Finite Group Databasebeta

← #14329519

Subgroups — #14329519

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91825C211yesmaximal, characteristic
2C2×D4×Dic15C211yesmaximal, characteristic
3C2×C4×Dic30C211yesmaximal, characteristic, complemented
4C3×C5×#390C211yesmaximal, characteristic
5C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
6#11751074C211yesmaximal, characteristic, complemented
7#11751047C212yesmaximal, complemented
8#11751046C212yesmaximal, complemented
9#11751045C212yesmaximal, complemented
10C22⋊C4×Dic15C212yesmaximal, complemented
11C4×#6712C211yesmaximal, characteristic, complemented
12#10608038—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C23×Dic15C2212yes—
15C22×Dic30C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C22⋊C4C2212yes—
18C2×C3×C5×C22⋊C4C2212yes—
19C2×C3×C42×C5C2211yescharacteristic
20C2×#6719C2212yes—
21C2×#6719C2212yes—
22C2×#6719C2212yes—
23#91825C418yescomplemented
24D4×Dic15—24no—
25C2×C4⋊Dic15C2211yescharacteristic
26C2×#6712C2211yescharacteristic
27C2×#6712C2211yescharacteristic
28C2×#6712C2212yes—
29C2×C4×Dic15C2211yescharacteristic
30C2×C4×Dic15C2212yes—
31C2×C4×Dic15C2212yes—
32C2×C4×Dic15C2211yescharacteristic
33C4×Dic30—24no—
34#91115C2211yescharacteristic
35#91115C2211yescharacteristic
36#91115C2212yes—
37#91115C2212yes—
38#91115C2212yes—
39#76307—51nomaximal, Hall
40C2×#4082—31no—
41C2×D4×Dic5—31no—
42C2×C4×Dic10—31no—
43C5×#390D311yescharacteristic
44C4⋊C4×Dic5—31no—
45#65263—31no—
46#65235—32no—
47#65234—32no—
48#65233—32no—
49C22⋊C4×Dic5—32no—
50C4×#683—31no—
51C24×C3×C5C2312yes—
52C2×C3×C5×D4—22no—
53C2×C3×C5×D4—22no—
54C2×C3×C5×D4C2×C414yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C22×Dic15—24no—
60C22×Dic15—24no—
61C22×Dic15—42no—
62C22×Dic15C2311yescharacteristic
63C22×Dic15C2311yescharacteristic
64C22×Dic15C2311yescharacteristic
65C22×Dic15C2312yes—
66C22×Dic15C2312yes—
67C22×Dic15C2311yescharacteristic
68C2×Dic30—22no—
69C2×Dic30—22no—
70C2×Dic30C2×C414yescomplemented
71C3×C5×C4⋊C4—22no—
72C3×C5×C22⋊C4—24no—
73C3×C5×C22⋊C4—42no—
74C3×C42×C5—22no—
75#6719—24no—
76#6719C2×C418yescomplemented
77#6719C2×C418yescomplemented
78C4⋊Dic15D414yes—
79#6712—22no—
80#6712—24no—
81#6712—22no—
82C4×Dic15—22no—
83C4×Dic15—24no—
84C4×Dic15—42no—
85C4×Dic15C2×C414yescomplemented
86C2×#936—51no—
87C2×D4×Dic3—51no—
88C2×C4×Dic6—51no—
89C3×#390D511yescharacteristic
90Dic3×C4⋊C4—51no—
91#4973—51no—
92#4946—52no—
93#4945—52no—
94#4944—52no—
95Dic3×C22⋊C4—52no—
96C4×Dic3⋊C4—51no—
97C22×C5×D4C2×D311yescharacteristic
98C23×Dic5—32no—
99C22×Dic10—31no—
100C2×C5×C4⋊C4C2×D311yescharacteristic
101C2×C5×C22⋊C4C2×D312yes—
102C2×C5×C22⋊C4C2×D312yes—
103C2×C42×C5C2×D311yescharacteristic
104C2×C22⋊Dic5—32no—
105C2×C22⋊Dic5—32no—
106C2×C22⋊Dic5—32no—
107#4082—38no—
108D4×Dic5—64no—
109C2×C4⋊Dic5—31no—
110C2×#683—32no—
111C2×#683—31no—
112C2×#683—31no—
113C2×C4×Dic5—31no—
114C2×C4×Dic5—32no—
115C2×C4×Dic5—31no—
116C2×C4×Dic5—32no—
117C4×Dic10—64no—
118#3963—31no—
119#3963—31no—
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#2367—151noSylow
124C23×C3×C5—22no—
125C23×C3×C5—22no—
126C23×C3×C5—22no—
127C23×C3×C5—22no—
128C23×C3×C5—22no—
129C23×C3×C5—22no—
130C23×C3×C5C22×C414yes—
131C23×C3×C5C2411yescharacteristic
132C3×C5×D4—28no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—21no—
141C2×C3×C4×C5C22×C412yes—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5#4112yes—
144C2×C3×C4×C5#4112yes—
145C2×Dic15—22no—
146C2×Dic15—21no—
147C2×Dic15—21no—
148C2×Dic15—21no—
149C2×Dic15—22no—
150C2×Dic15—22no—
151C2×Dic15—21no—
152C2×Dic15—22no—
153C2×Dic15—21no—
154C2×Dic15—21no—
155C2×Dic15—21no—
156C2×Dic15—21no—
157C2×Dic15—21no—
158C2×Dic15—48no—
159C2×Dic15C22×C412yes—
160C2×Dic15C22×C412yes—
161C2×Dic15C22×C414yes—
162C2×Dic15C2×D414yes—
163C2×Dic15#4112yes—
164Dic30—28no—
165C22×C3×D4C2×D511yescharacteristic
166C23×Dic3—52no—
167C22×Dic6—51no—
168C2×C3×C4⋊C4C2×D511yescharacteristic
169C2×C3×C22⋊C4C2×D512yes—
170C2×C3×C22⋊C4C2×D512yes—
171C2×C3×C42C2×D511yescharacteristic
172C2×#217—52no—
173C2×#217—52no—
174C2×#217—52no—
175#936—58no—
176D4×Dic3—104no—
177C2×C4⋊Dic3—51no—
178C2×Dic3⋊C4—51no—
179C2×Dic3⋊C4—51no—
180C2×Dic3⋊C4—52no—
181C2×C4×Dic3—51no—
182C2×C4×Dic3—51no—
183C2×C4×Dic3—52no—
184C2×C4×Dic3—52no—
185C4×Dic6—104no—
186C3⋊#95—51no—
187C3⋊#95—51no—
188C3⋊#95—52no—
189C3⋊#95—52no—
190C3⋊#95—52no—
191C24×C5C22×D312yes—
192C2×C5×D4—22no—
193C2×C5×D4—22no—
194C2×C5×D4C4×D314yes—
195C22×C4×C5C22×D311yescharacteristic
196C22×C4×C5C22×D312yes—
197C22×C4×C5C22×D311yescharacteristic
198C22×C4×C5C22×D311yescharacteristic
199C22×Dic5—32no—
200C22×Dic5—31no—
201C22×Dic5—31no—
202C22×Dic5—32no—
203C22×Dic5—31no—
204C22×Dic5—31no—
205C22×Dic5—64no—
206C22×Dic5—64no—
207C22×Dic5—122no—
208C2×Dic10—34no—
209C2×Dic10—62no—
210C2×Dic10—62no—
211C5×C4⋊C4—22no—
212C5×C22⋊C4—24no—
213C5×C22⋊C4—42no—
214C42×C5—22no—
215C22⋊Dic5—38no—
216C22⋊Dic5—38no—
217C22⋊Dic5—64no—
218C4⋊Dic5—34no—
219#683—64no—
220#683—62no—
221#683—62no—
222C4×Dic5—34no—
223C4×Dic5—64no—
224C4×Dic5—62no—
225C4×Dic5—122no—
226C2×#124—151no—
227C2×C4×Q8—151no—
228C2×C4×D4—151no—
229#391—151no—
230#390D1511yescharacteristic
231#388—152no—
232#388—152no—
233#385—152no—
234C4×C4⋊C4—151no—
235C4×C4⋊C4—151no—
236C4×C22⋊C4—152no—
237C22×C3×C5—24no—
238C22×C3×C5—24no—
239C22×C3×C5—24no—
240C22×C3×C5—42no—
241C22×C3×C5—42no—
242C22×C3×C5—42no—
243C22×C3×C5—42no—
244C22×C3×C5C23×C411yescharacteristic
245C22×C3×C5C22×D411yescharacteristic
246C22×C3×C5C2×#4111yescharacteristic
247C22×C3×C5C2×#4111yescharacteristic
248C22×C3×C5C2×#4111yescharacteristic
249C22×C3×C5H(5,𝔽2)11yescharacteristic
250C22×C3×C52−1+411yescharacteristic
251C3×C4×C5—22no—
252C3×C4×C5—41no—
253C3×C4×C5—42no—
254C3×C4×C5C4×D414yes—
255Dic15—24no—
256Dic15—22no—
257Dic15—22no—
258Dic15—22no—
259Dic15—24no—
260Dic15—41no—
261C24×C3C22×D512yes—
262C2×C3×D4—22no—
263C2×C3×D4—22no—
264C2×C3×D4C4×D514yes—
265C22×C3×C4C22×D512yes—
266C22×C3×C4C22×D511yescharacteristic
267C22×C3×C4C22×D511yescharacteristic
268C22×C3×C4C22×D511yescharacteristic
269C22×Dic3—51no—
270C22×Dic3—51no—
271C22×Dic3—51no—
272C22×Dic3—52no—
273C22×Dic3—52no—
274C22×Dic3—51no—
275C22×Dic3—104no—
276C22×Dic3—104no—
277C22×Dic3—202no—
278C2×Dic6—54no—
279C2×Dic6—102no—
280C2×Dic6—102no—
281C3×C4⋊C4—22no—
282C3×C22⋊C4—24no—
283C3×C22⋊C4—42no—
284C3×C42—22no—
285#217—58no—
286#217—58no—
287#217—104no—
288C4⋊Dic3—54no—
289Dic3⋊C4—104no—
290Dic3⋊C4—102no—
291Dic3⋊C4—102no—
292C4×Dic3—54no—
293C4×Dic3—104no—
294C4×Dic3—102no—
295C4×Dic3—202no—
296C23×C5—22no—
297C23×C5—22no—
298C23×C5—22no—
299C23×C5—22no—
300C23×C5—22no—
301C23×C5—22no—
302C23×C5C2×C4×D314yes—
303C23×C5C23×D311yescharacteristic
304C5×D4—28no—
305C2×C4×C5—21no—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—22no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5C2×C4×D312yes—
314C2×C4×C5D3×D412yes—
315C2×C4×C5#23712yes—
316C2×C4×C5#23712yes—
317C2×Dic5—32no—
318C2×Dic5—32no—
319C2×Dic5—34no—
320C2×Dic5—34no—
321C2×Dic5—32no—
322C2×Dic5—62no—
323C2×Dic5—61no—
324C2×Dic5—61no—
325C2×Dic5—62no—
326C2×Dic5—61no—
327C2×Dic5—62no—
328C2×Dic5—61no—
329C2×Dic5—61no—
330C2×Dic5—62no—
331C2×Dic5—61no—
332C2×Dic5—61no—
333C2×Dic5—61no—
334C2×Dic5—61no—
335C2×Dic5—128no—
336Dic10—68no—
337C22×Q8—151no—
338C22×D4C2×D1511yescharacteristic
339C23×C4—152no—
340#124—158no—
341C4×Q8—304no—
342C4×D4—304no—
343C2×C4⋊C4—151no—
344C2×C4⋊C4—152no—
345C2×C4⋊C4—151no—
346C2×C4⋊C4—151no—
347C2×C4⋊C4C2×D1511yescharacteristic
348C2×C22⋊C4—152no—
349C2×C22⋊C4—152no—
350C2×C22⋊C4—152no—
351C2×C22⋊C4C2×D1512yes—
352C2×C22⋊C4C2×D1512yes—
353C2×C42—151no—
354C2×C42—151no—
355C2×C42—152no—
356C2×C42—152no—
357C2×C42C2×D1511yescharacteristic
358#95—151no—
359#95—152no—
360#95—152no—
361#95—151no—
362#95—152no—
363C2×C3×C5—44no—
364C2×C3×C5C2×C4×D411yescharacteristic
365C2×C3×C5C4×#4111yescharacteristic
366C2×C3×C5#52011yescharacteristic
367C2×C3×C5#54611yescharacteristic
368C2×C3×C5#54811yescharacteristic
369C2×C3×C5#55611yescharacteristic
370C2×C3×C5#55911yescharacteristic
371C23×C3—22no—
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—22no—
377C23×C3C2×C4×D514yes—
378C23×C3C23×D511yescharacteristic
379C3×D4—28no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4C2×C4×D512yes—
389C2×C3×C4D4×D512yes—
390C2×C3×C4#71112yes—
391C2×C3×C4#71112yes—
392C2×Dic3—52no—
393C2×Dic3—52no—
394C2×Dic3—54no—
395C2×Dic3—54no—
396C2×Dic3—52no—
397C2×Dic3—102no—
398C2×Dic3—101no—
399C2×Dic3—101no—
400C2×Dic3—102no—
401C2×Dic3—101no—
402C2×Dic3—102no—
403C2×Dic3—101no—
404C2×Dic3—101no—
405C2×Dic3—102no—
406C2×Dic3—101no—
407C2×Dic3—101no—
408C2×Dic3—101no—
409C2×Dic3—101no—
410C2×Dic3—208no—
411Dic6—108no—
412C22×C5—24no—
413C22×C5—24no—
414C22×C5—24no—
415C22×C5—42no—
416C22×C5—42no—
417C22×C5—42no—
418C22×C5—42no—
419C22×C5C22×C4×D311yescharacteristic
420C22×C5C2×D3×D411yescharacteristic
421C22×C5C2×#23711yescharacteristic
422C22×C5C2×#23711yescharacteristic
423C22×C5#100411yescharacteristic
424C22×C5D3×#4111yescharacteristic
425C22×C5#101011yescharacteristic
426C4×C5—22no—
427C4×C5—42no—
428C4×C5—41no—
429C4×C5#67⋊C414yes—
430Dic5—64no—
431Dic5—62no—
432Dic5—62no—
433Dic5—64no—
434Dic5—62no—
435Dic5—121no—
436C24C22×D1512yes—
437C2×Q8—154no—
438C2×Q8—302no—
439C2×Q8—302no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4C4×D1514yes—
443C22×C4—151no—
444C22×C4—151no—
445C22×C4—151no—
446C22×C4—152no—
447C22×C4—152no—
448C22×C4—151no—
449C22×C4—304no—
450C22×C4—304no—
451C22×C4—602no—
452C22×C4C22×D1512yes—
453C22×C4C22×D1511yescharacteristic
454C22×C4C22×D1511yescharacteristic
455C22×C4C22×D1511yescharacteristic
456C4⋊C4—22no—
457C4⋊C4—154no—
458C4⋊C4—304no—
459C4⋊C4—302no—
460C4⋊C4—302no—
461C22⋊C4—24no—
462C22⋊C4—42no—
463C22⋊C4—158no—
464C22⋊C4—158no—
465C22⋊C4—304no—
466C42—22no—
467C42—154no—
468C42—304no—
469C42—302no—
470C42—602no—
471C3×C5#236711yescharacteristic, complemented, Hall
472C22×C3—24no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3C22×C4×D511yescharacteristic
480C22×C3C2×D4×D511yescharacteristic
481C22×C3C2×#71111yescharacteristic
482C22×C3C2×#71111yescharacteristic
483C22×C3#414411yescharacteristic
484C22×C3D5×#4111yescharacteristic
485C22×C3#415011yescharacteristic
486C3×C4—22no—
487C3×C4—41no—
488C3×C4—42no—
489C3×C4#402714yes—
490Dic3—104no—
491Dic3—102no—
492Dic3—102no—
493Dic3—104no—
494Dic3—102no—
495Dic3—201no—
496C2×C5—44no—
497C2×C5C2×#67⋊C411yescharacteristic
498C2×C5C4×#23711yescharacteristic
499C2×C5#554611yescharacteristic
500C2×C5#559811yescharacteristic
501C2×C5#561211yescharacteristic
502C2×C5#567111yescharacteristic
503C2×C5#571311yescharacteristic
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23C2×C4×D1514yes—
511C23C23×D1511yescharacteristic
512Q8—308no—
513D4—28no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—152no—
523C2×C4—152no—
524C2×C4—154no—
525C2×C4—154no—
526C2×C4—152no—
527C2×C4—302no—
528C2×C4—301no—
529C2×C4—301no—
530C2×C4—302no—
531C2×C4—301no—
532C2×C4—302no—
533C2×C4—301no—
534C2×C4—301no—
535C2×C4—302no—
536C2×C4—301no—
537C2×C4—301no—
538C2×C4—301no—
539C2×C4—301no—
540C2×C4—608no—
541C2×C4C2×C4×D1512yes—
542C2×C4D4×D1512yes—
543C2×C4#681912yes—
544C2×C4#681912yes—
545C2×C3—44no—
546C2×C3C2×#402711yescharacteristic
547C2×C3C4×#71111yescharacteristic
548C2×C3#6588011yescharacteristic
549C2×C3#6593211yescharacteristic
550C2×C3#6594611yescharacteristic
551C2×C3#6600511yescharacteristic
552C2×C3#6604711yescharacteristic
553C5#7630711yescharacteristic, complemented, Sylow
554C22—24no—
555C22—24no—
556C22—24no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22C22×C4×D1511yescharacteristic
562C22C2×D4×D1511yescharacteristic
563C22C2×#681911yescharacteristic
564C22C2×#681911yescharacteristic
565C22#9209511yescharacteristic
566C22#41×D1511yescharacteristic
567C22#9210111yescharacteristic
568C4—22no—
569C4—41no—
570C4—42no—
571C4—302no—
572C4—302no—
573C4—302no—
574C4—304no—
575C4—304no—
576C4—601no—
577C4#9177014yes—
578C3#1060803811yescharacteristic, complemented, Sylow
579C2—44no—
580C2C2×#9177011yescharacteristic
581C2C4×#681911yescharacteristic
582C2#1175630511yescharacteristic
583C2#1175635711yescharacteristic
584C2#1175637111yescharacteristic
585C2#1175643011yescharacteristic
586C2#1175647211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#390C2
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×C5×D4C22
hypercenter#390D15
hypocenterC3×C5#2367
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
3. C2×C4×Dic30C2semidirect
5. C4⋊C4×Dic15C2semidirect
6. #11751074C2semidirect
7. #11751047C2semidirect
8. #11751046C2semidirect
9. #11751045C2semidirect
10. C22⋊C4×Dic15C2semidirect
11. C4×#6712C2semidirect
23. #91825C4semidirect
70. C2×Dic30C2×C4semidirect
76. #6719C2×C4semidirect
77. #6719C2×C4semidirect
85. C4×Dic15C2×C4semidirect
471. C3×C5#2367semidirect
553. C5#76307semidirect
578. C3#10608038semidirect

Maximal subgroups

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

subgroupindex
C2×#918252
C2×D4×Dic152
C2×C4×Dic302
C3×C5×#3902
C4⋊C4×Dic152
#117510742
#117510472
#117510462
#117510452
C22⋊C4×Dic152
C4×#67122
#106080383
#763075