Finite Group Databasebeta

← #14301396

Subgroups — #14301396

Order 1920.

664 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#11756504C211yesmaximal, characteristic, complemented
2#11754572C211yesmaximal, characteristic
3C5×#5139C211yesmaximal, characteristic, complemented
4#11748626C211yesmaximal, characteristic, complemented
5#11748425C211yesmaximal, characteristic, complemented
6C3×#64775C211yesmaximal, characteristic, complemented
7#11745977C211yesmaximal, characteristic, complemented
8#10601210—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C5×#235C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13#91796—22no—
14M16×D15—22no—
15#91790—22no—
16C5×#926C2211yescharacteristic
17C3×#4072C2211yescharacteristic
18C2×C3×C4×Dic5C2211yescharacteristic
19#91566—22no—
20#91451C411yescharacteristic, complemented
21C4×#1203—22no—
22#91376C411yescharacteristic, complemented
23#91375C411yescharacteristic, complemented
24#91373C411yescharacteristic, complemented
25C5×#825—22no—
26#90997—22no—
27#90979—22no—
28#68777—51nomaximal, Hall
29#66166—31no—
30#66079—31no—
31C5×#421—31no—
32#65525—31no—
33#65295—31no—
34#64775D311yescharacteristic, complemented
35#64752—31no—
36C22×C3×C4×C5C2311yescharacteristic
37C2×#1220D412yes—
38#6817—21no—
39#6817—21no—
40#6817—21no—
41#6817—21no—
42C2×D60C2×C411yescharacteristic
43C2×C4×D15D412yes—
44C2×Dic30C2×C411yescharacteristic
45C2×C5×#67—21no—
46C5×#235—21no—
47C5×#235—21no—
48C5×#235—21no—
49C5×#235—21no—
50C2×C5×D12C2×C411yescharacteristic
51C2×C4×C5×D3—21no—
52C2×C5×Dic6C2×C411yescharacteristic
53C22×C3×Dic5—21no—
54C2×#1203—22no—
55C3×C5×M16—21no—
56C3×C5×M16D412yescomplemented
57C2×C3×C5×C8—21no—
58#6710—21no—
59#6710D412yescomplemented
60C2×C3⋊#168—21no—
61#6705—24no—
62C8×D15—24no—
63C5×#217—21no—
64C5×C4⋊Dic3—21no—
65C4×C5×Dic3—21no—
66C3×C22⋊Dic5—21no—
67C3×C4⋊Dic5—21no—
68C3×C4×Dic5—21no—
69C3×C4×Dic5—21no—
70C3×C4×Dic5C2×C411yescharacteristic
71C3×C4×Dic5C2×C411yescharacteristic
72#6668—22no—
73#6667—22no—
74(C2×D3)⋊Dic5—22no—
75#5745—51no—
76#5477—51no—
77#5139D511yescharacteristic, complemented
78#5006—51no—
79#5005—51no—
80C3×#339—51no—
81#4523—51no—
82C2×C5×#41—31no—
83C2×#709—31no—
84C2×C5×M16C2×D311yescharacteristic
85C5×#117—31no—
86#4093—31no—
87#4090—31no—
88#4086—31no—
89#4085—62no—
90#4082—31no—
91C4×C22⋊D5—62no—
92#4072C2×D311yescharacteristic
93C2×C4×Dic5C2×D311yescharacteristic
94C2×#681—31no—
95#4053—62no—
96D5×M16—62no—
97#4047—62no—
98C5×#104—62no—
99#3969—62no—
100#3938—62no—
101#1884—151noSylow
102C23×C3×C5C2×D411yescharacteristic
103C22×D15C22⋊C412yes—
104C22×C5×D3—21no—
105C2×C3×C4×C5C22×C411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111#1220—22no—
112#1220—22no—
113C2×Dic15C22⋊C412yes—
114D60—21no—
115D60—21no—
116C4×D15—22no—
117C4×D15—22no—
118Dic30—21no—
119Dic30—21no—
120C5×#67—41no—
121C5×#67—41no—
122C2×C5×Dic3—21no—
123C2×C5×Dic3—41no—
124C5×D12—21no—
125C5×D12—21no—
126C4×C5×D3—41no—
127C4×C5×D3—41no—
128C5×Dic6—21no—
129C5×Dic6—21no—
130C2×C3×Dic5—21no—
131C2×C3×Dic5—21no—
132C2×C3×Dic5—42no—
133C2×C3×Dic5—41no—
134#1203—44no—
135C3×C5×C8—22no—
136C3⋊#168—22no—
137C2×#235—51no—
138C2×#235C2×D511yescharacteristic
139C2×C3×M16C2×D511yescharacteristic
140C3×#117—51no—
141C2×C3×C42—51no—
142#930—102no—
143#926C2×D511yescharacteristic
144C2×#208—51no—
145#907—102no—
146D3×M16—102no—
147#901—102no—
148#875—51no—
149#875—51no—
150#874—51no—
151C4×D12—102no—
152#869—51no—
153#825—22no—
154#806—102no—
155#805—102no—
156C5×#41—61no—
157C5×#41—61no—
158C5×#41—61no—
159C5×#41—61no—
160C2×C5×Q8—31no—
161C2×C5×D4—31no—
162C2×C5×D4—61no—
163C22×C4×C5—61no—
164C22×C4×C5C22×D311yescharacteristic
165C2×C22⋊D5—32no—
166C2×C22⋊D5—62no—
167C22×Dic5—21no—
168#709—61no—
169#709—61no—
170#709—61no—
171#709—61no—
172C2×D20—31no—
173C2×C4×D5—32no—
174C2×Dic10—31no—
175C5×M16—21no—
176C5×M16D1212yescomplemented
177C2×C5×C8—21no—
178C5×C4⋊C4—61no—
179C5×C22⋊C4—61no—
180C42×C5—61no—
181C22⋊Dic5—21no—
182C22⋊Dic5—62no—
183#685—62no—
184C4⋊Dic5—21no—
185#683—62no—
186C4×Dic5—21no—
187C4×Dic5—21no—
188C4×Dic5C4×D311yescharacteristic
189C4×Dic5C4×D311yescharacteristic
190#681—32no—
191#681—61no—
192C2×#168—61no—
193#676—64no—
194C8×D5—64no—
195#568—151no—
196#532—151no—
197#421—31no—
198#421—151no—
199#410—151no—
200#339—51no—
201#339—151no—
202C22×C3×C5—41no—
203C22×C3×C5C2×C22⋊C411yescharacteristic
204C22×C3×C5#12111yescharacteristic
205C22×C3×C5#12111yescharacteristic
206C2×D15—22no—
207C2×D15—42no—
208C2×C5×D3—42no—
209C2×C5×D3—41no—
210C3×C4×C5C4×D411yescharacteristic
211C3×C4×C5C4×D411yescharacteristic
212C3×C4×C5#12011yescharacteristic
213C3×C4×C5#12311yescharacteristic
214Dic15—22no—
215C3×Dic5—22no—
216C3×Dic5—41no—
217C3×Dic5—81no—
218C5×Dic3—41no—
219C5×Dic3—81no—
220C22×C3×C4—101no—
221C22×C3×C4C22×D511yescharacteristic
222C2×#67—21no—
223C2×#67—52no—
224#235—21no—
225#235—21no—
226#235—21no—
227#235—21no—
228#235—101no—
229#235—101no—
230#235—101no—
231#235—101no—
232C2×D12—51no—
233C2×D12—102no—
234C2×D12C4×D511yescharacteristic
235C2×C4×D3—21no—
236C2×C4×D3—52no—
237C2×Dic6—51no—
238C2×Dic6C4×D511yescharacteristic
239C3×M16—21no—
240C3×M16C22⋊D512yescomplemented
241C2×C3×C8—21no—
242C3×C4⋊C4—101no—
243C3×C22⋊C4—101no—
244C3×C42—51no—
245C3×C42—51no—
246C3×C42—101no—
247C3×C42—101no—
248#217—21no—
249(C2×D3)⋊C4—102no—
250(C2×D3)⋊C4—102no—
251C4⋊Dic3—21no—
252C4⋊Dic3—102no—
253C4×Dic3—21no—
254#208—52no—
255#208—101no—
256C2×C3⋊C8—101no—
257ΓL(1,𝔽25)—104no—
258D3×C8—104no—
259C23×C5—61no—
260C23×C5D3×D411yescharacteristic
261C22×D5—32no—
262C5×Q8—61no—
263C5×Q8—61no—
264C5×D4—61no—
265C5×D4—61no—
266C5×D4—121no—
267C5×D4—121no—
268C2×C4×C5—61no—
269C2×C4×C5—121no—
270C2×C4×C5—121no—
271C2×C4×C5—121no—
272C2×C4×C5C2×C4×D311yescharacteristic
273C2×C4×C5C2×D1211yescharacteristic
274C2×C4×C5#23511yescharacteristic
275C2×C4×C5D3×D411yescharacteristic
276C2×C4×C5D3×D411yescharacteristic
277C2×C4×C5#23911yescharacteristic
278C22⋊D5—62no—
279C22⋊D5—62no—
280C22⋊D5—124no—
281C2×Dic5—21no—
282C2×Dic5—21no—
283C2×Dic5—32no—
284C2×Dic5—42no—
285C2×Dic5—41no—
286D20—61no—
287D20—61no—
288C4×D5—62no—
289C4×D5—62no—
290Dic10—61no—
291Dic10—61no—
292C5×C8—22no—
293#168—62no—
294C2×#41—31no—
295C2×#41—151no—
296#131—302no—
297#131—302no—
298C2×M16—151no—
299C2×M16—302no—
300C2×M16D3×D511yescharacteristic
301C4⋊Q8—151no—
302#127—151no—
303#124—151no—
304#124—151no—
305#121—302no—
306C4×D4—302no—
307#117—31no—
308#117—51no—
309C2×C42—51no—
310#104—62no—
311#104—302no—
312#99—302no—
313C2×C3×C5—21no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5#38611yescharacteristic
317D15—42no—
318C5×D3—81no—
319C23×C3D4×D511yescharacteristic
320C22×D3—21no—
321C22×D3—52no—
322C2×C3×C4—101no—
323C2×C3×C4—101no—
324C2×C3×C4—202no—
325C2×C3×C4—201no—
326C2×C3×C4C2×C4×D511yescharacteristic
327C2×C3×C4#70911yescharacteristic
328C2×C3×C4D4×D511yescharacteristic
329C2×C3×C4D4×D511yescharacteristic
330C2×C3×C4#71111yescharacteristic
331C2×C3×C4C2×C22⋊D511yescharacteristic
332#67—41no—
333#67—41no—
334#67—102no—
335#67—102no—
336C2×Dic3—21no—
337C2×Dic3—41no—
338C2×Dic3—52no—
339D12—21no—
340D12—21no—
341D12—101no—
342D12—101no—
343D12—204no—
344C4×D3—41no—
345C4×D3—41no—
346C4×D3—102no—
347C4×D3—102no—
348Dic6—21no—
349Dic6—21no—
350Dic6—101no—
351Dic6—101no—
352C3×C8—22no—
353C3⋊C8—102no—
354C22×C5—41no—
355C22×C5—122no—
356C22×C5—121no—
357C22×C5D3×C22⋊C411yescharacteristic
358C22×C5#88411yescharacteristic
359C22×C5#89511yescharacteristic
360C2×D5—62no—
361C2×D5—122no—
362C4×C5—121no—
363C4×C5—241no—
364C4×C5C4×D1211yescharacteristic
365C4×C5#88211yescharacteristic
366C4×C5#89311yescharacteristic
367C4×C5#89411yescharacteristic
368Dic5—22no—
369Dic5—41no—
370Dic5—62no—
371Dic5—81no—
372#41—61no—
373#41—61no—
374#41—61no—
375#41—61no—
376#41—301no—
377#41—301no—
378#41—301no—
379#41—301no—
380C2×Q8—31no—
381C2×Q8—151no—
382C2×D4—31no—
383C2×D4—61no—
384C2×D4—152no—
385C2×D4—151no—
386C2×D4—302no—
387C22×C4—61no—
388C22×C4—101no—
389C22×C4—152no—
390C22×C4C2×D3×D511yescharacteristic
391M16—21no—
392M16—152no—
393M16—304no—
394M16—301no—
395M16#120312yescomplemented
396C2×C8—21no—
397C2×C8—304no—
398C2×C8—301no—
399C4⋊C4—61no—
400C4⋊C4—101no—
401C4⋊C4—302no—
402C22⋊C4—61no—
403C22⋊C4—101no—
404C22⋊C4—302no—
405C22⋊C4—302no—
406C42—51no—
407C42—51no—
408C42—61no—
409C42—101no—
410C42—101no—
411C3×C5#188411yescharacteristic, complemented, Hall
412C22×C3—41no—
413C22×C3D5×C22⋊C411yescharacteristic
414C22×C3#403011yescharacteristic
415C22×C3#408411yescharacteristic
416C2×D3—42no—
417C2×D3—41no—
418C2×D3—102no—
419C2×D3—202no—
420C3×C4—102no—
421C3×C4—201no—
422C3×C4—401no—
423C3×C4#402711yescharacteristic
424C3×C4#402911yescharacteristic
425C3×C4C4×C22⋊D511yescharacteristic
426C3×C4#408311yescharacteristic
427Dic3—41no—
428Dic3—81no—
429Dic3—102no—
430C2×C5—21no—
431C2×C5—21no—
432C2×C5—21no—
433C2×C5—241no—
434C2×C5#466511yescharacteristic
435D5—122no—
436C23—61no—
437C23—152no—
438C23#677711yescharacteristic
439Q8—61no—
440Q8—61no—
441Q8—301no—
442Q8—301no—
443D4—61no—
444D4—61no—
445D4—121no—
446D4—121no—
447D4—302no—
448D4—302no—
449D4—301no—
450D4—301no—
451D4—604no—
452C2×C4—61no—
453C2×C4—101no—
454C2×C4—101no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—152no—
459C2×C4—202no—
460C2×C4—201no—
461C2×C4—302no—
462C2×C4—302no—
463C2×C4#676811yescharacteristic
464C2×C4#677111yescharacteristic
465C2×C4C4×D3×D511yescharacteristic
466C2×C4C2×#120311yescharacteristic
467C2×C4#679011yescharacteristic
468C2×C4#679011yescharacteristic
469C8—22no—
470C8—302no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3#6524511yescharacteristic
475D3—81no—
476D3—202no—
477C5#6877711yescharacteristic, complemented, Sylow
478C22—41no—
479C22—122no—
480C22—121no—
481C22—302no—
482C22—602no—
483C22#9146111yescharacteristic
484C22#9146611yescharacteristic
485C22#9147311yescharacteristic
486C4—102no—
487C4—121no—
488C4—201no—
489C4—241no—
490C4—302no—
491C4—401no—
492C4#9135511yescharacteristic
493C4#9140611yescharacteristic
494C4C4×#120311yescharacteristic
495C4#9157311yescharacteristic
496C3#1060121011yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—21no—
500C2—241no—
501C2—602no—
502C2#1174891811yescharacteristic
subgroupG/H
centerC4#91573
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC2×C3×C5#386
prime radical#11754572C2
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1884
LCSG ▷ C2×C3×C4×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
1. #11756504C2semidirect
3. C5×#5139C2semidirect
4. #11748626C2semidirect
5. #11748425C2semidirect
6. C3×#64775C2semidirect
7. #11745977C2semidirect
20. #91451C4semidirect
22. #91376C4semidirect
23. #91375C4semidirect
24. #91373C4semidirect
34. #64775D3semidirect
56. C3×C5×M16D4semidirect
59. #6710D4semidirect
77. #5139D5semidirect
176. C5×M16D12semidirect
240. C3×M16C22⋊D5semidirect
395. M16#1203semidirect
411. C3×C5#1884semidirect
477. C5#68777semidirect
496. C3#10601210semidirect

Maximal subgroups

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

subgroupindex
#117565042
#117545722
C5×#51392
#117486262
#117484252
C3×#647752
#117459772
#106012103
#687775