Finite Group Databasebeta

← #12158263

Subgroups — #12158263

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10606055C211yesmaximal, characteristic
2#10605447C211yesmaximal, characteristic, complemented
3C5×#1947C211yesmaximal, characteristic, complemented
4#10601434C212yesmaximal
5#10601408C212yesmaximal, complemented
6#10601272C211yesmaximal, characteristic
7#10601261C211yesmaximal, characteristic, complemented
8#10600370C211yesmaximal, characteristic, complemented
9C4×#3941C211yesmaximal, characteristic
10#10599694C212yesmaximal
11#10599580C212yesmaximal, complemented
12C2×#4038—21no—
13C2×C4×D20C2211yescharacteristic
14C2×#4016—21no—
15C2×C5×#107C2212yes—
16C2×C5×C4⋊C8C2211yescharacteristic
17C5×#384C2211yescharacteristic
18C4×C5×C4⋊C4C2211yescharacteristic
19C2×#3953C2212yes—
20C2×#3946C2212yes—
21#65279—22no—
22#65277C2212yes—
23#65276—21no—
24#65265C2212yes—
25C4⋊C4×Dic5C2212yes—
26#65262C2212yes—
27C2×#3941C2211yescharacteristic
28C2×#3941C2211yescharacteristic
29C2×#3939C2212yes—
30C2×#3939C2212yes—
31#65224C2211yescharacteristic
32C4×C4⋊Dic5C2211yescharacteristic
33#65217—21no—
34C2×C4×#168C2211yescharacteristic
35#64946—22no—
36C5×#340C2212yes—
37#64771C2212yes—
38#64770C2212yes—
39#64749C2211yescharacteristic
40#64749C2211yescharacteristic
41#26086—51nomaximal, Sylow
42C22×D20C2311yescharacteristic
43C22×C4×D5—21no—
44C22×C5×C8C2312yes—
45C2×C5×C4⋊C4—21no—
46C2×C5×C4⋊C4—21no—
47C2×C5×C4⋊C4C2311yescharacteristic
48C2×C5×C4⋊C4C2311yescharacteristic
49C2×C5×C4⋊C4C2312yes—
50C2×C42×C5—21no—
51C2×C42×C5C2311yescharacteristic
52C2×#685—21no—
53C2×#685—22no—
54C2×#685—22no—
55C2×C4⋊Dic5—21no—
56C2×C4⋊Dic5—21no—
57C2×C4⋊Dic5D411yescharacteristic
58C2×C4⋊Dic5D411yescharacteristic
59C2×C4⋊Dic5C2312yes—
60C2×C4⋊Dic5C2311yescharacteristic
61C2×#683—22no—
62C2×#683—22no—
63C2×#683—22no—
64C2×C4×Dic5—22no—
65C2×C4×Dic5C2312yes—
66C22×#168C2311yescharacteristic
67C22×#168C2311yescharacteristic
68#4038—44no—
69C4×D20—22no—
70C4×D20Q814yes—
71#4016—44no—
72C5×#107—42no—
73C5×C4⋊C8—22no—
74C5×#95—21no—
75#3963—22no—
76#3963—22no—
77#3963D412yes—
78#3953—42no—
79#3946—24no—
80#3941—22no—
81#3941—22no—
82#3939—24no—
83#3939—42no—
84C4×#168—22no—
85#2552—51no—
86#2354—51no—
87#2093—52no—
88#2067—52no—
89#1947—52no—
90#1947D511yescharacteristic, complemented
91#1939—51no—
92#1918—52no—
93#1845—51no—
94#1802—51no—
95C4×#102—51no—
96C23×D5—21no—
97C22×C4×C5—21no—
98C22×C4×C5—21no—
99C22×C4×C5—22no—
100C22×C4×C5C2×D411yescharacteristic
101C22×C4×C5C2×D411yescharacteristic
102C22×C4×C5C2411yescharacteristic
103C22×Dic5—21no—
104C22×Dic5—22no—
105C22×Dic5C2×D412yes—
106C22×Dic5C2×D412yes—
107C2×D20—21no—
108C2×D20—21no—
109C2×D20—21no—
110C2×D20—21no—
111C2×D20—21no—
112C2×D20C2×Q812yes—
113C2×C4×D5—24no—
114C2×C4×D5—42no—
115C2×C4×D5—42no—
116C2×C5×C8—22no—
117C2×C5×C8—22no—
118C2×C5×C8—22no—
119C5×C4⋊C4—22no—
120C5×C4⋊C4—21no—
121C5×C4⋊C4—21no—
122C5×C4⋊C4—22no—
123C5×C4⋊C4—21no—
124C5×C4⋊C4—21no—
125C5×C4⋊C4—42no—
126C5×C4⋊C4—81no—
127C42×C5—42no—
128C42×C5C2×Q812yes—
129C42×C5#4112yes—
130#685—24no—
131#685—44no—
132#685—82no—
133C4⋊Dic5—22no—
134C4⋊Dic5—21no—
135C4⋊Dic5—22no—
136C4⋊Dic5—22no—
137C4⋊Dic5—42no—
138C4⋊Dic5—81no—
139C4⋊Dic5D814yes—
140C4⋊Dic5C2×Q812yes—
141#683—44no—
142#683—44no—
143#683—82no—
144C4×Dic5—22no—
145C4×Dic5—22no—
146C4×Dic5—44no—
147C2×#168—21no—
148C2×#168—21no—
149C2×#168—21no—
150C2×#168—21no—
151C2×#168#4112yes—
152C2×#168#4112yes—
153C2×#125—101no—
154C2×C4×D4—51no—
155C2×#117—101no—
156C2×#107—52no—
157C2×#107—52no—
158C2×#107C2×D512yes—
159C2×C4⋊C8—52no—
160C2×C4⋊C8C2×D511yescharacteristic
161C2×#102—51no—
162C2×#102—51no—
163C2×#102—52no—
164C2×C4×C8—51no—
165#398—102no—
166#390—52no—
167#389—52no—
168#389—101no—
169#388—102no—
170#384—52no—
171#384—51no—
172#384C2×D511yescharacteristic
173#380—101no—
174C4×C4⋊C4—51no—
175C4×C4⋊C4—52no—
176C4×C4⋊C4C2×D511yescharacteristic
177#340—52no—
178#340—51no—
179#340—51no—
180#340C2×D512yes—
181#336—52no—
182C23×C5C22×D411yescharacteristic
183C22×D5—22no—
184C22×D5—41no—
185C22×D5—41no—
186C22×D5—41no—
187C22×D5—41no—
188C22×D5—41no—
189C22×D5—41no—
190C2×C4×C5—21no—
191C2×C4×C5—21no—
192C2×C4×C5—41no—
193C2×C4×C5—42no—
194C2×C4×C5—41no—
195C2×C4×C5—42no—
196C2×C4×C5—41no—
197C2×C4×C5—41no—
198C2×C4×C5—41no—
199C2×C4×C5—42no—
200C2×C4×C5—41no—
201C2×C4×C5C22⋊Q812yes—
202C2×C4×C5C22⋊Q812yes—
203C2×C4×C5C2×D812yes—
204C2×C4×C5#13712yes—
205C2×C4×C5C22×Q811yescharacteristic
206C2×C4×C5C2×#4111yescharacteristic
207C2×C4×C5C2×#4111yescharacteristic
208C2×C4×C5C2×#4111yescharacteristic
209C2×C4×C5H(5,𝔽2)11yescharacteristic
210C2×C4×C52−1+411yescharacteristic
211C2×Dic5—22no—
212C2×Dic5—22no—
213C2×Dic5—22no—
214C2×Dic5—22no—
215C2×Dic5—22no—
216C2×Dic5—22no—
217C2×Dic5—41no—
218C2×Dic5—41no—
219C2×Dic5—42no—
220C2×Dic5—42no—
221C2×Dic5—42no—
222C2×Dic5C2×D814yes—
223D20—22no—
224D20—22no—
225D20—41no—
226D20—41no—
227C4×D5—48no—
228C5×C8—42no—
229#168—22no—
230#168—22no—
231C22×D4—51no—
232C23×C4—101no—
233C22×C8—51no—
234C22×C8—51no—
235C22×C8C22×D512yes—
236#125—204no—
237C4×D4—54no—
238C4×D4—102no—
239#117—204no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—51no—
243C2×C4⋊C4—51no—
244C2×C4⋊C4—51no—
245C2×C4⋊C4—52no—
246C2×C4⋊C4—102no—
247C2×C4⋊C4—102no—
248C2×C4⋊C4—101no—
249C2×C4⋊C4—102no—
250C2×C4⋊C4—101no—
251C2×C4⋊C4C22×D511yescharacteristic
252C2×C4⋊C4C22×D511yescharacteristic
253C2×C4⋊C4C22×D512yes—
254C2×C22⋊C4—101no—
255C2×C22⋊C4—102no—
256C2×C22⋊C4—102no—
257C2×C42—21no—
258C2×C42—52no—
259C2×C42—102no—
260C2×C42C22×D511yescharacteristic
261#107—42no—
262#107—104no—
263#107—202no—
264C4⋊C8—22no—
265C4⋊C8—104no—
266#102—102no—
267#102—102no—
268#102—202no—
269C4×C8—102no—
270#95—21no—
271#95—52no—
272#95—102no—
273#95—102no—
274C22×C5C2×C22⋊Q811yescharacteristic
275C22×C5#54711yescharacteristic
276C22×C5#55011yescharacteristic
277C22×C5C22×D811yescharacteristic
278C22×C5C2×#13711yescharacteristic
279C22×C5#57611yescharacteristic
280C22×C5#57811yescharacteristic
281C2×D5—42no—
282C2×D5—42no—
283C2×D5—42no—
284C2×D5—81no—
285C2×D5—81no—
286C2×D5—81no—
287C2×D5—81no—
288C4×C5—22no—
289C4×C5—81no—
290C4×C5—82no—
291C4×C5—81no—
292C4×C5#47414yes—
293C4×C5#53311yescharacteristic
294C4×C5#54411yescharacteristic
295C4×C5#55211yescharacteristic
296C4×C5#55311yescharacteristic
297Dic5—24no—
298Dic5—42no—
299Dic5—42no—
300Dic5—82no—
301C24—101no—
302C2×D4—52no—
303C2×D4—101no—
304C2×D4—101no—
305C2×D4—101no—
306C2×D4—101no—
307C2×D4—101no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—52no—
312C22×C4—52no—
313C22×C4—101no—
314C22×C4—104no—
315C22×C4—102no—
316C22×C4—202no—
317C22×C4—202no—
318C22×C4D4×D511yescharacteristic
319C22×C4D4×D511yescharacteristic
320C22×C4C23×D511yescharacteristic
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—52no—
325C2×C8—52no—
326C2×C8—101no—
327C2×C8—101no—
328C2×C8—101no—
329C2×C8—101no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—42no—
337C4⋊C4—54no—
338C4⋊C4—52no—
339C4⋊C4—81no—
340C4⋊C4—101no—
341C4⋊C4—102no—
342C4⋊C4—102no—
343C4⋊C4—102no—
344C4⋊C4—204no—
345C4⋊C4—204no—
346C4⋊C4—202no—
347C4⋊C4—401no—
348C4⋊C4—402no—
349C22⋊C4—104no—
350C22⋊C4—204no—
351C22⋊C4—402no—
352C42—42no—
353C42—102no—
354C42—102no—
355C42—204no—
356C42#70912yes—
357C42Q8×D512yes—
358C2×C5#244111yescharacteristic
359C2×C5C2×#47411yescharacteristic
360C2×C5#308311yescharacteristic
361C2×C5#333411yescharacteristic
362C2×C5#335711yescharacteristic
363C2×C5#339111yescharacteristic
364C2×C5#339811yescharacteristic
365D5—82no—
366C23—102no—
367C23—201no—
368C23—201no—
369C23—201no—
370C23—201no—
371C23—201no—
372C23—201no—
373C23C2×D4×D511yescharacteristic
374D4—102no—
375D4—102no—
376D4—201no—
377D4—201no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—54no—
390C2×C4—102no—
391C2×C4—102no—
392C2×C4—102no—
393C2×C4—102no—
394C2×C4—102no—
395C2×C4—102no—
396C2×C4—201no—
397C2×C4—201no—
398C2×C4—208no—
399C2×C4—202no—
400C2×C4—202no—
401C2×C4—202no—
402C2×C4#404212yes—
403C2×C4#404212yes—
404C2×C4D5×D812yes—
405C2×C4#406412yes—
406C2×C4C2×#70911yescharacteristic
407C2×C4#414411yescharacteristic
408C2×C4C2×Q8×D511yescharacteristic
409C2×C4#414711yescharacteristic
410C2×C4D5×#4111yescharacteristic
411C2×C4D5×#4111yescharacteristic
412C8—42no—
413C8—102no—
414C8—102no—
415C5#2608611yescharacteristic, complemented, Sylow
416C22—202no—
417C22—202no—
418C22—202no—
419C22—401no—
420C22—401no—
421C22—401no—
422C22—401no—
423C22C2×#404211yescharacteristic
424C22#6593311yescharacteristic
425C22#6596511yescharacteristic
426C22C2×D5×D811yescharacteristic
427C22C2×#406411yescharacteristic
428C22#6610811yescharacteristic
429C22#6611311yescharacteristic
430C4—22no—
431C4—81no—
432C4—82no—
433C4—81no—
434C4—104no—
435C4—202no—
436C4—202no—
437C4—402no—
438C4#6517914yes—
439C4#6589211yescharacteristic
440C4#6591511yescharacteristic
441C4#6602211yescharacteristic
442C4#6606311yescharacteristic
443C2—402no—
444C2#1060795511yescharacteristic
445C2C2×#6517911yescharacteristic
446C2#1061097411yescharacteristic
447C2#1061205511yescharacteristic
448C2#1061220411yescharacteristic
449C2#1061271711yescharacteristic
450C2#1061286111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#1947C2
socleC23×C5C22×D4
prime radicalC22×D20C23
hypercenter#1947D5
hypocenterC5#26086
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
2. #10605447C2semidirect
3. C5×#1947C2semidirect
5. #10601408C2semidirect
7. #10601261C2semidirect
8. #10600370C2semidirect
11. #10599580C2semidirect
90. #1947D5semidirect
415. C5#26086semidirect

Maximal subgroups

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

subgroupindex
#106060552
#106054472
C5×#19472
#106014342
#106014082
#106012722
#106012612
#106003702
C4×#39412
#105996942
#105995802
#260865