Finite Group Databasebeta

← #14323140

Subgroups — #14323140

Order 1920.

1080 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1220C211yesmaximal, characteristic
2C3×C5×#381C211yesmaximal, characteristic, complemented
3#11751313C211yesmaximal, characteristic, complemented
4#11751058C212yesmaximal
5#11751046C212yesmaximal, complemented
6#11750771C212yesmaximal, complemented
7#11750768C211yesmaximal, characteristic, complemented
8#11750765C211yesmaximal, characteristic, complemented
9#11750750C211yesmaximal, characteristic, complemented
10#11750749C211yesmaximal, characteristic, complemented
11#11750748C212yesmaximal, complemented
12#10605786—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×#1220C2211yescharacteristic
15C22×C4×D15C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2212yes—
17C2×#6719C2212yes—
18C2×#6719C2211yescharacteristic
19C4×#1220C418yescomplemented
20C2×#6714C2212yes—
21C2×#6714C2212yes—
22C2×#6714C2211yescharacteristic
23C2×C4⋊Dic15C2212yes—
24C2×#6712C2212yes—
25C2×#6712C2212yes—
26C2×#6712C2211yescharacteristic
27C2×C4×Dic15C2212yes—
28C2×C4×Dic15C2212yes—
29C2×C4×Dic15C2211yescharacteristic
30C3×C5×#95C2212yes—
31C3×C5×#95C2211yescharacteristic
32C3×C5×#95C2211yescharacteristic
33#91115C2212yes—
34#91115C2211yescharacteristic
35#91115C2212yes—
36#91115C2211yescharacteristic
37#74055—51nomaximal, Hall
38C2×C4×C22⋊D5—31no—
39C5×#381D311yescharacteristic, complemented
40#65501—31no—
41#65246—32no—
42#65234—32no—
43#64959—32no—
44#64956—31no—
45#64953—31no—
46#64938—31no—
47#64937—31no—
48#64936—32no—
49C24×C3×C5C2311yescharacteristic
50C23×D15C2311yescharacteristic
51C22×C3×C4×C5—42no—
52C22×C3×C4×C5C2×C414yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C2×#1220—24no—
58C2×#1220C2×C414yes—
59C22×Dic15C2312yes—
60C22×Dic15C2312yes—
61C22×Dic15C2311yescharacteristic
62C22×Dic15C2311yescharacteristic
63C22×Dic15C2311yescharacteristic
64C2×C4×D15—42no—
65C2×C4×D15C2×C414yes—
66C3×C5×C22⋊C4—42no—
67#6719—42no—
68#6719C2×C414yes—
69#6714—42no—
70#6714—42no—
71#6714C2×C414yes—
72C4⋊Dic15—42no—
73#6712—24no—
74#6712—42no—
75#6712C2×C414yes—
76C4×Dic15—24no—
77C4×Dic15—42no—
78C4×Dic15C2×C414yes—
79C2×C4×#67—51no—
80C3×#381D511yescharacteristic, complemented
81#5212—51no—
82#4957—52no—
83#4945—52no—
84#4670—52no—
85#4667—51no—
86#4664—51no—
87#4649—51no—
88#4648—51no—
89#4647—52no—
90C23×C4×C5C2×D311yescharacteristic
91C22×C22⋊D5—31no—
92C22×C4×D5—31no—
93C2×C5×C22⋊C4C2×D312yes—
94C2×C22⋊Dic5—31no—
95C2×C22⋊Dic5—32no—
96C4×C22⋊D5—38no—
97C2×#685—31no—
98C2×#685—32no—
99C2×#685—32no—
100C2×C4⋊Dic5—32no—
101C2×#683—31no—
102C2×#683—32no—
103C2×#683—32no—
104C2×C4×Dic5—31no—
105C2×C4×Dic5—32no—
106C2×C4×Dic5—32no—
107C5×#95C2×D312yes—
108C5×#95C2×D311yescharacteristic
109C5×#95C2×D311yescharacteristic
110#3963—32no—
111#3963—31no—
112#3963—32no—
113#3963—31no—
114#2319—151noSylow
115C23×C3×C5—22no—
116C23×C3×C5—21no—
117C23×C3×C5—22no—
118C23×C3×C5—21no—
119C23×C3×C5C22×C412yes—
120C23×C3×C5C2411yescharacteristic
121C22×D15—22no—
122C22×D15—21no—
123C22×D15—22no—
124C22×D15—21no—
125C22×D15C22×C412yes—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—44no—
135C2×C3×C4×C5C22×C412yes—
136C2×C3×C4×C5C22×C412yes—
137#1220—28no—
138C2×Dic15—21no—
139C2×Dic15—22no—
140C2×Dic15—21no—
141C2×Dic15—22no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15—22no—
147C2×Dic15C22×C412yes—
148C2×Dic15C22×C412yes—
149C2×Dic15C22×C412yes—
150C2×Dic15#4114yes—
151C4×D15—44no—
152C23×C3×C4C2×D511yescharacteristic
153C22×#67—51no—
154C22×C4×D3—51no—
155C2×C3×C22⋊C4C2×D512yes—
156C2×#217—51no—
157C2×#217—52no—
158C4×#67—58no—
159C2×(C2×D3)⋊C4—51no—
160C2×(C2×D3)⋊C4—52no—
161C2×(C2×D3)⋊C4—52no—
162C2×C4⋊Dic3—52no—
163C2×Dic3⋊C4—51no—
164C2×Dic3⋊C4—52no—
165C2×Dic3⋊C4—52no—
166C2×C4×Dic3—51no—
167C2×C4×Dic3—52no—
168C2×C4×Dic3—52no—
169C3×#95C2×D512yes—
170C3×#95C2×D511yescharacteristic
171C3×#95C2×D511yescharacteristic
172C3⋊#95—52no—
173C3⋊#95—51no—
174C3⋊#95—52no—
175C3⋊#95—51no—
176C24×C5C22×D311yescharacteristic
177C23×D5—31no—
178C22×C4×C5—42no—
179C22×C4×C5C4×D314yes—
180C22×C4×C5C22×D312yes—
181C22×C4×C5C22×D312yes—
182C22×C4×C5C22×D311yescharacteristic
183C22×C4×C5C22×D311yescharacteristic
184C2×C22⋊D5—34no—
185C2×C22⋊D5—64no—
186C22×Dic5—31no—
187C22×Dic5—31no—
188C22×Dic5—31no—
189C22×Dic5—32no—
190C22×Dic5—32no—
191C2×C4×D5—34no—
192C2×C4×D5—122no—
193C5×C22⋊C4—42no—
194C22⋊Dic5—34no—
195C22⋊Dic5—122no—
196#685—34no—
197#685—122no—
198#685—122no—
199C4⋊Dic5—122no—
200#683—34no—
201#683—64no—
202#683—122no—
203C4×Dic5—34no—
204C4×Dic5—64no—
205C4×Dic5—122no—
206C2×C4×D4—151no—
207#390—152no—
208#389—152no—
209#388—152no—
210#388—152no—
211#381—151no—
212#381D1511yescharacteristic, complemented
213#380—151no—
214#380—151no—
215C4×C4⋊C4—151no—
216#376—151no—
217C22×C3×C5—22no—
218C22×C3×C5—22no—
219C22×C3×C5—22no—
220C22×C3×C5—42no—
221C22×C3×C5—42no—
222C22×C3×C5C23×C411yescharacteristic
223C22×C3×C5C2×#4112yes—
224C22×C3×C5H(5,𝔽2)12yes—
225C22×C3×C5H(5,𝔽2)11yescharacteristic
226C22×C3×C52−1+411yescharacteristic
227C2×D15—22no—
228C2×D15—22no—
229C2×D15—22no—
230C2×D15—42no—
231C2×D15—42no—
232C3×C4×C5—22no—
233C3×C4×C5—22no—
234C3×C4×C5—42no—
235C3×C4×C5—42no—
236Dic15—22no—
237Dic15—22no—
238Dic15—42no—
239Dic15—42no—
240Dic15#11714yes—
241C24×C3C22×D511yescharacteristic
242C23×D3—51no—
243C22×C3×C4—42no—
244C22×C3×C4C4×D514yes—
245C22×C3×C4C22×D512yes—
246C22×C3×C4C22×D512yes—
247C22×C3×C4C22×D511yescharacteristic
248C22×C3×C4C22×D511yescharacteristic
249C2×#67—54no—
250C2×#67—104no—
251C22×Dic3—51no—
252C22×Dic3—51no—
253C22×Dic3—51no—
254C22×Dic3—52no—
255C22×Dic3—52no—
256C2×C4×D3—54no—
257C2×C4×D3—202no—
258C3×C22⋊C4—42no—
259#217—54no—
260#217—202no—
261(C2×D3)⋊C4—54no—
262(C2×D3)⋊C4—202no—
263(C2×D3)⋊C4—202no—
264C4⋊Dic3—202no—
265Dic3⋊C4—54no—
266Dic3⋊C4—104no—
267Dic3⋊C4—202no—
268C4×Dic3—54no—
269C4×Dic3—104no—
270C4×Dic3—202no—
271C23×C5—22no—
272C23×C5—22no—
273C23×C5—21no—
274C23×C5—21no—
275C23×C5C2×C4×D312yes—
276C23×C5C23×D311yescharacteristic
277C22×D5—32no—
278C22×D5—62no—
279C22×D5—62no—
280C22×D5—61no—
281C22×D5—61no—
282C2×C4×C5—22no—
283C2×C4×C5—22no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—44no—
291C2×C4×C5C2×C4×D312yes—
292C2×C4×C5C2×C4×D312yes—
293C22⋊D5—68no—
294C2×Dic5—32no—
295C2×Dic5—34no—
296C2×Dic5—32no—
297C2×Dic5—32no—
298C2×Dic5—62no—
299C2×Dic5—61no—
300C2×Dic5—61no—
301C2×Dic5—62no—
302C2×Dic5—62no—
303C2×Dic5—62no—
304C2×Dic5—62no—
305C2×Dic5—62no—
306C2×Dic5—62no—
307C4×D5—124no—
308C22×D4—151no—
309C23×C4—151no—
310C23×C4C2×D1511yescharacteristic
311C4×D4—158no—
312C2×C4⋊C4—151no—
313C2×C4⋊C4—152no—
314C2×C4⋊C4—152no—
315C2×C4⋊C4—152no—
316C2×C22⋊C4—151no—
317C2×C22⋊C4—151no—
318C2×C22⋊C4—152no—
319C2×C22⋊C4—152no—
320C2×C22⋊C4—152no—
321C2×C22⋊C4C2×D1512yes—
322C2×C42—151no—
323C2×C42—152no—
324C2×C42—152no—
325#95—151no—
326#95—152no—
327#95—151no—
328#95—152no—
329#95C2×D1511yescharacteristic
330#95C2×D1511yescharacteristic
331#95C2×D1512yes—
332C2×C3×C5—42no—
333C2×C3×C5C2×#11711yescharacteristic
334C2×C3×C5#51811yescharacteristic
335C2×C3×C5#52011yescharacteristic
336C2×C3×C5#54012yes—
337C2×C3×C5#54212yes—
338D15—42no—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3C2×C4×D512yes—
344C23×C3C23×D511yescharacteristic
345C22×D3—52no—
346C22×D3—102no—
347C22×D3—102no—
348C22×D3—101no—
349C22×D3—101no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—44no—
359C2×C3×C4C2×C4×D512yes—
360C2×C3×C4C2×C4×D512yes—
361#67—108no—
362C2×Dic3—52no—
363C2×Dic3—54no—
364C2×Dic3—52no—
365C2×Dic3—52no—
366C2×Dic3—102no—
367C2×Dic3—101no—
368C2×Dic3—101no—
369C2×Dic3—102no—
370C2×Dic3—102no—
371C2×Dic3—102no—
372C2×Dic3—102no—
373C2×Dic3—102no—
374C2×Dic3—102no—
375C4×D3—204no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5—22no—
379C22×C5—42no—
380C22×C5—42no—
381C22×C5C22×C4×D311yescharacteristic
382C22×C5#100412yes—
383C22×C5D3×#4112yes—
384C22×C5#100911yescharacteristic
385C22×C5#101011yescharacteristic
386C2×D5—62no—
387C2×D5—62no—
388C2×D5—62no—
389C2×D5—122no—
390C2×D5—122no—
391C4×C5—22no—
392C4×C5—22no—
393C4×C5—42no—
394C4×C5—42no—
395Dic5—34no—
396Dic5—62no—
397Dic5—62no—
398Dic5—122no—
399Dic5—122no—
400C24—151no—
401C24C22×D1511yescharacteristic
402C2×D4—154no—
403C2×D4—304no—
404C22×C4—42no—
405C22×C4—151no—
406C22×C4—151no—
407C22×C4—151no—
408C22×C4—154no—
409C22×C4—152no—
410C22×C4—152no—
411C22×C4—602no—
412C22×C4C4×D1514yes—
413C22×C4C22×D1512yes—
414C22×C4C22×D1512yes—
415C22×C4C22×D1511yescharacteristic
416C22×C4C22×D1511yescharacteristic
417C4⋊C4—154no—
418C4⋊C4—304no—
419C4⋊C4—602no—
420C4⋊C4—602no—
421C22⋊C4—42no—
422C22⋊C4—154no—
423C22⋊C4—154no—
424C22⋊C4—602no—
425C22⋊C4—602no—
426C22⋊C4—602no—
427C42—154no—
428C42—304no—
429C42—602no—
430C3×C5#231911yescharacteristic, complemented, Hall
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3C22×C4×D511yescharacteristic
437C22×C3#414412yes—
438C22×C3D5×#4112yes—
439C22×C3#414911yescharacteristic
440C22×C3#415011yescharacteristic
441C2×D3—102no—
442C2×D3—102no—
443C2×D3—102no—
444C2×D3—202no—
445C2×D3—202no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—42no—
449C3×C4—42no—
450Dic3—54no—
451Dic3—102no—
452Dic3—102no—
453Dic3—202no—
454Dic3—202no—
455C2×C5—42no—
456C2×C5#548611yescharacteristic
457C2×C5D3×#11711yescharacteristic
458C2×C5#552311yescharacteristic
459C2×C5#566212yes—
460C2×C5#566312yes—
461D5—122no—
462C23—22no—
463C23—22no—
464C23—21no—
465C23—21no—
466C23—152no—
467C23—302no—
468C23—302no—
469C23—301no—
470C23—301no—
471C23C2×C4×D1512yes—
472C23C23×D1511yescharacteristic
473D4—308no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—44no—
483C2×C4—152no—
484C2×C4—154no—
485C2×C4—152no—
486C2×C4—152no—
487C2×C4—302no—
488C2×C4—301no—
489C2×C4—301no—
490C2×C4—302no—
491C2×C4—302no—
492C2×C4—302no—
493C2×C4—302no—
494C2×C4—302no—
495C2×C4—302no—
496C2×C4—604no—
497C2×C4C2×C4×D1512yes—
498C2×C4C2×C4×D1512yes—
499C2×C3—42no—
500C2×C3#6582011yescharacteristic
501C2×C3D5×#11711yescharacteristic
502C2×C3#6585711yescharacteristic
503C2×C3#6599612yes—
504C2×C3#6599712yes—
505D3—202no—
506C5#7405511yescharacteristic, complemented, Sylow
507C22—22no—
508C22—22no—
509C22—22no—
510C22—42no—
511C22—42no—
512C22—302no—
513C22—302no—
514C22—302no—
515C22—602no—
516C22—602no—
517C22C22×C4×D1511yescharacteristic
518C22#9209512yes—
519C22#41×D1512yes—
520C22#9210011yescharacteristic
521C22#9210111yescharacteristic
522C4—22no—
523C4—22no—
524C4—42no—
525C4—42no—
526C4—154no—
527C4—302no—
528C4—302no—
529C4—602no—
530C4—602no—
531C3#1060578611yescharacteristic, complemented, Sylow
532C2—42no—
533C2—602no—
534C2#1175624511yescharacteristic
535C2D15×#11711yescharacteristic
536C2#1175628211yescharacteristic
537C2#1175642112yes—
538C2#1175642212yes—
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#381C2
socle, cosocleC23×C3×C5C24
prime radicalC22×#1220C22
hypercenter#381D15
hypocenterC3×C5#2319
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C3×C5×#381C2semidirect
3. #11751313C2semidirect
5. #11751046C2semidirect
6. #11750771C2semidirect
7. #11750768C2semidirect
8. #11750765C2semidirect
9. #11750750C2semidirect
10. #11750749C2semidirect
11. #11750748C2semidirect
19. C4×#1220C4semidirect
39. C5×#381D3semidirect
80. C3×#381D5semidirect
212. #381D15semidirect
430. C3×C5#2319semidirect
506. C5#74055semidirect
531. C3#10605786semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#12202
C3×C5×#3812
#117513132
#117510582
#117510462
#117507712
#117507682
#117507652
#117507502
#117507492
#117507482
#106057863
#740555