Finite Group Databasebeta

← #14327081

Subgroups — #14327081

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×C4×C5×D12C211yesmaximal, characteristic
2C2×#91435C211yesmaximal, characteristic
3C2×#91435C211yesmaximal, characteristic
4C2×D3×#683C212yesmaximal, complemented
5C4×#6712C211yesmaximal, characteristic, complemented
6C3×#65217C211yesmaximal, characteristic, complemented
7#11748893C212yesmaximal, complemented
8#11748887C212yesmaximal, complemented
9#11748866C211yesmaximal, characteristic, complemented
10#11748865C211yesmaximal, characteristic, complemented
11#11748716C212yesmaximal, complemented
12D4×#683—31nomaximal, Hall
13C22×C5×D12C2211yescharacteristic
14C22×C4×C5×D3C2212yes—
15C22×D3×Dic5C2212yes—
16C22×D3×Dic5C2212yes—
17C2×C3×C42×C5C2211yescharacteristic
18C2×#6712C2211yescharacteristic
19C2×#6712C2211yescharacteristic
20C2×C4×Dic15C2211yescharacteristic
21C2×C4×Dic15C2211yescharacteristic
22C2×C5×(C2×D3)⋊C4C2212yes—
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C4×C5×D12—24no—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×#683C2212yes—
28C2×#6669C2212yes—
29C2×#6669C2212yes—
30C2×#6668C2212yes—
31C2×(C2×D3)⋊Dic5C2212yes—
32C2×(C2×D3)⋊Dic5C2212yes—
33#91435—24no—
34#91435C418yescomplemented
35D3×#683—28no—
36#91115C2211yescharacteristic
37#91115C2211yescharacteristic
38C3×#3963C2212yes—
39#90994C2212yes—
40#76202—51nomaximal, Hall
41C2×C4×C5×D4—31no—
42C2×D4×Dic5—31no—
43C2×D4×Dic5—31no—
44C22×#683—32no—
45#65499—32no—
46#65265—31no—
47#65262—31no—
48#65235—32no—
49#65231—32no—
50C4×#683—31no—
51#65217D311yescharacteristic, complemented
52C23×C5×D3C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×Dic15C2311yescharacteristic
57C22×Dic15C2311yescharacteristic
58C22×Dic15C2311yescharacteristic
59C22×Dic15C2311yescharacteristic
60C22×C5×Dic3C2312yes—
61C2×C5×D12C2×C414yes—
62C2×C5×D12D414yes—
63C2×C5×D12Q814yes—
64C2×C4×C5×D3—24no—
65C2×C4×C5×D3—24no—
66C2×C4×C5×D3—42no—
67C22×C3×Dic5C2312yes—
68C22×C3×Dic5C2312yes—
69C2×D3×Dic5—24no—
70C2×D3×Dic5—24no—
71C2×D3×Dic5—24no—
72C2×D3×Dic5—24no—
73C2×D3×Dic5—42no—
74C2×D3×Dic5C2×C418yescomplemented
75C3×C42×C5—22no—
76#6712—22no—
77#6712D414yescomplemented
78C4×Dic15—22no—
79C4×Dic15C2×C414yescomplemented
80C5×(C2×D3)⋊C4—24no—
81C5×C4⋊Dic3—22no—
82C3×C4⋊Dic5—22no—
83C3×C4⋊Dic5C2×C414yescomplemented
84C3×#683—24no—
85#6669—24no—
86#6669—42no—
87#6668—24no—
88(C2×D3)⋊Dic5—24no—
89(C2×D3)⋊Dic5C2×C418yescomplemented
90C2×#893—51no—
91C2×#893—51no—
92C2×D3×C4⋊C4—52no—
93C2×C4×D12D511yescharacteristic
94C3×#389—51no—
95#4989—52no—
96#4984—51no—
97#4975—51no—
98C4×Dic3⋊C4—51no—
99#4668—52no—
100#4667—52no—
101C22×C5×D4—31no—
102C23×C4×C5—32no—
103C23×Dic5—32no—
104C23×Dic5—32no—
105C4×C5×D4—64no—
106C2×C5×C4⋊C4—31no—
107C2×C5×C22⋊C4—32no—
108C2×C42×C5C2×D311yescharacteristic
109C2×C22⋊Dic5—32no—
110C2×C22⋊Dic5—32no—
111D4×Dic5—38no—
112D4×Dic5—64no—
113C2×C4⋊Dic5C2×D311yescharacteristic
114C2×C4⋊Dic5C2×D311yescharacteristic
115C2×#683—31no—
116C2×#683—31no—
117C2×#683—32no—
118C2×#683—32no—
119C2×#683—32no—
120C2×#683—68no—
121C2×#683C2×D312yes—
122C2×C4×Dic5—31no—
123C2×C4×Dic5—31no—
124#3963—31no—
125#3963—31no—
126#3963—32no—
127#3963C2×D312yes—
128D4×C4⋊C4—151noSylow
129C23×C3×C5C2411yescharacteristic
130C22×C5×D3—22no—
131C22×C5×D3—22no—
132C22×C5×D3—22no—
133C22×C5×D3—22no—
134C22×C5×D3C22×C414yes—
135C22×C5×D3C2×D414yes—
136C22×C5×D3C2×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—
161C5×D12C4⋊C4116yescomplemented
162C4×C5×D3—48no—
163C2×C3×Dic5—22no—
164C2×C3×Dic5—22no—
165C2×C3×Dic5—22no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—22no—
168C2×C3×Dic5C22×C414yes—
169D3×Dic5—216no—
170D3×Dic5—48no—
171C22×D12C2×D511yescharacteristic
172C22×C4×D3—52no—
173C22×C4×D3—52no—
174C22×C4×D3C2×D512yes—
175C2×C3×C4⋊C4—51no—
176C2×C3×C4⋊C4—51no—
177C2×C3×C4⋊C4—52no—
178C2×C3×C42C2×D511yescharacteristic
179C2×(C2×D3)⋊C4—52no—
180C2×(C2×D3)⋊C4—52no—
181C2×(C2×D3)⋊C4C2×D512yes—
182C2×C4⋊Dic3—52no—
183C2×C4⋊Dic3C2×D511yescharacteristic
184C2×Dic3⋊C4—51no—
185C2×Dic3⋊C4—51no—
186C2×Dic3⋊C4—52no—
187C2×Dic3⋊C4—52no—
188C2×C4×Dic3—51no—
189C2×C4×Dic3—51no—
190#893—58no—
191#893—104no—
192D3×C4⋊C4—108no—
193C4×D12—24no—
194C3×#95—52no—
195C3⋊#95—51no—
196C3⋊#95—51no—
197C3⋊#95—52no—
198C24×C5—32no—
199C2×C5×D4—34no—
200C2×C5×D4—34no—
201C2×C5×D4—34no—
202C22×C4×C5—32no—
203C22×C4×C5—64no—
204C22×C4×C5—64no—
205C22×C4×C5—122no—
206C22×C4×C5C22×D311yescharacteristic
207C22×C4×C5C22×D311yescharacteristic
208C22×C4×C5C22×D311yescharacteristic
209C22×Dic5—31no—
210C22×Dic5—31no—
211C22×Dic5—31no—
212C22×Dic5—38no—
213C22×Dic5—31no—
214C22×Dic5—64no—
215C22×Dic5—64no—
216C22×Dic5—64no—
217C22×Dic5—64no—
218C22×Dic5—122no—
219C22×Dic5C22×D312yes—
220C22×Dic5C22×D312yes—
221C5×C4⋊C4—62no—
222C5×C22⋊C4—64no—
223C42×C5—22no—
224C22⋊Dic5—38no—
225C22⋊Dic5—64no—
226C4⋊Dic5—22no—
227C4⋊Dic5C4×D314yescomplemented
228#683—24no—
229#683—34no—
230#683—64no—
231#683—64no—
232#683—62no—
233#683—122no—
234C4×Dic5—34no—
235C4×Dic5—62no—
236C2×C4×D4—31no—
237C2×C4×D4—151no—
238C2×C4×D4—151no—
239C22×C4⋊C4—152no—
240#389—51no—
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×C5×D3—24no—
255C2×C5×D3—24no—
256C2×C5×D3—24no—
257C2×C5×D3—24no—
258C2×C5×D3—24no—
259C2×C5×D3—24no—
260C2×C5×D3C2×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—24no—
269C3×Dic5—42no—
270C5×Dic3—42no—
271C23×D3C22×D512yes—
272C22×C3×C4—52no—
273C22×C3×C4—52no—
274C22×C3×C4C22×D511yescharacteristic
275C22×C3×C4C22×D511yescharacteristic
276C22×C3×C4C22×D511yescharacteristic
277C22×Dic3—51no—
278C22×Dic3—51no—
279C22×Dic3—51no—
280C22×Dic3—51no—
281C22×Dic3C22×D512yes—
282C2×D12Dic1014yes—
283C2×D12C4×D514yes—
284C2×D12C22⋊D514yes—
285C2×C4×D3—24no—
286C2×C4×D3—24no—
287C2×C4×D3—42no—
288C2×C4×D3—58no—
289C2×C4×D3—104no—
290C2×C4×D3—104no—
291C2×C4×D3—104no—
292C2×C4×D3—104no—
293C2×C4×D3—202no—
294C3×C4⋊C4—54no—
295C3×C4⋊C4—104no—
296C3×C4⋊C4—102no—
297C3×C42—22no—
298(C2×D3)⋊C4—24no—
299(C2×D3)⋊C4—58no—
300(C2×D3)⋊C4—104no—
301C4⋊Dic3—22no—
302C4⋊Dic3—104no—
303Dic3⋊C4—54no—
304Dic3⋊C4—104no—
305Dic3⋊C4—102no—
306Dic3⋊C4—202no—
307C4×Dic3—54no—
308C4×Dic3—102no—
309C23×C5—34no—
310C23×C5—34no—
311C23×C5—34no—
312C23×C5—62no—
313C23×C5—62no—
314C23×C5—62no—
315C23×C5—62no—
316C23×C5C23×D311yescharacteristic
317C5×D4—316no—
318C2×C4×C5—21no—
319C2×C4×C5—21no—
320C2×C4×C5—21no—
321C2×C4×C5—21no—
322C2×C4×C5—21no—
323C2×C4×C5—62no—
324C2×C4×C5—62no—
325C2×C4×C5—62no—
326C2×C4×C5—128no—
327C2×C4×C5C2×C4×D312yes—
328C2×C4×C5D3×D412yes—
329C2×C4×C5D3×D412yes—
330C2×C4×C5D3×Q812yes—
331C2×Dic5—22no—
332C2×Dic5—22no—
333C2×Dic5—22no—
334C2×Dic5—22no—
335C2×Dic5—22no—
336C2×Dic5—32no—
337C2×Dic5—32no—
338C2×Dic5—32no—
339C2×Dic5—32no—
340C2×Dic5—32no—
341C2×Dic5—616no—
342C2×Dic5—61no—
343C2×Dic5—61no—
344C2×Dic5—61no—
345C2×Dic5—61no—
346C2×Dic5—61no—
347C2×Dic5—61no—
348C2×Dic5—61no—
349C2×Dic5—128no—
350C2×Dic5C2×C4×D314yes—
351C22×D4—31no—
352C23×C4—32no—
353C23×C4—152no—
354C23×C4—152no—
355C4×D4—64no—
356C4×D4—158no—
357C4×D4—304no—
358C2×C4⋊C4—31no—
359C2×C4⋊C4—51no—
360C2×C4⋊C4—52no—
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—32no—
369C2×C22⋊C4—152no—
370C2×C22⋊C4—152no—
371C2×C42—151no—
372C2×C42—151no—
373C2×C42D3×D511yescharacteristic
374#95—52no—
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
385C5×D3—28no—
386C23×C3C23×D511yescharacteristic
387C22×D3—22no—
388C22×D3—22no—
389C22×D3—22no—
390C22×D3—22no—
391C22×D3C2×Dic1014yes—
392C22×D3C2×C4×D514yes—
393C22×D3C2×C22⋊D514yes—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—54no—
400C2×C3×C4—102no—
401C2×C3×C4—102no—
402C2×C3×C4—102no—
403C2×C3×C4—102no—
404C2×C3×C4—102no—
405C2×C3×C4C2×Dic1012yes—
406C2×C3×C4C2×C4×D512yes—
407C2×C3×C4D4×D512yes—
408C2×C3×C4C2×C22⋊D512yes—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—52no—
413C2×Dic3—52no—
414C2×Dic3—52no—
415C2×Dic3—52no—
416C2×Dic3—52no—
417C2×Dic3—101no—
418C2×Dic3—101no—
419C2×Dic3—101no—
420C2×Dic3—101no—
421C2×Dic3—101no—
422C2×Dic3—101no—
423C2×Dic3—101no—
424D12#683116yescomplemented
425C4×D3—48no—
426C4×D3—1016no—
427C4×D3—208no—
428C22×C5—38no—
429C22×C5—64no—
430C22×C5—64no—
431C22×C5—64no—
432C22×C5—64no—
433C22×C5—64no—
434C22×C5—64no—
435C22×C5C22×C4×D311yescharacteristic
436C22×C5C2×D3×D411yescharacteristic
437C22×C5C2×D3×D411yescharacteristic
438C22×C5C2×D3×Q811yescharacteristic
439C22×C5#100711yescharacteristic
440C22×C5D3×#4111yescharacteristic
441C22×C5#100911yescharacteristic
442C4×C5—22no—
443C4×C5—41no—
444C4×C5—122no—
445C4×C5D3×C4⋊C414yescomplemented
446Dic5—24no—
447Dic5—34no—
448Dic5—42no—
449Dic5—62no—
450Dic5—62no—
451Dic5—121no—
452C24—32no—
453C2×D4—34no—
454C2×D4—34no—
455C2×D4—34no—
456C22×C4—32no—
457C22×C4—52no—
458C22×C4—52no—
459C22×C4—64no—
460C22×C4—64no—
461C22×C4—122no—
462C22×C4—158no—
463C22×C4—151no—
464C22×C4—151no—
465C22×C4—151no—
466C22×C4—151no—
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—54no—
476C4⋊C4—62no—
477C4⋊C4—104no—
478C4⋊C4—102no—
479C4⋊C4—154no—
480C4⋊C4—302no—
481C4⋊C4—304no—
482C4⋊C4—304no—
483C4⋊C4—602no—
484C22⋊C4—64no—
485C22⋊C4—158no—
486C22⋊C4—304no—
487C42—22no—
488C42—154no—
489C42—302no—
490C3×C5D4×C4⋊C411yescharacteristic, complemented, Hall
491C22×C3C22×Dic1011yescharacteristic
492C22×C3C22×C4×D511yescharacteristic
493C22×C3C2×D4×D511yescharacteristic
494C22×C3#414411yescharacteristic
495C22×C3D5×#4111yescharacteristic
496C22×C3#415011yescharacteristic
497C22×C3C22×C22⋊D511yescharacteristic
498C2×D3—24no—
499C2×D3—24no—
500C2×D3—24no—
501C2×D3—24no—
502C2×D3—24no—
503C2×D3—24no—
504C2×D3C2×#68318yescomplemented
505C3×C4—22no—
506C3×C4—41no—
507C3×C4—104no—
508C3×C4—202no—
509C3×C4C2×#68314yescomplemented
510Dic3—42no—
511Dic3—54no—
512Dic3—102no—
513Dic3—102no—
514Dic3—201no—
515C2×C5—68no—
516C2×C5C2×D3×C4⋊C411yescharacteristic
517C2×C5C4×D3×D411yescharacteristic
518C2×C5#557411yescharacteristic
519C2×C5#560911yescharacteristic
520C2×C5#563111yescharacteristic
521C2×C5#569011yescharacteristic
522C2×C5#572711yescharacteristic
523C23—34no—
524C23—34no—
525C23—34no—
526C23—62no—
527C23—62no—
528C23—62no—
529C23—62no—
530C23C22×D3×D511yescharacteristic
531D4—316no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—54no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—102no—
542C2×C4—102no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—128no—
547C2×C4—152no—
548C2×C4—152no—
549C2×C4—152no—
550C2×C4—152no—
551C2×C4—152no—
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×C4D3×Dic1012yes—
562C2×C4C4×D3×D512yes—
563C2×C4#677712yes—
564C2×C4D3×C22⋊D512yes—
565C2×C3D4×Dic1011yescharacteristic
566C2×C3#6587311yescharacteristic
567C2×C3C4×D4×D511yescharacteristic
568C2×C3#6588011yescharacteristic
569C2×C3C22×#68311yescharacteristic
570C2×C3D4×C22⋊D511yescharacteristic
571C2×C3#6615811yescharacteristic
572D3—28no—
573C5#7620211yescharacteristic, complemented, Sylow
574C22—38no—
575C22—64no—
576C22—64no—
577C22—64no—
578C22—64no—
579C22—64no—
580C22—64no—
581C22C2×D3×Dic1011yescharacteristic
582C22#9200411yescharacteristic
583C22C2×C4×D3×D511yescharacteristic
584C22C2×#677711yescharacteristic
585C22#9201611yescharacteristic
586C22#9201811yescharacteristic
587C22C2×D3×C22⋊D511yescharacteristic
588C4—22no—
589C4—41no—
590C4—104no—
591C4—122no—
592C4—154no—
593C4—202no—
594C4—302no—
595C4—302no—
596C4—601no—
597C4D3×#68314yescomplemented
598C3D4×#68311yescharacteristic, complemented, Sylow
599C2—68no—
600C2#1175252511yescharacteristic
601C2#1175255811yescharacteristic
602C2#1175262711yescharacteristic
603C2C4×#677711yescharacteristic
604C2C2×D3×#68311yescharacteristic
605C2#1175435011yescharacteristic
606C2#1175465711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C5×D12C22
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×D3×#683C2semidirect
5. C4×#6712C2semidirect
6. C3×#65217C2semidirect
7. #11748893C2semidirect
8. #11748887C2semidirect
9. #11748866C2semidirect
10. #11748865C2semidirect
11. #11748716C2semidirect
34. #91435C4semidirect
51. #65217D3semidirect
74. C2×D3×Dic5C2×C4semidirect
77. #6712D4semidirect
79. C4×Dic15C2×C4semidirect
83. C3×C4⋊Dic5C2×C4semidirect
89. (C2×D3)⋊Dic5C2×C4semidirect
161. C5×D12C4⋊C4semidirect
227. C4⋊Dic5C4×D3semidirect
260. C2×C5×D3C2×C4⋊C4semidirect
263. C3×C4×C5C2×C4⋊C4semidirect
267. Dic15C4×D4semidirect
424. D12#683semidirect
445. C4×C5D3×C4⋊C4semidirect
490. C3×C5D4×C4⋊C4semidirect
504. C2×D3C2×#683semidirect
509. C3×C4C2×#683semidirect
573. C5#76202semidirect
597. C4D3×#683semidirect
598. C3D4×#683semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×D122
C2×#914352
C2×D3×#6832
C4×#67122
C3×#652172
#117488932
#117488872
#117488662
#117488652
#117487162
D4×#6833
#762025