Finite Group Databasebeta

← #12110501

Subgroups — #12110501

Order 1280.

1296 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C5×#121C211yesmaximal, characteristic, complemented
2C2×#65325C211yesmaximal, characteristic
3C2×#65228C211yesmaximal, characteristic, complemented
4#10602470C211yesmaximal, characteristic, complemented
5#10602406C212yesmaximal, complemented
6#10602322C211yesmaximal, characteristic, complemented
7#10600441C212yesmaximal, complemented
8#10600410C212yesmaximal, complemented
9#10600314C211yesmaximal, characteristic, complemented
10#10600298C211yesmaximal, characteristic, complemented
11#10600256C212yesmaximal, complemented
12C23×D20C2211yescharacteristic
13C2×C5×#121C2211yescharacteristic
14C2×C4×C5×D4—21no—
15C2×C4×C5×D4C2211yescharacteristic, complemented
16C22×C42×C5C2211yescharacteristic
17C22×#686C2212yes—
18C22×#685—21no—
19C22×#685—21no—
20C22×C4⋊Dic5C2211yescharacteristic
21C5×#390C2211yescharacteristic, complemented
22C5×#389C2211yescharacteristic, complemented
23C5×#388—21no—
24C5×#386—21no—
25C5×#380C2211yescharacteristic
26C4×C5×C4⋊C4C2211yescharacteristic, complemented
27C4×C5×C22⋊C4—21no—
28C2×#3964C2212yescomplemented
29C2×C22⋊#168C2211yescharacteristic
30C2×C22⋊#168C2211yescharacteristic
31#65325C418yescomplemented
32C2×#3941C2212yes—
33C2×#3941C2212yes—
34C2×#3941C2212yes—
35C2×#3940C2212yescomplemented
36#65228—21no—
37#65228—21no—
38#65228C2212yes—
39#65228C2212yes—
40C2×#3935C2212yescomplemented
41#64749C2212yescomplemented
42#64749C2212yescomplemented
43#64744C2212yescomplemented
44#22046—51nomaximal, Sylow
45C24×D5—21no—
46C22×C5×D4—21no—
47C22×C5×D4C2311yescharacteristic
48C23×C4×C5—21no—
49C23×C4×C5D411yescharacteristic
50C23×C4×C5D411yescharacteristic
51C23×C4×C5C2311yescharacteristic
52C23×Dic5—21no—
53C22×D20—21no—
54C22×D20—22no—
55C22×D20—42no—
56C22×D20—42no—
57C22×D20C2×C414yescomplemented
58C22×D20C2312yes—
59C5×#121—22no—
60C5×#121C2×C414yes—
61C4×C5×D4—42no—
62C4×C5×D4—44no—
63C2×C5×C4⋊C4—21no—
64C2×C5×C4⋊C4C2311yescharacteristic
65C2×C5×C4⋊C4C2311yescharacteristic
66C2×C5×C4⋊C4C2311yescharacteristic
67C2×C5×C22⋊C4—21no—
68C2×C5×C22⋊C4—21no—
69C2×C5×C22⋊C4—21no—
70C2×C5×C22⋊C4—21no—
71C2×C5×C22⋊C4—21no—
72C2×C42×C5—21no—
73C2×C42×C5—42no—
74C2×C42×C5D411yescharacteristic
75C2×C42×C5D411yescharacteristic
76C2×C42×C5C2311yescharacteristic
77C2×C42×C5C2311yescharacteristic
78C2×#686—42no—
79C2×#686—42no—
80C2×#686C2×C418yescomplemented
81C2×#685—22no—
82C2×#685—22no—
83C2×#685—22no—
84C2×#685—22no—
85C2×#685—44no—
86C2×#685—44no—
87C2×C4⋊Dic5—21no—
88C2×C4⋊Dic5—42no—
89C2×C4⋊Dic5C2312yes—
90C22×#168C2312yes—
91C22×#168C2312yes—
92C5×#95—21no—
93C5×#95—21no—
94C5×#95—21no—
95#3964—42no—
96C22⋊#168—22no—
97C22⋊#168C2×C414yescomplemented
98#3941—42no—
99#3941—42no—
100#3941C2×C418yescomplemented
101#3940—42no—
102#3935—42no—
103C2×#447—51no—
104C4×#121D511yescharacteristic, complemented
105C2×#390—51no—
106#1972—52no—
107#1936—52no—
108#1926—52no—
109#1879—52no—
110#1874—51no—
111#1853—51no—
112#1786—51no—
113#1749—51no—
114C24×C5—21no—
115C24×C5C2×D411yescharacteristic
116C23×D5—24no—
117C23×D5—22no—
118C23×D5—42no—
119C23×D5—42no—
120C23×D5—42no—
121C23×D5—42no—
122C23×D5—42no—
123C23×D5—42no—
124C2×C5×D4—24no—
125C2×C5×D4—21no—
126C2×C5×D4—22no—
127C2×C5×D4—22no—
128C2×C5×D4—42no—
129C2×C5×D4—42no—
130C2×C5×D4C22×C412yes—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—21no—
136C22×C4×C5—22no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—41no—
140C22×C4×C5—41no—
141C22×C4×C5—41no—
142C22×C4×C5—42no—
143C22×C4×C5—42no—
144C22×C4×C5—42no—
145C22×C4×C5C22×C412yes—
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5#4112yes—
152C22×C4×C5C2411yescharacteristic
153C22×Dic5—22no—
154C22×Dic5—42no—
155C22×Dic5—42no—
156C22×Dic5—42no—
157C2×D20—22no—
158C2×D20—22no—
159C2×D20—22no—
160C2×D20—22no—
161C2×D20—24no—
162C2×D20—22no—
163C2×D20—48no—
164C2×D20—42no—
165C2×D20—42no—
166C2×D20—48no—
167C2×D20C22×C414yes—
168C5×C4⋊C4—21no—
169C5×C4⋊C4—22no—
170C5×C4⋊C4—22no—
171C5×C4⋊C4—42no—
172C5×C4⋊C4C22×C412yes—
173C5×C22⋊C4—24no—
174C5×C22⋊C4—42no—
175C5×C22⋊C4—42no—
176C5×C22⋊C4—42no—
177C5×C22⋊C4—42no—
178C42×C5—22no—
179C42×C5—22no—
180C42×C5—41no—
181C42×C5—41no—
182C42×C5—42no—
183#686—48no—
184#685—44no—
185#685—44no—
186#685—44no—
187#685—44no—
188C4⋊Dic5—22no—
189C4⋊Dic5—22no—
190C4⋊Dic5—42no—
191C2×#168—22no—
192C2×#168—22no—
193C2×#168—22no—
194C2×#168—22no—
195C2×#168—22no—
196C2×#168C22×C414yes—
197C23×D4—51no—
198C22×D8—52no—
199C2×#121C2×D511yescharacteristic
200C2×C4×D4—21no—
201C2×C4×D4C2×D511yescharacteristic, complemented
202C22×C4⋊C4—51no—
203C22×C22⋊C4—101no—
204C22×C22⋊C4—101no—
205C22×C42C2×D511yescharacteristic
206#447—58no—
207C2×#106—52no—
208C2×#102—52no—
209C2×#102—52no—
210C2×#102—52no—
211C2×#102—52no—
212C2×C22⋊C8—51no—
213C2×C22⋊C8—51no—
214C2×#97—52no—
215#390—52no—
216#390—52no—
217#390—101no—
218#390—101no—
219#390C2×D511yescharacteristic, complemented
220#389C2×D511yescharacteristic, complemented
221#388—21no—
222#386—21no—
223#380C2×D511yescharacteristic
224C4×C4⋊C4C2×D511yescharacteristic, complemented
225C4×C22⋊C4—21no—
226#340—52no—
227#340—52no—
228#336—52no—
229C23×C5—21no—
230C23×C5—21no—
231C23×C5—21no—
232C23×C5—21no—
233C23×C5—22no—
234C23×C5—41no—
235C23×C5—41no—
236C23×C5—41no—
237C23×C5—41no—
238C23×C5—41no—
239C23×C5—41no—
240C23×C5C4×D412yes—
241C23×C5AGL(1,ℤ/8)12yes—
242C23×C5AGL(1,ℤ/8)12yes—
243C23×C5C22×D411yescharacteristic
244C22×D5—24no—
245C22×D5—48no—
246C22×D5—48no—
247C22×D5—42no—
248C22×D5—42no—
249C22×D5—42no—
250C22×D5—42no—
251C22×D5—42no—
252C22×D5—42no—
253C22×D5—48no—
254C22×D5—84no—
255C22×D5—84no—
256C22×D5—84no—
257C22×D5—84no—
258C5×D4—24no—
259C5×D4—48no—
260C5×D4—42no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—24no—
266C2×C4×C5—24no—
267C2×C4×C5—22no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—44no—
278C2×C4×C5—44no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—84no—
288C2×C4×C5C4×D412yes—
289C2×C4×C5AGL(1,ℤ/8)12yes—
290C2×C4×C5C23×C411yescharacteristic
291C2×C4×C5C2×#4111yescharacteristic
292C2×C4×C5H(5,𝔽2)12yes—
293C2×C4×C5H(5,𝔽2)12yes—
294C2×Dic5—42no—
295C2×Dic5—42no—
296C2×Dic5—42no—
297C2×Dic5—84no—
298D20—24no—
299D20—24no—
300D20—48no—
301D20—42no—
302D20—42no—
303#168—24no—
304#168—42no—
305C25—101no—
306C22×D4—21no—
307C22×D4—54no—
308C22×D4—52no—
309C22×D4—102no—
310C22×D4—101no—
311C22×D4—202no—
312C22×D4—202no—
313C22×D4C22×D511yescharacteristic
314C23×C4—21no—
315C23×C4—101no—
316C23×C4C22⋊D511yescharacteristic
317C23×C4C22⋊D511yescharacteristic
318C23×C4C22×D511yescharacteristic
319C2×D8—58no—
320C2×D8—202no—
321C2×D8—202no—
322C22×C8—52no—
323C22×C8—52no—
324#121—22no—
325#121C4×D514yes—
326C4×D4—42no—
327C4×D4—44no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—52no—
330C2×C4⋊C4—101no—
331C2×C4⋊C4—202no—
332C2×C4⋊C4C22×D511yescharacteristic
333C2×C4⋊C4C22×D511yescharacteristic
334C2×C4⋊C4C22×D511yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—102no—
341C2×C22⋊C4—102no—
342C2×C22⋊C4—102no—
343C2×C22⋊C4—102no—
344C2×C22⋊C4—204no—
345C2×C22⋊C4—204no—
346C2×C42—21no—
347C2×C42—42no—
348C2×C42C22⋊D511yescharacteristic
349C2×C42C22⋊D511yescharacteristic
350C2×C42C22×D511yescharacteristic
351C2×C42C22×D511yescharacteristic
352#106—202no—
353#102—58no—
354#102—202no—
355#102—202no—
356#102—202no—
357C22⋊C8—54no—
358C22⋊C8—102no—
359#97—202no—
360#95—21no—
361#95—21no—
362#95—21no—
363C22×C5—22no—
364C22×C5—22no—
365C22×C5—22no—
366C22×C5—22no—
367C22×C5—22no—
368C22×C5—22no—
369C22×C5—42no—
370C22×C5—42no—
371C22×C5—42no—
372C22×C5—81no—
373C22×C5—81no—
374C22×C5—81no—
375C22×C5—81no—
376C22×C5#44214yes—
377C22×C5C2×C4×D411yescharacteristic
378C22×C5#53411yescharacteristic
379C22×C5#53511yescharacteristic
380C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
381C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
382C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
383C22×C5#57511yescharacteristic
384C2×D5—44no—
385C2×D5—44no—
386C2×D5—44no—
387C2×D5—816no—
388C2×D5—82no—
389C2×D5—82no—
390C2×D5—82no—
391C2×D5—82no—
392C4×C5—22no—
393C4×C5—22no—
394C4×C5—41no—
395C4×C5—41no—
396C4×C5—41no—
397C4×C5—42no—
398C4×C5—81no—
399C4×C5—81no—
400C4×C5#51812yes—
401C4×C5#53812yes—
402Dic5—82no—
403C24—21no—
404C24—102no—
405C24—104no—
406C24—202no—
407C24—202no—
408C24—202no—
409C24—202no—
410C24—202no—
411C24—202no—
412C24C2×C22⋊D511yescharacteristic
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—54no—
420C2×D4—102no—
421C2×D4—102no—
422C2×D4—102no—
423C2×D4—102no—
424C2×D4—102no—
425C2×D4—104no—
426C2×D4—208no—
427C2×D4—202no—
428C2×D4—202no—
429C2×D4—208no—
430C2×D4C2×C4×D512yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—102no—
446C22×C4—202no—
447C22×C4—202no—
448C22×C4—202no—
449C22×C4C2×C4×D512yes—
450C22×C4#70912yes—
451C22×C4C2×C22⋊D511yescharacteristic
452C22×C4C2×C22⋊D511yescharacteristic
453C22×C4C2×C22⋊D511yescharacteristic
454C22×C4C2×C22⋊D511yescharacteristic
455C22×C4C2×C22⋊D511yescharacteristic
456C22×C4C23×D511yescharacteristic
457D8—208no—
458C2×C8—54no—
459C2×C8—102no—
460C2×C8—102no—
461C2×C8—102no—
462C2×C8—102no—
463C2×C8—102no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—102no—
469C4⋊C4—102no—
470C4⋊C4—202no—
471C4⋊C4C2×C4×D512yes—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—204no—
478C22⋊C4—204no—
479C22⋊C4—204no—
480C22⋊C4—204no—
481C42—22no—
482C42—22no—
483C42—41no—
484C42—41no—
485C42—42no—
486C2×C5—24no—
487C2×C5—82no—
488C2×C5#232111yescharacteristic
489C2×C5C2×#44211yescharacteristic
490C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
491C2×C5#319211yescharacteristic
492C2×C5#319411yescharacteristic
493C2×C5#321311yescharacteristic
494C2×C5#321711yescharacteristic
495D5—84no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—104no—
508C23—208no—
509C23—208no—
510C23—208no—
511C23—202no—
512C23—202no—
513C23—202no—
514C23—202no—
515C23—202no—
516C23—202no—
517C23—404no—
518C23—404no—
519C23—404no—
520C23—404no—
521C23C4×C22⋊D512yes—
522C23#407812yes—
523C23#409512yes—
524C23C22×C22⋊D511yescharacteristic
525D4—24no—
526D4—48no—
527D4—42no—
528D4—104no—
529D4—104no—
530D4—208no—
531D4—202no—
532D4—202no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—84no—
560C2×C4—202no—
561C2×C4—202no—
562C2×C4—202no—
563C2×C4—404no—
564C2×C4C4×C22⋊D512yes—
565C2×C4#409512yes—
566C2×C4C22×C4×D511yescharacteristic
567C2×C4C2×#70911yescharacteristic
568C2×C4#414412yes—
569C2×C4#414912yes—
570C8—104no—
571C8—202no—
572C5#2204611yescharacteristic, complemented, Sylow
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—204no—
587C22—204no—
588C22—204no—
589C22—4016no—
590C22—402no—
591C22—402no—
592C22—402no—
593C22—402no—
594C22#6530314yes—
595C22C2×C4×C22⋊D511yescharacteristic
596C22C2×#407811yescharacteristic
597C22#6613811yescharacteristic
598C22C2×#409511yescharacteristic
599C22C2×#409511yescharacteristic
600C22#6615611yescharacteristic
601C22#6616411yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—402no—
611C4#6587912yes—
612C4#6589412yes—
613C2—24no—
614C2—82no—
615C2—404no—
616C2C2×#6530311yescharacteristic
617C2#1061412611yescharacteristic
618C2C4×#409511yescharacteristic
619C2#1061435811yescharacteristic
620C2#1061466211yescharacteristic
621C2#1061487711yescharacteristic
622C2#1061503711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC4×C5×#121C2
socleC23×C5C22×D4
prime radicalC2×#65325C2
cosocleC22×C4×C5C24
hypercenterC4×#121D5
hypocenterC5#22046
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C4×C5×#121C2semidirect
3. C2×#65228C2semidirect
4. #10602470C2semidirect
5. #10602406C2semidirect
6. #10602322C2semidirect
7. #10600441C2semidirect
8. #10600410C2semidirect
9. #10600314C2semidirect
10. #10600298C2semidirect
11. #10600256C2semidirect
15. C2×C4×C5×D4C22semidirect
21. C5×#390C22semidirect
22. C5×#389C22semidirect
26. C4×C5×C4⋊C4C22semidirect
28. C2×#3964C22semidirect
31. #65325C4semidirect
35. C2×#3940C22semidirect
40. C2×#3935C22semidirect
41. #64749C22semidirect
42. #64749C22semidirect
43. #64744C22semidirect
57. C22×D20C2×C4semidirect
80. C2×#686C2×C4semidirect
97. C22⋊#168C2×C4semidirect
100. #3941C2×C4semidirect
104. C4×#121D5semidirect
201. C2×C4×D4C2×D5semidirect
219. #390C2×D5semidirect
220. #389C2×D5semidirect
224. C4×C4⋊C4C2×D5semidirect
572. C5#22046semidirect

Maximal subgroups

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

subgroupindex
C4×C5×#1212
C2×#653252
C2×#652282
#106024702
#106024062
#106023222
#106004412
#106004102
#106003142
#106002982
#106002562
#220465