Finite Group Databasebeta

← #10617809⋊C3 #14514051

Subgroups — #10617809⋊C3 #14514051

Order 1920.

558 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 79 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#66168⋊C3C211yesmaximal, characteristic, complemented
4#65910⋊C3C211yesmaximal, characteristic, complemented
5C4×D5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#65908⋊C3C211yesmaximal, characteristic, complemented
7SL(2,𝔽3)×Dic10C211yesmaximal, characteristic, complemented
8#10617809C311yesmaximal, characteristic, complemented, Hall
9C2×C5×#41⋊C3C2211yescharacteristic
10C2×D5×SL(2,𝔽3)C2211yescharacteristic
11C2×C5⋊(#41⋊C3)C2211yescharacteristic
12C3×#4042—41nomaximal
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14C4×C5×SL(2,𝔽3)C2211yescharacteristic
15Dic5×SL(2,𝔽3)C2211yescharacteristic
16Dic5×SL(2,𝔽3)C2211yescharacteristic
17#3522⋊C3—51nomaximal, Hall
18C5×#520C2×C311yescharacteristic, complemented
19#66168C2×C311yescharacteristic, complemented
20#66165C2×C311yescharacteristic
21#65910C2×C311yescharacteristic
22#65908C2×C311yescharacteristic, complemented
23C4×Q8×D5C2×C311yescharacteristic
24Q8×Dic10C2×C311yescharacteristic, complemented
25#65895—31no—
26#65893—31no—
27#65885—31no—
28#65882—31no—
29#65868—31no—
30#65862—31no—
31#65845—31no—
32#65841—31no—
33C2×C3×C4×D5—41no—
34C2×C3×Dic10—41no—
35C5×#41⋊C3—21no—
36C2×C5×SL(2,𝔽3)C2311yescharacteristic
37D5×SL(2,𝔽3)—21no—
38C5⋊(#41⋊C3)—21no—
39C3×C5×C4⋊C4—41no—
40C3×#685—41no—
41C3×#685—41no—
42C3×#683—41no—
43C3×#683—41no—
44Q8×SL(2,𝔽3)—51no—
45C4⋊(#41⋊C3)—51no—
46#548⋊C3—51no—
47#548⋊C3—51no—
48#520⋊C3—51no—
49#520⋊C3D511yescharacteristic, complemented
50C2×C4×SL(2,𝔽3)—51no—
51C2×C5×#41C22×C311yescharacteristic
52C2×#713C22×C311yescharacteristic
53C2×Q8×D5C22×C311yescharacteristic
54C2×#709—32no—
55C4×C5×Q8C22×C311yescharacteristic
56C4×C5×Q8C22×C311yescharacteristic
57C4×C5×D4—31no—
58C4×C5×D4—31no—
59C5×#117—31no—
60C2×C5×C4⋊C4—31no—
61Q8×Dic5C22×C311yescharacteristic
62Q8×Dic5C22×C311yescharacteristic
63#4086—31no—
64#4084—31no—
65#4084—31no—
66#4082—31no—
67#4076—31no—
68#4075—32no—
69C4×C22⋊D5—32no—
70C4×C22⋊D5—31no—
71C4×C22⋊D5—31no—
72#4070—32no—
73#4070—31no—
74#4044—31no—
75#4043—31no—
76#4042—32no—
77#4042A411yescharacteristic
78#4041—31no—
79#4040—31no—
80#4039—31no—
81D5×C4⋊C4—31no—
82D5×C4⋊C4—31no—
83#4035—31no—
84#4034—31no—
85#4034—31no—
86#4033—31no—
87#4031—32no—
88#4031—31no—
89#4030—31no—
90#4029—32no—
91#4029—31no—
92#4025—31no—
93#4021—31no—
94C4×D20—31no—
95C4×D20—31no—
96#4018—31no—
97C42×D5—31no—
98#4015—31no—
99C4×Dic10—31no—
100C4×Dic10—31no—
101#3522—51noSylow
102C22×C3×D5—41no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C2×C3×Dic5—41no—
107C2×C3×Dic5—41no—
108C2×C3×Dic5—41no—
109C3×C4×D5—82no—
110C3×Dic10—82no—
111C5×SL(2,𝔽3)#4111yescharacteristic, complemented
112C2×#41⋊C3—51no—
113C2×#41⋊C3C2×D511yescharacteristic
114C22×SL(2,𝔽3)—51no—
115C3×C22⋊Q8—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—32no—
128C2×C22⋊D5—31no—
129#713—21no—
130#713—61no—
131Q8×D5—21no—
132Q8×D5—61no—
133#709—124no—
134C2×D20—31no—
135C2×D20—31no—
136C2×C4×D5—31no—
137C2×C4×D5—31no—
138C2×C4×D5—31no—
139C2×C4×D5—32no—
140C2×C4×D5C2×A411yescharacteristic
141C2×Dic10—31no—
142C2×Dic10—32no—
143C2×Dic10C2×A411yescharacteristic
144C5×C4⋊C4—31no—
145C5×C4⋊C4—31no—
146C5×C4⋊C4—31no—
147C5×C4⋊C4C2×A411yescharacteristic
148C5×C22⋊C4—31no—
149C5×C22⋊C4—31no—
150C42×C5—31no—
151C42×C5—31no—
152C22⋊Dic5—32no—
153C22⋊Dic5—31no—
154C22⋊Dic5—31no—
155#685—32no—
156#685—31no—
157#685—31no—
158#685—31no—
159#685—31no—
160#685C2×A411yescharacteristic
161#685C2×A411yescharacteristic
162C4⋊Dic5—31no—
163C4⋊Dic5—31no—
164C4⋊Dic5—31no—
165C4⋊Dic5—31no—
166#683—32no—
167#683—31no—
168#683—31no—
169#683—31no—
170#683—31no—
171#683C2×A411yescharacteristic
172#683C2×A411yescharacteristic
173C4×Dic5—31no—
174C4×Dic5—31no—
175C4×Dic5—31no—
176C4×Dic5—31no—
177#559—151no—
178Q82—51no—
179#554—151no—
180#550—51no—
181#548—51no—
182#548—51no—
183#547—151no—
184#547—151no—
185#542—151no—
186#542—151no—
187#532—151no—
188#529—151no—
189#520—51no—
190#520C3×D511yescharacteristic, complemented
191C2×C4×Q8—51no—
192C22×C3×C5—41no—
193C2×C3×D5—42no—
194C2×C3×D5—81no—
195C2×C3×D5—81no—
196C3×C4×C5—81no—
197C3×C4×C5—81no—
198C3×C4×C5—81no—
199C3×Dic5—42no—
200C3×Dic5—81no—
201C3×Dic5—81no—
202C2×C3×Q8—201no—
203C22×C3×C4—201no—
204#41⋊C3—21no—
205#41⋊C3—101no—
206C2×SL(2,𝔽3)—101no—
207C2×SL(2,𝔽3)C22×D511yescharacteristic
208C3×C4⋊C4—41no—
209C3×C4⋊C4—201no—
210C3×C4⋊C4—201no—
211C3×C22⋊C4—201no—
212C3×C22⋊C4—201no—
213C23×C5—31no—
214C22×D5—31no—
215C22×D5C22×A411yescharacteristic
216C5×Q8—31no—
217C5×Q8C3×#4111yescharacteristic
218C5×D4—61no—
219C5×D4—61no—
220C2×C4×C5—31no—
221C2×C4×C5—31no—
222C2×C4×C5—31no—
223C2×C4×C5—61no—
224C2×C4×C5—61no—
225C2×C4×C5—122no—
226C2×C4×C5C22×A411yescharacteristic
227C2×C4×C5C22×A411yescharacteristic
228C2×C4×C5C22×A411yescharacteristic
229C22⋊D5—62no—
230C22⋊D5—62no—
231C22⋊D5—122no—
232C2×Dic5—31no—
233C2×Dic5—31no—
234C2×Dic5—31no—
235C2×Dic5C22×A411yescharacteristic
236C2×Dic5C22×A411yescharacteristic
237C2×Dic5C22×A411yescharacteristic
238D20—61no—
239D20—61no—
240D20—62no—
241C4×D5—22no—
242C4×D5—61no—
243C4×D5—61no—
244C4×D5—61no—
245C4×D5—61no—
246C4×D5—62no—
247C4×D5—122no—
248Dic10—22no—
249Dic10—61no—
250Dic10—61no—
251Dic10—122no—
252C2×#41—51no—
253C2×#41—152no—
254C2×#41C2×C3×D511yescharacteristic
255C22×Q8—51no—
256C4⋊Q8—151no—
257C4⋊Q8—151no—
258C4⋊Q8—151no—
259#127—151no—
260#126—151no—
261#126—151no—
262#125—151no—
263#124—151no—
264#124—151no—
265#124—152no—
266#124—151no—
267#123—151no—
268#123—152no—
269#123—151no—
270#123—152no—
271C22⋊Q8—51no—
272C22⋊Q8—152no—
273C22⋊Q8—151no—
274C22⋊Q8—152no—
275C22⋊Q8—151no—
276#121—151no—
277#121—151no—
278#121—151no—
279#121—151no—
280#121—151no—
281C4×Q8—51no—
282C4×Q8—51no—
283C4×Q8—151no—
284C4×Q8—151no—
285C4×Q8—151no—
286C4×Q8C2×C3×D511yescharacteristic
287C4×Q8C2×C3×D511yescharacteristic
288C4×D4—31no—
289C4×D4—31no—
290C4×D4—151no—
291C4×D4—151no—
292C4×D4—151no—
293C4×D4—151no—
294C4×D4—152no—
295C4×D4—151no—
296#117—31no—
297#117—151no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—151no—
300C2×C4⋊C4—151no—
301C2×C42—151no—
302C2×C3×C5—41no—
303C2×C3×C5—41no—
304C2×C3×C5—41no—
305C3×D5—82no—
306C23×C3—201no—
307C3×Q8—402no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—201no—
312C2×C3×C4—201no—
313C2×C3×C4—201no—
314C2×C3×C4—402no—
315SL(2,𝔽3)#70911yescharacteristic, complemented
316C22×C5—61no—
317C22×C5—61no—
318C22×C5—61no—
319C22×C5C23×A411yescharacteristic
320C2×D5—21no—
321C2×D5—21no—
322C2×D5—61no—
323C2×D5—61no—
324C2×D5—61no—
325C2×D5#99412yes—
326C4×C5—21no—
327C4×C5—21no—
328C4×C5—21no—
329C4×C5—31no—
330C4×C5—31no—
331C4×C5—61no—
332C4×C5—61no—
333Dic5—21no—
334Dic5—21no—
335Dic5—61no—
336Dic5—61no—
337Dic5—61no—
338Dic52−1+4⋊C312yes—
339#41—21no—
340#41—61no—
341#41—101no—
342#41—301no—
343#41—604no—
344C2×Q8—51no—
345C2×Q8—101no—
346C2×Q8—152no—
347C2×Q8—151no—
348C2×Q8—301no—
349C2×Q8C22×C3×D511yescharacteristic
350C2×D4—31no—
351C2×D4—151no—
352C2×D4—151no—
353C2×D4—151no—
354C2×D4—151no—
355C2×D4—152no—
356C22×C4—32no—
357C22×C4—31no—
358C22×C4—51no—
359C22×C4—151no—
360C22×C4—151no—
361C22×C4—151no—
362C22×C4—152no—
363C4⋊C4—31no—
364C4⋊C4—31no—
365C4⋊C4—31no—
366C4⋊C4—51no—
367C4⋊C4—51no—
368C4⋊C4—151no—
369C4⋊C4—151no—
370C4⋊C4—151no—
371C4⋊C4—151no—
372C4⋊C4—151no—
373C4⋊C4—151no—
374C4⋊C4—152no—
375C4⋊C4—151no—
376C4⋊C4—151no—
377C4⋊C4D5×A411yescharacteristic
378C22⋊C4—31no—
379C22⋊C4—31no—
380C22⋊C4—51no—
381C22⋊C4—51no—
382C22⋊C4—151no—
383C22⋊C4—152no—
384C22⋊C4—151no—
385C22⋊C4—151no—
386C22⋊C4—152no—
387C22⋊C4—151no—
388C22⋊C4—151no—
389C22⋊C4—151no—
390C42—31no—
391C42—31no—
392C42—151no—
393C42—151no—
394C42—151no—
395C42—151no—
396C3×C5—41noHall
397C22×C3—41no—
398C22×C3—202no—
399C22×C3—401no—
400C22×C3—401no—
401C3×C4—81no—
402C3×C4—81no—
403C3×C4—81no—
404C3×C4—202no—
405C3×C4—401no—
406C3×C4—401no—
407C2×C5—121no—
408C2×C5A4×#4111yescharacteristic
409C2×C5C2×#99411yescharacteristic
410C2×C5C2×2−1+4⋊C311yescharacteristic
411D5—22no—
412D5—121no—
413C23—31no—
414C23—51no—
415C23—151no—
416Q8—31no—
417Q8—102no—
418Q8—301no—
419Q8—301no—
420Q8—602no—
421Q8C3×#70911yescharacteristic
422D4—61no—
423D4—61no—
424D4—301no—
425D4—301no—
426D4—302no—
427D4—302no—
428D4—302no—
429D4—602no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—51no—
434C2×C4—51no—
435C2×C4—51no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—102no—
439C2×C4—122no—
440C2×C4—151no—
441C2×C4—151no—
442C2×C4—151no—
443C2×C4—301no—
444C2×C4—301no—
445C2×C4—301no—
446C2×C4—301no—
447C2×C4—302no—
448C2×C4—602no—
449C2×C4C2×D5×A411yescharacteristic
450C2×C4C2×D5×A411yescharacteristic
451C2×C4C2×D5×A411yescharacteristic
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—402no—
456C5#3522⋊C311yescharacteristic, complemented, Sylow
457C22—52no—
458C22—61no—
459C22—61no—
460C22—61no—
461C22—101no—
462C22—101no—
463C22—301no—
464C22—301no—
465C22—301no—
466C22C22×D5×A411yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—31no—
471C4—31no—
472C4—52no—
473C4—61no—
474C4—61no—
475C4—101no—
476C4—101no—
477C4—301no—
478C4—301no—
479C4—301no—
480C3—41noSylow
481C2—102no—
482C2—121no—
483C2—601no—
484C2A4×#70911yescharacteristic
485C2D5×#99411yescharacteristic
486C2D5×2−1+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. #66168⋊C3C2semidirect
4. #65910⋊C3C2semidirect
5. C4×D5×SL(2,𝔽3)C2semidirect
6. #65908⋊C3C2semidirect
7. SL(2,𝔽3)×Dic10C2semidirect
8. #10617809C3semidirect
18. C5×#520C2×C3semidirect
19. #66168C2×C3semidirect
22. #65908C2×C3semidirect
24. Q8×Dic10C2×C3semidirect
49. #520⋊C3D5semidirect
111. C5×SL(2,𝔽3)#41semidirect
190. #520C3×D5semidirect
315. SL(2,𝔽3)#709semidirect
456. C5#3522⋊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
#66168⋊C32
#65910⋊C32
C4×D5×SL(2,𝔽3)2
#65908⋊C32
SL(2,𝔽3)×Dic102
#106178093
C3×#40424
#3522⋊C35