Finite Group Databasebeta

← #14415537

Subgroups — #14415537

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
1C2×#91796C211yesmaximal, characteristic, complemented
2C2×C3×D5×M16C211yesmaximal, characteristic, complemented
3#11754564C211yesmaximal, characteristic
4C5×#5122C211yesmaximal, characteristic, complemented
5#11748441C211yesmaximal, characteristic, complemented
6#11748350C211yesmaximal, characteristic, complemented
7C2×#90955C211yesmaximal, characteristic, complemented
8#11747725C212yesmaximal, complemented
9#11747715C212yesmaximal, complemented
10#11747705C212yesmaximal, complemented
11C8×#1202C212yesmaximal, complemented
12#10610286—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×C3×C4×D5C2211yescharacteristic
15C2×C3×C5×M16C2211yescharacteristic
16C22×C3⋊#168C2211yescharacteristic
17#91796—24no—
18C2×#6705C2212yescomplemented
19C2×C8×D15C2212yescomplemented
20C5×#926C2211yescharacteristic
21C22×C5×C3⋊C8C2211yescharacteristic
22C2×C3×#681C2211yescharacteristic
23C3×D5×M16—24no—
24C2×C3×#676C2212yes—
25C2×C3×C8×D5C2212yes—
26#91564C412yes—
27#91546C412yes—
28#91450C411yescharacteristic
29#91450C411yescharacteristic
30C4×#1202C2212yes—
31C4×#1202C2212yes—
32#91360C411yescharacteristic
33#91360C411yescharacteristic
34C5×#815C2212yescomplemented
35C5×C8×Dic3C2212yescomplemented
36Dic3⋊#168C2212yescomplemented
37Dic3⋊#168C2212yescomplemented
38#90957C2212yescomplemented
39#90957C2212yescomplemented
40#90955C2211yescharacteristic
41#90955C2211yescharacteristic
42#90955C2211yescharacteristic
43#90955C2211yescharacteristic
44#83757—51nomaximal, Hall
45C2×#4053—31no—
46C2×D5×M16D311yescharacteristic, complemented
47#65858—31no—
48C5×#405—31no—
49#65399—31no—
50C2×#3952—31no—
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—
73C3×C5×M16D414yescomplemented
74C2×C3×C5×C8C2312yes—
75C2×C3⋊#168—22no—
76C2×C3⋊#168C2311yescharacteristic
77C2×C3⋊#168C2311yescharacteristic
78#6705—24no—
79C8×D15—24no—
80C5×#217C2×C412yes—
81C5×C4⋊Dic3C2×C411yescharacteristic
82C5×C4⋊Dic3C2×C411yescharacteristic
83C4×C5×Dic3C2312yes—
84C2×C5×C3⋊C8—22no—
85C2×C5×C3⋊C8C2311yescharacteristic
86C2×C5×C3⋊C8C2311yescharacteristic
87C3×#681—22no—
88C2×C3×#168C2312yes—
89C3×#676—24no—
90C3×C8×D5—24no—
91Dic3⋊Dic5C2×C412yes—
92Dic3⋊Dic5C2×C412yes—
93#6667C2×C412yes—
94#6667C2×C412yes—
95#6665C2×C412yes—
96#6665C2×C412yes—
97C22×C3×M16—51no—
98#5791—51no—
99C2×#907—51no—
100C2×#830—51no—
101#5129—51no—
102#5128—52no—
103#5127—52no—
104#5122D511yescharacteristic, complemented
105#5119—51no—
106#5111—52no—
107C8×#67—52no—
108C2×#709—31no—
109C22×C4×D5C2×D311yescharacteristic
110C2×C5×M16C2×D311yescharacteristic
111C22×C5×C8—31no—
112C5×#117—31no—
113C2×#681C2×D311yescharacteristic
114C22×#168—31no—
115#4053—64no—
116D5×M16—24no—
117C2×#676—32no—
118C2×#676C2×D312yes—
119C2×C8×D5—32no—
120C2×C8×D5C2×D312yes—
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—31no—
132#3952—31no—
133#3952—31no—
134#3952—31no—
135#3952—32no—
136#3952—32no—
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×C8C2×D414yes—
171C3⋊#168—21no—
172C3⋊#168—21no—
173C3×#168—22no—
174C5×C3⋊C8—21no—
175C5×C3⋊C8—21no—
176C23×C3×C4—51no—
177C2×#235—51no—
178C2×C3×M16—51no—
179C2×C3×M16—52no—
180C2×C3×M16—104no—
181C2×C3×M16C2×D511yescharacteristic
182C22×C3×C8—52no—
183#953—52no—
184#930—51no—
185#930—51no—
186#929—52no—
187C4×#67—52no—
188C4×#67—52no—
189#926C2×D511yescharacteristic
190#924—51no—
191#924—51no—
192C22×C3⋊C8—51no—
193C22×C3⋊C8C2×D511yescharacteristic
194#907—104no—
195C2×ΓL(1,𝔽25)—52no—
196C2×D3×C8—52no—
197#830—51no—
198#830—51no—
199#830—51no—
200#830—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#6714yescomplemented
225C2×C5×C8—31no—
226C2×C5×C8—31no—
227C2×C5×C8—62no—
228C2×C5×C8C22×D312yes—
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—22no—
240C2×#168—31no—
241C2×#168—31no—
242C2×#168—62no—
243C2×#168C22×D312yes—
244#676—24no—
245#676—64no—
246C8×D5—24no—
247C8×D5—64no—
248C2×#131—151no—
249C22×M16—51no—
250#525—151no—
251#436—152no—
252#435—152no—
253#435—152no—
254C8×D4—152no—
255#424—151no—
256#408—151no—
257C2×C22⋊C8—151no—
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—
298C3×M16—102no—
299C3×M16—104no—
300C3×M16D2014yescomplemented
301C2×C3×C8—52no—
302C2×C3×C8—104no—
303C2×C3×C8C22×D512yes—
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—
314C2×C3⋊C8—22no—
315C2×C3⋊C8—51no—
316C2×C3⋊C8—51no—
317C2×C3⋊C8—102no—
318C2×C3⋊C8C22×D511yescharacteristic
319C2×C3⋊C8C22×D511yescharacteristic
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×C5#23512yes—
333C2×C4×C5C2×#6712yes—
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—61no—
353C5×C8—61no—
354C5×C8C2×#6714yes—
355#168—22no—
356#168—61no—
357#168—61no—
358C2×#41—151no—
359C23×C4—51no—
360#131—304no—
361C2×M16—51no—
362C2×M16—52no—
363C2×M16—104no—
364C2×M16—152no—
365C2×M16D3×D511yescharacteristic
366C22×C8—31no—
367C22×C8—52no—
368C22×C8—151no—
369C22×C8—152no—
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—152no—
382C22⋊C8—152no—
383C22⋊C8—151no—
384C22⋊C8—151no—
385C22⋊C8—151no—
386C22⋊C8—151no—
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—102no—
427C3×C8C2×D2014yes—
428C3⋊C8—21no—
429C3⋊C8—21no—
430C3⋊C8—101no—
431C3⋊C8—101no—
432C22×C5—21no—
433C22×C5—21no—
434C22×C5C4×#6712yes—
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#90712yescomplemented
446C4×C5—62no—
447C4×C5—62no—
448C4×C5C4×#6712yes—
449C4×C5C2×#23511yescharacteristic
450C4×C5C22×#6711yescharacteristic
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—102no—
472M16—104no—
473M16—304no—
474M16#120214yescomplemented
475C2×C8—31no—
476C2×C8—31no—
477C2×C8—52no—
478C2×C8—62no—
479C2×C8—104no—
480C2×C8—151no—
481C2×C8—151no—
482C2×C8—302no—
483C2×C8—304no—
484C2×C8C2×D3×D512yes—
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×#90711yescharacteristic
520C2×C5#574511yescharacteristic
521C2×C5C2×C4×#6711yescharacteristic
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×C4#677112yes—
552C2×C4C4×D3×D511yescharacteristic
553C2×C4C2×#120212yes—
554C2×C4C22×D3×D511yescharacteristic
555C8—61no—
556C8—61no—
557C8—102no—
558C8—301no—
559C8—301no—
560C8C2×#120214yes—
561C2×C3—22no—
562C2×C3—102no—
563C2×C3—201no—
564C2×C3C2×C4×D2011yescharacteristic
565C2×C3D5×#13111yescharacteristic
566C2×C3#6608411yescharacteristic
567D3—202no—
568C5#8375711yescharacteristic, 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—
581C22C4×#120212yes—
582C22C2×C4×D3×D511yescharacteristic
583C4—62no—
584C4—62no—
585C4—101no—
586C4—101no—
587C4—302no—
588C4C4×#120212yes—
589C4C2×#677111yescharacteristic
590C4C22×#120211yescharacteristic
591C3#1061028611yescharacteristic, complemented, Sylow
592C2—22no—
593C2—102no—
594C2—201no—
595C2—602no—
596C2D5×#90711yescharacteristic
597C2#1175401411yescharacteristic
598C2C2×C4×#120211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D3×D5
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×M16C22
prime radical#11754564C2
cosocleC2×C3×C4×C5C24
hypercenterC2×M16D3×D5
hypocenterC3×C5#2933
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#91796C2semidirect
2. C2×C3×D5×M16C2semidirect
4. C5×#5122C2semidirect
5. #11748441C2semidirect
6. #11748350C2semidirect
7. C2×#90955C2semidirect
8. #11747725C2semidirect
9. #11747715C2semidirect
10. #11747705C2semidirect
11. C8×#1202C2semidirect
18. C2×#6705C22semidirect
19. C2×C8×D15C22semidirect
34. C5×#815C22semidirect
35. C5×C8×Dic3C22semidirect
36. Dic3⋊#168C22semidirect
37. Dic3⋊#168C22semidirect
38. #90957C22semidirect
39. #90957C22semidirect
46. C2×D5×M16D3semidirect
73. C3×C5×M16D4semidirect
104. #5122D5semidirect
203. #815C2×D5semidirect
206. C8×Dic3C2×D5semidirect
224. C5×M16#67semidirect
272. C2×C3×D5#131semidirect
300. C3×M16D20semidirect
445. C2×D5#907semidirect
474. M16#1202semidirect
495. C3×C5#2933semidirect
568. C5#83757semidirect
591. C3#10610286semidirect

Maximal subgroups

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

subgroupindex
C2×#917962
C2×C3×D5×M162
#117545642
C5×#51222
#117484412
#117483502
C2×#909552
#117477252
#117477152
#117477052
C8×#12022
#106102863
#837575