Finite Group Databasebeta

← #12077796

Subgroups — #12077796

Order 1280.

966 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#554C211yesmaximal, characteristic
2#10614953C211yesmaximal, characteristic, complemented
3#10614178C211yesmaximal, characteristic, complemented
4#10614027C211yesmaximal, characteristic
5#10602719C214yesmaximal, complemented
6#10602717C214yesmaximal, complemented
7#10602567C211yesmaximal, characteristic, complemented
8#10602415C211yesmaximal, characteristic, complemented
9#10602390C211yesmaximal, characteristic
10C5×#554—22no—
11C5×#554—22no—
12C5×#554—22no—
13C5×#554C2214yes—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C5×#125—21no—
16C2×C5×C22⋊Q8—21no—
17C2×C5×C22⋊Q8—22no—
18C2×C4×C5×Q8C2211yescharacteristic
19C2×C4×C5×D4—21no—
20C2×C4×C5×D4C2211yescharacteristic
21C22×C5×C4⋊C4—21no—
22#65804C2211yescharacteristic
23#65517—21no—
24#65517—22no—
25#65504—21no—
26#65381C2212yescomplemented
27#65379—22no—
28#65370C2212yescomplemented
29#65320C2212yescomplemented
30C4×#689—22no—
31#65310C2212yescomplemented
32#65298—22no—
33#65297C2212yes—
34#65296C2212yes—
35#65259—21no—
36#65253—21no—
37#65214C2211yescharacteristic
38#65214C2211yescharacteristic
39#65213C2211yescharacteristic
40#64766C2214yescomplemented
41Q8⋊#168C2214yescomplemented
42D4⋊#168C2214yes—
43#18749—51nomaximal, Sylow
44C22×C5×Q8D411yescharacteristic
45C22×C5×Q8D412yes—
46C22×C5×D4D411yescharacteristic
47C23×C4×C5—22no—
48C23×C4×C5—21no—
49C5×C4⋊Q8—21no—
50C5×C4⋊Q8—21no—
51C5×C4⋊Q8D412yes—
52C5×C4⋊Q8C2312yes—
53C5×#125—24no—
54C5×#125—24no—
55C5×C22⋊Q8—24no—
56C5×C22⋊Q8—28no—
57C5×C22⋊Q8—24no—
58C5×C22⋊Q8—28no—
59C4×C5×Q8—21no—
60C4×C5×Q8—21no—
61C4×C5×Q8D412yes—
62C4×C5×Q8C2312yes—
63C4×C5×D4—24no—
64C4×C5×D4—24no—
65C4×C5×D4—21no—
66C4×C5×D4—21no—
67C4×C5×D4D412yes—
68C4×C5×D4C2312yes—
69C2×C5×C4⋊C4—22no—
70C2×C5×C4⋊C4—24no—
71C2×C5×C4⋊C4—22no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—24no—
76C2×C5×C4⋊C4D411yescharacteristic
77C2×C5×C4⋊C4D411yescharacteristic
78C2×C5×C22⋊C4—22no—
79C2×C5×C22⋊C4—21no—
80C2×C42×C5—21no—
81C2×C42×C5C2311yescharacteristic
82C2×#689—22no—
83C2×#689—24no—
84#4072—21no—
85#4070—21no—
86#4070—21no—
87C2×#681—21no—
88#4016—21no—
89#4015C2311yescharacteristic
90#4015C2311yescharacteristic
91C4×Dic10—21no—
92C4×Dic10—21no—
93#3967—22no—
94#3967—24no—
95#3964—22no—
96#3964—22no—
97C22⋊#168—21no—
98C22⋊#168—21no—
99#3942—22no—
100#3940—22no—
101#3939—22no—
102#3939—22no—
103C4⋊#168—21no—
104C4⋊#168—21no—
105C4⋊#168C2312yes—
106C4⋊#168C2312yes—
107C4×#168—21no—
108C4×#168C2312yes—
109C2×#554D511yescharacteristic
110#3209—51no—
111#3104—51no—
112#3077—51no—
113#1644—54no—
114#1632—54no—
115#1548—51no—
116#1487—51no—
117#1478—51no—
118C24×C5—21no—
119C2×C5×Q8—21no—
120C2×C5×Q8—22no—
121C2×C5×Q8—42no—
122C2×C5×Q8—82no—
123C2×C5×Q8C2×D412yes—
124C2×C5×Q8C2×D414yes—
125C2×C5×D4—21no—
126C2×C5×D4—22no—
127C2×C5×D4Q1614yescomplemented
128C2×C5×D4C2×D412yes—
129C22×C4×C5—21no—
130C22×C4×C5—21no—
131C22×C4×C5—21no—
132C22×C4×C5—22no—
133C22×C4×C5—28no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—44no—
137C22×C4×C5—84no—
138C22×C4×C5C2×D411yescharacteristic
139C22×C4×C5C2×D411yescharacteristic
140C22×C4×C5C2×D411yescharacteristic
141C2×Dic10—21no—
142C2×Dic10—21no—
143C5×C4⋊C4—28no—
144C5×C4⋊C4—28no—
145C5×C4⋊C4—21no—
146C5×C4⋊C4—22no—
147C5×C4⋊C4—22no—
148C5×C4⋊C4—22no—
149C5×C4⋊C4—21no—
150C5×C4⋊C4—22no—
151C5×C4⋊C4—22no—
152C5×C4⋊C4—22no—
153C5×C4⋊C4C2×D412yes—
154C5×C4⋊C4C2×D412yes—
155C5×C22⋊C4—28no—
156C5×C22⋊C4—22no—
157C5×C22⋊C4—22no—
158C42×C5—22no—
159C42×C5—22no—
160C42×C5C2×D411yescharacteristic
161C42×C5C2×D411yescharacteristic
162C42×C5C2×D411yescharacteristic
163C42×C5C2411yescharacteristic
164C22⋊Dic5—41no—
165#689—44no—
166#689—84no—
167C4⋊Dic5—21no—
168C4⋊Dic5—21no—
169C4⋊Dic5—21no—
170C4⋊Dic5—21no—
171#683—41no—
172#683—41no—
173C4×Dic5—41no—
174#681—42no—
175C2×#168—22no—
176C2×#168—22no—
177C2×#168—22no—
178#554—22no—
179#554—22no—
180#554—22no—
181#554C2×D514yes—
182#533—51no—
183C2×C4⋊Q8C2×D511yescharacteristic
184C2×#125—21no—
185C2×C22⋊Q8—21no—
186C2×C22⋊Q8—22no—
187C2×C4×Q8C2×D511yescharacteristic
188C2×C4×D4—21no—
189C2×C4×D4C2×D511yescharacteristic
190C22×C4⋊C4—21no—
191#494—52no—
192#487—52no—
193#484—101no—
194#478—102no—
195#477—52no—
196#476—52no—
197#474—52no—
198#462—52no—
199#462—102no—
200#451—101no—
201#451—102no—
202C4×Q16—102no—
203#431—51no—
204#431—51no—
205#428—101no—
206#423—51no—
207#418—101no—
208#333—54no—
209Q8⋊C8—54no—
210D4⋊C8—54no—
211C23×C5—22no—
212C23×C5—24no—
213C23×C5—42no—
214C23×C5—42no—
215C23×C5#12011yescharacteristic
216C5×Q8—22no—
217C5×Q8—22no—
218C5×Q8—44no—
219C5×Q8—42no—
220C5×Q8—82no—
221C5×D4—22no—
222C5×D4—24no—
223C5×D4C2×Q1614yes—
224C2×C4×C5—22no—
225C2×C4×C5—22no—
226C2×C4×C5—22no—
227C2×C4×C5—22no—
228C2×C4×C5—24no—
229C2×C4×C5—41no—
230C2×C4×C5—44no—
231C2×C4×C5—42no—
232C2×C4×C5—42no—
233C2×C4×C5—44no—
234C2×C4×C5—42no—
235C2×C4×C5—44no—
236C2×C4×C5—88no—
237C2×C4×C5#12011yescharacteristic
238C2×C4×C5#12011yescharacteristic
239C2×C4×C5#12011yescharacteristic
240C2×C4×C5C2×Q1612yes—
241C2×C4×C5#13712yes—
242C2×C4×C5C22×D411yescharacteristic
243C2×C4×C5C22×D411yescharacteristic
244C2×C4×C5C22×D411yescharacteristic
245C2×Dic5—41no—
246C2×Dic5—41no—
247Dic10—42no—
248Dic10—42no—
249#168—42no—
250#168—42no—
251#168—42no—
252C22×Q8C22⋊D511yescharacteristic
253C22×Q8C22⋊D512yes—
254C22×D4C22⋊D511yescharacteristic
255C23×C4—22no—
256C23×C4—21no—
257C2×Q16—102no—
258C2×Q16—104no—
259C2×M16—101no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—51no—
263C4⋊Q8—51no—
264C4⋊Q8C22⋊D512yes—
265C4⋊Q8C22×D512yes—
266#125—24no—
267#125—24no—
268#125—101no—
269C22⋊Q8—24no—
270C22⋊Q8—28no—
271C22⋊Q8—24no—
272C22⋊Q8—28no—
273C22⋊Q8—101no—
274C22⋊Q8—101no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8—101no—
278C4×Q8—101no—
279C4×Q8C22⋊D512yes—
280C4×Q8C22×D512yes—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4C22⋊D512yes—
286C4×D4C22×D512yes—
287#117—101no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4C22⋊D511yescharacteristic
296C2×C4⋊C4C22⋊D511yescharacteristic
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C42—21no—
300C2×C42C22×D511yescharacteristic
301#107—102no—
302#107—102no—
303#106—102no—
304C4⋊C8—52no—
305C4⋊C8—52no—
306C4⋊C8—101no—
307C4⋊C8—101no—
308#103—102no—
309#103—104no—
310#103—102no—
311#102—102no—
312#102—102no—
313C22⋊C8—101no—
314C22⋊C8—101no—
315C4×C8—52no—
316C4×C8—101no—
317C22×C5—21no—
318C22×C5—21no—
319C22×C5—24no—
320C22×C5—48no—
321C22×C5—42no—
322C22×C5—42no—
323C22×C5#45112yes—
324C22×C5C2×#12011yescharacteristic
325C4×C5—21no—
326C4×C5—21no—
327C4×C5—21no—
328C4×C5—42no—
329C4×C5—42no—
330C4×C5—42no—
331C4×C5—44no—
332C4×C5#45112yes—
333C4×C5C22×Q1611yescharacteristic
334C4×C5C2×#13711yescharacteristic
335C4×C5#57512yes—
336Dic5—81no—
337Dic5—81no—
338C24—21no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—42no—
342C2×Q8—82no—
343C2×Q8—101no—
344C2×Q8—101no—
345C2×Q8C2×C22⋊D512yes—
346C2×Q8C2×C22⋊D514yes—
347C2×D4—21no—
348C2×D4—22no—
349C2×D4#68914yescomplemented
350C2×D4C2×C22⋊D512yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—44no—
359C22×C4—84no—
360C22×C4C2×C22⋊D511yescharacteristic
361C22×C4C2×C22⋊D511yescharacteristic
362C22×C4C2×C22⋊D511yescharacteristic
363Q16—204no—
364Q16—404no—
365M16—202no—
366C2×C8—102no—
367C2×C8—102no—
368C2×C8—102no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—101no—
380C4⋊C4—101no—
381C4⋊C4—101no—
382C4⋊C4—101no—
383C4⋊C4—201no—
384C4⋊C4—201no—
385C4⋊C4C2×C22⋊D512yes—
386C4⋊C4C2×C22⋊D512yes—
387C22⋊C4—28no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—201no—
391C42—22no—
392C42—22no—
393C42—201no—
394C42C2×C22⋊D511yescharacteristic
395C42C2×C22⋊D511yescharacteristic
396C42C2×C22⋊D511yescharacteristic
397C42C23×D511yescharacteristic
398C2×C5—22no—
399C2×C5—44no—
400C2×C5C2×#45111yescharacteristic
401C2×C5#300511yescharacteristic
402C2×C5#301911yescharacteristic
403C23—22no—
404C23—24no—
405C23—42no—
406C23—42no—
407C23#409911yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—44no—
411Q8—42no—
412Q8—82no—
413Q8—202no—
414Q8—202no—
415D4—22no—
416D4—24no—
417D4C2×#68914yes—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—88no—
431C2×C4—201no—
432C2×C4—201no—
433C2×C4#408812yes—
434C2×C4C2×#68912yes—
435C2×C4#409911yescharacteristic
436C2×C4#409911yescharacteristic
437C2×C4#409911yescharacteristic
438C2×C4C22×C22⋊D511yescharacteristic
439C2×C4C22×C22⋊D511yescharacteristic
440C2×C4C22×C22⋊D511yescharacteristic
441C8—202no—
442C8—202no—
443C8—202no—
444C5#1874911yescharacteristic, complemented, Sylow
445C22—21no—
446C22—21no—
447C22—24no—
448C22—48no—
449C22—42no—
450C22—42no—
451C22#6551712yes—
452C22C2×#409911yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—44no—
460C4—401no—
461C4—401no—
462C4#6551712yes—
463C4C2×#408811yescharacteristic
464C4C22×#68911yescharacteristic
465C4#6615612yes—
466C2—22no—
467C2—44no—
468C2C2×#6551711yescharacteristic
469C2#1061581211yescharacteristic
470C2#1061583511yescharacteristic
subgroupG/H
centerC22C2×#4099
commutator, cosocleC42×C5C24
FrattiniC42C23×D5
FittingC2×C5×#554C2
socleC22×C5C2×#120
prime radicalC22×C5×D4D4
hypercenterC2×#554D5
hypocenterC5#18749
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. #10614953C2semidirect
3. #10614178C2semidirect
5. #10602719C2semidirect
6. #10602717C2semidirect
7. #10602567C2semidirect
8. #10602415C2semidirect
26. #65381C22semidirect
28. #65370C22semidirect
29. #65320C22semidirect
31. #65310C22semidirect
40. #64766C22semidirect
41. Q8⋊#168C22semidirect
127. C2×C5×D4Q16semidirect
349. C2×D4#689semidirect
444. C5#18749semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5542
#106149532
#106141782
#106140272
#106027192
#106027172
#106025672
#106024152
#106023902
#187495