Finite Group Databasebeta

← C5×#3486⋊C3 #14511343

Subgroups — C5×#3486⋊C3 #14511343

Order 1920.

804 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 128 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
2C23×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C5×D4×SL(2,𝔽3)C212yesmaximal, complemented
4C5×#548⋊C3C212yesmaximal, complemented
5C5×#519⋊C3C211yesmaximal, characteristic, complemented
6C5×#3486C311yesmaximal, characteristic, complemented, Hall
7C5×2−1+4⋊C3—22no—
8C2×C5×#41⋊C3C2211yescharacteristic, complemented
9C22×C5×SL(2,𝔽3)—22no—
10C22×C5×SL(2,𝔽3)C2212yes—
11C22×C5×SL(2,𝔽3)C2211yescharacteristic
12C22×C5×SL(2,𝔽3)C2211yescharacteristic, complemented
13C3×C5×#120—41nomaximal
14C4×C5×SL(2,𝔽3)C2212yescomplemented
15#3486⋊C3C511yesmaximal, characteristic, direct product factor, Hall
16C2×C5×2−1+4C2×C311yescharacteristic, complemented
17C23×C5×Q8C2×C311yescharacteristic, complemented
18C5×D4×Q8—32no—
19C5×D4×Q8C2×C312yescomplemented
20C5×#548—32no—
21C5×#548C2×C312yescomplemented
22C5×#536—31no—
23C5×#536—31no—
24C5×#525—31no—
25C5×#525—31no—
26C5×#519C2×C311yescharacteristic, complemented
27C24×C3×C5—41no—
28C2×C3×C5×D4—41no—
29C2×C3×C5×D4—42no—
30C5×#41⋊C3—21no—
31C2×C5×SL(2,𝔽3)—22no—
32C2×C5×SL(2,𝔽3)—21no—
33C2×C5×SL(2,𝔽3)D412yescomplemented
34C2×C5×SL(2,𝔽3)C2311yescharacteristic
35C3×C5×C22⋊C4—41no—
36C3×C5×C22⋊C4—42no—
37C2×2−1+4⋊C3C2×C511yescharacteristic, complemented
38C23×SL(2,𝔽3)C2×C511yescharacteristic, complemented
39D4×SL(2,𝔽3)C2×C512yescomplemented
40#548⋊C3C2×C512yescomplemented
41#519⋊C3C2×C511yescharacteristic, complemented
42C5×2−1+4—22no—
43C5×2−1+4—62no—
44C2×C5×#41—31no—
45C2×C5×#41—31no—
46C2×C5×#41—31no—
47C2×C5×#41C22×C311yescharacteristic, complemented
48C22×C5×Q8—22no—
49C22×C5×Q8—31no—
50C22×C5×Q8—31no—
51C22×C5×Q8—32no—
52C22×C5×Q8—62no—
53C22×C5×Q8C22×C312yes—
54C22×C5×Q8C22×C311yescharacteristic
55C22×C5×Q8C22×C311yescharacteristic, complemented
56C23×C4×C5—31no—
57C5×C4⋊Q8—32no—
58C5×C4⋊Q8—32no—
59C5×#124—32no—
60C5×#124—32no—
61C5×#123—32no—
62C5×#123—31no—
63C5×#123—31no—
64C5×C22⋊Q8—34no—
65C5×C22⋊Q8—32no—
66C5×C22⋊Q8—32no—
67C5×C22⋊Q8—32no—
68C5×C22⋊Q8—32no—
69C5×#121—32no—
70C5×#121—32no—
71C5×#120—31no—
72C5×#120A411yescharacteristic
73C4×C5×Q8—32no—
74C4×C5×Q8C22×C312yescomplemented
75C4×C5×D4—34no—
76C4×C5×D4—32no—
77C4×C5×D4—32no—
78C5×#117—31no—
79C5×#117—31no—
80#3486C3×C511yescharacteristic, complemented, Sylow
81C23×C3×C5—41no—
82C23×C3×C5—42no—
83C23×C3×C5—41no—
84C23×C3×C5—82no—
85C23×C3×C5—84no—
86C3×C5×D4—82no—
87C3×C5×D4—84no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5—42no—
90C5×SL(2,𝔽3)C2×D411yescharacteristic
912−1+4⋊C3—22no—
92C2×#41⋊C3C22×C511yescharacteristic, complemented
93C22×SL(2,𝔽3)—22no—
94C22×SL(2,𝔽3)C22×C512yes—
95C22×SL(2,𝔽3)C22×C511yescharacteristic
96C22×SL(2,𝔽3)C22×C511yescharacteristic, complemented
97C3×#120—41no—
98C4×SL(2,𝔽3)C22×C512yescomplemented
99C24×C5C2×A411yescharacteristic
100C5×#41—21no—
101C5×#41—62no—
102C5×#41—62no—
103C5×#41—62no—
104C5×#41—62no—
105C5×#41—62no—
106C5×#41—62no—
107C5×#41—61no—
108C2×C5×Q8—22no—
109C2×C5×Q8—21no—
110C2×C5×Q8—32no—
111C2×C5×Q8—31no—
112C2×C5×Q8—31no—
113C2×C5×Q8—32no—
114C2×C5×Q8—32no—
115C2×C5×Q8—31no—
116C2×C5×Q8—62no—
117C2×C5×Q8—62no—
118C2×C5×Q8—62no—
119C2×C5×Q8—64no—
120C2×C5×Q8—62no—
121C2×C5×Q8—62no—
122C2×C5×Q8—61no—
123C2×C5×Q8—64no—
124C2×C5×Q8C3×D412yescomplemented
125C2×C5×Q8C23×C311yescharacteristic
126C2×C5×D4—31no—
127C2×C5×D4—32no—
128C2×C5×D4—31no—
129C2×C5×D4—31no—
130C2×C5×D4C2×A412yes—
131C2×C5×D4C2×A411yescharacteristic
132C22×C4×C5—31no—
133C22×C4×C5—32no—
134C22×C4×C5—32no—
135C22×C4×C5—32no—
136C22×C4×C5—31no—
137C22×C4×C5—31no—
138C22×C4×C5—31no—
139C22×C4×C5—62no—
140C22×C4×C5—62no—
141C5×C4⋊C4—32no—
142C5×C4⋊C4—34no—
143C5×C4⋊C4—32no—
144C5×C4⋊C4—32no—
145C5×C4⋊C4—32no—
146C5×C22⋊C4—34no—
147C5×C22⋊C4—31no—
148C5×C22⋊C4—32no—
149C5×C22⋊C4—32no—
150C5×C22⋊C4—32no—
151C5×C22⋊C4C2×A411yescharacteristic
152C5×C22⋊C4C2×A412yes—
153C42×C5—32no—
154C42×C5—32no—
155C2×2−1+4C2×C3×C511yescharacteristic, complemented
156C23×Q8C2×C3×C511yescharacteristic, complemented
157D4×Q8—32no—
158D4×Q8C2×C3×C512yescomplemented
159#548—32no—
160#548C2×C3×C512yescomplemented
161#536—31no—
162#536—31no—
163#525—31no—
164#525—31no—
165#519C2×C3×C511yescharacteristic, complemented
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—81no—
173C22×C3×C5—82no—
174C22×C3×C5—88no—
175C3×C4×C5—81no—
176C3×C4×C5—82no—
177C24×C3—41no—
178C2×C3×D4—41no—
179C2×C3×D4—42no—
180#41⋊C3—21no—
181C2×SL(2,𝔽3)—22no—
182C2×SL(2,𝔽3)—21no—
183C2×SL(2,𝔽3)C5×D412yescomplemented
184C2×SL(2,𝔽3)C23×C511yescharacteristic
185C3×C22⋊C4—41no—
186C3×C22⋊C4—42no—
187C23×C5—22no—
188C23×C5—24no—
189C23×C5—31no—
190C23×C5C22×A411yescharacteristic
191C23×C5C22×A411yescharacteristic
192C23×C5C22×A412yes—
193C5×Q8—32no—
194C5×Q8—31no—
195C5×Q8—32no—
196C5×Q8—62no—
197C5×Q8—62no—
198C5×Q8—61no—
199C5×Q8—62no—
200C5×Q8—61no—
201C5×Q8—64no—
202C5×Q8C2×C3×D411yescharacteristic
203C5×D4—22no—
204C5×D4—24no—
205C5×D4—62no—
206C5×D4—61no—
207C5×D4—61no—
208C5×D4—62no—
209C5×D4—64no—
210C2×C4×C5—32no—
211C2×C4×C5—32no—
212C2×C4×C5—31no—
213C2×C4×C5—31no—
214C2×C4×C5—32no—
215C2×C4×C5—32no—
216C2×C4×C5—31no—
217C2×C4×C5—61no—
218C2×C4×C5—61no—
219C2×C4×C5—62no—
220C2×C4×C5—64no—
221C2×C4×C5—62no—
222C2×C4×C5—61no—
223C2×C4×C5—61no—
224C2×C4×C5—64no—
225C2×C4×C5—62no—
226C2×C4×C5C22×A412yes—
227C2×C4×C5C22×A411yescharacteristic
2282−1+4—22no—
2292−1+4—62no—
230C2×#41—31no—
231C2×#41—31no—
232C2×#41—31no—
233C2×#41C22×C3×C511yescharacteristic, complemented
234C22×Q8—22no—
235C22×Q8—31no—
236C22×Q8—31no—
237C22×Q8—32no—
238C22×Q8—62no—
239C22×Q8C22×C3×C512yes—
240C22×Q8C22×C3×C511yescharacteristic
241C22×Q8C22×C3×C511yescharacteristic, complemented
242C23×C4—31no—
243C4⋊Q8—32no—
244C4⋊Q8—32no—
245#124—32no—
246#124—32no—
247#123—32no—
248#123—31no—
249#123—31no—
250C22⋊Q8—34no—
251C22⋊Q8—32no—
252C22⋊Q8—32no—
253C22⋊Q8—32no—
254C22⋊Q8—32no—
255#121—32no—
256#121—32no—
257#120—31no—
258#120C5×A411yescharacteristic
259C4×Q8—32no—
260C4×Q8C22×C3×C512yescomplemented
261C4×D4—34no—
262C4×D4—32no—
263C4×D4—32no—
264#117—31no—
265#117—31no—
266C2×C3×C5—41no—
267C2×C3×C5—42no—
268C2×C3×C5—82no—
269C2×C3×C5—84no—
270C2×C3×C5—161no—
271C23×C3—41no—
272C23×C3—42no—
273C23×C3—41no—
274C23×C3—82no—
275C23×C3—84no—
276C3×D4—82no—
277C3×D4—84no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280SL(2,𝔽3)C2×C5×D411yescharacteristic
281C22×C5—28no—
282C22×C5—22no—
283C22×C5—22no—
284C22×C5—22no—
285C22×C5—21no—
286C22×C5—22no—
287C22×C5—61no—
288C22×C5—62no—
289C22×C5D4×A412yes—
290C22×C52−1+4⋊C314yescomplemented
291C22×C5C23×A411yescharacteristic
292C4×C5—21no—
293C4×C5—22no—
294C4×C5—32no—
295C4×C5—31no—
296C4×C5—31no—
297C4×C5—61no—
298C4×C5—62no—
299C4×C5—62no—
300C24C2×C5×A411yescharacteristic
301#41—21no—
302#41—62no—
303#41—62no—
304#41—62no—
305#41—62no—
306#41—62no—
307#41—62no—
308#41—61no—
309C2×Q8—22no—
310C2×Q8—21no—
311C2×Q8—32no—
312C2×Q8—31no—
313C2×Q8—31no—
314C2×Q8—32no—
315C2×Q8—32no—
316C2×Q8—31no—
317C2×Q8—62no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8—64no—
321C2×Q8—62no—
322C2×Q8—62no—
323C2×Q8—61no—
324C2×Q8—64no—
325C2×Q8C3×C5×D412yescomplemented
326C2×Q8C23×C3×C511yescharacteristic
327C2×D4—31no—
328C2×D4—32no—
329C2×D4—31no—
330C2×D4—31no—
331C2×D4C2×C5×A412yes—
332C2×D4C2×C5×A411yescharacteristic
333C22×C4—31no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—32no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—62no—
341C22×C4—62no—
342C4⋊C4—32no—
343C4⋊C4—34no—
344C4⋊C4—32no—
345C4⋊C4—32no—
346C4⋊C4—32no—
347C22⋊C4—34no—
348C22⋊C4—31no—
349C22⋊C4—32no—
350C22⋊C4—32no—
351C22⋊C4—32no—
352C22⋊C4C2×C5×A411yescharacteristic
353C22⋊C4C2×C5×A412yes—
354C42—32no—
355C42—32no—
356C3×C5—41noHall
357C22×C3—42no—
358C22×C3—44no—
359C22×C3—41no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—82no—
364C22×C3—81no—
365C22×C3—88no—
366C3×C4—82no—
367C3×C4—81no—
368C2×C5—22no—
369C2×C5—24no—
370C2×C5—41no—
371C2×C5—121no—
372C2×C5C2×D4×A411yescharacteristic
373C2×C5C2×2−1+4⋊C312yes—
374C23—22no—
375C23—24no—
376C23—31no—
377C23C22×C5×A411yescharacteristic
378C23C22×C5×A411yescharacteristic
379C23C22×C5×A412yes—
380Q8—32no—
381Q8—31no—
382Q8—32no—
383Q8—62no—
384Q8—62no—
385Q8—61no—
386Q8—62no—
387Q8—61no—
388Q8—64no—
389Q8C2×C3×C5×D411yescharacteristic
390D4—22no—
391D4—24no—
392D4—62no—
393D4—61no—
394D4—61no—
395D4—62no—
396D4—64no—
397C2×C4—32no—
398C2×C4—32no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—31no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—64no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—64no—
412C2×C4—62no—
413C2×C4C22×C5×A412yes—
414C2×C4C22×C5×A411yescharacteristic
415C2×C3—41no—
416C2×C3—42no—
417C2×C3—82no—
418C2×C3—84no—
419C2×C3—161no—
420C5#3486⋊C311yescharacteristic, direct product factor, Sylow
421C22—28no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—21no—
426C22—22no—
427C22—61no—
428C22—62no—
429C22C5×D4×A412yes—
430C22C5×2−1+4⋊C314yescomplemented
431C22C23×C5×A411yescharacteristic
432C4—21no—
433C4—22no—
434C4—31no—
435C4—32no—
436C4—31no—
437C4—61no—
438C4—62no—
439C4—62no—
440C3—41noSylow
441C2—22no—
442C2—24no—
443C2—41no—
444C2—121no—
445C2C2×C5×D4×A411yescharacteristic
446C2C2×C5×2−1+4⋊C312yes—
subgroupG/H
center, socleC22×C5C23×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC5×#3486C3
hypercenterC5×#120A4
hypocenterQ8C2×C3×C5×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C5 ◁ C5×#120

Complements

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

NH ≅ G/Ntype
1. C2×C5×2−1+4⋊C3C2semidirect
2. C23×C5×SL(2,𝔽3)C2semidirect
3. C5×D4×SL(2,𝔽3)C2semidirect
4. C5×#548⋊C3C2semidirect
5. C5×#519⋊C3C2semidirect
6. C5×#3486C3semidirect
8. C2×C5×#41⋊C3C22semidirect
12. C22×C5×SL(2,𝔽3)C22semidirect
14. C4×C5×SL(2,𝔽3)C22semidirect
15. #3486⋊C3C5direct (N × H)
16. C2×C5×2−1+4C2×C3semidirect
17. C23×C5×Q8C2×C3semidirect
19. C5×D4×Q8C2×C3semidirect
21. C5×#548C2×C3semidirect
26. C5×#519C2×C3semidirect
33. C2×C5×SL(2,𝔽3)D4semidirect
37. C2×2−1+4⋊C3C2×C5semidirect
38. C23×SL(2,𝔽3)C2×C5semidirect
39. D4×SL(2,𝔽3)C2×C5semidirect
40. #548⋊C3C2×C5semidirect
41. #519⋊C3C2×C5semidirect
47. C2×C5×#41C22×C3semidirect
55. C22×C5×Q8C22×C3semidirect
74. C4×C5×Q8C22×C3semidirect
80. #3486C3×C5semidirect
92. C2×#41⋊C3C22×C5semidirect
96. C22×SL(2,𝔽3)C22×C5semidirect
98. C4×SL(2,𝔽3)C22×C5semidirect
124. C2×C5×Q8C3×D4semidirect
155. C2×2−1+4C2×C3×C5semidirect
156. C23×Q8C2×C3×C5semidirect
158. D4×Q8C2×C3×C5semidirect
160. #548C2×C3×C5semidirect
165. #519C2×C3×C5semidirect
183. C2×SL(2,𝔽3)C5×D4semidirect
233. C2×#41C22×C3×C5semidirect
241. C22×Q8C22×C3×C5semidirect
260. C4×Q8C22×C3×C5semidirect
290. C22×C52−1+4⋊C3semidirect
325. C2×Q8C3×C5×D4semidirect
420. C5#3486⋊C3direct (N × H)
430. 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
C23×C5×SL(2,𝔽3)2
C5×D4×SL(2,𝔽3)2
C5×#548⋊C32
C5×#519⋊C32
C5×#34863
C3×C5×#1204
#3486⋊C35