Finite Group Databasebeta

← #14298386

Subgroups — #14298386

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91437C211yesmaximal, characteristic
2#11751227C211yesmaximal, characteristic
3#11748351C211yesmaximal, characteristic
4C5×#4554C211yesmaximal, characteristic, complemented
5C3×#64774C211yesmaximal, characteristic, complemented
6#11745986C211yesmaximal, characteristic, complemented
7#11745985C211yesmaximal, characteristic, complemented
8#10599783—31nomaximal, Hall
9C22×C4×D15C2211yescharacteristic
10C22×#1200C412yes—
11C2×C3×C5×M16C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×C5×C4⋊Dic3C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C2×C3×C4⋊Dic5C2211yescharacteristic
16C2×C3×#681C2211yescharacteristic
17C2×#6669C412yes—
18#91437—21no—
19#91437—21no—
20#91437—21no—
21#91437—21no—
22#91104—21no—
23#91104—21no—
24#90957—21no—
25#90957—21no—
26#68990—51nomaximal, Hall
27C2×D5×C4⋊C4—31no—
28#65415—31no—
29#65415—31no—
30C5×#343—31no—
31#64774—31no—
32#64774D311yescharacteristic, complemented
33#64750—31no—
34C23×D15C2×C411yescharacteristic
35C22×C3×C4×C5C2311yescharacteristic
36C22×Dic15C2×C411yescharacteristic
37C2×C4×D15—21no—
38C2×C4×D15—21no—
39C2×C4×D15—21no—
40C2×C4×D15—21no—
41C2×C4×D15D411yescharacteristic
42C2×C4×D15D411yescharacteristic
43C2×C4×D15D411yescharacteristic
44C2×C4×D15D411yescharacteristic
45C22×C5×Dic3C2×C412yes—
46C22×C3×Dic5C2×C412yes—
47C2×#1200—22no—
48C2×#1200—22no—
49C2×#1200—44no—
50C3×C5×M16—21no—
51C3×C5×M16—21no—
52C2×C3×C5×C8—21no—
53#6710—21no—
54#6710—21no—
55C2×C3⋊#168—21no—
56C5×C4⋊Dic3—21no—
57C5×C4⋊Dic3—21no—
58C5×#208—21no—
59C5×#208—21no—
60C2×C5×C3⋊C8—21no—
61C3×C4⋊Dic5—21no—
62C3×C4⋊Dic5—21no—
63C3×#681—21no—
64C3×#681—21no—
65C2×C3×#168—21no—
66#6669—22no—
67#6669—22no—
68C2×D3×C4⋊C4—51no—
69#5126—51no—
70#5126—51no—
71C3×#343—51no—
72#4554—51no—
73#4554D511yescharacteristic, complemented
74#4530—51no—
75C22×C4×D5—31no—
76C22×C4×D5—32no—
77C2×C5×M16—31no—
78C2×C5×M16C2×D311yescharacteristic
79C2×C5×C4⋊C4—31no—
80C2×C4⋊Dic5C2×D311yescharacteristic
81C2×#683—32no—
82C2×#681—31no—
83C2×#681C2×D311yescharacteristic
84D5×C4⋊C4—61no—
85D5×C4⋊C4—61no—
86D5×C4⋊C4—61no—
87D5×C4⋊C4—61no—
88#3952—61no—
89#3952—61no—
90#3952—61no—
91#3952—61no—
92#1821—151noSylow
93C23×C3×C5C22×C411yescharacteristic
94C22×D15—21no—
95C22×D15—21no—
96C22×D15—22no—
97C22×D15—22no—
98C22×D15C22⋊C411yescharacteristic
99C22×D15C22⋊C411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5#4111yescharacteristic
103C2×C3×C4×C5#4111yescharacteristic
104C2×C3×C4×C5#4111yescharacteristic
105C2×C3×C4×C5#4111yescharacteristic
106C2×Dic15—21no—
107C2×Dic15—21no—
108C2×Dic15C22⋊C411yescharacteristic
109C2×Dic15C22⋊C411yescharacteristic
110C4×D15—21no—
111C4×D15—21no—
112C4×D15—21no—
113C4×D15—21no—
114C4×D15—21no—
115C4×D15—21no—
116C4×D15—21no—
117C4×D15—21no—
118C2×C5×Dic3—22no—
119C2×C5×Dic3—42no—
120C2×C3×Dic5—22no—
121C2×C3×Dic5—42no—
122#1200—44no—
123#1200—44no—
124C3×C5×C8—41no—
125C3⋊#168—41no—
126C3×#168—41no—
127C5×C3⋊C8—41no—
128C22×C4×D3—51no—
129C22×C4×D3—52no—
130C2×C3×M16—51no—
131C2×C3×M16C2×D511yescharacteristic
132C2×C3×C4⋊C4—51no—
133C2×C4⋊Dic3C2×D511yescharacteristic
134C2×Dic3⋊C4—52no—
135C2×#208—51no—
136C2×#208C2×D511yescharacteristic
137D3×C4⋊C4—101no—
138D3×C4⋊C4—101no—
139D3×C4⋊C4—101no—
140D3×C4⋊C4—101no—
141(C2×D3)⋊C8—101no—
142(C2×D3)⋊C8—101no—
143(C2×D3)⋊C8—101no—
144(C2×D3)⋊C8—101no—
145C23×D5—31no—
146C22×C4×C5—32no—
147C22×C4×C5C22×D311yescharacteristic
148C22×Dic5—31no—
149C22×Dic5C4×D312yes—
150C2×C4×D5—31no—
151C2×C4×D5—31no—
152C2×C4×D5—31no—
153C2×C4×D5—31no—
154C2×C4×D5—61no—
155C2×C4×D5—61no—
156C2×C4×D5—61no—
157C2×C4×D5—61no—
158C2×C4×D5—62no—
159C2×C4×D5—62no—
160C2×C4×D5—124no—
161C5×M16—21no—
162C5×M16—21no—
163C5×M16—61no—
164C5×M16—61no—
165C2×C5×C8—21no—
166C2×C5×C8—61no—
167C5×C4⋊C4—61no—
168C5×C4⋊C4—61no—
169C4⋊Dic5—21no—
170C4⋊Dic5—21no—
171#683—62no—
172#683—62no—
173#681—21no—
174#681—21no—
175#681—61no—
176#681—61no—
177C2×#168—21no—
178C2×#168—61no—
179C22×C4⋊C4—151no—
180#407—151no—
181#407—151no—
182#343—31no—
183#343—51no—
184#343—151no—
185#343—151no—
186C22×C3×C5—21no—
187C22×C3×C5—21no—
188C22×C3×C5C2×C22⋊C411yescharacteristic
189C22×C3×C5#11711yescharacteristic
190C22×C3×C5#11711yescharacteristic
191C2×D15—24no—
192C2×D15—24no—
193C2×D15—21no—
194C2×D15—21no—
195C2×D15—21no—
196C2×D15—21no—
197C2×D15#10012yes—
198C2×D15#10112yes—
199C3×C4×C5#12311yescharacteristic
200C3×C4×C5#12311yescharacteristic
201C3×C4×C5#12311yescharacteristic
202C3×C4×C5#12311yescharacteristic
203Dic15—21no—
204Dic15—21no—
205C3×Dic5—42no—
206C5×Dic3—42no—
207C23×D3—51no—
208C22×C3×C4—52no—
209C22×C3×C4C22×D511yescharacteristic
210C22×Dic3—51no—
211C22×Dic3C4×D512yes—
212C2×C4×D3—51no—
213C2×C4×D3—51no—
214C2×C4×D3—51no—
215C2×C4×D3—51no—
216C2×C4×D3—101no—
217C2×C4×D3—101no—
218C2×C4×D3—101no—
219C2×C4×D3—101no—
220C2×C4×D3—102no—
221C2×C4×D3—102no—
222C2×C4×D3—204no—
223C3×M16—21no—
224C3×M16—21no—
225C3×M16—101no—
226C3×M16—101no—
227C2×C3×C8—21no—
228C2×C3×C8—101no—
229C3×C4⋊C4—101no—
230C3×C4⋊C4—101no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233Dic3⋊C4—102no—
234Dic3⋊C4—102no—
235#208—21no—
236#208—21no—
237#208—101no—
238#208—101no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—101no—
241C23×C5C2×C4×D311yescharacteristic
242C22×D5—31no—
243C22×D5—31no—
244C22×D5—62no—
245C22×D5—62no—
246C22×D5—61no—
247C22×D5—61no—
248C2×C4×C5—62no—
249C2×C4×C5—122no—
250C2×C4×C5#23511yescharacteristic
251C2×C4×C5#23511yescharacteristic
252C2×C4×C5D3×D411yescharacteristic
253C2×C4×C5D3×D411yescharacteristic
254C2×C4×C5#23711yescharacteristic
255C2×C4×C5#23911yescharacteristic
256C2×Dic5—22no—
257C2×Dic5—31no—
258C2×Dic5—31no—
259C2×Dic5—42no—
260C2×Dic5—61no—
261C2×Dic5—61no—
262C4×D5—61no—
263C4×D5—61no—
264C4×D5—61no—
265C4×D5—61no—
266C4×D5—61no—
267C4×D5—61no—
268C4×D5—61no—
269C4×D5—61no—
270C4×D5—124no—
271C4×D5—124no—
272C5×C8—41no—
273C5×C8—121no—
274#168—41no—
275#168—121no—
276C23×C4—152no—
277C23×C4—151no—
278C2×M16—31no—
279C2×M16—51no—
280C2×M16—151no—
281C2×M16D3×D511yescharacteristic
282C2×C4⋊C4—31no—
283C2×C4⋊C4—51no—
284C2×C4⋊C4—152no—
285C2×C4⋊C4—301no—
286C2×C4⋊C4—301no—
287C2×C4⋊C4—301no—
288C2×C4⋊C4—301no—
289C22⋊C8—301no—
290C22⋊C8—301no—
291C22⋊C8—301no—
292C22⋊C8—301no—
293C2×C3×C5—21no—
294C2×C3×C5—21no—
295C2×C3×C5#38111yescharacteristic
296C2×C3×C5C2×#10011yescharacteristic
297C2×C3×C5C2×#10111yescharacteristic
298D15—22no—
299D15—22no—
300C23×C3C2×C4×D511yescharacteristic
301C22×D3—51no—
302C22×D3—51no—
303C22×D3—102no—
304C22×D3—102no—
305C22×D3—101no—
306C22×D3—101no—
307C2×C3×C4—102no—
308C2×C3×C4—202no—
309C2×C3×C4#70911yescharacteristic
310C2×C3×C4#70911yescharacteristic
311C2×C3×C4D4×D511yescharacteristic
312C2×C3×C4D4×D511yescharacteristic
313C2×C3×C4#71111yescharacteristic
314C2×C3×C4#71311yescharacteristic
315C2×Dic3—22no—
316C2×Dic3—42no—
317C2×Dic3—51no—
318C2×Dic3—51no—
319C2×Dic3—101no—
320C2×Dic3—101no—
321C4×D3—101no—
322C4×D3—101no—
323C4×D3—101no—
324C4×D3—101no—
325C4×D3—101no—
326C4×D3—101no—
327C4×D3—101no—
328C4×D3—101no—
329C4×D3—204no—
330C4×D3—204no—
331C3×C8—41no—
332C3×C8—201no—
333C3⋊C8—41no—
334C3⋊C8—201no—
335C22×C5—21no—
336C22×C5—21no—
337C22×C5#87211yescharacteristic
338C22×C5D3×C22⋊C411yescharacteristic
339C22×C5#89211yescharacteristic
340C2×D5—32no—
341C2×D5—32no—
342C2×D5—64no—
343C2×D5—64no—
344C2×D5—61no—
345C2×D5—61no—
346C2×D5—61no—
347C2×D5—61no—
348C4×C5—122no—
349C4×C5#88311yescharacteristic
350C4×C5#88311yescharacteristic
351C4×C5#89411yescharacteristic
352C4×C5#89411yescharacteristic
353Dic5—42no—
354Dic5—61no—
355Dic5—61no—
356C24—151no—
357C22×C4—32no—
358C22×C4—52no—
359C22×C4—151no—
360C22×C4—151no—
361C22×C4—151no—
362C22×C4—151no—
363C22×C4—151no—
364C22×C4—302no—
365C22×C4—302no—
366C22×C4—301no—
367C22×C4—301no—
368C22×C4—301no—
369C22×C4—301no—
370C22×C4—604no—
371C22×C4C2×D3×D511yescharacteristic
372M16—21no—
373M16—21no—
374M16—61no—
375M16—61no—
376M16—101no—
377M16—101no—
378M16—301no—
379M16—301no—
380C2×C8—21no—
381C2×C8—61no—
382C2×C8—101no—
383C2×C8—301no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—101no—
387C4⋊C4—101no—
388C4⋊C4—302no—
389C4⋊C4—302no—
390C3×C5#182111yescharacteristic, complemented, Hall
391C22×C3—21no—
392C22×C3—21no—
393C22×C3#401811yescharacteristic
394C22×C3D5×C22⋊C411yescharacteristic
395C22×C3#403811yescharacteristic
396C2×D3—52no—
397C2×D3—52no—
398C2×D3—104no—
399C2×D3—104no—
400C2×D3—101no—
401C2×D3—101no—
402C2×D3—101no—
403C2×D3—101no—
404C3×C4—202no—
405C3×C4#402911yescharacteristic
406C3×C4#402911yescharacteristic
407C3×C4#404011yescharacteristic
408C3×C4#404011yescharacteristic
409Dic3—42no—
410Dic3—101no—
411Dic3—101no—
412C2×C5—21no—
413C2×C5—21no—
414C2×C5#466411yescharacteristic
415C2×C5D3×#10011yescharacteristic
416C2×C5D3×#10111yescharacteristic
417D5—62no—
418D5—62no—
419C23—151no—
420C23—151no—
421C23—302no—
422C23—302no—
423C23—301no—
424C23—301no—
425C23C4×D3×D511yescharacteristic
426C2×C4—62no—
427C2×C4—102no—
428C2×C4—122no—
429C2×C4—151no—
430C2×C4—151no—
431C2×C4—202no—
432C2×C4—301no—
433C2×C4—301no—
434C2×C4—301no—
435C2×C4—301no—
436C2×C4—301no—
437C2×C4—301no—
438C2×C4—301no—
439C2×C4—301no—
440C2×C4—301no—
441C2×C4—301no—
442C2×C4—604no—
443C2×C4—604no—
444C2×C4#676911yescharacteristic
445C2×C4#677311yescharacteristic
446C2×C4#677711yescharacteristic
447C2×C4#677911yescharacteristic
448C2×C4#678211yescharacteristic
449C2×C4#679011yescharacteristic
450C8—41no—
451C8—121no—
452C8—201no—
453C8—601no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3#6495311yescharacteristic
457C2×C3D5×#10011yescharacteristic
458C2×C3D5×#10111yescharacteristic
459D3—102no—
460D3—102no—
461C5#6899011yescharacteristic, complemented, Sylow
462C22—21no—
463C22—21no—
464C22—152no—
465C22—152no—
466C22—304no—
467C22—304no—
468C22—301no—
469C22—301no—
470C22—301no—
471C22—301no—
472C22#9140211yescharacteristic
473C22#9140311yescharacteristic
474C22#9147311yescharacteristic
475C4—122no—
476C4—202no—
477C4—301no—
478C4—301no—
479C4#9135611yescharacteristic
480C4#9143311yescharacteristic
481C4#9143811yescharacteristic
482C4#9156211yescharacteristic
483C3#1059978311yescharacteristic, complemented, Sylow
484C2—21no—
485C2—21no—
486C2—302no—
487C2—302no—
488C2#1174822411yescharacteristic
489C2#1174822711yescharacteristic
490C2#1174888911yescharacteristic
subgroupG/H
centerC22#91473
commutatorC23×C3×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×D15C2×C4
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1821
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. C5×#4554C2semidirect
5. C3×#64774C2semidirect
6. #11745986C2semidirect
7. #11745985C2semidirect
32. #64774D3semidirect
73. #4554D5semidirect
390. C3×C5#1821semidirect
461. C5#68990semidirect
483. C3#10599783semidirect

Maximal subgroups

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

subgroupindex
C2×#914372
#117512272
#117483512
C5×#45542
C3×#647742
#117459862
#117459852
#105997833
#689905