Finite Group Databasebeta

← #14424095

Subgroups — #14424095

Order 1920.

1508 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6819C211yesmaximal, characteristic
2C2×#91804C212yesmaximal, complemented
3D15×C22⋊Q8C211yesmaximal, characteristic, complemented
4C3×C5×#450C211yesmaximal, characteristic, complemented
5#11751319C211yesmaximal, characteristic, complemented
6#11751274C212yesmaximal, complemented
7#11751151C211yesmaximal, characteristic, complemented
8#11750884C212yesmaximal, complemented
9#11750872C212yesmaximal, complemented
10#11750840C211yesmaximal, characteristic, complemented
11#11750827C211yesmaximal, characteristic, complemented
12#10611878—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C22×#1220—21no—
15C23×Dic15—21no—
16C2×Q8×D15C2211yescharacteristic, complemented
17C2×#6819—28no—
18C2×#6819—21no—
19C2×#6819—21no—
20C2×#6819C2212yes—
21C2×#6819C2212yes—
22C22×C4×D15C2211yescharacteristic
23C22×Dic30C2211yescharacteristic
24C2×C3×C5×SD16C2212yescomplemented
25C3×C5×C22⋊Q8C2211yescharacteristic
26#91835C2211yescharacteristic, complemented
27C2×#6718C2212yescomplemented
28C2×#6716C2212yescomplemented
29#91813C2211yescharacteristic
30#91804—28no—
31C2×Dic60C2212yescomplemented
32C2×#6705C2212yescomplemented
33#91786C2211yescharacteristic, complemented
34#91785—21no—
35C4⋊C4×D15—21no—
36C4⋊C4×D15C2211yescharacteristic, complemented
37C22⋊C4×D15—21no—
38#91767—21no—
39C3×C5×#102C2212yescomplemented
40C3×C5×C22⋊C8C2211yescharacteristic
41#91116C2212yescomplemented
42#91114C2211yescharacteristic
43#91104C2212yescomplemented
44#91100C2212yescomplemented
45#91094C2212yescomplemented
46#80147—51nomaximal, Hall
47C22×#711—31no—
48C2×#4061—32no—
49D5×C22⋊Q8—31no—
50C5×#450D311yescharacteristic, complemented
51#65507—31no—
52#65462—32no—
53#65339—31no—
54#65072—32no—
55#65060—32no—
56#65028—31no—
57#65015—31no—
58C24×C3×C5—21no—
59C23×D15D411yescharacteristic
60C2×C3×C5×Q8C2311yescharacteristic
61C2×C3×C5×D4—24no—
62C2×C3×C5×D4—21no—
63C2×C3×C5×D4C2312yes—
64C22×C3×C4×C5C2311yescharacteristic
65C2×#1220—28no—
66C2×#1220—22no—
67C2×#1220—22no—
68C22×Dic15—28no—
69C22×Dic15—21no—
70C22×Dic15—22no—
71C22×Dic15—22no—
72C22×Dic15D411yescharacteristic
73Q8×D15—24no—
74#6819—28no—
75#6819—28no—
76#6819—24no—
77#6819—24no—
78#6819—24no—
79#6819D418yes—
80C2×C4×D15—24no—
81C2×C4×D15—21no—
82C2×C4×D15—21no—
83C2×C4×D15D411yescharacteristic
84C2×C4×D15D411yescharacteristic
85C2×C4×D15C2311yescharacteristic
86C2×C4×D15C2311yescharacteristic
87C2×Dic30—21no—
88C2×Dic30—24no—
89C2×Dic30—21no—
90C2×Dic30C2311yescharacteristic
91C2×Dic30C2312yes—
92C3×C5×SD16—24no—
93C2×C3×C5×C8C2312yes—
94C3×C5×C4⋊C4—21no—
95C3×C5×C4⋊C4C2311yescharacteristic
96C3×C5×C22⋊C4—21no—
97#6719—21no—
98#6718—24no—
99#6716—24no—
100#6714—21no—
101#6714—21no—
102C4⋊Dic15—21no—
103C4⋊Dic15C2311yescharacteristic
104#6712—21no—
105#6712—21no—
106#6712—21no—
107C2×C3⋊#168C2312yes—
108Dic60—24no—
109#6705—24no—
110C22×#237—51no—
111C2×#915—52no—
112D3×C22⋊Q8—51no—
113C3×#450D511yescharacteristic, complemented
114#5218—51no—
115#5173—52no—
116#5050—51no—
117#4783—52no—
118#4771—52no—
119#4739—51no—
120#4726—51no—
121C22×C5×D4C2×D311yescharacteristic
122C22×C22⋊D5—61no—
123C23×Dic5—61no—
124C2×Q8×D5—31no—
125C2×#711—32no—
126C2×#711—32no—
127C2×#711—61no—
128C2×#711—61no—
129C2×#711—68no—
130C22×C4×D5—31no—
131C22×Dic10—31no—
132C2×C5×SD16C2×D312yescomplemented
133C5×C22⋊Q8C2×D311yescharacteristic
134#4092—31no—
135C2×#689—32no—
136C2×#687—32no—
137#4070—31no—
138#4061—68no—
139C2×Dic20—32no—
140C2×#676—32no—
141#4043—31no—
142#4042—61no—
143D5×C4⋊C4—31no—
144D5×C4⋊C4—61no—
145D5×C22⋊C4—61no—
146#4024—61no—
147C5×#102C2×D312yescomplemented
148C5×C22⋊C8C2×D311yescharacteristic
149#3964—32no—
150C22⋊#168—31no—
151#3952—32no—
152#3948—32no—
153#3942—32no—
154#3007—151noSylow
155C23×C3×C5—24no—
156C23×C3×C5—22no—
157C23×C3×C5—42no—
158C23×C3×C5—42no—
159C23×C3×C5C2×D411yescharacteristic
160C22×D15—22no—
161C22×D15—22no—
162C22×D15—41no—
163C22×D15C2×D411yescharacteristic
164C22×D15C2×D411yescharacteristic
165C3×C5×Q8—22no—
166C3×C5×D4—24no—
167C3×C5×D4—22no—
168C3×C5×D4C2×D414yes—
169C2×C3×C4×C5—22no—
170C2×C3×C4×C5—21no—
171C2×C3×C4×C5—21no—
172C2×C3×C4×C5C2×D411yescharacteristic
173C2×C3×C4×C5C2411yescharacteristic
174#1220—28no—
175#1220—28no—
176C2×Dic15—24no—
177C2×Dic15—28no—
178C2×Dic15—22no—
179C2×Dic15—22no—
180C2×Dic15—28no—
181C2×Dic15—21no—
182C2×Dic15—21no—
183C2×Dic15C2×D411yescharacteristic
184C2×Dic15C2×D411yescharacteristic
185C4×D15—22no—
186C4×D15—22no—
187C4×D15—22no—
188C4×D15—42no—
189C4×D15—42no—
190C4×D15C2×D414yes—
191Dic30—24no—
192Dic30—22no—
193Dic30—22no—
194Dic30—42no—
195Dic30C2×D414yes—
196C3×C5×C8—22no—
197C3⋊#168—22no—
198C22×C3×D4C2×D511yescharacteristic
199C22×#67—101no—
200C23×Dic3—101no—
201C2×D3×Q8—51no—
202C2×#237—52no—
203C2×#237—52no—
204C2×#237—101no—
205C2×#237—101no—
206C2×#237—108no—
207C22×C4×D3—51no—
208C22×Dic6—51no—
209C2×C3×SD16C2×D512yescomplemented
210C3×C22⋊Q8C2×D511yescharacteristic
211#946—51no—
212C2×#216—52no—
213C2×#214—52no—
214#924—51no—
215#915—108no—
216C2×Dic12—52no—
217C2×ΓL(1,𝔽25)—52no—
218#897—51no—
219#896—101no—
220D3×C4⋊C4—51no—
221D3×C4⋊C4—101no—
222D3×C22⋊C4—101no—
223#878—101no—
224C3×#102C2×D512yescomplemented
225C3×C22⋊C8C2×D511yescharacteristic
226#832—52no—
227#830—51no—
228(C2×D3)⋊C8—52no—
229#816—52no—
230#810—52no—
231C24×C5—21no—
232C23×D5—31no—
233C2×C5×Q8C22×D311yescharacteristic
234C2×C5×D4—24no—
235C2×C5×D4—21no—
236C2×C5×D4C22×D312yes—
237C22×C4×C5C22×D311yescharacteristic
238C2×C22⋊D5—62no—
239C2×C22⋊D5—62no—
240C2×C22⋊D5—68no—
241C22×Dic5—31no—
242C22×Dic5—62no—
243C22×Dic5—62no—
244C22×Dic5—68no—
245C22×Dic5—61no—
246Q8×D5—64no—
247#711—38no—
248#711—64no—
249#711—64no—
250#711—68no—
251#711—68no—
252#711—64no—
253C2×C4×D5—31no—
254C2×C4×D5—31no—
255C2×C4×D5—31no—
256C2×C4×D5—31no—
257C2×C4×D5—64no—
258C2×C4×D5—61no—
259C2×C4×D5—61no—
260C2×Dic10—32no—
261C2×Dic10—31no—
262C2×Dic10—64no—
263C2×Dic10—61no—
264C2×Dic10—61no—
265C5×SD16—24no—
266C2×C5×C8C22×D312yes—
267C5×C4⋊C4—21no—
268C5×C4⋊C4C22×D311yescharacteristic
269C5×C22⋊C4—21no—
270C22⋊Dic5—61no—
271#689—64no—
272#687—64no—
273#685—61no—
274#685—61no—
275C4⋊Dic5—31no—
276C4⋊Dic5—61no—
277#683—61no—
278#683—61no—
279#683—61no—
280C2×#168—32no—
281Dic20—64no—
282#676—64no—
283C22×#41—151no—
284C2×#137—152no—
285C2×C22⋊Q8—151no—
286#452—152no—
287#452—152no—
288#451—151no—
289#451—151no—
290#450—151no—
291#450D1511yescharacteristic, complemented
292#417—152no—
293#407—151no—
294C22×C3×C5—21no—
295C22×C3×C5—21no—
296C22×C3×C5—24no—
297C22×C3×C5—48no—
298C22×C3×C5—42no—
299C22×C3×C5—42no—
300C22×C3×C5#13712yescomplemented
301C22×C3×C5C22×D411yescharacteristic
302C2×D15—21no—
303C2×D15—21no—
304C2×D15—21no—
305C2×D15—21no—
306C2×D15—21no—
307C2×D15—44no—
308C2×D15#12012yes—
309C3×C4×C5—21no—
310C3×C4×C5—41no—
311C3×C4×C5—41no—
312C3×C4×C5C22×D412yes—
313Dic15—24no—
314Dic15—21no—
315Dic15—41no—
316Dic15—41no—
317Dic15#12012yes—
318C24×C3—21no—
319C23×D3—51no—
320C2×C3×Q8C22×D511yescharacteristic
321C2×C3×D4—24no—
322C2×C3×D4—21no—
323C2×C3×D4C22×D512yes—
324C22×C3×C4C22×D511yescharacteristic
325C2×#67—102no—
326C2×#67—102no—
327C2×#67—108no—
328C22×Dic3—51no—
329C22×Dic3—102no—
330C22×Dic3—102no—
331C22×Dic3—108no—
332C22×Dic3—101no—
333D3×Q8—104no—
334#237—58no—
335#237—104no—
336#237—104no—
337#237—108no—
338#237—108no—
339#237—104no—
340C2×C4×D3—51no—
341C2×C4×D3—51no—
342C2×C4×D3—51no—
343C2×C4×D3—51no—
344C2×C4×D3—104no—
345C2×C4×D3—101no—
346C2×C4×D3—101no—
347C2×Dic6—52no—
348C2×Dic6—51no—
349C2×Dic6—104no—
350C2×Dic6—101no—
351C2×Dic6—101no—
352C3×SD16—24no—
353C2×C3×C8C22×D512yes—
354C3×C4⋊C4—21no—
355C3×C4⋊C4C22×D511yescharacteristic
356C3×C22⋊C4—21no—
357#217—101no—
358#216—104no—
359#214—104no—
360(C2×D3)⋊C4—101no—
361(C2×D3)⋊C4—101no—
362C4⋊Dic3—51no—
363C4⋊Dic3—101no—
364Dic3⋊C4—101no—
365Dic3⋊C4—101no—
366Dic3⋊C4—101no—
367C2×C3⋊C8—52no—
368Dic12—104no—
369ΓL(1,𝔽25)—104no—
370C23×C5—24no—
371C23×C5—22no—
372C23×C5—42no—
373C23×C5—42no—
374C23×C5D3×D411yescharacteristic
375C22×D5—31no—
376C22×D5—31no—
377C22×D5—62no—
378C22×D5—62no—
379C22×D5—121no—
380C5×Q8—22no—
381C5×D4—24no—
382C5×D4—22no—
383C5×D4D3×D414yes—
384C2×C4×C5—22no—
385C2×C4×C5—21no—
386C2×C4×C5—21no—
387C2×C4×C5D3×D411yescharacteristic
388C2×C4×C5C23×D311yescharacteristic
389C22⋊D5—68no—
390C22⋊D5—68no—
391C2×Dic5—31no—
392C2×Dic5—31no—
393C2×Dic5—62no—
394C2×Dic5—68no—
395C2×Dic5—64no—
396C2×Dic5—68no—
397C2×Dic5—62no—
398C2×Dic5—61no—
399C2×Dic5—61no—
400C4×D5—34no—
401C4×D5—62no—
402C4×D5—62no—
403C4×D5—62no—
404C4×D5—122no—
405C4×D5—122no—
406Dic10—34no—
407Dic10—64no—
408Dic10—62no—
409Dic10—62no—
410Dic10—122no—
411C5×C8—22no—
412#168—62no—
413C2×#41—152no—
414C2×#41—152no—
415C2×#41—301no—
416C2×#41—301no—
417C2×#41—308no—
418C22×Q8—151no—
419C22×Q8—151no—
420C22×D4—301no—
421C22×D4C2×D1511yescharacteristic
422C23×C4—151no—
423C23×C4—301no—
424#137—308no—
425C2×Q16—152no—
426C2×Q16—152no—
427C2×SD16—152no—
428C2×SD16C2×D1512yescomplemented
429C2×M16—152no—
430C22⋊Q8—151no—
431C22⋊Q8—151no—
432C22⋊Q8—151no—
433C22⋊Q8—301no—
434C22⋊Q8—301no—
435C22⋊Q8C2×D1511yescharacteristic
436C2×C4⋊C4—151no—
437C2×C4⋊C4—301no—
438C2×C22⋊C4—301no—
439#103—152no—
440#103—152no—
441#102—152no—
442#102C2×D1512yescomplemented
443C22⋊C8—151no—
444C22⋊C8—152no—
445C22⋊C8C2×D1511yescharacteristic
446C2×C3×C5—22no—
447C2×C3×C5—44no—
448C2×C3×C5C2×#12011yescharacteristic
449C2×C3×C5C2×#13711yescharacteristic
450C2×C3×C5#57511yescharacteristic
451D15—22no—
452D15—41no—
453C23×C3—24no—
454C23×C3—22no—
455C23×C3—42no—
456C23×C3—42no—
457C23×C3D4×D511yescharacteristic
458C22×D3—51no—
459C22×D3—51no—
460C22×D3—102no—
461C22×D3—102no—
462C22×D3—201no—
463C3×Q8—22no—
464C3×D4—24no—
465C3×D4—22no—
466C3×D4D4×D514yes—
467C2×C3×C4—22no—
468C2×C3×C4—21no—
469C2×C3×C4—21no—
470C2×C3×C4D4×D511yescharacteristic
471C2×C3×C4C23×D511yescharacteristic
472#67—108no—
473#67—108no—
474C2×Dic3—51no—
475C2×Dic3—51no—
476C2×Dic3—102no—
477C2×Dic3—108no—
478C2×Dic3—104no—
479C2×Dic3—108no—
480C2×Dic3—102no—
481C2×Dic3—101no—
482C2×Dic3—101no—
483C4×D3—54no—
484C4×D3—102no—
485C4×D3—102no—
486C4×D3—102no—
487C4×D3—202no—
488C4×D3—202no—
489Dic6—54no—
490Dic6—104no—
491Dic6—102no—
492Dic6—102no—
493Dic6—202no—
494C3×C8—22no—
495C3⋊C8—102no—
496C22×C5—21no—
497C22×C5—21no—
498C22×C5—24no—
499C22×C5—48no—
500C22×C5—42no—
501C22×C5—42no—
502C22×C5#91512yescomplemented
503C22×C5C2×D3×D411yescharacteristic
504C2×D5—32no—
505C2×D5—61no—
506C2×D5—61no—
507C2×D5—61no—
508C2×D5—61no—
509C2×D5—61no—
510C2×D5—124no—
511C4×C5—21no—
512C4×C5—41no—
513C4×C5—41no—
514C4×C5C2×D3×D412yes—
515Dic5—32no—
516Dic5—64no—
517Dic5—61no—
518Dic5—121no—
519Dic5—121no—
520C24—21no—
521C24—151no—
522#41—158no—
523#41—304no—
524#41—304no—
525#41—308no—
526#41—308no—
527#41—304no—
528C2×Q8—152no—
529C2×Q8—151no—
530C2×Q8—301no—
531C2×Q8—304no—
532C2×Q8—304no—
533C2×Q8—301no—
534C2×Q8C22×D1511yescharacteristic
535C2×D4—24no—
536C2×D4—21no—
537C2×D4—302no—
538C2×D4—302no—
539C2×D4—308no—
540C2×D4C22×D1512yes—
541C22×C4—151no—
542C22×C4—151no—
543C22×C4—151no—
544C22×C4—151no—
545C22×C4—151no—
546C22×C4—304no—
547C22×C4—301no—
548C22×C4—301no—
549C22×C4—302no—
550C22×C4—302no—
551C22×C4—308no—
552C22×C4—301no—
553C22×C4C22×D1511yescharacteristic
554Q16—304no—
555Q16—304no—
556SD16—24no—
557SD16—304no—
558M16—304no—
559C2×C8—152no—
560C2×C8C22×D1512yes—
561C4⋊C4—21no—
562C4⋊C4—151no—
563C4⋊C4—301no—
564C4⋊C4—301no—
565C4⋊C4—301no—
566C4⋊C4—301no—
567C4⋊C4C22×D1511yescharacteristic
568C22⋊C4—21no—
569C22⋊C4—301no—
570C22⋊C4—301no—
571C22⋊C4—301no—
572C3×C5#300711yescharacteristic, complemented, Hall
573C22×C3—21no—
574C22×C3—21no—
575C22×C3—24no—
576C22×C3—48no—
577C22×C3—42no—
578C22×C3—42no—
579C22×C3#406112yescomplemented
580C22×C3C2×D4×D511yescharacteristic
581C2×D3—52no—
582C2×D3—101no—
583C2×D3—101no—
584C2×D3—101no—
585C2×D3—101no—
586C2×D3—101no—
587C2×D3—204no—
588C3×C4—21no—
589C3×C4—41no—
590C3×C4—41no—
591C3×C4C2×D4×D512yes—
592Dic3—52no—
593Dic3—104no—
594Dic3—101no—
595Dic3—201no—
596Dic3—201no—
597C2×C5—22no—
598C2×C5—44no—
599C2×C5D3×#12011yescharacteristic
600C2×C5C2×#91511yescharacteristic
601C2×C5#577311yescharacteristic
602D5—62no—
603D5—121no—
604C23—24no—
605C23—22no—
606C23—42no—
607C23—42no—
608C23—151no—
609C23—151no—
610C23—302no—
611C23—302no—
612C23—601no—
613C23D4×D1511yescharacteristic
614Q8—22no—
615Q8—154no—
616Q8—304no—
617Q8—302no—
618Q8—302no—
619Q8—602no—
620D4—24no—
621D4—22no—
622D4—308no—
623D4—308no—
624D4D4×D1514yes—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—151no—
629C2×C4—154no—
630C2×C4—151no—
631C2×C4—302no—
632C2×C4—302no—
633C2×C4—302no—
634C2×C4—308no—
635C2×C4—304no—
636C2×C4—308no—
637C2×C4—302no—
638C2×C4—302no—
639C2×C4—301no—
640C2×C4—301no—
641C2×C4—602no—
642C2×C4—602no—
643C2×C4D4×D1511yescharacteristic
644C2×C4C23×D1511yescharacteristic
645C8—22no—
646C8—302no—
647C2×C3—22no—
648C2×C3—44no—
649C2×C3D5×#12011yescharacteristic
650C2×C3C2×#406111yescharacteristic
651C2×C3#6610711yescharacteristic
652D3—102no—
653D3—201no—
654C5#8014711yescharacteristic, complemented, Sylow
655C22—21no—
656C22—21no—
657C22—24no—
658C22—48no—
659C22—42no—
660C22—42no—
661C22—152no—
662C22—301no—
663C22—301no—
664C22—301no—
665C22—301no—
666C22—301no—
667C22—604no—
668C22#9180412yescomplemented
669C22C2×D4×D1511yescharacteristic
670C4—21no—
671C4—41no—
672C4—41no—
673C4—152no—
674C4—301no—
675C4—304no—
676C4—601no—
677C4—601no—
678C4C2×D4×D1512yes—
679C3#1061187811yescharacteristic, complemented, Sylow
680C2—22no—
681C2—44no—
682C2—302no—
683C2—601no—
684C2D15×#12011yescharacteristic
685C2C2×#9180411yescharacteristic
686C2#1175653211yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#450C2
socleC22×C3×C5C22×D4
prime radicalC22×#6819C2
hypercenter#450D15
hypocenterC3×C5#3007
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#91804C2semidirect
3. D15×C22⋊Q8C2semidirect
4. C3×C5×#450C2semidirect
5. #11751319C2semidirect
6. #11751274C2semidirect
7. #11751151C2semidirect
8. #11750884C2semidirect
9. #11750872C2semidirect
10. #11750840C2semidirect
11. #11750827C2semidirect
16. C2×Q8×D15C22semidirect
24. C2×C3×C5×SD16C22semidirect
26. #91835C22semidirect
27. C2×#6718C22semidirect
28. C2×#6716C22semidirect
31. C2×Dic60C22semidirect
32. C2×#6705C22semidirect
33. #91786C22semidirect
36. C4⋊C4×D15C22semidirect
39. C3×C5×#102C22semidirect
41. #91116C22semidirect
43. #91104C22semidirect
44. #91100C22semidirect
45. #91094C22semidirect
50. C5×#450D3semidirect
113. C3×#450D5semidirect
132. C2×C5×SD16C2×D3semidirect
147. C5×#102C2×D3semidirect
209. C2×C3×SD16C2×D5semidirect
224. C3×#102C2×D5semidirect
291. #450D15semidirect
300. C22×C3×C5#137semidirect
428. C2×SD16C2×D15semidirect
442. #102C2×D15semidirect
502. C22×C5#915semidirect
572. C3×C5#3007semidirect
579. C22×C3#4061semidirect
654. C5#80147semidirect
668. C22#91804semidirect
679. C3#10611878semidirect

Maximal subgroups

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

subgroupindex
C22×#68192
C2×#918042
D15×C22⋊Q82
C3×C5×#4502
#117513192
#117512742
#117511512
#117508842
#117508722
#117508402
#117508272
#106118783
#801475