Finite Group Databasebeta

← #14322709

Subgroups — #14322709

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91815C211yesmaximal, characteristic
2C3×C5×#380C211yesmaximal, characteristic
3#11751313C211yesmaximal, characteristic
4#11751078C211yesmaximal, characteristic, complemented
5#11751076C211yesmaximal, characteristic, complemented
6C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
7C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
8#11751046C212yesmaximal
9#11751045C212yesmaximal
10#11750755C212yesmaximal, complemented
11#11750754C212yesmaximal, complemented
12#10605599—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2212yes—
17C2×#6719C2212yes—
18C2×#6719C2211yescharacteristic
19C2×#6719C2211yescharacteristic
20#91815C418yescomplemented
21C2×C4⋊Dic15C2211yescharacteristic
22C2×C4⋊Dic15C2211yescharacteristic
23C2×C4⋊Dic15C2211yescharacteristic
24C2×C4⋊Dic15C2211yescharacteristic
25C2×#6712C2212yes—
26C2×#6712C2212yes—
27C2×C4×Dic15C2212yes—
28C2×C4×Dic15C2212yes—
29C2×C4×Dic15C2211yescharacteristic
30C2×C4×Dic15C2211yescharacteristic
31C3×C5×#95C2212yes—
32#91115C2212yes—
33#91115C2212yes—
34#91115C2212yes—
35#91115C2212yes—
36#91115C2212yes—
37#73868—51nomaximal, Hall
38C2×#4072—31no—
39C5×#380D311yescharacteristic
40#65501—31no—
41#65266—31no—
42#65265—31no—
43C4⋊C4×Dic5—31no—
44C4⋊C4×Dic5—31no—
45#65234—32no—
46#65233—32no—
47#64943—32no—
48#64942—32no—
49C24×C3×C5C2311yescharacteristic
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—22no—
52C22×C3×C4×C5C2×C414yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15C2312yes—
58C22×Dic15C2312yes—
59C22×Dic15C2311yescharacteristic
60C22×Dic15C2311yescharacteristic
61C22×Dic15C2311yescharacteristic
62C22×Dic15C2311yescharacteristic
63C3×C5×C4⋊C4—22no—
64C3×C5×C4⋊C4—22no—
65C3×C5×C22⋊C4—42no—
66#6719—42no—
67#6719C2×C414yes—
68#6719C2×C414yes—
69C4⋊Dic15—22no—
70C4⋊Dic15—22no—
71C4⋊Dic15C2×C414yes—
72C4⋊Dic15C2×C414yes—
73#6712—42no—
74#6712—42no—
75C4×Dic15—42no—
76C4×Dic15—42no—
77C4×Dic15C2×C414yes—
78C4×Dic15C2×C414yes—
79C2×#926—51no—
80C3×#380D511yescharacteristic
81#5212—51no—
82#4977—51no—
83#4975—51no—
84Dic3×C4⋊C4—51no—
85Dic3×C4⋊C4—51no—
86#4945—52no—
87#4944—52no—
88#4654—52no—
89#4653—52no—
90C23×C4×C5C2×D311yescharacteristic
91C2×C5×C4⋊C4C2×D311yescharacteristic
92C2×C5×C4⋊C4C2×D311yescharacteristic
93C2×C5×C22⋊C4C2×D312yes—
94C2×C22⋊Dic5—31no—
95C2×C22⋊Dic5—31no—
96C2×C22⋊Dic5—32no—
97#4072—38no—
98C2×C4⋊Dic5—31no—
99C2×C4⋊Dic5—31no—
100C2×C4⋊Dic5—31no—
101C2×C4⋊Dic5—31no—
102C2×#683—32no—
103C2×#683—32no—
104C2×C4×Dic5—31no—
105C2×C4×Dic5—31no—
106C2×C4×Dic5—32no—
107C2×C4×Dic5—32no—
108C5×#95C2×D312yes—
109#3963—32no—
110#3963—32no—
111#3963—32no—
112#3963—32no—
113#3963—32no—
114#2314—151noSylow
115C23×C3×C5—21no—
116C23×C3×C5—21no—
117C23×C3×C5—21no—
118C23×C3×C5—21no—
119C23×C3×C5—21no—
120C23×C3×C5—21no—
121C23×C3×C5C22×C412yes—
122C23×C3×C5C2411yescharacteristic
123C2×C3×C4×C5—21no—
124C2×C3×C4×C5—21no—
125C2×C3×C4×C5—28no—
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×C5C22×C412yes—
133C2×C3×C4×C5C22×C412yes—
134C2×C3×C4×C5#4112yes—
135C2×C3×C4×C5#4112yes—
136C2×Dic15—21no—
137C2×Dic15—21no—
138C2×Dic15—21no—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—21no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15—22no—
147C2×Dic15—22no—
148C2×Dic15—22no—
149C2×Dic15—22no—
150C2×Dic15C22×C412yes—
151C2×Dic15C22×C412yes—
152C2×Dic15C22×C412yes—
153C2×Dic15C22×C412yes—
154C23×C3×C4C2×D511yescharacteristic
155C2×C3×C4⋊C4C2×D511yescharacteristic
156C2×C3×C4⋊C4C2×D511yescharacteristic
157C2×C3×C22⋊C4C2×D512yes—
158C2×#217—51no—
159C2×#217—51no—
160C2×#217—52no—
161#926—58no—
162C2×C4⋊Dic3—51no—
163C2×C4⋊Dic3—51no—
164C2×C4⋊Dic3—51no—
165C2×C4⋊Dic3—51no—
166C2×Dic3⋊C4—52no—
167C2×Dic3⋊C4—52no—
168C2×C4×Dic3—51no—
169C2×C4×Dic3—51no—
170C2×C4×Dic3—52no—
171C2×C4×Dic3—52no—
172C3×#95C2×D512yes—
173C3⋊#95—52no—
174C3⋊#95—52no—
175C3⋊#95—52no—
176C3⋊#95—52no—
177C3⋊#95—52no—
178C24×C5C22×D311yescharacteristic
179C22×C4×C5—22no—
180C22×C4×C5—22no—
181C22×C4×C5C4×D314yes—
182C22×C4×C5C22×D312yes—
183C22×C4×C5C22×D312yes—
184C22×C4×C5C22×D311yescharacteristic
185C22×C4×C5C22×D311yescharacteristic
186C22×Dic5—31no—
187C22×Dic5—31no—
188C22×Dic5—31no—
189C22×Dic5—31no—
190C22×Dic5—32no—
191C22×Dic5—32no—
192C5×C4⋊C4—22no—
193C5×C4⋊C4—22no—
194C5×C22⋊C4—42no—
195C22⋊Dic5—34no—
196C22⋊Dic5—34no—
197C22⋊Dic5—122no—
198C4⋊Dic5—34no—
199C4⋊Dic5—34no—
200C4⋊Dic5—62no—
201C4⋊Dic5—62no—
202#683—122no—
203#683—122no—
204C4×Dic5—34no—
205C4×Dic5—34no—
206C4×Dic5—122no—
207C4×Dic5—122no—
208C2×#117—151no—
209#388—152no—
210#388—152no—
211#387—152no—
212#387—152no—
213#384—151no—
214#382—151no—
215#380—151no—
216#380D1511yescharacteristic
217C4×C4⋊C4—151no—
218C4×C4⋊C4—151no—
219C22×C3×C5—22no—
220C22×C3×C5—22no—
221C22×C3×C5—22no—
222C22×C3×C5—41no—
223C22×C3×C5—41no—
224C22×C3×C5—41no—
225C22×C3×C5—41no—
226C22×C3×C5C23×C411yescharacteristic
227C22×C3×C5C2×#4111yescharacteristic
228C22×C3×C5C2×#4111yescharacteristic
229C22×C3×C5H(5,𝔽2)11yescharacteristic
230C22×C3×C5H(5,𝔽2)11yescharacteristic
231C22×C3×C52−1+411yescharacteristic
232C22×C3×C52−1+411yescharacteristic
233C3×C4×C5—22no—
234C3×C4×C5—42no—
235C3×C4×C5—42no—
236C3×C4×C5#11714yes—
237Dic15—22no—
238Dic15—22no—
239Dic15—22no—
240Dic15—22no—
241Dic15—42no—
242Dic15—42no—
243C24×C3C22×D511yescharacteristic
244C22×C3×C4—22no—
245C22×C3×C4—22no—
246C22×C3×C4C4×D514yes—
247C22×C3×C4C22×D512yes—
248C22×C3×C4C22×D512yes—
249C22×C3×C4C22×D511yescharacteristic
250C22×C3×C4C22×D511yescharacteristic
251C22×Dic3—51no—
252C22×Dic3—51no—
253C22×Dic3—51no—
254C22×Dic3—51no—
255C22×Dic3—52no—
256C22×Dic3—52no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—22no—
259C3×C22⋊C4—42no—
260#217—54no—
261#217—54no—
262#217—202no—
263C4⋊Dic3—54no—
264C4⋊Dic3—54no—
265C4⋊Dic3—102no—
266C4⋊Dic3—102no—
267Dic3⋊C4—202no—
268Dic3⋊C4—202no—
269C4×Dic3—54no—
270C4×Dic3—54no—
271C4×Dic3—202no—
272C4×Dic3—202no—
273C23×C5—21no—
274C23×C5—21no—
275C23×C5—21no—
276C23×C5—21no—
277C23×C5—21no—
278C23×C5—21no—
279C23×C5C2×C4×D312yes—
280C23×C5C23×D311yescharacteristic
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—28no—
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×C5C2×C4×D312yes—
291C2×C4×C5C2×C4×D312yes—
292C2×C4×C5#23712yes—
293C2×C4×C5#23712yes—
294C2×Dic5—32no—
295C2×Dic5—32no—
296C2×Dic5—32no—
297C2×Dic5—32no—
298C2×Dic5—61no—
299C2×Dic5—61no—
300C2×Dic5—61no—
301C2×Dic5—61no—
302C2×Dic5—61no—
303C2×Dic5—61no—
304C2×Dic5—61no—
305C2×Dic5—61no—
306C2×Dic5—62no—
307C2×Dic5—62no—
308C2×Dic5—62no—
309C2×Dic5—62no—
310C2×Dic5—62no—
311C2×Dic5—62no—
312C23×C4C2×D1511yescharacteristic
313#117—158no—
314C2×C4⋊C4—151no—
315C2×C4⋊C4—151no—
316C2×C4⋊C4—151no—
317C2×C4⋊C4—151no—
318C2×C4⋊C4—152no—
319C2×C4⋊C4—152no—
320C2×C4⋊C4C2×D1511yescharacteristic
321C2×C4⋊C4C2×D1511yescharacteristic
322C2×C22⋊C4—151no—
323C2×C22⋊C4—151no—
324C2×C22⋊C4—152no—
325C2×C22⋊C4C2×D1512yes—
326C2×C42—151no—
327C2×C42—151no—
328C2×C42—152no—
329C2×C42—152no—
330#95—152no—
331#95—152no—
332#95—152no—
333#95—152no—
334#95—152no—
335#95C2×D1512yes—
336C2×C3×C5—42no—
337C2×C3×C5C2×#11711yescharacteristic
338C2×C3×C5#52011yescharacteristic
339C2×C3×C5#52011yescharacteristic
340C2×C3×C5#54011yescharacteristic
341C2×C3×C5#54111yescharacteristic
342C2×C3×C5#54211yescharacteristic
343C2×C3×C5#54211yescharacteristic
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3C2×C4×D512yes—
351C23×C3C23×D511yescharacteristic
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—28no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4C2×C4×D512yes—
362C2×C3×C4C2×C4×D512yes—
363C2×C3×C4#71112yes—
364C2×C3×C4#71112yes—
365C2×Dic3—52no—
366C2×Dic3—52no—
367C2×Dic3—52no—
368C2×Dic3—52no—
369C2×Dic3—101no—
370C2×Dic3—101no—
371C2×Dic3—101no—
372C2×Dic3—101no—
373C2×Dic3—101no—
374C2×Dic3—101no—
375C2×Dic3—101no—
376C2×Dic3—101no—
377C2×Dic3—102no—
378C2×Dic3—102no—
379C2×Dic3—102no—
380C2×Dic3—102no—
381C2×Dic3—102no—
382C2×Dic3—102no—
383C22×C5—22no—
384C22×C5—22no—
385C22×C5—22no—
386C22×C5—41no—
387C22×C5—41no—
388C22×C5—41no—
389C22×C5—41no—
390C22×C5C22×C4×D311yescharacteristic
391C22×C5C2×#23711yescharacteristic
392C22×C5C2×#23711yescharacteristic
393C22×C5#100411yescharacteristic
394C22×C5#100711yescharacteristic
395C22×C5#100911yescharacteristic
396C22×C5#101011yescharacteristic
397C4×C5—22no—
398C4×C5—42no—
399C4×C5—42no—
400C4×C5#87714yes—
401Dic5—62no—
402Dic5—62no—
403Dic5—62no—
404Dic5—62no—
405Dic5—122no—
406Dic5—122no—
407C24C22×D1511yescharacteristic
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—151no—
411C22×C4—151no—
412C22×C4—151no—
413C22×C4—151no—
414C22×C4—152no—
415C22×C4—152no—
416C22×C4C4×D1514yes—
417C22×C4C22×D1512yes—
418C22×C4C22×D1512yes—
419C22×C4C22×D1511yescharacteristic
420C22×C4C22×D1511yescharacteristic
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—154no—
424C4⋊C4—154no—
425C4⋊C4—302no—
426C4⋊C4—302no—
427C4⋊C4—602no—
428C4⋊C4—602no—
429C22⋊C4—42no—
430C22⋊C4—154no—
431C22⋊C4—154no—
432C22⋊C4—602no—
433C42—154no—
434C42—154no—
435C42—602no—
436C42—602no—
437C3×C5#231411yescharacteristic, complemented, Hall
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3C22×C4×D511yescharacteristic
446C22×C3C2×#71111yescharacteristic
447C22×C3C2×#71111yescharacteristic
448C22×C3#414411yescharacteristic
449C22×C3#414711yescharacteristic
450C22×C3#414911yescharacteristic
451C22×C3#415011yescharacteristic
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4#402314yes—
456Dic3—102no—
457Dic3—102no—
458Dic3—102no—
459Dic3—102no—
460Dic3—202no—
461Dic3—202no—
462C2×C5—42no—
463C2×C5C2×#87711yescharacteristic
464C2×C5#550411yescharacteristic
465C2×C5#552311yescharacteristic
466C2×C5#560311yescharacteristic
467C2×C5#561911yescharacteristic
468C2×C5#562511yescharacteristic
469C2×C5#564311yescharacteristic
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23C2×C4×D1512yes—
477C23C23×D1511yescharacteristic
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—28no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—152no—
488C2×C4—152no—
489C2×C4—152no—
490C2×C4—152no—
491C2×C4—301no—
492C2×C4—301no—
493C2×C4—301no—
494C2×C4—301no—
495C2×C4—301no—
496C2×C4—301no—
497C2×C4—301no—
498C2×C4—301no—
499C2×C4—302no—
500C2×C4—302no—
501C2×C4—302no—
502C2×C4—302no—
503C2×C4—302no—
504C2×C4—302no—
505C2×C4C2×C4×D1512yes—
506C2×C4C2×C4×D1512yes—
507C2×C4#681912yes—
508C2×C4#681912yes—
509C2×C3—42no—
510C2×C3C2×#402311yescharacteristic
511C2×C3#6583811yescharacteristic
512C2×C3#6585711yescharacteristic
513C2×C3#6593711yescharacteristic
514C2×C3#6595311yescharacteristic
515C2×C3#6595911yescharacteristic
516C2×C3#6597711yescharacteristic
517C5#7386811yescharacteristic, complemented, Sylow
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22C22×C4×D1511yescharacteristic
526C22C2×#681911yescharacteristic
527C22C2×#681911yescharacteristic
528C22#9209511yescharacteristic
529C22#9209811yescharacteristic
530C22#9210011yescharacteristic
531C22#9210111yescharacteristic
532C4—22no—
533C4—42no—
534C4—42no—
535C4—302no—
536C4—302no—
537C4—302no—
538C4—302no—
539C4—602no—
540C4—602no—
541C4#9176614yes—
542C3#1060559911yescharacteristic, complemented, Sylow
543C2—42no—
544C2C2×#9176611yescharacteristic
545C2#1175626311yescharacteristic
546C2#1175628211yescharacteristic
547C2#1175636211yescharacteristic
548C2#1175637811yescharacteristic
549C2#1175638411yescharacteristic
550C2#1175640211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#380C2
socle, cosocleC23×C3×C5C24
prime radicalC24×C3×C5C23
hypercenter#380D15
hypocenterC3×C5#2314
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
4. #11751078C2semidirect
5. #11751076C2semidirect
6. C4⋊C4×Dic15C2semidirect
7. C4⋊C4×Dic15C2semidirect
10. #11750755C2semidirect
11. #11750754C2semidirect
20. #91815C4semidirect
437. C3×C5#2314semidirect
517. C5#73868semidirect
542. C3#10605599semidirect

Maximal subgroups

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

subgroupindex
C2×#918152
C3×C5×#3802
#117513132
#117510782
#117510762
C4⋊C4×Dic152
#117510462
#117510452
#117507552
#117507542
#106055993
#738685