Finite Group Databasebeta

← #14323222

Subgroups — #14323222

Order 1920.

1080 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×#91782C211yesmaximal, characteristic
2C5×#4984C211yesmaximal, characteristic, complemented
3C3×#65274C211yesmaximal, characteristic, complemented
4#11748799C212yesmaximal, complemented
5#11748789C211yesmaximal, characteristic
6#11748768C211yesmaximal, characteristic
7#11748699C211yesmaximal, characteristic, complemented
8#11748685C212yesmaximal, complemented
9#11748678C212yesmaximal, complemented
10#11748611C211yesmaximal, characteristic, complemented
11#11748606C212yesmaximal, complemented
12#10607700—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×#6714C2212yes—
17C2×C4×Dic15C2211yescharacteristic
18#91782C418yescomplemented
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C5×C4⋊Dic3C2212yes—
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×C4⋊Dic5C2212yes—
24C2×C3×C4×Dic5C2211yescharacteristic
25C2×#6667C2212yes—
26C2×#6667C2212yes—
27C2×#6667C2212yes—
28C2×#6667C2212yes—
29C2×Dic3×Dic5C2212yes—
30C5×C3⋊#95C2212yes—
31C3×#3963C2212yes—
32#90994C2211yescharacteristic
33#90994C2211yescharacteristic
34#90994C2211yescharacteristic
35#90994C2211yescharacteristic
36#90994C2212yes—
37#75969—51nomaximal, Hall
38C2×#4039—31no—
39C5×#389—31no—
40#65279—32no—
41#65277—31no—
42#65276—32no—
43#65274D311yescharacteristic, complemented
44C4⋊C4×Dic5—31no—
45#65228—31no—
46#65221—31no—
47#64959—32no—
48#64953—32no—
49C23×D15C2312yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15C2311yescharacteristic
53C22×Dic15C2311yescharacteristic
54C2×D60—22no—
55C2×D60—22no—
56C2×D60C2×C414yes—
57C2×C4×D15—42no—
58C2×C4×D15—42no—
59C2×C4×D15C2×C418yes—
60C22×C5×Dic3C2311yescharacteristic
61C22×C5×Dic3C2311yescharacteristic
62C22×C5×Dic3C2312yes—
63C22×C3×Dic5C2311yescharacteristic
64C22×C3×Dic5C2311yescharacteristic
65C22×C3×Dic5C2312yes—
66C3×C5×C4⋊C4C2×C414yes—
67#6714C2×C418yes—
68C4×Dic15C2×C414yes—
69C5×C4⋊Dic3—22no—
70C5×C4⋊Dic3—42no—
71C4×C5×Dic3—22no—
72C3×C4⋊Dic5—22no—
73C3×C4⋊Dic5—42no—
74C3×C4×Dic5—22no—
75#6667—42no—
76#6667—42no—
77#6667—42no—
78#6667—42no—
79Dic3×Dic5—42no—
80C2×#893—51no—
81C3×#389—51no—
82#4990—52no—
83#4988—51no—
84#4987—52no—
85#4984D511yescharacteristic, complemented
86Dic3×C4⋊C4—51no—
87#4939—51no—
88#4932—51no—
89#4670—52no—
90#4664—52no—
91C22×D20—31no—
92C22×C4×D5—32no—
93C2×C5×C4⋊C4—31no—
94C2×C5×C4⋊C4—32no—
95C2×C5×C4⋊C4C2×D311yescharacteristic
96C2×C42×C5—31no—
97C2×#685—32no—
98C2×#685—32no—
99C2×#685—32no—
100C2×#685—32no—
101C2×#685—32no—
102C2×C4⋊Dic5C2×D311yescharacteristic
103C2×C4⋊Dic5C2×D312yes—
104C2×C4×Dic5—31no—
105C2×C4×Dic5—32no—
106C2×C4×Dic5C2×D311yescharacteristic
107#4039—38no—
108C5×#95—32no—
109#3963—31no—
110#3963—32no—
111#3963—31no—
112#3963—31no—
113#3963—31no—
114#3963C2×D312yes—
115#2319—151noSylow
116C23×C3×C5C2411yescharacteristic
117C22×D15—22no—
118C22×D15—22no—
119C22×D15—22no—
120C22×D15—22no—
121C22×D15—22no—
122C22×D15—22no—
123C22×D15C22×C414yes—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5C22×C414yes—
127C2×C3×C4×C5C22×C412yes—
128C2×C3×C4×C5#4112yes—
129C2×C3×C4×C5#4112yes—
130C2×Dic15—21no—
131C2×Dic15—21no—
132C2×Dic15—21no—
133C2×Dic15—21no—
134C2×Dic15C22×C412yes—
135C2×Dic15C22×C412yes—
136D60—28no—
137C4×D15—48no—
138C2×C5×Dic3—21no—
139C2×C5×Dic3—21no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—21no—
143C2×C5×Dic3—21no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—21no—
146C2×C5×Dic3—21no—
147C2×C3×Dic5—21no—
148C2×C3×Dic5—21no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5—21no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5—21no—
155C2×C3×Dic5—21no—
156C22×D12—51no—
157C22×C4×D3—52no—
158C2×C3×C4⋊C4—51no—
159C2×C3×C4⋊C4—52no—
160C2×C3×C4⋊C4C2×D511yescharacteristic
161C2×C3×C42—51no—
162C2×(C2×D3)⋊C4—52no—
163C2×(C2×D3)⋊C4—52no—
164C2×(C2×D3)⋊C4—52no—
165C2×(C2×D3)⋊C4—52no—
166C2×(C2×D3)⋊C4—52no—
167C2×C4⋊Dic3C2×D511yescharacteristic
168C2×C4⋊Dic3C2×D512yes—
169C2×C4×Dic3—51no—
170C2×C4×Dic3—52no—
171C2×C4×Dic3C2×D511yescharacteristic
172#893—58no—
173C3×#95—52no—
174C3⋊#95—51no—
175C3⋊#95—52no—
176C3⋊#95—51no—
177C3⋊#95—51no—
178C3⋊#95—51no—
179C3⋊#95C2×D512yes—
180C23×D5—32no—
181C22×C4×C5—31no—
182C22×C4×C5—31no—
183C22×C4×C5—32no—
184C22×C4×C5C22×D312yes—
185C22×C4×C5C22×D311yescharacteristic
186C22×Dic5—31no—
187C22×Dic5—31no—
188C22×Dic5C22×D311yescharacteristic
189C22×Dic5C22×D311yescharacteristic
190C22×Dic5C22×D312yes—
191C2×D20—34no—
192C2×D20—62no—
193C2×D20—62no—
194C2×C4×D5—38no—
195C2×C4×D5—122no—
196C2×C4×D5—122no—
197C5×C4⋊C4—62no—
198C5×C4⋊C4—122no—
199C5×C4⋊C4C4×D314yes—
200C42×C5—62no—
201#685—38no—
202#685—122no—
203#685—122no—
204#685—122no—
205#685—122no—
206C4⋊Dic5—22no—
207C4⋊Dic5—42no—
208C4×Dic5—22no—
209C4×Dic5—34no—
210C4×Dic5—122no—
211C2×C4×D4—151no—
212#390—151no—
213#390—151no—
214#389—31no—
215#389—51no—
216#388—152no—
217#388—152no—
218#381—152no—
219#380—152no—
220C4×C4⋊C4—151no—
221#376—151no—
222C22×C3×C5C23×C411yescharacteristic
223C22×C3×C5C2×#4111yescharacteristic
224C22×C3×C5C2×#4111yescharacteristic
225C22×C3×C5H(5,𝔽2)11yescharacteristic
226C22×C3×C5H(5,𝔽2)11yescharacteristic
227C22×C3×C5H(5,𝔽2)11yescharacteristic
228C22×C3×C52−1+411yescharacteristic
229C2×D15—24no—
230C2×D15—24no—
231C2×D15—24no—
232C2×D15—42no—
233C2×D15—42no—
234C2×D15—42no—
235C2×D15—42no—
236C3×C4×C5—24no—
237C3×C4×C5#11714yes—
238Dic15—22no—
239Dic15—22no—
240C3×Dic5—41no—
241C3×Dic5—42no—
242C3×Dic5—41no—
243C5×Dic3—41no—
244C5×Dic3—42no—
245C5×Dic3—41no—
246C23×D3—52no—
247C22×C3×C4—51no—
248C22×C3×C4—51no—
249C22×C3×C4—52no—
250C22×C3×C4C22×D512yes—
251C22×C3×C4C22×D511yescharacteristic
252C22×Dic3—51no—
253C22×Dic3—51no—
254C22×Dic3C22×D511yescharacteristic
255C22×Dic3C22×D511yescharacteristic
256C22×Dic3C22×D512yes—
257C2×D12—54no—
258C2×D12—102no—
259C2×D12—102no—
260C2×C4×D3—58no—
261C2×C4×D3—202no—
262C2×C4×D3—202no—
263C3×C4⋊C4—102no—
264C3×C4⋊C4—202no—
265C3×C4⋊C4C4×D514yes—
266C3×C42—102no—
267(C2×D3)⋊C4—58no—
268(C2×D3)⋊C4—202no—
269(C2×D3)⋊C4—202no—
270(C2×D3)⋊C4—202no—
271(C2×D3)⋊C4—202no—
272C4⋊Dic3—22no—
273C4⋊Dic3—42no—
274C4×Dic3—22no—
275C4×Dic3—54no—
276C4×Dic3—202no—
277C23×C5C23×D311yescharacteristic
278C22×D5—34no—
279C22×D5—62no—
280C22×D5—62no—
281C22×D5—62no—
282C22×D5—62no—
283C22×D5—62no—
284C22×D5—62no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—61no—
288C2×C4×C5—61no—
289C2×C4×C5—61no—
290C2×C4×C5—62no—
291C2×C4×C5—61no—
292C2×C4×C5—62no—
293C2×C4×C5—61no—
294C2×C4×C5—62no—
295C2×C4×C5—61no—
296C2×C4×C5C2×C4×D314yes—
297C2×C4×C5C2×C4×D312yes—
298C2×C4×C5#23712yes—
299C2×C4×C5#23912yes—
300C2×Dic5—21no—
301C2×Dic5—21no—
302C2×Dic5—22no—
303C2×Dic5—21no—
304C2×Dic5—22no—
305C2×Dic5—21no—
306C2×Dic5—22no—
307C2×Dic5—21no—
308C2×Dic5—21no—
309C2×Dic5—32no—
310C2×Dic5—32no—
311C2×Dic5—61no—
312C2×Dic5—61no—
313C2×Dic5—61no—
314C2×Dic5—61no—
315D20—68no—
316C4×D5—128no—
317C22×D4—151no—
318C23×C4—152no—
319C4×D4—158no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—51no—
323C2×C4⋊C4—52no—
324C2×C4⋊C4D3×D511yescharacteristic
325C2×C22⋊C4—152no—
326C2×C22⋊C4—152no—
327C2×C22⋊C4—152no—
328C2×C22⋊C4—152no—
329C2×C22⋊C4—152no—
330C2×C42—31no—
331C2×C42—51no—
332C2×C42—151no—
333C2×C42—152no—
334#95—32no—
335#95—52no—
336#95—151no—
337#95—151no—
338#95—151no—
339#95—151no—
340#95—152no—
341C2×C3×C5C2×#11711yescharacteristic
342C2×C3×C5#51811yescharacteristic
343C2×C3×C5#52011yescharacteristic
344C2×C3×C5#54011yescharacteristic
345C2×C3×C5#54011yescharacteristic
346C2×C3×C5#54211yescharacteristic
347C2×C3×C5#54211yescharacteristic
348D15—44no—
349C23×C3C23×D511yescharacteristic
350C22×D3—54no—
351C22×D3—102no—
352C22×D3—102no—
353C22×D3—102no—
354C22×D3—102no—
355C22×D3—102no—
356C22×D3—102no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—101no—
360C2×C3×C4—101no—
361C2×C3×C4—101no—
362C2×C3×C4—102no—
363C2×C3×C4—101no—
364C2×C3×C4—102no—
365C2×C3×C4—101no—
366C2×C3×C4—102no—
367C2×C3×C4—101no—
368C2×C3×C4C2×C4×D514yes—
369C2×C3×C4C2×C4×D512yes—
370C2×C3×C4#71112yes—
371C2×C3×C4#71312yes—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—52no—
382C2×Dic3—52no—
383C2×Dic3—101no—
384C2×Dic3—101no—
385C2×Dic3—101no—
386C2×Dic3—101no—
387D12—108no—
388C4×D3—208no—
389C22×C5C22×C4×D311yescharacteristic
390C22×C5C2×#23711yescharacteristic
391C22×C5#100411yescharacteristic
392C22×C5C2×#23911yescharacteristic
393C22×C5#100711yescharacteristic
394C22×C5#100911yescharacteristic
395C22×C5#100911yescharacteristic
396C2×D5—64no—
397C2×D5—64no—
398C2×D5—64no—
399C2×D5—122no—
400C2×D5—122no—
401C2×D5—122no—
402C2×D5—122no—
403C4×C5—24no—
404C4×C5—121no—
405C4×C5—122no—
406C4×C5—121no—
407C4×C5#89214yes—
408Dic5—41no—
409Dic5—41no—
410Dic5—42no—
411Dic5—62no—
412Dic5—62no—
413C24—152no—
414C2×D4—154no—
415C2×D4—302no—
416C2×D4—302no—
417C22×C4—31no—
418C22×C4—32no—
419C22×C4—31no—
420C22×C4—51no—
421C22×C4—52no—
422C22×C4—51no—
423C22×C4—151no—
424C22×C4—151no—
425C22×C4—158no—
426C22×C4—602no—
427C22×C4—602no—
428C22×C4C2×D3×D512yes—
429C22×C4C2×D3×D511yescharacteristic
430C4⋊C4—62no—
431C4⋊C4—102no—
432C4⋊C4—122no—
433C4⋊C4—202no—
434C4⋊C4#120014yes—
435C22⋊C4—158no—
436C22⋊C4—602no—
437C22⋊C4—602no—
438C22⋊C4—602no—
439C22⋊C4—602no—
440C42—62no—
441C42—102no—
442C42—154no—
443C42—602no—
444C3×C5#231911yescharacteristic, complemented, Hall
445C22×C3C22×C4×D511yescharacteristic
446C22×C3C2×#71111yescharacteristic
447C22×C3#414411yescharacteristic
448C22×C3C2×#71311yescharacteristic
449C22×C3#414711yescharacteristic
450C22×C3#414911yescharacteristic
451C22×C3#414911yescharacteristic
452C2×D3—104no—
453C2×D3—104no—
454C2×D3—104no—
455C2×D3—202no—
456C2×D3—202no—
457C2×D3—202no—
458C2×D3—202no—
459C3×C4—24no—
460C3×C4—201no—
461C3×C4—201no—
462C3×C4—202no—
463C3×C4#403814yes—
464Dic3—41no—
465Dic3—42no—
466Dic3—41no—
467Dic3—102no—
468Dic3—102no—
469C2×C5C2×#89211yescharacteristic
470C2×C5#554511yescharacteristic
471C2×C5#557411yescharacteristic
472C2×C5#561911yescharacteristic
473C2×C5#564311yescharacteristic
474C2×C5#569711yescharacteristic
475C2×C5#573311yescharacteristic
476D5—124no—
477C23—154no—
478C23—302no—
479C23—302no—
480C23—302no—
481C23—302no—
482C23—302no—
483C23—302no—
484C23C22×D3×D511yescharacteristic
485D4—308no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—62no—
491C2×C4—61no—
492C2×C4—62no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—62no—
496C2×C4—61no—
497C2×C4—101no—
498C2×C4—101no—
499C2×C4—102no—
500C2×C4—102no—
501C2×C4—101no—
502C2×C4—101no—
503C2×C4—101no—
504C2×C4—102no—
505C2×C4—101no—
506C2×C4—152no—
507C2×C4—152no—
508C2×C4—301no—
509C2×C4—301no—
510C2×C4—301no—
511C2×C4—301no—
512C2×C4—608no—
513C2×C4#676612yes—
514C2×C4#676812yes—
515C2×C4C2×#120014yes—
516C2×C4C2×#120012yes—
517C2×C3C2×#403811yescharacteristic
518C2×C3#6587911yescharacteristic
519C2×C3#6590811yescharacteristic
520C2×C3#6595311yescharacteristic
521C2×C3#6597711yescharacteristic
522C2×C3#6603211yescharacteristic
523C2×C3#6606711yescharacteristic
524D3—204no—
525C5#7596911yescharacteristic, complemented, Sylow
526C22—304no—
527C22—304no—
528C22—304no—
529C22—602no—
530C22—602no—
531C22—602no—
532C22—602no—
533C22C2×#676611yescharacteristic
534C22C2×#676811yescharacteristic
535C22#9202611yescharacteristic
536C22#9202711yescharacteristic
537C22#9202911yescharacteristic
538C22#9203511yescharacteristic
539C22C22×#120011yescharacteristic
540C4—24no—
541C4—121no—
542C4—121no—
543C4—122no—
544C4—201no—
545C4—201no—
546C4—202no—
547C4—302no—
548C4—302no—
549C4#9134714yes—
550C3#1060770011yescharacteristic, complemented, Sylow
551C2—604no—
552C2#1175337211yescharacteristic
553C2#1175337511yescharacteristic
554C2#1175364911yescharacteristic
555C2#1175366111yescharacteristic
556C2C2×#9134711yescharacteristic
557C2#1175492211yescharacteristic
558C2#1175514911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2319
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#4984C2semidirect
3. C3×#65274C2semidirect
4. #11748799C2semidirect
7. #11748699C2semidirect
8. #11748685C2semidirect
9. #11748678C2semidirect
10. #11748611C2semidirect
11. #11748606C2semidirect
18. #91782C4semidirect
43. #65274D3semidirect
85. #4984D5semidirect
444. C3×C5#2319semidirect
525. C5#75969semidirect
550. C3#10607700semidirect

Maximal subgroups

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

subgroupindex
C2×#917822
C5×#49842
C3×#652742
#117487992
#117487892
#117487682
#117486992
#117486852
#117486782
#117486112
#117486062
#106077003
#759695