Finite Group Databasebeta

← #12067014

Subgroups — #12067014

Order 1280.

1070 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10618181C211yesmaximal, characteristic
2#10611017C211yesmaximal, characteristic
3#10605959C211yesmaximal, characteristic
4C5×#1518C211yesmaximal, characteristic, complemented
5#10602628C211yesmaximal, characteristic, complemented
6#10601585C211yesmaximal, characteristic, complemented
7#10600339C211yesmaximal, characteristic, complemented
8#10599132C214yesmaximal, complemented
9D5×#328C214yesmaximal, complemented
10C5×#533C2211yescharacteristic
11#66064C412yescomplemented
12#66061C412yescomplemented
13#66024—22no—
14#65968—22no—
15D5×C22⋊Q8C2214yes—
16#65913—22no—
17#65913C412yescomplemented
18Q8×D20C412yescomplemented
19#65855—21no—
20#65853—21no—
21C2×#4039—21no—
22C2×C4×D20C2211yescharacteristic
23#65804C2211yescharacteristic
24C5×#431C2211yescharacteristic
25C5×#382C2211yescharacteristic
26#65262—21no—
27C4×C4⋊Dic5C2211yescharacteristic
28#65214C2211yescharacteristic
29#65123—22no—
30D5×C22⋊C8C2214yes—
31C8×D20—22no—
32#64957—22no—
33#64956C2212yes—
34D5×#95C2212yes—
35C5×#328C2214yes—
36#64758C2212yes—
37#64758C2212yes—
38#64687C2212yes—
39#64687C2212yes—
40#64672C2214yes—
41#17614—51nomaximal, Sylow
42C2×Q8×D5C2×C414yescomplemented
43C22×D20D411yescharacteristic
44C22×C4×D5—22no—
45C22×C4×D5C2312yes—
46C5×C4⋊Q8C2×C411yescharacteristic, complemented
47C5×C4⋊Q8C2×C411yescharacteristic, complemented
48C5×#125—21no—
49C5×C22⋊Q8—21no—
50C5×C22⋊Q8C2312yes—
51C4×C5×Q8—21no—
52C4×C5×Q8C2×C412yescomplemented
53C5×#117—21no—
54C2×C5×C4⋊C4—21no—
55C2×C42×C5C2311yescharacteristic
56#4092—24no—
57#4092C2×C414yescomplemented
58C2×#685—22no—
59C2×#685D412yes—
60#4072—21no—
61C2×C4⋊Dic5—21no—
62C2×C4⋊Dic5D411yescharacteristic
63#4070—21no—
64#4070C2312yes—
65C2×#683—22no—
66C2×C4×Dic5—21no—
67C2×C8×D5—24no—
68#4043—24no—
69#4043C2×C414yescomplemented
70#4042—24no—
71#4042—24no—
72#4042—24no—
73#4039—22no—
74#4039—22no—
75#4039—44no—
76D5×C4⋊C4—24no—
77D5×C4⋊C4C2×C414yescomplemented
78#4036—22no—
79#4034—22no—
80D5×C22⋊C4—24no—
81#4024—24no—
82C4×D20—22no—
83C4×D20—22no—
84C4×D20C2×C412yes—
85C4×D20C2×C412yes—
86C4×D20D412yes—
87C4×D20D412yes—
88#4016—21no—
89#4016—22no—
90#4015—22no—
91#4015C2×C411yescharacteristic, complemented
92#4015C2×C411yescharacteristic, complemented
93C4×Dic10—21no—
94C4×Dic10C2×C412yescomplemented
95C5×C4⋊C8—21no—
96C5×C22⋊C8C2312yes—
97C4×C5×C8—21no—
98C5×#95—21no—
99C5×#95C2312yes—
100#3963—22no—
101#3963C2311yescharacteristic
102#3963C2311yescharacteristic
103C22⋊#168C2312yes—
104#3952—24no—
105C4⋊#168—21no—
106C4×#168—21no—
107#3501—51no—
108#2972—51no—
109#2349—51no—
110#1520—54no—
111#1518—51no—
112#1518D511yescharacteristic, complemented
113#1503—51no—
114#1503—51no—
115C2×#328—54no—
116C23×D5C2×D412yes—
117C2×C5×Q8—21no—
118C2×C5×Q8C22×C412yes—
119C22×C4×C5—22no—
120C22×C4×C5C2×D411yescharacteristic
121C22×C4×C5C2×D411yescharacteristic
122C22×C4×C5C2411yescharacteristic
123C22×Dic5—21no—
124C22×Dic5—21no—
125C22×Dic5C2×D412yes—
126Q8×D5—84no—
127C2×D20C22⋊C412yes—
128C2×D20C22⋊C412yes—
129C2×D20SD1614yes—
130C2×D20Q1614yes—
131C2×C4×D5—24no—
132C2×C4×D5—24no—
133C2×C4×D5—44no—
134C2×C4×D5—44no—
135C2×C4×D5—82no—
136C2×C4×D5—82no—
137C2×C4×D5C22×C414yes—
138C2×C4×D5C2×D414yes—
139C2×Dic10—21no—
140C2×Dic10—24no—
141C2×Dic10C22×C412yes—
142C2×C5×C8—22no—
143C5×C4⋊C4—21no—
144C5×C4⋊C4—21no—
145C5×C4⋊C4—21no—
146C5×C4⋊C4—22no—
147C5×C4⋊C4—22no—
148C5×C4⋊C4—42no—
149C5×C4⋊C4C22×C412yes—
150C5×C22⋊C4—22no—
151C42×C5—22no—
152C42×C5C22×C411yescharacteristic
153C42×C5C22×C411yescharacteristic
154C42×C5C2×D411yescharacteristic
155C42×C5C2×D411yescharacteristic
156C22⋊Dic5—22no—
157#685—24no—
158#685—24no—
159#685—44no—
160#685C22⋊C414yes—
161#685C22⋊C414yes—
162C4⋊Dic5—21no—
163C4⋊Dic5—21no—
164C4⋊Dic5—21no—
165C4⋊Dic5—24no—
166C4⋊Dic5—42no—
167C4⋊Dic5C22⋊C412yes—
168C4⋊Dic5C22⋊C412yes—
169C4⋊Dic5C22×C412yes—
170#683—22no—
171#683—22no—
172#683—24no—
173#683—24no—
174#683—82no—
175C4×Dic5—22no—
176C4×Dic5—42no—
177C2×#168—22no—
178C8×D5—84no—
179#554—52no—
180#554—52no—
181#554—102no—
182#554—102no—
183D4×Q8—52no—
184D4×Q8—52no—
185#544—101no—
186#543—102no—
187#533—51no—
188#533C2×D511yescharacteristic
189C2×C22⋊Q8—54no—
190#518—101no—
191C2×C4×D4—51no—
192C2×C4×D4—101no—
193C8×D4—102no—
194C8×D4—102no—
195#431—51no—
196#431C2×D511yescharacteristic
197C2×C22⋊C8—54no—
198#389—101no—
199#385—102no—
200#382C2×D511yescharacteristic
201#380—52no—
202C4×C4⋊C4—51no—
203C2×#95—52no—
204#328—52no—
205#328—52no—
206#328—52no—
207#328—52no—
208#328—54no—
209#328C2×D514yes—
210C23×C5C22×D411yescharacteristic
211C22×D5—42no—
212C22×D5—42no—
213C22×D5C2×C22⋊C414yes—
214C22×D5C2×SD1614yes—
215C22×D5C2×Q1614yes—
216C5×Q8—42no—
217C5×Q8—81no—
218C2×C4×C5—22no—
219C2×C4×C5—22no—
220C2×C4×C5—41no—
221C2×C4×C5—41no—
222C2×C4×C5—42no—
223C2×C4×C5—42no—
224C2×C4×C5—42no—
225C2×C4×C5C2×C22⋊C411yescharacteristic
226C2×C4×C5C2×C22⋊C411yescharacteristic
227C2×C4×C5C2×C22⋊C411yescharacteristic
228C2×C4×C5C2×C22⋊C411yescharacteristic
229C2×C4×C5C2×SD1612yes—
230C2×C4×C5C2×Q1612yes—
231C2×C4×C5C23×C411yescharacteristic
232C2×C4×C5C22×D411yescharacteristic
233C2×Dic5—22no—
234C2×Dic5—22no—
235C2×Dic5—22no—
236C2×Dic5—42no—
237C2×Dic5—41no—
238C2×Dic5—41no—
239C2×Dic5—41no—
240C2×Dic5—41no—
241C2×Dic5—41no—
242C2×Dic5C2×C22⋊C414yes—
243D20#10318yes—
244D20#10318yes—
245C4×D5—44no—
246C4×D5—44no—
247C4×D5—84no—
248C4×D5—84no—
249C4×D5—88no—
250Dic10—42no—
251Dic10—84no—
252Dic10—81no—
253C5×C8—42no—
254#168—42no—
255C22×Q8—54no—
256C22×D4—51no—
257C23×C4—52no—
258C23×C4—102no—
259C22×C8—104no—
260C4⋊Q8—51no—
261C4⋊Q8—51no—
262C4⋊Q8—102no—
263C4⋊Q8—102no—
264C4⋊Q8C4×D511yescharacteristic, complemented
265C4⋊Q8C4×D511yescharacteristic, complemented
266#125—21no—
267#125—102no—
268#125—102no—
269#125—101no—
270C22⋊Q8—21no—
271C22⋊Q8—54no—
272C22⋊Q8—52no—
273C22⋊Q8—54no—
274C22⋊Q8—104no—
275C22⋊Q8—104no—
276C22⋊Q8—104no—
277C22⋊Q8—104no—
278C22⋊Q8—104no—
279C22⋊Q8—104no—
280C22⋊Q8—101no—
281C22⋊Q8C22×D512yes—
282C4×Q8—21no—
283C4×Q8—52no—
284C4×Q8—101no—
285C4×Q8C4×D512yescomplemented
286C4×D4—52no—
287C4×D4—52no—
288C4×D4—52no—
289C4×D4—52no—
290C4×D4—102no—
291C4×D4—102no—
292C4×D4—102no—
293C4×D4—102no—
294C4×D4—204no—
295#117—21no—
296#117—101no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—51no—
299C2×C4⋊C4—54no—
300C2×C4⋊C4—104no—
301C2×C4⋊C4—101no—
302C2×C4⋊C4—102no—
303C2×C22⋊C4—52no—
304C2×C22⋊C4—104no—
305C2×C22⋊C4—102no—
306C2×C42—101no—
307C2×C42C22×D511yescharacteristic
308C4⋊C8—21no—
309C4⋊C8—101no—
310C22⋊C8—52no—
311C22⋊C8—104no—
312C22⋊C8C22×D512yes—
313C4×C8—21no—
314C4×C8—101no—
315#95—21no—
316#95—51no—
317#95—51no—
318#95—102no—
319#95C22×D512yes—
320C22×C5—21no—
321C22×C5—21no—
322C22×C5C22×C22⋊C411yescharacteristic
323C22×C5C22×SD1611yescharacteristic
324C22×C5C22×Q1611yescharacteristic
325C2×D5—24no—
326C2×D5—24no—
327C2×D5—44no—
328C2×D5—44no—
329C2×D5C2×#10318yes—
330C4×C5—21no—
331C4×C5—21no—
332C4×C5—21no—
333C4×C5—21no—
334C4×C5—42no—
335C4×C5—42no—
336C4×C5—82no—
337C4×C5C2×#10312yes—
338C4×C5C2×#10312yes—
339Dic5—24no—
340Dic5—42no—
341Dic5—42no—
342Dic5—42no—
343Dic5—81no—
344C24—52no—
345C2×Q8—21no—
346C2×Q8—52no—
347C2×Q8—104no—
348C2×Q8—101no—
349C2×Q8—404no—
350C2×Q8C2×C4×D512yes—
351C2×D4—52no—
352C2×D4—52no—
353C2×D4—54no—
354C2×D4—54no—
355C22×C4—22no—
356C22×C4—52no—
357C22×C4—54no—
358C22×C4—54no—
359C22×C4—104no—
360C22×C4—104no—
361C22×C4—101no—
362C22×C4—101no—
363C22×C4—204no—
364C22×C4—204no—
365C22×C4—402no—
366C22×C4—402no—
367C22×C4D4×D511yescharacteristic
368C22×C4D4×D511yescharacteristic
369C22×C4C23×D511yescharacteristic
370C2×C8—22no—
371C2×C8—102no—
372C2×C8—404no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—52no—
380C4⋊C4—52no—
381C4⋊C4—52no—
382C4⋊C4—104no—
383C4⋊C4—104no—
384C4⋊C4—104no—
385C4⋊C4—102no—
386C4⋊C4—101no—
387C4⋊C4—101no—
388C4⋊C4—102no—
389C4⋊C4—101no—
390C4⋊C4—202no—
391C4⋊C4—402no—
392C4⋊C4C2×C4×D512yes—
393C22⋊C4—22no—
394C22⋊C4—54no—
395C22⋊C4—54no—
396C22⋊C4—104no—
397C22⋊C4—104no—
398C22⋊C4—102no—
399C22⋊C4—204no—
400C42—22no—
401C42—102no—
402C42—202no—
403C42C2×C4×D511yescharacteristic
404C42C2×C4×D511yescharacteristic
405C42D4×D511yescharacteristic
406C42D4×D511yescharacteristic
407C2×C5—21no—
408C2×C5—21no—
409C2×C5#288311yescharacteristic
410C2×C5C22×#10311yescharacteristic
411C2×C5#290111yescharacteristic
412D5—28no—
413C23—54no—
414C23—54no—
415C23—54no—
416C23—202no—
417C23—202no—
418C23C2×D4×D511yescharacteristic
419Q8—42no—
420Q8—81no—
421Q8—202no—
422Q8—401no—
423Q8—404no—
424D4—58no—
425D4—58no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—54no—
434C2×C4—102no—
435C2×C4—102no—
436C2×C4—102no—
437C2×C4—204no—
438C2×C4—204no—
439C2×C4—202no—
440C2×C4—201no—
441C2×C4—201no—
442C2×C4—201no—
443C2×C4—201no—
444C2×C4—201no—
445C2×C4—404no—
446C2×C4—404no—
447C2×C4—408no—
448C2×C4D5×C22⋊C411yescharacteristic
449C2×C4D5×C22⋊C411yescharacteristic
450C2×C4D5×C22⋊C411yescharacteristic
451C2×C4D5×C22⋊C411yescharacteristic
452C2×C4D5×SD1612yes—
453C2×C4D5×Q1612yes—
454C2×C4C22×C4×D511yescharacteristic
455C2×C4C2×D4×D511yescharacteristic
456C8—42no—
457C8—202no—
458C5#1761411yescharacteristic, complemented, Sylow
459C22—21no—
460C22—21no—
461C22—58no—
462C22—104no—
463C22—104no—
464C22—204no—
465C22—204no—
466C22C2×D5×C22⋊C411yescharacteristic
467C22C2×D5×SD1611yescharacteristic
468C22C2×D5×Q1611yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—104no—
477C4—202no—
478C4—202no—
479C4—202no—
480C4—401no—
481C4D5×#10312yes—
482C4D5×#10312yes—
483C2—21no—
484C2—21no—
485C2—108no—
486C2#1061042511yescharacteristic
487C2C2×D5×#10311yescharacteristic
488C2#1061072311yescharacteristic
subgroupG/H
centerC22C2×D5×C22⋊C4
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1518C2
socleC22×C5C22×C22⋊C4
prime radicalC22×D20D4
cosocleC22×C4×C5C24
hypercenter#1518D5
hypocenterC5#17614
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
4. C5×#1518C2semidirect
5. #10602628C2semidirect
6. #10601585C2semidirect
7. #10600339C2semidirect
8. #10599132C2semidirect
9. D5×#328C2semidirect
11. #66064C4semidirect
12. #66061C4semidirect
17. #65913C4semidirect
18. Q8×D20C4semidirect
42. C2×Q8×D5C2×C4semidirect
46. C5×C4⋊Q8C2×C4semidirect
47. C5×C4⋊Q8C2×C4semidirect
52. C4×C5×Q8C2×C4semidirect
57. #4092C2×C4semidirect
69. #4043C2×C4semidirect
77. D5×C4⋊C4C2×C4semidirect
91. #4015C2×C4semidirect
92. #4015C2×C4semidirect
94. C4×Dic10C2×C4semidirect
112. #1518D5semidirect
264. C4⋊Q8C4×D5semidirect
265. C4⋊Q8C4×D5semidirect
285. C4×Q8C4×D5semidirect
458. C5#17614semidirect

Maximal subgroups

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

subgroupindex
#106181812
#106110172
#106059592
C5×#15182
#106026282
#106015852
#106003392
#105991322
D5×#3282
#176145