Finite Group Databasebeta

← #14328115

Subgroups — #14328115

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
1C2×#91826C211yesmaximal, characteristic
2C2×C4×#1220C212yesmaximal, complemented
3C2×C22⋊C4×D15C212yesmaximal, complemented
4C3×C5×#386C211yesmaximal, characteristic, complemented
5#11751322C211yesmaximal, characteristic, complemented
6#11751057C212yesmaximal, complemented
7#11751046C212yesmaximal, complemented
8#11750769C212yesmaximal, complemented
9#11750767C212yesmaximal, complemented
10#10606611—31nomaximal, Hall
11C24×D15C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic, complemented
14C22×#1220C2212yes—
15C22×C4×D15C2212yes—
16C22×C4×D15C2212yes—
17C2×C3×C5×C22⋊C4C2212yes—
18C2×C3×C5×C22⋊C4C2211yescharacteristic, complemented
19C2×#6719C2212yes—
20C2×#6719C2211yescharacteristic
21C2×#6719C2212yescomplemented
22#91826C418yescomplemented
23C4×#1220—28no—
24C2×#6714C2212yes—
25C2×#6714C2212yes—
26C2×#6714C2212yes—
27C2×C4⋊Dic15C2211yescharacteristic, complemented
28C2×#6712C2212yescomplemented
29C2×C4×Dic15C2212yescomplemented
30C2×C4×Dic15C2211yescharacteristic, complemented
31C22⋊C4×D15—28no—
32C3×C5×#95C2212yescomplemented
33#91115C2212yescomplemented
34#91115C2212yescomplemented
35#74880—51nomaximal, Hall
36C2×#4083—31no—
37C2×C4×C22⋊D5—32no—
38C2×D5×C22⋊C4—32no—
39C5×#386D311yescharacteristic, complemented
40#65510—31no—
41#65245—32no—
42#65234—32no—
43#64957—32no—
44#64955—32no—
45C24×C3×C5C2311yescharacteristic
46C24×C3×C5C2311yescharacteristic
47C23×D15—24no—
48C23×D15—24no—
49C23×D15—41no—
50C23×D15—41no—
51C23×D15C2×C414yescomplemented
52C23×D15C2312yes—
53C2×C3×C5×D4—41no—
54C2×C3×C5×D4—41no—
55C2×C3×C5×D4C2×C414yescomplemented
56C22×C3×C4×C5—24no—
57C22×C3×C4×C5—41no—
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2311yescharacteristic
60C22×C3×C4×C5C2312yes—
61C2×#1220—24no—
62C2×#1220—24no—
63C2×#1220C2×C418yescomplemented
64C22×Dic15C2312yes—
65C22×Dic15C2312yes—
66C22×Dic15C2312yes—
67C2×C4×D15—24no—
68C2×C4×D15—24no—
69C2×C4×D15—24no—
70C2×C4×D15—24no—
71C2×C4×D15—42no—
72C2×C4×D15—42no—
73C3×C5×C22⋊C4—24no—
74C3×C5×C22⋊C4—41no—
75#6719—24no—
76#6719—24no—
77#6719C2×C414yescomplemented
78#6714—24no—
79#6714C2×C418yescomplemented
80#6714D418yescomplemented
81C4⋊Dic15—41no—
82#6712—24no—
83C4×Dic15—24no—
84C4×Dic15—41no—
85C2×#937—51no—
86C2×C4×#67—52no—
87C2×D3×C22⋊C4—52no—
88C3×#386D511yescharacteristic, complemented
89#5221—51no—
90#4956—52no—
91#4945—52no—
92#4668—52no—
93#4666—52no—
94C24×D5—31no—
95C22×C5×D4C2×D311yescharacteristic
96C23×C4×C5C2×D311yescharacteristic, complemented
97C22×C22⋊D5—32no—
98C22×C4×D5—32no—
99C22×C4×D5—32no—
100C2×C5×C22⋊C4C2×D312yes—
101C2×C5×C22⋊C4C2×D311yescharacteristic, complemented
102C2×C22⋊Dic5—31no—
103C2×C22⋊Dic5—32no—
104C2×C22⋊Dic5—32no—
105#4083—38no—
106C4×C22⋊D5—68no—
107C2×#685—32no—
108C2×#685—32no—
109C2×#685—32no—
110C2×C4⋊Dic5—31no—
111C2×#683—32no—
112C2×C4×Dic5—32no—
113C2×C4×Dic5—31no—
114D5×C22⋊C4—68no—
115C5×#95C2×D312yescomplemented
116#3963—32no—
117#3963—32no—
118#2358—151noSylow
119C23×C3×C5—22no—
120C23×C3×C5—22no—
121C23×C3×C5—22no—
122C23×C3×C5—22no—
123C23×C3×C5—21no—
124C23×C3×C5—21no—
125C23×C3×C5—21no—
126C23×C3×C5—21no—
127C23×C3×C5C22×C412yes—
128C23×C3×C5C22×C412yes—
129C23×C3×C5C2411yescharacteristic
130C22×D15—22no—
131C22×D15—22no—
132C22×D15—22no—
133C22×D15—22no—
134C22×D15—216no—
135C22×D15—44no—
136C22×D15—44no—
137C22×D15—48no—
138C22×D15—44no—
139C22×D15C22×C414yes—
140C22×D15C2×D414yes—
141C22×D15#4114yes—
142C3×C5×D4—44no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5—21no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5—22no—
150C2×C3×C4×C5—44no—
151C2×C3×C4×C5C22×C412yes—
152C2×C3×C4×C5C2×D414yes—
153#1220—216no—
154C2×Dic15—22no—
155C2×Dic15—22no—
156C2×Dic15—22no—
157C2×Dic15—22no—
158C2×Dic15—22no—
159C2×Dic15—22no—
160C2×Dic15—22no—
161C2×Dic15C22×C414yes—
162C2×Dic15C2×D414yes—
163C4×D15—48no—
164C4×D15—48no—
165C24×D3—51no—
166C22×C3×D4C2×D511yescharacteristic
167C23×C3×C4C2×D511yescharacteristic, complemented
168C22×#67—52no—
169C22×C4×D3—52no—
170C22×C4×D3—52no—
171C2×C3×C22⋊C4C2×D512yes—
172C2×C3×C22⋊C4C2×D511yescharacteristic, complemented
173C2×#217—51no—
174C2×#217—52no—
175C2×#217—52no—
176#937—58no—
177C4×#67—108no—
178C2×(C2×D3)⋊C4—52no—
179C2×(C2×D3)⋊C4—52no—
180C2×(C2×D3)⋊C4—52no—
181C2×C4⋊Dic3—51no—
182C2×Dic3⋊C4—52no—
183C2×C4×Dic3—52no—
184C2×C4×Dic3—51no—
185D3×C22⋊C4—108no—
186C3×#95C2×D512yescomplemented
187C3⋊#95—52no—
188C3⋊#95—52no—
189C24×C5C22×D311yescharacteristic
190C24×C5C22×D311yescharacteristic
191C23×D5—32no—
192C23×D5—34no—
193C23×D5—64no—
194C23×D5—64no—
195C23×D5—121no—
196C23×D5—121no—
197C2×C5×D4—41no—
198C2×C5×D4—41no—
199C2×C5×D4C4×D314yescomplemented
200C22×C4×C5—24no—
201C22×C4×C5—41no—
202C22×C4×C5C22×D312yes—
203C22×C4×C5C22×D311yescharacteristic
204C22×C4×C5C22×D312yes—
205C2×C22⋊D5—38no—
206C2×C22⋊D5—64no—
207C2×C22⋊D5—64no—
208C22×Dic5—32no—
209C22×Dic5—32no—
210C22×Dic5—32no—
211C2×C4×D5—64no—
212C2×C4×D5—64no—
213C2×C4×D5—64no—
214C2×C4×D5—64no—
215C2×C4×D5—122no—
216C2×C4×D5—122no—
217C5×C22⋊C4—24no—
218C5×C22⋊C4—41no—
219C22⋊Dic5—34no—
220C22⋊Dic5—64no—
221C22⋊Dic5—64no—
222#685—38no—
223#685—38no—
224#685—64no—
225C4⋊Dic5—121no—
226#683—64no—
227C4×Dic5—64no—
228C4×Dic5—121no—
229C2×#120—151no—
230C2×C4×D4—152no—
231C22×C22⋊C4—152no—
232#390—151no—
233#388—152no—
234#386—152no—
235#386D1511yescharacteristic, complemented
236#385—152no—
237C4×C22⋊C4—152no—
238C22×C3×C5—24no—
239C22×C3×C5—24no—
240C22×C3×C5—22no—
241C22×C3×C5—22no—
242C22×C3×C5—41no—
243C22×C3×C5—41no—
244C22×C3×C5—42no—
245C22×C3×C5—42no—
246C22×C3×C5—41no—
247C22×C3×C5—41no—
248C22×C3×C5C23×C411yescharacteristic
249C22×C3×C5C22×D412yes—
250C22×C3×C5C2×#4112yes—
251C22×C3×C5H(5,𝔽2)11yescharacteristic
252C22×C3×C5H(5,𝔽2)11yescharacteristic
253C2×D15—24no—
254C2×D15—24no—
255C2×D15—24no—
256C2×D15—24no—
257C2×D15—24no—
258C2×D15—24no—
259C2×D15—416no—
260C2×D15C4×D418yescomplemented
261C3×C4×C5—22no—
262C3×C4×C5—24no—
263C3×C4×C5—42no—
264Dic15—24no—
265Dic15—24no—
266Dic15—42no—
267C24×C3C22×D511yescharacteristic
268C24×C3C22×D511yescharacteristic
269C23×D3—52no—
270C23×D3—54no—
271C23×D3—104no—
272C23×D3—104no—
273C23×D3—201no—
274C23×D3—201no—
275C2×C3×D4—41no—
276C2×C3×D4—41no—
277C2×C3×D4C4×D514yescomplemented
278C22×C3×C4—24no—
279C22×C3×C4—41no—
280C22×C3×C4C22×D512yes—
281C22×C3×C4C22×D511yescharacteristic
282C22×C3×C4C22×D512yes—
283C2×#67—58no—
284C2×#67—104no—
285C2×#67—104no—
286C22×Dic3—52no—
287C22×Dic3—52no—
288C22×Dic3—52no—
289C2×C4×D3—104no—
290C2×C4×D3—104no—
291C2×C4×D3—104no—
292C2×C4×D3—104no—
293C2×C4×D3—202no—
294C2×C4×D3—202no—
295C3×C22⋊C4—24no—
296C3×C22⋊C4—41no—
297#217—54no—
298#217—104no—
299#217—104no—
300(C2×D3)⋊C4—58no—
301(C2×D3)⋊C4—58no—
302(C2×D3)⋊C4—104no—
303C4⋊Dic3—201no—
304Dic3⋊C4—104no—
305C4×Dic3—104no—
306C4×Dic3—201no—
307C23×C5—22no—
308C23×C5—22no—
309C23×C5—21no—
310C23×C5—21no—
311C23×C5—21no—
312C23×C5—21no—
313C23×C5—22no—
314C23×C5—22no—
315C23×C5C2×C4×D312yes—
316C23×C5C2×C4×D312yes—
317C23×C5C23×D311yescharacteristic
318C22×D5—34no—
319C22×D5—34no—
320C22×D5—34no—
321C22×D5—616no—
322C22×D5—62no—
323C22×D5—62no—
324C22×D5—62no—
325C22×D5—62no—
326C22×D5—124no—
327C22×D5—124no—
328C22×D5—124no—
329C22×D5—128no—
330C5×D4—44no—
331C2×C4×C5—22no—
332C2×C4×C5—21no—
333C2×C4×C5—21no—
334C2×C4×C5—22no—
335C2×C4×C5—22no—
336C2×C4×C5—22no—
337C2×C4×C5—22no—
338C2×C4×C5—44no—
339C2×C4×C5C2×C4×D312yes—
340C2×C4×C5D3×D414yes—
341C22⋊D5—616no—
342C2×Dic5—34no—
343C2×Dic5—34no—
344C2×Dic5—62no—
345C2×Dic5—62no—
346C2×Dic5—62no—
347C2×Dic5—62no—
348C2×Dic5—62no—
349C2×Dic5—62no—
350C2×Dic5—62no—
351C4×D5—128no—
352C4×D5—128no—
353C25—151no—
354C22×D4—152no—
355C22×D4C2×D1511yescharacteristic
356C23×C4—152no—
357C23×C4—152no—
358C23×C4C2×D1511yescharacteristic, complemented
359#120—158no—
360C4×D4—308no—
361C2×C4⋊C4—152no—
362C2×C4⋊C4—151no—
363C2×C22⋊C4—151no—
364C2×C22⋊C4—152no—
365C2×C22⋊C4—152no—
366C2×C22⋊C4—152no—
367C2×C22⋊C4—152no—
368C2×C22⋊C4—152no—
369C2×C22⋊C4—308no—
370C2×C22⋊C4C2×D1512yes—
371C2×C22⋊C4C2×D1511yescharacteristic, complemented
372C2×C42—152no—
373C2×C42—151no—
374#95—152no—
375#95—152no—
376#95C2×D1512yescomplemented
377C2×C3×C5—42no—
378C2×C3×C5—42no—
379C2×C3×C5C2×C4×D412yes—
380C2×C3×C5#51811yescharacteristic
381C2×C3×C5D4211yescharacteristic
382C2×C3×C5#54612yes—
383C2×C3×C5#55111yescharacteristic
384D15—28no—
385C23×C3—22no—
386C23×C3—22no—
387C23×C3—21no—
388C23×C3—21no—
389C23×C3—21no—
390C23×C3—21no—
391C23×C3—22no—
392C23×C3—22no—
393C23×C3C2×C4×D512yes—
394C23×C3C2×C4×D512yes—
395C23×C3C23×D511yescharacteristic
396C22×D3—54no—
397C22×D3—54no—
398C22×D3—54no—
399C22×D3—1016no—
400C22×D3—102no—
401C22×D3—102no—
402C22×D3—102no—
403C22×D3—102no—
404C22×D3—204no—
405C22×D3—204no—
406C22×D3—208no—
407C22×D3—204no—
408C3×D4—44no—
409C2×C3×C4—22no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—44no—
417C2×C3×C4C2×C4×D512yes—
418C2×C3×C4D4×D514yes—
419#67—1016no—
420C2×Dic3—54no—
421C2×Dic3—54no—
422C2×Dic3—102no—
423C2×Dic3—102no—
424C2×Dic3—102no—
425C2×Dic3—102no—
426C2×Dic3—102no—
427C2×Dic3—102no—
428C2×Dic3—102no—
429C4×D3—208no—
430C4×D3—208no—
431C22×C5—22no—
432C22×C5—22no—
433C22×C5—24no—
434C22×C5—24no—
435C22×C5—41no—
436C22×C5—41no—
437C22×C5—41no—
438C22×C5—41no—
439C22×C5—42no—
440C22×C5—42no—
441C22×C5C22×C4×D311yescharacteristic
442C22×C5C2×D3×D412yes—
443C22×C5#100411yescharacteristic
444C22×C5#100411yescharacteristic
445C22×C5D3×#4112yes—
446C2×D5—38no—
447C2×D5—64no—
448C2×D5—64no—
449C2×D5—64no—
450C2×D5—64no—
451C2×D5—64no—
452C2×D5—64no—
453C2×D5—1216no—
454C4×C5—22no—
455C4×C5—24no—
456C4×C5—42no—
457Dic5—64no—
458Dic5—64no—
459Dic5—122no—
460C24—152no—
461C24—154no—
462C24—304no—
463C24—304no—
464C24—601no—
465C24—601no—
466C24C22×D1511yescharacteristic
467C24C22×D1511yescharacteristic
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—158no—
471C2×D4—304no—
472C2×D4—304no—
473C2×D4C4×D1514yescomplemented
474C22×C4—24no—
475C22×C4—41no—
476C22×C4—152no—
477C22×C4—152no—
478C22×C4—152no—
479C22×C4—304no—
480C22×C4—304no—
481C22×C4—304no—
482C22×C4—304no—
483C22×C4—602no—
484C22×C4—602no—
485C22×C4C22×D1512yes—
486C22×C4C22×D1511yescharacteristic
487C22×C4C22×D1512yes—
488C4⋊C4—304no—
489C4⋊C4—601no—
490C22⋊C4—24no—
491C22⋊C4—41no—
492C22⋊C4—154no—
493C22⋊C4—158no—
494C22⋊C4—158no—
495C22⋊C4—304no—
496C22⋊C4—304no—
497C22⋊C4—304no—
498C42—304no—
499C42—601no—
500C3×C5#235811yescharacteristic, complemented, Hall
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—24no—
504C22×C3—24no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3C22×C4×D511yescharacteristic
512C22×C3C2×D4×D512yes—
513C22×C3#414411yescharacteristic
514C22×C3#414411yescharacteristic
515C22×C3D5×#4112yes—
516C2×D3—58no—
517C2×D3—104no—
518C2×D3—104no—
519C2×D3—104no—
520C2×D3—104no—
521C2×D3—104no—
522C2×D3—104no—
523C2×D3—2016no—
524C3×C4—22no—
525C3×C4—24no—
526C3×C4—42no—
527Dic3—104no—
528Dic3—104no—
529Dic3—202no—
530C2×C5—42no—
531C2×C5—42no—
532C2×C5#548611yescharacteristic
533C2×C5C4×D3×D412yes—
534C2×C5#559011yescharacteristic
535C2×C5#561212yes—
536C2×C5#565911yescharacteristic
537D5—68no—
538C23—22no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—154no—
547C23—154no—
548C23—154no—
549C23—3016no—
550C23—302no—
551C23—302no—
552C23—302no—
553C23—302no—
554C23—604no—
555C23—604no—
556C23—608no—
557C23—604no—
558C23C2×C4×D1512yes—
559C23C2×C4×D1512yes—
560C23C23×D1511yescharacteristic
561D4—44no—
562D4—3016no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—44no—
571C2×C4—154no—
572C2×C4—154no—
573C2×C4—302no—
574C2×C4—302no—
575C2×C4—302no—
576C2×C4—302no—
577C2×C4—302no—
578C2×C4—302no—
579C2×C4—302no—
580C2×C4—608no—
581C2×C4—608no—
582C2×C4C2×C4×D1512yes—
583C2×C4D4×D1514yes—
584C2×C3—42no—
585C2×C3—42no—
586C2×C3#6582011yescharacteristic
587C2×C3C4×D4×D512yes—
588C2×C3#6592411yescharacteristic
589C2×C3#6594612yes—
590C2×C3#6599311yescharacteristic
591D3—108no—
592C5#7488011yescharacteristic, complemented, Sylow
593C22—24no—
594C22—24no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—42no—
601C22—41no—
602C22—41no—
603C22—158no—
604C22—304no—
605C22—304no—
606C22—304no—
607C22—304no—
608C22—304no—
609C22—304no—
610C22—6016no—
611C22C22×C4×D1511yescharacteristic
612C22C2×D4×D1512yes—
613C22#9209511yescharacteristic
614C22#9209511yescharacteristic
615C22#41×D1512yes—
616C4—24no—
617C4—22no—
618C4—42no—
619C4—304no—
620C4—304no—
621C4—602no—
622C3#1060661111yescharacteristic, complemented, Sylow
623C2—42no—
624C2—42no—
625C2—308no—
626C2#1175624511yescharacteristic
627C2C4×D4×D1512yes—
628C2#1175634911yescharacteristic
629C2#1175637112yes—
630C2#1175641811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#386C2
socle, cosocleC23×C3×C5C24
prime radicalC2×#91826C2
hypercenter#386D15
hypocenterC3×C5#2358
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×#1220C2semidirect
3. C2×C22⋊C4×D15C2semidirect
4. C3×C5×#386C2semidirect
5. #11751322C2semidirect
6. #11751057C2semidirect
7. #11751046C2semidirect
8. #11750769C2semidirect
9. #11750767C2semidirect
13. C23×C3×C4×C5C22semidirect
18. C2×C3×C5×C22⋊C4C22semidirect
21. C2×#6719C22semidirect
22. #91826C4semidirect
27. C2×C4⋊Dic15C22semidirect
28. C2×#6712C22semidirect
29. C2×C4×Dic15C22semidirect
30. C2×C4×Dic15C22semidirect
32. C3×C5×#95C22semidirect
33. #91115C22semidirect
34. #91115C22semidirect
39. C5×#386D3semidirect
51. C23×D15C2×C4semidirect
55. C2×C3×C5×D4C2×C4semidirect
63. C2×#1220C2×C4semidirect
77. #6719C2×C4semidirect
79. #6714C2×C4semidirect
80. #6714D4semidirect
88. C3×#386D5semidirect
96. C23×C4×C5C2×D3semidirect
101. C2×C5×C22⋊C4C2×D3semidirect
115. C5×#95C2×D3semidirect
167. C23×C3×C4C2×D5semidirect
172. C2×C3×C22⋊C4C2×D5semidirect
186. C3×#95C2×D5semidirect
199. C2×C5×D4C4×D3semidirect
235. #386D15semidirect
260. C2×D15C4×D4semidirect
277. C2×C3×D4C4×D5semidirect
358. C23×C4C2×D15semidirect
371. C2×C22⋊C4C2×D15semidirect
376. #95C2×D15semidirect
473. C2×D4C4×D15semidirect
500. C3×C5#2358semidirect
592. C5#74880semidirect
622. C3#10606611semidirect

Maximal subgroups

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

subgroupindex
C2×#918262
C2×C4×#12202
C2×C22⋊C4×D152
C3×C5×#3862
#117513222
#117510572
#117510462
#117507692
#117507672
#106066113
#748805