Finite Group Databasebeta

← C5×#3585⋊C3 #14511378

Subgroups — C5×#3585⋊C3 #14511378

Order 1920.

678 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C5×#5945C211yesmaximal, characteristic, complemented
2C2×C5×#994C211yesmaximal, characteristic, complemented
3C5×#577⋊C3C212yesmaximal, complemented
4C5×#576⋊C3C212yesmaximal, complemented
5C5×#568⋊C3C211yesmaximal, characteristic, complemented
6C5×#3585C311yesmaximal, characteristic, complemented, Hall
7C5×2−1+4⋊C3—21no—
8C5×2−1+4⋊C3C2211yescharacteristic, complemented
9C5×#994—21no—
10C5×#994C2212yescomplemented
11C5×#994C2211yescharacteristic, complemented
12C2×C5×#41⋊C3—21no—
13C2×C5×#41⋊C3—21no—
14C2×C5×#41⋊C3C2211yescharacteristic
15C3×C5×#137—41nomaximal
16C5×#131⋊C3C2212yescomplemented
17#3585⋊C3C511yesmaximal, characteristic, direct product factor, Hall
18C5×#585C2×C311yescharacteristic
19C2×C5×H(5,𝔽2)C2×C311yescharacteristic, complemented
20C5×#577—32no—
21C5×#577C2×C312yescomplemented
22C5×#576—32no—
23C5×#576C2×C312yescomplemented
24C5×#575—31no—
25C5×#575—31no—
26C2×C5×AGL(1,ℤ/8)—31no—
27C2×C5×AGL(1,ℤ/8)—31no—
28C5×#568C2×C311yescharacteristic, complemented
29C3×C5×#41—41no—
30C2×C3×C5×Q8—41no—
31C5×#41⋊C3—22no—
32C5×#41⋊C3—21no—
33C5×#41⋊C3D411yescharacteristic
34C5×#41⋊C3C2311yescharacteristic
35C2×C5×SL(2,𝔽3)—21no—
36C2×C5×SL(2,𝔽3)D411yescharacteristic
37C3×C5×Q16—42no—
38C3×C5×SD16—42no—
39C3×C5×M16—41no—
40#5945C2×C511yescharacteristic, complemented
41C2×#994C2×C511yescharacteristic, complemented
42#577⋊C3C2×C512yescomplemented
43#576⋊C3C2×C512yescomplemented
44#568⋊C3C2×C511yescharacteristic, complemented
45C5×2−1+4—21no—
46C5×2−1+4—31no—
47C5×2−1+4C22×C311yescharacteristic
48C5×H(5,𝔽2)—21no—
49C5×H(5,𝔽2)—32no—
50C5×H(5,𝔽2)—31no—
51C5×H(5,𝔽2)—61no—
52C5×H(5,𝔽2)—61no—
53C5×H(5,𝔽2)C22×C311yescharacteristic
54C5×H(5,𝔽2)C22×C312yes—
55C2×C5×#41—21no—
56C2×C5×#41—21no—
57C2×C5×#41—31no—
58C2×C5×#41—31no—
59C2×C5×#41—31no—
60C2×C5×#41—61no—
61C2×C5×#41C22×C311yescharacteristic
62C22×C5×D4—31no—
63C22×C5×D4—61no—
64C5×#137—31no—
65C5×#137—32no—
66C5×#137—32no—
67C5×#137A411yescharacteristic
68C5×AGL(1,ℤ/8)—32no—
69C5×AGL(1,ℤ/8)—34no—
70C5×AGL(1,ℤ/8)—31no—
71C5×AGL(1,ℤ/8)—32no—
72C5×AGL(1,ℤ/8)—32no—
73C5×AGL(1,ℤ/8)—32no—
74C5×AGL(1,ℤ/8)—32no—
75C5×AGL(1,ℤ/8)—31no—
76C5×#135—34no—
77C5×#135—32no—
78C5×#135—32no—
79C2×C5×SD16—32no—
80C2×C5×SD16—32no—
81C2×C5×D8—32no—
82C2×C5×D8—32no—
83C5×#131—32no—
84C5×#131C22×C312yes—
85C2×C5×M16—31no—
86C2×C5×M16—31no—
87#3585C3×C511yescharacteristic, complemented, Sylow
88C3×C5×Q8—42no—
89C3×C5×Q8—41no—
90C3×C5×Q8—81no—
91C3×C5×D4—41no—
92C3×C5×D4—81no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5—81no—
95C2×C3×C4×C5—81no—
96C5×SL(2,𝔽3)C2×D411yescharacteristic
97C3×C5×C8—42no—
982−1+4⋊C3—21no—
992−1+4⋊C3C22×C511yescharacteristic, complemented
100#994—21no—
101#994C22×C512yescomplemented
102#994C22×C511yescharacteristic, complemented
103C2×#41⋊C3—21no—
104C2×#41⋊C3—21no—
105C2×#41⋊C3C22×C511yescharacteristic
106C3×#137—41no—
107#131⋊C3C22×C512yescomplemented
108C24×C5—61no—
109C5×#41—22no—
110C5×#41—21no—
111C5×#41—31no—
112C5×#41—31no—
113C5×#41—32no—
114C5×#41—31no—
115C5×#41—32no—
116C5×#41—32no—
117C5×#41—31no—
118C5×#41—31no—
119C5×#41—61no—
120C5×#41—61no—
121C5×#41—61no—
122C5×#41—61no—
123C5×#41—61no—
124C5×#41—61no—
125C5×#41—62no—
126C5×#41—61no—
127C5×#41—61no—
128C5×#41—61no—
129C5×#41—62no—
130C5×#41C3×D411yescharacteristic
131C5×#41C2×A411yescharacteristic
132C5×#41C23×C311yescharacteristic
133C2×C5×Q8—21no—
134C2×C5×Q8—31no—
135C2×C5×Q8—31no—
136C2×C5×Q8—61no—
137C2×C5×Q8C3×D411yescharacteristic
138C2×C5×Q8C2×A411yescharacteristic
139C2×C5×D4—31no—
140C2×C5×D4—32no—
141C2×C5×D4—31no—
142C2×C5×D4—31no—
143C2×C5×D4—32no—
144C2×C5×D4—31no—
145C2×C5×D4—61no—
146C2×C5×D4—61no—
147C2×C5×D4—61no—
148C2×C5×D4—61no—
149C2×C5×D4—61no—
150C2×C5×D4—61no—
151C2×C5×D4—64no—
152C2×C5×D4—62no—
153C2×C5×D4—62no—
154C2×C5×D4—61no—
155C2×C5×D4—62no—
156C2×C5×D4—61no—
157C22×C4×C5—31no—
158C22×C4×C5—61no—
159C22×C4×C5—61no—
160C22×C4×C5—61no—
161C5×Q16—32no—
162C5×Q16C2×A412yes—
163C5×SD16—34no—
164C5×SD16—32no—
165C5×SD16—32no—
166C5×SD16—32no—
167C5×SD16C2×A412yes—
168C5×D8—34no—
169C5×D8—32no—
170C5×D8—32no—
171C5×M16—31no—
172C5×M16—32no—
173C5×M16—32no—
174C5×M16C2×A411yescharacteristic
175C2×C5×C8—32no—
176C2×C5×C8—32no—
177#585C2×C3×C511yescharacteristic
178C2×H(5,𝔽2)C2×C3×C511yescharacteristic, complemented
179#577—32no—
180#577C2×C3×C512yescomplemented
181#576—32no—
182#576C2×C3×C512yescomplemented
183#575—31no—
184#575—31no—
185C2×AGL(1,ℤ/8)—31no—
186C2×AGL(1,ℤ/8)—31no—
187#568C2×C3×C511yescharacteristic, complemented
188C22×C3×C5—41no—
189C22×C3×C5—81no—
190C3×C4×C5—41no—
191C3×C4×C5—41no—
192C3×C4×C5—81no—
193C3×C4×C5—82no—
194C3×#41—41no—
195C2×C3×Q8—41no—
196#41⋊C3—22no—
197#41⋊C3—21no—
198#41⋊C3C5×D411yescharacteristic
199#41⋊C3C23×C511yescharacteristic
200C2×SL(2,𝔽3)—21no—
201C2×SL(2,𝔽3)C5×D411yescharacteristic
202C3×Q16—42no—
203C3×SD16—42no—
204C3×M16—41no—
205C23×C5—31no—
206C23×C5—61no—
207C23×C5—62no—
208C23×C5—61no—
209C23×C5—62no—
210C23×C5—61no—
211C23×C5—242no—
212C5×Q8—21no—
213C5×Q8—31no—
214C5×Q8—31no—
215C5×Q8—61no—
216C5×Q8—61no—
217C5×Q8C2×C3×D411yescharacteristic
218C5×Q8C22×A411yescharacteristic
219C5×Q8C22×A412yes—
220C5×D4—21no—
221C5×D4—31no—
222C5×D4—31no—
223C5×D4—32no—
224C5×D4—31no—
225C5×D4—32no—
226C5×D4—62no—
227C5×D4—62no—
228C5×D4—61no—
229C5×D4—61no—
230C5×D4—62no—
231C5×D4—61no—
232C5×D4—62no—
233C5×D4—61no—
234C5×D4—62no—
235C5×D4—61no—
236C5×D4—61no—
237C5×D4C22×A411yescharacteristic
238C2×C4×C5—21no—
239C2×C4×C5—21no—
240C2×C4×C5—31no—
241C2×C4×C5—31no—
242C2×C4×C5—31no—
243C2×C4×C5—31no—
244C2×C4×C5—31no—
245C2×C4×C5—62no—
246C2×C4×C5—62no—
247C2×C4×C5—61no—
248C2×C4×C5—61no—
249C2×C4×C5—61no—
250C2×C4×C5—62no—
251C2×C4×C5—61no—
252C2×C4×C5—61no—
253C2×C4×C5C22×A411yescharacteristic
254C5×C8—32no—
255C5×C8C22×A412yes—
2562−1+4—21no—
2572−1+4—31no—
2582−1+4C22×C3×C511yescharacteristic
259H(5,𝔽2)—21no—
260H(5,𝔽2)—32no—
261H(5,𝔽2)—31no—
262H(5,𝔽2)—61no—
263H(5,𝔽2)—61no—
264H(5,𝔽2)C22×C3×C511yescharacteristic
265H(5,𝔽2)C22×C3×C512yes—
266C2×#41—21no—
267C2×#41—21no—
268C2×#41—31no—
269C2×#41—31no—
270C2×#41—31no—
271C2×#41—61no—
272C2×#41C22×C3×C511yescharacteristic
273C22×D4—31no—
274C22×D4—61no—
275#137—31no—
276#137—32no—
277#137—32no—
278#137C5×A411yescharacteristic
279AGL(1,ℤ/8)—32no—
280AGL(1,ℤ/8)—34no—
281AGL(1,ℤ/8)—31no—
282AGL(1,ℤ/8)—32no—
283AGL(1,ℤ/8)—32no—
284AGL(1,ℤ/8)—32no—
285AGL(1,ℤ/8)—32no—
286AGL(1,ℤ/8)—31no—
287#135—34no—
288#135—32no—
289#135—32no—
290C2×SD16—32no—
291C2×SD16—32no—
292C2×D8—32no—
293C2×D8—32no—
294#131—32no—
295#131C22×C3×C512yes—
296C2×M16—31no—
297C2×M16—31no—
298C2×C3×C5—41no—
299C2×C3×C5—81no—
300C2×C3×C5—161no—
301C3×Q8—42no—
302C3×Q8—41no—
303C3×Q8—81no—
304C3×D4—41no—
305C3×D4—81no—
306C2×C3×C4—41no—
307C2×C3×C4—81no—
308C2×C3×C4—81no—
309SL(2,𝔽3)C2×C5×D411yescharacteristic
310C3×C8—42no—
311C22×C5—21no—
312C22×C5—31no—
313C22×C5—31no—
314C22×C5—61no—
315C22×C5—62no—
316C22×C5—121no—
317C22×C5—122no—
318C22×C5—241no—
319C22×C5—241no—
320C22×C5—242no—
321C22×C5—242no—
322C22×C5D4×A411yescharacteristic
323C4×C5—21no—
324C4×C5—22no—
325C4×C5—31no—
326C4×C5—31no—
327C4×C5—61no—
328C4×C5D4×A411yescharacteristic
329C4×C5C23×A411yescharacteristic
330C24—61no—
331#41—22no—
332#41—21no—
333#41—31no—
334#41—31no—
335#41—32no—
336#41—31no—
337#41—32no—
338#41—32no—
339#41—31no—
340#41—31no—
341#41—61no—
342#41—61no—
343#41—61no—
344#41—61no—
345#41—61no—
346#41—61no—
347#41—62no—
348#41—61no—
349#41—61no—
350#41—61no—
351#41—62no—
352#41C3×C5×D411yescharacteristic
353#41C2×C5×A411yescharacteristic
354#41C23×C3×C511yescharacteristic
355C2×Q8—21no—
356C2×Q8—31no—
357C2×Q8—31no—
358C2×Q8—61no—
359C2×Q8C3×C5×D411yescharacteristic
360C2×Q8C2×C5×A411yescharacteristic
361C2×D4—31no—
362C2×D4—32no—
363C2×D4—31no—
364C2×D4—31no—
365C2×D4—32no—
366C2×D4—31no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—61no—
373C2×D4—64no—
374C2×D4—62no—
375C2×D4—62no—
376C2×D4—61no—
377C2×D4—62no—
378C2×D4—61no—
379C22×C4—31no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—61no—
383Q16—32no—
384Q16C2×C5×A412yes—
385SD16—34no—
386SD16—32no—
387SD16—32no—
388SD16—32no—
389SD16C2×C5×A412yes—
390D8—34no—
391D8—32no—
392D8—32no—
393M16—31no—
394M16—32no—
395M16—32no—
396M16C2×C5×A411yescharacteristic
397C2×C8—32no—
398C2×C8—32no—
399C3×C5—41noHall
400C22×C3—41no—
401C22×C3—81no—
402C3×C4—41no—
403C3×C4—41no—
404C3×C4—81no—
405C3×C4—82no—
406C2×C5—21no—
407C2×C5—41no—
408C2×C5—61no—
409C2×C5—61no—
410C2×C5—122no—
411C2×C5—121no—
412C2×C5C2×D4×A411yescharacteristic
413C23—31no—
414C23—61no—
415C23—62no—
416C23—61no—
417C23—62no—
418C23—61no—
419C23—242no—
420Q8—21no—
421Q8—31no—
422Q8—31no—
423Q8—61no—
424Q8—61no—
425Q8C2×C3×C5×D411yescharacteristic
426Q8C22×C5×A411yescharacteristic
427Q8C22×C5×A412yes—
428D4—21no—
429D4—31no—
430D4—31no—
431D4—32no—
432D4—31no—
433D4—32no—
434D4—62no—
435D4—62no—
436D4—61no—
437D4—61no—
438D4—62no—
439D4—61no—
440D4—62no—
441D4—61no—
442D4—62no—
443D4—61no—
444D4—61no—
445D4C22×C5×A411yescharacteristic
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—31no—
449C2×C4—31no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—62no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4C22×C5×A411yescharacteristic
462C8—32no—
463C8C22×C5×A412yes—
464C2×C3—41no—
465C2×C3—81no—
466C2×C3—161no—
467C5#3585⋊C311yescharacteristic, direct product factor, Sylow
468C22—21no—
469C22—31no—
470C22—31no—
471C22—61no—
472C22—62no—
473C22—121no—
474C22—122no—
475C22—241no—
476C22—242no—
477C22—241no—
478C22—242no—
479C22C5×D4×A411yescharacteristic
480C4—21no—
481C4—22no—
482C4—31no—
483C4—31no—
484C4—61no—
485C4C5×D4×A411yescharacteristic
486C4C23×C5×A411yescharacteristic
487C3—41noSylow
488C2—21no—
489C2—41no—
490C2—61no—
491C2—61no—
492C2—122no—
493C2—121no—
494C2C2×C5×D4×A411yescharacteristic
subgroupG/H
center, socleC2×C5C2×D4×A4
commutator, cosocle#41C23×C3×C5
FrattiniC4C23×C5×A4
FittingC5×#3585C3
hypercenterC5×#137A4
hypocenterQ8C2×C3×C5×D4
LCSG ▷ #41 ▷ Q8
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#137

Complements

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

NH ≅ G/Ntype
1. C5×#5945C2semidirect
2. C2×C5×#994C2semidirect
3. C5×#577⋊C3C2semidirect
4. C5×#576⋊C3C2semidirect
5. C5×#568⋊C3C2semidirect
6. C5×#3585C3semidirect
8. C5×2−1+4⋊C3C22semidirect
10. C5×#994C22semidirect
11. C5×#994C22semidirect
16. C5×#131⋊C3C22semidirect
17. #3585⋊C3C5direct (N × H)
19. C2×C5×H(5,𝔽2)C2×C3semidirect
21. C5×#577C2×C3semidirect
23. C5×#576C2×C3semidirect
28. C5×#568C2×C3semidirect
40. #5945C2×C5semidirect
41. C2×#994C2×C5semidirect
42. #577⋊C3C2×C5semidirect
43. #576⋊C3C2×C5semidirect
44. #568⋊C3C2×C5semidirect
87. #3585C3×C5semidirect
99. 2−1+4⋊C3C22×C5semidirect
101. #994C22×C5semidirect
102. #994C22×C5semidirect
107. #131⋊C3C22×C5semidirect
178. C2×H(5,𝔽2)C2×C3×C5semidirect
180. #577C2×C3×C5semidirect
182. #576C2×C3×C5semidirect
187. #568C2×C3×C5semidirect
467. C5#3585⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#59452
C2×C5×#9942
C5×#577⋊C32
C5×#576⋊C32
C5×#568⋊C32
C5×#35853
C3×C5×#1374
#3585⋊C35