Finite Group Databasebeta

← #14301718

Subgroups — #14301718

Order 1920.

684 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754577C211yesmaximal, characteristic
2#11754120C211yesmaximal, characteristic, complemented
3C5×#5126C211yesmaximal, characteristic, complemented
4C3×#65421C211yesmaximal, characteristic, complemented
5#11748346C211yesmaximal, characteristic, complemented
6#11746380C211yesmaximal, characteristic, complemented
7#11745990C211yesmaximal, characteristic, complemented
8#10600785—31nomaximal, Hall
9C22×C4×C5×D3C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91568C412yes—
17#91536C412yes—
18#91456—21no—
19C4×#1201—22no—
20#91369—21no—
21#91284—22no—
22#91283—22no—
23D5×#208—22no—
24C5×(C2×D3)⋊C8—22no—
25C3×#3956—21no—
26C3×#3955—21no—
27(C2×D3)⋊#168—22no—
28#70136—51nomaximal, Hall
29#66125—31no—
30#66079—31no—
31C5×#407—31no—
32#65496—31no—
33#65421D311yescharacteristic, complemented
34#64779—31no—
35#64776—31no—
36C23×C5×D3C2×C411yescharacteristic
37C22×C3×C4×C5C2311yescharacteristic
38C22×C5×Dic3C2×C411yescharacteristic
39C2×C4×C5×D3—21no—
40C2×C4×C5×D3—21no—
41C2×C4×C5×D3—21no—
42C2×C4×C5×D3—21no—
43C2×C4×C5×D3—21no—
44C2×C4×C5×D3—21no—
45C2×C3×C22⋊D5C2×C412yes—
46C3×#709—21no—
47C3×#709—21no—
48C3×#709—21no—
49C3×#709—21no—
50C2×C3×D20D411yescharacteristic
51C2×C3×C4×D5D412yes—
52C2×C3×Dic10D411yescharacteristic
53C2×#1201—22no—
54C3×C5×M16—21no—
55C3×C5×M16—21no—
56C2×C3×C5×C8—21no—
57#6719C2×C412yes—
58C4⋊Dic15—21no—
59C4×Dic15—21no—
60#6710—21no—
61#6710—21no—
62C2×C3⋊#168D412yescomplemented
63C5×#208—21no—
64C5×#208—21no—
65C2×C5×C3⋊C8D412yescomplemented
66C3×#681—21no—
67C3×#681—21no—
68C2×C3×#168—21no—
69#6669—22no—
70(C2×D3)⋊Dic5—22no—
71#6665—22no—
72#6649—24no—
73D5×C3⋊C8—24no—
74#5828—51no—
75#5819—51no—
76C3×#413—51no—
77#5126—51no—
78#5126D511yescharacteristic, complemented
79#4559—51no—
80#4559—51no—
81C23×C4×C5—31no—
82C2×#709C2×D311yescharacteristic
83C2×C5×M16—31no—
84C2×C5×M16C2×D311yescharacteristic
85#4099—32no—
86#4076—61no—
87#4075—32no—
88C4×C22⋊D5—62no—
89#4072—31no—
90#4070—61no—
91C2×#681—31no—
92C2×#681C2×D311yescharacteristic
93#4053—62no—
94D5×M16—62no—
95#4047—62no—
96C5×C22⋊C8—62no—
97C22⋊#168—62no—
98#3956—21no—
99#3955—21no—
100#1891—151noSylow
101C23×C3×C5C22×C411yescharacteristic
102C22×C5×D3—21no—
103C22×C5×D3—21no—
104C22×C5×D3—21no—
105C22×C5×D3—42no—
106C22×C3×D5C22⋊C412yes—
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5C2×D411yescharacteristic
111C2×C3×C4×C5#4111yescharacteristic
112C2×C3×C4×C5#4111yescharacteristic
113C2×Dic15—22no—
114C2×C5×Dic3—21no—
115C2×C5×Dic3—21no—
116C2×C5×Dic3—21no—
117C4×C5×D3—41no—
118C4×C5×D3—41no—
119C4×C5×D3—41no—
120C4×C5×D3—41no—
121C3×C22⋊D5—22no—
122C3×C22⋊D5—22no—
123C2×C3×Dic5C22⋊C412yes—
124C3×D20—21no—
125C3×D20—21no—
126C3×C4×D5—22no—
127C3×C4×D5—22no—
128C3×Dic10—21no—
129C3×Dic10—21no—
130#1201—44no—
131C3×C5×C8—41no—
132C3⋊#168—22no—
133C3×#168—41no—
134C5×C3⋊C8—22no—
135C2×C3×#41—51no—
136C22×C4×D3C2×D511yescharacteristic
137C2×C3×M16—51no—
138C2×C3×M16C2×D511yescharacteristic
139#948—102no—
140#948—102no—
141#946—101no—
142#938—101no—
143#937—52no—
144#935—52no—
145C4×#67—102no—
146#926—51no—
147C2×#208—51no—
148C2×#208—102no—
149C2×#208C2×D511yescharacteristic
150C3×#101—101no—
151C3×#100—101no—
152(C2×D3)⋊C8—22no—
153(C2×D3)⋊C8—102no—
154C24×C5—31no—
155C22×C4×C5—31no—
156C22×C4×C5—61no—
157C22×C4×C5—61no—
158C22×C4×C5—61no—
159C22×C4×C5—61no—
160C22×C4×C5—61no—
161C22×C4×C5—61no—
162C22×C4×C5C22×D311yescharacteristic
163C2×C22⋊D5—62no—
164C2×C22⋊D5C4×D312yes—
165#709—21no—
166#709—21no—
167#709—21no—
168#709—21no—
169C2×D20#6711yescharacteristic
170C2×C4×D5D1212yes—
171C2×Dic10#6711yescharacteristic
172C5×M16—21no—
173C5×M16—21no—
174C5×M16—61no—
175C5×M16—61no—
176C2×C5×C8—21no—
177C2×C5×C8—32no—
178C22⋊Dic5—32no—
179C22⋊Dic5—62no—
180#685—62no—
181C4⋊Dic5—61no—
182#683—62no—
183C4×Dic5—61no—
184#681—21no—
185#681—21no—
186#681—61no—
187#681—61no—
188C2×#168—21no—
189C2×#168—32no—
190#676—64no—
191C8×D5—64no—
192#568—151no—
193#525—151no—
194#413—51no—
195#407—31no—
196#407—151no—
197#344—151no—
198#344—151no—
199C22×C3×C5—41no—
200C22×C3×C5C2×C22⋊C411yescharacteristic
201C22×C3×C5C4×D411yescharacteristic
202C22×C3×C5C4×D411yescharacteristic
203C2×C5×D3—22no—
204C2×C5×D3—22no—
205C2×C5×D3—22no—
206C2×C5×D3—41no—
207C2×C5×D3—41no—
208C2×C5×D3—41no—
209C2×C5×D3—41no—
210C2×C3×D5—22no—
211C2×C3×D5—42no—
212C3×C4×C5#12011yescharacteristic
213C3×C4×C5#12111yescharacteristic
214C3×C4×C5#12111yescharacteristic
215C3×C4×C5#12311yescharacteristic
216Dic15—42no—
217C3×Dic5—22no—
218C5×Dic3—41no—
219C23×D3C4×D511yescharacteristic
220C3×#41—101no—
221C3×#41—101no—
222C3×#41—101no—
223C3×#41—101no—
224C2×C3×Q8—51no—
225C2×C3×D4—52no—
226C2×C3×D4—51no—
227C22×C3×C4—52no—
228C22×C3×C4C22×D511yescharacteristic
229C2×#67—102no—
230C22×Dic3C4×D511yescharacteristic
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C3×M16—21no—
238C3×M16—21no—
239C3×M16—101no—
240C3×M16—101no—
241C2×C3×C8—21no—
242C2×C3×C8—101no—
243#217—52no—
244#217—102no—
245(C2×D3)⋊C4—102no—
246C4⋊Dic3—101no—
247Dic3⋊C4—102no—
248C4×Dic3—101no—
249#208—21no—
250#208—21no—
251#208—104no—
252#208—101no—
253#208—101no—
254C2×C3⋊C8—52no—
255C2×C3⋊C8—104no—
256C2×C3⋊C8C22⋊D512yescomplemented
257C23×C5—61no—
258C23×C5—61no—
259C23×C5—61no—
260C23×C5—122no—
261C23×C5C2×C4×D311yescharacteristic
262C22×D5(C2×D3)⋊C412yes—
263C2×C4×C5—61no—
264C2×C4×C5—61no—
265C2×C4×C5—61no—
266C2×C4×C5—121no—
267C2×C4×C5—121no—
268C2×C4×C5—121no—
269C2×C4×C5—121no—
270C2×C4×C5C2×D1211yescharacteristic
271C2×C4×C5D3×D411yescharacteristic
272C2×C4×C5D3×D411yescharacteristic
273C2×C4×C5#23711yescharacteristic
274C2×C4×C5#23711yescharacteristic
275C2×C4×C5C2×#6711yescharacteristic
276C22⋊D5—22no—
277C22⋊D5—22no—
278C22⋊D5—124no—
279C2×Dic5—62no—
280C2×Dic5(C2×D3)⋊C412yes—
281D20—21no—
282D20—21no—
283C4×D5—22no—
284C4×D5—22no—
285Dic10—21no—
286Dic10—21no—
287C5×C8—41no—
288C5×C8—62no—
289#168—41no—
290#168—62no—
291C2×#41—51no—
292C23×C4—31no—
293#131—302no—
294#131—302no—
295C2×M16—31no—
296C2×M16—51no—
297C2×M16—151no—
298C2×M16—302no—
299C2×M16D3×D511yescharacteristic
300#123—152no—
301C22⋊Q8—301no—
302#121—301no—
303#120—152no—
304C4×D4—302no—
305#117—151no—
306#101—101no—
307#100—101no—
308C22⋊C8—62no—
309C22⋊C8—302no—
310C2×C3×C5—21no—
311C2×C3×C5—21no—
312C2×C3×C5—21no—
313C2×C3×C5#38611yescharacteristic
314C3×D5—42no—
315C5×D3—42no—
316C23×C3—52no—
317C23×C3C2×C4×D511yescharacteristic
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—42no—
322C3×Q8—101no—
323C3×Q8—101no—
324C3×D4—102no—
325C3×D4—102no—
326C3×D4—101no—
327C3×D4—101no—
328C2×C3×C4—52no—
329C2×C3×C4—102no—
330C2×C3×C4—102no—
331C2×C3×C4#70911yescharacteristic
332C2×C3×C4D4×D511yescharacteristic
333C2×C3×C4D4×D511yescharacteristic
334C2×C3×C4D4×D511yescharacteristic
335C2×C3×C4#71111yescharacteristic
336C2×C3×C4C2×C22⋊D511yescharacteristic
337#67—204no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—102no—
342C4×D3—41no—
343C4×D3—41no—
344C4×D3—41no—
345C4×D3—41no—
346C3×C8—41no—
347C3×C8—201no—
348C3⋊C8—22no—
349C3⋊C8—102no—
350C22×C5—41no—
351C22×C5—62no—
352C22×C5—62no—
353C22×C5—62no—
354C22×C5—121no—
355C22×C5—121no—
356C22×C5—121no—
357C22×C5—121no—
358C22×C5#67⋊C411yescharacteristic
359C22×C5#67⋊C411yescharacteristic
360C22×C5C2×(C2×D3)⋊C411yescharacteristic
361C2×D5—22no—
362C2×D5—42no—
363C4×C5—121no—
364C4×C5#88211yescharacteristic
365C4×C5#88611yescharacteristic
366C4×C5#93911yescharacteristic
367C4×C5#93911yescharacteristic
368Dic5—22no—
369Dic5—122no—
370C24—31no—
371#41—101no—
372#41—101no—
373#41—101no—
374#41—101no—
375C2×Q8—51no—
376C2×D4—52no—
377C2×D4—51no—
378C2×D4—302no—
379C22×C4—31no—
380C22×C4—52no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4C2×D3×D511yescharacteristic
388M16—21no—
389M16—21no—
390M16—61no—
391M16—61no—
392M16—101no—
393M16—101no—
394M16—304no—
395M16—301no—
396M16—301no—
397C2×C8—21no—
398C2×C8—32no—
399C2×C8—101no—
400C2×C8—152no—
401C2×C8—304no—
402C4⋊C4—302no—
403C4⋊C4—301no—
404C22⋊C4—152no—
405C22⋊C4—302no—
406C22⋊C4—302no—
407C42—301no—
408C3×C5#189111yescharacteristic, complemented, Hall
409C22×C3—41no—
410C22×C3—102no—
411C22×C3—202no—
412C22×C3D5×C22⋊C411yescharacteristic
413C22×C3#402711yescharacteristic
414C22×C3C4×C22⋊D511yescharacteristic
415C2×D3—22no—
416C2×D3—22no—
417C2×D3—22no—
418C2×D3—41no—
419C2×D3—41no—
420C2×D3—41no—
421C2×D3—41no—
422C3×C4—102no—
423C3×C4#402911yescharacteristic
424C3×C4#403011yescharacteristic
425C3×C4#408311yescharacteristic
426C3×C4#408411yescharacteristic
427Dic3—41no—
428Dic3—202no—
429C2×C5—21no—
430C2×C5—21no—
431C2×C5—21no—
432C2×C5—122no—
433C2×C5#495811yescharacteristic
434D5—42no—
435C23—52no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23—122no—
440C23C4×D3×D511yescharacteristic
441Q8—101no—
442Q8—101no—
443D4—102no—
444D4—102no—
445D4—101no—
446D4—101no—
447D4—604no—
448C2×C4—52no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—102no—
453C2×C4—102no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—302no—
459C2×C4#676511yescharacteristic
460C2×C4D5×D1211yescharacteristic
461C2×C4#677711yescharacteristic
462C2×C4#678011yescharacteristic
463C2×C4D5×#6711yescharacteristic
464C2×C4D3×C22⋊D511yescharacteristic
465C8—41no—
466C8—62no—
467C8—201no—
468C8—302no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—202no—
473C2×C3#6524511yescharacteristic
474D3—42no—
475C5#7013611yescharacteristic, complemented, Sylow
476C22—41no—
477C22—62no—
478C22—62no—
479C22—62no—
480C22—102no—
481C22—121no—
482C22—121no—
483C22—121no—
484C22—121no—
485C22—202no—
486C22#9144111yescharacteristic
487C22#9144211yescharacteristic
488C22D5×(C2×D3)⋊C411yescharacteristic
489C4—102no—
490C4—121no—
491C4—602no—
492C4#9140811yescharacteristic
493C4#9141411yescharacteristic
494C4#9147511yescharacteristic
495C4#9154911yescharacteristic
496C3#1060078511yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—21no—
500C2—122no—
501C2—202no—
502C2#1174890611yescharacteristic
subgroupG/H
centerC4#91475
commutatorC23×C3×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC2×C3×C5#386
prime radical#11754577C2
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1891
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #11754120C2semidirect
3. C5×#5126C2semidirect
4. C3×#65421C2semidirect
5. #11748346C2semidirect
6. #11746380C2semidirect
7. #11745990C2semidirect
33. #65421D3semidirect
62. C2×C3⋊#168D4semidirect
65. C2×C5×C3⋊C8D4semidirect
78. #5126D5semidirect
256. C2×C3⋊C8C22⋊D5semidirect
408. C3×C5#1891semidirect
475. C5#70136semidirect
496. C3#10600785semidirect

Maximal subgroups

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

subgroupindex
#117545772
#117541202
C5×#51262
C3×#654212
#117483462
#117463802
#117459902
#106007853
#701365