Finite Group Databasebeta

← #14322408

Subgroups — #14322408

Order 1920.

2320 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1202C211yesmaximal, characteristic
2C2×C22⋊C4×D15C211yesmaximal, characteristic, complemented
3C2×#91561C211yesmaximal, characteristic, complemented
4C5×#4943C211yesmaximal, characteristic, complemented
5C3×#65244C211yesmaximal, characteristic, complemented
6#11749488C211yesmaximal, characteristic, complemented
7#11749472C211yesmaximal, characteristic, complemented
8#11748917C214yesmaximal, complemented
9#11748700C214yesmaximal, complemented
10#10605850—31nomaximal, Hall
11C24×D15C2211yescharacteristic
12C24×C3×D5C2211yescharacteristic
13C22×C4×D15C2212yescomplemented
14C23×C5×Dic3C2211yescharacteristic
15C22×#1202—24no—
16C22×#1202—24no—
17C22×#1202C418yescomplemented
18C22×#1202C2212yes—
19C22×#1202C2212yes—
20C22×#1200C2212yescomplemented
21C2×C3×C5×C22⋊C4C2211yescharacteristic
22C2×#6719C2211yescharacteristic
23C2×#6714C2212yescomplemented
24C22⋊C4×D15—24no—
25C2×C5×#217C2211yescharacteristic
26C2×C3×C22⋊Dic5C2211yescharacteristic
27C2×C3×#685C2214yescomplemented
28#91561—24no—
29C2×#6667C2212yescomplemented
30C2×#6665C2214yescomplemented
31C5×C3⋊#95C2214yescomplemented
32#90994C2214yescomplemented
33#73849—51nomaximal, Hall
34C23×D20—31no—
35C2×D5×C22⋊C4—31no—
36C2×D5×C22⋊C4—31no—
37C5×#381—31no—
38#65500—31no—
39#65244—31no—
40#65244D311yescharacteristic, complemented
41#64954—34no—
42#64954—34no—
43C24×C3×C5C2311yescharacteristic
44C23×D15—44no—
45C23×D15C2×C414yes—
46C23×D15D414yes—
47C23×D15D414yes—
48C23×D15C2311yescharacteristic
49C23×D15C2311yescharacteristic
50C23×C3×D5—22no—
51C23×C3×D5—22no—
52C23×C3×D5—44no—
53C23×C3×D5C2×C414yescomplemented
54C23×C3×D5C2312yes—
55C22×C3×C4×C5C2312yes—
56C22×Dic15C2312yes—
57C2×C4×D15—24no—
58C2×C4×D15—44no—
59C22×C5×Dic3—22no—
60C22×C5×Dic3—22no—
61C22×C5×Dic3C2×C414yescomplemented
62C22×C5×Dic3C2312yes—
63C22×C5×Dic3C2312yes—
64C22×C3×Dic5C2312yes—
65C2×#1202—232no—
66C2×#1202—24no—
67C2×#1202—24no—
68C2×#1202—24no—
69C2×#1202—24no—
70C2×#1202C2×C418yescomplemented
71C2×#1202C2×C418yescomplemented
72C2×#1200—24no—
73C2×#1200—44no—
74C3×C5×C22⋊C4—22no—
75#6719—22no—
76#6714—24no—
77C5×#217—22no—
78C3×C22⋊Dic5—22no—
79C3×#685—44no—
80#6667—24no—
81#6665—44no—
82C23×#67—51no—
83C2×D3×C22⋊C4—51no—
84C2×D3×C22⋊C4—51no—
85C3×#379—51no—
86#5247—51no—
87#4956—54no—
88#4956—54no—
89#4943—51no—
90#4943D511yescharacteristic, complemented
91C24×D5—31no—
92C24×D5C2×D311yescharacteristic
93C23×C4×C5—31no—
94C22×D20—38no—
95C22×D20—32no—
96C22×D20—32no—
97C22×D20—64no—
98C22×D20—64no—
99C22×C4×D5—32no—
100C22×C4×D5—32no—
101C2×C5×C22⋊C4—31no—
102C2×C5×C22⋊C4C2×D311yescharacteristic
103C2×C22⋊Dic5—31no—
104C2×C22⋊Dic5C2×D311yescharacteristic
105C2×#685—32no—
106C2×#685—32no—
107C2×#685—34no—
108C2×#685C2×D314yescomplemented
109D5×C22⋊C4—64no—
110D5×C22⋊C4—64no—
111C5×#95—34no—
112#3963—34no—
113#2308—151noSylow
114C23×C3×C5—24no—
115C23×C3×C5C22×C412yes—
116C23×C3×C5C2×D412yes—
117C23×C3×C5C2×D412yes—
118C23×C3×C5C2411yescharacteristic
119C22×D15—24no—
120C22×D15—24no—
121C22×D15—48no—
122C22×D15—48no—
123C22×D15—48no—
124C22×D15C22⋊C4116yescomplemented
125C22×D15C22×C412yes—
126C22×D15C22×C412yes—
127C22×D15C2×D412yes—
128C22×D15C2×D412yes—
129C22×D15C2×D412yes—
130C22×D15C2×D412yes—
131C22×C3×D5—28no—
132C22×C3×D5—22no—
133C22×C3×D5—22no—
134C22×C3×D5—24no—
135C22×C3×D5—24no—
136C22×C3×D5—48no—
137C22×C3×D5—48no—
138C22×C3×D5—48no—
139C22×C3×D5C22×C414yes—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5—24no—
142C2×Dic15—22no—
143C2×Dic15—24no—
144C4×D15—48no—
145C2×C5×Dic3—28no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—24no—
150C2×C5×Dic3C22×C414yes—
151C2×C3×Dic5—22no—
152C2×C3×Dic5—24no—
153#1202—216no—
154#1202—216no—
155#1200—48no—
156C25×C3—51no—
157C24×D3—51no—
158C22×#67—58no—
159C22×#67—52no—
160C22×#67—52no—
161C22×#67—104no—
162C22×#67—104no—
163C23×Dic3C2×D511yescharacteristic
164C22×C4×D3—52no—
165C22×C4×D3—52no—
166C2×C3×C22⋊C4—51no—
167C2×C3×C22⋊C4—54no—
168C2×C3×C22⋊C4C2×D511yescharacteristic
169C2×#217—51no—
170C2×#217—54no—
171C2×#217C2×D511yescharacteristic
172C2×(C2×D3)⋊C4—52no—
173C2×(C2×D3)⋊C4—52no—
174D3×C22⋊C4—104no—
175D3×C22⋊C4—104no—
176C3⋊#95—54no—
177C3⋊#95C2×D514yescomplemented
178C24×C5C22×D311yescharacteristic
179C23×D5—22no—
180C23×D5—22no—
181C23×D5—31no—
182C23×D5—31no—
183C23×D5—34no—
184C23×D5—34no—
185C23×D5—34no—
186C23×D5—44no—
187C23×D5—124no—
188C23×D5C4×D314yescomplemented
189C23×D5C22×D312yes—
190C22×C4×C5—34no—
191C22×C4×C5—32no—
192C22×C4×C5—32no—
193C22×C4×C5—62no—
194C22×C4×C5—62no—
195C22×C4×C5C22×D312yes—
196C22×Dic5—32no—
197C22×Dic5C22×D312yes—
198C2×D20—38no—
199C2×D20—38no—
200C2×D20—632no—
201C2×D20—64no—
202C2×D20—64no—
203C2×D20—64no—
204C2×D20—64no—
205C2×C4×D5—64no—
206C2×C4×D5—64no—
207C2×C4×D5—124no—
208C2×C4×D5—124no—
209C5×C22⋊C4—22no—
210C5×C22⋊C4—62no—
211C22⋊Dic5—22no—
212C22⋊Dic5—62no—
213#685—44no—
214#685—64no—
215#685—64no—
216#685—124no—
217C23×D4—151no—
218C22×C22⋊C4—151no—
219C22×C22⋊C4—151no—
220#386—154no—
221#386—154no—
222#381—31no—
223#381—151no—
224#379—51no—
225#379—151no—
226C22×C3×C5—24no—
227C22×C3×C5—24no—
228C22×C3×C5—24no—
229C22×C3×C5C2×C22⋊C414yes—
230C22×C3×C5C23×C411yescharacteristic
231C22×C3×C5C22×D411yescharacteristic
232C22×C3×C5C22×D411yescharacteristic
233C22×C3×C5H(5,𝔽2)14yes—
234C2×D15—24no—
235C2×D15—24no—
236C2×D15—24no—
237C2×D15—24no—
238C2×D15—24no—
239C2×D15—24no—
240C2×D15—416no—
241C2×D15C2×C22⋊C414yes—
242C2×D15C2×C22⋊C414yes—
243C2×C3×D5—24no—
244C2×C3×D5—24no—
245C2×C3×D5—24no—
246C2×C3×D5—416no—
247C2×C3×D5—44no—
248C2×C3×D5—44no—
249C3×C4×C5—42no—
250Dic15—42no—
251C3×Dic5—42no—
252C5×Dic3—24no—
253C5×Dic3—42no—
254C24×C3—52no—
255C24×C3—54no—
256C24×C3—102no—
257C24×C3—102no—
258C24×C3—204no—
259C24×C3C22×D511yescharacteristic
260C23×D3—51no—
261C23×D3—51no—
262C23×D3—54no—
263C23×D3—54no—
264C23×D3—54no—
265C23×D3—204no—
266C22×C3×C4—52no—
267C22×C3×C4C22×D512yes—
268C2×#67—58no—
269C2×#67—58no—
270C2×#67—104no—
271C2×#67—104no—
272C2×#67—104no—
273C2×#67—104no—
274C2×#67—1032no—
275C22×Dic3—22no—
276C22×Dic3—22no—
277C22×Dic3—52no—
278C22×Dic3C4×D514yescomplemented
279C22×Dic3C22×D512yes—
280C22×Dic3C22×D512yes—
281C2×C4×D3—104no—
282C2×C4×D3—104no—
283C2×C4×D3—204no—
284C2×C4×D3—204no—
285C3×C22⋊C4—22no—
286C3×C22⋊C4—102no—
287C3×C22⋊C4—204no—
288#217—22no—
289#217—102no—
290#217—204no—
291(C2×D3)⋊C4—104no—
292(C2×D3)⋊C4—104no—
293C23×C5—24no—
294C23×C5C2×C4×D312yes—
295C23×C5D3×D412yes—
296C23×C5D3×D412yes—
297C23×C5C23×D311yescharacteristic
298C22×D5—28no—
299C22×D5—22no—
300C22×D5—22no—
301C22×D5—24no—
302C22×D5—24no—
303C22×D5—32no—
304C22×D5—32no—
305C22×D5—32no—
306C22×D5—32no—
307C22×D5—32no—
308C22×D5—32no—
309C22×D5—316no—
310C22×D5—48no—
311C22×D5—48no—
312C22×D5—48no—
313C22×D5—64no—
314C22×D5—64no—
315C22×D5—128no—
316C22×D5—128no—
317C22×D5—128no—
318C22×D5C2×C4×D314yes—
319C2×C4×C5—22no—
320C2×C4×C5—24no—
321C2×C4×C5—34no—
322C2×C4×C5—68no—
323C2×C4×C5—62no—
324C2×C4×C5—62no—
325C2×C4×C5—62no—
326C2×C4×C5—64no—
327C2×Dic5—22no—
328C2×Dic5—24no—
329C2×Dic5—62no—
330C2×Dic5—64no—
331D20—616no—
332D20—616no—
333C4×D5—128no—
334C4×D5—128no—
335C25—51no—
336C25—151no—
337C22×D4—152no—
338C22×D4—152no—
339C22×D4—158no—
340C22×D4—304no—
341C22×D4—304no—
342C23×C4—31no—
343C23×C4—152no—
344C23×C4—152no—
345C2×C22⋊C4—31no—
346C2×C22⋊C4—51no—
347C2×C22⋊C4—54no—
348C2×C22⋊C4—152no—
349C2×C22⋊C4—151no—
350C2×C22⋊C4—152no—
351C2×C22⋊C4—154no—
352C2×C22⋊C4—304no—
353C2×C22⋊C4—304no—
354C2×C22⋊C4D3×D511yescharacteristic
355#95—34no—
356#95—154no—
357C2×C3×C5—24no—
358C2×C3×C5C22×C22⋊C411yescharacteristic
359C2×C3×C5#51812yes—
360C2×C3×C5#53412yes—
361C2×C3×C5#53412yes—
362D15—24no—
363D15—24no—
364C3×D5—44no—
365C23×C3—24no—
366C23×C3—54no—
367C23×C3—102no—
368C23×C3—102no—
369C23×C3—104no—
370C23×C3—104no—
371C23×C3—108no—
372C23×C3—208no—
373C23×C3—208no—
374C23×C3—208no—
375C23×C3C2×C4×D512yes—
376C23×C3D4×D512yes—
377C23×C3D4×D512yes—
378C23×C3C23×D511yescharacteristic
379C22×D3—52no—
380C22×D3—52no—
381C22×D3—52no—
382C22×D3—52no—
383C22×D3—52no—
384C22×D3—52no—
385C22×D3—516no—
386C22×D3—104no—
387C22×D3—104no—
388C22×D3—208no—
389C22×D3—208no—
390C22×D3—208no—
391C2×C3×C4—22no—
392C2×C3×C4—24no—
393C2×C3×C4—102no—
394C2×C3×C4—104no—
395#67—1016no—
396#67—1016no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—28no—
400C2×Dic3—22no—
401C2×Dic3—24no—
402C2×Dic3—102no—
403C2×Dic3—104no—
404C2×Dic3C2×C4×D514yes—
405C4×D3—208no—
406C4×D3—208no—
407C22×C5—24no—
408C22×C5—24no—
409C22×C5—24no—
410C22×C5D3×C22⋊C414yes—
411C22×C5C22×C4×D311yescharacteristic
412C22×C5C2×D3×D411yescharacteristic
413C22×C5C2×D3×D411yescharacteristic
414C22×C5#100414yes—
415C2×D5—24no—
416C2×D5—24no—
417C2×D5—24no—
418C2×D5—34no—
419C2×D5—34no—
420C2×D5—416no—
421C2×D5—44no—
422C2×D5—44no—
423C2×D5—64no—
424C2×D5—64no—
425C2×D5—64no—
426C2×D5—64no—
427C2×D5—64no—
428C2×D5—64no—
429C2×D5—1216no—
430C4×C5—42no—
431C4×C5—64no—
432C4×C5—122no—
433Dic5—42no—
434Dic5—122no—
435C24—52no—
436C24—54no—
437C24—102no—
438C24—102no—
439C24—151no—
440C24—151no—
441C24—154no—
442C24—154no—
443C24—154no—
444C24—204no—
445C24—604no—
446C24C2×D3×D511yescharacteristic
447C2×D4—158no—
448C2×D4—158no—
449C2×D4—3032no—
450C2×D4—304no—
451C2×D4—304no—
452C2×D4—304no—
453C2×D4—304no—
454C22×C4—34no—
455C22×C4—32no—
456C22×C4—32no—
457C22×C4—52no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—152no—
461C22×C4—304no—
462C22×C4—304no—
463C22×C4—604no—
464C22×C4—604no—
465C22×C4C2×D3×D512yes—
466C22⋊C4—22no—
467C22⋊C4—62no—
468C22⋊C4—102no—
469C22⋊C4—204no—
470C22⋊C4—304no—
471C22⋊C4—304no—
472C22⋊C4—302no—
473C22⋊C4—604no—
474C3×C5#230811yescharacteristic, complemented, Hall
475C22×C3—24no—
476C22×C3—24no—
477C22×C3—24no—
478C22×C3—104no—
479C22×C3—104no—
480C22×C3—104no—
481C22×C3—2016no—
482C22×C3—204no—
483C22×C3—204no—
484C22×C3D5×C22⋊C414yes—
485C22×C3C22×C4×D511yescharacteristic
486C22×C3C2×D4×D511yescharacteristic
487C22×C3C2×D4×D511yescharacteristic
488C22×C3#414914yes—
489C2×D3—54no—
490C2×D3—54no—
491C2×D3—104no—
492C2×D3—104no—
493C2×D3—104no—
494C2×D3—104no—
495C2×D3—104no—
496C2×D3—104no—
497C2×D3—2016no—
498C3×C4—42no—
499C3×C4—202no—
500Dic3—24no—
501Dic3—42no—
502Dic3—202no—
503C2×C5—24no—
504C2×C5C2×D3×C22⋊C411yescharacteristic
505C2×C5#548612yes—
506C2×C5#558912yes—
507C2×C5#558912yes—
508D5—44no—
509D5—64no—
510D5—64no—
511C23—24no—
512C23—54no—
513C23—108no—
514C23—102no—
515C23—104no—
516C23—104no—
517C23—102no—
518C23—152no—
519C23—152no—
520C23—152no—
521C23—152no—
522C23—152no—
523C23—152no—
524C23—1516no—
525C23—208no—
526C23—208no—
527C23—208no—
528C23—304no—
529C23—304no—
530C23—608no—
531C23—608no—
532C23—608no—
533C23C4×D3×D512yes—
534C23#677712yes—
535C23#679012yes—
536C23C22×D3×D511yescharacteristic
537D4—3016no—
538D4—3016no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—34no—
542C2×C4—68no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—64no—
547C2×C4—102no—
548C2×C4—104no—
549C2×C4—304no—
550C2×C4—302no—
551C2×C4—608no—
552C2×C4—608no—
553C2×C3—24no—
554C2×C3—204no—
555C2×C3C2×D5×C22⋊C411yescharacteristic
556C2×C3#6585512yes—
557C2×C3#6598712yes—
558C2×C3#6598712yes—
559D3—104no—
560D3—104no—
561C5#7384911yescharacteristic, complemented, Sylow
562C22—24no—
563C22—24no—
564C22—24no—
565C22—104no—
566C22—104no—
567C22—104no—
568C22—154no—
569C22—154no—
570C22—2016no—
571C22—204no—
572C22—204no—
573C22—304no—
574C22—304no—
575C22—304no—
576C22—304no—
577C22—304no—
578C22—304no—
579C22—6016no—
580C22#9147314yes—
581C22C2×C4×D3×D511yescharacteristic
582C22C2×#677711yescharacteristic
583C22#9202614yes—
584C22C2×#679011yescharacteristic
585C4—42no—
586C4—64no—
587C4—122no—
588C4—202no—
589C4—602no—
590C3#1060585011yescharacteristic, complemented, Sylow
591C2—24no—
592C2—204no—
593C2—304no—
594C2—304no—
595C2#1175306612yes—
596C2#1175327512yes—
597C2C2×#9147311yescharacteristic
598C2#1175495712yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C22⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC23×#1202C2
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#2308
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×D15C2semidirect
3. C2×#91561C2semidirect
4. C5×#4943C2semidirect
5. C3×#65244C2semidirect
6. #11749488C2semidirect
7. #11749472C2semidirect
8. #11748917C2semidirect
9. #11748700C2semidirect
13. C22×C4×D15C22semidirect
17. C22×#1202C4semidirect
20. C22×#1200C22semidirect
23. C2×#6714C22semidirect
27. C2×C3×#685C22semidirect
29. C2×#6667C22semidirect
30. C2×#6665C22semidirect
31. C5×C3⋊#95C22semidirect
32. #90994C22semidirect
40. #65244D3semidirect
53. C23×C3×D5C2×C4semidirect
61. C22×C5×Dic3C2×C4semidirect
70. C2×#1202C2×C4semidirect
71. C2×#1202C2×C4semidirect
90. #4943D5semidirect
108. C2×#685C2×D3semidirect
124. C22×D15C22⋊C4semidirect
177. C3⋊#95C2×D5semidirect
188. C23×D5C4×D3semidirect
278. C22×Dic3C4×D5semidirect
474. C3×C5#2308semidirect
561. C5#73849semidirect
590. C3#10605850semidirect

Maximal subgroups

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

subgroupindex
C23×#12022
C2×C22⋊C4×D152
C2×#915612
C5×#49432
C3×#652442
#117494882
#117494722
#117489172
#117487002
#106058503
#738495