Finite Group Databasebeta

← #14415035

Subgroups — #14415035

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×#6738C211yesmaximal, characteristic, complemented
2C2×#91927C211yesmaximal, characteristic, complemented
3#11752872C211yesmaximal, characteristic
4#11752355C211yesmaximal, characteristic, complemented
5C3×#65691C211yesmaximal, characteristic, complemented
6#11752003C211yesmaximal, characteristic, complemented
7#11752001C211yesmaximal, characteristic, complemented
8#11751895C212yesmaximal, complemented
9#11751894C212yesmaximal, complemented
10#11751893C212yesmaximal, complemented
11#11751876C212yesmaximal, complemented
12#10617289—31nomaximal, Hall
13C22×#1200C2211yescharacteristic
14C2×#6782C2211yescharacteristic
15C2×#6762C2211yescharacteristic
16C22×C3⋊#170C2211yescharacteristic
17C2×C3×#704C2211yescharacteristic
18C22×C3×#170C2211yescharacteristic
19C2×#6740C2211yescharacteristic, complemented
20C2×#6739C2211yescharacteristic, complemented
21C2×#6738—24no—
22C2×#6738C2212yes—
23C2×#6738C2212yes—
24C2×#6738C2211yescharacteristic, complemented
25#91927—24no—
26C2×D3×#170C2211yescharacteristic, complemented
27#91771C411yescharacteristic
28C5×#886C411yescharacteristic
29C3×#4023C2211yescharacteristic
30#91573C411yescharacteristic
31#91542C411yescharacteristic
32#91420C412yes—
33#91419C412yes—
34#91418C2212yes—
35#91405C2212yes—
36#91233C2212yescomplemented
37C4⋊(C3⋊#170)C2212yescomplemented
38C3×#4002C2212yescomplemented
39C3×C4×#170C2212yescomplemented
40#91171C2212yes—
41#91171C2212yes—
42#91171C2212yescomplemented
43(C2×D3)⋊#170C2212yescomplemented
44#78553—51nomaximal, Hall
45C2×#4131—31no—
46C22×D5⋊C8—31no—
47#65990—31no—
48#65775—31no—
49#65755—31no—
50#65704—32no—
51#65702—32no—
52#65696—31no—
53#65694—32no—
54#65692—32no—
55#65691D311yescharacteristic, complemented
56C23×D15C2×C411yescharacteristic
57C2×#1220C2×C411yescharacteristic
58C2×D60C2×C412yes—
59C2×C5×#67C2×C411yescharacteristic
60C22×C5×Dic3C2×C411yescharacteristic
61C22×C3×Dic5C2311yescharacteristic
62C2×#1204C2×C411yescharacteristic
63C2×#1203C2×C412yes—
64C2×#1203C2×C411yescharacteristic
65C2×#1200—22no—
66C2×#1200—22no—
67C2×#1200C2×C412yes—
68C2×#1200C2312yes—
69C2×#1200C2311yescharacteristic
70#6782—22no—
71#6782—22no—
72C2×D3×Dic5C2311yescharacteristic
73#6762—22no—
74C2×C3⋊#170—22no—
75C2×C3⋊#170C2311yescharacteristic
76C2×C3⋊#170C2312yes—
77C2×C3⋊#170C2311yescharacteristic
78C3×#704—22no—
79C2×C3×#170D414yescomplemented
80C2×C3×#170C2312yes—
81C2×C3×#170C2311yescharacteristic
82C2×C3×#170C2311yescharacteristic
83#6740—22no—
84#6739—22no—
85#6738—22no—
86#6738—24no—
87#6738—24no—
88D3×#170—22no—
89C3×C5×C22⋊C4C2×C411yescharacteristic
90#6714C2×C412yes—
91C5×(C2×D3)⋊C4C2×C412yes—
92C5×C4⋊Dic3C2×C412yes—
93C3×C22⋊Dic5C2×C411yescharacteristic
94C3×#683C2×C412yes—
95C3×C4×Dic5C2312yes—
96#6668C2×C412yes—
97#6667C2×C412yes—
98#6667C2×C412yes—
99C2×#901—51no—
100C22×D3×C8—51no—
101#5524—51no—
102C3×#405—51no—
103#5129—51no—
104#5126—51no—
105#4999—51no—
106#4706—52no—
107#4706—52no—
108C8×D12—52no—
109C8×D12—52no—
110C2×#711—31no—
111C22×C4×D5—31no—
112C2×#704—31no—
113C2×#704—31no—
114C2×#704C2×D311yescharacteristic
115C22×#170—31no—
116C22×#170—31no—
117C22×#170C2×D311yescharacteristic
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#4023C2×D311yescharacteristic
133C22⋊#170—32no—
134#4004—32no—
135#4003—32no—
136#4003—32no—
137#4003—32no—
138#4002C2×D312yescomplemented
139C4⋊#170—32no—
140C4×#170C2×D312yescomplemented
141#2929—151noSylow
142C23×C3×C5C22×C411yescharacteristic
143C22×D15—22no—
144C22×D15—24no—
145C22×D15C22×C411yescharacteristic
146C22×D15C22×C412yes—
147C22×C5×D3C22×C411yescharacteristic
148C2×C3×C4×C5C22×C412yes—
149#1220—22no—
150C2×Dic15C22×C411yescharacteristic
151D60—24no—
152C5×#67—22no—
153C2×C5×Dic3—22no—
154C2×C5×Dic3C22×C411yescharacteristic
155C2×C5×Dic3C22×C412yes—
156C2×C3×Dic5C22×C412yes—
157C2×C3×Dic5C22×C411yescharacteristic
158C2×C3×Dic5C2×D412yes—
159C2×C3×Dic5#4112yes—
160C2×C3×Dic5C2411yescharacteristic
161#1204—22no—
162#1203—24no—
163#1203—22no—
164#1200—24no—
165#1200—22no—
166#1200—21no—
167#1200—21no—
168#1200—22no—
169D3×Dic5—21no—
170D3×Dic5—21no—
171C3⋊#170—21no—
172C3⋊#170—22no—
173C3⋊#170—21no—
174C3×#170—22no—
175C3×#170C2×D412yes—
176C3×#170C2×D412yes—
177C2×#235—51no—
178C22×C4×D3—51no—
179C2×C3×M16—51no—
180C22×C3×C8—51no—
181C3×#117—51no—
182C2×#208—51no—
183C22×C3⋊C8—51no—
184#901—104no—
185C2×ΓL(1,𝔽25)—51no—
186C2×ΓL(1,𝔽25)—51no—
187C2×D3×C8—51no—
188C2×D3×C8—52no—
189C2×D3×C8—52no—
190C2×D3×C8—51no—
191C2×D3×C8—104no—
192#897—52no—
193#895—52no—
194#886—51no—
195#886AGL(1,𝔽5)11yescharacteristic
196#882—51no—
197#882—51no—
198C4×D12—52no—
199C4×D12—52no—
200C3×#97—52no—
201C3×C4×C8—52no—
202(C2×D3)⋊C8—52no—
203(C2×D3)⋊C8—52no—
204(C2×D3)⋊C8—52no—
205(C2×D3)⋊C8—52no—
206C4⋊(C3⋊C8)—52no—
207C4⋊(C3⋊C8)—52no—
208C23×D5—31no—
209C2×C5×D4—31no—
210C22×C4×C5—31no—
211C2×C22⋊D5—31no—
212C2×C22⋊D5—31no—
213C2×C22⋊D5—32no—
214C22×Dic5—31no—
215C22×Dic5C22×D311yescharacteristic
216#711—62no—
217#711—62no—
218C2×D20—32no—
219C2×C4×D5—32no—
220C2×C4×D5—32no—
221C2×C4×D5—31no—
222C2×C4×D5—62no—
223C2×C4×D5—62no—
224C2×Dic10—31no—
225#704—22no—
226#704—62no—
227#704—62no—
228C2×#170—32no—
229C2×#170—31no—
230C2×#170—31no—
231C2×#170—62no—
232C2×#170—62no—
233C2×#170D1214yescomplemented
234C2×#170C22×D312yes—
235C2×#170C22×D311yescharacteristic
236C2×#170C22×D311yescharacteristic
237#700—62no—
238D5⋊C8—64no—
239D5⋊C8—64no—
240D5⋊C8—62no—
241C5×C4⋊C4—32no—
242C5×C22⋊C4—32no—
243C5×C22⋊C4C4×D311yescharacteristic
244C22⋊Dic5C4×D311yescharacteristic
245#685—32no—
246#685—32no—
247#685—32no—
248#683—32no—
249#683C4×D312yes—
250C4×Dic5C22×D312yes—
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—151no—
261#405—51no—
262C22×C3×C5—22no—
263C22×C3×C5C4×D412yes—
264C22×C3×C5C23×C411yescharacteristic
265C2×D15—28no—
266C2×D15—22no—
267C2×D15—22no—
268C2×D15—21no—
269C2×D15—22no—
270C2×D15#13114yescomplemented
271C2×C5×D3—21no—
272C2×C5×D3—22no—
273C3×C4×C5—22no—
274Dic15—21no—
275C3×Dic5—22no—
276C3×Dic5C4×D412yes—
277C3×Dic5C22×D411yescharacteristic
278C3×Dic5C2×#4111yescharacteristic
279C5×Dic3—22no—
280C5×Dic3—21no—
281C23×D3—51no—
282C22×C3×C4—51no—
283C2×#67—51no—
284C2×#67C2×AGL(1,𝔽5)11yescharacteristic
285C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
286#235—102no—
287#235—102no—
288C2×D12—52no—
289C2×D12—51no—
290C2×D12—52no—
291C2×C4×D3—51no—
292C2×C4×D3—52no—
293C2×C4×D3—51no—
294C2×C4×D3—52no—
295C2×C4×D3—102no—
296C2×C4×D3—102no—
297C2×Dic6—51no—
298C3×M16—102no—
299C2×C3×C8—51no—
300C2×C3×C8—51no—
301C2×C3×C8—54no—
302C2×C3×C8—52no—
303C3×C4⋊C4—52no—
304C3×C22⋊C4—51no—
305C3×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
306C3×C42—52no—
307(C2×D3)⋊C4—52no—
308(C2×D3)⋊C4—52no—
309(C2×D3)⋊C4—52no—
310(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
311C4⋊Dic3—52no—
312C4⋊Dic3C2×AGL(1,𝔽5)12yes—
313#208—102no—
314C2×C3⋊C8—52no—
315C2×C3⋊C8—51no—
316C2×C3⋊C8—51no—
317C2×C3⋊C8—102no—
318ΓL(1,𝔽25)—102no—
319ΓL(1,𝔽25)—102no—
320D3×C8—102no—
321D3×C8—104no—
322D3×C8—104no—
323D3×C8—102no—
324C23×C5—31no—
325C23×C5C2×C4×D311yescharacteristic
326C22×D5—32no—
327C22×D5—31no—
328C22×D5—64no—
329C22×D5—62no—
330C5×D4—62no—
331C2×C4×C5—32no—
332C2×C4×C5—31no—
333C2×C4×C5—62no—
334C2×C4×C5C2×C4×D312yes—
335C22⋊D5—64no—
336C22⋊D5—62no—
337C22⋊D5—62no—
338C2×Dic5—31no—
339C2×Dic5—61no—
340C2×Dic5—61no—
341C2×Dic5C2×C4×D312yes—
342C2×Dic5C2×C4×D311yescharacteristic
343C2×Dic5C2×D1212yes—
344C2×Dic5#23512yes—
345C2×Dic5C23×D311yescharacteristic
346D20—64no—
347C4×D5—64no—
348C4×D5—62no—
349C4×D5—61no—
350C4×D5—62no—
351C4×D5—61no—
352Dic10—62no—
353#170—22no—
354#170—62no—
355#170—61no—
356#170—61no—
357#170C2×D1212yes—
358#170C2×D1212yes—
359C2×#41—151no—
360C23×C4—151no—
361#131—304no—
362C2×M16—51no—
363C2×M16—151no—
364C2×M16—151no—
365C2×M16—151no—
366C22×C8—51no—
367C22×C8—152no—
368C22×C8—152no—
369C22×C8—151no—
370C22×C8—151no—
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—51no—
382C4⋊C8—152no—
383C4⋊C8—152no—
384C22⋊C8—152no—
385C22⋊C8—152no—
386C22⋊C8—152no—
387C22⋊C8—152no—
388#97—52no—
389C4×C8—52no—
390C2×C3×C5—22no—
391C2×C3×C5C2×C4×D411yescharacteristic
392C2×C3×C5C2×#13112yes—
393D15—24no—
394D15—41no—
395C5×D3—41no—
396C23×C3C22×AGL(1,𝔽5)11yescharacteristic
397C22×D3—52no—
398C22×D3—51no—
399C22×D3—104no—
400C22×D3—102no—
401C22×D3C22×AGL(1,𝔽5)11yescharacteristic
402C2×C3×C4—52no—
403C2×C3×C4—52no—
404C2×C3×C4—51no—
405C2×C3×C4—51no—
406C2×C3×C4—52no—
407C2×C3×C4C22×AGL(1,𝔽5)12yes—
408#67—22no—
409#67—102no—
410C2×Dic3—22no—
411C2×Dic3—51no—
412C2×Dic3C22×AGL(1,𝔽5)12yes—
413C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
414D12—104no—
415D12—102no—
416D12—104no—
417C4×D3—102no—
418C4×D3—101no—
419C4×D3—104no—
420C4×D3—102no—
421C4×D3—101no—
422C4×D3—101no—
423C4×D3—101no—
424Dic6—102no—
425C3×C8—52no—
426C3×C8—52no—
427C3×C8—102no—
428C3⋊C8—102no—
429C3⋊C8—101no—
430C3⋊C8—101no—
431C22×C5—22no—
432C22×C5—61no—
433C22×C5—62no—
434C22×C5C4×D1212yes—
435C22×C5C22×C4×D311yescharacteristic
436C2×D5—34no—
437C2×D5—68no—
438C2×D5—62no—
439C2×D5—62no—
440C2×D5—61no—
441C2×D5—62no—
442C4×C5—22no—
443C4×C5—61no—
444C4×C5—62no—
445Dic5—22no—
446Dic5—61no—
447Dic5C4×D1212yes—
448Dic5C22×D1211yescharacteristic
449Dic5C2×#23511yescharacteristic
450C24—151no—
451#41—302no—
452#41—302no—
453C2×Q8—151no—
454C2×D4—31no—
455C2×D4—152no—
456C2×D4—151no—
457C2×D4—151no—
458C2×D4—152no—
459C22×C4—31no—
460C22×C4—51no—
461C22×C4—152no—
462C22×C4—151no—
463C22×C4—152no—
464C22×C4—151no—
465C22×C4—302no—
466C22×C4—302no—
467M16—102no—
468M16—302no—
469M16—302no—
470M16—302no—
471C2×C8—51no—
472C2×C8—51no—
473C2×C8—54no—
474C2×C8—52no—
475C2×C8—152no—
476C2×C8—151no—
477C2×C8—151no—
478C2×C8—304no—
479C2×C8—304no—
480C2×C8—302no—
481C2×C8—302no—
482C2×C8—302no—
483C4⋊C4—32no—
484C4⋊C4—52no—
485C4⋊C4—152no—
486C22⋊C4—32no—
487C22⋊C4—51no—
488C22⋊C4—152no—
489C22⋊C4—152no—
490C22⋊C4—152no—
491C22⋊C4D3×AGL(1,𝔽5)11yescharacteristic
492C42—52no—
493C3×C5#292911yescharacteristic, complemented, Hall
494C22×C3—22no—
495C22×C3D4×AGL(1,𝔽5)12yes—
496C22×C3C23×AGL(1,𝔽5)11yescharacteristic
497C2×D3—21no—
498C2×D3—22no—
499C2×D3—54no—
500C2×D3—108no—
501C2×D3—102no—
502C2×D3—102no—
503C2×D3—101no—
504C2×D3—102no—
505C3×C4—22no—
506C3×C4—52no—
507C3×C4—51no—
508C3×C4—51no—
509C3×C4—102no—
510Dic3—21no—
511Dic3—22no—
512Dic3—101no—
513C2×C5—22no—
514C2×C5—121no—
515C2×C5C2×C4×D1211yescharacteristic
516C2×C5D3×#13112yes—
517D5—64no—
518D5—121no—
519C23—31no—
520C23—152no—
521C23—151no—
522C23—304no—
523C23—302no—
524C23C2×D3×AGL(1,𝔽5)11yescharacteristic
525Q8—302no—
526D4—62no—
527D4—304no—
528D4—302no—
529D4—304no—
530D4—302no—
531C2×C4—32no—
532C2×C4—31no—
533C2×C4—52no—
534C2×C4—52no—
535C2×C4—51no—
536C2×C4—51no—
537C2×C4—52no—
538C2×C4—62no—
539C2×C4—151no—
540C2×C4—302no—
541C2×C4—301no—
542C2×C4—302no—
543C2×C4—304no—
544C2×C4—301no—
545C2×C4—301no—
546C2×C4—301no—
547C2×C4C2×D3×AGL(1,𝔽5)12yes—
548C8—52no—
549C8—52no—
550C8—102no—
551C8—302no—
552C8—301no—
553C8—301no—
554C2×C3—22no—
555C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
556C2×C3D5⋊#13112yes—
557D3—41no—
558D3—104no—
559D3—201no—
560C5#7855311yescharacteristic, complemented, Sylow
561C22—22no—
562C22—61no—
563C22—62no—
564C22—154no—
565C22—308no—
566C22—302no—
567C22—302no—
568C22—301no—
569C22—302no—
570C22AGL(1,𝔽5)×D1212yes—
571C22C22×D3×AGL(1,𝔽5)11yescharacteristic
572C4—22no—
573C4—51no—
574C4—51no—
575C4—52no—
576C4—62no—
577C4—61no—
578C4—102no—
579C4—301no—
580C3#1061728911yescharacteristic, complemented, Sylow
581C2—22no—
582C2—121no—
583C2—304no—
584C2—601no—
585C2C2×AGL(1,𝔽5)×D1211yescharacteristic
586C2#1175684012yes—
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C5×C22⋊C4C2×C4
prime radical#11752872C2
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×#6738C2semidirect
2. C2×#91927C2semidirect
4. #11752355C2semidirect
5. C3×#65691C2semidirect
6. #11752003C2semidirect
7. #11752001C2semidirect
8. #11751895C2semidirect
9. #11751894C2semidirect
10. #11751893C2semidirect
11. #11751876C2semidirect
19. C2×#6740C22semidirect
20. C2×#6739C22semidirect
24. C2×#6738C22semidirect
26. C2×D3×#170C22semidirect
36. #91233C22semidirect
37. C4⋊(C3⋊#170)C22semidirect
38. C3×#4002C22semidirect
39. C3×C4×#170C22semidirect
42. #91171C22semidirect
43. (C2×D3)⋊#170C22semidirect
55. #65691D3semidirect
79. C2×C3×#170D4semidirect
138. #4002C2×D3semidirect
140. C4×#170C2×D3semidirect
233. C2×#170D12semidirect
270. C2×D15#131semidirect
493. C3×C5#2929semidirect
560. C5#78553semidirect
580. C3#10617289semidirect

Maximal subgroups

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

subgroupindex
C22×#67382
C2×#919272
#117528722
#117523552
C3×#656912
#117520032
#117520012
#117518952
#117518942
#117518932
#117518762
#106172893
#785535