Finite Group Databasebeta

← #14322537

Subgroups — #14322537

Order 1920.

1760 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6819C211yesmaximal, characteristic
2C22×#6669C211yesmaximal, characteristic, complemented
3C2×#91532C211yesmaximal, characteristic, complemented
4C5×#5220C211yesmaximal, characteristic, complemented
5C3×#65509C211yesmaximal, characteristic, complemented
6#11749516C212yesmaximal, complemented
7#11749498C211yesmaximal, characteristic, complemented
8#11749494C211yesmaximal, characteristic, complemented
9#11749483C212yesmaximal, complemented
10#11748863C212yesmaximal, complemented
11#11748694C212yesmaximal, complemented
12#10606393—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C22×#1220C2211yescharacteristic
15C22×#1220C2211yescharacteristic
16C23×Dic15C2211yescharacteristic
17C23×Dic15C2211yescharacteristic
18C2×#6819—24no—
19C2×#6819—24no—
20C2×#6819C418yescomplemented
21C22×C4×D15C2211yescharacteristic
22C22×Dic30C2211yescharacteristic
23C23×C5×Dic3C2211yescharacteristic
24C23×C3×Dic5C2211yescharacteristic
25C2×C5×#217C2211yescharacteristic
26C2×C5×#217C2212yes—
27C2×C3×C22⋊Dic5C2211yescharacteristic
28C2×C3×C22⋊Dic5C2212yes—
29C2×#6669—24no—
30C2×#6669C2212yes—
31C2×#6669C2212yes—
32C2×#6669C2212yes—
33C2×#6667C2212yes—
34C2×#6667C2212yes—
35#91532—24no—
36C2×Dic3×Dic5C2212yes—
37C5×C3⋊#95C2212yes—
38C3×#3963C2212yes—
39#90994C2212yes—
40#90994C2212yes—
41#74662—51nomaximal, Hall
42C22×#711—31no—
43C22×#683—31no—
44C2×#4023—31no—
45C5×#386—31no—
46#65509D311yescharacteristic, complemented
47#65502—31no—
48#65247—31no—
49#65246—32no—
50#65231—32no—
51#64956—32no—
52#64935—32no—
53C24×C3×C5C2311yescharacteristic
54C24×C3×C5C2311yescharacteristic
55C23×D15C2311yescharacteristic
56C2×C3×C5×D4—22no—
57C2×C3×C5×D4—22no—
58C2×C3×C5×D4C2×C414yes—
59C22×C3×C4×C5C2311yescharacteristic
60C2×#1220—22no—
61C2×#1220—22no—
62C2×#1220—22no—
63C2×#1220—22no—
64C2×#1220C2×C414yes—
65C2×#1220C2×C414yescomplemented
66C22×Dic15—22no—
67C22×Dic15—22no—
68C22×Dic15C2×C414yes—
69C22×Dic15C2×C414yescomplemented
70C22×Dic15D414yes—
71C22×Dic15D414yes—
72C22×Dic15C2311yescharacteristic
73C22×Dic15C2311yescharacteristic
74C22×Dic15C2311yescharacteristic
75#6819—232no—
76C2×C4×D15—22no—
77C2×C4×D15—22no—
78C2×C4×D15C2×C414yescomplemented
79C2×Dic30—22no—
80C2×Dic30—22no—
81C2×Dic30C2×C414yescomplemented
82C22×C5×Dic3—22no—
83C22×C5×Dic3—42no—
84C22×C5×Dic3C2312yes—
85C22×C5×Dic3C2312yes—
86C22×C3×Dic5—22no—
87C22×C3×Dic5—42no—
88C22×C3×Dic5C2312yes—
89C22×C3×Dic5C2312yes—
90C5×#217—22no—
91C5×#217—42no—
92C3×C22⋊Dic5—22no—
93C3×C22⋊Dic5—42no—
94#6669—24no—
95#6669—24no—
96#6669—24no—
97#6667—42no—
98#6667—42no—
99Dic3×Dic5—24no—
100C22×#237—51no—
101C22×Dic3⋊C4—51no—
102C2×#877—51no—
103C3×#386—51no—
104#5220D511yescharacteristic, complemented
105#5213—51no—
106#4958—51no—
107#4957—52no—
108#4942—52no—
109#4667—52no—
110#4646—52no—
111C22×C5×D4C2×D311yescharacteristic
112C23×C4×C5—31no—
113C22×C22⋊D5—31no—
114C22×C22⋊D5—31no—
115C23×Dic5—31no—
116C23×Dic5—31no—
117C23×Dic5C2×D311yescharacteristic
118C2×#711—38no—
119C2×#711—64no—
120C2×#711—64no—
121C22×C4×D5—31no—
122C22×Dic10—31no—
123C2×C5×C22⋊C4—31no—
124C2×C5×C22⋊C4—32no—
125C2×C22⋊Dic5C2×D311yescharacteristic
126C2×C22⋊Dic5C2×D312yes—
127C2×#685—32no—
128C2×#685—32no—
129C2×#683—32no—
130C2×#683—32no—
131C2×#683—32no—
132C2×#683—64no—
133C2×C4×Dic5—32no—
134#4023—64no—
135C5×#95—32no—
136#3963—32no—
137#3963—32no—
138#3963C2×D312yes—
139#2311—151noSylow
140C23×C3×C5—22no—
141C23×C3×C5—22no—
142C23×C3×C5—21no—
143C23×C3×C5—21no—
144C23×C3×C5—22no—
145C23×C3×C5—22no—
146C23×C3×C5C22×C412yes—
147C23×C3×C5C22×C412yes—
148C23×C3×C5C2×D412yes—
149C23×C3×C5C2×D412yes—
150C23×C3×C5C2411yescharacteristic
151C22×D15—21no—
152C22×D15—21no—
153C22×D15—22no—
154C22×D15—22no—
155C22×D15C22×C412yes—
156C3×C5×D4—28no—
157C2×C3×C4×C5—21no—
158C2×C3×C4×C5—21no—
159C2×C3×C4×C5C22×C412yes—
160#1220—28no—
161#1220—28no—
162C2×Dic15—21no—
163C2×Dic15—21no—
164C2×Dic15—28no—
165C2×Dic15C22⋊C4116yescomplemented
166C2×Dic15C22×C412yes—
167C2×Dic15C22×C412yes—
168C2×Dic15C22×C412yes—
169C2×Dic15C2×D412yes—
170C2×Dic15C2×D412yes—
171C2×Dic15C2×D412yes—
172C2×Dic15C2×D412yes—
173C4×D15—28no—
174Dic30—28no—
175C2×C5×Dic3—22no—
176C2×C5×Dic3—22no—
177C2×C5×Dic3—22no—
178C2×C5×Dic3—22no—
179C2×C5×Dic3—22no—
180C2×C5×Dic3—22no—
181C2×C5×Dic3—44no—
182C2×C3×Dic5—22no—
183C2×C3×Dic5—22no—
184C2×C3×Dic5—22no—
185C2×C3×Dic5—22no—
186C2×C3×Dic5—22no—
187C2×C3×Dic5—22no—
188C2×C3×Dic5—44no—
189C22×C3×D4C2×D511yescharacteristic
190C23×C3×C4—51no—
191C22×#67—51no—
192C22×#67—51no—
193C23×Dic3—51no—
194C23×Dic3—51no—
195C23×Dic3C2×D511yescharacteristic
196C2×#237—58no—
197C2×#237—104no—
198C2×#237—104no—
199C22×C4×D3—51no—
200C22×Dic6—51no—
201C2×C3×C22⋊C4—51no—
202C2×C3×C22⋊C4—52no—
203C2×#217C2×D511yescharacteristic
204C2×#217C2×D512yes—
205C2×(C2×D3)⋊C4—52no—
206C2×(C2×D3)⋊C4—52no—
207C2×Dic3⋊C4—52no—
208C2×Dic3⋊C4—52no—
209C2×Dic3⋊C4—52no—
210C2×Dic3⋊C4—104no—
211C2×C4×Dic3—52no—
212#877—104no—
213C3×#95—52no—
214C3⋊#95—52no—
215C3⋊#95—52no—
216C3⋊#95C2×D512yes—
217C24×C5C22×D311yescharacteristic
218C24×C5C22×D311yescharacteristic
219C23×D5—31no—
220C2×C5×D4—22no—
221C2×C5×D4—22no—
222C2×C5×D4C4×D314yes—
223C22×C4×C5—32no—
224C22×C4×C5—32no—
225C22×C4×C5—62no—
226C22×C4×C5—122no—
227C22×C4×C5C22×D311yescharacteristic
228C2×C22⋊D5—34no—
229C2×C22⋊D5—34no—
230C2×C22⋊D5—62no—
231C2×C22⋊D5—62no—
232C2×C22⋊D5—62no—
233C2×C22⋊D5—62no—
234C22×Dic5—22no—
235C22×Dic5—34no—
236C22×Dic5—34no—
237C22×Dic5—34no—
238C22×Dic5—31no—
239C22×Dic5—31no—
240C22×Dic5—34no—
241C22×Dic5—31no—
242C22×Dic5—42no—
243C22×Dic5—62no—
244C22×Dic5—62no—
245C22×Dic5C22×D312yes—
246C22×Dic5C22×D312yes—
247#711—632no—
248C2×C4×D5—34no—
249C2×C4×D5—62no—
250C2×C4×D5—62no—
251C2×Dic10—34no—
252C2×Dic10—62no—
253C2×Dic10—62no—
254C5×C22⋊C4—62no—
255C5×C22⋊C4—122no—
256C22⋊Dic5—22no—
257C22⋊Dic5—42no—
258#685—122no—
259#685—122no—
260#683—64no—
261#683—64no—
262#683—64no—
263C4×Dic5—64no—
264C22×#41—151no—
265C2×#117—151no—
266C22×C4⋊C4—151no—
267#391—152no—
268#390—152no—
269#386—31no—
270#386—51no—
271#386—151no—
272#386—151no—
273#380—152no—
274#380—152no—
275C22×C3×C5—24no—
276C22×C3×C5—22no—
277C22×C3×C5—22no—
278C22×C3×C5—24no—
279C22×C3×C5—22no—
280C22×C3×C5—24no—
281C22×C3×C5—41no—
282C22×C3×C5—41no—
283C22×C3×C5—42no—
284C22×C3×C5C2×C22⋊C414yes—
285C22×C3×C5C23×C411yescharacteristic
286C22×C3×C5C22×D411yescharacteristic
287C22×C3×C5C22×D411yescharacteristic
288C22×C3×C5H(5,𝔽2)12yes—
289C22×C3×C52−1+412yes—
290C2×D15—22no—
291C2×D15—24no—
292C2×D15—41no—
293C2×D15—41no—
294C2×D15—42no—
295C3×C4×C5—22no—
296Dic15—22no—
297Dic15C2×C22⋊C414yes—
298Dic15C2×C22⋊C414yes—
299C3×Dic5—42no—
300C3×Dic5—42no—
301C5×Dic3—42no—
302C5×Dic3—42no—
303C24×C3C22×D511yescharacteristic
304C24×C3C22×D511yescharacteristic
305C23×D3—51no—
306C2×C3×D4—22no—
307C2×C3×D4—22no—
308C2×C3×D4C4×D514yes—
309C22×C3×C4—52no—
310C22×C3×C4—52no—
311C22×C3×C4—102no—
312C22×C3×C4—202no—
313C22×C3×C4C22×D511yescharacteristic
314C2×#67—54no—
315C2×#67—54no—
316C2×#67—102no—
317C2×#67—102no—
318C2×#67—102no—
319C2×#67—102no—
320C22×Dic3—22no—
321C22×Dic3—42no—
322C22×Dic3—54no—
323C22×Dic3—54no—
324C22×Dic3—54no—
325C22×Dic3—51no—
326C22×Dic3—51no—
327C22×Dic3—54no—
328C22×Dic3—51no—
329C22×Dic3—102no—
330C22×Dic3—102no—
331C22×Dic3C22×D512yes—
332C22×Dic3C22×D512yes—
333#237—1032no—
334C2×C4×D3—54no—
335C2×C4×D3—102no—
336C2×C4×D3—102no—
337C2×Dic6—54no—
338C2×Dic6—102no—
339C2×Dic6—102no—
340C3×C22⋊C4—102no—
341C3×C22⋊C4—202no—
342#217—22no—
343#217—42no—
344(C2×D3)⋊C4—202no—
345(C2×D3)⋊C4—202no—
346Dic3⋊C4—104no—
347Dic3⋊C4—104no—
348Dic3⋊C4—104no—
349C4×Dic3—104no—
350C23×C5—22no—
351C23×C5—22no—
352C23×C5—21no—
353C23×C5—21no—
354C23×C5—22no—
355C23×C5—22no—
356C23×C5C2×C4×D312yes—
357C23×C5C2×C4×D312yes—
358C23×C5D3×D412yes—
359C23×C5D3×D412yes—
360C23×C5C23×D311yescharacteristic
361C22×D5—32no—
362C22×D5—61no—
363C22×D5—61no—
364C22×D5—62no—
365C22×D5—62no—
366C5×D4—28no—
367C2×C4×C5—21no—
368C2×C4×C5—21no—
369C2×C4×C5—62no—
370C2×C4×C5—62no—
371C2×C4×C5—62no—
372C2×C4×C5—62no—
373C2×C4×C5—62no—
374C2×C4×C5—62no—
375C2×C4×C5—124no—
376C2×C4×C5C2×C4×D312yes—
377C22⋊D5—68no—
378C22⋊D5—68no—
379C2×Dic5—22no—
380C2×Dic5—22no—
381C2×Dic5—22no—
382C2×Dic5—22no—
383C2×Dic5—22no—
384C2×Dic5—22no—
385C2×Dic5—316no—
386C2×Dic5—32no—
387C2×Dic5—32no—
388C2×Dic5—32no—
389C2×Dic5—32no—
390C2×Dic5—32no—
391C2×Dic5—32no—
392C2×Dic5—32no—
393C2×Dic5—44no—
394C2×Dic5—68no—
395C2×Dic5—61no—
396C2×Dic5—61no—
397C4×D5—68no—
398Dic10—68no—
399C2×#41—158no—
400C2×#41—304no—
401C2×#41—304no—
402C22×Q8—151no—
403C22×D4—151no—
404C22×D4—151no—
405C22×D4D3×D511yescharacteristic
406C23×C4—31no—
407C23×C4—51no—
408C23×C4—151no—
409C23×C4—151no—
410C23×C4—151no—
411#117—304no—
412C2×C4⋊C4—152no—
413C2×C4⋊C4—152no—
414C2×C4⋊C4—152no—
415C2×C4⋊C4—304no—
416C2×C22⋊C4—31no—
417C2×C22⋊C4—32no—
418C2×C22⋊C4—51no—
419C2×C22⋊C4—52no—
420C2×C22⋊C4—152no—
421C2×C22⋊C4—152no—
422C2×C42—152no—
423#95—32no—
424#95—52no—
425#95—152no—
426#95—152no—
427C2×C3×C5—24no—
428C2×C3×C5—42no—
429C2×C3×C5C22×C22⋊C411yescharacteristic
430C2×C3×C5#52012yes—
431C2×C3×C5#53511yescharacteristic
432C2×C3×C5#53611yescharacteristic
433C2×C3×C5#53712yes—
434D15—42no—
435C23×C3—22no—
436C23×C3—22no—
437C23×C3—21no—
438C23×C3—21no—
439C23×C3—22no—
440C23×C3—22no—
441C23×C3C2×C4×D512yes—
442C23×C3C2×C4×D512yes—
443C23×C3D4×D512yes—
444C23×C3D4×D512yes—
445C23×C3C23×D511yescharacteristic
446C22×D3—52no—
447C22×D3—101no—
448C22×D3—101no—
449C22×D3—102no—
450C22×D3—102no—
451C3×D4—28no—
452C2×C3×C4—21no—
453C2×C3×C4—21no—
454C2×C3×C4—102no—
455C2×C3×C4—102no—
456C2×C3×C4—102no—
457C2×C3×C4—102no—
458C2×C3×C4—102no—
459C2×C3×C4—102no—
460C2×C3×C4—204no—
461C2×C3×C4C2×C4×D512yes—
462#67—108no—
463#67—108no—
464C2×Dic3—22no—
465C2×Dic3—22no—
466C2×Dic3—22no—
467C2×Dic3—22no—
468C2×Dic3—22no—
469C2×Dic3—22no—
470C2×Dic3—44no—
471C2×Dic3—52no—
472C2×Dic3—52no—
473C2×Dic3—52no—
474C2×Dic3—52no—
475C2×Dic3—52no—
476C2×Dic3—52no—
477C2×Dic3—516no—
478C2×Dic3—52no—
479C2×Dic3—108no—
480C2×Dic3—101no—
481C2×Dic3—101no—
482C4×D3—108no—
483Dic6—108no—
484C22×C5—22no—
485C22×C5—24no—
486C22×C5—22no—
487C22×C5—24no—
488C22×C5—22no—
489C22×C5—24no—
490C22×C5—41no—
491C22×C5—41no—
492C22×C5—42no—
493C22×C5D3×C22⋊C414yes—
494C22×C5C22×C4×D311yescharacteristic
495C22×C5C2×D3×D411yescharacteristic
496C22×C5C2×D3×D411yescharacteristic
497C22×C5#100412yes—
498C22×C5#101012yes—
499C2×D5—62no—
500C2×D5—64no—
501C2×D5—121no—
502C2×D5—121no—
503C2×D5—122no—
504C4×C5—22no—
505C4×C5—122no—
506C4×C5—122no—
507Dic5—34no—
508Dic5—34no—
509Dic5—42no—
510Dic5—42no—
511Dic5—62no—
512C24—151no—
513C24C2×D3×D511yescharacteristic
514C24C2×D3×D511yescharacteristic
515#41—3032no—
516C2×Q8—154no—
517C2×Q8—302no—
518C2×Q8—302no—
519C2×D4—22no—
520C2×D4—22no—
521C2×D4—154no—
522C2×D4—154no—
523C2×D4—302no—
524C2×D4—302no—
525C2×D4—302no—
526C2×D4—302no—
527C2×D4#120014yes—
528C22×C4—32no—
529C22×C4—32no—
530C22×C4—52no—
531C22×C4—52no—
532C22×C4—62no—
533C22×C4—102no—
534C22×C4—122no—
535C22×C4—154no—
536C22×C4—154no—
537C22×C4—154no—
538C22×C4—151no—
539C22×C4—151no—
540C22×C4—154no—
541C22×C4—151no—
542C22×C4—154no—
543C22×C4—202no—
544C22×C4—302no—
545C22×C4—302no—
546C22×C4—302no—
547C22×C4—302no—
548C22×C4C2×D3×D511yescharacteristic
549C4⋊C4—304no—
550C4⋊C4—304no—
551C4⋊C4—304no—
552C22⋊C4—62no—
553C22⋊C4—102no—
554C22⋊C4—122no—
555C22⋊C4—202no—
556C22⋊C4—602no—
557C22⋊C4—602no—
558C42—304no—
559C3×C5#231111yescharacteristic, complemented, Hall
560C22×C3—24no—
561C22×C3—22no—
562C22×C3—24no—
563C22×C3—22no—
564C22×C3—22no—
565C22×C3—24no—
566C22×C3—41no—
567C22×C3—42no—
568C22×C3—41no—
569C22×C3D5×C22⋊C414yes—
570C22×C3C22×C4×D511yescharacteristic
571C22×C3C2×D4×D511yescharacteristic
572C22×C3C2×D4×D511yescharacteristic
573C22×C3#414412yes—
574C22×C3#415012yes—
575C2×D3—102no—
576C2×D3—104no—
577C2×D3—201no—
578C2×D3—202no—
579C2×D3—201no—
580C3×C4—22no—
581C3×C4—202no—
582C3×C4—202no—
583Dic3—42no—
584Dic3—42no—
585Dic3—54no—
586Dic3—54no—
587Dic3—102no—
588C2×C5—24no—
589C2×C5—42no—
590C2×C5C2×D3×C22⋊C411yescharacteristic
591C2×C5#554612yes—
592C2×C5#559211yescharacteristic
593C2×C5#559712yes—
594C2×C5#564811yescharacteristic
595D5—122no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—21no—
601C23—21no—
602C23—152no—
603C23—302no—
604C23—302no—
605C23—301no—
606C23—301no—
607C23C2×#120012yes—
608C23C2×#120012yes—
609C23#679012yes—
610C23#679012yes—
611C23C22×D3×D511yescharacteristic
612Q8—308no—
613D4—28no—
614D4—308no—
615D4—308no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—102no—
625C2×C4—102no—
626C2×C4—102no—
627C2×C4—102no—
628C2×C4—102no—
629C2×C4—102no—
630C2×C4—124no—
631C2×C4—152no—
632C2×C4—152no—
633C2×C4—152no—
634C2×C4—152no—
635C2×C4—152no—
636C2×C4—152no—
637C2×C4—1516no—
638C2×C4—152no—
639C2×C4—204no—
640C2×C4—308no—
641C2×C4—308no—
642C2×C4—301no—
643C2×C4—301no—
644C2×C4C2×#120012yes—
645C2×C3—24no—
646C2×C3—42no—
647C2×C3C2×D5×C22⋊C411yescharacteristic
648C2×C3#6588012yes—
649C2×C3#6592611yescharacteristic
650C2×C3#6593112yes—
651C2×C3#6598211yescharacteristic
652D3—202no—
653C5#7466211yescharacteristic, complemented, Sylow
654C22—22no—
655C22—24no—
656C22—24no—
657C22—22no—
658C22—22no—
659C22—24no—
660C22—41no—
661C22—42no—
662C22—41no—
663C22—302no—
664C22—304no—
665C22—601no—
666C22—602no—
667C22—601no—
668C22#9156114yes—
669C22#9202012yes—
670C22#9202512yes—
671C22C22×#120011yescharacteristic
672C22C2×#679011yescharacteristic
673C22C2×#679011yescharacteristic
674C4—22no—
675C4—122no—
676C4—122no—
677C4—154no—
678C4—154no—
679C4—202no—
680C4—202no—
681C4—302no—
682C3#1060639311yescharacteristic, complemented, Sylow
683C2—24no—
684C2—42no—
685C2—602no—
686C2#1175482112yes—
687C2#1175491612yes—
688C2#1175495311yescharacteristic
689C2#1175500911yescharacteristic
690C2C2×#9156111yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×D4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6819C2
hypercenterC22×D4D3×D5
hypocenterC3×C5#2311
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C22×#6669C2semidirect
3. C2×#91532C2semidirect
4. C5×#5220C2semidirect
5. C3×#65509C2semidirect
6. #11749516C2semidirect
7. #11749498C2semidirect
8. #11749494C2semidirect
9. #11749483C2semidirect
10. #11748863C2semidirect
11. #11748694C2semidirect
20. C2×#6819C4semidirect
46. #65509D3semidirect
65. C2×#1220C2×C4semidirect
69. C22×Dic15C2×C4semidirect
78. C2×C4×D15C2×C4semidirect
81. C2×Dic30C2×C4semidirect
104. #5220D5semidirect
165. C2×Dic15C22⋊C4semidirect
559. C3×C5#2311semidirect
653. C5#74662semidirect
682. C3#10606393semidirect

Maximal subgroups

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

subgroupindex
C22×#68192
C22×#66692
C2×#915322
C5×#52202
C3×#655092
#117495162
#117494982
#117494942
#117494832
#117488632
#117486942
#106063933
#746625