Finite Group Databasebeta

← #12117785

Subgroups — #12117785

Order 1280.

846 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65433C211yesmaximal, characteristic
2#10615516C211yesmaximal, characteristic, complemented
3#10610877C211yesmaximal, characteristic, complemented
4#10610600C212yesmaximal
5C5×#1816C211yesmaximal, characteristic, complemented
6#10601514C211yesmaximal, characteristic, complemented
7#10601495C211yesmaximal, characteristic, complemented
8#10601400C211yesmaximal, characteristic, complemented
9#10601091C212yesmaximal, complemented
10#10599796C212yesmaximal, complemented
11#10599777C212yesmaximal, complemented
12C2×C5×#131C2211yescharacteristic
13#66161C2212yes—
14C2×#4096C2211yescharacteristic
15C2×#4047C2211yescharacteristic
16#65838C2212yes—
17C2×C5×#108C2211yescharacteristic
18C2×C5×#107C2211yescharacteristic
19C5×#417C2212yes—
20C2×#3969C2211yescharacteristic
21#65433C418yescomplemented
22C2×#3957C2211yescharacteristic
23#65422C2212yes—
24C2×#3954C2211yescharacteristic
25#65411C2212yes—
26#65410—22no—
27#65410C2212yes—
28#65395—21no—
29#65395C2211yescharacteristic
30C2×#3950—21no—
31C2×#3950C2211yescharacteristic
32C2×C8×Dic5C2211yescharacteristic
33#65266C2212yes—
34#65253C2212yes—
35C2×#3940C2212yes—
36#65170—22no—
37#65169—21no—
38D5×#107—21no—
39#65078—22no—
40#65077—22no—
41#65074—22no—
42#65047—22no—
43C5×#343C2212yes—
44#64775C2212yes—
45#64774C2212yes—
46#22691—51nomaximal, Sylow
47C2×C5×#41C2311yescharacteristic
48C2×#709C2311yescharacteristic
49C5×#131—22no—
50C5×#131C2×C414yescomplemented
51C2×C5×M16—21no—
52C2×C5×M16C2312yes—
53C2×C5×M16C2311yescharacteristic
54C22×C5×C8—21no—
55C22×C5×C8C2311yescharacteristic
56C2×C5×C4⋊C4C2312yes—
57#4096C2×C414yes—
58#4096Q814yes—
59Q8×Dic5—22no—
60C2×#689D411yescharacteristic
61C2×#688D411yescharacteristic
62D4×Dic5—22no—
63D4×Dic5—42no—
64C2×#687D411yescharacteristic
65C2×#686D411yescharacteristic
66C4×C22⋊D5—42no—
67#4072—22no—
68#4072C2312yes—
69C2×C4⋊Dic5—22no—
70C2×C4⋊Dic5C2312yes—
71C2×C4×Dic5C2311yescharacteristic
72C2×#681C2311yescharacteristic
73C22×#168C2311yescharacteristic
74#4047—22no—
75#4047C2×C414yescomplemented
76C2×#676—21no—
77C2×C8×D5—21no—
78#4039—22no—
79#4038—22no—
80D5×C4⋊C4—22no—
81#4033—22no—
82C5×#108—22no—
83C5×#107—21no—
84C5×#107—21no—
85C5×#103—22no—
86C5×#102—22no—
87#3969C2×C414yescomplemented
88#3963—42no—
89#3957C2×C414yescomplemented
90#3954C2×C414yescomplemented
91#3953—22no—
92#3950—21no—
93#3950—21no—
94#3950—22no—
95#3950D411yescharacteristic
96#3950D411yescharacteristic
97#3949—22no—
98#3949D412yes—
99#3948—22no—
100#3947—22no—
101C8×Dic5—22no—
102C8×Dic5C2×C414yescomplemented
103#3940—22no—
104#3940—22no—
105#3940—22no—
106#3939—22no—
107C2×#443—51no—
108#2952—52no—
109#2913—51no—
110#2913—51no—
111#1951—52no—
112#1860—52no—
113#1860—52no—
114#1846—51no—
115#1816—51no—
116#1816D511yescharacteristic, complemented
117#1772—51no—
118C5×#41—21no—
119C5×#41—21no—
120C5×#41C22×C412yes—
121C5×#41C2×Q812yes—
122C2×C5×Q8C2×D411yescharacteristic
123C2×C5×D4—21no—
124C2×C5×D4C2×D411yescharacteristic
125C22×C4×C5—21no—
126C22×C4×C5—22no—
127C22×C4×C5C2411yescharacteristic
128C2×C22⋊D5—21no—
129C22×Dic5—21no—
130C22×Dic5—22no—
131C22×Dic5—42no—
132#709—21no—
133#709—21no—
134#709C22×C412yes—
135#709C2×Q812yes—
136C2×D20C2×D411yescharacteristic
137C2×C4×D5—21no—
138C2×C4×D5—42no—
139C2×Dic10C2×D411yescharacteristic
140C5×M16—21no—
141C5×M16—24no—
142C5×M16—22no—
143C5×M16—22no—
144C5×M16C22×C412yes—
145C2×C5×C8—21no—
146C2×C5×C8—24no—
147C2×C5×C8—22no—
148C2×C5×C8C22×C412yes—
149C2×C5×C8C2×D411yescharacteristic
150C2×C5×C8C2×D411yescharacteristic
151C2×C5×C8#4112yes—
152C5×C4⋊C4—22no—
153C5×C4⋊C4—22no—
154C22⋊Dic5—22no—
155C22⋊Dic5—42no—
156#689C4⋊C414yes—
157#688C4⋊C414yes—
158#687C4⋊C414yes—
159#686C4⋊C414yes—
160#685—42no—
161C4⋊Dic5—22no—
162C4⋊Dic5—22no—
163C4⋊Dic5—42no—
164C4⋊Dic5C2×D412yes—
165C4⋊Dic5C2×D412yes—
166#683—42no—
167C4×Dic5—22no—
168C4×Dic5—21no—
169C4×Dic5—42no—
170C4×Dic5—42no—
171C4×Dic5C22×C412yes—
172#681—21no—
173#681C22×C412yes—
174C2×#168—21no—
175C2×#168C22×C412yes—
176C2×#168C2×D411yescharacteristic
177C2×#168C2×D411yescharacteristic
178C2×#168C2×Q812yes—
179#676—24no—
180C8×D5—24no—
181C2×#135—51no—
182C2×#131—51no—
183C2×#131C2×D511yescharacteristic
184#520—52no—
185#520—52no—
186#443—58no—
187#442—102no—
188#441—102no—
189C4×SD16—102no—
190C4×SD16—102no—
191C2×#108—51no—
192C2×#108C2×D511yescharacteristic
193#428—52no—
194#428—52no—
195#428—102no—
196#428—102no—
197#427—51no—
198#427—101no—
199#427—101no—
200C2×#107—51no—
201C2×#107—101no—
202C2×#107—101no—
203C2×#107C2×D511yescharacteristic
204C2×#106—52no—
205C2×#104—51no—
206C2×#104—51no—
207#417—52no—
208#417C2×D512yes—
209#405—52no—
210C2×C4×C8—51no—
211#382—52no—
212#343—52no—
213#343C2×D512yes—
214#339—52no—
215C23×C5—21no—
216C23×C5H(5,𝔽2)11yescharacteristic
217C22×D5—21no—
218C5×Q8—21no—
219C5×Q8C2×C4⋊C412yes—
220C5×D4—22no—
221C5×D4—22no—
222C5×D4—21no—
223C5×D4C2×C4⋊C412yes—
224C2×C4×C5—21no—
225C2×C4×C5—22no—
226C2×C4×C5—22no—
227C2×C4×C5—42no—
228C2×C4×C5—82no—
229C2×C4×C5C23×C411yescharacteristic
230C2×C4×C5C22×D411yescharacteristic
231C2×C4×C5C22×D411yescharacteristic
232C2×C4×C5C22×Q811yescharacteristic
233C2×C4×C5C2×#4111yescharacteristic
234C2×C4×C52−1+411yescharacteristic
235C22⋊D5—22no—
236C22⋊D5—22no—
237C2×Dic5—21no—
238C2×Dic5—22no—
239C2×Dic5—22no—
240C2×Dic5—22no—
241C2×Dic5—41no—
242C2×Dic5—41no—
243C2×Dic5—42no—
244C2×Dic5—82no—
245C2×Dic5—162no—
246D20—21no—
247D20C2×C4⋊C412yes—
248C4×D5—22no—
249C4×D5—22no—
250C4×D5—162no—
251Dic10—21no—
252Dic10C2×C4⋊C412yes—
253C5×C8—22no—
254C5×C8—42no—
255C5×C8C4×D412yes—
256C5×C8C4×D412yes—
257#168—22no—
258#168C2×C4⋊C412yes—
259#168C2×C4⋊C412yes—
260C2×#41—51no—
261C2×#41C22×D511yescharacteristic
262#135—54no—
263#135—54no—
264C2×Q16—51no—
265C2×SD16—51no—
266C2×SD16—51no—
267C2×D8—51no—
268#131—22no—
269#131—54no—
270#131—102no—
271#131C4×D514yescomplemented
272C2×M16—21no—
273C2×M16—51no—
274C2×M16—101no—
275C2×M16C22×D512yes—
276C2×M16C22×D511yescharacteristic
277C22×C8—21no—
278C22×C8—51no—
279C22×C8—101no—
280C22×C8C22×D511yescharacteristic
281C4×Q8—102no—
282C4×Q8—102no—
283C4×D4—102no—
284C4×D4—102no—
285C4×D4—202no—
286C4×D4—202no—
287#117—52no—
288#117—102no—
289#117—102no—
290C2×C4⋊C4—52no—
291C2×C4⋊C4—102no—
292C2×C4⋊C4—102no—
293C2×C4⋊C4C22×D512yes—
294C2×C42—51no—
295#108—22no—
296#108—54no—
297#107—21no—
298#107—21no—
299#107—51no—
300#107—51no—
301#107—102no—
302#107—102no—
303#107—101no—
304#107—101no—
305#106—52no—
306#106—102no—
307#106—102no—
308#106—102no—
309#106—102no—
310#104—54no—
311#104—54no—
312#103—22no—
313#103—102no—
314#102—22no—
315#102—102no—
316#97—102no—
317C4×C8—54no—
318C4×C8—102no—
319#95—202no—
320C22×C5—21no—
321C22×C5—21no—
322C22×C5—22no—
323C22×C5—41no—
324C22×C5—41no—
325C22×C5#52011yescharacteristic
326C22×C5D4211yescharacteristic
327C22×C5#55411yescharacteristic
328C2×D5—22no—
329C2×D5—41no—
330C2×D5—41no—
331C4×C5—22no—
332C4×C5—82no—
333C4×C5C22×C4⋊C411yescharacteristic
334C4×C5C2×C4×D411yescharacteristic
335C4×C5#54711yescharacteristic
336C4×C5D4×Q811yescharacteristic
337Dic5—22no—
338Dic5—42no—
339Dic5—42no—
340Dic5—42no—
341Dic5—82no—
342#41—21no—
343#41—21no—
344#41—52no—
345#41—52no—
346#41—101no—
347#41—101no—
348#41C2×C4×D512yes—
349#41Q8×D512yes—
350C2×Q8—51no—
351C2×Q8D4×D511yescharacteristic
352C2×D4—21no—
353C2×D4—51no—
354C2×D4—101no—
355C2×D4D4×D511yescharacteristic
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—101no—
359C22×C4—101no—
360C22×C4—102no—
361C22×C4—202no—
362C22×C4—202no—
363C22×C4C23×D511yescharacteristic
364Q16—54no—
365SD16—54no—
366SD16—54no—
367D8—54no—
368M16—21no—
369M16—24no—
370M16—22no—
371M16—22no—
372M16—52no—
373M16—101no—
374M16—104no—
375M16C2×C4×D512yes—
376C2×C8—21no—
377C2×C8—24no—
378C2×C8—22no—
379C2×C8—52no—
380C2×C8—51no—
381C2×C8—51no—
382C2×C8—52no—
383C2×C8—101no—
384C2×C8—104no—
385C2×C8C2×C4×D512yes—
386C2×C8D4×D511yescharacteristic
387C2×C8D4×D511yescharacteristic
388C2×C8#71112yes—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—52no—
392C4⋊C4—52no—
393C4⋊C4—102no—
394C4⋊C4—102no—
395C4⋊C4—202no—
396C4⋊C4—202no—
397C22⋊C4—102no—
398C22⋊C4—202no—
399C22⋊C4—202no—
400C42—52no—
401C42—102no—
402C42—101no—
403C42—202no—
404C42—202no—
405C2×C5—21no—
406C2×C5—21no—
407C2×C5—42no—
408C2×C5D4×C4⋊C411yescharacteristic
409C2×C5#329011yescharacteristic
410C2×C5#329011yescharacteristic
411D5—42no—
412C23—21no—
413C23—101no—
414C23#414911yescharacteristic
415Q8—21no—
416Q8—52no—
417Q8—101no—
418Q8D5×C4⋊C412yes—
419D4—22no—
420D4—22no—
421D4—21no—
422D4—52no—
423D4—102no—
424D4—102no—
425D4—101no—
426D4D5×C4⋊C412yes—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—101no—
433C2×C4—102no—
434C2×C4—102no—
435C2×C4—102no—
436C2×C4—102no—
437C2×C4—102no—
438C2×C4—201no—
439C2×C4—201no—
440C2×C4—202no—
441C2×C4—402no—
442C2×C4—802no—
443C2×C4—802no—
444C2×C4C22×C4×D511yescharacteristic
445C2×C4C2×D4×D511yescharacteristic
446C2×C4C2×D4×D511yescharacteristic
447C2×C4C2×#71111yescharacteristic
448C2×C4C2×Q8×D511yescharacteristic
449C2×C4#415011yescharacteristic
450C8—22no—
451C8—42no—
452C8—52no—
453C8—52no—
454C8—102no—
455C8#402712yes—
456C8#402712yes—
457C5#2269111yescharacteristic, complemented, Sylow
458C22—21no—
459C22—21no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—102no—
464C22—201no—
465C22—201no—
466C22#6585711yescharacteristic
467C22#6594311yescharacteristic
468C22#6596911yescharacteristic
469C4—22no—
470C4—82no—
471C4—102no—
472C4—202no—
473C4—202no—
474C4—202no—
475C4—402no—
476C4C2×#402711yescharacteristic
477C4C2×D5×C4⋊C411yescharacteristic
478C4#6594511yescharacteristic
479C4#6596611yescharacteristic
480C2—21no—
481C2—21no—
482C2—42no—
483C2—202no—
484C2#1060574111yescharacteristic
485C2#1061267111yescharacteristic
486C2#1061267711yescharacteristic
subgroupG/H
centerC22#65943
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1816C2
socleC22×C5D42
prime radicalC2×#4096C22
cosocleC22×C4×C5C24
hypercenter#1816D5
hypocenterC5#22691
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1816

Complements

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

NH ≅ G/Ntype
2. #10615516C2semidirect
3. #10610877C2semidirect
5. C5×#1816C2semidirect
6. #10601514C2semidirect
7. #10601495C2semidirect
8. #10601400C2semidirect
9. #10601091C2semidirect
10. #10599796C2semidirect
11. #10599777C2semidirect
21. #65433C4semidirect
50. C5×#131C2×C4semidirect
75. #4047C2×C4semidirect
87. #3969C2×C4semidirect
89. #3957C2×C4semidirect
90. #3954C2×C4semidirect
102. C8×Dic5C2×C4semidirect
116. #1816D5semidirect
271. #131C4×D5semidirect
457. C5#22691semidirect

Maximal subgroups

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

subgroupindex
C2×#654332
#106155162
#106108772
#106106002
C5×#18162
#106015142
#106014952
#106014002
#106010912
#105997962
#105997772
#226915