Finite Group Databasebeta

← #14430227

Subgroups — #14430227

Order 1920.

776 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C5×#41C211yesmaximal, characteristic
2#11756229C211yesmaximal, characteristic, complemented
3#11751334C212yesmaximal
4#11751123C212yesmaximal, complemented
5#11751122C211yesmaximal, characteristic, complemented
6#11751121C211yesmaximal, characteristic, complemented
7#11751110C212yesmaximal, complemented
8#11751109C211yesmaximal, characteristic, complemented
9#11751108C211yesmaximal, characteristic, complemented
10#11751099C212yesmaximal, complemented
11C2×C4⋊(C3⋊#168)C211yesmaximal, characteristic, complemented
12#10614238—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C3×C4×C5×Q8C2211yescharacteristic
15C3×C4×C5×Q8C2211yescharacteristic
16C3×C4×C5×D4—21no—
17C3×C4×C5×D4—21no—
18C3×C4×C5×D4C2211yescharacteristic
19C3×C4×C5×D4C2211yescharacteristic
20C3×C5×#117—21no—
21C3×C5×#117C2211yescharacteristic
22C2×C3×C42×C5—21no—
23C2×C3×C42×C5C2211yescharacteristic
24#91815C2212yes—
25#91813—21no—
26#91813—21no—
27C22×C3⋊#168C2212yes—
28#91759C2212yes—
29#91758C2211yescharacteristic
30#91758C2211yescharacteristic
31C4×Dic30—21no—
32C4×Dic30—21no—
33#91119C2212yes—
34#91119C2212yes—
35#91116C2212yes—
36#91116C2212yes—
37#91092C2212yes—
38#91092C2212yes—
39#91091C2212yes—
40#91091C2212yes—
41C4⋊(C3⋊#168)C2211yescharacteristic
42C4⋊(C3⋊#168)C2211yescharacteristic
43C4⋊(C3⋊#168)C2212yes—
44#82507—51nomaximal, Hall
45C4×C5×#41D311yescharacteristic
46#65804—31no—
47#65522—32no—
48#65311—32no—
49#65310—31no—
50#65309—31no—
51#65298—32no—
52#65297—31no—
53#65296—31no—
54#65287—32no—
55C2×C4⋊#168—31no—
56C3×C5×#41—22no—
57C3×C5×#41Q814yescomplemented
58C2×C3×C5×Q8C2311yescharacteristic
59C2×C3×C5×D4—21no—
60C2×C3×C5×D4C2311yescharacteristic
61C22×C3×C4×C5—21no—
62C22×C3×C4×C5—21no—
63C22×C3×C4×C5—21no—
64C22×C3×C4×C5D411yescharacteristic
65C22×C3×C4×C5D411yescharacteristic
66C22×C3×C4×C5C2311yescharacteristic
67C2×Dic30—21no—
68C2×Dic30—21no—
69C3×C5×C4⋊C4—21no—
70C3×C5×C4⋊C4—21no—
71C3×C5×C4⋊C4C2311yescharacteristic
72C3×C5×C4⋊C4C2311yescharacteristic
73C3×C5×C22⋊C4—21no—
74C3×C5×C22⋊C4—21no—
75C3×C5×C22⋊C4—21no—
76C3×C42×C5—21no—
77C3×C42×C5—21no—
78C3×C42×C5—21no—
79C3×C42×C5D411yescharacteristic
80C3×C42×C5D411yescharacteristic
81C3×C42×C5C2311yescharacteristic
82C3×C42×C5C2311yescharacteristic
83#6719—22no—
84C4⋊Dic15—21no—
85C4⋊Dic15—21no—
86C4⋊Dic15C2312yes—
87C4⋊Dic15C2312yes—
88#6712—22no—
89#6712—22no—
90C4×Dic15—22no—
91C2×C3⋊#168—24no—
92C2×C3⋊#168C2312yes—
93C2×C3⋊#168C2312yes—
94C3×C4×#41D511yescharacteristic
95#5470—51no—
96#5233—52no—
97#5022—52no—
98#5021—51no—
99#5020—51no—
100#5009—52no—
101#5008—51no—
102#5007—51no—
103#4998—52no—
104C2×C4⋊(C3⋊C8)—51no—
105C2×C5×#41C2×D311yescharacteristic
106C4×C5×Q8C2×D311yescharacteristic
107C4×C5×Q8C2×D311yescharacteristic
108C4×C5×D4—21no—
109C4×C5×D4—21no—
110C4×C5×D4C2×D311yescharacteristic
111C4×C5×D4C2×D311yescharacteristic
112C5×#117—21no—
113C5×#117C2×D311yescharacteristic
114C2×C42×C5—21no—
115C2×C42×C5C2×D311yescharacteristic
116#4072—32no—
117#4070—61no—
118#4070—61no—
119C22×#168—32no—
120#4016—32no—
121#4015—31no—
122#4015—31no—
123C4×Dic10—61no—
124C4×Dic10—61no—
125#3967—32no—
126#3967—32no—
127#3964—32no—
128#3964—32no—
129#3940—32no—
130#3940—32no—
131#3939—32no—
132#3939—32no—
133C4⋊#168—31no—
134C4⋊#168—31no—
135C4⋊#168—32no—
136#3076—151noSylow
137C23×C3×C5—21no—
138C23×C3×C5C2×D411yescharacteristic
139C3×C5×Q8—21no—
140C3×C5×Q8C2×Q812yes—
141C3×C5×D4—24no—
142C3×C5×D4—21no—
143C3×C5×D4C2×Q812yes—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5—24no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5—21no—
148C2×C3×C4×C5—21no—
149C2×C3×C4×C5—41no—
150C2×C3×C4×C5—42no—
151C2×C3×C4×C5—42no—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5C2×D411yescharacteristic
154C2×C3×C4×C5C2×D411yescharacteristic
155C2×C3×C4×C5C2×D411yescharacteristic
156C2×C3×C4×C5C2×D411yescharacteristic
157C2×C3×C4×C5C2×Q812yes—
158C2×C3×C4×C5#4112yes—
159C2×C3×C4×C5C2411yescharacteristic
160C2×Dic15—22no—
161C2×Dic15—22no—
162Dic30—42no—
163Dic30—42no—
164C3⋊#168—22no—
165C3⋊#168—22no—
166C2×C3×#41C2×D511yescharacteristic
167C3×C4×Q8C2×D511yescharacteristic
168C3×C4×Q8C2×D511yescharacteristic
169C3×C4×D4—21no—
170C3×C4×D4—21no—
171C3×C4×D4C2×D511yescharacteristic
172C3×C4×D4C2×D511yescharacteristic
173C3×#117—21no—
174C3×#117C2×D511yescharacteristic
175C2×C3×C42—21no—
176C2×C3×C42C2×D511yescharacteristic
177#926—52no—
178#924—101no—
179#924—101no—
180C22×C3⋊C8—52no—
181#870—52no—
182#869—51no—
183#869—51no—
184C4×Dic6—101no—
185C4×Dic6—101no—
186#835—52no—
187#835—52no—
188#832—52no—
189#832—52no—
190#808—52no—
191#808—52no—
192#807—52no—
193#807—52no—
194C4⋊(C3⋊C8)—51no—
195C4⋊(C3⋊C8)—51no—
196C4⋊(C3⋊C8)—52no—
197C5×#41—22no—
198C5×#41Dic614yescomplemented
199C2×C5×Q8C22×D311yescharacteristic
200C2×C5×D4—21no—
201C2×C5×D4C22×D311yescharacteristic
202C22×C4×C5—21no—
203C22×C4×C5—21no—
204C22×C4×C5—21no—
205C22×C4×C5#6711yescharacteristic
206C22×C4×C5#6711yescharacteristic
207C22×C4×C5C22×D311yescharacteristic
208C2×Dic10—61no—
209C2×Dic10—61no—
210C5×C4⋊C4—21no—
211C5×C4⋊C4—21no—
212C5×C4⋊C4C22×D311yescharacteristic
213C5×C4⋊C4C22×D311yescharacteristic
214C5×C22⋊C4—21no—
215C5×C22⋊C4—21no—
216C5×C22⋊C4—21no—
217C42×C5—21no—
218C42×C5—21no—
219C42×C5—21no—
220C42×C5#6711yescharacteristic
221C42×C5#6711yescharacteristic
222C42×C5C22×D311yescharacteristic
223C42×C5C22×D311yescharacteristic
224C22⋊Dic5—62no—
225C4⋊Dic5—32no—
226C4⋊Dic5—32no—
227C4⋊Dic5—61no—
228C4⋊Dic5—61no—
229#683—62no—
230#683—62no—
231C4×Dic5—62no—
232C2×#168—32no—
233C2×#168—32no—
234C2×#168—64no—
235#533—151no—
236C4×#41D1511yescharacteristic
237#479—152no—
238#478—152no—
239#477—151no—
240#476—151no—
241#475—151no—
242#474—151no—
243#427—152no—
244C2×C4⋊C8—151no—
245#416—152no—
246C22×C3×C5—21no—
247C22×C3×C5—21no—
248C22×C3×C5—22no—
249C22×C3×C5—41no—
250C22×C3×C5—41no—
251C22×C3×C5C22⋊Q812yes—
252C22×C3×C5C22×D411yescharacteristic
253C3×C4×C5—22no—
254C3×C4×C5—21no—
255C3×C4×C5—21no—
256C3×C4×C5—21no—
257C3×C4×C5—41no—
258C3×C4×C5—41no—
259C3×C4×C5C22⋊Q812yes—
260C3×C4×C5#13512yes—
261C3×C4×C5C22×Q811yescharacteristic
262C3×C4×C5C2×#4111yescharacteristic
263Dic15—42no—
264Dic15—42no—
265C3×#41—22no—
266C3×#41Dic1014yescomplemented
267C2×C3×Q8C22×D511yescharacteristic
268C2×C3×D4—21no—
269C2×C3×D4C22×D511yescharacteristic
270C22×C3×C4—21no—
271C22×C3×C4—21no—
272C22×C3×C4—21no—
273C22×C3×C4C22⋊D511yescharacteristic
274C22×C3×C4C22⋊D511yescharacteristic
275C22×C3×C4C22×D511yescharacteristic
276C2×Dic6—101no—
277C2×Dic6—101no—
278C3×C4⋊C4—21no—
279C3×C4⋊C4—21no—
280C3×C4⋊C4C22×D511yescharacteristic
281C3×C4⋊C4C22×D511yescharacteristic
282C3×C22⋊C4—21no—
283C3×C22⋊C4—21no—
284C3×C22⋊C4—21no—
285C3×C42—21no—
286C3×C42—21no—
287C3×C42—21no—
288C3×C42C22⋊D511yescharacteristic
289C3×C42C22⋊D511yescharacteristic
290C3×C42C22×D511yescharacteristic
291C3×C42C22×D511yescharacteristic
292#217—102no—
293C4⋊Dic3—52no—
294C4⋊Dic3—52no—
295C4⋊Dic3—101no—
296C4⋊Dic3—101no—
297Dic3⋊C4—102no—
298Dic3⋊C4—102no—
299C4×Dic3—102no—
300C2×C3⋊C8—52no—
301C2×C3⋊C8—52no—
302C2×C3⋊C8—104no—
303C23×C5—21no—
304C23×C5C2×#6711yescharacteristic
305C5×Q8—21no—
306C5×Q8C2×Dic612yes—
307C5×D4—24no—
308C5×D4—21no—
309C5×D4C2×Dic612yes—
310C2×C4×C5—21no—
311C2×C4×C5—24no—
312C2×C4×C5—22no—
313C2×C4×C5—21no—
314C2×C4×C5—21no—
315C2×C4×C5—41no—
316C2×C4×C5—42no—
317C2×C4×C5—42no—
318C2×C4×C5C2×Dic612yes—
319C2×C4×C5#23512yes—
320C2×C4×C5C2×#6711yescharacteristic
321C2×C4×C5C2×#6711yescharacteristic
322C2×C4×C5C2×#6711yescharacteristic
323C2×C4×C5C2×#6711yescharacteristic
324C2×C4×C5C2×#6711yescharacteristic
325C2×C4×C5C23×D311yescharacteristic
326C2×Dic5—62no—
327C2×Dic5—62no—
328Dic10—122no—
329Dic10—122no—
330#168—62no—
331#168—62no—
332C2×#41C2×D1511yescharacteristic
333C22×C8—152no—
334C4⋊Q8—151no—
335C4⋊Q8—151no—
336#125—152no—
337C22⋊Q8—301no—
338C22⋊Q8—301no—
339C4×Q8—301no—
340C4×Q8—301no—
341C4×Q8C2×D1511yescharacteristic
342C4×Q8C2×D1511yescharacteristic
343C4×D4—21no—
344C4×D4—21no—
345C4×D4C2×D1511yescharacteristic
346C4×D4C2×D1511yescharacteristic
347#117—21no—
348#117—152no—
349#117C2×D1511yescharacteristic
350C2×C42—21no—
351C2×C42C2×D1511yescharacteristic
352#107—152no—
353#107—152no—
354#106—152no—
355#106—152no—
356C4⋊C8—151no—
357C4⋊C8—151no—
358C4⋊C8—152no—
359#103—152no—
360#103—152no—
361#102—152no—
362#102—152no—
363C2×C3×C5—22no—
364C2×C3×C5—42no—
365C2×C3×C5C2×C22⋊Q811yescharacteristic
366C2×C3×C5C2×#13511yescharacteristic
367C2×C3×C5#57511yescharacteristic
368C23×C3—21no—
369C23×C3C2×C22⋊D511yescharacteristic
370C3×Q8—21no—
371C3×Q8C2×Dic1012yes—
372C3×D4—24no—
373C3×D4—21no—
374C3×D4C2×Dic1012yes—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—41no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4C2×Dic1012yes—
384C2×C3×C4#70912yes—
385C2×C3×C4C2×C22⋊D511yescharacteristic
386C2×C3×C4C2×C22⋊D511yescharacteristic
387C2×C3×C4C2×C22⋊D511yescharacteristic
388C2×C3×C4C2×C22⋊D511yescharacteristic
389C2×C3×C4C2×C22⋊D511yescharacteristic
390C2×C3×C4C23×D511yescharacteristic
391C2×Dic3—102no—
392C2×Dic3—102no—
393Dic6—202no—
394Dic6—202no—
395C3⋊C8—102no—
396C3⋊C8—102no—
397C22×C5—21no—
398C22×C5—21no—
399C22×C5—22no—
400C22×C5—41no—
401C22×C5—41no—
402C22×C5#92412yes—
403C22×C5C22×#6711yescharacteristic
404C4×C5—22no—
405C4×C5—21no—
406C4×C5—21no—
407C4×C5—21no—
408C4×C5—41no—
409C4×C5—41no—
410C4×C5#92412yes—
411C4×C5#95012yes—
412C4×C5C22×Dic611yescharacteristic
413C4×C5C2×#23511yescharacteristic
414Dic5—122no—
415Dic5—122no—
416#41—22no—
417#41Dic3014yescomplemented
418C2×Q8—301no—
419C2×Q8—301no—
420C2×Q8C22×D1511yescharacteristic
421C2×D4—21no—
422C2×D4C22×D1511yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4#122011yescharacteristic
427C22×C4#122011yescharacteristic
428C22×C4C22×D1511yescharacteristic
429C2×C8—152no—
430C2×C8—152no—
431C2×C8—304no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—152no—
435C4⋊C4—152no—
436C4⋊C4—302no—
437C4⋊C4—302no—
438C4⋊C4—301no—
439C4⋊C4—301no—
440C4⋊C4C22×D1511yescharacteristic
441C4⋊C4C22×D1511yescharacteristic
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C22⋊C4—302no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—302no—
450C42#122011yescharacteristic
451C42#122011yescharacteristic
452C42C22×D1511yescharacteristic
453C42C22×D1511yescharacteristic
454C3×C5#307611yescharacteristic, complemented, Hall
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—22no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3#407012yes—
461C22×C3C22×C22⋊D511yescharacteristic
462C3×C4—22no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4#407012yes—
469C3×C4#409612yes—
470C3×C4C22×Dic1011yescharacteristic
471C3×C4C2×#70911yescharacteristic
472Dic3—202no—
473Dic3—202no—
474C2×C5—22no—
475C2×C5—42no—
476C2×C5C2×#92411yescharacteristic
477C2×C5C2×#95011yescharacteristic
478C2×C5#582211yescharacteristic
479C23—21no—
480C23C2×#122011yescharacteristic
481Q8—21no—
482Q8—602no—
483Q8—602no—
484Q8C2×Dic3012yes—
485D4—24no—
486D4—21no—
487D4C2×Dic3012yes—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—302no—
497C2×C4—302no—
498C2×C4C2×Dic3012yes—
499C2×C4#681712yes—
500C2×C4C2×#122011yescharacteristic
501C2×C4C2×#122011yescharacteristic
502C2×C4C2×#122011yescharacteristic
503C2×C4C2×#122011yescharacteristic
504C2×C4C2×#122011yescharacteristic
505C2×C4C23×D1511yescharacteristic
506C8—302no—
507C8—302no—
508C2×C3—22no—
509C2×C3—42no—
510C2×C3C2×#407011yescharacteristic
511C2×C3C2×#409611yescharacteristic
512C2×C3#6615611yescharacteristic
513C5#8250711yescharacteristic, complemented, Sylow
514C22—21no—
515C22—21no—
516C22—22no—
517C22—41no—
518C22—41no—
519C22#9181312yes—
520C22C22×#122011yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—22no—
525C4—41no—
526C4—41no—
527C4—602no—
528C4—602no—
529C4#9181312yes—
530C4#9183912yes—
531C4C22×Dic3011yescharacteristic
532C4C2×#681711yescharacteristic
533C3#1061423811yescharacteristic, complemented, Sylow
534C2—22no—
535C2—42no—
536C2C2×#9181311yescharacteristic
537C2C2×#9183911yescharacteristic
538C2#1175658111yescharacteristic
subgroupG/H
centerC2×C4C2×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C4×C5×#41C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×C5×#41C22
hypercenterC4×#41D15
hypocenterC3×C5#3076
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. #11756229C2semidirect
4. #11751123C2semidirect
5. #11751122C2semidirect
6. #11751121C2semidirect
7. #11751110C2semidirect
8. #11751109C2semidirect
9. #11751108C2semidirect
10. #11751099C2semidirect
11. C2×C4⋊(C3⋊#168)C2semidirect
57. C3×C5×#41Q8semidirect
198. C5×#41Dic6semidirect
266. C3×#41Dic10semidirect
417. #41Dic30semidirect
454. C3×C5#3076semidirect
513. C5#82507semidirect
533. C3#10614238semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C5×#412
#117562292
#117513342
#117511232
#117511222
#117511212
#117511102
#117511092
#117511082
#117510992
C2×C4⋊(C3⋊#168)2
#106142383
#825075