Finite Group Databasebeta

← #14321718

Subgroups — #14321718

Order 1920.

1376 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Dic5C211yesmaximal, characteristic, complemented
2C2×#91552C211yesmaximal, characteristic
3C2×#91546C211yesmaximal, characteristic
4#11751314C211yesmaximal, characteristic
5C5×#5210C211yesmaximal, characteristic, complemented
6Dic5×#217C211yesmaximal, characteristic, complemented
7#11749471C211yesmaximal, characteristic, complemented
8#11748875C212yesmaximal, complemented
9#11748816C212yesmaximal, complemented
10#11748727C212yesmaximal, complemented
11Dic5×Dic3⋊C4C212yesmaximal, complemented
12#10606111—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×#1220C2211yescharacteristic
15C23×C3×Dic5C2211yescharacteristic
16C23×C3×Dic5C2211yescharacteristic
17C22×#1200C2211yescharacteristic
18C2×#6719C2211yescharacteristic
19C2×#6714C2212yes—
20C2×C5×#217C2211yescharacteristic
21C2×C5×#217C2211yescharacteristic
22C2×C5×Dic3⋊C4C2212yes—
23C2×C3×C4×Dic5—24no—
24C2×C3×C4×Dic5C2212yes—
25C2×C3×C4×Dic5C2212yes—
26#91552—24no—
27#91546C418yescomplemented
28C2×Dic3⋊Dic5C2212yes—
29C2×#6669C2211yescharacteristic
30C2×#6667C2211yescharacteristic
31C2×#6667C2212yes—
32C2×Dic3×Dic5C2212yes—
33C2×Dic3×Dic5C2211yescharacteristic
34#91115C2212yes—
35C5×C3⋊#95C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39C4×#929—51nomaximal, Hall
40C22×C4×Dic5D311yescharacteristic, complemented
41C2×#4032—31no—
42C2×#4027—31no—
43C5×#385—31no—
44#65502—31no—
45#65279—32no—
46C4⋊C4×Dic5—32no—
47#65232—31no—
48C22⋊C4×Dic5—31no—
49#64955—32no—
50#64943—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15C2311yescharacteristic
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—24no—
55C22×C3×C4×C5C2312yes—
56C2×#1220—22no—
57C2×#1220—22no—
58C2×#1220C2×C414yes—
59C22×Dic15C2312yes—
60C22×Dic15C2311yescharacteristic
61C22×C5×Dic3C2311yescharacteristic
62C22×C5×Dic3C2311yescharacteristic
63C22×C5×Dic3C2312yes—
64C22×C3×Dic5—22no—
65C22×C3×Dic5—22no—
66C22×C3×Dic5—24no—
67C22×C3×Dic5C2×C414yescomplemented
68C22×C3×Dic5D414yes—
69C22×C3×Dic5C2312yes—
70C22×C3×Dic5C2311yescharacteristic
71C22×C3×Dic5C2311yescharacteristic
72C2×#1200—22no—
73C2×#1200—24no—
74#6719—22no—
75#6714—24no—
76C5×#217—22no—
77C5×#217C2×C414yescomplemented
78C5×Dic3⋊C4—24no—
79C3×C4×Dic5—24no—
80C3×C4×Dic5—24no—
81Dic3⋊Dic5C2×C418yescomplemented
82#6669—22no—
83#6667—22no—
84#6667C2×C418yescomplemented
85Dic3×Dic5—22no—
86Dic3×Dic5—24no—
87C22×C3×C42—51no—
88C2×#929—51no—
89C2×C4×#67—51no—
90#5213—51no—
91#5211—51no—
92#5210D511yescharacteristic, complemented
93C4×#217—51no—
94#4940—52no—
95C4×(C2×D3)⋊C4—52no—
96#4933—52no—
97C4×Dic3⋊C4—52no—
98C23×C4×C5C2×D311yescharacteristic
99C22×C22⋊D5—31no—
100C23×Dic5C2×D311yescharacteristic
101C23×Dic5C2×D311yescharacteristic
102C22×C4×D5—31no—
103C2×C5×C4⋊C4—32no—
104C2×C5×C22⋊C4—31no—
105C2×C5×C22⋊C4—31no—
106C2×C22⋊Dic5—31no—
107C2×#685—32no—
108C2×#685—32no—
109C2×#685—31no—
110C2×C4⋊Dic5—32no—
111C2×#683—31no—
112C2×C4×Dic5—24no—
113C2×C4×Dic5—31no—
114C2×C4×Dic5—32no—
115C2×C4×Dic5C2×D312yes—
116C2×C4×Dic5C2×D312yes—
117#4032—38no—
118#4027—64no—
119C5×#95—32no—
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#3963—32no—
124C4×#123—151noSylow
125C23×C3×C5—22no—
126C23×C3×C5—22no—
127C23×C3×C5C22×C412yes—
128C23×C3×C5C2×D412yes—
129C23×C3×C5#4112yes—
130C23×C3×C5C2411yescharacteristic
131C22×D15—21no—
132C22×D15—22no—
133C22×D15—22no—
134C22×D15—21no—
135C22×D15C22×C412yes—
136C2×C3×C4×C5—28no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5#4114yes—
140#1220—28no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15C22×C412yes—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3C22×C414yes—
154C2×C3×Dic5—28no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—28no—
159C2×C3×Dic5C22×C414yes—
160C2×C3×Dic5C2×D412yes—
161C2×C3×Dic5C2×D412yes—
162C2×C3×Dic5#4114yes—
163#1200—44no—
164C23×C3×C4—51no—
165C23×C3×C4—51no—
166C23×C3×C4C2×D511yescharacteristic
167C22×#67—51no—
168C22×C4×D3—51no—
169C2×C3×C42—52no—
170C2×C3×C42—52no—
171C2×C3×C42—104no—
172C2×#217—51no—
173C2×#217C2×D511yescharacteristic
174C2×#217C2×D511yescharacteristic
175#929—58no—
176C4×#67—104no—
177C2×(C2×D3)⋊C4—52no—
178C2×(C2×D3)⋊C4—52no—
179C2×(C2×D3)⋊C4—51no—
180C2×Dic3⋊C4—52no—
181C2×Dic3⋊C4—51no—
182C2×Dic3⋊C4C2×D512yes—
183C2×C4×Dic3—51no—
184C2×C4×Dic3—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C3⋊#95—52no—
189C3⋊#95C2×D512yes—
190C24×C5C22×D311yescharacteristic
191C23×D5—31no—
192C22×C4×C5—22no—
193C22×C4×C5—24no—
194C22×C4×C5—32no—
195C22×C4×C5—31no—
196C22×C4×C5—31no—
197C22×C4×C5C22×D312yes—
198C2×C22⋊D5—34no—
199C2×C22⋊D5—62no—
200C2×C22⋊D5—62no—
201C22×Dic5—22no—
202C22×Dic5—22no—
203C22×Dic5—24no—
204C22×Dic5—31no—
205C22×Dic5—32no—
206C22×Dic5C4×D314yescomplemented
207C22×Dic5#6714yes—
208C22×Dic5C22×D312yes—
209C22×Dic5C22×D311yescharacteristic
210C22×Dic5C22×D311yescharacteristic
211C2×C4×D5—64no—
212C2×C4×D5—62no—
213C5×C4⋊C4—64no—
214C5×C22⋊C4—34no—
215C5×C22⋊C4—62no—
216C22⋊Dic5—62no—
217#685—38no—
218#685—64no—
219#685—62no—
220C4⋊Dic5—38no—
221#683—62no—
222C4×Dic5—24no—
223C4×Dic5—24no—
224C4×Dic5—64no—
225C4×Dic5—62no—
226C2×#123—151no—
227C2×C4×D4—151no—
228C22×C42—51no—
229#388—152no—
230#387—152no—
231#386—151no—
232#385—31no—
233#381—151no—
234C4×C4⋊C4—152no—
235C4×C22⋊C4—152no—
236C4×C22⋊C4—151no—
237C22×C3×C5—24no—
238C22×C3×C5—22no—
239C22×C3×C5—22no—
240C22×C3×C5—24no—
241C22×C3×C5C4×D414yes—
242C22×C3×C5C23×C411yescharacteristic
243C22×C3×C5C22×D411yescharacteristic
244C22×C3×C5C2×#4112yes—
245C22×C3×C5C2×#4111yescharacteristic
246C22×C3×C5C2×#4112yes—
247C2×D15—22no—
248C2×D15—24no—
249C2×D15—41no—
250C2×D15—41no—
251C2×D15—42no—
252C3×C4×C5—24no—
253Dic15—22no—
254Dic15—42no—
255C3×Dic5—24no—
256C3×Dic5—22no—
257C3×Dic5#12314yescomplemented
258C5×Dic3—24no—
259C5×Dic3—41no—
260C5×Dic3—41no—
261C24×C3C22×D511yescharacteristic
262C23×D3—51no—
263C22×C3×C4—22no—
264C22×C3×C4—24no—
265C22×C3×C4—51no—
266C22×C3×C4—54no—
267C22×C3×C4—54no—
268C22×C3×C4—52no—
269C22×C3×C4—51no—
270C22×C3×C4—104no—
271C22×C3×C4—102no—
272C22×C3×C4—102no—
273C22×C3×C4C22×D512yes—
274C2×#67—54no—
275C2×#67—102no—
276C2×#67—102no—
277C22×Dic3—51no—
278C22×Dic3—52no—
279C22×Dic3C22×D511yescharacteristic
280C22×Dic3C22×D511yescharacteristic
281C22×Dic3C22×D512yes—
282C2×C4×D3—102no—
283C2×C4×D3—104no—
284C3×C42—104no—
285C3×C42—104no—
286#217—22no—
287#217—102no—
288#217C4×D514yescomplemented
289(C2×D3)⋊C4—58no—
290(C2×D3)⋊C4—102no—
291(C2×D3)⋊C4—104no—
292Dic3⋊C4—24no—
293Dic3⋊C4—58no—
294Dic3⋊C4—102no—
295C4×Dic3—104no—
296C4×Dic3—102no—
297C23×C5—22no—
298C23×C5—22no—
299C23×C5C2×C4×D312yes—
300C23×C5#23512yes—
301C23×C5C2×#6712yes—
302C23×C5C23×D311yescharacteristic
303C22×D5—32no—
304C22×D5—62no—
305C22×D5—62no—
306C22×D5—61no—
307C22×D5—61no—
308C2×C4×C5—28no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—34no—
312C2×C4×C5—62no—
313C2×C4×C5—62no—
314C2×C4×C5—62no—
315C2×C4×C5—62no—
316C2×C4×C5—61no—
317C2×C4×C5—61no—
318C2×C4×C5#23514yes—
319C22⋊D5—68no—
320C2×Dic5—28no—
321C2×Dic5—22no—
322C2×Dic5—22no—
323C2×Dic5—22no—
324C2×Dic5—28no—
325C2×Dic5—32no—
326C2×Dic5—61no—
327C2×Dic5—61no—
328C2×Dic5—62no—
329C2×Dic5—62no—
330C2×Dic5—62no—
331C2×Dic5C2×C4×D314yes—
332C2×Dic5#23514yes—
333C2×Dic5C2×#6712yes—
334C2×Dic5C2×#6712yes—
335C4×D5—124no—
336C22×D4—151no—
337C23×C4—51no—
338C23×C4—51no—
339C23×C4—151no—
340C23×C4D3×D511yescharacteristic
341#123—158no—
342C4×D4—304no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—152no—
345C2×C4⋊C4—151no—
346C2×C22⋊C4—31no—
347C2×C22⋊C4—31no—
348C2×C22⋊C4—152no—
349C2×C22⋊C4—151no—
350C2×C22⋊C4—151no—
351C2×C22⋊C4—152no—
352C2×C42—52no—
353C2×C42—52no—
354C2×C42—104no—
355C2×C42—151no—
356C2×C42—152no—
357#95—32no—
358#95—152no—
359#95—152no—
360#95—152no—
361#95—152no—
362C2×C3×C5—24no—
363C2×C3×C5C2×C4×D411yescharacteristic
364C2×C3×C5C4×#4112yes—
365C2×C3×C5C2×#12311yescharacteristic
366C2×C3×C5#52511yescharacteristic
367C2×C3×C5#52912yes—
368D15—42no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3C2×C4×D512yes—
372C23×C3D4×D512yes—
373C23×C3#71312yes—
374C23×C3C23×D511yescharacteristic
375C22×D3—52no—
376C22×D3—101no—
377C22×D3—102no—
378C22×D3—101no—
379C22×D3—102no—
380C2×C3×C4—28no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—52no—
384C2×C3×C4—54no—
385C2×C3×C4—54no—
386C2×C3×C4—52no—
387C2×C3×C4—108no—
388C2×C3×C4—102no—
389C2×C3×C4—102no—
390C2×C3×C4—102no—
391C2×C3×C4—108no—
392C2×C3×C4#71114yes—
393#67—108no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—52no—
401C2×Dic3—101no—
402C2×Dic3—101no—
403C2×Dic3—102no—
404C2×Dic3—102no—
405C2×Dic3—102no—
406C2×Dic3C2×C4×D514yes—
407C4×D3—204no—
408C22×C5—24no—
409C22×C5—24no—
410C22×C5—22no—
411C22×C5—22no—
412C22×C5C4×#6714yes—
413C22×C5C22×C4×D311yescharacteristic
414C22×C5C2×#23511yescharacteristic
415C22×C5C2×#23512yes—
416C22×C5C2×#23512yes—
417C22×C5C22×#6711yescharacteristic
418C2×D5—64no—
419C2×D5—62no—
420C2×D5—122no—
421C2×D5—121no—
422C2×D5—121no—
423C4×C5—24no—
424C4×C5—64no—
425C4×C5—121no—
426C4×C5—121no—
427Dic5—24no—
428Dic5—22no—
429Dic5—62no—
430Dic5—122no—
431Dic5#92914yescomplemented
432C24—151no—
433C24C2×D3×D511yescharacteristic
434C2×D4—154no—
435C2×D4—302no—
436C2×D4—302no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—32no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—51no—
443C22×C4—54no—
444C22×C4—54no—
445C22×C4—52no—
446C22×C4—51no—
447C22×C4—104no—
448C22×C4—102no—
449C22×C4—102no—
450C22×C4—151no—
451C22×C4—152no—
452C22×C4—302no—
453C22×C4—304no—
454C22×C4C2×D3×D512yes—
455C4⋊C4—64no—
456C4⋊C4—158no—
457C4⋊C4—302no—
458C22⋊C4—34no—
459C22⋊C4—62no—
460C22⋊C4—158no—
461C22⋊C4—302no—
462C22⋊C4—302no—
463C22⋊C4—304no—
464C42—104no—
465C42—104no—
466C42—302no—
467C42—304no—
468C3×C5C4×#12311yescharacteristic, complemented, Hall
469C22×C3—24no—
470C22×C3—22no—
471C22×C3—24no—
472C22×C3—22no—
473C22×C3#403914yes—
474C22×C3C22×C4×D511yescharacteristic
475C22×C3C2×D4×D511yescharacteristic
476C22×C3C2×#71112yes—
477C22×C3C2×#71311yescharacteristic
478C22×C3D5×#4112yes—
479C2×D3—102no—
480C2×D3—104no—
481C2×D3—201no—
482C2×D3—202no—
483C2×D3—201no—
484C3×C4—24no—
485C3×C4—54no—
486C3×C4—104no—
487C3×C4—102no—
488Dic3—24no—
489Dic3—41no—
490Dic3—41no—
491Dic3—102no—
492Dic3—202no—
493C2×C5—24no—
494C2×C5C4×#23512yes—
495C2×C5#547912yes—
496C2×C5C2×C4×#6711yescharacteristic
497C2×C5C2×#92911yescharacteristic
498C2×C5#579111yescharacteristic
499D5—122no—
500C23—22no—
501C23—22no—
502C23—152no—
503C23—301no—
504C23—302no—
505C23—302no—
506C23—301no—
507C23#677312yes—
508C23C4×D3×D512yes—
509C23D5×#6712yes—
510C23C22×D3×D511yescharacteristic
511D4—308no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—34no—
516C2×C4—54no—
517C2×C4—52no—
518C2×C4—54no—
519C2×C4—52no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—108no—
527C2×C4—108no—
528C2×C4—102no—
529C2×C4—102no—
530C2×C4—102no—
531C2×C4—152no—
532C2×C4—301no—
533C2×C4—301no—
534C2×C4—302no—
535C2×C4—302no—
536C2×C4—302no—
537C2×C4—604no—
538C2×C4#678214yes—
539C2×C3—24no—
540C2×C3C2×#403911yescharacteristic
541C2×C3C4×#71112yes—
542C2×C3#6592011yescharacteristic
543C2×C3#6595812yes—
544C2×C3D5×#12311yescharacteristic
545D3—202no—
546C5C4×#92911yescharacteristic, complemented, Sylow
547C22—24no—
548C22—22no—
549C22—22no—
550C22—24no—
551C22—302no—
552C22—304no—
553C22—601no—
554C22—601no—
555C22—602no—
556C22#9144414yes—
557C22C2×#677311yescharacteristic
558C22C2×C4×D3×D511yescharacteristic
559C22D5×#23512yes—
560C22C2×#678212yes—
561C22C2×D5×#6711yescharacteristic
562C4—24no—
563C4—54no—
564C4—64no—
565C4—104no—
566C4—102no—
567C4—121no—
568C4—121no—
569C4—302no—
570C4—602no—
571C3#1060611111yescharacteristic, complemented, Sylow
572C2—24no—
573C2—602no—
574C2C4×#678212yes—
575C2#1175441512yes—
576C2C2×#9144411yescharacteristic
577C2D5×#92911yescharacteristic
578C2#1175538511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1220C22
hypercenterC23×C4D3×D5
hypocenterC3×C5C4×#123
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×Dic5C2semidirect
5. C5×#5210C2semidirect
6. Dic5×#217C2semidirect
7. #11749471C2semidirect
8. #11748875C2semidirect
9. #11748816C2semidirect
10. #11748727C2semidirect
11. Dic5×Dic3⋊C4C2semidirect
27. #91546C4semidirect
40. C22×C4×Dic5D3semidirect
67. C22×C3×Dic5C2×C4semidirect
77. C5×#217C2×C4semidirect
81. Dic3⋊Dic5C2×C4semidirect
84. #6667C2×C4semidirect
92. #5210D5semidirect
206. C22×Dic5C4×D3semidirect
257. C3×Dic5#123semidirect
288. #217C4×D5semidirect
431. Dic5#929semidirect
468. C3×C5C4×#123semidirect
546. C5C4×#929semidirect
571. C3#10606111semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×Dic52
C2×#915522
C2×#915462
#117513142
C5×#52102
Dic5×#2172
#117494712
#117488752
#117488162
#117487272
Dic5×Dic3⋊C42
#106061113
C4×#9295