Finite Group Databasebeta

← #12152659

Subgroups — #12152659

Order 1280.

1126 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#552C211yesmaximal, characteristic
2C2×#65358C211yesmaximal, characteristic, complemented
3C2×#65320C211yesmaximal, characteristic, complemented
4#10614110C211yesmaximal, characteristic
5#10602531C211yesmaximal, characteristic, complemented
6#10602493C212yesmaximal, complemented
7#10602493C212yesmaximal, complemented
8#10602424C211yesmaximal, characteristic, complemented
9#10602408C211yesmaximal, characteristic
10#10602378C212yesmaximal, complemented
11#10602352C212yesmaximal, complemented
12C5×#552—24no—
13C5×#552—24no—
14C2×C5×#126—21no—
15C2×C5×#125—21no—
16C2×C5×#125C2211yescharacteristic
17C2×C5×#123—21no—
18C2×C5×C22⋊Q8C2211yescharacteristic
19C2×C5×C22⋊Q8C2211yescharacteristic
20C2×C4×C5×Q8C2211yescharacteristic
21C2×C4×C5×D4C2211yescharacteristic
22C2×C5×#117—21no—
23C2×C5×#117C2211yescharacteristic
24C22×C5×C4⋊C4C2211yescharacteristic
25C22×#689C2212yes—
26C2×#4070C2212yes—
27C2×#4070C2212yes—
28C2×#4015C2211yescharacteristic
29C2×#3967C2212yes—
30#65510—21no—
31C2×C22⋊#168C2212yes—
32C2×C22⋊#168C2212yes—
33#65358—24no—
34#65320—24no—
35#65265—21no—
36C2×#3942C2212yes—
37C2×#3942C2212yes—
38C2×#3942C2212yes—
39#65238—21no—
40#65238—21no—
41C4×C4⋊Dic5C2211yescharacteristic
42C2×C4⋊#168C2211yescharacteristic
43C2×#3935C2211yescharacteristic
44#64749C2212yes—
45#64749C2212yes—
46#64749C2212yes—
47#25584—51nomaximal, Sylow
48C22×C5×Q8C2311yescharacteristic
49C22×C5×D4D411yescharacteristic
50C23×C4×C5—21no—
51C23×C4×C5C2311yescharacteristic
52C23×C4×C5C2311yescharacteristic
53C22×Dic10C2312yes—
54C5×#126—28no—
55C5×#125—28no—
56C5×#125—22no—
57C5×#125D414yes—
58C5×#123—28no—
59C5×C22⋊Q8—22no—
60C5×C22⋊Q8—22no—
61C5×C22⋊Q8—22no—
62C5×C22⋊Q8—22no—
63C4×C5×Q8—22no—
64C4×C5×Q8D414yes—
65C4×C5×D4—22no—
66C4×C5×D4—22no—
67C5×#117—28no—
68C5×#117—22no—
69C5×#117—22no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—22no—
77C2×C5×C4⋊C4—22no—
78C2×C5×C4⋊C4D411yescharacteristic
79C2×C5×C4⋊C4C2311yescharacteristic
80C2×C5×C4⋊C4C2311yescharacteristic
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C5×C22⋊C4—21no—
83C2×C5×C22⋊C4—21no—
84C2×C5×C22⋊C4—21no—
85C2×C5×C22⋊C4D411yescharacteristic
86C2×C5×C22⋊C4D411yescharacteristic
87C2×C42×C5—21no—
88C2×C42×C5—21no—
89C2×C42×C5C2311yescharacteristic
90C2×C22⋊Dic5—22no—
91C2×C22⋊Dic5—22no—
92C2×#689—24no—
93C2×#689—42no—
94C2×#689—42no—
95C2×C4⋊Dic5—21no—
96C2×C4⋊Dic5—21no—
97C2×C4⋊Dic5C2312yes—
98#4070—24no—
99#4070—24no—
100#4070—42no—
101#4070—42no—
102C2×#683—22no—
103C2×#683—22no—
104C2×C4×Dic5—21no—
105C22×#168C2312yes—
106C22×#168C2312yes—
107#4015—22no—
108#4015—22no—
109#3967—24no—
110#3963—21no—
111C22⋊#168—42no—
112C22⋊#168—42no—
113#3942—24no—
114#3942—24no—
115#3942—24no—
116C4⋊#168—22no—
117#3935—22no—
118C2×#552D511yescharacteristic
119C2×#491—51no—
120C2×#462—51no—
121#2542—51no—
122#2001—52no—
123#2001—52no—
124#1989—51no—
125#1955—51no—
126#1873—52no—
127#1872—52no—
128#1763—51no—
129C24×C5C2×D411yescharacteristic
130C24×C5C2×D411yescharacteristic
131C2×C5×Q8—21no—
132C2×C5×Q8—22no—
133C2×C5×Q8—22no—
134C2×C5×Q8C2×D412yes—
135C2×C5×D4—22no—
136C2×C5×D4—22no—
137C2×C5×D4—41no—
138C22×C4×C5—24no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—22no—
148C22×C4×C5—22no—
149C22×C4×C5—22no—
150C22×C4×C5—22no—
151C22×C4×C5—42no—
152C22×C4×C5—42no—
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2411yescharacteristic
158C22×Dic5—22no—
159C22×Dic5—22no—
160C2×Dic10—22no—
161C2×Dic10—22no—
162C2×Dic10—22no—
163C2×Dic10—22no—
164C2×Dic10—22no—
165C2×Dic10—22no—
166C5×C4⋊C4—24no—
167C5×C4⋊C4—24no—
168C5×C4⋊C4—24no—
169C5×C4⋊C4—24no—
170C5×C4⋊C4—24no—
171C5×C4⋊C4—24no—
172C5×C4⋊C4—22no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4C2×D412yes—
177C5×C4⋊C4C2×D412yes—
178C5×C4⋊C4C2×D412yes—
179C5×C22⋊C4—24no—
180C5×C22⋊C4—24no—
181C5×C22⋊C4—24no—
182C5×C22⋊C4—22no—
183C5×C22⋊C4—22no—
184C42×C5—24no—
185C42×C5—24no—
186C42×C5C2×D412yes—
187C42×C5C2×D412yes—
188C22⋊Dic5—44no—
189C22⋊Dic5—44no—
190#689—48no—
191C4⋊Dic5—22no—
192C4⋊Dic5—42no—
193C4⋊Dic5—42no—
194C4⋊Dic5#4114yes—
195#683—44no—
196#683—44no—
197C4×Dic5—42no—
198C2×#168—22no—
199C2×#168—22no—
200C2×#168—22no—
201C2×#168—22no—
202C2×#168—22no—
203C2×#168—22no—
204C22×Q16—52no—
205#552—24no—
206#552—24no—
207C2×C4⋊Q8—51no—
208C2×#126—21no—
209C2×#125—21no—
210C2×#125C2×D511yescharacteristic
211C2×#123—21no—
212C2×C22⋊Q8—52no—
213C2×C22⋊Q8—52no—
214C2×C22⋊Q8C2×D511yescharacteristic
215C2×C22⋊Q8C2×D511yescharacteristic
216C2×C4×Q8C2×D511yescharacteristic
217C2×C4×D4C2×D511yescharacteristic
218C2×#117—21no—
219C2×#117C2×D511yescharacteristic
220C22×C4⋊C4C2×D511yescharacteristic
221#491—104no—
222#462—104no—
223C2×C4⋊C8—51no—
224C2×#103—52no—
225C2×#103—52no—
226C2×#103—52no—
227C2×#103—52no—
228C2×C22⋊C8—52no—
229C2×C22⋊C8—52no—
230C2×#97—51no—
231#399—101no—
232#399—101no—
233#390—101no—
234#384—101no—
235C4×C4⋊C4—51no—
236#340—52no—
237#340—52no—
238#340—52no—
239C23×C5—21no—
240C23×C5—22no—
241C23×C5—21no—
242C23×C5—22no—
243C23×C5—21no—
244C23×C5—22no—
245C23×C5—21no—
246C23×C5—21no—
247C23×C5—22no—
248C23×C5#13712yes—
249C23×C5C22×D411yescharacteristic
250C5×Q8—22no—
251C5×Q8—22no—
252C5×Q8—42no—
253C5×D4—44no—
254C2×C4×C5—21no—
255C2×C4×C5—21no—
256C2×C4×C5—22no—
257C2×C4×C5—21no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—21no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—48no—
266C2×C4×C5—42no—
267C2×C4×C5—41no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—42no—
272C2×C4×C5—41no—
273C2×C4×C5—41no—
274C2×C4×C5—41no—
275C2×C4×C5—41no—
276C2×C4×C5—42no—
277C2×C4×C5—42no—
278C2×C4×C5#12012yes—
279C2×C4×C5#12012yes—
280C2×C4×C5C22×D411yescharacteristic
281C2×C4×C5C22×D411yescharacteristic
282C2×C4×C5C2×#4112yes—
283C2×C4×C5H(5,𝔽2)12yes—
284C2×Dic5—24no—
285C2×Dic5—42no—
286C2×Dic5—42no—
287C2×Dic5—42no—
288C2×Dic5—42no—
289C2×Dic5—42no—
290Dic10—42no—
291Dic10—42no—
292Dic10—42no—
293Dic10—42no—
294#168—42no—
295#168—42no—
296C22×Q8—52no—
297C22×Q8C22×D511yescharacteristic
298C22×D4C22⋊D511yescharacteristic
299C23×C4—21no—
300C23×C4C22×D511yescharacteristic
301C23×C4C22×D511yescharacteristic
302C2×Q16—104no—
303C2×Q16—202no—
304C2×Q16—202no—
305C22×C8—52no—
306C22×C8—52no—
307C4⋊Q8—102no—
308C4⋊Q8—102no—
309#126—28no—
310#125—28no—
311#125—22no—
312#125C22⋊D514yes—
313#123—28no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—104no—
319C22⋊Q8—104no—
320C22⋊Q8—202no—
321C22⋊Q8—202no—
322C4×Q8—22no—
323C4×Q8C22⋊D514yes—
324C4×D4—22no—
325C4×D4—22no—
326#117—28no—
327#117—22no—
328#117—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—52no—
338C2×C4⋊C4—102no—
339C2×C4⋊C4—102no—
340C2×C4⋊C4—101no—
341C2×C4⋊C4—101no—
342C2×C4⋊C4C22⋊D511yescharacteristic
343C2×C4⋊C4C22×D511yescharacteristic
344C2×C4⋊C4C22×D511yescharacteristic
345C2×C4⋊C4C22×D511yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—102no—
350C2×C22⋊C4—102no—
351C2×C22⋊C4C22⋊D511yescharacteristic
352C2×C22⋊C4C22⋊D511yescharacteristic
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—101no—
356C2×C42C22×D511yescharacteristic
357C4⋊C8—102no—
358#103—104no—
359#103—104no—
360#103—104no—
361#103—104no—
362C22⋊C8—202no—
363C22⋊C8—202no—
364#97—102no—
365#95—101no—
366C22×C5—22no—
367C22×C5—24no—
368C22×C5—41no—
369C22×C5—41no—
370C22×C5—42no—
371C22×C5—41no—
372C22×C5—41no—
373C22×C5—41no—
374C22×C5—42no—
375C22×C5—42no—
376C22×C5C2×#12011yescharacteristic
377C22×C5#54612yes—
378C22×C5C2×#13711yescharacteristic
379C22×C5#57511yescharacteristic
380C22×C5#57812yes—
381C4×C5—22no—
382C4×C5—22no—
383C4×C5—41no—
384C4×C5—41no—
385C4×C5—42no—
386C4×C5—42no—
387C4×C5—42no—
388C4×C5—42no—
389C4×C5#53211yescharacteristic
390C4×C5#53511yescharacteristic
391C4×C5#55012yes—
392Dic5—44no—
393Dic5—82no—
394C24C2×C22⋊D511yescharacteristic
395C24C2×C22⋊D511yescharacteristic
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—102no—
400C2×Q8—102no—
401C2×Q8—102no—
402C2×Q8—102no—
403C2×Q8—102no—
404C2×Q8—102no—
405C2×Q8C2×C22⋊D512yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—41no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—102no—
425C22×C4—102no—
426C22×C4C2×C22⋊D511yescharacteristic
427C22×C4C2×C22⋊D511yescharacteristic
428C22×C4C2×C22⋊D511yescharacteristic
429C22×C4C2×C22⋊D511yescharacteristic
430C22×C4C23×D511yescharacteristic
431Q16—208no—
432C2×C8—102no—
433C2×C8—102no—
434C2×C8—102no—
435C2×C8—102no—
436C2×C8—102no—
437C2×C8—102no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—54no—
449C4⋊C4—102no—
450C4⋊C4—204no—
451C4⋊C4—204no—
452C4⋊C4—202no—
453C4⋊C4—202no—
454C4⋊C4C2×C22⋊D512yes—
455C4⋊C4C2×C22⋊D512yes—
456C4⋊C4C2×C22⋊D512yes—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—204no—
463C22⋊C4—204no—
464C42—24no—
465C42—24no—
466C42—202no—
467C42C2×C22⋊D512yes—
468C42C2×C22⋊D512yes—
469C2×C5—42no—
470C2×C5—81no—
471C2×C5#243111yescharacteristic
472C2×C5#300711yescharacteristic
473C2×C5#301111yescharacteristic
474C2×C5#331212yes—
475C2×C5#331512yes—
476C23—21no—
477C23—22no—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—21no—
485C23#409712yes—
486C23C22×C22⋊D511yescharacteristic
487Q8—22no—
488Q8—22no—
489Q8—42no—
490Q8—202no—
491Q8—202no—
492Q8—202no—
493Q8—202no—
494D4—44no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—48no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—104no—
520C2×C4—202no—
521C2×C4—202no—
522C2×C4—202no—
523C2×C4—202no—
524C2×C4—202no—
525C2×C4#409912yes—
526C2×C4#409912yes—
527C2×C4D5×#4112yes—
528C2×C4#414912yes—
529C2×C4C22×C22⋊D511yescharacteristic
530C2×C4C22×C22⋊D511yescharacteristic
531C8—202no—
532C8—202no—
533C5#2558411yescharacteristic, complemented, Sylow
534C22—22no—
535C22—24no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—41no—
544C22#6615611yescharacteristic
545C22C2×#409711yescharacteristic
546C22#6616312yes—
547C22#6617212yes—
548C22C2×#409911yescharacteristic
549C4—22no—
550C4—22no—
551C4—42no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—204no—
558C4—402no—
559C4#6616411yescharacteristic
560C4#6616611yescharacteristic
561C4#6616812yes—
562C2—42no—
563C2—81no—
564C2#1061533112yes—
565C2#1061533412yes—
566C2#1061581511yescharacteristic
567C2#1061582311yescharacteristic
568C2#1061584811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×#552C2
socleC23×C5C22×D4
prime radicalC22×C5×D4D4
hypercenterC2×#552D5
hypocenterC5#25584
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
2. C2×#65358C2semidirect
3. C2×#65320C2semidirect
5. #10602531C2semidirect
6. #10602493C2semidirect
7. #10602493C2semidirect
8. #10602424C2semidirect
10. #10602378C2semidirect
11. #10602352C2semidirect
533. C5#25584semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5522
C2×#653582
C2×#653202
#106141102
#106025312
#106024932
#106024242
#106024082
#106023782
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