Finite Group Databasebeta

← #14415540

Subgroups — #14415540

Order 1920.

1004 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
1#11754564C211yesmaximal, characteristic
2C2×#91290C211yesmaximal, characteristic, complemented
3C2×D5×#208C211yesmaximal, characteristic, complemented
4#11751211C211yesmaximal, characteristic, complemented
5C5×#5101C211yesmaximal, characteristic, complemented
6C2×C3×#3952C211yesmaximal, characteristic, complemented
7#11748439C211yesmaximal, characteristic, complemented
8#11748046C212yesmaximal, complemented
9#11748042C212yesmaximal, complemented
10#11747728C212yesmaximal, complemented
11#11747716C212yesmaximal, complemented
12#10610286—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×C3×C4×D5C2211yescharacteristic
15C22×C3×C5×C8C2211yescharacteristic
16C2×#6710C2211yescharacteristic
17C5×#926C2211yescharacteristic
18C2×C5×#208C2211yescharacteristic
19C22×C3×#168C2211yescharacteristic
20#91564C412yes—
21#91546C412yes—
22#91450C411yescharacteristic
23#91450C411yescharacteristic
24C4×#1202C2212yes—
25C4×#1202C2212yes—
26#91360C411yescharacteristic
27#91360C411yescharacteristic
28C2×#6651C2212yescomplemented
29C2×#6649C2212yes—
30#91290—24no—
31C2×#6648C2212yescomplemented
32D5×#208—24no—
33C2×D5×C3⋊C8C2212yes—
34#91104C2212yescomplemented
35#91104C2212yescomplemented
36C5×#815C2212yescomplemented
37C5×C8×Dic3C2212yescomplemented
38C3×#3952C2211yescharacteristic
39C3×#3952C2211yescharacteristic
40C3×#3952C2211yescharacteristic
41C3×#3952C2211yescharacteristic
42Dic3⋊#168C2212yescomplemented
43Dic3⋊#168C2212yescomplemented
44#78596—51nomaximal, Hall
45C2×#4053—31no—
46C2×D5×M16—31no—
47#65858—31no—
48C5×#405—31no—
49#65399—31no—
50C2×#3952D311yescharacteristic, complemented
51#65288—31no—
52#64995—32no—
53#64995—32no—
54#64977—32no—
55C8×D20—32no—
56C23×C3×D5C2×C411yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C2×#1220C2×C412yes—
59#6817—22no—
60#6817—22no—
61C2×D60C2×C411yescharacteristic
62C2×C4×D15C2312yes—
63C2×Dic30C2×C411yescharacteristic
64C22×C3×Dic5C2×C411yescharacteristic
65C2×C3×C4×D5—22no—
66C2×C3×C4×D5—22no—
67C2×C3×C4×D5C2×C411yescharacteristic
68C2×C3×C4×D5C2×C411yescharacteristic
69C2×C3×C4×D5C2311yescharacteristic
70C2×C3×C4×D5C2311yescharacteristic
71C2×#1202C2×C412yes—
72C2×#1202C2×C412yes—
73C2×C3×C5×C8—22no—
74C2×C3×C5×C8C2311yescharacteristic
75C2×C3×C5×C8C2311yescharacteristic
76#6710—22no—
77C2×C3⋊#168C2312yes—
78C5×#217C2×C412yes—
79C5×C4⋊Dic3C2×C411yescharacteristic
80C5×C4⋊Dic3C2×C411yescharacteristic
81C4×C5×Dic3C2312yes—
82C5×#208D414yescomplemented
83C2×C5×C3⋊C8C2312yes—
84C2×C3×#168—22no—
85C2×C3×#168C2311yescharacteristic
86C2×C3×#168C2311yescharacteristic
87Dic3⋊Dic5C2×C412yes—
88Dic3⋊Dic5C2×C412yes—
89#6667C2×C412yes—
90#6667C2×C412yes—
91#6665C2×C412yes—
92#6665C2×C412yes—
93#6651—24no—
94#6649—24no—
95#6648—24no—
96D5×C3⋊C8—24no—
97#5791—51no—
98C22×#208—51no—
99C2×#901—51no—
100C2×C3×C22⋊C8—51no—
101#5110—51no—
102#5102—51no—
103#5101D511yescharacteristic, complemented
104#4730—52no—
105#4729—52no—
106#4728—52no—
107#67⋊C8—52no—
108C2×#709—31no—
109C22×C4×D5C2×D311yescharacteristic
110C2×C5×M16—31no—
111C22×C5×C8C2×D311yescharacteristic
112C5×#117—31no—
113C2×#681—31no—
114C22×#168C2×D311yescharacteristic
115#4053—64no—
116D5×M16—64no—
117C2×#676—32no—
118C2×#676—32no—
119C2×C8×D5—32no—
120C2×C8×D5—32no—
121#4043—31no—
122#4043—31no—
123#4041—31no—
124#4041—31no—
125#4032—32no—
126#4028—32no—
127C4×D20—32no—
128C4×D20—32no—
129C5×#97—32no—
130C4×C5×C8—32no—
131#3952—32no—
132#3952—32no—
133#3952C2×D311yescharacteristic
134#3952C2×D311yescharacteristic
135#3952C2×D311yescharacteristic
136#3952C2×D311yescharacteristic
137C4⋊#168—32no—
138C4⋊#168—32no—
139#2933—151noSylow
140C23×C3×C5C22×C411yescharacteristic
141C22×D15C22×C412yes—
142C22×C3×D5—22no—
143C22×C3×D5—22no—
144C22×C3×D5—41no—
145C22×C3×D5C22×C411yescharacteristic
146C22×C3×D5C22×C411yescharacteristic
147C2×C3×C4×C5C22×C411yescharacteristic
148C2×C3×C4×C5C2×D412yes—
149C2×C3×C4×C5#4112yes—
150C2×C3×C4×C5C2411yescharacteristic
151#1220—24no—
152C2×Dic15C22×C412yes—
153D60—22no—
154C4×D15—22no—
155C4×D15—22no—
156Dic30—22no—
157C2×C5×Dic3C22×C412yes—
158C2×C5×Dic3C22×C412yes—
159C2×C3×Dic5—22no—
160C2×C3×Dic5C22×C411yescharacteristic
161C2×C3×Dic5C22×C411yescharacteristic
162C3×C4×D5—22no—
163C3×C4×D5—22no—
164C3×C4×D5—21no—
165C3×C4×D5—21no—
166C3×C4×D5—21no—
167C3×C4×D5—21no—
168#1202—24no—
169#1202—42no—
170C3×C5×C8—21no—
171C3×C5×C8—21no—
172C3⋊#168—22no—
173C3×#168—21no—
174C3×#168—21no—
175C5×C3⋊C8C2×D414yes—
176C23×C3×C4—51no—
177C2×#235—51no—
178C22×C3×C8—51no—
179C22×C3×C8C2×D511yescharacteristic
180#953—52no—
181#930—51no—
182#930—51no—
183#929—52no—
184C4×#67—52no—
185C4×#67—52no—
186#926C2×D511yescharacteristic
187#924—51no—
188#924—51no—
189C2×#208—51no—
190C2×#208—52no—
191C2×#208—104no—
192C2×#208C2×D511yescharacteristic
193C22×C3⋊C8—52no—
194#901—104no—
195C2×ΓL(1,𝔽25)—52no—
196C2×D3×C8—52no—
197C3×C22⋊C8—51no—
198C3×C22⋊C8—51no—
199C3×C22⋊C8—51no—
200C3×C22⋊C8—51no—
201(C2×D3)⋊C8—52no—
202(C2×D3)⋊C8—52no—
203#815C2×D512yescomplemented
204Dic3⋊C8—52no—
205Dic3⋊C8—52no—
206C8×Dic3C2×D512yescomplemented
207C23×D5C4×D311yescharacteristic
208C22×C4×C5C22×D311yescharacteristic
209C2×C22⋊D5—32no—
210C22×Dic5C4×D311yescharacteristic
211#709—62no—
212#709—62no—
213C2×D20—32no—
214C2×D20—31no—
215C2×D20—32no—
216C2×C4×D5—22no—
217C2×C4×D5—22no—
218C2×C4×D5—32no—
219C2×C4×D5C4×D311yescharacteristic
220C2×C4×D5C4×D311yescharacteristic
221C2×C4×D5C22×D311yescharacteristic
222C2×C4×D5C22×D311yescharacteristic
223C2×Dic10—31no—
224C5×M16—34no—
225C2×C5×C8—22no—
226C2×C5×C8—32no—
227C2×C5×C8C22×D311yescharacteristic
228C2×C5×C8C22×D311yescharacteristic
229C5×C4⋊C4—31no—
230C5×C4⋊C4—31no—
231C5×C22⋊C4—32no—
232C42×C5—32no—
233#685—32no—
234#685—32no—
235#685—32no—
236#685—32no—
237C4⋊Dic5—32no—
238C4⋊Dic5—32no—
239#681—62no—
240C2×#168—22no—
241C2×#168—32no—
242C2×#168C22×D311yescharacteristic
243C2×#168C22×D311yescharacteristic
244#676—64no—
245#676—64no—
246C8×D5—64no—
247C8×D5—64no—
248C2×#131—151no—
249C22×M16—151no—
250#525—151no—
251#436—152no—
252#435—152no—
253#435—152no—
254C8×D4—152no—
255#424—151no—
256#408—151no—
257C2×C22⋊C8—51no—
258#405—31no—
259C22×C3×C5—21no—
260C22×C3×C5—21no—
261C22×C3×C5C4×D412yes—
262C22×C3×C5C23×C411yescharacteristic
263C2×D15—22no—
264C2×D15—22no—
265C2×D15—22no—
266C2×C3×D5—21no—
267C2×C3×D5—21no—
268C2×C3×D5—21no—
269C2×C3×D5—21no—
270C2×C3×D5—21no—
271C2×C3×D5—44no—
272C2×C3×D5#13112yescomplemented
273C3×C4×C5C4×D412yes—
274C3×C4×C5C22×D411yescharacteristic
275C3×C4×C5C2×#4111yescharacteristic
276Dic15—22no—
277C3×Dic5—21no—
278C3×Dic5—21no—
279C5×Dic3—22no—
280C5×Dic3—22no—
281C24×C3—51no—
282C22×C3×C4—51no—
283C22×C3×C4—51no—
284C22×C3×C4—51no—
285C22×C3×C4—51no—
286C22×C3×C4—51no—
287C22×C3×C4—102no—
288C22×C3×C4—102no—
289C22×C3×C4C22×D511yescharacteristic
290C2×#67—52no—
291C2×#67—52no—
292C2×#67—52no—
293#235—102no—
294#235—102no—
295C2×D12—51no—
296C2×C4×D3—52no—
297C2×Dic6—51no—
298C2×C3×C8—22no—
299C2×C3×C8—51no—
300C2×C3×C8—51no—
301C2×C3×C8—102no—
302C2×C3×C8C22×D511yescharacteristic
303C2×C3×C8C22×D511yescharacteristic
304#217—52no—
305#217—52no—
306#217C4×D512yes—
307(C2×D3)⋊C4—52no—
308(C2×D3)⋊C4—52no—
309C4⋊Dic3C4×D511yescharacteristic
310C4⋊Dic3C4×D511yescharacteristic
311Dic3⋊C4—52no—
312Dic3⋊C4—52no—
313C4×Dic3C22×D512yes—
314#208—102no—
315#208—104no—
316#208D2014yescomplemented
317C2×C3⋊C8—52no—
318C2×C3⋊C8—104no—
319C2×C3⋊C8C22×D512yes—
320ΓL(1,𝔽25)—104no—
321D3×C8—104no—
322C23×C5C2×C4×D311yescharacteristic
323C22×D5—22no—
324C22×D5—22no—
325C22×D5—32no—
326C22×D5—41no—
327C22×D5C2×C4×D311yescharacteristic
328C22×D5C2×C4×D311yescharacteristic
329C2×C4×C5—32no—
330C2×C4×C5—32no—
331C2×C4×C5C2×C4×D311yescharacteristic
332C2×C4×C5D3×D412yes—
333C2×C4×C5#23712yes—
334C2×C4×C5C23×D311yescharacteristic
335C22⋊D5—64no—
336C2×Dic5—22no—
337C2×Dic5—32no—
338C2×Dic5C2×C4×D311yescharacteristic
339C2×Dic5C2×C4×D311yescharacteristic
340D20—64no—
341D20—62no—
342D20—122no—
343C4×D5—22no—
344C4×D5—22no—
345C4×D5—21no—
346C4×D5—21no—
347C4×D5—21no—
348C4×D5—21no—
349C4×D5—62no—
350C4×D5—62no—
351Dic10—62no—
352C5×C8—21no—
353C5×C8—21no—
354C5×C8—34no—
355#168—21no—
356#168—21no—
357#168—62no—
358C2×#41—151no—
359C23×C4—51no—
360#131—304no—
361C2×M16—31no—
362C2×M16—152no—
363C2×M16—152no—
364C2×M16—151no—
365C2×M16—304no—
366C22×C8—51no—
367C22×C8—152no—
368C22×C8—152no—
369C22×C8D3×D511yescharacteristic
370#123—152no—
371C22⋊Q8—151no—
372C22⋊Q8—151no—
373#121—151no—
374#121—151no—
375#120—152no—
376C4×D4—152no—
377C4×D4—152no—
378#117—31no—
379C4⋊C8—152no—
380C4⋊C8—152no—
381C22⋊C8—51no—
382C22⋊C8—51no—
383C22⋊C8—51no—
384C22⋊C8—51no—
385C22⋊C8—152no—
386C22⋊C8—152no—
387#97—32no—
388C4×C8—32no—
389C2×C3×C5—22no—
390C2×C3×C5C2×C4×D411yescharacteristic
391C2×C3×C5C2×#13111yescharacteristic
392C2×C3×C5#56811yescharacteristic
393D15—42no—
394C3×D5—22no—
395C3×D5—41no—
396C23×C3—51no—
397C23×C3—51no—
398C23×C3—102no—
399C23×C3—102no—
400C23×C3—201no—
401C23×C3C2×C4×D511yescharacteristic
402C22×D3—52no—
403C2×C3×C4—51no—
404C2×C3×C4—51no—
405C2×C3×C4—102no—
406C2×C3×C4—101no—
407C2×C3×C4—101no—
408C2×C3×C4—101no—
409C2×C3×C4—101no—
410C2×C3×C4—102no—
411C2×C3×C4—102no—
412C2×C3×C4C2×C4×D511yescharacteristic
413C2×C3×C4C2×D2012yes—
414C2×C3×C4#70912yes—
415C2×C3×C4C23×D511yescharacteristic
416#67—104no—
417#67—104no—
418#67—202no—
419C2×Dic3—52no—
420C2×Dic3C2×C4×D512yes—
421C2×Dic3C2×C4×D512yes—
422D12—102no—
423C4×D3—102no—
424C4×D3—102no—
425Dic6—102no—
426C3×C8—21no—
427C3×C8—21no—
428C3×C8—101no—
429C3×C8—101no—
430C3⋊C8—102no—
431C3⋊C8C2×D2014yes—
432C22×C5—21no—
433C22×C5—21no—
434C22×C5#67⋊C412yes—
435C22×C5C22×C4×D311yescharacteristic
436C2×D5—21no—
437C2×D5—21no—
438C2×D5—21no—
439C2×D5—21no—
440C2×D5—21no—
441C2×D5—44no—
442C2×D5—62no—
443C2×D5—62no—
444C2×D5—62no—
445C2×D5#90112yescomplemented
446C4×C5—62no—
447C4×C5—62no—
448C4×C5#67⋊C412yes—
449C4×C5C2×D3×D411yescharacteristic
450C4×C5C2×#23711yescharacteristic
451Dic5—21no—
452Dic5—21no—
453Dic5—62no—
454C24—51no—
455#41—302no—
456#41—302no—
457C2×Q8—151no—
458C2×D4—152no—
459C2×D4—152no—
460C2×D4—152no—
461C2×D4—151no—
462C22×C4—51no—
463C22×C4—51no—
464C22×C4—51no—
465C22×C4—51no—
466C22×C4—51no—
467C22×C4—102no—
468C22×C4—102no—
469C22×C4—152no—
470C22×C4C2×D3×D511yescharacteristic
471M16—34no—
472M16—304no—
473M16—304no—
474M16—302no—
475C2×C8—22no—
476C2×C8—32no—
477C2×C8—51no—
478C2×C8—51no—
479C2×C8—102no—
480C2×C8—152no—
481C2×C8—304no—
482C2×C8—304no—
483C2×C8C2×D3×D511yescharacteristic
484C2×C8C2×D3×D511yescharacteristic
485C4⋊C4—31no—
486C4⋊C4—31no—
487C4⋊C4—152no—
488C4⋊C4—152no—
489C22⋊C4—32no—
490C22⋊C4—152no—
491C22⋊C4—152no—
492C22⋊C4—152no—
493C22⋊C4—152no—
494C42—32no—
495C3×C5#293311yescharacteristic, complemented, Hall
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—52no—
499C22×C3—101no—
500C22×C3—101no—
501C22×C3—101no—
502C22×C3—101no—
503C22×C3—101no—
504C22×C3—204no—
505C22×C3C4×D2012yes—
506C22×C3C22×C4×D511yescharacteristic
507C2×D3—102no—
508C2×D3—102no—
509C2×D3—102no—
510C3×C4—101no—
511C3×C4—101no—
512C3×C4C4×D2012yes—
513C3×C4C22×D2011yescharacteristic
514C3×C4C2×#70911yescharacteristic
515Dic3—22no—
516Dic3—22no—
517Dic3—102no—
518C2×C5—22no—
519C2×C5C2×#67⋊C411yescharacteristic
520C2×C5C2×#90111yescharacteristic
521C2×C5#574511yescharacteristic
522D5—22no—
523D5—41no—
524D5—122no—
525C23—51no—
526C23—51no—
527C23—102no—
528C23—102no—
529C23—152no—
530C23—201no—
531C23C4×D3×D511yescharacteristic
532Q8—302no—
533D4—304no—
534D4—304no—
535D4—302no—
536D4—602no—
537C2×C4—32no—
538C2×C4—32no—
539C2×C4—51no—
540C2×C4—51no—
541C2×C4—102no—
542C2×C4—102no—
543C2×C4—102no—
544C2×C4—101no—
545C2×C4—101no—
546C2×C4—101no—
547C2×C4—101no—
548C2×C4—152no—
549C2×C4—302no—
550C2×C4—302no—
551C2×C4C4×D3×D511yescharacteristic
552C2×C4D3×D2012yes—
553C2×C4#677912yes—
554C2×C4C22×D3×D511yescharacteristic
555C8—21no—
556C8—21no—
557C8—34no—
558C8—101no—
559C8—101no—
560C8—302no—
561C2×C3—22no—
562C2×C3—102no—
563C2×C3—201no—
564C2×C3C2×C4×D2011yescharacteristic
565C2×C3D5×#13111yescharacteristic
566C2×C3#6608411yescharacteristic
567D3—202no—
568C5#7859611yescharacteristic, complemented, Sylow
569C22—21no—
570C22—21no—
571C22—52no—
572C22—101no—
573C22—101no—
574C22—101no—
575C22—101no—
576C22—101no—
577C22—204no—
578C22—302no—
579C22—302no—
580C22—302no—
581C22#9139512yes—
582C22C2×C4×D3×D511yescharacteristic
583C4—62no—
584C4—62no—
585C4—101no—
586C4—101no—
587C4—302no—
588C4#9139512yes—
589C4C2×D3×D2011yescharacteristic
590C4C2×#677911yescharacteristic
591C3#1061028611yescharacteristic, complemented, Sylow
592C2—22no—
593C2—102no—
594C2—201no—
595C2—602no—
596C2D5×#90111yescharacteristic
597C2#1175394911yescharacteristic
598C2C2×#9139511yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D3×D5
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C5×C8C22
prime radical#11754564C2
cosocleC2×C3×C4×C5C24
hypercenterC22×C8D3×D5
hypocenterC3×C5#2933
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C2×#91290C2semidirect
3. C2×D5×#208C2semidirect
4. #11751211C2semidirect
5. C5×#5101C2semidirect
6. C2×C3×#3952C2semidirect
7. #11748439C2semidirect
8. #11748046C2semidirect
9. #11748042C2semidirect
10. #11747728C2semidirect
11. #11747716C2semidirect
28. C2×#6651C22semidirect
31. C2×#6648C22semidirect
34. #91104C22semidirect
35. #91104C22semidirect
36. C5×#815C22semidirect
37. C5×C8×Dic3C22semidirect
42. Dic3⋊#168C22semidirect
43. Dic3⋊#168C22semidirect
50. C2×#3952D3semidirect
82. C5×#208D4semidirect
103. #5101D5semidirect
203. #815C2×D5semidirect
206. C8×Dic3C2×D5semidirect
272. C2×C3×D5#131semidirect
316. #208D20semidirect
445. C2×D5#901semidirect
495. C3×C5#2933semidirect
568. C5#78596semidirect
591. C3#10610286semidirect

Maximal subgroups

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

subgroupindex
#117545642
C2×#912902
C2×D5×#2082
#117512112
C5×#51012
C2×C3×#39522
#117484392
#117480462
#117480422
#117477282
#117477162
#106102863
#785965