C2×C5×D4—23no—
188C2×C5×D4—33no—
189C2×C5×D4—312no—
190C2×C5×D4—63no—
191C2×C5×D4—624no—
192C2×C5×D4—63no—
193C2×C5×D4—66no—
194C2×C5×D4#6713yes—
195C2×C5×D4C22×D311yescharacteristic
196C22×C4×C5—23no—
197C22×C4×C5—23no—
198C22×C4×C5—63no—
199C22×C4×C5C22×D313yes—
200C5×Q16—312no—
201C5×SD16—312no—
202C5×SD16—324no—
203C5×SD16—34no—
204C5×D8—312no—
205C5×M16—312no—
206C2×C5×C8—31no—
207C2×C5×C8—33no—
208C2×C5×C8—312no—
209C2×2−1+4C5×D311yescharacteristic, complemented
210C2×H(5,𝔽2)—31no—
211#577—316no—
212C2×#137—33no—
213C2×AGL(1,ℤ/8)—33no—
214C2×#135—33no—
215C22×SD16—33no—
216C2×#131—31no—
217C22×C3×C5—26no—
218C22×C3×C5—22no—
219C22×C3×C5—42no—
220C22×C3×C5C22×D416yes—
221C22×C3×C5C22×D411yescharacteristic
222C2×C5×D3—26no—
223C2×C5×D3—46no—
224C2×C5×D3—812no—
225C3×C4×C5—26no—
226C3×C4×C5C22×D411yescharacteristic
227C3×C4×C5C22×D416yes—
228C3×C4×C5C2511yescharacteristic
229C5×Dic3—22no—
230C23×D3—23no—
231C3×#41—224no—
232C3×#41—26no—
233C3×#41C5×D414yes—
234C3×#41C23×C514yes—
235C3×#41C23×C5112yescomplemented
236C2×C3×Q8—28no—
237C2×C3×Q8—23no—
238C2×C3×Q8—26no—
239C2×C3×Q8C5×D411yescharacteristic
240C2×C3×Q8C23×C513yes—
241C2×C3×Q8C23×C5112yescomplemented
242C2×C3×D4—23no—
243C2×C3×D4C5×D413yes—
244C2×C3×D4C23×C511yescharacteristic
245C22×C3×C4—23no—
246C22×C3×C4—23no—
247C22×C3×C4C23×C513yes—
248C2×#67—23no—
249#239—28no—
250D3×D4—224no—
251#235—26no—
252#235C23×C5112yes—
253C2×D12—23no—
254C2×D12—26no—
255C2×D12C23×C513yes—
256C2×D12C23×C5112yes—
257C2×C4×D3—23no—
258C2×Dic6C23×C511yescharacteristic
259#216C23×C5112yescomplemented
260#215C23×C5112yescomplemented
261#215C23×C5124yescomplemented
262#214C23×C514yescomplemented
263#213C23×C5112yescomplemented
264#208C23×C5112yescomplemented
265C2×C3⋊C8C23×C511yescharacteristic
266C2×C3⋊C8C23×C513yes—
267C2×C3⋊C8C23×C5112yescomplemented
268C23×C5—21no—
269C23×C5—63no—
270C23×C5—612no—
271C23×C5—246no—
272C23×C5C2×#6713yes—
273C5×Q8—23no—
274C5×Q8—212no—
275C5×Q8—32no—
276C5×Q8—61no—
277C5×Q8C2×#6714yes—
278C5×Q8C23×D316yes—
279C5×D4—212no—
280C5×D4—21no—
281C5×D4—36no—
282C5×D4—612no—
283C5×D4—612no—
284C5×D4—63no—
285C5×D4C2×#67112yes—
286C5×D4C23×D312yes—
287C2×C4×C5—212no—
288C2×C4×C5—23no—
289C2×C4×C5—212no—
290C2×C4×C5—61no—
291C2×C4×C5—612no—
292C2×C4×C5C2×#6716yes—
293C2×C4×C5C2×#6713yes—
294C2×C4×C5C23×D316yes—
295C2×C4×C5C23×D311yescharacteristic
296C5×C8—32no—
297C5×C8—36no—
2982−1+4—24no—
2992−1+4C2×C5×D318yescomplemented
300H(5,𝔽2)—38no—
301H(5,𝔽2)—64no—
302C2×#41—23no—
303C2×#41—33no—
304C2×#41—61no—
305C2×#41C2×C5×D311yescharacteristic
306C2×#41C2×C5×D313yescomplemented
307C22×Q8—21no—
308C22×Q8C2×C5×D313yescomplemented
309C22×D4—33no—
310C22×D4—63no—
311#137—324no—
312AGL(1,ℤ/8)—324no—
313#135—324no—
314C2×Q16—33no—
315C2×SD16—33no—
316C2×SD16—36no—
317C2×SD16—324no—
318C2×SD16—31no—
319C2×D8—33no—
320#131—38no—
321C2×M16—33no—
322C22×C8—33no—
323C2×C3×C5—26no—
324C2×C3×C5—42no—
325C2×C3×C5#57712yescomplemented
326C2×C3×C5C23×D411yescharacteristic
327C5×D3—46no—
328C23×C3—21no—
329C23×C3C2×C5×D413yes—
330C22×D3—23no—
331C22×D3—212no—
332C22×D3—86no—
333C3×Q8—23no—
334C3×Q8—212no—
335C3×Q8C2×C5×D414yes—
336C3×Q8C24×C516yes—
337C3×D4—212no—
338C3×D4—21no—
339C3×D4C2×C5×D4112yes—
340C3×D4C24×C512yes—
341C2×C3×C4—23no—
342C2×C3×C4—212no—
343C2×C3×C4—212no—
344C2×C3×C4C2×C5×D413yes—
345C2×C3×C4C2×C5×D416yes—
346C2×C3×C4C24×C511yescharacteristic
347C2×C3×C4C24×C516yes—
348#67—212no—
349C2×Dic3—21no—
350D12—23no—
351D12—212no—
352D12C24×C516yes—
353C4×D3—212no—
354Dic6—21no—
355Dic6C24×C512yes—
356C3⋊C8C24×C512yes—
357C3⋊C8C24×C516yes—
358C22×C5—26no—
359C22×C5—22no—
360C22×C5—42no—
361C22×C5—66no—
362C22×C5—126no—
363C22×C5—2412no—
364C22×C5C22×#6716yes—
365C22×C5C22×#6711yescharacteristic
366C4×C5—26no—
367C4×C5—62no—
368C4×C5C22×#6716yes—
369C4×C5C22×#6711yescharacteristic
370C4×C5C24×D311yescharacteristic
371C24—63no—
372#41—224no—
373#41—26no—
374#41—312no—
375#41—68no—
376#41—66no—
377#41C5×#6714yes—
378#41C22×C5×D314yes—
379#41C22×C5×D3112yescomplemented
380C2×Q8—28no—
381C2×Q8—23no—
382C2×Q8—26no—
383C2×Q8—31no—
384C2×Q8C5×#6711yescharacteristic
385C2×Q8C22×C5×D313yes—
386C2×Q8C22×C5×D3112yescomplemented
387C2×D4—23no—
388C2×D4—33no—
389C2×D4—312no—
390C2×D4—63no—
391C2×D4—624no—
392C2×D4—63no—
393C2×D4—66no—
394C2×D4C5×#6713yes—
395C2×D4C22×C5×D311yescharacteristic
396C22×C4—23no—
397C22×C4—23no—
398C22×C4—63no—
399C22×C4C22×C5×D313yes—
400Q16—312no—
401SD16—312no—
402SD16—324no—
403SD16—34no—
404D8—312no—
405M16—312no—
406C2×C8—31no—
407C2×C8—33no—
408C2×C8—312no—
409C3×C5C2×#57711yescharacteristic, complemented, Hall
410C22×C3—26no—
411C22×C3—22no—
412C22×C3—42no—
413C22×C3C22×C5×D416yes—
414C22×C3C22×C5×D411yescharacteristic
415C2×D3—26no—
416C2×D3—46no—
417C2×D3—812no—
418C3×C4—26no—
419C3×C4C22×C5×D411yescharacteristic
420C3×C4C22×C5×D416yes—
421C3×C4C25×C511yescharacteristic
422Dic3—22no—
423C2×C5—26no—
424C2×C5—42no—
425C2×C5—126no—
426C2×C5#583712yesdirect product factor
427C2×C5C23×#6711yescharacteristic
428C23—21no—
429C23—63no—
430C23—612no—
431C23—246no—
432C23C2×C5×#6713yes—
433Q8—23no—
434Q8—212no—
435Q8—32no—
436Q8—61no—
437Q8C2×C5×#6714yes—
438Q8C23×C5×D316yes—
439D4—212no—
440D4—21no—
441D4—36no—
442D4—612no—
443D4—612no—
444D4—63no—
445D4C2×C5×#67112yes—
446D4C23×C5×D312yes—
447C2×C4—23no—
448C2×C4—212no—
449C2×C4—212no—
450C2×C4—61no—
451C2×C4—612no—
452C2×C4C2×C5×#6713yes—
453C2×C4C2×C5×#6716yes—
454C2×C4C23×C5×D311yescharacteristic
455C2×C4C23×C5×D316yes—
456C8—32no—
457C8—36no—
458C2×C3—26no—
459C2×C3—42no—
460C2×C3C5×#57712yescomplemented
461C2×C3C23×C5×D411yescharacteristic
462D3—46no—
463C5C2×#583711yescharacteristic, direct product factor, Sylow
464C22—26no—
465C22—22no—
466C22—42no—
467C22—66no—
468C22—126no—
469C22—2412no—
470C22C22×C5×#6716yes—
471C22C22×C5×#6711yescharacteristic
472C4—26no—
473C4—62no—
474C4C22×C5×#6716yes—
475C4C22×C5×#6711yescharacteristic
476C4C24×C5×D311yescharacteristic
477C3C2×C5×#57711yescharacteristic, complemented, Sylow
478C2—26no—
479C2—42no—
480C2—126no—
481C2C5×#583712yesdirect product factor
482C2C23×C5×#6711yescharacteristic
subgroupG/H
centerC22×C5C22×#67
commutator, cosocleC3×C4C25×C5
FrattiniC4C24×C5×D3
FittingC2×C3×C5×2−1+4C2
socleC22×C3×C5C22×D4
hypercenterC2×C5×2−1+4D3
hypocenterC3C2×C5×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×2−1+4C2semidirect
2. C2×C5×#1009C2semidirect
3. C5×#5837C2direct (N × H)
4. C2×C5×#950C2semidirect
5. C2×C5×#949C2semidirect
6. C2×C5×#948C2semidirect
7. C2×C5×#942C2semidirect
8. C22×C5×#215C2semidirect
11. C3×C5×2−1+4C22semidirect
14. C2×C3×C5×#41C22semidirect
16. C22×C3×C5×Q8C22semidirect
18. C5×#1009C22semidirect
23. C5×#950C22semidirect
24. C5×#949C22semidirect
25. C5×#948C22semidirect
26. C2×C5×#216C22semidirect
27. C5×#942C22semidirect
28. C2×C5×#215C22semidirect
29. C2×C5×#215C22semidirect
30. C2×C5×#215C22semidirect
31. C2×C5×#214C22semidirect
32. C2×C5×#213C22semidirect
33. C2×C5×#208C22semidirect
34. C22×C5×C3⋊C8C22semidirect
35. C2×#5837C5direct (N × H)
36. C2×C5×2−1+4D3semidirect
49. C3×C5×#41C23semidirect
55. C2×C3×C5×Q8C23semidirect
73. C5×#216C23semidirect
74. C5×#215C23semidirect
75. C5×#215C23semidirect
76. C5×#214C23semidirect
77. C5×#213C23semidirect
78. C5×#208C23semidirect
81. C2×C5×C3⋊C8C23semidirect
82. C2×C3×2−1+4C2×C5semidirect
83. C2×#1009C2×C5semidirect
84. #5837C2×C5direct (N × H)
85. C2×#950C2×C5semidirect
86. C2×#949C2×C5semidirect
87. C2×#948C2×C5semidirect
88. C2×#942C2×C5semidirect
89. C22×#215C2×C5semidirect
91. C5×2−1+4C2×D3semidirect
98. C2×C5×#41C2×D3semidirect
100. C22×C5×Q8C2×D3semidirect
147. C3×2−1+4C22×C5semidirect
150. C2×C3×#41C22×C5semidirect
152. C22×C3×Q8C22×C5semidirect
154. #1009C22×C5semidirect
159. #950C22×C5semidirect
160. #949C22×C5semidirect
161. #948C22×C5semidirect
162. C2×#216C22×C5semidirect
163. #942C22×C5semidirect
164. C2×#215C22×C5semidirect
165. C2×#215C22×C5semidirect
166. C2×#215C22×C5semidirect
167. C2×#214C22×C5semidirect
168. C2×#213C22×C5semidirect
169. C2×#208C22×C5semidirect
170. C22×C3⋊C8C22×C5semidirect
179. C5×#41C22×D3semidirect
186. C2×C5×Q8C22×D3semidirect
209. C2×2−1+4C5×D3semidirect
235. C3×#41C23×C5semidirect
241. C2×C3×Q8C23×C5semidirect
259. #216C23×C5semidirect
260. #215C23×C5semidirect
261. #215C23×C5semidirect
262. #214C23×C5semidirect
263. #213C23×C5semidirect
264. #208C23×C5semidirect
267. C2×C3⋊C8C23×C5semidirect
299. 2−1+4C2×C5×D3semidirect
306. C2×#41C2×C5×D3semidirect
308. C22×Q8C2×C5×D3semidirect
325. C2×C3×C5#577semidirect
379. #41C22×C5×D3semidirect
386. C2×Q8C22×C5×D3semidirect
409. C3×C5C2×#577semidirect
426. C2×C5#5837direct (N × H)
460. C2×C3C5×#577semidirect
463. C5C2×#5837direct (N × H)
477. C3C2×C5×#577semidirect
481. C2C5×#5837direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×2−1+42
C2×C5×#10092
C5×#58372
C2×C5×#9502
C2×C5×#9492
C2×C5×#9482
C2×C5×#9422
C22×C5×#2152
C2×C5×#5773
C2×#58375