Finite Group Databasebeta

← #14329500

Subgroups — #14329500

Order 1920.

1664 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#940C211yesmaximal, characteristic
2C2×#67⋊Dic5C212yesmaximal, complemented
3C2×Dic5×#67C212yesmaximal, complemented
4#11751321C211yesmaximal, characteristic, complemented
5C3×#65509C211yesmaximal, characteristic, complemented
6#11749482C212yesmaximal, complemented
7#11749482C212yesmaximal, complemented
8#11748901C212yesmaximal, complemented
9#11748667C212yesmaximal, complemented
10#10614904—31nomaximal, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C23×Dic15C2211yescharacteristic
13C22×C5×#67C2212yes—
14C22×C5×#67C2212yes—
15C22×C5×D12C2211yescharacteristic
16C23×C3×Dic5C2211yescharacteristic
17C22×D3×Dic5C2212yes—
18C2×#6719C2212yes—
19C2×#6719C2211yescharacteristic
20C5×#940C418yescomplemented
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C3×C22⋊Dic5C2212yes—
23C2×C3×C22⋊Dic5C2211yescharacteristic
24#67⋊Dic5—28no—
25Dic5×#67—28no—
26C2×Dic3⋊Dic5C2212yes—
27C2×Dic3⋊Dic5C2212yes—
28C2×(C2×D3)⋊Dic5C2212yes—
29C2×(C2×D3)⋊Dic5C2212yes—
30C2×(C2×D3)⋊Dic5C2212yes—
31C2×Dic3×Dic5C2212yes—
32C2×Dic3×Dic5C2212yes—
33#91115C2212yes—
34C3×#3963C2212yes—
35#74679—51nomaximal, Hall
36C2×C5×#127—31no—
37C2×D4×Dic5—32no—
38C2×D4×Dic5—32no—
39#65510—32no—
40#65510—32no—
41#65509—32no—
42#65509—31no—
43#65509D311yescharacteristic, complemented
44C4×C4⋊Dic5—32no—
45C24×C3×C5C2311yescharacteristic
46C24×C3×C5C2311yescharacteristic
47C23×C5×D3C2312yes—
48C2×C3×C5×D4—41no—
49C2×C3×C5×D4—41no—
50C2×C3×C5×D4C2×C414yescomplemented
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15—24no—
53C22×Dic15—41no—
54C22×Dic15C2312yes—
55C22×Dic15C2312yes—
56C2×C5×#67—24no—
57C2×C5×#67—24no—
58C2×C5×#67—24no—
59C2×C5×#67—24no—
60C2×C5×#67C2×C418yescomplemented
61C2×C5×#67C2×C418yescomplemented
62C22×C5×Dic3C2312yes—
63C2×C5×D12—41no—
64C2×C5×D12—41no—
65C2×C5×D12C2×C414yescomplemented
66C22×C3×Dic5—24no—
67C22×C3×Dic5—41no—
68C22×C3×Dic5C2312yes—
69C22×C3×Dic5C2312yes—
70C2×D3×Dic5—24no—
71C2×D3×Dic5—24no—
72C2×D3×Dic5—42no—
73#6719—24no—
74#6719—41no—
75C4×C5×Dic3C2×C414yescomplemented
76C3×C22⋊Dic5—24no—
77C3×C22⋊Dic5—41no—
78Dic3⋊Dic5—24no—
79Dic3⋊Dic5D418yescomplemented
80(C2×D3)⋊Dic5—24no—
81(C2×D3)⋊Dic5—24no—
82(C2×D3)⋊Dic5—42no—
83Dic3×Dic5—24no—
84Dic3×Dic5—24no—
85C2×#940D511yescharacteristic
86C2×C4×#67—52no—
87C2×#67⋊C4—52no—
88C3×#386—51no—
89#5220—51no—
90#4957—52no—
91#4957—52no—
92#4665—52no—
93#4649—52no—
94C22×C5×D4—32no—
95C22×C5×D4—32no—
96C22×C5×D4—31no—
97C22×C5×D4C2×D311yescharacteristic
98C23×Dic5—31no—
99C23×Dic5—32no—
100C23×Dic5C2×D311yescharacteristic
101C5×#127—38no—
102C2×C42×C5—31no—
103C2×C22⋊Dic5—32no—
104C2×C22⋊Dic5—32no—
105C2×C22⋊Dic5—32no—
106C2×C22⋊Dic5—31no—
107C2×C22⋊Dic5—32no—
108C2×C22⋊Dic5C2×D312yes—
109C2×C22⋊Dic5C2×D311yescharacteristic
110D4×Dic5—68no—
111D4×Dic5—68no—
112C2×C4⋊Dic5—32no—
113C2×C4⋊Dic5—32no—
114C2×C4×Dic5—32no—
115C2×C4×Dic5—32no—
116#3963—32no—
117#3963C2×D312yes—
118#2366—151noSylow
119C23×C3×C5—22no—
120C23×C3×C5—22no—
121C23×C3×C5—22no—
122C23×C3×C5—22no—
123C23×C3×C5—21no—
124C23×C3×C5—21no—
125C23×C3×C5—21no—
126C23×C3×C5—21no—
127C23×C3×C5C22×C412yes—
128C23×C3×C5C22×C412yes—
129C23×C3×C5C2411yescharacteristic
130C22×C5×D3—22no—
131C22×C5×D3—22no—
132C22×C5×D3—22no—
133C22×C5×D3—22no—
134C22×C5×D3—22no—
135C22×C5×D3—22no—
136C22×C5×D3C22×C414yes—
137C3×C5×D4—44no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5C22×C412yes—
141C2×Dic15—22no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15—44no—
147C2×Dic15C2×D414yes—
148C5×#67—216no—
149C5×#67—216no—
150C2×C5×Dic3C22×C414yes—
151C2×C5×Dic3C2×D414yes—
152C2×C5×Dic3#4114yes—
153C5×D12—44no—
154C2×C3×Dic5—22no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5—44no—
160C2×C3×Dic5C2×D414yes—
161D3×Dic5—48no—
162C22×C3×D4C2×D511yescharacteristic
163C23×C3×C4—51no—
164C22×#67C2×D512yes—
165C22×#67C2×D512yes—
166C23×Dic3—51no—
167C22×D12C2×D511yescharacteristic
168C22×C4×D3—52no—
169C2×C3×C22⋊C4—52no—
170C2×C3×C22⋊C4—51no—
171C2×#217—52no—
172C2×#217—51no—
173#940Dic518yescomplemented
174C4×#67—108no—
175C2×(C2×D3)⋊C4—52no—
176C2×(C2×D3)⋊C4—52no—
177C2×(C2×D3)⋊C4—52no—
178C2×Dic3⋊C4—52no—
179C2×Dic3⋊C4—52no—
180C2×C4×Dic3—52no—
181C2×C4×Dic3—52no—
182C2×C4×Dic3C2×D511yescharacteristic
183#67⋊C4—108no—
184C3×#95—52no—
185C3⋊#95—52no—
186C24×C5—32no—
187C24×C5C22×D311yescharacteristic
188C24×C5C22×D311yescharacteristic
189C2×C5×D4—38no—
190C2×C5×D4—38no—
191C2×C5×D4—34no—
192C2×C5×D4—41no—
193C2×C5×D4—41no—
194C2×C5×D4—64no—
195C2×C5×D4—64no—
196C2×C5×D4—64no—
197C2×C5×D4—64no—
198C2×C5×D4—121no—
199C2×C5×D4—121no—
200C2×C5×D4C4×D314yescomplemented
201C22×C4×C5—32no—
202C22×C4×C5C22×D311yescharacteristic
203C22×Dic5—24no—
204C22×Dic5—32no—
205C22×Dic5—32no—
206C22×Dic5—41no—
207C22×Dic5—64no—
208C22×Dic5—64no—
209C22×Dic5—64no—
210C22×Dic5—122no—
211C22×Dic5—121no—
212C22×Dic5C22×D312yes—
213C22×Dic5C22×D312yes—
214C42×C5—34no—
215C22⋊Dic5—24no—
216C22⋊Dic5—41no—
217C22⋊Dic5—64no—
218C22⋊Dic5—64no—
219C22⋊Dic5—64no—
220C22⋊Dic5—122no—
221C22⋊Dic5—121no—
222C4⋊Dic5—38no—
223C4⋊Dic5—64no—
224C4×Dic5—64no—
225C4×Dic5—64no—
226C2×#127—31no—
227C2×C4×D4—152no—
228C2×C4×D4—152no—
229#390—152no—
230#390—152no—
231#386—51no—
232#386—152no—
233#386—151no—
234C4×C4⋊C4—152no—
235C22×C3×C5—24no—
236C22×C3×C5—24no—
237C22×C3×C5—22no—
238C22×C3×C5—22no—
239C22×C3×C5—41no—
240C22×C3×C5—41no—
241C22×C3×C5—42no—
242C22×C3×C5—42no—
243C22×C3×C5—41no—
244C22×C3×C5—41no—
245C22×C3×C5C23×C411yescharacteristic
246C22×C3×C5C22×D412yes—
247C22×C3×C5C2×#4112yes—
248C22×C3×C5H(5,𝔽2)11yescharacteristic
249C22×C3×C5H(5,𝔽2)11yescharacteristic
250C2×C5×D3—24no—
251C2×C5×D3—24no—
252C2×C5×D3—24no—
253C2×C5×D3—42no—
254C2×C5×D3—42no—
255C2×C5×D3—42no—
256C2×C5×D3—42no—
257C3×C4×C5—22no—
258Dic15—24no—
259Dic15—42no—
260C3×Dic5—24no—
261C3×Dic5—42no—
262C5×Dic3C4×D418yescomplemented
263C24×C3C22×D511yescharacteristic
264C24×C3C22×D511yescharacteristic
265C23×D3C22×D512yes—
266C2×C3×D4—41no—
267C2×C3×D4—41no—
268C2×C3×D4C2×Dic514yescomplemented
269C22×C3×C4—52no—
270C22×C3×C4—52no—
271C22×C3×C4—104no—
272C22×C3×C4—201no—
273C22×C3×C4C22×D511yescharacteristic
274C2×#67—24no—
275C2×#67—24no—
276C2×#67—24no—
277C2×#67—24no—
278C2×#67C2×Dic518yescomplemented
279C2×#67C2×Dic518yescomplemented
280C22×Dic3—52no—
281C22×Dic3—52no—
282C22×Dic3—104no—
283C22×Dic3—201no—
284C22×Dic3C22×D512yes—
285C2×D12—41no—
286C2×D12—41no—
287C2×D12C2×Dic514yescomplemented
288C2×C4×D3—104no—
289C2×C4×D3—104no—
290C2×C4×D3—202no—
291C3×C22⋊C4—104no—
292C3×C22⋊C4—201no—
293#217—104no—
294#217—201no—
295(C2×D3)⋊C4—104no—
296(C2×D3)⋊C4—104no—
297(C2×D3)⋊C4—202no—
298Dic3⋊C4—58no—
299Dic3⋊C4—104no—
300C4×Dic3—104no—
301C4×Dic3—104no—
302C4×Dic3C2×Dic514yescomplemented
303C23×C5—22no—
304C23×C5—22no—
305C23×C5—21no—
306C23×C5—21no—
307C23×C5—21no—
308C23×C5—21no—
309C23×C5—22no—
310C23×C5—22no—
311C23×C5—34no—
312C23×C5—62no—
313C23×C5—62no—
314C23×C5—62no—
315C23×C5—62no—
316C23×C5—62no—
317C23×C5—62no—
318C23×C5C2×C4×D312yes—
319C23×C5C2×C4×D312yes—
320C23×C5C23×D311yescharacteristic
321C5×D4—44no—
322C5×D4—616no—
323C5×D4—616no—
324C5×D4—124no—
325C2×C4×C5—21no—
326C2×C4×C5—21no—
327C2×C4×C5—34no—
328C2×C4×C5—34no—
329C2×C4×C5—34no—
330C2×C4×C5C2×C4×D312yes—
331C2×Dic5—22no—
332C2×Dic5—22no—
333C2×Dic5—22no—
334C2×Dic5—22no—
335C2×Dic5—22no—
336C2×Dic5—34no—
337C2×Dic5—44no—
338C2×Dic5—62no—
339C2×Dic5—62no—
340C2×Dic5—62no—
341C2×Dic5—62no—
342C2×Dic5—62no—
343C2×Dic5—128no—
344C2×Dic5—124no—
345C2×Dic5D3×D414yes—
346C22×D4—32no—
347C22×D4—32no—
348C22×D4—31no—
349C22×D4D3×D511yescharacteristic
350C23×C4—51no—
351C23×C4—152no—
352C23×C4—151no—
353#127—38no—
354C4×D4—308no—
355C4×D4—308no—
356C2×C4⋊C4—152no—
357C2×C4⋊C4—152no—
358C2×C22⋊C4—52no—
359C2×C22⋊C4—51no—
360C2×C22⋊C4—152no—
361C2×C22⋊C4—152no—
362C2×C22⋊C4—152no—
363C2×C22⋊C4—152no—
364C2×C22⋊C4—151no—
365C2×C42—31no—
366C2×C42—152no—
367C2×C42—152no—
368#95—52no—
369#95—152no—
370C2×C3×C5—42no—
371C2×C3×C5—42no—
372C2×C3×C5C2×C4×D412yes—
373C2×C3×C5#51811yescharacteristic
374C2×C3×C5D4211yescharacteristic
375C2×C3×C5#55012yes—
376C2×C3×C5#55911yescharacteristic
377C5×D3—44no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—22no—
381C23×C3—22no—
382C23×C3—21no—
383C23×C3—21no—
384C23×C3—21no—
385C23×C3—21no—
386C23×C3C22×Dic512yes—
387C23×C3C22×Dic512yes—
388C23×C3C23×D511yescharacteristic
389C22×D3—22no—
390C22×D3—22no—
391C22×D3—22no—
392C22×D3—22no—
393C22×D3—22no—
394C22×D3—22no—
395C22×D3C22×Dic514yes—
396C3×D4—44no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—54no—
400C2×C3×C4—102no—
401C2×C3×C4—102no—
402C2×C3×C4—102no—
403C2×C3×C4—102no—
404C2×C3×C4—102no—
405C2×C3×C4—204no—
406C2×C3×C4C22×Dic512yes—
407#67—216no—
408#67—216no—
409C2×Dic3—54no—
410C2×Dic3—102no—
411C2×Dic3—102no—
412C2×Dic3—102no—
413C2×Dic3—102no—
414C2×Dic3—102no—
415C2×Dic3—204no—
416C2×Dic3D4×D514yes—
417C2×Dic3#71114yes—
418C2×Dic3C22×Dic514yes—
419D12—44no—
420C4×D3—208no—
421C22×C5—22no—
422C22×C5—22no—
423C22×C5—24no—
424C22×C5—24no—
425C22×C5—41no—
426C22×C5—41no—
427C22×C5—41no—
428C22×C5—41no—
429C22×C5—42no—
430C22×C5—42no—
431C22×C5—64no—
432C22×C5—64no—
433C22×C5—64no—
434C22×C5—122no—
435C22×C5—122no—
436C22×C5—122no—
437C22×C5—122no—
438C22×C5C22×C4×D311yescharacteristic
439C22×C5C2×D3×D412yes—
440C22×C5#100411yescharacteristic
441C22×C5#100411yescharacteristic
442C22×C5D3×#4112yes—
443C4×C5—22no—
444C4×C5—38no—
445Dic5—24no—
446Dic5—42no—
447Dic5—64no—
448Dic5—122no—
449C24—32no—
450C24C2×D3×D511yescharacteristic
451C24C2×D3×D511yescharacteristic
452C2×D4—38no—
453C2×D4—38no—
454C2×D4—34no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—64no—
458C2×D4—64no—
459C2×D4—64no—
460C2×D4—64no—
461C2×D4—121no—
462C2×D4—121no—
463C2×D4D3×Dic514yescomplemented
464C22×C4—32no—
465C22×C4—52no—
466C22×C4—52no—
467C22×C4—104no—
468C22×C4—152no—
469C22×C4—152no—
470C22×C4—201no—
471C22×C4—304no—
472C22×C4—304no—
473C22×C4—304no—
474C22×C4—602no—
475C22×C4—601no—
476C22×C4C2×D3×D511yescharacteristic
477C4⋊C4—158no—
478C4⋊C4—304no—
479C22⋊C4—104no—
480C22⋊C4—201no—
481C22⋊C4—304no—
482C22⋊C4—304no—
483C22⋊C4—304no—
484C22⋊C4—602no—
485C22⋊C4—601no—
486C42—34no—
487C42—304no—
488C42—304no—
489C3×C5#236611yescharacteristic, complemented, Hall
490C22×C3—24no—
491C22×C3—24no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3C2×D4×D512yes—
501C22×C3C2×#71112yes—
502C22×C3#414411yescharacteristic
503C22×C3#414411yescharacteristic
504C22×C3C23×Dic511yescharacteristic
505C2×D3—24no—
506C2×D3—24no—
507C2×D3—24no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—42no—
512C3×C4—22no—
513C3×C4—104no—
514C3×C4—202no—
515Dic3—104no—
516Dic3—202no—
517Dic3D4×Dic518yescomplemented
518C2×C5—42no—
519C2×C5—42no—
520C2×C5—124no—
521C2×C5#548611yescharacteristic
522C2×C5C4×D3×D412yes—
523C2×C5#559011yescharacteristic
524C2×C5#559912yes—
525C2×C5#566411yescharacteristic
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—34no—
535C23—62no—
536C23—62no—
537C23—62no—
538C23—62no—
539C23—62no—
540C23—62no—
541C23C2×D3×Dic512yes—
542C23C2×D3×Dic512yes—
543C23C22×D3×D511yescharacteristic
544D4—44no—
545D4—616no—
546D4—616no—
547D4—124no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—34no—
551C2×C4—34no—
552C2×C4—34no—
553C2×C4—54no—
554C2×C4—102no—
555C2×C4—102no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—102no—
559C2×C4—154no—
560C2×C4—204no—
561C2×C4—302no—
562C2×C4—302no—
563C2×C4—302no—
564C2×C4—302no—
565C2×C4—302no—
566C2×C4—608no—
567C2×C4—604no—
568C2×C4C2×D3×Dic512yes—
569C2×C3—42no—
570C2×C3—42no—
571C2×C3#6604711yescharacteristic
572C2×C3#6605111yescharacteristic
573C2×C3#6605212yes—
574C2×C3C2×D4×Dic512yes—
575C2×C3#6613211yescharacteristic
576D3—44no—
577C5#7467911yescharacteristic, complemented, Sylow
578C22—24no—
579C22—24no—
580C22—22no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—42no—
587C22—42no—
588C22—64no—
589C22—64no—
590C22—64no—
591C22—122no—
592C22—122no—
593C22—122no—
594C22—122no—
595C22D3×D4×D512yes—
596C22D3×#71112yes—
597C22C22×D3×Dic511yescharacteristic
598C22#9204911yescharacteristic
599C22#9204911yescharacteristic
600C4—22no—
601C4—38no—
602C4—104no—
603C4—202no—
604C4—304no—
605C4—602no—
606C3#1061490411yescharacteristic, complemented, Sylow
607C2—42no—
608C2—42no—
609C2—124no—
610C2D3×D4×Dic512yes—
611C2#1175488911yescharacteristic
612C2#1175492812yes—
613C2#1175504211yescharacteristic
614C2#1175541011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×D4C22
socle, cosocleC23×C3×C5C24
prime radicalC2×C5×#940C2
hypercenterC22×D4D3×D5
hypocenterC3×C5#2366
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. C2×#67⋊Dic5C2semidirect
3. C2×Dic5×#67C2semidirect
4. #11751321C2semidirect
5. C3×#65509C2semidirect
6. #11749482C2semidirect
7. #11749482C2semidirect
8. #11748901C2semidirect
9. #11748667C2semidirect
20. C5×#940C4semidirect
43. #65509D3semidirect
50. C2×C3×C5×D4C2×C4semidirect
60. C2×C5×#67C2×C4semidirect
61. C2×C5×#67C2×C4semidirect
65. C2×C5×D12C2×C4semidirect
75. C4×C5×Dic3C2×C4semidirect
79. Dic3⋊Dic5D4semidirect
173. #940Dic5semidirect
200. C2×C5×D4C4×D3semidirect
262. C5×Dic3C4×D4semidirect
268. C2×C3×D4C2×Dic5semidirect
278. C2×#67C2×Dic5semidirect
279. C2×#67C2×Dic5semidirect
287. C2×D12C2×Dic5semidirect
302. C4×Dic3C2×Dic5semidirect
463. C2×D4D3×Dic5semidirect
489. C3×C5#2366semidirect
517. Dic3D4×Dic5semidirect
577. C5#74679semidirect
606. C3#10614904semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9402
C2×#67⋊Dic52
C2×Dic5×#672
#117513212
C3×#655092
#117494822
#117489012
#117486672
#106149043
#746795