Finite Group Databasebeta

← #10617802⋊C3 #14514059

Subgroups — #10617802⋊C3 #14514059

Order 1920.

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

normal subgroups only

This group has 754 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×#4150⋊C3C211yesmaximal, characteristic, complemented
2C2×D5×#41⋊C3C211yesmaximal, characteristic, complemented
3C5×#520⋊C3C211yesmaximal, characteristic, complemented
4#65910⋊C3C212yesmaximal, complemented
5SL(2,𝔽3)×D20C212yesmaximal, complemented
6#10617802C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3C2211yescharacteristic
8#4150⋊C3—22no—
9D5×#41⋊C3—22no—
10C2×D5×SL(2,𝔽3)C2211yescharacteristic
11C2×D5×SL(2,𝔽3)C2212yescomplemented
12C2×C5⋊(#41⋊C3)C2211yescharacteristic
13C3×#4041—41nomaximal
14C4×C5×SL(2,𝔽3)C2212yescomplemented
15#3489⋊C3—51nomaximal, Hall
16C2×#4150C2×C311yescharacteristic, complemented
17C2×D5×#41C2×C311yescharacteristic, complemented
18C5×#520C2×C311yescharacteristic, complemented
19#65910C2×C312yescomplemented
20Q8×D20C2×C312yescomplemented
21#65889—32no—
22#65888—32no—
23#65860—31no—
24#65858—31no—
25#65844—31no—
26C2×#4043—31no—
27C2×C3×D20—41no—
28C2×C3×D20—42no—
29C2×C3×C4×D5—41no—
30C5×#41⋊C3D412yescomplemented
31C2×C5×SL(2,𝔽3)C2311yescharacteristic
32D5×SL(2,𝔽3)—21no—
33D5×SL(2,𝔽3)—22no—
34C5⋊(#41⋊C3)—21no—
35C3×C5×C4⋊C4—41no—
36C3×#685—42no—
37C2×2−1+4⋊C3—51no—
38C22×#41⋊C3—51no—
39D4×SL(2,𝔽3)—52no—
40#548⋊C3—52no—
41#520⋊C3D511yescharacteristic, complemented
42C2×C5×#41C22×C311yescharacteristic
43#4150—22no—
44#4150—62no—
45D5×#41—22no—
46D5×#41—62no—
47C2×#713C22×C311yescharacteristic
48C2×Q8×D5C22×C311yescharacteristic
49C2×Q8×D5C22×C312yes—
50C2×#711—32no—
51C2×#711—31no—
52C2×D4×D5—31no—
53C2×#709—31no—
54C2×#709—31no—
55C22×C4×D5—31no—
56C22×Dic10—31no—
57C4×C5×Q8C22×C312yes—
58C4×C5×D4—32no—
59C5×#117—31no—
60C2×C5×C4⋊C4—31no—
61#4076—34no—
62C2×#685—32no—
63C2×C4⋊Dic5—32no—
64#4043—32no—
65#4043—31no—
66#4043—31no—
67#4043—32no—
68#4043—32no—
69#4043—32no—
70#4041—32no—
71#4041—31no—
72#4041A411yescharacteristic
73#4032—34no—
74#4032—32no—
75#4028—32no—
76#4021—32no—
77C4×D20—32no—
78C4×D20—32no—
79#4015—32no—
80#3489—51noSylow
81C22×C3×D5—41no—
82C22×C3×D5—42no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—42no—
85C2×C3×Dic5—41no—
86C3×D20—82no—
87C3×D20—84no—
88C3×C4×D5—82no—
89C5×SL(2,𝔽3)C2×D411yescharacteristic
902−1+4⋊C3—102no—
91C2×#41⋊C3—51no—
92C2×#41⋊C3—102no—
93C2×#41⋊C3C2×D511yescharacteristic
94C22×SL(2,𝔽3)—51no—
95C22×SL(2,𝔽3)—52no—
96C3×#121—201no—
97C4×SL(2,𝔽3)C2×D512yescomplemented
98C23×D5—31no—
99C5×#41—32no—
100C5×#41C3×D412yescomplemented
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—32no—
108C22×Dic5—31no—
109#713—21no—
110#713—61no—
111Q8×D5—21no—
112Q8×D5—22no—
113Q8×D5—61no—
114Q8×D5—62no—
115#711—62no—
116#711—64no—
117#711—64no—
118#711—62no—
119D4×D5—62no—
120D4×D5—62no—
121#709—62no—
122#709—62no—
123#709—62no—
124#709—62no—
125C2×D20—31no—
126C2×D20—32no—
127C2×D20C2×A411yescharacteristic
128C2×D20C2×A412yes—
129C2×C4×D5—31no—
130C2×C4×D5—32no—
131C2×C4×D5—31no—
132C2×C4×D5—31no—
133C2×C4×D5—62no—
134C2×C4×D5—62no—
135C2×C4×D5C2×A411yescharacteristic
136C2×Dic10—31no—
137C2×Dic10—32no—
138C2×Dic10—31no—
139C2×Dic10—62no—
140C2×Dic10—62no—
141C5×C4⋊C4—32no—
142C5×C4⋊C4—31no—
143C5×C4⋊C4C2×A411yescharacteristic
144C5×C22⋊C4—32no—
145C42×C5—32no—
146#685—34no—
147#685—32no—
148#685—32no—
149#685—32no—
150#685C2×A412yes—
151C4⋊Dic5—34no—
152C4⋊Dic5—32no—
153C4⋊Dic5—32no—
154C2×2−1+4—51no—
155C22×#41—51no—
156D4×Q8—52no—
157#548—52no—
158#547—152no—
159#546—152no—
160#537—151no—
161#536—151no—
162#525—151no—
163C2×C22⋊Q8—151no—
164#520C3×D511yescharacteristic, complemented
165C22×C3×C5—41no—
166C2×C3×D5—42no—
167C2×C3×D5—81no—
168C2×C3×D5—81no—
169C2×C3×D5—82no—
170C2×C3×D5—82no—
171C2×C3×D5—82no—
172C3×C4×C5—42no—
173C3×C4×C5—82no—
174C3×Dic5—81no—
175C2×C3×D4—201no—
176C2×C3×D4—202no—
177C22×C3×C4—201no—
178#41⋊C3—101no—
179#41⋊C3D2012yescomplemented
180C2×SL(2,𝔽3)—101no—
181C2×SL(2,𝔽3)—102no—
182C2×SL(2,𝔽3)C22×D511yescharacteristic
183C3×C4⋊C4—41no—
184C3×C22⋊C4—202no—
185C23×C5—31no—
186C22×D5—31no—
187C22×D5—62no—
188C22×D5—62no—
189C22×D5—121no—
190C22×D5C22×A412yes—
191C22×D5C22×A411yescharacteristic
192C5×Q8—31no—
193C5×Q8C2×C3×D411yescharacteristic
194C5×D4—32no—
195C5×D4—32no—
196C2×C4×C5—31no—
197C2×C4×C5—32no—
198C2×C4×C5—32no—
199C2×C4×C5—32no—
200C2×C4×C5—122no—
201C2×C4×C5C22×A412yes—
202C2×C4×C5C22×A411yescharacteristic
203C22⋊D5—64no—
204C22⋊D5—64no—
205C2×Dic5—31no—
206C2×Dic5—32no—
207C2×Dic5—62no—
208C2×Dic5—64no—
209C2×Dic5C22×A411yescharacteristic
210D20—24no—
211D20—22no—
212D20—61no—
213D20—61no—
214D20—122no—
215C4×D5—22no—
216C4×D5—61no—
217C4×D5—61no—
218C4×D5—61no—
219C4×D5—61no—
220C4×D5—62no—
221C4×D5—62no—
222C4×D5—62no—
223Dic10—62no—
224Dic10—62no—
225Dic10—62no—
226Dic10—61no—
227Dic10—61no—
2282−1+4—102no—
2292−1+4—302no—
230C2×#41—51no—
231C2×#41—102no—
232C2×#41—151no—
233C2×#41—151no—
234C2×#41—152no—
235C2×#41—151no—
236C2×#41—302no—
237C2×#41C2×C3×D511yescharacteristic
238C22×Q8—52no—
239C22×Q8—51no—
240C22×Q8—151no—
241C22×D4—151no—
242C23×C4—151no—
243C4⋊Q8—152no—
244#124—152no—
245#123—152no—
246#123—154no—
247C22⋊Q8—151no—
248C22⋊Q8—152no—
249C22⋊Q8—152no—
250C22⋊Q8—151no—
251C22⋊Q8—152no—
252C22⋊Q8—152no—
253#121—51no—
254#121—151no—
255#121—152no—
256#121—154no—
257#120—152no—
258C4×Q8C2×C3×D512yes—
259C4×D4—32no—
260C4×D4—152no—
261C4×D4—152no—
262#117—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—152no—
265C2×C22⋊C4—152no—
266C2×C3×C5—41no—
267C2×C3×C5—41no—
268C2×C3×C5—41no—
269C3×D5—82no—
270C3×D5—162no—
271C23×C3—201no—
272C23×C3—202no—
273C3×D4—402no—
274C3×D4—404no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—201no—
278C2×C3×C4—402no—
279SL(2,𝔽3)C2×D2011yescharacteristic
280C22×C5—32no—
281C22×C5—61no—
282C22×C5—61no—
283C22×C5C23×A411yescharacteristic
284C2×D5—21no—
285C2×D5—21no—
286C2×D5—22no—
287C2×D5—22no—
288C2×D5—22no—
289C2×D5—61no—
290C2×D5—61no—
291C2×D5—61no—
292C2×D5—124no—
293C2×D52−1+4⋊C312yescomplemented
294C4×C5—22no—
295C4×C5—31no—
296C4×C5—31no—
297C4×C5—62no—
298C4×C5D4×A412yes—
299Dic5—21no—
300Dic5—62no—
301Dic5—61no—
302C24—151no—
303#41—32no—
304#41—101no—
305#41—302no—
306#41—302no—
307#41—302no—
308#41—302no—
309#41—302no—
310#41—304no—
311#41—304no—
312#41—302no—
313#41—301no—
314#41C3×D2012yescomplemented
315C2×Q8—101no—
316C2×Q8—102no—
317C2×Q8—152no—
318C2×Q8—151no—
319C2×Q8—151no—
320C2×Q8—302no—
321C2×Q8—302no—
322C2×Q8—301no—
323C2×Q8—302no—
324C2×Q8C22×C3×D511yescharacteristic
325C2×D4—31no—
326C2×D4—52no—
327C2×D4—51no—
328C2×D4—151no—
329C2×D4—152no—
330C2×D4—152no—
331C2×D4—152no—
332C2×D4—302no—
333C2×D4—302no—
334C22×C4—31no—
335C22×C4—32no—
336C22×C4—51no—
337C22×C4—151no—
338C22×C4—152no—
339C22×C4—151no—
340C22×C4—151no—
341C22×C4—152no—
342C22×C4—151no—
343C22×C4—302no—
344C22×C4—302no—
345C4⋊C4—32no—
346C4⋊C4—31no—
347C4⋊C4—152no—
348C4⋊C4—154no—
349C4⋊C4—152no—
350C4⋊C4D5×A411yescharacteristic
351C22⋊C4—32no—
352C22⋊C4—52no—
353C22⋊C4—152no—
354C22⋊C4—152no—
355C22⋊C4—154no—
356C22⋊C4—152no—
357C42—32no—
358C3×C5—41noHall
359C22×C3—41no—
360C22×C3—202no—
361C22×C3—401no—
362C22×C3—401no—
363C22×C3—402no—
364C22×C3—402no—
365C22×C3—402no—
366C3×C4—42no—
367C3×C4—82no—
368C3×C4—401no—
369C2×C5—62no—
370C2×C5C2×D4×A411yescharacteristic
371C2×C5C2×2−1+4⋊C311yescharacteristic
372C2×C5#594511yescharacteristic
373D5—22no—
374D5—42no—
375D5—121no—
376C23—31no—
377C23—51no—
378C23—52no—
379C23—151no—
380C23—302no—
381C23—302no—
382C23—601no—
383Q8—31no—
384Q8—302no—
385Q8—301no—
386Q8—302no—
387Q8—301no—
388Q8—302no—
389Q8C2×C3×D2011yescharacteristic
390D4—32no—
391D4—32no—
392D4—104no—
393D4—102no—
394D4—301no—
395D4—301no—
396D4—304no—
397D4—304no—
398D4—602no—
399C2×C4—31no—
400C2×C4—32no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—51no—
404C2×C4—102no—
405C2×C4—122no—
406C2×C4—152no—
407C2×C4—151no—
408C2×C4—301no—
409C2×C4—301no—
410C2×C4—301no—
411C2×C4—301no—
412C2×C4—302no—
413C2×C4—302no—
414C2×C4—302no—
415C2×C4—304no—
416C2×C4—302no—
417C2×C4C2×D5×A412yes—
418C2×C4C2×D5×A411yescharacteristic
419C2×C3—41no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—402no—
423C2×C3—802no—
424C5#3489⋊C311yescharacteristic, complemented, Sylow
425C22—32no—
426C22—52no—
427C22—61no—
428C22—61no—
429C22—101no—
430C22—101no—
431C22—102no—
432C22—102no—
433C22—102no—
434C22—301no—
435C22—301no—
436C22—301no—
437C22—604no—
438C22C22×D5×A411yescharacteristic
439C4—22no—
440C4—31no—
441C4—31no—
442C4—62no—
443C4—101no—
444C4—301no—
445C4—302no—
446C4A4×D2012yes—
447C3—41noSylow
448C2—62no—
449C2—102no—
450C2—202no—
451C2—601no—
452C2C2×A4×D2011yescharacteristic
453C2#1175694411yescharacteristic
454C2D5×2−1+4⋊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×#4150⋊C3C2semidirect
2. C2×D5×#41⋊C3C2semidirect
3. C5×#520⋊C3C2semidirect
4. #65910⋊C3C2semidirect
5. SL(2,𝔽3)×D20C2semidirect
6. #10617802C3semidirect
11. C2×D5×SL(2,𝔽3)C22semidirect
14. C4×C5×SL(2,𝔽3)C22semidirect
16. C2×#4150C2×C3semidirect
17. C2×D5×#41C2×C3semidirect
18. C5×#520C2×C3semidirect
19. #65910C2×C3semidirect
20. Q8×D20C2×C3semidirect
30. C5×#41⋊C3D4semidirect
41. #520⋊C3D5semidirect
97. C4×SL(2,𝔽3)C2×D5semidirect
100. C5×#41C3×D4semidirect
164. #520C3×D5semidirect
179. #41⋊C3D20semidirect
293. C2×D52−1+4⋊C3semidirect
314. #41C3×D20semidirect
424. C5#3489⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#4150⋊C32
C2×D5×#41⋊C32
C5×#520⋊C32
#65910⋊C32
SL(2,𝔽3)×D202
#106178023
C3×#40414
#3489⋊C35