Finite Group Databasebeta

← #14412574

Subgroups — #14412574

Order 1920.

1544 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6773C211yesmaximal, characteristic
2C2×C3×D5×M16C211yesmaximal, characteristic, complemented
3D5×#926C211yesmaximal, characteristic, complemented
4#11751232C211yesmaximal, characteristic, complemented
5C5×#5120C211yesmaximal, characteristic, complemented
6#11748409C211yesmaximal, characteristic, complemented
7#11748396C211yesmaximal, characteristic, complemented
8#11747642C214yesmaximal, complemented
9#11747606C214yesmaximal, complemented
10#10610687—31nomaximal, Hall
11C22×D60C2211yescharacteristic
12C22×C5×Dic6C2211yescharacteristic
13C22×C3×C4×D5C2211yescharacteristic
14C22×#1202—21no—
15C22×#1200—21no—
16C2×#6773—21no—
17C2×#6773—21no—
18C2×#6773—24no—
19C2×#6773C418yescomplemented
20C2×#6773C2212yes—
21C2×#6773C2212yes—
22C2×C3×C5×M16C2211yescharacteristic
23#91815C2211yescharacteristic
24C5×#926C2211yescharacteristic
25C2×C3×#681C2211yescharacteristic
26C3×D5×M16—24no—
27C2×C3×#676C2212yescomplemented
28C2×C3×C8×D5C2212yescomplemented
29D5×#217—21no—
30#91528—21no—
31D5×C4⋊Dic3C2212yescomplemented
32C4×D5×Dic3—21no—
33#91358C2212yescomplemented
34#91357—21no—
35#91105C2214yescomplemented
36C5×#816C2214yescomplemented
37#90973C2214yescomplemented
38#90969C2214yescomplemented
39#83768—51nomaximal, Hall
40C22×#713—31no—
41C2×D5×M16D311yescharacteristic, complemented
42D5×#117—31no—
43C5×#419—31no—
44#65514—31no—
45#65420—31no—
46#65290—31no—
47#65093—34no—
48#65092—34no—
49C23×D15—21no—
50C23×C3×D5D411yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C2×D60—21no—
53C2×D60—22no—
54C2×D60C2×C414yescomplemented
55C2×D60C2312yes—
56C22×C5×Dic3—21no—
57C2×C5×Dic6—21no—
58C2×C5×Dic6—22no—
59C2×C5×Dic6C2×C414yescomplemented
60C2×C5×Dic6C2312yes—
61C22×C3×Dic5D411yescharacteristic
62C2×C3×C4×D5C2×C414yes—
63C2×C3×C4×D5D411yescharacteristic
64C2×C3×C4×D5D411yescharacteristic
65C2×C3×C4×D5D414yes—
66C2×C3×C4×D5C2311yescharacteristic
67C2×C3×C4×D5C2311yescharacteristic
68C2×#1202—28no—
69C2×#1202—22no—
70C2×#1202—22no—
71C2×#1200—28no—
72C2×#1200—22no—
73C2×#1200—22no—
74C2×D5×Dic3—22no—
75#6773—28no—
76#6773—28no—
77#6773—24no—
78#6773—24no—
79#6773C2×C418yes—
80#6773C2×C418yescomplemented
81C3×C5×M16—22no—
82C2×C3×C5×C8C2312yes—
83#6719—21no—
84C4⋊Dic15C2312yes—
85C4×Dic15—21no—
86C5×#217—21no—
87C5×C4⋊Dic3C2312yes—
88C4×C5×Dic3—21no—
89C3×#681—22no—
90C2×C3×#168C2312yes—
91C3×#676—24no—
92C3×C8×D5—24no—
93#6668—22no—
94#6665—22no—
95Dic3×Dic5—22no—
96C22×#235—51no—
97C22×C3×M16—51no—
98C2×#926—51no—
99#5133—54no—
100#5131—51no—
101#5131—51no—
102#5120—51no—
103#5120D511yescharacteristic, complemented
104#5113—54no—
105C22×C5×Q8—31no—
106C2×#713—38no—
107C2×#713—32no—
108C2×#713—32no—
109C2×#713—61no—
110C2×#713—61no—
111C2×#713—64no—
112C22×D20—31no—
113C22×D20—61no—
114C22×C4×D5—61no—
115C22×C4×D5C2×D311yescharacteristic
116C2×C5×M16C2×D311yescharacteristic
117C5×#117—31no—
118#4072—31no—
119C2×#681C2×D311yescharacteristic
120D5×M16—24no—
121C2×#676C2×D312yescomplemented
122C2×C8×D5C2×D312yescomplemented
123#4038—32no—
124D5×C4⋊C4—32no—
125D5×C22⋊C4—61no—
126#4023—61no—
127#4018—61no—
128C42×D5—61no—
129C5×#103—34no—
130#3967—34no—
131#3953—34no—
132#3941—34no—
133#2896—151noSylow
134C23×C3×C5C2×D411yescharacteristic
135C22×D15—24no—
136C22×D15—22no—
137C22×D15—44no—
138C22×C3×D5—42no—
139C22×C3×D5C22⋊C414yes—
140C22×C3×D5C2×D411yescharacteristic
141C22×C3×D5C2×D411yescharacteristic
142C2×C3×C4×C5C22×C412yes—
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2411yescharacteristic
146C2×Dic15—22no—
147D60—24no—
148D60—22no—
149D60C22×C414yes—
150C2×C5×Dic3—24no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—22no—
153C5×Dic6—24no—
154C5×Dic6—22no—
155C5×Dic6C22×C414yes—
156C2×C3×Dic5C22⋊C414yes—
157C2×C3×Dic5C2×D411yescharacteristic
158C2×C3×Dic5C2×D411yescharacteristic
159C3×C4×D5C22⋊C414yes—
160C3×C4×D5C22⋊C414yes—
161C3×C4×D5C22×C412yes—
162C3×C4×D5C22×C412yes—
163C3×C4×D5C2×D412yes—
164C3×C4×D5C2×D412yes—
165#1202—28no—
166#1202—28no—
167#1200—28no—
168#1200—28no—
169D5×Dic3—44no—
170C3×C5×C8—22no—
171C3×#168—22no—
172C23×C3×C4—51no—
173C22×#67—101no—
174C2×#235—58no—
175C2×#235—52no—
176C2×#235—52no—
177C2×#235—101no—
178C2×#235—101no—
179C2×#235—104no—
180C22×D12—51no—
181C22×C4×D3—101no—
182C22×Dic6C2×D511yescharacteristic
183C2×C3×M16—51no—
184C2×C3×M16—52no—
185C2×C3×M16—104no—
186C2×C3×M16C2×D511yescharacteristic
187C22×C3×C8—52no—
188C2×#217—101no—
189#926—51no—
190#926—52no—
191#926—101no—
192#926—101no—
193#926C2×D511yescharacteristic
194C2×C4⋊Dic3—52no—
195C2×C4×Dic3—101no—
196#821—54no—
197#821—54no—
198#816—54no—
199#816C2×D514yescomplemented
200C23×D5—61no—
201C23×D5D1211yescharacteristic
202C2×C5×Q8—34no—
203C2×C5×Q8—32no—
204C2×C5×Q8—61no—
205C2×C5×Q8—62no—
206C22×C4×C5—61no—
207C22×C4×C5C22×D311yescharacteristic
208C22×Dic5D1211yescharacteristic
209#713—38no—
210#713—38no—
211#713—68no—
212#713—68no—
213#713—64no—
214#713—64no—
215C2×D20—34no—
216C2×D20—32no—
217C2×D20—68no—
218C2×D20—62no—
219C2×D20—62no—
220C2×D20—61no—
221C2×D20—62no—
222C2×C4×D5—68no—
223C2×C4×D5—62no—
224C2×C4×D5—62no—
225C2×C4×D5—62no—
226C2×C4×D5C4×D314yes—
227C2×C4×D5D1211yescharacteristic
228C2×C4×D5D1211yescharacteristic
229C2×C4×D5#6714yes—
230C2×C4×D5C22×D311yescharacteristic
231C2×C4×D5C22×D311yescharacteristic
232C5×M16—22no—
233C2×C5×C8C22×D312yes—
234C5×C4⋊C4—32no—
235C5×C22⋊C4—61no—
236C42×C5—61no—
237C22⋊Dic5—61no—
238#685—62no—
239C4⋊Dic5—32no—
240#683—62no—
241C4×Dic5—61no—
242C4×Dic5—62no—
243#681—22no—
244C2×#168C22×D312yes—
245#676—24no—
246C8×D5—24no—
247C22×#41—151no—
248C22×M16—51no—
249C2×#117—151no—
250#419—31no—
251#419—151no—
252#418—151no—
253#418—151no—
254#417—154no—
255#416—154no—
256C22×C3×C5—22no—
257C22×C3×C5C2×C22⋊C412yes—
258C22×C3×C5C22×D411yescharacteristic
259C2×D15—24no—
260C2×D15—48no—
261C2×D15—44no—
262C2×C3×D5—22no—
263C2×C3×D5—22no—
264C2×C3×D5—44no—
265C2×C3×D5C2×C22⋊C412yes—
266C2×C3×D5C2×C22⋊C412yes—
267C3×C4×C5C2×C22⋊C412yes—
268C3×C4×C5C23×C411yescharacteristic
269C3×C4×C5C22×D411yescharacteristic
270Dic15—42no—
271C3×Dic5C2×C22⋊C412yes—
272C3×Dic5C2×C22⋊C412yes—
273C5×Dic3—24no—
274C5×Dic3—42no—
275C24×C3—51no—
276C23×D3—101no—
277C22×C3×C4—51no—
278C22×C3×C4—54no—
279C22×C3×C4—51no—
280C22×C3×C4—51no—
281C22×C3×C4—54no—
282C22×C3×C4—51no—
283C22×C3×C4—51no—
284C22×C3×C4C22×D511yescharacteristic
285C2×#67—102no—
286C2×#67—102no—
287C2×#67—108no—
288C22×Dic3—21no—
289C22×Dic3—102no—
290#235—58no—
291#235—58no—
292#235—104no—
293#235—104no—
294#235—108no—
295#235—108no—
296C2×D12—54no—
297C2×D12—52no—
298C2×D12—101no—
299C2×D12—102no—
300C2×C4×D3—102no—
301C2×C4×D3—102no—
302C2×C4×D3—108no—
303C2×Dic6—21no—
304C2×Dic6—22no—
305C2×Dic6C4×D514yescomplemented
306C2×Dic6C22×D512yes—
307C3×M16—22no—
308C3×M16—104no—
309C3×M16—102no—
310C2×C3×C8—52no—
311C2×C3×C8—104no—
312C2×C3×C8C22×D512yes—
313#217—21no—
314#217—102no—
315#217—101no—
316C4⋊Dic3—52no—
317C4⋊Dic3—102no—
318C4⋊Dic3C22×D512yes—
319C4×Dic3—21no—
320C4×Dic3—102no—
321C4×Dic3—101no—
322C23×C5C2×D1211yescharacteristic
323C22×D5—42no—
324C22×D5—64no—
325C22×D5—62no—
326C22×D5—124no—
327C22×D5(C2×D3)⋊C414yes—
328C22×D5C2×D1211yescharacteristic
329C22×D5C2×D1211yescharacteristic
330C5×Q8—34no—
331C5×Q8—64no—
332C5×Q8—62no—
333C2×C4×C5—64no—
334C2×C4×C5—62no—
335C2×C4×C5—62no—
336C2×C4×C5C2×C4×D312yes—
337C2×C4×C5C2×D1211yescharacteristic
338C2×C4×C5C2×#6712yes—
339C2×C4×C5C23×D311yescharacteristic
340C2×Dic5—62no—
341C2×Dic5(C2×D3)⋊C414yes—
342C2×Dic5C2×D1211yescharacteristic
343C2×Dic5C2×D1211yescharacteristic
344D20—34no—
345D20—68no—
346D20—68no—
347D20—64no—
348D20—62no—
349C4×D5—68no—
350C4×D5—68no—
351C4×D5—124no—
352C4×D5(C2×D3)⋊C414yes—
353C4×D5(C2×D3)⋊C414yes—
354C4×D5C2×C4×D312yes—
355C4×D5C2×C4×D312yes—
356C4×D5C2×#6712yes—
357C4×D5C2×#6712yes—
358C5×C8—22no—
359#168—22no—
360C2×#41—152no—
361C2×#41—152no—
362C2×#41—158no—
363C2×#41—301no—
364C2×#41—301no—
365C2×#41—304no—
366C22×Q8—31no—
367C22×D4—151no—
368C22×D4—301no—
369C23×C4—51no—
370C23×C4—301no—
371C2×M16—51no—
372C2×M16—52no—
373C2×M16—104no—
374C2×M16D3×D511yescharacteristic
375C22×C8—52no—
376#117—31no—
377#117—152no—
378#117—151no—
379#117—301no—
380#117—301no—
381C2×C4⋊C4—152no—
382C2×C22⋊C4—301no—
383C2×C42—301no—
384#103—34no—
385#103—154no—
386#102—154no—
387#102—154no—
388C2×C3×C5—22no—
389C2×C3×C5C22×C22⋊C411yescharacteristic
390C2×C3×C5#57512yes—
391D15—44no—
392C3×D5—22no—
393C3×D5—22no—
394C23×C3—51no—
395C23×C3—51no—
396C23×C3—54no—
397C23×C3—202no—
398C23×C3D4×D511yescharacteristic
399C22×D3—102no—
400C22×D3—104no—
401C22×D3—204no—
402C2×C3×C4—51no—
403C2×C3×C4—51no—
404C2×C3×C4—54no—
405C2×C3×C4—52no—
406C2×C3×C4—52no—
407C2×C3×C4—52no—
408C2×C3×C4—52no—
409C2×C3×C4—54no—
410C2×C3×C4—54no—
411C2×C3×C4C2×C4×D512yes—
412C2×C3×C4D4×D511yescharacteristic
413C2×C3×C4D4×D512yes—
414C2×C3×C4C23×D511yescharacteristic
415#67—108no—
416#67—108no—
417C2×Dic3—22no—
418C2×Dic3—24no—
419C2×Dic3—22no—
420C2×Dic3—102no—
421C2×Dic3—204no—
422D12—54no—
423D12—102no—
424D12—104no—
425C4×D3—108no—
426C4×D3—108no—
427Dic6—22no—
428Dic6—24no—
429Dic6C2×C4×D514yes—
430C3×C8—22no—
431C3×C8—102no—
432C22×C5—22no—
433C22×C5C2×(C2×D3)⋊C412yes—
434C22×C5C22×D1211yescharacteristic
435C2×D5—22no—
436C2×D5—22no—
437C2×D5—44no—
438C2×D5—64no—
439C2×D5—128no—
440C2×D5—124no—
441C2×D5C2×(C2×D3)⋊C412yes—
442C2×D5C2×(C2×D3)⋊C412yes—
443C4×C5—64no—
444C4×C5—122no—
445C4×C5C2×(C2×D3)⋊C412yes—
446C4×C5C22×C4×D311yescharacteristic
447C4×C5C22×#6711yescharacteristic
448Dic5—122no—
449Dic5C2×(C2×D3)⋊C412yes—
450Dic5C2×(C2×D3)⋊C412yes—
451C24—51no—
452C24—301no—
453#41—158no—
454#41—158no—
455#41—308no—
456#41—308no—
457#41—304no—
458#41—304no—
459C2×Q8—34no—
460C2×Q8—32no—
461C2×Q8—61no—
462C2×Q8—62no—
463C2×D4—154no—
464C2×D4—152no—
465C2×D4—308no—
466C2×D4—302no—
467C2×D4—302no—
468C2×D4—301no—
469C2×D4—302no—
470C22×C4—51no—
471C22×C4—54no—
472C22×C4—51no—
473C22×C4—51no—
474C22×C4—54no—
475C22×C4—51no—
476C22×C4—51no—
477C22×C4—61no—
478C22×C4—308no—
479C22×C4—302no—
480C22×C4—302no—
481C22×C4—302no—
482C22×C4C2×D3×D511yescharacteristic
483M16—22no—
484M16—104no—
485M16—102no—
486C2×C8—52no—
487C2×C8—104no—
488C2×C8C2×D3×D512yes—
489C4⋊C4—32no—
490C4⋊C4—152no—
491C4⋊C4—302no—
492C22⋊C4—61no—
493C22⋊C4—302no—
494C22⋊C4—301no—
495C42—61no—
496C42—302no—
497C42—301no—
498C3×C5#289611yescharacteristic, complemented, Hall
499C22×C3—22no—
500C22×C3—52no—
501C22×C3—52no—
502C22×C3—102no—
503C22×C3—102no—
504C22×C3—204no—
505C22×C3D5×C22⋊C412yes—
506C22×C3C2×D4×D511yescharacteristic
507C2×D3—104no—
508C2×D3—208no—
509C2×D3—204no—
510C3×C4—52no—
511C3×C4—52no—
512C3×C4D5×C22⋊C412yes—
513C3×C4C22×C4×D511yescharacteristic
514C3×C4C2×D4×D511yescharacteristic
515Dic3—24no—
516Dic3—42no—
517Dic3—202no—
518C2×C5—22no—
519C2×C5#574812yes—
520C2×C5C22×(C2×D3)⋊C411yescharacteristic
521D5—22no—
522D5—22no—
523D5—124no—
524C23—51no—
525C23—51no—
526C23—54no—
527C23—202no—
528C23—304no—
529C23—302no—
530C23—604no—
531C23D5×D1211yescharacteristic
532Q8—34no—
533Q8—64no—
534Q8—62no—
535D4—154no—
536D4—308no—
537D4—308no—
538D4—304no—
539D4—302no—
540C2×C4—54no—
541C2×C4—51no—
542C2×C4—51no—
543C2×C4—54no—
544C2×C4—54no—
545C2×C4—52no—
546C2×C4—52no—
547C2×C4—52no—
548C2×C4—52no—
549C2×C4—64no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—308no—
553C2×C4—308no—
554C2×C4—302no—
555C2×C4—604no—
556C2×C4C4×D3×D512yes—
557C2×C4D5×D1211yescharacteristic
558C2×C4D5×#6712yes—
559C2×C4C22×D3×D511yescharacteristic
560C8—22no—
561C8—102no—
562C2×C3—22no—
563C2×C3—102no—
564C2×C3—102no—
565C2×C3C2×D5×C22⋊C411yescharacteristic
566C2×C3#6611112yes—
567D3—204no—
568C5#8376811yescharacteristic, complemented, Sylow
569C22—22no—
570C22—52no—
571C22—52no—
572C22—102no—
573C22—102no—
574C22—204no—
575C22—304no—
576C22—608no—
577C22—604no—
578C22D5×(C2×D3)⋊C412yes—
579C22C2×D5×D1211yescharacteristic
580C4—52no—
581C4—52no—
582C4—64no—
583C4—122no—
584C4—602no—
585C4D5×(C2×D3)⋊C412yes—
586C4C2×C4×D3×D511yescharacteristic
587C4C2×D5×#6711yescharacteristic
588C3#1061068711yescharacteristic, complemented, Sylow
589C2—22no—
590C2—102no—
591C2—102no—
592C2—604no—
593C2#1175402112yes—
594C2C2×D5×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22C2×D5×D12
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC22×#6773C2
cosocleC2×C3×C4×C5C24
hypercenterC2×M16D3×D5
hypocenterC3×C5#2896
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×C3×D5×M16C2semidirect
3. D5×#926C2semidirect
4. #11751232C2semidirect
5. C5×#5120C2semidirect
6. #11748409C2semidirect
7. #11748396C2semidirect
8. #11747642C2semidirect
9. #11747606C2semidirect
19. C2×#6773C4semidirect
27. C2×C3×#676C22semidirect
28. C2×C3×C8×D5C22semidirect
31. D5×C4⋊Dic3C22semidirect
33. #91358C22semidirect
35. #91105C22semidirect
36. C5×#816C22semidirect
37. #90973C22semidirect
38. #90969C22semidirect
41. C2×D5×M16D3semidirect
54. C2×D60C2×C4semidirect
59. C2×C5×Dic6C2×C4semidirect
80. #6773C2×C4semidirect
103. #5120D5semidirect
121. C2×#676C2×D3semidirect
122. C2×C8×D5C2×D3semidirect
199. #816C2×D5semidirect
305. C2×Dic6C4×D5semidirect
498. C3×C5#2896semidirect
568. C5#83768semidirect
588. C3#10610687semidirect

Maximal subgroups

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

subgroupindex
C22×#67732
C2×C3×D5×M162
D5×#9262
#117512322
C5×#51202
#117484092
#117483962
#117476422
#117476062
#106106873
#837685