Finite Group Databasebeta

← #14326229

Subgroups — #14326229

Order 1920.

1584 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×C22⋊C4C211yesmaximal, characteristic
2C22×#6719C211yesmaximal, characteristic
3C2×#91766C211yesmaximal, characteristic, complemented
4#11751348C211yesmaximal, characteristic
5#11751311C212yesmaximal, complemented
6C4×#6719C212yesmaximal, complemented
7#11751046C212yesmaximal, complemented
8#11751045C212yesmaximal, complemented
9#11751044C211yesmaximal, characteristic, complemented
10#11751043C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic15C211yesmaximal, characteristic, complemented
12#10613829—31nomaximal, Hall
13C25×C3×C5C2211yescharacteristic
14C23×C3×C4×C5C2212yes—
15C23×Dic15C2211yescharacteristic
16C23×Dic15C2211yescharacteristic
17C2×C3×C5×C22⋊C4—24no—
18C2×C3×C5×C22⋊C4C2211yescharacteristic
19C2×C3×C5×C22⋊C4C2212yes—
20C2×C3×C5×C22⋊C4C2211yescharacteristic
21C2×#6719C418yescomplemented
22C2×#6719C2211yescharacteristic
23C2×#6719C2211yescharacteristic
24C2×#6719C2212yes—
25C2×#6719C2212yes—
26C2×#6719C2211yescharacteristic
27C2×#6719C2211yescharacteristic
28C2×#6719C2211yescharacteristic
29C2×#6719C2211yescharacteristic
30C2×#6712C2212yescomplemented
31C2×#6712C2212yescomplemented
32C2×C4×Dic15C2212yescomplemented
33C2×C4×Dic15C2212yescomplemented
34#91766—24no—
35#91115C2212yescomplemented
36#91115C2212yescomplemented
37#91115C2212yescomplemented
38#91115C2212yescomplemented
39#82098—51nomaximal, Hall
40C22×C5×C22⋊C4D311yescharacteristic
41C22×C22⋊Dic5—31no—
42C2×#4023—31no—
43#65536—31no—
44#65499—32no—
45C4×C22⋊Dic5—32no—
46#65234—32no—
47#65233—32no—
48#65232—31no—
49#65231—31no—
50C22⋊C4×Dic5—31no—
51C24×C3×C5—22no—
52C24×C3×C5—22no—
53C24×C3×C5—24no—
54C24×C3×C5—42no—
55C24×C3×C5C2×C414yes—
56C24×C3×C5C2311yescharacteristic
57C24×C3×C5C2311yescharacteristic
58C24×C3×C5C2311yescharacteristic
59C22×C3×C4×C5—24no—
60C22×C3×C4×C5—24no—
61C22×C3×C4×C5—42no—
62C22×C3×C4×C5C2312yes—
63C22×C3×C4×C5C2312yes—
64C22×Dic15—22no—
65C22×Dic15—22no—
66C22×Dic15—42no—
67C22×Dic15C2×C414yescomplemented
68C22×Dic15C2×C414yescomplemented
69C22×Dic15C2312yes—
70C22×Dic15C2312yes—
71C22×Dic15C2311yescharacteristic
72C22×Dic15C2311yescharacteristic
73C22×Dic15C2311yescharacteristic
74C22×Dic15C2311yescharacteristic
75C3×C5×C22⋊C4—22no—
76C3×C5×C22⋊C4—24no—
77C3×C5×C22⋊C4D414yes—
78#6719—22no—
79#6719—22no—
80#6719—24no—
81#6719—42no—
82#6719C2×C414yes—
83#6719C2×C414yes—
84#6719C2×C414yescomplemented
85#6719C2×C414yescomplemented
86#6712—24no—
87#6712—42no—
88C4×Dic15—24no—
89C4×Dic15—42no—
90C22×C3×C22⋊C4D511yescharacteristic
91C22×#217—51no—
92C2×#877—51no—
93#5247—51no—
94#5210—52no—
95C4×#217—52no—
96#4945—52no—
97#4944—52no—
98#4943—51no—
99#4942—51no—
100Dic3×C22⋊C4—51no—
101C25×C5C2×D311yescharacteristic
102C23×C4×C5C2×D312yes—
103C23×Dic5—31no—
104C23×Dic5—31no—
105C2×C5×C22⋊C4—24no—
106C2×C5×C22⋊C4C2×D311yescharacteristic
107C2×C5×C22⋊C4C2×D312yes—
108C2×C5×C22⋊C4C2×D311yescharacteristic
109C2×C22⋊Dic5—38no—
110C2×C22⋊Dic5—31no—
111C2×C22⋊Dic5—31no—
112C2×C22⋊Dic5—31no—
113C2×C22⋊Dic5—31no—
114C2×C22⋊Dic5—31no—
115C2×C22⋊Dic5—31no—
116C2×C22⋊Dic5—32no—
117C2×C22⋊Dic5—32no—
118C2×#683—32no—
119C2×#683—32no—
120C2×C4×Dic5—32no—
121C2×C4×Dic5—32no—
122#4023—64no—
123#3963—32no—
124#3963—32no—
125#3963—32no—
126#3963—32no—
127#2342—151noSylow
128C23×C3×C5—21no—
129C23×C3×C5—22no—
130C23×C3×C5—21no—
131C23×C3×C5—22no—
132C23×C3×C5—22no—
133C23×C3×C5—28no—
134C23×C3×C5—28no—
135C23×C3×C5—21no—
136C23×C3×C5—22no—
137C23×C3×C5—21no—
138C23×C3×C5—22no—
139C23×C3×C5—44no—
140C23×C3×C5—48no—
141C23×C3×C5—48no—
142C23×C3×C5C22×C412yes—
143C23×C3×C5C22×C412yes—
144C23×C3×C5C22×C412yes—
145C23×C3×C5C2×D412yes—
146C23×C3×C5#4114yes—
147C23×C3×C5#4112yes—
148C23×C3×C5C2411yescharacteristic
149C2×C3×C4×C5—22no—
150C2×C3×C4×C5—22no—
151C2×C3×C4×C5—22no—
152C2×C3×C4×C5—22no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—48no—
155C2×C3×C4×C5C2×D414yes—
156C2×Dic15—28no—
157C2×Dic15—21no—
158C2×Dic15—21no—
159C2×Dic15—21no—
160C2×Dic15—21no—
161C2×Dic15—22no—
162C2×Dic15—22no—
163C2×Dic15—22no—
164C2×Dic15—22no—
165C2×Dic15—22no—
166C2×Dic15—22no—
167C2×Dic15—22no—
168C2×Dic15—22no—
169C2×Dic15—44no—
170C2×Dic15C22×C412yes—
171C2×Dic15C22×C412yes—
172C2×Dic15C22×C412yes—
173C2×Dic15C22×C412yes—
174C25×C3C2×D511yescharacteristic
175C23×C3×C4C2×D512yes—
176C23×Dic3—51no—
177C23×Dic3—51no—
178C2×C3×C22⋊C4—24no—
179C2×C3×C22⋊C4C2×D511yescharacteristic
180C2×C3×C22⋊C4C2×D512yes—
181C2×C3×C22⋊C4C2×D511yescharacteristic
182C2×#217—58no—
183C2×#217—51no—
184C2×#217—51no—
185C2×#217—51no—
186C2×#217—51no—
187C2×#217—51no—
188C2×#217—51no—
189C2×#217—52no—
190C2×#217—52no—
191C2×Dic3⋊C4—52no—
192C2×Dic3⋊C4—52no—
193C2×C4×Dic3—52no—
194C2×C4×Dic3—52no—
195#877—104no—
196C3⋊#95—52no—
197C3⋊#95—52no—
198C3⋊#95—52no—
199C3⋊#95—52no—
200C24×C5—22no—
201C24×C5—24no—
202C24×C5—22no—
203C24×C5—42no—
204C24×C5C4×D314yes—
205C24×C5C22×D311yescharacteristic
206C24×C5C22×D311yescharacteristic
207C24×C5C22×D311yescharacteristic
208C22×C4×C5—24no—
209C22×C4×C5—24no—
210C22×C4×C5—42no—
211C22×C4×C5C22×D312yes—
212C22×C4×C5C22×D312yes—
213C22×Dic5—34no—
214C22×Dic5—31no—
215C22×Dic5—31no—
216C22×Dic5—31no—
217C22×Dic5—31no—
218C22×Dic5—34no—
219C22×Dic5—32no—
220C22×Dic5—32no—
221C22×Dic5—62no—
222C22×Dic5—62no—
223C22×Dic5—122no—
224C5×C22⋊C4—22no—
225C5×C22⋊C4—24no—
226C5×C22⋊C4#6714yes—
227C22⋊Dic5—34no—
228C22⋊Dic5—34no—
229C22⋊Dic5—34no—
230C22⋊Dic5—34no—
231C22⋊Dic5—62no—
232C22⋊Dic5—64no—
233C22⋊Dic5—62no—
234C22⋊Dic5—122no—
235#683—64no—
236#683—122no—
237C4×Dic5—64no—
238C4×Dic5—122no—
239C2×#117—151no—
240C22×C22⋊C4—151no—
241C22×C22⋊C4D1511yescharacteristic
242#388—152no—
243#388—152no—
244#385—152no—
245#381—151no—
246#380—151no—
247#379—151no—
248C4×C22⋊C4—152no—
249C4×C22⋊C4—151no—
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—24no—
255C22×C3×C5—24no—
256C22×C3×C5—24no—
257C22×C3×C5—24no—
258C22×C3×C5—22no—
259C22×C3×C5—22no—
260C22×C3×C5—22no—
261C22×C3×C5—416no—
262C22×C3×C5—42no—
263C22×C3×C5—42no—
264C22×C3×C5#11714yescomplemented
265C22×C3×C5C4×D414yes—
266C22×C3×C5C23×C411yescharacteristic
267C22×C3×C5C22×D411yescharacteristic
268C22×C3×C5C2×#4111yescharacteristic
269C22×C3×C5C2×#4112yes—
270C22×C3×C5H(5,𝔽2)12yes—
271C3×C4×C5—24no—
272C3×C4×C5—42no—
273Dic15—22no—
274Dic15—22no—
275Dic15—22no—
276Dic15—22no—
277Dic15—42no—
278Dic15—42no—
279C24×C3—22no—
280C24×C3—24no—
281C24×C3—22no—
282C24×C3—42no—
283C24×C3C4×D514yes—
284C24×C3C22×D511yescharacteristic
285C24×C3C22×D511yescharacteristic
286C24×C3C22×D511yescharacteristic
287C22×C3×C4—24no—
288C22×C3×C4—24no—
289C22×C3×C4—42no—
290C22×C3×C4C22×D512yes—
291C22×C3×C4C22×D512yes—
292C22×Dic3—54no—
293C22×Dic3—51no—
294C22×Dic3—51no—
295C22×Dic3—51no—
296C22×Dic3—51no—
297C22×Dic3—54no—
298C22×Dic3—52no—
299C22×Dic3—52no—
300C22×Dic3—102no—
301C22×Dic3—102no—
302C22×Dic3—202no—
303C3×C22⋊C4—22no—
304C3×C22⋊C4—24no—
305C3×C22⋊C4C22⋊D514yes—
306#217—54no—
307#217—54no—
308#217—54no—
309#217—54no—
310#217—102no—
311#217—104no—
312#217—102no—
313#217—202no—
314Dic3⋊C4—104no—
315Dic3⋊C4—202no—
316C4×Dic3—104no—
317C4×Dic3—202no—
318C23×C5—21no—
319C23×C5—22no—
320C23×C5—22no—
321C23×C5—22no—
322C23×C5—21no—
323C23×C5—28no—
324C23×C5—28no—
325C23×C5—21no—
326C23×C5—22no—
327C23×C5—22no—
328C23×C5—21no—
329C23×C5—44no—
330C23×C5—48no—
331C23×C5—48no—
332C23×C5C2×C4×D312yes—
333C23×C5C2×C4×D312yes—
334C23×C5C2×C4×D312yes—
335C23×C5#23512yes—
336C23×C5#23714yes—
337C23×C5C2×#6712yes—
338C23×C5C23×D311yescharacteristic
339C2×C4×C5—22no—
340C2×C4×C5—22no—
341C2×C4×C5—22no—
342C2×C4×C5—22no—
343C2×C4×C5—22no—
344C2×C4×C5—48no—
345C2×C4×C5C2×#6714yes—
346C2×Dic5—32no—
347C2×Dic5—32no—
348C2×Dic5—32no—
349C2×Dic5—32no—
350C2×Dic5—68no—
351C2×Dic5—61no—
352C2×Dic5—61no—
353C2×Dic5—62no—
354C2×Dic5—62no—
355C2×Dic5—61no—
356C2×Dic5—61no—
357C2×Dic5—62no—
358C2×Dic5—62no—
359C2×Dic5—62no—
360C2×Dic5—62no—
361C2×Dic5—62no—
362C2×Dic5—62no—
363C2×Dic5—124no—
364C25C2×D1511yescharacteristic
365C23×C4—151no—
366C23×C4—151no—
367C23×C4C2×D1512yes—
368#117—304no—
369C2×C4⋊C4—152no—
370C2×C4⋊C4—152no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—158no—
373C2×C22⋊C4—151no—
374C2×C22⋊C4—151no—
375C2×C22⋊C4—151no—
376C2×C22⋊C4—151no—
377C2×C22⋊C4—151no—
378C2×C22⋊C4—151no—
379C2×C22⋊C4—152no—
380C2×C22⋊C4—152no—
381C2×C22⋊C4C2×D1511yescharacteristic
382C2×C22⋊C4C2×D1512yes—
383C2×C22⋊C4C2×D1511yescharacteristic
384C2×C42—152no—
385C2×C42—152no—
386#95—152no—
387#95—152no—
388#95—152no—
389#95—152no—
390C2×C3×C5—24no—
391C2×C3×C5—24no—
392C2×C3×C5—42no—
393C2×C3×C5C2×#11711yescharacteristic
394C2×C3×C5C2×C4×D411yescharacteristic
395C2×C3×C5#51811yescharacteristic
396C2×C3×C5#54612yes—
397C2×C3×C5#55112yes—
398C23×C3—21no—
399C23×C3—22no—
400C23×C3—22no—
401C23×C3—22no—
402C23×C3—21no—
403C23×C3—21no—
404C23×C3—22no—
405C23×C3—22no—
406C23×C3—21no—
407C23×C3—28no—
408C23×C3—28no—
409C23×C3—44no—
410C23×C3—48no—
411C23×C3—48no—
412C23×C3C2×C4×D512yes—
413C23×C3C2×C4×D512yes—
414C23×C3C2×C4×D512yes—
415C23×C3#70912yes—
416C23×C3#71114yes—
417C23×C3C2×C22⋊D512yes—
418C23×C3C23×D511yescharacteristic
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—48no—
425C2×C3×C4C2×C22⋊D514yes—
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—101no—
435C2×Dic3—101no—
436C2×Dic3—108no—
437C2×Dic3—102no—
438C2×Dic3—102no—
439C2×Dic3—102no—
440C2×Dic3—102no—
441C2×Dic3—102no—
442C2×Dic3—102no—
443C2×Dic3—204no—
444C22×C5—22no—
445C22×C5—22no—
446C22×C5—22no—
447C22×C5—24no—
448C22×C5—24no—
449C22×C5—22no—
450C22×C5—24no—
451C22×C5—24no—
452C22×C5—22no—
453C22×C5—22no—
454C22×C5—22no—
455C22×C5—416no—
456C22×C5—42no—
457C22×C5—42no—
458C22×C5#87714yescomplemented
459C22×C5C4×#6714yes—
460C22×C5C22×C4×D311yescharacteristic
461C22×C5C2×#23511yescharacteristic
462C22×C5C2×#23712yes—
463C22×C5#100412yes—
464C22×C5C22×#6711yescharacteristic
465C4×C5—24no—
466C4×C5—42no—
467Dic5—62no—
468Dic5—62no—
469Dic5—62no—
470Dic5—62no—
471Dic5—122no—
472Dic5—122no—
473C24—22no—
474C24—24no—
475C24—22no—
476C24—42no—
477C24C4×D1514yes—
478C24C22×D1511yescharacteristic
479C24C22×D1511yescharacteristic
480C24C22×D1511yescharacteristic
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—42no—
484C22×C4—154no—
485C22×C4—151no—
486C22×C4—151no—
487C22×C4—151no—
488C22×C4—151no—
489C22×C4—154no—
490C22×C4—152no—
491C22×C4—152no—
492C22×C4—302no—
493C22×C4—302no—
494C22×C4—602no—
495C22×C4C22×D1512yes—
496C22×C4C22×D1512yes—
497C4⋊C4—304no—
498C4⋊C4—602no—
499C22⋊C4—22no—
500C22⋊C4—24no—
501C22⋊C4—154no—
502C22⋊C4—154no—
503C22⋊C4—154no—
504C22⋊C4—154no—
505C22⋊C4—302no—
506C22⋊C4—304no—
507C22⋊C4—302no—
508C22⋊C4—602no—
509C22⋊C4#122014yes—
510C42—304no—
511C42—602no—
512C3×C5#234211yescharacteristic, complemented, Hall
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—24no—
517C22×C3—24no—
518C22×C3—22no—
519C22×C3—24no—
520C22×C3—24no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—22no—
524C22×C3—416no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3#402314yescomplemented
528C22×C3C4×C22⋊D514yes—
529C22×C3C22×C4×D511yescharacteristic
530C22×C3C2×#70911yescharacteristic
531C22×C3C2×#71112yes—
532C22×C3#414412yes—
533C22×C3C22×C22⋊D511yescharacteristic
534C3×C4—24no—
535C3×C4—42no—
536Dic3—102no—
537Dic3—102no—
538Dic3—102no—
539Dic3—102no—
540Dic3—202no—
541Dic3—202no—
542C2×C5—24no—
543C2×C5—24no—
544C2×C5—42no—
545C2×C5C2×#87711yescharacteristic
546C2×C5#548611yescharacteristic
547C2×C5#549512yes—
548C2×C5C2×C4×#6711yescharacteristic
549C2×C5#580612yes—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—21no—
555C23—28no—
556C23—28no—
557C23—21no—
558C23—22no—
559C23—22no—
560C23—21no—
561C23—44no—
562C23—48no—
563C23—48no—
564C23C2×C4×D1512yes—
565C23C2×C4×D1512yes—
566C23C2×C4×D1512yes—
567C23#681712yes—
568C23#681914yes—
569C23C2×#122012yes—
570C23C23×D1511yescharacteristic
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—48no—
577C2×C4—152no—
578C2×C4—152no—
579C2×C4—152no—
580C2×C4—152no—
581C2×C4—308no—
582C2×C4—301no—
583C2×C4—301no—
584C2×C4—302no—
585C2×C4—302no—
586C2×C4—301no—
587C2×C4—301no—
588C2×C4—302no—
589C2×C4—302no—
590C2×C4—302no—
591C2×C4—302no—
592C2×C4—302no—
593C2×C4—302no—
594C2×C4—604no—
595C2×C4C2×#122014yes—
596C2×C3—24no—
597C2×C3—24no—
598C2×C3—42no—
599C2×C3C2×#402311yescharacteristic
600C2×C3#6582011yescharacteristic
601C2×C3#6582912yes—
602C2×C3C2×C4×C22⋊D511yescharacteristic
603C2×C3#6614012yes—
604C5#8209811yescharacteristic, complemented, Sylow
605C22—22no—
606C22—24no—
607C22—24no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—416no—
617C22—42no—
618C22—42no—
619C22#9176614yescomplemented
620C22C4×#122014yes—
621C22C22×C4×D1511yescharacteristic
622C22C2×#681711yescharacteristic
623C22C2×#681912yes—
624C22#9209512yes—
625C22C22×#122011yescharacteristic
626C4—24no—
627C4—42no—
628C4—302no—
629C4—302no—
630C4—302no—
631C4—302no—
632C4—602no—
633C4—602no—
634C3#1061382911yescharacteristic, complemented, Sylow
635C2—24no—
636C2—24no—
637C2—42no—
638C2C2×#9176611yescharacteristic
639C2#1175624511yescharacteristic
640C2#1175625412yes—
641C2C2×C4×#122011yescharacteristic
642C2#1175656512yes—
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×C22⋊C4C2
socle, cosocleC23×C3×C5C24
prime radicalC25×C3×C5C22
hypercenterC22×C22⋊C4D15
hypocenterC3×C5#2342
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#91766C2semidirect
5. #11751311C2semidirect
6. C4×#6719C2semidirect
7. #11751046C2semidirect
8. #11751045C2semidirect
9. #11751044C2semidirect
10. #11751043C2semidirect
11. C22⋊C4×Dic15C2semidirect
21. C2×#6719C4semidirect
30. C2×#6712C22semidirect
31. C2×#6712C22semidirect
32. C2×C4×Dic15C22semidirect
33. C2×C4×Dic15C22semidirect
35. #91115C22semidirect
36. #91115C22semidirect
37. #91115C22semidirect
38. #91115C22semidirect
67. C22×Dic15C2×C4semidirect
68. C22×Dic15C2×C4semidirect
84. #6719C2×C4semidirect
85. #6719C2×C4semidirect
264. C22×C3×C5#117semidirect
458. C22×C5#877semidirect
512. C3×C5#2342semidirect
527. C22×C3#4023semidirect
604. C5#82098semidirect
619. C22#91766semidirect
634. C3#10613829semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C5×C22⋊C42
C22×#67192
C2×#917662
#117513482
#117513112
C4×#67192
#117510462
#117510452
#117510442
#117510432
C22⋊C4×Dic152
#106138293
#820985