Finite Group Databasebeta

← #14327083

Subgroups — #14327083

Order 1920.

1552 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D20C211yesmaximal, characteristic
2C2×#91434C211yesmaximal, characteristic
3C2×#91434C211yesmaximal, characteristic
4C2×D5×Dic3⋊C4C212yesmaximal, complemented
5C4×#6712C211yesmaximal, characteristic, complemented
6C5×#4928C211yesmaximal, characteristic, complemented
7#11748890C212yesmaximal, complemented
8#11748885C212yesmaximal, complemented
9#11748868C211yesmaximal, characteristic, complemented
10#11748866C211yesmaximal, characteristic, complemented
11#11748703C212yesmaximal, complemented
12#10607933—31nomaximal, Hall
13C22×C3×D20C2211yescharacteristic
14C22×C3×C4×D5C2212yes—
15C22×D5×Dic3C2212yes—
16C22×D5×Dic3C2212yes—
17C2×C3×C42×C5C2211yescharacteristic
18C2×#6712C2211yescharacteristic
19C2×#6712C2211yescharacteristic
20C2×C4×Dic15C2211yescharacteristic
21C2×C4×Dic15C2211yescharacteristic
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C5×Dic3⋊C4C2212yes—
25C2×C3×#685C2212yes—
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C3×C4×D20—24no—
28C2×Dic3⋊Dic5C2212yes—
29C2×#6669C2212yes—
30C2×#6669C2212yes—
31C2×#6665C2212yes—
32C2×#6665C2212yes—
33#91434—24no—
34#91434C418yescomplemented
35D5×Dic3⋊C4—28no—
36#91115C2211yescharacteristic
37#91115C2211yescharacteristic
38C5×C3⋊#95C2212yes—
39#90994C2212yes—
40D4×Dic3⋊C4—51nomaximal, Hall
41C2×#4039—31no—
42C2×#4039—31no—
43C2×D5×C4⋊C4—32no—
44C2×C4×D20D311yescharacteristic
45C5×#389—31no—
46#65278—32no—
47#65274—31no—
48#65265—31no—
49C4×#683—31no—
50#64957—32no—
51#64956—32no—
52C23×C3×D5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×Dic15C2311yescharacteristic
57C22×Dic15C2311yescharacteristic
58C22×Dic15C2311yescharacteristic
59C22×Dic15C2311yescharacteristic
60C22×C5×Dic3C2312yes—
61C22×C5×Dic3C2312yes—
62C22×C3×Dic5C2312yes—
63C2×C3×D20C2×C414yes—
64C2×C3×D20D414yes—
65C2×C3×D20Q814yes—
66C2×C3×C4×D5—24no—
67C2×C3×C4×D5—24no—
68C2×C3×C4×D5—42no—
69C2×D5×Dic3—24no—
70C2×D5×Dic3—24no—
71C2×D5×Dic3—24no—
72C2×D5×Dic3—24no—
73C2×D5×Dic3—42no—
74C2×D5×Dic3C2×C418yescomplemented
75C3×C42×C5—22no—
76#6712—22no—
77#6712D414yescomplemented
78C4×Dic15—22no—
79C4×Dic15C2×C414yescomplemented
80C5×C4⋊Dic3—22no—
81C5×C4⋊Dic3C2×C414yescomplemented
82C5×Dic3⋊C4—24no—
83C3×#685—24no—
84C3×C4⋊Dic5—22no—
85Dic3⋊Dic5—24no—
86#6669—24no—
87#6669—42no—
88#6665—24no—
89#6665C2×C418yescomplemented
90C2×C3×C4×D4—51no—
91C2×D4×Dic3—51no—
92C2×D4×Dic3—51no—
93C22×Dic3⋊C4—52no—
94#5210—52no—
95#4975—51no—
96#4972—51no—
97#4946—52no—
98#4942—52no—
99C4×Dic3⋊C4—51no—
100#4928D511yescharacteristic, complemented
101C22×D20C2×D311yescharacteristic
102C22×C4×D5—32no—
103C22×C4×D5—32no—
104C22×C4×D5C2×D312yes—
105C2×C5×C4⋊C4—31no—
106C2×C5×C4⋊C4—32no—
107C2×C5×C4⋊C4—31no—
108C2×C42×C5C2×D311yescharacteristic
109C2×#685—32no—
110C2×#685—32no—
111C2×#685C2×D312yes—
112C2×C4⋊Dic5—32no—
113C2×C4⋊Dic5C2×D311yescharacteristic
114C2×#683—31no—
115C2×#683—31no—
116C2×#683—32no—
117C2×#683—32no—
118C2×C4×Dic5—31no—
119C2×C4×Dic5—31no—
120#4039—38no—
121#4039—64no—
122D5×C4⋊C4—68no—
123C4×D20—24no—
124C5×#95—32no—
125#3963—31no—
126#3963—31no—
127#3963—32no—
128D4×C4⋊C4—151noSylow
129C23×C3×C5C2411yescharacteristic
130C22×C3×D5—22no—
131C22×C3×D5—22no—
132C22×C3×D5—22no—
133C22×C3×D5—22no—
134C22×C3×D5C22×C414yes—
135C22×C3×D5C2×D414yes—
136C22×C3×D5C2×Q814yes—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5—21no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5C22×C412yes—
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2×Q812yes—
146C2×Dic15—21no—
147C2×Dic15—21no—
148C2×Dic15—21no—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—21no—
152C2×Dic15—21no—
153C2×Dic15C22×C412yes—
154C2×Dic15C22×C412yes—
155C2×Dic15C2×D412yes—
156C2×Dic15C2×D412yes—
157C2×Dic15#4112yes—
158C2×C5×Dic3—22no—
159C2×C5×Dic3—22no—
160C2×C5×Dic3—22no—
161C2×C5×Dic3—22no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3C22×C414yes—
164C2×C3×Dic5—22no—
165C2×C3×Dic5—22no—
166C2×C3×Dic5—22no—
167C3×D20C4⋊C4116yescomplemented
168C3×C4×D5—48no—
169D5×Dic3—216no—
170D5×Dic3—48no—
171C22×C3×D4—51no—
172C23×C3×C4—52no—
173C23×Dic3—52no—
174C23×Dic3—52no—
175C3×C4×D4—104no—
176C2×C3×C4⋊C4—51no—
177C2×C3×C22⋊C4—52no—
178C2×C3×C42C2×D511yescharacteristic
179C2×#217—52no—
180C2×#217—52no—
181D4×Dic3—58no—
182D4×Dic3—104no—
183C2×C4⋊Dic3C2×D511yescharacteristic
184C2×C4⋊Dic3C2×D511yescharacteristic
185C2×Dic3⋊C4—51no—
186C2×Dic3⋊C4—51no—
187C2×Dic3⋊C4—52no—
188C2×Dic3⋊C4—52no—
189C2×Dic3⋊C4—52no—
190C2×Dic3⋊C4—108no—
191C2×Dic3⋊C4C2×D512yes—
192C2×C4×Dic3—51no—
193C2×C4×Dic3—51no—
194C3⋊#95—51no—
195C3⋊#95—51no—
196C3⋊#95—52no—
197C3⋊#95C2×D512yes—
198C23×D5C22×D312yes—
199C22×C4×C5—32no—
200C22×C4×C5—32no—
201C22×C4×C5C22×D311yescharacteristic
202C22×C4×C5C22×D311yescharacteristic
203C22×C4×C5C22×D311yescharacteristic
204C22×Dic5—31no—
205C22×Dic5—31no—
206C22×Dic5—31no—
207C22×Dic5—31no—
208C22×Dic5C22×D312yes—
209C2×D20Dic614yes—
210C2×D20C4×D314yes—
211C2×D20#6714yes—
212C2×C4×D5—24no—
213C2×C4×D5—24no—
214C2×C4×D5—38no—
215C2×C4×D5—42no—
216C2×C4×D5—64no—
217C2×C4×D5—64no—
218C2×C4×D5—64no—
219C2×C4×D5—64no—
220C2×C4×D5—122no—
221C5×C4⋊C4—34no—
222C5×C4⋊C4—64no—
223C5×C4⋊C4—62no—
224C42×C5—22no—
225#685—24no—
226#685—38no—
227#685—64no—
228C4⋊Dic5—22no—
229C4⋊Dic5—64no—
230#683—34no—
231#683—64no—
232#683—62no—
233#683—122no—
234C4×Dic5—34no—
235C4×Dic5—62no—
236C2×C4×D4—51no—
237C2×C4×D4—151no—
238C2×C4×D4—151no—
239C22×C4⋊C4—152no—
240#389—31no—
241#389—151no—
242#385—152no—
243#385—152no—
244#384—151no—
245#380—152no—
246C4×C4⋊C4—151no—
247C22×C3×C5C23×C411yescharacteristic
248C22×C3×C5C22×D411yescharacteristic
249C22×C3×C5C22×D411yescharacteristic
250C22×C3×C5C22×Q811yescharacteristic
251C22×C3×C5C2×#4111yescharacteristic
252C22×C3×C5H(5,𝔽2)11yescharacteristic
253C22×C3×C52−1+411yescharacteristic
254C2×C3×D5—24no—
255C2×C3×D5—24no—
256C2×C3×D5—24no—
257C2×C3×D5—24no—
258C2×C3×D5—24no—
259C2×C3×D5—24no—
260C2×C3×D5C2×C4⋊C418yescomplemented
261C3×C4×C5—22no—
262C3×C4×C5—41no—
263C3×C4×C5C2×C4⋊C414yescomplemented
264Dic15—22no—
265Dic15—22no—
266Dic15—41no—
267Dic15C4×D414yescomplemented
268C3×Dic5—42no—
269C5×Dic3—24no—
270C5×Dic3—42no—
271C24×C3—52no—
272C2×C3×D4—54no—
273C2×C3×D4—54no—
274C2×C3×D4—54no—
275C22×C3×C4—52no—
276C22×C3×C4—104no—
277C22×C3×C4—104no—
278C22×C3×C4—202no—
279C22×C3×C4C22×D511yescharacteristic
280C22×C3×C4C22×D511yescharacteristic
281C22×C3×C4C22×D511yescharacteristic
282C22×Dic3—51no—
283C22×Dic3—51no—
284C22×Dic3—51no—
285C22×Dic3—58no—
286C22×Dic3—51no—
287C22×Dic3—104no—
288C22×Dic3—104no—
289C22×Dic3—104no—
290C22×Dic3—104no—
291C22×Dic3—202no—
292C22×Dic3C22×D512yes—
293C22×Dic3C22×D512yes—
294C3×C4⋊C4—102no—
295C3×C22⋊C4—104no—
296C3×C42—22no—
297#217—58no—
298#217—104no—
299C4⋊Dic3—22no—
300C4⋊Dic3C4×D514yescomplemented
301Dic3⋊C4—24no—
302Dic3⋊C4—54no—
303Dic3⋊C4—104no—
304Dic3⋊C4—104no—
305Dic3⋊C4—102no—
306Dic3⋊C4—202no—
307C4×Dic3—54no—
308C4×Dic3—102no—
309C23×C5C23×D311yescharacteristic
310C22×D5—22no—
311C22×D5—22no—
312C22×D5—22no—
313C22×D5—22no—
314C22×D5C2×Dic614yes—
315C22×D5C2×C4×D314yes—
316C22×D5C2×#6714yes—
317C2×C4×C5—21no—
318C2×C4×C5—21no—
319C2×C4×C5—21no—
320C2×C4×C5—21no—
321C2×C4×C5—21no—
322C2×C4×C5—34no—
323C2×C4×C5—62no—
324C2×C4×C5—62no—
325C2×C4×C5—62no—
326C2×C4×C5—62no—
327C2×C4×C5—62no—
328C2×C4×C5C2×Dic612yes—
329C2×C4×C5C2×C4×D312yes—
330C2×C4×C5D3×D412yes—
331C2×C4×C5C2×#6712yes—
332C2×Dic5—22no—
333C2×Dic5—22no—
334C2×Dic5—22no—
335C2×Dic5—32no—
336C2×Dic5—32no—
337C2×Dic5—32no—
338C2×Dic5—32no—
339C2×Dic5—32no—
340C2×Dic5—61no—
341C2×Dic5—61no—
342C2×Dic5—61no—
343C2×Dic5—61no—
344C2×Dic5—61no—
345C2×Dic5—61no—
346C2×Dic5—61no—
347D20Dic3⋊C4116yescomplemented
348C4×D5—48no—
349C4×D5—616no—
350C4×D5—128no—
351C22×D4—51no—
352C23×C4—52no—
353C23×C4—152no—
354C23×C4—152no—
355C4×D4—104no—
356C4×D4—158no—
357C4×D4—304no—
358C2×C4⋊C4—31no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—31no—
361C2×C4⋊C4—51no—
362C2×C4⋊C4—151no—
363C2×C4⋊C4—152no—
364C2×C4⋊C4—152no—
365C2×C4⋊C4—151no—
366C2×C4⋊C4—152no—
367C2×C4⋊C4—308no—
368C2×C22⋊C4—52no—
369C2×C22⋊C4—152no—
370C2×C22⋊C4—152no—
371C2×C42—151no—
372C2×C42—151no—
373C2×C42D3×D511yescharacteristic
374#95—32no—
375#95—152no—
376#95—151no—
377#95—151no—
378C2×C3×C5C22×C4⋊C411yescharacteristic
379C2×C3×C5C2×C4×D411yescharacteristic
380C2×C3×C5#52011yescharacteristic
381C2×C3×C5D4211yescharacteristic
382C2×C3×C5#54711yescharacteristic
383C2×C3×C5D4×Q811yescharacteristic
384C2×C3×C5#55411yescharacteristic
385C3×D5—28no—
386C23×C3—54no—
387C23×C3—54no—
388C23×C3—54no—
389C23×C3—102no—
390C23×C3—102no—
391C23×C3—102no—
392C23×C3—102no—
393C23×C3C23×D511yescharacteristic
394C3×D4—516no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—102no—
401C2×C3×C4—102no—
402C2×C3×C4—102no—
403C2×C3×C4—208no—
404C2×C3×C4C2×C4×D512yes—
405C2×C3×C4D4×D512yes—
406C2×C3×C4D4×D512yes—
407C2×C3×C4Q8×D512yes—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—52no—
414C2×Dic3—52no—
415C2×Dic3—52no—
416C2×Dic3—52no—
417C2×Dic3—52no—
418C2×Dic3—1016no—
419C2×Dic3—101no—
420C2×Dic3—101no—
421C2×Dic3—101no—
422C2×Dic3—101no—
423C2×Dic3—101no—
424C2×Dic3—101no—
425C2×Dic3—101no—
426C2×Dic3—208no—
427C2×Dic3C2×C4×D514yes—
428C22×C5C22×Dic611yescharacteristic
429C22×C5C22×C4×D311yescharacteristic
430C22×C5C2×D3×D411yescharacteristic
431C22×C5#100411yescharacteristic
432C22×C5D3×#4111yescharacteristic
433C22×C5#101011yescharacteristic
434C22×C5C22×#6711yescharacteristic
435C2×D5—24no—
436C2×D5—24no—
437C2×D5—24no—
438C2×D5—24no—
439C2×D5—24no—
440C2×D5—24no—
441C2×D5C2×Dic3⋊C418yescomplemented
442C4×C5—22no—
443C4×C5—41no—
444C4×C5—64no—
445C4×C5—122no—
446C4×C5C2×Dic3⋊C414yescomplemented
447Dic5—34no—
448Dic5—42no—
449Dic5—62no—
450Dic5—62no—
451Dic5—121no—
452C24—52no—
453C2×D4—54no—
454C2×D4—54no—
455C2×D4—54no—
456C22×C4—32no—
457C22×C4—32no—
458C22×C4—52no—
459C22×C4—104no—
460C22×C4—104no—
461C22×C4—158no—
462C22×C4—151no—
463C22×C4—151no—
464C22×C4—151no—
465C22×C4—151no—
466C22×C4—202no—
467C22×C4—304no—
468C22×C4—304no—
469C22×C4—304no—
470C22×C4—304no—
471C22×C4—602no—
472C22×C4C2×D3×D511yescharacteristic
473C22×C4C2×D3×D511yescharacteristic
474C22×C4C2×D3×D511yescharacteristic
475C4⋊C4—34no—
476C4⋊C4—64no—
477C4⋊C4—62no—
478C4⋊C4—102no—
479C4⋊C4—154no—
480C4⋊C4—302no—
481C4⋊C4—304no—
482C4⋊C4—304no—
483C4⋊C4—602no—
484C22⋊C4—104no—
485C22⋊C4—158no—
486C22⋊C4—304no—
487C42—22no—
488C42—154no—
489C42—302no—
490C3×C5D4×C4⋊C411yescharacteristic, complemented, Hall
491C22×C3—58no—
492C22×C3—104no—
493C22×C3—104no—
494C22×C3—104no—
495C22×C3—104no—
496C22×C3—104no—
497C22×C3—104no—
498C22×C3C22×C4×D511yescharacteristic
499C22×C3C2×D4×D511yescharacteristic
500C22×C3C2×D4×D511yescharacteristic
501C22×C3C2×Q8×D511yescharacteristic
502C22×C3#414711yescharacteristic
503C22×C3D5×#4111yescharacteristic
504C22×C3#414911yescharacteristic
505C3×C4—22no—
506C3×C4—41no—
507C3×C4—202no—
508C3×C4D5×C4⋊C414yescomplemented
509Dic3—24no—
510Dic3—42no—
511Dic3—54no—
512Dic3—102no—
513Dic3—102no—
514Dic3—201no—
515C2×C5D4×Dic611yescharacteristic
516C2×C5#553911yescharacteristic
517C2×C5C4×D3×D411yescharacteristic
518C2×C5#554611yescharacteristic
519C2×C5C22×Dic3⋊C411yescharacteristic
520C2×C5D4×#6711yescharacteristic
521C2×C5#582411yescharacteristic
522D5—28no—
523C23—54no—
524C23—54no—
525C23—54no—
526C23—102no—
527C23—102no—
528C23—102no—
529C23—102no—
530C23C22×D3×D511yescharacteristic
531D4—516no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—34no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—152no—
547C2×C4—152no—
548C2×C4—152no—
549C2×C4—152no—
550C2×C4—152no—
551C2×C4—208no—
552C2×C4—3016no—
553C2×C4—301no—
554C2×C4—301no—
555C2×C4—301no—
556C2×C4—301no—
557C2×C4—301no—
558C2×C4—301no—
559C2×C4—301no—
560C2×C4—608no—
561C2×C4D5×Dic612yes—
562C2×C4C4×D3×D512yes—
563C2×C4#677712yes—
564C2×C4D5×#6712yes—
565C2×C3—108no—
566C2×C3C2×D5×C4⋊C411yescharacteristic
567C2×C3C4×D4×D511yescharacteristic
568C2×C3#6590811yescharacteristic
569C2×C3#6594311yescharacteristic
570C2×C3#6596511yescharacteristic
571C2×C3#6602411yescharacteristic
572C2×C3#6606111yescharacteristic
573C5D4×Dic3⋊C411yescharacteristic, complemented, Sylow
574C22—58no—
575C22—104no—
576C22—104no—
577C22—104no—
578C22—104no—
579C22—104no—
580C22—104no—
581C22C2×D5×Dic611yescharacteristic
582C22#9200011yescharacteristic
583C22C2×C4×D3×D511yescharacteristic
584C22C2×#677711yescharacteristic
585C22#9201611yescharacteristic
586C22#9201711yescharacteristic
587C22C2×D5×#6711yescharacteristic
588C4—22no—
589C4—41no—
590C4—64no—
591C4—122no—
592C4—154no—
593C4—202no—
594C4—302no—
595C4—302no—
596C4—601no—
597C4D5×Dic3⋊C414yescomplemented
598C3#1060793311yescharacteristic, complemented, Sylow
599C2—108no—
600C2#1175252011yescharacteristic
601C2#1175255211yescharacteristic
602C2#1175261811yescharacteristic
603C2C4×#677711yescharacteristic
604C2C2×D5×Dic3⋊C411yescharacteristic
605C2#1175430911yescharacteristic
606C2#1175464011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×D20C22
hypercenterC2×C42D3×D5
hypocenterC3×C5D4×C4⋊C4
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
4. C2×D5×Dic3⋊C4C2semidirect
5. C4×#6712C2semidirect
6. C5×#4928C2semidirect
7. #11748890C2semidirect
8. #11748885C2semidirect
9. #11748868C2semidirect
10. #11748866C2semidirect
11. #11748703C2semidirect
34. #91434C4semidirect
74. C2×D5×Dic3C2×C4semidirect
77. #6712D4semidirect
79. C4×Dic15C2×C4semidirect
81. C5×C4⋊Dic3C2×C4semidirect
89. #6665C2×C4semidirect
100. #4928D5semidirect
167. C3×D20C4⋊C4semidirect
260. C2×C3×D5C2×C4⋊C4semidirect
263. C3×C4×C5C2×C4⋊C4semidirect
267. Dic15C4×D4semidirect
300. C4⋊Dic3C4×D5semidirect
347. D20Dic3⋊C4semidirect
441. C2×D5C2×Dic3⋊C4semidirect
446. C4×C5C2×Dic3⋊C4semidirect
490. C3×C5D4×C4⋊C4semidirect
508. C3×C4D5×C4⋊C4semidirect
573. C5D4×Dic3⋊C4semidirect
597. C4D5×Dic3⋊C4semidirect
598. C3#10607933semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D202
C2×#914342
C2×D5×Dic3⋊C42
C4×#67122
C5×#49282
#117488902
#117488852
#117488682
#117488662
#117487032
#106079333
D4×Dic3⋊C45