Finite Group Databasebeta

← #12129301

Subgroups — #12129301

Order 1280.

880 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C5×#121C211yesmaximal, characteristic
2#10613793C211yesmaximal, characteristic, complemented
3#10602473C212yesmaximal
4C4×#3964C212yesmaximal, complemented
5C4×C22⋊#168C211yesmaximal, characteristic, complemented
6#10600449C211yesmaximal, characteristic, complemented
7#10600433C211yesmaximal, characteristic, complemented
8#10600405C211yesmaximal, characteristic, complemented
9#10600401C211yesmaximal, characteristic, complemented
10#10600313C212yesmaximal, complemented
11C4×#3940C212yesmaximal, complemented
12C2×C5×#121C2211yescharacteristic
13C2×C4×C5×D4—21no—
14C2×C4×C5×D4C2211yescharacteristic
15C22×C42×C5C2211yescharacteristic
16C2×#4072—21no—
17C5×#390C2211yescharacteristic
18C5×#389C2211yescharacteristic
19C5×#388—21no—
20C5×#386—21no—
21C5×#380C2211yescharacteristic
22C4×C5×C4⋊C4C2211yescharacteristic
23C4×C5×C22⋊C4—21no—
24C2×#3964C2212yes—
25C2×#3964C2212yes—
26#65501—21no—
27#65501C2212yes—
28C2×C22⋊#168C2212yes—
29C2×#3940C2212yes—
30C2×#3940C2212yes—
31#65225—21no—
32#65224C2211yescharacteristic
33#65224C2211yescharacteristic
34C4×C4⋊Dic5—21no—
35C4×C4⋊Dic5C2212yes—
36C2×C4×#168C2212yes—
37#64749C2212yes—
38#64749C2212yes—
39#64749C2212yes—
40#64749C2212yes—
41#64744C2211yescharacteristic
42#64744C2211yescharacteristic
43#23755—51nomaximal, Sylow
44C22×C5×D4—21no—
45C22×C5×D4C2311yescharacteristic
46C23×C4×C5—21no—
47C23×C4×C5D411yescharacteristic
48C23×C4×C5D411yescharacteristic
49C23×C4×C5C2311yescharacteristic
50C5×#121—22no—
51C5×#121—22no—
52C4×C5×D4—24no—
53C4×C5×D4—44no—
54C2×C5×C4⋊C4—21no—
55C2×C5×C4⋊C4C2311yescharacteristic
56C2×C5×C4⋊C4C2311yescharacteristic
57C2×C5×C4⋊C4C2311yescharacteristic
58C2×C5×C22⋊C4—21no—
59C2×C5×C22⋊C4—21no—
60C2×C5×C22⋊C4—21no—
61C2×C5×C22⋊C4—21no—
62C2×C5×C22⋊C4—21no—
63C2×C42×C5—24no—
64C2×C42×C5—21no—
65C2×C42×C5D411yescharacteristic
66C2×C42×C5D411yescharacteristic
67C2×C42×C5C2311yescharacteristic
68C2×C42×C5C2311yescharacteristic
69C2×C22⋊Dic5—22no—
70#4072—44no—
71C2×C4⋊Dic5—21no—
72C2×C4⋊Dic5—21no—
73C2×C4⋊Dic5—21no—
74C2×C4⋊Dic5—21no—
75C2×C4⋊Dic5C2312yes—
76C2×C4⋊Dic5C2312yes—
77C2×C4×Dic5—22no—
78C22×#168C2312yes—
79C22×#168C2312yes—
80C5×#95—21no—
81C5×#95—21no—
82C5×#95—21no—
83#3964—24no—
84#3964—24no—
85#3963—22no—
86#3963—22no—
87C22⋊#168—24no—
88#3940—24no—
89#3940—24no—
90C4×#168—24no—
91C4×#121D511yescharacteristic
92#2285—51no—
93#2100—51no—
94#2094—51no—
95#2084—51no—
96#2077—52no—
97#2071—51no—
98#2034—52no—
99C4×#106—52no—
100C4×#102—52no—
101C4×C22⋊C8—51no—
102C24×C5—21no—
103C24×C5C2×D411yescharacteristic
104C2×C5×D4—24no—
105C2×C5×D4—22no—
106C2×C5×D4—22no—
107C2×C5×D4—21no—
108C2×C5×D4—42no—
109C2×C5×D4—42no—
110C2×C5×D4#4112yes—
111C22×C4×C5—22no—
112C22×C4×C5—22no—
113C22×C4×C5—21no—
114C22×C4×C5—22no—
115C22×C4×C5—22no—
116C22×C4×C5—22no—
117C22×C4×C5—21no—
118C22×C4×C5—21no—
119C22×C4×C5—22no—
120C22×C4×C5—21no—
121C22×C4×C5—21no—
122C22×C4×C5—41no—
123C22×C4×C5—42no—
124C22×C4×C5—42no—
125C22×C4×C5—42no—
126C22×C4×C5SD1614yes—
127C22×C4×C5C2×D411yescharacteristic
128C22×C4×C5C2×D411yescharacteristic
129C22×C4×C5C2×D411yescharacteristic
130C22×C4×C5C2×D411yescharacteristic
131C22×C4×C5C2×D411yescharacteristic
132C22×C4×C5C2411yescharacteristic
133C22×Dic5—22no—
134C22×Dic5—22no—
135C5×C4⋊C4—21no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—22no—
138C5×C4⋊C4—42no—
139C5×C4⋊C4#4112yes—
140C5×C22⋊C4—24no—
141C5×C22⋊C4—42no—
142C5×C22⋊C4—42no—
143C5×C22⋊C4—42no—
144C5×C22⋊C4—42no—
145C42×C5—22no—
146C42×C5—22no—
147C42×C5—22no—
148C42×C5—42no—
149C42×C5#4114yes—
150C22⋊Dic5—44no—
151C4⋊Dic5—22no—
152C4⋊Dic5—22no—
153C4⋊Dic5—22no—
154C4⋊Dic5—22no—
155C4⋊Dic5—42no—
156C4⋊Dic5—42no—
157C4⋊Dic5—42no—
158C4⋊Dic5—42no—
159C4×Dic5—44no—
160C2×#168—22no—
161C2×#168—22no—
162C2×#168—22no—
163C2×#168—22no—
164C2×#168—22no—
165C2×#168#4114yes—
166C2×#121C2×D511yescharacteristic
167C2×C4×D4—21no—
168C2×C4×D4C2×D511yescharacteristic
169C2×#117—101no—
170C22×C42C2×D511yescharacteristic
171C2×#106—52no—
172C2×#106—52no—
173C2×#102—52no—
174C2×#102—52no—
175C2×C22⋊C8—52no—
176C2×C4×C8—52no—
177#390C2×D511yescharacteristic
178#389C2×D511yescharacteristic
179#388—21no—
180#386—21no—
181#384—51no—
182#384—51no—
183#382—101no—
184#380—52no—
185#380—101no—
186#380C2×D511yescharacteristic
187C4×C4⋊C4—52no—
188C4×C4⋊C4—101no—
189C4×C4⋊C4C2×D511yescharacteristic
190C4×C22⋊C4—21no—
191#340—52no—
192#340—52no—
193#340—52no—
194#340—52no—
195#336—51no—
196#336—51no—
197C23×C5—21no—
198C23×C5—21no—
199C23×C5—21no—
200C23×C5—21no—
201C23×C5—21no—
202C23×C5—21no—
203C23×C5—22no—
204C23×C5—41no—
205C23×C5—41no—
206C23×C5—41no—
207C23×C5—41no—
208C23×C5—41no—
209C23×C5—41no—
210C23×C5C2×SD1612yes—
211C23×C5C22×D411yescharacteristic
212C5×D4—24no—
213C5×D4—48no—
214C5×D4—42no—
215C2×C4×C5—21no—
216C2×C4×C5—21no—
217C2×C4×C5—21no—
218C2×C4×C5—21no—
219C2×C4×C5—21no—
220C2×C4×C5—21no—
221C2×C4×C5—28no—
222C2×C4×C5—21no—
223C2×C4×C5—21no—
224C2×C4×C5—22no—
225C2×C4×C5—21no—
226C2×C4×C5—44no—
227C2×C4×C5—41no—
228C2×C4×C5—41no—
229C2×C4×C5—42no—
230C2×C4×C5—42no—
231C2×C4×C5—41no—
232C2×C4×C5—41no—
233C2×C4×C5—41no—
234C2×C4×C5—41no—
235C2×C4×C5—41no—
236C2×C4×C5—41no—
237C2×C4×C5—84no—
238C2×C4×C5#12312yes—
239C2×C4×C5#12312yes—
240C2×C4×C5C2×SD1612yes—
241C2×C4×C5C2×SD1612yes—
242C2×C4×C5#13512yes—
243C2×C4×C5AGL(1,ℤ/8)12yes—
244C2×C4×C5C2×#4111yescharacteristic
245C2×C4×C5C2×#4111yescharacteristic
246C2×C4×C5C2×#4112yes—
247C2×C4×C5C2×#4112yes—
248C2×Dic5—42no—
249C2×Dic5—42no—
250C2×Dic5—42no—
251C2×Dic5—42no—
252C2×Dic5—42no—
253C2×Dic5—42no—
254#168—24no—
255#168—42no—
256C22×D4—21no—
257C22×D4C22×D511yescharacteristic
258C23×C4—21no—
259C23×C4C22⋊D511yescharacteristic
260C23×C4C22⋊D511yescharacteristic
261C23×C4C22×D511yescharacteristic
262C22×C8—52no—
263C22×C8—52no—
264#121—22no—
265#121—22no—
266C4×D4—24no—
267C4×D4—44no—
268#117—204no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—52no—
271C2×C4⋊C4—52no—
272C2×C4⋊C4—101no—
273C2×C4⋊C4—101no—
274C2×C4⋊C4—101no—
275C2×C4⋊C4—101no—
276C2×C4⋊C4C22×D511yescharacteristic
277C2×C4⋊C4C22×D511yescharacteristic
278C2×C4⋊C4C22×D511yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—102no—
285C2×C42—24no—
286C2×C42—21no—
287C2×C42—102no—
288C2×C42C22⋊D511yescharacteristic
289C2×C42C22⋊D511yescharacteristic
290C2×C42C22×D511yescharacteristic
291C2×C42C22×D511yescharacteristic
292#106—104no—
293#106—104no—
294#102—104no—
295#102—104no—
296C22⋊C8—104no—
297C4×C8—104no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—102no—
302#95—102no—
303C22×C5—22no—
304C22×C5—22no—
305C22×C5—22no—
306C22×C5—41no—
307C22×C5—41no—
308C22×C5—41no—
309C22×C5—41no—
310C22×C5—42no—
311C22×C5—42no—
312C22×C5—42no—
313C22×C5—81no—
314C22×C5—81no—
315C22×C5—81no—
316C22×C5—81no—
317C22×C5C2×#12311yescharacteristic
318C22×C5#52511yescharacteristic
319C22×C5#53211yescharacteristic
320C22×C5C22×SD1611yescharacteristic
321C22×C5C2×#13511yescharacteristic
322C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
323C22×C5#57511yescharacteristic
324C4×C5—22no—
325C4×C5—22no—
326C4×C5—22no—
327C4×C5—42no—
328C4×C5—41no—
329C4×C5—81no—
330C4×C5—81no—
331C4×C5#48114yes—
332C4×C5#52912yes—
333C4×C5#52912yes—
334Dic5—82no—
335Dic5—82no—
336C24—21no—
337C24C2×C22⋊D511yescharacteristic
338C2×D4—24no—
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4#70912yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—102no—
361C22×C4—102no—
362C22×C4#68714yes—
363C22×C4C2×C22⋊D511yescharacteristic
364C22×C4C2×C22⋊D511yescharacteristic
365C22×C4C2×C22⋊D511yescharacteristic
366C22×C4C2×C22⋊D511yescharacteristic
367C22×C4C2×C22⋊D511yescharacteristic
368C22×C4C23×D511yescharacteristic
369C2×C8—54no—
370C2×C8—102no—
371C2×C8—102no—
372C2×C8—102no—
373C2×C8—102no—
374C2×C8—102no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—102no—
380C4⋊C4—102no—
381C4⋊C4—102no—
382C4⋊C4—102no—
383C4⋊C4—202no—
384C4⋊C4—202no—
385C4⋊C4—202no—
386C4⋊C4—202no—
387C4⋊C4#70912yes—
388C22⋊C4—24no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—204no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—42no—
398C42—204no—
399C42#71114yes—
400C2×C5—42no—
401C2×C5—82no—
402C2×C5#239511yescharacteristic
403C2×C5C2×#48111yescharacteristic
404C2×C5#309111yescharacteristic
405C2×C5#310111yescharacteristic
406C2×C5#310611yescharacteristic
407C2×C5#316711yescharacteristic
408C2×C5#317111yescharacteristic
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23C2×#68712yes—
423C23C22×C22⋊D511yescharacteristic
424D4—24no—
425D4—48no—
426D4—42no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—28no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—84no—
450C2×C4—202no—
451C2×C4—202no—
452C2×C4—202no—
453C2×C4—202no—
454C2×C4—202no—
455C2×C4—202no—
456C2×C4C2×#68712yes—
457C2×C4C2×#68712yes—
458C2×C4#408112yes—
459C2×C4#408112yes—
460C2×C4#409512yes—
461C2×C4#409612yes—
462C2×C4C2×#70911yescharacteristic
463C2×C4C2×#70911yescharacteristic
464C2×C4C2×#71112yes—
465C2×C4D5×#4112yes—
466C8—104no—
467C8—202no—
468C5#2375511yescharacteristic, complemented, Sylow
469C22—22no—
470C22—22no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22#6612511yescharacteristic
484C22C22×#68711yescharacteristic
485C22C2×#408111yescharacteristic
486C22C2×#409511yescharacteristic
487C22C2×#409611yescharacteristic
488C22#6615611yescharacteristic
489C22#6616611yescharacteristic
490C4—22no—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—41no—
495C4—81no—
496C4—81no—
497C4—402no—
498C4—402no—
499C4#6532314yes—
500C4#6587212yes—
501C4#6589112yes—
502C2—42no—
503C2—82no—
504C2#1061404511yescharacteristic
505C2#1061407711yescharacteristic
506C2#1061426011yescharacteristic
507C2C2×#6532311yescharacteristic
508C2#1061464411yescharacteristic
509C2#1061486611yescharacteristic
510C2#1061502111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC4×C5×#121C2
socleC23×C5C22×D4
prime radicalC2×C5×#121C22
hypercenterC4×#121D5
hypocenterC5#23755
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. #10613793C2semidirect
4. C4×#3964C2semidirect
5. C4×C22⋊#168C2semidirect
6. #10600449C2semidirect
7. #10600433C2semidirect
8. #10600405C2semidirect
9. #10600401C2semidirect
10. #10600313C2semidirect
11. C4×#3940C2semidirect
468. C5#23755semidirect

Maximal subgroups

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

subgroupindex
C4×C5×#1212
#106137932
#106024732
C4×#39642
C4×C22⋊#1682
#106004492
#106004332
#106004052
#106004012
#106003132
C4×#39402
#237555