Finite Group Databasebeta

← #14321159

Subgroups — #14321159

Order 1920.

1872 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6821C211yesmaximal, characteristic
2C2×#91781C211yesmaximal, characteristic, complemented
3C2×C42×D15C211yesmaximal, characteristic, complemented
4C3×C5×#391C211yesmaximal, characteristic, complemented
5#11751330C211yesmaximal, characteristic, complemented
6#11751089C211yesmaximal, characteristic, complemented
7#11751040C211yesmaximal, characteristic, complemented
8#11750767C214yesmaximal, complemented
9#11750766C214yesmaximal, complemented
10#10608209—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C2×#6821—24no—
13C2×#6821—24no—
14C2×#6821C418yescomplemented
15C22×D60C2212yes—
16C22×D60C2211yescharacteristic
17C22×C4×D15C2212yes—
18C22×C4×D15C2212yes—
19C22×C4×D15C2211yescharacteristic
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C3×C42×C5C2211yescharacteristic
22C2×#6714C2212yes—
23C2×#6714C2214yes—
24C2×#6714C2214yes—
25C2×C4⋊Dic15C2211yescharacteristic
26C2×C4×Dic15C2212yes—
27C2×C4×Dic15C2212yes—
28C2×C4×Dic15C2211yescharacteristic
29#91781—24no—
30C42×D15—24no—
31C3×C5×#95C2214yes—
32#91115C2214yes—
33#76478—51nomaximal, Hall
34C22×#713—31no—
35C2×#4038—31no—
36C2×C42×D5—31no—
37C5×#391D311yescharacteristic, complemented
38#65519—31no—
39#65277—31no—
40#65228—31no—
41#64955—34no—
42#64954—34no—
43C23×D15C2312yes—
44C23×D15C2311yescharacteristic
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8—22no—
47C2×C3×C5×Q8C2×C414yescomplemented
48C22×C3×C4×C5C2312yes—
49C22×C3×C4×C5C2312yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15C2312yes—
53C22×Dic15C2312yes—
54C22×Dic15C2311yescharacteristic
55#6821—232no—
56C2×D60—24no—
57C2×D60—24no—
58C2×D60—22no—
59C2×D60—22no—
60C2×D60C2×C418yes—
61C2×D60C2×C414yescomplemented
62C2×C4×D15—24no—
63C2×C4×D15—24no—
64C2×C4×D15—24no—
65C2×C4×D15—24no—
66C2×C4×D15—24no—
67C2×C4×D15C2×C414yes—
68C2×C4×D15C2×C418yescomplemented
69C2×C4×D15D414yes—
70C2×C4×D15D414yes—
71C3×C5×C4⋊C4—22no—
72C3×C42×C5—22no—
73#6714—24no—
74#6714—28no—
75#6714—28no—
76C4⋊Dic15—22no—
77C4×Dic15—22no—
78C4×Dic15—24no—
79C4×Dic15—24no—
80C22×#239—51no—
81C2×#892—51no—
82C2×C42×D3—51no—
83C3×#391D511yescharacteristic, complemented
84#5229—51no—
85#4988—51no—
86#4939—51no—
87#4666—54no—
88#4665—54no—
89C22×C5×Q8C2×D311yescharacteristic
90C2×#713—38no—
91C2×#713—64no—
92C2×#713—64no—
93C22×D20—32no—
94C22×D20—31no—
95C22×C4×D5—31no—
96C22×C4×D5—32no—
97C22×C4×D5—32no—
98C2×C5×C4⋊C4C2×D311yescharacteristic
99C2×C42×C5C2×D311yescharacteristic
100C2×#685—32no—
101C2×#685—34no—
102C2×#685—34no—
103C2×C4⋊Dic5—31no—
104C2×C4×Dic5—32no—
105C2×C4×Dic5—31no—
106C2×C4×Dic5—32no—
107#4038—64no—
108C42×D5—64no—
109C5×#95C2×D314yes—
110#3963—34no—
111#2297—151noSylow
112C23×C3×C5C2411yescharacteristic
113C22×D15—22no—
114C22×D15—22no—
115C22×D15—22no—
116C22×D15—22no—
117C22×D15—24no—
118C22×D15—24no—
119C22×D15C22×C414yes—
120C22×D15C22×C412yes—
121C22×D15C2×D412yes—
122C22×D15C2×D412yes—
123C3×C5×Q8—28no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—24no—
130C2×C3×C4×C5C22×C414yes—
131C2×C3×C4×C5C22×C412yes—
132C2×C3×C4×C5C2×D412yes—
133C2×C3×C4×C5C2×D412yes—
134C2×C3×C4×C5#4114yescomplemented
135C2×Dic15—22no—
136C2×Dic15—22no—
137C2×Dic15—22no—
138C2×Dic15—24no—
139C2×Dic15C22×C412yes—
140C2×Dic15C2×D412yes—
141C2×Dic15C2×D412yes—
142C2×Dic15#4114yescomplemented
143D60—216no—
144D60—28no—
145C4×D15—216no—
146C4×D15—216no—
147C4×D15C22⋊C4116yescomplemented
148C22×C3×Q8C2×D511yescharacteristic
149C2×#239—58no—
150C2×#239—104no—
151C2×#239—104no—
152C22×D12—52no—
153C22×D12—51no—
154C22×C4×D3—51no—
155C22×C4×D3—52no—
156C22×C4×D3—52no—
157C2×C3×C4⋊C4C2×D511yescharacteristic
158C2×C3×C42C2×D511yescharacteristic
159C2×(C2×D3)⋊C4—52no—
160C2×(C2×D3)⋊C4—54no—
161C2×(C2×D3)⋊C4—54no—
162C2×C4⋊Dic3—51no—
163C2×C4×Dic3—52no—
164C2×C4×Dic3—52no—
165C2×C4×Dic3—51no—
166#892—104no—
167C42×D3—104no—
168C3×#95C2×D514yes—
169C3⋊#95—54no—
170C23×D5—31no—
171C23×D5—32no—
172C2×C5×Q8—22no—
173C2×C5×Q8—22no—
174C2×C5×Q8C4×D314yescomplemented
175C22×C4×C5C22×D312yes—
176C22×C4×C5C22×D312yes—
177C22×C4×C5C22×D312yes—
178C22×C4×C5C22×D311yescharacteristic
179C22×Dic5—31no—
180C22×Dic5—32no—
181C22×Dic5—32no—
182#713—632no—
183C2×D20—38no—
184C2×D20—34no—
185C2×D20—64no—
186C2×D20—64no—
187C2×D20—62no—
188C2×D20—62no—
189C2×C4×D5—34no—
190C2×C4×D5—34no—
191C2×C4×D5—34no—
192C2×C4×D5—38no—
193C2×C4×D5—64no—
194C2×C4×D5—64no—
195C2×C4×D5—64no—
196C2×C4×D5—64no—
197C2×C4×D5—64no—
198C5×C4⋊C4—22no—
199C42×C5—22no—
200#685—68no—
201#685—64no—
202#685—68no—
203C4⋊Dic5—62no—
204C4×Dic5—64no—
205C4×Dic5—64no—
206C4×Dic5—62no—
207C22×#41—151no—
208C2×#117—151no—
209C22×C42—151no—
210#391—151no—
211#391D1511yescharacteristic, complemented
212#390—151no—
213#390—151no—
214#386—154no—
215C4×C22⋊C4—154no—
216C22×C3×C5C23×C411yescharacteristic
217C22×C3×C5C22×D411yescharacteristic
218C22×C3×C5C22×D411yescharacteristic
219C22×C3×C5C2×#4112yes—
220C22×C3×C5C2×#4112yes—
221C2×D15—24no—
222C2×D15—24no—
223C2×D15—22no—
224C2×D15—22no—
225C2×D15—24no—
226C2×D15—28no—
227C2×D15—44no—
228C2×D15—42no—
229C2×D15—42no—
230C2×D15C2×C22⋊C414yes—
231C3×C4×C5—24no—
232C3×C4×C5—24no—
233C3×C4×C5—42no—
234C3×C4×C5C2×C22⋊C414yes—
235Dic15—24no—
236Dic15—42no—
237Dic15C2×C22⋊C414yes—
238C23×D3—51no—
239C23×D3—52no—
240C2×C3×Q8—22no—
241C2×C3×Q8—22no—
242C2×C3×Q8C4×D514yescomplemented
243C22×C3×C4C22×D512yes—
244C22×C3×C4C22×D512yes—
245C22×C3×C4C22×D512yes—
246C22×C3×C4C22×D511yescharacteristic
247C22×Dic3—51no—
248C22×Dic3—52no—
249C22×Dic3—52no—
250#239—1032no—
251C2×D12—58no—
252C2×D12—54no—
253C2×D12—104no—
254C2×D12—104no—
255C2×D12—102no—
256C2×D12—102no—
257C2×C4×D3—54no—
258C2×C4×D3—54no—
259C2×C4×D3—54no—
260C2×C4×D3—58no—
261C2×C4×D3—104no—
262C2×C4×D3—104no—
263C2×C4×D3—104no—
264C2×C4×D3—104no—
265C2×C4×D3—104no—
266C3×C4⋊C4—22no—
267C3×C42—22no—
268(C2×D3)⋊C4—108no—
269(C2×D3)⋊C4—104no—
270(C2×D3)⋊C4—108no—
271C4⋊Dic3—102no—
272C4×Dic3—104no—
273C4×Dic3—104no—
274C4×Dic3—102no—
275C23×C5C23×D311yescharacteristic
276C22×D5—32no—
277C22×D5—32no—
278C22×D5—32no—
279C22×D5—34no—
280C22×D5—62no—
281C22×D5—62no—
282C22×D5—62no—
283C22×D5—62no—
284C22×D5—64no—
285C22×D5—64no—
286C5×Q8—28no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—24no—
293C2×C4×C5C2×C4×D314yes—
294C2×C4×C5C2×C4×D312yes—
295C2×C4×C5D3×D412yes—
296C2×C4×C5D3×D412yes—
297C2×C4×C5#23914yescomplemented
298C2×Dic5—32no—
299C2×Dic5—32no—
300C2×Dic5—32no—
301C2×Dic5—34no—
302C2×Dic5—62no—
303C2×Dic5—62no—
304C2×Dic5—62no—
305C2×Dic5—64no—
306D20—616no—
307D20—68no—
308C4×D5—316no—
309C4×D5—616no—
310C4×D5—616no—
311C2×#41—158no—
312C2×#41—304no—
313C2×#41—304no—
314C22×Q8C2×D1511yescharacteristic
315C22×D4—152no—
316C22×D4—151no—
317C23×C4—151no—
318C23×C4—152no—
319C23×C4—152no—
320#117—304no—
321C2×C4⋊C4—151no—
322C2×C4⋊C4C2×D1511yescharacteristic
323C2×C22⋊C4—152no—
324C2×C22⋊C4—154no—
325C2×C22⋊C4—154no—
326C2×C42—152no—
327C2×C42—152no—
328C2×C42—151no—
329C2×C42—304no—
330C2×C42C2×D1511yescharacteristic
331#95—154no—
332#95C2×D1514yes—
333C2×C3×C5C22×C22⋊C411yescharacteristic
334C2×C3×C5C4×#4112yes—
335C2×C3×C5#52512yes—
336C2×C3×C5#53211yescharacteristic
337C2×C3×C5#53211yescharacteristic
338D15—24no—
339D15—44no—
340C23×C3C23×D511yescharacteristic
341C22×D3—52no—
342C22×D3—52no—
343C22×D3—52no—
344C22×D3—54no—
345C22×D3—102no—
346C22×D3—102no—
347C22×D3—102no—
348C22×D3—102no—
349C22×D3—104no—
350C22×D3—104no—
351C3×Q8—28no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4C2×C4×D514yes—
359C2×C3×C4C2×C4×D512yes—
360C2×C3×C4D4×D512yes—
361C2×C3×C4D4×D512yes—
362C2×C3×C4#71314yescomplemented
363C2×Dic3—52no—
364C2×Dic3—52no—
365C2×Dic3—52no—
366C2×Dic3—54no—
367C2×Dic3—102no—
368C2×Dic3—102no—
369C2×Dic3—102no—
370C2×Dic3—104no—
371D12—1016no—
372D12—108no—
373C4×D3—516no—
374C4×D3—1016no—
375C4×D3—1016no—
376C22×C5C22×C4×D311yescharacteristic
377C22×C5C2×D3×D411yescharacteristic
378C22×C5C2×D3×D411yescharacteristic
379C22×C5C2×#23912yes—
380C22×C5D3×#4112yes—
381C2×D5—34no—
382C2×D5—62no—
383C2×D5—62no—
384C2×D5—64no—
385C2×D5—64no—
386C2×D5—64no—
387C2×D5—68no—
388C2×D5—122no—
389C2×D5—122no—
390C2×D5—124no—
391C4×C5—24no—
392C4×C5—24no—
393C4×C5—42no—
394C4×C5D3×C22⋊C414yes—
395Dic5—34no—
396Dic5—64no—
397Dic5—122no—
398C24—151no—
399C24—152no—
400#41—3032no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8C4×D1514yescomplemented
404C2×D4—158no—
405C2×D4—154no—
406C2×D4—304no—
407C2×D4—304no—
408C2×D4—302no—
409C2×D4—302no—
410C22×C4—151no—
411C22×C4—154no—
412C22×C4—154no—
413C22×C4—154no—
414C22×C4—152no—
415C22×C4—158no—
416C22×C4—152no—
417C22×C4—304no—
418C22×C4—304no—
419C22×C4—304no—
420C22×C4—304no—
421C22×C4—304no—
422C22×C4C22×D1512yes—
423C22×C4C22×D1512yes—
424C22×C4C22×D1512yes—
425C22×C4C22×D1511yescharacteristic
426C4⋊C4—22no—
427C4⋊C4—302no—
428C22⋊C4—308no—
429C22⋊C4—304no—
430C22⋊C4—308no—
431C42—22no—
432C42—304no—
433C42—304no—
434C42—302no—
435C3×C5#229711yescharacteristic, complemented, Hall
436C22×C3C22×C4×D511yescharacteristic
437C22×C3C2×D4×D511yescharacteristic
438C22×C3C2×D4×D511yescharacteristic
439C22×C3C2×#71312yes—
440C22×C3D5×#4112yes—
441C2×D3—54no—
442C2×D3—102no—
443C2×D3—104no—
444C2×D3—104no—
445C2×D3—102no—
446C2×D3—104no—
447C2×D3—108no—
448C2×D3—202no—
449C2×D3—204no—
450C2×D3—202no—
451C3×C4—24no—
452C3×C4—24no—
453C3×C4—42no—
454C3×C4D5×C22⋊C414yes—
455Dic3—54no—
456Dic3—104no—
457Dic3—202no—
458C2×C5C2×D3×C22⋊C411yescharacteristic
459C2×C5C4×#23912yes—
460C2×C5#562712yes—
461C2×C5#566811yescharacteristic
462C2×C5#572511yescharacteristic
463D5—64no—
464D5—124no—
465C23—152no—
466C23—152no—
467C23—152no—
468C23—154no—
469C23—302no—
470C23—302no—
471C23—302no—
472C23—302no—
473C23—304no—
474C23—304no—
475C23C23×D1511yescharacteristic
476Q8—28no—
477D4—3016no—
478D4—308no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—152no—
486C2×C4—152no—
487C2×C4—152no—
488C2×C4—1516no—
489C2×C4—154no—
490C2×C4—3016no—
491C2×C4—3016no—
492C2×C4—302no—
493C2×C4—302no—
494C2×C4—302no—
495C2×C4—304no—
496C2×C4C2×C4×D1514yes—
497C2×C4C2×C4×D1512yes—
498C2×C4D4×D1512yes—
499C2×C4D4×D1512yes—
500C2×C4#682114yescomplemented
501C2×C3C2×D5×C22⋊C411yescharacteristic
502C2×C3C4×#71312yes—
503C2×C3#6596112yes—
504C2×C3#6600211yescharacteristic
505C2×C3#6605911yescharacteristic
506D3—104no—
507D3—204no—
508C5#7647811yescharacteristic, complemented, Sylow
509C22—154no—
510C22—304no—
511C22—304no—
512C22—302no—
513C22—302no—
514C22—308no—
515C22—304no—
516C22—604no—
517C22—602no—
518C22—602no—
519C22C22×C4×D1511yescharacteristic
520C22C2×D4×D1511yescharacteristic
521C22C2×D4×D1511yescharacteristic
522C22C2×#682112yes—
523C22#41×D1512yes—
524C4—24no—
525C4—24no—
526C4—42no—
527C4—154no—
528C4—304no—
529C4—602no—
530C4C22⋊C4×D1514yes—
531C3#1060820911yescharacteristic, complemented, Sylow
532C2—304no—
533C2—604no—
534C2C2×C22⋊C4×D1511yescharacteristic
535C2C4×#682112yes—
536C2#1175638612yes—
537C2#1175642711yescharacteristic
538C2#1175648411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#391C2
socle, cosocleC23×C3×C5C24
prime radicalC22×#6821C2
hypercenter#391D15
hypocenterC3×C5#2297
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#91781C2semidirect
3. C2×C42×D15C2semidirect
4. C3×C5×#391C2semidirect
5. #11751330C2semidirect
6. #11751089C2semidirect
7. #11751040C2semidirect
8. #11750767C2semidirect
9. #11750766C2semidirect
14. C2×#6821C4semidirect
37. C5×#391D3semidirect
47. C2×C3×C5×Q8C2×C4semidirect
61. C2×D60C2×C4semidirect
68. C2×C4×D15C2×C4semidirect
83. C3×#391D5semidirect
134. C2×C3×C4×C5#41semidirect
142. C2×Dic15#41semidirect
147. C4×D15C22⋊C4semidirect
174. C2×C5×Q8C4×D3semidirect
211. #391D15semidirect
242. C2×C3×Q8C4×D5semidirect
297. C2×C4×C5#239semidirect
362. C2×C3×C4#713semidirect
403. C2×Q8C4×D15semidirect
435. C3×C5#2297semidirect
500. C2×C4#6821semidirect
508. C5#76478semidirect
531. C3#10608209semidirect

Maximal subgroups

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

subgroupindex
C22×#68212
C2×#917812
C2×C42×D152
C3×C5×#3912
#117513302
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