Finite Group Databasebeta

← C2×C5×#5817 #14482811

Subgroups — C2×C5×#5817 #14482811

Order 1920.

3240 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
1C23×C3×C5×Q8C211yesmaximal, characteristic, complemented
2C22×C5×#239C211yesmaximal, characteristic, complemented
3C5×#5817C2116yesmaximal, direct product factor
4C2×C5×#947C213yesmaximal, complemented
5C2×C5×#946C213yesmaximal, complemented
6C2×C5×Q8×Dic3C211yesmaximal, characteristic, complemented
7C2×C5×#930C213yesmaximal, complemented
8C2×C4×C5×#67C213yesmaximal, complemented
9C2×C5×#548—31nomaximal, Hall
10C22×C3×C5×Q8—28no—
11C22×C3×C5×Q8C2213yes—
12C22×C3×C5×Q8C2211yescharacteristic
13C22×C3×C5×Q8C2218yescomplemented
14C23×C3×C4×C5C2213yes—
15C22×C5×#67C2213yes—
16C2×C5×#239—28no—
17C2×C5×#239C2218yescomplemented
18C22×C5×D12C2213yes—
19C22×C4×C5×D3C2213yes—
20C2×C5×#217C2211yescharacteristic
21C5×#947C22124yescomplemented
22C5×#946C22124yescomplemented
23C5×Q8×Dic3C2218yescomplemented
24C5×#930C22124yescomplemented
25C4×C5×#67C22124yescomplemented
26C2×C5×(C2×D3)⋊C4C2213yes—
27C2×C5×(C2×D3)⋊C4C2216yes—
28C2×C5×C4⋊Dic3C2213yes—
29C2×C5×Dic3⋊C4C2213yes—
30C2×C4×C5×Dic3C2213yes—
31C2×#5817C511yesmaximal, characteristic, direct product factor, Hall
32C22×C5×#41—31no—
33C23×C5×Q8D311yescharacteristic, complemented
34C5×#548—316no—
35C2×C5×#124—33no—
36C2×C5×C22⋊Q8—33no—
37C2×C5×#121—33no—
38C2×C4×C5×Q8—31no—
39C2×C4×C5×D4—33no—
40C24×C3×C5C2311yescharacteristic
41C23×C5×D3C2313yes—
42C2×C3×C5×Q8—232no—
43C2×C3×C5×Q8—212no—
44C2×C3×C5×Q8D418yescomplemented
45C2×C3×C5×Q8C23112yes—
46C2×C3×C5×Q8C2314yes—
47C22×C3×C4×C5—212no—
48C22×C3×C4×C5C2313yes—
49C22×C3×C4×C5C23112yes—
50C22×C3×C4×C5C2313yes—
51C2×C5×#67—212no—
52C2×C5×#67C23112yes—
53C22×C5×Dic3C2311yescharacteristic
54C22×C5×Dic3C2313yes—
55C22×C5×Dic3C2311yescharacteristic
56C5×#239—232no—
57C2×C5×D12—212no—
58C2×C5×D12C23112yes—
59C2×C4×C5×D3—212no—
60C2×C4×C5×D3C23112yes—
61C5×#217C2314yes—
62C5×(C2×D3)⋊C4C23112yes—
63C5×(C2×D3)⋊C4C23124yes—
64C5×C4⋊Dic3C23112yes—
65C5×Dic3⋊C4C23112yes—
66C4×C5×Dic3C23112yes—
67C23×C3×Q8C2×C511yescharacteristic, complemented
68C22×#239C2×C511yescharacteristic, complemented
69#5817C2×C5116yesdirect product factor
70C2×#947C2×C513yescomplemented
71C2×#946C2×C513yescomplemented
72C2×Q8×Dic3C2×C511yescharacteristic, complemented
73C2×#930C2×C513yescomplemented
74C2×C4×#67C2×C513yescomplemented
75C2×C5×#41—38no—
76C2×C5×#41—68no—
77C22×C5×Q8—28no—
78C22×C5×Q8C2×D313yes—
79C22×C5×Q8C2×D311yescharacteristic
80C22×C5×Q8C2×D318yescomplemented
81C22×C5×D4—33no—
82C22×C5×D4—33no—
83C23×C4×C5—33no—
84C23×C4×C5C2×D313yes—
85C5×#124—324no—
86C5×C22⋊Q8—324no—
87C5×#121—324no—
88C4×C5×Q8—38no—
89C4×C5×D4—324no—
90C2×C5×C4⋊C4—33no—
91C2×C5×C4⋊C4—33no—
92C2×C5×C22⋊C4—31no—
93C2×C5×C22⋊C4—33no—
94C2×C5×C22⋊C4—36no—
95C2×C42×C5—33no—
96C2×#548—31noSylow
97C23×C3×C5—22no—
98C23×C3×C5—22no—
99C23×C3×C5#4114yescomplemented
100C23×C3×C5C2412yes—
101C23×C3×C5C2411yescharacteristic
102C22×C5×D3—26no—
103C22×C5×D3—26no—
104C22×C5×D3—26no—
105C22×C5×D3C2416yes—
106C3×C5×Q8—224no—
107C3×C5×Q8C2×D4116yescomplemented
108C2×C3×C4×C5—26no—
109C2×C3×C4×C5—224no—
110C2×C3×C4×C5C2×D4112yes—
111C2×C3×C4×C5C2416yes—
112C2×C3×C4×C5C2416yes—
113C5×#67—224no—
114C2×C5×Dic3—22no—
115C2×C5×Dic3—26no—
116C2×C5×Dic3—22no—
117C2×C5×Dic3C2412yes—
118C2×C5×Dic3C2416yes—
119C2×C5×Dic3C2412yes—
120C5×D12—224no—
121C4×C5×D3—224no—
122C22×C3×Q8—28no—
123C22×C3×Q8C22×C513yes—
124C22×C3×Q8C22×C511yescharacteristic
125C22×C3×Q8C22×C518yescomplemented
126C23×C3×C4C22×C513yes—
127C22×#67C22×C513yes—
128C2×#239—28no—
129C2×#239C22×C518yescomplemented
130C22×D12C22×C513yes—
131C22×C4×D3C22×C513yes—
132C2×#217C22×C511yescharacteristic
133#947C22×C5124yescomplemented
134#946C22×C5124yescomplemented
135Q8×Dic3C22×C518yescomplemented
136#930C22×C5124yescomplemented
137C4×#67C22×C5124yescomplemented
138C2×(C2×D3)⋊C4C22×C513yes—
139C2×(C2×D3)⋊C4C22×C516yes—
140C2×C4⋊Dic3C22×C513yes—
141C2×Dic3⋊C4C22×C513yes—
142C2×C4×Dic3C22×C513yes—
143C24×C5—33no—
144C24×C5C22×D311yescharacteristic
145C5×#41—632no—
146C2×C5×Q8—232no—
147C2×C5×Q8—212no—
148C2×C5×Q8#6718yescomplemented
149C2×C5×Q8C22×D3112yes—
150C2×C5×Q8C22×D314yes—
151C2×C5×D4—312no—
152C2×C5×D4—312no—
153C2×C5×D4—612no—
154C2×C5×D4—612no—
155C22×C4×C5—212no—
156C22×C4×C5—31no—
157C22×C4×C5—33no—
158C22×C4×C5—31no—
159C22×C4×C5—312no—
160C22×C4×C5—612no—
161C22×C4×C5C22×D3112yes—
162C22×C4×C5C22×D313yes—
163C22×C4×C5C22×D313yes—
164C5×C4⋊C4—312no—
165C5×C4⋊C4—312no—
166C5×C22⋊C4—34no—
167C5×C22⋊C4—312no—
168C5×C22⋊C4—324no—
169C42×C5—312no—
170C22×#41—31no—
171C23×Q8C5×D311yescharacteristic, complemented
172#548—316no—
173C2×#124—33no—
174C2×C22⋊Q8—33no—
175C2×#121—33no—
176C2×C4×Q8—31no—
177C2×C4×D4—33no—
178C22×C3×C5—28no—
179C22×C3×C5—22no—
180C22×C3×C5—22no—
181C22×C3×C5C22×D412yes—
182C22×C3×C5C2×#4112yes—
183C22×C3×C5C2×#4114yescomplemented
184C22×C3×C52−1+412yes—
185C22×C3×C5C2511yescharacteristic
186C2×C5×D3—26no—
187C2×C5×D3—26no—
188C2×C5×D3—26no—
189C2×C5×D3—412no—
190C3×C4×C5—26no—
191C3×C4×C5C22×D4112yes—
192C5×Dic3—22no—
193C5×Dic3—26no—
194C5×Dic3—22no—
195C24×C3C23×C511yescharacteristic
196C23×D3C23×C513yes—
197C2×C3×Q8—232no—
198C2×C3×Q8—212no—
199C2×C3×Q8C5×D418yescomplemented
200C2×C3×Q8C23×C5112yes—
201C2×C3×Q8C23×C514yes—
202C22×C3×C4—212no—
203C22×C3×C4C23×C513yes—
204C22×C3×C4C23×C5112yes—
205C22×C3×C4C23×C513yes—
206C2×#67—212no—
207C2×#67C23×C5112yes—
208C22×Dic3C23×C511yescharacteristic
209C22×Dic3C23×C513yes—
210C22×Dic3C23×C511yescharacteristic
211#239—232no—
212C2×D12—212no—
213C2×D12C23×C5112yes—
214C2×C4×D3—212no—
215C2×C4×D3C23×C5112yes—
216#217C23×C514yes—
217(C2×D3)⋊C4C23×C5112yes—
218(C2×D3)⋊C4C23×C5124yes—
219C4⋊Dic3C23×C5112yes—
220Dic3⋊C4C23×C5112yes—
221C4×Dic3C23×C5112yes—
222C23×C5—22no—
223C23×C5—22no—
224C23×C5—36no—
225C23×C5—66no—
226C23×C5—66no—
227C23×C5—66no—
228C23×C5#23914yescomplemented
229C23×C5C23×D312yes—
230C23×C5C23×D311yescharacteristic
231C5×Q8—224no—
232C5×Q8C2×#67116yescomplemented
233C5×D4—624no—
234C5×D4—624no—
235C2×C4×C5—26no—
236C2×C4×C5—224no—
237C2×C4×C5—32no—
238C2×C4×C5—36no—
239C2×C4×C5—32no—
240C2×C4×C5—66no—
241C2×C4×C5—62no—
242C2×C4×C5—62no—
243C2×C4×C5—624no—
244C2×C4×C5C2×#67112yes—
245C2×C4×C5C23×D316yes—
246C2×C4×C5C23×D316yes—
247C2×#41—38no—
248C2×#41—68no—
249C22×Q8—28no—
250C22×Q8C2×C5×D313yes—
251C22×Q8C2×C5×D311yescharacteristic
252C22×Q8C2×C5×D318yescomplemented
253C22×D4—33no—
254C22×D4—33no—
255C23×C4—33no—
256C23×C4C2×C5×D313yes—
257#124—324no—
258C22⋊Q8—324no—
259#121—324no—
260C4×Q8—38no—
261C4×D4—324no—
262C2×C4⋊C4—33no—
263C2×C4⋊C4—33no—
264C2×C22⋊C4—31no—
265C2×C22⋊C4—33no—
266C2×C22⋊C4—36no—
267C2×C42—33no—
268C2×C3×C5—24no—
269C2×C3×C5#54814yescomplemented
270C2×C3×C5C23×D411yescharacteristic
271C2×C3×C5C22×#4111yescharacteristic
272C2×C3×C5C2×2−1+411yescharacteristic
273C5×D3—46no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3C5×#4114yescomplemented
277C23×C3C24×C512yes—
278C23×C3C24×C511yescharacteristic
279C22×D3—26no—
280C22×D3—26no—
281C22×D3—26no—
282C22×D3C24×C516yes—
283C3×Q8—224no—
284C3×Q8C2×C5×D4116yescomplemented
285C2×C3×C4—26no—
286C2×C3×C4—224no—
287C2×C3×C4C2×C5×D4112yes—
288C2×C3×C4C24×C516yes—
289C2×C3×C4C24×C516yes—
290#67—224no—
291C2×Dic3—22no—
292C2×Dic3—26no—
293C2×Dic3—22no—
294C2×Dic3C24×C512yes—
295C2×Dic3C24×C516yes—
296C2×Dic3C24×C512yes—
297D12—224no—
298C4×D3—224no—
299C22×C5—22no—
300C22×C5—22no—
301C22×C5—28no—
302C22×C5—66no—
303C22×C5—66no—
304C22×C5—66no—
305C22×C5—1212no—
306C22×C5C2×#23912yes—
307C22×C5C2×#23914yescomplemented
308C22×C5#100712yes—
309C22×C5C22×#6712yes—
310C22×C5C24×D311yescharacteristic
311C4×C5—26no—
312C4×C5—66no—
313C4×C5—62no—
314C4×C5—62no—
315C4×C5C22×#67112yes—
316C24—33no—
317C24C22×C5×D311yescharacteristic
318#41—632no—
319C2×Q8—232no—
320C2×Q8—212no—
321C2×Q8C5×#6718yescomplemented
322C2×Q8C22×C5×D3112yes—
323C2×Q8C22×C5×D314yes—
324C2×D4—312no—
325C2×D4—312no—
326C2×D4—612no—
327C2×D4—612no—
328C22×C4—212no—
329C22×C4—31no—
330C22×C4—33no—
331C22×C4—31no—
332C22×C4—312no—
333C22×C4—612no—
334C22×C4C22×C5×D313yes—
335C22×C4C22×C5×D3112yes—
336C22×C4C22×C5×D313yes—
337C4⋊C4—312no—
338C4⋊C4—312no—
339C22⋊C4—34no—
340C22⋊C4—312no—
341C22⋊C4—324no—
342C42—312no—
343C3×C5C2×#54811yescharacteristic, complemented, Hall
344C22×C3—28no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3C22×C5×D412yes—
348C22×C3C2×C5×#4112yes—
349C22×C3C2×C5×#4114yescomplemented
350C22×C3C5×2−1+412yes—
351C22×C3C25×C511yescharacteristic
352C2×D3—26no—
353C2×D3—26no—
354C2×D3—26no—
355C2×D3—412no—
356C3×C4—26no—
357C3×C4C22×C5×D4112yes—
358Dic3—22no—
359Dic3—26no—
360Dic3—22no—
361C2×C5—24no—
362C2×C5—126no—
363C2×C5#581714yesdirect product factor
364C2×C5C22×#23911yescharacteristic
365C2×C5C2×#100711yescharacteristic
366C2×C5C23×#6711yescharacteristic
367C23—22no—
368C23—22no—
369C23—36no—
370C23—66no—
371C23—66no—
372C23—66no—
373C23C5×#23914yescomplemented
374C23C23×C5×D312yes—
375C23C23×C5×D311yescharacteristic
376Q8—224no—
377Q8C2×C5×#67116yescomplemented
378D4—624no—
379D4—624no—
380C2×C4—26no—
381C2×C4—224no—
382C2×C4—32no—
383C2×C4—36no—
384C2×C4—32no—
385C2×C4—66no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—624no—
389C2×C4C2×C5×#67112yes—
390C2×C4C23×C5×D316yes—
391C2×C4C23×C5×D316yes—
392C2×C3—24no—
393C2×C3C5×#54814yescomplemented
394C2×C3C23×C5×D411yescharacteristic
395C2×C3C22×C5×#4111yescharacteristic
396C2×C3C2×C5×2−1+411yescharacteristic
397D3—46no—
398C5C2×#581711yescharacteristic, direct product factor, Sylow
399C22—22no—
400C22—22no—
401C22—28no—
402C22—66no—
403C22—66no—
404C22—66no—
405C22—1212no—
406C22C2×C5×#23912yes—
407C22C2×C5×#23914yescomplemented
408C22C5×#100712yes—
409C22C22×C5×#6712yes—
410C22C24×C5×D311yescharacteristic
411C4—26no—
412C4—66no—
413C4—62no—
414C4—62no—
415C4C22×C5×#67112yes—
416C3C2×C5×#54811yescharacteristic, complemented, Sylow
417C2—24no—
418C2—126no—
419C2C5×#581714yesdirect product factor
420C2C22×C5×#23911yescharacteristic
421C2C2×C5×#100711yescharacteristic
422C2C23×C5×#6711yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutator, cosocleC22×C3C25×C5
FrattiniC22C24×C5×D3
FittingC23×C3×C5×Q8C2
socleC23×C3×C5C24
hypercenterC23×C5×Q8D3
hypocenterC3C2×C5×#548
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C23×C5×Q8

Complements

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

NH ≅ G/Ntype
1. C23×C3×C5×Q8C2semidirect
2. C22×C5×#239C2semidirect
3. C5×#5817C2direct (N × H)
4. C2×C5×#947C2semidirect
5. C2×C5×#946C2semidirect
6. C2×C5×Q8×Dic3C2semidirect
7. C2×C5×#930C2semidirect
8. C2×C4×C5×#67C2semidirect
13. C22×C3×C5×Q8C22semidirect
17. C2×C5×#239C22semidirect
21. C5×#947C22semidirect
22. C5×#946C22semidirect
23. C5×Q8×Dic3C22semidirect
24. C5×#930C22semidirect
25. C4×C5×#67C22semidirect
31. C2×#5817C5direct (N × H)
33. C23×C5×Q8D3semidirect
44. C2×C3×C5×Q8D4semidirect
67. C23×C3×Q8C2×C5semidirect
68. C22×#239C2×C5semidirect
69. #5817C2×C5direct (N × H)
70. C2×#947C2×C5semidirect
71. C2×#946C2×C5semidirect
72. C2×Q8×Dic3C2×C5semidirect
73. C2×#930C2×C5semidirect
74. C2×C4×#67C2×C5semidirect
80. C22×C5×Q8C2×D3semidirect
99. C23×C3×C5#41semidirect
107. C3×C5×Q8C2×D4semidirect
125. C22×C3×Q8C22×C5semidirect
129. C2×#239C22×C5semidirect
133. #947C22×C5semidirect
134. #946C22×C5semidirect
135. Q8×Dic3C22×C5semidirect
136. #930C22×C5semidirect
137. C4×#67C22×C5semidirect
148. C2×C5×Q8#67semidirect
171. C23×Q8C5×D3semidirect
183. C22×C3×C5C2×#41semidirect
199. C2×C3×Q8C5×D4semidirect
228. C23×C5#239semidirect
232. C5×Q8C2×#67semidirect
252. C22×Q8C2×C5×D3semidirect
269. C2×C3×C5#548semidirect
276. C23×C3C5×#41semidirect
284. C3×Q8C2×C5×D4semidirect
307. C22×C5C2×#239semidirect
321. C2×Q8C5×#67semidirect
343. C3×C5C2×#548semidirect
349. C22×C3C2×C5×#41semidirect
363. C2×C5#5817direct (N × H)
373. C23C5×#239semidirect
377. Q8C2×C5×#67semidirect
393. C2×C3C5×#548semidirect
398. C5C2×#5817direct (N × H)
407. C22C2×C5×#239semidirect
416. C3C2×C5×#548semidirect
419. C2C5×#5817direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C5×Q82
C22×C5×#2392
C5×#58172
C2×C5×#9472
C2×C5×#9462
C2×C5×Q8×Dic32
C2×C5×#9302
C2×C4×C5×#672
C2×C5×#5483
C2×#58175