Finite Group Databasebeta

← #14301720

Subgroups — #14301720

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#11754563C211yesmaximal, characteristic
2#11754125C211yesmaximal, characteristic, complemented
3C5×#5132C211yesmaximal, characteristic, complemented
4C3×#65415C211yesmaximal, characteristic, complemented
5#11748341C211yesmaximal, characteristic, complemented
6#11746380C211yesmaximal, characteristic, complemented
7#11745991C211yesmaximal, characteristic, complemented
8#10600929—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C22×C3×C4×D5C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91563C412yes—
17#91539C412yes—
18#91449—21no—
19C4×#1201—22no—
20#91359—21no—
21#91288—22no—
22D3×#681—22no—
23#91286—22no—
24C5×#824—21no—
25C5×#823—21no—
26C3×#3952—22no—
27#90955—22no—
28#69992—51nomaximal, Hall
29#66162—31no—
30#66153—31no—
31C5×#413—31no—
32#65415—31no—
33#65415D311yescharacteristic, complemented
34#64779—31no—
35#64779—31no—
36C23×C3×D5C2×C411yescharacteristic
37C22×C3×C4×C5C2311yescharacteristic
38C2×C5×#67C2×C412yes—
39C5×#235—21no—
40C5×#235—21no—
41C5×#235—21no—
42C5×#235—21no—
43C2×C5×D12D411yescharacteristic
44C2×C4×C5×D3D412yes—
45C2×C5×Dic6D411yescharacteristic
46C22×C3×Dic5C2×C411yescharacteristic
47C2×C3×C4×D5—21no—
48C2×C3×C4×D5—21no—
49C2×C3×C4×D5—21no—
50C2×C3×C4×D5—21no—
51C2×C3×C4×D5—21no—
52C2×C3×C4×D5—21no—
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⋊C8—21no—
66C3×#681—21no—
67C3×#681—21no—
68C2×C3×#168D412yescomplemented
69#6669—22no—
70(C2×D3)⋊Dic5—22no—
71#6665—22no—
72#6650—24no—
73D3×#168—24no—
74#5791—51no—
75#5745—51no—
76C3×#407—51no—
77#5207—51no—
78#5132D511yescharacteristic, complemented
79#4559—51no—
80#4556—51no—
81C2×C5×#41—31no—
82C22×C4×D5C2×D311yescharacteristic
83C2×C5×M16—31no—
84C2×C5×M16C2×D311yescharacteristic
85#4094—62no—
86#4094—62no—
87#4092—61no—
88#4084—61no—
89#4083—32no—
90#4081—32no—
91C4×C22⋊D5—62no—
92#4072—31no—
93C2×#681—31no—
94C2×#681—62no—
95C2×#681C2×D311yescharacteristic
96C5×#101—61no—
97C5×#100—61no—
98#3952—22no—
99#3952—62no—
100#1891—151noSylow
101C23×C3×C5C22×C411yescharacteristic
102C22×C5×D3C22⋊C412yes—
103C22×C3×D5—21no—
104C22×C3×D5—21no—
105C22×C3×D5—21no—
106C22×C3×D5—42no—
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—
114C5×#67—22no—
115C5×#67—22no—
116C2×C5×Dic3C22⋊C412yes—
117C5×D12—21no—
118C5×D12—21no—
119C4×C5×D3—22no—
120C4×C5×D3—22no—
121C5×Dic6—21no—
122C5×Dic6—21no—
123C2×C3×Dic5—21no—
124C2×C3×Dic5—21no—
125C2×C3×Dic5—21no—
126C3×C4×D5—41no—
127C3×C4×D5—41no—
128C3×C4×D5—41no—
129C3×C4×D5—41no—
130#1201—44no—
131C3×C5×C8—41no—
132C3⋊#168—22no—
133C3×#168—22no—
134C5×C3⋊C8—41no—
135C23×C3×C4—51no—
136C2×#235C2×D511yescharacteristic
137C2×C3×M16—51no—
138C2×C3×M16C2×D511yescharacteristic
139#953—52no—
140#930—101no—
141#929—52no—
142C4×#67—102no—
143#926—51no—
144#924—101no—
145C2×#208—51no—
146C2×#208C2×D511yescharacteristic
147#907—102no—
148D3×M16—102no—
149#901—102no—
150C3×C22⋊C8—102no—
151#830—102no—
152#824—21no—
153#823—21no—
154C23×D5C4×D311yescharacteristic
155C5×#41—61no—
156C5×#41—61no—
157C5×#41—61no—
158C5×#41—61no—
159C2×C5×Q8—31no—
160C2×C5×D4—32no—
161C2×C5×D4—31no—
162C22×C4×C5—32no—
163C22×C4×C5C22×D311yescharacteristic
164C2×C22⋊D5—62no—
165C22×Dic5C4×D311yescharacteristic
166C2×C4×D5—21no—
167C2×C4×D5—21no—
168C2×C4×D5—21no—
169C2×C4×D5—21no—
170C2×C4×D5—21no—
171C2×C4×D5—21no—
172C5×M16—21no—
173C5×M16—21no—
174C5×M16—61no—
175C5×M16—61no—
176C2×C5×C8—21no—
177C2×C5×C8—61no—
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—64no—
187#681—61no—
188#681—61no—
189C2×#168—32no—
190C2×#168—64no—
191C2×#168#6712yescomplemented
192#568—151no—
193#525—151no—
194#413—31no—
195#407—51no—
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—42no—
205C2×C3×D5—22no—
206C2×C3×D5—22no—
207C2×C3×D5—22no—
208C2×C3×D5—41no—
209C2×C3×D5—41no—
210C2×C3×D5—41no—
211C2×C3×D5—41no—
212C3×C4×C5#12011yescharacteristic
213C3×C4×C5#12111yescharacteristic
214C3×C4×C5#12111yescharacteristic
215C3×C4×C5#12311yescharacteristic
216Dic15—42no—
217C3×Dic5—41no—
218C5×Dic3—22no—
219C24×C3—51no—
220C22×C3×C4—51no—
221C22×C3×C4—101no—
222C22×C3×C4—101no—
223C22×C3×C4—101no—
224C22×C3×C4—101no—
225C22×C3×C4—101no—
226C22×C3×C4—101no—
227C22×C3×C4C22×D511yescharacteristic
228C2×#67—102no—
229C2×#67C4×D512yes—
230#235—21no—
231#235—21no—
232#235—21no—
233#235—21no—
234C2×D12C22⋊D511yescharacteristic
235C2×C4×D3D2012yes—
236C2×Dic6C22⋊D511yescharacteristic
237C3×M16—21no—
238C3×M16—21no—
239C3×M16—101no—
240C3×M16—101no—
241C2×C3×C8—21no—
242C2×C3×C8—52no—
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—101no—
252#208—101no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—52no—
255ΓL(1,𝔽25)—104no—
256D3×C8—104no—
257C23×C5—32no—
258C23×C5C2×C4×D311yescharacteristic
259C22×D5—21no—
260C22×D5—21no—
261C22×D5—21no—
262C22×D5—42no—
263C5×Q8—61no—
264C5×Q8—61no—
265C5×D4—62no—
266C5×D4—62no—
267C5×D4—61no—
268C5×D4—61no—
269C2×C4×C5—32no—
270C2×C4×C5—62no—
271C2×C4×C5—62no—
272C2×C4×C5#23511yescharacteristic
273C2×C4×C5D3×D411yescharacteristic
274C2×C4×C5D3×D411yescharacteristic
275C2×C4×C5D3×D411yescharacteristic
276C2×C4×C5#23711yescharacteristic
277C2×C4×C5C2×#6711yescharacteristic
278C22⋊D5—124no—
279C2×Dic5—21no—
280C2×Dic5—21no—
281C2×Dic5—21no—
282C2×Dic5—62no—
283C4×D5—41no—
284C4×D5—41no—
285C4×D5—41no—
286C4×D5—41no—
287C5×C8—41no—
288C5×C8—121no—
289#168—22no—
290#168—62no—
291C2×#41—31no—
292C23×C4—51no—
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—61no—
307#100—61no—
308C22⋊C8—102no—
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—101no—
317C23×C3—101no—
318C23×C3—101no—
319C23×C3—202no—
320C23×C3C2×C4×D511yescharacteristic
321C22×D3#68512yes—
322C2×C3×C4—101no—
323C2×C3×C4—101no—
324C2×C3×C4—101no—
325C2×C3×C4—201no—
326C2×C3×C4—201no—
327C2×C3×C4—201no—
328C2×C3×C4—201no—
329C2×C3×C4C2×D2011yescharacteristic
330C2×C3×C4D4×D511yescharacteristic
331C2×C3×C4D4×D511yescharacteristic
332C2×C3×C4#71111yescharacteristic
333C2×C3×C4#71111yescharacteristic
334C2×C3×C4C2×C22⋊D511yescharacteristic
335#67—22no—
336#67—22no—
337#67—204no—
338C2×Dic3—102no—
339C2×Dic3#68512yes—
340D12—21no—
341D12—21no—
342C4×D3—22no—
343C4×D3—22no—
344Dic6—21no—
345Dic6—21no—
346C3×C8—41no—
347C3×C8—102no—
348C3⋊C8—41no—
349C3⋊C8—102no—
350C22×C5—41no—
351C22×C5—62no—
352C22×C5—122no—
353C22×C5D3×C22⋊C411yescharacteristic
354C22×C5#67⋊C411yescharacteristic
355C22×C5C4×#6711yescharacteristic
356C2×D5—22no—
357C2×D5—22no—
358C2×D5—22no—
359C2×D5—41no—
360C2×D5—41no—
361C2×D5—41no—
362C2×D5—41no—
363C4×C5—62no—
364C4×C5#88311yescharacteristic
365C4×C5#88411yescharacteristic
366C4×C5#93711yescharacteristic
367C4×C5#93811yescharacteristic
368Dic5—41no—
369Dic5—122no—
370C24—51no—
371#41—61no—
372#41—61no—
373#41—61no—
374#41—61no—
375C2×Q8—31no—
376C2×D4—32no—
377C2×D4—31no—
378C2×D4—302no—
379C22×C4—32no—
380C22×C4—51no—
381C22×C4—101no—
382C22×C4—101no—
383C22×C4—101no—
384C22×C4—101no—
385C22×C4—101no—
386C22×C4—101no—
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—52no—
399C2×C8—61no—
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—102no—
412C22×C3—102no—
413C22×C3—201no—
414C22×C3—201no—
415C22×C3—201no—
416C22×C3—201no—
417C22×C3#402711yescharacteristic
418C22×C3#402711yescharacteristic
419C22×C3C2×#68511yescharacteristic
420C2×D3—22no—
421C2×D3—42no—
422C3×C4—201no—
423C3×C4#402811yescharacteristic
424C3×C4#403211yescharacteristic
425C3×C4#408511yescharacteristic
426C3×C4#408511yescharacteristic
427Dic3—22no—
428Dic3—202no—
429C2×C5—21no—
430C2×C5—21no—
431C2×C5—21no—
432C2×C5—122no—
433C2×C5#495611yescharacteristic
434D5—42no—
435C23—32no—
436C23—101no—
437C23—101no—
438C23—101no—
439C23—202no—
440C23C4×D3×D511yescharacteristic
441Q8—61no—
442Q8—61no—
443D4—62no—
444D4—62no—
445D4—61no—
446D4—61no—
447D4—604no—
448C2×C4—32no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—101no—
452C2×C4—101no—
453C2×C4—101no—
454C2×C4—201no—
455C2×C4—201no—
456C2×C4—201no—
457C2×C4—201no—
458C2×C4—302no—
459C2×C4#677211yescharacteristic
460C2×C4D3×D2011yescharacteristic
461C2×C4#677711yescharacteristic
462C2×C4#678011yescharacteristic
463C2×C4D5×#6711yescharacteristic
464C2×C4D3×C22⋊D511yescharacteristic
465C8—41no—
466C8—102no—
467C8—121no—
468C8—302no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—202no—
473C2×C3#6524711yescharacteristic
474D3—42no—
475C5#6999211yescharacteristic, complemented, Sylow
476C22—41no—
477C22—62no—
478C22—102no—
479C22—102no—
480C22—102no—
481C22—122no—
482C22—201no—
483C22—201no—
484C22—201no—
485C22—201no—
486C22#9144011yescharacteristic
487C22#9144211yescharacteristic
488C22D3×#68511yescharacteristic
489C4—62no—
490C4—201no—
491C4—602no—
492C4#9145311yescharacteristic
493C4#9145711yescharacteristic
494C4#9147411yescharacteristic
495C4#9156511yescharacteristic
496C3#1060092911yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—21no—
500C2—122no—
501C2—202no—
502C2#1174891111yescharacteristic
subgroupG/H
centerC4#91474
commutatorC23×C3×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC2×C3×C5#386
prime radical#11754563C2
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. #11754125C2semidirect
3. C5×#5132C2semidirect
4. C3×#65415C2semidirect
5. #11748341C2semidirect
6. #11746380C2semidirect
7. #11745991C2semidirect
33. #65415D3semidirect
62. C2×C3⋊#168D4semidirect
68. C2×C3×#168D4semidirect
78. #5132D5semidirect
191. C2×#168#67semidirect
408. C3×C5#1891semidirect
475. C5#69992semidirect
496. C3#10600929semidirect

Maximal subgroups

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

subgroupindex
#117545632
#117541252
C5×#51322
C3×#654152
#117483412
#117463802
#117459912
#106009293
#699925