Finite Group Databasebeta

← #14331612

Subgroups — #14331612

Order 1920.

1888 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6819C211yesmaximal, characteristic, complemented
2C2×#91782C211yesmaximal, characteristic, complemented
3C2×#91770C212yesmaximal, complemented
4C3×C5×#394C211yesmaximal, characteristic, complemented
5#11751324C211yesmaximal, characteristic, complemented
6#11751080C211yesmaximal, characteristic, complemented
7#11751061C212yesmaximal, complemented
8#11751060C212yesmaximal, complemented
9C22⋊C4×Dic15C212yesmaximal, complemented
10#11750774C211yesmaximal, characteristic, complemented
11#11750767C211yesmaximal, characteristic, complemented
12#10608620—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C22×#1220C2212yes—
15C22×#1220C2212yes—
16C23×Dic15C2212yes—
17C2×#6819—28no—
18C2×#6819—24no—
19C22×D60C2211yescharacteristic
20C22×C4×D15C2211yescharacteristic
21C22×C4×D15C2211yescharacteristic
22C22×Dic30C2211yescharacteristic
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×C3×C5×C22⋊C4C2212yes—
25C2×C3×C5×C22⋊C4C2212yes—
26C2×#6719C2212yes—
27C2×#6714C2212yes—
28C2×#6714C2212yes—
29C2×#6714C2211yescharacteristic
30C2×#6714C2211yescharacteristic
31C2×#6712C2212yes—
32C2×C4×Dic15C2212yes—
33C2×C4×Dic15C2211yescharacteristic
34#91782—24no—
35#91770—28no—
36C3×C5×#95C2211yescharacteristic
37#91115C2211yescharacteristic
38#91115C2212yes—
39#91115C2212yes—
40#76889—51nomaximal, Hall
41C22×#711—31no—
42C2×#4039—31no—
43C2×#4027—32no—
44C5×#394D311yescharacteristic, complemented
45#65512—31no—
46#65268—31no—
47#65249—32no—
48#65248—32no—
49C22⋊C4×Dic5—32no—
50#64962—31no—
51#64955—31no—
52C24×C3×C5C2312yes—
53C23×D15C2311yescharacteristic
54C23×D15C2311yescharacteristic
55C2×C3×C5×D4—24no—
56C2×C3×C5×D4—22no—
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C22×C3×C4×C5C2311yescharacteristic
61C2×#1220—24no—
62C2×#1220—24no—
63C2×#1220—24no—
64C2×#1220—24no—
65C2×#1220—24no—
66C2×#1220—24no—
67C22×Dic15—24no—
68C22×Dic15—24no—
69C22×Dic15D418yescomplemented
70C22×Dic15C2311yescharacteristic
71C22×Dic15C2312yes—
72C22×Dic15C2312yes—
73C22×Dic15C2311yescharacteristic
74#6819—232no—
75C2×D60—22no—
76C2×D60—22no—
77C2×D60—22no—
78C2×C4×D15—24no—
79C2×C4×D15—24no—
80C2×C4×D15—22no—
81C2×C4×D15D414yescomplemented
82C2×Dic30—22no—
83C2×Dic30—24no—
84C3×C5×C4⋊C4—22no—
85C3×C5×C22⋊C4—24no—
86C3×C5×C22⋊C4—24no—
87#6719—24no—
88#6714—22no—
89#6714—22no—
90#6714—24no—
91#6714—24no—
92#6712—24no—
93C4×Dic15—24no—
94C4×Dic15—22no—
95C22×#237—51no—
96C2×#893—51no—
97C2×#67⋊C4—52no—
98C3×#394D511yescharacteristic, complemented
99#5223—51no—
100#4979—51no—
101#4960—52no—
102#4959—52no—
103Dic3×C22⋊C4—52no—
104#4673—51no—
105#4666—51no—
106C22×C5×D4C2×D311yescharacteristic
107C22×C22⋊D5—32no—
108C22×C22⋊D5—32no—
109C23×Dic5—32no—
110C2×#711—64no—
111C2×#711—68no—
112C22×D20—31no—
113C22×C4×D5—31no—
114C22×C4×D5—31no—
115C22×Dic10—31no—
116C2×C5×C4⋊C4C2×D311yescharacteristic
117C2×C5×C22⋊C4C2×D312yes—
118C2×C5×C22⋊C4C2×D312yes—
119C2×C22⋊Dic5—32no—
120C2×#685—32no—
121C2×#685—31no—
122C2×#685—31no—
123C2×#685—32no—
124C2×#683—32no—
125C2×C4×Dic5—32no—
126C2×C4×Dic5—31no—
127#4039—64no—
128#4027—68no—
129C5×#95C2×D311yescharacteristic
130#3963—31no—
131#3963—32no—
132#3963—32no—
133#2388—151noSylow
134C23×C3×C5—22no—
135C23×C3×C5—22no—
136C23×C3×C5—22no—
137C23×C3×C5—22no—
138C23×C3×C5—22no—
139C23×C3×C5—22no—
140C23×C3×C5C2×D414yes—
141C23×C3×C5C2411yescharacteristic
142C22×D15—21no—
143C22×D15—22no—
144C22×D15—22no—
145C22×D15—22no—
146C22×D15—21no—
147C22×D15—21no—
148C22×D15—21no—
149C22×D15—22no—
150C22×D15C2×D412yes—
151C22×D15#4112yescomplemented
152C3×C5×D4—28no—
153C2×C3×C4×C5—21no—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5—21no—
156C2×C3×C4×C5—22no—
157C2×C3×C4×C5—22no—
158C2×C3×C4×C5—21no—
159C2×C3×C4×C5—22no—
160C2×C3×C4×C5—22no—
161C2×C3×C4×C5C2×D412yes—
162C2×C3×C4×C5#4112yescomplemented
163#1220—216no—
164#1220—216no—
165C2×Dic15—22no—
166C2×Dic15—22no—
167C2×Dic15—22no—
168C2×Dic15—22no—
169C2×Dic15—216no—
170C2×Dic15—21no—
171C2×Dic15—22no—
172C2×Dic15C2×D414yes—
173C2×Dic15C2×D414yes—
174C2×Dic15C2×D412yes—
175C2×Dic15#4114yescomplemented
176D60—28no—
177C4×D15—28no—
178C4×D15—44no—
179Dic30—28no—
180C22×C3×D4C2×D511yescharacteristic
181C22×#67—52no—
182C22×#67—52no—
183C23×Dic3—52no—
184C2×#237—104no—
185C2×#237—108no—
186C22×D12—51no—
187C22×C4×D3—51no—
188C22×C4×D3—51no—
189C22×Dic6—51no—
190C2×C3×C4⋊C4C2×D511yescharacteristic
191C2×C3×C22⋊C4C2×D512yes—
192C2×C3×C22⋊C4C2×D512yes—
193C2×#217—52no—
194C2×(C2×D3)⋊C4—52no—
195C2×(C2×D3)⋊C4—52no—
196C2×(C2×D3)⋊C4—51no—
197C2×(C2×D3)⋊C4—51no—
198C2×Dic3⋊C4—52no—
199C2×C4×Dic3—52no—
200C2×C4×Dic3—51no—
201#893—104no—
202#67⋊C4—108no—
203C3×#95C2×D511yescharacteristic
204C3⋊#95—51no—
205C3⋊#95—52no—
206C3⋊#95—52no—
207C24×C5C22×D312yes—
208C23×D5—31no—
209C23×D5—31no—
210C2×C5×D4—24no—
211C2×C5×D4—22no—
212C22×C4×C5C22×D311yescharacteristic
213C22×C4×C5C22×D312yes—
214C22×C4×C5C22×D311yescharacteristic
215C22×C4×C5C22×D311yescharacteristic
216C2×C22⋊D5—64no—
217C2×C22⋊D5—64no—
218C2×C22⋊D5—64no—
219C2×C22⋊D5—64no—
220C2×C22⋊D5—64no—
221C2×C22⋊D5—64no—
222C22×Dic5—31no—
223C22×Dic5—38no—
224C22×Dic5—32no—
225C22×Dic5—32no—
226C22×Dic5—31no—
227C22×Dic5—64no—
228C22×Dic5—64no—
229#711—632no—
230C2×D20—62no—
231C2×D20—62no—
232C2×D20—62no—
233C2×C4×D5—34no—
234C2×C4×D5—64no—
235C2×C4×D5—64no—
236C2×C4×D5—62no—
237C2×Dic10—64no—
238C2×Dic10—62no—
239C5×C4⋊C4—22no—
240C5×C22⋊C4—24no—
241C5×C22⋊C4—24no—
242C22⋊Dic5—64no—
243#685—64no—
244#685—64no—
245#685—62no—
246#685—62no—
247#683—64no—
248C4×Dic5—64no—
249C4×Dic5—62no—
250C22×#41—151no—
251C2×C4×D4—151no—
252C2×C4×D4—152no—
253#395—151no—
254#394—152no—
255#394D1511yescharacteristic, complemented
256#393—151no—
257#392—151no—
258#392—152no—
259C4×C22⋊C4—152no—
260C4×C22⋊C4—151no—
261C22×C3×C5—24no—
262C22×C3×C5—24no—
263C22×C3×C5—24no—
264C22×C3×C5—42no—
265C22×C3×C5—42no—
266C22×C3×C5—42no—
267C22×C3×C5—42no—
268C22×C3×C5C22×D411yescharacteristic
269C22×C3×C5C22×D412yes—
270C22×C3×C5C2×#4111yescharacteristic
271C22×C3×C5C2×#4111yescharacteristic
272C22×C3×C5C2×#4112yes—
273C2×D15—22no—
274C2×D15—22no—
275C2×D15—22no—
276C2×D15—24no—
277C2×D15—22no—
278C2×D15—41no—
279C2×D15—42no—
280C2×D15—42no—
281C2×D15—42no—
282C2×D15—41no—
283C3×C4×C5—22no—
284C3×C4×C5—22no—
285C3×C4×C5—41no—
286C3×C4×C5—42no—
287Dic15—24no—
288Dic15—24no—
289Dic15—41no—
290Dic15#12014yes—
291C24×C3C22×D512yes—
292C23×D3—51no—
293C23×D3—51no—
294C2×C3×D4—24no—
295C2×C3×D4—22no—
296C22×C3×C4C22×D512yes—
297C22×C3×C4C22×D511yescharacteristic
298C22×C3×C4C22×D511yescharacteristic
299C22×C3×C4C22×D511yescharacteristic
300C2×#67—104no—
301C2×#67—104no—
302C2×#67—104no—
303C2×#67—104no—
304C2×#67—104no—
305C2×#67—104no—
306C22×Dic3—51no—
307C22×Dic3—58no—
308C22×Dic3—52no—
309C22×Dic3—52no—
310C22×Dic3—51no—
311C22×Dic3—104no—
312C22×Dic3—104no—
313#237—1032no—
314C2×D12—102no—
315C2×D12—102no—
316C2×D12—102no—
317C2×C4×D3—54no—
318C2×C4×D3—104no—
319C2×C4×D3—104no—
320C2×C4×D3—102no—
321C2×Dic6—104no—
322C2×Dic6—102no—
323C3×C4⋊C4—22no—
324C3×C22⋊C4—24no—
325C3×C22⋊C4—24no—
326#217—104no—
327(C2×D3)⋊C4—104no—
328(C2×D3)⋊C4—104no—
329(C2×D3)⋊C4—102no—
330(C2×D3)⋊C4—102no—
331Dic3⋊C4—104no—
332C4×Dic3—104no—
333C4×Dic3—102no—
334C23×C5—22no—
335C23×C5—22no—
336C23×C5—22no—
337C23×C5—22no—
338C23×C5—22no—
339C23×C5—22no—
340C23×C5D3×D414yes—
341C23×C5C23×D311yescharacteristic
342C22×D5—32no—
343C22×D5—32no—
344C22×D5—61no—
345C22×D5—62no—
346C22×D5—62no—
347C22×D5—61no—
348C22×D5—61no—
349C22×D5—62no—
350C22×D5—61no—
351C22×D5—62no—
352C5×D4—28no—
353C2×C4×C5—21no—
354C2×C4×C5—22no—
355C2×C4×C5—21no—
356C2×C4×C5—22no—
357C2×C4×C5—22no—
358C2×C4×C5—21no—
359C2×C4×C5—22no—
360C2×C4×C5—22no—
361C2×C4×C5D3×D412yes—
362C2×C4×C5#23712yescomplemented
363C22⋊D5—616no—
364C22⋊D5—616no—
365C2×Dic5—32no—
366C2×Dic5—34no—
367C2×Dic5—34no—
368C2×Dic5—34no—
369C2×Dic5—616no—
370C2×Dic5—62no—
371C2×Dic5—62no—
372C2×Dic5—62no—
373C2×Dic5—62no—
374C2×Dic5—61no—
375C2×Dic5—62no—
376D20—68no—
377C4×D5—68no—
378C4×D5—124no—
379Dic10—68no—
380C2×#41—308no—
381C2×#41—304no—
382C22×Q8—151no—
383C22×D4—152no—
384C22×D4—152no—
385C22×D4—151no—
386C22×D4C2×D1511yescharacteristic
387C23×C4—152no—
388C23×C4—151no—
389C23×C4—151no—
390C4×D4—308no—
391C4×D4—304no—
392C2×C4⋊C4—152no—
393C2×C4⋊C4C2×D1511yescharacteristic
394C2×C22⋊C4—152no—
395C2×C22⋊C4—152no—
396C2×C22⋊C4—151no—
397C2×C22⋊C4—151no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4C2×D1512yes—
400C2×C22⋊C4C2×D1512yes—
401C2×C42—152no—
402C2×C42—151no—
403#95—151no—
404#95—152no—
405#95—152no—
406#95C2×D1511yescharacteristic
407C2×C3×C5—44no—
408C2×C3×C5C2×#12011yescharacteristic
409C2×C3×C5#52511yescharacteristic
410C2×C3×C5#52512yes—
411C2×C3×C5#53212yes—
412C2×C3×C5#53211yescharacteristic
413D15—42no—
414D15—42no—
415C23×C3—22no—
416C23×C3—22no—
417C23×C3—22no—
418C23×C3—22no—
419C23×C3—22no—
420C23×C3—22no—
421C23×C3D4×D514yes—
422C23×C3C23×D511yescharacteristic
423C22×D3—52no—
424C22×D3—52no—
425C22×D3—101no—
426C22×D3—102no—
427C22×D3—102no—
428C22×D3—101no—
429C22×D3—101no—
430C22×D3—102no—
431C22×D3—101no—
432C22×D3—102no—
433C3×D4—28no—
434C2×C3×C4—21no—
435C2×C3×C4—22no—
436C2×C3×C4—21no—
437C2×C3×C4—22no—
438C2×C3×C4—22no—
439C2×C3×C4—21no—
440C2×C3×C4—22no—
441C2×C3×C4—22no—
442C2×C3×C4D4×D512yes—
443C2×C3×C4#71112yescomplemented
444#67—1016no—
445#67—1016no—
446C2×Dic3—52no—
447C2×Dic3—54no—
448C2×Dic3—54no—
449C2×Dic3—54no—
450C2×Dic3—1016no—
451C2×Dic3—102no—
452C2×Dic3—102no—
453C2×Dic3—102no—
454C2×Dic3—102no—
455C2×Dic3—101no—
456C2×Dic3—102no—
457D12—108no—
458C4×D3—108no—
459C4×D3—204no—
460Dic6—108no—
461C22×C5—24no—
462C22×C5—24no—
463C22×C5—24no—
464C22×C5—42no—
465C22×C5—42no—
466C22×C5—42no—
467C22×C5—42no—
468C22×C5C2×D3×D412yes—
469C22×C5C2×D3×D411yescharacteristic
470C22×C5C2×#23711yescharacteristic
471C22×C5D3×#4111yescharacteristic
472C22×C5D3×#4112yes—
473C2×D5—64no—
474C2×D5—62no—
475C2×D5—62no—
476C2×D5—62no—
477C2×D5—62no—
478C2×D5—122no—
479C2×D5—121no—
480C2×D5—121no—
481C2×D5—122no—
482C2×D5—122no—
483C4×C5—22no—
484C4×C5—22no—
485C4×C5—41no—
486C4×C5—42no—
487Dic5—34no—
488Dic5—64no—
489Dic5—64no—
490Dic5—121no—
491C24—151no—
492C24—151no—
493C24C22×D1512yes—
494#41—3032no—
495C2×Q8—304no—
496C2×Q8—302no—
497C2×D4—24no—
498C2×D4—22no—
499C2×D4—302no—
500C2×D4—302no—
501C2×D4—304no—
502C2×D4—304no—
503C2×D4—304no—
504C2×D4—304no—
505C2×D4—304no—
506C2×D4—304no—
507C2×D4—302no—
508C22×C4—151no—
509C22×C4—158no—
510C22×C4—152no—
511C22×C4—152no—
512C22×C4—154no—
513C22×C4—151no—
514C22×C4—304no—
515C22×C4—304no—
516C22×C4—302no—
517C22×C4—304no—
518C22×C4—304no—
519C22×C4C22×D1512yes—
520C22×C4C22×D1511yescharacteristic
521C22×C4C22×D1511yescharacteristic
522C22×C4C22×D1511yescharacteristic
523C4⋊C4—22no—
524C4⋊C4—304no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—304no—
528C22⋊C4—304no—
529C22⋊C4—302no—
530C22⋊C4—304no—
531C22⋊C4—302no—
532C42—302no—
533C42—304no—
534C3×C5#238811yescharacteristic, complemented, Hall
535C22×C3—24no—
536C22×C3—24no—
537C22×C3—24no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3—42no—
542C22×C3C2×D4×D512yes—
543C22×C3C2×D4×D511yescharacteristic
544C22×C3C2×#71111yescharacteristic
545C22×C3D5×#4112yes—
546C22×C3D5×#4111yescharacteristic
547C2×D3—102no—
548C2×D3—102no—
549C2×D3—102no—
550C2×D3—104no—
551C2×D3—102no—
552C2×D3—201no—
553C2×D3—202no—
554C2×D3—202no—
555C2×D3—202no—
556C2×D3—201no—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—41no—
560C3×C4—42no—
561Dic3—54no—
562Dic3—104no—
563Dic3—104no—
564Dic3—201no—
565C2×C5—44no—
566C2×C5D3×#12011yescharacteristic
567C2×C5#559612yes—
568C2×C5#560611yescharacteristic
569C2×C5#565612yes—
570C2×C5#566811yescharacteristic
571D5—122no—
572D5—122no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—152no—
580C23—152no—
581C23—301no—
582C23—302no—
583C23—302no—
584C23—301no—
585C23—301no—
586C23—302no—
587C23—301no—
588C23—302no—
589C23D4×D1514yes—
590C23C23×D1511yescharacteristic
591Q8—308no—
592D4—28no—
593D4—308no—
594D4—3016no—
595D4—3016no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—154no—
605C2×C4—152no—
606C2×C4—154no—
607C2×C4—154no—
608C2×C4—308no—
609C2×C4—3016no—
610C2×C4—302no—
611C2×C4—302no—
612C2×C4—302no—
613C2×C4—302no—
614C2×C4—301no—
615C2×C4—302no—
616C2×C4—604no—
617C2×C4D4×D1512yes—
618C2×C4#681912yescomplemented
619C2×C3—44no—
620C2×C3D5×#12011yescharacteristic
621C2×C3#6593012yes—
622C2×C3#6594011yescharacteristic
623C2×C3#6599012yes—
624C2×C3#6600211yescharacteristic
625D3—202no—
626D3—202no—
627C5#7688911yescharacteristic, complemented, Sylow
628C22—24no—
629C22—24no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—302no—
636C22—302no—
637C22—302no—
638C22—304no—
639C22—302no—
640C22—601no—
641C22—602no—
642C22—602no—
643C22—602no—
644C22—601no—
645C22C2×D4×D1512yes—
646C22C2×D4×D1511yescharacteristic
647C22C2×#681911yescharacteristic
648C22#41×D1512yes—
649C22#41×D1511yescharacteristic
650C4—22no—
651C4—22no—
652C4—41no—
653C4—42no—
654C4—154no—
655C4—304no—
656C4—304no—
657C4—601no—
658C3#1060862011yescharacteristic, complemented, Sylow
659C2—44no—
660C2—602no—
661C2—602no—
662C2D15×#12011yescharacteristic
663C2#1175635512yes—
664C2#1175636511yescharacteristic
665C2#1175641512yes—
666C2#1175642711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutator, socle, cosocleC23×C3×C5C24
FittingC3×C5×#394C2
hypercenter#394D15
hypocenterC3×C5#2388
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#6819C2semidirect
2. C2×#91782C2semidirect
3. C2×#91770C2semidirect
4. C3×C5×#394C2semidirect
5. #11751324C2semidirect
6. #11751080C2semidirect
7. #11751061C2semidirect
8. #11751060C2semidirect
9. C22⋊C4×Dic15C2semidirect
10. #11750774C2semidirect
11. #11750767C2semidirect
44. C5×#394D3semidirect
69. C22×Dic15D4semidirect
81. C2×C4×D15D4semidirect
98. C3×#394D5semidirect
151. C22×D15#41semidirect
162. C2×C3×C4×C5#41semidirect
175. C2×Dic15#41semidirect
255. #394D15semidirect
362. C2×C4×C5#237semidirect
443. C2×C3×C4#711semidirect
534. C3×C5#2388semidirect
618. C2×C4#6819semidirect
627. C5#76889semidirect
658. C3#10608620semidirect

Maximal subgroups

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

subgroupindex
C22×#68192
C2×#917822
C2×#917702
C3×C5×#3942
#117513242
#117510802
#117510612
#117510602
C22⋊C4×Dic152
#117507742
#117507672
#106086203
#768895