Finite Group Databasebeta

← #14424730

Subgroups — #14424730

Order 1920.

1528 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92100C211yesmaximal, characteristic, complemented
2C5×#5833C211yesmaximal, characteristic, complemented
3C2×#91305C211yesmaximal, characteristic, complemented
4C2×#91303C211yesmaximal, characteristic, complemented
5C3×#65523C211yesmaximal, characteristic, complemented
6#11748566C212yesmaximal, complemented
7#11748554C212yesmaximal, complemented
8#11748526C212yesmaximal, complemented
9#11748525C212yesmaximal, complemented
10C2×#90969C211yesmaximal, characteristic, complemented
11#11748394C211yesmaximal, characteristic, complemented
12#10615155—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14#92100—24no—
15#92100—24no—
16C2×#6821—21no—
17C2×D4×D15—21no—
18C2×D4×D15—22no—
19C2×#6817C2212yes—
20C2×#6817C2211yescharacteristic
21C22×D60C2212yes—
22C22×D60C2211yescharacteristic
23C2×C5×#235C2211yescharacteristic, complemented
24C5×#946—21no—
25C5×#939—22no—
26C5×#938—21no—
27C5×#930C2212yescomplemented
28C2×C5×C4⋊Dic3C2211yescharacteristic, complemented
29C5×#924C2212yescomplemented
30C2×C3×#681C2211yescharacteristic, complemented
31C22×C3×#168C2211yescharacteristic, complemented
32C2×#6663C2211yescharacteristic, complemented
33C2×#6660C2211yescharacteristic, complemented
34C2×#6660C2212yescomplemented
35C2×#6659C2211yescharacteristic, complemented
36#91305—24no—
37#91303—24no—
38C2×#6654C2211yescharacteristic, complemented
39C2×#6654C2212yescomplemented
40C3×C22⋊#168C2212yescomplemented
41C3×C22⋊#168C2212yescomplemented
42#90975C2212yescomplemented
43#90969C2212yescomplemented
44#90969C2212yescomplemented
45#90969C2212yescomplemented
46#78672—51nomaximal, Hall
47C2×#4149—31no—
48C5×#537—31no—
49C2×#4096—31no—
50C2×#4095—31no—
51#65523D311yescharacteristic, complemented
52#65336—32no—
53#65335—32no—
54#65326—32no—
55#65325—32no—
56#65255—31no—
57C2×#3941—31no—
58C23×D15—22no—
59C23×D15—21no—
60C3×C5×#41—22no—
61C3×C5×#41—22no—
62C2×C3×C5×Q8D411yescharacteristic
63C2×C3×C5×D4D411yescharacteristic
64C2×C3×C5×D4D412yes—
65C22×C3×C4×C5C2312yes—
66C22×C3×C4×C5C2311yescharacteristic
67C2×#1220—22no—
68C2×#1220—21no—
69#6821—28no—
70D4×D15—216no—
71D4×D15—28no—
72#6817—22no—
73#6817—24no—
74#6817—24no—
75#6817D414yescomplemented
76C2×D60—22no—
77C2×D60—21no—
78C2×D60—22no—
79C2×D60—24no—
80C2×D60—24no—
81C2×D60D414yescomplemented
82C2×D60C2311yescharacteristic
83C2×D60C2312yes—
84C2×C4×D15—22no—
85C2×C4×D15—21no—
86C2×Dic30C2311yescharacteristic
87C2×C5×#67—21no—
88C2×C5×#67—22no—
89C22×C5×Dic3—21no—
90C5×#235—22no—
91C5×#235—22no—
92C2×C5×D12C2311yescharacteristic
93C2×C4×C5×D3—21no—
94C2×C5×Dic6C2311yescharacteristic
95C5×#217—22no—
96C5×(C2×D3)⋊C4—22no—
97C5×C4⋊Dic3—21no—
98C5×C4⋊Dic3C2312yes—
99C5×Dic3⋊C4—22no—
100C3×#681—22no—
101C2×C3×#168—22no—
102C2×C3×#168C2311yescharacteristic
103C2×C3×#168C2311yescharacteristic
104C2×C3×#168C2312yes—
105#6663—22no—
106#6660—22no—
107#6660—24no—
108#6659—22no—
109#6654—22no—
110#6654—24no—
111C2×#1009—51no—
112#5833D511yescharacteristic, complemented
113C2×#908—51no—
114C2×#904—51no—
115C3×#408—51no—
116#5133—51no—
117C2×#821—51no—
118#4734—52no—
119#4733—52no—
120#4732—52no—
121#4731—52no—
122C2×C5×#41—31no—
123C2×C5×#41C2×D311yescharacteristic
124#4149—64no—
125#4149—64no—
126C2×#713—61no—
127C2×D4×D5—62no—
128C2×D4×D5—61no—
129C2×#709—31no—
130C2×#709—32no—
131C22×D20—31no—
132C22×D20—32no—
133C5×C22⋊Q8—32no—
134C5×C22⋊Q8—61no—
135C5×#121—32no—
136C5×#121—61no—
137C5×#121—62no—
138C2×C5×C4⋊C4—31no—
139#4096—64no—
140#4095—64no—
141C2×#689—31no—
142C2×#688—31no—
143C2×#688—32no—
144C2×#687—31no—
145C2×#686—31no—
146C2×#686—32no—
147C2×#681C2×D311yescharacteristic, complemented
148C22×#168C2×D311yescharacteristic, complemented
149C22⋊#168C2×D312yescomplemented
150C22⋊#168C2×D312yescomplemented
151#3942—32no—
152#3941—32no—
153#3941—32no—
154#3941—32no—
155#3012—151noSylow
156C23×C3×C5C2×D412yes—
157C23×C3×C5C2×D411yescharacteristic
158C22×D15—22no—
159C22×D15—21no—
160C22×D15—28no—
161C22×D15—24no—
162C22×D15—81no—
163C22×D15—82no—
164C22×D15—81no—
165C22×D15—82no—
166C22×C5×D3—21no—
167C3×C5×Q8—22no—
168C3×C5×D4—24no—
169C3×C5×D4—22no—
170C2×C3×C4×C5—22no—
171C2×C3×C4×C5—22no—
172C2×C3×C4×C5C2×D412yes—
173C2×C3×C4×C5C2×D411yescharacteristic
174C2×C3×C4×C5C2×D412yes—
175C2×C3×C4×C5C2×D412yes—
176C2×C3×C4×C5C2411yescharacteristic
177#1220—28no—
178#1220—24no—
179C2×Dic15—21no—
180D60—28no—
181D60—24no—
182D60—21no—
183D60—22no—
184D60C2×D412yes—
185D60C2×D414yes—
186C4×D15—28no—
187C4×D15—24no—
188Dic30—21no—
189Dic30C2×D412yes—
190C5×#67—42no—
191C5×#67—82no—
192C2×C5×Dic3—21no—
193C2×C5×Dic3—22no—
194C2×C5×Dic3—81no—
195C5×D12—21no—
196C5×D12—21no—
197C4×C5×D3—42no—
198C5×Dic6—21no—
199C5×Dic6—21no—
200C3×#168—21no—
201C3×#168—22no—
202C3×#168—21no—
203C2×C3×#41C2×D511yescharacteristic
204#1009—104no—
205#1009—104no—
206C2×#239—101no—
207C2×D3×D4—101no—
208C2×D3×D4—102no—
209C2×#235—51no—
210C2×#235—52no—
211C2×#235C2×D511yescharacteristic, complemented
212C22×D12—51no—
213C22×D12—52no—
214C2×C3×M16—51no—
215C22×C3×C8—51no—
216#946—21no—
217#939—22no—
218#938—21no—
219#930C2×D512yescomplemented
220C2×C4⋊Dic3C2×D511yescharacteristic, complemented
221#924C2×D512yescomplemented
222#908—104no—
223C2×Dic12—51no—
224#904—104no—
225C2×D24—51no—
226C2×D24—52no—
227C2×#204—51no—
228C2×#204—51no—
229C2×#204—52no—
230C3×C22⋊C8—52no—
231C3×C22⋊C8—52no—
232#821—52no—
233#821—52no—
234#821—52no—
235#816—52no—
236C23×D5—62no—
237C23×D5—61no—
238C5×#41—22no—
239C5×#41—22no—
240C5×#41—62no—
241C5×#41—62no—
242C2×C5×Q8—31no—
243C2×C5×Q8D1211yescharacteristic
244C2×C5×D4—31no—
245C2×C5×D4—61no—
246C2×C5×D4—62no—
247C2×C5×D4D1212yes—
248C2×C5×D4D1211yescharacteristic
249C22×C4×C5—61no—
250C22×C4×C5—61no—
251C22×C4×C5C22×D312yes—
252C22×C4×C5C22×D311yescharacteristic
253C2×C22⋊D5—62no—
254C2×C22⋊D5—61no—
255#713—68no—
256D4×D5—616no—
257D4×D5—68no—
258#709—34no—
259#709—64no—
260#709—64no—
261#709—62no—
262C2×D20—34no—
263C2×D20—32no—
264C2×D20—31no—
265C2×D20—64no—
266C2×D20—64no—
267C2×D20—62no—
268C2×D20—61no—
269C2×D20—62no—
270C2×C4×D5—62no—
271C2×C4×D5—61no—
272C2×Dic10—31no—
273C5×C4⋊C4—32no—
274C5×C4⋊C4—61no—
275C5×C4⋊C4—62no—
276C5×C22⋊C4—62no—
277C5×C22⋊C4—62no—
278#689—62no—
279#688—62no—
280#688—64no—
281#687—62no—
282#686—62no—
283#686—64no—
284#681—22no—
285C2×#168—22no—
286C2×#168C22×D311yescharacteristic
287C2×#168C22×D311yescharacteristic
288C2×#168C22×D312yes—
289C2×H(5,𝔽2)—151no—
290C2×AGL(1,ℤ/8)—151no—
291C2×#135—151no—
292#537—31no—
293#452—152no—
294#450—152no—
295#449—152no—
296#447—152no—
297#417—151no—
298C2×#102—151no—
299#408—51no—
300C22×C3×C5—21no—
301C22×C3×C5—21no—
302C22×C3×C5—22no—
303C22×C3×C5—22no—
304C22×C3×C5—22no—
305C22×C3×C5#12012yes—
306C22×C3×C5C22×D411yescharacteristic
307C2×D15—24no—
308C2×D15—22no—
309C2×D15—42no—
310C2×D15—41no—
311C2×D15—42no—
312C2×D15—41no—
313C2×D15—84no—
314C2×D15—88no—
315C2×C5×D3—41no—
316C2×C5×D3—41no—
317C2×C5×D3—41no—
318C3×C4×C5—22no—
319C3×C4×C5#12012yes—
320C3×C4×C5C22×D411yescharacteristic
321C3×C4×C5C22×D411yescharacteristic
322Dic15—22no—
323C5×Dic3—41no—
324C5×Dic3—42no—
325C23×D3—102no—
326C23×D3—101no—
327C3×#41—22no—
328C3×#41—22no—
329C2×C3×Q8C22⋊D511yescharacteristic
330C2×C3×D4C22⋊D511yescharacteristic
331C2×C3×D4C22⋊D512yes—
332C22×C3×C4C22×D512yes—
333C22×C3×C4C22×D511yescharacteristic
334C2×#67—21no—
335C2×#67—22no—
336C2×#67—102no—
337C2×#67—101no—
338C22×Dic3—21no—
339#239—108no—
340D3×D4—1016no—
341D3×D4—108no—
342#235—22no—
343#235—22no—
344#235—54no—
345#235—104no—
346#235—104no—
347#235—102no—
348C2×D12—54no—
349C2×D12—52no—
350C2×D12—51no—
351C2×D12—104no—
352C2×D12—104no—
353C2×D12—102no—
354C2×D12—102no—
355C2×D12—101no—
356C2×D12C22×D511yescharacteristic
357C2×C4×D3—21no—
358C2×C4×D3—102no—
359C2×C4×D3—101no—
360C2×Dic6—51no—
361C2×Dic6C22×D511yescharacteristic
362C3×M16—102no—
363C2×C3×C8—52no—
364C2×C3×C8—51no—
365C2×C3×C8—51no—
366C2×C3×C8—102no—
367#217—22no—
368(C2×D3)⋊C4—22no—
369C4⋊Dic3—21no—
370C4⋊Dic3C22×D512yes—
371Dic3⋊C4—22no—
372Dic12—102no—
373D24—102no—
374D24—104no—
375#204—102no—
376#204—104no—
377#204—102no—
378C23×C5—61no—
379C23×C5C2×D1212yes—
380C23×C5C2×D1211yescharacteristic
381C22×D5—62no—
382C22×D5—61no—
383C22×D5—68no—
384C22×D5—64no—
385C22×D5—241no—
386C22×D5—242no—
387C22×D5—241no—
388C22×D5—242no—
389C5×Q8—22no—
390C5×Q8—61no—
391C5×Q8—61no—
392C5×D4—24no—
393C5×D4—22no—
394C5×D4—61no—
395C5×D4—61no—
396C5×D4—122no—
397C5×D4—242no—
398C2×C4×C5—22no—
399C2×C4×C5—22no—
400C2×C4×C5—62no—
401C2×C4×C5—61no—
402C2×C4×C5—122no—
403C2×C4×C5—241no—
404C2×C4×C5C2×D1212yes—
405C2×C4×C5C2×D1211yescharacteristic
406C2×C4×C5D3×D412yes—
407C2×C4×C5D3×D412yes—
408C2×C4×C5C23×D311yescharacteristic
409C22⋊D5—68no—
410C22⋊D5—64no—
411C2×Dic5—61no—
412D20—34no—
413D20—32no—
414D20—68no—
415D20—64no—
416D20—62no—
417D20—61no—
418C4×D5—68no—
419C4×D5—64no—
420Dic10—32no—
421Dic10—61no—
422#168—21no—
423#168—21no—
424#168—22no—
425H(5,𝔽2)—304no—
426H(5,𝔽2)—304no—
427C2×#41—31no—
428C2×#41—151no—
429C2×#41—152no—
430C2×#41—301no—
431C2×#41D3×D511yescharacteristic
432C22×D4—151no—
433C22×D4—152no—
434C22×D4—302no—
435C22×D4—301no—
436AGL(1,ℤ/8)—304no—
437#135—304no—
438C2×Q16—151no—
439C2×SD16—151no—
440C2×SD16—151no—
441C2×SD16—152no—
442C2×D8—151no—
443C2×D8—152no—
444C2×M16—51no—
445C22×C8—51no—
446C22⋊Q8—32no—
447C22⋊Q8—61no—
448#121—32no—
449#121—61no—
450#121—62no—
451C2×C4⋊C4—31no—
452#103—152no—
453#102—152no—
454#102—152no—
455#102—152no—
456C22⋊C8—52no—
457C22⋊C8—52no—
458C2×C3×C5—22no—
459C2×C3×C5—42no—
460C2×C3×C5C2×#12011yescharacteristic
461C2×C3×C5#57611yescharacteristic
462C2×C3×C5#57711yescharacteristic
463D15—44no—
464D15—42no—
465C5×D3—81no—
466C23×C3C2×C22⋊D512yes—
467C23×C3C2×C22⋊D511yescharacteristic
468C22×D3—21no—
469C22×D3—102no—
470C22×D3—101no—
471C22×D3—108no—
472C22×D3—104no—
473C22×D3—401no—
474C22×D3—402no—
475C22×D3—401no—
476C22×D3—402no—
477C3×Q8—22no—
478C3×D4—24no—
479C3×D4—22no—
480C2×C3×C4—22no—
481C2×C3×C4—22no—
482C2×C3×C4D4×D512yes—
483C2×C3×C4D4×D512yes—
484C2×C3×C4C2×C22⋊D512yes—
485C2×C3×C4C2×C22⋊D511yescharacteristic
486C2×C3×C4C23×D511yescharacteristic
487#67—42no—
488#67—82no—
489#67—108no—
490#67—104no—
491C2×Dic3—21no—
492C2×Dic3—22no—
493C2×Dic3—81no—
494C2×Dic3—101no—
495D12—21no—
496D12—21no—
497D12—54no—
498D12—52no—
499D12—108no—
500D12—102no—
501D12—101no—
502D12—104no—
503C4×D3—42no—
504C4×D3—108no—
505C4×D3—104no—
506Dic6—21no—
507Dic6—21no—
508Dic6—52no—
509Dic6—101no—
510C3×C8—101no—
511C3×C8—102no—
512C3×C8—101no—
513C22×C5—21no—
514C22×C5—21no—
515C22×C5—22no—
516C22×C5—22no—
517C22×C5—22no—
518C22×C5—121no—
519C22×C5—121no—
520C22×C5—121no—
521C22×C5#88212yes—
522C22×C5C22×D1211yescharacteristic
523C2×D5—64no—
524C2×D5—62no—
525C2×D5—122no—
526C2×D5—121no—
527C2×D5—122no—
528C2×D5—121no—
529C2×D5—244no—
530C2×D5—248no—
531C4×C5—22no—
532C4×C5—121no—
533C4×C5—122no—
534C4×C5#88212yes—
535C4×C5C2×D3×D411yescharacteristic
536C4×C5C2×D3×D411yescharacteristic
537Dic5—62no—
538C24—302no—
539C24—301no—
540#41—22no—
541#41—22no—
542#41—62no—
543#41—62no—
544#41—154no—
545#41—304no—
546#41—304no—
547#41—308no—
548#41—302no—
549C2×Q8—31no—
550C2×Q8—151no—
551C2×Q8#120311yescharacteristic
552C2×D4—31no—
553C2×D4—61no—
554C2×D4—62no—
555C2×D4—154no—
556C2×D4—152no—
557C2×D4—151no—
558C2×D4—304no—
559C2×D4—304no—
560C2×D4—3016no—
561C2×D4—308no—
562C2×D4—302no—
563C2×D4—301no—
564C2×D4—302no—
565C2×D4—301no—
566C2×D4—302no—
567C2×D4#120312yes—
568C2×D4#120311yescharacteristic
569C22×C4—61no—
570C22×C4—61no—
571C22×C4—302no—
572C22×C4—301no—
573C22×C4C2×D3×D512yes—
574C22×C4C2×D3×D511yescharacteristic
575Q16—302no—
576SD16—302no—
577SD16—302no—
578SD16—304no—
579D8—302no—
580D8—304no—
581M16—102no—
582C2×C8—51no—
583C2×C8—51no—
584C2×C8—52no—
585C2×C8—102no—
586C4⋊C4—32no—
587C4⋊C4—61no—
588C4⋊C4—62no—
589C22⋊C4—62no—
590C22⋊C4—62no—
591C3×C5#301211yescharacteristic, complemented, Hall
592C22×C3—21no—
593C22×C3—21no—
594C22×C3—22no—
595C22×C3—22no—
596C22×C3—22no—
597C22×C3#408312yes—
598C22×C3C22×C22⋊D511yescharacteristic
599C2×D3—41no—
600C2×D3—41no—
601C2×D3—41no—
602C2×D3—104no—
603C2×D3—102no—
604C2×D3—202no—
605C2×D3—201no—
606C2×D3—202no—
607C2×D3—201no—
608C2×D3—404no—
609C2×D3—408no—
610C3×C4—22no—
611C3×C4#408312yes—
612C3×C4C2×D4×D511yescharacteristic
613C3×C4C2×D4×D511yescharacteristic
614Dic3—41no—
615Dic3—42no—
616Dic3—102no—
617C2×C5—22no—
618C2×C5—42no—
619C2×C5—241no—
620C2×C5C2×#88211yescharacteristic
621C2×C5#575111yescharacteristic
622C2×C5#575211yescharacteristic
623D5—124no—
624D5—122no—
625C23—61no—
626C23—308no—
627C23—302no—
628C23—301no—
629C23—304no—
630C23—1201no—
631C23—1201no—
632C23—1202no—
633C23—1202no—
634C23C2×#120312yes—
635C23C2×#120311yescharacteristic
636Q8—22no—
637Q8—61no—
638Q8—61no—
639Q8—152no—
640Q8—301no—
641D4—24no—
642D4—22no—
643D4—61no—
644D4—61no—
645D4—122no—
646D4—154no—
647D4—152no—
648D4—242no—
649D4—308no—
650D4—308no—
651D4—304no—
652D4—301no—
653D4—302no—
654D4—304no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—62no—
658C2×C4—61no—
659C2×C4—122no—
660C2×C4—241no—
661C2×C4—308no—
662C2×C4—301no—
663C2×C4—304no—
664C2×C4C2×#120312yes—
665C2×C4C2×#120311yescharacteristic
666C2×C4#679012yes—
667C2×C4#679012yes—
668C2×C4C22×D3×D511yescharacteristic
669C8—101no—
670C8—101no—
671C8—102no—
672C2×C3—22no—
673C2×C3—42no—
674C2×C3C2×#408311yescharacteristic
675C2×C3#6616911yescharacteristic
676C2×C3#6617111yescharacteristic
677D3—81no—
678D3—204no—
679D3—202no—
680C5#7867211yescharacteristic, complemented, Sylow
681C22—21no—
682C22—21no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—121no—
687C22—121no—
688C22—121no—
689C22—304no—
690C22—302no—
691C22—602no—
692C22—601no—
693C22—602no—
694C22—601no—
695C22—1204no—
696C22—1208no—
697C22#9157312yes—
698C22C22×#120311yescharacteristic
699C4—22no—
700C4—122no—
701C4—121no—
702C4—302no—
703C4#9157312yes—
704C4C2×#679011yescharacteristic
705C4C2×#679011yescharacteristic
706C3#1061515511yescharacteristic, complemented, Sylow
707C2—22no—
708C2—42no—
709C2—241no—
710C2—604no—
711C2—602no—
712C2#1175530511yescharacteristic
713C2#1175530811yescharacteristic
714C2C2×#9157311yescharacteristic
subgroupG/H
centerC22C22×#1203
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
hypercenterC2×#41D3×D5
hypocenterC3×C5#3012
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#92100C2semidirect
2. C5×#5833C2semidirect
3. C2×#91305C2semidirect
4. C2×#91303C2semidirect
5. C3×#65523C2semidirect
6. #11748566C2semidirect
7. #11748554C2semidirect
8. #11748526C2semidirect
9. #11748525C2semidirect
10. C2×#90969C2semidirect
11. #11748394C2semidirect
23. C2×C5×#235C22semidirect
27. C5×#930C22semidirect
28. C2×C5×C4⋊Dic3C22semidirect
29. C5×#924C22semidirect
30. C2×C3×#681C22semidirect
31. C22×C3×#168C22semidirect
32. C2×#6663C22semidirect
33. C2×#6660C22semidirect
34. C2×#6660C22semidirect
35. C2×#6659C22semidirect
38. C2×#6654C22semidirect
39. C2×#6654C22semidirect
40. C3×C22⋊#168C22semidirect
41. C3×C22⋊#168C22semidirect
42. #90975C22semidirect
43. #90969C22semidirect
44. #90969C22semidirect
45. #90969C22semidirect
51. #65523D3semidirect
75. #6817D4semidirect
81. C2×D60D4semidirect
112. #5833D5semidirect
147. C2×#681C2×D3semidirect
148. C22×#168C2×D3semidirect
149. C22⋊#168C2×D3semidirect
150. C22⋊#168C2×D3semidirect
211. C2×#235C2×D5semidirect
219. #930C2×D5semidirect
220. C2×C4⋊Dic3C2×D5semidirect
221. #924C2×D5semidirect
591. C3×C5#3012semidirect
680. C5#78672semidirect
706. C3#10615155semidirect

Maximal subgroups

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

subgroupindex
C2×#921002
C5×#58332
C2×#913052
C2×#913032
C3×#655232
#117485662
#117485542
#117485262
#117485252
C2×#909692
#117483942
#106151553
#786725