Finite Group Databasebeta

← #14280006

Subgroups — #14280006

Order 1920.

1656 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×C22⋊C4C211yesmaximal, characteristic
2C22×#6719C212yesmaximal
3C22×C4×Dic15C211yesmaximal, characteristic, complemented
4C2×#91115C211yesmaximal, characteristic, complemented
5C2×#91115C212yesmaximal, complemented
6#10602338—31nomaximal, Hall
7C25×C3×C5C2211yescharacteristic
8C23×C3×C4×C5C2211yescharacteristic
9C23×C3×C4×C5C2211yescharacteristic
10C23×Dic15C2212yes—
11C23×Dic15C2212yes—
12C2×C3×C5×C22⋊C4—22no—
13C2×C3×C5×C22⋊C4—22no—
14C2×C3×C5×C22⋊C4C414yescomplemented
15C2×#6719—24no—
16C2×#6719—24no—
17C2×#6719C418yescomplemented
18C2×C4×Dic15—22no—
19C2×C4×Dic15—24no—
20#91115—22no—
21#91115—24no—
22#71545—51nomaximal, Hall
23C22×C5×C22⋊C4D311yescharacteristic
24C22×C22⋊Dic5—32no—
25C22×C4×Dic5—31no—
26C2×#3963—32no—
27C2×#3963—31no—
28C24×C3×C5—22no—
29C24×C3×C5—22no—
30C24×C3×C5—21no—
31C24×C3×C5—21no—
32C24×C3×C5—22no—
33C24×C3×C5—22no—
34C24×C3×C5—22no—
35C24×C3×C5C2×C414yes—
36C24×C3×C5C2×C412yes—
37C24×C3×C5C2311yescharacteristic
38C22×C3×C4×C5—22no—
39C22×C3×C4×C5—22no—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5—22no—
43C22×C3×C4×C5—22no—
44C22×C3×C4×C5C2×C412yes—
45C22×C3×C4×C5C2×C412yes—
46C22×C3×C4×C5D412yes—
47C22×C3×C4×C5D412yes—
48C22×Dic15—22no—
49C22×Dic15—22no—
50C22×Dic15—22no—
51C22×Dic15—22no—
52C22×Dic15—22no—
53C22×Dic15—22no—
54C22×Dic15—22no—
55C22×Dic15—22no—
56C22×Dic15—22no—
57C22×Dic15—22no—
58C22×Dic15C2×C414yes—
59C22×Dic15C2×C414yes—
60C22×Dic15D414yes—
61C22×Dic15D414yes—
62C3×C5×C22⋊C4—28no—
63#6719—216no—
64C4×Dic15—28no—
65C22×C3×C22⋊C4D511yescharacteristic
66C22×#217—52no—
67C22×C4×Dic3—51no—
68C2×C3⋊#95—52no—
69C2×C3⋊#95—51no—
70C25×C5C2×D311yescharacteristic
71C23×C4×C5C2×D311yescharacteristic
72C23×C4×C5C2×D311yescharacteristic
73C23×Dic5—32no—
74C23×Dic5—32no—
75C2×C5×C22⋊C4—22no—
76C2×C5×C22⋊C4—22no—
77C2×C5×C22⋊C4Dic314yescomplemented
78C2×C22⋊Dic5—38no—
79C2×C22⋊Dic5—64no—
80C2×C22⋊Dic5—64no—
81C2×C4×Dic5—64no—
82C2×C4×Dic5—62no—
83#3963—64no—
84#3963—62no—
85#1437—151noSylow
86C23×C3×C5—24no—
87C23×C3×C5—22no—
88C23×C3×C5—24no—
89C23×C3×C5—22no—
90C23×C3×C5—24no—
91C23×C3×C5—24no—
92C23×C3×C5—24no—
93C23×C3×C5—24no—
94C23×C3×C5—24no—
95C23×C3×C5—24no—
96C23×C3×C5—41no—
97C23×C3×C5—42no—
98C23×C3×C5—42no—
99C23×C3×C5—42no—
100C23×C3×C5—42no—
101C23×C3×C5—41no—
102C23×C3×C5—42no—
103C23×C3×C5—41no—
104C23×C3×C5—42no—
105C23×C3×C5—41no—
106C23×C3×C5C4214yescomplemented
107C23×C3×C5C22×C412yes—
108C23×C3×C5C22×C411yescharacteristic
109C23×C3×C5C2×D412yes—
110C23×C3×C5C2×D411yescharacteristic
111C23×C3×C5C2×D412yes—
112C23×C3×C5C2×D411yescharacteristic
113C23×C3×C5#4112yes—
114C23×C3×C5#4112yes—
115C23×C3×C5#4112yes—
116C2×C3×C4×C5—24no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—24no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—24no—
122C2×C3×C4×C5—42no—
123C2×C3×C4×C5—41no—
124C2×C3×C4×C5—41no—
125C2×C3×C4×C5C22⋊C414yescomplemented
126C2×Dic15—24no—
127C2×Dic15—24no—
128C2×Dic15—24no—
129C2×Dic15—24no—
130C2×Dic15—24no—
131C2×Dic15—24no—
132C2×Dic15—24no—
133C2×Dic15—24no—
134C2×Dic15—24no—
135C2×Dic15—42no—
136C2×Dic15—42no—
137C2×Dic15—42no—
138C2×Dic15—42no—
139C2×Dic15C22⋊C418yescomplemented
140C25×C3C2×D511yescharacteristic
141C23×C3×C4C2×D511yescharacteristic
142C23×C3×C4C2×D511yescharacteristic
143C23×Dic3—52no—
144C23×Dic3—52no—
145C2×C3×C22⋊C4—22no—
146C2×C3×C22⋊C4—22no—
147C2×C3×C22⋊C4Dic514yescomplemented
148C2×#217—58no—
149C2×#217—104no—
150C2×#217—104no—
151C2×C4×Dic3—104no—
152C2×C4×Dic3—102no—
153C3⋊#95—104no—
154C3⋊#95—102no—
155C24×C5—22no—
156C24×C5—22no—
157C24×C5—22no—
158C24×C5—22no—
159C24×C5—22no—
160C24×C5—21no—
161C24×C5—21no—
162C24×C5C4×D314yes—
163C24×C5C2×Dic312yes—
164C24×C5C22×D311yescharacteristic
165C22×C4×C5—22no—
166C22×C4×C5—22no—
167C22×C4×C5—22no—
168C22×C4×C5—22no—
169C22×C4×C5—21no—
170C22×C4×C5—21no—
171C22×C4×C5C2×Dic312yes—
172C22×C4×C5C2×Dic312yes—
173C22×C4×C5#6712yes—
174C22×C4×C5#6712yes—
175C22×Dic5—34no—
176C22×Dic5—34no—
177C22×Dic5—34no—
178C22×Dic5—34no—
179C22×Dic5—62no—
180C22×Dic5—62no—
181C22×Dic5—62no—
182C22×Dic5—62no—
183C22×Dic5—62no—
184C22×Dic5—62no—
185C22×Dic5—62no—
186C22×Dic5—62no—
187C22×Dic5—62no—
188C22×Dic5—62no—
189C5×C22⋊C4—28no—
190C22⋊Dic5—616no—
191C4×Dic5—68no—
192C22×C22⋊C4—152no—
193C22×C22⋊C4D1511yescharacteristic
194C22×C42—151no—
195C2×#95—152no—
196C2×#95—151no—
197C22×C3×C5—28no—
198C22×C3×C5—24no—
199C22×C3×C5—44no—
200C22×C3×C5—42no—
201C22×C3×C5—44no—
202C22×C3×C5—44no—
203C22×C3×C5—44no—
204C22×C3×C5—44no—
205C22×C3×C5—42no—
206C22×C3×C5C2×C4211yescharacteristic
207C22×C3×C5C2×C22⋊C411yescharacteristic
208C22×C3×C5C2×C22⋊C412yes—
209C22×C3×C5#11712yes—
210C22×C3×C5#11711yescharacteristic
211C22×C3×C5C4×D412yes—
212C22×C3×C5C4×D412yes—
213C22×C3×C5C4×D412yes—
214C22×C3×C5C4×D412yes—
215C22×C3×C5C4×D412yes—
216C22×C3×C5C4×D412yes—
217C22×C3×C5#12011yescharacteristic
218C22×C3×C5#12112yes—
219C22×C3×C5#12112yes—
220C22×C3×C5#12112yes—
221C22×C3×C5#12312yes—
222C22×C3×C5#12311yescharacteristic
223C22×C3×C5#12412yes—
224C22×C3×C5#12411yescharacteristic
225C22×C3×C5#12612yes—
226C22×C3×C5#12711yescharacteristic
227C3×C4×C5—24no—
228C3×C4×C5—42no—
229Dic15—28no—
230Dic15—44no—
231C24×C3—22no—
232C24×C3—22no—
233C24×C3—22no—
234C24×C3—22no—
235C24×C3—22no—
236C24×C3—21no—
237C24×C3—21no—
238C24×C3C4×D514yes—
239C24×C3C2×Dic512yes—
240C24×C3C22×D511yescharacteristic
241C22×C3×C4—22no—
242C22×C3×C4—22no—
243C22×C3×C4—22no—
244C22×C3×C4—22no—
245C22×C3×C4—21no—
246C22×C3×C4—21no—
247C22×C3×C4C2×Dic512yes—
248C22×C3×C4C2×Dic512yes—
249C22×C3×C4C22⋊D512yes—
250C22×C3×C4C22⋊D512yes—
251C22×Dic3—54no—
252C22×Dic3—54no—
253C22×Dic3—54no—
254C22×Dic3—54no—
255C22×Dic3—102no—
256C22×Dic3—102no—
257C22×Dic3—102no—
258C22×Dic3—102no—
259C22×Dic3—102no—
260C22×Dic3—102no—
261C22×Dic3—102no—
262C22×Dic3—102no—
263C22×Dic3—102no—
264C22×Dic3—102no—
265C3×C22⋊C4—28no—
266#217—1016no—
267C4×Dic3—108no—
268C23×C5—24no—
269C23×C5—24no—
270C23×C5—24no—
271C23×C5—24no—
272C23×C5—22no—
273C23×C5—24no—
274C23×C5—24no—
275C23×C5—24no—
276C23×C5—24no—
277C23×C5—22no—
278C23×C5—42no—
279C23×C5—42no—
280C23×C5—42no—
281C23×C5—42no—
282C23×C5—41no—
283C23×C5—41no—
284C23×C5—41no—
285C23×C5—42no—
286C23×C5—42no—
287C23×C5—41no—
288C23×C5C4×Dic314yescomplemented
289C23×C5C2×C4×D312yes—
290C23×C5#23512yes—
291C23×C5D3×D412yes—
292C23×C5D3×D412yes—
293C23×C5#23712yes—
294C23×C5#23712yes—
295C23×C5C22×Dic311yescharacteristic
296C23×C5C2×#6711yescharacteristic
297C23×C5C2×#6711yescharacteristic
298C2×C4×C5—24no—
299C2×C4×C5—22no—
300C2×C4×C5—22no—
301C2×C4×C5—24no—
302C2×C4×C5—24no—
303C2×C4×C5—22no—
304C2×C4×C5—41no—
305C2×C4×C5—42no—
306C2×C4×C5—41no—
307C2×C4×C5#21714yescomplemented
308C2×Dic5—38no—
309C2×Dic5—64no—
310C2×Dic5—64no—
311C2×Dic5—64no—
312C2×Dic5—64no—
313C2×Dic5—64no—
314C2×Dic5—64no—
315C2×Dic5—64no—
316C2×Dic5—64no—
317C2×Dic5—64no—
318C2×Dic5—122no—
319C2×Dic5—122no—
320C2×Dic5—122no—
321C2×Dic5—122no—
322C25C2×D1511yescharacteristic
323C23×C4—152no—
324C23×C4—152no—
325C23×C4C2×D1511yescharacteristic
326C23×C4C2×D1511yescharacteristic
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—158no—
330C2×C22⋊C4—304no—
331C2×C22⋊C4—304no—
332C2×C22⋊C4Dic1514yescomplemented
333C2×C42—304no—
334C2×C42—302no—
335#95—304no—
336#95—302no—
337C2×C3×C5—44no—
338C2×C3×C5C4×C22⋊C412yes—
339C2×C3×C5C4×C22⋊C411yescharacteristic
340C2×C3×C5#38612yes—
341C2×C3×C5#38611yescharacteristic
342C2×C3×C5#38812yes—
343C2×C3×C5#38812yes—
344C2×C3×C5#38812yes—
345C2×C3×C5#39012yes—
346C2×C3×C5#39011yescharacteristic
347C23×C3—22no—
348C23×C3—24no—
349C23×C3—24no—
350C23×C3—24no—
351C23×C3—24no—
352C23×C3—22no—
353C23×C3—24no—
354C23×C3—24no—
355C23×C3—24no—
356C23×C3—24no—
357C23×C3—42no—
358C23×C3—42no—
359C23×C3—42no—
360C23×C3—42no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—42no—
365C23×C3—42no—
366C23×C3—41no—
367C23×C3C4×Dic514yescomplemented
368C23×C3C2×C4×D512yes—
369C23×C3#70912yes—
370C23×C3D4×D512yes—
371C23×C3D4×D512yes—
372C23×C3#71112yes—
373C23×C3#71112yes—
374C23×C3C22×Dic511yescharacteristic
375C23×C3C2×C22⋊D511yescharacteristic
376C23×C3C2×C22⋊D511yescharacteristic
377C2×C3×C4—24no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—24no—
381C2×C3×C4—22no—
382C2×C3×C4—24no—
383C2×C3×C4—41no—
384C2×C3×C4—42no—
385C2×C3×C4—41no—
386C2×C3×C4C22⋊Dic514yescomplemented
387C2×Dic3—58no—
388C2×Dic3—104no—
389C2×Dic3—104no—
390C2×Dic3—104no—
391C2×Dic3—104no—
392C2×Dic3—104no—
393C2×Dic3—104no—
394C2×Dic3—104no—
395C2×Dic3—104no—
396C2×Dic3—104no—
397C2×Dic3—202no—
398C2×Dic3—202no—
399C2×Dic3—202no—
400C2×Dic3—202no—
401C22×C5—28no—
402C22×C5—24no—
403C22×C5—44no—
404C22×C5—44no—
405C22×C5—44no—
406C22×C5—44no—
407C22×C5—44no—
408C22×C5—42no—
409C22×C5—42no—
410C22×C5#87712yes—
411C22×C5#87912yes—
412C22×C5D3×C22⋊C412yes—
413C22×C5#67⋊C412yes—
414C22×C5#67⋊C412yes—
415C22×C5#88312yes—
416C22×C5#88412yes—
417C22×C5#88512yes—
418C22×C5C2×C4×Dic311yescharacteristic
419C22×C5#92611yescharacteristic
420C22×C5C4×#6712yes—
421C22×C5C4×#6712yes—
422C22×C5D4×Dic312yes—
423C22×C5D4×Dic312yes—
424C22×C5#93511yescharacteristic
425C22×C5#93611yescharacteristic
426C22×C5#93711yescharacteristic
427C22×C5#93812yes—
428C22×C5#93912yes—
429C22×C5#94011yescharacteristic
430C22×C5C2×#21711yescharacteristic
431C4×C5—24no—
432C4×C5—42no—
433Dic5—68no—
434Dic5—124no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—21no—
441C24—21no—
442C24C4×D1514yes—
443C24C2×Dic1512yes—
444C24C22×D1511yescharacteristic
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—154no—
452C22×C4—154no—
453C22×C4—154no—
454C22×C4—154no—
455C22×C4—302no—
456C22×C4—302no—
457C22×C4—302no—
458C22×C4—302no—
459C22×C4—302no—
460C22×C4—302no—
461C22×C4—302no—
462C22×C4—302no—
463C22×C4—302no—
464C22×C4—302no—
465C22×C4C2×Dic1512yes—
466C22×C4C2×Dic1512yes—
467C22×C4#122012yes—
468C22×C4#122012yes—
469C22⋊C4—28no—
470C22⋊C4—3016no—
471C42—308no—
472C3×C5#143711yescharacteristic, complemented, Hall
473C22×C3—28no—
474C22×C3—24no—
475C22×C3—44no—
476C22×C3—44no—
477C22×C3—44no—
478C22×C3—44no—
479C22×C3—44no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3#402312yes—
483C22×C3#402512yes—
484C22×C3D5×C22⋊C412yes—
485C22×C3#402712yes—
486C22×C3#402712yes—
487C22×C3#402912yes—
488C22×C3#403012yes—
489C22×C3#403112yes—
490C22×C3C2×C4×Dic511yescharacteristic
491C22×C3#407211yescharacteristic
492C22×C3C4×C22⋊D512yes—
493C22×C3C4×C22⋊D512yes—
494C22×C3D4×Dic512yes—
495C22×C3D4×Dic512yes—
496C22×C3#408111yescharacteristic
497C22×C3#408211yescharacteristic
498C22×C3#408311yescharacteristic
499C22×C3#408412yes—
500C22×C3#408512yes—
501C22×C3#408611yescharacteristic
502C22×C3C2×C22⋊Dic511yescharacteristic
503C3×C4—24no—
504C3×C4—42no—
505Dic3—108no—
506Dic3—204no—
507C2×C5—44no—
508C2×C5Dic3×C22⋊C412yes—
509C2×C5#494412yes—
510C2×C5#494512yes—
511C2×C5#495112yes—
512C2×C5#495612yes—
513C2×C5#495712yes—
514C2×C5C4×#21711yescharacteristic
515C2×C5#522011yescharacteristic
516C2×C5#522111yescharacteristic
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23C4×Dic1514yescomplemented
538C23C2×C4×D1512yes—
539C23#681712yes—
540C23D4×D1512yes—
541C23D4×D1512yes—
542C23#681912yes—
543C23#681912yes—
544C23C22×Dic1511yescharacteristic
545C23C2×#122011yescharacteristic
546C23C2×#122011yescharacteristic
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—158no—
557C2×C4—304no—
558C2×C4—304no—
559C2×C4—304no—
560C2×C4—304no—
561C2×C4—304no—
562C2×C4—304no—
563C2×C4—304no—
564C2×C4—304no—
565C2×C4—304no—
566C2×C4—602no—
567C2×C4—602no—
568C2×C4—602no—
569C2×C4—602no—
570C2×C4#671914yescomplemented
571C2×C3—44no—
572C2×C3C22⋊C4×Dic512yes—
573C2×C3#6523312yes—
574C2×C3#6523412yes—
575C2×C3#6524012yes—
576C2×C3#6524512yes—
577C2×C3#6524612yes—
578C2×C3C4×C22⋊Dic511yescharacteristic
579C2×C3#6550911yescharacteristic
580C2×C3#6551011yescharacteristic
581C5#7154511yescharacteristic, complemented, Sylow
582C22—28no—
583C22—24no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22#9176612yes—
592C22#9176812yes—
593C22C22⋊C4×D1512yes—
594C22#9177012yes—
595C22#9177012yes—
596C22#9177212yes—
597C22#9177312yes—
598C22#9177412yes—
599C22C2×C4×Dic1511yescharacteristic
600C22#9181511yescharacteristic
601C22C4×#122012yes—
602C22C4×#122012yes—
603C22D4×Dic1512yes—
604C22D4×Dic1512yes—
605C22#9182411yescharacteristic
606C22#9182511yescharacteristic
607C22#9182611yescharacteristic
608C22#9182712yes—
609C22#9182812yes—
610C22#9182911yescharacteristic
611C22C2×#671911yescharacteristic
612C4—24no—
613C4—42no—
614C4—308no—
615C4—604no—
616C3#1060233811yescharacteristic, complemented, Sylow
617C2—44no—
618C2C22⋊C4×Dic1512yes—
619C2#1175104512yes—
620C2#1175104612yes—
621C2#1175105212yes—
622C2#1175105712yes—
623C2#1175105812yes—
624C2C4×#671911yescharacteristic
625C2#1175132111yescharacteristic
626C2#1175132211yescharacteristic
subgroupG/H
center, FrattiniC24C22×D15
commutatorC22×C3×C5C2×C42
FittingC22×C3×C5×C22⋊C4C2
socle, cosocleC24×C3×C5C23
prime radicalC25×C3×C5C22
hypercenterC22×C22⋊C4D15
hypocenterC3×C5#1437
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×C4×Dic15C2semidirect
4. C2×#91115C2semidirect
5. C2×#91115C2semidirect
14. C2×C3×C5×C22⋊C4C4semidirect
17. C2×#6719C4semidirect
77. C2×C5×C22⋊C4Dic3semidirect
106. C23×C3×C5C42semidirect
125. C2×C3×C4×C5C22⋊C4semidirect
139. C2×Dic15C22⋊C4semidirect
147. C2×C3×C22⋊C4Dic5semidirect
288. C23×C5C4×Dic3semidirect
307. C2×C4×C5#217semidirect
332. C2×C22⋊C4Dic15semidirect
367. C23×C3C4×Dic5semidirect
386. C2×C3×C4C22⋊Dic5semidirect
472. C3×C5#1437semidirect
537. C23C4×Dic15semidirect
570. C2×C4#6719semidirect
581. C5#71545semidirect
616. C3#10602338semidirect

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
C22×C4×Dic152
C2×#911152
#106023383
#715455