Finite Group Databasebeta

← #14424668

Subgroups — #14424668

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×#92095C211yesmaximal, characteristic, complemented
2C2×#91821C211yesmaximal, characteristic, complemented
3C2×#91793C211yesmaximal, characteristic, complemented
4#11756269C211yesmaximal, characteristic, complemented
5C2×C3×C5×#102C211yesmaximal, characteristic, complemented
6#11751211C211yesmaximal, characteristic, complemented
7#11751067C211yesmaximal, characteristic, complemented
8#11750884C212yesmaximal, complemented
9#11750882C212yesmaximal, complemented
10#11750877C212yesmaximal, complemented
11#11750874C212yesmaximal, complemented
12#10610478—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C22×#1220—22no—
15#92095—24no—
16#92095—24no—
17C2×#6819—21no—
18C2×#6819C2212yes—
19C2×D4×D15—21no—
20C2×D4×D15C2212yes—
21C2×#6817C2211yescharacteristic
22C2×#6817C2211yescharacteristic
23C2×#6817C2211yescharacteristic, complemented
24C22×C3×C5×C8C2211yescharacteristic, complemented
25C2×C3×C5×C4⋊C4C2211yescharacteristic, complemented
26C2×#6716C2212yescomplemented
27#91821—24no—
28C2×#6715C2212yescomplemented
29#91819—22no—
30C2×#6710C2211yescharacteristic, complemented
31C2×Dic60C2211yescharacteristic, complemented
32#91793—24no—
33C2×D120C2211yescharacteristic, complemented
34C2×#6706C2211yescharacteristic, complemented
35C2×#6706C2211yescharacteristic, complemented
36#91786—21no—
37#91786C2212yescomplemented
38#91784—21no—
39#91784C2212yescomplemented
40C3×C5×#102C2212yescomplemented
41C3×C5×#102C2212yescomplemented
42#91104C2212yescomplemented
43#91104C2212yescomplemented
44#91094C2212yescomplemented
45#91093C2212yescomplemented
46#78747—51nomaximal, Hall
47C2×#4144—31no—
48C2×#4078—31no—
49C2×#4050—31no—
50#65844—31no—
51C2×C5×#102D311yescharacteristic, complemented
52#65399—31no—
53#65255—31no—
54#65072—32no—
55#65070—32no—
56#65065—32no—
57#65062—32no—
58C24×C3×C5—21no—
59C23×D15—22no—
60C2×C3×C5×D4—21no—
61C2×C3×C5×D4—22no—
62C2×C3×C5×D4D414yescomplemented
63C2×C3×C5×D4C2312yes—
64C22×C3×C4×C5—21no—
65C22×C3×C4×C5C2311yescharacteristic
66C2×#1220—216no—
67C2×#1220—21no—
68C2×#1220—22no—
69C2×#1220—21no—
70C2×#1220—22no—
71C2×#1220D412yes—
72C22×Dic15—22no—
73#6819—28no—
74#6819—24no—
75#6819—24no—
76D4×D15—28no—
77D4×D15—24no—
78D4×D15—24no—
79#6817—22no—
80#6817—22no—
81#6817—22no—
82#6817—22no—
83#6817—22no—
84#6817D414yescomplemented
85C2×D60—22no—
86C2×D60D411yescharacteristic
87C2×D60C2311yescharacteristic
88C2×D60C2311yescharacteristic
89C2×C4×D15—21no—
90C2×C4×D15—21no—
91C2×C4×D15C2312yes—
92C2×Dic30D411yescharacteristic
93C2×Dic30C2311yescharacteristic
94C2×Dic30C2311yescharacteristic
95C2×C3×C5×C8—22no—
96C2×C3×C5×C8C2311yescharacteristic
97C2×C3×C5×C8C2311yescharacteristic
98C3×C5×C4⋊C4—21no—
99C3×C5×C4⋊C4C2312yes—
100#6716—24no—
101#6715—24no—
102#6714—22no—
103#6714—22no—
104C4⋊Dic15—22no—
105#6710—22no—
106C2×C3⋊#168C2312yes—
107Dic60—22no—
108D120—22no—
109#6706—22no—
110#6706—22no—
111C2×#1004—51no—
112C2×#932—51no—
113C2×#904—51no—
114#5510—51no—
115C2×C3×#102D511yescharacteristic, complemented
116#5110—51no—
117#4966—51no—
118#4783—52no—
119#4781—52no—
120#4776—52no—
121#4773—52no—
122C22×C5×D4C2×D311yescharacteristic
123C22×C22⋊D5—62no—
124#4144—64no—
125#4144—64no—
126C2×#711—32no—
127C2×#711—61no—
128C2×D4×D5—32no—
129C2×D4×D5—61no—
130C2×#709—31no—
131C2×#709—31no—
132C2×#709—31no—
133C22×C5×C8C2×D311yescharacteristic, complemented
134C2×C5×C4⋊C4C2×D311yescharacteristic, complemented
135C2×#687—32no—
136#4078—64no—
137C2×#686—32no—
138#4076—62no—
139C2×#681—31no—
140C2×Dic20—31no—
141#4050—64no—
142C2×D40—31no—
143C2×#677—31no—
144C2×#677—31no—
145#4043—32no—
146#4043—61no—
147#4041—32no—
148#4041—61no—
149C5×#102C2×D312yescomplemented
150C5×#102C2×D312yescomplemented
151#3952—32no—
152#3952—32no—
153#3942—32no—
154#3941—32no—
155#3012—151noSylow
156C23×C3×C5—22no—
157C23×C3×C5—24no—
158C23×C3×C5—81no—
159C23×C3×C5—81no—
160C23×C3×C5C2×D411yescharacteristic
161C22×D15—21no—
162C22×D15—28no—
163C22×D15—21no—
164C22×D15—82no—
165C22×D15—82no—
166C22×D15C2×D412yes—
167C3×C5×D4—24no—
168C3×C5×D4—22no—
169C3×C5×D4C2×D414yes—
170C2×C3×C4×C5—22no—
171C2×C3×C4×C5—81no—
172C2×C3×C4×C5C2×D411yescharacteristic
173C2×C3×C4×C5C2×D412yes—
174C2×C3×C4×C5C2×D412yes—
175C2×C3×C4×C5C2411yescharacteristic
176#1220—28no—
177#1220—28no—
178#1220—24no—
179#1220—24no—
180#1220—42no—
181C2×Dic15—28no—
182C2×Dic15—21no—
183C2×Dic15—21no—
184C2×Dic15C2×D412yes—
185D60—22no—
186D60—21no—
187D60—21no—
188D60—21no—
189D60—82no—
190D60C2×D412yes—
191C4×D15—24no—
192C4×D15—22no—
193C4×D15—22no—
194C4×D15—42no—
195Dic30—22no—
196Dic30—21no—
197Dic30—21no—
198Dic30—21no—
199Dic30C2×D412yes—
200C3×C5×C8—21no—
201C3×C5×C8—21no—
202C3⋊#168—22no—
203C22×C3×D4C2×D511yescharacteristic
204C22×#67—102no—
205#1004—104no—
206#1004—104no—
207C2×#237—52no—
208C2×#237—101no—
209C2×D3×D4—52no—
210C2×D3×D4—101no—
211C2×#235—51no—
212C2×#235—51no—
213C2×#235—51no—
214C22×C3×C8C2×D511yescharacteristic, complemented
215C2×C3×C4⋊C4C2×D511yescharacteristic, complemented
216C2×#214—52no—
217#932—104no—
218C2×#213—52no—
219#930—102no—
220C2×#208—51no—
221C2×Dic12—51no—
222#904—104no—
223C2×D24—51no—
224C2×#204—51no—
225C2×#204—51no—
226#897—52no—
227#897—101no—
228#895—52no—
229#895—101no—
230C3×#102C2×D512yescomplemented
231C3×#102C2×D512yescomplemented
232(C2×D3)⋊C8—52no—
233(C2×D3)⋊C8—52no—
234#810—52no—
235#809—52no—
236C24×C5—21no—
237C23×D5—62no—
238C2×C5×D4—21no—
239C2×C5×D4—22no—
240C2×C5×D4D1214yescomplemented
241C2×C5×D4C22×D312yes—
242C22×C4×C5—21no—
243C22×C4×C5C22×D311yescharacteristic
244C2×C22⋊D5—32no—
245C2×C22⋊D5—62no—
246C2×C22⋊D5—62no—
247C2×C22⋊D5—616no—
248C2×C22⋊D5—61no—
249C2×C22⋊D5—61no—
250C22×Dic5—62no—
251#711—64no—
252#711—68no—
253#711—64no—
254D4×D5—64no—
255D4×D5—68no—
256D4×D5—64no—
257#709—34no—
258#709—62no—
259#709—62no—
260#709—62no—
261#709—62no—
262#709—62no—
263C2×D20—31no—
264C2×D20—31no—
265C2×D20—31no—
266C2×D20—62no—
267C2×C4×D5—32no—
268C2×C4×D5—61no—
269C2×C4×D5—61no—
270C2×Dic10—31no—
271C2×Dic10—31no—
272C2×Dic10—31no—
273C2×C5×C8—22no—
274C2×C5×C8C22×D311yescharacteristic
275C2×C5×C8C22×D311yescharacteristic
276C5×C4⋊C4—21no—
277C5×C4⋊C4C22×D312yes—
278#687—64no—
279#686—64no—
280#685—62no—
281#685—62no—
282C4⋊Dic5—62no—
283#681—62no—
284C2×#168—32no—
285Dic20—62no—
286D40—62no—
287#677—62no—
288#677—62no—
289C2×H(5,𝔽2)—151no—
290C2×AGL(1,ℤ/8)—151no—
291C2×#135—151no—
292#537—151no—
293#452—152no—
294#450—152no—
295#449—152no—
296#447—152no—
297#417—151no—
298C2×#102D1511yescharacteristic, complemented
299#408—151no—
300C22×C3×C5—21no—
301C22×C3×C5—21no—
302C22×C3×C5—24no—
303C22×C3×C5—42no—
304C22×C3×C5—42no—
305C22×C3×C5—84no—
306C22×C3×C5#12012yes—
307C22×C3×C5C22×D411yescharacteristic
308C2×D15—22no—
309C2×D15—22no—
310C2×D15—22no—
311C2×D15—22no—
312C2×D15—41no—
313C2×D15—41no—
314C2×D15—41no—
315C2×D15—41no—
316C2×D15—41no—
317C2×D15—88no—
318C3×C4×C5—42no—
319C3×C4×C5#12012yes—
320C3×C4×C5C22×D411yescharacteristic
321C3×C4×C5C22×D411yescharacteristic
322Dic15—22no—
323Dic15—22no—
324Dic15—41no—
325C24×C3—21no—
326C23×D3—102no—
327C2×C3×D4—21no—
328C2×C3×D4—22no—
329C2×C3×D4D2014yescomplemented
330C2×C3×D4C22×D512yes—
331C22×C3×C4—21no—
332C22×C3×C4C22×D511yescharacteristic
333C2×#67—52no—
334C2×#67—102no—
335C2×#67—102no—
336C2×#67—1016no—
337C2×#67—101no—
338C2×#67—101no—
339C22×Dic3—102no—
340#237—104no—
341#237—108no—
342#237—104no—
343D3×D4—104no—
344D3×D4—108no—
345D3×D4—104no—
346#235—54no—
347#235—102no—
348#235—102no—
349#235—102no—
350#235—102no—
351#235—102no—
352C2×D12—51no—
353C2×D12—51no—
354C2×D12—51no—
355C2×D12—102no—
356C2×C4×D3—52no—
357C2×C4×D3—101no—
358C2×C4×D3—101no—
359C2×Dic6—51no—
360C2×Dic6—51no—
361C2×Dic6—51no—
362C2×C3×C8—22no—
363C2×C3×C8C22×D511yescharacteristic
364C2×C3×C8C22×D511yescharacteristic
365C3×C4⋊C4—21no—
366C3×C4⋊C4C22×D512yes—
367#214—104no—
368#213—104no—
369(C2×D3)⋊C4—102no—
370(C2×D3)⋊C4—102no—
371C4⋊Dic3—102no—
372#208—102no—
373C2×C3⋊C8—52no—
374Dic12—102no—
375D24—102no—
376#204—102no—
377#204—102no—
378C23×C5—22no—
379C23×C5—24no—
380C23×C5—81no—
381C23×C5—81no—
382C23×C5D3×D411yescharacteristic
383C22×D5—32no—
384C22×D5—61no—
385C22×D5—68no—
386C22×D5—61no—
387C22×D5—242no—
388C22×D5—242no—
389C5×D4—24no—
390C5×D4—22no—
391C5×D4C2×D1214yes—
392C2×C4×C5—22no—
393C2×C4×C5—81no—
394C2×C4×C5C2×D1212yes—
395C2×C4×C5D3×D411yescharacteristic
396C2×C4×C5D3×D412yes—
397C2×C4×C5C23×D311yescharacteristic
398C22⋊D5—68no—
399C22⋊D5—68no—
400C22⋊D5—64no—
401C22⋊D5—64no—
402C22⋊D5—122no—
403C2×Dic5—32no—
404C2×Dic5—68no—
405C2×Dic5—61no—
406C2×Dic5—61no—
407D20—32no—
408D20—62no—
409D20—61no—
410D20—61no—
411D20—61no—
412D20—242no—
413C4×D5—64no—
414C4×D5—62no—
415C4×D5—62no—
416C4×D5—122no—
417Dic10—32no—
418Dic10—62no—
419Dic10—61no—
420Dic10—61no—
421Dic10—61no—
422C5×C8—21no—
423C5×C8—21no—
424#168—62no—
425H(5,𝔽2)—304no—
426H(5,𝔽2)—304no—
427C2×#41—151no—
428C2×#41—152no—
429C2×#41—151no—
430C2×#41—151no—
431C2×#41—301no—
432C22×D4—152no—
433C22×D4—301no—
434C22×D4—302no—
435C22×D4C2×D1511yescharacteristic
436AGL(1,ℤ/8)—304no—
437#135—304no—
438C2×Q16—151no—
439C2×SD16—152no—
440C2×SD16—151no—
441C2×SD16—151no—
442C2×D8—152no—
443C2×D8—151no—
444C2×M16—151no—
445C22×C8C2×D1511yescharacteristic, complemented
446C22⋊Q8—152no—
447C22⋊Q8—301no—
448#121—152no—
449#121—301no—
450#121—302no—
451C2×C4⋊C4C2×D1511yescharacteristic, complemented
452#103—152no—
453#102—152no—
454#102C2×D1512yescomplemented
455#102C2×D1512yescomplemented
456C22⋊C8—152no—
457C22⋊C8—152no—
458C2×C3×C5—22no—
459C2×C3×C5—44no—
460C2×C3×C5C2×#12011yescharacteristic
461C2×C3×C5#57611yescharacteristic
462C2×C3×C5#57711yescharacteristic
463D15—42no—
464D15—42no—
465D15—81no—
466C23×C3—22no—
467C23×C3—24no—
468C23×C3—81no—
469C23×C3—81no—
470C23×C3D4×D511yescharacteristic
471C22×D3—52no—
472C22×D3—101no—
473C22×D3—108no—
474C22×D3—101no—
475C22×D3—402no—
476C22×D3—402no—
477C3×D4—22no—
478C3×D4—24no—
479C3×D4C2×D2014yes—
480C2×C3×C4—22no—
481C2×C3×C4—81no—
482C2×C3×C4C2×D2012yes—
483C2×C3×C4D4×D512yes—
484C2×C3×C4D4×D511yescharacteristic
485C2×C3×C4C23×D511yescharacteristic
486#67—108no—
487#67—108no—
488#67—104no—
489#67—104no—
490#67—202no—
491C2×Dic3—52no—
492C2×Dic3—108no—
493C2×Dic3—101no—
494C2×Dic3—101no—
495D12—52no—
496D12—101no—
497D12—101no—
498D12—101no—
499D12—102no—
500D12—402no—
501C4×D3—102no—
502C4×D3—102no—
503C4×D3—104no—
504C4×D3—202no—
505Dic6—52no—
506Dic6—101no—
507Dic6—102no—
508Dic6—101no—
509Dic6—101no—
510C3×C8—21no—
511C3×C8—21no—
512C3⋊C8—102no—
513C22×C5—21no—
514C22×C5—21no—
515C22×C5—24no—
516C22×C5—42no—
517C22×C5—42no—
518C22×C5—84no—
519C22×C5#88212yes—
520C22×C5C2×D3×D411yescharacteristic
521C2×D5—62no—
522C2×D5—62no—
523C2×D5—62no—
524C2×D5—62no—
525C2×D5—121no—
526C2×D5—121no—
527C2×D5—121no—
528C2×D5—121no—
529C2×D5—121no—
530C2×D5—248no—
531C4×C5—42no—
532C4×C5#88212yes—
533C4×C5C22×D1211yescharacteristic
534C4×C5C2×D3×D411yescharacteristic
535Dic5—62no—
536Dic5—62no—
537Dic5—121no—
538C24—21no—
539C24—302no—
540#41—154no—
541#41—304no—
542#41—308no—
543#41—304no—
544#41—302no—
545#41—302no—
546#41—302no—
547#41—302no—
548#41—302no—
549C2×Q8—151no—
550C2×Q8—151no—
551C2×Q8—151no—
552C2×D4—21no—
553C2×D4—22no—
554C2×D4—152no—
555C2×D4—151no—
556C2×D4—151no—
557C2×D4—151no—
558C2×D4—302no—
559C2×D4—302no—
560C2×D4—304no—
561C2×D4—308no—
562C2×D4—302no—
563C2×D4—3016no—
564C2×D4—301no—
565C2×D4—304no—
566C2×D4—301no—
567C2×D4D6014yescomplemented
568C2×D4C22×D1512yes—
569C22×C4—21no—
570C22×C4—152no—
571C22×C4—302no—
572C22×C4—301no—
573C22×C4—301no—
574C22×C4C22×D1511yescharacteristic
575Q16—302no—
576SD16—302no—
577SD16—302no—
578SD16—304no—
579D8—302no—
580D8—304no—
581M16—302no—
582C2×C8—22no—
583C2×C8—152no—
584C2×C8C22×D1511yescharacteristic
585C2×C8C22×D1511yescharacteristic
586C4⋊C4—21no—
587C4⋊C4—302no—
588C4⋊C4C22×D1512yes—
589C22⋊C4—302no—
590C22⋊C4—302no—
591C3×C5#301211yescharacteristic, complemented, Hall
592C22×C3—21no—
593C22×C3—21no—
594C22×C3—24no—
595C22×C3—42no—
596C22×C3—42no—
597C22×C3—84no—
598C22×C3#402812yes—
599C22×C3C2×D4×D511yescharacteristic
600C2×D3—102no—
601C2×D3—102no—
602C2×D3—102no—
603C2×D3—102no—
604C2×D3—201no—
605C2×D3—201no—
606C2×D3—201no—
607C2×D3—201no—
608C2×D3—201no—
609C2×D3—408no—
610C3×C4—42no—
611C3×C4#402812yes—
612C3×C4C22×D2011yescharacteristic
613C3×C4C2×D4×D511yescharacteristic
614Dic3—102no—
615Dic3—102no—
616Dic3—201no—
617C2×C5—22no—
618C2×C5—44no—
619C2×C5C2×#88211yescharacteristic
620C2×C5#575711yescharacteristic
621C2×C5#576211yescharacteristic
622D5—122no—
623D5—122no—
624D5—241no—
625C23—22no—
626C23—24no—
627C23—81no—
628C23—81no—
629C23—152no—
630C23—308no—
631C23—301no—
632C23—301no—
633C23—1202no—
634C23—1202no—
635C23D4×D1511yescharacteristic
636Q8—152no—
637Q8—302no—
638Q8—301no—
639Q8—301no—
640Q8—301no—
641D4—24no—
642D4—22no—
643D4—152no—
644D4—308no—
645D4—308no—
646D4—304no—
647D4—304no—
648D4—302no—
649D4—301no—
650D4—301no—
651D4—301no—
652D4—602no—
653D4—1202no—
654D4C2×D6014yes—
655C2×C4—22no—
656C2×C4—81no—
657C2×C4—152no—
658C2×C4—308no—
659C2×C4—301no—
660C2×C4—304no—
661C2×C4—302no—
662C2×C4—302no—
663C2×C4—301no—
664C2×C4—602no—
665C2×C4C2×D6012yes—
666C2×C4D4×D1511yescharacteristic
667C2×C4D4×D1512yes—
668C2×C4C23×D1511yescharacteristic
669C8—21no—
670C8—21no—
671C8—302no—
672C2×C3—22no—
673C2×C3—44no—
674C2×C3C2×#402811yescharacteristic
675C2×C3#6609111yescharacteristic
676C2×C3#6609611yescharacteristic
677D3—202no—
678D3—202no—
679D3—401no—
680C5#7874711yescharacteristic, complemented, Sylow
681C22—21no—
682C22—21no—
683C22—24no—
684C22—42no—
685C22—42no—
686C22—84no—
687C22—302no—
688C22—302no—
689C22—302no—
690C22—302no—
691C22—601no—
692C22—601no—
693C22—601no—
694C22—601no—
695C22—601no—
696C22—1208no—
697C22#9177112yes—
698C22C2×D4×D1511yescharacteristic
699C4—42no—
700C4—302no—
701C4—302no—
702C4—601no—
703C4#9177112yes—
704C4C22×D6011yescharacteristic
705C4C2×D4×D1511yescharacteristic
706C3#1061047811yescharacteristic, complemented, Sylow
707C2—22no—
708C2—44no—
709C2—602no—
710C2—602no—
711C2—1201no—
712C2C2×#9177111yescharacteristic
713C2#1175651611yescharacteristic
714C2#1175652111yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×#102C2
socleC22×C3×C5C22×D4
hypercenterC2×#102D15
hypocenterC3×C5#3012
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#92095C2semidirect
2. C2×#91821C2semidirect
3. C2×#91793C2semidirect
4. #11756269C2semidirect
5. C2×C3×C5×#102C2semidirect
6. #11751211C2semidirect
7. #11751067C2semidirect
8. #11750884C2semidirect
9. #11750882C2semidirect
10. #11750877C2semidirect
11. #11750874C2semidirect
23. C2×#6817C22semidirect
24. C22×C3×C5×C8C22semidirect
25. C2×C3×C5×C4⋊C4C22semidirect
26. C2×#6716C22semidirect
28. C2×#6715C22semidirect
30. C2×#6710C22semidirect
31. C2×Dic60C22semidirect
33. C2×D120C22semidirect
34. C2×#6706C22semidirect
35. C2×#6706C22semidirect
37. #91786C22semidirect
39. #91784C22semidirect
40. C3×C5×#102C22semidirect
41. C3×C5×#102C22semidirect
42. #91104C22semidirect
43. #91104C22semidirect
44. #91094C22semidirect
45. #91093C22semidirect
51. C2×C5×#102D3semidirect
62. C2×C3×C5×D4D4semidirect
84. #6817D4semidirect
115. C2×C3×#102D5semidirect
133. C22×C5×C8C2×D3semidirect
134. C2×C5×C4⋊C4C2×D3semidirect
149. C5×#102C2×D3semidirect
150. C5×#102C2×D3semidirect
214. C22×C3×C8C2×D5semidirect
215. C2×C3×C4⋊C4C2×D5semidirect
230. C3×#102C2×D5semidirect
231. C3×#102C2×D5semidirect
240. C2×C5×D4D12semidirect
298. C2×#102D15semidirect
329. C2×C3×D4D20semidirect
445. C22×C8C2×D15semidirect
451. C2×C4⋊C4C2×D15semidirect
454. #102C2×D15semidirect
455. #102C2×D15semidirect
567. C2×D4D60semidirect
591. C3×C5#3012semidirect
680. C5#78747semidirect
706. C3#10610478semidirect

Maximal subgroups

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

subgroupindex
C2×#920952
C2×#918212
C2×#917932
#117562692
C2×C3×C5×#1022
#117512112
#117510672
#117508842
#117508822
#117508772
#117508742
#106104783
#787475