Finite Group Databasebeta

← #14424320

Subgroups — #14424320

Order 1920.

1504 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×H(5,𝔽2)C211yesmaximal, characteristic
2#11756584C211yesmaximal, characteristic, complemented
3C2×#91840C211yesmaximal, characteristic, complemented
4C22×#6716C211yesmaximal, characteristic, complemented
5#11751340C214yesmaximal, complemented
6#11751335C211yesmaximal, characteristic, complemented
7#11751334C211yesmaximal, characteristic, complemented
8#11751319C214yesmaximal, complemented
9#11751316C211yesmaximal, characteristic, complemented
10#10615836—31nomaximal, Hall
11C3×C5×H(5,𝔽2)—24no—
12C3×C5×H(5,𝔽2)—24no—
13C2×C3×C5×#41—21no—
14C2×C3×C5×#41C2212yes—
15C2×C3×C5×#41C2211yescharacteristic
16C2×C3×C5×#41C2211yescharacteristic
17C22×C3×C5×D4—22no—
18C22×C3×C5×D4—21no—
19C22×C3×C5×D4C2212yes—
20C22×C3×C5×D4C2211yescharacteristic
21C22×Dic30C2211yescharacteristic, complemented
22#91840—24no—
23#91833—21no—
24C2×#6718C2212yescomplemented
25#91825—21no—
26#91824—22no—
27C2×#6716—24no—
28C2×#6716C2214yescomplemented
29C2×#6716C2212yescomplemented
30#91815C2211yescharacteristic, complemented
31#91813C2214yescomplemented
32C2×#6710C2211yescharacteristic, complemented
33C22×C3⋊#168C2211yescharacteristic, complemented
34#91119C2212yescomplemented
35#91116C2214yescomplemented
36#91116C2212yescomplemented
37#91114C2214yescomplemented
38#84105—51nomaximal, Hall
39C2×C5×H(5,𝔽2)D311yescharacteristic
40#66159—31no—
41C2×#4097—31no—
42C22×#687—31no—
43#65528—34no—
44#65523—31no—
45#65522—31no—
46#65507—34no—
47#65504—31no—
48C24×C3×C5—22no—
49C24×C3×C5—21no—
50C3×C5×#41—28no—
51C3×C5×#41—24no—
52C3×C5×#41—24no—
53C3×C5×#41—22no—
54C3×C5×#41—22no—
55C3×C5×#41—22no—
56C3×C5×#41D414yes—
57C2×C3×C5×Q8D411yescharacteristic
58C2×C3×C5×Q8C2311yescharacteristic
59C2×C3×C5×D4—216no—
60C2×C3×C5×D4—28no—
61C2×C3×C5×D4—22no—
62C2×C3×C5×D4—21no—
63C2×C3×C5×D4—24no—
64C2×C3×C5×D4—24no—
65C2×C3×C5×D4—22no—
66C2×C3×C5×D4—21no—
67C2×C3×C5×D4—22no—
68C2×C3×C5×D4D412yes—
69C2×C3×C5×D4D411yescharacteristic
70C2×C3×C5×D4D414yes—
71C2×C3×C5×D4C2311yescharacteristic
72C2×C3×C5×D4C2312yes—
73C22×C3×C4×C5—22no—
74C22×C3×C4×C5—21no—
75C22×C3×C4×C5C2312yes—
76C22×C3×C4×C5C2311yescharacteristic
77C22×Dic15—21no—
78C2×Dic30—22no—
79C2×Dic30—22no—
80C2×Dic30—21no—
81C2×Dic30C2312yes—
82#6719—24no—
83#6719—21no—
84#6718—24no—
85#6716—28no—
86#6716—24no—
87C4⋊Dic15C2312yes—
88#6712—24no—
89C4×Dic15—21no—
90#6710—22no—
91C2×C3⋊#168—22no—
92C2×C3⋊#168C2312yes—
93C2×C3⋊#168C2312yes—
94C2×C3×H(5,𝔽2)D511yescharacteristic
95#5825—51no—
96C2×#951—51no—
97C22×#214—51no—
98#5239—54no—
99#5234—51no—
100#5233—51no—
101#5218—54no—
102#5215—51no—
103C5×H(5,𝔽2)—24no—
104C5×H(5,𝔽2)—24no—
105C2×C5×#41—21no—
106C2×C5×#41C2×D312yes—
107C2×C5×#41C2×D311yescharacteristic
108C2×C5×#41C2×D311yescharacteristic
109C22×C5×D4—22no—
110C22×C5×D4—21no—
111C22×C5×D4C2×D312yes—
112C22×C5×D4C2×D311yescharacteristic
113C22×Dic10—31no—
114#4097—64no—
115#4090—61no—
116C2×#689—32no—
117#4082—61no—
118#4081—62no—
119C2×#687—34no—
120C2×#687—32no—
121C2×#687—64no—
122#4072—31no—
123#4070—34no—
124C2×#681—31no—
125C22×#168—31no—
126#3967—32no—
127#3964—34no—
128#3964—32no—
129C22⋊#168—34no—
130#3009—151noSylow
131C23×C3×C5—28no—
132C23×C3×C5—22no—
133C23×C3×C5—21no—
134C23×C3×C5—24no—
135C23×C3×C5—82no—
136C23×C3×C5—84no—
137C23×C3×C5C2×D412yes—
138C23×C3×C5C2×D411yescharacteristic
139C3×C5×Q8—22no—
140C3×C5×Q8—21no—
141C3×C5×Q8C2×D412yes—
142C3×C5×D4—28no—
143C3×C5×D4—28no—
144C3×C5×D4—24no—
145C3×C5×D4—24no—
146C3×C5×D4—22no—
147C3×C5×D4—24no—
148C3×C5×D4—22no—
149C3×C5×D4—21no—
150C3×C5×D4C2×D412yes—
151C3×C5×D4C2×D414yes—
152C2×C3×C4×C5—28no—
153C2×C3×C4×C5—21no—
154C2×C3×C4×C5—24no—
155C2×C3×C4×C5—22no—
156C2×C3×C4×C5—22no—
157C2×C3×C4×C5C2×D412yes—
158C2×C3×C4×C5C2×D411yescharacteristic
159C2×C3×C4×C5C2×D412yes—
160C2×C3×C4×C5C2×D412yes—
161C2×C3×C4×C5C2411yescharacteristic
162C2×Dic15—22no—
163C2×Dic15—22no—
164C2×Dic15—42no—
165Dic30—22no—
166Dic30—22no—
167Dic30—42no—
168C3⋊#168—22no—
169C3⋊#168—22no—
170C3×H(5,𝔽2)—24no—
171C3×H(5,𝔽2)—24no—
172C2×C3×#41—21no—
173C2×C3×#41C2×D512yes—
174C2×C3×#41C2×D511yescharacteristic
175C2×C3×#41C2×D511yescharacteristic
176C22×C3×D4—22no—
177C22×C3×D4—21no—
178C22×C3×D4C2×D512yes—
179C22×C3×D4C2×D511yescharacteristic
180C22×Dic6—51no—
181#951—104no—
182#944—101no—
183C2×#216—52no—
184#936—101no—
185#935—102no—
186C2×#214—52no—
187C2×#214—54no—
188C2×#214—104no—
189#926—51no—
190#924—54no—
191C2×#208—51no—
192C22×C3⋊C8—51no—
193#835—52no—
194#832—54no—
195#832—52no—
196#830—54no—
197C24×C5—22no—
198C24×C5—21no—
199C5×#41—28no—
200C5×#41—24no—
201C5×#41—24no—
202C5×#41—22no—
203C5×#41—22no—
204C5×#41—22no—
205C5×#41#6714yes—
206C2×C5×Q8#6711yescharacteristic
207C2×C5×Q8C22×D311yescharacteristic
208C2×C5×D4—216no—
209C2×C5×D4—28no—
210C2×C5×D4—22no—
211C2×C5×D4—21no—
212C2×C5×D4—24no—
213C2×C5×D4—24no—
214C2×C5×D4—22no—
215C2×C5×D4—21no—
216C2×C5×D4—22no—
217C2×C5×D4#6712yes—
218C2×C5×D4#6711yescharacteristic
219C2×C5×D4#6714yes—
220C2×C5×D4C22×D311yescharacteristic
221C2×C5×D4C22×D312yes—
222C22×C4×C5—22no—
223C22×C4×C5—21no—
224C22×C4×C5C22×D312yes—
225C22×C4×C5C22×D311yescharacteristic
226C22×Dic5—61no—
227C2×Dic10—32no—
228C2×Dic10—62no—
229C2×Dic10—62no—
230C2×Dic10—61no—
231C22⋊Dic5—64no—
232C22⋊Dic5—61no—
233#689—64no—
234#687—64no—
235#687—68no—
236C4⋊Dic5—32no—
237#683—64no—
238C4×Dic5—61no—
239#681—62no—
240C2×#168—32no—
241C2×#168—32no—
242C2×#168—62no—
243C2×H(5,𝔽2)D1511yescharacteristic
244C2×#137—151no—
245C22×SD16—151no—
246#536—151no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—151no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—24no—
255C22×C3×C5—22no—
256C22×C3×C5—24no—
257C22×C3×C5—44no—
258C22×C3×C5—42no—
259C22×C3×C5—84no—
260C22×C3×C5—88no—
261C22×C3×C5#12012yes—
262C22×C3×C5C22×D411yescharacteristic
263C3×C4×C5—22no—
264C3×C4×C5—22no—
265C3×C4×C5#12012yes—
266C3×C4×C5C22×D411yescharacteristic
267C3×C4×C5C22×D411yescharacteristic
268Dic15—42no—
269Dic15—42no—
270C24×C3—22no—
271C24×C3—21no—
272C3×#41—28no—
273C3×#41—24no—
274C3×#41—24no—
275C3×#41—22no—
276C3×#41—22no—
277C3×#41—22no—
278C3×#41C22⋊D514yes—
279C2×C3×Q8C22⋊D511yescharacteristic
280C2×C3×Q8C22×D511yescharacteristic
281C2×C3×D4—216no—
282C2×C3×D4—28no—
283C2×C3×D4—22no—
284C2×C3×D4—21no—
285C2×C3×D4—24no—
286C2×C3×D4—24no—
287C2×C3×D4—22no—
288C2×C3×D4—21no—
289C2×C3×D4—22no—
290C2×C3×D4C22⋊D512yes—
291C2×C3×D4C22⋊D511yescharacteristic
292C2×C3×D4C22⋊D514yes—
293C2×C3×D4C22×D511yescharacteristic
294C2×C3×D4C22×D512yes—
295C22×C3×C4—22no—
296C22×C3×C4—21no—
297C22×C3×C4C22×D512yes—
298C22×C3×C4C22×D511yescharacteristic
299C22×Dic3—101no—
300C2×Dic6—52no—
301C2×Dic6—102no—
302C2×Dic6—102no—
303C2×Dic6—101no—
304#217—104no—
305#217—101no—
306#216—104no—
307#214—108no—
308#214—104no—
309C4⋊Dic3—52no—
310Dic3⋊C4—104no—
311C4×Dic3—101no—
312#208—102no—
313C2×C3⋊C8—52no—
314C2×C3⋊C8—52no—
315C2×C3⋊C8—102no—
316C23×C5—28no—
317C23×C5—22no—
318C23×C5—21no—
319C23×C5—24no—
320C23×C5—82no—
321C23×C5—84no—
322C23×C5C2×#6712yes—
323C23×C5C2×#6711yescharacteristic
324C5×Q8—22no—
325C5×Q8—21no—
326C5×Q8C2×#6712yes—
327C5×D4—28no—
328C5×D4—28no—
329C5×D4—24no—
330C5×D4—24no—
331C5×D4—22no—
332C5×D4—24no—
333C5×D4—22no—
334C5×D4—21no—
335C5×D4C2×#6712yes—
336C5×D4C2×#6714yes—
337C2×C4×C5—28no—
338C2×C4×C5—21no—
339C2×C4×C5—24no—
340C2×C4×C5—22no—
341C2×C4×C5—22no—
342C2×C4×C5C2×#6712yes—
343C2×C4×C5C2×#6711yescharacteristic
344C2×C4×C5C2×#6712yes—
345C2×C4×C5C2×#6712yes—
346C2×C4×C5C23×D311yescharacteristic
347C2×Dic5—62no—
348C2×Dic5—62no—
349C2×Dic5—122no—
350Dic10—62no—
351Dic10—62no—
352Dic10—122no—
353#168—62no—
354#168—62no—
355H(5,𝔽2)—24no—
356H(5,𝔽2)—24no—
357C2×#41—21no—
358C2×#41C2×D1512yes—
359C2×#41C2×D1511yescharacteristic
360C2×#41C2×D1511yescharacteristic
361C22×Q8—151no—
362C22×D4—22no—
363C22×D4—21no—
364C22×D4C2×D1512yes—
365C22×D4C2×D1511yescharacteristic
366#137—304no—
367C2×Q16—152no—
368C2×SD16—154no—
369C2×SD16—152no—
370C2×SD16—304no—
371C2×M16—151no—
372C22×C8—151no—
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—151no—
378#103—152no—
379#102—152no—
380#102—154no—
381C22⋊C8—154no—
382C2×C3×C5—22no—
383C2×C3×C5—42no—
384C2×C3×C5—44no—
385C2×C3×C5—42no—
386C2×C3×C5C2×#12011yescharacteristic
387C2×C3×C5#57712yes—
388C23×C3—28no—
389C23×C3—22no—
390C23×C3—21no—
391C23×C3—24no—
392C23×C3—82no—
393C23×C3—84no—
394C23×C3C2×C22⋊D512yes—
395C23×C3C2×C22⋊D511yescharacteristic
396C3×Q8—22no—
397C3×Q8—21no—
398C3×Q8C2×C22⋊D512yes—
399C3×D4—28no—
400C3×D4—28no—
401C3×D4—24no—
402C3×D4—24no—
403C3×D4—22no—
404C3×D4—24no—
405C3×D4—22no—
406C3×D4—21no—
407C3×D4C2×C22⋊D512yes—
408C3×D4C2×C22⋊D514yes—
409C2×C3×C4—28no—
410C2×C3×C4—21no—
411C2×C3×C4—24no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4C2×C22⋊D512yes—
415C2×C3×C4C2×C22⋊D511yescharacteristic
416C2×C3×C4C2×C22⋊D512yes—
417C2×C3×C4C2×C22⋊D512yes—
418C2×C3×C4C23×D511yescharacteristic
419C2×Dic3—102no—
420C2×Dic3—102no—
421C2×Dic3—202no—
422Dic6—102no—
423Dic6—102no—
424Dic6—202no—
425C3⋊C8—102no—
426C3⋊C8—102no—
427C22×C5—22no—
428C22×C5—24no—
429C22×C5—22no—
430C22×C5—24no—
431C22×C5—22no—
432C22×C5—44no—
433C22×C5—42no—
434C22×C5—84no—
435C22×C5—88no—
436C22×C5#95312yes—
437C22×C5C22×#6711yescharacteristic
438C4×C5—22no—
439C4×C5—22no—
440C4×C5#95312yes—
441C4×C5C22×#6711yescharacteristic
442C4×C5C22×#6711yescharacteristic
443Dic5—122no—
444Dic5—122no—
445C24—22no—
446C24—21no—
447#41—28no—
448#41—24no—
449#41—24no—
450#41—22no—
451#41—22no—
452#41—22no—
453#41#122014yes—
454C2×Q8—152no—
455C2×Q8—301no—
456C2×Q8—302no—
457C2×Q8—302no—
458C2×Q8#122011yescharacteristic
459C2×Q8C22×D1511yescharacteristic
460C2×D4—216no—
461C2×D4—28no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—22no—
469C2×D4#122012yes—
470C2×D4#122011yescharacteristic
471C2×D4#122014yes—
472C2×D4C22×D1511yescharacteristic
473C2×D4C22×D1512yes—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—301no—
477C22×C4C22×D1512yes—
478C22×C4C22×D1511yescharacteristic
479Q16—304no—
480SD16—308no—
481SD16—304no—
482M16—302no—
483C2×C8—152no—
484C2×C8—152no—
485C2×C8—302no—
486C4⋊C4—152no—
487C4⋊C4—304no—
488C22⋊C4—301no—
489C22⋊C4—304no—
490C42—301no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—24no—
494C22×C3—22no—
495C22×C3—24no—
496C22×C3—22no—
497C22×C3—44no—
498C22×C3—42no—
499C22×C3—84no—
500C22×C3—88no—
501C22×C3#409912yes—
502C22×C3C22×C22⋊D511yescharacteristic
503C3×C4—22no—
504C3×C4—22no—
505C3×C4#409912yes—
506C3×C4C22×C22⋊D511yescharacteristic
507C3×C4C22×C22⋊D511yescharacteristic
508Dic3—202no—
509Dic3—202no—
510C2×C5—22no—
511C2×C5—42no—
512C2×C5—44no—
513C2×C5—42no—
514C2×C5#583612yes—
515C2×C5C2×#95311yescharacteristic
516C23—28no—
517C23—22no—
518C23—21no—
519C23—24no—
520C23—82no—
521C23—84no—
522C23C2×#122012yes—
523C23C2×#122011yescharacteristic
524Q8—22no—
525Q8—21no—
526Q8—302no—
527Q8—302no—
528Q8—602no—
529Q8C2×#122012yes—
530D4—28no—
531D4—28no—
532D4—24no—
533D4—24no—
534D4—22no—
535D4—24no—
536D4—22no—
537D4—21no—
538D4C2×#122012yes—
539D4C2×#122014yes—
540C2×C4—28no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—302no—
546C2×C4—302no—
547C2×C4—602no—
548C2×C4C2×#122012yes—
549C2×C4C2×#122011yescharacteristic
550C2×C4C2×#122012yes—
551C2×C4C2×#122012yes—
552C2×C4C23×D1511yescharacteristic
553C8—302no—
554C8—302no—
555C2×C3—22no—
556C2×C3—42no—
557C2×C3—44no—
558C2×C3—42no—
559C2×C3#6617012yes—
560C2×C3C2×#409911yescharacteristic
561C5#8410511yescharacteristic, complemented, Sylow
562C22—22no—
563C22—24no—
564C22—22no—
565C22—24no—
566C22—22no—
567C22—44no—
568C22—42no—
569C22—84no—
570C22—88no—
571C22#9184212yes—
572C22C22×#122011yescharacteristic
573C4—22no—
574C4—22no—
575C4—602no—
576C4—602no—
577C4#9184212yes—
578C4C22×#122011yescharacteristic
579C4C22×#122011yescharacteristic
580C3#1061583611yescharacteristic, complemented, Sylow
581C2—22no—
582C2—42no—
583C2—44no—
584C2—42no—
585C2#1175659512yes—
586C2C2×#9184211yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
Fitting, prime radicalC2×C3×C5×H(5,𝔽2)C2
socleC22×C3×C5C22×D4
hypercenterC2×H(5,𝔽2)D15
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. #11756584C2semidirect
3. C2×#91840C2semidirect
4. C22×#6716C2semidirect
5. #11751340C2semidirect
6. #11751335C2semidirect
7. #11751334C2semidirect
8. #11751319C2semidirect
9. #11751316C2semidirect
21. C22×Dic30C22semidirect
24. C2×#6718C22semidirect
28. C2×#6716C22semidirect
29. C2×#6716C22semidirect
30. #91815C22semidirect
31. #91813C22semidirect
32. C2×#6710C22semidirect
33. C22×C3⋊#168C22semidirect
34. #91119C22semidirect
35. #91116C22semidirect
36. #91116C22semidirect
37. #91114C22semidirect
491. C3×C5#3009semidirect
561. C5#84105semidirect
580. C3#10615836semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×H(5,𝔽2)2
#117565842
C2×#918402
C22×#67162
#117513402
#117513352
#117513342
#117513192
#117513162
#106158363
#841055