Finite Group Databasebeta

← #13089019

Subgroups — #13089019

Order 1440.

624 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10624230C211yesmaximal, characteristic, complemented
2#10624210C211yesmaximal, characteristic, complemented
3#10624207C211yesmaximal, characteristic, complemented
4C5×#3804C211yesmaximal, characteristic, complemented
5#10624165C211yesmaximal, characteristic, complemented
6C2×C3×#1204C211yesmaximal, characteristic, complemented
7C3×#6782C211yesmaximal, characteristic, complemented
8#10624056C211yesmaximal, characteristic, complemented
9#10624055C211yesmaximal, characteristic, complemented
10#10624054C211yesmaximal, characteristic, complemented
11#10624051C211yesmaximal, characteristic, complemented
12#10624040C211yesmaximal, characteristic, complemented
13#10624039C211yesmaximal, characteristic, complemented
14#10624038C211yesmaximal, characteristic, complemented
15D3×#1204C211yesmaximal, characteristic, complemented
16#92020—31nomaximal
17#92004—31nomaximal
18#67184C2211yescharacteristic
19C5×#638C2211yescharacteristic
20C3×#1220C2211yescharacteristic
21C2×C3×Dic15C2211yescharacteristic
22C3×C5×#67C2211yescharacteristic
23C2×C3×C5×Dic3C2211yescharacteristic
24C2×C32×Dic5C2211yescharacteristic
25#67158C2211yescharacteristic
26#67156C2211yescharacteristic
27#67155C2211yescharacteristic
28#67154C2211yescharacteristic
29#67153C2211yescharacteristic
30#67153C2211yescharacteristic
31#67152C2211yescharacteristic
32#67151C2211yescharacteristic
33#67149C2211yescharacteristic
34#67149C2211yescharacteristic
35D3×Dic15C2211yescharacteristic
36Dic3×D15C2211yescharacteristic
37C5×#627C2211yescharacteristic
38C5×#626C2211yescharacteristic
39C5×#624C2211yescharacteristic
40C5×D3×Dic3C2211yescharacteristic
41#67141C2211yescharacteristic
42#67140C2211yescharacteristic
43#67137C2211yescharacteristic
44C3⋊D3×Dic5C2211yescharacteristic
45C3×#1204C2211yescharacteristic
46C3×#1204C2211yescharacteristic
47C3×#1204C2211yescharacteristic
48C3×#1204C2211yescharacteristic
49C3×#1204C2211yescharacteristic
50C3×#1203C2211yescharacteristic
51C3×#1200C2211yescharacteristic
52C3×D3×Dic5C2211yescharacteristic
53#64297—51nomaximal, Hall
54#6819—31no—
55#6817—31no—
56C5×#237—31no—
57C5×#235—31no—
58C3×#711—31no—
59C2×C3×Dic10—31no—
60C2×#1204—31no—
61C2×#1204D311yescharacteristic, complemented
62#6782—31no—
63#6782—31no—
64#6782—31no—
65#6782—31no—
66#6782—31no—
67#6782—61no—
68#6782D311yescharacteristic, complemented
69#6780—31no—
70#6780—31no—
71#6779—31no—
72#6779—31no—
73#6771—31no—
74D15⋊Q8—31no—
75D15⋊Q8—31no—
76D15⋊Q8—31no—
77#6769—31no—
78#6769—31no—
79#6768—31no—
80#6768—31no—
81D3×Dic10—31no—
82D3×Dic10—31no—
83D3×Dic10—31no—
84D5×Dic6—31no—
85C22×C32×C5C2311yescharacteristic
86C2×C3⋊D15C2311yescharacteristic
87C2×C5×C3⋊D3C2311yescharacteristic
88C2×C3×D15C2311yescharacteristic
89C2×C3×C5×D3C2311yescharacteristic
90C3⋊Dic15C2311yescharacteristic
91C5×C3⋊Dic3C2311yescharacteristic
92C3×Dic15C2311yescharacteristic
93C3×Dic15C2311yescharacteristic
94C3×Dic15C2311yescharacteristic
95C3×C5×Dic3C2311yescharacteristic
96C3×C5×Dic3C2311yescharacteristic
97C3×C5×Dic3C2311yescharacteristic
98C32×Dic5C2311yescharacteristic
99C32×Dic5C2311yescharacteristic
100#4150—91noHall
101#3828—51no—
102C3×#235—51no—
103C2×C3×Dic6—51no—
104#3804—51no—
105#3804—51no—
106#3804—51no—
107#3804D511yescharacteristic, complemented
108#3799—51no—
109#3799—51no—
110#3797—51no—
111#3797—51no—
112#3796—51no—
113#3796—51no—
114D3×Dic6—51no—
115D3×Dic6—51no—
116C3×C5×D4—31no—
117C2×C3×C4×C5—31no—
118#1220—31no—
119#1220—31no—
120#1220—31no—
121#1220—61no—
122#1220C2×D311yescharacteristic
123C2×Dic15—31no—
124C2×Dic15C2×D311yescharacteristic
125D60—31no—
126C4×D15—31no—
127C4×D15—31no—
128C4×D15—31no—
129Dic30—31no—
130Dic30—31no—
131C5×#67—31no—
132C5×#67—31no—
133C5×#67—31no—
134C5×#67—61no—
135C5×#67C2×D311yescharacteristic
136C2×C5×Dic3—31no—
137C2×C5×Dic3C2×D311yescharacteristic
138C5×D12—31no—
139C4×C5×D3—31no—
140C4×C5×D3—31no—
141C4×C5×D3—31no—
142C5×Dic6—31no—
143C5×Dic6—31no—
144C3×C22⋊D5—31no—
145C3×C22⋊D5—31no—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—31no—
148C2×C3×Dic5—31no—
149C2×C3×Dic5C2×D311yescharacteristic
150C2×C3×Dic5C2×D311yescharacteristic
151C3×C4×D5—31no—
152C3×Dic10—31no—
153C3×Dic10—31no—
154C3×Dic10—31no—
155C3×Dic10—31no—
156C3×Dic10—31no—
157#1204—31no—
158#1204—31no—
159#1204—31no—
160#1204—31no—
161#1204—31no—
162#1204—31no—
163#1204—61no—
164#1204C2×D311yescharacteristic
165#1204C2×D311yescharacteristic
166#1204C2×D311yescharacteristic
167#1204C2×D311yescharacteristic
168#1204C2×D311yescharacteristic
169#1203—31no—
170#1203—31no—
171#1203—31no—
172#1203—31no—
173#1203—61no—
174#1203C2×D311yescharacteristic
175#1202—31no—
176#1201—31no—
177#1200—31no—
178#1200—31no—
179#1200—31no—
180#1200—31no—
181#1200—31no—
182#1200—61no—
183#1200C2×D311yescharacteristic
184D3×Dic5—31no—
185D3×Dic5—31no—
186D3×Dic5—31no—
187D3×Dic5—31no—
188D3×Dic5—31no—
189D3×Dic5—61no—
190D3×Dic5C2×D311yescharacteristic
191D5×Dic3—31no—
192D5×Dic3—31no—
193#1010—151no—
194#1007—151no—
195C2×C32×C5—21no—
196C2×C32×C5C2411yescharacteristic
197C3⋊D15—21no—
198C5×C3⋊D3—21no—
199C3×D15—21no—
200C3×C5×D3—21no—
201C5×#41—91no—
202Q8×D5—91no—
203Q8×D5—91no—
204#711—31no—
205#711—91no—
206#711—91no—
207#711—91no—
208#711—91no—
209#711—91no—
210#709—91no—
211#709—91no—
212#709—91no—
213C2×Dic10—31no—
214C2×Dic10—91no—
215C2×Dic10—91no—
216C2×C32×C4—51no—
217#638—51no—
218#638C2×D511yescharacteristic
219C4⋊(C3⋊D3)—51no—
220C4×C3⋊D3—51no—
221C4×C3⋊D3—51no—
222#634—51no—
223C3×#67—51no—
224C3×#67C2×D511yescharacteristic
225C2×C3×Dic3—51no—
226C2×C3×Dic3C2×D511yescharacteristic
227C3×D12—51no—
228C3×C4×D3—51no—
229C3×C4×D3—51no—
230C3×Dic6—51no—
231C3×Dic6—51no—
232C3×Dic6—51no—
233C3×Dic6—51no—
234C3×Dic6—51no—
235#627—51no—
236#627—51no—
237#627—51no—
238#627C2×D511yescharacteristic
239#626—51no—
240#626—51no—
241#626—51no—
242#626C2×D511yescharacteristic
243#624—51no—
244#624—51no—
245#624—51no—
246#624C2×D511yescharacteristic
247D3×Dic3—51no—
248D3×Dic3—51no—
249D3×Dic3—51no—
250D3×Dic3C2×D511yescharacteristic
251C22×C3×C5—21no—
252C22×C3×C5—31no—
253C22×C3×C5C22×D311yescharacteristic
254C22×C3×C5C22×D311yescharacteristic
255C2×D15—31no—
256C2×D15—31no—
257C2×D15—61no—
258C2×D15C22×D311yescharacteristic
259C2×C5×D3—31no—
260C2×C5×D3—31no—
261C2×C5×D3—61no—
262C2×C5×D3C22×D311yescharacteristic
263C2×C3×D5—31no—
264C3×C4×C5—31no—
265C3×C4×C5—31no—
266C3×C4×C5—31no—
267Dic15—31no—
268Dic15—31no—
269Dic15—61no—
270Dic15C22×D311yescharacteristic
271Dic15C22×D311yescharacteristic
272Dic15C22×D311yescharacteristic
273C3×Dic5—21no—
274C3×Dic5—21no—
275C3×Dic5—31no—
276C3×Dic5—31no—
277C3×Dic5—31no—
278C3×Dic5C22×D311yescharacteristic
279C3×Dic5C22×D311yescharacteristic
280C3×Dic5C22×D311yescharacteristic
281C3×Dic5C22×D311yescharacteristic
282C5×Dic3—31no—
283C5×Dic3—31no—
284C5×Dic3—61no—
285C5×Dic3C22×D311yescharacteristic
286C5×Dic3C22×D311yescharacteristic
287C5×Dic3C22×D311yescharacteristic
288C3×#41—151no—
289C2×C3×Q8—151no—
290#239—151no—
291#239—151no—
292#239—151no—
293#239—151no—
294D3×Q8—151no—
295D3×Q8—151no—
296D3×Q8—151no—
297D3×Q8—151no—
298D3×Q8—151no—
299D3×Q8—151no—
300#237—31no—
301#237—151no—
302#237—151no—
303#237—151no—
304#237—151no—
305#237—151no—
306#235—31no—
307#235—51no—
308#235—151no—
309#235—151no—
310#235—151no—
311#235—151no—
312#235—151no—
313#235—151no—
314#235—151no—
315#235—301no—
316C2×Dic6—51no—
317C2×Dic6—151no—
318C2×Dic6—151no—
319C32×C52−1+411yescharacteristic, complemented, Hall
320C5×Q8—91no—
321C5×D4—31no—
322C5×D4—91no—
323C5×D4—91no—
324C2×C4×C5—31no—
325C2×C4×C5—91no—
326C2×C4×C5—91no—
327C22⋊D5—31no—
328C22⋊D5—31no—
329C22⋊D5—91no—
330C22⋊D5—91no—
331C22⋊D5—91no—
332C22⋊D5—91no—
333C2×Dic5—31no—
334C2×Dic5—31no—
335C2×Dic5—91no—
336C2×Dic5—91no—
337C2×Dic5—91no—
338C2×Dic5D3211yescharacteristic
339D20—91no—
340C4×D5—31no—
341C4×D5—91no—
342C4×D5—91no—
343C4×D5—91no—
344C4×D5—91no—
345C4×D5—91no—
346Dic10—31no—
347Dic10—31no—
348Dic10—31no—
349Dic10—31no—
350Dic10—31no—
351Dic10—91no—
352Dic10—91no—
353Dic10—91no—
354Dic10—91no—
355C22×C32C22×D511yescharacteristic
356C2×C3⋊D3—51no—
357C2×C3⋊D3C22×D511yescharacteristic
358C2×C3×D3—51no—
359C2×C3×D3C22×D511yescharacteristic
360C32×C4—51no—
361C32×C4—51no—
362C3⋊Dic3—51no—
363C3⋊Dic3C22×D511yescharacteristic
364C3×Dic3—51no—
365C3×Dic3—51no—
366C3×Dic3—51no—
367C3×Dic3C22×D511yescharacteristic
368C3×Dic3C22×D511yescharacteristic
369C3×Dic3C22×D511yescharacteristic
3702−1+4—451noSylow
371C2×C3×C5—21no—
372C2×C3×C5—21no—
373C2×C3×C5—21no—
374C2×C3×C5—41no—
375C2×C3×C5—61no—
376C2×C3×C5C23×D311yescharacteristic
377C2×C3×C5C23×D311yescharacteristic
378D15—21no—
379D15—61no—
380D15—61no—
381D15—121no—
382C3×D5—61no—
383C5×D3—21no—
384C5×D3—61no—
385C5×D3—61no—
386C5×D3—121no—
387C3×Q8—151no—
388C3×Q8—151no—
389C3×Q8—151no—
390C3×Q8—151no—
391C3×Q8—151no—
392C3×D4—31no—
393C3×D4—151no—
394C3×D4—151no—
395C2×C3×C4—31no—
396C2×C3×C4—51no—
397C2×C3×C4—51no—
398C2×C3×C4—101no—
399C2×C3×C4—151no—
400C2×C3×C4—151no—
401C2×C3×C4—151no—
402#67—31no—
403#67—31no—
404#67—31no—
405#67—51no—
406#67—61no—
407#67—151no—
408#67—151no—
409#67—151no—
410#67—151no—
411#67—151no—
412#67—301no—
413#67D3×D511yescharacteristic
414C2×Dic3—31no—
415C2×Dic3—51no—
416C2×Dic3—151no—
417C2×Dic3—151no—
418C2×Dic3—151no—
419C2×Dic3D3×D511yescharacteristic
420D12—31no—
421D12—51no—
422D12—151no—
423D12—151no—
424D12—151no—
425D12—151no—
426D12—151no—
427D12—301no—
428C4×D3—31no—
429C4×D3—31no—
430C4×D3—31no—
431C4×D3—51no—
432C4×D3—51no—
433C4×D3—151no—
434C4×D3—151no—
435C4×D3—151no—
436C4×D3—151no—
437C4×D3—151no—
438C4×D3—151no—
439C4×D3—151no—
440C4×D3—151no—
441C4×D3—151no—
442C4×D3—151no—
443C4×D3—151no—
444C4×D3—151no—
445C4×D3—151no—
446C4×D3—301no—
447C4×D3—301no—
448Dic6—31no—
449Dic6—31no—
450Dic6—51no—
451Dic6—51no—
452Dic6—51no—
453Dic6—51no—
454Dic6—51no—
455Dic6—151no—
456Dic6—151no—
457Dic6—151no—
458Dic6—151no—
459Dic6—151no—
460Dic6—151no—
461Dic6—151no—
462Dic6—151no—
463Dic6—301no—
464C22×C5—31no—
465C22×C5—91no—
466C22×C5C2×D3211yescharacteristic
467C2×D5—31no—
468C2×D5—91no—
469C4×C5—31no—
470C4×C5—31no—
471C4×C5—31no—
472C4×C5—91no—
473Dic5—31no—
474Dic5—31no—
475Dic5—31no—
476Dic5—91no—
477Dic5C2×D3211yescharacteristic
478Dic5C2×D3211yescharacteristic
479C2×C32—21no—
480C2×C32C23×D511yescharacteristic
481C3⋊D3—21no—
482C3⋊D3—101no—
483C3×D3—21no—
484C3×D3—101no—
485#41—91no—
486#41—151no—
487#41—451no—
488#41—451no—
489#41—451no—
490#41—451no—
491#41—451no—
492#41—451no—
493#41—451no—
494#41—451no—
495C2×Q8—151no—
496C2×Q8—451no—
497C2×Q8—451no—
498C2×Q8—451no—
499C2×Q8—451no—
500C3×C5—21no—
501C3×C5#100711yescharacteristic, complemented
502C3×C5#101011yescharacteristic, complemented
503C22×C3—21no—
504C22×C3—31no—
505C22×C3—151no—
506C22×C3C2×D3×D511yescharacteristic
507C22×C3C2×D3×D511yescharacteristic
508C2×D3—31no—
509C2×D3—31no—
510C2×D3—51no—
511C2×D3—61no—
512C2×D3—151no—
513C2×D3—151no—
514C2×D3—301no—
515C2×D3C2×D3×D511yescharacteristic
516C3×C4—31no—
517C3×C4—31no—
518C3×C4—31no—
519C3×C4—51no—
520C3×C4—51no—
521C3×C4—51no—
522C3×C4—51no—
523C3×C4—101no—
524C3×C4—101no—
525C3×C4—151no—
526C3×C4—151no—
527C3×C4—151no—
528Dic3—31no—
529Dic3—31no—
530Dic3—51no—
531Dic3—51no—
532Dic3—51no—
533Dic3—61no—
534Dic3—151no—
535Dic3—151no—
536Dic3—301no—
537Dic3C2×D3×D511yescharacteristic
538Dic3C2×D3×D511yescharacteristic
539Dic3C2×D3×D511yescharacteristic
540C2×C5—21no—
541C2×C5—61no—
542C2×C5—181no—
543C2×C5C22×D3211yescharacteristic
544D5—61no—
545D5—181no—
546C32#415011yescharacteristic, complemented, Sylow
547Q8—91no—
548Q8—151no—
549Q8—151no—
550Q8—151no—
551Q8—151no—
552Q8—151no—
553Q8—451no—
554Q8—451no—
555Q8—451no—
556Q8—451no—
557D4—31no—
558D4—91no—
559D4—91no—
560D4—151no—
561D4—151no—
562D4—451no—
563D4—451no—
564D4—451no—
565D4—451no—
566D4—451no—
567C2×C4—31no—
568C2×C4—51no—
569C2×C4—91no—
570C2×C4—91no—
571C2×C4—151no—
572C2×C4—151no—
573C2×C4—151no—
574C2×C4—451no—
575C2×C4—451no—
576C2×C4—451no—
577C2×C4—451no—
578C2×C4—451no—
579C2×C4—451no—
580C2×C4—451no—
581C2×C4—451no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—41no—
586C2×C3—61no—
587C2×C3—301no—
588C2×C3C22×D3×D511yescharacteristic
589C2×C3C22×D3×D511yescharacteristic
590D3—21no—
591D3—61no—
592D3—61no—
593D3—101no—
594D3—121no—
595D3—301no—
596D3—301no—
597D3—601no—
598C5#6429711yescharacteristic, complemented, Sylow
599C22—31no—
600C22—91no—
601C22—151no—
602C22—451no—
603C22D32×D511yescharacteristic
604C4—31no—
605C4—31no—
606C4—31no—
607C4—51no—
608C4—51no—
609C4—91no—
610C4—151no—
611C4—151no—
612C4—151no—
613C4—451no—
614C3—21no—
615C3#9200411yescharacteristic, complemented
616C3#9202011yescharacteristic, complemented
617C2—21no—
618C2—61no—
619C2—181no—
620C2—301no—
621C2—901no—
622C2C2×D32×D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D5
commutator, socle, cosocleC2×C32×C5C24
FittingC22×C32×C5C23
hypercenterC22D32×D5
hypocenterC32×C52−1+4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #10624230C2semidirect
2. #10624210C2semidirect
3. #10624207C2semidirect
4. C5×#3804C2semidirect
5. #10624165C2semidirect
6. C2×C3×#1204C2semidirect
7. C3×#6782C2semidirect
8. #10624056C2semidirect
9. #10624055C2semidirect
10. #10624054C2semidirect
11. #10624051C2semidirect
12. #10624040C2semidirect
13. #10624039C2semidirect
14. #10624038C2semidirect
15. D3×#1204C2semidirect
61. C2×#1204D3semidirect
68. #6782D3semidirect
107. #3804D5semidirect
319. C32×C52−1+4semidirect
501. C3×C5#1007semidirect
502. C3×C5#1010semidirect
546. C32#4150semidirect
598. C5#64297semidirect
615. C3#92004semidirect
616. C3#92020semidirect

Maximal subgroups

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

subgroupindex
#106242302
#106242102
#106242072
C5×#38042
#106241652
C2×C3×#12042
C3×#67822
#106240562
#106240552
#106240542
#106240512
#106240402
#106240392
#106240382
D3×#12042
#920203
#920043
#642975