Finite Group Databasebeta

← #14326728

Subgroups — #14326728

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×#91437C211yesmaximal, characteristic
2C4×#6714C211yesmaximal, characteristic
3C5×#4928C211yesmaximal, characteristic, complemented
4C3×C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
5#11748882C212yesmaximal
6#11748875C212yesmaximal
7#11748868C211yesmaximal, characteristic, complemented
8#11748815C212yesmaximal, complemented
9#11748799C211yesmaximal, characteristic
10#11748699C211yesmaximal, characteristic, complemented
11#11748662C212yesmaximal, complemented
12#10607932—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×#1200C2212yes—
15C2×C3×C42×C5C2211yescharacteristic
16C2×#6714C2211yescharacteristic
17C2×#6714C2211yescharacteristic
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C5×C4⋊Dic3C2211yescharacteristic
21C2×C5×Dic3⋊C4C2212yes—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×C4⋊Dic5C2211yescharacteristic
24C2×C3×C4×Dic5C2212yes—
25C2×Dic3⋊Dic5C2212yes—
26C2×#6669C2212yes—
27C2×#6667C2212yes—
28C2×Dic3×Dic5C2212yes—
29#91437C418yescomplemented
30#91115C2211yescharacteristic
31#91115C2211yescharacteristic
32C5×C3⋊#95C2212yes—
33C3×#3963C2212yes—
34#90994C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#73768—51nomaximal, Hall
38C2×D5×C4⋊C4—31no—
39C5×#389—31no—
40#65276—31no—
41#65276—32no—
42#65274—31no—
43#65267—32no—
44C4⋊C4×Dic5—31no—
45C4×#685—31no—
46C4×C4⋊Dic5D311yescharacteristic, complemented
47#64955—32no—
48#64944—32no—
49C23×D15C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C22×Dic15C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C2×C4×D15—22no—
57C2×C4×D15—22no—
58C2×C4×D15C2×C414yes—
59C22×C5×Dic3C2312yes—
60C22×C5×Dic3C2312yes—
61C22×C3×Dic5C2312yes—
62C22×C3×Dic5C2312yes—
63C2×#1200—42no—
64C2×#1200—42no—
65C2×#1200C2×C418yes—
66C3×C42×C5—22no—
67#6714—22no—
68#6714Q814yes—
69C4×Dic15—22no—
70C5×C4⋊Dic3—22no—
71C5×C4⋊Dic3C2×C414yes—
72C5×Dic3⋊C4—42no—
73C3×C4⋊Dic5—22no—
74C3×C4⋊Dic5C2×C414yes—
75C3×C4×Dic5—42no—
76Dic3⋊Dic5—42no—
77#6669C2×C418yes—
78#6667—24no—
79Dic3×Dic5—42no—
80C2×D3×C4⋊C4—51no—
81C3×C4×C4⋊C4—51no—
82#4987—51no—
83Dic3×C4⋊C4—51no—
84#4972—51no—
85C4×(C2×D3)⋊C4—51no—
86C4×(C2×D3)⋊C4—52no—
87#4933—52no—
88#4928D511yescharacteristic, complemented
89#4667—52no—
90#4652—52no—
91C22×C4×D5—31no—
92C22×C4×D5—32no—
93C2×C5×C4⋊C4—31no—
94C2×C5×C4⋊C4—31no—
95C2×C5×C4⋊C4—32no—
96C2×C42×C5C2×D311yescharacteristic
97C2×#685—31no—
98C2×#685—31no—
99C2×#685—32no—
100C2×C4⋊Dic5—32no—
101C2×C4⋊Dic5C2×D311yescharacteristic
102C2×C4⋊Dic5C2×D311yescharacteristic
103C2×#683—32no—
104C2×C4×Dic5—31no—
105C2×C4×Dic5—32no—
106C2×C4×Dic5C2×D312yes—
107D5×C4⋊C4—38no—
108C5×#95—32no—
109#3963—31no—
110#3963—31no—
111#3963—32no—
112#3963—32no—
113#3963—32no—
114#3963C2×D312yes—
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—21no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5—21no—
129C2×C3×C4×C5C22×C412yes—
130C2×C3×C4×C5C2×Q812yes—
131C2×C3×C4×C5#4112yes—
132C2×C3×C4×C5#4112yes—
133C2×Dic15—21no—
134C2×Dic15—21no—
135C2×Dic15—21no—
136C2×Dic15—21no—
137C2×Dic15—21no—
138C2×Dic15—21no—
139C2×Dic15—21no—
140C2×Dic15C22×C412yes—
141C2×Dic15C2×Q812yes—
142C4×D15—28no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—22no—
148C2×C5×Dic3C22×C414yes—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—22no—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5C22×C414yes—
155#1200—48no—
156C22×C4×D3—51no—
157C22×C4×D3—52no—
158C2×C3×C4⋊C4—51no—
159C2×C3×C4⋊C4—51no—
160C2×C3×C42—52no—
161C2×C3×C42C2×D511yescharacteristic
162C2×(C2×D3)⋊C4—51no—
163C2×(C2×D3)⋊C4—51no—
164C2×(C2×D3)⋊C4—52no—
165C2×C4⋊Dic3C2×D511yescharacteristic
166C2×C4⋊Dic3C2×D511yescharacteristic
167C2×Dic3⋊C4—52no—
168C2×Dic3⋊C4—52no—
169C2×Dic3⋊C4C2×D512yes—
170C2×C4×Dic3—51no—
171C2×C4×Dic3—52no—
172D3×C4⋊C4—58no—
173C3×#95—52no—
174C3⋊#95—51no—
175C3⋊#95—51no—
176C3⋊#95—52no—
177C3⋊#95—52no—
178C3⋊#95—52no—
179C3⋊#95C2×D512yes—
180C23×D5—31no—
181C22×C4×C5—32no—
182C22×C4×C5—32no—
183C22×C4×C5C22×D311yescharacteristic
184C22×C4×C5C22×D311yescharacteristic
185C22×C4×C5C22×D311yescharacteristic
186C22×Dic5—31no—
187C22×Dic5—31no—
188C22×Dic5—31no—
189C22×Dic5C22×D312yes—
190C22×Dic5C22×D312yes—
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—34no—
198C5×C4⋊C4—62no—
199C5×C4⋊C4—122no—
200C42×C5—22no—
201#685—34no—
202#685—64no—
203#685—62no—
204C4⋊Dic5—22no—
205C4⋊Dic5—122no—
206C4⋊Dic5C4×D314yes—
207#683—38no—
208C4×Dic5—42no—
209C4×Dic5—62no—
210C4×Dic5—122no—
211C22×C4⋊C4—151no—
212#389—31no—
213#389—151no—
214#387—152no—
215#387—152no—
216#380—152no—
217#380—151no—
218C4×C4⋊C4—51no—
219C4×C4⋊C4—151no—
220C4×C22⋊C4—152no—
221C4×C22⋊C4—151no—
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—22no—
237C3×C4×C5—41no—
238C3×C4×C5#11714yes—
239Dic15—22no—
240Dic15—22no—
241Dic15—41no—
242C3×Dic5—24no—
243C3×Dic5—42no—
244C5×Dic3—24no—
245C5×Dic3—42no—
246C23×D3—51no—
247C22×C3×C4—52no—
248C22×C3×C4—52no—
249C22×C3×C4C22×D511yescharacteristic
250C22×C3×C4C22×D511yescharacteristic
251C22×C3×C4C22×D511yescharacteristic
252C22×Dic3—51no—
253C22×Dic3—51no—
254C22×Dic3—51no—
255C22×Dic3C22×D512yes—
256C22×Dic3C22×D512yes—
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—54no—
264C3×C4⋊C4—102no—
265C3×C42—22no—
266C3×C42—202no—
267(C2×D3)⋊C4—54no—
268(C2×D3)⋊C4—104no—
269(C2×D3)⋊C4—102no—
270C4⋊Dic3—22no—
271C4⋊Dic3C4×D514yes—
272Dic3⋊C4—42no—
273Dic3⋊C4—58no—
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—21no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—34no—
291C2×C4×C5—62no—
292C2×C4×C5—62no—
293C2×C4×C5—62no—
294C2×C4×C5—62no—
295C2×C4×C5—62no—
296C2×C4×C5C2×C4×D312yes—
297C2×C4×C5#23512yes—
298C2×C4×C5#23512yes—
299C2×C4×C5D3×Q812yes—
300C2×Dic5—22no—
301C2×Dic5—22no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—32no—
306C2×Dic5—32no—
307C2×Dic5—61no—
308C2×Dic5—61no—
309C2×Dic5—61no—
310C2×Dic5—61no—
311C2×Dic5—61no—
312C2×Dic5—61no—
313C2×Dic5—61no—
314C2×Dic5C2×C4×D314yes—
315C4×D5—68no—
316C4×D5—128no—
317C23×C4—152no—
318C23×C4—151no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—51no—
323C2×C4⋊C4—51no—
324C2×C4⋊C4—158no—
325C2×C4⋊C4—152no—
326C2×C4⋊C4—152no—
327C2×C22⋊C4—151no—
328C2×C22⋊C4—152no—
329C2×C22⋊C4—151no—
330C2×C42—52no—
331C2×C42—151no—
332C2×C42—152no—
333C2×C42D3×D511yescharacteristic
334#95—32no—
335#95—52no—
336#95—152no—
337#95—152no—
338#95—152no—
339#95—151no—
340#95—151no—
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—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—54no—
363C2×C3×C4—102no—
364C2×C3×C4—102no—
365C2×C3×C4—102no—
366C2×C3×C4—102no—
367C2×C3×C4—102no—
368C2×C3×C4C2×C4×D512yes—
369C2×C3×C4#71112yes—
370C2×C3×C4Q8×D512yes—
371C2×C3×C4#71312yes—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—52no—
378C2×Dic3—52no—
379C2×Dic3—101no—
380C2×Dic3—101no—
381C2×Dic3—101no—
382C2×Dic3—101no—
383C2×Dic3—101no—
384C2×Dic3—101no—
385C2×Dic3—101no—
386C2×Dic3C2×C4×D514yes—
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—41no—
405C4×C5—64no—
406C4×C5—122no—
407C4×C5#87214yes—
408Dic5—24no—
409Dic5—42no—
410Dic5—62no—
411Dic5—62no—
412Dic5—121no—
413C24—151no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4—52no—
417C22×C4—52no—
418C22×C4—158no—
419C22×C4—151no—
420C22×C4—151no—
421C22×C4—154no—
422C22×C4—151no—
423C22×C4—302no—
424C22×C4—302no—
425C22×C4—602no—
426C22×C4—602no—
427C22×C4C2×D3×D511yescharacteristic
428C22×C4C2×D3×D511yescharacteristic
429C22×C4C2×D3×D511yescharacteristic
430C4⋊C4—34no—
431C4⋊C4—54no—
432C4⋊C4—62no—
433C4⋊C4—102no—
434C4⋊C4—122no—
435C4⋊C4—158no—
436C4⋊C4—602no—
437C22⋊C4—154no—
438C22⋊C4—302no—
439C22⋊C4—304no—
440C42—22no—
441C42—202no—
442C42—302no—
443C42—602no—
444C3×C5#234511yescharacteristic, complemented, Hall
445C22×C3C22×C4×D511yescharacteristic
446C22×C3C2×#71111yescharacteristic
447C22×C3#414411yescharacteristic
448C22×C3C2×Q8×D511yescharacteristic
449C22×C3C2×#71311yescharacteristic
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—41no—
461C3×C4—104no—
462C3×C4—202no—
463C3×C4#403814yes—
464Dic3—24no—
465Dic3—42no—
466Dic3—102no—
467Dic3—102no—
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—
484C23C22×D3×D511yescharacteristic
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—34no—
491C2×C4—54no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—102no—
498C2×C4—102no—
499C2×C4—102no—
500C2×C4—102no—
501C2×C4—102no—
502C2×C4—152no—
503C2×C4—152no—
504C2×C4—301no—
505C2×C4—301no—
506C2×C4—301no—
507C2×C4—301no—
508C2×C4—308no—
509C2×C4—301no—
510C2×C4—301no—
511C2×C4—301no—
512C2×C4—608no—
513C2×C4D15⋊Q812yes—
514C2×C4#677312yes—
515C2×C4C4×D3×D512yes—
516C2×C4#678212yes—
517C2×C3C2×#403811yescharacteristic
518C2×C3#6587911yescharacteristic
519C2×C3C4×Q8×D511yescharacteristic
520C2×C3#6594811yescharacteristic
521C2×C3#6596911yescharacteristic
522C2×C3#6602911yescharacteristic
523C2×C3#6606411yescharacteristic
524D3—104no—
525C5#7376811yescharacteristic, complemented, Sylow
526C22—154no—
527C22—302no—
528C22—302no—
529C22—302no—
530C22—302no—
531C22—302no—
532C22—302no—
533C22C2×D15⋊Q811yescharacteristic
534C22C2×#677311yescharacteristic
535C22C2×C4×D3×D511yescharacteristic
536C22#9201611yescharacteristic
537C22#9201711yescharacteristic
538C22#9201811yescharacteristic
539C22C2×#678211yescharacteristic
540C4—22no—
541C4—41no—
542C4—64no—
543C4—104no—
544C4—122no—
545C4—202no—
546C4—302no—
547C4—302no—
548C4—601no—
549C4#9140214yes—
550C3#1060793211yescharacteristic, complemented, Sylow
551C2—304no—
552C2C4×D15⋊Q811yescharacteristic
553C2#1175265411yescharacteristic
554C2#1175270511yescharacteristic
555C2#1175272911yescharacteristic
556C2C2×#9140211yescharacteristic
557C2#1175446711yescharacteristic
558C2#1175449811yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#2345
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
3. C5×#4928C2semidirect
4. C3×C4×C4⋊Dic5C2semidirect
7. #11748868C2semidirect
8. #11748815C2semidirect
10. #11748699C2semidirect
11. #11748662C2semidirect
29. #91437C4semidirect
46. C4×C4⋊Dic5D3semidirect
88. #4928D5semidirect
444. C3×C5#2345semidirect
525. C5#73768semidirect
550. C3#10607932semidirect

Maximal subgroups

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

subgroupindex
C2×#914372
C4×#67142
C5×#49282
C3×C4×C4⋊Dic52
#117488822
#117488752
#117488682
#117488152
#117487992
#117486992
#117486622
#106079323
#737685