Finite Group Databasebeta

← C5×#3516⋊C3 #14511354

Subgroups — C5×#3516⋊C3 #14511354

Order 1920.

534 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#557⋊C3C211yesmaximal, characteristic, complemented
2C5×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C5×#548⋊C3C211yesmaximal, characteristic, complemented
4C5×#548⋊C3C211yesmaximal, characteristic, complemented
5C5×#520⋊C3C211yesmaximal, characteristic, complemented
6C5×#519⋊C3C211yesmaximal, characteristic, complemented
7C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
8C5×#3516C311yesmaximal, characteristic, complemented, Hall
9C2×C5×#41⋊C3C2211yescharacteristic
10C22×C5×SL(2,𝔽3)C2211yescharacteristic
11C22×C5×SL(2,𝔽3)C2211yescharacteristic
12C3×C5×#123—41nomaximal
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14C4×C5×SL(2,𝔽3)C2211yescharacteristic
15C4×C5×SL(2,𝔽3)C2211yescharacteristic
16C4×C5×SL(2,𝔽3)C2211yescharacteristic
17#3516⋊C3C511yesmaximal, characteristic, direct product factor, Hall
18C5×#557C2×C311yescharacteristic, complemented
19C5×#556—31no—
20C5×#554—31no—
21C5×#553—31no—
22C5×#552—31no—
23C5×D4×Q8C2×C311yescharacteristic
24C5×#548C2×C311yescharacteristic, complemented
25C5×#548C2×C311yescharacteristic, complemented
26C5×#542—31no—
27C5×#541—31no—
28C5×#529—31no—
29C5×#529—31no—
30C5×#520C2×C311yescharacteristic, complemented
31C5×#519C2×C311yescharacteristic, complemented
32C2×C4×C5×Q8C2×C311yescharacteristic, complemented
33C2×C3×C5×D4—41no—
34C22×C3×C4×C5—41no—
35C5×#41⋊C3—21no—
36C2×C5×SL(2,𝔽3)—21no—
37C2×C5×SL(2,𝔽3)—21no—
38C2×C5×SL(2,𝔽3)C2311yescharacteristic
39C3×C5×C4⋊C4—41no—
40C3×C5×C4⋊C4—41no—
41C3×C5×C22⋊C4—41no—
42C3×C5×C22⋊C4—41no—
43C3×C5×C22⋊C4—41no—
44#557⋊C3C2×C511yescharacteristic, complemented
45D4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
46#548⋊C3C2×C511yescharacteristic, complemented
47#548⋊C3C2×C511yescharacteristic, complemented
48#520⋊C3C2×C511yescharacteristic, complemented
49#519⋊C3C2×C511yescharacteristic, complemented
50C2×C4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
51C2×C5×#41C22×C311yescharacteristic
52C22×C5×Q8C22×C311yescharacteristic
53C22×C5×Q8C22×C311yescharacteristic
54C5×C4⋊Q8—31no—
55C5×C4⋊Q8—31no—
56C5×#126—32no—
57C5×#126—32no—
58C5×#126—31no—
59C5×#126—31no—
60C5×#125—31no—
61C5×#125—31no—
62C5×#125—32no—
63C5×#125—31no—
64C5×#124—31no—
65C5×#124—31no—
66C5×#124—31no—
67C5×#123—32no—
68C5×#123—31no—
69C5×#123A411yescharacteristic
70C5×C22⋊Q8—32no—
71C5×C22⋊Q8—31no—
72C5×C22⋊Q8—31no—
73C5×C22⋊Q8—31no—
74C5×C22⋊Q8—31no—
75C5×C22⋊Q8—31no—
76C5×#121—31no—
77C5×#121—31no—
78C5×#121—31no—
79C4×C5×Q8—31no—
80C4×C5×Q8—31no—
81C4×C5×Q8—31no—
82C4×C5×Q8—31no—
83C4×C5×Q8—31no—
84C4×C5×Q8C22×C311yescharacteristic
85C4×C5×Q8C22×C311yescharacteristic
86C4×C5×Q8C22×C311yescharacteristic
87C4×C5×Q8C22×C311yescharacteristic
88C4×C5×D4—32no—
89C4×C5×D4—31no—
90C4×C5×D4—31no—
91C4×C5×D4—31no—
92C4×C5×D4—31no—
93C4×C5×D4—31no—
94C5×#117—32no—
95C5×#117—31no—
96C5×#117—31no—
97C5×#117—31no—
98C2×C5×C4⋊C4—31no—
99C2×C5×C4⋊C4—31no—
100C2×C42×C5—31no—
101#3516C3×C511yescharacteristic, complemented, Sylow
102C23×C3×C5—41no—
103C23×C3×C5—41no—
104C3×C5×D4—82no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5—82no—
111C5×SL(2,𝔽3)#4111yescharacteristic
112C2×#41⋊C3C22×C511yescharacteristic
113C22×SL(2,𝔽3)C22×C511yescharacteristic
114C22×SL(2,𝔽3)C22×C511yescharacteristic
115C3×#123—41no—
116C4×SL(2,𝔽3)C22×C511yescharacteristic
117C4×SL(2,𝔽3)C22×C511yescharacteristic
118C4×SL(2,𝔽3)C22×C511yescharacteristic
119C4×SL(2,𝔽3)C22×C511yescharacteristic
120C5×#41—21no—
121C5×#41—61no—
122C2×C5×Q8—21no—
123C2×C5×Q8—21no—
124C2×C5×Q8—31no—
125C2×C5×Q8—31no—
126C2×C5×Q8—31no—
127C2×C5×Q8—61no—
128C2×C5×Q8—61no—
129C2×C5×Q8C23×C311yescharacteristic
130C2×C5×D4—31no—
131C2×C5×D4—31no—
132C2×C5×D4—31no—
133C2×C5×D4C2×A411yescharacteristic
134C22×C4×C5—31no—
135C22×C4×C5—31no—
136C22×C4×C5—32no—
137C22×C4×C5—31no—
138C22×C4×C5—31no—
139C22×C4×C5C2×A411yescharacteristic
140C5×C4⋊C4—31no—
141C5×C4⋊C4—31no—
142C5×C4⋊C4—31no—
143C5×C4⋊C4—31no—
144C5×C4⋊C4—32no—
145C5×C4⋊C4—32no—
146C5×C4⋊C4—32no—
147C5×C4⋊C4—31no—
148C5×C4⋊C4—31no—
149C5×C4⋊C4—31no—
150C5×C4⋊C4—31no—
151C5×C4⋊C4—31no—
152C5×C4⋊C4—31no—
153C5×C4⋊C4—31no—
154C5×C4⋊C4—31no—
155C5×C4⋊C4C2×A411yescharacteristic
156C5×C4⋊C4C2×A411yescharacteristic
157C5×C22⋊C4—32no—
158C5×C22⋊C4—32no—
159C5×C22⋊C4—31no—
160C5×C22⋊C4—31no—
161C5×C22⋊C4—31no—
162C5×C22⋊C4—31no—
163C5×C22⋊C4—31no—
164C5×C22⋊C4—31no—
165C5×C22⋊C4—31no—
166C5×C22⋊C4C2×A411yescharacteristic
167C5×C22⋊C4C2×A411yescharacteristic
168C5×C22⋊C4C2×A411yescharacteristic
169C42×C5—32no—
170C42×C5—31no—
171C42×C5—31no—
172C42×C5—31no—
173C42×C5—31no—
174C42×C5—31no—
175C42×C5—31no—
176#557C2×C3×C511yescharacteristic, complemented
177#556—31no—
178#554—31no—
179#553—31no—
180#552—31no—
181D4×Q8C2×C3×C511yescharacteristic
182#548C2×C3×C511yescharacteristic, complemented
183#548C2×C3×C511yescharacteristic, complemented
184#542—31no—
185#541—31no—
186#529—31no—
187#529—31no—
188#520C2×C3×C511yescharacteristic, complemented
189#519C2×C3×C511yescharacteristic, complemented
190C2×C4×Q8C2×C3×C511yescharacteristic, complemented
191C22×C3×C5—42no—
192C22×C3×C5—41no—
193C22×C3×C5—81no—
194C22×C3×C5—81no—
195C22×C3×C5—81no—
196C22×C3×C5—81no—
197C22×C3×C5—81no—
198C3×C4×C5—81no—
199C3×C4×C5—81no—
200C3×C4×C5—81no—
201C3×C4×C5—81no—
202C3×C4×C5—81no—
203C2×C3×D4—41no—
204C22×C3×C4—41no—
205#41⋊C3—21no—
206C2×SL(2,𝔽3)—21no—
207C2×SL(2,𝔽3)—21no—
208C2×SL(2,𝔽3)C23×C511yescharacteristic
209C3×C4⋊C4—41no—
210C3×C4⋊C4—41no—
211C3×C22⋊C4—41no—
212C3×C22⋊C4—41no—
213C3×C22⋊C4—41no—
214C23×C5—31no—
215C23×C5C22×A411yescharacteristic
216C23×C5C22×A411yescharacteristic
217C5×Q8—31no—
218C5×Q8—61no—
219C5×Q8—61no—
220C5×Q8—61no—
221C5×Q8—61no—
222C5×Q8—121no—
223C5×Q8C3×#4111yescharacteristic
224C5×D4—22no—
225C5×D4—61no—
226C5×D4—61no—
227C5×D4—62no—
228C5×D4—121no—
229C2×C4×C5—22no—
230C2×C4×C5—31no—
231C2×C4×C5—31no—
232C2×C4×C5—31no—
233C2×C4×C5—31no—
234C2×C4×C5—31no—
235C2×C4×C5—31no—
236C2×C4×C5—31no—
237C2×C4×C5—61no—
238C2×C4×C5—61no—
239C2×C4×C5—61no—
240C2×C4×C5—61no—
241C2×C4×C5—61no—
242C2×C4×C5—61no—
243C2×C4×C5—121no—
244C2×C4×C5—122no—
245C2×C4×C5C22×A411yescharacteristic
246C2×C4×C5C22×A411yescharacteristic
247C2×C4×C5C22×A411yescharacteristic
248C2×C4×C5C22×A411yescharacteristic
249C2×C4×C5C22×A411yescharacteristic
250C2×#41C22×C3×C511yescharacteristic
251C22×Q8C22×C3×C511yescharacteristic
252C22×Q8C22×C3×C511yescharacteristic
253C4⋊Q8—31no—
254C4⋊Q8—31no—
255#126—32no—
256#126—32no—
257#126—31no—
258#126—31no—
259#125—31no—
260#125—31no—
261#125—32no—
262#125—31no—
263#124—31no—
264#124—31no—
265#124—31no—
266#123—32no—
267#123—31no—
268#123C5×A411yescharacteristic
269C22⋊Q8—32no—
270C22⋊Q8—31no—
271C22⋊Q8—31no—
272C22⋊Q8—31no—
273C22⋊Q8—31no—
274C22⋊Q8—31no—
275#121—31no—
276#121—31no—
277#121—31no—
278C4×Q8—31no—
279C4×Q8—31no—
280C4×Q8—31no—
281C4×Q8—31no—
282C4×Q8—31no—
283C4×Q8C22×C3×C511yescharacteristic
284C4×Q8C22×C3×C511yescharacteristic
285C4×Q8C22×C3×C511yescharacteristic
286C4×Q8C22×C3×C511yescharacteristic
287C4×D4—32no—
288C4×D4—31no—
289C4×D4—31no—
290C4×D4—31no—
291C4×D4—31no—
292C4×D4—31no—
293#117—32no—
294#117—31no—
295#117—31no—
296#117—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C42—31no—
300C2×C3×C5—41no—
301C2×C3×C5—41no—
302C2×C3×C5—41no—
303C2×C3×C5—82no—
304C2×C3×C5—161no—
305C23×C3—41no—
306C23×C3—41no—
307C3×D4—82no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—82no—
314SL(2,𝔽3)C5×#4111yescharacteristic
315C22×C5—21no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5—21no—
319C22×C5—21no—
320C22×C5—61no—
321C22×C5—61no—
322C22×C5—61no—
323C22×C52−1+4⋊C312yes—
324C22×C5C23×A411yescharacteristic
325C4×C5—21no—
326C4×C5—21no—
327C4×C5—21no—
328C4×C5—21no—
329C4×C5—21no—
330C4×C5—31no—
331C4×C5—31no—
332C4×C5—61no—
333C4×C5—61no—
334C4×C5—61no—
335C4×C5—61no—
336C4×C5—61no—
337C4×C5—61no—
338#41—21no—
339#41—61no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—31no—
343C2×Q8—31no—
344C2×Q8—31no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8C23×C3×C511yescharacteristic
348C2×D4—31no—
349C2×D4—31no—
350C2×D4—31no—
351C2×D4C2×C5×A411yescharacteristic
352C22×C4—31no—
353C22×C4—31no—
354C22×C4—32no—
355C22×C4—31no—
356C22×C4—31no—
357C22×C4C2×C5×A411yescharacteristic
358C4⋊C4—31no—
359C4⋊C4—31no—
360C4⋊C4—31no—
361C4⋊C4—31no—
362C4⋊C4—32no—
363C4⋊C4—32no—
364C4⋊C4—32no—
365C4⋊C4—31no—
366C4⋊C4—31no—
367C4⋊C4—31no—
368C4⋊C4—31no—
369C4⋊C4—31no—
370C4⋊C4—31no—
371C4⋊C4—31no—
372C4⋊C4—31no—
373C4⋊C4C2×C5×A411yescharacteristic
374C4⋊C4C2×C5×A411yescharacteristic
375C22⋊C4—32no—
376C22⋊C4—32no—
377C22⋊C4—31no—
378C22⋊C4—31no—
379C22⋊C4—31no—
380C22⋊C4—31no—
381C22⋊C4—31no—
382C22⋊C4—31no—
383C22⋊C4—31no—
384C22⋊C4C2×C5×A411yescharacteristic
385C22⋊C4C2×C5×A411yescharacteristic
386C22⋊C4C2×C5×A411yescharacteristic
387C42—32no—
388C42—31no—
389C42—31no—
390C42—31no—
391C42—31no—
392C42—31no—
393C42—31no—
394C3×C5—41noHall
395C22×C3—42no—
396C22×C3—41no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C3×C4—81no—
403C3×C4—81no—
404C3×C4—81no—
405C3×C4—81no—
406C3×C4—81no—
407C2×C5—22no—
408C2×C5—41no—
409C2×C5—121no—
410C2×C5A4×#4111yescharacteristic
411C2×C5C2×2−1+4⋊C311yescharacteristic
412C2×C5#594511yescharacteristic
413C23—31no—
414C23C22×C5×A411yescharacteristic
415C23C22×C5×A411yescharacteristic
416Q8—31no—
417Q8—61no—
418Q8—61no—
419Q8—61no—
420Q8—61no—
421Q8—121no—
422Q8C3×C5×#4111yescharacteristic
423D4—22no—
424D4—61no—
425D4—61no—
426D4—62no—
427D4—121no—
428C2×C4—22no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—121no—
443C2×C4—122no—
444C2×C4C22×C5×A411yescharacteristic
445C2×C4C22×C5×A411yescharacteristic
446C2×C4C22×C5×A411yescharacteristic
447C2×C4C22×C5×A411yescharacteristic
448C2×C4C22×C5×A411yescharacteristic
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—82no—
453C2×C3—161no—
454C5#3516⋊C311yescharacteristic, direct product factor, Sylow
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—61no—
461C22—61no—
462C22—61no—
463C22C5×2−1+4⋊C312yes—
464C22C23×C5×A411yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—31no—
471C4—31no—
472C4—61no—
473C4—61no—
474C4—61no—
475C4—61no—
476C4—61no—
477C4—61no—
478C3—41noSylow
479C2—22no—
480C2—41no—
481C2—121no—
482C2C5×A4×#4111yescharacteristic
483C2C2×C5×2−1+4⋊C311yescharacteristic
484C2C5×#594511yescharacteristic
subgroupG/H
center, socleC22×C5C23×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC5×#3516C3
hypercenterC5×#123A4
hypocenterQ8C3×C5×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C5 ◁ C5×#123

Complements

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

NH ≅ G/Ntype
1. C5×#557⋊C3C2semidirect
2. C5×D4×SL(2,𝔽3)C2semidirect
3. C5×#548⋊C3C2semidirect
4. C5×#548⋊C3C2semidirect
5. C5×#520⋊C3C2semidirect
6. C5×#519⋊C3C2semidirect
7. C2×C4×C5×SL(2,𝔽3)C2semidirect
8. C5×#3516C3semidirect
17. #3516⋊C3C5direct (N × H)
18. C5×#557C2×C3semidirect
24. C5×#548C2×C3semidirect
25. C5×#548C2×C3semidirect
30. C5×#520C2×C3semidirect
31. C5×#519C2×C3semidirect
32. C2×C4×C5×Q8C2×C3semidirect
44. #557⋊C3C2×C5semidirect
45. D4×SL(2,𝔽3)C2×C5semidirect
46. #548⋊C3C2×C5semidirect
47. #548⋊C3C2×C5semidirect
48. #520⋊C3C2×C5semidirect
49. #519⋊C3C2×C5semidirect
50. C2×C4×SL(2,𝔽3)C2×C5semidirect
101. #3516C3×C5semidirect
176. #557C2×C3×C5semidirect
182. #548C2×C3×C5semidirect
183. #548C2×C3×C5semidirect
188. #520C2×C3×C5semidirect
189. #519C2×C3×C5semidirect
190. C2×C4×Q8C2×C3×C5semidirect
454. C5#3516⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#557⋊C32
C5×D4×SL(2,𝔽3)2
C5×#548⋊C32
C5×#520⋊C32
C5×#519⋊C32
C2×C4×C5×SL(2,𝔽3)2
C5×#35163
C3×C5×#1234
#3516⋊C35