Finite Group Databasebeta

← #10619196⋊C3 #14514236

Subgroups — #10619196⋊C3 #14514236

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×C5×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×D5×#41⋊C3C211yesmaximal, characteristic, complemented
3#66161⋊C3C211yesmaximal, characteristic, complemented
4SL(2,𝔽3)×C22⋊D5C212yesmaximal, complemented
5#66151⋊C3C212yesmaximal, complemented
6#10619196C311yesmaximal, characteristic, complemented, Hall
7C5×2−1+4⋊C3—22no—
8C2×C5×#41⋊C3C2211yescharacteristic
9C22×C5×SL(2,𝔽3)C2212yescomplemented
10D5×#41⋊C3—22no—
11C2×D5×SL(2,𝔽3)C2211yescharacteristic
12C2×C5⋊(#41⋊C3)C2211yescharacteristic
13C3×#4084—41nomaximal
14Dic5×SL(2,𝔽3)C2212yescomplemented
15#3489⋊C3—51nomaximal, Hall
16C2×C5×2−1+4C2×C311yescharacteristic, complemented
17C2×D5×#41C2×C311yescharacteristic, complemented
18#66167—31no—
19#66163—32no—
20#66162—31no—
21#66161C2×C311yescharacteristic, complemented
22#66158—32no—
23#66151C2×C312yescomplemented
24#66150—31no—
25Q8×C22⋊D5C2×C312yescomplemented
26C2×#4092—31no—
27C2×C3×C5×D4—41no—
28C2×C3×C22⋊D5—42no—
29C2×C3×C4×D5—41no—
30C5×#41⋊C3D412yescomplemented
31C2×C5×SL(2,𝔽3)—22no—
32C2×C5×SL(2,𝔽3)C2311yescharacteristic
33D5×SL(2,𝔽3)—21no—
34C5⋊(#41⋊C3)—21no—
35C3×C22⋊Dic5—42no—
36C3×C4⋊Dic5—41no—
37C2×2−1+4⋊C3D511yescharacteristic, complemented
38C22×#41⋊C3—51no—
39D4×SL(2,𝔽3)—52no—
40#548⋊C3—52no—
41#520⋊C3—51no—
42C5×2−1+4—22no—
43C5×2−1+4—62no—
44C2×C5×#41—31no—
45C2×C5×#41—32no—
46C2×C5×#41C22×C311yescharacteristic
47C22×C5×Q8—31no—
48C22×C5×Q8C22×C312yes—
49D5×#41—22no—
50D5×#41—62no—
51C2×#713C22×C311yescharacteristic
52C2×Q8×D5C22×C311yescharacteristic
53C2×#711—31no—
54C2×D4×D5—31no—
55C2×#709—31no—
56C22×C4×D5—31no—
57#4093—32no—
58#4092—32no—
59#4092—31no—
60#4092—32no—
61#4092—31no—
62#4092—32no—
63Q8×Dic5C22×C312yes—
64#4090—32no—
65#4085—34no—
66#4084—31no—
67#4084A411yescharacteristic
68#4083—32no—
69#4081—32no—
70D4×Dic5—32no—
71#4076—32no—
72#4075—34no—
73C4×C22⋊D5—32no—
74C4×C22⋊D5—32no—
75C2×#685—32no—
76#4072—31no—
77C2×C4⋊Dic5—31no—
78#4070—32no—
79C2×#683—32no—
80#3489—51noSylow
81C23×C3×C5—42no—
82C22×C3×D5—41no—
83C3×C5×D4—82no—
84C2×C3×C4×C5—41no—
85C3×C22⋊D5—84no—
86C2×C3×Dic5—42no—
87C2×C3×Dic5—41no—
88C3×C4×D5—82no—
89C5×SL(2,𝔽3)C2×D411yescharacteristic
902−1+4⋊C3—22no—
91C2×#41⋊C3—51no—
92C2×#41⋊C3—102no—
93C2×#41⋊C3C2×D511yescharacteristic
94C22×SL(2,𝔽3)—51no—
95C22×SL(2,𝔽3)C2×D512yescomplemented
96C3×#121—201no—
97C4×SL(2,𝔽3)—52no—
98C23×D5—31no—
99C5×#41—32no—
100C5×#41—62no—
101C5×#41—62no—
102C5×#41—64no—
103C5×#41—64no—
104C5×#41C3×D412yescomplemented
105C2×C5×Q8—22no—
106C2×C5×Q8—31no—
107C2×C5×Q8—32no—
108C2×C5×Q8—62no—
109C2×C5×Q8—62no—
110C2×C5×Q8—62no—
111C2×C5×Q8C23×C311yescharacteristic
112C2×C5×D4—31no—
113C2×C5×D4—32no—
114C2×C5×D4C2×A411yescharacteristic
115C22×C4×C5—31no—
116C22×C4×C5—32no—
117C22×C4×C5—32no—
118C2×C22⋊D5—32no—
119C2×C22⋊D5—32no—
120C2×C22⋊D5C2×A412yes—
121C22×Dic5—32no—
122C22×Dic5—31no—
123#713—21no—
124#713—61no—
125Q8×D5—21no—
126Q8×D5—61no—
127#711—62no—
128#711—62no—
129D4×D5—62no—
130D4×D5—62no—
131#709—62no—
132#709—62no—
133C2×D20—31no—
134C2×C4×D5—31no—
135C2×C4×D5—31no—
136C2×C4×D5—31no—
137C2×C4×D5—62no—
138C2×C4×D5—62no—
139C2×C4×D5C2×A411yescharacteristic
140C2×Dic10—31no—
141C22⋊Dic5—32no—
142C22⋊Dic5—32no—
143C22⋊Dic5C2×A412yes—
144#685—34no—
145#685—32no—
146#685—32no—
147C4⋊Dic5—31no—
148C4⋊Dic5—32no—
149C4⋊Dic5C2×A411yescharacteristic
150#683—34no—
151#683—32no—
152#683—32no—
153C4×Dic5—32no—
154C2×2−1+4C3×D511yescharacteristic, complemented
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#520—51no—
165C22×C3×C5—41no—
166C22×C3×C5—82no—
167C22×C3×C5—82no—
168C22×C3×C5—82no—
169C2×C3×D5—42no—
170C2×C3×D5—81no—
171C2×C3×D5—81no—
172C3×C4×C5—42no—
173C3×Dic5—81no—
174C3×Dic5—82no—
175C2×C3×D4—41no—
176C2×C3×D4—202no—
177C22×C3×C4—201no—
178#41⋊C3—101no—
179#41⋊C3C22⋊D512yescomplemented
180C2×SL(2,𝔽3)—22no—
181C2×SL(2,𝔽3)—101no—
182C2×SL(2,𝔽3)C22×D511yescharacteristic
183C3×C4⋊C4—201no—
184C3×C22⋊C4—202no—
185C23×C5—31no—
186C23×C5C22×A412yes—
187C22×D5—31no—
188C22×D5—62no—
189C22×D5—62no—
190C22×D5—121no—
191C22×D5C22×A411yescharacteristic
192C5×Q8—31no—
193C5×Q8—62no—
194C5×Q8—62no—
195C5×Q8—62no—
196C5×Q8C2×C3×D411yescharacteristic
197C5×D4—22no—
198C5×D4—32no—
199C5×D4—32no—
200C5×D4—64no—
201C2×C4×C5—31no—
202C2×C4×C5—32no—
203C2×C4×C5—32no—
204C2×C4×C5—32no—
205C2×C4×C5—62no—
206C2×C4×C5—62no—
207C2×C4×C5—64no—
208C2×C4×C5C22×A411yescharacteristic
209C22⋊D5—24no—
210C22⋊D5—64no—
211C22⋊D5—122no—
212C2×Dic5—32no—
213C2×Dic5—31no—
214C2×Dic5—62no—
215C2×Dic5—122no—
216C2×Dic5C22×A412yes—
217C2×Dic5C22×A411yescharacteristic
218D20—61no—
219D20—61no—
220C4×D5—22no—
221C4×D5—61no—
222C4×D5—61no—
223C4×D5—61no—
224C4×D5—61no—
225C4×D5—62no—
226Dic10—61no—
227Dic10—61no—
2282−1+4—22no—
2292−1+4—62no—
230C2×#41—31no—
231C2×#41—32no—
232C2×#41—51no—
233C2×#41—102no—
234C2×#41—151no—
235C2×#41—151no—
236C2×#41—302no—
237C2×#41C2×C3×D511yescharacteristic
238C22×Q8—31no—
239C22×Q8—51no—
240C22×Q8C2×C3×D512yes—
241C22×D4—151no—
242C23×C4—151no—
243C4⋊Q8—152no—
244#124—152no—
245#123—152no—
246#123—154no—
247C22⋊Q8—151no—
248C22⋊Q8—151no—
249C22⋊Q8—152no—
250C22⋊Q8—152no—
251C22⋊Q8—152no—
252C22⋊Q8—152no—
253#121—51no—
254#121—152no—
255#121—154no—
256#121—151no—
257#120—152no—
258C4×Q8—52no—
259C4×D4—152no—
260C4×D4—152no—
261C4×D4—152no—
262#117—151no—
263C2×C4⋊C4—152no—
264C2×C4⋊C4—151no—
265C2×C22⋊C4—152no—
266C2×C3×C5—41no—
267C2×C3×C5—41no—
268C2×C3×C5—41no—
269C2×C3×C5—162no—
270C3×D5—82no—
271C23×C3—42no—
272C23×C3—201no—
273C3×D4—82no—
274C3×D4—404no—
275C2×C3×C4—41no—
276C2×C3×C4—202no—
277C2×C3×C4—201no—
278C2×C3×C4—402no—
279SL(2,𝔽3)C2×C22⋊D511yescharacteristic
280C22×C5—22no—
281C22×C5—22no—
282C22×C5—22no—
283C22×C5—32no—
284C22×C5—61no—
285C22×C5—61no—
286C22×C5C23×A411yescharacteristic
287C2×D5—21no—
288C2×D5—21no—
289C2×D5—61no—
290C2×D5—61no—
291C2×D5—61no—
292C2×D5—124no—
293C2×D52−1+4⋊C312yescomplemented
294C4×C5—31no—
295C4×C5—31no—
296C4×C5—62no—
297C4×C5D4×A412yes—
298Dic5—21no—
299Dic5—22no—
300Dic5—61no—
301Dic5—62no—
302C24—151no—
303#41—32no—
304#41—62no—
305#41—62no—
306#41—64no—
307#41—64no—
308#41—101no—
309#41—302no—
310#41—302no—
311#41—302no—
312#41—302no—
313#41—301no—
314#41C3×C22⋊D512yescomplemented
315C2×Q8—22no—
316C2×Q8—31no—
317C2×Q8—32no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8—62no—
321C2×Q8—101no—
322C2×Q8—151no—
323C2×Q8—301no—
324C2×Q8C22×C3×D511yescharacteristic
325C2×D4—31no—
326C2×D4—32no—
327C2×D4—52no—
328C2×D4—151no—
329C2×D4—152no—
330C2×D4—152no—
331C2×D4—302no—
332C2×D4—302no—
333C2×D4D5×A411yescharacteristic
334C22×C4—31no—
335C22×C4—32no—
336C22×C4—32no—
337C22×C4—51no—
338C22×C4—151no—
339C22×C4—151no—
340C22×C4—151no—
341C22×C4—152no—
342C22×C4—151no—
343C22×C4—302no—
344C22×C4—302no—
345C4⋊C4—51no—
346C4⋊C4—152no—
347C4⋊C4—152no—
348C4⋊C4—152no—
349C4⋊C4—154no—
350C4⋊C4—151no—
351C22⋊C4—52no—
352C22⋊C4—152no—
353C22⋊C4—152no—
354C22⋊C4—154no—
355C22⋊C4—152no—
356C22⋊C4—152no—
357C42—152no—
358C3×C5—41noHall
359C22×C3—41no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—202no—
364C22×C3—401no—
365C22×C3—401no—
366C3×C4—42no—
367C3×C4—401no—
368C3×C4—402no—
369C2×C5—42no—
370C2×C5—62no—
371C2×C5C2×D4×A411yescharacteristic
372C2×C5C2×2−1+4⋊C311yescharacteristic
373C2×C5#594511yescharacteristic
374D5—22no—
375D5—121no—
376C23—31no—
377C23—51no—
378C23—151no—
379C23—302no—
380C23—302no—
381C23—601no—
382C23C2×D5×A412yes—
383Q8—31no—
384Q8—62no—
385Q8—62no—
386Q8—62no—
387Q8—301no—
388Q8—301no—
389Q8C2×C3×C22⋊D511yescharacteristic
390D4—22no—
391D4—32no—
392D4—32no—
393D4—64no—
394D4—104no—
395D4—301no—
396D4—301no—
397D4—304no—
398D4—602no—
399C2×C4—31no—
400C2×C4—32no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—52no—
404C2×C4—51no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—102no—
409C2×C4—152no—
410C2×C4—151no—
411C2×C4—301no—
412C2×C4—301no—
413C2×C4—301no—
414C2×C4—301no—
415C2×C4—302no—
416C2×C4—302no—
417C2×C4—602no—
418C2×C4C2×D5×A411yescharacteristic
419C2×C3—41no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—162no—
423C2×C3—402no—
424C5#3489⋊C311yescharacteristic, complemented, Sylow
425C22—22no—
426C22—22no—
427C22—22no—
428C22—32no—
429C22—52no—
430C22—61no—
431C22—61no—
432C22—101no—
433C22—101no—
434C22—301no—
435C22—301no—
436C22—301no—
437C22—604no—
438C22C22×D5×A411yescharacteristic
439C4—31no—
440C4—31no—
441C4—62no—
442C4—102no—
443C4—101no—
444C4—302no—
445C4—301no—
446C4A4×C22⋊D512yes—
447C3—41noSylow
448C2—42no—
449C2—62no—
450C2—102no—
451C2—601no—
452C2D5×2−1+4⋊C311yescharacteristic
453C2#66287⋊C311yescharacteristic
454C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC2×C5×2−1+4C2×C3
socleC22×C5C23×A4
hypercenterC2×D4D5×A4
hypocenterC5×Q8C2×C3×D4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×2−1+4⋊C3C2semidirect
2. C2×D5×#41⋊C3C2semidirect
3. #66161⋊C3C2semidirect
4. SL(2,𝔽3)×C22⋊D5C2semidirect
5. #66151⋊C3C2semidirect
6. #10619196C3semidirect
9. C22×C5×SL(2,𝔽3)C22semidirect
14. Dic5×SL(2,𝔽3)C22semidirect
16. C2×C5×2−1+4C2×C3semidirect
17. C2×D5×#41C2×C3semidirect
21. #66161C2×C3semidirect
23. #66151C2×C3semidirect
25. Q8×C22⋊D5C2×C3semidirect
30. C5×#41⋊C3D4semidirect
37. C2×2−1+4⋊C3D5semidirect
95. C22×SL(2,𝔽3)C2×D5semidirect
104. C5×#41C3×D4semidirect
154. C2×2−1+4C3×D5semidirect
179. #41⋊C3C22⋊D5semidirect
293. C2×D52−1+4⋊C3semidirect
314. #41C3×C22⋊D5semidirect
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×C5×2−1+4⋊C32
C2×D5×#41⋊C32
#66161⋊C32
SL(2,𝔽3)×C22⋊D52
#66151⋊C32
#106191963
C3×#40844
#3489⋊C35