Finite Group Databasebeta

← #12149148

Subgroups — #12149148

Order 1280.

838 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
1C5×#2365C211yesmaximal, characteristic, complemented
2C2×#65372C211yesmaximal, characteristic
3C2×#65370C211yesmaximal, characteristic, complemented
4#10605538C211yesmaximal, characteristic
5#10602426C212yesmaximal
6#10602420C212yesmaximal, complemented
7#10600473C212yesmaximal
8#10600464C212yesmaximal, complemented
9#10600282C211yesmaximal, characteristic, complemented
10#10600277C211yesmaximal, characteristic, complemented
11C4×#3941C211yesmaximal, characteristic
12C2×C5×C4⋊Q8C2211yescharacteristic
13C2×C4×C5×Q8C2212yes—
14C22×#688C2212yes—
15C2×C4×D20C2211yescharacteristic
16C2×#4015C2211yescharacteristic
17C5×#391C2212yes—
18C5×#389—21no—
19C5×#387—22no—
20C4×C5×C4⋊C4—21no—
21C4×C5×C4⋊C4C2211yescharacteristic
22C4×C5×C4⋊C4C2211yescharacteristic
23C2×#3967C2212yes—
24C2×#3967C2212yes—
25C2×#3967C2212yes—
26#65372—24no—
27#65370—24no—
28#65276—21no—
29#65276—21no—
30C2×#3941C2211yescharacteristic
31C2×#3941C2211yescharacteristic
32#65228C2212yes—
33#65225C2212yes—
34C2×C4⋊#168C2211yescharacteristic
35C2×C4⋊#168C2212yes—
36C2×C4×#168C2211yescharacteristic
37#64962—22no—
38#64749C2211yescharacteristic
39#64749C2212yes—
40#64749C2211yescharacteristic
41#64749C2212yes—
42#64744C2212yes—
43#25319—51nomaximal, Sylow
44C22×C5×Q8C2312yes—
45C22×D20C2311yescharacteristic
46C22×C4×D5—21no—
47C22×Dic10—21no—
48C5×C4⋊Q8—22no—
49C5×C4⋊Q8D414yes—
50C4×C5×Q8—24no—
51C4×C5×Q8—42no—
52C2×C5×C4⋊C4—21no—
53C2×C5×C4⋊C4—21no—
54C2×C5×C4⋊C4—22no—
55C2×C5×C4⋊C4D411yescharacteristic
56C2×C5×C4⋊C4D411yescharacteristic
57C2×C5×C4⋊C4C2311yescharacteristic
58C2×C5×C4⋊C4C2311yescharacteristic
59C2×C42×C5—21no—
60C2×C42×C5—22no—
61C2×C42×C5C2312yes—
62C2×C42×C5C2311yescharacteristic
63C2×#688—24no—
64C2×#688—24no—
65C2×#688—42no—
66C2×#685—21no—
67C2×#685—22no—
68C2×#685—22no—
69C2×C4⋊Dic5—21no—
70C2×C4⋊Dic5C2312yes—
71C2×C4⋊Dic5C2311yescharacteristic
72C22×#168C2312yes—
73C22×#168C2311yescharacteristic
74C22×#168C2311yescharacteristic
75C4×D20—22no—
76C4×D20—22no—
77#4015—22no—
78#4015—22no—
79C5×#95—22no—
80C5×#95—21no—
81C5×#95—22no—
82C5×#95D412yes—
83#3967—24no—
84#3967—24no—
85#3967—24no—
86#3963—22no—
87#3963—22no—
88#3941—22no—
89#3941—22no—
90C4⋊#168—22no—
91C4⋊#168—42no—
92C4×#168—22no—
93C2×#487—51no—
94C2×#461—51no—
95#2563—51no—
96#2365D511yescharacteristic, complemented
97#2074—52no—
98#2055—52no—
99#1987—51no—
100#1939—51no—
101#1880—52no—
102#1805—52no—
103C4×#102—51no—
104C23×D5—21no—
105C2×C5×Q8—22no—
106C2×C5×Q8—22no—
107C2×C5×Q8—22no—
108C2×C5×Q8—22no—
109C2×C5×Q8—22no—
110C2×C5×Q8C2×D414yes—
111C22×C4×C5—21no—
112C22×C4×C5—22no—
113C22×C4×C5—21no—
114C22×C4×C5C2×D412yes—
115C22×C4×C5C2×D412yes—
116C22×C4×C5C2×D411yescharacteristic
117C22×C4×C5C2×D411yescharacteristic
118C22×C4×C5C2411yescharacteristic
119C22×Dic5—22no—
120C22×Dic5—21no—
121C2×D20—21no—
122C2×D20—21no—
123C2×D20—21no—
124C2×D20—21no—
125C2×D20—21no—
126C2×D20#4112yes—
127C2×C4×D5—42no—
128C2×C4×D5—42no—
129C2×C4×D5—42no—
130C2×Dic10—42no—
131C2×Dic10—81no—
132C2×Dic10—81no—
133C5×C4⋊C4—24no—
134C5×C4⋊C4—24no—
135C5×C4⋊C4—22no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4—21no—
139C5×C4⋊C4—21no—
140C5×C4⋊C4—44no—
141C5×C4⋊C4#4112yes—
142C42×C5—22no—
143C42×C5—22no—
144C42×C5—44no—
145C42×C5—42no—
146C42×C5C2×D412yes—
147C42×C5#4112yes—
148#688—48no—
149#685—42no—
150#685—82no—
151#685—82no—
152C4⋊Dic5—22no—
153C4⋊Dic5—21no—
154C4⋊Dic5—22no—
155C4⋊Dic5—21no—
156C4⋊Dic5—42no—
157C2×#168—21no—
158C2×#168—21no—
159C2×#168—22no—
160C2×#168—21no—
161C2×#168—21no—
162C2×#168—22no—
163C2×#168—21no—
164C2×#168—22no—
165C2×#168—21no—
166C22×SD16—52no—
167C2×C4⋊Q8—51no—
168C2×C4⋊Q8C2×D511yescharacteristic
169C2×C4×Q8C2×D512yes—
170C2×C4×D4—51no—
171#487—104no—
172#461—104no—
173C2×C4⋊C8—51no—
174C2×C4⋊C8—52no—
175C2×#103—52no—
176C2×#103—52no—
177C2×#103—52no—
178C2×#102—51no—
179C2×#102—51no—
180C2×C4×C8—51no—
181#393—102no—
182#391C2×D512yes—
183#390—52no—
184#389—21no—
185#387—22no—
186#382—52no—
187#380—101no—
188#380—101no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4C2×D511yescharacteristic
191C4×C4⋊C4C2×D511yescharacteristic
192#340—52no—
193#340—51no—
194#340—52no—
195#340—51no—
196#336—52no—
197C23×C5C22×D411yescharacteristic
198C22×D5—22no—
199C22×D5—41no—
200C22×D5—41no—
201C22×D5—41no—
202C22×D5—41no—
203C22×D5—41no—
204C22×D5—41no—
205C5×Q8—24no—
206C5×Q8—24no—
207C5×Q8—42no—
208C5×Q8—42no—
209C2×C4×C5—21no—
210C2×C4×C5—21no—
211C2×C4×C5—24no—
212C2×C4×C5—22no—
213C2×C4×C5—22no—
214C2×C4×C5—22no—
215C2×C4×C5—22no—
216C2×C4×C5—22no—
217C2×C4×C5—22no—
218C2×C4×C5—22no—
219C2×C4×C5—42no—
220C2×C4×C5—42no—
221C2×C4×C5—41no—
222C2×C4×C5—41no—
223C2×C4×C5—41no—
224C2×C4×C5—41no—
225C2×C4×C5—41no—
226C2×C4×C5#12112yes—
227C2×C4×C5#12112yes—
228C2×C4×C5#13512yes—
229C2×C4×C5#13712yes—
230C2×C4×C5C22×D411yescharacteristic
231C2×C4×C5C2×#4111yescharacteristic
232C2×C4×C5C2×#4111yescharacteristic
233C2×C4×C5C2×#4111yescharacteristic
234C2×C4×C5H(5,𝔽2)11yescharacteristic
235C2×C4×C52−1+411yescharacteristic
236C2×Dic5—42no—
237C2×Dic5—41no—
238C2×Dic5—41no—
239C2×Dic5—41no—
240C2×Dic5—42no—
241C2×Dic5—42no—
242D20—22no—
243D20—22no—
244D20—41no—
245D20—41no—
246C4×D5—84no—
247Dic10—84no—
248#168—41no—
249#168—42no—
250#168—41no—
251C22×Q8—101no—
252C22×Q8C22×D512yes—
253C22×D4—51no—
254C23×C4—101no—
255C2×SD16—104no—
256C2×SD16—104no—
257C2×SD16—202no—
258C22×C8—52no—
259C22×C8—51no—
260C22×C8—51no—
261C4⋊Q8—22no—
262C4⋊Q8—102no—
263C4⋊Q8—102no—
264C4⋊Q8D2014yes—
265C4×Q8—24no—
266C4×Q8—42no—
267C4×D4—102no—
268C4×D4—102no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—51no—
273C2×C4⋊C4—52no—
274C2×C4⋊C4—101no—
275C2×C4⋊C4C22⋊D511yescharacteristic
276C2×C4⋊C4C22⋊D511yescharacteristic
277C2×C4⋊C4C22×D511yescharacteristic
278C2×C4⋊C4C22×D511yescharacteristic
279C2×C22⋊C4—101no—
280C2×C22⋊C4—102no—
281C2×C22⋊C4—102no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42C22×D512yes—
285C2×C42C22×D511yescharacteristic
286C4⋊C8—102no—
287C4⋊C8—202no—
288#103—104no—
289#103—104no—
290#103—104no—
291#102—102no—
292#102—102no—
293C4×C8—102no—
294#95—22no—
295#95—21no—
296#95—22no—
297#95—102no—
298#95—102no—
299#95C22⋊D512yes—
300C22×C5C2×#12111yescharacteristic
301C22×C5#54611yescharacteristic
302C22×C5#54811yescharacteristic
303C22×C5C2×#13511yescharacteristic
304C22×C5C2×#13711yescharacteristic
305C22×C5#57711yescharacteristic
306C22×C5#57711yescharacteristic
307C2×D5—42no—
308C2×D5—42no—
309C2×D5—42no—
310C2×D5—81no—
311C2×D5—81no—
312C2×D5—81no—
313C2×D5—81no—
314C4×C5—22no—
315C4×C5—42no—
316C4×C5—44no—
317C4×C5—42no—
318C4×C5—42no—
319C4×C5—81no—
320C4×C5#46414yes—
321C4×C5#52911yescharacteristic
322C4×C5#54211yescharacteristic
323C4×C5#54611yescharacteristic
324C4×C5#54711yescharacteristic
325Dic5—81no—
326Dic5—82no—
327C24—101no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—202no—
334C2×Q8—401no—
335C2×Q8—401no—
336C2×Q8C2×D2014yes—
337C2×D4—52no—
338C2×D4—101no—
339C2×D4—101no—
340C2×D4—101no—
341C2×D4—101no—
342C2×D4—101no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—101no—
347C22×C4—102no—
348C22×C4—202no—
349C22×C4—202no—
350C22×C4—202no—
351C22×C4C2×C22⋊D512yes—
352C22×C4C2×C22⋊D512yes—
353C22×C4C2×C22⋊D511yescharacteristic
354C22×C4C2×C22⋊D511yescharacteristic
355C22×C4C23×D511yescharacteristic
356SD16—208no—
357C2×C8—101no—
358C2×C8—101no—
359C2×C8—102no—
360C2×C8—101no—
361C2×C8—101no—
362C2×C8—102no—
363C2×C8—101no—
364C2×C8—102no—
365C2×C8—101no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—44no—
374C4⋊C4—101no—
375C4⋊C4—102no—
376C4⋊C4—101no—
377C4⋊C4—102no—
378C4⋊C4—202no—
379C4⋊C4#71112yes—
380C22⋊C4—202no—
381C22⋊C4—402no—
382C22⋊C4—402no—
383C42—22no—
384C42—22no—
385C42—44no—
386C42—42no—
387C42C2×D2012yes—
388C42#70912yes—
389C2×C5#242711yescharacteristic
390C2×C5C2×#46411yescharacteristic
391C2×C5#304311yescharacteristic
392C2×C5#330311yescharacteristic
393C2×C5#332911yescharacteristic
394C2×C5#336011yescharacteristic
395C2×C5#337211yescharacteristic
396D5—82no—
397C23—102no—
398C23—201no—
399C23—201no—
400C23—201no—
401C23—201no—
402C23—201no—
403C23—201no—
404C23C22×C22⋊D511yescharacteristic
405Q8—24no—
406Q8—24no—
407Q8—42no—
408Q8—42no—
409Q8—404no—
410D4—102no—
411D4—102no—
412D4—201no—
413D4—201no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—201no—
432C2×C4—202no—
433C2×C4—201no—
434C2×C4—201no—
435C2×C4—202no—
436C2×C4—202no—
437C2×C4—404no—
438C2×C4#407612yes—
439C2×C4#407612yes—
440C2×C4#408812yes—
441C2×C4#409612yes—
442C2×C4C22×D2011yescharacteristic
443C2×C4C2×#70911yescharacteristic
444C2×C4C2×#71111yescharacteristic
445C2×C4#414411yescharacteristic
446C2×C4D5×#4111yescharacteristic
447C2×C4#415011yescharacteristic
448C8—202no—
449C8—201no—
450C8—201no—
451C5#2531911yescharacteristic, complemented, Sylow
452C22—202no—
453C22—202no—
454C22—202no—
455C22—401no—
456C22—401no—
457C22—401no—
458C22—401no—
459C22C2×#407611yescharacteristic
460C22#6614011yescharacteristic
461C22C2×#408811yescharacteristic
462C22C2×#409611yescharacteristic
463C22#6616511yescharacteristic
464C22#6617111yescharacteristic
465C22#6617111yescharacteristic
466C4—22no—
467C4—44no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—81no—
472C4—402no—
473C4—401no—
474C4#6531714yes—
475C4#6588811yescharacteristic
476C4#6588911yescharacteristic
477C4#6589111yescharacteristic
478C4#6589511yescharacteristic
479C2—402no—
480C2#1061413911yescharacteristic
481C2C2×#6531711yescharacteristic
482C2#1061428211yescharacteristic
483C2#1061524211yescharacteristic
484C2#1061528611yescharacteristic
485C2#1061539811yescharacteristic
486C2#1061542311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2365C2
socleC23×C5C22×D4
prime radicalC22×D20C23
hypercenter#2365D5
hypocenterC5#25319
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
1. C5×#2365C2semidirect
3. C2×#65370C2semidirect
6. #10602420C2semidirect
8. #10600464C2semidirect
9. #10600282C2semidirect
10. #10600277C2semidirect
96. #2365D5semidirect
451. C5#25319semidirect

Maximal subgroups

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

subgroupindex
C5×#23652
C2×#653722
C2×#653702
#106055382
#106024262
#106024202
#106004732
#106004642
#106002822
#106002772
C4×#39412
#253195