Finite Group Databasebeta

← #10619152⋊C3 #14514203

Subgroups — #10619152⋊C3 #14514203

Order 1920.

804 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C23×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#4147⋊C3C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×C22⋊D5C212yesmaximal, complemented
4#66146⋊C3C211yesmaximal, characteristic, complemented
5#66151⋊C3C212yesmaximal, complemented
6#10619152C311yesmaximal, characteristic, complemented, Hall
7C22×C5×SL(2,𝔽3)—22no—
8C22×C5×SL(2,𝔽3)C2212yes—
9C22×C5×SL(2,𝔽3)C2211yescharacteristic
10#4147⋊C3—22no—
11C2×D5×SL(2,𝔽3)C2211yescharacteristic, complemented
12C2×C5⋊(#41⋊C3)C2211yescharacteristic, complemented
13C3×#4099—41nomaximal
14Dic5×SL(2,𝔽3)C2212yescomplemented
15#3486⋊C3—51nomaximal, Hall
16C23×C5×Q8C2×C311yescharacteristic, complemented
17C2×#4147C2×C311yescharacteristic, complemented
18#66151—32no—
19#66151C2×C312yescomplemented
20#66150—31no—
21#66150—31no—
22Q8×C22⋊D5—32no—
23Q8×C22⋊D5C2×C312yescomplemented
24#66146C2×C311yescharacteristic, complemented
25#66125—31no—
26#66125—31no—
27C24×C3×C5—41no—
28C2×C3×C22⋊D5—42no—
29C2×C3×C22⋊D5—41no—
30C2×C5×SL(2,𝔽3)—22no—
31C2×C5×SL(2,𝔽3)D412yescomplemented
32C2×C5×SL(2,𝔽3)C2311yescharacteristic
33D5×SL(2,𝔽3)—21no—
34C5⋊(#41⋊C3)—21no—
35C3×C22⋊Dic5—42no—
36C3×C22⋊Dic5—41no—
37C2×2−1+4⋊C3—51no—
38C23×SL(2,𝔽3)D511yescharacteristic, complemented
39D4×SL(2,𝔽3)—52no—
40#548⋊C3—52no—
41#519⋊C3—51no—
42C22×C5×Q8—22no—
43C22×C5×Q8—32no—
44C22×C5×Q8—31no—
45C22×C5×Q8—62no—
46C22×C5×Q8C22×C312yes—
47C22×C5×Q8C22×C311yescharacteristic
48C23×C4×C5—31no—
49#4147—22no—
50#4147—62no—
51C2×#713—31no—
52C2×#713C22×C311yescharacteristic, complemented
53C2×Q8×D5—31no—
54C2×Q8×D5C22×C311yescharacteristic, complemented
55C2×#709—31no—
56C2×#709—31no—
57#4099—31no—
58#4099A411yescharacteristic
59#4093—32no—
60#4093—32no—
61#4092—34no—
62#4092—32no—
63#4092—32no—
64Q8×Dic5—32no—
65Q8×Dic5C22×C312yescomplemented
66#4090—32no—
67#4090—32no—
68#4076—32no—
69#4076—32no—
70#4075—32no—
71#4075—31no—
72#4075—31no—
73C4×C22⋊D5—34no—
74C4×C22⋊D5—32no—
75C4×C22⋊D5—32no—
76#4072—31no—
77#4072—31no—
78#4070—32no—
79#4070—32no—
80#3486—51noSylow
81C23×C3×C5—41no—
82C23×C3×C5—42no—
83C23×C3×C5—82no—
84C23×C3×C5—84no—
85C22×C3×D5—41no—
86C3×C22⋊D5—82no—
87C3×C22⋊D5—84no—
88C2×C3×Dic5—42no—
89C2×C3×Dic5—41no—
90C5×SL(2,𝔽3)C2×D411yescharacteristic
912−1+4⋊C3—102no—
92C2×#41⋊C3—51no—
93C22×SL(2,𝔽3)—22no—
94C22×SL(2,𝔽3)—51no—
95C22×SL(2,𝔽3)C2×D512yes—
96C22×SL(2,𝔽3)C2×D511yescharacteristic
97C3×#120—201no—
98C4×SL(2,𝔽3)—52no—
99C24×C5C2×A411yescharacteristic
100C2×C5×Q8—22no—
101C2×C5×Q8—32no—
102C2×C5×Q8—31no—
103C2×C5×Q8—32no—
104C2×C5×Q8—32no—
105C2×C5×Q8—62no—
106C2×C5×Q8—62no—
107C2×C5×Q8—64no—
108C2×C5×Q8—62no—
109C2×C5×Q8—64no—
110C2×C5×Q8C3×D412yescomplemented
111C2×C5×Q8C23×C311yescharacteristic
112C22×C4×C5—31no—
113C22×C4×C5—32no—
114C22×C4×C5—32no—
115C22×C4×C5—31no—
116C22×C4×C5—62no—
117C22×C4×C5—62no—
118C2×C22⋊D5—31no—
119C2×C22⋊D5—32no—
120C2×C22⋊D5C2×A411yescharacteristic
121C2×C22⋊D5C2×A412yes—
122#713—21no—
123#713—62no—
124#713—62no—
125#713—61no—
126Q8×D5—21no—
127Q8×D5—62no—
128Q8×D5—62no—
129Q8×D5—61no—
130#709—62no—
131#709—62no—
132#709—62no—
133#709—62no—
134C2×D20—31no—
135C2×D20—31no—
136C2×C4×D5—32no—
137C2×C4×D5—31no—
138C2×C4×D5—31no—
139C2×Dic10—31no—
140C2×Dic10—31no—
141C22⋊Dic5—32no—
142C22⋊Dic5—31no—
143C22⋊Dic5C2×A412yes—
144C22⋊Dic5C2×A411yescharacteristic
145#685—34no—
146#685—32no—
147#685—32no—
148C4⋊Dic5—32no—
149C4⋊Dic5—32no—
150#683—34no—
151#683—32no—
152#683—32no—
153C4×Dic5—32no—
154C4×Dic5—32no—
155C2×2−1+4—51no—
156C23×Q8C3×D511yescharacteristic, complemented
157D4×Q8—52no—
158D4×Q8—152no—
159#548—52no—
160#548—152no—
161#536—151no—
162#536—151no—
163#525—151no—
164#525—151no—
165#519—51no—
166C22×C3×C5—42no—
167C22×C3×C5—44no—
168C22×C3×C5—41no—
169C22×C3×C5—82no—
170C22×C3×C5—82no—
171C22×C3×C5—82no—
172C22×C3×C5—88no—
173C2×C3×D5—81no—
174C2×C3×D5—82no—
175C3×Dic5—81no—
176C3×Dic5—82no—
177C24×C3—41no—
178C2×C3×D4—201no—
179C2×C3×D4—202no—
180#41⋊C3—101no—
181C2×SL(2,𝔽3)—22no—
182C2×SL(2,𝔽3)—101no—
183C2×SL(2,𝔽3)C22⋊D512yescomplemented
184C2×SL(2,𝔽3)C22×D511yescharacteristic
185C3×C22⋊C4—201no—
186C3×C22⋊C4—202no—
187C23×C5—22no—
188C23×C5—24no—
189C23×C5C22×A411yescharacteristic
190C23×C5C22×A412yes—
191C22×D5—31no—
192C22×D5C22×A411yescharacteristic
193C5×Q8—32no—
194C5×Q8—31no—
195C5×Q8—32no—
196C5×Q8—62no—
197C5×Q8—62no—
198C5×Q8—64no—
199C5×Q8C2×C3×D411yescharacteristic
200C2×C4×C5—32no—
201C2×C4×C5—32no—
202C2×C4×C5—31no—
203C2×C4×C5—31no—
204C2×C4×C5—32no—
205C2×C4×C5—62no—
206C2×C4×C5—62no—
207C2×C4×C5—64no—
208C2×C4×C5—62no—
209C22⋊D5—22no—
210C22⋊D5—24no—
211C22⋊D5—62no—
212C22⋊D5—64no—
213C2×Dic5—32no—
214C2×Dic5—31no—
215C2×Dic5C22×A412yes—
216C2×Dic5C22×A411yescharacteristic
217D20—62no—
218D20—61no—
219D20—61no—
220C4×D5—61no—
221C4×D5—61no—
222C4×D5—61no—
223C4×D5—61no—
224C4×D5—64no—
225Dic10—62no—
226Dic10—61no—
227Dic10—61no—
2282−1+4—102no—
2292−1+4—302no—
230C2×#41—51no—
231C2×#41—151no—
232C2×#41—151no—
233C2×#41—151no—
234C22×Q8—22no—
235C22×Q8—31no—
236C22×Q8—32no—
237C22×Q8—51no—
238C22×Q8—62no—
239C22×Q8—151no—
240C22×Q8C2×C3×D512yes—
241C22×Q8C2×C3×D511yescharacteristic
242C23×C4—31no—
243C4⋊Q8—152no—
244C4⋊Q8—152no—
245#124—152no—
246#124—152no—
247#123—151no—
248#123—151no—
249#123—152no—
250C22⋊Q8—152no—
251C22⋊Q8—154no—
252C22⋊Q8—152no—
253C22⋊Q8—152no—
254C22⋊Q8—152no—
255#121—152no—
256#121—152no—
257#120—51no—
258#120—151no—
259C4×Q8—52no—
260C4×Q8—152no—
261C4×D4—152no—
262C4×D4—152no—
263C4×D4—154no—
264#117—151no—
265#117—151no—
266C2×C3×C5—41no—
267C2×C3×C5—42no—
268C2×C3×C5—82no—
269C2×C3×C5—84no—
270C3×D5—161no—
271C23×C3—42no—
272C23×C3—41no—
273C23×C3—82no—
274C23×C3—84no—
275C23×C3—201no—
276C3×D4—402no—
277C3×D4—404no—
278C2×C3×C4—202no—
279C2×C3×C4—201no—
280SL(2,𝔽3)C2×C22⋊D511yescharacteristic
281C22×C5—28no—
282C22×C5—22no—
283C22×C5—22no—
284C22×C5—22no—
285C22×C5D4×A412yes—
286C22×C52−1+4⋊C314yescomplemented
287C22×C5C23×A411yescharacteristic
288C2×D5—21no—
289C2×D5—22no—
290C2×D5—61no—
291C2×D5—62no—
292C4×C5—32no—
293C4×C5—31no—
294C4×C5—31no—
295C4×C5—62no—
296Dic5—21no—
297Dic5—22no—
298Dic5—61no—
299Dic5—62no—
300C24D5×A411yescharacteristic
301#41—101no—
302#41—302no—
303#41—302no—
304#41—302no—
305#41—302no—
306#41—302no—
307#41—302no—
308#41—301no—
309C2×Q8—22no—
310C2×Q8—32no—
311C2×Q8—31no—
312C2×Q8—32no—
313C2×Q8—32no—
314C2×Q8—62no—
315C2×Q8—62no—
316C2×Q8—64no—
317C2×Q8—62no—
318C2×Q8—64no—
319C2×Q8—101no—
320C2×Q8—151no—
321C2×Q8—151no—
322C2×Q8—302no—
323C2×Q8—302no—
324C2×Q8—301no—
325C2×Q8C3×C22⋊D512yescomplemented
326C2×Q8C22×C3×D511yescharacteristic
327C2×D4—52no—
328C2×D4—51no—
329C2×D4—151no—
330C2×D4—151no—
331C2×D4—152no—
332C2×D4—151no—
333C22×C4—31no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—31no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—151no—
340C22×C4—151no—
341C22×C4—152no—
342C4⋊C4—152no—
343C4⋊C4—152no—
344C4⋊C4—152no—
345C4⋊C4—152no—
346C4⋊C4—154no—
347C22⋊C4—51no—
348C22⋊C4—52no—
349C22⋊C4—152no—
350C22⋊C4—152no—
351C22⋊C4—154no—
352C22⋊C4—151no—
353C22⋊C4—152no—
354C42—152no—
355C42—152no—
356C3×C5—41noHall
357C22×C3—44no—
358C22×C3—42no—
359C22×C3—41no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—88no—
364C22×C3—402no—
365C22×C3—401no—
366C3×C4—402no—
367C3×C4—401no—
368C2×C5—22no—
369C2×C5—24no—
370C2×C5C2×D4×A411yescharacteristic
371C2×C5C2×2−1+4⋊C312yes—
372D5—41no—
373D5—121no—
374C23—24no—
375C23—22no—
376C23—51no—
377C23—151no—
378C23C2×D5×A412yes—
379C23C2×D5×A411yescharacteristic
380Q8—32no—
381Q8—31no—
382Q8—32no—
383Q8—62no—
384Q8—62no—
385Q8—64no—
386Q8—302no—
387Q8—301no—
388Q8—301no—
389Q8C2×C3×C22⋊D511yescharacteristic
390D4—104no—
391D4—102no—
392D4—302no—
393D4—301no—
394D4—301no—
395D4—304no—
396D4—302no—
397C2×C4—32no—
398C2×C4—32no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—32no—
402C2×C4—52no—
403C2×C4—51no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—64no—
407C2×C4—62no—
408C2×C4—152no—
409C2×C4—151no—
410C2×C4—301no—
411C2×C4—301no—
412C2×C4—301no—
413C2×C4—301no—
414C2×C4—304no—
415C2×C3—42no—
416C2×C3—41no—
417C2×C3—84no—
418C2×C3—82no—
419C2×C3—801no—
420C5#3486⋊C311yescharacteristic, complemented, Sylow
421C22—28no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—102no—
426C22—101no—
427C22—302no—
428C22—301no—
429C22#4147⋊C314yescomplemented
430C22A4×C22⋊D512yes—
431C22C22×D5×A411yescharacteristic
432C4—31no—
433C4—32no—
434C4—31no—
435C4—62no—
436C4—102no—
437C4—101no—
438C4—302no—
439C4—301no—
440C3—41noSylow
441C2—24no—
442C2—22no—
443C2—201no—
444C2—601no—
445C2C2×#4147⋊C312yes—
446C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC23×C5×Q8C2×C3
socleC22×C5C23×A4
hypercenterC24D5×A4
hypocenterC5×Q8C2×C3×D4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C5×SL(2,𝔽3)C2semidirect
2. C2×#4147⋊C3C2semidirect
3. SL(2,𝔽3)×C22⋊D5C2semidirect
4. #66146⋊C3C2semidirect
5. #66151⋊C3C2semidirect
6. #10619152C3semidirect
11. C2×D5×SL(2,𝔽3)C22semidirect
12. C2×C5⋊(#41⋊C3)C22semidirect
14. Dic5×SL(2,𝔽3)C22semidirect
16. C23×C5×Q8C2×C3semidirect
17. C2×#4147C2×C3semidirect
19. #66151C2×C3semidirect
23. Q8×C22⋊D5C2×C3semidirect
24. #66146C2×C3semidirect
31. C2×C5×SL(2,𝔽3)D4semidirect
38. C23×SL(2,𝔽3)D5semidirect
52. C2×#713C22×C3semidirect
54. C2×Q8×D5C22×C3semidirect
65. Q8×Dic5C22×C3semidirect
110. C2×C5×Q8C3×D4semidirect
156. C23×Q8C3×D5semidirect
183. C2×SL(2,𝔽3)C22⋊D5semidirect
286. C22×C52−1+4⋊C3semidirect
325. C2×Q8C3×C22⋊D5semidirect
420. C5#3486⋊C3semidirect
429. C22#4147⋊C3semidirect

Maximal subgroups

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

subgroupindex
C23×C5×SL(2,𝔽3)2
C2×#4147⋊C32
SL(2,𝔽3)×C22⋊D52
#66146⋊C32
#66151⋊C32
#106191523
C3×#40994
#3486⋊C35