Finite Group Databasebeta

← C5×#3489⋊C3 #14511346

Subgroups — C5×#3489⋊C3 #14511346

Order 1920.

754 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 124 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
2C22×C5×#41⋊C3C211yesmaximal, characteristic, complemented
3C5×D4×SL(2,𝔽3)C212yesmaximal, complemented
4C5×#548⋊C3C212yesmaximal, complemented
5C5×#520⋊C3C211yesmaximal, characteristic, complemented
6C5×#3489C311yesmaximal, characteristic, complemented, Hall
7C5×2−1+4⋊C3—22no—
8C2×C5×#41⋊C3—22no—
9C2×C5×#41⋊C3C2211yescharacteristic
10C2×C5×#41⋊C3C2211yescharacteristic
11C22×C5×SL(2,𝔽3)C2211yescharacteristic
12C22×C5×SL(2,𝔽3)C2212yescomplemented
13C3×C5×#121—41nomaximal
14C4×C5×SL(2,𝔽3)C2212yescomplemented
15#3489⋊C3C511yesmaximal, characteristic, direct product factor, Hall
16C2×C5×2−1+4C2×C311yescharacteristic, complemented
17C22×C5×#41C2×C311yescharacteristic, complemented
18C5×D4×Q8C2×C312yescomplemented
19C5×#548C2×C312yescomplemented
20C5×#547—32no—
21C5×#546—32no—
22C5×#537—31no—
23C5×#536—31no—
24C5×#525—31no—
25C2×C5×C22⋊Q8—31no—
26C5×#520C2×C311yescharacteristic, complemented
27C2×C3×C5×D4—41no—
28C2×C3×C5×D4—42no—
29C22×C3×C4×C5—41no—
30C5×#41⋊C3—21no—
31C5×#41⋊C3D412yescomplemented
32C2×C5×SL(2,𝔽3)—21no—
33C2×C5×SL(2,𝔽3)—22no—
34C2×C5×SL(2,𝔽3)C2311yescharacteristic
35C3×C5×C4⋊C4—41no—
36C3×C5×C22⋊C4—42no—
37C2×2−1+4⋊C3C2×C511yescharacteristic, complemented
38C22×#41⋊C3C2×C511yescharacteristic, complemented
39D4×SL(2,𝔽3)C2×C512yescomplemented
40#548⋊C3C2×C512yescomplemented
41#520⋊C3C2×C511yescharacteristic, complemented
42C5×2−1+4—22no—
43C5×2−1+4—62no—
44C2×C5×#41—22no—
45C2×C5×#41—31no—
46C2×C5×#41—32no—
47C2×C5×#41—31no—
48C2×C5×#41—31no—
49C2×C5×#41—62no—
50C2×C5×#41C22×C311yescharacteristic
51C2×C5×#41C22×C311yescharacteristic
52C22×C5×Q8—31no—
53C22×C5×Q8C22×C311yescharacteristic
54C22×C5×Q8C22×C312yes—
55C22×C5×D4—31no—
56C23×C4×C5—31no—
57C5×C4⋊Q8—32no—
58C5×#124—32no—
59C5×#123—34no—
60C5×#123—32no—
61C5×C22⋊Q8—32no—
62C5×C22⋊Q8—31no—
63C5×C22⋊Q8—31no—
64C5×C22⋊Q8—32no—
65C5×C22⋊Q8—32no—
66C5×C22⋊Q8—32no—
67C5×#121—32no—
68C5×#121—34no—
69C5×#121—31no—
70C5×#121A411yescharacteristic
71C5×#120—32no—
72C4×C5×Q8C22×C312yes—
73C4×C5×D4—32no—
74C4×C5×D4—32no—
75C4×C5×D4—32no—
76C5×#117—31no—
77C2×C5×C4⋊C4—32no—
78C2×C5×C4⋊C4—31no—
79C2×C5×C22⋊C4—32no—
80#3489C3×C511yescharacteristic, complemented, Sylow
81C23×C3×C5—41no—
82C23×C3×C5—42no—
83C3×C5×D4—82no—
84C3×C5×D4—84no—
85C2×C3×C4×C5—41no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5—42no—
88C2×C3×C4×C5—82no—
89C5×SL(2,𝔽3)C2×D411yescharacteristic
902−1+4⋊C3—22no—
91C2×#41⋊C3—22no—
92C2×#41⋊C3C22×C511yescharacteristic
93C2×#41⋊C3C22×C511yescharacteristic
94C22×SL(2,𝔽3)C22×C511yescharacteristic
95C22×SL(2,𝔽3)C22×C512yescomplemented
96C3×#121—41no—
97C4×SL(2,𝔽3)C22×C512yescomplemented
98C24×C5—31no—
99C5×#41—21no—
100C5×#41—32no—
101C5×#41—62no—
102C5×#41—62no—
103C5×#41—62no—
104C5×#41—62no—
105C5×#41—62no—
106C5×#41—64no—
107C5×#41—64no—
108C5×#41—62no—
109C5×#41—61no—
110C5×#41C3×D412yescomplemented
111C2×C5×Q8—21no—
112C2×C5×Q8—22no—
113C2×C5×Q8—31no—
114C2×C5×Q8—32no—
115C2×C5×Q8—31no—
116C2×C5×Q8—62no—
117C2×C5×Q8—62no—
118C2×C5×Q8—61no—
119C2×C5×Q8—62no—
120C2×C5×Q8C23×C311yescharacteristic
121C2×C5×D4—31no—
122C2×C5×D4—31no—
123C2×C5×D4—32no—
124C2×C5×D4—32no—
125C2×C5×D4—32no—
126C2×C5×D4—62no—
127C2×C5×D4—62no—
128C2×C5×D4C2×A412yes—
129C2×C5×D4C2×A411yescharacteristic
130C22×C4×C5—31no—
131C22×C4×C5—32no—
132C22×C4×C5—32no—
133C22×C4×C5—31no—
134C22×C4×C5—32no—
135C22×C4×C5—31no—
136C22×C4×C5—31no—
137C22×C4×C5—31no—
138C22×C4×C5—62no—
139C22×C4×C5—62no—
140C22×C4×C5C2×A411yescharacteristic
141C5×C4⋊C4—32no—
142C5×C4⋊C4—34no—
143C5×C4⋊C4—32no—
144C5×C4⋊C4—31no—
145C5×C4⋊C4—32no—
146C5×C4⋊C4C2×A411yescharacteristic
147C5×C22⋊C4—34no—
148C5×C22⋊C4—32no—
149C5×C22⋊C4—32no—
150C5×C22⋊C4—32no—
151C5×C22⋊C4—32no—
152C5×C22⋊C4C2×A412yes—
153C42×C5—32no—
154C2×2−1+4C2×C3×C511yescharacteristic, complemented
155C22×#41C2×C3×C511yescharacteristic, complemented
156D4×Q8C2×C3×C512yescomplemented
157#548C2×C3×C512yescomplemented
158#547—32no—
159#546—32no—
160#537—31no—
161#536—31no—
162#525—31no—
163C2×C22⋊Q8—31no—
164#520C2×C3×C511yescharacteristic, complemented
165C22×C3×C5—42no—
166C22×C3×C5—41no—
167C22×C3×C5—81no—
168C22×C3×C5—81no—
169C22×C3×C5—82no—
170C22×C3×C5—82no—
171C22×C3×C5—82no—
172C3×C4×C5—42no—
173C3×C4×C5—81no—
174C3×C4×C5—82no—
175C2×C3×D4—41no—
176C2×C3×D4—42no—
177C22×C3×C4—41no—
178#41⋊C3—21no—
179#41⋊C3C5×D412yescomplemented
180C2×SL(2,𝔽3)—21no—
181C2×SL(2,𝔽3)—22no—
182C2×SL(2,𝔽3)C23×C511yescharacteristic
183C3×C4⋊C4—41no—
184C3×C22⋊C4—42no—
185C23×C5—31no—
186C23×C5—31no—
187C23×C5—62no—
188C23×C5—62no—
189C23×C5—121no—
190C23×C5C22×A412yes—
191C23×C5C22×A411yescharacteristic
192C5×Q8—31no—
193C5×Q8—62no—
194C5×Q8—61no—
195C5×Q8—62no—
196C5×Q8—61no—
197C5×Q8—62no—
198C5×Q8C2×C3×D411yescharacteristic
199C5×D4—24no—
200C5×D4—22no—
201C5×D4—32no—
202C5×D4—32no—
203C5×D4—61no—
204C5×D4—61no—
205C5×D4—64no—
206C5×D4—64no—
207C5×D4—122no—
208C2×C4×C5—22no—
209C2×C4×C5—31no—
210C2×C4×C5—32no—
211C2×C4×C5—32no—
212C2×C4×C5—32no—
213C2×C4×C5—32no—
214C2×C4×C5—31no—
215C2×C4×C5—61no—
216C2×C4×C5—61no—
217C2×C4×C5—62no—
218C2×C4×C5—61no—
219C2×C4×C5—61no—
220C2×C4×C5—62no—
221C2×C4×C5—62no—
222C2×C4×C5—64no—
223C2×C4×C5—62no—
224C2×C4×C5—122no—
225C2×C4×C5C22×A412yes—
226C2×C4×C5C22×A411yescharacteristic
227C2×C4×C5C22×A411yescharacteristic
2282−1+4—22no—
2292−1+4—62no—
230C2×#41—22no—
231C2×#41—31no—
232C2×#41—32no—
233C2×#41—31no—
234C2×#41—31no—
235C2×#41—62no—
236C2×#41C22×C3×C511yescharacteristic
237C2×#41C22×C3×C511yescharacteristic
238C22×Q8—31no—
239C22×Q8C22×C3×C512yes—
240C22×Q8C22×C3×C511yescharacteristic
241C22×D4—31no—
242C23×C4—31no—
243C4⋊Q8—32no—
244#124—32no—
245#123—34no—
246#123—32no—
247C22⋊Q8—32no—
248C22⋊Q8—31no—
249C22⋊Q8—31no—
250C22⋊Q8—32no—
251C22⋊Q8—32no—
252C22⋊Q8—32no—
253#121—32no—
254#121—34no—
255#121—31no—
256#121C5×A411yescharacteristic
257#120—32no—
258C4×Q8C22×C3×C512yes—
259C4×D4—32no—
260C4×D4—32no—
261C4×D4—32no—
262#117—31no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—31no—
265C2×C22⋊C4—32no—
266C2×C3×C5—41no—
267C2×C3×C5—41no—
268C2×C3×C5—41no—
269C2×C3×C5—82no—
270C2×C3×C5—162no—
271C23×C3—41no—
272C23×C3—42no—
273C3×D4—82no—
274C3×D4—84no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4—82no—
279SL(2,𝔽3)C2×C5×D411yescharacteristic
280C22×C5—21no—
281C22×C5—21no—
282C22×C5—22no—
283C22×C5—22no—
284C22×C5—22no—
285C22×C5—32no—
286C22×C5—61no—
287C22×C5—61no—
288C22×C5—61no—
289C22×C5—61no—
290C22×C5—61no—
291C22×C5—124no—
292C22×C52−1+4⋊C312yescomplemented
293C22×C5C23×A411yescharacteristic
294C4×C5—21no—
295C4×C5—22no—
296C4×C5—31no—
297C4×C5—31no—
298C4×C5—61no—
299C4×C5—62no—
300C4×C5—62no—
301C4×C5D4×A412yes—
302C24—31no—
303#41—21no—
304#41—32no—
305#41—62no—
306#41—62no—
307#41—62no—
308#41—62no—
309#41—62no—
310#41—64no—
311#41—64no—
312#41—62no—
313#41—61no—
314#41C3×C5×D412yescomplemented
315C2×Q8—21no—
316C2×Q8—22no—
317C2×Q8—31no—
318C2×Q8—32no—
319C2×Q8—31no—
320C2×Q8—62no—
321C2×Q8—62no—
322C2×Q8—61no—
323C2×Q8—62no—
324C2×Q8C23×C3×C511yescharacteristic
325C2×D4—31no—
326C2×D4—31no—
327C2×D4—32no—
328C2×D4—32no—
329C2×D4—32no—
330C2×D4—62no—
331C2×D4—62no—
332C2×D4C2×C5×A412yes—
333C2×D4C2×C5×A411yescharacteristic
334C22×C4—31no—
335C22×C4—32no—
336C22×C4—32no—
337C22×C4—31no—
338C22×C4—32no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4C2×C5×A411yescharacteristic
345C4⋊C4—32no—
346C4⋊C4—34no—
347C4⋊C4—32no—
348C4⋊C4—31no—
349C4⋊C4—32no—
350C4⋊C4C2×C5×A411yescharacteristic
351C22⋊C4—34no—
352C22⋊C4—32no—
353C22⋊C4—32no—
354C22⋊C4—32no—
355C22⋊C4—32no—
356C22⋊C4C2×C5×A412yes—
357C42—32no—
358C3×C5—41noHall
359C22×C3—42no—
360C22×C3—41no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3—82no—
364C22×C3—82no—
365C22×C3—82no—
366C3×C4—42no—
367C3×C4—81no—
368C3×C4—82no—
369C2×C5—22no—
370C2×C5—42no—
371C2×C5—62no—
372C2×C5—121no—
373C2×C5C2×D4×A411yescharacteristic
374C2×C5C2×2−1+4⋊C311yescharacteristic
375C2×C5#594511yescharacteristic
376C23—31no—
377C23—31no—
378C23—62no—
379C23—62no—
380C23—121no—
381C23C22×C5×A412yes—
382C23C22×C5×A411yescharacteristic
383Q8—31no—
384Q8—62no—
385Q8—61no—
386Q8—62no—
387Q8—61no—
388Q8—62no—
389Q8C2×C3×C5×D411yescharacteristic
390D4—24no—
391D4—22no—
392D4—32no—
393D4—32no—
394D4—61no—
395D4—61no—
396D4—64no—
397D4—64no—
398D4—122no—
399C2×C4—22no—
400C2×C4—31no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—32no—
405C2×C4—31no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—62no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—64no—
414C2×C4—62no—
415C2×C4—122no—
416C2×C4C22×C5×A412yes—
417C2×C4C22×C5×A411yescharacteristic
418C2×C4C22×C5×A411yescharacteristic
419C2×C3—41no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—82no—
423C2×C3—162no—
424C5#3489⋊C311yescharacteristic, direct product factor, Sylow
425C22—21no—
426C22—21no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—32no—
431C22—61no—
432C22—61no—
433C22—61no—
434C22—61no—
435C22—61no—
436C22—124no—
437C22C5×2−1+4⋊C312yescomplemented
438C22C23×C5×A411yescharacteristic
439C4—21no—
440C4—22no—
441C4—31no—
442C4—31no—
443C4—61no—
444C4—62no—
445C4—62no—
446C4C5×D4×A412yes—
447C3—41noSylow
448C2—22no—
449C2—42no—
450C2—62no—
451C2—121no—
452C2C2×C5×D4×A411yescharacteristic
453C2C2×C5×2−1+4⋊C311yescharacteristic
454C2C5×#594511yescharacteristic
subgroupG/H
center, socleC22×C5C23×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC5×#3489C3
hypercenterC5×#121A4
hypocenterQ8C2×C3×C5×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C5 ◁ C5×#121

Complements

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

NH ≅ G/Ntype
1. C2×C5×2−1+4⋊C3C2semidirect
2. C22×C5×#41⋊C3C2semidirect
3. C5×D4×SL(2,𝔽3)C2semidirect
4. C5×#548⋊C3C2semidirect
5. C5×#520⋊C3C2semidirect
6. C5×#3489C3semidirect
12. C22×C5×SL(2,𝔽3)C22semidirect
14. C4×C5×SL(2,𝔽3)C22semidirect
15. #3489⋊C3C5direct (N × H)
16. C2×C5×2−1+4C2×C3semidirect
17. C22×C5×#41C2×C3semidirect
18. C5×D4×Q8C2×C3semidirect
19. C5×#548C2×C3semidirect
26. C5×#520C2×C3semidirect
31. C5×#41⋊C3D4semidirect
37. C2×2−1+4⋊C3C2×C5semidirect
38. C22×#41⋊C3C2×C5semidirect
39. D4×SL(2,𝔽3)C2×C5semidirect
40. #548⋊C3C2×C5semidirect
41. #520⋊C3C2×C5semidirect
80. #3489C3×C5semidirect
95. C22×SL(2,𝔽3)C22×C5semidirect
97. C4×SL(2,𝔽3)C22×C5semidirect
110. C5×#41C3×D4semidirect
154. C2×2−1+4C2×C3×C5semidirect
155. C22×#41C2×C3×C5semidirect
156. D4×Q8C2×C3×C5semidirect
157. #548C2×C3×C5semidirect
164. #520C2×C3×C5semidirect
179. #41⋊C3C5×D4semidirect
292. C22×C52−1+4⋊C3semidirect
314. #41C3×C5×D4semidirect
424. C5#3489⋊C3direct (N × H)
437. C22C5×2−1+4⋊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
C22×C5×#41⋊C32
C5×D4×SL(2,𝔽3)2
C5×#548⋊C32
C5×#520⋊C32
C5×#34893
C3×C5×#1214
#3489⋊C35