Finite Group Databasebeta

← #14328118

Subgroups — #14328118

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×C3×C5×#120C211yesmaximal, characteristic
2C22×#6719C212yesmaximal, complemented
3C2×D4×Dic15C212yesmaximal, complemented
4#11751322C211yesmaximal, characteristic, complemented
5#11751321C211yesmaximal, characteristic, complemented
6#11751321C212yesmaximal, complemented
7#11751052C212yesmaximal, complemented
8#11751051C212yesmaximal, complemented
9C22⋊C4×Dic15C212yesmaximal, complemented
10#10614310—31nomaximal, Hall
11C25×C3×C5C2211yescharacteristic
12C22×C3×C5×D4C2212yes—
13C22×C3×C5×D4C2211yescharacteristic
14C23×Dic15C2212yes—
15C23×Dic15C2212yes—
16C23×Dic15C2211yescharacteristic, complemented
17C3×C5×#120C418yescomplemented
18C2×C3×C5×C22⋊C4C2212yes—
19C2×C3×C5×C22⋊C4C2211yescharacteristic
20C2×#6719—28no—
21C2×#6719C2212yes—
22C2×#6719C2212yes—
23C2×#6719C2212yes—
24C2×#6719C2212yes—
25C2×#6719C2212yescomplemented
26C2×#6719C2211yescharacteristic, complemented
27D4×Dic15—28no—
28C2×C4⋊Dic15C2212yescomplemented
29C2×C4⋊Dic15C2211yescharacteristic, complemented
30C2×C4×Dic15C2212yescomplemented
31C2×C4×Dic15C2211yescharacteristic, complemented
32#91115C2212yescomplemented
33#91115C2212yescomplemented
34#91115C2212yescomplemented
35#82579—51nomaximal, Hall
36C2×C5×#120D311yescharacteristic
37C22×C22⋊Dic5—32no—
38C2×D4×Dic5—32no—
39#65510—31no—
40#65509—32no—
41#65509—31no—
42#65240—32no—
43#65239—32no—
44C22⋊C4×Dic5—32no—
45C24×C3×C5—24no—
46C24×C3×C5—24no—
47C24×C3×C5—41no—
48C24×C3×C5—41no—
49C24×C3×C5C2×C414yescomplemented
50C24×C3×C5C2311yescharacteristic
51C24×C3×C5C2311yescharacteristic
52C24×C3×C5C2312yes—
53C2×C3×C5×D4—24no—
54C2×C3×C5×D4—24no—
55C2×C3×C5×D4—41no—
56C2×C3×C5×D4—41no—
57C2×C3×C5×D4C2×C418yescomplemented
58C2×C3×C5×D4C2×C414yescomplemented
59C22×C3×C4×C5C2312yes—
60C22×C3×C4×C5C2311yescharacteristic
61C22×Dic15—24no—
62C22×Dic15—24no—
63C22×Dic15—24no—
64C22×Dic15—24no—
65C22×Dic15—24no—
66C22×Dic15—42no—
67C22×Dic15—42no—
68C22×Dic15—41no—
69C22×Dic15C2312yes—
70C22×Dic15C2312yes—
71C22×Dic15C2312yes—
72C22×Dic15C2312yes—
73C3×C5×C22⋊C4C2×C414yescomplemented
74C3×C5×C22⋊C4C2×C418yescomplemented
75#6719—24no—
76#6719—24no—
77#6719—24no—
78#6719—24no—
79#6719—41no—
80#6719D418yescomplemented
81C4⋊Dic15—24no—
82C4⋊Dic15—41no—
83C4×Dic15—24no—
84C4×Dic15—41no—
85C2×C3×#120D511yescharacteristic
86C22×#217—52no—
87C2×D4×Dic3—52no—
88#5221—51no—
89#5220—51no—
90#5220—52no—
91#4951—52no—
92#4950—52no—
93Dic3×C22⋊C4—52no—
94C25×C5C2×D311yescharacteristic
95C22×C5×D4C2×D312yes—
96C22×C5×D4C2×D311yescharacteristic
97C23×Dic5—31no—
98C23×Dic5—32no—
99C23×Dic5—32no—
100C5×#120Dic318yescomplemented
101C2×C5×C22⋊C4C2×D312yes—
102C2×C5×C22⋊C4C2×D311yescharacteristic
103C2×C22⋊Dic5—32no—
104C2×C22⋊Dic5—32no—
105C2×C22⋊Dic5—32no—
106C2×C22⋊Dic5—32no—
107C2×C22⋊Dic5—32no—
108C2×C22⋊Dic5—31no—
109C2×C22⋊Dic5—68no—
110D4×Dic5—68no—
111C2×C4⋊Dic5—32no—
112C2×C4⋊Dic5—31no—
113C2×C4×Dic5—32no—
114C2×C4×Dic5—31no—
115#3963—32no—
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—216no—
124C23×C3×C5—22no—
125C23×C3×C5—22no—
126C23×C3×C5—22no—
127C23×C3×C5—22no—
128C23×C3×C5—21no—
129C23×C3×C5—21no—
130C23×C3×C5—21no—
131C23×C3×C5—21no—
132C23×C3×C5—44no—
133C23×C3×C5—44no—
134C23×C3×C5—48no—
135C23×C3×C5—44no—
136C23×C3×C5C22×C412yes—
137C23×C3×C5C22×C412yes—
138C23×C3×C5C22×C414yes—
139C23×C3×C5C2×D414yes—
140C23×C3×C5#4114yes—
141C23×C3×C5C2411yescharacteristic
142C3×C5×D4—216no—
143C3×C5×D4—44no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5—21no—
147C2×C3×C4×C5—21no—
148C2×C3×C4×C5C22×C414yes—
149C2×C3×C4×C5C22×C412yes—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15—22no—
154C2×Dic15—22no—
155C2×Dic15—22no—
156C2×Dic15—22no—
157C2×Dic15—22no—
158C2×Dic15—22no—
159C2×Dic15—22no—
160C2×Dic15—44no—
161C2×Dic15—48no—
162C2×Dic15—48no—
163C2×Dic15C2×D414yes—
164C2×Dic15C2×D414yes—
165C25×C3C2×D511yescharacteristic
166C22×C3×D4C2×D512yes—
167C22×C3×D4C2×D511yescharacteristic
168C23×Dic3—52no—
169C23×Dic3—52no—
170C23×Dic3—51no—
171C3×#120Dic518yescomplemented
172C2×C3×C22⋊C4C2×D512yes—
173C2×C3×C22⋊C4C2×D511yescharacteristic
174C2×#217—52no—
175C2×#217—52no—
176C2×#217—52no—
177C2×#217—52no—
178C2×#217—52no—
179C2×#217—51no—
180C2×#217—108no—
181D4×Dic3—108no—
182C2×C4⋊Dic3—52no—
183C2×C4⋊Dic3—51no—
184C2×C4×Dic3—52no—
185C2×C4×Dic3—51no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C3⋊#95—52no—
189C24×C5—24no—
190C24×C5—24no—
191C24×C5—41no—
192C24×C5—41no—
193C24×C5C2×Dic314yescomplemented
194C24×C5C22×D311yescharacteristic
195C24×C5C22×D311yescharacteristic
196C24×C5C22×D312yes—
197C2×C5×D4—24no—
198C2×C5×D4—24no—
199C2×C5×D4—41no—
200C2×C5×D4—41no—
201C2×C5×D4C2×Dic318yescomplemented
202C2×C5×D4C2×Dic314yescomplemented
203C22×C4×C5C22×D312yes—
204C22×C4×C5C22×D311yescharacteristic
205C22×Dic5—32no—
206C22×Dic5—32no—
207C22×Dic5—32no—
208C22×Dic5—32no—
209C22×Dic5—64no—
210C22×Dic5—64no—
211C22×Dic5—64no—
212C22×Dic5—64no—
213C22×Dic5—64no—
214C22×Dic5—122no—
215C22×Dic5—121no—
216C22×Dic5—122no—
217C5×C22⋊C4C2×Dic314yescomplemented
218C5×C22⋊C4C2×Dic318yescomplemented
219C22⋊Dic5—38no—
220C22⋊Dic5—64no—
221C22⋊Dic5—64no—
222C22⋊Dic5—64no—
223C22⋊Dic5—64no—
224C22⋊Dic5—121no—
225C4⋊Dic5—64no—
226C4⋊Dic5—121no—
227C4×Dic5—64no—
228C4×Dic5—121no—
229C2×#120D1511yescharacteristic
230C2×C4×D4—152no—
231C22×C22⋊C4—152no—
232#390—151no—
233#388—152no—
234#386—151no—
235#386—152no—
236#385—152no—
237C4×C22⋊C4—152no—
238C22×C3×C5—24no—
239C22×C3×C5—24no—
240C22×C3×C5—24no—
241C22×C3×C5—24no—
242C22×C3×C5—24no—
243C22×C3×C5—24no—
244C22×C3×C5—24no—
245C22×C3×C5—24no—
246C22×C3×C5—22no—
247C22×C3×C5—22no—
248C22×C3×C5—416no—
249C22×C3×C5—41no—
250C22×C3×C5—41no—
251C22×C3×C5—42no—
252C22×C3×C5—42no—
253C22×C3×C5—41no—
254C22×C3×C5—41no—
255C22×C3×C5C4×D418yescomplemented
256C22×C3×C5C23×C411yescharacteristic
257C22×C3×C5C22×D412yes—
258C22×C3×C5C2×#4112yes—
259C22×C3×C5H(5,𝔽2)11yescharacteristic
260C22×C3×C5H(5,𝔽2)11yescharacteristic
261C3×C4×C5—24no—
262C3×C4×C5—22no—
263Dic15—24no—
264Dic15—24no—
265Dic15—42no—
266Dic15—42no—
267C24×C3—24no—
268C24×C3—24no—
269C24×C3—41no—
270C24×C3—41no—
271C24×C3C2×Dic514yescomplemented
272C24×C3C22×D511yescharacteristic
273C24×C3C22×D511yescharacteristic
274C24×C3C22×D512yes—
275C2×C3×D4—24no—
276C2×C3×D4—24no—
277C2×C3×D4—41no—
278C2×C3×D4—41no—
279C2×C3×D4C2×Dic518yescomplemented
280C2×C3×D4C2×Dic514yescomplemented
281C22×C3×C4C22×D512yes—
282C22×C3×C4C22×D511yescharacteristic
283C22×Dic3—52no—
284C22×Dic3—52no—
285C22×Dic3—52no—
286C22×Dic3—52no—
287C22×Dic3—104no—
288C22×Dic3—104no—
289C22×Dic3—104no—
290C22×Dic3—104no—
291C22×Dic3—104no—
292C22×Dic3—202no—
293C22×Dic3—201no—
294C22×Dic3—202no—
295C3×C22⋊C4C2×Dic514yescomplemented
296C3×C22⋊C4C2×Dic518yescomplemented
297#217—58no—
298#217—104no—
299#217—104no—
300#217—104no—
301#217—104no—
302#217—201no—
303C4⋊Dic3—104no—
304C4⋊Dic3—201no—
305C4×Dic3—104no—
306C4×Dic3—201no—
307C23×C5—22no—
308C23×C5—22no—
309C23×C5—22no—
310C23×C5—22no—
311C23×C5—216no—
312C23×C5—22no—
313C23×C5—22no—
314C23×C5—22no—
315C23×C5—22no—
316C23×C5—21no—
317C23×C5—21no—
318C23×C5—21no—
319C23×C5—21no—
320C23×C5—44no—
321C23×C5—44no—
322C23×C5—44no—
323C23×C5—48no—
324C23×C5D3×D414yes—
325C23×C5#23714yes—
326C23×C5C22×Dic312yes—
327C23×C5C22×Dic312yes—
328C23×C5C22×Dic314yes—
329C23×C5C23×D311yescharacteristic
330C5×D4—216no—
331C5×D4—44no—
332C2×C4×C5—22no—
333C2×C4×C5—22no—
334C2×C4×C5—21no—
335C2×C4×C5—21no—
336C2×C4×C5C22×Dic314yes—
337C2×C4×C5C22×Dic312yes—
338C2×Dic5—34no—
339C2×Dic5—34no—
340C2×Dic5—62no—
341C2×Dic5—62no—
342C2×Dic5—62no—
343C2×Dic5—62no—
344C2×Dic5—62no—
345C2×Dic5—62no—
346C2×Dic5—62no—
347C2×Dic5—62no—
348C2×Dic5—62no—
349C2×Dic5—62no—
350C2×Dic5—124no—
351C2×Dic5—128no—
352C2×Dic5—128no—
353C25C2×D1511yescharacteristic
354C22×D4C2×D1512yes—
355C22×D4C2×D1511yescharacteristic
356C23×C4—152no—
357C23×C4—152no—
358C23×C4—151no—
359#120Dic1518yescomplemented
360C4×D4—308no—
361C2×C4⋊C4—152no—
362C2×C4⋊C4—151no—
363C2×C22⋊C4—152no—
364C2×C22⋊C4—152no—
365C2×C22⋊C4—152no—
366C2×C22⋊C4—152no—
367C2×C22⋊C4—152no—
368C2×C22⋊C4—151no—
369C2×C22⋊C4—308no—
370C2×C22⋊C4C2×D1512yes—
371C2×C22⋊C4C2×D1511yescharacteristic
372C2×C42—152no—
373C2×C42—151no—
374#95—152no—
375#95—152no—
376#95—152no—
377C2×C3×C5—28no—
378C2×C3×C5—42no—
379C2×C3×C5—42no—
380C2×C3×C5C2×C4×D412yes—
381C2×C3×C5#51811yescharacteristic
382C2×C3×C5D4211yescharacteristic
383C2×C3×C5#54612yes—
384C2×C3×C5#55111yescharacteristic
385C23×C3—22no—
386C23×C3—22no—
387C23×C3—22no—
388C23×C3—22no—
389C23×C3—22no—
390C23×C3—22no—
391C23×C3—22no—
392C23×C3—22no—
393C23×C3—21no—
394C23×C3—21no—
395C23×C3—21no—
396C23×C3—21no—
397C23×C3—216no—
398C23×C3—44no—
399C23×C3—44no—
400C23×C3—44no—
401C23×C3—48no—
402C23×C3D4×D514yes—
403C23×C3#71114yes—
404C23×C3C22×Dic512yes—
405C23×C3C22×Dic512yes—
406C23×C3C22×Dic514yes—
407C23×C3C23×D511yescharacteristic
408C3×D4—216no—
409C3×D4—44no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4C22×Dic514yes—
415C2×C3×C4C22×Dic512yes—
416C2×Dic3—54no—
417C2×Dic3—54no—
418C2×Dic3—102no—
419C2×Dic3—102no—
420C2×Dic3—102no—
421C2×Dic3—102no—
422C2×Dic3—102no—
423C2×Dic3—102no—
424C2×Dic3—102no—
425C2×Dic3—102no—
426C2×Dic3—102no—
427C2×Dic3—102no—
428C2×Dic3—204no—
429C2×Dic3—208no—
430C2×Dic3—208no—
431C22×C5—24no—
432C22×C5—24no—
433C22×C5—24no—
434C22×C5—24no—
435C22×C5—24no—
436C22×C5—24no—
437C22×C5—24no—
438C22×C5—24no—
439C22×C5—22no—
440C22×C5—22no—
441C22×C5—416no—
442C22×C5—41no—
443C22×C5—41no—
444C22×C5—41no—
445C22×C5—41no—
446C22×C5—42no—
447C22×C5—42no—
448C22×C5D4×Dic318yescomplemented
449C22×C5C2×D3×D412yes—
450C22×C5C2×#23712yes—
451C22×C5#100411yescharacteristic
452C22×C5#100411yescharacteristic
453C22×C5C23×Dic311yescharacteristic
454C4×C5—24no—
455C4×C5—22no—
456Dic5—64no—
457Dic5—64no—
458Dic5—122no—
459Dic5—122no—
460C24—24no—
461C24—24no—
462C24—41no—
463C24—41no—
464C24C2×Dic1514yescomplemented
465C24C22×D1511yescharacteristic
466C24C22×D1511yescharacteristic
467C24C22×D1512yes—
468C2×D4—24no—
469C2×D4—24no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4C2×Dic1518yescomplemented
473C2×D4C2×Dic1514yescomplemented
474C22×C4—152no—
475C22×C4—152no—
476C22×C4—152no—
477C22×C4—152no—
478C22×C4—304no—
479C22×C4—304no—
480C22×C4—304no—
481C22×C4—304no—
482C22×C4—304no—
483C22×C4—602no—
484C22×C4—601no—
485C22×C4—602no—
486C22×C4C22×D1512yes—
487C22×C4C22×D1511yescharacteristic
488C4⋊C4—304no—
489C4⋊C4—601no—
490C22⋊C4—158no—
491C22⋊C4—304no—
492C22⋊C4—304no—
493C22⋊C4—304no—
494C22⋊C4—304no—
495C22⋊C4—601no—
496C22⋊C4C2×Dic1514yescomplemented
497C22⋊C4C2×Dic1518yescomplemented
498C42—304no—
499C42—601no—
500C3×C5#235811yescharacteristic, complemented, Hall
501C22×C3—24no—
502C22×C3—24no—
503C22×C3—24no—
504C22×C3—24no—
505C22×C3—24no—
506C22×C3—24no—
507C22×C3—24no—
508C22×C3—24no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—416no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3D4×Dic518yescomplemented
519C22×C3C2×D4×D512yes—
520C22×C3C2×#71112yes—
521C22×C3#414411yescharacteristic
522C22×C3#414411yescharacteristic
523C22×C3C23×Dic511yescharacteristic
524C3×C4—24no—
525C3×C4—22no—
526Dic3—104no—
527Dic3—104no—
528Dic3—202no—
529Dic3—202no—
530C2×C5—28no—
531C2×C5—42no—
532C2×C5—42no—
533C2×C5#558711yescharacteristic
534C2×C5#559011yescharacteristic
535C2×C5#559112yes—
536C2×C5C2×D4×Dic312yes—
537C2×C5#579811yescharacteristic
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—216no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—48no—
555C23D4×D1514yes—
556C23#681914yes—
557C23C22×Dic1512yes—
558C23C22×Dic1512yes—
559C23C22×Dic1514yes—
560C23C23×D1511yescharacteristic
561D4—216no—
562D4—44no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—154no—
568C2×C4—154no—
569C2×C4—302no—
570C2×C4—302no—
571C2×C4—302no—
572C2×C4—302no—
573C2×C4—302no—
574C2×C4—302no—
575C2×C4—302no—
576C2×C4—302no—
577C2×C4—302no—
578C2×C4—302no—
579C2×C4—604no—
580C2×C4—608no—
581C2×C4—608no—
582C2×C4C22×Dic1514yes—
583C2×C4C22×Dic1512yes—
584C2×C3—28no—
585C2×C3—42no—
586C2×C3—42no—
587C2×C3#6592111yescharacteristic
588C2×C3#6592411yescharacteristic
589C2×C3#6592512yes—
590C2×C3C2×D4×Dic512yes—
591C2×C3#6613211yescharacteristic
592C5#8257911yescharacteristic, complemented, Sylow
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—22no—
602C22—22no—
603C22—416no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—42no—
610C22D4×Dic1518yescomplemented
611C22C2×D4×D1512yes—
612C22C2×#681912yes—
613C22#9209511yescharacteristic
614C22#9209511yescharacteristic
615C22C23×Dic1511yescharacteristic
616C4—24no—
617C4—22no—
618C4—304no—
619C4—304no—
620C4—602no—
621C4—602no—
622C3#1061431011yescharacteristic, complemented, Sylow
623C2—28no—
624C2—42no—
625C2—42no—
626C2#1175634611yescharacteristic
627C2#1175634911yescharacteristic
628C2#1175635012yes—
629C2C2×D4×Dic1512yes—
630C2#1175655711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
Fitting, prime radicalC2×C3×C5×#120C2
socle, cosocleC23×C3×C5C24
hypercenterC2×#120D15
hypocenterC3×C5#2358
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. C22×#6719C2semidirect
3. C2×D4×Dic15C2semidirect
4. #11751322C2semidirect
5. #11751321C2semidirect
6. #11751321C2semidirect
7. #11751052C2semidirect
8. #11751051C2semidirect
9. C22⋊C4×Dic15C2semidirect
16. C23×Dic15C22semidirect
17. C3×C5×#120C4semidirect
25. C2×#6719C22semidirect
26. C2×#6719C22semidirect
28. C2×C4⋊Dic15C22semidirect
29. C2×C4⋊Dic15C22semidirect
30. C2×C4×Dic15C22semidirect
31. C2×C4×Dic15C22semidirect
32. #91115C22semidirect
33. #91115C22semidirect
34. #91115C22semidirect
49. C24×C3×C5C2×C4semidirect
57. C2×C3×C5×D4C2×C4semidirect
58. C2×C3×C5×D4C2×C4semidirect
73. C3×C5×C22⋊C4C2×C4semidirect
74. C3×C5×C22⋊C4C2×C4semidirect
80. #6719D4semidirect
100. C5×#120Dic3semidirect
171. C3×#120Dic5semidirect
193. C24×C5C2×Dic3semidirect
201. C2×C5×D4C2×Dic3semidirect
202. C2×C5×D4C2×Dic3semidirect
217. C5×C22⋊C4C2×Dic3semidirect
218. C5×C22⋊C4C2×Dic3semidirect
255. C22×C3×C5C4×D4semidirect
271. C24×C3C2×Dic5semidirect
279. C2×C3×D4C2×Dic5semidirect
280. C2×C3×D4C2×Dic5semidirect
295. C3×C22⋊C4C2×Dic5semidirect
296. C3×C22⋊C4C2×Dic5semidirect
359. #120Dic15semidirect
448. C22×C5D4×Dic3semidirect
464. C24C2×Dic15semidirect
472. C2×D4C2×Dic15semidirect
473. C2×D4C2×Dic15semidirect
496. C22⋊C4C2×Dic15semidirect
497. C22⋊C4C2×Dic15semidirect
500. C3×C5#2358semidirect
518. C22×C3D4×Dic5semidirect
592. C5#82579semidirect
610. C22D4×Dic15semidirect
622. C3#10614310semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1202
C22×#67192
C2×D4×Dic152
#117513222
#117513212
#117510522
#117510512
C22⋊C4×Dic152
#106143103
#825795