Finite Group Databasebeta

← #10617774⋊C3 #14514045

Subgroups — #10617774⋊C3 #14514045

Order 1920.

744 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×#4147⋊C3C211yesmaximal, characteristic
2C5×#519⋊C3C211yesmaximal, characteristic, complemented
3C2×Dic5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#65906⋊C3C211yesmaximal, characteristic, complemented
5C4×D5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#65908⋊C3C211yesmaximal, characteristic, complemented
7C4×C5⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
8#10617774C311yesmaximal, characteristic, complemented, Hall
9C22×C5×SL(2,𝔽3)C2211yescharacteristic
10#4147⋊C3C414yescomplemented
11C2×D5×SL(2,𝔽3)C2211yescharacteristic
12C2×C5⋊(#41⋊C3)C2211yescharacteristic
13C3×#4027—41nomaximal
14C4×C5×SL(2,𝔽3)C2211yescharacteristic
15C4×C5×SL(2,𝔽3)C2211yescharacteristic
16Dic5×SL(2,𝔽3)C2211yescharacteristic
17Dic5×SL(2,𝔽3)C2211yescharacteristic
18C4×2−1+4⋊C3—51nomaximal, Hall
19C2×#4147C2×C311yescharacteristic
20C5×#519C2×C311yescharacteristic, complemented
21C2×Q8×Dic5C2×C311yescharacteristic, complemented
22#65908—31no—
23#65908C2×C311yescharacteristic, complemented
24C4×#713—31no—
25C4×#713C2×C311yescharacteristic, complemented
26#65906—31no—
27#65906C2×C311yescharacteristic, complemented
28C4×Q8×D5—31no—
29C4×Q8×D5C2×C311yescharacteristic, complemented
30#65857—31no—
31#65857—31no—
32#65856—31no—
33#65856—31no—
34C2×C3×C22⋊D5—41no—
35C22×C3×Dic5—41no—
36C2×C3×C4×D5—41no—
37C2×C5×SL(2,𝔽3)C2×C412yescomplemented
38C2×C5×SL(2,𝔽3)C2311yescharacteristic
39D5×SL(2,𝔽3)C2×C412yescomplemented
40C5⋊(#41⋊C3)C2×C412yescomplemented
41C3×C5×C22⋊C4—41no—
42C3×#685—41no—
43C3×#683—41no—
44C3×C4×Dic5—41no—
45C2×2−1+4⋊C3—51no—
46#520⋊C3—51no—
47#519⋊C3—51no—
48#519⋊C3D511yescharacteristic, complemented
49C4×#41⋊C3—51no—
50C2×C4×SL(2,𝔽3)—51no—
51C2×C4×SL(2,𝔽3)—51no—
52C22×C5×Q8C22×C311yescharacteristic
53#4147—34no—
54#4147C3×C414yescomplemented
55C2×#713—31no—
56C2×#713C22×C311yescharacteristic
57C2×Q8×D5—31no—
58C2×Q8×D5C22×C311yescharacteristic
59C2×#709—31no—
60C2×#709—31no—
61C4×C5×Q8—31no—
62C4×C5×Q8—31no—
63C4×C5×Q8C22×C311yescharacteristic
64C4×C5×Q8C22×C311yescharacteristic
65C5×#117—31no—
66C5×#117—31no—
67Q8×Dic5—31no—
68Q8×Dic5—31no—
69Q8×Dic5C22×C311yescharacteristic
70Q8×Dic5C22×C311yescharacteristic
71C2×C4⋊Dic5—31no—
72C2×C4×Dic5—31no—
73#4039—31no—
74#4039—31no—
75#4039—31no—
76#4039—31no—
77#4038—32no—
78#4038—31no—
79#4038—31no—
80D5×C4⋊C4—31no—
81D5×C4⋊C4—32no—
82D5×C4⋊C4—31no—
83#4033—31no—
84#4033—31no—
85#4033—31no—
86#4033—31no—
87#4027—32no—
88#4027—31no—
89#4027—31no—
90#4027—31no—
91#4027A411yescharacteristic
92C4×D20—32no—
93C4×D20—31no—
94C4×D20—31no—
95#4018—32no—
96#4018—31no—
97#4018—31no—
98C42×D5—32no—
99C42×D5—31no—
100C42×D5—31no—
101C4×Dic10—32no—
102C4×Dic10—31no—
103C4×Dic10—31no—
104C4×2−1+4—51noSylow
105C23×C3×C5—41no—
106C22×C3×D5—41no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5—41no—
109C3×C22⋊D5—44no—
110C2×C3×Dic5—41no—
111C2×C3×Dic5—41no—
112C2×C3×Dic5—41no—
113C2×C3×Dic5—82no—
114C3×C4×D5—82no—
115C5×SL(2,𝔽3)C22×C411yescharacteristic
1162−1+4⋊C3—54no—
117C2×#41⋊C3—51no—
118C22×SL(2,𝔽3)—51no—
119C22×SL(2,𝔽3)C2×D511yescharacteristic
120C3×C4×D4—201no—
121C4×SL(2,𝔽3)—51no—
122C4×SL(2,𝔽3)—51no—
123C4×SL(2,𝔽3)C2×D511yescharacteristic
124C4×SL(2,𝔽3)C2×D511yescharacteristic
125C2×C5×Q8—31no—
126C2×C5×Q8—32no—
127C2×C5×Q8C2×C3×C412yescomplemented
128C2×C5×Q8C23×C311yescharacteristic
129C22×C4×C5—31no—
130C2×C22⋊D5—31no—
131C2×C22⋊D5C2×A411yescharacteristic
132C22×Dic5—31no—
133C22×Dic5C2×A411yescharacteristic
134#713—34no—
135#713—34no—
136#713—32no—
137#713C2×C3×C412yescomplemented
138Q8×D5—34no—
139Q8×D5—32no—
140Q8×D5—34no—
141Q8×D5C2×C3×C412yescomplemented
142#709—34no—
143#709—34no—
144#709—34no—
145#709—34no—
146C2×D20—31no—
147C2×D20—31no—
148C2×C4×D5—32no—
149C2×C4×D5—31no—
150C2×C4×D5—31no—
151C2×C4×D5—31no—
152C2×C4×D5—32no—
153C2×C4×D5—31no—
154C2×C4×D5—31no—
155C2×C4×D5C2×A411yescharacteristic
156C2×Dic10—31no—
157C2×Dic10—31no—
158C5×C4⋊C4—31no—
159C5×C4⋊C4—31no—
160C5×C4⋊C4—31no—
161C5×C4⋊C4—31no—
162C5×C22⋊C4—31no—
163C5×C22⋊C4C2×A411yescharacteristic
164C42×C5—32no—
165C42×C5—31no—
166C42×C5—31no—
167#685—32no—
168#685—31no—
169#685—31no—
170#685—31no—
171#685C2×A411yescharacteristic
172C4⋊Dic5—32no—
173C4⋊Dic5—31no—
174C4⋊Dic5—31no—
175#683—32no—
176#683—31no—
177#683—31no—
178#683—31no—
179#683C2×A411yescharacteristic
180C4×Dic5—32no—
181C4×Dic5—31no—
182C4×Dic5—31no—
183C4×Dic5—31no—
184C4×Dic5—31no—
185C4×Dic5—31no—
186C4×Dic5—31no—
187C4×Dic5—31no—
188C4×Dic5C2×A411yescharacteristic
189C2×2−1+4—51no—
190#520—51no—
191#520—151no—
192#520—151no—
193#520—151no—
194#519—51no—
195#519—151no—
196#519C3×D511yescharacteristic, complemented
197C4×#41—51no—
198C4×#41—151no—
199C4×#41—151no—
200C4×#41—151no—
201C2×C4×Q8—51no—
202C2×C4×Q8—51no—
203C2×C4×Q8—151no—
204C22×C3×C5—42no—
205C22×C3×C5—41no—
206C22×C3×C5—81no—
207C22×C3×C5—81no—
208C2×C3×D5—42no—
209C2×C3×D5—81no—
210C2×C3×D5—81no—
211C3×C4×C5—81no—
212C3×C4×C5—81no—
213C3×Dic5—42no—
214C3×Dic5—42no—
215C3×Dic5—81no—
216C2×C3×D4—201no—
217C22×C3×C4—201no—
218C22×C3×C4—201no—
219#41⋊C3—52no—
220C2×SL(2,𝔽3)—52no—
221C2×SL(2,𝔽3)C4×D512yescomplemented
222C2×SL(2,𝔽3)C22×D511yescharacteristic
223C3×C4⋊C4—201no—
224C3×C22⋊C4—41no—
225C3×C22⋊C4—201no—
226C3×C42—201no—
227C23×C5C22×A411yescharacteristic
228C22×D5—31no—
229C22×D5C22×A411yescharacteristic
230C5×Q8—32no—
231C5×Q8—31no—
232C5×Q8—32no—
233C5×Q8C22×C3×C411yescharacteristic
234C2×C4×C5—32no—
235C2×C4×C5—32no—
236C2×C4×C5—31no—
237C2×C4×C5—31no—
238C2×C4×C5—31no—
239C2×C4×C5—31no—
240C2×C4×C5C22×A411yescharacteristic
241C2×C4×C5C22×A411yescharacteristic
242C22⋊D5—34no—
243C22⋊D5C4×A414yes—
244C2×Dic5—22no—
245C2×Dic5—31no—
246C2×Dic5—31no—
247C2×Dic5—31no—
248C2×Dic5—121no—
249C2×Dic5C22×A411yescharacteristic
250C2×Dic5C22×A411yescharacteristic
251C2×Dic5C22×A411yescharacteristic
252D20—32no—
253D20—32no—
254D20—34no—
255C4×D5—22no—
256C4×D5—38no—
257C4×D5—32no—
258C4×D5—32no—
259C4×D5—32no—
260C4×D5—32no—
261C4×D5—62no—
262C4×D5—121no—
263C4×D5—121no—
264C4×D5—122no—
265Dic10—32no—
266Dic10—34no—
267Dic10—32no—
2682−1+4—54no—
2692−1+4—154no—
270C2×#41—51no—
271C2×#41—151no—
272C2×#41—151no—
273C2×#41—151no—
274C22×Q8—51no—
275C22×Q8—151no—
276C22×Q8C2×C3×D511yescharacteristic
277C4×Q8—31no—
278C4×Q8—31no—
279C4×Q8—51no—
280C4×Q8—51no—
281C4×Q8—152no—
282C4×Q8—151no—
283C4×Q8—151no—
284C4×Q8—151no—
285C4×Q8—151no—
286C4×Q8—151no—
287C4×Q8—151no—
288C4×Q8—151no—
289C4×Q8—151no—
290C4×Q8C2×C3×D511yescharacteristic
291C4×Q8C2×C3×D511yescharacteristic
292C4×D4—51no—
293C4×D4—152no—
294C4×D4—151no—
295C4×D4—151no—
296C4×D4—151no—
297C4×D4—151no—
298C4×D4—152no—
299C4×D4—151no—
300C4×D4—151no—
301C4×D4—151no—
302C4×D4—151no—
303C4×D4—151no—
304#117—31no—
305#117—31no—
306#117—151no—
307#117—151no—
308#117—152no—
309#117—152no—
310#117—151no—
311#117—151no—
312C2×C4⋊C4—151no—
313C2×C4⋊C4—152no—
314C2×C4⋊C4—151no—
315C2×C4⋊C4—151no—
316C2×C42—151no—
317C2×C42—151no—
318C2×C42—152no—
319C2×C42—151no—
320C2×C3×C5—41no—
321C2×C3×C5—41no—
322C2×C3×C5—41no—
323C2×C3×C5—82no—
324C3×D5—82no—
325C23×C3—41no—
326C23×C3—201no—
327C3×D4—204no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—201no—
331C2×C3×C4—201no—
332C2×C3×C4—201no—
333C2×C3×C4—402no—
334C2×C3×C4—402no—
335SL(2,𝔽3)C2×C4×D511yescharacteristic
336C22×C5—21no—
337C22×C5—21no—
338C22×C5C2×C4×A412yes—
339C22×C5C23×A411yescharacteristic
340C2×D5—21no—
341C2×D5—21no—
342C2×D5—32no—
343C2×D5—61no—
344C2×D5—61no—
345C2×D5C2×C4×A412yes—
346C4×C5—21no—
347C4×C5—21no—
348C4×C5—31no—
349C4×C5—32no—
350C4×C5—31no—
351C4×C5—61no—
352C4×C5—61no—
353Dic5—21no—
354Dic5—32no—
355Dic5—61no—
356Dic5—61no—
357Dic5C2×C4×A412yes—
358Dic52−1+4⋊C312yescomplemented
359#41—52no—
360#41—154no—
361#41—154no—
362#41—154no—
363#41—154no—
364#41—154no—
365#41—154no—
366#41—152no—
367C2×Q8—31no—
368C2×Q8—32no—
369C2×Q8—52no—
370C2×Q8—151no—
371C2×Q8—151no—
372C2×Q8—154no—
373C2×Q8—154no—
374C2×Q8—152no—
375C2×Q8C3×C4×D512yescomplemented
376C2×Q8C22×C3×D511yescharacteristic
377C2×D4—51no—
378C2×D4—151no—
379C2×D4—151no—
380C2×D4—151no—
381C22×C4—31no—
382C22×C4—51no—
383C22×C4—51no—
384C22×C4—151no—
385C22×C4—151no—
386C22×C4—151no—
387C22×C4—152no—
388C22×C4—151no—
389C22×C4—152no—
390C22×C4—151no—
391C22×C4—151no—
392C4⋊C4—31no—
393C4⋊C4—31no—
394C4⋊C4—31no—
395C4⋊C4—31no—
396C4⋊C4—51no—
397C4⋊C4—151no—
398C4⋊C4—151no—
399C4⋊C4—151no—
400C4⋊C4—151no—
401C4⋊C4—152no—
402C4⋊C4—152no—
403C4⋊C4—151no—
404C22⋊C4—31no—
405C22⋊C4—51no—
406C22⋊C4—151no—
407C22⋊C4—151no—
408C22⋊C4—152no—
409C22⋊C4—151no—
410C22⋊C4D5×A411yescharacteristic
411C42—32no—
412C42—31no—
413C42—31no—
414C42—51no—
415C42—151no—
416C42—151no—
417C42—151no—
418C42—151no—
419C42—151no—
420C42—151no—
421C42—152no—
422C42—151no—
423C3×C5—41noHall
424C22×C3—42no—
425C22×C3—41no—
426C22×C3—81no—
427C22×C3—81no—
428C22×C3—202no—
429C22×C3—401no—
430C22×C3—401no—
431C3×C4—81no—
432C3×C4—81no—
433C3×C4—202no—
434C3×C4—202no—
435C3×C4—401no—
436C2×C5—22no—
437C2×C5C22×C4×A411yescharacteristic
438C2×C5C2×2−1+4⋊C311yescharacteristic
439C2×C5#594511yescharacteristic
440D5—22no—
441D5—62no—
442C23—51no—
443C23—151no—
444C23C2×D5×A411yescharacteristic
445Q8—32no—
446Q8—31no—
447Q8—32no—
448Q8—154no—
449Q8—152no—
450Q8—152no—
451Q8C2×C3×C4×D511yescharacteristic
452D4—54no—
453D4—154no—
454D4—152no—
455D4—152no—
456D4—154no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—31no—
460C2×C4—31no—
461C2×C4—31no—
462C2×C4—31no—
463C2×C4—51no—
464C2×C4—51no—
465C2×C4—51no—
466C2×C4—102no—
467C2×C4—102no—
468C2×C4—152no—
469C2×C4—152no—
470C2×C4—152no—
471C2×C4—152no—
472C2×C4—158no—
473C2×C4—151no—
474C2×C4—151no—
475C2×C4—151no—
476C2×C4—302no—
477C2×C4—601no—
478C2×C4—601no—
479C2×C4—602no—
480C2×C4—601no—
481C2×C4C2×D5×A411yescharacteristic
482C2×C4C2×D5×A411yescharacteristic
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—82no—
487C2×C3—402no—
488C5C4×2−1+4⋊C311yescharacteristic, complemented, Sylow
489C22—21no—
490C22—21no—
491C22—52no—
492C22—101no—
493C22—101no—
494C22—152no—
495C22—301no—
496C22—301no—
497C22C4×D5×A412yes—
498C22C22×D5×A411yescharacteristic
499C4—21no—
500C4—21no—
501C4—31no—
502C4—32no—
503C4—31no—
504C4—52no—
505C4—52no—
506C4—61no—
507C4—61no—
508C4—101no—
509C4—152no—
510C4—301no—
511C4—301no—
512C3—41noSylow
513C2—22no—
514C2—102no—
515C2—302no—
516C2C2×C4×D5×A411yescharacteristic
517C2D5×2−1+4⋊C311yescharacteristic
518C2#66287⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, hypocenterC5×Q8C22×C3×C4
FittingC5×#519C2×C3
socleC22×C5C23×A4
prime radicalC2×#4147⋊C3C2
cosocleC2×C5×Q8C23×C3
hypercenterC22⋊C4D5×A4
LCSG ▷ C5×Q8
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#519⋊C3C2semidirect
3. C2×Dic5×SL(2,𝔽3)C2semidirect
4. #65906⋊C3C2semidirect
5. C4×D5×SL(2,𝔽3)C2semidirect
6. #65908⋊C3C2semidirect
7. C4×C5⋊(#41⋊C3)C2semidirect
8. #10617774C3semidirect
10. #4147⋊C3C4semidirect
20. C5×#519C2×C3semidirect
21. C2×Q8×Dic5C2×C3semidirect
23. #65908C2×C3semidirect
25. C4×#713C2×C3semidirect
27. #65906C2×C3semidirect
29. C4×Q8×D5C2×C3semidirect
37. C2×C5×SL(2,𝔽3)C2×C4semidirect
39. D5×SL(2,𝔽3)C2×C4semidirect
40. C5⋊(#41⋊C3)C2×C4semidirect
48. #519⋊C3D5semidirect
54. #4147C3×C4semidirect
127. C2×C5×Q8C2×C3×C4semidirect
137. #713C2×C3×C4semidirect
141. Q8×D5C2×C3×C4semidirect
196. #519C3×D5semidirect
221. C2×SL(2,𝔽3)C4×D5semidirect
358. Dic52−1+4⋊C3semidirect
375. C2×Q8C3×C4×D5semidirect
488. C5C4×2−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#4147⋊C32
C5×#519⋊C32
C2×Dic5×SL(2,𝔽3)2
#65906⋊C32
C4×D5×SL(2,𝔽3)2
#65908⋊C32
C4×C5⋊(#41⋊C3)2
#106177743
C3×#40274
C4×2−1+4⋊C35