Finite Group Databasebeta

← #14322575

Subgroups — #14322575

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×C3×#713C211yesmaximal, characteristic
2C2×D5×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#91357C211yesmaximal, characteristic, complemented
4#11751330C211yesmaximal, characteristic, complemented
5C5×#5229C211yesmaximal, characteristic, complemented
6#11748896C214yesmaximal, complemented
7#11748796C211yesmaximal, characteristic, complemented
8#11748785C211yesmaximal, characteristic, complemented
9#11748738C214yesmaximal, complemented
10#10608213—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C2×C3×#713—24no—
13C2×C3×#713—24no—
14C2×C3×#713C418yescomplemented
15C22×C3×D20C2212yes—
16C22×C3×D20C2211yescharacteristic
17C22×C3×C4×D5C2212yes—
18C22×C3×C4×D5C2211yescharacteristic
19C22×D5×Dic3C2212yes—
20C2×C4⋊Dic15C2211yescharacteristic
21C2×C4×Dic15C2211yescharacteristic
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C4×C5×Dic3C2211yescharacteristic
24C2×#6668C2212yes—
25C2×#6668C2212yes—
26C2×#6665C2212yes—
27C2×#6665C2214yes—
28C2×#6665C2214yes—
29D5×C4⋊Dic3—24no—
30#91357—24no—
31#91115C2214yes—
32C5×C3⋊#95C2214yes—
33#84087—51nomaximal, Hall
34C22×#713D311yescharacteristic
35C2×D5×C4⋊C4—31no—
36C2×#4018—31no—
37C5×#391—31no—
38#65519—31no—
39#65277—31no—
40#65228—31no—
41#64956—34no—
42#64954—34no—
43C23×C3×D5C2312yes—
44C23×C3×D5C2311yescharacteristic
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8—22no—
47C2×C3×C5×Q8C2×C414yescomplemented
48C22×C3×C4×C5C2312yes—
49C22×C3×C4×C5C2311yescharacteristic
50C22×Dic15C2312yes—
51C22×Dic15C2312yes—
52C22×C5×Dic3C2312yes—
53C22×C5×Dic3C2312yes—
54C22×C3×Dic5C2311yescharacteristic
55C3×#713—232no—
56C2×C3×D20—24no—
57C2×C3×D20—24no—
58C2×C3×D20—22no—
59C2×C3×D20—22no—
60C2×C3×D20C2×C418yes—
61C2×C3×D20C2×C414yescomplemented
62C2×C3×C4×D5—24no—
63C2×C3×C4×D5—24no—
64C2×C3×C4×D5C2×C414yes—
65C2×C3×C4×D5C2×C418yescomplemented
66C2×C3×C4×D5D414yes—
67C2×C3×C4×D5D414yes—
68C2×D5×Dic3—24no—
69C2×D5×Dic3—44no—
70C4⋊Dic15—22no—
71C4×Dic15—22no—
72C5×C4⋊Dic3—22no—
73C4×C5×Dic3—22no—
74#6668—24no—
75#6668—24no—
76#6665—24no—
77#6665—44no—
78#6665—44no—
79C22×C3×#41—51no—
80C2×#926—51no—
81C22×C4⋊Dic3—51no—
82#5229—51no—
83#5229D511yescharacteristic, complemented
84#5221—51no—
85#5221—51no—
86#5220—54no—
87#5212—54no—
88C22×C5×Q8C2×D311yescharacteristic
89C2×#713—24no—
90C2×#713—24no—
91C2×#713Dic318yescomplemented
92C22×D20C2×D312yes—
93C22×D20C2×D311yescharacteristic
94C22×C4×D5—32no—
95C22×C4×D5C2×D312yes—
96C22×C4×D5C2×D311yescharacteristic
97C2×C5×C4⋊C4—31no—
98C2×C42×C5—31no—
99C2×#685—32no—
100C2×#685—34no—
101C2×#685—34no—
102C2×C4⋊Dic5—31no—
103C2×#683—32no—
104C2×#683—32no—
105C2×C4×Dic5—31no—
106D5×C4⋊C4—64no—
107#4018—64no—
108C5×#95—34no—
109#3963—34no—
110#2311—151noSylow
111C23×C3×C5C2411yescharacteristic
112C22×C3×D5—22no—
113C22×C3×D5—22no—
114C22×C3×D5—22no—
115C22×C3×D5—22no—
116C22×C3×D5—24no—
117C22×C3×D5—24no—
118C22×C3×D5C22×C414yes—
119C22×C3×D5C22×C412yes—
120C22×C3×D5C2×D412yes—
121C22×C3×D5C2×D412yes—
122C3×C5×Q8—28no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5C22×C414yes—
126C2×C3×C4×C5C22×C412yes—
127C2×C3×C4×C5C2×D412yes—
128C2×C3×C4×C5C2×D412yes—
129C2×Dic15—22no—
130C2×Dic15—22no—
131C2×Dic15—22no—
132C2×Dic15—24no—
133C2×Dic15—22no—
134C2×C5×Dic3—22no—
135C2×C5×Dic3—22no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3—24no—
138C2×C5×Dic3—22no—
139C2×C3×Dic5C22×C412yes—
140C2×C3×Dic5C2×D412yes—
141C2×C3×Dic5C2×D412yes—
142C3×D20—216no—
143C3×D20—28no—
144C3×C4×D5—216no—
145C3×C4×D5C22⋊C4116yescomplemented
146D5×Dic3—48no—
147C2×C3×#41—58no—
148C2×C3×#41—104no—
149C2×C3×#41—104no—
150C22×C3×Q8C2×D511yescharacteristic
151C22×C3×D4—52no—
152C22×C3×D4—51no—
153C23×C3×C4—51no—
154C23×C3×C4—52no—
155C23×Dic3—52no—
156C2×#217—52no—
157C2×#217—54no—
158C2×#217—54no—
159#926—104no—
160C2×C4⋊Dic3—51no—
161C2×C4⋊Dic3—52no—
162C2×C4⋊Dic3—52no—
163C2×C4⋊Dic3—104no—
164C2×C4⋊Dic3C2×D511yescharacteristic
165C2×C4×Dic3—51no—
166C2×C4×Dic3C2×D511yescharacteristic
167C3⋊#95—54no—
168C3⋊#95C2×D514yes—
169C23×D5C22×D312yes—
170C23×D5C22×D311yescharacteristic
171C2×C5×Q8—22no—
172C2×C5×Q8—22no—
173C2×C5×Q8C2×Dic314yescomplemented
174C22×C4×C5—32no—
175C22×C4×C5—32no—
176C22×C4×C5C22×D312yes—
177C22×C4×C5C22×D311yescharacteristic
178C22×Dic5—32no—
179C22×Dic5—32no—
180C22×Dic5C22×D311yescharacteristic
181#713—232no—
182C2×D20—24no—
183C2×D20—24no—
184C2×D20—22no—
185C2×D20—22no—
186C2×D20C2×Dic318yes—
187C2×D20C2×Dic314yescomplemented
188C2×C4×D5—24no—
189C2×C4×D5—24no—
190C2×C4×D5—64no—
191C2×C4×D5—124no—
192C2×C4×D5C2×Dic314yes—
193C2×C4×D5C2×Dic318yescomplemented
194C2×C4×D5#6714yes—
195C2×C4×D5#6714yes—
196C5×C4⋊C4—62no—
197C42×C5—62no—
198#685—64no—
199#685—124no—
200#685—124no—
201C4⋊Dic5—62no—
202#683—64no—
203#683—64no—
204C4×Dic5—62no—
205C22×#41—51no—
206C2×#117—151no—
207C22×C4⋊C4—151no—
208#391—31no—
209#391—151no—
210#390—151no—
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×C3×D5—24no—
220C2×C3×D5—22no—
221C2×C3×D5—22no—
222C2×C3×D5—24no—
223C2×C3×D5—24no—
224C2×C3×D5—28no—
225C2×C3×D5—42no—
226C2×C3×D5—42no—
227C2×C3×D5—44no—
228C2×C3×D5C2×C22⋊C414yes—
229C3×C4×C5—24no—
230C3×C4×C5C2×C22⋊C414yes—
231Dic15—42no—
232Dic15—42no—
233C3×Dic5C2×C22⋊C414yes—
234C5×Dic3—42no—
235C5×Dic3—42no—
236C24×C3—51no—
237C24×C3—52no—
238C3×#41—1032no—
239C2×C3×Q8—22no—
240C2×C3×Q8—22no—
241C2×C3×Q8C4×D514yescomplemented
242C2×C3×D4—58no—
243C2×C3×D4—54no—
244C2×C3×D4—104no—
245C2×C3×D4—104no—
246C2×C3×D4—102no—
247C2×C3×D4—102no—
248C22×C3×C4—51no—
249C22×C3×C4—54no—
250C22×C3×C4—54no—
251C22×C3×C4—54no—
252C22×C3×C4—58no—
253C22×C3×C4—104no—
254C22×C3×C4—104no—
255C22×C3×C4C22×D512yes—
256C22×C3×C4C22×D511yescharacteristic
257C22×Dic3—52no—
258C22×Dic3—52no—
259C22×Dic3—104no—
260C22×Dic3—204no—
261C22×Dic3C22×D512yes—
262C22×Dic3C22×D512yes—
263#217—104no—
264#217—204no—
265#217—204no—
266C4⋊Dic3—22no—
267C4⋊Dic3—104no—
268C4⋊Dic3—104no—
269C4⋊Dic3—102no—
270C4×Dic3—22no—
271C4×Dic3—102no—
272C23×C5C23×D311yescharacteristic
273C22×D5—22no—
274C22×D5—22no—
275C22×D5—24no—
276C22×D5—24no—
277C22×D5—22no—
278C22×D5—22no—
279C22×D5C22×Dic314yes—
280C22×D5C22×Dic312yes—
281C22×D5C2×#6712yes—
282C22×D5C2×#6712yes—
283C5×Q8—28no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—62no—
287C2×C4×C5—64no—
288C2×C4×C5—62no—
289C2×C4×C5—62no—
290C2×C4×C5—62no—
291C2×C4×C5C22×Dic314yes—
292C2×C4×C5C22×Dic312yes—
293C2×C4×C5C2×#6712yes—
294C2×C4×C5C2×#6712yes—
295C2×Dic5—62no—
296C2×Dic5—64no—
297C2×Dic5—62no—
298C2×Dic5—62no—
299C2×Dic5—62no—
300C2×Dic5C22×Dic312yes—
301C2×Dic5C2×#6712yes—
302C2×Dic5C2×#6712yes—
303D20—216no—
304D20—28no—
305C4×D5—216no—
306C4×D5—128no—
307C4×D5#217116yescomplemented
308C2×#41—58no—
309C2×#41—104no—
310C2×#41—104no—
311C22×Q8D3×D511yescharacteristic
312C22×D4—52no—
313C22×D4—51no—
314C23×C4—51no—
315C23×C4—52no—
316C23×C4—152no—
317#117—304no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—152no—
320C2×C4⋊C4—152no—
321C2×C4⋊C4—151no—
322C2×C4⋊C4—304no—
323C2×C22⋊C4—152no—
324C2×C22⋊C4—154no—
325C2×C22⋊C4—154no—
326C2×C42—31no—
327C2×C42—151no—
328#95—34no—
329#95—154no—
330C2×C3×C5C22×C22⋊C411yescharacteristic
331C2×C3×C5#52012yes—
332C2×C3×C5#53511yescharacteristic
333C2×C3×C5#53611yescharacteristic
334C2×C3×C5#53712yes—
335C3×D5—24no—
336C3×D5—44no—
337C23×C3—52no—
338C23×C3—52no—
339C23×C3—52no—
340C23×C3—54no—
341C23×C3—102no—
342C23×C3—102no—
343C23×C3—102no—
344C23×C3—102no—
345C23×C3—104no—
346C23×C3—104no—
347C23×C3C23×D511yescharacteristic
348C3×Q8—28no—
349C3×D4—1016no—
350C3×D4—108no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—52no—
354C2×C3×C4—52no—
355C2×C3×C4—52no—
356C2×C3×C4—516no—
357C2×C3×C4—1016no—
358C2×C3×C4C2×C4×D514yes—
359C2×C3×C4C2×C4×D512yes—
360C2×C3×C4D4×D512yes—
361C2×C3×C4D4×D512yes—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—24no—
367C2×Dic3—102no—
368C2×Dic3—102no—
369C2×Dic3—102no—
370C2×Dic3—102no—
371C2×Dic3—104no—
372C2×Dic3—208no—
373C22×C5#100912yes—
374C22×C5#101012yes—
375C22×C5C23×Dic311yescharacteristic
376C22×C5C22×#6711yescharacteristic
377C22×C5C22×#6711yescharacteristic
378C2×D5—22no—
379C2×D5—24no—
380C2×D5—22no—
381C2×D5—24no—
382C2×D5—28no—
383C2×D5—24no—
384C2×D5—42no—
385C2×D5—42no—
386C2×D5—44no—
387C2×D5C2×#21714yes—
388C4×C5—24no—
389C4×C5—122no—
390C4×C5—122no—
391C4×C5C2×#21714yes—
392Dic5—122no—
393Dic5—122no—
394Dic5C2×#21714yes—
395C24—51no—
396C24—52no—
397#41—1032no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8D5×Dic314yescomplemented
401C2×D4—58no—
402C2×D4—54no—
403C2×D4—104no—
404C2×D4—104no—
405C2×D4—102no—
406C2×D4—102no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—51no—
410C22×C4—54no—
411C22×C4—54no—
412C22×C4—54no—
413C22×C4—58no—
414C22×C4—104no—
415C22×C4—104no—
416C22×C4—152no—
417C22×C4—152no—
418C22×C4—304no—
419C22×C4—604no—
420C22×C4C2×D3×D512yes—
421C22×C4C2×D3×D511yescharacteristic
422C4⋊C4—62no—
423C4⋊C4—304no—
424C4⋊C4—304no—
425C4⋊C4—302no—
426C22⋊C4—304no—
427C22⋊C4—604no—
428C22⋊C4—604no—
429C42—62no—
430C42—302no—
431C3×C5#231111yescharacteristic, complemented, Hall
432C22×C3—54no—
433C22×C3—102no—
434C22×C3—104no—
435C22×C3—104no—
436C22×C3—102no—
437C22×C3—104no—
438C22×C3—108no—
439C22×C3—202no—
440C22×C3—204no—
441C22×C3—202no—
442C22×C3C22×C4×D511yescharacteristic
443C22×C3C2×D4×D511yescharacteristic
444C22×C3C2×D4×D511yescharacteristic
445C22×C3#414712yes—
446C22×C3#414912yes—
447C3×C4—24no—
448C3×C4—54no—
449C3×C4D5×C22⋊C414yes—
450Dic3—42no—
451Dic3—42no—
452Dic3—202no—
453Dic3—202no—
454C2×C5#582511yescharacteristic
455C2×C5#582712yes—
456C2×C5#583011yescharacteristic
457C2×C5#583312yes—
458C2×C5C22×#21711yescharacteristic
459D5—24no—
460D5—44no—
461C23—52no—
462C23—52no—
463C23—52no—
464C23—54no—
465C23—102no—
466C23—102no—
467C23—104no—
468C23—104no—
469C23—102no—
470C23—102no—
471C23C22×D3×D511yescharacteristic
472Q8—28no—
473D4—1016no—
474D4—108no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—52no—
478C2×C4—52no—
479C2×C4—52no—
480C2×C4—516no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—64no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—1016no—
487C2×C4—302no—
488C2×C4—302no—
489C2×C4—304no—
490C2×C4—302no—
491C2×C4—302no—
492C2×C4—608no—
493C2×C4C2×D5×Dic314yes—
494C2×C4C2×D5×Dic312yes—
495C2×C4D5×#6712yes—
496C2×C4D5×#6712yes—
497C2×C3—104no—
498C2×C3—204no—
499C2×C3C2×D5×C22⋊C411yescharacteristic
500C2×C3#6590812yes—
501C2×C3#6596312yes—
502C2×C3#6600811yescharacteristic
503C2×C3#6605811yescharacteristic
504C5#8408711yescharacteristic, complemented, Sylow
505C22—54no—
506C22—102no—
507C22—102no—
508C22—104no—
509C22—104no—
510C22—104no—
511C22—108no—
512C22—202no—
513C22—202no—
514C22—204no—
515C22#9202812yes—
516C22#9203512yes—
517C22C22×D5×Dic311yescharacteristic
518C22C2×D5×#6711yescharacteristic
519C22C2×D5×#6711yescharacteristic
520C4—24no—
521C4—54no—
522C4—122no—
523C4—122no—
524C4—602no—
525C4—602no—
526C4D5×#21714yes—
527C3#1060821311yescharacteristic, complemented, Sylow
528C2—104no—
529C2—204no—
530C2#1175515511yescharacteristic
531C2#1175521712yes—
532C2#1175522312yes—
533C2#1175527211yescharacteristic
534C2C2×D5×#21711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×Q8C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×#713C2
hypercenterC22×Q8D3×D5
hypocenterC3×C5#2311
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×D5×C4⋊Dic3C2semidirect
3. C2×#91357C2semidirect
4. #11751330C2semidirect
5. C5×#5229C2semidirect
6. #11748896C2semidirect
7. #11748796C2semidirect
8. #11748785C2semidirect
9. #11748738C2semidirect
14. C2×C3×#713C4semidirect
47. C2×C3×C5×Q8C2×C4semidirect
61. C2×C3×D20C2×C4semidirect
65. C2×C3×C4×D5C2×C4semidirect
83. #5229D5semidirect
91. C2×#713Dic3semidirect
145. C3×C4×D5C22⋊C4semidirect
173. C2×C5×Q8C2×Dic3semidirect
187. C2×D20C2×Dic3semidirect
193. C2×C4×D5C2×Dic3semidirect
241. C2×C3×Q8C4×D5semidirect
307. C4×D5#217semidirect
400. C2×Q8D5×Dic3semidirect
431. C3×C5#2311semidirect
504. C5#84087semidirect
527. C3#10608213semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#7132
C2×D5×C4⋊Dic32
C2×#913572
#117513302
C5×#52292
#117488962
#117487962
#117487852
#117487382
#106082133
#840875