Finite Group Databasebeta

← C2×#11751148 #14423495

Subgroups — C2×#11751148 #14423495

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6817C211yesmaximal, characteristic
2C2×C3×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
3C22×#6718C211yesmaximal, characteristic, complemented
4C22×#6717C211yesmaximal, characteristic, complemented
5C2×#91114C211yesmaximal, characteristic, complemented
6#11751148C218yesmaximal, direct product factor
7C2×#91094C211yesmaximal, characteristic, complemented
8C2×#91093C211yesmaximal, characteristic, complemented
9C2×#65336—31nomaximal, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C23×C3×C4×C5C2211yescharacteristic
12C22×#1220—21no—
13C2×#6817—22no—
14C2×#6817—24no—
15C2×#6817—24no—
16C2×#6817C2214yes—
17C22×D60C2211yescharacteristic
18C22×C4×D15—21no—
19C22×Dic30C2211yescharacteristic
20C3×C5×C22⋊Q8—22no—
21C3×C5×C22⋊Q8C2214yescomplemented
22C2×C3×C5×C4⋊C4—21no—
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×C3×C5×C22⋊C4—21no—
25C2×#6718—24no—
26C2×#6718C2214yescomplemented
27C2×#6717—24no—
28C2×#6717C2214yescomplemented
29C22×C3⋊#168C2211yescharacteristic
30C22×C3⋊#168C2211yescharacteristic
31#91114C2214yescomplemented
32#91094C2214yescomplemented
33#91093C2214yescomplemented
34C2×#5047—51nomaximal, Hall
35C22×#709—31no—
36C2×C5×C22⋊Q8D311yescharacteristic, complemented
37C22×#689—31no—
38C22×#688—31no—
39C2×C22⋊#168—31no—
40#65336—38no—
41C2×#3942—31no—
42C2×#3941—31no—
43C24×C3×C5D411yescharacteristic
44C23×D15—21no—
45C2×C3×C5×Q8—21no—
46C2×C3×C5×Q8—22no—
47C2×C3×C5×Q8—22no—
48C2×C3×C5×Q8C2312yes—
49C22×C3×C4×C5—22no—
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—21no—
52C22×C3×C4×C5—21no—
53C22×C3×C4×C5D411yescharacteristic
54C22×C3×C4×C5D412yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C2×#1220—28no—
58C2×#1220—24no—
59C22×Dic15—21no—
60#6817—216no—
61#6817—28no—
62#6817—28no—
63C2×D60—21no—
64C2×D60—22no—
65C2×D60D414yes—
66C2×D60C2312yes—
67C2×C4×D15—28no—
68C2×C4×D15—24no—
69C2×Dic30—21no—
70C2×Dic30—22no—
71C2×Dic30D414yes—
72C2×Dic30C2312yes—
73C3×C5×C4⋊C4—24no—
74C3×C5×C4⋊C4—21no—
75C3×C5×C4⋊C4C2312yes—
76C3×C5×C22⋊C4—24no—
77#6718—28no—
78#6717—28no—
79C2×C3⋊#168—22no—
80C2×C3⋊#168—22no—
81C2×C3⋊#168C2312yes—
82C2×C3⋊#168C2312yes—
83C22×#235—51no—
84C2×C3×C22⋊Q8D511yescharacteristic, complemented
85C22×#216—51no—
86C22×#215—51no—
87C2×#830—51no—
88#5047—58no—
89C2×#810—51no—
90C2×#809—51no—
91C22×C5×Q8C2×D311yescharacteristic
92C23×C4×C5C2×D311yescharacteristic
93C22×C22⋊D5—61no—
94C2×#709—34no—
95C2×#709—64no—
96C2×#709—64no—
97C2×#709—62no—
98C22×D20—31no—
99C22×C4×D5—61no—
100C22×Dic10—31no—
101C5×C22⋊Q8—22no—
102C5×C22⋊Q8C2×D314yescomplemented
103C2×C5×C4⋊C4—21no—
104C2×C5×C4⋊C4C2×D311yescharacteristic
105C2×C5×C22⋊C4—21no—
106C2×#689—34no—
107C2×#689—64no—
108C2×#688—34no—
109C2×#688—64no—
110C22×#168—31no—
111C22×#168—31no—
112C22⋊#168—34no—
113#3942—34no—
114#3941—34no—
115C2×#452—151noSylow
116C23×C3×C5—22no—
117C23×C3×C5—22no—
118C23×C3×C5—22no—
119C23×C3×C5C2×D412yes—
120C23×C3×C5C2×D411yescharacteristic
121C22×D15—24no—
122C22×D15—22no—
123C22×D15—42no—
124C22×D15—42no—
125C3×C5×Q8—22no—
126C3×C5×Q8—22no—
127C3×C5×Q8—42no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—24no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—42no—
135C2×C3×C4×C5—42no—
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D412yes—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5C2411yescharacteristic
141#1220—216no—
142C2×Dic15—24no—
143C2×Dic15—22no—
144D60—24no—
145D60—22no—
146D60C2×D414yes—
147C4×D15—216no—
148Dic30—24no—
149Dic30—22no—
150Dic30C2×D414yes—
151C3⋊#168—22no—
152C3⋊#168—22no—
153C22×C3×Q8C2×D511yescharacteristic
154C23×C3×C4C2×D511yescharacteristic
155C22×#67—101no—
156C2×#235—54no—
157C2×#235—104no—
158C2×#235—104no—
159C2×#235—102no—
160C22×D12—51no—
161C22×C4×D3—101no—
162C22×Dic6—51no—
163C3×C22⋊Q8—22no—
164C3×C22⋊Q8C2×D514yescomplemented
165C2×C3×C4⋊C4—21no—
166C2×C3×C4⋊C4C2×D511yescharacteristic
167C2×C3×C22⋊C4—21no—
168C2×#216—54no—
169C2×#216—104no—
170C2×#215—54no—
171C2×#215—104no—
172C22×C3⋊C8—51no—
173C22×C3⋊C8—51no—
174#830—54no—
175#810—54no—
176#809—54no—
177C24×C5#6711yescharacteristic
178C23×D5—61no—
179C2×C5×Q8—21no—
180C2×C5×Q8—22no—
181C2×C5×Q8—22no—
182C2×C5×Q8C22×D312yes—
183C22×C4×C5—22no—
184C22×C4×C5—22no—
185C22×C4×C5—21no—
186C22×C4×C5—21no—
187C22×C4×C5#6711yescharacteristic
188C22×C4×C5#6712yes—
189C22×C4×C5C22×D312yes—
190C22×C4×C5C22×D311yescharacteristic
191C2×C22⋊D5—68no—
192C2×C22⋊D5—64no—
193C22×Dic5—61no—
194#709—68no—
195#709—68no—
196#709—616no—
197C2×D20—34no—
198C2×D20—32no—
199C2×D20—62no—
200C2×D20—61no—
201C2×C4×D5—68no—
202C2×C4×D5—64no—
203C2×Dic10—34no—
204C2×Dic10—32no—
205C2×Dic10—62no—
206C2×Dic10—61no—
207C5×C4⋊C4—24no—
208C5×C4⋊C4—21no—
209C5×C4⋊C4C22×D312yes—
210C5×C22⋊C4—24no—
211#689—68no—
212#688—68no—
213C2×#168—32no—
214C2×#168—32no—
215C2×#168—62no—
216C2×#168—62no—
217C22×#41—151no—
218C22×Q16—151no—
219C22×SD16—151no—
220C2×C22⋊Q8D1511yescharacteristic, complemented
221#452—158no—
222C2×#103—151no—
223C2×#102—151no—
224C2×C22⋊C8—151no—
225C22×C3×C5—22no—
226C22×C3×C5—22no—
227C22×C3×C5—22no—
228C22×C3×C5—44no—
229C22×C3×C5#12012yes—
230C22×C3×C5#13512yes—
231C22×C3×C5#13712yes—
232C22×C3×C5C22×D411yescharacteristic
233C2×D15—24no—
234C2×D15—42no—
235C2×D15—42no—
236C2×D15—48no—
237C3×C4×C5—22no—
238C3×C4×C5—42no—
239C3×C4×C5—42no—
240C3×C4×C5#12012yes—
241C3×C4×C5C22×D411yescharacteristic
242C3×C4×C5C22×D411yescharacteristic
243Dic15—24no—
244C24×C3C22⋊D511yescharacteristic
245C23×D3—101no—
246C2×C3×Q8—21no—
247C2×C3×Q8—22no—
248C2×C3×Q8—22no—
249C2×C3×Q8C22×D512yes—
250C22×C3×C4—22no—
251C22×C3×C4—22no—
252C22×C3×C4—21no—
253C22×C3×C4—21no—
254C22×C3×C4C22⋊D511yescharacteristic
255C22×C3×C4C22⋊D512yes—
256C22×C3×C4C22×D512yes—
257C22×C3×C4C22×D511yescharacteristic
258C2×#67—108no—
259C2×#67—104no—
260C22×Dic3—101no—
261#235—108no—
262#235—108no—
263#235—1016no—
264C2×D12—54no—
265C2×D12—52no—
266C2×D12—102no—
267C2×D12—101no—
268C2×C4×D3—108no—
269C2×C4×D3—104no—
270C2×Dic6—54no—
271C2×Dic6—52no—
272C2×Dic6—102no—
273C2×Dic6—101no—
274C3×C4⋊C4—24no—
275C3×C4⋊C4—21no—
276C3×C4⋊C4C22×D512yes—
277C3×C22⋊C4—24no—
278#216—108no—
279#215—108no—
280C2×C3⋊C8—52no—
281C2×C3⋊C8—52no—
282C2×C3⋊C8—102no—
283C2×C3⋊C8—102no—
284C23×C5—22no—
285C23×C5—22no—
286C23×C5—22no—
287C23×C5C2×#6712yes—
288C23×C5C2×#6711yescharacteristic
289C22×D5—64no—
290C22×D5—62no—
291C22×D5—122no—
292C22×D5—122no—
293C5×Q8—22no—
294C5×Q8—22no—
295C5×Q8—42no—
296C2×C4×C5—22no—
297C2×C4×C5—24no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—22no—
301C2×C4×C5—22no—
302C2×C4×C5—42no—
303C2×C4×C5—42no—
304C2×C4×C5D3×D412yes—
305C2×C4×C5D3×D412yes—
306C2×C4×C5C2×#6712yes—
307C2×C4×C5C2×#6711yescharacteristic
308C2×C4×C5C23×D311yescharacteristic
309C22⋊D5—616no—
310C2×Dic5—64no—
311C2×Dic5—62no—
312D20—34no—
313D20—64no—
314D20—62no—
315C4×D5—616no—
316Dic10—34no—
317Dic10—64no—
318Dic10—62no—
319#168—62no—
320#168—62no—
321C2×#41—154no—
322C2×#41—304no—
323C2×#41—304no—
324C2×#41—302no—
325C22×Q8—151no—
326C22×Q8C2×D1511yescharacteristic
327C22×D4—151no—
328C22×D4—301no—
329C23×C4—301no—
330C23×C4C2×D1511yescharacteristic
331C2×Q16—154no—
332C2×Q16—304no—
333C2×SD16—154no—
334C2×SD16—304no—
335C22×C8—151no—
336C22×C8—151no—
337C22⋊Q8—22no—
338C22⋊Q8C2×D1514yescomplemented
339C2×C4⋊C4—21no—
340C2×C4⋊C4C2×D1511yescharacteristic
341C2×C22⋊C4—21no—
342#103—154no—
343#102—154no—
344C22⋊C8—154no—
345C2×C3×C5—42no—
346C2×C3×C5#45214yescomplemented
347C2×C3×C5C2×#12011yescharacteristic
348C2×C3×C5C2×#13511yescharacteristic
349C2×C3×C5C2×#13711yescharacteristic
350D15—44no—
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3C2×C22⋊D512yes—
355C23×C3C2×C22⋊D511yescharacteristic
356C22×D3—102no—
357C22×D3—104no—
358C22×D3—202no—
359C22×D3—202no—
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—42no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4D4×D512yes—
372C2×C3×C4D4×D512yes—
373C2×C3×C4C2×C22⋊D512yes—
374C2×C3×C4C2×C22⋊D511yescharacteristic
375C2×C3×C4C23×D511yescharacteristic
376#67—1016no—
377C2×Dic3—102no—
378C2×Dic3—104no—
379D12—54no—
380D12—102no—
381D12—104no—
382C4×D3—1016no—
383Dic6—54no—
384Dic6—102no—
385Dic6—104no—
386C3⋊C8—102no—
387C3⋊C8—102no—
388C22×C5—22no—
389C22×C5—22no—
390C22×C5—22no—
391C22×C5—44no—
392C22×C5#93712yes—
393C22×C5#94212yes—
394C22×C5#95012yes—
395C22×C5C22×#6711yescharacteristic
396C2×D5—64no—
397C2×D5—122no—
398C2×D5—128no—
399C2×D5—122no—
400C4×C5—22no—
401C4×C5—42no—
402C4×C5—42no—
403C4×C5#93712yes—
404C4×C5C2×D3×D411yescharacteristic
405C4×C5C2×D3×D411yescharacteristic
406Dic5—64no—
407C24—301no—
408C24#122011yescharacteristic
409#41—308no—
410#41—308no—
411#41—3016no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—154no—
416C2×Q8—152no—
417C2×Q8—302no—
418C2×Q8—301no—
419C2×Q8C22×D1512yes—
420C2×D4—154no—
421C2×D4—152no—
422C2×D4—308no—
423C2×D4—304no—
424C2×D4—302no—
425C2×D4—301no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—308no—
431C22×C4—301no—
432C22×C4—304no—
433C22×C4#122012yes—
434C22×C4#122011yescharacteristic
435C22×C4C22×D1512yes—
436C22×C4C22×D1511yescharacteristic
437Q16—308no—
438SD16—308no—
439C2×C8—152no—
440C2×C8—152no—
441C2×C8—302no—
442C2×C8—302no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4C22×D1512yes—
446C22⋊C4—24no—
447C3×C5C2×#45211yescharacteristic, complemented, Hall
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—44no—
452C22×C3#408312yes—
453C22×C3#408812yes—
454C22×C3#409612yes—
455C22×C3C22×C22⋊D511yescharacteristic
456C2×D3—104no—
457C2×D3—208no—
458C2×D3—202no—
459C2×D3—202no—
460C3×C4—22no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4#408312yes—
464C3×C4C2×D4×D511yescharacteristic
465C3×C4C2×D4×D511yescharacteristic
466Dic3—104no—
467C2×C5—42no—
468C2×C5#504714yescomplemented
469C2×C5C2×#93711yescharacteristic
470C2×C5C2×#94211yescharacteristic
471C2×C5C2×#95011yescharacteristic
472D5—124no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—304no—
477C23—302no—
478C23—602no—
479C23—602no—
480C23C2×#122012yes—
481C23C2×#122011yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—42no—
485Q8—154no—
486Q8—304no—
487Q8—302no—
488D4—154no—
489D4—3016no—
490D4—304no—
491D4—302no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—3016no—
501C2×C4—304no—
502C2×C4—302no—
503C2×C4D4×D1512yes—
504C2×C4D4×D1512yes—
505C2×C4C2×#122012yes—
506C2×C4C2×#122011yescharacteristic
507C2×C4C23×D1511yescharacteristic
508C8—302no—
509C8—302no—
510C2×C3—42no—
511C2×C3#6533614yescomplemented
512C2×C3C2×#408311yescharacteristic
513C2×C3C2×#408811yescharacteristic
514C2×C3C2×#409611yescharacteristic
515D3—204no—
516C5C2×#504711yescharacteristic, complemented, Sylow
517C22—22no—
518C22—22no—
519C22—22no—
520C22—44no—
521C22—304no—
522C22—602no—
523C22—602no—
524C22—608no—
525C22#9182612yes—
526C22#9183112yes—
527C22#9183912yes—
528C22C22×#122011yescharacteristic
529C4—22no—
530C4—42no—
531C4—42no—
532C4—304no—
533C4#9182612yes—
534C4C2×D4×D1511yescharacteristic
535C4C2×D4×D1511yescharacteristic
536C3C2×#6533611yescharacteristic, complemented, Sylow
537C2—42no—
538C2—604no—
539C2#1175114814yesdirect product factor
540C2C2×#9182611yescharacteristic
541C2C2×#9183111yescharacteristic
542C2C2×#9183911yescharacteristic
subgroupG/H
centerC23C2×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×C22⋊Q8C2
socleC23×C3×C5C2×D4
prime radicalC22×#6817C2
hypercenterC2×C22⋊Q8D15
hypocenterC3×C5C2×#452
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×C22⋊Q8C2semidirect
3. C22×#6718C2semidirect
4. C22×#6717C2semidirect
5. C2×#91114C2semidirect
6. #11751148C2direct (N × H)
7. C2×#91094C2semidirect
8. C2×#91093C2semidirect
21. C3×C5×C22⋊Q8C22semidirect
26. C2×#6718C22semidirect
28. C2×#6717C22semidirect
31. #91114C22semidirect
32. #91094C22semidirect
33. #91093C22semidirect
36. C2×C5×C22⋊Q8D3semidirect
84. C2×C3×C22⋊Q8D5semidirect
102. C5×C22⋊Q8C2×D3semidirect
164. C3×C22⋊Q8C2×D5semidirect
220. C2×C22⋊Q8D15semidirect
338. C22⋊Q8C2×D15semidirect
346. C2×C3×C5#452semidirect
447. C3×C5C2×#452semidirect
468. C2×C5#5047semidirect
511. C2×C3#65336semidirect
516. C5C2×#5047semidirect
536. C3C2×#65336semidirect
539. C2#11751148direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#68172
C2×C3×C5×C22⋊Q82
C22×#67182
C22×#67172
C2×#911142
#117511482
C2×#910942
C2×#910932
C2×#653363
C2×#50475