Finite Group Databasebeta

← #72259883

Subgroups — #72259883

Order 4800.

896 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 126 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
1#72259867C212yesmaximal, complemented
2C5×#90977C212yesmaximal, complemented
3#72229453C211yesmaximal, characteristic, complemented
4C3×#11717471C211yesmaximal, characteristic
5C4×Dic3×C5⋊D5C211yesmaximal, characteristic, complemented
6#13100574—31nomaximal, Hall
7C2×C3×C4×C5⋊D5C2211yescharacteristic
8C3×C5×#709C2212yes—
9C2×Dic3×C5⋊D5—21no—
10C3×#87886—21no—
11C3×#87880—21no—
12C3×#87877C411yescharacteristic, complemented
13C3×C4×D52—21no—
14C3×#87873—21no—
15C3×#87872C411yescharacteristic, complemented
16C3×#87871C412yescomplemented
17C4×C3⋊#1039C2211yescharacteristic, complemented
18C5×#6710C2212yescomplemented
19C4×C52×Dic3C2211yescharacteristic, complemented
20#11925152—21no—
21#11925150—21no—
22Dic3×#1039—21no—
23D5×#837—52nomaximal
24#11717471D311yescharacteristic
25C42×C5⋊D5—31no—
26#11716936—31no—
27C5×#3937—32no—
28#11716609—32no—
29C22×C3×C5⋊D5D411yescharacteristic
30C2×C3×D52—21no—
31C2×C3×C4×C52C2311yescharacteristic
32C2×C3⋊#1039—21no—
33C2×C52×Dic3—21no—
34C2×C3×#1039D411yescharacteristic
35C3×C4×C5⋊D5C2×C411yescharacteristic
36C3×C4×C5⋊D5C2×C411yescharacteristic
37C3×C4×C5⋊D5D411yescharacteristic
38C3×C4×C5⋊D5D411yescharacteristic
39C3×C5×C22⋊D5—22no—
40C3×C5×D20C2×C412yescomplemented
41C3×C4×C5×D5—22no—
42C3×C5×Dic10C2×C412yescomplemented
43Dic3×C5⋊D5—24no—
44C3×#6039—21no—
45C3×#6038—22no—
46C3×#6037—21no—
47C3×#6036—21no—
48C3×D5×Dic5—22no—
49C5×C3⋊#168—22no—
50C3×D5×#41—52no—
51C4×D5×Dic3—52no—
52C4×D5×Dic3—102no—
53D5×#208—52no—
54#91121—52no—
55C5×#837—52no—
56#90977D512yescomplemented
57#90976—52no—
58C2×C4×C5⋊D5—61no—
59C2×C4×C5⋊D5C2×D311yescharacteristic
60C5×#709C2×D312yes—
61#87886—21no—
62#87880—21no—
63#87877Dic311yescharacteristic, complemented
64C4×D52—21no—
65#87873—21no—
66#87872Dic311yescharacteristic, complemented
67#87871Dic312yescomplemented
68C42×C52—31no—
69C4×#1039—31no—
70C4×#1039—61no—
71C5×#681—32no—
72#87769—61no—
73#87767—61no—
74D5×#104—152no—
75C22×C3×C52C2×D411yescharacteristic
76C2×C3×C5⋊D5—22no—
77C2×C3×C5⋊D5C22⋊C411yescharacteristic
78C2×C3×C5⋊D5C22⋊C411yescharacteristic
79C2×C3×C5×D5—22no—
80C3×D52—42no—
81C3×C4×C52C22×C411yescharacteristic
82C3×C4×C52C2×D411yescharacteristic
83C3⋊#1039—22no—
84C52×Dic3—22no—
85C3×#1039C22⋊C411yescharacteristic
86C3×#1039C22⋊C411yescharacteristic
87C3×C5×Dic5—22no—
88C3×C5×#41—52no—
89C3×#713—52no—
90C3×Q8×D5—52no—
91C3×#711—52no—
92C3×#711—102no—
93C3×D4×D5—52no—
94C3×D4×D5—102no—
95C3×#709—102no—
96C3×#709C2×D512yes—
97C2×C3×C4×D5—52no—
98C2×C3×C4×D5—102no—
99C2×C3×C4×D5—102no—
100C2×D5×Dic3—102no—
101C2×D5×Dic3—104no—
102C4×Dic15—52no—
103C4×Dic15—102no—
104#6710C2×D512yescomplemented
105C4×C5×Dic3—22no—
106C4×C5×Dic3C2×D512yescomplemented
107C5×#208—52no—
108Dic3×Dic5—102no—
109Dic3×Dic5—104no—
110#6649—102no—
111D5×C3⋊C8—102no—
112C22×C5⋊D5#6711yescharacteristic
113C2×D52—21no—
114C2×C4×C52—61no—
115C2×C4×C52C22×D311yescharacteristic
116C2×#1039—61no—
117C2×#1039#6711yescharacteristic
118C4×C5⋊D5—64no—
119C4×C5⋊D5C2×Dic311yescharacteristic
120C4×C5⋊D5C2×Dic311yescharacteristic
121C4×C5⋊D5#6711yescharacteristic
122C4×C5⋊D5#6711yescharacteristic
123C5×C22⋊D5—22no—
124C5×D20C2×Dic312yescomplemented
125C4×C5×D5—22no—
126C5×Dic10C2×Dic312yescomplemented
127#6039—21no—
128#6038—22no—
129#6037—21no—
130#6036—21no—
131D5×Dic5—22no—
132C5×#168—62no—
133C2×#837—251noHall
134D5×#41—52no—
135D5×M16—152no—
136C42×D5—152no—
137C42×D5—302no—
138C5×#104—152no—
139#3969—152no—
140#3957—152no—
141#3937—32no—
142C2×C3×C52—21no—
143C2×C3×C52C2×C22⋊C411yescharacteristic
144C3×C5⋊D5—21no—
145C3×C5⋊D5#10412yescomplemented
146C3×C5×D5—42no—
147C22×C3×D5—52no—
148C22×C3×D5—102no—
149C22×C3×D5—102no—
150C3×C5×Q8—52no—
151C3×C5×D4—52no—
152C3×C5×D4—102no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—102no—
155C2×C3×C4×C5C22×D512yes—
156C2×Dic15—104no—
157C2×Dic15—102no—
158C2×C5×Dic3—24no—
159C2×C5×Dic3—22no—
160C3×C22⋊D5—22no—
161C3×C22⋊D5—102no—
162C3×C22⋊D5—102no—
163C2×C3×Dic5—52no—
164C2×C3×Dic5—102no—
165C2×C3×Dic5—102no—
166C3×D20—102no—
167C3×D20C4×D512yescomplemented
168C3×C4×D5—22no—
169C3×C4×D5—52no—
170C3×C4×D5—52no—
171C3×C4×D5—52no—
172C3×C4×D5—52no—
173C3×C4×D5—102no—
174C3×C4×D5—102no—
175C3×C4×D5—102no—
176C3×C4×D5—102no—
177C3×C4×D5—102no—
178C3×C4×D5—102no—
179C3×Dic10—102no—
180C3×Dic10C4×D512yescomplemented
181D5×Dic3—108no—
182D5×Dic3—1016no—
183C3⋊#168—22no—
184C5×C3⋊C8—102no—
185C22×C52C2×#6711yescharacteristic
186C2×C5⋊D5—22no—
187C2×C5⋊D5#21711yescharacteristic
188C2×C5⋊D5#21711yescharacteristic
189C2×C5×D5—22no—
190D52—42no—
191C4×C52—62no—
192C4×C52C22×Dic311yescharacteristic
193C4×C52C2×#6711yescharacteristic
194#1039—62no—
195#1039#21711yescharacteristic
196#1039#21711yescharacteristic
197C5×Dic5—22no—
198C2×C3×#41—251no—
199C2×C4×Dic3—251no—
200C2×#208—251no—
201#837—52no—
202#837—252no—
203C5×#41—52no—
204#713—52no—
205Q8×D5—52no—
206#711—52no—
207#711—102no—
208D4×D5—52no—
209D4×D5—102no—
210#709—102no—
211#709D3×D512yes—
212C2×C4×D5—52no—
213C2×C4×D5—102no—
214C2×C4×D5—102no—
215C2×C4×D5—302no—
216C2×C4×D5—304no—
217C5×M16—152no—
218C42×C5—32no—
219C42×C5—62no—
220C4×Dic5—152no—
221C4×Dic5—302no—
222C4×Dic5—302no—
223C4×Dic5—304no—
224#681—32no—
225#676—302no—
226C8×D5—302no—
227C3×C52C2×#10411yescharacteristic, complemented, Hall
228C2×#104—751noSylow
229C22×C3×C5—22no—
230C22×C3×C5—102no—
231C22×C3×C5D4×D512yes—
232C2×C3×D5—22no—
233C2×C3×D5—52no—
234C2×C3×D5—52no—
235C2×C3×D5—102no—
236C2×C3×D5—102no—
237C2×C3×D5—104no—
238C2×C3×D5—108no—
239C2×C3×D5—204no—
240C3×C4×C5—22no—
241C3×C4×C5—22no—
242C3×C4×C5—102no—
243C3×C4×C5C2×C4×D512yes—
244C3×C4×C5D4×D512yes—
245Dic15—108no—
246Dic15—104no—
247C3×Dic5—22no—
248C3×Dic5—52no—
249C3×Dic5—52no—
250C3×Dic5—102no—
251C3×Dic5—102no—
252C5×Dic3—28no—
253C5×Dic3—24no—
254C2×C52—21no—
255C2×C52C2×#21711yescharacteristic
256C5⋊D5—21no—
257C5⋊D5#83712yescomplemented
258C5×D5—42no—
259C3×#41—52no—
260C3×#41—252no—
261C3×#41—501no—
262C3×#41—501no—
263C2×C3×Q8—251no—
264C2×C3×D4—251no—
265C2×C3×D4—501no—
266C22×C3×C4—251no—
267C22×C3×C4—501no—
268C22×Dic3—501no—
269C4×Dic3—251no—
270C4×Dic3—501no—
271C4×Dic3D5211yescharacteristic, complemented
272#208—52no—
273#208—501no—
274C2×C3⋊C8—501no—
275C22×D5—52no—
276C22×D5—102no—
277C22×D5—102no—
278C5×Q8—52no—
279C5×D4—52no—
280C5×D4—102no—
281C2×C4×C5—22no—
282C2×C4×C5—64no—
283C2×C4×C5—62no—
284C2×C4×C5—102no—
285C2×C4×C5C2×D3×D512yes—
286C22⋊D5—22no—
287C22⋊D5—102no—
288C22⋊D5—102no—
289C2×Dic5—52no—
290C2×Dic5—102no—
291C2×Dic5—102no—
292C2×Dic5—304no—
293C2×Dic5—302no—
294D20—102no—
295D20D5×Dic312yescomplemented
296C4×D5—22no—
297C4×D5—52no—
298C4×D5—52no—
299C4×D5—52no—
300C4×D5—52no—
301C4×D5—102no—
302C4×D5—102no—
303C4×D5—102no—
304C4×D5—102no—
305C4×D5—102no—
306C4×D5—102no—
307C4×D5—308no—
308C4×D5—3016no—
309Dic10—102no—
310Dic10D5×Dic312yescomplemented
311C5×C8—302no—
312#168—62no—
313C2×#41—251no—
314C2×M16—751no—
315C2×C42—751no—
316#104—152no—
317#104—752no—
318C2×C3×C5—24no—
319C2×C3×C5—22no—
320C2×C3×C5—22no—
321C2×C3×C5—202no—
322C2×C3×C5D5×C22⋊C412yes—
323C3×D5—42no—
324C3×D5—54no—
325C3×D5—104no—
326C3×D5—104no—
327C3×D5—102no—
328C52C2×#83711yescharacteristic, complemented, Sylow
329C23×C3—251no—
330C23×C3—501no—
331C3×Q8—52no—
332C3×Q8—501no—
333C3×D4—52no—
334C3×D4—102no—
335C3×D4—502no—
336C3×D4—501no—
337C2×C3×C4—102no—
338C2×C3×C4—251no—
339C2×C3×C4—251no—
340C2×C3×C4—251no—
341C2×C3×C4—251no—
342C2×C3×C4—251no—
343C2×C3×C4—501no—
344C2×C3×C4—502no—
345C2×C3×C4C2×D5211yescharacteristic
346C2×Dic3—21no—
347C2×Dic3—504no—
348C2×Dic3—501no—
349C3⋊C8—102no—
350C22×C5—22no—
351C22×C5—102no—
352C22×C5D5×#6712yes—
353C2×D5—22no—
354C2×D5—52no—
355C2×D5—52no—
356C2×D5—102no—
357C2×D5—102no—
358C2×D5—104no—
359C2×D5—108no—
360C2×D5—204no—
361C4×C5—22no—
362C4×C5—22no—
363C4×C5—68no—
364C4×C5—64no—
365C4×C5—102no—
366C4×C5C2×D5×Dic312yes—
367C4×C5D5×#6712yes—
368Dic5—22no—
369Dic5—52no—
370Dic5—52no—
371Dic5—102no—
372Dic5—102no—
373Dic5—308no—
374Dic5—304no—
375#41—52no—
376#41—252no—
377#41—501no—
378#41—501no—
379C2×Q8—251no—
380C2×D4—251no—
381C2×D4—501no—
382C22×C4—251no—
383C22×C4—501no—
384C22×C4—1501no—
385M16—152no—
386M16—1501no—
387C2×C8—1501no—
388C42—31no—
389C42—751no—
390C42—1501no—
391C3×C5—22no—
392C3×C5D5×#10412yescomplemented
393C22×C3—102no—
394C22×C3—251no—
395C22×C3—251no—
396C22×C3—502no—
397C22×C3—1002no—
398C22×C3#8788611yescharacteristic
399C3×C4—102no—
400C3×C4—251no—
401C3×C4—251no—
402C3×C4C4×D5211yescharacteristic
403C3×C4#8787711yescharacteristic
404Dic3—22no—
405Dic3—502no—
406C2×C5—24no—
407C2×C5—22no—
408C2×C5—22no—
409C2×C5—202no—
410C2×C5D5×#21712yes—
411D5—42no—
412D5—54no—
413D5—104no—
414D5—104no—
415D5—102no—
416C23—251no—
417C23—501no—
418Q8—52no—
419Q8—501no—
420D4—52no—
421D4—102no—
422D4—502no—
423D4—501no—
424C2×C4—61no—
425C2×C4—102no—
426C2×C4—251no—
427C2×C4—251no—
428C2×C4—251no—
429C2×C4—251no—
430C2×C4—251no—
431C2×C4—501no—
432C2×C4—502no—
433C2×C4—1504no—
434C2×C4—1501no—
435C2×C4D3×D5211yescharacteristic
436C8—302no—
437C2×C3—21no—
438C2×C3—202no—
439C2×C3—252no—
440C2×C3—501no—
441C2×C3#1171703911yescharacteristic
442C5—22no—
443C5D5×#83712yescomplemented
444C22—102no—
445C22—251no—
446C22—251no—
447C22—502no—
448C22—1002no—
449C22#1192541911yescharacteristic
450C4—62no—
451C4—102no—
452C4—251no—
453C4—251no—
454C4—1502no—
455C4D52×Dic311yescharacteristic
456C4#1192542011yescharacteristic
457C3#1310057411yescharacteristic, complemented, Sylow
458C2—21no—
459C2—202no—
460C2—252no—
461C2—501no—
462C2#7225987111yescharacteristic
subgroupG/H
centerC4#11925420
commutatorC3×C4×C52C22×C4
Frattini, hypercenterC2×C4D3×D52
Fitting, cosocleC2×C3×C4×C52C23
socleC2×C3×C52C2×C22⋊C4
prime radicalC3×#11717471C2
hypocenterC3×C52C2×#104
LCSG ▷ C3×C4×C52 ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #72259867C2semidirect
2. C5×#90977C2semidirect
3. #72229453C2semidirect
5. C4×Dic3×C5⋊D5C2semidirect
12. C3×#87877C4semidirect
15. C3×#87872C4semidirect
16. C3×#87871C4semidirect
17. C4×C3⋊#1039C22semidirect
18. C5×#6710C22semidirect
19. C4×C52×Dic3C22semidirect
40. C3×C5×D20C2×C4semidirect
42. C3×C5×Dic10C2×C4semidirect
56. #90977D5semidirect
63. #87877Dic3semidirect
66. #87872Dic3semidirect
67. #87871Dic3semidirect
104. #6710C2×D5semidirect
106. C4×C5×Dic3C2×D5semidirect
124. C5×D20C2×Dic3semidirect
126. C5×Dic10C2×Dic3semidirect
145. C3×C5⋊D5#104semidirect
167. C3×D20C4×D5semidirect
180. C3×Dic10C4×D5semidirect
227. C3×C52C2×#104semidirect
257. C5⋊D5#837semidirect
271. C4×Dic3D52semidirect
295. D20D5×Dic3semidirect
310. Dic10D5×Dic3semidirect
328. C52C2×#837semidirect
392. C3×C5D5×#104semidirect
443. C5D5×#837semidirect
457. C3#13100574semidirect

Maximal subgroups

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

subgroupindex
#722598672
C5×#909772
#722294532
C3×#117174712
C4×Dic3×C5⋊D52
#131005743
D5×#8375