Finite Group Databasebeta

← #14302021

Subgroups — #14302021

Order 1920.

716 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65858C211yesmaximal, characteristic
2C2×#91357C211yesmaximal, characteristic, complemented
3C2×#90977C211yesmaximal, characteristic, complemented
4C2×#90976C211yesmaximal, characteristic, complemented
5#11748341C211yesmaximal, characteristic, complemented
6#11746353C211yesmaximal, characteristic, complemented
7C5×#4532C211yesmaximal, characteristic, complemented
8#10599550—31nomaximal, Hall
9C2×C3×#709C2211yescharacteristic
10C22×C3×C4×D5C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C2×C4×Dic15C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C2×C4×C5×Dic3C2211yescharacteristic
15C2×C5×#208C2211yescharacteristic
16C3×#4043C412yescomplemented
17C3×#4041C412yescomplemented
18C3×#4032—21no—
19C3×#4028—21no—
20C3×C4×D20—22no—
21C2×#6668—21no—
22C2×#6665—21no—
23#91357—22no—
24#91357—22no—
25#90977—22no—
26#90976—22no—
27#90955—22no—
28#71712—51nomaximal, Hall
29#65858D311yescharacteristic
30C2×#4018—31no—
31C2×#3957—31no—
32#65415—31no—
33C2×#3937—31no—
34C5×#339—31no—
35#64752—31no—
36C23×C3×D5D411yescharacteristic
37C22×C3×C4×C5C2311yescharacteristic
38C22×Dic15—21no—
39C22×C5×Dic3—21no—
40C2×C3×C22⋊D5—21no—
41C22×C3×Dic5D411yescharacteristic
42C3×#709—21no—
43C3×#709—21no—
44C3×#709—21no—
45C3×#709—21no—
46C2×C3×D20—22no—
47C2×C3×D20C2×C411yescharacteristic
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×C3×C4×D5C2×C412yes—
54C2×C3×C4×D5D412yes—
55C2×C3×Dic10C2×C411yescharacteristic
56C3×C5×C4⋊C4C2×C412yes—
57C3×C5×C22⋊C4—21no—
58C3×C42×C5—21no—
59C4×Dic15—21no—
60C4×Dic15D412yescomplemented
61#6710—21no—
62#6710—21no—
63C2×C3⋊#168—21no—
64C4×C5×Dic3—21no—
65C4×C5×Dic3D412yescomplemented
66C5×#208—21no—
67C5×#208—21no—
68C2×C5×C3⋊C8—21no—
69C3×#685—22no—
70C3×#685—22no—
71C3×C4⋊Dic5—22no—
72#6668—24no—
73#6665—24no—
74C3×#525—51no—
75C2×#926—51no—
76C2×#837—51no—
77C2×#837—51no—
78#5207—51no—
79#4532—51no—
80#4532D511yescharacteristic, complemented
81C2×#709C2×D311yescharacteristic
82C22×C4×D5C2×D311yescharacteristic
83C2×C5×M16—31no—
84C5×#117C2×D311yescharacteristic
85C2×C42×C5—31no—
86C2×#685—61no—
87C2×#683—61no—
88C2×C4×Dic5—31no—
89C2×#681—31no—
90#4043Dic312yescomplemented
91#4041Dic312yescomplemented
92#4032—21no—
93#4028—21no—
94C4×D20—22no—
95#4018—62no—
96#4018—62no—
97#3957—62no—
98#3952—62no—
99#3937—62no—
100#1897—151noSylow
101C23×C3×C5C2×D411yescharacteristic
102C22×C3×D5—21no—
103C22×C3×D5—21no—
104C22×C3×D5—21no—
105C22×C3×D5—42no—
106C22×C3×D5C22⋊C412yes—
107C2×C3×C4×C5—22no—
108C2×C3×C4×C5C22×C411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5C2×D411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C3×C4×C5#4111yescharacteristic
113C2×C3×C4×C5#4111yescharacteristic
114C2×Dic15—22no—
115C2×Dic15—42no—
116C2×C5×Dic3—22no—
117C2×C5×Dic3—42no—
118C3×C22⋊D5—41no—
119C3×C22⋊D5—41no—
120C2×C3×Dic5—21no—
121C2×C3×Dic5—21no—
122C2×C3×Dic5—21no—
123C2×C3×Dic5C22⋊C412yes—
124C3×D20—21no—
125C3×D20—21no—
126C3×D20—44no—
127C3×C4×D5—22no—
128C3×C4×D5—22no—
129C3×C4×D5—22no—
130C3×C4×D5—22no—
131C3×C4×D5—41no—
132C3×C4×D5—41no—
133C3×Dic10—21no—
134C3×Dic10—21no—
135C3⋊#168—41no—
136C5×C3⋊C8—41no—
137C2×C3×#41—51no—
138C23×C3×C4—51no—
139C3×#123—101no—
140C3×C22⋊Q8—52no—
141C3×#121—52no—
142C3×#120—101no—
143C3×C4×D4—102no—
144C3×#117C2×D511yescharacteristic
145C2×#217—101no—
146#926—102no—
147#926—102no—
148C2×C4⋊Dic3—101no—
149C2×C4×Dic3—51no—
150C2×C4×Dic3C2×D511yescharacteristic
151C2×#208—51no—
152C2×#208C2×D511yescharacteristic
153#837—102no—
154#837—102no—
155#830—102no—
156C23×D5#6711yescharacteristic
157C22×C4×C5—61no—
158C22×C4×C5C22×D311yescharacteristic
159C2×C22⋊D5—21no—
160C22×Dic5—61no—
161C22×Dic5#6711yescharacteristic
162#709—21no—
163#709—21no—
164#709—21no—
165#709—21no—
166C2×D20—22no—
167C2×D20C2×Dic311yescharacteristic
168C2×C4×D5—21no—
169C2×C4×D5—21no—
170C2×C4×D5—21no—
171C2×C4×D5—21no—
172C2×C4×D5—21no—
173C2×C4×D5C2×Dic312yes—
174C2×C4×D5#6712yes—
175C2×Dic10C2×Dic311yescharacteristic
176C5×M16—61no—
177C5×M16—61no—
178C2×C5×C8—61no—
179C5×C4⋊C4C2×Dic312yes—
180C5×C22⋊C4—21no—
181C42×C5—21no—
182C42×C5—32no—
183C42×C5—61no—
184#685—22no—
185#685—22no—
186#685—64no—
187C4⋊Dic5—22no—
188#683—64no—
189C4×Dic5—32no—
190C4×Dic5—61no—
191#681—61no—
192#681—61no—
193C2×#168—61no—
194#525—51no—
195C2×#117—151no—
196C2×#104—151no—
197C2×#104—151no—
198#407—151no—
199#339—31no—
200#339—151no—
201C22×C3×C5—22no—
202C22×C3×C5C2×C22⋊C411yescharacteristic
203C22×C3×C5#12011yescharacteristic
204C22×C3×C5#12311yescharacteristic
205C2×C3×D5—22no—
206C2×C3×D5—22no—
207C2×C3×D5—22no—
208C2×C3×D5—42no—
209C2×C3×D5—42no—
210C2×C3×D5—41no—
211C2×C3×D5—42no—
212C3×C4×C5—42no—
213C3×C4×C5C4×D411yescharacteristic
214C3×C4×C5C4×D411yescharacteristic
215C3×C4×C5#12111yescharacteristic
216C3×C4×C5#12111yescharacteristic
217Dic15—42no—
218C3×Dic5—22no—
219C3×Dic5—41no—
220C5×Dic3—42no—
221C24×C3—51no—
222C3×#41—101no—
223C3×#41—101no—
224C3×#41—101no—
225C3×#41—101no—
226C2×C3×Q8—51no—
227C2×C3×D4—51no—
228C2×C3×D4—102no—
229C2×C3×D4—101no—
230C22×C3×C4—51no—
231C22×C3×C4—52no—
232C22×C3×C4—52no—
233C22×C3×C4—101no—
234C22×C3×C4—101no—
235C22×C3×C4—101no—
236C22×C3×C4—101no—
237C22×C3×C4—101no—
238C22×C3×C4C22×D511yescharacteristic
239C22×Dic3—21no—
240C22×Dic3—101no—
241C3×C4⋊C4—102no—
242C3×C4⋊C4C4×D512yes—
243C3×C22⋊C4—21no—
244C3×C22⋊C4—102no—
245C3×C22⋊C4—102no—
246C3×C42—21no—
247#217—104no—
248C4⋊Dic3—104no—
249C4×Dic3—21no—
250C4×Dic3—52no—
251C4×Dic3—101no—
252C4×Dic3D2012yescomplemented
253#208—21no—
254#208—21no—
255#208—101no—
256#208—101no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—101no—
259C23×C5C2×#6711yescharacteristic
260C22×D5—21no—
261C22×D5—21no—
262C22×D5—21no—
263C22×D5—42no—
264C22×D5#21712yes—
265C2×C4×C5—22no—
266C2×C4×C5—62no—
267C2×C4×C5—122no—
268C2×C4×C5D3×D411yescharacteristic
269C2×C4×C5D3×D411yescharacteristic
270C2×C4×C5#23711yescharacteristic
271C2×C4×C5#23711yescharacteristic
272C2×C4×C5C22×Dic311yescharacteristic
273C2×C4×C5C2×#6711yescharacteristic
274C22⋊D5—41no—
275C22⋊D5—41no—
276C2×Dic5—21no—
277C2×Dic5—21no—
278C2×Dic5—21no—
279C2×Dic5—62no—
280C2×Dic5—122no—
281C2×Dic5#21712yes—
282D20—21no—
283D20—21no—
284D20—44no—
285C4×D5—22no—
286C4×D5—22no—
287C4×D5—22no—
288C4×D5—22no—
289C4×D5—41no—
290C4×D5—41no—
291Dic10—21no—
292Dic10—21no—
293C5×C8—121no—
294#168—121no—
295C2×#41—51no—
296C23×C4—51no—
297C2×M16—31no—
298C2×M16—151no—
299#123—101no—
300C22⋊Q8—52no—
301#121—52no—
302#120—101no—
303C4×D4—102no—
304#117—302no—
305#117—302no—
306#117D3×D511yescharacteristic
307C2×C4⋊C4—301no—
308C2×C22⋊C4—301no—
309C2×C42—31no—
310C2×C42—151no—
311#104—302no—
312#104—302no—
313C22⋊C8—302no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5#38611yescharacteristic
317C2×C3×C5#42112yes—
318C3×D5—42no—
319C3×D5—81no—
320C23×C3—52no—
321C23×C3—101no—
322C23×C3—101no—
323C23×C3—101no—
324C23×C3—202no—
325C23×C3D4×D511yescharacteristic
326C3×Q8—101no—
327C3×Q8—101no—
328C3×D4—101no—
329C3×D4—101no—
330C3×D4—204no—
331C3×D4—201no—
332C3×D4—201no—
333C2×C3×C4—22no—
334C2×C3×C4—52no—
335C2×C3×C4—101no—
336C2×C3×C4—101no—
337C2×C3×C4—102no—
338C2×C3×C4—102no—
339C2×C3×C4—102no—
340C2×C3×C4—102no—
341C2×C3×C4—101no—
342C2×C3×C4—201no—
343C2×C3×C4—201no—
344C2×C3×C4C2×C4×D511yescharacteristic
345C2×C3×C4C2×D2011yescharacteristic
346C2×C3×C4#70911yescharacteristic
347C2×C3×C4D4×D511yescharacteristic
348C2×C3×C4D4×D511yescharacteristic
349C2×C3×C4#71311yescharacteristic
350C2×Dic3—22no—
351C2×Dic3—42no—
352C2×Dic3—102no—
353C2×Dic3—202no—
354C3⋊C8—41no—
355C3⋊C8—201no—
356C22×C5—22no—
357C22×C5#93511yescharacteristic
358C22×C5#93711yescharacteristic
359C22×C5C2×#21711yescharacteristic
360C2×D5—22no—
361C2×D5—22no—
362C2×D5—22no—
363C2×D5—42no—
364C2×D5—42no—
365C2×D5—41no—
366C2×D5—42no—
367C4×C5—42no—
368C4×C5—122no—
369C4×C5D4×Dic311yescharacteristic
370C4×C5D4×Dic311yescharacteristic
371C4×C5#93911yescharacteristic
372C4×C5#93911yescharacteristic
373Dic5—22no—
374Dic5—41no—
375Dic5—122no—
376C24—51no—
377#41—101no—
378#41—101no—
379#41—101no—
380#41—101no—
381C2×Q8—51no—
382C2×D4—51no—
383C2×D4—102no—
384C2×D4—101no—
385C22×C4—51no—
386C22×C4—52no—
387C22×C4—52no—
388C22×C4—61no—
389C22×C4—101no—
390C22×C4—101no—
391C22×C4—101no—
392C22×C4—101no—
393C22×C4—101no—
394C22×C4—301no—
395C22×C4C2×D3×D511yescharacteristic
396M16—61no—
397M16—61no—
398M16—301no—
399M16—301no—
400C2×C8—61no—
401C2×C8—301no—
402C4⋊C4—102no—
403C4⋊C4—304no—
404C4⋊C4D5×Dic312yes—
405C22⋊C4—21no—
406C22⋊C4—102no—
407C22⋊C4—102no—
408C22⋊C4—304no—
409C42—21no—
410C42—32no—
411C42—61no—
412C42—152no—
413C42—301no—
414C3×C5#189711yescharacteristic, complemented, Hall
415C22×C3—22no—
416C22×C3—102no—
417C22×C3—102no—
418C22×C3—102no—
419C22×C3—202no—
420C22×C3—202no—
421C22×C3—202no—
422C22×C3—201no—
423C22×C3D5×C22⋊C411yescharacteristic
424C22×C3#402811yescharacteristic
425C22×C3#404011yescharacteristic
426C3×C4—42no—
427C3×C4—102no—
428C3×C4—201no—
429C3×C4C4×D2011yescharacteristic
430C3×C4#403011yescharacteristic
431C3×C4#403911yescharacteristic
432C3×C4#404111yescharacteristic
433Dic3—42no—
434Dic3—202no—
435C2×C5—21no—
436C2×C5—21no—
437C2×C5#522011yescharacteristic
438C2×C5#523612yes—
439D5—42no—
440D5—81no—
441C23—52no—
442C23—101no—
443C23—101no—
444C23—101no—
445C23—202no—
446C23D5×#6711yescharacteristic
447Q8—101no—
448Q8—101no—
449D4—101no—
450D4—101no—
451D4—204no—
452D4—201no—
453D4—201no—
454C2×C4—22no—
455C2×C4—52no—
456C2×C4—62no—
457C2×C4—101no—
458C2×C4—101no—
459C2×C4—102no—
460C2×C4—102no—
461C2×C4—102no—
462C2×C4—102no—
463C2×C4—101no—
464C2×C4—122no—
465C2×C4—201no—
466C2×C4—201no—
467C2×C4—302no—
468C2×C4—602no—
469C2×C4#676511yescharacteristic
470C2×C4#676611yescharacteristic
471C2×C4D3×D2011yescharacteristic
472C2×C4#677711yescharacteristic
473C2×C4C2×D5×Dic311yescharacteristic
474C2×C4D5×#6711yescharacteristic
475C8—121no—
476C8—601no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—202no—
480C2×C3—401no—
481C2×C3#6495411yescharacteristic
482C2×C3#6511012yes—
483C5#7171211yescharacteristic, complemented, Sylow
484C22—22no—
485C22—102no—
486C22—102no—
487C22—102no—
488C22—202no—
489C22—202no—
490C22—201no—
491C22—202no—
492C22#9140711yescharacteristic
493C22#9147411yescharacteristic
494C22D5×#21711yescharacteristic
495C4—42no—
496C4—102no—
497C4—122no—
498C4—201no—
499C4—602no—
500C4#9140811yescharacteristic
501C4#9140911yescharacteristic
502C4Dic3×D2011yescharacteristic
503C4#9143411yescharacteristic
504C3#1059955011yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—202no—
508C2—401no—
509C2#1174889611yescharacteristic
510C2#1174942712yes—
subgroupG/H
centerC2×C4D5×#67
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65858C2
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1897
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C2×#91357C2semidirect
3. C2×#90977C2semidirect
4. C2×#90976C2semidirect
5. #11748341C2semidirect
6. #11746353C2semidirect
7. C5×#4532C2semidirect
16. C3×#4043C4semidirect
17. C3×#4041C4semidirect
60. C4×Dic15D4semidirect
65. C4×C5×Dic3D4semidirect
80. #4532D5semidirect
90. #4043Dic3semidirect
91. #4041Dic3semidirect
252. C4×Dic3D20semidirect
414. C3×C5#1897semidirect
483. C5#71712semidirect
504. C3#10599550semidirect

Maximal subgroups

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

subgroupindex
C3×#658582
C2×#913572
C2×#909772
C2×#909762
#117483412
#117463532
C5×#45322
#105995503
#717125