Finite Group Databasebeta

← #14322564

Subgroups — #14322564

Order 1920.

1760 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6768C211yesmaximal, characteristic
2C2×#91761C211yesmaximal, characteristic, complemented
3C2×#91437C211yesmaximal, characteristic, complemented
4C5×#4939C211yesmaximal, characteristic, complemented
5C3×#65218C211yesmaximal, characteristic, complemented
6#11748919C214yesmaximal, complemented
7#11748887C214yesmaximal, complemented
8#11748797C211yesmaximal, characteristic, complemented
9#11748610C211yesmaximal, characteristic, complemented
10#10608025—31nomaximal, Hall
11C22×C4×D15C2211yescharacteristic
12C22×C5×D12C2211yescharacteristic
13C22×C3×Dic10C2211yescharacteristic
14C22×#1203C2212yes—
15C22×#1200C2212yes—
16C22×D3×Dic5C2212yes—
17C2×#6768—24no—
18C2×#6768—24no—
19C2×#6768C418yescomplemented
20C2×C3×C42×C5C2211yescharacteristic
21C2×#6714C2212yes—
22C2×#6712C2212yes—
23C2×C4×Dic15C2211yescharacteristic
24#91761—24no—
25C2×C5×(C2×D3)⋊C4C2214yes—
26C2×C5×C4⋊Dic3C2211yescharacteristic
27C2×C3×C4⋊Dic5C2211yescharacteristic
28C2×#6669C2212yes—
29C2×(C2×D3)⋊Dic5C2214yes—
30#91437—24no—
31C3×#3963C2214yes—
32#90994C2214yes—
33#76482—51nomaximal, Hall
34C22×#711—31no—
35C2×D5×C4⋊C4—31no—
36C2×#4018—31no—
37C5×#390—31no—
38#65510—31no—
39#65263—31no—
40#65245—34no—
41#65231—34no—
42#65218D311yescharacteristic, complemented
43C23×D15C2311yescharacteristic
44C23×C5×D3C2312yes—
45C22×C3×C4×C5C2312yes—
46C22×C3×C4×C5C2311yescharacteristic
47C22×Dic15C2312yes—
48C22×Dic15C2311yescharacteristic
49C2×C4×D15C2×C414yes—
50C2×C4×D15D414yes—
51C2×C4×D15D414yes—
52C22×C5×Dic3C2312yes—
53C2×C5×D12—22no—
54C2×C5×D12—22no—
55C2×C5×D12C2×C414yescomplemented
56C22×C3×Dic5C2312yes—
57C22×C3×Dic5C2312yes—
58C2×C3×Dic10—22no—
59C2×C3×Dic10—22no—
60C2×C3×Dic10C2×C414yescomplemented
61C2×#1203—24no—
62C2×#1203—24no—
63C2×#1203C2×C418yes—
64C2×#1200—24no—
65C2×#1200—44no—
66C2×D3×Dic5—24no—
67C2×D3×Dic5—24no—
68C2×D3×Dic5C2×C418yescomplemented
69#6768—232no—
70C3×C42×C5—22no—
71#6714—24no—
72#6712—24no—
73C4×Dic15—22no—
74C5×(C2×D3)⋊C4—44no—
75C5×C4⋊Dic3—22no—
76C3×C4⋊Dic5—22no—
77#6669—24no—
78(C2×D3)⋊Dic5—44no—
79C22×#239—51no—
80C2×D3×C4⋊C4—51no—
81C2×#872—51no—
82C3×#391—51no—
83#5229—51no—
84#4988—51no—
85#4939D511yescharacteristic, complemented
86#4667—54no—
87#4665—54no—
88C22×C5×D4—31no—
89C22×C22⋊D5—32no—
90C23×Dic5—32no—
91C2×#711—38no—
92C2×#711—64no—
93C2×#711—64no—
94C22×C4×D5—31no—
95C22×C4×D5—32no—
96C22×Dic10C2×D311yescharacteristic
97C2×C5×C4⋊C4—31no—
98C2×C5×C22⋊C4—34no—
99C2×C42×C5C2×D311yescharacteristic
100C2×C22⋊Dic5—34no—
101C2×#685—32no—
102C2×C4⋊Dic5C2×D311yescharacteristic
103C2×#683—32no—
104C2×#683—32no—
105C2×C4×Dic5—31no—
106D5×C4⋊C4—64no—
107#4018—64no—
108#3963—34no—
109#3963C2×D314yes—
110#2311—151noSylow
111C23×C3×C5C2411yescharacteristic
112C22×D15—22no—
113C22×D15—22no—
114C22×D15C22×C412yes—
115C22×D15C2×D412yes—
116C22×D15C2×D412yes—
117C22×C5×D3—22no—
118C22×C5×D3—22no—
119C22×C5×D3—24no—
120C22×C5×D3—24no—
121C22×C5×D3C22×C414yes—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5C22×C412yes—
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5C2×D412yes—
128C2×Dic15—22no—
129C2×Dic15—22no—
130C2×Dic15—22no—
131C2×Dic15C22×C412yes—
132C2×Dic15C2×D412yes—
133C2×Dic15C2×D412yes—
134C4×D15C22⋊C4116yescomplemented
135C2×C5×Dic3—22no—
136C2×C5×Dic3—24no—
137C5×D12—28no—
138C2×C3×Dic5—22no—
139C2×C3×Dic5—22no—
140C2×C3×Dic5—22no—
141C2×C3×Dic5—24no—
142C2×C3×Dic5C22×C414yes—
143C3×Dic10—28no—
144#1203—216no—
145#1200—48no—
146D3×Dic5—216no—
147C22×C3×Q8—51no—
148C2×#239—58no—
149C2×#239—104no—
150C2×#239—104no—
151C22×D12—52no—
152C22×D12C2×D511yescharacteristic
153C22×C4×D3—51no—
154C22×C4×D3—52no—
155C22×C4×D3—52no—
156C2×C3×C4⋊C4—51no—
157C2×C3×C42C2×D511yescharacteristic
158C2×(C2×D3)⋊C4—52no—
159C2×(C2×D3)⋊C4—54no—
160C2×(C2×D3)⋊C4C2×D514yes—
161C2×C4⋊Dic3C2×D511yescharacteristic
162C2×Dic3⋊C4—52no—
163C2×Dic3⋊C4—52no—
164C2×C4×Dic3—51no—
165D3×C4⋊C4—104no—
166#872—104no—
167C3×#95—54no—
168C3⋊#95—54no—
169C24×C5—32no—
170C23×D5—31no—
171C2×C5×D4—34no—
172C2×C5×D4—62no—
173C2×C5×D4—62no—
174C22×C4×C5—32no—
175C22×C4×C5C22×D312yes—
176C22×C4×C5C22×D311yescharacteristic
177C2×C22⋊D5—38no—
178C2×C22⋊D5—64no—
179C2×C22⋊D5—64no—
180C22×Dic5—31no—
181C22×Dic5—38no—
182C22×Dic5—32no—
183C22×Dic5—64no—
184C22×Dic5—64no—
185C22×Dic5C22×D312yes—
186C22×Dic5C22×D312yes—
187#711—632no—
188C2×C4×D5—34no—
189C2×C4×D5—34no—
190C2×C4×D5—34no—
191C2×C4×D5—64no—
192C2×C4×D5—124no—
193C2×Dic10—22no—
194C2×Dic10—22no—
195C2×Dic10C4×D314yescomplemented
196C5×C4⋊C4—62no—
197C5×C22⋊C4—124no—
198C42×C5—22no—
199C22⋊Dic5—124no—
200#685—64no—
201C4⋊Dic5—22no—
202#683—64no—
203#683—64no—
204C4×Dic5—62no—
205C22×#41—151no—
206C2×#117—151no—
207C22×C4⋊C4—151no—
208#391—51no—
209#391—151no—
210#390—31no—
211#390—151no—
212#386—154no—
213#380—154no—
214C22×C3×C5C23×C411yescharacteristic
215C22×C3×C5C22×D411yescharacteristic
216C22×C3×C5C22×D411yescharacteristic
217C22×C3×C5H(5,𝔽2)12yes—
218C22×C3×C52−1+412yes—
219C2×D15—24no—
220C2×D15—22no—
221C2×D15—22no—
222C2×D15—24no—
223C2×D15C2×C22⋊C414yes—
224C2×C5×D3—24no—
225C2×C5×D3—28no—
226C2×C5×D3—42no—
227C2×C5×D3—42no—
228C2×C5×D3—44no—
229C3×C4×C5—42no—
230C3×C4×C5C2×C22⋊C414yes—
231Dic15—42no—
232Dic15C2×C22⋊C414yes—
233C3×Dic5—24no—
234C3×Dic5—42no—
235C5×Dic3—42no—
236C23×D3—51no—
237C23×D3C22×D512yes—
238C2×C3×Q8—54no—
239C2×C3×Q8—102no—
240C2×C3×Q8—102no—
241C22×C3×C4—52no—
242C22×C3×C4—52no—
243C22×C3×C4C22×D512yes—
244C22×C3×C4C22×D511yescharacteristic
245C22×Dic3—51no—
246C22×Dic3—52no—
247C22×Dic3C22×D512yes—
248#239—1032no—
249C2×D12—22no—
250C2×D12—22no—
251C2×D12—58no—
252C2×D12—104no—
253C2×D12—104no—
254C2×D12C4×D514yescomplemented
255C2×C4×D3—54no—
256C2×C4×D3—54no—
257C2×C4×D3—54no—
258C2×C4×D3—58no—
259C2×C4×D3—104no—
260C2×C4×D3—104no—
261C2×C4×D3—104no—
262C2×C4×D3—204no—
263C3×C4⋊C4—102no—
264C3×C42—22no—
265(C2×D3)⋊C4—44no—
266(C2×D3)⋊C4—104no—
267(C2×D3)⋊C4—204no—
268C4⋊Dic3—22no—
269Dic3⋊C4—104no—
270Dic3⋊C4—104no—
271C4×Dic3—102no—
272C23×C5—34no—
273C23×C5—62no—
274C23×C5—62no—
275C23×C5—64no—
276C23×C5—64no—
277C23×C5C23×D311yescharacteristic
278C22×D5—32no—
279C22×D5—32no—
280C22×D5—32no—
281C22×D5—62no—
282C22×D5—62no—
283C5×D4—68no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—62no—
288C2×C4×C5—64no—
289C2×C4×C5C2×C4×D312yes—
290C2×C4×C5D3×D412yes—
291C2×C4×C5D3×D412yes—
292C22⋊D5—616no—
293C2×Dic5—22no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—24no—
297C2×Dic5—32no—
298C2×Dic5—32no—
299C2×Dic5—32no—
300C2×Dic5—616no—
301C2×Dic5—62no—
302C2×Dic5—62no—
303C2×Dic5—62no—
304C2×Dic5C2×C4×D314yes—
305C4×D5—316no—
306C4×D5—128no—
307Dic10—28no—
308C2×#41—158no—
309C2×#41—304no—
310C2×#41—304no—
311C22×Q8—51no—
312C22×D4—31no—
313C22×D4—152no—
314C23×C4—152no—
315C23×C4—152no—
316C23×C4—151no—
317#117—304no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—51no—
320C2×C4⋊C4—152no—
321C2×C4⋊C4—152no—
322C2×C4⋊C4—304no—
323C2×C22⋊C4—34no—
324C2×C22⋊C4—152no—
325C2×C22⋊C4—154no—
326C2×C42—151no—
327C2×C42D3×D511yescharacteristic
328#95—54no—
329#95—154no—
330C2×C3×C5C22×C22⋊C411yescharacteristic
331C2×C3×C5#52012yes—
332C2×C3×C5#53511yescharacteristic
333C2×C3×C5#53611yescharacteristic
334C2×C3×C5#53712yes—
335D15—24no—
336C5×D3—44no—
337C23×C3C23×D511yescharacteristic
338C22×D3—22no—
339C22×D3—22no—
340C22×D3—24no—
341C22×D3—24no—
342C22×D3—52no—
343C22×D3—52no—
344C22×D3—52no—
345C22×D3—102no—
346C22×D3—102no—
347C22×D3C2×C4×D514yes—
348C3×Q8—108no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—54no—
353C2×C3×C4—102no—
354C2×C3×C4—102no—
355C2×C3×C4—102no—
356C2×C3×C4—104no—
357C2×C3×C4C2×C4×D512yes—
358C2×C3×C4D4×D512yes—
359C2×C3×C4D4×D512yes—
360C2×Dic3—22no—
361C2×Dic3—24no—
362C2×Dic3—52no—
363C2×Dic3—52no—
364C2×Dic3—52no—
365C2×Dic3—102no—
366C2×Dic3—102no—
367C2×Dic3—102no—
368D12—28no—
369D12—1016no—
370C4×D3—516no—
371C4×D3—1016no—
372C4×D3—208no—
373C22×C5—64no—
374C22×C5—68no—
375C22×C5—122no—
376C22×C5—122no—
377C22×C5—124no—
378C22×C5C22×C4×D311yescharacteristic
379C22×C5C2×D3×D411yescharacteristic
380C22×C5C2×D3×D411yescharacteristic
381C22×C5#100712yes—
382C22×C5#100912yes—
383C2×D5—34no—
384C2×D5—62no—
385C2×D5—62no—
386C2×D5—64no—
387C2×D5—64no—
388C4×C5—42no—
389C4×C5—122no—
390C4×C5D3×C22⋊C414yes—
391Dic5—24no—
392Dic5—34no—
393Dic5—42no—
394Dic5—122no—
395C24—32no—
396C24—151no—
397#41—3032no—
398C2×Q8—54no—
399C2×Q8—102no—
400C2×Q8—102no—
401C2×D4—34no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—158no—
405C2×D4—304no—
406C2×D4—304no—
407C22×C4—32no—
408C22×C4—52no—
409C22×C4—52no—
410C22×C4—158no—
411C22×C4—151no—
412C22×C4—154no—
413C22×C4—154no—
414C22×C4—154no—
415C22×C4—152no—
416C22×C4—304no—
417C22×C4—304no—
418C22×C4—304no—
419C22×C4—604no—
420C22×C4C2×D3×D512yes—
421C22×C4C2×D3×D511yescharacteristic
422C4⋊C4—62no—
423C4⋊C4—102no—
424C4⋊C4—304no—
425C4⋊C4—304no—
426C22⋊C4—124no—
427C22⋊C4—304no—
428C22⋊C4—604no—
429C42—22no—
430C42—302no—
431C3×C5#231111yescharacteristic, complemented, Hall
432C22×C3C22×C4×D511yescharacteristic
433C22×C3C2×D4×D511yescharacteristic
434C22×C3C2×D4×D511yescharacteristic
435C22×C3#414412yes—
436C22×C3#415012yes—
437C2×D3—24no—
438C2×D3—28no—
439C2×D3—42no—
440C2×D3—44no—
441C2×D3—42no—
442C2×D3—54no—
443C2×D3—102no—
444C2×D3—104no—
445C2×D3—104no—
446C2×D3—102no—
447C3×C4—42no—
448C3×C4—104no—
449C3×C4—202no—
450C3×C4D5×C22⋊C414yes—
451Dic3—42no—
452Dic3—54no—
453Dic3—202no—
454C2×C5—124no—
455C2×C5C2×D3×C22⋊C411yescharacteristic
456C2×C5#557412yes—
457C2×C5#562912yes—
458C2×C5#567411yescharacteristic
459C2×C5#572411yescharacteristic
460D5—64no—
461C23—34no—
462C23—64no—
463C23—62no—
464C23—64no—
465C23—62no—
466C23—152no—
467C23—152no—
468C23—152no—
469C23—302no—
470C23—302no—
471C23C22×D3×D511yescharacteristic
472Q8—108no—
473D4—68no—
474D4—3016no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—54no—
479C2×C4—62no—
480C2×C4—64no—
481C2×C4—102no—
482C2×C4—102no—
483C2×C4—102no—
484C2×C4—104no—
485C2×C4—152no—
486C2×C4—152no—
487C2×C4—152no—
488C2×C4—1516no—
489C2×C4—3016no—
490C2×C4—302no—
491C2×C4—302no—
492C2×C4—302no—
493C2×C4—608no—
494C2×C4C4×D3×D512yes—
495C2×C4#677712yes—
496C2×C4#679012yes—
497C2×C3C2×D5×C22⋊C411yescharacteristic
498C2×C3#6588012yes—
499C2×C3#6594212yes—
500C2×C3#6600911yescharacteristic
501C2×C3#6604911yescharacteristic
502D3—44no—
503D3—104no—
504C5#7648211yescharacteristic, complemented, Sylow
505C22—68no—
506C22—64no—
507C22—122no—
508C22—122no—
509C22—124no—
510C22—154no—
511C22—302no—
512C22—304no—
513C22—302no—
514C22—304no—
515C22#9200412yes—
516C22C2×C4×D3×D511yescharacteristic
517C22C2×#677711yescharacteristic
518C22#9201812yes—
519C22C2×#679011yescharacteristic
520C4—42no—
521C4—104no—
522C4—122no—
523C4—154no—
524C4—202no—
525C4—602no—
526C4#9147314yes—
527C3#1060802511yescharacteristic, complemented, Sylow
528C2—124no—
529C2—304no—
530C2#1175252512yes—
531C2#1175256011yescharacteristic
532C2#1175275311yescharacteristic
533C2#1175438012yes—
534C2C2×#9147311yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6768C2
hypercenterC2×C42D3×D5
hypocenterC3×C5#2311
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×#91761C2semidirect
3. C2×#91437C2semidirect
4. C5×#4939C2semidirect
5. C3×#65218C2semidirect
6. #11748919C2semidirect
7. #11748887C2semidirect
8. #11748797C2semidirect
9. #11748610C2semidirect
19. C2×#6768C4semidirect
42. #65218D3semidirect
55. C2×C5×D12C2×C4semidirect
60. C2×C3×Dic10C2×C4semidirect
68. C2×D3×Dic5C2×C4semidirect
85. #4939D5semidirect
134. C4×D15C22⋊C4semidirect
195. C2×Dic10C4×D3semidirect
254. C2×D12C4×D5semidirect
431. C3×C5#2311semidirect
504. C5#76482semidirect
527. C3#10608025semidirect

Maximal subgroups

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

subgroupindex
C22×#67682
C2×#917612
C2×#914372
C5×#49392
C3×#652182
#117489192
#117488872
#117487972
#117486102
#106080253
#764825