Finite Group Databasebeta

← C2×#11755309 #14510113

Subgroups — C2×#11755309 #14510113

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×#92101C211yesmaximal, characteristic, complemented
2C2×C3×#4149C211yesmaximal, characteristic, complemented
3C2×C5×#948C211yesmaximal, characteristic, complemented
4#11755309C2116yesmaximal, direct product factor
5C2×#91313C213yesmaximal, complemented
6C2×#91312C213yesmaximal, complemented
7C2×#91311C213yesmaximal, complemented
8C22×#6657C213yesmaximal, complemented
9C2×#66085—31nomaximal, Hall
10C2×C3×C5×#41C2211yescharacteristic
11#92101—24no—
12#92101C2218yescomplemented
13C2×Q8×D15—21no—
14C2×#6819—23no—
15C2×#6817C2213yescomplemented
16C22×Dic30C2213yescomplemented
17C3×#4149—24no—
18C3×#4149C2218yescomplemented
19C2×C3×#713—21no—
20C2×C3×D4×D5—23no—
21C2×C3×#709C2213yes—
22C22×C3×D20C2213yes—
23C5×#948C2218yescomplemented
24C2×C5×#208C2213yescomplemented
25C22×C5×C3⋊C8C2213yescomplemented
26C2×#6662C2213yescomplemented
27C2×#6658C2211yescharacteristic, complemented
28#91313C22124yescomplemented
29#91312C22124yescomplemented
30#91311C22124yescomplemented
31C2×#6657C2213yescomplemented
32C2×#6657C2216yescomplemented
33C2×#6657C22124yescomplemented
34C2×#6653C2213yescomplemented
35C2×#5836—51nomaximal, Hall
36C2×#4150—31no—
37C2×#4149D311yescharacteristic, complemented
38C2×C5×#131—31no—
39#66085—316no—
40C2×#4055—33no—
41C2×#4054—33no—
42C2×#4050—33no—
43C22×#677—33no—
44C23×C3×D5—23no—
45C3×C5×#41D414yes—
46C3×C5×#41C2314yes—
47C2×C3×C5×Q8D411yescharacteristic
48C2×C3×C5×D4D413yes—
49C22×C3×C4×C5C2313yes—
50C2×#1220—23no—
51C22×Dic15—23no—
52Q8×D15—28no—
53#6819—224no—
54#6817—26no—
55#6817C23112yescomplemented
56C2×D60C2311yescharacteristic
57C2×C4×D15—23no—
58C2×Dic30—23no—
59C2×Dic30—26no—
60C2×Dic30C2313yes—
61C2×Dic30C23112yescomplemented
62C2×C3×C22⋊D5—23no—
63C3×#713—28no—
64C3×D4×D5—224no—
65C3×#709—26no—
66C3×#709C23112yes—
67C2×C3×D20—23no—
68C2×C3×D20—26no—
69C2×C3×D20C2313yes—
70C2×C3×D20C23112yes—
71C2×C3×C4×D5—23no—
72C2×C3×Dic10C2311yescharacteristic
73C5×#208C23112yescomplemented
74C2×C5×C3⋊C8C2311yescharacteristic
75C2×C5×C3⋊C8C2313yes—
76C2×C5×C3⋊C8C23112yescomplemented
77#6662C23112yescomplemented
78#6658C2314yescomplemented
79#6657C23112yescomplemented
80#6657C23124yescomplemented
81#6653C23112yescomplemented
82C2×C3×H(5,𝔽2)—51no—
83C2×#1010—51no—
84#5836—516no—
85C2×#951—53no—
86C2×#950—53no—
87C2×#948D511yescharacteristic, complemented
88C22×#214—53no—
89C2×#932—53no—
90C2×C5×#41C2×D311yescharacteristic
91#4150—38no—
92#4150—64no—
93#4149—24no—
94#4149C2×D318yescomplemented
95C2×#713—21no—
96C2×Q8×D5—61no—
97C2×#711—63no—
98C2×D4×D5—23no—
99C2×#709—33no—
100C2×#709C2×D313yes—
101C22×D20C2×D313yes—
102C22×Dic10—33no—
103C5×#131—38no—
104C2×C5×M16—33no—
105C22×C5×C8—33no—
106#4055—324no—
107#4054—324no—
108C2×Dic20—33no—
109#4050—324no—
110C2×D40—33no—
111C2×#677—33no—
112C2×#677—36no—
113C2×#677—324no—
114C2×#677—31no—
115C2×#577—151noSylow
116C23×C3×C5C2×D413yes—
117C22×D15—21no—
118C22×C3×D5—23no—
119C22×C3×D5—212no—
120C22×C3×D5—86no—
121C3×C5×Q8C2×D414yes—
122C3×C5×D4C2×D4112yes—
123C2×C3×C4×C5C2×D413yes—
124C2×C3×C4×C5C2×D416yes—
125C2×C3×C4×C5C2411yescharacteristic
126C2×C3×C4×C5C2416yes—
127#1220—212no—
128C2×Dic15—23no—
129C2×Dic15—212no—
130D60—21no—
131D60C2412yes—
132C4×D15—212no—
133Dic30—23no—
134Dic30—212no—
135Dic30C2416yes—
136C3×C22⋊D5—212no—
137C2×C3×Dic5—21no—
138C3×D20—23no—
139C3×D20—212no—
140C3×D20C2416yes—
141C3×C4×D5—212no—
142C3×Dic10—21no—
143C3×Dic10C2412yes—
144C5×C3⋊C8C2412yes—
145C5×C3⋊C8C2416yes—
146C3×H(5,𝔽2)—58no—
147C3×H(5,𝔽2)—104no—
148C2×C3×#41—53no—
149C2×C3×#41—101no—
150C2×C3×#41C2×D511yescharacteristic
151C22×C3×D4—53no—
152C22×C3×D4—103no—
153#1010—58no—
154#1010—104no—
155C2×D3×Q8—101no—
156C2×#237—103no—
157C2×#235—53no—
158C22×Dic6—53no—
159#951—524no—
160#950—524no—
161#948C2×D518yescomplemented
162C2×#216—53no—
163C2×#215—51no—
164C2×#214—56no—
165C2×#214—53no—
166C2×#214—524no—
167#932—524no—
168C2×#213—53no—
169C2×#208C2×D513yescomplemented
170C22×C3⋊C8C2×D513yescomplemented
171C23×D5—23no—
172C5×#41#6714yes—
173C5×#41C22×D314yes—
174C2×C5×Q8#6711yescharacteristic
175C2×C5×D4#6713yes—
176C22×C4×C5C22×D313yes—
177C2×C22⋊D5—23no—
178C2×C22⋊D5—63no—
179C22×Dic5—63no—
180#713—28no—
181Q8×D5—68no—
182#711—624no—
183D4×D5—224no—
184#709—26no—
185#709—312no—
186#709—66no—
187#709C22×D3112yes—
188C2×D20—23no—
189C2×D20—26no—
190C2×D20—31no—
191C2×D20C22×D313yes—
192C2×D20C22×D3112yes—
193C2×C4×D5—23no—
194C2×C4×D5—63no—
195C2×Dic10—33no—
196C2×Dic10—312no—
197C2×Dic10—63no—
198C2×Dic10—66no—
199C2×Dic10C22×D311yescharacteristic
200C5×M16—312no—
201C2×C5×C8—31no—
202C2×C5×C8—33no—
203C2×C5×C8—312no—
204Dic20—312no—
205D40—312no—
206#677—312no—
207#677—324no—
208#677—34no—
209C2×2−1+4—151no—
210C2×H(5,𝔽2)—51no—
211#577—1516no—
212C2×#137—153no—
213C2×AGL(1,ℤ/8)—153no—
214C2×#135—153no—
215C22×SD16—153no—
216C2×#131—31no—
217C22×C3×C5—26no—
218C22×C3×C5C22×D416yes—
219C22×C3×C5C22×D411yescharacteristic
220C2×D15—22no—
221C2×D15—42no—
222C2×C3×D5—26no—
223C2×C3×D5—46no—
224C2×C3×D5—812no—
225C3×C4×C5C22×D411yescharacteristic
226C3×C4×C5C22×D416yes—
227C3×C4×C5C2511yescharacteristic
228Dic15—26no—
229C3×Dic5—22no—
230C24×C3—103no—
231C3×#41—512no—
232C3×#41—108no—
233C3×#41—106no—
234C3×#41D2014yes—
235C3×#41C22×D514yes—
236C2×C3×Q8—51no—
237C2×C3×Q8D2011yescharacteristic
238C2×C3×D4—53no—
239C2×C3×D4—512no—
240C2×C3×D4—103no—
241C2×C3×D4—1024no—
242C2×C3×D4—103no—
243C2×C3×D4—106no—
244C2×C3×D4D2013yes—
245C22×C3×C4—103no—
246C22×C3×C4C22×D513yes—
247C2×#67—103no—
248C22×Dic3—103no—
249D3×Q8—108no—
250#237—1024no—
251#235—512no—
252#235—106no—
253C2×D12—51no—
254C2×C4×D3—103no—
255C2×Dic6—53no—
256C2×Dic6—512no—
257C2×Dic6—103no—
258C2×Dic6—106no—
259#216—512no—
260#215—54no—
261#214—524no—
262#214—512no—
263#213—512no—
264#208C22×D5112yescomplemented
265C2×C3⋊C8C22×D511yescharacteristic
266C2×C3⋊C8C22×D513yes—
267C2×C3⋊C8C22×D5112yescomplemented
268C23×C5C2×#6713yes—
269C22×D5—23no—
270C22×D5—212no—
271C22×D5—61no—
272C22×D5—86no—
273C5×Q8C2×#6714yes—
274C5×D4C2×#67112yes—
275C2×C4×C5C2×#6713yes—
276C2×C4×C5C2×#6716yes—
277C2×C4×C5C23×D311yescharacteristic
278C2×C4×C5C23×D316yes—
279C22⋊D5—212no—
280C22⋊D5—612no—
281C2×Dic5—21no—
282C2×Dic5—63no—
283C2×Dic5—612no—
284D20—23no—
285D20—212no—
286D20—32no—
287D20—61no—
288D20C23×D316yes—
289C4×D5—212no—
290C4×D5—612no—
291Dic10—21no—
292Dic10—36no—
293Dic10—63no—
294Dic10—612no—
295Dic10C23×D312yes—
296C5×C8—32no—
297C5×C8—36no—
2982−1+4—158no—
2992−1+4—304no—
300H(5,𝔽2)—58no—
301H(5,𝔽2)—104no—
302C2×#41—53no—
303C2×#41—101no—
304C2×#41—153no—
305C2×#41—303no—
306C2×#41D3×D511yescharacteristic
307C22×Q8—153no—
308C22×Q8—301no—
309C22×D4—53no—
310C22×D4—103no—
311#137—1524no—
312AGL(1,ℤ/8)—1524no—
313#135—1524no—
314C2×Q16—153no—
315C2×SD16—153no—
316C2×SD16—156no—
317C2×SD16—1524no—
318C2×SD16—151no—
319C2×D8—153no—
320#131—38no—
321C2×M16—33no—
322C22×C8—33no—
323C2×C3×C5—26no—
324C2×C3×C5#57712yescomplemented
325C2×C3×C5C23×D411yescharacteristic
326D15—42no—
327C3×D5—46no—
328C23×C3—103no—
329C23×C3—1012no—
330C23×C3—406no—
331C23×C3C2×D2013yes—
332C22×D3—101no—
333C3×Q8—52no—
334C3×Q8—101no—
335C3×Q8C2×D2014yes—
336C3×D4—56no—
337C3×D4—1012no—
338C3×D4—103no—
339C3×D4—1012no—
340C3×D4C2×D20112yes—
341C2×C3×C4—101no—
342C2×C3×C4—1012no—
343C2×C3×C4C2×D2013yes—
344C2×C3×C4C2×D2016yes—
345C2×C3×C4C23×D511yescharacteristic
346C2×C3×C4C23×D516yes—
347#67—1012no—
348C2×Dic3—103no—
349C2×Dic3—1012no—
350D12—52no—
351D12—101no—
352C4×D3—1012no—
353Dic6—56no—
354Dic6—103no—
355Dic6—1012no—
356C3⋊C8C23×D512yes—
357C3⋊C8C23×D516yes—
358C22×C5—26no—
359C22×C5C22×#6716yes—
360C22×C5C22×#6711yescharacteristic
361C2×D5—26no—
362C2×D5—46no—
363C2×D5—62no—
364C2×D5—812no—
365C2×D5—122no—
366C4×C5C22×#6711yescharacteristic
367C4×C5C22×#6716yes—
368C4×C5C24×D311yescharacteristic
369Dic5—22no—
370Dic5—66no—
371C24—103no—
372#41—512no—
373#41—108no—
374#41—106no—
375#41—1512no—
376#41—3024no—
377#41—306no—
378#41#120214yes—
379#41C2×D3×D514yes—
380C2×Q8—51no—
381C2×Q8—153no—
382C2×Q8—1512no—
383C2×Q8—308no—
384C2×Q8—303no—
385C2×Q8—306no—
386C2×Q8#120211yescharacteristic
387C2×D4—53no—
388C2×D4—512no—
389C2×D4—103no—
390C2×D4—1024no—
391C2×D4—103no—
392C2×D4—106no—
393C2×D4—151no—
394C2×D4—303no—
395C2×D4#120213yes—
396C22×C4—103no—
397C22×C4—303no—
398C22×C4—303no—
399C22×C4C2×D3×D513yes—
400Q16—1512no—
401SD16—1512no—
402SD16—1524no—
403SD16—154no—
404D8—1512no—
405M16—312no—
406C2×C8—31no—
407C2×C8—33no—
408C2×C8—312no—
409C3×C5C2×#57711yescharacteristic, complemented, Hall
410C22×C3—26no—
411C22×C3—106no—
412C22×C3—206no—
413C22×C3—4012no—
414C22×C3C22×D2016yes—
415C22×C3C22×D2011yescharacteristic
416C2×D3—102no—
417C2×D3—202no—
418C3×C4—102no—
419C3×C4C22×D2011yescharacteristic
420C3×C4C22×D2016yes—
421C3×C4C24×D511yescharacteristic
422Dic3—106no—
423C2×C5—26no—
424C2×C5#583612yescomplemented
425C2×C5C23×#6711yescharacteristic
426D5—46no—
427D5—122no—
428C23—103no—
429C23—1012no—
430C23—301no—
431C23—406no—
432C23C2×#120213yes—
433Q8—52no—
434Q8—101no—
435Q8—156no—
436Q8—3012no—
437Q8—303no—
438Q8C2×#120214yes—
439D4—56no—
440D4—1012no—
441D4—1012no—
442D4—103no—
443D4—152no—
444D4—3012no—
445D4—301no—
446D4C2×#1202112yes—
447C2×C4—101no—
448C2×C4—1012no—
449C2×C4—3012no—
450C2×C4—303no—
451C2×C4—3012no—
452C2×C4C2×#120216yes—
453C2×C4C2×#120213yes—
454C2×C4C22×D3×D516yes—
455C2×C4C22×D3×D511yescharacteristic
456C8—32no—
457C8—36no—
458C2×C3—26no—
459C2×C3—206no—
460C2×C3#6608512yescomplemented
461C2×C3C23×D2011yescharacteristic
462D3—202no—
463C5C2×#583611yescharacteristic, complemented, Sylow
464C22—26no—
465C22—106no—
466C22—206no—
467C22—302no—
468C22—4012no—
469C22—602no—
470C22C22×#120216yes—
471C22C22×#120211yescharacteristic
472C4—102no—
473C4—306no—
474C4C22×#120216yes—
475C4C22×#120211yescharacteristic
476C4C23×D3×D511yescharacteristic
477C3C2×#6608511yescharacteristic, complemented, Sylow
478C2—26no—
479C2—206no—
480C2—602no—
481C2#1175530912yesdirect product factor
482C2C23×#120211yescharacteristic
subgroupG/H
centerC22C22×#1202
commutator, cosocleC3×C4×C5C25
FrattiniC4C23×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
hypercenterC2×#41D3×D5
hypocenterC3×C5C2×#577
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#92101C2semidirect
2. C2×C3×#4149C2semidirect
3. C2×C5×#948C2semidirect
4. #11755309C2direct (N × H)
5. C2×#91313C2semidirect
6. C2×#91312C2semidirect
7. C2×#91311C2semidirect
8. C22×#6657C2semidirect
12. #92101C22semidirect
15. C2×#6817C22semidirect
16. C22×Dic30C22semidirect
18. C3×#4149C22semidirect
23. C5×#948C22semidirect
24. C2×C5×#208C22semidirect
25. C22×C5×C3⋊C8C22semidirect
26. C2×#6662C22semidirect
27. C2×#6658C22semidirect
28. #91313C22semidirect
29. #91312C22semidirect
30. #91311C22semidirect
31. C2×#6657C22semidirect
32. C2×#6657C22semidirect
33. C2×#6657C22semidirect
34. C2×#6653C22semidirect
37. C2×#4149D3semidirect
55. #6817C23semidirect
61. C2×Dic30C23semidirect
73. C5×#208C23semidirect
76. C2×C5×C3⋊C8C23semidirect
77. #6662C23semidirect
78. #6658C23semidirect
79. #6657C23semidirect
80. #6657C23semidirect
81. #6653C23semidirect
87. C2×#948D5semidirect
94. #4149C2×D3semidirect
161. #948C2×D5semidirect
169. C2×#208C2×D5semidirect
170. C22×C3⋊C8C2×D5semidirect
264. #208C22×D5semidirect
267. C2×C3⋊C8C22×D5semidirect
324. C2×C3×C5#577semidirect
409. C3×C5C2×#577semidirect
424. C2×C5#5836semidirect
460. C2×C3#66085semidirect
463. C5C2×#5836semidirect
477. C3C2×#66085semidirect
481. C2#11755309direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#921012
C2×C3×#41492
C2×C5×#9482
#117553092
C2×#913132
C2×#913122
C2×#913112
C22×#66572
C2×#660853
C2×#58365