Finite Group Databasebeta

← #10617801⋊C3 #14514060

Subgroups — #10617801⋊C3 #14514060

Order 1920.

766 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×D5×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×C5⋊#994C211yesmaximal, characteristic, complemented
3C5×#520⋊C3C211yesmaximal, characteristic, complemented
4#65910⋊C3C212yesmaximal, complemented
5C4⋊(C5⋊(#41⋊C3))C212yesmaximal, complemented
6#10617801C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3C2211yescharacteristic
8D5×#41⋊C3—22no—
9C5⋊#994—22no—
10C2×D5×SL(2,𝔽3)C2211yescharacteristic, complemented
11C2×C5⋊(#41⋊C3)C2212yescomplemented
12C2×C5⋊(#41⋊C3)C2211yescharacteristic, complemented
13C3×#4043—41nomaximal
14C4×C5×SL(2,𝔽3)C2212yescomplemented
15#3488⋊C3—51nomaximal, Hall
16C2×#4149C2×C311yescharacteristic, complemented
17C2×D5×#41C2×C311yescharacteristic
18C5×#520C2×C311yescharacteristic, complemented
19#65911C2×C312yescomplemented
20#65910C2×C312yes—
21#65888—32no—
22D4×D20—32no—
23#65859—31no—
24#65858—31no—
25#65844—31no—
26C2×#4041—31no—
27C2×C3×C4×D5—41no—
28C2×C3×Dic10—41no—
29C5×#41⋊C3D412yescomplemented
30C2×C5×SL(2,𝔽3)C2311yescharacteristic
31D5×SL(2,𝔽3)—21no—
32C5⋊(#41⋊C3)—22no—
33C5⋊(#41⋊C3)—21no—
34C3×C5×C4⋊C4—41no—
35C3×#685—42no—
36C3×C4⋊Dic5—42no—
37C2×#994—51no—
38C22×#41⋊C3—51no—
39C4⋊(#41⋊C3)—52no—
40#548⋊C3—52no—
41#520⋊C3D511yescharacteristic, complemented
42C2×C5×#41C22×C311yescharacteristic
43#4149—22no—
44#4149—62no—
45D5×#41—22no—
46D5×#41—62no—
47C2×#713C22×C312yes—
48C2×#713C22×C311yescharacteristic
49C2×Q8×D5C22×C311yescharacteristic
50C2×#711—31no—
51C2×D4×D5—31no—
52C2×D4×D5—32no—
53C2×#709—31no—
54C2×#709—31no—
55C22×D20—32no—
56C22×D20—31no—
57C22×C4×D5—31no—
58C4×C5×Q8C22×C312yes—
59C4×C5×D4—32no—
60C5×#117—31no—
61C2×C5×C4⋊C4—31no—
62#4076—34no—
63C2×#685—32no—
64#4043—31no—
65#4043—32no—
66#4043A411yescharacteristic
67#4041—32no—
68#4041—32no—
69#4041—31no—
70#4041—31no—
71#4041—32no—
72#4041—32no—
73#4032—32no—
74#4028—34no—
75#4028—32no—
76#4021—32no—
77#4020—32no—
78C4×D20—32no—
79C4×D20—32no—
80#3488—51noSylow
81C22×C3×D5—41no—
82C2×C3×C4×C5—41no—
83C2×C3×C4×C5—42no—
84C2×C3×Dic5—41no—
85C2×C3×Dic5—42no—
86C3×C4×D5—82no—
87C3×Dic10—82no—
88C5×SL(2,𝔽3)C2×D411yescharacteristic, complemented
89#994—102no—
90C2×#41⋊C3—51no—
91C2×#41⋊C3—52no—
92C2×#41⋊C3—102no—
93C2×#41⋊C3C2×D511yescharacteristic
94C22×SL(2,𝔽3)—51no—
95C3×C22⋊Q8—201no—
96C4×SL(2,𝔽3)C2×D512yescomplemented
97C23×D5—32no—
98C23×D5—31no—
99C5×#41—32no—
100C5×#41C3×D412yes—
101C2×C5×Q8C23×C311yescharacteristic
102C2×C5×D4—31no—
103C22×C4×C5—31no—
104C22×C4×C5—32no—
105C2×C22⋊D5—32no—
106C2×C22⋊D5—32no—
107C22×Dic5—31no—
108#713—21no—
109#713—22no—
110#713—61no—
111#713—62no—
112Q8×D5—21no—
113Q8×D5—61no—
114#711—62no—
115#711—62no—
116D4×D5—62no—
117D4×D5—62no—
118D4×D5—64no—
119D4×D5—64no—
120#709—62no—
121#709—62no—
122#709—62no—
123#709—62no—
124C2×D20—32no—
125C2×D20—31no—
126C2×D20—32no—
127C2×D20—31no—
128C2×D20—34no—
129C2×D20—32no—
130C2×D20—62no—
131C2×D20—62no—
132C2×D20—124no—
133C2×C4×D5—31no—
134C2×C4×D5—31no—
135C2×C4×D5—31no—
136C2×C4×D5—32no—
137C2×C4×D5—62no—
138C2×C4×D5—62no—
139C2×C4×D5C2×A411yescharacteristic
140C2×Dic10—31no—
141C2×Dic10C2×A411yescharacteristic
142C5×C4⋊C4—32no—
143C5×C4⋊C4—31no—
144C5×C4⋊C4C2×A411yescharacteristic
145C5×C22⋊C4—32no—
146C42×C5—32no—
147#685—34no—
148#685—32no—
149#685—32no—
150#685—32no—
151#685C2×A412yes—
152C4⋊Dic5—32no—
153C4⋊Dic5C2×A412yes—
154C2×H(5,𝔽2)—51no—
155C22×#41—51no—
156#550—52no—
157#548—52no—
158#546—152no—
159D42—152no—
160#537—151no—
161#535—151no—
162#525—151no—
163C2×#121—151no—
164#520C3×D511yescharacteristic, complemented
165C22×C3×C5—41no—
166C2×C3×D5—42no—
167C2×C3×D5—81no—
168C2×C3×D5—81no—
169C3×C4×C5—42no—
170C3×C4×C5—82no—
171C3×Dic5—82no—
172C3×Dic5—81no—
173C2×C3×Q8—201no—
174C22×C3×C4—201no—
175#41⋊C3—101no—
176#41⋊C3—102no—
177#41⋊C3D2012yescomplemented
178C2×SL(2,𝔽3)—101no—
179C2×SL(2,𝔽3)C22×D511yescharacteristic
180C3×C4⋊C4—41no—
181C3×C4⋊C4—202no—
182C3×C22⋊C4—202no—
183C23×C5—31no—
184C22×D5—32no—
185C22×D5—31no—
186C22×D5—62no—
187C22×D5—64no—
188C22×D5—62no—
189C22×D5—121no—
190C22×D5—122no—
191C22×D5—122no—
192C22×D5C22×A411yescharacteristic
193C5×Q8—31no—
194C5×Q8C2×C3×D411yescharacteristic
195C5×D4—32no—
196C5×D4—32no—
197C2×C4×C5—31no—
198C2×C4×C5—32no—
199C2×C4×C5—32no—
200C2×C4×C5—32no—
201C2×C4×C5—122no—
202C2×C4×C5C22×A412yes—
203C2×C4×C5C22×A411yescharacteristic
204C22⋊D5—64no—
205C22⋊D5—64no—
206C2×Dic5—31no—
207C2×Dic5—62no—
208C2×Dic5C22×A412yes—
209C2×Dic5C22×A411yescharacteristic
210D20—62no—
211D20—62no—
212D20—61no—
213D20—61no—
214D20—64no—
215D20—62no—
216D20—122no—
217D20—124no—
218C4×D5—22no—
219C4×D5—61no—
220C4×D5—61no—
221C4×D5—62no—
222C4×D5—62no—
223C4×D5—61no—
224C4×D5—61no—
225C4×D5—62no—
226Dic10—22no—
227Dic10—61no—
228Dic10—61no—
229H(5,𝔽2)—102no—
230H(5,𝔽2)—302no—
231C2×#41—52no—
232C2×#41—51no—
233C2×#41—102no—
234C2×#41—151no—
235C2×#41—151no—
236C2×#41—151no—
237C2×#41—302no—
238C2×#41C2×C3×D511yescharacteristic
239C22×Q8—51no—
240C22×D4—152no—
241C22×D4—151no—
242C22×D4—151no—
243C22×D4—152no—
244C23×C4—151no—
245#127—152no—
246#124—152no—
247#123—152no—
248C22⋊Q8—51no—
249C22⋊Q8—151no—
250C22⋊Q8—152no—
251#121—151no—
252#121—152no—
253#121—152no—
254#121—154no—
255#121—151no—
256#121—152no—
257#121—152no—
258#120—152no—
259#120—154no—
260C4×Q8C2×C3×D512yes—
261C4×D4—32no—
262C4×D4—152no—
263C4×D4—152no—
264#117—31no—
265C2×C4⋊C4—31no—
266C2×C22⋊C4—152no—
267C2×C3×C5—41no—
268C2×C3×C5—41no—
269C2×C3×C5—41no—
270C3×D5—82no—
271C23×C3—201no—
272C3×Q8—402no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—202no—
276C2×C3×C4—201no—
277C2×C3×C4—402no—
278SL(2,𝔽3)C2×D2011yescharacteristic, complemented
279C22×C5—32no—
280C22×C5—61no—
281C22×C5—61no—
282C22×C5C23×A411yescharacteristic
283C2×D5—21no—
284C2×D5—21no—
285C2×D5—61no—
286C2×D5—62no—
287C2×D5—62no—
288C2×D5—61no—
289C2×D5—61no—
290C2×D5—62no—
291C2×D5—124no—
292C2×D5—244no—
293C2×D5#99412yescomplemented
294C4×C5—22no—
295C4×C5—31no—
296C4×C5—31no—
297C4×C5—62no—
298C4×C5D4×A412yes—
299Dic5—22no—
300Dic5—21no—
301Dic5—61no—
302C24—151no—
303C24—152no—
304#41—32no—
305#41—101no—
306#41—102no—
307#41—302no—
308#41—302no—
309#41—302no—
310#41—302no—
311#41—302no—
312#41—302no—
313#41—301no—
314#41—302no—
315#41C3×D2012yes—
316C2×Q8—51no—
317C2×Q8—101no—
318C2×Q8—151no—
319C2×Q8—301no—
320C2×Q8C22×C3×D511yescharacteristic
321C2×D4—31no—
322C2×D4—152no—
323C2×D4—152no—
324C2×D4—151no—
325C2×D4—154no—
326C2×D4—152no—
327C2×D4—152no—
328C2×D4—151no—
329C2×D4—152no—
330C2×D4—302no—
331C2×D4—302no—
332C2×D4—302no—
333C2×D4—302no—
334C2×D4—304no—
335C2×D4—304no—
336C2×D4—604no—
337C22×C4—31no—
338C22×C4—32no—
339C22×C4—51no—
340C22×C4—151no—
341C22×C4—152no—
342C22×C4—151no—
343C22×C4—151no—
344C22×C4—151no—
345C22×C4—302no—
346C22×C4—302no—
347C4⋊C4—32no—
348C4⋊C4—31no—
349C4⋊C4—52no—
350C4⋊C4—152no—
351C4⋊C4D5×A411yescharacteristic
352C22⋊C4—32no—
353C22⋊C4—52no—
354C22⋊C4—152no—
355C22⋊C4—152no—
356C22⋊C4—154no—
357C22⋊C4—152no—
358C42—32no—
359C3×C5—41noHall
360C22×C3—41no—
361C22×C3—202no—
362C22×C3—401no—
363C22×C3—401no—
364C3×C4—42no—
365C3×C4—82no—
366C3×C4—401no—
367C3×C4—402no—
368C2×C5—62no—
369C2×C5C2×D4×A411yescharacteristic
370C2×C5C2×#99411yescharacteristic
371C2×C5#594511yescharacteristic
372D5—22no—
373D5—121no—
374D5—122no—
375C23—31no—
376C23—51no—
377C23—152no—
378C23—151no—
379C23—302no—
380C23—304no—
381C23—302no—
382C23—601no—
383C23—602no—
384C23—602no—
385Q8—31no—
386Q8—102no—
387Q8—301no—
388Q8—301no—
389Q8C2×C3×D2011yescharacteristic
390D4—32no—
391D4—32no—
392D4—302no—
393D4—302no—
394D4—301no—
395D4—301no—
396D4—304no—
397D4—304no—
398D4—304no—
399D4—302no—
400D4—602no—
401D4—604no—
402C2×C4—31no—
403C2×C4—32no—
404C2×C4—32no—
405C2×C4—32no—
406C2×C4—52no—
407C2×C4—51no—
408C2×C4—102no—
409C2×C4—122no—
410C2×C4—151no—
411C2×C4—301no—
412C2×C4—301no—
413C2×C4—302no—
414C2×C4—302no—
415C2×C4—301no—
416C2×C4—301no—
417C2×C4—302no—
418C2×C4—302no—
419C2×C4C2×D5×A412yes—
420C2×C4C2×D5×A411yescharacteristic
421C2×C3—41no—
422C2×C3—41no—
423C2×C3—41no—
424C2×C3—402no—
425C5#3488⋊C311yescharacteristic, complemented, Sylow
426C22—32no—
427C22—52no—
428C22—61no—
429C22—61no—
430C22—101no—
431C22—101no—
432C22—301no—
433C22—302no—
434C22—302no—
435C22—301no—
436C22—301no—
437C22—302no—
438C22—604no—
439C22—1204no—
440C22C22×D5×A411yescharacteristic
441C4—22no—
442C4—31no—
443C4—31no—
444C4—62no—
445C4—101no—
446C4—102no—
447C4—301no—
448C4A4×D2012yes—
449C3—41noSylow
450C2—62no—
451C2—102no—
452C2—601no—
453C2—602no—
454C2C2×A4×D2011yescharacteristic
455C2D5×#99411yescharacteristic
456C2#66287⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC5×#520C2×C3
socleC22×C5C23×A4
hypercenterC4⋊C4D5×A4
hypocenterC5×Q8C2×C3×D4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D5×#41⋊C3C2semidirect
2. C2×C5⋊#994C2semidirect
3. C5×#520⋊C3C2semidirect
4. #65910⋊C3C2semidirect
5. C4⋊(C5⋊(#41⋊C3))C2semidirect
6. #10617801C3semidirect
10. C2×D5×SL(2,𝔽3)C22semidirect
11. C2×C5⋊(#41⋊C3)C22semidirect
12. C2×C5⋊(#41⋊C3)C22semidirect
14. C4×C5×SL(2,𝔽3)C22semidirect
16. C2×#4149C2×C3semidirect
18. C5×#520C2×C3semidirect
19. #65911C2×C3semidirect
29. C5×#41⋊C3D4semidirect
41. #520⋊C3D5semidirect
88. C5×SL(2,𝔽3)C2×D4semidirect
96. C4×SL(2,𝔽3)C2×D5semidirect
164. #520C3×D5semidirect
177. #41⋊C3D20semidirect
278. SL(2,𝔽3)C2×D20semidirect
293. C2×D5#994semidirect
425. C5#3488⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D5×#41⋊C32
C2×C5⋊#9942
C5×#520⋊C32
#65910⋊C32
C4⋊(C5⋊(#41⋊C3))2
#106178013
C3×#40434
#3488⋊C35