Finite Group Databasebeta

← #14326673

Subgroups — #14326673

Order 1920.

1096 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D15C211yesmaximal, characteristic
2C3×C4×C5×C4⋊C4C211yesmaximal, characteristic, complemented
3#11751088C211yesmaximal, characteristic
4C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
5#11751073C211yesmaximal, characteristic, complemented
6C4×#6714C211yesmaximal, characteristic
7C4×#6714C212yesmaximal
8#11751034C212yesmaximal, complemented
9#11751029C211yesmaximal, characteristic, complemented
10#11750768C212yesmaximal
11#11750753C212yesmaximal, complemented
12#10605499—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×C4×D15C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C42×C5C2211yescharacteristic
18C2×C3×C42×C5C2212yes—
19C2×#6714C2211yescharacteristic
20C2×#6714C2212yes—
21C2×#6714C2211yescharacteristic
22C2×C4⋊Dic15C2211yescharacteristic
23C2×C4⋊Dic15C2211yescharacteristic
24C2×#6712C2212yes—
25C2×#6712C2212yes—
26C2×#6712C2212yes—
27C2×C4×Dic15C2211yescharacteristic
28C2×C4×Dic15C2212yes—
29C4⋊C4×D15C418yescomplemented
30C3×C5×#95C2212yes—
31#91115C2211yescharacteristic
32#91115C2212yes—
33#91115C2212yes—
34#91115C2212yes—
35#91115C2212yes—
36#91115C2211yescharacteristic
37#73768—51nomaximal, Hall
38C2×D5×C4⋊C4—31no—
39C4×C5×C4⋊C4D311yescharacteristic, complemented
40#65276—31no—
41C4⋊C4×Dic5—31no—
42#65262—31no—
43C4×#685—31no—
44C4×#685—32no—
45#65222—32no—
46#65217—31no—
47#64956—32no—
48#64941—32no—
49C23×D15C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2312yes—
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2312yes—
57C22×Dic15C2312yes—
58C22×Dic15C2311yescharacteristic
59C22×Dic15C2311yescharacteristic
60C2×C4×D15—22no—
61C2×C4×D15—22no—
62C2×C4×D15—42no—
63C2×C4×D15—42no—
64C2×C4×D15C2×C414yes—
65C2×C4×D15C2×C418yes—
66C3×C5×C4⋊C4—22no—
67C3×C5×C4⋊C4C2×C414yes—
68C3×C42×C5—22no—
69C3×C42×C5—42no—
70#6714—22no—
71#6714—24no—
72#6714Q814yes—
73C4⋊Dic15—22no—
74C4⋊Dic15C2×C414yes—
75#6712—42no—
76#6712—42no—
77#6712C2×C418yes—
78C4×Dic15—22no—
79C4×Dic15—42no—
80C2×D3×C4⋊C4—51no—
81C3×C4×C4⋊C4D511yescharacteristic, complemented
82#4987—51no—
83Dic3×C4⋊C4—51no—
84#4972—51no—
85C4×(C2×D3)⋊C4—51no—
86C4×(C2×D3)⋊C4—52no—
87#4933—52no—
88#4928—51no—
89#4667—52no—
90#4652—52no—
91C22×C4×D5—31no—
92C22×C4×D5—32no—
93C2×C5×C4⋊C4C2×D311yescharacteristic
94C2×C5×C4⋊C4C2×D311yescharacteristic
95C2×C42×C5C2×D311yescharacteristic
96C2×C42×C5C2×D312yes—
97C2×#685—31no—
98C2×#685—31no—
99C2×#685—32no—
100C2×C4⋊Dic5—31no—
101C2×C4⋊Dic5—31no—
102C2×#683—32no—
103C2×#683—32no—
104C2×#683—32no—
105C2×C4×Dic5—31no—
106C2×C4×Dic5—32no—
107D5×C4⋊C4—38no—
108C5×#95C2×D312yes—
109#3963—31no—
110#3963—32no—
111#3963—32no—
112#3963—32no—
113#3963—32no—
114#3963—31no—
115#2345—151noSylow
116C23×C3×C5C2411yescharacteristic
117C22×D15—21no—
118C22×D15—21no—
119C22×D15—21no—
120C22×D15—21no—
121C22×D15C22×C412yes—
122C22×D15C2×Q812yes—
123C22×D15#4112yes—
124C2×C3×C4×C5—21no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—21no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—21no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5C22×C414yes—
135C2×C3×C4×C5C22×C412yes—
136C2×C3×C4×C5C2×Q812yes—
137C2×C3×C4×C5#4112yes—
138C2×C3×C4×C5#4112yes—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—21no—
146C2×Dic15—22no—
147C2×Dic15—21no—
148C2×Dic15—22no—
149C2×Dic15—22no—
150C2×Dic15—21no—
151C2×Dic15C22×C412yes—
152C2×Dic15C22×C414yes—
153C2×Dic15C2×Q812yes—
154C4×D15—28no—
155C4×D15—48no—
156C22×C4×D3—51no—
157C22×C4×D3—52no—
158C2×C3×C4⋊C4C2×D511yescharacteristic
159C2×C3×C4⋊C4C2×D511yescharacteristic
160C2×C3×C42C2×D511yescharacteristic
161C2×C3×C42C2×D512yes—
162C2×(C2×D3)⋊C4—51no—
163C2×(C2×D3)⋊C4—51no—
164C2×(C2×D3)⋊C4—52no—
165C2×C4⋊Dic3—51no—
166C2×C4⋊Dic3—51no—
167C2×Dic3⋊C4—52no—
168C2×Dic3⋊C4—52no—
169C2×Dic3⋊C4—52no—
170C2×C4×Dic3—51no—
171C2×C4×Dic3—52no—
172D3×C4⋊C4—58no—
173C3×#95C2×D512yes—
174C3⋊#95—51no—
175C3⋊#95—52no—
176C3⋊#95—52no—
177C3⋊#95—52no—
178C3⋊#95—52no—
179C3⋊#95—51no—
180C23×D5—31no—
181C22×C4×C5C22×D312yes—
182C22×C4×C5C22×D311yescharacteristic
183C22×C4×C5C22×D312yes—
184C22×C4×C5C22×D311yescharacteristic
185C22×C4×C5C22×D311yescharacteristic
186C22×Dic5—31no—
187C22×Dic5—32no—
188C22×Dic5—31no—
189C22×Dic5—31no—
190C22×Dic5—32no—
191C2×C4×D5—34no—
192C2×C4×D5—38no—
193C2×C4×D5—62no—
194C2×C4×D5—62no—
195C2×C4×D5—122no—
196C2×C4×D5—122no—
197C5×C4⋊C4—22no—
198C5×C4⋊C4C4×D314yes—
199C42×C5—22no—
200C42×C5—42no—
201#685—34no—
202#685—62no—
203#685—64no—
204C4⋊Dic5—34no—
205C4⋊Dic5—62no—
206#683—38no—
207#683—122no—
208#683—122no—
209C4×Dic5—62no—
210C4×Dic5—122no—
211C22×C4⋊C4—151no—
212#389—151no—
213#389—151no—
214#387—152no—
215#387—152no—
216#380—151no—
217#380—152no—
218C4×C4⋊C4—151no—
219C4×C4⋊C4D1511yescharacteristic, complemented
220C4×C22⋊C4—151no—
221C4×C22⋊C4—152no—
222C22×C3×C5C23×C411yescharacteristic
223C22×C3×C5C22×Q811yescharacteristic
224C22×C3×C5C2×#4111yescharacteristic
225C22×C3×C5C2×#4111yescharacteristic
226C22×C3×C5C2×#4111yescharacteristic
227C22×C3×C5H(5,𝔽2)11yescharacteristic
228C22×C3×C5H(5,𝔽2)11yescharacteristic
229C2×D15—22no—
230C2×D15—22no—
231C2×D15—22no—
232C2×D15—22no—
233C2×D15—22no—
234C2×D15—22no—
235C2×D15C4×Q814yes—
236C3×C4×C5—24no—
237C3×C4×C5—22no—
238C3×C4×C5—41no—
239C3×C4×C5—42no—
240C3×C4×C5#11714yes—
241Dic15—22no—
242Dic15—24no—
243Dic15—22no—
244Dic15—42no—
245Dic15—41no—
246C23×D3—51no—
247C22×C3×C4C22×D512yes—
248C22×C3×C4C22×D511yescharacteristic
249C22×C3×C4C22×D512yes—
250C22×C3×C4C22×D511yescharacteristic
251C22×C3×C4C22×D511yescharacteristic
252C22×Dic3—51no—
253C22×Dic3—51no—
254C22×Dic3—52no—
255C22×Dic3—51no—
256C22×Dic3—52no—
257C2×C4×D3—54no—
258C2×C4×D3—58no—
259C2×C4×D3—102no—
260C2×C4×D3—102no—
261C2×C4×D3—202no—
262C2×C4×D3—202no—
263C3×C4⋊C4—22no—
264C3×C4⋊C4C4×D514yes—
265C3×C42—22no—
266C3×C42—42no—
267(C2×D3)⋊C4—54no—
268(C2×D3)⋊C4—102no—
269(C2×D3)⋊C4—104no—
270C4⋊Dic3—54no—
271C4⋊Dic3—102no—
272Dic3⋊C4—58no—
273Dic3⋊C4—202no—
274Dic3⋊C4—202no—
275C4×Dic3—102no—
276C4×Dic3—202no—
277C23×C5C23×D311yescharacteristic
278C22×D5—32no—
279C22×D5—32no—
280C22×D5—32no—
281C22×D5—61no—
282C22×D5—61no—
283C22×D5—61no—
284C22×D5—61no—
285C2×C4×C5—21no—
286C2×C4×C5—21no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—21no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—21no—
293C2×C4×C5—22no—
294C2×C4×C5—21no—
295C2×C4×C5C2×C4×D314yes—
296C2×C4×C5C2×C4×D312yes—
297C2×C4×C5#23512yes—
298C2×C4×C5#23512yes—
299C2×C4×C5D3×Q812yes—
300C2×Dic5—32no—
301C2×Dic5—34no—
302C2×Dic5—32no—
303C2×Dic5—61no—
304C2×Dic5—61no—
305C2×Dic5—61no—
306C2×Dic5—62no—
307C2×Dic5—61no—
308C2×Dic5—62no—
309C2×Dic5—61no—
310C2×Dic5—62no—
311C2×Dic5—61no—
312C2×Dic5—62no—
313C2×Dic5—61no—
314C2×Dic5—62no—
315C4×D5—68no—
316C4×D5—128no—
317C23×C4—151no—
318C23×C4—152no—
319C2×C4⋊C4—152no—
320C2×C4⋊C4—151no—
321C2×C4⋊C4—158no—
322C2×C4⋊C4—151no—
323C2×C4⋊C4—152no—
324C2×C4⋊C4—152no—
325C2×C4⋊C4C2×D1511yescharacteristic
326C2×C4⋊C4C2×D1511yescharacteristic
327C2×C22⋊C4—151no—
328C2×C22⋊C4—152no—
329C2×C22⋊C4—151no—
330C2×C42—151no—
331C2×C42—152no—
332C2×C42C2×D1511yescharacteristic
333C2×C42C2×D1512yes—
334#95—151no—
335#95—151no—
336#95—152no—
337#95—152no—
338#95—152no—
339#95—152no—
340#95C2×D1512yes—
341C2×C3×C5C2×#11711yescharacteristic
342C2×C3×C5C2×C4×Q811yescharacteristic
343C2×C3×C5#51811yescharacteristic
344C2×C3×C5#55311yescharacteristic
345C2×C3×C5#55311yescharacteristic
346C2×C3×C5#55411yescharacteristic
347C2×C3×C5#55411yescharacteristic
348D15—24no—
349C23×C3C23×D511yescharacteristic
350C22×D3—52no—
351C22×D3—52no—
352C22×D3—52no—
353C22×D3—101no—
354C22×D3—101no—
355C22×D3—101no—
356C22×D3—101no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4C2×C4×D514yes—
368C2×C3×C4C2×C4×D512yes—
369C2×C3×C4#70912yes—
370C2×C3×C4#70912yes—
371C2×C3×C4Q8×D512yes—
372C2×Dic3—52no—
373C2×Dic3—54no—
374C2×Dic3—52no—
375C2×Dic3—101no—
376C2×Dic3—101no—
377C2×Dic3—101no—
378C2×Dic3—102no—
379C2×Dic3—101no—
380C2×Dic3—102no—
381C2×Dic3—101no—
382C2×Dic3—102no—
383C2×Dic3—101no—
384C2×Dic3—102no—
385C2×Dic3—101no—
386C2×Dic3—102no—
387C4×D3—108no—
388C4×D3—208no—
389C22×C5C22×C4×D311yescharacteristic
390C22×C5C2×#23511yescharacteristic
391C22×C5C2×#23511yescharacteristic
392C22×C5#100411yescharacteristic
393C22×C5C2×D3×Q811yescharacteristic
394C22×C5D3×#4111yescharacteristic
395C22×C5#100911yescharacteristic
396C2×D5—34no—
397C2×D5—62no—
398C2×D5—62no—
399C2×D5—62no—
400C2×D5—62no—
401C2×D5—62no—
402C2×D5—62no—
403C4×C5—22no—
404C4×C5—24no—
405C4×C5—41no—
406C4×C5—42no—
407C4×C5#87214yes—
408Dic5—62no—
409Dic5—64no—
410Dic5—62no—
411Dic5—122no—
412Dic5—121no—
413C24—151no—
414C22×C4—151no—
415C22×C4—151no—
416C22×C4—152no—
417C22×C4—154no—
418C22×C4—158no—
419C22×C4—151no—
420C22×C4—152no—
421C22×C4—302no—
422C22×C4—302no—
423C22×C4—602no—
424C22×C4—602no—
425C22×C4C22×D1512yes—
426C22×C4C22×D1511yescharacteristic
427C22×C4C22×D1512yes—
428C22×C4C22×D1511yescharacteristic
429C22×C4C22×D1511yescharacteristic
430C4⋊C4—22no—
431C4⋊C4—158no—
432C4⋊C4—154no—
433C4⋊C4—302no—
434C4⋊C4—602no—
435C4⋊C4—602no—
436C4⋊C4C4×D1514yes—
437C22⋊C4—154no—
438C22⋊C4—302no—
439C22⋊C4—304no—
440C42—22no—
441C42—42no—
442C42—302no—
443C42—602no—
444C3×C5#234511yescharacteristic, complemented, Hall
445C22×C3C22×C4×D511yescharacteristic
446C22×C3C2×#70911yescharacteristic
447C22×C3C2×#70911yescharacteristic
448C22×C3#414411yescharacteristic
449C22×C3C2×Q8×D511yescharacteristic
450C22×C3D5×#4111yescharacteristic
451C22×C3#414911yescharacteristic
452C2×D3—54no—
453C2×D3—102no—
454C2×D3—102no—
455C2×D3—102no—
456C2×D3—102no—
457C2×D3—102no—
458C2×D3—102no—
459C3×C4—22no—
460C3×C4—24no—
461C3×C4—41no—
462C3×C4—42no—
463C3×C4#401814yes—
464Dic3—102no—
465Dic3—104no—
466Dic3—102no—
467Dic3—202no—
468Dic3—201no—
469C2×C5C2×#87211yescharacteristic
470C2×C5#551111yescharacteristic
471C2×C5#553011yescharacteristic
472C2×C5#554511yescharacteristic
473C2×C5#555811yescharacteristic
474C2×C5C4×D3×Q811yescharacteristic
475C2×C5#557911yescharacteristic
476D5—64no—
477C23—152no—
478C23—152no—
479C23—152no—
480C23—301no—
481C23—301no—
482C23—301no—
483C23—301no—
484C23C23×D1511yescharacteristic
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—154no—
496C2×C4—152no—
497C2×C4—152no—
498C2×C4—301no—
499C2×C4—301no—
500C2×C4—301no—
501C2×C4—302no—
502C2×C4—301no—
503C2×C4—302no—
504C2×C4—308no—
505C2×C4—301no—
506C2×C4—302no—
507C2×C4—301no—
508C2×C4—302no—
509C2×C4—301no—
510C2×C4—302no—
511C2×C4—608no—
512C2×C4C2×C4×D1514yes—
513C2×C4C2×C4×D1512yes—
514C2×C4#681712yes—
515C2×C4#681712yes—
516C2×C4Q8×D1512yes—
517C2×C3C2×#401811yescharacteristic
518C2×C3#6584511yescharacteristic
519C2×C3#6586411yescharacteristic
520C2×C3#6587911yescharacteristic
521C2×C3#6589211yescharacteristic
522C2×C3C4×Q8×D511yescharacteristic
523C2×C3#6591311yescharacteristic
524D3—104no—
525C5#7376811yescharacteristic, complemented, Sylow
526C22—154no—
527C22—302no—
528C22—302no—
529C22—302no—
530C22—302no—
531C22—302no—
532C22—302no—
533C22C22×C4×D1511yescharacteristic
534C22C2×#681711yescharacteristic
535C22C2×#681711yescharacteristic
536C22#9209511yescharacteristic
537C22C2×Q8×D1511yescharacteristic
538C22#41×D1511yescharacteristic
539C22#9210011yescharacteristic
540C4—24no—
541C4—22no—
542C4—41no—
543C4—42no—
544C4—304no—
545C4—302no—
546C4—302no—
547C4—602no—
548C4—601no—
549C4#9176114yes—
550C3#1060549911yescharacteristic, complemented, Sylow
551C2—304no—
552C2C2×#9176111yescharacteristic
553C2#1175627011yescharacteristic
554C2#1175628911yescharacteristic
555C2#1175630411yescharacteristic
556C2#1175631711yescharacteristic
557C2C4×Q8×D1511yescharacteristic
558C2#1175633811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C4×C5×C4⋊C4C2
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
hypercenterC4×C4⋊C4D15
hypocenterC3×C5#2345
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×C5×C4⋊C4C2semidirect
4. C4⋊C4×Dic15C2semidirect
5. #11751073C2semidirect
8. #11751034C2semidirect
9. #11751029C2semidirect
11. #11750753C2semidirect
29. C4⋊C4×D15C4semidirect
39. C4×C5×C4⋊C4D3semidirect
81. C3×C4×C4⋊C4D5semidirect
219. C4×C4⋊C4D15semidirect
444. C3×C5#2345semidirect
525. C5#73768semidirect
550. C3#10605499semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D152
C3×C4×C5×C4⋊C42
#117510882
C4⋊C4×Dic152
#117510732
C4×#67142
#117510342
#117510292
#117507682
#117507532
#106054993
#737685