Finite Group Databasebeta

← #14326296

Subgroups — #14326296

Order 1920.

1584 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#877C211yesmaximal, characteristic, complemented
2C2×#91561C211yesmaximal, characteristic
3C22×#6667C211yesmaximal, characteristic
4#11751056C211yesmaximal, characteristic
5C3×#65231C211yesmaximal, characteristic, complemented
6Dic5×#217C211yesmaximal, characteristic, complemented
7#11749472C211yesmaximal, characteristic, complemented
8#11748894C212yesmaximal, complemented
9#11748881C212yesmaximal, complemented
10#11748728C212yesmaximal, complemented
11#11748669C212yesmaximal, complemented
12#10614002—31nomaximal, Hall
13C24×D15C2211yescharacteristic
14C23×C5×Dic3C2211yescharacteristic
15C23×C3×Dic5C2211yescharacteristic
16C22×#1200C2212yes—
17C2×C3×C5×C22⋊C4C2211yescharacteristic
18C2×#6719C2211yescharacteristic
19C2×#6714C2212yes—
20C2×#6714C2212yes—
21C2×C5×#217C2211yescharacteristic
22C2×C5×Dic3⋊C4C2212yescomplemented
23C2×C4×C5×Dic3C2212yescomplemented
24C5×#877—24no—
25C2×C3×C22⋊Dic5C2211yescharacteristic
26C2×C3×#683C2212yescomplemented
27#91561—24no—
28C2×#6667C418yescomplemented
29C2×#6667C2212yes—
30C2×#6667C2211yescharacteristic
31C2×#6667C2211yescharacteristic
32C2×#6667C2211yescharacteristic
33C2×#6667C2211yescharacteristic
34C2×Dic3×Dic5C2212yescomplemented
35C3×#3963C2212yescomplemented
36#90994C2212yescomplemented
37#90994C2212yescomplemented
38#90994C2212yescomplemented
39#74009—51nomaximal, Hall
40C2×C5×#117—31no—
41C22×#685—31no—
42C2×D5×C22⋊C4—31no—
43#65500—31no—
44#65279—32no—
45#65278—32no—
46#65244—31no—
47#65231D311yescharacteristic, complemented
48C22⋊C4×Dic5—31no—
49#65229—32no—
50C4×#685—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15—22no—
53C23×D15—22no—
54C23×D15—24no—
55C23×D15—42no—
56C23×D15C2×C414yes—
57C23×D15C2311yescharacteristic
58C23×D15C2311yescharacteristic
59C22×C3×C4×C5C2312yes—
60C22×Dic15C2312yes—
61C22×C5×Dic3—22no—
62C22×C5×Dic3—22no—
63C22×C5×Dic3C2×C414yescomplemented
64C22×C5×Dic3C2311yescharacteristic
65C22×C5×Dic3C2311yescharacteristic
66C22×C5×Dic3C2312yes—
67C22×C3×Dic5—42no—
68C22×C3×Dic5C2×C414yescomplemented
69C22×C3×Dic5C2312yes—
70C22×C3×Dic5C2311yescharacteristic
71C22×C3×Dic5C2311yescharacteristic
72C2×#1200—24no—
73C2×#1200—24no—
74C2×#1200—42no—
75C3×C5×C22⋊C4—22no—
76#6719—22no—
77#6714—24no—
78#6714—42no—
79C5×#217D414yes—
80C5×Dic3⋊C4—24no—
81C4×C5×Dic3—24no—
82C3×C22⋊Dic5—22no—
83C3×#683—42no—
84#6667—24no—
85#6667C2×C414yes—
86#6667C2×C414yes—
87#6667C2×C414yescomplemented
88#6667C2×C414yescomplemented
89Dic3×Dic5—42no—
90C22×(C2×D3)⋊C4—51no—
91C2×D3×C22⋊C4—51no—
92C2×#877D511yescharacteristic, complemented
93C3×#380—51no—
94C4×#217—51no—
95#4990—52no—
96#4989—52no—
97#4955—51no—
98#4943—51no—
99#4669—52no—
100#4666—52no—
101C24×D5—31no—
102C23×C4×C5—31no—
103C23×Dic5C2×D311yescharacteristic
104C22×C4×D5—32no—
105C5×#117—64no—
106C2×C5×C4⋊C4—32no—
107C2×C5×C22⋊C4—31no—
108C2×C5×C22⋊C4C2×D311yescharacteristic
109C2×C42×C5—32no—
110C2×C22⋊Dic5—31no—
111C2×C22⋊Dic5C2×D311yescharacteristic
112C2×#685—38no—
113C2×#685—31no—
114C2×#685—31no—
115C2×#685—31no—
116C2×#685—31no—
117C2×#685—32no—
118C2×#685—32no—
119C2×#685—32no—
120C2×#683C2×D312yescomplemented
121C2×C4×Dic5—32no—
122D5×C22⋊C4—64no—
123#3963—32no—
124#3963—32no—
125#3963—32no—
126#3963C2×D312yescomplemented
127#2342—151noSylow
128C23×C3×C5—22no—
129C23×C3×C5—22no—
130C23×C3×C5C22×C412yes—
131C23×C3×C5C2×D412yes—
132C23×C3×C5#4112yes—
133C23×C3×C5C2411yescharacteristic
134C22×D15—21no—
135C22×D15—21no—
136C22×D15—22no—
137C22×D15—28no—
138C22×D15—21no—
139C22×D15—22no—
140C22×D15—21no—
141C22×D15—22no—
142C22×D15—28no—
143C22×D15—44no—
144C22×D15—48no—
145C22×D15—48no—
146C22×D15C22×C412yes—
147C22×D15C22×C412yes—
148C22×D15#4114yes—
149C2×C3×C4×C5—22no—
150C2×C3×C4×C5—22no—
151C2×C3×C4×C5—22no—
152C2×Dic15—22no—
153C2×Dic15—22no—
154C2×Dic15—22no—
155C2×C5×Dic3—28no—
156C2×C5×Dic3—21no—
157C2×C5×Dic3—21no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3—21no—
160C2×C5×Dic3—21no—
161C2×C5×Dic3—22no—
162C2×C5×Dic3C22×C412yes—
163C2×C5×Dic3C22×C412yes—
164C2×C5×Dic3C2×D414yes—
165C2×C3×Dic5—22no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—22no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5—44no—
171C2×C3×Dic5C22×C412yes—
172C2×C3×Dic5C22×C412yes—
173#1200—48no—
174C24×D3—51no—
175C23×C3×C4—51no—
176C23×Dic3C2×D511yescharacteristic
177C22×C4×D3—52no—
178C2×C3×C4⋊C4—52no—
179C2×C3×C22⋊C4—51no—
180C2×C3×C22⋊C4C2×D511yescharacteristic
181C2×#217—51no—
182C2×#217C2×D511yescharacteristic
183C2×(C2×D3)⋊C4—58no—
184C2×(C2×D3)⋊C4—51no—
185C2×(C2×D3)⋊C4—51no—
186C2×(C2×D3)⋊C4—51no—
187C2×(C2×D3)⋊C4—51no—
188C2×(C2×D3)⋊C4—52no—
189C2×(C2×D3)⋊C4—52no—
190C2×(C2×D3)⋊C4—52no—
191C2×Dic3⋊C4C2×D512yescomplemented
192C2×C4×Dic3—52no—
193C2×C4×Dic3C2×D512yescomplemented
194D3×C22⋊C4—104no—
195#877—24no—
196C3×#95—52no—
197C3⋊#95—52no—
198C3⋊#95—52no—
199C3⋊#95—52no—
200C24×C5C22×D311yescharacteristic
201C23×D5—31no—
202C23×D5—31no—
203C23×D5—34no—
204C23×D5—64no—
205C23×D5—62no—
206C23×D5—62no—
207C23×D5—122no—
208C22×C4×C5—34no—
209C22×C4×C5—31no—
210C22×C4×C5—32no—
211C22×C4×C5—31no—
212C22×C4×C5—62no—
213C22×C4×C5—62no—
214C22×C4×C5C22×D312yes—
215C22×Dic5—32no—
216C22×Dic5—42no—
217C22×Dic5C4×D314yescomplemented
218C22×Dic5C22×D312yes—
219C22×Dic5C22×D311yescharacteristic
220C22×Dic5C22×D311yescharacteristic
221C2×C4×D5—64no—
222C2×C4×D5—64no—
223C2×C4×D5—122no—
224C5×C4⋊C4—64no—
225C5×C22⋊C4—22no—
226C5×C22⋊C4—34no—
227C42×C5—64no—
228C22⋊Dic5—22no—
229C22⋊Dic5—62no—
230#685—34no—
231#685—34no—
232#685—34no—
233#685—34no—
234#685—64no—
235#685—64no—
236#685—122no—
237#683—42no—
238C4×Dic5—122no—
239C2×#117—31no—
240C22×C22⋊C4—151no—
241C22×C22⋊C4—151no—
242#388—152no—
243#388—152no—
244#385—152no—
245#381—151no—
246#380—51no—
247#379—151no—
248C4×C22⋊C4—152no—
249C4×C22⋊C4—151no—
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—24no—
253C22×C3×C5—24no—
254C22×C3×C5C4×D414yes—
255C22×C3×C5C23×C411yescharacteristic
256C22×C3×C5C22×D411yescharacteristic
257C22×C3×C5C2×#4111yescharacteristic
258C22×C3×C5C2×#4112yes—
259C22×C3×C5H(5,𝔽2)12yes—
260C2×D15—22no—
261C2×D15—22no—
262C2×D15—24no—
263C2×D15—24no—
264C2×D15—22no—
265C2×D15—22no—
266C2×D15—22no—
267C2×D15—416no—
268C2×D15—42no—
269C2×D15—42no—
270C2×D15#11714yescomplemented
271C3×C4×C5—42no—
272Dic15—42no—
273C3×Dic5—22no—
274C3×Dic5—22no—
275C3×Dic5—42no—
276C5×Dic3—22no—
277C5×Dic3—22no—
278C5×Dic3—24no—
279C24×C3C22×D511yescharacteristic
280C23×D3—51no—
281C23×D3—51no—
282C23×D3—54no—
283C23×D3—104no—
284C23×D3—102no—
285C23×D3—102no—
286C23×D3—202no—
287C22×C3×C4—51no—
288C22×C3×C4—51no—
289C22×C3×C4—54no—
290C22×C3×C4—52no—
291C22×C3×C4—202no—
292C22×C3×C4C22×D512yes—
293C22×Dic3—22no—
294C22×Dic3—22no—
295C22×Dic3—52no—
296C22×Dic3C4×D514yescomplemented
297C22×Dic3C22×D511yescharacteristic
298C22×Dic3C22×D511yescharacteristic
299C22×Dic3C22×D512yes—
300C2×C4×D3—104no—
301C2×C4×D3—104no—
302C2×C4×D3—202no—
303C3×C4⋊C4—202no—
304C3×C22⋊C4—22no—
305C3×C22⋊C4—102no—
306#217—102no—
307#217C22⋊D514yes—
308(C2×D3)⋊C4—54no—
309(C2×D3)⋊C4—54no—
310(C2×D3)⋊C4—54no—
311(C2×D3)⋊C4—54no—
312(C2×D3)⋊C4—104no—
313(C2×D3)⋊C4—104no—
314(C2×D3)⋊C4—202no—
315Dic3⋊C4—24no—
316C4×Dic3—24no—
317C4×Dic3—202no—
318C23×C5—22no—
319C23×C5—22no—
320C23×C5C2×C4×D312yes—
321C23×C5D3×D412yes—
322C23×C5#23912yes—
323C23×C5C23×D311yescharacteristic
324C22×D5—32no—
325C22×D5—34no—
326C22×D5—32no—
327C22×D5—68no—
328C22×D5—61no—
329C22×D5—62no—
330C22×D5—61no—
331C22×D5—68no—
332C22×D5—61no—
333C22×D5—62no—
334C22×D5—61no—
335C22×D5—62no—
336C22×D5—124no—
337C22×D5—128no—
338C22×D5—128no—
339C2×C4×C5—22no—
340C2×C4×C5—22no—
341C2×C4×C5—22no—
342C2×C4×C5—32no—
343C2×C4×C5—32no—
344C2×C4×C5—34no—
345C2×C4×C5—68no—
346C2×C4×C5—61no—
347C2×C4×C5—61no—
348C2×C4×C5—62no—
349C2×C4×C5—61no—
350C2×C4×C5—61no—
351C2×C4×C5—62no—
352C2×Dic5—22no—
353C2×Dic5—22no—
354C2×Dic5—22no—
355C2×Dic5—22no—
356C2×Dic5—22no—
357C2×Dic5—44no—
358C2×Dic5—62no—
359C2×Dic5—62no—
360C2×Dic5—62no—
361C2×Dic5C2×C4×D312yes—
362C2×Dic5C2×C4×D312yes—
363C4×D5—128no—
364C25—151no—
365C23×C4—31no—
366C23×C4—51no—
367C23×C4—152no—
368#117—64no—
369C2×C4⋊C4—32no—
370C2×C4⋊C4—52no—
371C2×C22⋊C4—31no—
372C2×C22⋊C4—51no—
373C2×C22⋊C4—151no—
374C2×C22⋊C4—151no—
375C2×C22⋊C4—151no—
376C2×C22⋊C4—151no—
377C2×C22⋊C4—158no—
378C2×C22⋊C4—152no—
379C2×C22⋊C4—151no—
380C2×C22⋊C4—152no—
381C2×C22⋊C4—152no—
382C2×C22⋊C4—304no—
383C2×C22⋊C4D3×D511yescharacteristic
384C2×C42—32no—
385C2×C42—152no—
386#95—52no—
387#95—152no—
388#95—152no—
389#95—152no—
390C2×C3×C5—24no—
391C2×C3×C5C2×#11711yescharacteristic
392C2×C3×C5C2×C4×D411yescharacteristic
393C2×C3×C5#51811yescharacteristic
394C2×C3×C5#54612yes—
395C2×C3×C5#55112yes—
396D15—24no—
397D15—42no—
398C23×C3—22no—
399C23×C3—22no—
400C23×C3C2×C4×D512yes—
401C23×C3#70912yes—
402C23×C3C2×C22⋊D512yes—
403C23×C3C23×D511yescharacteristic
404C22×D3—52no—
405C22×D3—54no—
406C22×D3—52no—
407C22×D3—108no—
408C22×D3—101no—
409C22×D3—102no—
410C22×D3—101no—
411C22×D3—101no—
412C22×D3—102no—
413C22×D3—102no—
414C22×D3—101no—
415C22×D3—108no—
416C22×D3—204no—
417C22×D3—208no—
418C22×D3—208no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—52no—
423C2×C3×C4—52no—
424C2×C3×C4—102no—
425C2×C3×C4—102no—
426C2×C3×C4—102no—
427C2×C3×C4—102no—
428C2×C3×C4—102no—
429C2×C3×C4—204no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—28no—
437C2×Dic3—102no—
438C2×Dic3—102no—
439C2×Dic3—102no—
440C2×Dic3C2×C4×D512yes—
441C2×Dic3C2×C4×D512yes—
442C2×Dic3C2×C22⋊D514yes—
443C4×D3—208no—
444C22×C5—24no—
445C22×C5—22no—
446C22×C5—22no—
447C22×C5—24no—
448C22×C5#89314yes—
449C22×C5C22×C4×D311yescharacteristic
450C22×C5C2×D3×D411yescharacteristic
451C22×C5#100412yes—
452C22×C5C2×#23911yescharacteristic
453C22×C5D3×#4112yes—
454C2×D5—34no—
455C2×D5—64no—
456C2×D5—62no—
457C2×D5—62no—
458C2×D5—64no—
459C2×D5—62no—
460C2×D5—62no—
461C2×D5—62no—
462C2×D5—1216no—
463C2×D5—122no—
464C2×D5—122no—
465C4×C5—42no—
466C4×C5—62no—
467C4×C5—62no—
468C4×C5—64no—
469Dic5—22no—
470Dic5—22no—
471Dic5—42no—
472Dic5—122no—
473C24—151no—
474C24—151no—
475C24—154no—
476C24—304no—
477C24—302no—
478C24—302no—
479C24—602no—
480C24C2×D3×D511yescharacteristic
481C22×C4—34no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—32no—
485C22×C4—51no—
486C22×C4—51no—
487C22×C4—54no—
488C22×C4—52no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—152no—
492C22×C4—202no—
493C22×C4—304no—
494C22×C4—304no—
495C22×C4—602no—
496C22×C4C2×D3×D512yes—
497C4⋊C4—64no—
498C4⋊C4—202no—
499C22⋊C4—22no—
500C22⋊C4—34no—
501C22⋊C4—102no—
502C22⋊C4—154no—
503C22⋊C4—154no—
504C22⋊C4—154no—
505C22⋊C4—154no—
506C22⋊C4—304no—
507C22⋊C4—304no—
508C22⋊C4—302no—
509C22⋊C4—602no—
510C42—64no—
511C42—602no—
512C3×C5#234211yescharacteristic, complemented, Hall
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—24no—
516C22×C3—24no—
517C22×C3C4×C22⋊D514yes—
518C22×C3C22×C4×D511yescharacteristic
519C22×C3C2×#70911yescharacteristic
520C22×C3D5×#4112yes—
521C22×C3#414912yes—
522C22×C3C22×C22⋊D511yescharacteristic
523C2×D3—54no—
524C2×D3—102no—
525C2×D3—104no—
526C2×D3—102no—
527C2×D3—104no—
528C2×D3—102no—
529C2×D3—102no—
530C2×D3—102no—
531C2×D3—2016no—
532C2×D3—202no—
533C2×D3—202no—
534C3×C4—42no—
535C3×C4—102no—
536C3×C4—102no—
537C3×C4—202no—
538Dic3—22no—
539Dic3—22no—
540Dic3—24no—
541Dic3—202no—
542C2×C5—24no—
543C2×C5#548611yescharacteristic
544C2×C5C2×#89311yescharacteristic
545C2×C5D3×#11711yescharacteristic
546C2×C5#561012yes—
547C2×C5#563712yes—
548D5—64no—
549D5—122no—
550C23—22no—
551C23—22no—
552C23—152no—
553C23—154no—
554C23—152no—
555C23—308no—
556C23—301no—
557C23—302no—
558C23—301no—
559C23—308no—
560C23—301no—
561C23—302no—
562C23—302no—
563C23—301no—
564C23—604no—
565C23—608no—
566C23—608no—
567C23#676912yes—
568C23C4×D3×D512yes—
569C23D3×C22⋊D512yes—
570C23C22×D3×D511yescharacteristic
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—34no—
577C2×C4—52no—
578C2×C4—52no—
579C2×C4—68no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—102no—
587C2×C4—102no—
588C2×C4—102no—
589C2×C4—102no—
590C2×C4—102no—
591C2×C4—204no—
592C2×C4—302no—
593C2×C4—302no—
594C2×C4—302no—
595C2×C4—608no—
596C2×C3—24no—
597C2×C3D5×#11711yescharacteristic
598C2×C3#6585511yescharacteristic
599C2×C3#6586112yes—
600C2×C3C2×C4×C22⋊D511yescharacteristic
601C2×C3#6616312yes—
602D3—104no—
603D3—202no—
604C5#7400911yescharacteristic, complemented, Sylow
605C22—24no—
606C22—24no—
607C22—22no—
608C22—22no—
609C22—154no—
610C22—302no—
611C22—304no—
612C22—304no—
613C22—302no—
614C22—302no—
615C22—302no—
616C22—302no—
617C22—6016no—
618C22—602no—
619C22—602no—
620C22#9144614yes—
621C22C2×#676911yescharacteristic
622C22C2×C4×D3×D511yescharacteristic
623C22#9202412yes—
624C22#9202612yes—
625C22C2×D3×C22⋊D511yescharacteristic
626C4—42no—
627C4—62no—
628C4—62no—
629C4—64no—
630C4—102no—
631C4—102no—
632C4—202no—
633C4—602no—
634C3#1061400211yescharacteristic, complemented, Sylow
635C2—24no—
636C2—304no—
637C2—602no—
638C2#1175282511yescharacteristic
639C2#1175303512yes—
640C2#1175306611yescharacteristic
641C2C2×#9144611yescharacteristic
642C2#1175506912yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C22⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC24×D15C22
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#2342
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#877C2semidirect
5. C3×#65231C2semidirect
6. Dic5×#217C2semidirect
7. #11749472C2semidirect
8. #11748894C2semidirect
9. #11748881C2semidirect
10. #11748728C2semidirect
11. #11748669C2semidirect
22. C2×C5×Dic3⋊C4C22semidirect
23. C2×C4×C5×Dic3C22semidirect
26. C2×C3×#683C22semidirect
28. C2×#6667C4semidirect
34. C2×Dic3×Dic5C22semidirect
35. C3×#3963C22semidirect
36. #90994C22semidirect
37. #90994C22semidirect
38. #90994C22semidirect
47. #65231D3semidirect
63. C22×C5×Dic3C2×C4semidirect
68. C22×C3×Dic5C2×C4semidirect
87. #6667C2×C4semidirect
88. #6667C2×C4semidirect
92. C2×#877D5semidirect
120. C2×#683C2×D3semidirect
126. #3963C2×D3semidirect
191. C2×Dic3⋊C4C2×D5semidirect
193. C2×C4×Dic3C2×D5semidirect
217. C22×Dic5C4×D3semidirect
270. C2×D15#117semidirect
296. C22×Dic3C4×D5semidirect
512. C3×C5#2342semidirect
604. C5#74009semidirect
634. C3#10614002semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#8772
C2×#915612
C22×#66672
#117510562
C3×#652312
Dic5×#2172
#117494722
#117488942
#117488812
#117487282
#117486692
#106140023
#740095