Finite Group Databasebeta

← #12254171

Subgroups — #12254171

Order 1280.

568 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3061C211yesmaximal, characteristic
2#10615516C211yesmaximal, characteristic, complemented
3#10602871C211yesmaximal, characteristic, complemented
4#10602870C211yesmaximal, characteristic, complemented
5#10602797C211yesmaximal, characteristic, complemented
6#10601873C212yesmaximal, complemented
7#10601871C212yesmaximal, complemented
8#10601863C212yesmaximal, complemented
9#10601862C212yesmaximal, complemented
10#10601389C211yesmaximal, characteristic, complemented
11C2×#64710C211yesmaximal, characteristic, complemented
12C2×C5×#135C2211yescharacteristic
13C2×C5×#131C2211yescharacteristic
14C5×#537—21no—
15#66161—21no—
16C5×#468—22no—
17C5×#467C2212yes—
18C5×#466C2212yes—
19C2×C5×#107C2211yescharacteristic
20C5×#417—21no—
21#65410—21no—
22#65410—22no—
23#65395C2211yescharacteristic
24#65395C2212yes—
25C2×#3950C2211yescharacteristic
26C2×#3950C2212yes—
27C2×#3944C2211yescharacteristic
28C22×#672C2211yescharacteristic
29#64784C2211yescharacteristic
30#64784C2211yescharacteristic
31#64784C2212yes—
32#64782C2211yescharacteristic
33#64782C2211yescharacteristic
34#64782C2212yes—
35C22⋊#672C2212yes—
36C22⋊#672C2212yes—
37#64710C2212yes—
38#64710C2212yes—
39#64710C2212yes—
40#64709C2212yes—
41#33653—51nomaximal, Sylow
42C2×C5×#41—21no—
43C2×C5×#41D411yescharacteristic
44C5×#135—22no—
45C5×#135—22no—
46C2×C5×Q16C2311yescharacteristic
47C2×C5×SD16—21no—
48C2×C5×D8C2311yescharacteristic
49C5×#131—22no—
50C5×#131—22no—
51C2×C5×M16D411yescharacteristic
52C2×C5×M16D412yes—
53C22×C5×C8C2312yes—
54C22×C5×C8C2311yescharacteristic
55C5×C22⋊Q8—22no—
56C5×C22⋊Q8—41no—
57C5×#121—22no—
58C5×#121—41no—
59C5×#121—42no—
60C2×C5×C4⋊C4—21no—
61Q8×Dic5—41no—
62D4×Dic5—42no—
63D4×Dic5—41no—
64#4072—22no—
65#4072—21no—
66C2×C4⋊Dic5—22no—
67C2×C4⋊Dic5—21no—
68C5×#107—21no—
69C5×#107C2312yes—
70C5×#103—22no—
71C5×#102—22no—
72#3950—22no—
73#3950—21no—
74#3950C2311yescharacteristic
75#3950C2312yes—
76#3950C2311yescharacteristic
77#3949—22no—
78#3949—21no—
79#3944—22no—
80C2×#672—22no—
81C2×#672C2311yescharacteristic
82C2×#672C2311yescharacteristic
83C2×#672C2312yes—
84#3061D511yescharacteristic
85#2913—51no—
86#2237—52no—
87#2235—52no—
88#2234—52no—
89#2233—52no—
90#2159—51no—
91C2×#367—51no—
92#2139—51no—
93#2138—51no—
94#2113—51no—
95C5×#41—22no—
96C5×#41—22no—
97C5×#41—42no—
98C5×#41—42no—
99C2×C5×Q8—21no—
100C2×C5×Q8D811yescharacteristic
101C2×C5×D4—21no—
102C2×C5×D4—41no—
103C2×C5×D4—42no—
104C2×C5×D4D812yes—
105C2×C5×D4D811yescharacteristic
106C22×C4×C5—41no—
107C22×C4×C5—41no—
108C22×C4×C5C2×D412yes—
109C22×C4×C5C2×D411yescharacteristic
110C22×Dic5—42no—
111C22×Dic5—41no—
112C5×Q16—21no—
113C5×Q16—21no—
114C5×SD16—42no—
115C5×D8—21no—
116C5×D8—21no—
117C5×M16—22no—
118C5×M16—24no—
119C2×C5×C8—22no—
120C2×C5×C8—22no—
121C2×C5×C8C2×D412yes—
122C2×C5×C8C2×D411yescharacteristic
123C2×C5×C8#4112yes—
124C2×C5×C8#4112yes—
125C2×C5×C8C2411yescharacteristic
126C5×C4⋊C4—22no—
127C5×C4⋊C4—41no—
128C5×C4⋊C4—42no—
129C5×C22⋊C4—42no—
130C5×C22⋊C4—42no—
131C22⋊Dic5—42no—
132C22⋊Dic5—41no—
133C4⋊Dic5—21no—
134C4⋊Dic5—22no—
135C4⋊Dic5—21no—
136C4⋊Dic5—42no—
137C4⋊Dic5—41no—
138C4×Dic5—42no—
139C4×Dic5—41no—
140#672—21no—
141#672—21no—
142#672—22no—
143C2×#135C2×D511yescharacteristic
144C2×#131C2×D511yescharacteristic
145#537—21no—
146#520—101no—
147C2×M32—51no—
148C22×C16—51no—
149#468—22no—
150#467C2×D512yes—
151#466C2×D512yes—
152#428—101no—
153#428—102no—
154#427—51no—
155#427—52no—
156C2×#107—51no—
157C2×#107—52no—
158C2×#107C2×D511yescharacteristic
159#417—21no—
160#367—52no—
161#367—52no—
162#367—52no—
163#366—52no—
164#358—51no—
165#358—51no—
166#358—52no—
167#357—51no—
168#357—51no—
169#357—52no—
170C22⋊C16—52no—
171C22⋊C16—52no—
172C23×C5—41no—
173C23×C5C2×D812yes—
174C23×C5C2×D811yescharacteristic
175C5×Q8—22no—
176C5×Q8—41no—
177C5×Q8—41no—
178C5×D4—24no—
179C5×D4—22no—
180C5×D4—41no—
181C5×D4—41no—
182C5×D4—82no—
183C5×D4—162no—
184C2×C4×C5—22no—
185C2×C4×C5—22no—
186C2×C4×C5—42no—
187C2×C4×C5—41no—
188C2×C4×C5—82no—
189C2×C4×C5—161no—
190C2×C4×C5#12312yes—
191C2×C4×C5C2×D812yes—
192C2×C4×C5C2×D811yescharacteristic
193C2×C4×C5#13712yes—
194C2×C4×C5C22×D411yescharacteristic
195C2×Dic5—41no—
196C2×Dic5—42no—
197C2×Dic5—41no—
198C2×Dic5—161no—
199C2×Dic5—162no—
200C5×C8—22no—
201C5×C8#12312yes—
202C5×C8C2×#4111yescharacteristic
203C5×C8C2×#4111yescharacteristic
204C2×#41—21no—
205C2×#41C22⋊D511yescharacteristic
206#135—22no—
207#135—22no—
208C2×Q16C22×D511yescharacteristic
209C2×SD16—21no—
210C2×D8C22×D511yescharacteristic
211#131—22no—
212#131—22no—
213C2×M16C22⋊D511yescharacteristic
214C2×M16C22⋊D512yes—
215C22×C8C22×D512yes—
216C22×C8C22×D511yescharacteristic
217C22⋊Q8—22no—
218C22⋊Q8—41no—
219#121—22no—
220#121—41no—
221#121—42no—
222C4×Q8—201no—
223C4×D4—202no—
224C4×D4—201no—
225#117—102no—
226#117—101no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—102no—
229C2×C4⋊C4—101no—
230M32—102no—
231C2×C16—52no—
232C2×C16—51no—
233C2×C16—51no—
234C2×C16—102no—
235#107—21no—
236#107—51no—
237#107—51no—
238#107—52no—
239#107—102no—
240#107—101no—
241#107C22×D512yes—
242#106—102no—
243#106—101no—
244#103—22no—
245#102—22no—
246C22×C5—21no—
247C22×C5—21no—
248C22×C5—22no—
249C22×C5—22no—
250C22×C5—22no—
251C22×C5—81no—
252C22×C5—81no—
253C22×C5—81no—
254C22×C5#48012yes—
255C22×C5C22×D811yescharacteristic
256C4×C5—22no—
257C4×C5—81no—
258C4×C5—82no—
259C4×C5#48012yes—
260C4×C5C2×#12311yescharacteristic
261C4×C5C2×#13711yescharacteristic
262Dic5—81no—
263Dic5—82no—
264Dic5—81no—
265#41—22no—
266#41—22no—
267#41—42no—
268#41—42no—
269C2×Q8—21no—
270C2×Q8#68611yescharacteristic
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4#68612yes—
275C2×D4#68611yescharacteristic
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—202no—
279C22×C4—201no—
280C22×C4C2×C22⋊D512yes—
281C22×C4C2×C22⋊D511yescharacteristic
282Q16—21no—
283Q16—21no—
284SD16—42no—
285D8—21no—
286D8—21no—
287M16—22no—
288M16—24no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8#71112yes—
292C2×C8#71112yes—
293C2×C8C2×C22⋊D512yes—
294C2×C8C2×C22⋊D511yescharacteristic
295C2×C8C23×D511yescharacteristic
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—101no—
300C4⋊C4—102no—
301C4⋊C4—101no—
302C4⋊C4—202no—
303C4⋊C4—201no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—202no—
307C22⋊C4—201no—
308C42—202no—
309C42—201no—
310C16—101no—
311C16—101no—
312C16—102no—
313C2×C5—22no—
314C2×C5—42no—
315C2×C5—161no—
316C2×C5C2×#48011yescharacteristic
317C2×C5#341611yescharacteristic
318C2×C5#341611yescharacteristic
319C23—41no—
320C23C2×#68612yes—
321C23C2×#68611yescharacteristic
322Q8—22no—
323Q8—41no—
324Q8—41no—
325D4—24no—
326D4—22no—
327D4—41no—
328D4—41no—
329D4—82no—
330D4—162no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—82no—
336C2×C4—161no—
337C2×C4—201no—
338C2×C4—202no—
339C2×C4—201no—
340C2×C4—801no—
341C2×C4—802no—
342C2×C4C2×#68612yes—
343C2×C4C2×#68611yescharacteristic
344C2×C4#408112yes—
345C2×C4#409712yes—
346C2×C4C22×C22⋊D511yescharacteristic
347C8—22no—
348C8#408112yes—
349C8C2×#71111yescharacteristic
350C8C2×#71111yescharacteristic
351C5#3365311yescharacteristic, complemented, Sylow
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#6532212yes—
361C22C22×#68611yescharacteristic
362C4—22no—
363C4—81no—
364C4—82no—
365C4—401no—
366C4—402no—
367C4—401no—
368C4#6532212yes—
369C4C2×#408111yescharacteristic
370C4C2×#409711yescharacteristic
371C2—22no—
372C2—42no—
373C2—161no—
374C2C2×#6532211yescharacteristic
375C2#1061573911yescharacteristic
376C2#1061574711yescharacteristic
subgroupG/H
centerC22C22×#686
commutator, cosocleC2×C5×C8C24
FrattiniC2×C8C23×D5
Fitting, prime radicalC5×#3061C2
socleC22×C5C22×D8
hypercenter#3061D5
hypocenterC5#33653
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3061

Complements

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

NH ≅ G/Ntype
2. #10615516C2semidirect
3. #10602871C2semidirect
4. #10602870C2semidirect
5. #10602797C2semidirect
6. #10601873C2semidirect
7. #10601871C2semidirect
8. #10601863C2semidirect
9. #10601862C2semidirect
10. #10601389C2semidirect
11. C2×#64710C2semidirect
351. C5#33653semidirect

Maximal subgroups

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

subgroupindex
C5×#30612
#106155162
#106028712
#106028702
#106027972
#106018732
#106018712
#106018632
#106018622
#106013892
C2×#647102
#336535