Finite Group Databasebeta

← #14415031

Subgroups — #14415031

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
1C2×#91990C211yesmaximal, characteristic, complemented
2C22×#6738C211yesmaximal, characteristic, complemented
3#11754800C211yesmaximal, characteristic
4C3×#65775C211yesmaximal, characteristic, complemented
5#11752043C212yesmaximal, complemented
6#11752041C212yesmaximal, complemented
7#11752011C211yesmaximal, characteristic, complemented
8#11752003C211yesmaximal, characteristic, complemented
9#11751995C211yesmaximal, characteristic, complemented
10#11751903C212yesmaximal, complemented
11#11751882C212yesmaximal, complemented
12#10617289—31nomaximal, Hall
13C2×C3×#711C2211yescharacteristic
14C22×#1200C2211yescharacteristic
15C2×#6762C2211yescharacteristic
16C2×#6762C2211yescharacteristic, complemented
17C22×C3⋊#170C2211yescharacteristic
18C22×C3⋊#170C2211yescharacteristic, complemented
19#91990—24no—
20C2×#6758C2211yescharacteristic, complemented
21C2×D5⋊(C3⋊C8)C2211yescharacteristic, complemented
22C2×C3×#704C2211yescharacteristic
23C22×C3×#170C2211yescharacteristic
24C2×#6738—24no—
25C2×#6738C2212yes—
26C2×#6738C2212yes—
27#91826C411yescharacteristic
28C5×#935C411yescharacteristic
29#91572C411yescharacteristic
30#91552C2212yes—
31#91536C411yescharacteristic
32#91533C412yes—
33#91528C2211yescharacteristic
34#91444C2212yes—
35#91411C412yes—
36C3×C22⋊#170C2212yescomplemented
37C3×#4003C2212yescomplemented
38#91177C2212yescomplemented
39Dic3⋊#170C2212yescomplemented
40#91171C2212yes—
41#91171C2212yes—
42#91169C2212yescomplemented
43Dic3×#170C2212yescomplemented
44#78655—51nomaximal, Hall
45C2×#4131—31no—
46C22×D5⋊C8—31no—
47#65990—31no—
48#65775D311yescharacteristic, complemented
49#65755—31no—
50#65704—32no—
51#65702—32no—
52#65696—31no—
53#65694—32no—
54#65692—32no—
55#65691—31no—
56C23×D15C2×C411yescharacteristic
57C2×C3×C5×D4C2×C411yescharacteristic
58C2×#1220C2×C412yes—
59C22×C5×Dic3C2×C411yescharacteristic
60C2×C3×C22⋊D5C2×C411yescharacteristic
61C2×C3×C22⋊D5C2×C411yescharacteristic
62C22×C3×Dic5C2311yescharacteristic
63C22×C3×Dic5C2311yescharacteristic
64C3×#711—22no—
65C3×#711—22no—
66C2×C3×C4×D5C2311yescharacteristic
67C2×C3×Dic10C2×C411yescharacteristic
68C2×#1202C2×C412yes—
69C2×#1200—22no—
70C2×#1200—22no—
71C2×#1200C2×C412yes—
72C2×#1200C2312yes—
73#6762—22no—
74#6762—22no—
75C2×C3⋊#170—22no—
76C2×C3⋊#170D414yescomplemented
77C2×C3⋊#170C2311yescharacteristic
78C2×C3⋊#170C2311yescharacteristic
79C2×C3⋊#170C2311yescharacteristic
80C2×C3⋊#170C2311yescharacteristic
81#6758—22no—
82D5⋊(C3⋊C8)—22no—
83C3×#704—22no—
84C2×C3×#170—22no—
85C2×C3×#170C2312yes—
86C2×C3×#170C2312yes—
87#6738—24no—
88#6738—24no—
89#6719C2×C411yescharacteristic
90#6714C2×C412yes—
91C5×#217C2×C411yescharacteristic
92C5×#217C2×C412yes—
93C5×Dic3⋊C4C2×C412yes—
94#6669C2×C412yes—
95#6668C2×C412yes—
96#6667C2×C412yes—
97#6665C2×C412yes—
98Dic3×Dic5C2312yes—
99#5828—51no—
100C2×#948—51no—
101C22×D3×C8—51no—
102C3×#408—51no—
103#5126—51no—
104#5122—51no—
105#5102—51no—
106#4728—52no—
107#4728—52no—
108#67⋊C8—52no—
109#67⋊C8—52no—
110C2×#711C2×D311yescharacteristic
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—31no—
121C2×D5⋊C8—32no—
122C2×D5⋊C8—32no—
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⋊#170C2×D312yescomplemented
134#4004—32no—
135#4003—32no—
136#4003—32no—
137#4003C2×D312yescomplemented
138#4002—32no—
139C4⋊#170—32no—
140C4×#170—32no—
141#2929—151noSylow
142C23×C3×C5C22×C411yescharacteristic
143C23×C3×C5C22×C411yescharacteristic
144C22×D15—22no—
145C22×D15—24no—
146C22×D15C22×C412yes—
147C22×C3×D5C22×C411yescharacteristic
148C3×C5×D4—22no—
149C2×C3×C4×C5C22×C411yescharacteristic
150#1220—24no—
151C2×Dic15C22×C412yes—
152C2×C5×Dic3—22no—
153C2×C5×Dic3C22×C412yes—
154C2×C5×Dic3C22×C412yes—
155C3×C22⋊D5—22no—
156C3×C22⋊D5—22no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5C22×C411yescharacteristic
160C2×C3×Dic5C22×C411yescharacteristic
161C2×C3×Dic5C2×D412yes—
162C2×C3×Dic5#4112yes—
163C2×C3×Dic5C2411yescharacteristic
164C3×C4×D5—21no—
165C3×C4×D5—21no—
166C3×Dic10—22no—
167#1202—24no—
168#1200—24no—
169#1200—22no—
170#1200—22no—
171C3⋊#170—21no—
172C3⋊#170—21no—
173C3⋊#170C2×D412yes—
174C3⋊#170C2×D412yes—
175C3×#170—22no—
176C3×#170—22no—
177C2×C3×#41—51no—
178C22×C4×D3—51no—
179C2×C3×M16—51no—
180C22×C3×C8—51no—
181#948—104no—
182#946—52no—
183#938—52no—
184#937—51no—
185#937—51no—
186#935—51no—
187#935AGL(1,𝔽5)11yescharacteristic
188C4×#67—52no—
189C4×#67—52no—
190#926—51no—
191C2×#208—51no—
192C2×#208—51no—
193C2×#208—51no—
194C22×C3⋊C8—51no—
195C22×C3⋊C8—51no—
196C22×C3⋊C8—51no—
197C2×D3×C8—52no—
198C2×D3×C8—52no—
199C2×D3×C8—104no—
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×D5—31no—
209C2×C5×D4C4×D311yescharacteristic
210C22×C4×C5—31no—
211C2×C22⋊D5—32no—
212C2×C22⋊D5C4×D311yescharacteristic
213C2×C22⋊D5C4×D311yescharacteristic
214C22×Dic5C22×D311yescharacteristic
215C22×Dic5C22×D311yescharacteristic
216#711—22no—
217#711—22no—
218C2×D20—32no—
219C2×C4×D5—32no—
220C2×C4×D5—32no—
221C2×C4×D5—62no—
222C2×C4×D5—62no—
223C2×C4×D5C22×D311yescharacteristic
224C2×Dic10C4×D311yescharacteristic
225#704—22no—
226#704—62no—
227#704—62no—
228C2×#170—22no—
229C2×#170—34no—
230C2×#170—31no—
231C2×#170—31no—
232C2×#170—31no—
233C2×#170—31no—
234C2×#170—62no—
235C2×#170C22×D312yes—
236C2×#170C22×D312yes—
237#700—62no—
238D5⋊C8—64no—
239D5⋊C8—64no—
240D5⋊C8—62no—
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—51no—
260#407—151no—
261#405—151no—
262C22×C3×C5—22no—
263C22×C3×C5—21no—
264C22×C3×C5—22no—
265C22×C3×C5C4×D412yes—
266C22×C3×C5C23×C411yescharacteristic
267C2×D15—28no—
268C2×D15—22no—
269C2×D15—22no—
270C2×D15#13114yescomplemented
271C2×C3×D5—21no—
272C2×C3×D5—22no—
273C3×C4×C5—21no—
274Dic15—22no—
275C3×Dic5—21no—
276C3×Dic5C4×D412yes—
277C3×Dic5C22×D411yescharacteristic
278C3×Dic5C2×#4111yescharacteristic
279C5×Dic3—22no—
280C5×Dic3—22no—
281C23×D3—51no—
282C3×#41—102no—
283C3×#41—102no—
284C2×C3×Q8—51no—
285C2×C3×D4—51no—
286C2×C3×D4—51no—
287C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
288C22×C3×C4—51no—
289C22×C3×C4—51no—
290C22×C3×C4—51no—
291C2×#67—52no—
292C2×#67—52no—
293C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
294C2×C4×D3—52no—
295C2×C4×D3—52no—
296C2×C4×D3—102no—
297C2×C4×D3—102no—
298C3×M16—102no—
299C2×C3×C8—52no—
300C2×C3×C8—52no—
301C2×C3×C8—102no—
302#217—51no—
303#217—52no—
304#217C2×AGL(1,𝔽5)11yescharacteristic
305#217C2×AGL(1,𝔽5)12yes—
306(C2×D3)⋊C4—52no—
307(C2×D3)⋊C4—52no—
308C4⋊Dic3—52no—
309Dic3⋊C4—52no—
310Dic3⋊C4C2×AGL(1,𝔽5)12yes—
311C4×Dic3—52no—
312#208—102no—
313#208—102no—
314#208—102no—
315C2×C3⋊C8—51no—
316C2×C3⋊C8—51no—
317C2×C3⋊C8—54no—
318C2×C3⋊C8—51no—
319C2×C3⋊C8—51no—
320C2×C3⋊C8—102no—
321C2×C3⋊C8—102no—
322D3×C8—104no—
323D3×C8—104no—
324C23×C5C2×C4×D311yescharacteristic
325C23×C5C2×C4×D311yescharacteristic
326C22×D5—32no—
327C22×D5—64no—
328C22×D5—62no—
329C22×D5C2×C4×D311yescharacteristic
330C5×D4—22no—
331C2×C4×C5—32no—
332C2×C4×C5—32no—
333C2×C4×C5—62no—
334C2×C4×C5C2×C4×D311yescharacteristic
335C22⋊D5—22no—
336C22⋊D5—22no—
337C22⋊D5—64no—
338C2×Dic5—21no—
339C2×Dic5—21no—
340C2×Dic5—32no—
341C2×Dic5C2×C4×D311yescharacteristic
342C2×Dic5C2×C4×D311yescharacteristic
343C2×Dic5D3×D412yes—
344C2×Dic5#23712yes—
345C2×Dic5C23×D311yescharacteristic
346D20—64no—
347C4×D5—21no—
348C4×D5—21no—
349C4×D5—64no—
350C4×D5—62no—
351C4×D5—62no—
352Dic10—22no—
353#170—22no—
354#170—22no—
355#170—32no—
356#170—32no—
357#170—61no—
358#170—61no—
359C2×#41—51no—
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—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×C5—41no—
392C2×C3×C5C2×C4×D411yescharacteristic
393C2×C3×C5C2×#13112yes—
394D15—24no—
395C3×D5—41no—
396C23×C3—51no—
397C23×C3C22×AGL(1,𝔽5)11yescharacteristic
398C23×C3C22×AGL(1,𝔽5)11yescharacteristic
399C22×D3—52no—
400C22×D3—104no—
401C22×D3—102no—
402C3×Q8—102no—
403C3×D4—22no—
404C3×D4—102no—
405C3×D4—102no—
406C2×C3×C4—52no—
407C2×C3×C4—52no—
408C2×C3×C4—51no—
409C2×C3×C4—51no—
410C2×C3×C4—51no—
411C2×C3×C4—101no—
412C2×C3×C4—101no—
413C2×C3×C4—101no—
414C2×C3×C4—101no—
415C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
416#67—104no—
417#67—104no—
418C2×Dic3—22no—
419C2×Dic3—52no—
420C2×Dic3C22×AGL(1,𝔽5)12yes—
421C2×Dic3C22×AGL(1,𝔽5)12yes—
422C4×D3—102no—
423C4×D3—104no—
424C4×D3—102no—
425C3×C8—102no—
426C3×C8—102no—
427C3⋊C8—52no—
428C3⋊C8—52no—
429C3⋊C8—101no—
430C3⋊C8—101no—
431C22×C5—22no—
432C22×C5—21no—
433C22×C5—22no—
434C22×C5#67⋊C412yes—
435C22×C5C22×C4×D311yescharacteristic
436C2×D5—21no—
437C2×D5—22no—
438C2×D5—34no—
439C2×D5—68no—
440C2×D5—62no—
441C2×D5—62no—
442C4×C5—21no—
443C4×C5—62no—
444C4×C5—62no—
445Dic5—21no—
446Dic5—62no—
447Dic5#67⋊C412yes—
448Dic5C2×D3×D411yescharacteristic
449Dic5C2×#23711yescharacteristic
450C24—151no—
451#41—102no—
452#41—102no—
453C2×Q8—51no—
454C2×D4—51no—
455C2×D4—51no—
456C2×D4—152no—
457C2×D4—152no—
458C2×D4D3×AGL(1,𝔽5)11yescharacteristic
459C22×C4—31no—
460C22×C4—51no—
461C22×C4—51no—
462C22×C4—51no—
463C22×C4—152no—
464C22×C4—152no—
465C22×C4—302no—
466C22×C4—302no—
467M16—102no—
468M16—302no—
469M16—302no—
470M16—302no—
471C2×C8—52no—
472C2×C8—52no—
473C2×C8—102no—
474C2×C8—151no—
475C2×C8—151no—
476C2×C8—154no—
477C2×C8—151no—
478C2×C8—151no—
479C2×C8—304no—
480C2×C8—304no—
481C2×C8—302no—
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—21no—
496C22×C3—22no—
497C22×C3—101no—
498C22×C3—102no—
499C22×C3D4×AGL(1,𝔽5)12yes—
500C22×C3C23×AGL(1,𝔽5)11yescharacteristic
501C2×D3—54no—
502C2×D3—108no—
503C2×D3—102no—
504C2×D3—102no—
505C3×C4—21no—
506C3×C4—52no—
507C3×C4—51no—
508C3×C4—51no—
509C3×C4—101no—
510Dic3—22no—
511Dic3—22no—
512Dic3—102no—
513C2×C5—22no—
514C2×C5—41no—
515C2×C5C2×#67⋊C411yescharacteristic
516C2×C5D3×#13112yes—
517D5—41no—
518D5—64no—
519C23—51no—
520C23—152no—
521C23—304no—
522C23—302no—
523C23C2×D3×AGL(1,𝔽5)11yescharacteristic
524C23C2×D3×AGL(1,𝔽5)11yescharacteristic
525Q8—102no—
526D4—22no—
527D4—102no—
528D4—102no—
529D4—304no—
530D4—304no—
531C2×C4—32no—
532C2×C4—32no—
533C2×C4—52no—
534C2×C4—52no—
535C2×C4—51no—
536C2×C4—51no—
537C2×C4—51no—
538C2×C4—62no—
539C2×C4—101no—
540C2×C4—101no—
541C2×C4—101no—
542C2×C4—101no—
543C2×C4—152no—
544C2×C4—302no—
545C2×C4—302no—
546C2×C4—304no—
547C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
548C8—102no—
549C8—102no—
550C8—152no—
551C8—152no—
552C8—301no—
553C8—301no—
554C2×C3—22no—
555C2×C3—41no—
556C2×C3—201no—
557C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
558C2×C3D5⋊#13112yes—
559D3—104no—
560C5#7865511yescharacteristic, complemented, Sylow
561C22—22no—
562C22—22no—
563C22—21no—
564C22—102no—
565C22—101no—
566C22—154no—
567C22—308no—
568C22—302no—
569C22—302no—
570C22#67⋊AGL(1,𝔽5)12yes—
571C22C22×D3×AGL(1,𝔽5)11yescharacteristic
572C4—21no—
573C4—51no—
574C4—51no—
575C4—52no—
576C4—62no—
577C4—62no—
578C4—101no—
579C4—302no—
580C3#1061728911yescharacteristic, complemented, Sylow
581C2—22no—
582C2—41no—
583C2—201no—
584C2—304no—
585C2#1175684012yes—
586C2C2×#67⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×D4C2×C4
prime radical#11754800C2
cosocleC2×C3×Dic5C24
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5#2929
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#91990C2semidirect
2. C22×#6738C2semidirect
4. C3×#65775C2semidirect
5. #11752043C2semidirect
6. #11752041C2semidirect
7. #11752011C2semidirect
8. #11752003C2semidirect
9. #11751995C2semidirect
10. #11751903C2semidirect
11. #11751882C2semidirect
16. C2×#6762C22semidirect
18. C22×C3⋊#170C22semidirect
20. C2×#6758C22semidirect
21. C2×D5⋊(C3⋊C8)C22semidirect
36. C3×C22⋊#170C22semidirect
37. C3×#4003C22semidirect
38. #91177C22semidirect
39. Dic3⋊#170C22semidirect
42. #91169C22semidirect
43. Dic3×#170C22semidirect
48. #65775D3semidirect
76. C2×C3⋊#170D4semidirect
133. C22⋊#170C2×D3semidirect
137. #4003C2×D3semidirect
270. C2×D15#131semidirect
493. C3×C5#2929semidirect
560. C5#78655semidirect
580. C3#10617289semidirect

Maximal subgroups

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

subgroupindex
C2×#919902
C22×#67382
#117548002
C3×#657752
#117520432
#117520412
#117520112
#117520032
#117519952
#117519032
#117518822
#106172893
#786555