Finite Group Databasebeta

← #14326052

Subgroups — #14326052

Order 1920.

1856 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6771C211yesmaximal, characteristic
2C2×#91395C212yesmaximal, complemented
3#11750765C211yesmaximal, characteristic, complemented
4C5×#4666C211yesmaximal, characteristic, complemented
5C3×#64953C211yesmaximal, characteristic, complemented
6#11748934C212yesmaximal, complemented
7#11748919C212yesmaximal, complemented
8#11748906C212yesmaximal, complemented
9#11748888C211yesmaximal, characteristic, complemented
10D5×C3⋊#95C211yesmaximal, characteristic, complemented
11#11748782C211yesmaximal, characteristic, complemented
12#10605304—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×C4×C5×D3C2211yescharacteristic
15C22×C3×C4×D5C2211yescharacteristic
16C22×#1204C2211yescharacteristic
17C22×#1203C2211yescharacteristic
18C22×#1202C2211yescharacteristic
19C22×#1201C2211yescharacteristic
20C22×#1200C2212yes—
21C22×D5×Dic3C2212yes—
22C2×#6771—28no—
23C2×#6771C418yescomplemented
24C2×#6714C2212yes—
25C2×C5×(C2×D3)⋊C4C2212yes—
26C2×C4×C5×Dic3C2212yes—
27C2×C3×#685C2212yes—
28C2×Dic3⋊Dic5C2212yes—
29C2×(C2×D3)⋊Dic5C2212yes—
30#91395—28no—
31C3×C5×#95C2211yescharacteristic
32#91115C2211yescharacteristic
33#91115C2212yes—
34C5×C3⋊#95C2211yescharacteristic
35C3×#3963C2211yescharacteristic
36C3×#3963C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#74121—51nomaximal, Hall
40C22×#709—31no—
41C2×C4×D20—32no—
42C4×C5×C22⋊C4—31no—
43#65501—31no—
44#65245—32no—
45#65245—32no—
46#65218—32no—
47#64953—31no—
48#64953D311yescharacteristic, complemented
49D5×#95—31no—
50D5×#95—31no—
51C23×D15C2311yescharacteristic
52C23×C5×D3C2311yescharacteristic
53C23×C3×D5C2311yescharacteristic
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×Dic15C2312yes—
57C22×Dic15C2311yescharacteristic
58C2×C4×D15—24no—
59C2×C4×D15C2×C414yescomplemented
60C22×C5×Dic3C2312yes—
61C22×C5×Dic3C2311yescharacteristic
62C2×C4×C5×D3—24no—
63C2×C4×C5×D3C2×C414yescomplemented
64C22×C3×Dic5C2312yes—
65C22×C3×Dic5C2311yescharacteristic
66C2×C3×C4×D5—24no—
67C2×C3×C4×D5C2×C414yescomplemented
68C2×#1204—24no—
69C2×#1204C2×C414yescomplemented
70C2×#1203—24no—
71C2×#1203C2×C414yescomplemented
72C2×#1202C2×C414yes—
73C2×#1202D418yes—
74C2×#1201—24no—
75C2×#1201C2×C414yescomplemented
76C2×#1200—24no—
77C2×#1200—24no—
78C2×#1200—42no—
79C2×D5×Dic3—24no—
80C2×D5×Dic3—24no—
81C2×D5×Dic3—42no—
82#6771—232no—
83#6714—24no—
84C5×(C2×D3)⋊C4—24no—
85C4×C5×Dic3—24no—
86C3×#685—24no—
87Dic3⋊Dic5—24no—
88(C2×D3)⋊Dic5—42no—
89C22×#235—51no—
90C2×#67⋊C4—52no—
91C3×#381—51no—
92C2×C3⋊#95—51no—
93#4958—52no—
94#4667—51no—
95#4666D511yescharacteristic, complemented
96#4665—52no—
97#4664—51no—
98D3×#95—51no—
99#4646—52no—
100C23×C4×C5—31no—
101C22×C22⋊D5—31no—
102C22×C22⋊D5—31no—
103C2×#709—38no—
104C2×#709—68no—
105C22×D20—31no—
106C22×C4×D5—31no—
107C22×C4×D5—32no—
108C22×C4×D5—32no—
109C22×C4×D5C2×D311yescharacteristic
110C22×Dic10—31no—
111C2×C5×C22⋊C4—32no—
112C2×C42×C5—32no—
113C2×C22⋊Dic5—32no—
114C2×#685—32no—
115C2×#685C2×D312yes—
116C2×C4⋊Dic5—32no—
117C4×D20—68no—
118C5×#95—31no—
119C5×#95C2×D311yescharacteristic
120#3963—31no—
121#3963—32no—
122#3963—32no—
123#3963—32no—
124#3963C2×D311yescharacteristic
125#3963C2×D312yes—
126#2339—151noSylow
127C23×C3×C5C2411yescharacteristic
128C22×D15—22no—
129C22×D15—21no—
130C22×D15—21no—
131C22×D15C22×C412yes—
132C22×D15C2×D414yes—
133C22×C5×D3—22no—
134C22×C5×D3—22no—
135C22×C5×D3—21no—
136C22×C5×D3—21no—
137C22×C5×D3C22×C412yes—
138C22×C3×D5—22no—
139C22×C3×D5—21no—
140C22×C3×D5—21no—
141C22×C3×D5C22×C412yes—
142C22×C3×D5C2×D414yes—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5C22×C412yes—
148C2×Dic15—22no—
149C2×Dic15—22no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15C22×C412yes—
153C4×D15—28no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3—22no—
156C2×C5×Dic3C22×C412yes—
157C2×C5×Dic3C2×D414yes—
158C2×C5×Dic3#4114yescomplemented
159C4×C5×D3—28no—
160C2×C3×Dic5—22no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5—22no—
163C2×C3×Dic5—22no—
164C2×C3×Dic5C22×C412yes—
165C3×C4×D5—28no—
166#1204—28no—
167#1203—28no—
168#1202C22⋊C4116yescomplemented
169#1201—28no—
170#1200—48no—
171D5×Dic3—48no—
172C23×C3×C4—51no—
173C22×#67—51no—
174C22×#67—51no—
175C23×Dic3—52no—
176C2×#235—58no—
177C2×#235—108no—
178C22×D12—51no—
179C22×C4×D3—51no—
180C22×C4×D3—52no—
181C22×C4×D3C2×D511yescharacteristic
182C22×Dic6—51no—
183C2×C3×C22⋊C4—52no—
184C2×(C2×D3)⋊C4—52no—
185C2×(C2×D3)⋊C4—52no—
186C2×(C2×D3)⋊C4C2×D512yes—
187C2×Dic3⋊C4—52no—
188C2×C4×Dic3C2×D512yes—
189#67⋊C4—108no—
190C3×#95—52no—
191C3×#95—51no—
192C3×#95C2×D511yescharacteristic
193C3⋊#95—52no—
194C3⋊#95—51no—
195C3⋊#95—52no—
196C3⋊#95—52no—
197C3⋊#95C2×D511yescharacteristic
198C24×C5—31no—
199C23×D5—31no—
200C23×D5C22×D311yescharacteristic
201C22×C4×C5—31no—
202C22×C4×C5—32no—
203C22×C4×C5—34no—
204C22×C4×C5—64no—
205C22×C4×C5C22×D312yes—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—34no—
208C2×C22⋊D5—34no—
209C2×C22⋊D5—64no—
210C2×C22⋊D5—64no—
211C22×Dic5—31no—
212C22×Dic5—32no—
213C22×Dic5C22×D312yes—
214C22×Dic5C22×D311yescharacteristic
215#709—632no—
216C2×D20—34no—
217C2×D20—38no—
218C2×C4×D5—24no—
219C2×C4×D5—34no—
220C2×C4×D5—64no—
221C2×C4×D5—64no—
222C2×C4×D5—64no—
223C2×C4×D5—64no—
224C2×C4×D5—64no—
225C2×C4×D5—122no—
226C2×C4×D5—122no—
227C2×C4×D5C4×D314yescomplemented
228C2×Dic10—34no—
229C2×Dic10—64no—
230C5×C22⋊C4—64no—
231C42×C5—64no—
232C22⋊Dic5—122no—
233#685—24no—
234#685—64no—
235C4⋊Dic5—64no—
236C22×#41—151no—
237C2×C4×D4—152no—
238#391—152no—
239#386—152no—
240#386—152no—
241#381—51no—
242#381—151no—
243#380—151no—
244C4×C22⋊C4—31no—
245C2×#95—151no—
246C2×#95—151no—
247C22×C3×C5C23×C411yescharacteristic
248C22×C3×C5C22×D412yes—
249C22×C3×C5C2×#4112yes—
250C22×C3×C5H(5,𝔽2)11yescharacteristic
251C22×C3×C52−1+411yescharacteristic
252C2×D15—22no—
253C2×D15—22no—
254C2×D15—24no—
255C2×D15—24no—
256C2×D15C2×C22⋊C414yes—
257C2×C5×D3—22no—
258C2×C5×D3—22no—
259C2×C5×D3—22no—
260C2×C5×D3—42no—
261C2×C5×D3—42no—
262C2×C3×D5—22no—
263C2×C3×D5—22no—
264C2×C3×D5—24no—
265C2×C3×D5—24no—
266C2×C3×D5C2×C22⋊C414yes—
267C3×C4×C5—22no—
268C3×C4×C5—42no—
269Dic15—22no—
270Dic15—42no—
271C3×Dic5—22no—
272C3×Dic5—42no—
273C5×Dic3—24no—
274C5×Dic3C2×C22⋊C414yes—
275C24×C3—51no—
276C23×D3—51no—
277C23×D3C22×D511yescharacteristic
278C22×C3×C4—51no—
279C22×C3×C4—54no—
280C22×C3×C4—52no—
281C22×C3×C4—104no—
282C22×C3×C4C22×D512yes—
283C22×C3×C4C22×D511yescharacteristic
284C2×#67—54no—
285C2×#67—54no—
286C2×#67—58no—
287C2×#67—104no—
288C22×Dic3—51no—
289C22×Dic3—52no—
290C22×Dic3—104no—
291C22×Dic3—104no—
292C22×Dic3—202no—
293C22×Dic3C22×D512yes—
294C22×Dic3C22×D511yescharacteristic
295#235—1032no—
296C2×D12—54no—
297C2×D12—104no—
298C2×C4×D3—24no—
299C2×C4×D3—54no—
300C2×C4×D3—104no—
301C2×C4×D3—104no—
302C2×C4×D3—104no—
303C2×C4×D3—202no—
304C2×C4×D3C4×D514yescomplemented
305C2×Dic6—54no—
306C2×Dic6—104no—
307C3×C22⋊C4—104no—
308(C2×D3)⋊C4—24no—
309(C2×D3)⋊C4—104no—
310(C2×D3)⋊C4—202no—
311Dic3⋊C4—104no—
312C4×Dic3—24no—
313C23×C5—32no—
314C23×C5—62no—
315C23×C5—62no—
316C23×C5—61no—
317C23×C5—61no—
318C23×C5C23×D311yescharacteristic
319C22×D5—22no—
320C22×D5—21no—
321C22×D5—21no—
322C22×D5—32no—
323C22×D5—34no—
324C22×D5—62no—
325C22×D5—61no—
326C22×D5—61no—
327C22×D5C2×C4×D312yes—
328C22×D5D3×D414yes—
329C2×C4×C5—22no—
330C2×C4×C5—22no—
331C2×C4×C5—22no—
332C2×C4×C5—22no—
333C2×C4×C5—32no—
334C2×C4×C5—34no—
335C2×C4×C5—34no—
336C2×C4×C5—68no—
337C2×C4×C5—62no—
338C2×C4×C5—62no—
339C2×C4×C5C2×C4×D312yes—
340C22⋊D5—68no—
341C22⋊D5—68no—
342C2×Dic5—22no—
343C2×Dic5—22no—
344C2×Dic5—22no—
345C2×Dic5—22no—
346C2×Dic5—32no—
347C2×Dic5—62no—
348C2×Dic5—62no—
349C2×Dic5—62no—
350C2×Dic5—62no—
351C2×Dic5C2×C4×D312yes—
352D20—316no—
353C4×D5—28no—
354C4×D5—68no—
355C4×D5—128no—
356C4×D5—128no—
357Dic10—68no—
358C2×#41—158no—
359C2×#41—308no—
360C22×Q8—151no—
361C22×D4—151no—
362C22×D4—151no—
363C22×D4—151no—
364C23×C4—31no—
365C23×C4—51no—
366C23×C4—152no—
367C23×C4—152no—
368C23×C4—151no—
369C4×D4—308no—
370C2×C4⋊C4—152no—
371C2×C22⋊C4—32no—
372C2×C22⋊C4—52no—
373C2×C22⋊C4—152no—
374C2×C22⋊C4—152no—
375C2×C42—32no—
376#95—31no—
377#95—51no—
378#95—52no—
379#95—152no—
380#95—152no—
381#95—151no—
382#95—152no—
383#95D3×D511yescharacteristic
384C2×C3×C5C22×C22⋊C411yescharacteristic
385C2×C3×C5C4×#4111yescharacteristic
386C2×C3×C5#52011yescharacteristic
387C2×C3×C5#54612yes—
388C2×C3×C5#54712yes—
389D15—24no—
390C3×D5—24no—
391C5×D3—42no—
392C23×C3—52no—
393C23×C3—54no—
394C23×C3—102no—
395C23×C3—101no—
396C23×C3—101no—
397C23×C3C23×D511yescharacteristic
398C22×D3—22no—
399C22×D3—22no—
400C22×D3—21no—
401C22×D3—21no—
402C22×D3—52no—
403C22×D3—54no—
404C22×D3—102no—
405C22×D3—101no—
406C22×D3—101no—
407C22×D3C2×C4×D512yes—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—52no—
413C2×C3×C4—108no—
414C2×C3×C4—102no—
415C2×C3×C4—102no—
416C2×C3×C4—102no—
417C2×C3×C4—102no—
418C2×C3×C4C2×C4×D512yes—
419#67—516no—
420#67—108no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—52no—
424C2×Dic3—102no—
425C2×Dic3—102no—
426C2×Dic3—102no—
427C2×Dic3—102no—
428C2×Dic3—208no—
429C2×Dic3C2×C4×D512yes—
430C2×Dic3#70914yescomplemented
431C2×Dic3D4×D514yes—
432D12—108no—
433C4×D3—28no—
434C4×D3—108no—
435C4×D3—208no—
436Dic6—108no—
437C22×C5—62no—
438C22×C5—62no—
439C22×C5—62no—
440C22×C5—122no—
441C22×C5—122no—
442C22×C5C22×C4×D311yescharacteristic
443C22×C5C2×D3×D412yes—
444C22×C5D3×#4112yes—
445C22×C5#100911yescharacteristic
446C22×C5#101011yescharacteristic
447C2×D5—24no—
448C2×D5—24no—
449C2×D5—22no—
450C2×D5—22no—
451C2×D5—34no—
452C2×D5—64no—
453C2×D5—64no—
454C2×D5—62no—
455C2×D5—62no—
456C2×D5D3×C22⋊C414yes—
457C4×C5—22no—
458C4×C5—34no—
459C4×C5—42no—
460C4×C5—64no—
461Dic5—22no—
462Dic5—42no—
463Dic5—62no—
464Dic5—122no—
465C24—31no—
466C24—51no—
467C24—151no—
468#41—3032no—
469C2×Q8—154no—
470C2×Q8—304no—
471C2×D4—158no—
472C2×D4—154no—
473C2×D4—154no—
474C2×D4—154no—
475C2×D4—304no—
476C2×D4—304no—
477C22×C4—31no—
478C22×C4—32no—
479C22×C4—34no—
480C22×C4—51no—
481C22×C4—54no—
482C22×C4—52no—
483C22×C4—64no—
484C22×C4—104no—
485C22×C4—151no—
486C22×C4—154no—
487C22×C4—152no—
488C22×C4—304no—
489C22×C4—304no—
490C22×C4—304no—
491C22×C4—304no—
492C22×C4—304no—
493C22×C4—602no—
494C22×C4—602no—
495C22×C4C2×D3×D512yes—
496C22×C4C2×D3×D511yescharacteristic
497C4⋊C4—304no—
498C22⋊C4—64no—
499C22⋊C4—104no—
500C22⋊C4—304no—
501C22⋊C4—602no—
502C42—64no—
503C3×C5#233911yescharacteristic, complemented, Hall
504C22×C3—54no—
505C22×C3—102no—
506C22×C3—102no—
507C22×C3—104no—
508C22×C3—104no—
509C22×C3C22×C4×D511yescharacteristic
510C22×C3C2×#70912yes—
511C22×C3C2×D4×D512yes—
512C22×C3#414911yescharacteristic
513C22×C3#415011yescharacteristic
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—42no—
518C2×D3—42no—
519C2×D3—54no—
520C2×D3—102no—
521C2×D3—102no—
522C2×D3—104no—
523C2×D3—104no—
524C3×C4—22no—
525C3×C4—42no—
526C3×C4—102no—
527C3×C4—202no—
528Dic3—24no—
529Dic3—102no—
530Dic3—202no—
531Dic3D5×C22⋊C414yes—
532C2×C5—122no—
533C2×C5C2×D3×C22⋊C411yescharacteristic
534C2×C5#552211yescharacteristic
535C2×C5#552311yescharacteristic
536C2×C5#565412yes—
537C2×C5#565512yes—
538D5—24no—
539D5—64no—
540C23—32no—
541C23—52no—
542C23—54no—
543C23—62no—
544C23—61no—
545C23—62no—
546C23—61no—
547C23—102no—
548C23—101no—
549C23—101no—
550C23—152no—
551C23—154no—
552C23—302no—
553C23—301no—
554C23—301no—
555C23C22×D3×D511yescharacteristic
556Q8—308no—
557D4—1516no—
558D4—308no—
559D4—308no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—32no—
565C2×C4—34no—
566C2×C4—34no—
567C2×C4—52no—
568C2×C4—68no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—108no—
572C2×C4—102no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—102no—
576C2×C4—152no—
577C2×C4—308no—
578C2×C4—302no—
579C2×C4—302no—
580C2×C4—302no—
581C2×C4—302no—
582C2×C4—608no—
583C2×C4—608no—
584C2×C4C4×D3×D512yes—
585C2×C3—104no—
586C2×C3C4×#70911yescharacteristic
587C2×C3C2×D5×C22⋊C411yescharacteristic
588C2×C3#6585711yescharacteristic
589C2×C3#6588412yes—
590C2×C3#6588512yes—
591D3—42no—
592D3—104no—
593C5#7412111yescharacteristic, complemented, Sylow
594C22—54no—
595C22—62no—
596C22—62no—
597C22—62no—
598C22—104no—
599C22—104no—
600C22—102no—
601C22—102no—
602C22—122no—
603C22—122no—
604C22—154no—
605C22—304no—
606C22—304no—
607C22—302no—
608C22—302no—
609C22C2×C4×D3×D511yescharacteristic
610C22D3×#70912yes—
611C22#9202011yescharacteristic
612C22D3×D4×D512yes—
613C22#9203511yescharacteristic
614C4—22no—
615C4—34no—
616C4—42no—
617C4—64no—
618C4—102no—
619C4—202no—
620C4—302no—
621C4—602no—
622C3#1060530411yescharacteristic, complemented, Sylow
623C2—104no—
624C2—122no—
625C2—304no—
626C2#1175266611yescharacteristic
627C2#1175307412yes—
628C2D3×D5×C22⋊C411yescharacteristic
629C2#1175367011yescharacteristic
630C2#1175385812yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC3×C5×#95C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6771C2
hypercenter#95D3×D5
hypocenterC3×C5#2339
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
2. C2×#91395C2semidirect
3. #11750765C2semidirect
4. C5×#4666C2semidirect
5. C3×#64953C2semidirect
6. #11748934C2semidirect
7. #11748919C2semidirect
8. #11748906C2semidirect
9. #11748888C2semidirect
10. D5×C3⋊#95C2semidirect
11. #11748782C2semidirect
23. C2×#6771C4semidirect
48. #64953D3semidirect
59. C2×C4×D15C2×C4semidirect
63. C2×C4×C5×D3C2×C4semidirect
67. C2×C3×C4×D5C2×C4semidirect
69. C2×#1204C2×C4semidirect
71. C2×#1203C2×C4semidirect
75. C2×#1201C2×C4semidirect
95. #4666D5semidirect
158. C2×C5×Dic3#41semidirect
168. #1202C22⋊C4semidirect
227. C2×C4×D5C4×D3semidirect
304. C2×C4×D3C4×D5semidirect
430. C2×Dic3#709semidirect
503. C3×C5#2339semidirect
593. C5#74121semidirect
622. C3#10605304semidirect

Maximal subgroups

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

subgroupindex
C22×#67712
C2×#913952
#117507652
C5×#46662
C3×#649532
#117489342
#117489192
#117489062
#117488882
D5×C3⋊#952
#117487822
#106053043
#741215