Finite Group Databasebeta

← Dic5×#924 #14321822

Subgroups — Dic5×#924 #14321822

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#924C211yesmaximal, characteristic
2C22×C3×C4×Dic5C211yesmaximal, characteristic
3C2×Dic5×Dic6C211yesmaximal, characteristic, complemented
4#11751313C211yesmaximal, characteristic
5#11749509C212yesmaximal
6Dic5×#217C212yesmaximal
7#11748842C211yesmaximal, characteristic, complemented
8Dic5×C4⋊Dic3C211yesmaximal, characteristic, complemented
9Dic5×Dic3⋊C4C212yesmaximal, complemented
10#11748589C211yesmaximal, characteristic, complemented
11#11748573C212yesmaximal, complemented
12Dic5×C22⋊Q8—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C23×C3×Dic5C2211yescharacteristic
16C23×C3×Dic5C2211yescharacteristic
17C2×#6719C2212yes—
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C4⋊Dic15C2211yescharacteristic
20C2×C5×#217C2212yes—
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C5×#924C418yescomplemented
23C2×C5×Dic3⋊C4C2212yes—
24C2×C3×C4×Dic5—24no—
25C2×C3×C4×Dic5C2211yescharacteristic
26C2×C3×C4×Dic5C2211yescharacteristic
27C2×C3×C4×Dic5C2211yescharacteristic
28C2×C3×C4×Dic5C2211yescharacteristic
29C2×Dic3⋊Dic5C2212yes—
30C2×Dic3⋊Dic5C2212yes—
31C2×Dic3×Dic5C2212yes—
32C2×Dic3×Dic5C2212yes—
33Dic5×Dic6—24no—
34#91115C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39C4×#924—51nomaximal, Hall
40C2×C5×C22⋊Q8—31no—
41C2×Q8×Dic5—31no—
42C22×C4×Dic5D311yescharacteristic
43#65519—31no—
44#65501—31no—
45#65274—31no—
46#65270—32no—
47C4⋊C4×Dic5—32no—
48C4⋊C4×Dic5—31no—
49#65239—32no—
50C22⋊C4×Dic5—32no—
51C24×C3×C5C2311yescharacteristic
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5C2×C414yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15C2312yes—
58C22×Dic15C2312yes—
59C22×C5×Dic3C2312yes—
60C22×C5×Dic3C2312yes—
61C2×C5×Dic6—22no—
62C2×C5×Dic6—22no—
63C2×C5×Dic6C2×C414yescomplemented
64C22×C3×Dic5—22no—
65C22×C3×Dic5—22no—
66C22×C3×Dic5—22no—
67C22×C3×Dic5—22no—
68C22×C3×Dic5—22no—
69C22×C3×Dic5Q814yescomplemented
70C22×C3×Dic5C2311yescharacteristic
71C22×C3×Dic5C2311yescharacteristic
72C22×C3×Dic5C2311yescharacteristic
73C22×C3×Dic5C2311yescharacteristic
74#6719—24no—
75C4⋊Dic15—22no—
76C4⋊Dic15—22no—
77C5×#217C2×C418yes—
78C5×C4⋊Dic3C2×C414yescomplemented
79C5×Dic3⋊C4C2×C418yescomplemented
80C3×C4×Dic5—22no—
81C3×C4×Dic5—22no—
82C3×C4×Dic5—22no—
83C3×C4×Dic5D414yes—
84Dic3⋊Dic5—24no—
85Dic3⋊Dic5—24no—
86Dic3×Dic5—24no—
87Dic3×Dic5—24no—
88C22×C3×C42—51no—
89C2×#924D511yescharacteristic
90C2×C4×Dic6—51no—
91#5212—51no—
92#5210—52no—
93C4×#217—52no—
94C4×C4⋊Dic3—51no—
95#4933—52no—
96C4×Dic3⋊C4—52no—
97#4929—51no—
98#4928—51no—
99C22×C5×Q8—31no—
100C23×C4×C5C2×D311yescharacteristic
101C23×Dic5C2×D311yescharacteristic
102C23×Dic5C2×D311yescharacteristic
103C5×C22⋊Q8—38no—
104C2×C5×C4⋊C4—32no—
105C2×C5×C4⋊C4—31no—
106C2×C5×C22⋊C4—32no—
107C2×C22⋊Dic5—32no—
108Q8×Dic5—64no—
109C2×C4⋊Dic5—31no—
110C2×C4⋊Dic5—31no—
111C2×C4⋊Dic5—32no—
112C2×C4⋊Dic5—32no—
113C2×C4×Dic5—24no—
114C2×C4×Dic5—32no—
115C2×C4×Dic5—32no—
116C2×C4×Dic5C2×D311yescharacteristic
117C2×C4×Dic5C2×D311yescharacteristic
118C2×C4×Dic5C2×D311yescharacteristic
119C2×C4×Dic5C2×D311yescharacteristic
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#3963—32no—
124#3963—32no—
125C4×C22⋊Q8—151noSylow
126C23×C3×C5—21no—
127C23×C3×C5—21no—
128C23×C3×C5—21no—
129C23×C3×C5—21no—
130C23×C3×C5C22×C412yes—
131C23×C3×C5C2×Q812yes—
132C23×C3×C5#4112yes—
133C23×C3×C5C2411yescharacteristic
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—28no—
137C2×C3×C4×C5C22×C412yes—
138C2×C3×C4×C5C22×C412yes—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5#4112yes—
141C2×Dic15—22no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15—22no—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3C22×C414yes—
152C2×C5×Dic3C22×C414yes—
153C5×Dic6—28no—
154C2×C3×Dic5—28no—
155C2×C3×Dic5—21no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5—21no—
161C2×C3×Dic5—28no—
162C2×C3×Dic5C2×D412yes—
163C2×C3×Dic5C2×D412yes—
164C2×C3×Dic5C2×Q812yes—
165C2×C3×Dic5C2×Q812yes—
166C2×C3×Dic5#4112yes—
167C2×C3×Dic5#4112yes—
168C23×C3×C4—51no—
169C23×C3×C4—51no—
170C23×C3×C4C2×D511yescharacteristic
171C22×Dic6C2×D511yescharacteristic
172C2×C3×C42—51no—
173C2×C3×C42—51no—
174C2×C3×C42—51no—
175C2×C3×C42—51no—
176C2×C3×C42—104no—
177C2×#217—52no—
178C2×#217C2×D512yes—
179C2×C4⋊Dic3—51no—
180C2×C4⋊Dic3—51no—
181C2×C4⋊Dic3C2×D511yescharacteristic
182#924Dic518yesdirect product factor
183C2×Dic3⋊C4—52no—
184C2×Dic3⋊C4—52no—
185C2×Dic3⋊C4C2×D512yes—
186C2×C4×Dic3—52no—
187C2×C4×Dic3—52no—
188C4×Dic6—104no—
189C3⋊#95—52no—
190C3⋊#95—52no—
191C3⋊#95—52no—
192C3⋊#95—52no—
193C3⋊#95—52no—
194C24×C5C22×D311yescharacteristic
195C2×C5×Q8—34no—
196C2×C5×Q8—62no—
197C2×C5×Q8—62no—
198C22×C4×C5—22no—
199C22×C4×C5—22no—
200C22×C4×C5—32no—
201C22×C4×C5—32no—
202C22×C4×C5C4×D314yes—
203C22×C4×C5C22×D311yescharacteristic
204C22×C4×C5C22×D311yescharacteristic
205C22×Dic5—22no—
206C22×Dic5—22no—
207C22×Dic5—22no—
208C22×Dic5—22no—
209C22×Dic5—22no—
210C22×Dic5—32no—
211C22×Dic5—32no—
212C22×Dic5Dic614yescomplemented
213C22×Dic5C22×D311yescharacteristic
214C22×Dic5C22×D311yescharacteristic
215C22×Dic5C22×D311yescharacteristic
216C22×Dic5C22×D311yescharacteristic
217C5×C4⋊C4—38no—
218C5×C4⋊C4—34no—
219C5×C22⋊C4—38no—
220C22⋊Dic5—64no—
221C4⋊Dic5—62no—
222C4⋊Dic5—62no—
223C4⋊Dic5—64no—
224C4⋊Dic5—64no—
225C4×Dic5—22no—
226C4×Dic5—22no—
227C4×Dic5—22no—
228C4×Dic5—64no—
229C4×Dic5—64no—
230C4×Dic5#6714yes—
231C2×C22⋊Q8—31no—
232C2×C4×Q8—151no—
233C22×C42—51no—
234#391—151no—
235#389—151no—
236#387—152no—
237#385—152no—
238#380—151no—
239C4×C4⋊C4—152no—
240C4×C4⋊C4—151no—
241C4×C22⋊C4—152no—
242C22×C3×C5—22no—
243C22×C3×C5—22no—
244C22×C3×C5—22no—
245C22×C3×C5—22no—
246C22×C3×C5—22no—
247C22×C3×C5—22no—
248C22×C3×C5C4×Q814yescomplemented
249C22×C3×C5C23×C411yescharacteristic
250C22×C3×C5C22×D411yescharacteristic
251C22×C3×C5C22×Q811yescharacteristic
252C22×C3×C5C2×#4111yescharacteristic
253C22×C3×C5C2×#4111yescharacteristic
254C22×C3×C5C2×#4111yescharacteristic
255C22×C3×C5C2×#4111yescharacteristic
256C3×C4×C5—22no—
257C3×C4×C5C4×D414yes—
258Dic15—42no—
259Dic15—42no—
260C3×Dic5—22no—
261C3×Dic5—22no—
262C3×Dic5—22no—
263C3×Dic5C22⋊Q814yescomplemented
264C5×Dic3—24no—
265C5×Dic3—24no—
266C24×C3C22×D511yescharacteristic
267C22×C3×C4—22no—
268C22×C3×C4—22no—
269C22×C3×C4—51no—
270C22×C3×C4—54no—
271C22×C3×C4—51no—
272C22×C3×C4—51no—
273C22×C3×C4—51no—
274C22×C3×C4—102no—
275C22×C3×C4—102no—
276C22×C3×C4—102no—
277C22×C3×C4—102no—
278C22×C3×C4—102no—
279C22×C3×C4C2×Dic514yes—
280C22×C3×C4C22×D511yescharacteristic
281C22×C3×C4C22×D511yescharacteristic
282C22×Dic3—52no—
283C22×Dic3—52no—
284C22×Dic3C22×D512yes—
285C22×Dic3C22×D512yes—
286C2×Dic6—22no—
287C2×Dic6—22no—
288C2×Dic6C2×Dic514yescomplemented
289C3×C42—54no—
290C3×C42—102no—
291C3×C42—102no—
292C3×C42—102no—
293#217—104no—
294#217C2×Dic518yes—
295C4⋊Dic3—102no—
296C4⋊Dic3—102no—
297C4⋊Dic3C2×Dic514yescomplemented
298Dic3⋊C4—104no—
299Dic3⋊C4—104no—
300Dic3⋊C4C2×Dic518yescomplemented
301C4×Dic3—104no—
302C4×Dic3—104no—
303C23×C5—21no—
304C23×C5—21no—
305C23×C5—21no—
306C23×C5—21no—
307C23×C5C2×Dic612yes—
308C23×C5C2×C4×D312yes—
309C23×C5#23512yes—
310C23×C5C23×D311yescharacteristic
311C5×Q8—68no—
312C2×C4×C5—21no—
313C2×C4×C5—21no—
314C2×C4×C5—28no—
315C2×C4×C5—34no—
316C2×C4×C5—34no—
317C2×C4×C5—62no—
318C2×C4×C5—62no—
319C2×C4×C5—62no—
320C2×C4×C5—62no—
321C2×C4×C5C2×C4×D312yes—
322C2×C4×C5C2×C4×D312yes—
323C2×C4×C5#23512yes—
324C2×C4×C5C2×#6712yes—
325C2×Dic5—28no—
326C2×Dic5—21no—
327C2×Dic5—21no—
328C2×Dic5—21no—
329C2×Dic5—21no—
330C2×Dic5—21no—
331C2×Dic5—21no—
332C2×Dic5—28no—
333C2×Dic5—62no—
334C2×Dic5—62no—
335C2×Dic5—62no—
336C2×Dic5—62no—
337C2×Dic5—62no—
338C2×Dic5—62no—
339C2×Dic5C2×Dic612yes—
340C2×Dic5C2×Dic612yes—
341C2×Dic5#23512yes—
342C2×Dic5#23512yes—
343C2×Dic5C2×#6712yes—
344C2×Dic5C2×#6712yes—
345C22×Q8—31no—
346C23×C4—51no—
347C23×C4—51no—
348C23×C4D3×D511yescharacteristic
349C22⋊Q8—38no—
350C4×Q8—304no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—31no—
353C2×C4⋊C4—152no—
354C2×C4⋊C4—152no—
355C2×C4⋊C4—151no—
356C2×C4⋊C4—151no—
357C2×C22⋊C4—32no—
358C2×C22⋊C4—152no—
359C2×C42—51no—
360C2×C42—51no—
361C2×C42—51no—
362C2×C42—51no—
363C2×C42—104no—
364C2×C42—152no—
365C2×C42—152no—
366#95—152no—
367#95—152no—
368#95—152no—
369#95—152no—
370#95—152no—
371C2×C3×C5—24no—
372C2×C3×C5C2×C4×D411yescharacteristic
373C2×C3×C5C2×C4×Q811yescharacteristic
374C2×C3×C5C4×#4111yescharacteristic
375C2×C3×C5C2×C22⋊Q811yescharacteristic
376C2×C3×C5#52511yescharacteristic
377C2×C3×C5#52911yescharacteristic
378C2×C3×C5#53311yescharacteristic
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—21no—
382C23×C3—21no—
383C23×C3Q8×D512yes—
384C23×C3#71312yes—
385C23×C3C22×Dic512yes—
386C23×C3C23×D511yescharacteristic
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—28no—
390C2×C3×C4—52no—
391C2×C3×C4—52no—
392C2×C3×C4—52no—
393C2×C3×C4—52no—
394C2×C3×C4—52no—
395C2×C3×C4—52no—
396C2×C3×C4—108no—
397C2×C3×C4—108no—
398C2×C3×C4—101no—
399C2×C3×C4—101no—
400C2×C3×C4—101no—
401C2×C3×C4—101no—
402C2×C3×C4—101no—
403C2×C3×C4—101no—
404C2×C3×C4D4×D512yes—
405C2×C3×C4#71112yes—
406C2×C3×C4C22×Dic512yes—
407C2×C3×C4C22×Dic512yes—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—102no—
413C2×Dic3—102no—
414C2×Dic3—102no—
415C2×Dic3—102no—
416C2×Dic3—102no—
417C2×Dic3—102no—
418C2×Dic3C22×Dic514yes—
419C2×Dic3C22×Dic514yes—
420Dic6—28no—
421C22×C5—22no—
422C22×C5—22no—
423C22×C5—22no—
424C22×C5—22no—
425C22×C5—22no—
426C22×C5—22no—
427C22×C5C4×Dic614yescomplemented
428C22×C5C22×Dic611yescharacteristic
429C22×C5C22×C4×D311yescharacteristic
430C22×C5C2×#23511yescharacteristic
431C22×C5C2×#23511yescharacteristic
432C22×C5C2×#23511yescharacteristic
433C22×C5C2×#23511yescharacteristic
434C22×C5C22×#6711yescharacteristic
435C4×C5—22no—
436C4×C5—64no—
437C4×C5—64no—
438C4×C5C4×#6714yes—
439Dic5—22no—
440Dic5—22no—
441Dic5—22no—
442Dic5—122no—
443Dic5—122no—
444Dic5#92414yesdirect product factor
445C24C2×D3×D511yescharacteristic
446C2×Q8—34no—
447C2×Q8—62no—
448C2×Q8—62no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—32no—
452C22×C4—32no—
453C22×C4—51no—
454C22×C4—54no—
455C22×C4—51no—
456C22×C4—51no—
457C22×C4—51no—
458C22×C4—102no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—102no—
462C22×C4—102no—
463C22×C4—152no—
464C22×C4—152no—
465C22×C4D3×Dic514yes—
466C22×C4C2×D3×D511yescharacteristic
467C22×C4C2×D3×D511yescharacteristic
468C4⋊C4—38no—
469C4⋊C4—34no—
470C4⋊C4—304no—
471C4⋊C4—304no—
472C4⋊C4—302no—
473C4⋊C4—302no—
474C22⋊C4—38no—
475C22⋊C4—304no—
476C42—54no—
477C42—102no—
478C42—102no—
479C42—102no—
480C42—304no—
481C42—304no—
482C3×C5C4×C22⋊Q811yescharacteristic, complemented, Hall
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3Q8×Dic514yescomplemented
490C22×C3C2×D4×D511yescharacteristic
491C22×C3C2×#71111yescharacteristic
492C22×C3C2×Q8×D511yescharacteristic
493C22×C3C2×#71311yescharacteristic
494C22×C3D5×#4111yescharacteristic
495C22×C3D5×#4111yescharacteristic
496C22×C3C23×Dic511yescharacteristic
497C3×C4—22no—
498C3×C4—54no—
499C3×C4—102no—
500C3×C4—102no—
501C3×C4—102no—
502C3×C4D4×Dic514yes—
503Dic3—24no—
504Dic3—24no—
505Dic3—202no—
506Dic3—202no—
507C2×C5—24no—
508C2×C5C2×C4×Dic611yescharacteristic
509C2×C5#547011yescharacteristic
510C2×C5C4×#23511yescharacteristic
511C2×C5#547911yescharacteristic
512C2×C5C2×#92411yescharacteristic
513C2×C5C2×C4×#6711yescharacteristic
514C2×C5#579111yescharacteristic
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23D5×Dic612yes—
520C23#677312yes—
521C23C2×D3×Dic512yes—
522C23C22×D3×D511yescharacteristic
523Q8—68no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—28no—
527C2×C4—34no—
528C2×C4—34no—
529C2×C4—52no—
530C2×C4—52no—
531C2×C4—52no—
532C2×C4—52no—
533C2×C4—52no—
534C2×C4—52no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—108no—
540C2×C4—108no—
541C2×C4—101no—
542C2×C4—101no—
543C2×C4—101no—
544C2×C4—101no—
545C2×C4—101no—
546C2×C4—101no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4—302no—
550C2×C4—302no—
551C2×C4—302no—
552C2×C4—302no—
553C2×C4C2×D3×Dic512yes—
554C2×C4C2×D3×Dic512yes—
555C2×C4#678212yes—
556C2×C4D5×#6712yes—
557C2×C3—24no—
558C2×C3#6595611yescharacteristic
559C2×C3#6595811yescharacteristic
560C2×C3D5×C22⋊Q811yescharacteristic
561C2×C3#6596111yescharacteristic
562C2×C3C2×D4×Dic511yescharacteristic
563C2×C3C2×Q8×Dic511yescharacteristic
564C2×C3#41×Dic511yescharacteristic
565C5C4×#92411yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22Dic5×Dic614yescomplemented
573C22C2×D5×Dic611yescharacteristic
574C22C2×#677311yescharacteristic
575C22D5×#23511yescharacteristic
576C22D5×#23511yescharacteristic
577C22C22×D3×Dic511yescharacteristic
578C22C2×#678211yescharacteristic
579C22C2×D5×#6711yescharacteristic
580C4—22no—
581C4—54no—
582C4—64no—
583C4—64no—
584C4—102no—
585C4—102no—
586C4—102no—
587C4—602no—
588C4—602no—
589C4Dic5×#6714yes—
590C3Dic5×C22⋊Q811yescharacteristic, complemented, Sylow
591C2—24no—
592C2#1175417411yescharacteristic
593C2C2×Dic5×Dic611yescharacteristic
594C2D5×#92411yescharacteristic
595C2Dic5×#23511yescharacteristic
596C2#1175441511yescharacteristic
597C2#1175452711yescharacteristic
598C2C2×Dic5×#6711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC24×C3×C5C23
hypercenterC23×C4D3×D5
hypocenterC3×C5C4×C22⋊Q8
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
3. C2×Dic5×Dic6C2semidirect
7. #11748842C2semidirect
8. Dic5×C4⋊Dic3C2semidirect
9. Dic5×Dic3⋊C4C2semidirect
10. #11748589C2semidirect
11. #11748573C2semidirect
22. C5×#924C4semidirect
63. C2×C5×Dic6C2×C4semidirect
69. C22×C3×Dic5Q8semidirect
78. C5×C4⋊Dic3C2×C4semidirect
79. C5×Dic3⋊C4C2×C4semidirect
182. #924Dic5direct (N × H)
212. C22×Dic5Dic6semidirect
248. C22×C3×C5C4×Q8semidirect
263. C3×Dic5C22⋊Q8semidirect
288. C2×Dic6C2×Dic5semidirect
297. C4⋊Dic3C2×Dic5semidirect
300. Dic3⋊C4C2×Dic5semidirect
427. C22×C5C4×Dic6semidirect
444. Dic5#924direct (N × H)
482. C3×C5C4×C22⋊Q8semidirect
489. C22×C3Q8×Dic5semidirect
565. C5C4×#924semidirect
572. C22Dic5×Dic6semidirect
590. C3Dic5×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9242
C22×C3×C4×Dic52
C2×Dic5×Dic62
#117513132
#117495092
Dic5×#2172
#117488422
Dic5×C4⋊Dic32
Dic5×Dic3⋊C42
#117485892
#117485732
Dic5×C22⋊Q83
C4×#9245