Finite Group Databasebeta

← #14321139

Subgroups — #14321139

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#892C211yesmaximal, characteristic
2C2×C4×D3×Dic5C211yesmaximal, characteristic
3C2×#91338C211yesmaximal, characteristic
4#11751085C211yesmaximal, characteristic, complemented
5C3×#65274C211yesmaximal, characteristic, complemented
6#11748903C212yesmaximal
7#11748897C212yesmaximal
8#11748868C211yesmaximal, characteristic, complemented
9#11748842C211yesmaximal, characteristic, complemented
10Dic3×C4⋊Dic5C212yesmaximal, complemented
11#11748667C212yesmaximal, complemented
12#10613958—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C22×D3×Dic5C2211yescharacteristic
15C22×D3×Dic5C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C4⋊Dic15C2212yes—
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C4×Dic15C2211yescharacteristic
20C2×C5×(C2×D3)⋊C4C2212yes—
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C4×C5×Dic3C2212yes—
23C5×#892C418yescomplemented
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×C4⋊Dic5C2212yes—
26C2×C3×C4×Dic5C2211yescharacteristic
27C2×Dic3⋊Dic5C2212yes—
28C2×Dic3⋊Dic5C2212yes—
29C2×(C2×D3)⋊Dic5C2212yes—
30C2×Dic3×Dic5C2211yescharacteristic
31C2×Dic3×Dic5C2211yescharacteristic
32C2×Dic3×Dic5C2212yes—
33C4×D3×Dic5—24no—
34#91338—24no—
35#91115C2212yes—
36C3×#3963C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#75963—51nomaximal, Hall
40C2×C5×#117—31no—
41C2×#4072—31no—
42C22×C4×Dic5—31no—
43#65274—31no—
44#65274—31no—
45#65274—31no—
46#65274D311yescharacteristic, complemented
47#65239—32no—
48#65239—32no—
49C4×C4⋊Dic5—32no—
50C4×C4⋊Dic5—32no—
51C23×C5×D3C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2312yes—
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2311yescharacteristic
57C22×C5×Dic3C2312yes—
58C22×C5×Dic3C2311yescharacteristic
59C2×C4×C5×D3C2×C414yes—
60C2×C4×C5×D3D414yes—
61C2×C4×C5×D3Q814yes—
62C22×C3×Dic5C2312yes—
63C22×C3×Dic5C2311yescharacteristic
64C22×C3×Dic5C2311yescharacteristic
65C2×D3×Dic5—22no—
66C2×D3×Dic5—22no—
67C2×D3×Dic5—22no—
68C2×D3×Dic5—22no—
69C2×D3×Dic5—22no—
70C2×D3×Dic5—22no—
71C3×C5×C4⋊C4C2×C414yescomplemented
72C4⋊Dic15—22no—
73C4⋊Dic15—24no—
74C4×Dic15—22no—
75C5×(C2×D3)⋊C4C2×C418yes—
76C5×C4⋊Dic3C2×C414yescomplemented
77C4×C5×Dic3C2×C418yescomplemented
78C3×C4⋊Dic5—22no—
79C3×C4⋊Dic5—24no—
80C3×C4×Dic5—22no—
81Dic3⋊Dic5—24no—
82Dic3⋊Dic5—24no—
83(C2×D3)⋊Dic5—24no—
84Dic3×Dic5—22no—
85Dic3×Dic5—22no—
86Dic3×Dic5—24no—
87C2×#892—51no—
88C2×#892D511yescharacteristic
89C2×C42×D3—51no—
90C3×#389—51no—
91#4989—52no—
92#4984—51no—
93Dic3×C4⋊C4—52no—
94#4972—51no—
95#4928—51no—
96#4668—52no—
97#4649—52no—
98C23×C4×C5—31no—
99C23×Dic5—31no—
100C23×Dic5—31no—
101C5×#117—38no—
102C2×C5×C4⋊C4—31no—
103C2×C5×C4⋊C4C2×D311yescharacteristic
104C2×C5×C22⋊C4—32no—
105C2×C42×C5—32no—
106C2×C22⋊Dic5—32no—
107#4072—64no—
108C2×C4⋊Dic5—31no—
109C2×C4⋊Dic5—32no—
110C2×C4⋊Dic5—32no—
111C2×C4⋊Dic5—32no—
112C2×C4⋊Dic5C2×D311yescharacteristic
113C2×C4⋊Dic5C2×D312yes—
114C2×C4×Dic5—31no—
115C2×C4×Dic5—31no—
116C2×C4×Dic5—31no—
117C2×C4×Dic5—32no—
118C2×C4×Dic5—64no—
119C2×C4×Dic5C2×D311yescharacteristic
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#3963C2×D312yes—
124#2296—151noSylow
125C23×C3×C5C2411yescharacteristic
126C22×C5×D3—21no—
127C22×C5×D3—21no—
128C22×C5×D3—21no—
129C22×C5×D3—21no—
130C22×C5×D3C22×C412yes—
131C22×C5×D3C2×D412yes—
132C22×C5×D3C2×Q812yes—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5C22×C414yes—
136C2×C3×C4×C5C22×C412yes—
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5C2×Q812yes—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15#4112yes—
147C2×Dic15#4112yes—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3C22×C414yes—
151C2×C5×Dic3C22×C412yes—
152C2×C5×Dic3C2×D412yes—
153C2×C5×Dic3C2×Q812yes—
154C4×C5×D3C4⋊C4116yescomplemented
155C2×C3×Dic5—21no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5—22no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5#4112yes—
163C2×C3×Dic5#4112yes—
164D3×Dic5—28no—
165D3×Dic5—28no—
166C22×C4×D3—51no—
167C22×C4×D3—51no—
168C22×C4×D3C2×D511yescharacteristic
169C2×C3×C4⋊C4—51no—
170C2×C3×C4⋊C4—52no—
171C2×C3×C4⋊C4C2×D511yescharacteristic
172C2×C3×C42—51no—
173C2×(C2×D3)⋊C4—52no—
174C2×(C2×D3)⋊C4C2×D512yes—
175C2×C4⋊Dic3—52no—
176C2×C4⋊Dic3—51no—
177C2×C4⋊Dic3C2×D511yescharacteristic
178C2×Dic3⋊C4—52no—
179C2×Dic3⋊C4—52no—
180C2×C4×Dic3—51no—
181C2×C4×Dic3—51no—
182C2×C4×Dic3—52no—
183C2×C4×Dic3—51no—
184C2×C4×Dic3C2×D512yes—
185#892—104no—
186#892Dic518yescomplemented
187C42×D3—104no—
188C3×#95—52no—
189C3⋊#95—52no—
190C3⋊#95—52no—
191C3⋊#95—52no—
192C24×C5—31no—
193C22×C4×C5—31no—
194C22×C4×C5—34no—
195C22×C4×C5—34no—
196C22×C4×C5—34no—
197C22×C4×C5—32no—
198C22×C4×C5C22×D312yes—
199C22×C4×C5C22×D311yescharacteristic
200C22×Dic5—31no—
201C22×Dic5—31no—
202C22×Dic5—32no—
203C22×Dic5—62no—
204C22×Dic5—62no—
205C22×Dic5—62no—
206C22×Dic5—62no—
207C22×Dic5—62no—
208C22×Dic5—62no—
209C22×Dic5C22×D312yes—
210C22×Dic5C22×D311yescharacteristic
211C22×Dic5C22×D311yescharacteristic
212C5×C4⋊C4—34no—
213C5×C4⋊C4C4×D314yescomplemented
214C5×C22⋊C4—38no—
215C42×C5—38no—
216C22⋊Dic5—64no—
217C4⋊Dic5—22no—
218C4⋊Dic5—24no—
219C4⋊Dic5—64no—
220C4⋊Dic5—64no—
221C4⋊Dic5—62no—
222C4⋊Dic5—64no—
223C4×Dic5—22no—
224C4×Dic5—62no—
225C4×Dic5—62no—
226C4×Dic5—64no—
227C4×Dic5—62no—
228C2×#117—31no—
229C2×#117—151no—
230C22×C42—151no—
231#389—51no—
232#389—151no—
233#389—151no—
234#389—151no—
235#385—152no—
236#385—152no—
237C4×C4⋊C4—152no—
238C4×C4⋊C4—152no—
239C22×C3×C5C23×C411yescharacteristic
240C22×C3×C5C22×D411yescharacteristic
241C22×C3×C5C22×Q811yescharacteristic
242C22×C3×C5C2×#4111yescharacteristic
243C22×C3×C5C2×#4111yescharacteristic
244C22×C3×C5C2×#4111yescharacteristic
245C22×C3×C5C2×#4111yescharacteristic
246C2×C5×D3—22no—
247C2×C5×D3—22no—
248C2×C5×D3—22no—
249C2×C5×D3—22no—
250C2×C5×D3—22no—
251C2×C5×D3—22no—
252C2×C5×D3C2×C4⋊C414yes—
253C3×C4×C5—24no—
254C3×C4×C5C2×C4⋊C414yes—
255Dic15—22no—
256Dic15—22no—
257Dic15—42no—
258C3×Dic5—22no—
259C3×Dic5—22no—
260C3×Dic5—42no—
261C5×Dic3—24no—
262C5×Dic3C2×C4⋊C414yes—
263C23×D3C22×D511yescharacteristic
264C22×C3×C4—51no—
265C22×C3×C4—51no—
266C22×C3×C4—52no—
267C22×C3×C4C22×D512yes—
268C22×C3×C4C22×D511yescharacteristic
269C22×Dic3—51no—
270C22×Dic3—51no—
271C22×Dic3—52no—
272C22×Dic3C22×D512yes—
273C22×Dic3C22×D511yescharacteristic
274C2×C4×D3—102no—
275C2×C4×D3—102no—
276C2×C4×D3—102no—
277C2×C4×D3—102no—
278C2×C4×D3—102no—
279C2×C4×D3—102no—
280C2×C4×D3Dic1014yes—
281C2×C4×D3D2014yes—
282C2×C4×D3C2×Dic514yes—
283C3×C4⋊C4—104no—
284C3×C4⋊C4—102no—
285C3×C4⋊C4C2×Dic514yescomplemented
286C3×C42—102no—
287(C2×D3)⋊C4—104no—
288(C2×D3)⋊C4C2×Dic518yes—
289C4⋊Dic3—104no—
290C4⋊Dic3—102no—
291C4⋊Dic3C2×Dic514yescomplemented
292Dic3⋊C4—104no—
293Dic3⋊C4—104no—
294C4×Dic3—102no—
295C4×Dic3—102no—
296C4×Dic3—102no—
297C4×Dic3—104no—
298C4×Dic3C2×Dic518yescomplemented
299C23×C5—32no—
300C23×C5—32no—
301C23×C5—32no—
302C23×C5—61no—
303C23×C5—61no—
304C23×C5—61no—
305C23×C5—61no—
306C23×C5C23×D311yescharacteristic
307C2×C4×C5—22no—
308C2×C4×C5—22no—
309C2×C4×C5—32no—
310C2×C4×C5—32no—
311C2×C4×C5—32no—
312C2×C4×C5—316no—
313C2×C4×C5—34no—
314C2×C4×C5—62no—
315C2×C4×C5—62no—
316C2×C4×C5C2×C4×D314yes—
317C2×C4×C5C2×C4×D312yes—
318C2×C4×C5D3×D412yes—
319C2×C4×C5D3×Q812yes—
320C2×Dic5—21no—
321C2×Dic5—21no—
322C2×Dic5—21no—
323C2×Dic5—21no—
324C2×Dic5—22no—
325C2×Dic5—22no—
326C2×Dic5—22no—
327C2×Dic5—32no—
328C2×Dic5—32no—
329C2×Dic5—68no—
330C2×Dic5—68no—
331C2×Dic5—61no—
332C2×Dic5—61no—
333C2×Dic5—61no—
334C2×Dic5—61no—
335C2×Dic5—62no—
336C2×Dic5—62no—
337C2×Dic5—62no—
338C2×Dic5#23712yes—
339C2×Dic5#23912yes—
340C23×C4—31no—
341C23×C4—151no—
342C23×C4—151no—
343#117—38no—
344#117—304no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—51no—
347C2×C4⋊C4—52no—
348C2×C4⋊C4—152no—
349C2×C4⋊C4—152no—
350C2×C4⋊C4—151no—
351C2×C4⋊C4—152no—
352C2×C4⋊C4D3×D511yescharacteristic
353C2×C22⋊C4—32no—
354C2×C22⋊C4—152no—
355C2×C42—32no—
356C2×C42—51no—
357C2×C42—152no—
358C2×C42—151no—
359C2×C42—151no—
360C2×C42—151no—
361C2×C42—304no—
362#95—52no—
363#95—152no—
364#95—152no—
365#95—152no—
366C2×C3×C5C22×C4⋊C411yescharacteristic
367C2×C3×C5C4×#4111yescharacteristic
368C2×C3×C5C4×#4111yescharacteristic
369C2×C3×C5#52511yescharacteristic
370C2×C3×C5#52511yescharacteristic
371C2×C3×C5#53311yescharacteristic
372C2×C3×C5#53311yescharacteristic
373C5×D3—24no—
374C23×C3C23×D511yescharacteristic
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—21no—
378C22×D3—21no—
379C22×D3C2×Dic1012yes—
380C22×D3C2×D2012yes—
381C22×D3C22×Dic512yes—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—52no—
385C2×C3×C4—52no—
386C2×C3×C4—101no—
387C2×C3×C4—101no—
388C2×C3×C4—101no—
389C2×C3×C4—101no—
390C2×C3×C4—102no—
391C2×C3×C4—102no—
392C2×C3×C4—102no—
393C2×C3×C4C2×Dic1012yes—
394C2×C3×C4C2×D2012yes—
395C2×C3×C4C22×Dic514yes—
396C2×C3×C4C22×Dic512yes—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—52no—
400C2×Dic3—52no—
401C2×Dic3—101no—
402C2×Dic3—101no—
403C2×Dic3—101no—
404C2×Dic3—101no—
405C2×Dic3—102no—
406C2×Dic3—102no—
407C2×Dic3—102no—
408C2×Dic3C2×Dic1012yes—
409C2×Dic3C2×D2012yes—
410C2×Dic3C22×Dic514yes—
411C2×Dic3C22×Dic512yes—
412C4×D3—108no—
413C4×D3—108no—
414C4×D3C4⋊Dic5116yescomplemented
415C22×C5—34no—
416C22×C5—62no—
417C22×C5—62no—
418C22×C5—62no—
419C22×C5—62no—
420C22×C5—62no—
421C22×C5—62no—
422C22×C5C22×C4×D311yescharacteristic
423C22×C5C2×D3×D411yescharacteristic
424C22×C5C2×#23711yescharacteristic
425C22×C5C2×D3×Q811yescharacteristic
426C22×C5C2×#23911yescharacteristic
427C22×C5D3×#4111yescharacteristic
428C22×C5D3×#4111yescharacteristic
429C4×C5—24no—
430C4×C5—34no—
431C4×C5—64no—
432C4×C5D3×C4⋊C414yes—
433Dic5—22no—
434Dic5—22no—
435Dic5—42no—
436Dic5—62no—
437Dic5—62no—
438Dic5—122no—
439C24—31no—
440C22×C4—31no—
441C22×C4—34no—
442C22×C4—34no—
443C22×C4—34no—
444C22×C4—32no—
445C22×C4—51no—
446C22×C4—51no—
447C22×C4—52no—
448C22×C4—151no—
449C22×C4—151no—
450C22×C4—152no—
451C22×C4—302no—
452C22×C4—302no—
453C22×C4—302no—
454C22×C4—302no—
455C22×C4—302no—
456C22×C4—302no—
457C22×C4C2×D3×D512yes—
458C22×C4C2×D3×D511yescharacteristic
459C4⋊C4—34no—
460C4⋊C4—104no—
461C4⋊C4—102no—
462C4⋊C4—304no—
463C4⋊C4—304no—
464C4⋊C4—304no—
465C4⋊C4—302no—
466C4⋊C4D3×Dic514yescomplemented
467C22⋊C4—38no—
468C22⋊C4—304no—
469C42—38no—
470C42—102no—
471C42—302no—
472C42—302no—
473C42—304no—
474C42—302no—
475C3×C5#229611yescharacteristic, complemented, Hall
476C22×C3C22×Dic1011yescharacteristic
477C22×C3C22×D2011yescharacteristic
478C22×C3D5×#4111yescharacteristic
479C22×C3D5×#4111yescharacteristic
480C22×C3D5×#4111yescharacteristic
481C22×C3D5×#4111yescharacteristic
482C22×C3C23×Dic511yescharacteristic
483C2×D3—22no—
484C2×D3—22no—
485C2×D3—22no—
486C2×D3—22no—
487C2×D3—22no—
488C2×D3—22no—
489C2×D3C2×C4⋊Dic514yes—
490C3×C4—24no—
491C3×C4—102no—
492C3×C4—102no—
493C3×C4—202no—
494C3×C4C2×C4⋊Dic514yes—
495Dic3—24no—
496Dic3—102no—
497Dic3—102no—
498Dic3—202no—
499Dic3C2×C4⋊Dic514yes—
500C2×C5—64no—
501C2×C5C2×D3×C4⋊C411yescharacteristic
502C2×C5C4×#23711yescharacteristic
503C2×C5C4×#23911yescharacteristic
504C2×C5#560611yescharacteristic
505C2×C5#562711yescharacteristic
506C2×C5#568811yescharacteristic
507C2×C5#572811yescharacteristic
508C23—32no—
509C23—32no—
510C23—32no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23C22×D3×D511yescharacteristic
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—32no—
519C2×C4—32no—
520C2×C4—32no—
521C2×C4—316no—
522C2×C4—34no—
523C2×C4—52no—
524C2×C4—52no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—101no—
528C2×C4—101no—
529C2×C4—101no—
530C2×C4—101no—
531C2×C4—102no—
532C2×C4—102no—
533C2×C4—102no—
534C2×C4—152no—
535C2×C4—152no—
536C2×C4—308no—
537C2×C4—308no—
538C2×C4—301no—
539C2×C4—301no—
540C2×C4—301no—
541C2×C4—301no—
542C2×C4—302no—
543C2×C4—302no—
544C2×C4—302no—
545C2×C4D3×Dic1012yes—
546C2×C4D3×D2012yes—
547C2×C4C2×D3×Dic514yes—
548C2×C4C2×D3×Dic512yes—
549C2×C3#6585111yescharacteristic
550C2×C3#6585111yescharacteristic
551C2×C3#6585811yescharacteristic
552C2×C3#6585811yescharacteristic
553C2×C3C22×C4⋊Dic511yescharacteristic
554C2×C3#41×Dic511yescharacteristic
555C2×C3#41×Dic511yescharacteristic
556D3—24no—
557C5#7596311yescharacteristic, complemented, Sylow
558C22—34no—
559C22—62no—
560C22—62no—
561C22—62no—
562C22—62no—
563C22—62no—
564C22—62no—
565C22C2×D3×Dic1011yescharacteristic
566C22C2×D3×D2011yescharacteristic
567C22D5×#23711yescharacteristic
568C22#9202411yescharacteristic
569C22D5×#23911yescharacteristic
570C22#9203411yescharacteristic
571C22C22×D3×Dic511yescharacteristic
572C4—24no—
573C4—34no—
574C4—64no—
575C4—102no—
576C4—102no—
577C4—202no—
578C4—302no—
579C4—302no—
580C4—602no—
581C4D3×C4⋊Dic514yes—
582C3#1061395811yescharacteristic, complemented, Sylow
583C2—64no—
584C2#1175351811yescharacteristic
585C2#1175361811yescharacteristic
586C2#1175387911yescharacteristic
587C2#1175388011yescharacteristic
588C2C2×D3×C4⋊Dic511yescharacteristic
589C2Dic5×#23711yescharacteristic
590C2Dic5×#23911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C5×D3C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2296
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
4. #11751085C2semidirect
5. C3×#65274C2semidirect
8. #11748868C2semidirect
9. #11748842C2semidirect
10. Dic3×C4⋊Dic5C2semidirect
11. #11748667C2semidirect
23. C5×#892C4semidirect
46. #65274D3semidirect
71. C3×C5×C4⋊C4C2×C4semidirect
76. C5×C4⋊Dic3C2×C4semidirect
77. C4×C5×Dic3C2×C4semidirect
154. C4×C5×D3C4⋊C4semidirect
186. #892Dic5semidirect
213. C5×C4⋊C4C4×D3semidirect
285. C3×C4⋊C4C2×Dic5semidirect
291. C4⋊Dic3C2×Dic5semidirect
298. C4×Dic3C2×Dic5semidirect
414. C4×D3C4⋊Dic5semidirect
466. C4⋊C4D3×Dic5semidirect
475. C3×C5#2296semidirect
557. C5#75963semidirect
582. C3#10613958semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#8922
C2×C4×D3×Dic52
C2×#913382
#117510852
C3×#652742
#117489032
#117488972
#117488682
#117488422
Dic3×C4⋊Dic52
#117486672
#106139583
#759635