Finite Group Databasebeta

← #14322824

Subgroups — #14322824

Order 1920.

1632 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×C22⋊C4×D15C211yesmaximal, characteristic
2C2×#91561C211yesmaximal, characteristic
3C22×#6667C211yesmaximal, characteristic
4C5×#4942C211yesmaximal, characteristic, complemented
5C3×#65231C211yesmaximal, characteristic, complemented
6#11749513C211yesmaximal, characteristic, complemented
7#11749506C211yesmaximal, characteristic, complemented
8#11748895C212yesmaximal, complemented
9#11748894C212yesmaximal, complemented
10#11748731C212yesmaximal, complemented
11#11748729C212yesmaximal, complemented
12#10605736—31nomaximal, Hall
13C24×D15C2211yescharacteristic
14C22×C4×D15C2212yes—
15C23×C5×Dic3C2211yescharacteristic
16C23×C3×Dic5C2211yescharacteristic
17C22×#1200C2212yes—
18C2×C3×C5×C22⋊C4C2211yescharacteristic
19C2×#6719C2211yescharacteristic
20C2×#6714C2212yes—
21C22⋊C4×D15—24no—
22C2×C5×#217C2211yescharacteristic
23C2×C5×Dic3⋊C4C2212yescomplemented
24C2×C3×C22⋊Dic5C2211yescharacteristic
25C2×C3×#683C2212yescomplemented
26#91561—24no—
27C2×Dic3⋊Dic5C2212yescomplemented
28C2×#6668C2212yescomplemented
29C2×#6667C418yescomplemented
30C2×#6667C2212yes—
31C2×#6667C2211yescharacteristic
32C2×#6667C2211yescharacteristic
33C2×#6667C2211yescharacteristic
34C2×#6667C2211yescharacteristic
35C5×C3⋊#95C2212yescomplemented
36C3×#3963C2212yescomplemented
37#90994C2212yescomplemented
38#90994C2212yescomplemented
39#74005—51nomaximal, Hall
40C22×#685—31no—
41C2×D5×C22⋊C4—31no—
42C2×D5×C22⋊C4—31no—
43C5×#380—31no—
44#65501—31no—
45#65278—32no—
46#65278—32no—
47#65231—31no—
48#65231D311yescharacteristic, complemented
49#64957—32no—
50#64957—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15—42no—
53C23×D15—42no—
54C23×D15C2×C414yes—
55C23×D15D414yes—
56C23×D15Q814yes—
57C23×D15C2311yescharacteristic
58C23×D15C2311yescharacteristic
59C22×C3×C4×C5C2312yes—
60C22×Dic15C2312yes—
61C2×C4×D15—24no—
62C2×C4×D15—42no—
63C2×C4×D15—42no—
64C22×C5×Dic3—42no—
65C22×C5×Dic3C2×C414yescomplemented
66C22×C5×Dic3C2312yes—
67C22×C5×Dic3C2311yescharacteristic
68C22×C5×Dic3C2311yescharacteristic
69C22×C3×Dic5—42no—
70C22×C3×Dic5C2×C414yescomplemented
71C22×C3×Dic5C2312yes—
72C22×C3×Dic5C2311yescharacteristic
73C22×C3×Dic5C2311yescharacteristic
74C2×#1200—24no—
75C2×#1200—42no—
76C2×#1200—42no—
77C3×C5×C22⋊C4—22no—
78#6719—22no—
79#6714—24no—
80C5×#217—22no—
81C5×Dic3⋊C4—42no—
82C3×C22⋊Dic5—22no—
83C3×#683—42no—
84Dic3⋊Dic5—42no—
85#6668—42no—
86#6667—24no—
87#6667C2×C414yescomplemented
88#6667C2×C414yescomplemented
89#6667C2×C414yescomplemented
90#6667C2×C414yescomplemented
91C22×(C2×D3)⋊C4—51no—
92C2×D3×C22⋊C4—51no—
93C2×D3×C22⋊C4—51no—
94C3×#380—51no—
95#5212—51no—
96#4989—52no—
97#4989—52no—
98#4942—51no—
99#4942D511yescharacteristic, complemented
100#4668—52no—
101#4668—52no—
102C24×D5—31no—
103C23×C4×C5—31no—
104C23×Dic5C2×D311yescharacteristic
105C22×C4×D5—32no—
106C22×C4×D5—32no—
107C2×C5×C4⋊C4—32no—
108C2×C5×C22⋊C4—31no—
109C2×C5×C22⋊C4C2×D311yescharacteristic
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×C4⋊Dic5—32no—
120C2×#683—32no—
121C2×#683C2×D312yescomplemented
122D5×C22⋊C4—64no—
123D5×C22⋊C4—64no—
124C5×#95—32no—
125#3963—32no—
126#3963—32no—
127#3963C2×D312yescomplemented
128#2315—151noSylow
129C23×C3×C5—22no—
130C23×C3×C5—22no—
131C23×C3×C5C22×C412yes—
132C23×C3×C5C2×D412yes—
133C23×C3×C5C2×Q812yes—
134C23×C3×C5C2411yescharacteristic
135C22×D15—22no—
136C22×D15—22no—
137C22×D15—22no—
138C22×D15—22no—
139C22×D15—44no—
140C22×D15—44no—
141C22×D15—48no—
142C22×D15—48no—
143C22×D15C4⋊C4116yescomplemented
144C22×D15C22×C412yes—
145C22×D15C22×C412yes—
146C22×D15C2×D412yes—
147C22×D15C2×D412yes—
148C22×D15C2×Q812yes—
149C22×D15C2×Q812yes—
150C2×C3×C4×C5—22no—
151C2×C3×C4×C5—22no—
152C2×C3×C4×C5—22no—
153C2×Dic15—22no—
154C2×Dic15—22no—
155C2×Dic15—22no—
156C4×D15—48no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3—22no—
160C2×C5×Dic3—22no—
161C2×C5×Dic3—22no—
162C2×C5×Dic3—44no—
163C2×C5×Dic3C22×C412yes—
164C2×C5×Dic3C22×C412yes—
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—
178C22×C4×D3—52no—
179C2×C3×C4⋊C4—52no—
180C2×C3×C22⋊C4—51no—
181C2×C3×C22⋊C4C2×D511yescharacteristic
182C2×#217—51no—
183C2×#217C2×D511yescharacteristic
184C2×(C2×D3)⋊C4—58no—
185C2×(C2×D3)⋊C4—51no—
186C2×(C2×D3)⋊C4—51no—
187C2×(C2×D3)⋊C4—51no—
188C2×(C2×D3)⋊C4—51no—
189C2×(C2×D3)⋊C4—52no—
190C2×(C2×D3)⋊C4—52no—
191C2×C4⋊Dic3—52no—
192C2×Dic3⋊C4—52no—
193C2×Dic3⋊C4C2×D512yescomplemented
194D3×C22⋊C4—104no—
195D3×C22⋊C4—104no—
196C3×#95—52no—
197C3⋊#95—52no—
198C3⋊#95—52no—
199C3⋊#95C2×D512yescomplemented
200C24×C5C22×D311yescharacteristic
201C23×D5—31no—
202C23×D5—31no—
203C23×D5—34no—
204C23×D5—34no—
205C23×D5—34no—
206C23×D5—122no—
207C23×D5—122no—
208C22×C4×C5—31no—
209C22×C4×C5—31no—
210C22×C4×C5—34no—
211C22×C4×C5—32no—
212C22×C4×C5—122no—
213C22×C4×C5C22×D312yes—
214C22×Dic5—32no—
215C22×Dic5—42no—
216C22×Dic5C4×D314yescomplemented
217C22×Dic5C22×D312yes—
218C22×Dic5C22×D311yescharacteristic
219C22×Dic5C22×D311yescharacteristic
220C2×C4×D5—64no—
221C2×C4×D5—64no—
222C2×C4×D5—122no—
223C2×C4×D5—122no—
224C2×C4×D5—122no—
225C2×C4×D5—122no—
226C5×C4⋊C4—122no—
227C5×C22⋊C4—22no—
228C5×C22⋊C4—62no—
229C22⋊Dic5—22no—
230C22⋊Dic5—62no—
231#685—34no—
232#685—34no—
233#685—34no—
234#685—34no—
235#685—64no—
236#685—64no—
237C4⋊Dic5—122no—
238#683—42no—
239#683—122no—
240C22×C22⋊C4—151no—
241C22×C22⋊C4—151no—
242C22×C22⋊C4—151no—
243#385—152no—
244#385—152no—
245#385—152no—
246#385—152no—
247#380—31no—
248#380—51no—
249#380—151no—
250#380—151no—
251C22×C3×C5—22no—
252C22×C3×C5—22no—
253C22×C3×C5—24no—
254C22×C3×C5—24no—
255C22×C3×C5C2×C4⋊C414yes—
256C22×C3×C5C23×C411yescharacteristic
257C22×C3×C5C22×D411yescharacteristic
258C22×C3×C5C22×Q811yescharacteristic
259C22×C3×C5H(5,𝔽2)12yes—
260C22×C3×C5H(5,𝔽2)12yes—
261C2×D15—22no—
262C2×D15—22no—
263C2×D15—22no—
264C2×D15—22no—
265C2×D15—24no—
266C2×D15—24no—
267C2×D15—24no—
268C2×D15—24no—
269C2×D15—416no—
270C2×D15C2×C4⋊C414yes—
271C2×D15C2×C4⋊C414yes—
272C3×C4×C5—42no—
273Dic15—42no—
274C3×Dic5—22no—
275C3×Dic5—22no—
276C3×Dic5—42no—
277C5×Dic3—22no—
278C5×Dic3—22no—
279C5×Dic3—42no—
280C24×C3C22×D511yescharacteristic
281C23×D3—51no—
282C23×D3—51no—
283C23×D3—54no—
284C23×D3—54no—
285C23×D3—54no—
286C23×D3—202no—
287C23×D3—202no—
288C22×C3×C4—51no—
289C22×C3×C4—51no—
290C22×C3×C4—54no—
291C22×C3×C4—52no—
292C22×C3×C4—202no—
293C22×C3×C4C22×D512yes—
294C22×Dic3—42no—
295C22×Dic3—52no—
296C22×Dic3C4×D514yescomplemented
297C22×Dic3C22×D512yes—
298C22×Dic3C22×D511yescharacteristic
299C22×Dic3C22×D511yescharacteristic
300C2×C4×D3—104no—
301C2×C4×D3—104no—
302C2×C4×D3—202no—
303C2×C4×D3—202no—
304C2×C4×D3—202no—
305C2×C4×D3—202no—
306C3×C4⋊C4—202no—
307C3×C22⋊C4—22no—
308C3×C22⋊C4—102no—
309#217—22no—
310#217—102no—
311(C2×D3)⋊C4—54no—
312(C2×D3)⋊C4—54no—
313(C2×D3)⋊C4—54no—
314(C2×D3)⋊C4—54no—
315(C2×D3)⋊C4—104no—
316(C2×D3)⋊C4—104no—
317C4⋊Dic3—202no—
318Dic3⋊C4—42no—
319Dic3⋊C4—202no—
320C23×C5—22no—
321C23×C5—22no—
322C23×C5C2×C4×D312yes—
323C23×C5D3×D412yes—
324C23×C5D3×Q812yes—
325C23×C5C23×D311yescharacteristic
326C22×D5—32no—
327C22×D5—32no—
328C22×D5—32no—
329C22×D5—32no—
330C22×D5—32no—
331C22×D5—32no—
332C22×D5—316no—
333C22×D5—62no—
334C22×D5—62no—
335C22×D5—62no—
336C22×D5—62no—
337C22×D5—128no—
338C22×D5—124no—
339C22×D5—124no—
340C22×D5—128no—
341C2×C4×C5—22no—
342C2×C4×C5—22no—
343C2×C4×C5—22no—
344C2×C4×C5—32no—
345C2×C4×C5—32no—
346C2×C4×C5—62no—
347C2×C4×C5—62no—
348C2×C4×C5—62no—
349C2×C4×C5—62no—
350C2×C4×C5—62no—
351C2×C4×C5—124no—
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—
364C4×D5—128no—
365C25—151no—
366C23×C4—31no—
367C23×C4—51no—
368C23×C4—152no—
369C23×C4—152no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—52no—
372C2×C4⋊C4—152no—
373C2×C4⋊C4—152no—
374C2×C22⋊C4—31no—
375C2×C22⋊C4—51no—
376C2×C22⋊C4—151no—
377C2×C22⋊C4—151no—
378C2×C22⋊C4—151no—
379C2×C22⋊C4—151no—
380C2×C22⋊C4—158no—
381C2×C22⋊C4—152no—
382C2×C22⋊C4—151no—
383C2×C22⋊C4—152no—
384C2×C22⋊C4—304no—
385C2×C22⋊C4—304no—
386C2×C22⋊C4D3×D511yescharacteristic
387#95—32no—
388#95—52no—
389#95—152no—
390#95—152no—
391C2×C3×C5—24no—
392C2×C3×C5C22×C4⋊C411yescharacteristic
393C2×C3×C5#51811yescharacteristic
394C2×C3×C5#51811yescharacteristic
395C2×C3×C5#53412yes—
396C2×C3×C5#54312yes—
397D15—24no—
398D15—24no—
399C23×C3—22no—
400C23×C3—22no—
401C23×C3C2×C4×D512yes—
402C23×C3D4×D512yes—
403C23×C3Q8×D512yes—
404C23×C3C23×D511yescharacteristic
405C22×D3—52no—
406C22×D3—52no—
407C22×D3—52no—
408C22×D3—52no—
409C22×D3—52no—
410C22×D3—52no—
411C22×D3—516no—
412C22×D3—102no—
413C22×D3—102no—
414C22×D3—102no—
415C22×D3—102no—
416C22×D3—204no—
417C22×D3—204no—
418C22×D3—208no—
419C22×D3—208no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—52no—
424C2×C3×C4—52no—
425C2×C3×C4—102no—
426C2×C3×C4—102no—
427C2×C3×C4—102no—
428C2×C3×C4—102no—
429C2×C3×C4—102no—
430C2×C3×C4—204no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—44no—
437C2×Dic3—102no—
438C2×Dic3—102no—
439C2×Dic3—102no—
440C2×Dic3C2×C4×D512yes—
441C2×Dic3C2×C4×D512yes—
442C4×D3—208no—
443C4×D3—208no—
444C22×C5—24no—
445C22×C5—22no—
446C22×C5—22no—
447C22×C5—24no—
448C22×C5D3×C4⋊C414yes—
449C22×C5C22×C4×D311yescharacteristic
450C22×C5C2×D3×D411yescharacteristic
451C22×C5#100412yes—
452C22×C5C2×D3×Q811yescharacteristic
453C22×C5#100912yes—
454C2×D5—34no—
455C2×D5—34no—
456C2×D5—64no—
457C2×D5—64no—
458C2×D5—62no—
459C2×D5—62no—
460C2×D5—62no—
461C2×D5—62no—
462C2×D5—64no—
463C2×D5—64no—
464C2×D5—1216no—
465C4×C5—42no—
466C4×C5—62no—
467C4×C5—62no—
468C4×C5—122no—
469Dic5—22no—
470Dic5—22no—
471Dic5—42no—
472Dic5—122no—
473C24—151no—
474C24—151no—
475C24—154no—
476C24—154no—
477C24—154no—
478C24—602no—
479C24—602no—
480C24C2×D3×D511yescharacteristic
481C22×C4—31no—
482C22×C4—31no—
483C22×C4—34no—
484C22×C4—32no—
485C22×C4—51no—
486C22×C4—51no—
487C22×C4—54no—
488C22×C4—52no—
489C22×C4—122no—
490C22×C4—152no—
491C22×C4—202no—
492C22×C4—304no—
493C22×C4—304no—
494C22×C4—602no—
495C22×C4—602no—
496C22×C4—602no—
497C22×C4—602no—
498C22×C4C2×D3×D512yes—
499C4⋊C4—122no—
500C4⋊C4—202no—
501C4⋊C4—602no—
502C4⋊C4—602no—
503C22⋊C4—22no—
504C22⋊C4—62no—
505C22⋊C4—102no—
506C22⋊C4—154no—
507C22⋊C4—154no—
508C22⋊C4—154no—
509C22⋊C4—154no—
510C22⋊C4—304no—
511C22⋊C4—304no—
512C22⋊C4—302no—
513C3×C5#231511yescharacteristic, complemented, Hall
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—24no—
517C22×C3—24no—
518C22×C3D5×C4⋊C414yes—
519C22×C3C22×C4×D511yescharacteristic
520C22×C3C2×D4×D511yescharacteristic
521C22×C3#414412yes—
522C22×C3C2×Q8×D511yescharacteristic
523C22×C3#414912yes—
524C2×D3—54no—
525C2×D3—54no—
526C2×D3—102no—
527C2×D3—102no—
528C2×D3—102no—
529C2×D3—102no—
530C2×D3—104no—
531C2×D3—104no—
532C2×D3—104no—
533C2×D3—104no—
534C2×D3—2016no—
535C3×C4—42no—
536C3×C4—102no—
537C3×C4—102no—
538C3×C4—202no—
539Dic3—22no—
540Dic3—22no—
541Dic3—42no—
542Dic3—202no—
543C2×C5—24no—
544C2×C5#548611yescharacteristic
545C2×C5C2×D3×C4⋊C411yescharacteristic
546C2×C5#552111yescharacteristic
547C2×C5#560512yes—
548C2×C5#563412yes—
549D5—64no—
550D5—64no—
551C23—22no—
552C23—22no—
553C23—152no—
554C23—152no—
555C23—152no—
556C23—152no—
557C23—152no—
558C23—152no—
559C23—1516no—
560C23—302no—
561C23—302no—
562C23—302no—
563C23—302no—
564C23—608no—
565C23—608no—
566C23—604no—
567C23—604no—
568C23D15⋊Q812yes—
569C23C4×D3×D512yes—
570C23#679012yes—
571C23C22×D3×D511yescharacteristic
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—52no—
578C2×C4—52no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—102no—
585C2×C4—102no—
586C2×C4—102no—
587C2×C4—102no—
588C2×C4—102no—
589C2×C4—124no—
590C2×C4—204no—
591C2×C4—302no—
592C2×C4—302no—
593C2×C4—302no—
594C2×C4—608no—
595C2×C4—608no—
596C2×C3—24no—
597C2×C3#6582011yescharacteristic
598C2×C3C2×D5×C4⋊C411yescharacteristic
599C2×C3#6585511yescharacteristic
600C2×C3#6593912yes—
601C2×C3#6596812yes—
602D3—104no—
603D3—104no—
604C5#7400511yescharacteristic, complemented, Sylow
605C22—24no—
606C22—24no—
607C22—22no—
608C22—22no—
609C22—154no—
610C22—154no—
611C22—304no—
612C22—304no—
613C22—304no—
614C22—304no—
615C22—302no—
616C22—302no—
617C22—302no—
618C22—302no—
619C22—6016no—
620C22#9140414yes—
621C22C2×D15⋊Q811yescharacteristic
622C22C2×C4×D3×D511yescharacteristic
623C22#9202612yes—
624C22#9202712yes—
625C22C2×#679011yescharacteristic
626C4—42no—
627C4—62no—
628C4—62no—
629C4—102no—
630C4—102no—
631C4—122no—
632C4—202no—
633C4—602no—
634C3#1060573611yescharacteristic, complemented, Sylow
635C2—24no—
636C2—304no—
637C2—304no—
638C2#1175306012yes—
639C2#1175306611yescharacteristic
640C2#1175306711yescharacteristic
641C2C2×#9140411yescharacteristic
642C2#1175507312yes—
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#2315
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
4. C5×#4942C2semidirect
5. C3×#65231C2semidirect
6. #11749513C2semidirect
7. #11749506C2semidirect
8. #11748895C2semidirect
9. #11748894C2semidirect
10. #11748731C2semidirect
11. #11748729C2semidirect
23. C2×C5×Dic3⋊C4C22semidirect
25. C2×C3×#683C22semidirect
27. C2×Dic3⋊Dic5C22semidirect
28. C2×#6668C22semidirect
29. C2×#6667C4semidirect
35. C5×C3⋊#95C22semidirect
36. C3×#3963C22semidirect
37. #90994C22semidirect
38. #90994C22semidirect
48. #65231D3semidirect
65. C22×C5×Dic3C2×C4semidirect
70. C22×C3×Dic5C2×C4semidirect
87. #6667C2×C4semidirect
88. #6667C2×C4semidirect
89. #6667C2×C4semidirect
90. #6667C2×C4semidirect
99. #4942D5semidirect
121. C2×#683C2×D3semidirect
127. #3963C2×D3semidirect
143. C22×D15C4⋊C4semidirect
193. C2×Dic3⋊C4C2×D5semidirect
199. C3⋊#95C2×D5semidirect
216. C22×Dic5C4×D3semidirect
296. C22×Dic3C4×D5semidirect
513. C3×C5#2315semidirect
604. C5#74005semidirect
634. C3#10605736semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D152
C2×#915612
C22×#66672
C5×#49422
C3×#652312
#117495132
#117495062
#117488952
#117488942
#117487312
#117487292
#106057363
#740055