Finite Group Databasebeta

← #10617797⋊C3 #14514053

Subgroups — #10617797⋊C3 #14514053

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#66168⋊C3C211yesmaximal, characteristic, complemented
3#66168⋊C3C211yesmaximal, characteristic, complemented
4C4⋊(C5⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
5#65903⋊C3C211yesmaximal, characteristic, complemented
6#65908⋊C3C211yesmaximal, characteristic, complemented
7C4×C5⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
8#10617797C311yesmaximal, characteristic, complemented, Hall
9C2×C5×#41⋊C3C2211yescharacteristic
10C2×C5⋊(#41⋊C3)C2211yescharacteristic
11C2×C5⋊(#41⋊C3)C2211yescharacteristic
12C3×#4035—41nomaximal
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14C4×C5×SL(2,𝔽3)C2211yescharacteristic
15Dic5×SL(2,𝔽3)C2211yescharacteristic
16Dic5×SL(2,𝔽3)C2211yescharacteristic
17#3517⋊C3—51nomaximal, Hall
18C5×#520C2×C311yescharacteristic
19#66168C2×C311yescharacteristic
20#66168C2×C311yescharacteristic
21#65911C2×C311yescharacteristic
22#65908C2×C311yescharacteristic
23C4×#713C2×C311yescharacteristic
24#65903C2×C311yescharacteristic
25#65896—31no—
26#65892—31no—
27#65887—31no—
28#65882—31no—
29#65869—31no—
30#65862—31no—
31#65848—31no—
32#65841—31no—
33C5×#41⋊C3—21no—
34C2×C5×SL(2,𝔽3)C2311yescharacteristic
35C5⋊(#41⋊C3)—21no—
36C5⋊(#41⋊C3)—21no—
37C3×C5×C4⋊C4—41no—
38C3×C4⋊Dic5—41no—
39C3×#683—41no—
40C3×#683—41no—
41C3×#683—41no—
42C3×#683—41no—
43C3×C4×Dic5—41no—
44#557⋊C3—51no—
45C4⋊(#41⋊C3)—51no—
46C4⋊(#41⋊C3)—51no—
47C4⋊(#41⋊C3)—51no—
48#520⋊C3—51no—
49#520⋊C3D511yescharacteristic, complemented
50C4×#41⋊C3—51no—
51C2×C5×#41C22×C311yescharacteristic
52C2×#713C22×C311yescharacteristic
53C2×#713C22×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#4086—31no—
65#4084—31no—
66#4084—31no—
67#4076—32no—
68#4076—31no—
69#4076—31no—
70#4075—32no—
71C4×C22⋊D5—32no—
72C4×C22⋊D5—31no—
73C4×C22⋊D5—31no—
74#4044—31no—
75#4042—31no—
76#4041—31no—
77#4041—31no—
78#4040—32no—
79#4040—31no—
80#4039—31no—
81#4039—31no—
82#4038—31no—
83D5×C4⋊C4—31no—
84#4036—31no—
85#4035A411yescharacteristic
86#4034—31no—
87#4031—31no—
88#4030—32no—
89#4030—32no—
90#4030—31no—
91#4029—31no—
92#4025—31no—
93#4020—31no—
94C4×D20—31no—
95C4×D20—31no—
96C4×D20—31no—
97#4018—31no—
98C42×D5—31no—
99#4016—31no—
100C4×Dic10—31no—
101#3517—51noSylow
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×Dic5—41no—
106C2×C3×Dic5—41no—
107C2×C3×Dic5—41no—
108C2×C3×Dic5—41no—
109C5×SL(2,𝔽3)#4111yescharacteristic, complemented
110C2×#41⋊C3—51no—
111C2×#41⋊C3—51no—
112C2×#41⋊C3C2×D511yescharacteristic
113C3×#125—201no—
114C4×SL(2,𝔽3)—51no—
115C4×SL(2,𝔽3)—51no—
116C4×SL(2,𝔽3)C2×D511yescharacteristic
117C4×SL(2,𝔽3)C2×D511yescharacteristic
118C5×#41—21no—
119C5×#41—61no—
120C2×C5×Q8C23×C311yescharacteristic
121C2×C5×D4—31no—
122C22×C4×C5—32no—
123C22×C4×C5—31no—
124C2×C22⋊D5—32no—
125C2×C22⋊D5—32no—
126C2×C22⋊D5—31no—
127C2×C22⋊D5—31no—
128#713—21no—
129#713—21no—
130#713—61no—
131#713—61no—
132#709—124no—
133C2×D20—31no—
134C2×D20—31no—
135C2×D20—32no—
136C2×D20—31no—
137C2×C4×D5—31no—
138C2×C4×D5—31no—
139C2×C4×D5—31no—
140C2×C4×D5—32no—
141C2×C4×D5—31no—
142C2×Dic10—31no—
143C5×C4⋊C4—31no—
144C5×C4⋊C4—31no—
145C5×C4⋊C4—31no—
146C5×C4⋊C4C2×A411yescharacteristic
147C5×C22⋊C4—31no—
148C5×C22⋊C4—31no—
149C42×C5—31no—
150C42×C5—31no—
151C22⋊Dic5—31no—
152C22⋊Dic5—31no—
153#685—32no—
154#685—31no—
155#685—31no—
156#685—31no—
157#685—31no—
158#685—31no—
159#685—31no—
160C4⋊Dic5—31no—
161C4⋊Dic5—31no—
162C4⋊Dic5—31no—
163C4⋊Dic5C2×A411yescharacteristic
164#683—32no—
165#683—31no—
166#683—31no—
167#683C2×A411yescharacteristic
168#683C2×A411yescharacteristic
169#683C2×A411yescharacteristic
170#683C2×A411yescharacteristic
171C4×Dic5—31no—
172C4×Dic5—31no—
173C4×Dic5—31no—
174C4×Dic5C2×A411yescharacteristic
175#557—51no—
176#553—151no—
177#553—151no—
178#550—51no—
179#550—51no—
180#550—51no—
181#550—151no—
182#547—151no—
183#540—151no—
184#540—151no—
185#532—151no—
186#529—151no—
187#520—51no—
188#520C3×D511yescharacteristic
189C4×#41—51no—
190C22×C3×C5—41no—
191C3×C4×C5—81no—
192C3×C4×C5—81no—
193C3×C4×C5—81no—
194C3×Dic5—42no—
195C3×Dic5—81no—
196C3×Dic5—81no—
197C3×Dic5—81no—
198#41⋊C3—21no—
199#41⋊C3—101no—
200#41⋊C3—101no—
201C2×SL(2,𝔽3)C22×D511yescharacteristic
202C3×C4⋊C4—41no—
203C3×C4⋊C4—201no—
204C3×C4⋊C4—201no—
205C3×C4⋊C4—201no—
206C3×C4⋊C4—201no—
207C3×C4⋊C4—201no—
208C3×C42—201no—
209C23×C5—31no—
210C22×D5—31no—
211C22×D5—31no—
212C5×Q8—31no—
213C5×Q8C3×#4111yescharacteristic
214C5×D4—61no—
215C5×D4—61no—
216C2×C4×C5—31no—
217C2×C4×C5—31no—
218C2×C4×C5—31no—
219C2×C4×C5—61no—
220C2×C4×C5—61no—
221C2×C4×C5—122no—
222C2×C4×C5C22×A411yescharacteristic
223C2×C4×C5C22×A411yescharacteristic
224C2×C4×C5C22×A411yescharacteristic
225C22⋊D5—62no—
226C22⋊D5—121no—
227C22⋊D5—122no—
228C22⋊D5—122no—
229C2×Dic5—31no—
230C2×Dic5—31no—
231C2×Dic5C22×A411yescharacteristic
232C2×Dic5C22×A411yescharacteristic
233C2×Dic5C22×A411yescharacteristic
234C2×Dic5C22×A411yescharacteristic
235D20—61no—
236D20—61no—
237D20—61no—
238D20—61no—
239D20—121no—
240D20—122no—
241C4×D5—61no—
242C4×D5—61no—
243C4×D5—61no—
244C4×D5—61no—
245C4×D5—62no—
246C4×D5—121no—
247C4×D5—122no—
248Dic10—62no—
249C2×#41—51no—
250C2×#41—51no—
251C2×#41—152no—
252C2×#41C2×C3×D511yescharacteristic
253C4⋊Q8—151no—
254#127—151no—
255#127—151no—
256#127—151no—
257#126—151no—
258#126—151no—
259#125—51no—
260#125—151no—
261#125—151no—
262#124—151no—
263#123—151no—
264#123—152no—
265#123—152no—
266#123—151no—
267C22⋊Q8—151no—
268#121—151no—
269#121—152no—
270#121—152no—
271#121—152no—
272#121—151no—
273#121—151no—
274#121—151no—
275#121—151no—
276#121—151no—
277#121—151no—
278C4×Q8—51no—
279C4×Q8—51no—
280C4×Q8—151no—
281C4×Q8C2×C3×D511yescharacteristic
282C4×Q8C2×C3×D511yescharacteristic
283C4×D4—31no—
284C4×D4—31no—
285C4×D4—151no—
286C4×D4—151no—
287C4×D4—151no—
288C4×D4—151no—
289C4×D4—151no—
290C4×D4—151no—
291C4×D4—152no—
292C4×D4—151no—
293#117—31no—
294#117—151no—
295#117—151no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—151no—
298C2×C42—151no—
299C2×C3×C5—41no—
300C2×C3×C5—41no—
301C2×C3×C5—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—201no—
306C2×C3×C4—201no—
307C2×C3×C4—201no—
308C2×C3×C4—201no—
309SL(2,𝔽3)#70911yescharacteristic, complemented
310C22×C5—61no—
311C22×C5—61no—
312C22×C5—61no—
313C22×C5C23×A411yescharacteristic
314C2×D5—61no—
315C2×D5—61no—
316C2×D5—61no—
317C2×D5—61no—
318C2×D5—61no—
319C2×D5—61no—
320C4×C5—21no—
321C4×C5—21no—
322C4×C5—21no—
323C4×C5—31no—
324C4×C5—31no—
325C4×C5—61no—
326C4×C5—61no—
327Dic5—21no—
328Dic5—21no—
329Dic5—21no—
330Dic5—61no—
331Dic5—61no—
332Dic5#99412yes—
333#41—21no—
334#41—61no—
335#41—101no—
336#41—101no—
337#41—301no—
338#41—301no—
339#41—604no—
340C2×Q8—151no—
341C2×Q8C22×C3×D511yescharacteristic
342C2×D4—31no—
343C2×D4—151no—
344C2×D4—151no—
345C2×D4—152no—
346C2×D4—151no—
347C2×D4—151no—
348C2×D4—152no—
349C2×D4—152no—
350C2×D4—151no—
351C22×C4—32no—
352C22×C4—31no—
353C22×C4—151no—
354C22×C4—151no—
355C22×C4—151no—
356C22×C4—152no—
357C22×C4—151no—
358C4⋊C4—31no—
359C4⋊C4—31no—
360C4⋊C4—31no—
361C4⋊C4—51no—
362C4⋊C4—51no—
363C4⋊C4—51no—
364C4⋊C4—51no—
365C4⋊C4—51no—
366C4⋊C4—151no—
367C4⋊C4—151no—
368C4⋊C4—151no—
369C4⋊C4—151no—
370C4⋊C4—152no—
371C4⋊C4—151no—
372C4⋊C4D5×A411yescharacteristic
373C22⋊C4—31no—
374C22⋊C4—31no—
375C22⋊C4—151no—
376C22⋊C4—151no—
377C22⋊C4—151no—
378C22⋊C4—152no—
379C22⋊C4—151no—
380C22⋊C4—151no—
381C22⋊C4—151no—
382C22⋊C4—151no—
383C22⋊C4—151no—
384C42—31no—
385C42—31no—
386C42—51no—
387C42—151no—
388C42—151no—
389C42—151no—
390C3×C5—41noHall
391C22×C3—41no—
392C3×C4—81no—
393C3×C4—81no—
394C3×C4—81no—
395C3×C4—202no—
396C3×C4—401no—
397C3×C4—401no—
398C3×C4—401no—
399C2×C5—121no—
400C2×C5A4×#4111yescharacteristic
401C2×C5C2×#99411yescharacteristic
402C2×C5#594511yescharacteristic
403D5—121no—
404D5—121no—
405C23—31no—
406C23—151no—
407C23—151no—
408Q8—31no—
409Q8—302no—
410Q8C3×#70911yescharacteristic
411D4—61no—
412D4—61no—
413D4—301no—
414D4—301no—
415D4—301no—
416D4—301no—
417D4—302no—
418D4—601no—
419D4—602no—
420D4—601no—
421D4—602no—
422D4—602no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—31no—
426C2×C4—51no—
427C2×C4—51no—
428C2×C4—51no—
429C2×C4—51no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—122no—
433C2×C4—151no—
434C2×C4—151no—
435C2×C4—301no—
436C2×C4—301no—
437C2×C4—301no—
438C2×C4—301no—
439C2×C4—302no—
440C2×C4—601no—
441C2×C4—602no—
442C2×C4C2×D5×A411yescharacteristic
443C2×C4C2×D5×A411yescharacteristic
444C2×C4C2×D5×A411yescharacteristic
445C2×C3—41no—
446C2×C3—41no—
447C2×C3—41no—
448C5#3517⋊C311yescharacteristic, complemented, Sylow
449C22—61no—
450C22—61no—
451C22—61no—
452C22—301no—
453C22—301no—
454C22—301no—
455C22—301no—
456C22—301no—
457C22—301no—
458C22C22×D5×A411yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—31no—
463C4—31no—
464C4—52no—
465C4—61no—
466C4—61no—
467C4—101no—
468C4—101no—
469C4—101no—
470C4—301no—
471C4—301no—
472C3—41noSylow
473C2—121no—
474C2—601no—
475C2—601no—
476C2A4×#70911yescharacteristic
477C2D5×#99411yescharacteristic
478C2#66287⋊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. #66168⋊C3C2semidirect
3. #66168⋊C3C2semidirect
4. C4⋊(C5⋊(#41⋊C3))C2semidirect
5. #65903⋊C3C2semidirect
6. #65908⋊C3C2semidirect
7. C4×C5⋊(#41⋊C3)C2semidirect
8. #10617797C3semidirect
49. #520⋊C3D5semidirect
109. C5×SL(2,𝔽3)#41semidirect
309. SL(2,𝔽3)#709semidirect
448. C5#3517⋊C3semidirect

Maximal subgroups

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

subgroupindex
C5×#520⋊C32
#66168⋊C32
C4⋊(C5⋊(#41⋊C3))2
#65903⋊C32
#65908⋊C32
C4×C5⋊(#41⋊C3)2
#106177973
C3×#40354
#3517⋊C35