Finite Group Databasebeta

← #10617807⋊C3 #14514058

Subgroups — #10617807⋊C3 #14514058

Order 1920.

534 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 77 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×#520⋊C3C211yesmaximal, characteristic, complemented
2#66165⋊C3C211yesmaximal, characteristic, complemented
3#66165⋊C3C211yesmaximal, characteristic, complemented
4#65906⋊C3C211yesmaximal, characteristic, complemented
5SL(2,𝔽3)×D20C211yesmaximal, characteristic, complemented
6C4×D5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
7#65903⋊C3C211yesmaximal, characteristic, complemented
8#10617807C311yesmaximal, characteristic, complemented, Hall
9C2×C5×#41⋊C3C2211yescharacteristic
10C2×D5×SL(2,𝔽3)C2211yescharacteristic
11C2×D5×SL(2,𝔽3)C2211yescharacteristic
12C3×#4040—41nomaximal
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14C4×C5×SL(2,𝔽3)C2211yescharacteristic
15Dic5×SL(2,𝔽3)C2211yescharacteristic
16Dic5×SL(2,𝔽3)C2211yescharacteristic
17#3516⋊C3—51nomaximal, Hall
18C5×#520C2×C311yescharacteristic, complemented
19#66165C2×C311yescharacteristic, complemented
20#66165C2×C311yescharacteristic, complemented
21Q8×D20C2×C311yescharacteristic
22#65906C2×C311yescharacteristic, complemented
23C4×Q8×D5C2×C311yescharacteristic, complemented
24#65903C2×C311yescharacteristic, complemented
25#65895—31no—
26#65892—31no—
27#65891—31no—
28#65876—31no—
29#65862—31no—
30#65852—31no—
31#65847—31no—
32#65845—31no—
33C2×C3×D20—41no—
34C2×C3×C4×D5—41no—
35C5×#41⋊C3—21no—
36C2×C5×SL(2,𝔽3)C2311yescharacteristic
37D5×SL(2,𝔽3)—21no—
38D5×SL(2,𝔽3)—21no—
39C3×C5×C4⋊C4—41no—
40C3×#685—41no—
41C3×#685—41no—
42C3×#685—41no—
43C3×#683—41no—
44#557⋊C3—51no—
45D4×SL(2,𝔽3)—51no—
46#548⋊C3—51no—
47#548⋊C3—51no—
48#520⋊C3D511yescharacteristic, complemented
49#519⋊C3—51no—
50C2×C4×SL(2,𝔽3)—51no—
51C2×C5×#41C22×C311yescharacteristic
52C2×Q8×D5C22×C311yescharacteristic
53C2×Q8×D5C22×C311yescharacteristic
54C4×C5×Q8C22×C311yescharacteristic
55C4×C5×Q8C22×C311yescharacteristic
56C4×C5×D4—31no—
57C4×C5×D4—31no—
58C5×#117—31no—
59C2×C5×C4⋊C4—31no—
60Q8×Dic5C22×C311yescharacteristic
61Q8×Dic5C22×C311yescharacteristic
62#4084—31no—
63#4084—31no—
64#4082—31no—
65#4082—31no—
66#4075—32no—
67C4×C22⋊D5—32no—
68C4×C22⋊D5—31no—
69C4×C22⋊D5—31no—
70#4072—32no—
71#4070—32no—
72#4070—31no—
73#4070—31no—
74#4044—31no—
75#4043—31no—
76#4043—31no—
77#4042—31no—
78#4040A411yescharacteristic
79#4038—31no—
80D5×C4⋊C4—31no—
81#4036—31no—
82#4035—32no—
83#4035—31no—
84#4034—31no—
85#4033—31no—
86#4033—31no—
87#4031—31no—
88#4030—31no—
89#4029—31no—
90#4025—32no—
91#4025—32no—
92#4025—31no—
93C4×D20—31no—
94#4018—31no—
95C42×D5—31no—
96#4016—31no—
97#4015—31no—
98C4×Dic10—31no—
99C4×Dic10—31no—
100C4×Dic10—31no—
101#3516—51noSylow
102C22×C3×D5—41no—
103C22×C3×D5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—41no—
107C2×C3×Dic5—41no—
108C2×C3×Dic5—41no—
109C3×D20—82no—
110C3×C4×D5—82no—
111C5×SL(2,𝔽3)#4111yescharacteristic
112C2×#41⋊C3C2×D511yescharacteristic
113C22×SL(2,𝔽3)—51no—
114C22×SL(2,𝔽3)—51no—
115C3×#123—201no—
116C4×SL(2,𝔽3)—51no—
117C4×SL(2,𝔽3)—51no—
118C4×SL(2,𝔽3)C2×D511yescharacteristic
119C4×SL(2,𝔽3)C2×D511yescharacteristic
120C5×#41—21no—
121C5×#41—61no—
122C2×C5×Q8C23×C311yescharacteristic
123C2×C5×D4—31no—
124C22×C4×C5—32no—
125C22×C4×C5—31no—
126C2×C22⋊D5—31no—
127C2×C22⋊D5—31no—
128Q8×D5—21no—
129Q8×D5—21no—
130Q8×D5—61no—
131Q8×D5—61no—
132C2×D20C2×A411yescharacteristic
133C2×C4×D5—31no—
134C2×C4×D5—31no—
135C2×C4×D5—31no—
136C2×C4×D5C2×A411yescharacteristic
137C2×Dic10—31no—
138C2×Dic10—31no—
139C2×Dic10—31no—
140C5×C4⋊C4—31no—
141C5×C4⋊C4—31no—
142C5×C4⋊C4—31no—
143C5×C4⋊C4C2×A411yescharacteristic
144C5×C22⋊C4—31no—
145C5×C22⋊C4—31no—
146C42×C5—31no—
147C42×C5—31no—
148C22⋊Dic5—32no—
149C22⋊Dic5—32no—
150C22⋊Dic5—31no—
151C22⋊Dic5—31no—
152#685—31no—
153#685—31no—
154#685—31no—
155#685C2×A411yescharacteristic
156#685C2×A411yescharacteristic
157#685C2×A411yescharacteristic
158C4⋊Dic5—32no—
159C4⋊Dic5—31no—
160C4⋊Dic5—31no—
161C4⋊Dic5—31no—
162C4⋊Dic5—31no—
163#683—32no—
164#683—32no—
165#683—31no—
166#683—31no—
167#683—31no—
168#683—31no—
169#683—31no—
170#683C2×A411yescharacteristic
171C4×Dic5—32no—
172C4×Dic5—31no—
173C4×Dic5—31no—
174C4×Dic5—31no—
175C4×Dic5—31no—
176#557—51no—
177#556—151no—
178#554—151no—
179#553—151no—
180#552—151no—
181D4×Q8—51no—
182#548—51no—
183#548—51no—
184#542—151no—
185#541—151no—
186#529—151no—
187#529—151no—
188#520C3×D511yescharacteristic, complemented
189#519—51no—
190C2×C4×Q8—51no—
191C22×C3×C5—41no—
192C2×C3×D5—42no—
193C2×C3×D5—81no—
194C2×C3×D5—81no—
195C2×C3×D5—81no—
196C2×C3×D5—81no—
197C2×C3×D5—81no—
198C3×C4×C5—81no—
199C3×C4×C5—81no—
200C3×C4×C5—81no—
201C3×Dic5—81no—
202C3×Dic5—81no—
203C2×C3×D4—201no—
204C22×C3×C4—201no—
205#41⋊C3—21no—
206C2×SL(2,𝔽3)—101no—
207C2×SL(2,𝔽3)—101no—
208C2×SL(2,𝔽3)C22×D511yescharacteristic
209C3×C4⋊C4—41no—
210C3×C4⋊C4—201no—
211C3×C22⋊C4—201no—
212C3×C22⋊C4—201no—
213C3×C22⋊C4—201no—
214C23×C5—31no—
215C22×D5C22×A411yescharacteristic
216C22×D5C22×A411yescharacteristic
217C5×Q8—31no—
218C5×Q8C3×#4111yescharacteristic
219C5×D4—61no—
220C5×D4—61no—
221C2×C4×C5—31no—
222C2×C4×C5—31no—
223C2×C4×C5—31no—
224C2×C4×C5—61no—
225C2×C4×C5—61no—
226C2×C4×C5—122no—
227C2×C4×C5C22×A411yescharacteristic
228C2×C4×C5C22×A411yescharacteristic
229C2×C4×C5C22×A411yescharacteristic
230C22⋊D5—62no—
231C22⋊D5—121no—
232C2×Dic5—31no—
233C2×Dic5—31no—
234C2×Dic5—31no—
235C2×Dic5—31no—
236C2×Dic5C22×A411yescharacteristic
237C2×Dic5C22×A411yescharacteristic
238D20—22no—
239C4×D5—22no—
240C4×D5—61no—
241C4×D5—61no—
242C4×D5—61no—
243C4×D5—61no—
244C4×D5—121no—
245Dic10—61no—
246Dic10—61no—
247Dic10—61no—
248Dic10—61no—
249Dic10—121no—
250C2×#41C2×C3×D511yescharacteristic
251C22×Q8—51no—
252C22×Q8—51no—
253C4⋊Q8—151no—
254C4⋊Q8—151no—
255#126—151no—
256#126—152no—
257#126—152no—
258#126—151no—
259#125—151no—
260#125—151no—
261#125—152no—
262#125—151no—
263#124—151no—
264#124—151no—
265#124—151no—
266#123—51no—
267#123—151no—
268#123—152no—
269C22⋊Q8—151no—
270C22⋊Q8—152no—
271C22⋊Q8—151no—
272C22⋊Q8—151no—
273C22⋊Q8—151no—
274C22⋊Q8—151no—
275#121—151no—
276#121—151no—
277#121—151no—
278C4×Q8—51no—
279C4×Q8—51no—
280C4×Q8—151no—
281C4×Q8—151no—
282C4×Q8—151no—
283C4×Q8—151no—
284C4×Q8—151no—
285C4×Q8C2×C3×D511yescharacteristic
286C4×Q8C2×C3×D511yescharacteristic
287C4×D4—31no—
288C4×D4—31no—
289C4×D4—151no—
290C4×D4—151no—
291C4×D4—151no—
292C4×D4—152no—
293#117—31no—
294#117—151no—
295#117—152no—
296#117—151no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—151no—
299C2×C42—151no—
300C2×C3×C5—41no—
301C2×C3×C5—41no—
302C2×C3×C5—41no—
303C3×D5—82no—
304C3×D5—161no—
305C23×C3—201no—
306C23×C3—201no—
307C3×D4—402no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—201no—
312C2×C3×C4—201no—
313C2×C3×C4—402no—
314SL(2,𝔽3)#70911yescharacteristic
315C22×C5—61no—
316C22×C5—61no—
317C22×C5—61no—
318C22×C5C23×A411yescharacteristic
319C2×D5—21no—
320C2×D5—21no—
321C2×D5—21no—
322C2×D5—21no—
323C2×D5—21no—
324C2×D52−1+4⋊C312yes—
325C4×C5—21no—
326C4×C5—21no—
327C4×C5—21no—
328C4×C5—31no—
329C4×C5—31no—
330C4×C5—61no—
331C4×C5—61no—
332Dic5—21no—
333Dic5—21no—
334Dic5—61no—
335Dic5—61no—
336Dic5—61no—
337Dic5—61no—
338#41—21no—
339#41—61no—
340C2×Q8—101no—
341C2×Q8—101no—
342C2×Q8—151no—
343C2×Q8—151no—
344C2×Q8—151no—
345C2×Q8—301no—
346C2×Q8—301no—
347C2×Q8C22×C3×D511yescharacteristic
348C2×D4—31no—
349C2×D4—51no—
350C2×D4—151no—
351C2×D4—151no—
352C22×C4—32no—
353C22×C4—31no—
354C22×C4—51no—
355C22×C4—151no—
356C22×C4—151no—
357C22×C4—151no—
358C4⋊C4—31no—
359C4⋊C4—31no—
360C4⋊C4—31no—
361C4⋊C4—51no—
362C4⋊C4—151no—
363C4⋊C4—151no—
364C4⋊C4—151no—
365C4⋊C4—151no—
366C4⋊C4—152no—
367C4⋊C4—152no—
368C4⋊C4—152no—
369C4⋊C4—151no—
370C4⋊C4—151no—
371C4⋊C4—151no—
372C4⋊C4—151no—
373C4⋊C4—151no—
374C4⋊C4D5×A411yescharacteristic
375C22⋊C4—31no—
376C22⋊C4—31no—
377C22⋊C4—51no—
378C22⋊C4—51no—
379C22⋊C4—51no—
380C22⋊C4—151no—
381C22⋊C4—151no—
382C22⋊C4—151no—
383C22⋊C4—151no—
384C22⋊C4—151no—
385C22⋊C4—152no—
386C22⋊C4—152no—
387C42—31no—
388C42—31no—
389C42—151no—
390C42—151no—
391C42—151no—
392C42—152no—
393C42—151no—
394C3×C5—41noHall
395C22×C3—41no—
396C22×C3—202no—
397C22×C3—401no—
398C22×C3—401no—
399C22×C3—401no—
400C22×C3—401no—
401C22×C3—401no—
402C3×C4—81no—
403C3×C4—81no—
404C3×C4—81no—
405C3×C4—401no—
406C3×C4—401no—
407C2×C5—121no—
408C2×C5A4×#4111yescharacteristic
409C2×C5C2×2−1+4⋊C311yescharacteristic
410C2×C5#594511yescharacteristic
411D5—22no—
412D5—41no—
413C23—31no—
414C23—51no—
415C23—51no—
416Q8—31no—
417Q8—301no—
418Q8—301no—
419Q8—301no—
420Q8—301no—
421Q8—601no—
422Q8C3×#70911yescharacteristic
423D4—61no—
424D4—61no—
425D4—102no—
426D4—302no—
427D4—601no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—51no—
432C2×C4—51no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—102no—
436C2×C4—122no—
437C2×C4—151no—
438C2×C4—151no—
439C2×C4—151no—
440C2×C4—151no—
441C2×C4—301no—
442C2×C4—301no—
443C2×C4—301no—
444C2×C4—301no—
445C2×C4—601no—
446C2×C4C2×D5×A411yescharacteristic
447C2×C4C2×D5×A411yescharacteristic
448C2×C4C2×D5×A411yescharacteristic
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—402no—
453C2×C3—801no—
454C5#3516⋊C311yescharacteristic, complemented, Sylow
455C22—52no—
456C22—61no—
457C22—61no—
458C22—61no—
459C22—101no—
460C22—101no—
461C22—101no—
462C22—101no—
463C22—101no—
464C22C22×D5×A411yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—31no—
469C4—31no—
470C4—61no—
471C4—61no—
472C4—101no—
473C4—101no—
474C4—301no—
475C4—301no—
476C4—301no—
477C4—301no—
478C3—41noSylow
479C2—102no—
480C2—121no—
481C2—201no—
482C2A4×#70911yescharacteristic
483C2#1175694411yescharacteristic
484C2D5×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC5×#520C2×C3
socleC22×C5C23×A4
hypercenterC4⋊C4D5×A4
hypocenterC5×Q8C3×#41
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C5×#520⋊C3C2semidirect
2. #66165⋊C3C2semidirect
3. #66165⋊C3C2semidirect
4. #65906⋊C3C2semidirect
5. SL(2,𝔽3)×D20C2semidirect
6. C4×D5×SL(2,𝔽3)C2semidirect
7. #65903⋊C3C2semidirect
8. #10617807C3semidirect
18. C5×#520C2×C3semidirect
19. #66165C2×C3semidirect
20. #66165C2×C3semidirect
22. #65906C2×C3semidirect
23. C4×Q8×D5C2×C3semidirect
24. #65903C2×C3semidirect
48. #520⋊C3D5semidirect
188. #520C3×D5semidirect
454. C5#3516⋊C3semidirect

Maximal subgroups

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

subgroupindex
C5×#520⋊C32
#66165⋊C32
#65906⋊C32
SL(2,𝔽3)×D202
C4×D5×SL(2,𝔽3)2
#65903⋊C32
#106178073
C3×#40404
#3516⋊C35