Finite Group Databasebeta

← #14301109

Subgroups — #14301109

Order 1920.

748 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6717C211yesmaximal, characteristic
2C2×C3×Q8×Dic5C211yesmaximal, characteristic, complemented
3C2×C5×#835C211yesmaximal, characteristic, complemented
4#11748768C211yesmaximal, characteristic
5C2×#90968C211yesmaximal, characteristic
6#11745979C211yesmaximal, characteristic, complemented
7#11745966C211yesmaximal, characteristic, complemented
8#10601033—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C22×D60C2211yescharacteristic
11C2×#6717—22no—
12C2×#6717—22no—
13C2×#6717C414yescomplemented
14C22×C3⋊#168C2211yescharacteristic
15C2×C5×C4⋊Dic3C2211yescharacteristic
16C22×C5×C3⋊C8C2211yescharacteristic
17C3×Q8×Dic5—22no—
18C3×Q8×Dic5—22no—
19C2×C3×C4⋊Dic5—21no—
20C2×C3×C4⋊Dic5C2211yescharacteristic
21C2×C3×C4×Dic5—21no—
22C2×C3×C4×Dic5C2211yescharacteristic
23C2×#6667—21no—
24C2×#6667—21no—
25C5×#835—22no—
26#90968—22no—
27#71633—51nomaximal, Hall
28C2×Q8×Dic5D311yescharacteristic, complemented
29C22×#688—31no—
30C2×C5×#103—31no—
31C2×#3953—31no—
32#65277—31no—
33#64770—31no—
34#64749—31no—
35C23×D15—21no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8C2×C412yes—
40C2×C3×C5×Q8D412yes—
41C2×C3×C5×Q8D412yes—
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5C2311yescharacteristic
44C2×D60—21no—
45C2×D60—21no—
46C2×D60—21no—
47C2×D60—21no—
48C2×D60—21no—
49C2×D60C2×C412yes—
50C22×C5×Dic3—21no—
51C22×C3×Dic5—21no—
52C22×C3×Dic5D411yescharacteristic
53C22×C3×Dic5D411yescharacteristic
54#6717—28no—
55C2×C3⋊#168—21no—
56C2×C3⋊#168—21no—
57C2×C3⋊#168C2×C412yes—
58C5×C4⋊Dic3—21no—
59C5×C4⋊Dic3—21no—
60C2×C5×C3⋊C8—21no—
61C2×C5×C3⋊C8—21no—
62C2×C5×C3⋊C8—21no—
63C3×C4⋊Dic5—24no—
64C3×C4⋊Dic5—21no—
65C3×C4⋊Dic5D412yes—
66C3×C4×Dic5—24no—
67C3×C4×Dic5—21no—
68C3×C4×Dic5—21no—
69#6667—42no—
70#6667—42no—
71C2×C3×C4×Q8—51no—
72C22×#215—51no—
73C2×#835D511yescharacteristic, complemented
74C2×#809—51no—
75#4939—51no—
76#4529—51no—
77#4524—51no—
78C22×C5×Q8C2×D311yescharacteristic
79C22×D20—31no—
80C22×C5×C8—31no—
81C2×C5×C4⋊C4—31no—
82Q8×Dic5—22no—
83Q8×Dic5—22no—
84C2×#688—34no—
85C2×#688—62no—
86C2×#688—62no—
87C2×#685—61no—
88C2×#685—61no—
89C2×C4⋊Dic5—21no—
90C2×C4⋊Dic5C2×D311yescharacteristic
91C2×C4×Dic5—21no—
92C2×C4×Dic5C2×D311yescharacteristic
93C22×#168—31no—
94C5×#103—62no—
95#3953—62no—
96#1880—151noSylow
97C23×C3×C5C2×D411yescharacteristic
98C22×D15—22no—
99C22×D15—41no—
100C22×D15—41no—
101C22×D15—41no—
102C22×D15—41no—
103C22×D15—41no—
104C22×D15—41no—
105C3×C5×Q8—22no—
106C3×C5×Q8—22no—
107C3×C5×Q8—22no—
108C3×C5×Q8C22⋊C414yescomplemented
109C2×C3×C4×C5—22no—
110C2×C3×C4×C5—22no—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5C22×C411yescharacteristic
113C2×C3×C4×C5C2×D411yescharacteristic
114C2×C3×C4×C5C2×D411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5#4111yescharacteristic
117C2×C3×C4×C5#4111yescharacteristic
118D60—22no—
119D60—22no—
120D60—41no—
121D60—41no—
122C2×C5×Dic3—41no—
123C2×C5×Dic3—41no—
124C2×C5×Dic3—41no—
125C2×C3×Dic5—21no—
126C2×C3×Dic5—21no—
127C2×C3×Dic5—21no—
128C2×C3×Dic5—22no—
129C2×C3×Dic5—21no—
130C2×C3×Dic5—21no—
131C2×C3×Dic5—41no—
132C2×C3×Dic5—41no—
133C2×C3×Dic5SD1612yescomplemented
134C3⋊#168—22no—
135C5×C3⋊C8—41no—
136C22×C3×Q8C2×D511yescharacteristic
137C22×D12—51no—
138C3×C4×Q8—102no—
139C3×C4×Q8—102no—
140C2×C3×C4⋊C4—51no—
141C2×C3×C4⋊C4—101no—
142C2×C3×C42—51no—
143C2×C3×C42—101no—
144C2×#215—54no—
145C2×#215—102no—
146C2×#215—102no—
147C2×(C2×D3)⋊C4—101no—
148C2×(C2×D3)⋊C4—101no—
149C2×C4⋊Dic3C2×D511yescharacteristic
150C22×C3⋊C8—51no—
151C22×C3⋊C8C2×D511yescharacteristic
152#835—22no—
153#809—102no—
154C23×D5—61no—
155C2×C5×Q8—21no—
156C2×C5×Q8—21no—
157C2×C5×Q8—21no—
158C2×C5×Q8C4×D312yes—
159C2×C5×Q8D1212yes—
160C2×C5×Q8#6712yes—
161C22×C4×C5—21no—
162C22×C4×C5—61no—
163C22×C4×C5C22×D311yescharacteristic
164C22×Dic5—21no—
165C22×Dic5#6711yescharacteristic
166C22×Dic5#6711yescharacteristic
167C2×D20—32no—
168C2×D20—61no—
169C2×D20—61no—
170C2×D20—61no—
171C2×D20—61no—
172C2×D20—61no—
173C2×C5×C8—61no—
174C2×C5×C8—61no—
175C2×C5×C8—61no—
176C5×C4⋊C4—61no—
177C5×C4⋊C4—61no—
178#688—68no—
179#685—122no—
180#685—122no—
181C4⋊Dic5—24no—
182C4⋊Dic5—21no—
183C4⋊Dic5#6712yes—
184C4×Dic5—24no—
185C4×Dic5—21no—
186C4×Dic5—21no—
187C2×#168—32no—
188C2×#168—61no—
189C2×#168—61no—
190C22×SD16—151no—
191C2×C4×Q8—51no—
192C2×#103—31no—
193C2×#102—151no—
194#390—151no—
195#340—151no—
196#336—151no—
197C22×C3×C5C4×D411yescharacteristic
198C22×C3×C5#12011yescharacteristic
199C22×C3×C5#12111yescharacteristic
200C22×C3×C5C2×SD1611yescharacteristic
201C22×C3×C5#13511yescharacteristic
202C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
203C22×C3×C5#13711yescharacteristic
204C2×D15—42no—
205C2×D15—42no—
206C2×D15—42no—
207C2×D15—81no—
208C2×D15—81no—
209C2×D15—81no—
210C2×D15—81no—
211C3×C4×C5—24no—
212C3×C4×C5C2×C22⋊C411yescharacteristic
213C3×C4×C5C4×D411yescharacteristic
214C3×C4×C5#12111yescharacteristic
215C3×C4×C5#12311yescharacteristic
216C3×Dic5—22no—
217C3×Dic5—41no—
218C3×Dic5—42no—
219C5×Dic3—81no—
220C23×D3—101no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—21no—
224C2×C3×Q8C4×D512yes—
225C2×C3×Q8D2012yes—
226C2×C3×Q8C22⋊D512yes—
227C22×C3×C4—21no—
228C22×C3×C4—51no—
229C22×C3×C4—51no—
230C22×C3×C4—101no—
231C22×C3×C4C22×D511yescharacteristic
232C22×Dic3—21no—
233C2×D12—52no—
234C2×D12—101no—
235C2×D12—101no—
236C2×D12—101no—
237C2×D12—101no—
238C2×D12—101no—
239C3×C4⋊C4—52no—
240C3×C4⋊C4—104no—
241C3×C4⋊C4—101no—
242C3×C42—104no—
243C3×C42—101no—
244C3×C42—101no—
245#215—108no—
246(C2×D3)⋊C4—202no—
247(C2×D3)⋊C4—202no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—52no—
254C2×C3⋊C8—101no—
255C2×C3⋊C8—101no—
256C23×C5C2×#6711yescharacteristic
257C22×D5—62no—
258C22×D5—121no—
259C22×D5—121no—
260C22×D5—121no—
261C22×D5—121no—
262C22×D5—121no—
263C22×D5—121no—
264C5×Q8—22no—
265C5×Q8—22no—
266C5×Q8—22no—
267C5×Q8(C2×D3)⋊C414yescomplemented
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—121no—
272C2×C4×C5—121no—
273C2×C4×C5—121no—
274C2×C4×C5C2×C4×D311yescharacteristic
275C2×C4×C5C2×D1211yescharacteristic
276C2×C4×C5#23511yescharacteristic
277C2×C4×C5#23511yescharacteristic
278C2×C4×C5C2×#6711yescharacteristic
279C2×C4×C5C2×#6711yescharacteristic
280C2×Dic5—21no—
281C2×Dic5—21no—
282C2×Dic5—21no—
283C2×Dic5—21no—
284C2×Dic5—21no—
285C2×Dic5—22no—
286C2×Dic5—41no—
287C2×Dic5—41no—
288C2×Dic5#21512yescomplemented
289D20—62no—
290D20—62no—
291D20—121no—
292D20—121no—
293C5×C8—121no—
294#168—62no—
295C22×Q8D3×D511yescharacteristic
296C22×D4—151no—
297C2×SD16—154no—
298C2×SD16—302no—
299C2×SD16—302no—
300C22×C8—31no—
301C22×C8—151no—
302C4×Q8—102no—
303C4×Q8—102no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—51no—
306C2×C4⋊C4—101no—
307C2×C22⋊C4—301no—
308C2×C22⋊C4—301no—
309C2×C42—51no—
310C2×C42—101no—
311#103—62no—
312#102—302no—
313C2×C3×C5#38611yescharacteristic
314C2×C3×C5C4×SD1611yescharacteristic
315C2×C3×C5#44011yescharacteristic
316C2×C3×C5#44811yescharacteristic
317C2×C3×C5#44911yescharacteristic
318C2×C3×C5#46011yescharacteristic
319C2×C3×C5#46411yescharacteristic
320D15—82no—
321C23×C3D4×D511yescharacteristic
322C22×D3—102no—
323C22×D3—201no—
324C22×D3—201no—
325C22×D3—201no—
326C22×D3—201no—
327C22×D3—201no—
328C22×D3—201no—
329C3×Q8—22no—
330C3×Q8—22no—
331C3×Q8—22no—
332C3×Q8#68514yescomplemented
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—52no—
337C2×C3×C4—101no—
338C2×C3×C4—101no—
339C2×C3×C4—102no—
340C2×C3×C4—101no—
341C2×C3×C4—101no—
342C2×C3×C4—101no—
343C2×C3×C4—201no—
344C2×C3×C4—201no—
345C2×C3×C4C2×C4×D511yescharacteristic
346C2×C3×C4C2×D2011yescharacteristic
347C2×C3×C4D4×D511yescharacteristic
348C2×C3×C4#71111yescharacteristic
349C2×C3×C4#71111yescharacteristic
350C2×C3×C4C2×C22⋊D511yescharacteristic
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354D12—102no—
355D12—102no—
356D12—201no—
357D12—201no—
358C3⋊C8—41no—
359C3⋊C8—102no—
360C22×C5C4×#6711yescharacteristic
361C22×C5#93011yescharacteristic
362C22×C5C2×#21511yescharacteristic
363C22×C5#94211yescharacteristic
364C22×C5#94911yescharacteristic
365C22×C5#95011yescharacteristic
366C22×C5#95311yescharacteristic
367C2×D5—122no—
368C2×D5—122no—
369C2×D5—122no—
370C2×D5—241no—
371C2×D5—241no—
372C2×D5—241no—
373C2×D5—241no—
374C4×C5—24no—
375C4×C5—241no—
376C4×C5C2×(C2×D3)⋊C411yescharacteristic
377C4×C5C4×#6711yescharacteristic
378C4×C5#92911yescharacteristic
379C4×C5#93011yescharacteristic
380Dic5—22no—
381Dic5—41no—
382Dic5—42no—
383C24—301no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8#120012yes—
388C2×Q8#120212yes—
389C2×Q8#120312yes—
390C2×D4—152no—
391C2×D4—301no—
392C2×D4—301no—
393C2×D4—301no—
394C2×D4—301no—
395C2×D4—301no—
396C22×C4—21no—
397C22×C4—51no—
398C22×C4—51no—
399C22×C4—61no—
400C22×C4—101no—
401C22×C4C2×D3×D511yescharacteristic
402SD16—308no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—152no—
407C2×C8—301no—
408C2×C8—301no—
409C4⋊C4—52no—
410C4⋊C4—61no—
411C4⋊C4—61no—
412C4⋊C4—104no—
413C4⋊C4—101no—
414C22⋊C4—602no—
415C22⋊C4—602no—
416C42—104no—
417C42—101no—
418C42—101no—
419C3×C5#188011yescharacteristic, complemented, Hall
420C22×C3#402711yescharacteristic
421C22×C3#402811yescharacteristic
422C22×C3D5×SD1611yescharacteristic
423C22×C3#406011yescharacteristic
424C22×C3#406411yescharacteristic
425C22×C3#406511yescharacteristic
426C22×C3#408511yescharacteristic
427C2×D3—202no—
428C2×D3—202no—
429C2×D3—202no—
430C2×D3—401no—
431C2×D3—401no—
432C2×D3—401no—
433C2×D3—401no—
434C3×C4—24no—
435C3×C4—102no—
436C3×C4—201no—
437C3×C4—202no—
438C3×C4#402711yescharacteristic
439C3×C4#403211yescharacteristic
440C3×C4C2×#68511yescharacteristic
441C3×C4#408511yescharacteristic
442Dic3—81no—
443C2×C5C4×#21511yescharacteristic
444C2×C5#502611yescharacteristic
445C2×C5#502711yescharacteristic
446C2×C5#502811yescharacteristic
447C2×C5#521311yescharacteristic
448C2×C5#522711yescharacteristic
449C2×C5#523811yescharacteristic
450D5—242no—
451C23—302no—
452C23—601no—
453C23—601no—
454C23—601no—
455C23—601no—
456C23—601no—
457C23—601no—
458C23D5×#6711yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—22no—
462Q8#666714yescomplemented
463D4—302no—
464D4—302no—
465D4—601no—
466D4—601no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—52no—
471C2×C4—101no—
472C2×C4—101no—
473C2×C4—101no—
474C2×C4—101no—
475C2×C4—102no—
476C2×C4—101no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—201no—
481C2×C4—201no—
482C2×C4#678211yescharacteristic
483C2×C4#678211yescharacteristic
484C2×C4C2×#120011yescharacteristic
485C2×C4C2×#120211yescharacteristic
486C2×C4C2×#120311yescharacteristic
487C2×C4D5×#6711yescharacteristic
488C8—121no—
489C8—302no—
490C2×C3#6507711yescharacteristic
491C2×C3#6509511yescharacteristic
492C2×C3#6510011yescharacteristic
493C2×C3#6510611yescharacteristic
494C2×C3#6524711yescharacteristic
495C2×C3#6545211yescharacteristic
496C2×C3#6547411yescharacteristic
497D3—402no—
498C5#7163311yescharacteristic, complemented, Sylow
499C22—602no—
500C22—602no—
501C22—602no—
502C22—1201no—
503C22—1201no—
504C22—1201no—
505C22—1201no—
506C22D5×#21511yescharacteristic
507C22#9150211yescharacteristic
508C22#9151511yescharacteristic
509C22#9151611yescharacteristic
510C22#9155211yescharacteristic
511C22#9156411yescharacteristic
512C22#9156611yescharacteristic
513C4—24no—
514C4—102no—
515C4—201no—
516C4—202no—
517C4—241no—
518C4C2×#666711yescharacteristic
519C4#9154611yescharacteristic
520C4#9155211yescharacteristic
521C4#9156611yescharacteristic
522C3#1060103311yescharacteristic, complemented, Sylow
523C2—1202no—
524C2#1174922111yescharacteristic
525C2#1174922511yescharacteristic
526C2#1174926811yescharacteristic
527C2#1174927211yescharacteristic
528C2#1174931411yescharacteristic
529C2#1174933811yescharacteristic
530C2#1174949411yescharacteristic
subgroupG/H
centerC23D5×#67
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC22×C3×C5×Q8C22
socleC23×C3×C5C2×D4
prime radicalC22×D60C22
cosocleC22×C3×C4×C5C23
hypercenterC22×Q8D3×D5
hypocenterC3×C5#1880
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q8×Dic5C2semidirect
3. C2×C5×#835C2semidirect
6. #11745979C2semidirect
7. #11745966C2semidirect
13. C2×#6717C4semidirect
28. C2×Q8×Dic5D3semidirect
73. C2×#835D5semidirect
108. C3×C5×Q8C22⋊C4semidirect
133. C2×C3×Dic5SD16semidirect
267. C5×Q8(C2×D3)⋊C4semidirect
288. C2×Dic5#215semidirect
332. C3×Q8#685semidirect
419. C3×C5#1880semidirect
462. Q8#6667semidirect
498. C5#71633semidirect
522. C3#10601033semidirect

Maximal subgroups

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

subgroupindex
C22×#67172
C2×C3×Q8×Dic52
C2×C5×#8352
#117487682
C2×#909682
#117459792
#117459662
#106010333
#716335