Finite Group Databasebeta

← #14417525

Subgroups — #14417525

Order 1920.

984 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
1#11756987C211yesmaximal, characteristic
2#11756812C211yesmaximal, characteristic, complemented
3C2×D5×#214C211yesmaximal, characteristic
4C3×#65767C211yesmaximal, characteristic
5#11751946C214yesmaximal, complemented
6#11751945C214yesmaximal, complemented
7#11751726C211yesmaximal, characteristic, complemented
8#11751712C211yesmaximal, characteristic, complemented
9#11751696C211yesmaximal, characteristic, complemented
10#10617428—31nomaximal, Hall
11C2×C3×D4×D5C2211yescharacteristic
12C2×D5×Dic6C2211yescharacteristic
13C2×#6763—22no—
14D4×ΓL(1,𝔽16)—22no—
15D4×ΓL(1,𝔽16)C2214yes—
16#91989C2211yescharacteristic
17#91989C2211yescharacteristic
18C3×#4130C2211yescharacteristic
19C3×D5⋊M16C2211yescharacteristic
20#91923—22no—
21ΓL(1,𝔽16)⋊Q8C2214yescomplemented
22AGL(1,𝔽5)×Dic6—22no—
23C2×#6716C411yescharacteristic, complemented
24C2×C5×#214C411yescharacteristic, complemented
25D5×#214C412yescomplemented
26D5×#214C412yescomplemented
27D5×#214C2212yes—
28D5×#214C2212yes—
29C2×#6657C411yescharacteristic, complemented
30C2×#6655C411yescharacteristic, complemented
31C2×D5×C3⋊C8C2211yescharacteristic
32C3×#4007C2214yes—
33#91160C2212yescomplemented
34#91159C2212yescomplemented
35#91154C2214yescomplemented
36#91150C2212yescomplemented
37ΓL(1,𝔽16)⋊C8C2212yescomplemented
38#78916—51nomaximal, Hall
39#66262—31no—
40#66258—31no—
41C2×D5×SD16—31no—
42#65782—31no—
43#65767D311yescharacteristic
44#65735—34no—
45SD16×AGL(1,𝔽5)—34no—
46#65721—31no—
47#65718—31no—
48C23×C3×D5—21no—
49C22×ΓL(1,𝔽16)—24no—
50C2×C3×C5×D4C2×C411yescharacteristic
51C2×Dic30C2×C411yescharacteristic, complemented
52C2×C5×Dic6C2×C411yescharacteristic, complemented
53C2×C3×C22⋊D5—21no—
54C3×D4×D5—21no—
55C3×D4×D5—21no—
56C3×D4×D5C2×C412yes—
57C3×D4×D5C2312yes—
58C2×C3×D20C2×C411yescharacteristic
59C2×C3×C4×D5C2311yescharacteristic
60C2×#1204—21no—
61C2×D5×Dic3—21no—
62D5×Dic6—21no—
63D5×Dic6—21no—
64D5×Dic6C2×C412yescomplemented
65D5×Dic6C2312yes—
66#6763—21no—
67#6763—24no—
68#6763D412yes—
69C4⋊ΓL(1,𝔽16)D412yes—
70C4⋊ΓL(1,𝔽16)C2312yes—
71C4×ΓL(1,𝔽16)—21no—
72C4×ΓL(1,𝔽16)C2312yes—
73C3×C22⋊AGL(1,𝔽5)—21no—
74C3×#704—21no—
75C3×C4⋊AGL(1,𝔽5)C2312yes—
76C3×C4×AGL(1,𝔽5)—21no—
77C3×#700—21no—
78C3×D5⋊C8C2312yes—
79Dic3⋊AGL(1,𝔽5)—24no—
80Dic3×AGL(1,𝔽5)—24no—
81#6716C2×C412yescomplemented
82#6716C2×C412yescomplemented
83C2×C3⋊#168C2×C411yescharacteristic, complemented
84C5×#214C2×C412yescomplemented
85C5×#214C2×C412yescomplemented
86C2×C5×C3⋊C8C2×C411yescharacteristic, complemented
87#6657C2×C412yescomplemented
88#6657C2×C412yescomplemented
89#6655C2×C412yescomplemented
90#6655C2×C412yescomplemented
91D5×C3⋊C8C2×C411yescharacteristic, complemented
92D5×C3⋊C8C2×C411yescharacteristic, complemented
93D5×C3⋊C8C2311yescharacteristic
94D5×C3⋊C8C2311yescharacteristic
95#5798—51no—
96C22×#214—51no—
97#5516—51no—
98C3×#418—51no—
99#5122—51no—
100#5120—51no—
101#4998—51no—
102#4758—54no—
103#4757—54no—
104C2×Q8×D5—31no—
105C2×D4×D5C2×D311yescharacteristic
106C2×C22⋊AGL(1,𝔽5)—62no—
107Q8×AGL(1,𝔽5)—34no—
108Q8×AGL(1,𝔽5)—62no—
109D4×AGL(1,𝔽5)—34no—
110D4×AGL(1,𝔽5)—62no—
111#4130—31no—
112#4130—31no—
113#4130—62no—
114#4130C2×D311yescharacteristic
115D5⋊M16C2×D311yescharacteristic
116C2×C5×SD16—31no—
117C2×#688—31no—
118C2×#687—31no—
119D5×SD16—32no—
120D5×SD16—32no—
121D5×SD16—32no—
122D5×SD16—32no—
123C2×#677—31no—
124C2×C8×D5—31no—
125#4009—34no—
126#4007C2×D314yes—
127#3994—32no—
128#3993—32no—
129#3992—32no—
130C8×AGL(1,𝔽5)—32no—
131#2949—151noSylow
132C23×C3×C5—21no—
133C22×C3×D5—21no—
134C22×C3×D5—22no—
135C22×C3×D5—22no—
136C22×C3×D5—82no—
137C22×C3×D5C2×D411yescharacteristic
138C2×ΓL(1,𝔽16)—22no—
139C2×ΓL(1,𝔽16)—84no—
140C2×ΓL(1,𝔽16)C2×D414yes—
141C2×C3×AGL(1,𝔽5)—22no—
142C3×C5×D4—21no—
143C3×C5×D4C22×C412yes—
144C2×C3×C4×C5C22×C411yescharacteristic
145C2×Dic15—21no—
146Dic30—21no—
147Dic30C22×C412yes—
148C2×C5×Dic3—21no—
149C5×Dic6—21no—
150C5×Dic6C22×C412yes—
151C3×C22⋊D5—22no—
152C3×C22⋊D5—22no—
153C2×C3×Dic5#4111yescharacteristic
154C3×D20—21no—
155C3×D20C22×C412yes—
156C3×C4×D5C22×C411yescharacteristic
157C3×C4×D5C2×D411yescharacteristic
158C3×C4×D5#4111yescharacteristic
159C3×C4×D5C2411yescharacteristic
160#1204—22no—
161#1204—22no—
162D5×Dic3—22no—
163D5×Dic3—22no—
164C3×#170—22no—
165C3⋊#168C22×C411yescharacteristic
166C3⋊#168C22×C411yescharacteristic
167C5×C3⋊C8C22×C411yescharacteristic
168C5×C3⋊C8C22×C411yescharacteristic
169C22×C3×D4—51no—
170C22×Dic6—51no—
171C2×C3×M16—51no—
172C3×#117—51no—
173C2×#217—102no—
174D4×Dic3—54no—
175D4×Dic3—102no—
176C2×#214—51no—
177C2×#214—51no—
178C2×#214—52no—
179C2×#214—52no—
180C2×#214—52no—
181C2×#214—52no—
182C2×#214—51no—
183C2×#214AGL(1,𝔽5)11yescharacteristic, complemented
184#926—51no—
185#926—51no—
186C22×C3⋊C8—51no—
187#887—54no—
188#877—102no—
189C4×Dic6—102no—
190C3×#102—54no—
191#816—54no—
192#815—52no—
193C8×Dic3—52no—
194#808—52no—
195#807—52no—
196C23×D5—21no—
197C22×AGL(1,𝔽5)—64no—
198C2×C5×Q8—31no—
199C2×C5×D4C4×D311yescharacteristic
200C2×C22⋊D5—21no—
201Q8×D5—32no—
202Q8×D5—32no—
203Q8×D5—61no—
204Q8×D5—61no—
205D4×D5—21no—
206D4×D5—21no—
207D4×D5C4×D312yes—
208D4×D5C22×D312yes—
209C2×D20C4×D311yescharacteristic
210C2×C4×D5—61no—
211C2×C4×D5C22×D311yescharacteristic
212C2×Dic10—31no—
213C2×Dic10—61no—
214C22⋊AGL(1,𝔽5)—21no—
215C22⋊AGL(1,𝔽5)—32no—
216C22⋊AGL(1,𝔽5)—64no—
217C22⋊AGL(1,𝔽5)—61no—
218#704—21no—
219C4⋊AGL(1,𝔽5)—32no—
220C4⋊AGL(1,𝔽5)—32no—
221C4⋊AGL(1,𝔽5)—64no—
222C4⋊AGL(1,𝔽5)C22×D312yes—
223C4×AGL(1,𝔽5)—21no—
224C4×AGL(1,𝔽5)—32no—
225C4×AGL(1,𝔽5)—64no—
226C4×AGL(1,𝔽5)—61no—
227#700—21no—
228D5⋊C8C22×D312yes—
229C5×SD16—32no—
230C5×SD16—32no—
231C2×C5×C8—31no—
232#688—32no—
233#688—32no—
234#687—32no—
235#687—32no—
236C2×#168—31no—
237#677—32no—
238#677—32no—
239C8×D5—31no—
240C8×D5—31no—
241C8×D5—31no—
242C8×D5—31no—
243C22×SD16—151no—
244#519—151no—
245#518—151no—
246#440—154no—
247C4×SD16—154no—
248#427—151no—
249#419—151no—
250#418—51no—
251#405—151no—
252C22×C3×C5—22no—
253C22×C3×C5—81no—
254C22×C3×C5C4×D411yescharacteristic
255C2×C3×D5—22no—
256C2×C3×D5—22no—
257C2×C3×D5—44no—
258C2×C3×D5—82no—
259C2×C3×D5—81no—
260C2×C3×D5C4×D411yescharacteristic
261C2×C3×D5C22×D411yescharacteristic
262ΓL(1,𝔽16)—24no—
263ΓL(1,𝔽16)—42no—
264C3×AGL(1,𝔽5)—42no—
265C3×C4×C5C4×D411yescharacteristic
266C3×C4×C5C23×C411yescharacteristic
267Dic15—22no—
268C3×Dic5C4×D411yescharacteristic
269C3×Dic5C2×#4111yescharacteristic
270C5×Dic3—22no—
271C24×C3—101no—
272C2×C3×D4—52no—
273C2×C3×D4—52no—
274C2×C3×D4—51no—
275C2×C3×D4—101no—
276C2×C3×D4—101no—
277C2×C3×D4—101no—
278C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
279C22×C3×C4—51no—
280C22×Dic3—101no—
281C22×Dic3—104no—
282C2×Dic6—51no—
283C2×Dic6—52no—
284C2×Dic6—52no—
285C2×Dic6—101no—
286C2×Dic6—101no—
287C2×Dic6—101no—
288C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic, complemented
289C3×M16—101no—
290C3×M16—101no—
291C2×C3×C8—52no—
292C3×C4⋊C4—52no—
293C3×C22⋊C4—101no—
294C3×C42—101no—
295#217—52no—
296#217—104no—
297#217—101no—
298#214—52no—
299#214—52no—
300#214—52no—
301#214—52no—
302#214—52no—
303#214—52no—
304#214C2×AGL(1,𝔽5)12yescomplemented
305#214C2×AGL(1,𝔽5)12yescomplemented
306C4⋊Dic3—52no—
307C4⋊Dic3—52no—
308Dic3⋊C4—104no—
309C4×Dic3—52no—
310C4×Dic3—104no—
311C4×Dic3—101no—
312C2×C3⋊C8—51no—
313C2×C3⋊C8—51no—
314C2×C3⋊C8—51no—
315C2×C3⋊C8—51no—
316C2×C3⋊C8—51no—
317C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic, complemented
318C23×C5—21no—
319C22×D5—21no—
320C22×D5—22no—
321C22×D5—22no—
322C22×D5—82no—
323C22×D5D3×D411yescharacteristic
324C2×AGL(1,𝔽5)—22no—
325C2×AGL(1,𝔽5)—34no—
326C2×AGL(1,𝔽5)—62no—
327C2×AGL(1,𝔽5)—244no—
328C5×Q8—32no—
329C5×Q8—61no—
330C5×D4—21no—
331C5×D4C2×C4×D312yes—
332C2×C4×C5—61no—
333C2×C4×C5C2×C4×D311yescharacteristic
334C22⋊D5—22no—
335C22⋊D5—22no—
336C2×Dic5—61no—
337C2×Dic5#23711yescharacteristic
338D20—21no—
339D20C2×C4×D312yes—
340C4×D5—62no—
341C4×D5—62no—
342C4×D5C2×C4×D311yescharacteristic
343C4×D5D3×D411yescharacteristic
344C4×D5#23711yescharacteristic
345C4×D5C23×D311yescharacteristic
346Dic10—32no—
347Dic10—61no—
348Dic10—62no—
349Dic10—62no—
350#170—22no—
351C5×C8—31no—
352C5×C8—31no—
353#168—31no—
354#168—31no—
355C22×Q8—151no—
356C22×D4—51no—
357C2×SD16—31no—
358C2×SD16—151no—
359C2×SD16—152no—
360C2×SD16—152no—
361C2×SD16—152no—
362C2×SD16—152no—
363C2×SD16—151no—
364C2×SD16—151no—
365C2×M16—51no—
366C22×C8—151no—
367C4×Q8—154no—
368C4×Q8—302no—
369C4×D4—154no—
370C4×D4—302no—
371#117—51no—
372#117—151no—
373#117—151no—
374#117—302no—
375C2×C22⋊C4—302no—
376#107—152no—
377#106—152no—
378#103—154no—
379#102—54no—
380#97—152no—
381C4×C8—152no—
382C2×C3×C5—21no—
383C2×C3×C5—42no—
384C2×C3×C5C2×C4×D411yescharacteristic
385C3×D5—21no—
386C3×D5—42no—
387C3×D5#57712yes—
388C23×C3—21no—
389C23×C3—51no—
390C23×C3—102no—
391C23×C3—102no—
392C23×C3—101no—
393C23×C3—402no—
394C3×D4—21no—
395C3×D4—52no—
396C3×D4—102no—
397C3×D4—102no—
398C3×D4—101no—
399C3×D4C22×AGL(1,𝔽5)12yes—
400C2×C3×C4—51no—
401C2×C3×C4—51no—
402C2×C3×C4—51no—
403C2×C3×C4—51no—
404C2×C3×C4—51no—
405C2×C3×C4—102no—
406C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
407C2×Dic3—21no—
408C2×Dic3—54no—
409C2×Dic3—102no—
410C2×Dic3—102no—
411C2×Dic3—101no—
412C2×Dic3—102no—
413C2×Dic3—404no—
414Dic6—21no—
415Dic6—52no—
416Dic6—102no—
417Dic6—102no—
418Dic6—101no—
419Dic6C22×AGL(1,𝔽5)12yes—
420C3×C8—102no—
421C3⋊C8—51no—
422C3⋊C8—51no—
423C3⋊C8C22×AGL(1,𝔽5)11yescharacteristic
424C3⋊C8C22×AGL(1,𝔽5)11yescharacteristic
425C22×C5—22no—
426C22×C5—81no—
427C22×C5#67⋊C411yescharacteristic
428C2×D5—22no—
429C2×D5—22no—
430C2×D5—44no—
431C2×D5—82no—
432C2×D5—81no—
433C2×D5#67⋊C411yescharacteristic
434C2×D5C2×D3×D411yescharacteristic
435AGL(1,𝔽5)—42no—
436AGL(1,𝔽5)—64no—
437AGL(1,𝔽5)—122no—
438C4×C5—62no—
439C4×C5#67⋊C411yescharacteristic
440C4×C5C22×C4×D311yescharacteristic
441Dic5—62no—
442Dic5#67⋊C411yescharacteristic
443Dic5C2×#23711yescharacteristic
444C24—101no—
445C2×Q8—31no—
446C2×Q8—152no—
447C2×Q8—152no—
448C2×Q8—151no—
449C2×Q8—301no—
450C2×Q8—301no—
451C2×Q8—301no—
452C2×D4—52no—
453C2×D4—52no—
454C2×D4—51no—
455C2×D4—101no—
456C2×D4—101no—
457C2×D4—101no—
458C2×D4D3×AGL(1,𝔽5)11yescharacteristic
459C22×C4—51no—
460C22×C4—301no—
461C22×C4—304no—
462SD16—32no—
463SD16—32no—
464SD16—152no—
465SD16—152no—
466SD16—152no—
467SD16—152no—
468SD16—152no—
469SD16—152no—
470M16—101no—
471M16—101no—
472C2×C8—31no—
473C2×C8—52no—
474C2×C8—151no—
475C2×C8—151no—
476C2×C8—151no—
477C2×C8—151no—
478C2×C8—151no—
479C4⋊C4—52no—
480C4⋊C4—152no—
481C4⋊C4—152no—
482C4⋊C4—304no—
483C22⋊C4—101no—
484C22⋊C4—152no—
485C22⋊C4—304no—
486C22⋊C4—301no—
487C42—101no—
488C42—152no—
489C42—301no—
490C42—304no—
491C3×C5#294911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—51no—
494C22×C3—51no—
495C22×C3—81no—
496C22×C3—102no—
497C22×C3—102no—
498C22×C3—204no—
499C22×C3—401no—
500C22×C3—402no—
501C22×C3D4×AGL(1,𝔽5)11yescharacteristic
502C3×C4—51no—
503C3×C4—51no—
504C3×C4—202no—
505C3×C4D4×AGL(1,𝔽5)11yescharacteristic
506C3×C4C23×AGL(1,𝔽5)11yescharacteristic
507Dic3—22no—
508Dic3—104no—
509Dic3—102no—
510Dic3—202no—
511C2×C5—21no—
512C2×C5—42no—
513C2×C5C2×#67⋊C411yescharacteristic
514D5—21no—
515D5—42no—
516D5#577512yes—
517C23—21no—
518C23—51no—
519C23—102no—
520C23—102no—
521C23—101no—
522C23—402no—
523Q8—32no—
524Q8—61no—
525Q8—152no—
526Q8—302no—
527Q8—302no—
528Q8—301no—
529D4—21no—
530D4—52no—
531D4—102no—
532D4—102no—
533D4—101no—
534D4C2×D3×AGL(1,𝔽5)12yes—
535C2×C4—51no—
536C2×C4—51no—
537C2×C4—51no—
538C2×C4—51no—
539C2×C4—51no—
540C2×C4—61no—
541C2×C4—102no—
542C2×C4—154no—
543C2×C4—302no—
544C2×C4—302no—
545C2×C4—301no—
546C2×C4—302no—
547C2×C4—1204no—
548C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
549C8—31no—
550C8—31no—
551C8—102no—
552C8—151no—
553C8—151no—
554C2×C3—21no—
555C2×C3—42no—
556C2×C3—52no—
557C2×C3—101no—
558C2×C3—202no—
559C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
560C5#7891611yescharacteristic, complemented, Sylow
561C22—22no—
562C22—51no—
563C22—51no—
564C22—81no—
565C22—102no—
566C22—102no—
567C22—204no—
568C22—401no—
569C22—402no—
570C22#67⋊AGL(1,𝔽5)11yescharacteristic
571C4—51no—
572C4—51no—
573C4—62no—
574C4—202no—
575C4—304no—
576C4—302no—
577C4—602no—
578C4#67⋊AGL(1,𝔽5)11yescharacteristic
579C4C22×D3×AGL(1,𝔽5)11yescharacteristic
580C3#1061742811yescharacteristic, complemented, Sylow
581C2—21no—
582C2—42no—
583C2—52no—
584C2—101no—
585C2—202no—
586C2C2×#67⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2C2×#67⋊AGL(1,𝔽5)
commutatorC3×C4×C5C23×C4
FrattiniC4C22×D3×AGL(1,𝔽5)
FittingC2×C3×C5×D4C2×C4
socleC2×C3×C5C2×C4×D4
prime radicalC2×C3×D4×D5C22
cosocleC3×C4×D5C24
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5#2949
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11756812C2semidirect
5. #11751946C2semidirect
6. #11751945C2semidirect
7. #11751726C2semidirect
8. #11751712C2semidirect
9. #11751696C2semidirect
21. ΓL(1,𝔽16)⋊Q8C22semidirect
23. C2×#6716C4semidirect
24. C2×C5×#214C4semidirect
25. D5×#214C4semidirect
26. D5×#214C4semidirect
29. C2×#6657C4semidirect
30. C2×#6655C4semidirect
33. #91160C22semidirect
34. #91159C22semidirect
35. #91154C22semidirect
36. #91150C22semidirect
37. ΓL(1,𝔽16)⋊C8C22semidirect
51. C2×Dic30C2×C4semidirect
52. C2×C5×Dic6C2×C4semidirect
64. D5×Dic6C2×C4semidirect
81. #6716C2×C4semidirect
82. #6716C2×C4semidirect
83. C2×C3⋊#168C2×C4semidirect
84. C5×#214C2×C4semidirect
85. C5×#214C2×C4semidirect
86. C2×C5×C3⋊C8C2×C4semidirect
87. #6657C2×C4semidirect
88. #6657C2×C4semidirect
89. #6655C2×C4semidirect
90. #6655C2×C4semidirect
91. D5×C3⋊C8C2×C4semidirect
92. D5×C3⋊C8C2×C4semidirect
183. C2×#214AGL(1,𝔽5)semidirect
288. C2×Dic6C2×AGL(1,𝔽5)semidirect
304. #214C2×AGL(1,𝔽5)semidirect
305. #214C2×AGL(1,𝔽5)semidirect
317. C2×C3⋊C8C2×AGL(1,𝔽5)semidirect
491. C3×C5#2949semidirect
560. C5#78916semidirect
580. C3#10617428semidirect

Maximal subgroups

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

subgroupindex
#117569872
#117568122
C2×D5×#2142
C3×#657672
#117519462
#117519452
#117517262
#117517122
#117516962
#106174283
#789165