Finite Group Databasebeta

← #14323205

Subgroups — #14323205

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×#1201C211yesmaximal, characteristic
2#11750749C211yesmaximal, characteristic, complemented
3C5×#4664C211yesmaximal, characteristic, complemented
4C3×#64953C211yesmaximal, characteristic, complemented
5#11748915C212yesmaximal
6#11748804C212yesmaximal, complemented
7#11748782C211yesmaximal, characteristic, complemented
8#11748738C211yesmaximal, characteristic, complemented
9#11748723C212yesmaximal, complemented
10#11748712C212yesmaximal, complemented
11#11748663C211yesmaximal, characteristic, complemented
12#10605786—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C22×C3×C4×D5C2211yescharacteristic
15C22×#1201C2211yescharacteristic
16C2×C4×Dic15C2212yes—
17C2×C4×Dic15C2211yescharacteristic
18C2×C5×(C2×D3)⋊C4C2212yes—
19C2×C3×#685C2212yes—
20C2×Dic3⋊Dic5C2212yes—
21C2×#6669C2212yes—
22C2×#6669C2211yescharacteristic
23C2×#6668C2212yes—
24C2×(C2×D3)⋊Dic5C2212yes—
25C2×(C2×D3)⋊Dic5C2211yescharacteristic
26C2×#6665C2212yes—
27C2×#6665C2211yescharacteristic
28C2×Dic3×Dic5C2212yes—
29C4×#1201C418yescomplemented
30C3×C5×#95C2211yescharacteristic
31#91115C2212yes—
32#91115C2211yescharacteristic
33C5×C3⋊#95C2211yescharacteristic
34C5×C3⋊#95C2212yes—
35C3×#3963C2211yescharacteristic
36C3×#3963C2212yes—
37#74055—51nomaximal, Hall
38C2×C4×C22⋊D5—31no—
39C5×#381—31no—
40#65501—31no—
41#65246—32no—
42#65234—32no—
43#64959—32no—
44#64956—31no—
45#64953D311yescharacteristic, complemented
46#64938—31no—
47#64937—31no—
48#64936—32no—
49C23×C5×D3C2311yescharacteristic
50C23×C3×D5C2311yescharacteristic
51C22×C3×C4×C5C2312yes—
52C22×C3×C4×C5C2311yescharacteristic
53C22×Dic15C2312yes—
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×C5×Dic3C2312yes—
57C22×C5×Dic3C2311yescharacteristic
58C2×C4×C5×D3—42no—
59C2×C4×C5×D3C2×C414yes—
60C22×C3×Dic5C2312yes—
61C22×C3×Dic5C2311yescharacteristic
62C2×C3×C4×D5—42no—
63C2×C3×C4×D5C2×C414yes—
64C2×#1201—24no—
65C2×#1201C2×C414yes—
66C4×Dic15—24no—
67C4×Dic15C2×C414yes—
68C5×(C2×D3)⋊C4—42no—
69C3×#685—42no—
70Dic3⋊Dic5—42no—
71#6669—24no—
72#6669C2×C414yes—
73#6668—42no—
74(C2×D3)⋊Dic5—42no—
75(C2×D3)⋊Dic5C2×C414yes—
76#6665—42no—
77#6665C2×C414yes—
78Dic3×Dic5—42no—
79C2×C4×#67—51no—
80C3×#381—51no—
81#5212—51no—
82#4957—52no—
83#4945—52no—
84#4670—52no—
85#4667—51no—
86#4664D511yescharacteristic, complemented
87#4649—51no—
88#4648—51no—
89#4647—52no—
90C23×C4×C5—31no—
91C22×C22⋊D5—31no—
92C22×C4×D5C2×D311yescharacteristic
93C2×C5×C22⋊C4—32no—
94C2×C22⋊Dic5—31no—
95C2×C22⋊Dic5—32no—
96C4×C22⋊D5—38no—
97C2×#685—31no—
98C2×#685—32no—
99C2×#685C2×D312yes—
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×#95—32no—
108C5×#95—31no—
109C5×#95C2×D311yescharacteristic
110#3963—31no—
111#3963—32no—
112#3963C2×D312yes—
113#3963C2×D311yescharacteristic
114#2319—151noSylow
115C23×C3×C5C2411yescharacteristic
116C22×C5×D3—22no—
117C22×C5×D3—21no—
118C22×C5×D3—22no—
119C22×C5×D3—21no—
120C22×C5×D3C22×C412yes—
121C22×C3×D5—22no—
122C22×C3×D5—21no—
123C22×C3×D5—22no—
124C22×C3×D5—21no—
125C22×C3×D5C22×C412yes—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5C22×C412yes—
131C2×Dic15—21no—
132C2×Dic15—21no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15C22×C412yes—
137C2×Dic15C22×C412yes—
138C2×Dic15#4114yes—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3C22×C412yes—
144C4×C5×D3—44no—
145C2×C3×Dic5—22no—
146C2×C3×Dic5—22no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5C22×C412yes—
150C3×C4×D5—44no—
151#1201—28no—
152C23×C3×C4—51no—
153C22×#67—51no—
154C22×C4×D3C2×D511yescharacteristic
155C2×C3×C22⋊C4—52no—
156C2×#217—51no—
157C2×#217—52no—
158C4×#67—58no—
159C2×(C2×D3)⋊C4—51no—
160C2×(C2×D3)⋊C4—52no—
161C2×(C2×D3)⋊C4C2×D512yes—
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×#95—52no—
170C3×#95—51no—
171C3×#95C2×D511yescharacteristic
172C3⋊#95—51no—
173C3⋊#95—52no—
174C3⋊#95C2×D512yes—
175C3⋊#95C2×D511yescharacteristic
176C24×C5—31no—
177C23×D5C22×D311yescharacteristic
178C22×C4×C5—31no—
179C22×C4×C5—34no—
180C22×C4×C5—32no—
181C22×C4×C5—122no—
182C22×C4×C5C22×D312yes—
183C22×C4×C5C22×D311yescharacteristic
184C2×C22⋊D5—34no—
185C2×C22⋊D5—64no—
186C22×Dic5—31no—
187C22×Dic5—31no—
188C22×Dic5—32no—
189C22×Dic5C22×D312yes—
190C22×Dic5C22×D311yescharacteristic
191C2×C4×D5—42no—
192C2×C4×D5C4×D314yes—
193C5×C22⋊C4—122no—
194C22⋊Dic5—34no—
195C22⋊Dic5—122no—
196#685—34no—
197#685—42no—
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—31no—
212#381—51no—
213#380—151no—
214#380—151no—
215C4×C4⋊C4—151no—
216#376—151no—
217C22×C3×C5C23×C411yescharacteristic
218C22×C3×C5C2×#4112yes—
219C22×C3×C5H(5,𝔽2)12yes—
220C22×C3×C5H(5,𝔽2)11yescharacteristic
221C22×C3×C52−1+411yescharacteristic
222C2×C5×D3—22no—
223C2×C5×D3—22no—
224C2×C5×D3—22no—
225C2×C5×D3—42no—
226C2×C5×D3—42no—
227C2×C3×D5—22no—
228C2×C3×D5—22no—
229C2×C3×D5—22no—
230C2×C3×D5—42no—
231C2×C3×D5—42no—
232C3×C4×C5—22no—
233C3×C4×C5—42no—
234Dic15—22no—
235Dic15—42no—
236Dic15#11714yes—
237C3×Dic5—22no—
238C3×Dic5—42no—
239C5×Dic3—22no—
240C5×Dic3—42no—
241C24×C3—51no—
242C23×D3C22×D511yescharacteristic
243C22×C3×C4—51no—
244C22×C3×C4—54no—
245C22×C3×C4—52no—
246C22×C3×C4—202no—
247C22×C3×C4C22×D512yes—
248C22×C3×C4C22×D511yescharacteristic
249C2×#67—54no—
250C2×#67—104no—
251C22×Dic3—51no—
252C22×Dic3—51no—
253C22×Dic3—52no—
254C22×Dic3C22×D512yes—
255C22×Dic3C22×D511yescharacteristic
256C2×C4×D3—42no—
257C2×C4×D3C4×D514yes—
258C3×C22⋊C4—202no—
259#217—54no—
260#217—202no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—54no—
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—32no—
272C23×C5—62no—
273C23×C5—62no—
274C23×C5—61no—
275C23×C5—61no—
276C23×C5C23×D311yescharacteristic
277C22×D5—22no—
278C22×D5—22no—
279C22×D5—21no—
280C22×D5—21no—
281C22×D5C2×C4×D312yes—
282C2×C4×C5—22no—
283C2×C4×C5—22no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—32no—
287C2×C4×C5—62no—
288C2×C4×C5—62no—
289C2×C4×C5—62no—
290C2×C4×C5—62no—
291C2×C4×C5—124no—
292C2×C4×C5C2×C4×D312yes—
293C22⋊D5—68no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—32no—
299C2×Dic5—34no—
300C2×Dic5—32no—
301C2×Dic5—61no—
302C2×Dic5—61no—
303C2×Dic5—62no—
304C2×Dic5—62no—
305C2×Dic5—62no—
306C2×Dic5C2×C4×D312yes—
307C4×D5—44no—
308C22×D4—151no—
309C23×C4—31no—
310C23×C4—51no—
311C4×D4—158no—
312C2×C4⋊C4—151no—
313C2×C4⋊C4—152no—
314C2×C4⋊C4—152no—
315C2×C4⋊C4—152no—
316C2×C22⋊C4—32no—
317C2×C22⋊C4—52no—
318C2×C22⋊C4—151no—
319C2×C22⋊C4—151no—
320C2×C22⋊C4—152no—
321C2×C22⋊C4—152no—
322C2×C42—151no—
323C2×C42—152no—
324C2×C42—152no—
325#95—31no—
326#95—32no—
327#95—51no—
328#95—52no—
329#95—151no—
330#95—152no—
331#95D3×D511yescharacteristic
332C2×C3×C5C2×#11711yescharacteristic
333C2×C3×C5#51811yescharacteristic
334C2×C3×C5#52011yescharacteristic
335C2×C3×C5#54012yes—
336C2×C3×C5#54212yes—
337C3×D5—42no—
338C5×D3—42no—
339C23×C3—52no—
340C23×C3—102no—
341C23×C3—102no—
342C23×C3—101no—
343C23×C3—101no—
344C23×C3C23×D511yescharacteristic
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3C2×C4×D512yes—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—52no—
355C2×C3×C4—102no—
356C2×C3×C4—102no—
357C2×C3×C4—102no—
358C2×C3×C4—102no—
359C2×C3×C4—204no—
360C2×C3×C4C2×C4×D512yes—
361#67—108no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—52no—
367C2×Dic3—54no—
368C2×Dic3—52no—
369C2×Dic3—101no—
370C2×Dic3—101no—
371C2×Dic3—102no—
372C2×Dic3—102no—
373C2×Dic3—102no—
374C2×Dic3C2×C4×D512yes—
375C4×D3—44no—
376C22×C5—62no—
377C22×C5—62no—
378C22×C5—62no—
379C22×C5—122no—
380C22×C5—122no—
381C22×C5C22×C4×D311yescharacteristic
382C22×C5#100412yes—
383C22×C5D3×#4112yes—
384C22×C5#100911yescharacteristic
385C22×C5#101011yescharacteristic
386C2×D5—22no—
387C2×D5—22no—
388C2×D5—22no—
389C2×D5—42no—
390C2×D5—42no—
391C4×C5—22no—
392C4×C5—42no—
393C4×C5—62no—
394C4×C5—122no—
395Dic5—22no—
396Dic5—34no—
397Dic5—42no—
398Dic5—62no—
399Dic5—122no—
400C24—31no—
401C24—51no—
402C2×D4—154no—
403C2×D4—304no—
404C22×C4—31no—
405C22×C4—34no—
406C22×C4—32no—
407C22×C4—51no—
408C22×C4—54no—
409C22×C4—52no—
410C22×C4—122no—
411C22×C4—151no—
412C22×C4—151no—
413C22×C4—152no—
414C22×C4—202no—
415C22×C4C2×D3×D512yes—
416C22×C4C2×D3×D511yescharacteristic
417C4⋊C4—154no—
418C4⋊C4—304no—
419C4⋊C4—602no—
420C4⋊C4—602no—
421C22⋊C4—122no—
422C22⋊C4—154no—
423C22⋊C4—154no—
424C22⋊C4—202no—
425C22⋊C4—602no—
426C22⋊C4—602no—
427C42—154no—
428C42—304no—
429C42—602no—
430C3×C5#231911yescharacteristic, complemented, Hall
431C22×C3—102no—
432C22×C3—102no—
433C22×C3—102no—
434C22×C3—202no—
435C22×C3—202no—
436C22×C3C22×C4×D511yescharacteristic
437C22×C3#414412yes—
438C22×C3D5×#4112yes—
439C22×C3#414911yescharacteristic
440C22×C3#415011yescharacteristic
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—42no—
445C2×D3—42no—
446C3×C4—22no—
447C3×C4—42no—
448C3×C4—102no—
449C3×C4—202no—
450Dic3—22no—
451Dic3—42no—
452Dic3—54no—
453Dic3—102no—
454Dic3—202no—
455C2×C5—122no—
456C2×C5#548611yescharacteristic
457C2×C5D3×#11711yescharacteristic
458C2×C5#552311yescharacteristic
459C2×C5#566212yes—
460C2×C5#566312yes—
461D5—42no—
462C23—32no—
463C23—52no—
464C23—62no—
465C23—61no—
466C23—62no—
467C23—61no—
468C23—102no—
469C23—101no—
470C23—102no—
471C23—101no—
472C23C22×D3×D511yescharacteristic
473D4—308no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—32no—
479C2×C4—52no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—102no—
485C2×C4—102no—
486C2×C4—102no—
487C2×C4—102no—
488C2×C4—124no—
489C2×C4—152no—
490C2×C4—154no—
491C2×C4—152no—
492C2×C4—204no—
493C2×C4—301no—
494C2×C4—301no—
495C2×C4—302no—
496C2×C4—302no—
497C2×C4—302no—
498C2×C4C4×D3×D512yes—
499C2×C3—202no—
500C2×C3#6582011yescharacteristic
501C2×C3D5×#11711yescharacteristic
502C2×C3#6585711yescharacteristic
503C2×C3#6599612yes—
504C2×C3#6599712yes—
505D3—42no—
506C5#7405511yescharacteristic, complemented, Sylow
507C22—62no—
508C22—62no—
509C22—62no—
510C22—102no—
511C22—102no—
512C22—102no—
513C22—122no—
514C22—122no—
515C22—202no—
516C22—202no—
517C22C2×C4×D3×D511yescharacteristic
518C22#9201612yes—
519C22#9202011yescharacteristic
520C22#9202512yes—
521C22#9203511yescharacteristic
522C4—22no—
523C4—42no—
524C4—62no—
525C4—102no—
526C4—122no—
527C4—154no—
528C4—202no—
529C4—302no—
530C4—602no—
531C3#1060578611yescharacteristic, complemented, Sylow
532C2—122no—
533C2—202no—
534C2#1175272311yescharacteristic
535C2#1175306511yescharacteristic
536C2#1175314412yes—
537C2#1175367011yescharacteristic
538C2#1175378512yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC3×C5×#95C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1201C22
hypercenter#95D3×D5
hypocenterC3×C5#2319
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
2. #11750749C2semidirect
3. C5×#4664C2semidirect
4. C3×#64953C2semidirect
6. #11748804C2semidirect
7. #11748782C2semidirect
8. #11748738C2semidirect
9. #11748723C2semidirect
10. #11748712C2semidirect
11. #11748663C2semidirect
29. C4×#1201C4semidirect
45. #64953D3semidirect
86. #4664D5semidirect
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×#12012
#117507492
C5×#46642
C3×#649532
#117489152
#117488042
#117487822
#117487382
#117487232
#117487122
#117486632
#106057863
#740555