Finite Group Databasebeta

← #14327375

Subgroups — #14327375

Order 1920.

1320 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×C4×#1220C211yesmaximal, characteristic
2C2×C5×#926C211yesmaximal, characteristic, complemented
3C2×#91437C211yesmaximal, characteristic, complemented
4C3×#65501C211yesmaximal, characteristic, complemented
5#11749520C212yesmaximal
6#11749481C212yesmaximal, complemented
7#11748881C212yesmaximal, complemented
8#11748847C212yesmaximal, complemented
9#11748799C211yesmaximal, characteristic, complemented
10#11748699C211yesmaximal, characteristic, complemented
11#11748623C211yesmaximal, characteristic, complemented
12#10614003—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×#1220C2211yescharacteristic
15C22×C4×D15C2211yescharacteristic
16C22×#1200C2212yes—
17C2×#6719C2211yescharacteristic
18C4×#1220C418yescomplemented
19C2×#6714C2211yescharacteristic
20C2×#6712C2211yescharacteristic
21C2×C4×Dic15C2211yescharacteristic
22C2×C5×#217C2212yes—
23C5×#926—24no—
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C5×C4⋊Dic3C2211yescharacteristic
26C2×C4×C5×Dic3C2212yes—
27C2×C3×C22⋊Dic5C2212yes—
28C2×C3×C4⋊Dic5C2211yescharacteristic
29C2×C3×C4⋊Dic5C2211yescharacteristic
30C2×#6669C2212yes—
31C2×#6667C2212yes—
32C2×Dic3×Dic5C2212yes—
33#91437—24no—
34C3×#3963C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#74012—51nomaximal, Hall
40C2×C5×#117—31no—
41C2×C4×C22⋊D5—31no—
42C2×D5×C4⋊C4—31no—
43#65501D311yescharacteristic, complemented
44#65276—31no—
45#65266—31no—
46C4⋊C4×Dic5—31no—
47#65245—32no—
48#65234—32no—
49C4×#685—32no—
50#65222—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15C2311yescharacteristic
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—22no—
55C22×C3×C4×C5C2×C414yescomplemented
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C2×#1220—22no—
59C2×#1220—22no—
60C2×#1220C2×C414yes—
61C22×Dic15C2311yescharacteristic
62C22×Dic15C2311yescharacteristic
63C22×Dic15C2311yescharacteristic
64C2×C4×D15—22no—
65C2×C4×D15—22no—
66C2×C4×D15C2×C414yes—
67C22×C5×Dic3C2312yes—
68C22×C5×Dic3C2312yes—
69C22×C3×Dic5C2312yes—
70C22×C3×Dic5C2312yes—
71C2×#1200—24no—
72C2×#1200—24no—
73C2×#1200—42no—
74#6719C2×C414yescomplemented
75#6714C2×C414yes—
76#6712C2×C414yescomplemented
77C4×Dic15C2×C414yescomplemented
78C5×#217—24no—
79C5×C4⋊Dic3—22no—
80C5×C4⋊Dic3D414yescomplemented
81C4×C5×Dic3—24no—
82C3×C22⋊Dic5—42no—
83C3×C4⋊Dic5—22no—
84C3×C4⋊Dic5—22no—
85#6669—24no—
86#6667—24no—
87Dic3×Dic5—42no—
88C2×C4×#67—51no—
89C2×#926D511yescharacteristic, complemented
90C2×D3×C4⋊C4—51no—
91C3×#380—51no—
92#4987—51no—
93#4977—51no—
94Dic3×C4⋊C4—51no—
95#4956—52no—
96#4944—52no—
97#4666—52no—
98#4652—52no—
99C23×C4×C5C2×D311yescharacteristic
100C22×C22⋊D5—31no—
101C22×C4×D5—31no—
102C22×C4×D5—32no—
103C5×#117—64no—
104C2×C5×C4⋊C4—31no—
105C2×C5×C4⋊C4—31no—
106C2×C5×C22⋊C4—32no—
107C2×C42×C5—32no—
108C2×C22⋊Dic5—31no—
109C2×C22⋊Dic5C2×D312yes—
110C4×C22⋊D5—38no—
111C2×#685—31no—
112C2×#685—32no—
113C2×C4⋊Dic5C2×D311yescharacteristic
114C2×C4⋊Dic5C2×D311yescharacteristic
115C2×#683—31no—
116C2×#683—32no—
117C2×C4×Dic5—31no—
118C2×C4×Dic5—32no—
119D5×C4⋊C4—64no—
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#3963—32no—
124#3963C2×D312yes—
125#2352—151noSylow
126C23×C3×C5—21no—
127C23×C3×C5—21no—
128C23×C3×C5—21no—
129C23×C3×C5—21no—
130C23×C3×C5—21no—
131C23×C3×C5—21no—
132C23×C3×C5C22×C412yes—
133C23×C3×C5C2411yescharacteristic
134C22×D15—21no—
135C22×D15—21no—
136C22×D15—21no—
137C22×D15—21no—
138C22×D15C22×C412yes—
139C22×D15#4112yes—
140C22×D15#4112yes—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—28no—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C22×C412yes—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5#4112yes—
148#1220—28no—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—21no—
152C2×Dic15—21no—
153C2×Dic15—21no—
154C2×Dic15—21no—
155C2×Dic15C22×C412yes—
156C2×Dic15C22×C412yes—
157C2×Dic15C22×C412yes—
158C4×D15—28no—
159C2×C5×Dic3—22no—
160C2×C5×Dic3—22no—
161C2×C5×Dic3—22no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—22no—
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—22no—
171#1200—48no—
172C23×C3×C4C2×D511yescharacteristic
173C22×#67—51no—
174C22×C4×D3—51no—
175C22×C4×D3—52no—
176C2×C3×C4⋊C4—51no—
177C2×C3×C4⋊C4—51no—
178C2×C3×C22⋊C4—52no—
179C2×#217—51no—
180C2×#217C2×D512yes—
181C4×#67—58no—
182C2×(C2×D3)⋊C4—51no—
183C2×(C2×D3)⋊C4—52no—
184#926—24no—
185C2×C4⋊Dic3C2×D511yescharacteristic
186C2×C4⋊Dic3C2×D511yescharacteristic
187C2×Dic3⋊C4—51no—
188C2×Dic3⋊C4—52no—
189C2×C4×Dic3—51no—
190C2×C4×Dic3—52no—
191C2×C4×Dic3C2×D512yes—
192D3×C4⋊C4—104no—
193C3×#95—52no—
194C3⋊#95—52no—
195C3⋊#95—52no—
196C3⋊#95—52no—
197C3⋊#95—52no—
198C24×C5C22×D311yescharacteristic
199C23×D5—31no—
200C22×C4×C5—22no—
201C22×C4×C5—22no—
202C22×C4×C5—32no—
203C22×C4×C5—32no—
204C22×C4×C5C4×D314yescomplemented
205C22×C4×C5C22×D311yescharacteristic
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—34no—
208C2×C22⋊D5—62no—
209C2×C22⋊D5—62no—
210C22×Dic5—31no—
211C22×Dic5—31no—
212C22×Dic5—31no—
213C22×Dic5C22×D312yes—
214C22×Dic5C22×D312yes—
215C2×C4×D5—34no—
216C2×C4×D5—62no—
217C2×C4×D5—62no—
218C2×C4×D5—64no—
219C2×C4×D5—64no—
220C2×C4×D5—122no—
221C5×C4⋊C4—34no—
222C5×C4⋊C4—62no—
223C5×C22⋊C4—64no—
224C42×C5—64no—
225C22⋊Dic5—34no—
226C22⋊Dic5—42no—
227#685—34no—
228#685—64no—
229C4⋊Dic5—22no—
230C4⋊Dic5—22no—
231#683—34no—
232#683—64no—
233C4×Dic5—34no—
234C4×Dic5—122no—
235C2×C4×D4—151no—
236C2×#117—31no—
237C22×C4⋊C4—151no—
238#388—152no—
239#387—152no—
240#386—152no—
241#382—151no—
242#380—51no—
243#380—151no—
244C4×C4⋊C4—151no—
245C4×C22⋊C4—152no—
246C22×C3×C5—22no—
247C22×C3×C5—22no—
248C22×C3×C5—22no—
249C22×C3×C5—41no—
250C22×C3×C5—41no—
251C22×C3×C5—41no—
252C22×C3×C5—41no—
253C22×C3×C5C23×C411yescharacteristic
254C22×C3×C5C22×D411yescharacteristic
255C22×C3×C5C2×#4111yescharacteristic
256C22×C3×C5C2×#4111yescharacteristic
257C22×C3×C5C2×#4111yescharacteristic
258C22×C3×C5H(5,𝔽2)11yescharacteristic
259C22×C3×C52−1+411yescharacteristic
260C2×D15—22no—
261C2×D15—22no—
262C2×D15—22no—
263C2×D15—22no—
264C2×D15—22no—
265C2×D15—22no—
266C2×D15#11714yescomplemented
267C3×C4×C5—22no—
268C3×C4×C5C4×D414yes—
269Dic15—22no—
270Dic15—22no—
271Dic15—22no—
272C3×Dic5—42no—
273C3×Dic5—42no—
274C5×Dic3—24no—
275C5×Dic3—42no—
276C24×C3C22×D511yescharacteristic
277C23×D3—51no—
278C22×C3×C4—22no—
279C22×C3×C4—22no—
280C22×C3×C4—52no—
281C22×C3×C4—52no—
282C22×C3×C4C4×D514yescomplemented
283C22×C3×C4C22×D511yescharacteristic
284C22×C3×C4C22×D511yescharacteristic
285C2×#67—54no—
286C2×#67—102no—
287C2×#67—102no—
288C22×Dic3—51no—
289C22×Dic3—51no—
290C22×Dic3—51no—
291C22×Dic3C22×D512yes—
292C22×Dic3C22×D512yes—
293C2×C4×D3—54no—
294C2×C4×D3—102no—
295C2×C4×D3—102no—
296C2×C4×D3—104no—
297C2×C4×D3—104no—
298C2×C4×D3—202no—
299C3×C4⋊C4—102no—
300C3×C4⋊C4—102no—
301C3×C22⋊C4—202no—
302#217—24no—
303#217—54no—
304(C2×D3)⋊C4—54no—
305(C2×D3)⋊C4—104no—
306C4⋊Dic3—22no—
307C4⋊Dic3C22⋊D514yescomplemented
308Dic3⋊C4—54no—
309Dic3⋊C4—104no—
310C4×Dic3—24no—
311C4×Dic3—54no—
312C4×Dic3—202no—
313C23×C5—21no—
314C23×C5—21no—
315C23×C5—21no—
316C23×C5—21no—
317C23×C5—21no—
318C23×C5—21no—
319C23×C5C2×C4×D312yes—
320C23×C5C23×D311yescharacteristic
321C22×D5—32no—
322C22×D5—32no—
323C22×D5—32no—
324C22×D5—61no—
325C22×D5—61no—
326C22×D5—61no—
327C22×D5—61no—
328C2×C4×C5—21no—
329C2×C4×C5—21no—
330C2×C4×C5—28no—
331C2×C4×C5—34no—
332C2×C4×C5—62no—
333C2×C4×C5—62no—
334C2×C4×C5—62no—
335C2×C4×C5—62no—
336C2×C4×C5—62no—
337C2×C4×C5C2×C4×D312yes—
338C2×C4×C5C2×C4×D312yes—
339C2×C4×C5D3×D412yes—
340C2×C4×C5#23712yes—
341C22⋊D5—68no—
342C2×Dic5—22no—
343C2×Dic5—22no—
344C2×Dic5—22no—
345C2×Dic5—22no—
346C2×Dic5—22no—
347C2×Dic5—22no—
348C2×Dic5—32no—
349C2×Dic5—32no—
350C2×Dic5—32no—
351C2×Dic5—61no—
352C2×Dic5—61no—
353C2×Dic5—61no—
354C2×Dic5—61no—
355C2×Dic5—61no—
356C2×Dic5—61no—
357C4×D5—68no—
358C4×D5—128no—
359C22×D4—151no—
360C23×C4—152no—
361C23×C4—151no—
362C23×C4D3×D511yescharacteristic
363C4×D4—158no—
364#117—64no—
365C2×C4⋊C4—31no—
366C2×C4⋊C4—31no—
367C2×C4⋊C4—51no—
368C2×C4⋊C4—51no—
369C2×C4⋊C4—151no—
370C2×C4⋊C4—152no—
371C2×C4⋊C4—304no—
372C2×C22⋊C4—32no—
373C2×C22⋊C4—52no—
374C2×C22⋊C4—151no—
375C2×C22⋊C4—151no—
376C2×C22⋊C4—152no—
377C2×C42—32no—
378C2×C42—151no—
379C2×C42—152no—
380#95—52no—
381#95—152no—
382#95—152no—
383#95—152no—
384#95—152no—
385C2×C3×C5—42no—
386C2×C3×C5C2×#11711yescharacteristic
387C2×C3×C5C2×C4×D411yescharacteristic
388C2×C3×C5#52011yescharacteristic
389C2×C3×C5#54611yescharacteristic
390C2×C3×C5#54811yescharacteristic
391C2×C3×C5#55211yescharacteristic
392C2×C3×C5#55311yescharacteristic
393D15—24no—
394C23×C3—21no—
395C23×C3—21no—
396C23×C3—21no—
397C23×C3—21no—
398C23×C3—21no—
399C23×C3—21no—
400C23×C3C2×C4×D512yes—
401C23×C3C23×D511yescharacteristic
402C22×D3—52no—
403C22×D3—52no—
404C22×D3—52no—
405C22×D3—101no—
406C22×D3—101no—
407C22×D3—101no—
408C22×D3—101no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—28no—
412C2×C3×C4—102no—
413C2×C3×C4—102no—
414C2×C3×C4—102no—
415C2×C3×C4—102no—
416C2×C3×C4—102no—
417C2×C3×C4—102no—
418C2×C3×C4C2×C4×D512yes—
419C2×C3×C4C2×C4×D512yes—
420C2×C3×C4#70912yes—
421C2×C3×C4C2×C22⋊D512yes—
422#67—108no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—52no—
429C2×Dic3—52no—
430C2×Dic3—52no—
431C2×Dic3—101no—
432C2×Dic3—101no—
433C2×Dic3—101no—
434C2×Dic3—101no—
435C2×Dic3—101no—
436C2×Dic3—101no—
437C2×Dic3C2×C22⋊D514yes—
438C4×D3—108no—
439C4×D3—208no—
440C22×C5—22no—
441C22×C5—22no—
442C22×C5—22no—
443C22×C5—41no—
444C22×C5—41no—
445C22×C5—41no—
446C22×C5—41no—
447C22×C5C22×C4×D311yescharacteristic
448C22×C5C2×D3×D411yescharacteristic
449C22×C5C2×#23711yescharacteristic
450C22×C5#100411yescharacteristic
451C22×C5#100711yescharacteristic
452C22×C5D3×#4111yescharacteristic
453C22×C5D3×#4111yescharacteristic
454C2×D5—34no—
455C2×D5—62no—
456C2×D5—62no—
457C2×D5—62no—
458C2×D5—62no—
459C2×D5—62no—
460C2×D5—62no—
461C4×C5—22no—
462C4×C5—64no—
463C4×C5—122no—
464C4×C5#67⋊C414yes—
465Dic5—42no—
466Dic5—42no—
467Dic5—62no—
468Dic5—62no—
469Dic5—62no—
470C24—151no—
471C24C2×D3×D511yescharacteristic
472C2×D4—154no—
473C2×D4—302no—
474C2×D4—302no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—32no—
478C22×C4—32no—
479C22×C4—52no—
480C22×C4—52no—
481C22×C4—151no—
482C22×C4—151no—
483C22×C4—151no—
484C22×C4—154no—
485C22×C4—304no—
486C22×C4—304no—
487C22×C4—302no—
488C22×C4—302no—
489C22×C4—602no—
490C22×C4#120014yescomplemented
491C22×C4C2×D3×D511yescharacteristic
492C22×C4C2×D3×D511yescharacteristic
493C4⋊C4—34no—
494C4⋊C4—62no—
495C4⋊C4—102no—
496C4⋊C4—102no—
497C4⋊C4—154no—
498C4⋊C4—304no—
499C22⋊C4—64no—
500C22⋊C4—154no—
501C22⋊C4—154no—
502C22⋊C4—202no—
503C22⋊C4—304no—
504C42—64no—
505C42—154no—
506C42—602no—
507C3×C5#235211yescharacteristic, complemented, Hall
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3C22×C4×D511yescharacteristic
516C22×C3C2×#70911yescharacteristic
517C22×C3D5×#4111yescharacteristic
518C22×C3D5×#4111yescharacteristic
519C22×C3#414911yescharacteristic
520C22×C3#415011yescharacteristic
521C22×C3C22×C22⋊D511yescharacteristic
522C2×D3—54no—
523C2×D3—102no—
524C2×D3—102no—
525C2×D3—102no—
526C2×D3—102no—
527C2×D3—102no—
528C2×D3—102no—
529C3×C4—22no—
530C3×C4—202no—
531C3×C4—202no—
532C3×C4C4×C22⋊D514yes—
533Dic3—24no—
534Dic3—42no—
535Dic3—102no—
536Dic3—102no—
537Dic3—102no—
538C2×C5—42no—
539C2×C5C2×#67⋊C411yescharacteristic
540C2×C5#550411yescharacteristic
541C2×C5D3×#11711yescharacteristic
542C2×C5#561211yescharacteristic
543C2×C5#561411yescharacteristic
544C2×C5#563311yescharacteristic
545C2×C5#563911yescharacteristic
546D5—64no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—152no—
554C23—152no—
555C23—152no—
556C23—301no—
557C23—301no—
558C23—301no—
559C23—301no—
560C23C2×#120012yes—
561C23C22×D3×D511yescharacteristic
562D4—308no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—34no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—102no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—102no—
576C2×C4—102no—
577C2×C4—102no—
578C2×C4—152no—
579C2×C4—152no—
580C2×C4—152no—
581C2×C4—301no—
582C2×C4—301no—
583C2×C4—301no—
584C2×C4—301no—
585C2×C4—301no—
586C2×C4—301no—
587C2×C4—308no—
588C2×C4—608no—
589C2×C4#677912yes—
590C2×C4C2×#120012yes—
591C2×C4C2×#120012yes—
592C2×C4D3×C22⋊D512yes—
593C2×C3—42no—
594C2×C3D5×#11711yescharacteristic
595C2×C3#6585711yescharacteristic
596C2×C3#6586311yescharacteristic
597C2×C3#6586411yescharacteristic
598C2×C3C2×C4×C22⋊D511yescharacteristic
599C2×C3#6616311yescharacteristic
600C2×C3#6616511yescharacteristic
601D3—104no—
602C5#7401211yescharacteristic, complemented, Sylow
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—154no—
611C22—302no—
612C22—302no—
613C22—302no—
614C22—302no—
615C22—302no—
616C22—302no—
617C22#9200411yescharacteristic
618C22#9201611yescharacteristic
619C22#9201611yescharacteristic
620C22#9201711yescharacteristic
621C22C2×#677911yescharacteristic
622C22C22×#120011yescharacteristic
623C22C2×D3×C22⋊D511yescharacteristic
624C4—22no—
625C4—64no—
626C4—122no—
627C4—202no—
628C4—202no—
629C4—302no—
630C4—302no—
631C4—302no—
632C4#9155614yes—
633C3#1061400311yescharacteristic, complemented, Sylow
634C2—42no—
635C2—304no—
636C2#1175445311yescharacteristic
637C2#1175446111yescharacteristic
638C2#1175446411yescharacteristic
639C2#1175449611yescharacteristic
640C2#1175454611yescharacteristic
641C2#1175465811yescharacteristic
642C2C2×#9155611yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1220C22
hypercenterC23×C4D3×D5
hypocenterC3×C5#2352
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×C5×#926C2semidirect
3. C2×#91437C2semidirect
4. C3×#65501C2semidirect
6. #11749481C2semidirect
7. #11748881C2semidirect
8. #11748847C2semidirect
9. #11748799C2semidirect
10. #11748699C2semidirect
11. #11748623C2semidirect
18. C4×#1220C4semidirect
43. #65501D3semidirect
55. C22×C3×C4×C5C2×C4semidirect
74. #6719C2×C4semidirect
76. #6712C2×C4semidirect
77. C4×Dic15C2×C4semidirect
80. C5×C4⋊Dic3D4semidirect
89. C2×#926D5semidirect
204. C22×C4×C5C4×D3semidirect
266. C2×D15#117semidirect
282. C22×C3×C4C4×D5semidirect
307. C4⋊Dic3C22⋊D5semidirect
490. C22×C4#1200semidirect
507. C3×C5#2352semidirect
602. C5#74012semidirect
633. C3#10614003semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#12202
C2×C5×#9262
C2×#914372
C3×#655012
#117495202
#117494812
#117488812
#117488472
#117487992
#117486992
#117486232
#106140033
#740125