Finite Group Databasebeta

← #12098903

Subgroups — #12098903

Order 1280.

1438 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4053C211yesmaximal, characteristic
2C2×C5×#428C211yesmaximal, characteristic, complemented
3C2×#65410C211yesmaximal, characteristic, complemented
4#10614001C211yesmaximal, characteristic
5#10613917C211yesmaximal, characteristic
6#10602373C211yesmaximal, characteristic, complemented
7C2×#64749C211yesmaximal, characteristic, complemented
8#10601424C212yesmaximal, complemented
9#10601423C212yesmaximal, complemented
10#10601371C212yesmaximal, complemented
11#10601370C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C22×C5×M16C2211yescharacteristic
14C2×C5×#117C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C2×#4072C2211yescharacteristic
17C22×C4⋊Dic5C2211yescharacteristic
18C23×#168C2211yescharacteristic
19C2×#4053—24no—
20C2×#4053—24no—
21C2×#4053C418yescomplemented
22C22×#676C2212yes—
23C22×C8×D5C2212yes—
24C5×#428—24no—
25C2×C5×#107C2212yes—
26C2×C5×#106C2212yes—
27#65502—22no—
28#65410—24no—
29C2×#3950C2212yes—
30C2×#3949C2212yes—
31#65277—21no—
32#65276C2212yes—
33#65276C2212yes—
34#65276C2212yes—
35#65276C2212yes—
36#65263—21no—
37#65247—22no—
38#65228—21no—
39#65218—21no—
40#64749C2212yes—
41#64749C2212yes—
42#64749C2211yescharacteristic
43#64749C2211yescharacteristic
44#64749C2211yescharacteristic
45#64749C2211yescharacteristic
46#20664—51nomaximal, Sylow
47C23×C4×C5—21no—
48C23×C4×C5C2311yescharacteristic
49C22×C22⋊D5D412yes—
50C23×Dic5—21no—
51C2×#709—22no—
52C2×#709—22no—
53C2×#709—22no—
54C2×#709—22no—
55C2×#709C2×C414yes—
56C2×#709Q814yes—
57C22×D20D411yescharacteristic
58C22×C4×D5C2312yes—
59C22×Dic10D411yescharacteristic
60C2×C5×M16C2×C414yes—
61C2×C5×M16D414yes—
62C2×C5×M16D414yes—
63C22×C5×C8C2312yes—
64C5×#117—22no—
65C5×#117—22no—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4—22no—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C4⋊C4C2312yes—
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C22⋊C4—21no—
73C2×C42×C5—21no—
74C2×C22⋊Dic5—21no—
75C2×#685—22no—
76C2×#685—22no—
77C2×#685—22no—
78C2×#685—22no—
79#4072—22no—
80#4072—22no—
81C2×C4⋊Dic5—22no—
82C2×C4⋊Dic5—22no—
83C2×C4⋊Dic5—21no—
84C2×C4⋊Dic5C2312yes—
85C2×C4⋊Dic5C2311yescharacteristic
86C2×C4⋊Dic5C2311yescharacteristic
87C2×C4×Dic5—21no—
88C22×#168—22no—
89C22×#168—22no—
90C22×#168C2×C414yes—
91C22×#168C2311yescharacteristic
92C22×#168C2311yescharacteristic
93#4053—232no—
94C2×#676—24no—
95C2×#676—24no—
96C2×#676C2×C418yescomplemented
97C2×C8×D5—24no—
98C2×C8×D5—24no—
99C2×C8×D5C2×C418yescomplemented
100C5×#107—24no—
101C5×#106—24no—
102#3963—22no—
103#3963—22no—
104#3963—22no—
105#3963—22no—
106#3950—24no—
107#3949—24no—
108C22×#131—51no—
109C2×#428—51no—
110C2×#428D511yescharacteristic, complemented
111#2311—51no—
112#2311—51no—
113#1809—52no—
114#1809—52no—
115#1808—52no—
116#1807—52no—
117#1735—51no—
118C2×#340—51no—
119C24×C5C2×D411yescharacteristic
120C23×D5C2×D412yes—
121C22×C4×C5—21no—
122C22×C4×C5—21no—
123C22×C4×C5—22no—
124C22×C4×C5—42no—
125C22×C4×C5—81no—
126C22×C4×C5—81no—
127C22×C4×C5C22×C412yes—
128C22×C4×C5C2×D412yes—
129C22×C4×C5C2×D411yescharacteristic
130C22×C4×C5C2×D412yes—
131C22×C4×C5C2×Q812yes—
132C22×C4×C5#4112yes—
133C22×C4×C5#4112yes—
134C22×C4×C5C2411yescharacteristic
135C2×C22⋊D5—24no—
136C2×C22⋊D5—24no—
137C2×C22⋊D5C4⋊C418yes—
138C22×Dic5—21no—
139C22×Dic5—21no—
140C22×Dic5—22no—
141C22×Dic5—42no—
142C22×Dic5—81no—
143C22×Dic5—81no—
144C22×Dic5C2×D412yes—
145#709—28no—
146#709—28no—
147#709—28no—
148#709—28no—
149C2×D20—22no—
150C2×D20—22no—
151C2×D20C4⋊C414yes—
152C2×C4×D5—22no—
153C2×C4×D5—22no—
154C2×C4×D5—22no—
155C2×C4×D5—22no—
156C2×C4×D5C22×C414yes—
157C2×C4×D5C2×Q814yes—
158C2×Dic10—22no—
159C2×Dic10—22no—
160C2×Dic10C4⋊C414yes—
161C5×M16C22⋊C4116yescomplemented
162C2×C5×C8C22×C414yes—
163C2×C5×C8C2×D414yes—
164C2×C5×C8C2×D414yes—
165C5×C4⋊C4—22no—
166C5×C4⋊C4—21no—
167C5×C4⋊C4—21no—
168C5×C4⋊C4—22no—
169C5×C4⋊C4—21no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—42no—
172C5×C22⋊C4—42no—
173C42×C5—42no—
174C22⋊Dic5—42no—
175#685—82no—
176#685—82no—
177#685—82no—
178#685—82no—
179C4⋊Dic5—21no—
180C4⋊Dic5—22no—
181C4⋊Dic5—21no—
182C4⋊Dic5—22no—
183C4⋊Dic5—21no—
184C4⋊Dic5—21no—
185C4⋊Dic5—42no—
186C4×Dic5—42no—
187C2×#168—28no—
188C2×#168—21no—
189C2×#168—21no—
190C2×#168—21no—
191C2×#168—21no—
192C2×#168C22×C412yes—
193C2×#168C22×C412yes—
194#676—216no—
195C8×D5—216no—
196C22×#41—51no—
197C2×#131—58no—
198C2×#131—104no—
199C2×#131—104no—
200C22×M16—52no—
201C22×M16C2×D511yescharacteristic
202C23×C8—51no—
203C23×C8—52no—
204C2×#117—51no—
205C2×#117C2×D511yescharacteristic
206C22×C4⋊C4—51no—
207C22×C4⋊C4C2×D511yescharacteristic
208#428—24no—
209#428—104no—
210C2×#107—52no—
211C2×#107C2×D512yes—
212C2×#106—52no—
213C2×#106C2×D512yes—
214#391—101no—
215#391—101no—
216#390—101no—
217#390—101no—
218#386—102no—
219#386—102no—
220#380—52no—
221#380—52no—
222#380—52no—
223#380—52no—
224#340—51no—
225#340—52no—
226#340—51no—
227#340—52no—
228#340—51no—
229#340—51no—
230C23×C5—21no—
231C23×C5—21no—
232C23×C5—21no—
233C23×C5—21no—
234C23×C5C2×C4⋊C412yes—
235C23×C5#12112yes—
236C23×C5#12112yes—
237C23×C5C22×D411yescharacteristic
238C22×D5—22no—
239C22×D5—22no—
240C22×D5—22no—
241C22×D5—22no—
242C22×D5—22no—
243C22×D5—22no—
244C22×D5C2×C4⋊C414yes—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—42no—
248C2×C4×C5—41no—
249C2×C4×C5—42no—
250C2×C4×C5—41no—
251C2×C4×C5—42no—
252C2×C4×C5—41no—
253C2×C4×C5—41no—
254C2×C4×C5—84no—
255C2×C4×C5C2×C22⋊C414yes—
256C2×C4×C5C2×C4⋊C412yes—
257C2×C4×C5#11714yes—
258C2×C4×C5#12112yes—
259C2×C4×C5#12112yes—
260C2×C4×C5C22⋊Q814yes—
261C2×C4×C5C22⋊Q814yes—
262C2×C4×C5C23×C411yescharacteristic
263C2×C4×C5C22×D411yescharacteristic
264C2×C4×C5C22×D411yescharacteristic
265C2×C4×C5C22×Q811yescharacteristic
266C2×C4×C5C2×#4111yescharacteristic
267C2×C4×C5C2×#4111yescharacteristic
268C22⋊D5—216no—
269C2×Dic5—22no—
270C2×Dic5—22no—
271C2×Dic5—41no—
272C2×Dic5—41no—
273C2×Dic5—42no—
274C2×Dic5—41no—
275C2×Dic5—42no—
276C2×Dic5—41no—
277C2×Dic5—42no—
278C2×Dic5—41no—
279C2×Dic5—41no—
280C2×Dic5—84no—
281C2×Dic5C2×C4⋊C414yes—
282D20—28no—
283C4×D5—24no—
284C4×D5—24no—
285C4×D5—24no—
286C4×D5—24no—
287Dic10—28no—
288C5×C8C2×C22⋊C418yes—
289#168—22no—
290#168—22no—
291C2×#41—54no—
292C2×#41—54no—
293C2×#41—102no—
294C2×#41—102no—
295C2×#41—102no—
296C2×#41—102no—
297C22×Q8—51no—
298C22×D4—52no—
299C22×D4—51no—
300C23×C4—21no—
301C23×C4—52no—
302C23×C4—101no—
303C23×C4C22×D511yescharacteristic
304#131—1032no—
305C2×M16—58no—
306C2×M16—104no—
307C2×M16—104no—
308C2×M16C4×D514yes—
309C2×M16D2014yes—
310C2×M16C22⋊D514yes—
311C22×C8—54no—
312C22×C8—51no—
313C22×C8—51no—
314C22×C8—58no—
315C22×C8—102no—
316C22×C8—102no—
317C22×C8—104no—
318C22×C8—104no—
319C22×C8C22×D512yes—
320#117—22no—
321#117—22no—
322#117—102no—
323#117—102no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—52no—
328C2×C4⋊C4—51no—
329C2×C4⋊C4—51no—
330C2×C4⋊C4—102no—
331C2×C4⋊C4—102no—
332C2×C4⋊C4—101no—
333C2×C4⋊C4C22×D512yes—
334C2×C4⋊C4C22×D511yescharacteristic
335C2×C4⋊C4C22×D511yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—101no—
338C2×C22⋊C4—102no—
339C2×C22⋊C4—102no—
340C2×C22⋊C4—102no—
341C2×C22⋊C4—102no—
342C2×C42—21no—
343C2×C42—101no—
344#107—24no—
345#107—104no—
346#106—24no—
347#106—104no—
348#95—102no—
349#95—102no—
350#95—102no—
351#95—102no—
352C22×C5—22no—
353C22×C5—22no—
354C22×C5—22no—
355C22×C5—22no—
356C22×C5—22no—
357C22×C5—22no—
358C22×C5#38014yes—
359C22×C5C22×C4⋊C411yescharacteristic
360C22×C5C2×#12111yescharacteristic
361C22×C5C2×#12111yescharacteristic
362C22×C5#57611yescharacteristic
363C22×C5#57711yescharacteristic
364C22×C5#57711yescharacteristic
365C22×C5#57811yescharacteristic
366C2×D5—24no—
367C2×D5—24no—
368C2×D5—24no—
369C2×D5—42no—
370C2×D5—42no—
371C2×D5—42no—
372C2×D5—42no—
373C4×C5—81no—
374C4×C5—82no—
375C4×C5—81no—
376C4×C5#38014yes—
377C4×C5C22×C22⋊C411yescharacteristic
378C4×C5C2×#11711yescharacteristic
379C4×C5C2×C22⋊Q811yescharacteristic
380C4×C5C2×C22⋊Q811yescharacteristic
381Dic5—24no—
382Dic5—81no—
383Dic5—82no—
384Dic5—81no—
385C24—52no—
386C24D4×D511yescharacteristic
387#41—108no—
388#41—108no—
389#41—108no—
390#41—108no—
391C2×Q8—54no—
392C2×Q8—102no—
393C2×Q8—102no—
394C2×D4—58no—
395C2×D4—54no—
396C2×D4—104no—
397C2×D4—104no—
398C2×D4—102no—
399C2×D4—102no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—42no—
404C22×C4—52no—
405C22×C4—54no—
406C22×C4—54no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—102no—
410C22×C4—102no—
411C22×C4—102no—
412C22×C4—102no—
413C22×C4—101no—
414C22×C4—101no—
415C22×C4—102no—
416C22×C4—202no—
417C22×C4—401no—
418C22×C4—401no—
419C22×C4C2×C4×D512yes—
420C22×C4C2×D2012yes—
421C22×C4D4×D511yescharacteristic
422C22×C4#71112yes—
423C22×C4Q8×D512yes—
424C22×C4#71312yes—
425C22×C4C2×C22⋊D512yes—
426C22×C4C23×D511yescharacteristic
427M16—1016no—
428M16#685116yescomplemented
429C2×C8—52no—
430C2×C8—52no—
431C2×C8—101no—
432C2×C8—101no—
433C2×C8—101no—
434C2×C8—101no—
435C2×C8—108no—
436C2×C8—1016no—
437C2×C8C2×C4×D514yes—
438C2×C8C2×D2014yes—
439C2×C8C2×C22⋊D514yes—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—42no—
447C4⋊C4—101no—
448C4⋊C4—102no—
449C4⋊C4—101no—
450C4⋊C4—102no—
451C4⋊C4—101no—
452C4⋊C4—101no—
453C4⋊C4—202no—
454C22⋊C4—42no—
455C22⋊C4—202no—
456C22⋊C4—402no—
457C22⋊C4—402no—
458C22⋊C4—402no—
459C22⋊C4—402no—
460C42—42no—
461C42—202no—
462C2×C5—24no—
463C2×C5C2×#38011yescharacteristic
464C2×C5#291211yescharacteristic
465C2×C5#291311yescharacteristic
466C2×C5#306211yescharacteristic
467C2×C5#306211yescharacteristic
468C2×C5#306311yescharacteristic
469C2×C5#306311yescharacteristic
470D5—44no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—54no—
476C23—102no—
477C23—102no—
478C23—102no—
479C23—102no—
480C23—102no—
481C23—102no—
482C23D5×C4⋊C412yes—
483C23#404112yes—
484C23#408412yes—
485C23C2×D4×D511yescharacteristic
486Q8—108no—
487D4—1016no—
488D4—108no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—54no—
499C2×C4—84no—
500C2×C4—102no—
501C2×C4—102no—
502C2×C4—104no—
503C2×C4—104no—
504C2×C4—104no—
505C2×C4—104no—
506C2×C4—201no—
507C2×C4—201no—
508C2×C4—202no—
509C2×C4—201no—
510C2×C4—202no—
511C2×C4—201no—
512C2×C4—202no—
513C2×C4—201no—
514C2×C4—201no—
515C2×C4—404no—
516C2×C4D5×C4⋊C412yes—
517C2×C4#403814yes—
518C2×C4#404112yes—
519C2×C4#404314yes—
520C2×C4C2×#68514yes—
521C2×C4#408412yes—
522C2×C4#409214yes—
523C2×C4C22×C4×D511yescharacteristic
524C2×C4C22×D2011yescharacteristic
525C2×C4C2×#71111yescharacteristic
526C2×C4C2×Q8×D511yescharacteristic
527C2×C4C2×#71311yescharacteristic
528C2×C4C22×C22⋊D511yescharacteristic
529C8—102no—
530C8—102no—
531C8C2×#68518yes—
532C5#2066411yescharacteristic, complemented, Sylow
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—104no—
540C22—104no—
541C22—104no—
542C22—202no—
543C22—202no—
544C22—202no—
545C22—202no—
546C22#6527614yes—
547C22C2×D5×C4⋊C411yescharacteristic
548C22C2×#404111yescharacteristic
549C22#6610811yescharacteristic
550C22#6610911yescharacteristic
551C22#6611211yescharacteristic
552C22#6611311yescharacteristic
553C22C2×#408411yescharacteristic
554C4—82no—
555C4—81no—
556C4—81no—
557C4—104no—
558C4—402no—
559C4—401no—
560C4—401no—
561C4#6527614yes—
562C4C2×#403811yescharacteristic
563C4C2×#404311yescharacteristic
564C4C22×#68511yescharacteristic
565C4C2×#409211yescharacteristic
566C2—24no—
567C2—204no—
568C2#1061094011yescharacteristic
569C2#1061094111yescharacteristic
570C2#1061094411yescharacteristic
571C2#1061094511yescharacteristic
572C2C2×#6527611yescharacteristic
573C2#1061568211yescharacteristic
574C2#1061571711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#428C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenterC2×#428D5
hypocenterC5#20664
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×C5×#428C2semidirect
3. C2×#65410C2semidirect
6. #10602373C2semidirect
7. C2×#64749C2semidirect
8. #10601424C2semidirect
9. #10601423C2semidirect
10. #10601371C2semidirect
11. #10601370C2semidirect
21. C2×#4053C4semidirect
96. C2×#676C2×C4semidirect
99. C2×C8×D5C2×C4semidirect
110. C2×#428D5semidirect
161. C5×M16C22⋊C4semidirect
428. M16#685semidirect
532. C5#20664semidirect

Maximal subgroups

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

subgroupindex
C22×#40532
C2×C5×#4282
C2×#654102
#106140012
#106139172
#106023732
C2×#647492
#106014242
#106014232
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