Finite Group Databasebeta

← #14417523

Subgroups — #14417523

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
1C3×#66262C211yesmaximal, characteristic, complemented
2#11756837C211yesmaximal, characteristic
3C2×D5×#215C211yesmaximal, characteristic
4#11752441C211yesmaximal, characteristic, complemented
5#11751964C214yesmaximal, complemented
6AGL(1,𝔽5)×#215C214yesmaximal, complemented
7#11751726C211yesmaximal, characteristic, complemented
8#11751706C211yesmaximal, characteristic
9#11751695C211yesmaximal, characteristic, complemented
10#10617428—31nomaximal, Hall
11C2×C3×Q8×D5C2211yescharacteristic
12C2×D5×D12C2211yescharacteristic
13#91989C2211yescharacteristic
14D5⋊#208C2211yescharacteristic
15C3×Q8×AGL(1,𝔽5)—22no—
16C3×Q8×AGL(1,𝔽5)C2214yescomplemented
17C3×#4130—22no—
18C3×#4130C2211yescharacteristic
19C3×#4130C2211yescharacteristic
20D3×C22⋊AGL(1,𝔽5)—22no—
21#91921—22no—
22AGL(1,𝔽5)×D12C2214yes—
23C2×#6717C411yescharacteristic, complemented
24C2×C5×#215C411yescharacteristic, complemented
25D5×#215C412yescomplemented
26D5×#215C412yescomplemented
27D5×#215C2212yes—
28D5×#215C2212yes—
29C2×#6658C411yescharacteristic, complemented
30C2×#6656C411yescharacteristic, complemented
31C2×D5×C3⋊C8C2211yescharacteristic
32#91238C2214yescomplemented
33#91160C2212yescomplemented
34#91159C2212yescomplemented
35#91151C2214yes—
36#91149C2212yescomplemented
37AGL(1,𝔽5)×C3⋊C8C2212yescomplemented
38#82548—51nomaximal, Hall
39#66262D311yescharacteristic, complemented
40#66258—31no—
41C2×D5×SD16—31no—
42#65782—31no—
43#65767—31no—
44#65735—34no—
45SD16×AGL(1,𝔽5)—34no—
46#65721—31no—
47#65718—31no—
48C22×D3×D5—21no—
49C2×D3×AGL(1,𝔽5)—24no—
50C2×C3×C5×Q8C2×C411yescharacteristic, complemented
51C2×D60C2×C411yescharacteristic
52C2×C5×D12C2×C411yescharacteristic
53C3×Q8×D5—21no—
54C3×Q8×D5—21no—
55C3×Q8×D5C2×C412yescomplemented
56C3×Q8×D5C2312yes—
57C2×C3×C4×D5—21no—
58C2×C3×C4×D5C2311yescharacteristic
59C2×C3×Dic10—21no—
60C2×C3×Dic10C2×C411yescharacteristic, complemented
61C2×#1203—21no—
62D5×D12—21no—
63D5×D12—21no—
64D5×D12C2×C412yes—
65D5×D12C2312yes—
66#6763—21no—
67#6762—21no—
68C4⋊ΓL(1,𝔽16)C2312yes—
69C4×ΓL(1,𝔽16)—21no—
70#6758—21no—
71D5⋊(C3⋊C8)C2312yes—
72C3×C22⋊AGL(1,𝔽5)—21no—
73C3×C22⋊AGL(1,𝔽5)D412yes—
74C3×C4⋊AGL(1,𝔽5)—24no—
75C3×C4⋊AGL(1,𝔽5)D412yes—
76C3×C4⋊AGL(1,𝔽5)C2312yes—
77C3×C4×AGL(1,𝔽5)—24no—
78C3×C4×AGL(1,𝔽5)—21no—
79C3×C4×AGL(1,𝔽5)C2312yes—
80(C2×D3)⋊AGL(1,𝔽5)—24no—
81#6717C2×C412yescomplemented
82#6717C2×C412yescomplemented
83C2×C3⋊#168C2×C411yescharacteristic, complemented
84C5×#215C2×C412yescomplemented
85C5×#215C2×C412yescomplemented
86C2×C5×C3⋊C8C2×C411yescharacteristic, complemented
87#6658C2×C412yescomplemented
88#6658C2×C412yescomplemented
89#6656C2×C412yescomplemented
90#6656C2×C412yescomplemented
91D5×C3⋊C8C2×C411yescharacteristic, complemented
92D5×C3⋊C8C2×C411yescharacteristic, complemented
93D5×C3⋊C8C2311yescharacteristic
94D5×C3⋊C8C2311yescharacteristic
95C3×#519—51no—
96C22×#215—51no—
97#5521—51no—
98#5225—51no—
99#5026—54no—
100C4×#215—54no—
101#5001—51no—
102#4998—51no—
103#4997—51no—
104C2×Q8×D5C2×D311yescharacteristic
105C2×D4×D5—31no—
106C2×C22⋊AGL(1,𝔽5)—62no—
107Q8×AGL(1,𝔽5)—22no—
108Q8×AGL(1,𝔽5)C2×D314yescomplemented
109D4×AGL(1,𝔽5)—34no—
110D4×AGL(1,𝔽5)—62no—
111#4130—22no—
112#4130—31no—
113#4130C2×D311yescharacteristic
114#4130C2×D311yescharacteristic
115D5⋊M16—31no—
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#4007—34no—
127#3994—32no—
128#3993—32no—
129#3992—32no—
130C8×AGL(1,𝔽5)—32no—
131#2949—151noSylow
132C22×D15—21no—
133C22×C5×D3—21no—
134C22×C3×D5C2×D411yescharacteristic
135C2×D3×D5—22no—
136C2×D3×D5—22no—
137C2×D3×D5—82no—
138C2×ΓL(1,𝔽16)—22no—
139C2×C3×AGL(1,𝔽5)—22no—
140C2×C3×AGL(1,𝔽5)C2×D414yes—
141D3×AGL(1,𝔽5)—84no—
142C3×C5×Q8—21no—
143C3×C5×Q8C22×C412yes—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5C22×C411yescharacteristic
146D60—21no—
147D60C22×C412yes—
148C5×D12—21no—
149C5×D12C22×C412yes—
150C2×C3×Dic5—21no—
151C2×C3×Dic5#4111yescharacteristic
152C3×C4×D5—22no—
153C3×C4×D5—22no—
154C3×C4×D5C22×C411yescharacteristic
155C3×C4×D5C2×D411yescharacteristic
156C3×C4×D5#4111yescharacteristic
157C3×C4×D5C2411yescharacteristic
158C3×Dic10—21no—
159C3×Dic10—22no—
160C3×Dic10—22no—
161C3×Dic10C22×C412yes—
162#1203—22no—
163#1203—22no—
164C3⋊#170—22no—
165C3⋊#168C22×C411yescharacteristic
166C3⋊#168C22×C411yescharacteristic
167C5×C3⋊C8C22×C411yescharacteristic
168C5×C3⋊C8C22×C411yescharacteristic
169C22×C3×Q8—51no—
170C22×D12—51no—
171C3×C4×Q8—54no—
172C3×C4×Q8—102no—
173C3×#117—51no—
174C3×#117—51no—
175C3×#117—102no—
176C2×#215—51no—
177C2×#215—51no—
178C2×#215—52no—
179C2×#215—52no—
180C2×#215—52no—
181C2×#215—52no—
182C2×#215—51no—
183C2×#215AGL(1,𝔽5)11yescharacteristic, complemented
184#926—51no—
185C2×#208—51no—
186C22×C3⋊C8—51no—
187#893—102no—
188D3×C22⋊C4—102no—
189C4×D12—54no—
190#835—54no—
191#809—54no—
192#808—52no—
193#807—52no—
194#803—52no—
195C4×C3⋊C8—52no—
196C23×D5—61no—
197C22×AGL(1,𝔽5)—64no—
198C2×C5×Q8C4×D311yescharacteristic, complemented
199C2×C5×D4—31no—
200C2×C22⋊D5—61no—
201Q8×D5—21no—
202Q8×D5—21no—
203Q8×D5C4×D312yescomplemented
204Q8×D5C22×D312yes—
205D4×D5—32no—
206D4×D5—32no—
207D4×D5—61no—
208D4×D5—61no—
209C2×D20—31no—
210C2×C4×D5—21no—
211C2×C4×D5C22×D311yescharacteristic
212C2×Dic10—21no—
213C2×Dic10C4×D311yescharacteristic, complemented
214C22⋊AGL(1,𝔽5)—21no—
215C22⋊AGL(1,𝔽5)—61no—
216C22⋊AGL(1,𝔽5)—64no—
217C22⋊AGL(1,𝔽5)#6712yes—
218#704—61no—
219C4⋊AGL(1,𝔽5)—24no—
220C4⋊AGL(1,𝔽5)—32no—
221C4⋊AGL(1,𝔽5)#6712yes—
222C4⋊AGL(1,𝔽5)C22×D312yes—
223C4×AGL(1,𝔽5)—24no—
224C4×AGL(1,𝔽5)—21no—
225C4×AGL(1,𝔽5)—61no—
226C4×AGL(1,𝔽5)C22×D312yes—
227#700—61no—
228D5⋊C8—32no—
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—51no—
245#518—151no—
246#440—154no—
247C4×SD16—154no—
248#427—151no—
249#419—151no—
250#418—151no—
251#405—151no—
252C22×C3×C5C4×D411yescharacteristic
253C2×D15—22no—
254C2×D15—81no—
255C2×C5×D3—22no—
256C2×C5×D3—81no—
257C2×C3×D5—22no—
258C2×C3×D5C4×D411yescharacteristic
259C2×C3×D5C22×D411yescharacteristic
260D3×D5—44no—
261D3×D5—82no—
262ΓL(1,𝔽16)—42no—
263C3×AGL(1,𝔽5)—24no—
264C3×AGL(1,𝔽5)—42no—
265C3×C4×C5—22no—
266C3×C4×C5C4×D411yescharacteristic
267C3×C4×C5C23×C411yescharacteristic
268C3×Dic5—22no—
269C3×Dic5C4×D411yescharacteristic
270C3×Dic5C2×#4111yescharacteristic
271C23×D3—101no—
272C2×C3×Q8—52no—
273C2×C3×Q8—52no—
274C2×C3×Q8—51no—
275C2×C3×Q8—101no—
276C2×C3×Q8—101no—
277C2×C3×Q8—101no—
278C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
279C22×C3×C4—51no—
280C22×C3×C4—101no—
281C2×D12—51no—
282C2×D12—52no—
283C2×D12—52no—
284C2×D12—101no—
285C2×D12—101no—
286C2×D12—101no—
287C2×D12C2×AGL(1,𝔽5)11yescharacteristic
288C2×C4×D3—104no—
289C3×C4⋊C4—52no—
290C3×C4⋊C4—52no—
291C3×C4⋊C4—104no—
292C3×C22⋊C4—52no—
293C3×C22⋊C4—101no—
294C3×C42—52no—
295C3×C42—101no—
296C3×C42—104no—
297#217—101no—
298#215—52no—
299#215—52no—
300#215—52no—
301#215—52no—
302#215—52no—
303#215—52no—
304#215C2×AGL(1,𝔽5)12yescomplemented
305#215C2×AGL(1,𝔽5)12yescomplemented
306(C2×D3)⋊C4—104no—
307C4⋊Dic3—52no—
308C4×Dic3—101no—
309#208—101no—
310#208—101no—
311C2×C3⋊C8—51no—
312C2×C3⋊C8—51no—
313C2×C3⋊C8—51no—
314C2×C3⋊C8—51no—
315C2×C3⋊C8—51no—
316C2×C3⋊C8—52no—
317C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic, complemented
318C23×C5—61no—
319C22×D5—61no—
320C22×D5—62no—
321C22×D5—62no—
322C22×D5—242no—
323C22×D5C2×#6711yescharacteristic
324C2×AGL(1,𝔽5)—22no—
325C2×AGL(1,𝔽5)—62no—
326C2×AGL(1,𝔽5)—244no—
327C2×AGL(1,𝔽5)C2×#6714yes—
328C5×Q8—21no—
329C5×Q8C2×C4×D312yes—
330C5×D4—32no—
331C5×D4—61no—
332C2×C4×C5—21no—
333C2×C4×C5C2×C4×D311yescharacteristic
334C22⋊D5—62no—
335C22⋊D5—62no—
336C2×Dic5—21no—
337C2×Dic5#23511yescharacteristic
338D20—32no—
339D20—61no—
340C4×D5—22no—
341C4×D5—22no—
342C4×D5C2×C4×D311yescharacteristic
343C4×D5#23511yescharacteristic
344C4×D5C2×#6711yescharacteristic
345C4×D5C23×D311yescharacteristic
346Dic10—21no—
347Dic10—22no—
348Dic10—22no—
349Dic10C2×C4×D312yes—
350#170—62no—
351C5×C8—31no—
352C5×C8—31no—
353#168—31no—
354#168—31no—
355C22×Q8—51no—
356C22×D4—151no—
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—151no—
366C22×C8—151no—
367C4×Q8—54no—
368C4×Q8—102no—
369C4×D4—154no—
370C4×D4—302no—
371#117—51no—
372#117—51no—
373#117—102no—
374#117—151no—
375C2×C22⋊C4—302no—
376#107—152no—
377#106—152no—
378#103—154no—
379#102—154no—
380#97—152no—
381C4×C8—152no—
382C2×C3×C5—21no—
383C2×C3×C5C2×C4×D411yescharacteristic
384D15—42no—
385C3×D5—21no—
386C3×D5#57712yes—
387C5×D3—42no—
388C23×C3—51no—
389C22×D3—21no—
390C22×D3—102no—
391C22×D3—102no—
392C22×D3—101no—
393C22×D3—402no—
394C3×Q8—21no—
395C3×Q8—52no—
396C3×Q8—102no—
397C3×Q8—102no—
398C3×Q8—101no—
399C3×Q8C22×AGL(1,𝔽5)12yes—
400C2×C3×C4—21no—
401C2×C3×C4—51no—
402C2×C3×C4—51no—
403C2×C3×C4—51no—
404C2×C3×C4—51no—
405C2×C3×C4—51no—
406C2×C3×C4—54no—
407C2×C3×C4—102no—
408C2×C3×C4—102no—
409C2×C3×C4—101no—
410C2×C3×C4—102no—
411C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
412C2×Dic3—102no—
413D12—21no—
414D12—52no—
415D12—102no—
416D12—102no—
417D12—101no—
418D12C22×AGL(1,𝔽5)12yes—
419C4×D3—404no—
420C3⋊C8—51no—
421C3⋊C8—51no—
422C3⋊C8—102no—
423C3⋊C8C22×AGL(1,𝔽5)11yescharacteristic
424C3⋊C8C22×AGL(1,𝔽5)11yescharacteristic
425C22×C5—62no—
426C22×C5—241no—
427C22×C5C4×#6711yescharacteristic
428C2×D5—22no—
429C2×D5—62no—
430C2×D5—124no—
431C2×D5—242no—
432C2×D5—241no—
433C2×D5C4×#6711yescharacteristic
434C2×D5C22×#6711yescharacteristic
435AGL(1,𝔽5)—24no—
436AGL(1,𝔽5)—42no—
437AGL(1,𝔽5)—122no—
438C4×C5—22no—
439C4×C5C4×#6711yescharacteristic
440C4×C5C22×C4×D311yescharacteristic
441Dic5—22no—
442Dic5C4×#6711yescharacteristic
443Dic5C2×#23511yescharacteristic
444C24—301no—
445C2×Q8—52no—
446C2×Q8—52no—
447C2×Q8—51no—
448C2×Q8—101no—
449C2×Q8—101no—
450C2×Q8—101no—
451C2×Q8D3×AGL(1,𝔽5)11yescharacteristic, complemented
452C2×D4—31no—
453C2×D4—152no—
454C2×D4—152no—
455C2×D4—151no—
456C2×D4—301no—
457C2×D4—301no—
458C2×D4—301no—
459C22×C4—51no—
460C22×C4—101no—
461C22×C4—304no—
462SD16—32no—
463SD16—32no—
464SD16—152no—
465SD16—152no—
466SD16—152no—
467SD16—152no—
468SD16—152no—
469SD16—152no—
470M16—301no—
471M16—301no—
472C2×C8—31no—
473C2×C8—151no—
474C2×C8—151no—
475C2×C8—151no—
476C2×C8—151no—
477C2×C8—151no—
478C2×C8—152no—
479C4⋊C4—52no—
480C4⋊C4—52no—
481C4⋊C4—104no—
482C4⋊C4—152no—
483C22⋊C4—52no—
484C22⋊C4—101no—
485C22⋊C4—304no—
486C22⋊C4—301no—
487C42—52no—
488C42—101no—
489C42—104no—
490C42—301no—
491C3×C5#294911yescharacteristic, complemented, Hall
492C22×C3—51no—
493C22×C3—51no—
494C22×C3—102no—
495C22×C3D4×AGL(1,𝔽5)11yescharacteristic
496C2×D3—22no—
497C2×D3—81no—
498C2×D3—102no—
499C2×D3—204no—
500C2×D3—402no—
501C2×D3—401no—
502C3×C4—22no—
503C3×C4—51no—
504C3×C4—51no—
505C3×C4—104no—
506C3×C4—102no—
507C3×C4—202no—
508C3×C4D4×AGL(1,𝔽5)11yescharacteristic
509C3×C4C23×AGL(1,𝔽5)11yescharacteristic
510Dic3—202no—
511C2×C5—21no—
512C2×C5—122no—
513C2×C5C2×C4×#6711yescharacteristic
514D5—21no—
515D5—122no—
516D5#583712yes—
517C23—51no—
518C23—61no—
519C23—302no—
520C23—302no—
521C23—301no—
522C23—1202no—
523Q8—21no—
524Q8—52no—
525Q8—102no—
526Q8—102no—
527Q8—101no—
528Q8C2×D3×AGL(1,𝔽5)12yes—
529D4—32no—
530D4—61no—
531D4—152no—
532D4—302no—
533D4—302no—
534D4—301no—
535C2×C4—21no—
536C2×C4—51no—
537C2×C4—51no—
538C2×C4—51no—
539C2×C4—51no—
540C2×C4—51no—
541C2×C4—54no—
542C2×C4—102no—
543C2×C4—102no—
544C2×C4—101no—
545C2×C4—102no—
546C2×C4—302no—
547C2×C4—1204no—
548C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
549C8—31no—
550C8—31no—
551C8—151no—
552C8—151no—
553C8—302no—
554C2×C3—21no—
555C2×C3—52no—
556C2×C3—101no—
557C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
558D3—42no—
559D3—202no—
560C5#8254811yescharacteristic, complemented, Sylow
561C22—51no—
562C22—51no—
563C22—62no—
564C22—102no—
565C22—241no—
566C22—302no—
567C22—604no—
568C22—1201no—
569C22—1202no—
570C22AGL(1,𝔽5)×#6711yescharacteristic
571C4—22no—
572C4—51no—
573C4—51no—
574C4—104no—
575C4—102no—
576C4—202no—
577C4—602no—
578C4AGL(1,𝔽5)×#6711yescharacteristic
579C4C22×D3×AGL(1,𝔽5)11yescharacteristic
580C3#1061742811yescharacteristic, complemented, Sylow
581C2—21no—
582C2—52no—
583C2—101no—
584C2—122no—
585C2—602no—
586C2C2×AGL(1,𝔽5)×#6711yescharacteristic
subgroupG/H
centerC2C2×AGL(1,𝔽5)×#67
commutatorC3×C4×C5C23×C4
FrattiniC4C22×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC2×C3×C5C2×C4×D4
prime radicalC2×D5×D12C22
cosocleC3×C4×D5C24
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#2949
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#66262C2semidirect
4. #11752441C2semidirect
5. #11751964C2semidirect
6. AGL(1,𝔽5)×#215C2semidirect
7. #11751726C2semidirect
9. #11751695C2semidirect
16. C3×Q8×AGL(1,𝔽5)C22semidirect
23. C2×#6717C4semidirect
24. C2×C5×#215C4semidirect
25. D5×#215C4semidirect
26. D5×#215C4semidirect
29. C2×#6658C4semidirect
30. C2×#6656C4semidirect
32. #91238C22semidirect
33. #91160C22semidirect
34. #91159C22semidirect
36. #91149C22semidirect
37. AGL(1,𝔽5)×C3⋊C8C22semidirect
39. #66262D3semidirect
50. C2×C3×C5×Q8C2×C4semidirect
55. C3×Q8×D5C2×C4semidirect
60. C2×C3×Dic10C2×C4semidirect
81. #6717C2×C4semidirect
82. #6717C2×C4semidirect
83. C2×C3⋊#168C2×C4semidirect
84. C5×#215C2×C4semidirect
85. C5×#215C2×C4semidirect
86. C2×C5×C3⋊C8C2×C4semidirect
87. #6658C2×C4semidirect
88. #6658C2×C4semidirect
89. #6656C2×C4semidirect
90. #6656C2×C4semidirect
91. D5×C3⋊C8C2×C4semidirect
92. D5×C3⋊C8C2×C4semidirect
108. Q8×AGL(1,𝔽5)C2×D3semidirect
183. C2×#215AGL(1,𝔽5)semidirect
198. C2×C5×Q8C4×D3semidirect
203. Q8×D5C4×D3semidirect
213. C2×Dic10C4×D3semidirect
278. C2×C3×Q8C2×AGL(1,𝔽5)semidirect
304. #215C2×AGL(1,𝔽5)semidirect
305. #215C2×AGL(1,𝔽5)semidirect
317. C2×C3⋊C8C2×AGL(1,𝔽5)semidirect
451. C2×Q8D3×AGL(1,𝔽5)semidirect
491. C3×C5#2949semidirect
560. C5#82548semidirect
580. C3#10617428semidirect

Maximal subgroups

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

subgroupindex
C3×#662622
#117568372
C2×D5×#2152
#117524412
#117519642
AGL(1,𝔽5)×#2152
#117517262
#117517062
#117516952
#106174283
#825485