Finite Group Databasebeta

← #12008748

Subgroups — #12008748

Order 1280.

738 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C5×#120C211yesmaximal, characteristic
2#10602396C211yesmaximal, characteristic, complemented
3#10602395C211yesmaximal, characteristic, complemented
4#10598127C212yesmaximal, complemented
5#10598113C212yesmaximal, complemented
6C24×C4×C5C411yescharacteristic
7C2×C5×#120C411yescharacteristic
8C2×C4×C5×D4—21no—
9C2×C4×C5×D4C2211yescharacteristic
10C5×#386—21no—
11C5×#386C411yescharacteristic
12C5×#385—21no—
13C5×#385C2211yescharacteristic
14C5×#379C2211yescharacteristic
15C4×C5×C22⋊C4—21no—
16C4×C5×C22⋊C4C411yescharacteristic
17#65496—21no—
18#65215—21no—
19#64748—21no—
20#64745—21no—
21#64745C2212yes—
22#64745C2212yes—
23#11525—51nomaximal, Sylow
24C25×C5C2×C411yescharacteristic
25C22×C5×D4—21no—
26C22×C5×D4C2×C411yescharacteristic
27C23×C4×C5—21no—
28C23×C4×C5—21no—
29C23×C4×C5—44no—
30C23×C4×C5—41no—
31C23×C4×C5—41no—
32C23×C4×C5C2×C411yescharacteristic
33C23×C4×C5C2×C411yescharacteristic
34C23×C4×C5D411yescharacteristic
35C5×#120—22no—
36C5×#120—22no—
37C4×C5×D4—41no—
38C4×C5×D4—41no—
39C4×C5×D4—44no—
40C2×C5×C4⋊C4—21no—
41C2×C5×C4⋊C4D411yescharacteristic
42C2×C5×C22⋊C4—21no—
43C2×C5×C22⋊C4—21no—
44C2×C5×C22⋊C4—21no—
45C2×C5×C22⋊C4—21no—
46C2×C5×C22⋊C4C2×C411yescharacteristic
47C2×C5×C22⋊C4C2×C411yescharacteristic
48C2×C5×C22⋊C4D411yescharacteristic
49C2×C5×C22⋊C4C2311yescharacteristic
50C2×C42×C5—21no—
51C2×C42×C5D411yescharacteristic
52C2×#681—41no—
53C5×#95—21no—
54C5×#95—21no—
55C5×#95—21no—
56C22⋊#168—22no—
57C22⋊#168—22no—
58C22⋊#168—41no—
59C4⋊#168—41no—
60#3935—41no—
61C4×#120D511yescharacteristic
62#1466—51no—
63#1465—51no—
64#1326—52no—
65#1316—52no—
66C24×C5—21no—
67C24×C5—21no—
68C24×C5—42no—
69C24×C5—41no—
70C24×C5—42no—
71C24×C5—41no—
72C24×C5C22⋊C411yescharacteristic
73C24×C5M1612yescomplemented
74C24×C5C22×C411yescharacteristic
75C2×C5×D4—24no—
76C2×C5×D4—21no—
77C2×C5×D4—21no—
78C2×C5×D4—41no—
79C2×C5×D4—41no—
80C2×C5×D4—42no—
81C2×C5×D4—42no—
82C22×C4×C5—24no—
83C22×C4×C5—21no—
84C22×C4×C5—21no—
85C22×C4×C5—21no—
86C22×C4×C5—21no—
87C22×C4×C5—21no—
88C22×C4×C5—21no—
89C22×C4×C5—21no—
90C22×C4×C5—42no—
91C22×C4×C5—42no—
92C22×C4×C5—42no—
93C22×C4×C5—42no—
94C22×C4×C5—42no—
95C22×C4×C5—42no—
96C22×C4×C5—42no—
97C22×C4×C5—44no—
98C22×C4×C5—44no—
99C22×C4×C5—44no—
100C22×C4×C5—41no—
101C22×C4×C5—41no—
102C22×C4×C5—41no—
103C22×C4×C5—44no—
104C22×C4×C5C22⋊C411yescharacteristic
105C22×C4×C5C22⋊C411yescharacteristic
106C22×C4×C5C22⋊C411yescharacteristic
107C22×C4×C5C2×D411yescharacteristic
108C22×C4×C5C2×D411yescharacteristic
109C5×C4⋊C4—41no—
110C5×C4⋊C4—42no—
111C5×C22⋊C4—24no—
112C5×C22⋊C4—21no—
113C5×C22⋊C4—41no—
114C5×C22⋊C4—41no—
115C5×C22⋊C4—42no—
116C5×C22⋊C4—42no—
117C5×C22⋊C4—42no—
118C5×C22⋊C4—41no—
119C5×C22⋊C4M1612yescomplemented
120C42×C5—21no—
121C42×C5—21no—
122C42×C5—42no—
123#681—82no—
124C2×#168—42no—
125C2×#168—42no—
126C24×C4Dic511yescharacteristic
127C2×#120Dic511yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4C2×D511yescharacteristic
130#432—101no—
131#407—101no—
132#386—21no—
133#386Dic511yescharacteristic
134#385—21no—
135#385C2×D511yescharacteristic
136#379C2×D511yescharacteristic
137C4×C22⋊C4—21no—
138C4×C22⋊C4Dic511yescharacteristic
139#324—101no—
140#323—52no—
141#323—52no—
142#323—101no—
143C23×C5—22no—
144C23×C5—22no—
145C23×C5—22no—
146C23×C5—21no—
147C23×C5—21no—
148C23×C5—21no—
149C23×C5—21no—
150C23×C5—21no—
151C23×C5—41no—
152C23×C5—44no—
153C23×C5—42no—
154C23×C5—41no—
155C23×C5—41no—
156C23×C5—41no—
157C23×C5—44no—
158C23×C5—42no—
159C23×C5—41no—
160C23×C5—41no—
161C23×C5—42no—
162C23×C5—42no—
163C23×C5—42no—
164C23×C5—48no—
165C23×C5—42no—
166C23×C5—41no—
167C23×C5—41no—
168C23×C5C2×C22⋊C411yescharacteristic
169C23×C5C2×M1611yescharacteristic
170C23×C5C2×M1611yescharacteristic
171C5×D4—48no—
172C5×D4—42no—
173C5×D4—42no—
174C2×C4×C5—22no—
175C2×C4×C5—22no—
176C2×C4×C5—21no—
177C2×C4×C5—22no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—21no—
183C2×C4×C5—44no—
184C2×C4×C5—42no—
185C2×C4×C5—41no—
186C2×C4×C5—41no—
187C2×C4×C5—41no—
188C2×C4×C5—48no—
189C2×C4×C5—42no—
190C2×C4×C5—41no—
191C2×C4×C5—42no—
192C2×C4×C5—44no—
193C2×C4×C5—42no—
194C2×C4×C5—41no—
195C2×C4×C5—41no—
196C2×C4×C5—41no—
197C2×C4×C5—42no—
198C2×C4×C5—41no—
199C2×C4×C5—41no—
200C2×C4×C5—41no—
201C2×C4×C5—81no—
202C2×C4×C5#9911yescharacteristic
203C2×C4×C5#9911yescharacteristic
204#168—82no—
205#168—82no—
206C25C2×Dic511yescharacteristic
207C22×D4—21no—
208C22×D4C2×Dic511yescharacteristic
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—44no—
212C23×C4—41no—
213C23×C4—41no—
214C23×C4C2×Dic511yescharacteristic
215C23×C4C2×Dic511yescharacteristic
216C23×C4C22⋊D511yescharacteristic
217C2×M16—201no—
218#120—22no—
219#120—22no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—44no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C22⋊D511yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4C2×Dic511yescharacteristic
230C2×C22⋊C4C2×Dic511yescharacteristic
231C2×C22⋊C4C22⋊D511yescharacteristic
232C2×C22⋊C4C22×D511yescharacteristic
233C2×C42—21no—
234C2×C42C22⋊D511yescharacteristic
235C4⋊C8—201no—
236C22⋊C8—102no—
237C22⋊C8—102no—
238C22⋊C8—201no—
239#97—201no—
240#95—21no—
241#95—21no—
242#95—21no—
243C22×C5—21no—
244C22×C5—21no—
245C22×C5—24no—
246C22×C5—21no—
247C22×C5—21no—
248C22×C5—41no—
249C22×C5—41no—
250C22×C5—42no—
251C22×C5—42no—
252C22×C5—42no—
253C22×C5—42no—
254C22×C5—42no—
255C22×C5—44no—
256C22×C5—44no—
257C22×C5—44no—
258C22×C5—44no—
259C22×C5—42no—
260C22×C5—41no—
261C22×C5—42no—
262C22×C5—41no—
263C22×C5—42no—
264C22×C5—41no—
265C22×C5—41no—
266C22×C5—41no—
267C22×C5—81no—
268C22×C5#40711yescharacteristic
269C22×C5C2×#9911yescharacteristic
270C22×C5#41311yescharacteristic
271C4×C5—21no—
272C4×C5—42no—
273C4×C5—41no—
274C4×C5—42no—
275C4×C5—41no—
276C4×C5—41no—
277C4×C5—41no—
278C4×C5—81no—
279C4×C5AΣL(1,𝔽16)12yes—
280C24—21no—
281C24—21no—
282C24—42no—
283C24—41no—
284C24—42no—
285C24—41no—
286C24#68112yescomplemented
287C24C22⋊Dic511yescharacteristic
288C24C22×Dic511yescharacteristic
289C2×D4—24no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—44no—
318C22×C4C22⋊Dic511yescharacteristic
319C22×C4C22⋊Dic511yescharacteristic
320C22×C4C22⋊Dic511yescharacteristic
321C22×C4C2×C22⋊D511yescharacteristic
322C22×C4C2×C22⋊D511yescharacteristic
323M16—402no—
324C2×C8—202no—
325C2×C8—202no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C22⋊C4—24no—
329C22⋊C4—21no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—41no—
336C22⋊C4#68112yescomplemented
337C42—21no—
338C42—21no—
339C42—42no—
340C2×C5—21no—
341C2×C5—21no—
342C2×C5—44no—
343C2×C5—41no—
344C2×C5—41no—
345C2×C5—81no—
346C2×C5#146511yescharacteristic
347C2×C5C2×AΣL(1,𝔽16)11yescharacteristic
348C2×C5#212011yescharacteristic
349C23—22no—
350C23—22no—
351C23—22no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—44no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—44no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—48no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23C2×#68111yescharacteristic
375C23C2×#68111yescharacteristic
376C23C2×C22⋊Dic511yescharacteristic
377D4—48no—
378D4—42no—
379D4—42no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4#396611yescharacteristic
409C2×C4#396611yescharacteristic
410C8—402no—
411C8—402no—
412C5#1152511yescharacteristic, complemented, Sylow
413C22—21no—
414C22—21no—
415C22—24no—
416C22—21no—
417C22—21no—
418C22—41no—
419C22—41no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—42no—
430C22—41no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—81no—
438C22#6549611yescharacteristic
439C22C2×#396611yescharacteristic
440C22#6552611yescharacteristic
441C4—21no—
442C4—42no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—42no—
447C4—41no—
448C4—81no—
449C4#6475612yes—
450C2—21no—
451C2—21no—
452C2—44no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2#1060239511yescharacteristic
457C2C2×#6475611yescharacteristic
458C2#1060250911yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC24×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC4×C5×#120C2
socleC22×C5C2×#99
prime radicalC2×C5×#120C4
cosocleC2×C5×C22⋊C4C23
hypercenterC4×#120D5
hypocenterC5#11525
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#120

Complements

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

NH ≅ G/Ntype
2. #10602396C2semidirect
3. #10602395C2semidirect
4. #10598127C2semidirect
5. #10598113C2semidirect
73. C24×C5M16semidirect
119. C5×C22⋊C4M16semidirect
286. C24#681semidirect
336. C22⋊C4#681semidirect
412. C5#11525semidirect

Maximal subgroups

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

subgroupindex
C4×C5×#1202
#106023962
#106023952
#105981272
#105981132
#115255