Finite Group Databasebeta

← #10617784⋊C3 #14514047

Subgroups — #10617784⋊C3 #14514047

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×#519⋊C3C211yesmaximal, characteristic, complemented
2SL(2,𝔽3)×C22⋊D5C211yesmaximal, characteristic, complemented
3#66151⋊C3C211yesmaximal, characteristic, complemented
4#65910⋊C3C211yesmaximal, characteristic, complemented
5C4×D5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#65904⋊C3C211yesmaximal, characteristic, complemented
7#65908⋊C3C211yesmaximal, characteristic, complemented
8#10617784C311yesmaximal, characteristic, complemented, Hall
9C22×C5×SL(2,𝔽3)C2211yescharacteristic
10C2×D5×SL(2,𝔽3)C2211yescharacteristic
11C2×C5⋊(#41⋊C3)C2211yescharacteristic
12C3×#4029—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×#519C2×C311yescharacteristic, complemented
19#66151C2×C311yescharacteristic, complemented
20Q8×C22⋊D5C2×C311yescharacteristic
21#65917—31no—
22#65914—31no—
23#65913—31no—
24#65910C2×C311yescharacteristic, complemented
25#65908C2×C311yescharacteristic, complemented
26C4×Q8×D5C2×C311yescharacteristic, complemented
27#65904C2×C311yescharacteristic, complemented
28#65902—31no—
29#65868—31no—
30#65864—31no—
31#65863—31no—
32#65862—31no—
33C2×C3×C22⋊D5—41no—
34C2×C3×C4×D5—41no—
35C2×C5×SL(2,𝔽3)—21no—
36C2×C5×SL(2,𝔽3)C2311yescharacteristic
37D5×SL(2,𝔽3)—21no—
38C5⋊(#41⋊C3)—21no—
39C3×C5×C22⋊C4—41no—
40C3×C22⋊Dic5—41no—
41C3×#685—41no—
42C3×C4⋊Dic5—41no—
43C3×#683—41no—
44#557⋊C3—51no—
45D4×SL(2,𝔽3)—51no—
46#548⋊C3—51no—
47#548⋊C3—51no—
48#520⋊C3—51no—
49#519⋊C3D511yescharacteristic, complemented
50C2×C4×SL(2,𝔽3)—51no—
51C22×C5×Q8C22×C311yescharacteristic
52C2×#713C22×C311yescharacteristic
53C2×Q8×D5C22×C311yescharacteristic
54C4×C5×Q8—31no—
55C4×C5×Q8—31no—
56C4×C5×Q8C22×C311yescharacteristic
57C4×C5×Q8C22×C311yescharacteristic
58C5×#117—31no—
59C5×#117—31no—
60#4093—31no—
61#4092—31no—
62#4092—31no—
63Q8×Dic5C22×C311yescharacteristic
64Q8×Dic5C22×C311yescharacteristic
65#4090—31no—
66#4076—31no—
67C4×C22⋊D5—31no—
68C4×C22⋊D5—31no—
69#4070—31no—
70#4044—32no—
71#4044—31no—
72#4043—31no—
73#4042—32no—
74#4042—31no—
75#4041—31no—
76#4040—31no—
77#4039—31no—
78D5×C4⋊C4—31no—
79D5×C4⋊C4—31no—
80#4036—31no—
81#4036—31no—
82#4035—31no—
83#4033—31no—
84#4031—31no—
85#4030—31no—
86#4029—32no—
87#4029A411yescharacteristic
88#4025—31no—
89#4022—32no—
90#4021—31no—
91C4×D20—32no—
92C4×D20—31no—
93C4×D20—31no—
94#4018—32no—
95#4018—31no—
96C42×D5—31no—
97#4016—32no—
98#4015—31no—
99C4×Dic10—31no—
100C4×Dic10—31no—
101#3516—51noSylow
102C23×C3×C5—41no—
103C22×C3×D5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C3×C22⋊D5—82no—
107C2×C3×Dic5—41no—
108C2×C3×Dic5—41no—
109C2×C3×Dic5—41no—
110C3×C4×D5—82no—
111C5×SL(2,𝔽3)#4111yescharacteristic
112C2×#41⋊C3—51no—
113C22×SL(2,𝔽3)—51no—
114C22×SL(2,𝔽3)C2×D511yescharacteristic
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
120C2×C5×Q8—21no—
121C2×C5×Q8—31no—
122C2×C5×Q8—61no—
123C2×C5×Q8C23×C311yescharacteristic
124C22×C4×C5—31no—
125C2×C22⋊D5—31no—
126C2×C22⋊D5C2×A411yescharacteristic
127#713—21no—
128#713—61no—
129Q8×D5—21no—
130Q8×D5—61no—
131C2×D20—31no—
132C2×D20—31no—
133C2×C4×D5—31no—
134C2×C4×D5—31no—
135C2×C4×D5—32no—
136C2×C4×D5—31no—
137C2×C4×D5C2×A411yescharacteristic
138C2×Dic10—31no—
139C2×Dic10—31no—
140C5×C4⋊C4—31no—
141C5×C4⋊C4—31no—
142C5×C4⋊C4—31no—
143C5×C4⋊C4—31no—
144C5×C22⋊C4—31no—
145C5×C22⋊C4C2×A411yescharacteristic
146C42×C5—32no—
147C42×C5—31no—
148C42×C5—31no—
149C22⋊Dic5—31no—
150C22⋊Dic5C2×A411yescharacteristic
151#685—32no—
152#685—32no—
153#685—31no—
154#685—31no—
155#685—31no—
156#685—31no—
157#685—31no—
158#685C2×A411yescharacteristic
159C4⋊Dic5—32no—
160C4⋊Dic5—31no—
161C4⋊Dic5—31no—
162C4⋊Dic5—31no—
163C4⋊Dic5C2×A411yescharacteristic
164#683—32no—
165#683—32no—
166#683—31no—
167#683—31no—
168#683—31no—
169#683—31no—
170#683—31no—
171#683C2×A411yescharacteristic
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#520—51no—
189#519C3×D511yescharacteristic, complemented
190C2×C4×Q8—51no—
191C22×C3×C5—41no—
192C22×C3×C5—81no—
193C22×C3×C5—81no—
194C22×C3×C5—81no—
195C2×C3×D5—42no—
196C2×C3×D5—81no—
197C2×C3×D5—81no—
198C3×C4×C5—81no—
199C3×C4×C5—81no—
200C3×Dic5—81no—
201C3×Dic5—81no—
202C3×Dic5—81no—
203C2×C3×D4—201no—
204C22×C3×C4—201no—
205#41⋊C3—101no—
206C2×SL(2,𝔽3)—21no—
207C2×SL(2,𝔽3)—101no—
208C2×SL(2,𝔽3)C22×D511yescharacteristic
209C3×C4⋊C4—201no—
210C3×C4⋊C4—201no—
211C3×C22⋊C4—41no—
212C3×C22⋊C4—201no—
213C3×C22⋊C4—201no—
214C23×C5C22×A411yescharacteristic
215C22×D5—31no—
216C22×D5C22×A411yescharacteristic
217C5×Q8—31no—
218C5×Q8—61no—
219C5×Q8—61no—
220C5×Q8C3×#4111yescharacteristic
221C2×C4×C5—31no—
222C2×C4×C5—31no—
223C2×C4×C5—31no—
224C2×C4×C5—31no—
225C2×C4×C5—61no—
226C2×C4×C5—61no—
227C2×C4×C5C22×A411yescharacteristic
228C2×C4×C5C22×A411yescharacteristic
229C22⋊D5—22no—
230C22⋊D5—121no—
231C2×Dic5—31no—
232C2×Dic5—31no—
233C2×Dic5—31no—
234C2×Dic5C22×A411yescharacteristic
235C2×Dic5C22×A411yescharacteristic
236C2×Dic5C22×A411yescharacteristic
237D20—61no—
238D20—61no—
239D20—62no—
240C4×D5—22no—
241C4×D5—61no—
242C4×D5—61no—
243C4×D5—61no—
244C4×D5—61no—
245C4×D5—121no—
246C4×D5—122no—
247Dic10—61no—
248Dic10—61no—
249Dic10—121no—
250C2×#41—51no—
251C22×Q8—51no—
252C22×Q8C2×C3×D511yescharacteristic
253C4⋊Q8—151no—
254C4⋊Q8—151no—
255#126—152no—
256#126—151no—
257#126—151no—
258#126—152no—
259#125—152no—
260#125—151no—
261#125—151no—
262#125—151no—
263#124—151no—
264#124—151no—
265#124—151no—
266#123—51no—
267#123—152no—
268#123—151no—
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—31no—
279C4×Q8—31no—
280C4×Q8—51no—
281C4×Q8—51no—
282C4×Q8—151no—
283C4×Q8—151no—
284C4×Q8—151no—
285C4×Q8C2×C3×D511yescharacteristic
286C4×Q8C2×C3×D511yescharacteristic
287C4×D4—151no—
288C4×D4—152no—
289C4×D4—151no—
290C4×D4—151no—
291C4×D4—151no—
292C4×D4—151no—
293#117—31no—
294#117—31no—
295#117—151no—
296#117—152no—
297C2×C4⋊C4—151no—
298C2×C4⋊C4—151no—
299C2×C42—151no—
300C2×C3×C5—41no—
301C2×C3×C5—41no—
302C2×C3×C5—41no—
303C2×C3×C5—161no—
304C3×D5—82no—
305C23×C3—41no—
306C23×C3—201no—
307C3×D4—402no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—201no—
311C2×C3×C4—201no—
312C2×C3×C4—201no—
313C2×C3×C4—402no—
314SL(2,𝔽3)#70911yescharacteristic
315C22×C5—21no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5C23×A411yescharacteristic
319C2×D5—21no—
320C2×D5—21no—
321C2×D5—61no—
322C2×D5—61no—
323C2×D5—61no—
324C2×D52−1+4⋊C312yes—
325C4×C5—21no—
326C4×C5—21no—
327C4×C5—31no—
328C4×C5—31no—
329C4×C5—61no—
330C4×C5—61no—
331C4×C5—61no—
332Dic5—21no—
333Dic5—21no—
334Dic5—21no—
335Dic5—61no—
336Dic5—61no—
337Dic5—61no—
338#41—101no—
339#41—301no—
340C2×Q8—21no—
341C2×Q8—31no—
342C2×Q8—61no—
343C2×Q8—101no—
344C2×Q8—151no—
345C2×Q8—151no—
346C2×Q8—301no—
347C2×Q8C22×C3×D511yescharacteristic
348C2×D4—51no—
349C2×D4—151no—
350C2×D4—151no—
351C2×D4—151no—
352C22×C4—31no—
353C22×C4—51no—
354C22×C4—151no—
355C22×C4—151no—
356C22×C4—151no—
357C22×C4—152no—
358C4⋊C4—31no—
359C4⋊C4—31no—
360C4⋊C4—31no—
361C4⋊C4—31no—
362C4⋊C4—51no—
363C4⋊C4—51no—
364C4⋊C4—151no—
365C4⋊C4—151no—
366C4⋊C4—151no—
367C4⋊C4—151no—
368C4⋊C4—151no—
369C4⋊C4—151no—
370C4⋊C4—151no—
371C4⋊C4—151no—
372C4⋊C4—152no—
373C4⋊C4—152no—
374C4⋊C4—152no—
375C22⋊C4—31no—
376C22⋊C4—51no—
377C22⋊C4—51no—
378C22⋊C4—151no—
379C22⋊C4—151no—
380C22⋊C4—151no—
381C22⋊C4—152no—
382C22⋊C4—152no—
383C22⋊C4—151no—
384C22⋊C4—151no—
385C22⋊C4—151no—
386C22⋊C4D5×A411yescharacteristic
387C42—32no—
388C42—31no—
389C42—31no—
390C42—151no—
391C42—151no—
392C42—151no—
393C42—151no—
394C3×C5—41noHall
395C22×C3—41no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—202no—
400C22×C3—401no—
401C22×C3—401no—
402C3×C4—81no—
403C3×C4—81no—
404C3×C4—401no—
405C3×C4—401no—
406C3×C4—401no—
407C2×C5—41no—
408C2×C5A4×#4111yescharacteristic
409C2×C5C2×2−1+4⋊C311yescharacteristic
410C2×C5#594511yescharacteristic
411D5—22no—
412D5—121no—
413C23—51no—
414C23—151no—
415C23C2×D5×A411yescharacteristic
416Q8—31no—
417Q8—61no—
418Q8—61no—
419Q8—301no—
420Q8—301no—
421Q8—601no—
422Q8C3×#70911yescharacteristic
423D4—102no—
424D4—301no—
425D4—301no—
426D4—302no—
427D4—601no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—51no—
433C2×C4—51no—
434C2×C4—51no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—102no—
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×C4—602no—
447C2×C4C2×D5×A411yescharacteristic
448C2×C4C2×D5×A411yescharacteristic
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—161no—
453C2×C3—402no—
454C5#3516⋊C311yescharacteristic, complemented, Sylow
455C22—21no—
456C22—21no—
457C22—21no—
458C22—52no—
459C22—101no—
460C22—101no—
461C22—301no—
462C22—301no—
463C22—301no—
464C22C22×D5×A411yescharacteristic
465C4—21no—
466C4—21no—
467C4—31no—
468C4—31no—
469C4—61no—
470C4—61no—
471C4—61no—
472C4—101no—
473C4—101no—
474C4—101no—
475C4—301no—
476C4—301no—
477C4—301no—
478C3—41noSylow
479C2—41no—
480C2—102no—
481C2—601no—
482C2A4×#70911yescharacteristic
483C2D5×2−1+4⋊C311yescharacteristic
484C2#66287⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC5×#519C2×C3
socleC22×C5C23×A4
hypercenterC22⋊C4D5×A4
hypocenterC5×Q8C3×#41
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C5×#519⋊C3C2semidirect
2. SL(2,𝔽3)×C22⋊D5C2semidirect
3. #66151⋊C3C2semidirect
4. #65910⋊C3C2semidirect
5. C4×D5×SL(2,𝔽3)C2semidirect
6. #65904⋊C3C2semidirect
7. #65908⋊C3C2semidirect
8. #10617784C3semidirect
18. C5×#519C2×C3semidirect
19. #66151C2×C3semidirect
24. #65910C2×C3semidirect
25. #65908C2×C3semidirect
26. C4×Q8×D5C2×C3semidirect
27. #65904C2×C3semidirect
49. #519⋊C3D5semidirect
189. #519C3×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×#519⋊C32
SL(2,𝔽3)×C22⋊D52
#66151⋊C32
#65910⋊C32
C4×D5×SL(2,𝔽3)2
#65904⋊C32
#65908⋊C32
#106177843
C3×#40294
#3516⋊C35