Finite Group Databasebeta

← #12156317

Subgroups — #12156317

Order 1280.

1176 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×#554C211yesmaximal, characteristic, complemented
2C2×#65356C211yesmaximal, characteristic, complemented
3#10614207C211yesmaximal, characteristic, complemented
4C2×#65304C211yesmaximal, characteristic
5#10602585C212yesmaximal, complemented
6#10602536C212yesmaximal, complemented
7#10602531C211yesmaximal, characteristic, complemented
8#10602428C211yesmaximal, characteristic, complemented
9#10602406C212yesmaximal, complemented
10#10602406C212yesmaximal, complemented
11#10602402C211yesmaximal, characteristic, complemented
12C5×#554—24no—
13C5×#554—24no—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C5×#125C2211yescharacteristic
16C2×C5×#125C2211yescharacteristic
17C2×C5×C22⋊Q8—21no—
18C2×C5×C22⋊Q8—21no—
19C2×C5×C22⋊Q8—21no—
20C2×C4×C5×Q8C2211yescharacteristic
21C2×C4×C5×D4C2211yescharacteristic
22C2×C4×C5×D4C2211yescharacteristic
23C2×C4×C5×D4C2211yescharacteristic
24C22×C5×C4⋊C4—21no—
25C2×#4093—21no—
26C22×#686C2212yes—
27C2×#4020C2211yescharacteristic
28C2×#3964C2212yes—
29#65356—24no—
30#65304—24no—
31#65280—21no—
32#65280—21no—
33#65277—21no—
34C2×#3941C2212yes—
35C2×#3941C2212yes—
36C2×#3941C2212yes—
37#65228C2212yes—
38#65228C2212yes—
39C4×C4⋊Dic5C2211yescharacteristic
40C2×C4⋊#168C2211yescharacteristic
41C2×#3935C2211yescharacteristic
42#64749C2212yes—
43#64749C2212yes—
44#64749C2212yes—
45#64744C2212yes—
46#64744C2212yes—
47#25946—51nomaximal, Sylow
48C22×C5×Q8—21no—
49C22×C5×Q8D411yescharacteristic
50C22×C5×D4C2311yescharacteristic
51C23×C4×C5—21no—
52C23×C4×C5—21no—
53C23×C4×C5—21no—
54C22×D20—21no—
55C22×D20—21no—
56C22×D20C2312yes—
57C5×C4⋊Q8—22no—
58C5×C4⋊Q8—22no—
59C5×#125—22no—
60C5×#125—22no—
61C5×#125—22no—
62C5×#125D414yes—
63C5×C22⋊Q8—28no—
64C5×C22⋊Q8—28no—
65C5×C22⋊Q8—28no—
66C4×C5×Q8—22no—
67C4×C5×Q8—22no—
68C4×C5×D4—22no—
69C4×C5×D4—22no—
70C4×C5×D4—22no—
71C4×C5×D4—22no—
72C4×C5×D4—22no—
73C4×C5×D4D414yes—
74C2×C5×C4⋊C4—28no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4D411yescharacteristic
81C2×C5×C4⋊C4D411yescharacteristic
82C2×C5×C4⋊C4D411yescharacteristic
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C4⋊C4C2311yescharacteristic
85C2×C5×C4⋊C4C2311yescharacteristic
86C2×C5×C22⋊C4—21no—
87C2×C5×C22⋊C4—21no—
88C2×C5×C22⋊C4—21no—
89C2×C42×C5C2311yescharacteristic
90C2×C42×C5C2311yescharacteristic
91C2×C42×C5C2311yescharacteristic
92#4093—44no—
93C2×#686—24no—
94C2×#686—24no—
95C2×#686—24no—
96C2×#685—22no—
97C2×#685—22no—
98C2×#685—22no—
99C2×#685—22no—
100C2×C4⋊Dic5C2312yes—
101C2×C4×Dic5—21no—
102C22×#168C2312yes—
103C22×#168C2312yes—
104#4020—22no—
105#4020—22no—
106#3964—24no—
107#3963—21no—
108#3941—24no—
109#3941—24no—
110#3941—24no—
111C4⋊#168—22no—
112#3935—22no—
113C2×#554D511yescharacteristic, complemented
114C2×#490—51no—
115C2×#459—51no—
116#2548—51no—
117#2025—52no—
118#2023—52no—
119#1989—51no—
120#1985—51no—
121#1879—52no—
122#1879—52no—
123#1762—51no—
124C24×C5—21no—
125C23×D5—22no—
126C2×C5×Q8—24no—
127C2×C5×Q8—22no—
128C2×C5×Q8—22no—
129C2×C5×Q8—41no—
130C2×C5×Q8—82no—
131C2×C5×D4—21no—
132C2×C5×D4—22no—
133C2×C5×D4C2×D412yes—
134C2×C5×D4#4114yes—
135C22×C4×C5—24no—
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—24no—
140C22×C4×C5—24no—
141C22×C4×C5—42no—
142C22×C4×C5—42no—
143C22×C4×C5—82no—
144C22×C4×C5—82no—
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2411yescharacteristic
152C22×Dic5—22no—
153C2×D20—22no—
154C2×D20—22no—
155C2×D20—22no—
156C2×D20—22no—
157C2×D20—22no—
158C2×D20—22no—
159C2×D20—42no—
160C2×D20—42no—
161C2×D20—42no—
162C2×D20—42no—
163C2×D20—82no—
164C5×C4⋊C4—24no—
165C5×C4⋊C4—24no—
166C5×C4⋊C4—24no—
167C5×C4⋊C4—22no—
168C5×C4⋊C4—24no—
169C5×C4⋊C4—24no—
170C5×C4⋊C4—22no—
171C5×C4⋊C4—22no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4C2×D412yes—
176C5×C4⋊C4C2×D412yes—
177C5×C4⋊C4C2×D412yes—
178C5×C22⋊C4—24no—
179C5×C22⋊C4—24no—
180C5×C22⋊C4—24no—
181C42×C5—21no—
182C42×C5—21no—
183C42×C5—21no—
184C42×C5—21no—
185C42×C5C2×D412yes—
186C42×C5C2×D412yes—
187#686—48no—
188#685—44no—
189#685—44no—
190#685—82no—
191#685—82no—
192C4⋊Dic5—22no—
193C4⋊Dic5—22no—
194C4×Dic5—42no—
195C2×#168—22no—
196C2×#168—22no—
197C2×#168—22no—
198C2×#168—22no—
199C2×#168—22no—
200C2×#168—22no—
201C22×D8—52no—
202#554—24no—
203#554—24no—
204C2×C4⋊Q8C2×D511yescharacteristic
205C2×#127—51no—
206C2×#125C2×D511yescharacteristic
207C2×#125C2×D511yescharacteristic
208C2×#124—101no—
209C2×C22⋊Q8—21no—
210C2×C22⋊Q8—21no—
211C2×C22⋊Q8—21no—
212C2×C4×Q8C2×D511yescharacteristic
213C2×C4×D4C2×D511yescharacteristic
214C2×C4×D4C2×D511yescharacteristic
215C2×C4×D4C2×D511yescharacteristic
216C22×C4⋊C4—21no—
217#490—104no—
218#459—104no—
219C2×C4⋊C8—51no—
220C2×#102—52no—
221C2×#102—52no—
222C2×#102—52no—
223C2×#102—52no—
224C2×#97—51no—
225#394—101no—
226#394—101no—
227#390—52no—
228#390—52no—
229#390—101no—
230C4×C4⋊C4—51no—
231#340—52no—
232#340—52no—
233#340—52no—
234#336—52no—
235#336—52no—
236C23×C5—22no—
237C23×C5—24no—
238C23×C5—41no—
239C23×C5—41no—
240C23×C5—42no—
241C23×C5C22×D411yescharacteristic
242C22×D5—42no—
243C22×D5—42no—
244C22×D5—42no—
245C22×D5—42no—
246C22×D5—42no—
247C22×D5—42no—
248C22×D5—42no—
249C5×Q8—44no—
250C5×Q8—84no—
251C5×D4—22no—
252C5×D4—24no—
253C5×D4#12414yescomplemented
254C2×C4×C5—21no—
255C2×C4×C5—21no—
256C2×C4×C5—21no—
257C2×C4×C5—21no—
258C2×C4×C5—22no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—48no—
270C2×C4×C5—41no—
271C2×C4×C5—41no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—84no—
275C2×C4×C5—84no—
276C2×C4×C5#12012yes—
277C2×C4×C5#12012yes—
278C2×C4×C5AGL(1,ℤ/8)12yes—
279C2×C4×C5C22×D411yescharacteristic
280C2×C4×C5C22×D411yescharacteristic
281C2×C4×C5C2×#4112yes—
282C2×C4×C52−1+412yes—
283C2×Dic5—42no—
284C2×Dic5—42no—
285C2×Dic5—42no—
286D20—42no—
287D20—42no—
288D20—42no—
289D20—42no—
290D20—84no—
291D20—84no—
292#168—42no—
293#168—42no—
294C22×Q8—21no—
295C22×Q8C22⋊D511yescharacteristic
296C22×D4—52no—
297C22×D4—101no—
298C22×D4—101no—
299C22×D4C22×D511yescharacteristic
300C23×C4—21no—
301C23×C4—21no—
302C23×C4—21no—
303C2×D8—104no—
304C2×D8—104no—
305C2×D8—104no—
306C22×C8—52no—
307C22×C8—52no—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310#127—102no—
311#127—102no—
312#125—22no—
313#125—22no—
314#125—22no—
315#125C22⋊D514yes—
316#124—204no—
317C22⋊Q8—28no—
318C22⋊Q8—28no—
319C22⋊Q8—28no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4C22⋊D514yes—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—52no—
335C2×C4⋊C4C22⋊D511yescharacteristic
336C2×C4⋊C4C22⋊D511yescharacteristic
337C2×C4⋊C4C22⋊D511yescharacteristic
338C2×C4⋊C4C22×D511yescharacteristic
339C2×C4⋊C4C22×D511yescharacteristic
340C2×C4⋊C4C22×D511yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—102no—
345C2×C22⋊C4—102no—
346C2×C22⋊C4—102no—
347C2×C22⋊C4—102no—
348C2×C42—101no—
349C2×C42C22×D511yescharacteristic
350C2×C42C22×D511yescharacteristic
351C2×C42C22×D511yescharacteristic
352C4⋊C8—102no—
353#102—104no—
354#102—104no—
355#102—104no—
356#102—104no—
357#97—102no—
358#95—101no—
359C22×C5—24no—
360C22×C5—44no—
361C22×C5—44no—
362C22×C5—42no—
363C22×C5—42no—
364C22×C5C2×#12011yescharacteristic
365C22×C5#54812yes—
366C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
367C22×C5#57511yescharacteristic
368C22×C5#57612yes—
369C2×D5—82no—
370C2×D5—82no—
371C2×D5—82no—
372C2×D5—82no—
373C2×D5—82no—
374C2×D5—82no—
375C2×D5—82no—
376C4×C5—22no—
377C4×C5—22no—
378C4×C5—22no—
379C4×C5—41no—
380C4×C5—42no—
381C4×C5—41no—
382C4×C5—41no—
383C4×C5—42no—
384C4×C5—42no—
385C4×C5C2×#12411yescharacteristic
386C4×C5#53611yescharacteristic
387C4×C5#54812yes—
388Dic5—82no—
389C24—21no—
390C24—102no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—41no—
395C2×Q8—82no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—102no—
399C2×D4—102no—
400C2×D4—102no—
401C2×D4—102no—
402C2×D4—102no—
403C2×D4—102no—
404C2×D4—202no—
405C2×D4—202no—
406C2×D4—202no—
407C2×D4—202no—
408C2×D4—402no—
409C2×D4#71314yes—
410C2×D4C2×C22⋊D512yes—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—102no—
422C22×C4C2×C22⋊D511yescharacteristic
423C22×C4C2×C22⋊D511yescharacteristic
424C22×C4C2×C22⋊D511yescharacteristic
425C22×C4C2×C22⋊D511yescharacteristic
426C22×C4C2×C22⋊D511yescharacteristic
427C22×C4C2×C22⋊D511yescharacteristic
428C22×C4C23×D511yescharacteristic
429D8—208no—
430C2×C8—102no—
431C2×C8—102no—
432C2×C8—102no—
433C2×C8—102no—
434C2×C8—102no—
435C2×C8—102no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—102no—
448C4⋊C4—102no—
449C4⋊C4C2×C22⋊D512yes—
450C4⋊C4C2×C22⋊D512yes—
451C4⋊C4C2×C22⋊D512yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—204no—
456C22⋊C4—204no—
457C22⋊C4—402no—
458C22⋊C4—402no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—202no—
464C42C2×C22⋊D512yes—
465C42C2×C22⋊D512yes—
466C2×C5—44no—
467C2×C5#243611yescharacteristic
468C2×C5#300611yescharacteristic
469C2×C5#300811yescharacteristic
470C2×C5#336612yes—
471C2×C5#337012yes—
472D5—162no—
473C23—24no—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—202no—
479C23—202no—
480C23—202no—
481C23—202no—
482C23—202no—
483C23—202no—
484C23—202no—
485C23C22×C22⋊D511yescharacteristic
486Q8—44no—
487Q8—84no—
488D4—22no—
489D4—24no—
490D4—202no—
491D4—202no—
492D4—202no—
493D4—202no—
494D4—404no—
495D4—404no—
496D4#409314yescomplemented
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—48no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—202no—
520C2×C4—202no—
521C2×C4—202no—
522C2×C4#409512yes—
523C2×C4#409912yes—
524C2×C4#409912yes—
525C2×C4C2×#71312yes—
526C2×C4#414712yes—
527C2×C4C22×C22⋊D511yescharacteristic
528C2×C4C22×C22⋊D511yescharacteristic
529C8—202no—
530C8—202no—
531C5#2594611yescharacteristic, complemented, Sylow
532C22—24no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—402no—
538C22—402no—
539C22—402no—
540C22—402no—
541C22—402no—
542C22—402no—
543C22—402no—
544C22#6615112yes—
545C22C2×#409511yescharacteristic
546C22#6615611yescharacteristic
547C22#6616912yes—
548C22C2×#409911yescharacteristic
549C4—22no—
550C4—22no—
551C4—22no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—402no—
559C4C2×#409311yescharacteristic
560C4#6615011yescharacteristic
561C4#6615112yes—
562C2—44no—
563C2—802no—
564C2#1061536912yes—
565C2#1061537012yes—
566C2#1061579711yescharacteristic
567C2#1061581311yescharacteristic
568C2#1061583211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×#554C2
socleC23×C5C22×D4
prime radicalC2×#65304C2
hypercenterC2×#554D5
hypocenterC5#25946
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×C5×#554C2semidirect
2. C2×#65356C2semidirect
3. #10614207C2semidirect
5. #10602585C2semidirect
6. #10602536C2semidirect
7. #10602531C2semidirect
8. #10602428C2semidirect
9. #10602406C2semidirect
10. #10602406C2semidirect
11. #10602402C2semidirect
113. C2×#554D5semidirect
253. C5×D4#124semidirect
496. D4#4093semidirect
531. C5#25946semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5542
C2×#653562
#106142072
C2×#653042
#106025852
#106025362
#106025312
#106024282
#106024062
#106024022
#259465