Finite Group Databasebeta

← #14415032

Subgroups — #14415032

Order 1920.

1040 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
1C22×D5⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
2C2×#91929C211yesmaximal, characteristic, complemented
3#11754316C211yesmaximal, characteristic
4C3×#65755C211yesmaximal, characteristic, complemented
5#11752014C211yesmaximal, characteristic, complemented
6#11752001C211yesmaximal, characteristic, complemented
7#11751995C211yesmaximal, characteristic, complemented
8#11751925C212yesmaximal, complemented
9#11751924C212yesmaximal, complemented
10#11751882C212yesmaximal, complemented
11#11751879C212yesmaximal, complemented
12#10617289—31nomaximal, Hall
13C22×C3×C4×D5C2211yescharacteristic
14C2×#6782C2211yescharacteristic
15C22×C3⋊#170C2211yescharacteristic
16C22×C3⋊#170C2211yescharacteristic
17C2×D5⋊(C3⋊C8)—24no—
18C2×D5⋊(C3⋊C8)C2212yes—
19C2×D5⋊(C3⋊C8)C2212yes—
20C2×C3×#704C2211yescharacteristic
21C2×C3×#704C2211yescharacteristic
22C2×#6740C2211yescharacteristic, complemented
23C2×#6739C2211yescharacteristic, complemented
24C2×#6738C2211yescharacteristic, complemented
25#91929—24no—
26C2×D3×#170C2211yescharacteristic, complemented
27#91818C411yescharacteristic
28C5×#929C411yescharacteristic
29#91564C411yescharacteristic
30#91563C411yescharacteristic
31#91528C2211yescharacteristic
32#91448C412yes—
33#91444C2212yes—
34#91440C2212yes—
35#91349C412yes—
36C3×#4003C2212yes—
37C3×#4003C2212yes—
38#91177C2212yescomplemented
39Dic3⋊#170C2212yescomplemented
40#91171C2212yescomplemented
41(C2×D3)⋊#170C2212yescomplemented
42#91169C2212yescomplemented
43Dic3×#170C2212yescomplemented
44#78626—51nomaximal, Hall
45C2×#4131—31no—
46C22×D5⋊C8—31no—
47#65990—31no—
48#65775—31no—
49#65755D311yescharacteristic, complemented
50#65704—32no—
51#65702—32no—
52#65696—31no—
53#65694—32no—
54#65692—32no—
55#65691—31no—
56C23×C3×D5C2×C411yescharacteristic
57C22×C3×C4×C5C2×C411yescharacteristic
58C2×#1220C2×C411yescharacteristic
59C2×C5×#67C2×C411yescharacteristic
60C22×C3×Dic5C2311yescharacteristic
61C2×C3×C4×D5—22no—
62C2×C3×C4×D5—22no—
63C2×C3×C4×D5C2×C412yes—
64C2×C3×C4×D5C2312yes—
65C2×#1204C2×C411yescharacteristic
66C2×#1203C2×C411yescharacteristic
67C2×#1202C2×C412yes—
68C2×#1201C2×C412yes—
69C2×#1200C2311yescharacteristic
70#6782—22no—
71#6782—22no—
72C2×D3×Dic5C2311yescharacteristic
73C2×C3⋊#170—22no—
74C2×C3⋊#170D414yescomplemented
75C2×C3⋊#170C2312yes—
76C2×C3⋊#170C2311yescharacteristic
77C2×C3⋊#170C2311yescharacteristic
78D5⋊(C3⋊C8)—24no—
79D5⋊(C3⋊C8)—24no—
80C3×#704—22no—
81C3×#704—22no—
82C2×C3×#170C2311yescharacteristic
83C2×C3×#170C2311yescharacteristic
84C2×C3×#170C2312yes—
85#6740—22no—
86#6739—22no—
87#6738—22no—
88D3×#170—22no—
89#6719C2×C411yescharacteristic
90#6714C2×C412yes—
91#6712C2×C412yes—
92C5×#217C2×C411yescharacteristic
93C5×(C2×D3)⋊C4C2×C412yes—
94C5×Dic3⋊C4C2×C412yes—
95#6668C2×C412yes—
96#6665C2×C412yes—
97#6665C2×C412yes—
98Dic3×Dic5C2312yes—
99#5791—51no—
100C23×C3⋊C8—51no—
101C2×#907—51no—
102C3×#407—51no—
103#5129—51no—
104#5122—51no—
105#5119—51no—
106#4730—52no—
107#4730—52no—
108#67⋊C8—52no—
109#67⋊C8—52no—
110C2×#711—31no—
111C22×C4×D5C2×D311yescharacteristic
112C2×#704—31no—
113C2×#704C2×D311yescharacteristic
114C2×#704C2×D311yescharacteristic
115C22×#170—31no—
116C22×#170—31no—
117C22×#170—31no—
118#4131—64no—
119C2×#700—31no—
120C2×D5⋊C8—32no—
121C2×D5⋊C8—32no—
122C2×D5⋊C8—31no—
123C2×D5⋊C8—64no—
124C5×#123—31no—
125#4083—31no—
126#4075—31no—
127#4042—32no—
128#4039—32no—
129#4030—32no—
130#4028—31no—
131#4027—32no—
132#4023—31no—
133C22⋊#170—32no—
134#4004—32no—
135#4003—32no—
136#4003C2×D312yes—
137#4003C2×D312yes—
138#4002—32no—
139C4⋊#170—32no—
140C4×#170—32no—
141#2929—151noSylow
142C23×C3×C5C22×C411yescharacteristic
143C22×D15C22×C411yescharacteristic
144C22×C5×D3C22×C411yescharacteristic
145C22×C3×D5—22no—
146C22×C3×D5—24no—
147C22×C3×D5C22×C412yes—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5C22×C412yes—
150#1220—22no—
151C2×Dic15C22×C412yes—
152C2×Dic15C22×C411yescharacteristic
153C5×#67—22no—
154C2×C5×Dic3C22×C412yes—
155C2×C5×Dic3C22×C411yescharacteristic
156C2×C3×Dic5C22×C411yescharacteristic
157C2×C3×Dic5C2×D412yes—
158C2×C3×Dic5#4112yes—
159C2×C3×Dic5C2411yescharacteristic
160C3×C4×D5—24no—
161C3×C4×D5—22no—
162C3×C4×D5—22no—
163#1204—22no—
164#1203—22no—
165#1202—24no—
166#1201—24no—
167#1200—21no—
168#1200—21no—
169D3×Dic5—21no—
170D3×Dic5—21no—
171C3⋊#170—22no—
172C3⋊#170C2×D412yes—
173C3⋊#170C2×D412yes—
174C3×#170—21no—
175C3×#170—21no—
176C3×#170—22no—
177C23×C3×C4—51no—
178C2×#235—51no—
179C2×C3×M16—51no—
180C2×C3×M16—51no—
181#953—51no—
182#953—51no—
183#930—52no—
184#929—51no—
185#929AGL(1,𝔽5)11yescharacteristic
186C4×#67—52no—
187C4×#67—52no—
188#926—51no—
189#924—52no—
190C22×C3⋊C8—51no—
191C22×C3⋊C8—51no—
192C22×C3⋊C8—52no—
193C22×C3⋊C8—52no—
194C22×C3⋊C8—104no—
195#907—104no—
196C2×ΓL(1,𝔽25)—51no—
197C2×ΓL(1,𝔽25)—51no—
198C2×D3×C8—51no—
199C2×D3×C8—51no—
200C3×C22⋊C8—52no—
201C3×C22⋊C8—52no—
202(C2×D3)⋊C8—52no—
203(C2×D3)⋊C8—52no—
204#815—52no—
205Dic3⋊C8—52no—
206Dic3⋊C8—52no—
207C8×Dic3—52no—
208C23×D5C4×D311yescharacteristic
209C2×C5×D4—31no—
210C22×C4×C5C4×D311yescharacteristic
211C2×C22⋊D5—31no—
212C2×C22⋊D5—32no—
213C2×C22⋊D5—31no—
214C22×Dic5—31no—
215C22×Dic5C22×D311yescharacteristic
216#711—62no—
217#711—62no—
218C2×D20—32no—
219C2×C4×D5—22no—
220C2×C4×D5—22no—
221C2×C4×D5—31no—
222C2×C4×D5C4×D312yes—
223C2×C4×D5C22×D312yes—
224C2×Dic10—31no—
225#704—22no—
226#704—22no—
227#704—62no—
228C2×#170—34no—
229C2×#170—31no—
230C2×#170—31no—
231C2×#170—32no—
232C2×#170—62no—
233C2×#170—62no—
234C2×#170C22×D311yescharacteristic
235C2×#170C22×D311yescharacteristic
236C2×#170C22×D312yes—
237#700—62no—
238D5⋊C8—62no—
239D5⋊C8—64no—
240D5⋊C8—64no—
241C5×C4⋊C4—32no—
242C5×C22⋊C4—31no—
243C5×C22⋊C4—32no—
244C22⋊Dic5—31no—
245#685—32no—
246#685—32no—
247#685—32no—
248#683—32no—
249#683—32no—
250C4×Dic5—32no—
251C2×#131—151no—
252C23×C8—151no—
253#525—151no—
254#435—152no—
255#435—152no—
256C8×D4—152no—
257C8×D4—152no—
258#424—151no—
259#408—151no—
260#407—51no—
261#405—151no—
262C22×C3×C5—22no—
263C22×C3×C5C4×D412yes—
264C22×C3×C5C23×C411yescharacteristic
265C2×D15—21no—
266C2×D15—22no—
267C2×C5×D3—21no—
268C2×C5×D3—22no—
269C2×C3×D5—28no—
270C2×C3×D5—22no—
271C2×C3×D5—22no—
272C2×C3×D5#13114yescomplemented
273C3×C4×C5—22no—
274Dic15—21no—
275Dic15—22no—
276C3×Dic5C4×D412yes—
277C3×Dic5C22×D411yescharacteristic
278C3×Dic5C2×#4111yescharacteristic
279C5×Dic3—21no—
280C5×Dic3—22no—
281C24×C3—51no—
282C22×C3×C4—51no—
283C22×C3×C4—52no—
284C22×C3×C4—52no—
285C22×C3×C4—102no—
286C22×C3×C4—102no—
287C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
288C2×#67—51no—
289C2×#67—52no—
290C2×#67—52no—
291C2×#67C2×AGL(1,𝔽5)11yescharacteristic
292#235—102no—
293#235—102no—
294C2×D12—51no—
295C2×C4×D3—51no—
296C2×C4×D3—51no—
297C2×Dic6—51no—
298C3×M16—102no—
299C3×M16—102no—
300C2×C3×C8—52no—
301C2×C3×C8—51no—
302C2×C3×C8—51no—
303#217—51no—
304#217—52no—
305#217—52no—
306#217C2×AGL(1,𝔽5)11yescharacteristic
307(C2×D3)⋊C4—52no—
308(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
309C4⋊Dic3—52no—
310Dic3⋊C4—52no—
311Dic3⋊C4C2×AGL(1,𝔽5)12yes—
312C4×Dic3—52no—
313C2×C3⋊C8—51no—
314C2×C3⋊C8—51no—
315C2×C3⋊C8—54no—
316C2×C3⋊C8—52no—
317C2×C3⋊C8—104no—
318C2×C3⋊C8—104no—
319C2×C3⋊C8—102no—
320ΓL(1,𝔽25)—102no—
321ΓL(1,𝔽25)—102no—
322D3×C8—102no—
323D3×C8—102no—
324C23×C5—31no—
325C23×C5C2×C4×D311yescharacteristic
326C22×D5—22no—
327C22×D5—24no—
328C22×D5—31no—
329C22×D5C2×C4×D312yes—
330C5×D4—62no—
331C2×C4×C5—22no—
332C2×C4×C5—32no—
333C2×C4×C5—31no—
334C2×C4×C5C2×C4×D312yes—
335C22⋊D5—62no—
336C22⋊D5—64no—
337C22⋊D5—62no—
338C2×Dic5—32no—
339C2×Dic5—31no—
340C2×Dic5—61no—
341C2×Dic5—61no—
342C2×Dic5C2×C4×D311yescharacteristic
343C2×Dic5D3×D412yes—
344C2×Dic5#23712yes—
345C2×Dic5C23×D311yescharacteristic
346D20—64no—
347C4×D5—24no—
348C4×D5—22no—
349C4×D5—22no—
350C4×D5—61no—
351C4×D5—61no—
352Dic10—62no—
353#170—21no—
354#170—21no—
355#170—22no—
356#170—32no—
357#170—32no—
358#170—62no—
359C2×#41—151no—
360C23×C4—51no—
361#131—304no—
362C2×M16—51no—
363C2×M16—51no—
364C2×M16—151no—
365C2×M16—151no—
366C22×C8—151no—
367C22×C8—151no—
368C22×C8—151no—
369C22×C8—152no—
370C22×C8—152no—
371C22×C8—151no—
372C22×C8—304no—
373#123—31no—
374#123—151no—
375C22⋊Q8—152no—
376#121—152no—
377#120—151no—
378#120—151no—
379C4×D4—152no—
380C4×D4—152no—
381#117—151no—
382C4⋊C8—152no—
383C4⋊C8—152no—
384C22⋊C8—52no—
385C22⋊C8—52no—
386C22⋊C8—152no—
387C22⋊C8—152no—
388#97—152no—
389C4×C8—152no—
390C2×C3×C5—22no—
391C2×C3×C5C2×C4×D411yescharacteristic
392C2×C3×C5C2×#13112yes—
393D15—41no—
394C3×D5—24no—
395C5×D3—41no—
396C23×C3—52no—
397C23×C3—104no—
398C23×C3—102no—
399C23×C3C22×AGL(1,𝔽5)11yescharacteristic
400C22×D3—51no—
401C22×D3C22×AGL(1,𝔽5)11yescharacteristic
402C2×C3×C4—22no—
403C2×C3×C4—52no—
404C2×C3×C4—52no—
405C2×C3×C4—51no—
406C2×C3×C4—51no—
407C2×C3×C4—102no—
408C2×C3×C4—104no—
409C2×C3×C4—102no—
410C2×C3×C4C22×AGL(1,𝔽5)12yes—
411#67—22no—
412#67—104no—
413#67—104no—
414#67—102no—
415C2×Dic3—52no—
416C2×Dic3—51no—
417C2×Dic3C22×AGL(1,𝔽5)12yes—
418C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
419D12—102no—
420C4×D3—101no—
421C4×D3—101no—
422C4×D3—101no—
423C4×D3—101no—
424Dic6—102no—
425C3×C8—101no—
426C3×C8—101no—
427C3×C8—102no—
428C3⋊C8—52no—
429C3⋊C8—52no—
430C3⋊C8—102no—
431C22×C5—22no—
432C22×C5—61no—
433C22×C5—62no—
434C22×C5#67⋊C412yes—
435C22×C5C22×C4×D311yescharacteristic
436C2×D5—28no—
437C2×D5—22no—
438C2×D5—22no—
439C2×D5—61no—
440C2×D5—62no—
441C2×D5#90714yescomplemented
442C4×C5—22no—
443C4×C5—61no—
444C4×C5—62no—
445Dic5—61no—
446Dic5—62no—
447Dic5#67⋊C412yes—
448Dic5C2×D3×D411yescharacteristic
449Dic5C2×#23711yescharacteristic
450C24—51no—
451#41—302no—
452#41—302no—
453C2×Q8—151no—
454C2×D4—31no—
455C2×D4—152no—
456C2×D4—151no—
457C2×D4—152no—
458C2×D4—151no—
459C22×C4—51no—
460C22×C4—52no—
461C22×C4—52no—
462C22×C4—102no—
463C22×C4—102no—
464C22×C4—151no—
465C22×C4—151no—
466C22×C4D3×AGL(1,𝔽5)11yescharacteristic
467M16—102no—
468M16—102no—
469M16—302no—
470M16—302no—
471C2×C8—52no—
472C2×C8—51no—
473C2×C8—51no—
474C2×C8—151no—
475C2×C8—151no—
476C2×C8—154no—
477C2×C8—152no—
478C2×C8—302no—
479C2×C8—302no—
480C2×C8—304no—
481C2×C8—304no—
482C2×C8—302no—
483C4⋊C4—32no—
484C4⋊C4—152no—
485C4⋊C4—152no—
486C22⋊C4—31no—
487C22⋊C4—32no—
488C22⋊C4—152no—
489C22⋊C4—152no—
490C22⋊C4—151no—
491C22⋊C4—152no—
492C42—152no—
493C3×C5#292911yescharacteristic, complemented, Hall
494C22×C3—22no—
495C22×C3—54no—
496C22×C3—108no—
497C22×C3—102no—
498C22×C3—102no—
499C22×C3D4×AGL(1,𝔽5)12yes—
500C22×C3C23×AGL(1,𝔽5)11yescharacteristic
501C2×D3—21no—
502C2×D3—22no—
503C2×D3—101no—
504C2×D3—102no—
505C3×C4—22no—
506C3×C4—52no—
507C3×C4—51no—
508C3×C4—51no—
509Dic3—21no—
510Dic3—22no—
511Dic3—101no—
512Dic3—102no—
513C2×C5—22no—
514C2×C5—121no—
515C2×C5C2×#67⋊C411yescharacteristic
516C2×C5C2×#90712yes—
517D5—24no—
518D5—121no—
519C23—31no—
520C23—52no—
521C23—104no—
522C23—102no—
523C23—151no—
524C23C2×D3×AGL(1,𝔽5)11yescharacteristic
525Q8—302no—
526D4—62no—
527D4—302no—
528D4—304no—
529D4—304no—
530D4—302no—
531C2×C4—22no—
532C2×C4—32no—
533C2×C4—31no—
534C2×C4—52no—
535C2×C4—52no—
536C2×C4—51no—
537C2×C4—51no—
538C2×C4—102no—
539C2×C4—102no—
540C2×C4—104no—
541C2×C4—152no—
542C2×C4—151no—
543C2×C4—301no—
544C2×C4—301no—
545C2×C4—301no—
546C2×C4—301no—
547C2×C4C2×D3×AGL(1,𝔽5)12yes—
548C8—101no—
549C8—101no—
550C8—102no—
551C8—152no—
552C8—152no—
553C8—302no—
554C2×C3—22no—
555C2×C3—104no—
556C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
557C2×C3D5⋊#13112yes—
558D3—41no—
559D3—201no—
560C5#7862611yescharacteristic, complemented, Sylow
561C22—22no—
562C22—54no—
563C22—62no—
564C22—61no—
565C22—108no—
566C22—102no—
567C22—102no—
568C22—302no—
569C22—301no—
570C22#67⋊AGL(1,𝔽5)12yes—
571C22C22×D3×AGL(1,𝔽5)11yescharacteristic
572C4—22no—
573C4—51no—
574C4—51no—
575C4—52no—
576C4—62no—
577C4—61no—
578C4—302no—
579C4—301no—
580C3#1061728911yescharacteristic, complemented, Sylow
581C2—22no—
582C2—104no—
583C2—121no—
584C2—601no—
585C2D5⋊#90712yes—
586C2C2×#67⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C4×C5C2×C4
prime radical#11754316C2
cosocleC2×C3×Dic5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2929
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. C22×D5⋊(C3⋊C8)C2semidirect
2. C2×#91929C2semidirect
4. C3×#65755C2semidirect
5. #11752014C2semidirect
6. #11752001C2semidirect
7. #11751995C2semidirect
8. #11751925C2semidirect
9. #11751924C2semidirect
10. #11751882C2semidirect
11. #11751879C2semidirect
22. C2×#6740C22semidirect
23. C2×#6739C22semidirect
24. C2×#6738C22semidirect
26. C2×D3×#170C22semidirect
38. #91177C22semidirect
39. Dic3⋊#170C22semidirect
40. #91171C22semidirect
41. (C2×D3)⋊#170C22semidirect
42. #91169C22semidirect
43. Dic3×#170C22semidirect
49. #65755D3semidirect
74. C2×C3⋊#170D4semidirect
272. C2×C3×D5#131semidirect
441. C2×D5#907semidirect
493. C3×C5#2929semidirect
560. C5#78626semidirect
580. C3#10617289semidirect

Maximal subgroups

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

subgroupindex
C22×D5⋊(C3⋊C8)2
C2×#919292
#117543162
C3×#657552
#117520142
#117520012
#117519952
#117519252
#117519242
#117518822
#117518792
#106172893
#786265