Finite Group Databasebeta

← #14323199

Subgroups — #14323199

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×C5×#67C211yesmaximal, characteristic
2#11751313C211yesmaximal, characteristic, complemented
3C3×#65500C211yesmaximal, characteristic, complemented
4#11749482C212yesmaximal
5#11749479C212yesmaximal, complemented
6#11748782C211yesmaximal, characteristic, complemented
7#11748777C211yesmaximal, characteristic, complemented
8#11748693C212yesmaximal, complemented
9#11748686C212yesmaximal, complemented
10#11748667C211yesmaximal, characteristic, complemented
11#11748643C211yesmaximal, characteristic, complemented
12#10614056—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C5×#67C2211yescharacteristic
15C22×C4×C5×D3C2211yescharacteristic
16C2×#6719C2212yes—
17C2×C4⋊Dic15C2212yes—
18C2×C5×#217C2211yescharacteristic
19C4×C5×#67C418yescomplemented
20C2×C5×(C2×D3)⋊C4C2211yescharacteristic
21C2×C5×Dic3⋊C4C2211yescharacteristic
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×C3×C22⋊Dic5C2212yes—
24C2×Dic3⋊Dic5C2212yes—
25C2×Dic3⋊Dic5C2212yes—
26C2×(C2×D3)⋊Dic5C2212yes—
27C2×(C2×D3)⋊Dic5C2212yes—
28C2×Dic3×Dic5C2212yes—
29C2×Dic3×Dic5C2212yes—
30#91115C2211yescharacteristic
31#91115C2211yescharacteristic
32C3×#3963C2211yescharacteristic
33C3×#3963C2211yescharacteristic
34C3×#3963C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#74055—51nomaximal, Hall
38C2×C4×C5×D4—31no—
39#65510—32no—
40#65501—31no—
41#65501—31no—
42#65500—31no—
43#65500D311yescharacteristic, complemented
44#65274—32no—
45#65240—32no—
46#65240—32no—
47C4×C4⋊Dic5—31no—
48#65221—31no—
49C24×C3×C5C2311yescharacteristic
50C23×C5×D3C2311yescharacteristic
51C22×C3×C4×C5—42no—
52C22×C3×C4×C5C2×C414yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2312yes—
56C22×Dic15C2312yes—
57C2×C5×#67—24no—
58C2×C5×#67C2×C414yes—
59C22×C5×Dic3C2311yescharacteristic
60C22×C5×Dic3C2311yescharacteristic
61C22×C5×Dic3C2311yescharacteristic
62C2×C4×C5×D3—42no—
63C2×C4×C5×D3C2×C414yes—
64C22×C3×Dic5C2312yes—
65C22×C3×Dic5C2312yes—
66#6719—42no—
67C4⋊Dic15—42no—
68C5×#217C2×C414yes—
69C5×(C2×D3)⋊C4C2×C414yes—
70C5×Dic3⋊C4C2×C414yes—
71C4×C5×Dic3C2×C414yes—
72C3×C22⋊Dic5—42no—
73Dic3⋊Dic5—24no—
74Dic3⋊Dic5—42no—
75(C2×D3)⋊Dic5—42no—
76(C2×D3)⋊Dic5—42no—
77Dic3×Dic5—24no—
78Dic3×Dic5—42no—
79C2×C4×#67D511yescharacteristic
80C3×#381—51no—
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—
90C22×C5×D4—31no—
91C23×C4×C5—31no—
92C23×C4×C5C2×D311yescharacteristic
93C4×C5×D4—38no—
94C2×C5×C4⋊C4—31no—
95C2×C5×C22⋊C4—31no—
96C2×C5×C22⋊C4—31no—
97C2×C42×C5—31no—
98C2×C22⋊Dic5—32no—
99C2×C22⋊Dic5—32no—
100C2×C22⋊Dic5—32no—
101C2×C22⋊Dic5C2×D312yes—
102C2×C4⋊Dic5—32no—
103C2×C4⋊Dic5—32no—
104C2×C4⋊Dic5—32no—
105C2×C4×Dic5—32no—
106C2×C4×Dic5—32no—
107#3963—31no—
108#3963—31no—
109#3963—32no—
110#3963—32no—
111#3963C2×D311yescharacteristic
112#3963C2×D311yescharacteristic
113#3963C2×D312yes—
114#2319—151noSylow
115C23×C3×C5—22no—
116C23×C3×C5—22no—
117C23×C3×C5—21no—
118C23×C3×C5—21no—
119C23×C3×C5C22×C412yes—
120C23×C3×C5C2411yescharacteristic
121C22×C5×D3—22no—
122C22×C5×D3—21no—
123C22×C5×D3—22no—
124C22×C5×D3—21no—
125C22×C5×D3C22×C412yes—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—44no—
129C2×C3×C4×C5C22×C412yes—
130C2×C3×C4×C5C22×C412yes—
131C2×Dic15—22no—
132C2×Dic15—22no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15—22no—
137C5×#67—28no—
138C2×C5×Dic3—21no—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3C22×C412yes—
142C2×C5×Dic3C22×C412yes—
143C2×C5×Dic3C22×C412yes—
144C2×C5×Dic3#4114yes—
145C4×C5×D3—44no—
146C2×C3×Dic5—22no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—22no—
152C23×C3×C4C2×D511yescharacteristic
153C22×#67C2×D511yescharacteristic
154C22×C4×D3C2×D511yescharacteristic
155C2×C3×C22⋊C4—52no—
156C2×#217—52no—
157C2×#217C2×D511yescharacteristic
158C4×#67Dic518yescomplemented
159C2×(C2×D3)⋊C4—52no—
160C2×(C2×D3)⋊C4—52no—
161C2×(C2×D3)⋊C4C2×D511yescharacteristic
162C2×C4⋊Dic3—52no—
163C2×Dic3⋊C4—52no—
164C2×Dic3⋊C4—52no—
165C2×Dic3⋊C4C2×D511yescharacteristic
166C2×C4×Dic3—52no—
167C2×C4×Dic3—52no—
168C2×C4×Dic3C2×D511yescharacteristic
169C3×#95—52no—
170C3×#95—51no—
171C3×#95—51no—
172C3⋊#95—51no—
173C3⋊#95—51no—
174C3⋊#95—52no—
175C3⋊#95—52no—
176C24×C5—31no—
177C24×C5C22×D311yescharacteristic
178C2×C5×D4—34no—
179C2×C5×D4—64no—
180C22×C4×C5—31no—
181C22×C4×C5—31no—
182C22×C4×C5—31no—
183C22×C4×C5—34no—
184C22×C4×C5—42no—
185C22×C4×C5—122no—
186C22×C4×C5C4×D314yes—
187C22×C4×C5C22×D311yescharacteristic
188C22×C4×C5C22×D311yescharacteristic
189C22×Dic5—32no—
190C22×Dic5—32no—
191C22×Dic5C22×D312yes—
192C22×Dic5C22×D312yes—
193C5×C4⋊C4—34no—
194C5×C22⋊C4—34no—
195C5×C22⋊C4—34no—
196C42×C5—34no—
197C22⋊Dic5—42no—
198C22⋊Dic5—122no—
199C22⋊Dic5—122no—
200C22⋊Dic5—122no—
201C4⋊Dic5—64no—
202C4⋊Dic5—122no—
203C4⋊Dic5—122no—
204C4×Dic5—64no—
205C4×Dic5—122no—
206C2×C4×D4—31no—
207#390—152no—
208#389—152no—
209#388—152no—
210#388—152no—
211#381—51no—
212#381—151no—
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×C5×D3—22no—
228C2×C5×D3—22no—
229C2×C5×D3—22no—
230C2×C5×D3—42no—
231C2×C5×D3—42no—
232C3×C4×C5—22no—
233C3×C4×C5—22no—
234Dic15—42no—
235Dic15—42no—
236C3×Dic5—42no—
237C3×Dic5—42no—
238C5×Dic3—22no—
239C5×Dic3—22no—
240C5×Dic3#11714yes—
241C24×C3C22×D511yescharacteristic
242C23×D3C22×D511yescharacteristic
243C22×C3×C4—42no—
244C22×C3×C4—52no—
245C22×C3×C4—52no—
246C22×C3×C4C2×Dic514yes—
247C22×C3×C4C22×D511yescharacteristic
248C22×C3×C4C22×D511yescharacteristic
249C2×#67—24no—
250C2×#67C2×Dic514yes—
251C22×Dic3—52no—
252C22×Dic3—52no—
253C22×Dic3C22×D511yescharacteristic
254C22×Dic3C22×D511yescharacteristic
255C22×Dic3C22×D511yescharacteristic
256C2×C4×D3—42no—
257C2×C4×D3C2×Dic514yes—
258C3×C22⋊C4—202no—
259#217—202no—
260#217C2×Dic514yes—
261(C2×D3)⋊C4—202no—
262(C2×D3)⋊C4—202no—
263(C2×D3)⋊C4C2×Dic514yes—
264C4⋊Dic3—202no—
265Dic3⋊C4—104no—
266Dic3⋊C4—202no—
267Dic3⋊C4C2×Dic514yes—
268C4×Dic3—104no—
269C4×Dic3—202no—
270C4×Dic3C2×Dic514yes—
271C23×C5—22no—
272C23×C5—22no—
273C23×C5—21no—
274C23×C5—21no—
275C23×C5—32no—
276C23×C5—62no—
277C23×C5—62no—
278C23×C5—61no—
279C23×C5—61no—
280C23×C5C2×C4×D312yes—
281C23×C5C23×D311yescharacteristic
282C5×D4—68no—
283C2×C4×C5—22no—
284C2×C4×C5—22no—
285C2×C4×C5—32no—
286C2×C4×C5—34no—
287C2×C4×C5—32no—
288C2×C4×C5—32no—
289C2×C4×C5—44no—
290C2×C4×C5—62no—
291C2×C4×C5—61no—
292C2×C4×C5—61no—
293C2×C4×C5—124no—
294C2×C4×C5C2×C4×D312yes—
295C2×C4×C5C2×C4×D312yes—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—22no—
302C2×Dic5—62no—
303C2×Dic5—62no—
304C2×Dic5—62no—
305C2×Dic5—62no—
306C2×Dic5—62no—
307C2×Dic5—62no—
308C22×D4—31no—
309C23×C4—31no—
310C23×C4D3×D511yescharacteristic
311C4×D4—38no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—152no—
314C2×C4⋊C4—152no—
315C2×C4⋊C4—152no—
316C2×C22⋊C4—31no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—52no—
319C2×C22⋊C4—152no—
320C2×C22⋊C4—152no—
321C2×C22⋊C4—152no—
322C2×C42—31no—
323C2×C42—152no—
324C2×C42—152no—
325#95—51no—
326#95—51no—
327#95—52no—
328#95—152no—
329#95—152no—
330#95—151no—
331#95—151no—
332C2×C3×C5—42no—
333C2×C3×C5C2×#11711yescharacteristic
334C2×C3×C5#51811yescharacteristic
335C2×C3×C5#52011yescharacteristic
336C2×C3×C5#54012yes—
337C2×C3×C5#54212yes—
338C5×D3—42no—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3C22×Dic512yes—
344C23×C3C23×D511yescharacteristic
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3C22×Dic512yes—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—44no—
353C2×C3×C4—102no—
354C2×C3×C4—102no—
355C2×C3×C4—102no—
356C2×C3×C4—102no—
357C2×C3×C4—102no—
358C2×C3×C4—102no—
359C2×C3×C4C22×Dic512yes—
360C2×C3×C4C22×Dic512yes—
361#67—28no—
362C2×Dic3—22no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—102no—
366C2×Dic3—102no—
367C2×Dic3—102no—
368C2×Dic3—102no—
369C2×Dic3—102no—
370C2×Dic3—102no—
371C2×Dic3#70914yes—
372C2×Dic3C22×Dic512yes—
373C2×Dic3C22×Dic512yes—
374C2×Dic3C22×Dic512yes—
375C4×D3—44no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5—22no—
379C22×C5—42no—
380C22×C5—42no—
381C22×C5—62no—
382C22×C5—62no—
383C22×C5—62no—
384C22×C5—122no—
385C22×C5—122no—
386C22×C5C22×C4×D311yescharacteristic
387C22×C5#100412yes—
388C22×C5D3×#4112yes—
389C22×C5#100911yescharacteristic
390C22×C5#101011yescharacteristic
391C4×C5—22no—
392C4×C5—22no—
393C4×C5—34no—
394C4×C5—62no—
395C4×C5—62no—
396Dic5—42no—
397Dic5—42no—
398Dic5—122no—
399Dic5—122no—
400C24—31no—
401C24C2×D3×D511yescharacteristic
402C2×D4—34no—
403C2×D4—64no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—34no—
408C22×C4—42no—
409C22×C4—52no—
410C22×C4—52no—
411C22×C4—122no—
412C22×C4—152no—
413C22×C4—152no—
414C22×C4D3×Dic514yes—
415C22×C4C2×D3×D511yescharacteristic
416C22×C4C2×D3×D511yescharacteristic
417C4⋊C4—34no—
418C4⋊C4—304no—
419C4⋊C4—602no—
420C4⋊C4—602no—
421C22⋊C4—34no—
422C22⋊C4—34no—
423C22⋊C4—202no—
424C22⋊C4—602no—
425C22⋊C4—602no—
426C22⋊C4—602no—
427C42—34no—
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×C3C2×#70912yes—
437C22×C3#414412yes—
438C22×C3#414911yescharacteristic
439C22×C3#415011yescharacteristic
440C22×C3C23×Dic511yescharacteristic
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—42no—
445C2×D3—42no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—202no—
449C3×C4—202no—
450Dic3—22no—
451Dic3—22no—
452Dic3—202no—
453Dic3—202no—
454Dic3#407214yes—
455C2×C5—42no—
456C2×C5—122no—
457C2×C5#548611yescharacteristic
458C2×C5D3×#11711yescharacteristic
459C2×C5#552311yescharacteristic
460C2×C5#566212yes—
461C2×C5#566312yes—
462C23—21no—
463C23—22no—
464C23—21no—
465C23—22no—
466C23—32no—
467C23—61no—
468C23—62no—
469C23—61no—
470C23—62no—
471C23C2×D3×Dic512yes—
472C23C22×D3×D511yescharacteristic
473D4—68no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—34no—
477C2×C4—32no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—44no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—62no—
484C2×C4—102no—
485C2×C4—102no—
486C2×C4—102no—
487C2×C4—102no—
488C2×C4—102no—
489C2×C4—102no—
490C2×C4—124no—
491C2×C4—302no—
492C2×C4—302no—
493C2×C4—302no—
494C2×C4—302no—
495C2×C4—302no—
496C2×C4—302no—
497C2×C4C2×D3×Dic512yes—
498C2×C4C2×D3×Dic512yes—
499C2×C3—42no—
500C2×C3#6589512yes—
501C2×C3#6589612yes—
502C2×C3C2×#407211yescharacteristic
503C2×C3#6613211yescharacteristic
504C2×C3#6616111yescharacteristic
505D3—42no—
506C5#7405511yescharacteristic, complemented, Sylow
507C22—22no—
508C22—22no—
509C22—22no—
510C22—42no—
511C22—42no—
512C22—62no—
513C22—62no—
514C22—62no—
515C22—122no—
516C22—122no—
517C22#9200111yescharacteristic
518C22D3×#70912yes—
519C22#9201911yescharacteristic
520C22C22×D3×Dic511yescharacteristic
521C22#9204912yes—
522C4—22no—
523C4—22no—
524C4—34no—
525C4—62no—
526C4—62no—
527C4—202no—
528C4—202no—
529C4—602no—
530C4—602no—
531C3#1061405611yescharacteristic, complemented, Sylow
532C2—42no—
533C2—122no—
534C2#1175442112yes—
535C2D3×#407211yescharacteristic
536C2#1175451311yescharacteristic
537C2#1175458312yes—
538C2#1175541011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C5×#67C22
hypercenterC23×C4D3×D5
hypocenterC3×C5#2319
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. #11751313C2semidirect
3. C3×#65500C2semidirect
5. #11749479C2semidirect
6. #11748782C2semidirect
7. #11748777C2semidirect
8. #11748693C2semidirect
9. #11748686C2semidirect
10. #11748667C2semidirect
11. #11748643C2semidirect
19. C4×C5×#67C4semidirect
43. #65500D3semidirect
158. C4×#67Dic5semidirect
430. C3×C5#2319semidirect
506. C5#74055semidirect
531. C3#10614056semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×#672
#117513132
C3×#655002
#117494822
#117494792
#117487822
#117487772
#117486932
#117486862
#117486672
#117486432
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