Finite Group Databasebeta

← #11753235

Subgroups — #11753235

Order 960.

856 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6790C211yesmaximal, characteristic, complemented
2C2×D5×#67C211yesmaximal, characteristic, complemented
3C22×#1202C211yesmaximal, characteristic, complemented
4C2×D3×D20C211yesmaximal, characteristic, complemented
5#91771C211yesmaximal, characteristic, complemented
6C5×#67⋊C4C211yesmaximal, characteristic, complemented
7C3×#4028C211yesmaximal, characteristic, complemented
8#91569C211yesmaximal, characteristic, complemented
9#91476C211yesmaximal, characteristic, complemented
10#91474C211yesmaximal, characteristic, complemented
11#91459C211yesmaximal, characteristic, complemented
12#91458C211yesmaximal, characteristic, complemented
13#91454C211yesmaximal, characteristic, complemented
14#91409C211yesmaximal, characteristic, complemented
15#91395C211yesmaximal, characteristic, complemented
16D4×D20—31nomaximal, Hall
17C23×D15C2211yescharacteristic, complemented
18C23×C3×D5C2211yescharacteristic, complemented
19C22×D3×D5C2211yescharacteristic, complemented
20C22×D3×D5C2211yescharacteristic, complemented
21C2×#1220C2211yescharacteristic, complemented
22C2×D60C2211yescharacteristic, complemented
23C2×D60C2211yescharacteristic, complemented
24C2×C5×#67C2211yescharacteristic
25C22×C5×Dic3C2211yescharacteristic, complemented
26C2×C4×C5×D3C2211yescharacteristic, complemented
27C2×C3×C22⋊D5C2211yescharacteristic, complemented
28C2×C3×D20C2211yescharacteristic, complemented
29C2×C3×D20C2211yescharacteristic, complemented
30#6790—24no—
31D5×#67—24no—
32C2×#1203C2211yescharacteristic, complemented
33C2×#1202—24no—
34C2×#1202C2211yescharacteristic
35C2×#1202C2211yescharacteristic, complemented
36C2×#1202C2211yescharacteristic, complemented
37C2×#1202C2211yescharacteristic, complemented
38C2×#1202C2211yescharacteristic, complemented
39C2×#1202C2211yescharacteristic, complemented
40C2×#1202C2211yescharacteristic, complemented
41C2×#1201C2211yescharacteristic, complemented
42C2×#1200C2211yescharacteristic, complemented
43C2×D5×Dic3C2211yescharacteristic, complemented
44D3×D20—24no—
45C3×C5×C22⋊C4C2211yescharacteristic, complemented
46#6714C2211yescharacteristic, complemented
47#6714C2211yescharacteristic, complemented
48C5×(C2×D3)⋊C4C2211yescharacteristic, complemented
49C5×Dic3⋊C4C2211yescharacteristic, complemented
50C4×C5×Dic3C2211yescharacteristic, complemented
51C3×#685C2211yescharacteristic, complemented
52C3×#685C2211yescharacteristic, complemented
53Dic3⋊Dic5C2211yescharacteristic, complemented
54#6667C2211yescharacteristic, complemented
55#6665C2211yescharacteristic, complemented
56#5590—51nomaximal, Hall
57C2×D4×D5—31no—
58C2×D4×D5—31no—
59C22×D20—31no—
60C22×D20—31no—
61C4×C5×D4—31no—
62#4076—31no—
63#4076—31no—
64#4041—31no—
65#4041—31no—
66#4028—31no—
67#4028—31no—
68#4028—31no—
69#4028D311yescharacteristic, complemented
70#4020—31no—
71C4×D20—31no—
72C23×C3×C5C2311yescharacteristic
73C22×D15—22no—
74C22×D15—22no—
75C22×D15—41no—
76C22×D15C2311yescharacteristic
77C22×D15C2311yescharacteristic
78C22×D15C2311yescharacteristic
79C22×C5×D3C2311yescharacteristic
80C22×C3×D5—22no—
81C22×C3×D5—22no—
82C22×C3×D5—41no—
83C22×C3×D5C2311yescharacteristic
84C22×C3×D5C2311yescharacteristic
85C22×C3×D5C2311yescharacteristic
86C2×D3×D5—22no—
87C2×D3×D5—22no—
88C2×D3×D5—22no—
89C2×D3×D5—22no—
90C2×D3×D5—41no—
91C2×D3×D5—41no—
92C2×C3×C4×C5C2311yescharacteristic
93C2×C3×C4×C5C2311yescharacteristic
94#1220—22no—
95C2×Dic15C2311yescharacteristic
96D60—22no—
97D60—41no—
98C5×#67D414yescomplemented
99C2×C5×Dic3—22no—
100C2×C5×Dic3C2311yescharacteristic
101C2×C5×Dic3C2311yescharacteristic
102C2×C5×Dic3C2311yescharacteristic
103C4×C5×D3—22no—
104C3×C22⋊D5—22no—
105C2×C3×Dic5C2311yescharacteristic
106C3×D20—22no—
107C3×D20—41no—
108#1203—22no—
109#1202—22no—
110#1202—22no—
111#1202—22no—
112#1202—22no—
113#1202—22no—
114#1202—22no—
115#1202D414yescomplemented
116#1201—22no—
117#1200—22no—
118D5×Dic3—22no—
119C22×#67—51no—
120C22×#67—51no—
121C2×D3×D4—51no—
122C2×D3×D4—51no—
123C3×#120—51no—
124#940—51no—
125#939—51no—
126#939—51no—
127#937—51no—
128#884—51no—
129#884—51no—
130#882—51no—
131#882—51no—
132#67⋊C4—51no—
133#67⋊C4D511yescharacteristic, complemented
134C23×D5—31no—
135C23×D5—31no—
136C23×D5—31no—
137C23×D5C2×D311yescharacteristic, complemented
138C2×C5×D4—31no—
139C22×C4×C5—31no—
140C22×C4×C5—31no—
141C2×C22⋊D5—31no—
142C2×C22⋊D5—31no—
143C2×C22⋊D5—31no—
144C2×C22⋊D5C2×D311yescharacteristic, complemented
145D4×D5—64no—
146D4×D5—64no—
147C2×D20—31no—
148C2×D20—31no—
149C2×D20—31no—
150C2×D20—31no—
151C2×D20—31no—
152C2×D20—31no—
153C2×D20—31no—
154C2×D20—31no—
155C2×D20—31no—
156C2×D20—64no—
157C2×D20—64no—
158C2×D20C2×D311yescharacteristic, complemented
159C2×D20C2×D311yescharacteristic, complemented
160C2×C4×D5—31no—
161C2×C4×D5—31no—
162C5×C4⋊C4—31no—
163C5×C22⋊C4—31no—
164C5×C22⋊C4C2×D311yescharacteristic, complemented
165C42×C5—31no—
166#685—31no—
167#685—31no—
168#685—31no—
169#685—31no—
170#685C2×D311yescharacteristic, complemented
171#685C2×D311yescharacteristic, complemented
172C4⋊Dic5—31no—
173D42—151noSylow
174C22×C3×C5—21no—
175C22×C3×C5—21no—
176C22×C3×C5C2×D412yescomplemented
177C22×C3×C5C2411yescharacteristic
178C2×D15—21no—
179C2×D15—21no—
180C2×D15—21no—
181C2×D15—21no—
182C2×D15—21no—
183C2×D15—21no—
184C2×D15—21no—
185C2×D15—21no—
186C2×D15—44no—
187C2×D15C2×D412yescomplemented
188C2×C5×D3—21no—
189C2×C5×D3—21no—
190C2×C5×D3C2×D412yescomplemented
191C2×C3×D5—21no—
192C2×C3×D5—21no—
193C2×C3×D5—21no—
194C2×C3×D5—21no—
195C2×C3×D5—21no—
196C2×C3×D5—21no—
197C2×C3×D5—21no—
198C2×C3×D5—21no—
199C2×C3×D5—44no—
200C2×C3×D5C2×D412yescomplemented
201D3×D5—44no—
202D3×D5—44no—
203C3×C4×C5—21no—
204C3×C4×C5—21no—
205Dic15—21no—
206C3×Dic5—21no—
207C5×Dic3—21no—
208C5×Dic3C2×D412yescomplemented
209C5×Dic3C2×D412yescomplemented
210C24×C3—51no—
211C23×D3—51no—
212C23×D3—51no—
213C23×D3—51no—
214C2×C3×D4—51no—
215C2×C3×D4—51no—
216C2×C3×D4—51no—
217C2×#67—51no—
218C2×#67—51no—
219C2×#67—51no—
220C2×#67—51no—
221C2×#67—51no—
222C2×#67—51no—
223C2×#67—51no—
224C2×#67—51no—
225C2×#67—51no—
226C2×#67—104no—
227C2×#67—104no—
228C2×#67C2×D511yescharacteristic
229C22×Dic3—51no—
230C22×Dic3C2×D511yescharacteristic, complemented
231D3×D4—104no—
232D3×D4—104no—
233C2×D12—51no—
234C2×D12—51no—
235C2×D12—51no—
236C2×C4×D3—51no—
237C2×C4×D3C2×D511yescharacteristic, complemented
238C3×C22⋊C4—51no—
239C3×C22⋊C4—51no—
240C3×C22⋊C4C2×D511yescharacteristic, complemented
241#217—51no—
242(C2×D3)⋊C4—51no—
243(C2×D3)⋊C4—51no—
244(C2×D3)⋊C4—51no—
245(C2×D3)⋊C4C2×D511yescharacteristic, complemented
246Dic3⋊C4—51no—
247Dic3⋊C4C2×D511yescharacteristic, complemented
248C4×Dic3C2×D511yescharacteristic, complemented
249C23×C5—31no—
250C23×C5C22×D311yescharacteristic
251C22×D5—22no—
252C22×D5—22no—
253C22×D5—31no—
254C22×D5—31no—
255C22×D5—31no—
256C22×D5—41no—
257C22×D5—62no—
258C22×D5—62no—
259C22×D5—62no—
260C22×D5—62no—
261C22×D5—62no—
262C22×D5—62no—
263C22×D5—121no—
264C22×D5—121no—
265C22×D5—121no—
266C22×D5C22×D311yescharacteristic
267C22×D5C22×D311yescharacteristic
268C22×D5C22×D311yescharacteristic
269C5×D4—34no—
270C2×C4×C5—31no—
271C2×C4×C5—31no—
272C2×C4×C5—31no—
273C2×C4×C5—62no—
274C2×C4×C5—62no—
275C2×C4×C5C22×D311yescharacteristic
276C2×C4×C5C22×D311yescharacteristic
277C22⋊D5—22no—
278C22⋊D5—62no—
279C22⋊D5—62no—
280C22⋊D5—62no—
281C2×Dic5—31no—
282C2×Dic5C22×D311yescharacteristic
283D20—22no—
284D20—34no—
285D20—41no—
286D20—62no—
287D20—62no—
288D20—62no—
289D20—62no—
290D20—62no—
291D20—62no—
292D20—62no—
293D20—121no—
294C4×D5—62no—
295C4×D5—62no—
296C22×D4—151no—
297C22×D4—151no—
298C22×D4—151no—
299C22×D4—151no—
300#127—151no—
301#121—151no—
302#121—151no—
303#121—151no—
304#121—151no—
305#120—51no—
306#120—151no—
307#120—151no—
308#120—151no—
309C4×D4—31no—
310C4×D4—151no—
311C2×C3×C5—22no—
312C2×C3×C5C22×D411yescharacteristic
313C2×C3×C5C22×D411yescharacteristic
314C2×C3×C5H(5,𝔽2)11yescharacteristic
315D15—22no—
316D15—41no—
317D15—41no—
318C3×D5—22no—
319C3×D5—41no—
320C3×D5—41no—
321C5×D3—22no—
322C23×C3—51no—
323C23×C3—51no—
324C23×C3—51no—
325C23×C3—102no—
326C23×C3—102no—
327C23×C3—201no—
328C23×C3C22×D511yescharacteristic
329C22×D3—51no—
330C22×D3—51no—
331C22×D3—51no—
332C22×D3—102no—
333C22×D3—102no—
334C22×D3—102no—
335C22×D3—102no—
336C22×D3—102no—
337C22×D3—102no—
338C22×D3—201no—
339C22×D3—201no—
340C22×D3—201no—
341C22×D3C22×D511yescharacteristic
342C3×D4—102no—
343C3×D4—102no—
344C3×D4—201no—
345C2×C3×C4—51no—
346C2×C3×C4C22×D511yescharacteristic
347C2×C3×C4C22×D511yescharacteristic
348#67—54no—
349#67—102no—
350#67—102no—
351#67—102no—
352#67—102no—
353#67—102no—
354#67—102no—
355#67—102no—
356#67—102no—
357#67D2014yescomplemented
358C2×Dic3—22no—
359C2×Dic3—51no—
360C2×Dic3—102no—
361C2×Dic3C22×D511yescharacteristic
362C2×Dic3C22×D511yescharacteristic
363C2×Dic3C22×D511yescharacteristic
364D12—102no—
365D12—102no—
366D12—201no—
367C4×D3—22no—
368C4×D3—102no—
369C22×C5—21no—
370C22×C5—21no—
371C22×C5—32no—
372C22×C5—61no—
373C22×C5—61no—
374C22×C5D3×D412yescomplemented
375C22×C5C23×D311yescharacteristic
376C2×D5—21no—
377C2×D5—21no—
378C2×D5—21no—
379C2×D5—21no—
380C2×D5—21no—
381C2×D5—21no—
382C2×D5—21no—
383C2×D5—21no—
384C2×D5—32no—
385C2×D5—44no—
386C2×D5—61no—
387C2×D5—61no—
388C2×D5—61no—
389C2×D5—61no—
390C2×D5—61no—
391C2×D5—61no—
392C2×D5—61no—
393C2×D5—61no—
394C2×D5—124no—
395C2×D5—124no—
396C2×D5—124no—
397C2×D5D3×D412yescomplemented
398C4×C5—21no—
399C4×C5—21no—
400C4×C5—32no—
401C4×C5—32no—
402C4×C5—61no—
403Dic5—21no—
404Dic5—61no—
405C24—51no—
406C24—151no—
407C24—151no—
408C24—151no—
409C2×D4—31no—
410C2×D4—51no—
411C2×D4—51no—
412C2×D4—51no—
413C2×D4—151no—
414C2×D4—151no—
415C2×D4—151no—
416C2×D4—151no—
417C2×D4—151no—
418C2×D4—151no—
419C2×D4—151no—
420C2×D4—151no—
421C2×D4—151no—
422C2×D4—151no—
423C2×D4—151no—
424C2×D4—151no—
425C2×D4—304no—
426C2×D4—304no—
427C2×D4—304no—
428C2×D4—304no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—151no—
432C22×C4—151no—
433C4⋊C4—31no—
434C4⋊C4—151no—
435C22⋊C4—31no—
436C22⋊C4—51no—
437C22⋊C4—51no—
438C22⋊C4—151no—
439C22⋊C4—151no—
440C22⋊C4—151no—
441C22⋊C4—151no—
442C22⋊C4D3×D511yescharacteristic, complemented
443C42—31no—
444C3×C5D4211yescharacteristic, complemented, Hall
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—52no—
448C22×C3—101no—
449C22×C3—101no—
450C22×C3—101no—
451C22×C3—101no—
452C22×C3—101no—
453C22×C3—101no—
454C22×C3—101no—
455C22×C3—101no—
456C22×C3—204no—
457C22×C3C2×D2012yescomplemented
458C22×C3C23×D511yescharacteristic
459C2×D3—21no—
460C2×D3—21no—
461C2×D3—52no—
462C2×D3—101no—
463C2×D3—101no—
464C2×D3—101no—
465C2×D3—101no—
466C2×D3—101no—
467C2×D3—101no—
468C2×D3—101no—
469C2×D3—101no—
470C2×D3—204no—
471C2×D3—204no—
472C2×D3—204no—
473C2×D3C2×D2012yescomplemented
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—101no—
477Dic3—21no—
478Dic3—101no—
479Dic3C2×D2012yescomplemented
480Dic3D4×D512yescomplemented
481C2×C5—22no—
482C2×C5—62no—
483C2×C5C2×D3×D411yescharacteristic
484C2×C5C2×D3×D411yescharacteristic
485C2×C5#100411yescharacteristic
486D5—22no—
487D5—41no—
488D5—41no—
489D5—62no—
490D5—121no—
491D5—121no—
492C23—31no—
493C23—51no—
494C23—51no—
495C23—51no—
496C23—102no—
497C23—102no—
498C23—151no—
499C23—151no—
500C23—151no—
501C23—201no—
502C23—302no—
503C23—302no—
504C23—302no—
505C23—302no—
506C23—302no—
507C23—302no—
508C23—601no—
509C23—601no—
510C23—601no—
511C23C2×D3×D511yescharacteristic
512D4—34no—
513D4—102no—
514D4—102no—
515D4—154no—
516D4—201no—
517D4—302no—
518D4—302no—
519D4—302no—
520D4—302no—
521D4—302no—
522D4—302no—
523D4—302no—
524D4—302no—
525D4—302no—
526D4—302no—
527D4—601no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—31no—
531C2×C4—51no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—151no—
535C2×C4—302no—
536C2×C4—302no—
537C2×C4C2×D3×D511yescharacteristic
538C2×C4C2×D3×D511yescharacteristic
539C2×C3—22no—
540C2×C3—102no—
541C2×C3—201no—
542C2×C3—201no—
543C2×C3C22×D2011yescharacteristic
544C2×C3C2×D4×D511yescharacteristic
545C2×C3#414911yescharacteristic
546D3—22no—
547D3—102no—
548D3—201no—
549D3—201no—
550C5#559011yescharacteristic, complemented, Sylow
551C22—21no—
552C22—21no—
553C22—32no—
554C22—52no—
555C22—61no—
556C22—61no—
557C22—101no—
558C22—101no—
559C22—101no—
560C22—101no—
561C22—101no—
562C22—101no—
563C22—101no—
564C22—101no—
565C22—152no—
566C22—204no—
567C22—301no—
568C22—301no—
569C22—301no—
570C22—301no—
571C22—301no—
572C22—301no—
573C22—301no—
574C22—301no—
575C22—604no—
576C22—604no—
577C22—604no—
578C22D3×D2012yescomplemented
579C22C22×D3×D511yescharacteristic
580C4—21no—
581C4—21no—
582C4—32no—
583C4—32no—
584C4—61no—
585C4—101no—
586C4—301no—
587C3D4×D2011yescharacteristic, complemented, Sylow
588C2—22no—
589C2—62no—
590C2—102no—
591C2—201no—
592C2—201no—
593C2—302no—
594C2—601no—
595C2—601no—
596C2C2×D3×D2011yescharacteristic
597C2D3×D4×D511yescharacteristic
598C2#9202611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C22⋊C4C22
hypercenterC22⋊C4D3×D5
hypocenterC3×C5D42
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#6790C2semidirect
2. C2×D5×#67C2semidirect
3. C22×#1202C2semidirect
4. C2×D3×D20C2semidirect
5. #91771C2semidirect
6. C5×#67⋊C4C2semidirect
7. C3×#4028C2semidirect
8. #91569C2semidirect
9. #91476C2semidirect
10. #91474C2semidirect
11. #91459C2semidirect
12. #91458C2semidirect
13. #91454C2semidirect
14. #91409C2semidirect
15. #91395C2semidirect
17. C23×D15C22semidirect
18. C23×C3×D5C22semidirect
19. C22×D3×D5C22semidirect
20. C22×D3×D5C22semidirect
21. C2×#1220C22semidirect
22. C2×D60C22semidirect
23. C2×D60C22semidirect
25. C22×C5×Dic3C22semidirect
26. C2×C4×C5×D3C22semidirect
27. C2×C3×C22⋊D5C22semidirect
28. C2×C3×D20C22semidirect
29. C2×C3×D20C22semidirect
32. C2×#1203C22semidirect
35. C2×#1202C22semidirect
36. C2×#1202C22semidirect
37. C2×#1202C22semidirect
38. C2×#1202C22semidirect
39. C2×#1202C22semidirect
40. C2×#1202C22semidirect
41. C2×#1201C22semidirect
42. C2×#1200C22semidirect
43. C2×D5×Dic3C22semidirect
45. C3×C5×C22⋊C4C22semidirect
46. #6714C22semidirect
47. #6714C22semidirect
48. C5×(C2×D3)⋊C4C22semidirect
49. C5×Dic3⋊C4C22semidirect
50. C4×C5×Dic3C22semidirect
51. C3×#685C22semidirect
52. C3×#685C22semidirect
53. Dic3⋊Dic5C22semidirect
54. #6667C22semidirect
55. #6665C22semidirect
69. #4028D3semidirect
98. C5×#67D4semidirect
115. #1202D4semidirect
133. #67⋊C4D5semidirect
137. C23×D5C2×D3semidirect
144. C2×C22⋊D5C2×D3semidirect
158. C2×D20C2×D3semidirect
159. C2×D20C2×D3semidirect
164. C5×C22⋊C4C2×D3semidirect
170. #685C2×D3semidirect
171. #685C2×D3semidirect
176. C22×C3×C5C2×D4semidirect
187. C2×D15C2×D4semidirect
190. C2×C5×D3C2×D4semidirect
200. C2×C3×D5C2×D4semidirect
208. C5×Dic3C2×D4semidirect
209. C5×Dic3C2×D4semidirect
230. C22×Dic3C2×D5semidirect
237. C2×C4×D3C2×D5semidirect
240. C3×C22⋊C4C2×D5semidirect
245. (C2×D3)⋊C4C2×D5semidirect
247. Dic3⋊C4C2×D5semidirect
248. C4×Dic3C2×D5semidirect
357. #67D20semidirect
374. C22×C5D3×D4semidirect
397. C2×D5D3×D4semidirect
442. C22⋊C4D3×D5semidirect
444. C3×C5D42semidirect
457. C22×C3C2×D20semidirect
473. C2×D3C2×D20semidirect
479. Dic3C2×D20semidirect
480. Dic3D4×D5semidirect
550. C5#5590semidirect
578. C22D3×D20semidirect
587. C3D4×D20semidirect

Maximal subgroups

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

subgroupindex
C2×#67902
C2×D5×#672
C22×#12022
C2×D3×D202
#917712
C5×#67⋊C42
C3×#40282
#915692
#914762
#914742
#914592
#914582
#914542
#914092
#913952
D4×D203
#55905