Finite Group Databasebeta

← #14327708

Subgroups — #14327708

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4038C211yesmaximal, characteristic, complemented
2C2×C4×#1201C211yesmaximal, characteristic
3C2×#91338C211yesmaximal, characteristic, complemented
4#11751076C211yesmaximal, characteristic, complemented
5C5×#4987C211yesmaximal, characteristic, complemented
6#11748916C212yesmaximal
7#11748759C211yesmaximal, characteristic, complemented
8#11748715C212yesmaximal, complemented
9#11748707C212yesmaximal, complemented
10#11748699C211yesmaximal, characteristic, complemented
11#11748640C212yesmaximal, complemented
12#10605616—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C22×C3×C4×D5C2211yescharacteristic
15C22×#1201C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C4⋊Dic15C2211yescharacteristic
18C2×#6712C2212yes—
19C2×#6712C2212yes—
20C2×C4×Dic15C2211yescharacteristic
21C2×C5×(C2×D3)⋊C4C2212yes—
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C3×#685C2212yes—
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×C4×Dic5C2212yes—
26C3×#4038—24no—
27C2×#6669C2211yescharacteristic
28C2×(C2×D3)⋊Dic5C2212yes—
29C2×(C2×D3)⋊Dic5C2211yescharacteristic
30C2×#6665C2212yes—
31C2×#6665C2211yescharacteristic
32C2×Dic3×Dic5C2212yes—
33C2×Dic3×Dic5C2212yes—
34C4×#1201C418yescomplemented
35#91338—24no—
36C5×C3⋊#95C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#82241—51nomaximal, Hall
40C2×C4×C22⋊D5—31no—
41C2×#4072—31no—
42C2×#4038D311yescharacteristic, complemented
43C5×#380—31no—
44#65276—31no—
45#65265—31no—
46C4⋊C4×Dic5—31no—
47#65246—32no—
48#65234—32no—
49#64958—32no—
50#64938—32no—
51C23×C5×D3C2311yescharacteristic
52C23×C3×D5C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2312yes—
56C22×Dic15C2311yescharacteristic
57C22×Dic15C2311yescharacteristic
58C22×C5×Dic3C2312yes—
59C22×C5×Dic3C2311yescharacteristic
60C2×C4×C5×D3—22no—
61C2×C4×C5×D3—22no—
62C2×C4×C5×D3C2×C414yes—
63C22×C3×Dic5C2311yescharacteristic
64C22×C3×Dic5C2312yes—
65C2×C3×C4×D5—22no—
66C2×C3×C4×D5—22no—
67C2×C3×C4×D5C2×C414yes—
68C2×#1201—22no—
69C2×#1201—22no—
70C2×#1201C2×C414yes—
71C3×C5×C4⋊C4—22no—
72C4⋊Dic15—22no—
73#6712—24no—
74#6712—42no—
75C4×Dic15C2×C414yes—
76C5×(C2×D3)⋊C4—42no—
77C5×C4⋊Dic3—22no—
78C3×#685—24no—
79C3×C4⋊Dic5D414yescomplemented
80C3×C4×Dic5—24no—
81#6669C2×C414yes—
82(C2×D3)⋊Dic5—24no—
83(C2×D3)⋊Dic5C2×C414yes—
84#6665—42no—
85#6665C2×C414yes—
86Dic3×Dic5—24no—
87Dic3×Dic5—24no—
88C2×C3×#117—51no—
89C2×C4×#67—51no—
90C2×#892—51no—
91#5212—51no—
92#4987D511yescharacteristic, complemented
93#4975—51no—
94Dic3×C4⋊C4—51no—
95#4957—52no—
96#4944—52no—
97#4940—52no—
98C4×Dic3⋊C4—52no—
99C23×C4×C5—31no—
100C22×C22⋊D5—31no—
101C22×C4×D5C2×D311yescharacteristic
102C2×C5×C4⋊C4—31no—
103C2×C5×C4⋊C4C2×D311yescharacteristic
104C2×C5×C22⋊C4—32no—
105C2×C22⋊Dic5—31no—
106C2×C22⋊Dic5—32no—
107C4×C22⋊D5—38no—
108C2×#685—31no—
109C2×#685—32no—
110C2×#685C2×D312yes—
111#4072—64no—
112C2×C4⋊Dic5—31no—
113C2×C4⋊Dic5C2×D311yescharacteristic
114C2×#683—31no—
115C2×#683—32no—
116C2×#683—32no—
117C2×C4×Dic5—31no—
118C2×C4×Dic5—32no—
119C2×C4×Dic5—32no—
120C2×C4×Dic5C2×D312yes—
121#4038—24no—
122C5×#95—32no—
123#3963—32no—
124#3963—32no—
125#2354—151noSylow
126C23×C3×C5C2411yescharacteristic
127C22×C5×D3—21no—
128C22×C5×D3—21no—
129C22×C5×D3—21no—
130C22×C5×D3—21no—
131C22×C5×D3—21no—
132C22×C5×D3—21no—
133C22×C5×D3C22×C412yes—
134C22×C3×D5—21no—
135C22×C3×D5—21no—
136C22×C3×D5—21no—
137C22×C3×D5—21no—
138C22×C3×D5—21no—
139C22×C3×D5—21no—
140C22×C3×D5C22×C412yes—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5#4112yes—
147C2×Dic15—21no—
148C2×Dic15—21no—
149C2×Dic15—22no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15C22×C412yes—
153C2×Dic15C22×C412yes—
154C2×Dic15#4112yes—
155C2×Dic15#4112yes—
156C2×C5×Dic3—21no—
157C2×C5×Dic3—21no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3—22no—
160C2×C5×Dic3—22no—
161C2×C5×Dic3C22×C412yes—
162C4×C5×D3—28no—
163C2×C3×Dic5—21no—
164C2×C3×Dic5—22no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5C22×C412yes—
168C2×C3×Dic5C2×D414yes—
169C3×C4×D5—28no—
170#1201—28no—
171C23×C3×C4—51no—
172C22×#67—51no—
173C22×C4×D3C2×D511yescharacteristic
174C3×#117—104no—
175C2×C3×C4⋊C4—51no—
176C2×C3×C4⋊C4C2×D511yescharacteristic
177C2×C3×C22⋊C4—52no—
178C2×C3×C42—52no—
179C2×#217—51no—
180C2×#217—52no—
181C4×#67—58no—
182C2×(C2×D3)⋊C4—51no—
183C2×(C2×D3)⋊C4—52no—
184C2×(C2×D3)⋊C4C2×D512yes—
185C2×C4⋊Dic3—51no—
186C2×C4⋊Dic3C2×D511yescharacteristic
187C2×Dic3⋊C4—51no—
188C2×Dic3⋊C4—52no—
189C2×Dic3⋊C4—52no—
190C2×C4×Dic3—51no—
191C2×C4×Dic3—52no—
192C2×C4×Dic3—52no—
193#892—104no—
194C3⋊#95—52no—
195C3⋊#95—52no—
196C3⋊#95C2×D512yes—
197C24×C5—31no—
198C23×D5C22×D311yescharacteristic
199C22×C4×C5—31no—
200C22×C4×C5—34no—
201C22×C4×C5—32no—
202C22×C4×C5—62no—
203C22×C4×C5—62no—
204C22×C4×C5C22×D312yes—
205C22×C4×C5C22×D311yescharacteristic
206C2×C22⋊D5—34no—
207C2×C22⋊D5—62no—
208C2×C22⋊D5—62no—
209C22×Dic5—31no—
210C22×Dic5—31no—
211C22×Dic5—32no—
212C22×Dic5C22×D311yescharacteristic
213C22×Dic5C22×D312yes—
214C2×C4×D5—22no—
215C2×C4×D5—22no—
216C2×C4×D5C4×D314yes—
217C5×C4⋊C4—22no—
218C5×C4⋊C4—62no—
219C5×C22⋊C4—122no—
220C22⋊Dic5—34no—
221C22⋊Dic5—64no—
222#685—24no—
223#685—34no—
224#685—122no—
225C4⋊Dic5—62no—
226C4⋊Dic5#6714yescomplemented
227#683—34no—
228#683—64no—
229#683—122no—
230C4×Dic5—24no—
231C4×Dic5—34no—
232C4×Dic5—64no—
233C4×Dic5—64no—
234C2×C4×D4—151no—
235C2×#117—51no—
236C2×#117—151no—
237#390—152no—
238#388—152no—
239#388—152no—
240#384—151no—
241#380—31no—
242#380—151no—
243C4×C4⋊C4—152no—
244C4×C4⋊C4—151no—
245C22×C3×C5C23×C411yescharacteristic
246C22×C3×C5C22×D411yescharacteristic
247C22×C3×C5C2×#4111yescharacteristic
248C22×C3×C5C2×#4111yescharacteristic
249C22×C3×C5C2×#4111yescharacteristic
250C22×C3×C5H(5,𝔽2)11yescharacteristic
251C22×C3×C52−1+411yescharacteristic
252C2×C5×D3—22no—
253C2×C5×D3—22no—
254C2×C5×D3—22no—
255C2×C5×D3—41no—
256C2×C5×D3—41no—
257C2×C5×D3—41no—
258C2×C5×D3—41no—
259C2×C3×D5—22no—
260C2×C3×D5—22no—
261C2×C3×D5—22no—
262C2×C3×D5—41no—
263C2×C3×D5—41no—
264C2×C3×D5—41no—
265C2×C3×D5—41no—
266C3×C4×C5—42no—
267C3×C4×C5C4×D414yes—
268Dic15—22no—
269Dic15—42no—
270Dic15#11714yescomplemented
271C3×Dic5—22no—
272C3×Dic5—24no—
273C5×Dic3—22no—
274C5×Dic3—42no—
275C24×C3—51no—
276C23×D3C22×D511yescharacteristic
277C22×C3×C4—51no—
278C22×C3×C4—52no—
279C22×C3×C4—54no—
280C22×C3×C4—102no—
281C22×C3×C4—102no—
282C22×C3×C4C22×D512yes—
283C22×C3×C4C22×D511yescharacteristic
284C2×#67—54no—
285C2×#67—102no—
286C2×#67—102no—
287C22×Dic3—51no—
288C22×Dic3—51no—
289C22×Dic3—52no—
290C22×Dic3C22×D512yes—
291C22×Dic3C22×D511yescharacteristic
292C2×C4×D3—22no—
293C2×C4×D3—22no—
294C2×C4×D3C4×D514yes—
295C3×C4⋊C4—22no—
296C3×C4⋊C4—54no—
297C3×C22⋊C4—104no—
298C3×C42—104no—
299#217—54no—
300#217—202no—
301(C2×D3)⋊C4—42no—
302(C2×D3)⋊C4—54no—
303(C2×D3)⋊C4—104no—
304C4⋊Dic3—22no—
305C4⋊Dic3—102no—
306Dic3⋊C4—54no—
307Dic3⋊C4—104no—
308Dic3⋊C4—202no—
309C4×Dic3—54no—
310C4×Dic3—104no—
311C4×Dic3—104no—
312C23×C5—32no—
313C23×C5—61no—
314C23×C5—61no—
315C23×C5—61no—
316C23×C5—61no—
317C23×C5—61no—
318C23×C5—61no—
319C23×C5C23×D311yescharacteristic
320C22×D5—21no—
321C22×D5—21no—
322C22×D5—21no—
323C22×D5—21no—
324C22×D5—21no—
325C22×D5—21no—
326C22×D5C2×C4×D312yes—
327C2×C4×C5—22no—
328C2×C4×C5—22no—
329C2×C4×C5—22no—
330C2×C4×C5—32no—
331C2×C4×C5—61no—
332C2×C4×C5—61no—
333C2×C4×C5—68no—
334C2×C4×C5—62no—
335C2×C4×C5—62no—
336C2×C4×C5—62no—
337C2×C4×C5C2×C4×D312yes—
338C2×C4×C5#23512yes—
339C2×C4×C5C2×#6712yes—
340C22⋊D5—68no—
341C2×Dic5—21no—
342C2×Dic5—22no—
343C2×Dic5—22no—
344C2×Dic5—21no—
345C2×Dic5—32no—
346C2×Dic5—32no—
347C2×Dic5—32no—
348C2×Dic5—32no—
349C2×Dic5—61no—
350C2×Dic5—61no—
351C2×Dic5—62no—
352C2×Dic5—62no—
353C2×Dic5—62no—
354C2×Dic5C2×C4×D312yes—
355C2×Dic5C2×#6714yes—
356C4×D5—28no—
357C22×D4—151no—
358C23×C4—31no—
359C23×C4—51no—
360C4×D4—158no—
361#117—104no—
362#117—304no—
363C2×C4⋊C4—31no—
364C2×C4⋊C4—51no—
365C2×C4⋊C4—151no—
366C2×C4⋊C4—151no—
367C2×C4⋊C4—152no—
368C2×C4⋊C4—152no—
369C2×C4⋊C4D3×D511yescharacteristic
370C2×C22⋊C4—32no—
371C2×C22⋊C4—52no—
372C2×C22⋊C4—151no—
373C2×C22⋊C4—151no—
374C2×C22⋊C4—152no—
375C2×C22⋊C4—152no—
376C2×C42—52no—
377C2×C42—151no—
378C2×C42—152no—
379C2×C42—152no—
380#95—32no—
381#95—152no—
382#95—152no—
383C2×C3×C5C2×#11711yescharacteristic
384C2×C3×C5C2×C4×D411yescharacteristic
385C2×C3×C5#52011yescharacteristic
386C2×C3×C5#54711yescharacteristic
387C2×C3×C5#55011yescharacteristic
388C2×C3×C5#55511yescharacteristic
389C2×C3×C5#55611yescharacteristic
390C3×D5—42no—
391C5×D3—42no—
392C23×C3—52no—
393C23×C3—101no—
394C23×C3—101no—
395C23×C3—101no—
396C23×C3—101no—
397C23×C3—101no—
398C23×C3—101no—
399C23×C3C23×D511yescharacteristic
400C22×D3—21no—
401C22×D3—21no—
402C22×D3—21no—
403C22×D3—21no—
404C22×D3—21no—
405C22×D3—21no—
406C22×D3C2×C4×D512yes—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—52no—
411C2×C3×C4—54no—
412C2×C3×C4—101no—
413C2×C3×C4—101no—
414C2×C3×C4—102no—
415C2×C3×C4—102no—
416C2×C3×C4—108no—
417C2×C3×C4C2×C4×D512yes—
418C2×C3×C4D4×D512yes—
419C2×C3×C4#71112yes—
420#67—108no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—52no—
427C2×Dic3—52no—
428C2×Dic3—52no—
429C2×Dic3—52no—
430C2×Dic3—101no—
431C2×Dic3—101no—
432C2×Dic3—102no—
433C2×Dic3—102no—
434C2×Dic3—102no—
435C2×Dic3C2×C4×D512yes—
436C4×D3—28no—
437C22×C5—62no—
438C22×C5—62no—
439C22×C5—62no—
440C22×C5—121no—
441C22×C5—121no—
442C22×C5—121no—
443C22×C5—121no—
444C22×C5C22×C4×D311yescharacteristic
445C22×C5C2×#23511yescharacteristic
446C22×C5D3×#4111yescharacteristic
447C22×C5D3×#4111yescharacteristic
448C22×C5#100911yescharacteristic
449C22×C5#101011yescharacteristic
450C22×C5C22×#6711yescharacteristic
451C2×D5—22no—
452C2×D5—22no—
453C2×D5—22no—
454C2×D5—41no—
455C2×D5—41no—
456C2×D5—41no—
457C2×D5—41no—
458C4×C5—42no—
459C4×C5—62no—
460C4×C5—122no—
461C4×C5C4×#6714yes—
462Dic5—22no—
463Dic5—24no—
464Dic5—34no—
465Dic5—62no—
466Dic5—122no—
467C24—31no—
468C24—51no—
469C2×D4—154no—
470C2×D4—302no—
471C2×D4—302no—
472C22×C4—31no—
473C22×C4—34no—
474C22×C4—32no—
475C22×C4—51no—
476C22×C4—52no—
477C22×C4—54no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—102no—
481C22×C4—102no—
482C22×C4—151no—
483C22×C4—151no—
484C22×C4—152no—
485C22×C4C2×D3×D512yes—
486C22×C4C2×D3×D511yescharacteristic
487C4⋊C4—22no—
488C4⋊C4—54no—
489C4⋊C4—62no—
490C4⋊C4—154no—
491C4⋊C4—304no—
492C4⋊C4—302no—
493C4⋊C4—602no—
494C22⋊C4—104no—
495C22⋊C4—122no—
496C22⋊C4—154no—
497C22⋊C4—154no—
498C22⋊C4—304no—
499C22⋊C4—602no—
500C42—104no—
501C42—154no—
502C42—304no—
503C42—304no—
504C3×C5#235411yescharacteristic, complemented, Hall
505C22×C3—102no—
506C22×C3—102no—
507C22×C3—102no—
508C22×C3—201no—
509C22×C3—201no—
510C22×C3—201no—
511C22×C3—201no—
512C22×C3C22×C4×D511yescharacteristic
513C22×C3C2×D4×D511yescharacteristic
514C22×C3C2×#71111yescharacteristic
515C22×C3#414411yescharacteristic
516C22×C3#414711yescharacteristic
517C22×C3D5×#4111yescharacteristic
518C22×C3D5×#4111yescharacteristic
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—41no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C3×C4—42no—
527C3×C4—102no—
528C3×C4—104no—
529C3×C4#402714yes—
530Dic3—22no—
531Dic3—42no—
532Dic3—54no—
533Dic3—102no—
534Dic3—202no—
535C2×C5—122no—
536C2×C5D3×#11711yescharacteristic
537C2×C5#552311yescharacteristic
538C2×C5#553211yescharacteristic
539C2×C5#553311yescharacteristic
540C2×C5C2×C4×#6711yescharacteristic
541C2×C5#582411yescharacteristic
542C2×C5#583411yescharacteristic
543D5—42no—
544C23—32no—
545C23—52no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—101no—
553C23—101no—
554C23—101no—
555C23—101no—
556C23—101no—
557C23—101no—
558C23C22×D3×D511yescharacteristic
559D4—308no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—32no—
564C2×C4—52no—
565C2×C4—54no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—68no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—101no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—101no—
576C2×C4—108no—
577C2×C4—152no—
578C2×C4—152no—
579C2×C4—152no—
580C2×C4—152no—
581C2×C4—301no—
582C2×C4—301no—
583C2×C4—302no—
584C2×C4—302no—
585C2×C4—302no—
586C2×C4#677212yes—
587C2×C4C4×D3×D512yes—
588C2×C4D5×#6712yes—
589C2×C3—202no—
590C2×C3C2×#402711yescharacteristic
591C2×C3#6583811yescharacteristic
592C2×C3D5×#11711yescharacteristic
593C2×C3#6593311yescharacteristic
594C2×C3#6595011yescharacteristic
595C2×C3#6596511yescharacteristic
596C2×C3#6597511yescharacteristic
597D3—42no—
598C5#8224111yescharacteristic, complemented, Sylow
599C22—62no—
600C22—62no—
601C22—62no—
602C22—102no—
603C22—102no—
604C22—102no—
605C22—121no—
606C22—121no—
607C22—121no—
608C22—121no—
609C22—201no—
610C22—201no—
611C22—201no—
612C22—201no—
613C22C2×#677211yescharacteristic
614C22C2×C4×D3×D511yescharacteristic
615C22#9202411yescharacteristic
616C22#9202711yescharacteristic
617C22#9202811yescharacteristic
618C22#9203411yescharacteristic
619C22C2×D5×#6711yescharacteristic
620C4—42no—
621C4—62no—
622C4—102no—
623C4—104no—
624C4—122no—
625C4—154no—
626C4—302no—
627C4—602no—
628C4#9144014yes—
629C3#1060561611yescharacteristic, complemented, Sylow
630C2—122no—
631C2—202no—
632C2#1175366911yescharacteristic
633C2#1175379611yescharacteristic
634C2#1175380411yescharacteristic
635C2#1175383511yescharacteristic
636C2C2×#9144011yescharacteristic
637C2#1175493811yescharacteristic
638C2#1175515611yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1201C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2354
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4038C2semidirect
3. C2×#91338C2semidirect
4. #11751076C2semidirect
5. C5×#4987C2semidirect
7. #11748759C2semidirect
8. #11748715C2semidirect
9. #11748707C2semidirect
10. #11748699C2semidirect
11. #11748640C2semidirect
34. C4×#1201C4semidirect
42. C2×#4038D3semidirect
79. C3×C4⋊Dic5D4semidirect
92. #4987D5semidirect
226. C4⋊Dic5#67semidirect
270. Dic15#117semidirect
504. C3×C5#2354semidirect
598. C5#82241semidirect
629. C3#10605616semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40382
C2×C4×#12012
C2×#913382
#117510762
C5×#49872
#117489162
#117487592
#117487152
#117487072
#117486992
#117486402
#106056163
#822415