Finite Group Databasebeta

← #12118492

Subgroups — #12118492

Order 1280.

902 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10614193C211yesmaximal, characteristic, complemented
2C2×C4×#688C211yesmaximal, characteristic
3#10606091C211yesmaximal, characteristic
4C5×#1803C211yesmaximal, characteristic, complemented
5#10601272C211yesmaximal, characteristic
6#10601259C211yesmaximal, characteristic, complemented
7#10601069C212yesmaximal
8#10601049C212yesmaximal, complemented
9#10600367C211yesmaximal, characteristic, complemented
10#10599582C212yesmaximal, complemented
11#10599575C212yesmaximal
12C2×C4×C5×Q8C2211yescharacteristic
13C22×#688C2211yescharacteristic
14C2×#4039C2212yes—
15C2×D5×C4⋊C4—21no—
16C2×C4×D20C2211yescharacteristic
17C2×C5×C4⋊C8C2211yescharacteristic
18C2×C5×#103C2212yes—
19C5×#384C2211yescharacteristic
20#65519C2212yes—
21C2×#3967C2211yescharacteristic
22C2×#3953C2212yes—
23C2×#3947C2212yes—
24C4×#688C418yescomplemented
25#65278—22no—
26#65276—21no—
27#65270—22no—
28C4⋊C4×Dic5C2212yes—
29#65262C2212yes—
30C2×#3941C2211yescharacteristic
31C2×#3940C2211yescharacteristic
32#65225—21no—
33#65224C2211yescharacteristic
34C4×C4⋊Dic5—21no—
35C4×C4⋊Dic5C2211yescharacteristic
36C2×C4×#168C2211yescharacteristic
37C5×#340C2212yes—
38#64771C2212yes—
39#64770C2212yes—
40#64749C2212yes—
41#64749C2212yes—
42#22762—51nomaximal, Sylow
43C22×C5×Q8C2311yescharacteristic
44C22×D20C2311yescharacteristic
45C22×C4×D5—21no—
46C22×C4×D5—22no—
47C22×C5×C8C2312yes—
48C4×C5×Q8—22no—
49C4×C5×Q8C2×C414yescomplemented
50C2×C5×C4⋊C4—21no—
51C2×C5×C4⋊C4—21no—
52C2×C5×C4⋊C4—21no—
53C2×C5×C4⋊C4C2312yes—
54C2×C5×C4⋊C4C2311yescharacteristic
55C2×C42×C5—21no—
56C2×C42×C5C2311yescharacteristic
57C2×#688—22no—
58C2×#688—22no—
59C2×#688C2×C414yes—
60C2×#685—21no—
61C2×#685—22no—
62C2×C4⋊Dic5—21no—
63C2×C4⋊Dic5—21no—
64C2×C4⋊Dic5D411yescharacteristic
65C2×C4⋊Dic5D411yescharacteristic
66C2×C4⋊Dic5C2312yes—
67C2×C4⋊Dic5C2311yescharacteristic
68C2×#683—22no—
69C2×C4×Dic5—22no—
70C2×C4×Dic5C2312yes—
71C22×#168C2311yescharacteristic
72C22×#168C2311yescharacteristic
73#4039—24no—
74#4039—42no—
75D5×C4⋊C4—44no—
76C4×D20—22no—
77C4×D20C2×C414yes—
78C5×C4⋊C8—22no—
79C5×#103—42no—
80#3967C2×C414yescomplemented
81#3963—22no—
82#3963—22no—
83#3963—22no—
84#3963—22no—
85#3963D412yes—
86#3953—24no—
87#3947—42no—
88#3941C2×C414yes—
89#3940C2×C414yescomplemented
90C4×#168C2×C414yescomplemented
91C2×C4×SD16—51no—
92#2355—51no—
93D4×C4⋊C4—51no—
94#1935—52no—
95#1920—52no—
96#1877—52no—
97#1834—51no—
98#1803D511yescharacteristic, complemented
99#1802—51no—
100#1770—51no—
101#1762—52no—
102C23×D5—21no—
103C2×C5×Q8—21no—
104C2×C5×Q8—21no—
105C2×C5×Q8—21no—
106C2×C5×Q8—21no—
107C2×C5×Q8—21no—
108C2×C5×Q8C22×C412yes—
109C22×C4×C5—21no—
110C22×C4×C5—21no—
111C22×C4×C5—22no—
112C22×C4×C5C2×D411yescharacteristic
113C22×C4×C5C2×D411yescharacteristic
114C22×C4×C5C2411yescharacteristic
115C22×Dic5—21no—
116C22×Dic5—22no—
117C22×Dic5C2×D412yes—
118C22×Dic5C2×D412yes—
119C2×D20—21no—
120C2×D20—21no—
121C2×D20—21no—
122C2×D20C22×C412yes—
123C2×D20#4112yes—
124C2×D20#4112yes—
125C2×C4×D5—24no—
126C2×C4×D5—42no—
127C2×C4×D5—42no—
128C2×C4×D5—44no—
129C2×C4×D5—44no—
130C2×C4×D5—82no—
131C2×C5×C8—22no—
132C2×C5×C8—22no—
133C2×C5×C8—22no—
134C5×C4⋊C4—24no—
135C5×C4⋊C4—21no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—22no—
138C5×C4⋊C4—42no—
139C5×C4⋊C4—81no—
140C5×C4⋊C4C22×C412yes—
141C42×C5—24no—
142C42×C5C22×C412yes—
143C42×C5#4112yes—
144#688—28no—
145#685—24no—
146#685—44no—
147C4⋊Dic5—22no—
148C4⋊Dic5—22no—
149C4⋊Dic5—21no—
150C4⋊Dic5—22no—
151C4⋊Dic5—22no—
152C4⋊Dic5—42no—
153C4⋊Dic5—81no—
154C4⋊Dic5C22×C412yes—
155#683—44no—
156C4×Dic5—22no—
157C4×Dic5—22no—
158C4×Dic5—82no—
159C2×#168—21no—
160C2×#168—21no—
161C2×#168—21no—
162C2×#168—21no—
163C2×#168C22×C412yes—
164C2×#168C22×C412yes—
165C22×SD16—51no—
166C2×C4×Q8C2×D511yescharacteristic
167C2×C4×D4—52no—
168C2×C4×D4—51no—
169C22×C4⋊C4—101no—
170C4×SD16—58no—
171C2×#106—51no—
172C2×C4⋊C8C2×D511yescharacteristic
173C2×#103—51no—
174C2×#103C2×D512yes—
175C2×#102—51no—
176C2×#102—52no—
177C2×#97—52no—
178C2×C4×C8—51no—
179#391—52no—
180#389—52no—
181#387—102no—
182#385—102no—
183#384—51no—
184#384C2×D511yescharacteristic
185#382—101no—
186#380—101no—
187C4×C4⋊C4—51no—
188C4×C4⋊C4—52no—
189C4×C4⋊C4—101no—
190#340—52no—
191#340—52no—
192#340—52no—
193#340C2×D512yes—
194#336—52no—
195C23×C5C22×D411yescharacteristic
196C22×D5—22no—
197C22×D5—22no—
198C22×D5—22no—
199C22×D5—41no—
200C22×D5—41no—
201C22×D5—41no—
202C22×D5—41no—
203C5×Q8—22no—
204C5×Q8—22no—
205C5×Q8—41no—
206C5×Q8—41no—
207C2×C4×C5—21no—
208C2×C4×C5—21no—
209C2×C4×C5—22no—
210C2×C4×C5—22no—
211C2×C4×C5—41no—
212C2×C4×C5—41no—
213C2×C4×C5—41no—
214C2×C4×C5—41no—
215C2×C4×C5—42no—
216C2×C4×C5—42no—
217C2×C4×C5—42no—
218C2×C4×C5C4×D412yes—
219C2×C4×C5C4×D412yes—
220C2×C4×C5AGL(1,ℤ/8)12yes—
221C2×C4×C5#13712yes—
222C2×C4×C5C23×C411yescharacteristic
223C2×C4×C5C2×#4111yescharacteristic
224C2×C4×C5C2×#4111yescharacteristic
225C2×C4×C5C2×#4111yescharacteristic
226C2×C4×C5H(5,𝔽2)11yescharacteristic
227C2×C4×C52−1+411yescharacteristic
228C2×Dic5—22no—
229C2×Dic5—22no—
230C2×Dic5—22no—
231C2×Dic5—22no—
232C2×Dic5—22no—
233C2×Dic5—22no—
234C2×Dic5—22no—
235C2×Dic5—41no—
236C2×Dic5—41no—
237C2×Dic5—42no—
238C2×Dic5—42no—
239C2×Dic5—42no—
240D20—22no—
241D20—22no—
242D20—22no—
243D20#11714yes—
244C4×D5—48no—
245C4×D5—88no—
246C5×C8—42no—
247#168—22no—
248#168—22no—
249C22×Q8C22×D511yescharacteristic
250C22×D4—51no—
251C23×C4—102no—
252C23×C4—101no—
253C2×SD16—54no—
254C2×SD16—102no—
255C2×SD16—102no—
256C22×C8—51no—
257C22×C8—51no—
258C22×C8C22×D512yes—
259C4×Q8—22no—
260C4×Q8C4×D514yescomplemented
261C4×D4—54no—
262C4×D4—104no—
263C4×D4—102no—
264C4×D4—202no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—51no—
269C2×C4⋊C4—51no—
270C2×C4⋊C4—51no—
271C2×C4⋊C4—52no—
272C2×C4⋊C4—102no—
273C2×C4⋊C4—101no—
274C2×C4⋊C4—101no—
275C2×C4⋊C4—204no—
276C2×C4⋊C4C22×D512yes—
277C2×C4⋊C4C22×D511yescharacteristic
278C2×C22⋊C4—101no—
279C2×C22⋊C4—102no—
280C2×C42—21no—
281C2×C42—52no—
282C2×C42—102no—
283C2×C42C22×D511yescharacteristic
284#106—54no—
285C4⋊C8—22no—
286#103—42no—
287#103—54no—
288#102—54no—
289#102—104no—
290#97—202no—
291C4×C8—54no—
292#95—52no—
293#95—102no—
294#95—102no—
295#95—102no—
296#95—102no—
297C22×C5C2×C4×D411yescharacteristic
298C22×C5#54611yescharacteristic
299C22×C5#54811yescharacteristic
300C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C5C2×#13711yescharacteristic
302C22×C5#57711yescharacteristic
303C22×C5#57711yescharacteristic
304C2×D5—24no—
305C2×D5—42no—
306C2×D5—42no—
307C2×D5—42no—
308C2×D5—42no—
309C2×D5—42no—
310C2×D5—42no—
311C4×C5—22no—
312C4×C5—42no—
313C4×C5—42no—
314C4×C5—82no—
315C4×C5#44014yes—
316C4×C5C2×#11711yescharacteristic
317C4×C5#52011yescharacteristic
318C4×C5#55211yescharacteristic
319C4×C5#55311yescharacteristic
320Dic5—42no—
321Dic5—42no—
322Dic5—42no—
323Dic5—82no—
324C24—101no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8C2×C4×D512yes—
331C2×D4—52no—
332C2×D4—52no—
333C2×D4—52no—
334C2×D4—101no—
335C2×D4—101no—
336C2×D4—101no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—52no—
341C22×C4—52no—
342C22×C4—101no—
343C22×C4—104no—
344C22×C4—102no—
345C22×C4—204no—
346C22×C4—204no—
347C22×C4—202no—
348C22×C4—202no—
349C22×C4—402no—
350C22×C4D4×D511yescharacteristic
351C22×C4D4×D511yescharacteristic
352C22×C4C23×D511yescharacteristic
353SD16—108no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—52no—
358C2×C8—52no—
359C2×C8—101no—
360C2×C8—101no—
361C2×C8—101no—
362C2×C8—101no—
363C4⋊C4—21no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—52no—
369C4⋊C4—81no—
370C4⋊C4—101no—
371C4⋊C4—102no—
372C4⋊C4—102no—
373C4⋊C4—102no—
374C4⋊C4—102no—
375C4⋊C4—204no—
376C4⋊C4—202no—
377C4⋊C4—401no—
378C4⋊C4C2×C4×D512yes—
379C22⋊C4—104no—
380C22⋊C4—204no—
381C42—24no—
382C42—102no—
383C42—102no—
384C42—402no—
385C42C2×C4×D512yes—
386C42#71112yes—
387C2×C5#235211yescharacteristic
388C2×C5C2×#44011yescharacteristic
389C2×C5#295211yescharacteristic
390C2×C5#332411yescharacteristic
391C2×C5#333111yescharacteristic
392C2×C5#337211yescharacteristic
393C2×C5#337311yescharacteristic
394D5—44no—
395C23—102no—
396C23—102no—
397C23—102no—
398C23—201no—
399C23—201no—
400C23—201no—
401C23—201no—
402C23C2×D4×D511yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—41no—
406Q8—41no—
407D4—54no—
408D4—102no—
409D4—102no—
410D4—102no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—102no—
423C2×C4—102no—
424C2×C4—102no—
425C2×C4—102no—
426C2×C4—102no—
427C2×C4—102no—
428C2×C4—102no—
429C2×C4—201no—
430C2×C4—201no—
431C2×C4—208no—
432C2×C4—202no—
433C2×C4—202no—
434C2×C4—202no—
435C2×C4—408no—
436C2×C4#402712yes—
437C2×C4#402712yes—
438C2×C4#406012yes—
439C2×C4#406412yes—
440C2×C4C22×C4×D511yescharacteristic
441C2×C4C2×#71111yescharacteristic
442C2×C4#414411yescharacteristic
443C2×C4#414711yescharacteristic
444C2×C4D5×#4111yescharacteristic
445C2×C4D5×#4111yescharacteristic
446C8—42no—
447C8—102no—
448C8—102no—
449C5#2276211yescharacteristic, complemented, Sylow
450C22—104no—
451C22—202no—
452C22—202no—
453C22—202no—
454C22—202no—
455C22—202no—
456C22—202no—
457C22C2×#402711yescharacteristic
458C22#6594611yescharacteristic
459C22#6596711yescharacteristic
460C22C2×#406011yescharacteristic
461C22C2×#406411yescharacteristic
462C22#6610911yescharacteristic
463C22#6611211yescharacteristic
464C4—22no—
465C4—42no—
466C4—42no—
467C4—82no—
468C4—202no—
469C4—202no—
470C4—202no—
471C4—402no—
472C4#6510614yes—
473C4#6583811yescharacteristic
474C4D5×#11711yescharacteristic
475C4#6594811yescharacteristic
476C4#6597311yescharacteristic
477C2—204no—
478C2#1060574311yescharacteristic
479C2C2×#6510611yescharacteristic
480C2#1061060011yescharacteristic
481C2#1061213011yescharacteristic
482C2#1061222711yescharacteristic
483C2#1061276911yescharacteristic
484C2#1061286611yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1803C2
socleC23×C5C22×D4
prime radicalC22×D20C23
cosocleC22×C4×C5C24
hypercenter#1803D5
hypocenterC5#22762
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. #10614193C2semidirect
4. C5×#1803C2semidirect
6. #10601259C2semidirect
8. #10601049C2semidirect
9. #10600367C2semidirect
10. #10599582C2semidirect
24. C4×#688C4semidirect
49. C4×C5×Q8C2×C4semidirect
80. #3967C2×C4semidirect
89. #3940C2×C4semidirect
90. C4×#168C2×C4semidirect
98. #1803D5semidirect
260. C4×Q8C4×D5semidirect
449. C5#22762semidirect

Maximal subgroups

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

subgroupindex
#106141932
C2×C4×#6882
#106060912
C5×#18032
#106012722
#106012592
#106010692
#106010492
#106003672
#105995822
#105995752
#227625