Finite Group Databasebeta

← #14327350

Subgroups — #14327350

Order 1920.

1320 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
1C2×#91781C211yesmaximal, characteristic, complemented
2C2×C4⋊C4×D15C211yesmaximal, characteristic, complemented
3C2×C4×D60C211yesmaximal, characteristic
4C3×C4×C5×C4⋊C4C211yesmaximal, characteristic, complemented
5#11751088C211yesmaximal, characteristic, complemented
6#11751088C211yesmaximal, characteristic, complemented
7#11751041C212yesmaximal, complemented
8C4×#6714C212yesmaximal, complemented
9#11751037C211yesmaximal, characteristic, complemented
10#11750766C212yesmaximal
11#11750754C212yesmaximal, complemented
12#10605502—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15C2212yes—
15C22×C4×D15C2211yescharacteristic
16C22×C4×D15C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C42×C5C2212yes—
20C2×C3×C42×C5C2211yescharacteristic
21C2×#6714C2212yes—
22C2×#6714C2212yes—
23C2×#6714C2212yes—
24C2×#6714C2211yescharacteristic
25C2×#6714C2211yescharacteristic
26C2×C4⋊Dic15C2211yescharacteristic
27C2×C4⋊Dic15C2211yescharacteristic
28C2×C4⋊Dic15C2211yescharacteristic
29C2×#6712C2212yes—
30C2×C4×Dic15C2212yes—
31#91781—24no—
32C4⋊C4×D15—24no—
33C4×D60C418yescomplemented
34C3×C5×#95C2212yes—
35#91115C2212yes—
36#91115C2212yes—
37#91115C2212yes—
38#91115C2212yes—
39#73771—51nomaximal, Hall
40C2×#4038—31no—
41C2×D5×C4⋊C4—31no—
42C2×C4×D20—31no—
43C4×C5×C4⋊C4D311yescharacteristic, complemented
44#65276—31no—
45#65276—31no—
46#65229—32no—
47C4×#685—32no—
48#65225—31no—
49#64954—32no—
50#64942—32no—
51C23×D15C2311yescharacteristic
52C23×D15C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15C2312yes—
59C22×Dic15C2312yes—
60C22×Dic15C2311yescharacteristic
61C22×Dic15C2311yescharacteristic
62C2×D60—22no—
63C2×D60—22no—
64C2×D60C2×C414yes—
65C2×C4×D15—22no—
66C2×C4×D15—22no—
67C2×C4×D15—22no—
68C2×C4×D15—22no—
69C2×C4×D15—24no—
70C2×C4×D15—24no—
71C2×C4×D15—42no—
72C2×C4×D15C2×C414yes—
73C2×C4×D15C2×C414yescomplemented
74C3×C5×C4⋊C4—22no—
75C3×C5×C4⋊C4D414yescomplemented
76C3×C42×C5—42no—
77C3×C42×C5C2×C414yescomplemented
78#6714—24no—
79#6714—24no—
80#6714—42no—
81#6714C2×C414yes—
82#6714C2×C414yescomplemented
83C4⋊Dic15—22no—
84C4⋊Dic15—22no—
85C4⋊Dic15C2×C414yescomplemented
86#6712—24no—
87C4×Dic15—24no—
88C2×#892—51no—
89C2×D3×C4⋊C4—51no—
90C2×C4×D12—51no—
91C3×C4×C4⋊C4D511yescharacteristic, complemented
92#4987—51no—
93#4987—51no—
94#4940—52no—
95C4×(C2×D3)⋊C4—52no—
96#4936—51no—
97#4665—52no—
98#4653—52no—
99C22×D20—31no—
100C22×C4×D5—31no—
101C22×C4×D5—31no—
102C22×C4×D5—32no—
103C2×C5×C4⋊C4C2×D311yescharacteristic
104C2×C5×C4⋊C4C2×D311yescharacteristic
105C2×C42×C5C2×D312yes—
106C2×C42×C5C2×D311yescharacteristic
107C2×#685—31no—
108C2×#685—31no—
109C2×#685—32no—
110C2×#685—32no—
111C2×#685—32no—
112C2×C4⋊Dic5—31no—
113C2×C4⋊Dic5—31no—
114C2×C4⋊Dic5—31no—
115C2×#683—32no—
116C2×C4×Dic5—32no—
117#4038—64no—
118D5×C4⋊C4—64no—
119C4×D20—38no—
120C5×#95C2×D312yes—
121#3963—32no—
122#3963—32no—
123#3963—32no—
124#3963—32no—
125#2352—151noSylow
126C23×C3×C5C2411yescharacteristic
127C22×D15—21no—
128C22×D15—21no—
129C22×D15—21no—
130C22×D15—21no—
131C22×D15—21no—
132C22×D15—21no—
133C22×D15—21no—
134C22×D15—21no—
135C22×D15—21no—
136C22×D15—21no—
137C22×D15C22×C412yes—
138C22×D15C22×C412yes—
139C22×D15#4112yes—
140C22×D15#4112yes—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—21no—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5—21no—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5—22no—
150C2×C3×C4×C5C22×C412yes—
151C2×C3×C4×C5C22×C412yes—
152C2×C3×C4×C5C22×C412yes—
153C2×C3×C4×C5C2×D414yes—
154C2×C3×C4×C5C2×D412yes—
155C2×C3×C4×C5#4112yes—
156C2×Dic15—21no—
157C2×Dic15—21no—
158C2×Dic15—21no—
159C2×Dic15—21no—
160C2×Dic15—22no—
161C2×Dic15—22no—
162C2×Dic15—22no—
163C2×Dic15—22no—
164C2×Dic15—22no—
165C2×Dic15—22no—
166C2×Dic15C22×C412yes—
167C2×Dic15C22×C412yes—
168D60—28no—
169C4×D15—28no—
170C4×D15—28no—
171C4×D15—48no—
172C22×D12—51no—
173C22×C4×D3—51no—
174C22×C4×D3—51no—
175C22×C4×D3—52no—
176C2×C3×C4⋊C4C2×D511yescharacteristic
177C2×C3×C4⋊C4C2×D511yescharacteristic
178C2×C3×C42C2×D512yes—
179C2×C3×C42C2×D511yescharacteristic
180C2×(C2×D3)⋊C4—51no—
181C2×(C2×D3)⋊C4—51no—
182C2×(C2×D3)⋊C4—52no—
183C2×(C2×D3)⋊C4—52no—
184C2×(C2×D3)⋊C4—52no—
185C2×C4⋊Dic3—51no—
186C2×C4⋊Dic3—51no—
187C2×C4⋊Dic3—51no—
188C2×Dic3⋊C4—52no—
189C2×C4×Dic3—52no—
190#892—104no—
191D3×C4⋊C4—104no—
192C4×D12—58no—
193C3×#95C2×D512yes—
194C3⋊#95—52no—
195C3⋊#95—52no—
196C3⋊#95—52no—
197C3⋊#95—52no—
198C23×D5—31no—
199C23×D5—31no—
200C22×C4×C5C22×D312yes—
201C22×C4×C5C22×D311yescharacteristic
202C22×C4×C5C22×D311yescharacteristic
203C22×C4×C5C22×D312yes—
204C22×C4×C5C22×D311yescharacteristic
205C22×Dic5—31no—
206C22×Dic5—31no—
207C22×Dic5—32no—
208C22×Dic5—32no—
209C2×D20—34no—
210C2×D20—62no—
211C2×D20—62no—
212C2×C4×D5—34no—
213C2×C4×D5—34no—
214C2×C4×D5—62no—
215C2×C4×D5—62no—
216C2×C4×D5—62no—
217C2×C4×D5—62no—
218C2×C4×D5—64no—
219C2×C4×D5—64no—
220C2×C4×D5—122no—
221C5×C4⋊C4—22no—
222C5×C4⋊C4D1214yescomplemented
223C42×C5—42no—
224C42×C5C4×D314yescomplemented
225#685—34no—
226#685—34no—
227#685—64no—
228#685—64no—
229#685—122no—
230C4⋊Dic5—34no—
231C4⋊Dic5—62no—
232C4⋊Dic5—62no—
233#683—64no—
234C4×Dic5—64no—
235C2×C4×D4—151no—
236C2×#117—151no—
237C22×C4⋊C4—151no—
238#388—152no—
239#387—152no—
240#386—152no—
241#382—151no—
242#380—151no—
243#380—151no—
244C4×C4⋊C4D1511yescharacteristic, complemented
245C4×C22⋊C4—152no—
246C22×C3×C5C23×C411yescharacteristic
247C22×C3×C5C22×D411yescharacteristic
248C22×C3×C5C2×#4111yescharacteristic
249C22×C3×C5C2×#4111yescharacteristic
250C22×C3×C5C2×#4111yescharacteristic
251C22×C3×C5H(5,𝔽2)11yescharacteristic
252C22×C3×C52−1+411yescharacteristic
253C2×D15—22no—
254C2×D15—22no—
255C2×D15—22no—
256C2×D15—22no—
257C2×D15—22no—
258C2×D15—22no—
259C2×D15—22no—
260C2×D15—22no—
261C2×D15—22no—
262C2×D15—41no—
263C2×D15—41no—
264C2×D15—41no—
265C2×D15—41no—
266C2×D15#11714yescomplemented
267C3×C4×C5—22no—
268C3×C4×C5—22no—
269C3×C4×C5—24no—
270C3×C4×C5—42no—
271C3×C4×C5C4×D414yes—
272Dic15—22no—
273Dic15—22no—
274Dic15—42no—
275Dic15—42no—
276C23×D3—51no—
277C23×D3—51no—
278C22×C3×C4C22×D512yes—
279C22×C3×C4C22×D511yescharacteristic
280C22×C3×C4C22×D511yescharacteristic
281C22×C3×C4C22×D512yes—
282C22×C3×C4C22×D511yescharacteristic
283C22×Dic3—51no—
284C22×Dic3—51no—
285C22×Dic3—52no—
286C22×Dic3—52no—
287C2×D12—54no—
288C2×D12—102no—
289C2×D12—102no—
290C2×C4×D3—54no—
291C2×C4×D3—54no—
292C2×C4×D3—102no—
293C2×C4×D3—102no—
294C2×C4×D3—102no—
295C2×C4×D3—102no—
296C2×C4×D3—104no—
297C2×C4×D3—104no—
298C2×C4×D3—202no—
299C3×C4⋊C4—22no—
300C3×C4⋊C4D2014yescomplemented
301C3×C42—42no—
302C3×C42C4×D514yescomplemented
303(C2×D3)⋊C4—54no—
304(C2×D3)⋊C4—54no—
305(C2×D3)⋊C4—104no—
306(C2×D3)⋊C4—104no—
307(C2×D3)⋊C4—202no—
308C4⋊Dic3—54no—
309C4⋊Dic3—102no—
310C4⋊Dic3—102no—
311Dic3⋊C4—104no—
312C4×Dic3—104no—
313C23×C5C23×D311yescharacteristic
314C22×D5—32no—
315C22×D5—32no—
316C22×D5—32no—
317C22×D5—32no—
318C22×D5—61no—
319C22×D5—61no—
320C22×D5—61no—
321C22×D5—61no—
322C22×D5—61no—
323C22×D5—61no—
324C22×D5—61no—
325C22×D5—61no—
326C22×D5—61no—
327C22×D5—61no—
328C2×C4×C5—21no—
329C2×C4×C5—22no—
330C2×C4×C5—21no—
331C2×C4×C5—22no—
332C2×C4×C5—21no—
333C2×C4×C5—21no—
334C2×C4×C5—22no—
335C2×C4×C5—22no—
336C2×C4×C5—22no—
337C2×C4×C5C2×C4×D312yes—
338C2×C4×C5C2×C4×D312yes—
339C2×C4×C5C2×C4×D312yes—
340C2×C4×C5C2×D1214yes—
341C2×C4×C5C2×D1212yes—
342C2×C4×C5#23512yes—
343C2×Dic5—32no—
344C2×Dic5—32no—
345C2×Dic5—61no—
346C2×Dic5—61no—
347C2×Dic5—61no—
348C2×Dic5—61no—
349C2×Dic5—62no—
350C2×Dic5—62no—
351C2×Dic5—62no—
352C2×Dic5—62no—
353C2×Dic5—62no—
354C2×Dic5—62no—
355D20—68no—
356C4×D5—68no—
357C4×D5—68no—
358C4×D5—128no—
359C22×D4—151no—
360C23×C4—151no—
361C23×C4—151no—
362C23×C4—152no—
363C4×D4—158no—
364#117—304no—
365C2×C4⋊C4—151no—
366C2×C4⋊C4—151no—
367C2×C4⋊C4—152no—
368C2×C4⋊C4—151no—
369C2×C4⋊C4—304no—
370C2×C4⋊C4C2×D1511yescharacteristic
371C2×C4⋊C4C2×D1511yescharacteristic
372C2×C22⋊C4—151no—
373C2×C22⋊C4—151no—
374C2×C22⋊C4—152no—
375C2×C22⋊C4—152no—
376C2×C22⋊C4—152no—
377C2×C42—152no—
378C2×C42C2×D1512yes—
379C2×C42C2×D1511yescharacteristic
380#95—152no—
381#95—152no—
382#95—152no—
383#95—152no—
384#95C2×D1512yes—
385C2×C3×C5C2×#11711yescharacteristic
386C2×C3×C5C2×C4×D411yescharacteristic
387C2×C3×C5#52011yescharacteristic
388C2×C3×C5#54611yescharacteristic
389C2×C3×C5#54811yescharacteristic
390C2×C3×C5#55211yescharacteristic
391C2×C3×C5#55311yescharacteristic
392D15—24no—
393D15—42no—
394C23×C3C23×D511yescharacteristic
395C22×D3—52no—
396C22×D3—52no—
397C22×D3—52no—
398C22×D3—52no—
399C22×D3—101no—
400C22×D3—101no—
401C22×D3—101no—
402C22×D3—101no—
403C22×D3—101no—
404C22×D3—101no—
405C22×D3—101no—
406C22×D3—101no—
407C22×D3—101no—
408C22×D3—101no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—21no—
412C2×C3×C4—22no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4C2×C4×D512yes—
419C2×C3×C4C2×C4×D512yes—
420C2×C3×C4C2×C4×D512yes—
421C2×C3×C4C2×D2014yes—
422C2×C3×C4C2×D2012yes—
423C2×C3×C4#70912yes—
424C2×Dic3—52no—
425C2×Dic3—52no—
426C2×Dic3—101no—
427C2×Dic3—101no—
428C2×Dic3—101no—
429C2×Dic3—101no—
430C2×Dic3—102no—
431C2×Dic3—102no—
432C2×Dic3—102no—
433C2×Dic3—102no—
434C2×Dic3—102no—
435C2×Dic3—102no—
436D12—108no—
437C4×D3—108no—
438C4×D3—108no—
439C4×D3—208no—
440C22×C5C22×C4×D311yescharacteristic
441C22×C5C22×D1211yescharacteristic
442C22×C5C2×#23511yescharacteristic
443C22×C5#100411yescharacteristic
444C22×C5#100711yescharacteristic
445C22×C5D3×#4111yescharacteristic
446C22×C5D3×#4111yescharacteristic
447C2×D5—34no—
448C2×D5—62no—
449C2×D5—62no—
450C2×D5—62no—
451C2×D5—62no—
452C2×D5—62no—
453C2×D5—62no—
454C2×D5—62no—
455C2×D5—62no—
456C2×D5—62no—
457C2×D5—121no—
458C2×D5—121no—
459C2×D5—121no—
460C2×D5—121no—
461C4×C5—22no—
462C4×C5—22no—
463C4×C5—24no—
464C4×C5—42no—
465C4×C5C4×D1214yes—
466Dic5—62no—
467Dic5—62no—
468Dic5—122no—
469Dic5—122no—
470C24—151no—
471C24—151no—
472C2×D4—154no—
473C2×D4—302no—
474C2×D4—302no—
475C22×C4—151no—
476C22×C4—151no—
477C22×C4—154no—
478C22×C4—154no—
479C22×C4—152no—
480C22×C4—152no—
481C22×C4—304no—
482C22×C4—302no—
483C22×C4—302no—
484C22×C4—302no—
485C22×C4—302no—
486C22×C4—304no—
487C22×C4—602no—
488C22×C4C22×D1512yes—
489C22×C4C22×D1511yescharacteristic
490C22×C4C22×D1511yescharacteristic
491C22×C4C22×D1512yes—
492C22×C4C22×D1511yescharacteristic
493C4⋊C4—22no—
494C4⋊C4—154no—
495C4⋊C4—304no—
496C4⋊C4—302no—
497C4⋊C4—302no—
498C4⋊C4D6014yescomplemented
499C22⋊C4—154no—
500C22⋊C4—154no—
501C22⋊C4—304no—
502C22⋊C4—304no—
503C22⋊C4—602no—
504C42—42no—
505C42—304no—
506C42C4×D1514yescomplemented
507C3×C5#235211yescharacteristic, complemented, Hall
508C22×C3C22×C4×D511yescharacteristic
509C22×C3C22×D2011yescharacteristic
510C22×C3C2×#70911yescharacteristic
511C22×C3#414411yescharacteristic
512C22×C3#414711yescharacteristic
513C22×C3D5×#4111yescharacteristic
514C22×C3D5×#4111yescharacteristic
515C2×D3—54no—
516C2×D3—102no—
517C2×D3—102no—
518C2×D3—102no—
519C2×D3—102no—
520C2×D3—102no—
521C2×D3—102no—
522C2×D3—102no—
523C2×D3—102no—
524C2×D3—102no—
525C2×D3—201no—
526C2×D3—201no—
527C2×D3—201no—
528C2×D3—201no—
529C3×C4—22no—
530C3×C4—22no—
531C3×C4—24no—
532C3×C4—42no—
533C3×C4C4×D2014yes—
534Dic3—102no—
535Dic3—102no—
536Dic3—202no—
537Dic3—202no—
538C2×C5C2×C4×D1211yescharacteristic
539C2×C5#550411yescharacteristic
540C2×C5D3×#11711yescharacteristic
541C2×C5#555411yescharacteristic
542C2×C5#555811yescharacteristic
543C2×C5#557611yescharacteristic
544C2×C5#558111yescharacteristic
545D5—64no—
546D5—122no—
547C23—152no—
548C23—152no—
549C23—152no—
550C23—152no—
551C23—301no—
552C23—301no—
553C23—301no—
554C23—301no—
555C23—301no—
556C23—301no—
557C23—301no—
558C23—301no—
559C23—301no—
560C23—301no—
561C23C23×D1511yescharacteristic
562D4—308no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—152no—
573C2×C4—152no—
574C2×C4—301no—
575C2×C4—301no—
576C2×C4—301no—
577C2×C4—301no—
578C2×C4—308no—
579C2×C4—308no—
580C2×C4—302no—
581C2×C4—302no—
582C2×C4—302no—
583C2×C4—302no—
584C2×C4—302no—
585C2×C4—302no—
586C2×C4—608no—
587C2×C4C2×C4×D1512yes—
588C2×C4C2×C4×D1512yes—
589C2×C4C2×C4×D1512yes—
590C2×C4C2×D6014yes—
591C2×C4C2×D6012yes—
592C2×C4#681712yes—
593C2×C3C2×C4×D2011yescharacteristic
594C2×C3#6583811yescharacteristic
595C2×C3D5×#11711yescharacteristic
596C2×C3#6588811yescharacteristic
597C2×C3#6589211yescharacteristic
598C2×C3#6591011yescharacteristic
599C2×C3#6591511yescharacteristic
600D3—104no—
601D3—202no—
602C5#7377111yescharacteristic, complemented, Sylow
603C22—154no—
604C22—302no—
605C22—302no—
606C22—302no—
607C22—302no—
608C22—302no—
609C22—302no—
610C22—302no—
611C22—302no—
612C22—302no—
613C22—601no—
614C22—601no—
615C22—601no—
616C22—601no—
617C22C22×C4×D1511yescharacteristic
618C22C22×D6011yescharacteristic
619C22C2×#681711yescharacteristic
620C22#9209511yescharacteristic
621C22#9209811yescharacteristic
622C22#41×D1511yescharacteristic
623C22#41×D1511yescharacteristic
624C4—24no—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—302no—
629C4—302no—
630C4—602no—
631C4—602no—
632C4C4×D6014yes—
633C3#1060550211yescharacteristic, complemented, Sylow
634C2—304no—
635C2—602no—
636C2C2×C4×D6011yescharacteristic
637C2#1175626311yescharacteristic
638C2D15×#11711yescharacteristic
639C2#1175631311yescharacteristic
640C2#1175631711yescharacteristic
641C2#1175633511yescharacteristic
642C2#1175634011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C4×C5×C4⋊C4C2
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC4×C4⋊C4D15
hypocenterC3×C5#2352
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#91781C2semidirect
2. C2×C4⋊C4×D15C2semidirect
4. C3×C4×C5×C4⋊C4C2semidirect
5. #11751088C2semidirect
6. #11751088C2semidirect
7. #11751041C2semidirect
8. C4×#6714C2semidirect
9. #11751037C2semidirect
11. #11750754C2semidirect
33. C4×D60C4semidirect
43. C4×C5×C4⋊C4D3semidirect
73. C2×C4×D15C2×C4semidirect
75. C3×C5×C4⋊C4D4semidirect
77. C3×C42×C5C2×C4semidirect
82. #6714C2×C4semidirect
85. C4⋊Dic15C2×C4semidirect
91. C3×C4×C4⋊C4D5semidirect
222. C5×C4⋊C4D12semidirect
224. C42×C5C4×D3semidirect
244. C4×C4⋊C4D15semidirect
266. C2×D15#117semidirect
300. C3×C4⋊C4D20semidirect
302. C3×C42C4×D5semidirect
498. C4⋊C4D60semidirect
506. C42C4×D15semidirect
507. C3×C5#2352semidirect
602. C5#73771semidirect
633. C3#10605502semidirect

Maximal subgroups

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

subgroupindex
C2×#917812
C2×C4⋊C4×D152
C2×C4×D602
C3×C4×C5×C4⋊C42
#117510882
#117510412
C4×#67142
#117510372
#117507662
#117507542
#106055023
#737715