Finite Group Databasebeta

← #14328023

Subgroups — #14328023

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1201C211yesmaximal, characteristic
2#11750750C211yesmaximal, characteristic, complemented
3C5×#4664C211yesmaximal, characteristic, complemented
4C3×#64953C211yesmaximal, characteristic, complemented
5#11748915C212yesmaximal
6#11748867C211yesmaximal, characteristic, complemented
7#11748720C212yesmaximal, complemented
8#11748719C211yesmaximal, characteristic, complemented
9#11748707C212yesmaximal, complemented
10#11748702C211yesmaximal, characteristic, complemented
11#11748663C212yesmaximal, complemented
12#10605792—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C22×C3×C4×D5C2211yescharacteristic
15C22×#1201C2211yescharacteristic
16C2×#6712C2212yes—
17C2×C4×Dic15C2212yes—
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×(C2×D3)⋊C4C2212yes—
20C2×C3×#685C2212yes—
21C2×#6669C2212yes—
22C2×#6669C2211yescharacteristic
23C2×(C2×D3)⋊Dic5C2212yes—
24C2×(C2×D3)⋊Dic5C2211yescharacteristic
25C2×#6665C2212yes—
26C2×#6665C2211yescharacteristic
27C2×Dic3×Dic5C2212yes—
28C2×Dic3×Dic5C2212yes—
29C4×#1201C418yescomplemented
30C3×C5×#95C2211yescharacteristic
31#91115C2211yescharacteristic
32C5×C3⋊#95C2211yescharacteristic
33C5×C3⋊#95C2212yes—
34C3×#3963C2211yescharacteristic
35C3×#3963C2212yes—
36#90994C2212yes—
37#74061—51nomaximal, Hall
38C2×C4×C22⋊D5—31no—
39C5×#381—31no—
40C4×C22⋊Dic5—31no—
41#65246—32no—
42#65233—32no—
43#64958—32no—
44#64955—31no—
45#64953D311yescharacteristic, complemented
46#64939—31no—
47#64938—31no—
48#64938—32no—
49C23×C5×D3C2311yescharacteristic
50C23×C3×D5C2311yescharacteristic
51C22×C3×C4×C5C2312yes—
52C22×C3×C4×C5C2311yescharacteristic
53C22×Dic15C2311yescharacteristic
54C22×Dic15C2312yes—
55C22×Dic15C2311yescharacteristic
56C22×C5×Dic3C2312yes—
57C22×C5×Dic3C2311yescharacteristic
58C2×C4×C5×D3—42no—
59C2×C4×C5×D3C2×C414yes—
60C22×C3×Dic5C2312yes—
61C22×C3×Dic5C2311yescharacteristic
62C2×C3×C4×D5—42no—
63C2×C3×C4×D5C2×C414yes—
64C2×#1201—22no—
65C2×#1201—22no—
66C2×#1201C2×C414yes—
67#6712—24no—
68C4×Dic15—24no—
69C4×Dic15C2×C414yes—
70C5×(C2×D3)⋊C4—42no—
71C3×#685—42no—
72#6669—24no—
73#6669C2×C414yes—
74(C2×D3)⋊Dic5—24no—
75(C2×D3)⋊Dic5C2×C414yes—
76#6665—24no—
77#6665C2×C414yes—
78Dic3×Dic5—42no—
79Dic3×Dic5—42no—
80C2×C4×#67—51no—
81C3×#381—51no—
82C4×#217—51no—
83#4957—52no—
84#4944—52no—
85#4669—52no—
86#4666—51no—
87#4664D511yescharacteristic, complemented
88#4650—51no—
89#4649—51no—
90#4649—52no—
91C23×C4×C5—31no—
92C22×C22⋊D5—31no—
93C22×C4×D5C2×D311yescharacteristic
94C2×C5×C22⋊C4—32no—
95C2×C22⋊Dic5—31no—
96C2×C22⋊Dic5—32no—
97C4×C22⋊D5—38no—
98C2×#685—31no—
99C2×#685—32no—
100C2×#685C2×D312yes—
101C2×#683—31no—
102C2×#683—32no—
103C2×#683—32no—
104C2×C4×Dic5—31no—
105C2×C4×Dic5—32no—
106C2×C4×Dic5—32no—
107C2×C4×Dic5—32no—
108C5×#95—32no—
109C5×#95—31no—
110C5×#95C2×D311yescharacteristic
111#3963—31no—
112#3963—32no—
113#3963C2×D312yes—
114#3963C2×D311yescharacteristic
115#2357—151noSylow
116C23×C3×C5C2411yescharacteristic
117C22×C5×D3—22no—
118C22×C5×D3—21no—
119C22×C5×D3—21no—
120C22×C5×D3—22no—
121C22×C5×D3C22×C412yes—
122C22×C3×D5—22no—
123C22×C3×D5—21no—
124C22×C3×D5—21no—
125C22×C3×D5—22no—
126C22×C3×D5C22×C412yes—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5C22×C412yes—
132C2×Dic15—22no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15C22×C412yes—
136C2×Dic15C22×C412yes—
137C2×Dic15#4114yes—
138C2×Dic15#4112yes—
139C2×Dic15#4112yes—
140C2×C5×Dic3—22no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3C22×C412yes—
145C4×C5×D3—44no—
146C2×C3×Dic5—22no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5C22×C412yes—
151C3×C4×D5—44no—
152#1201—28no—
153C23×C3×C4—51no—
154C22×#67—51no—
155C22×C4×D3C2×D511yescharacteristic
156C2×C3×C22⋊C4—52no—
157C2×#217—51no—
158C2×#217—52no—
159C4×#67—58no—
160C2×(C2×D3)⋊C4—51no—
161C2×(C2×D3)⋊C4—52no—
162C2×(C2×D3)⋊C4C2×D512yes—
163C2×Dic3⋊C4—51no—
164C2×Dic3⋊C4—52no—
165C2×Dic3⋊C4—52no—
166C2×C4×Dic3—51no—
167C2×C4×Dic3—52no—
168C2×C4×Dic3—52no—
169C2×C4×Dic3—52no—
170C3×#95—52no—
171C3×#95—51no—
172C3×#95C2×D511yescharacteristic
173C3⋊#95—51no—
174C3⋊#95—52no—
175C3⋊#95C2×D512yes—
176C3⋊#95C2×D511yescharacteristic
177C24×C5—31no—
178C23×D5C22×D311yescharacteristic
179C22×C4×C5—31no—
180C22×C4×C5—34no—
181C22×C4×C5—32no—
182C22×C4×C5—122no—
183C22×C4×C5C22×D312yes—
184C22×C4×C5C22×D311yescharacteristic
185C2×C22⋊D5—34no—
186C2×C22⋊D5—62no—
187C2×C22⋊D5—62no—
188C22×Dic5—31no—
189C22×Dic5—31no—
190C22×Dic5—32no—
191C22×Dic5C22×D312yes—
192C22×Dic5C22×D311yescharacteristic
193C2×C4×D5—42no—
194C2×C4×D5C4×D314yes—
195C5×C22⋊C4—122no—
196C22⋊Dic5—34no—
197C22⋊Dic5—64no—
198#685—34no—
199#685—42no—
200#685—64no—
201#683—34no—
202#683—64no—
203#683—64no—
204C4×Dic5—34no—
205C4×Dic5—64no—
206C4×Dic5—122no—
207C4×Dic5—122no—
208C2×C4×D4—151no—
209#390—152no—
210#388—152no—
211#388—152no—
212#382—151no—
213#381—31no—
214#381—51no—
215C4×C4⋊C4—152no—
216C4×C4⋊C4—151no—
217C4×C22⋊C4—151no—
218C4×C22⋊C4—151no—
219C22×C3×C5C23×C411yescharacteristic
220C22×C3×C5C2×#4111yescharacteristic
221C22×C3×C5C2×#4111yescharacteristic
222C22×C3×C5C2×#4112yes—
223C22×C3×C5H(5,𝔽2)12yes—
224C2×C5×D3—22no—
225C2×C5×D3—22no—
226C2×C5×D3—22no—
227C2×C5×D3—42no—
228C2×C5×D3—42no—
229C2×C3×D5—22no—
230C2×C3×D5—22no—
231C2×C3×D5—22no—
232C2×C3×D5—42no—
233C2×C3×D5—42no—
234C3×C4×C5—22no—
235C3×C4×C5—42no—
236Dic15—22no—
237Dic15—24no—
238Dic15#11714yes—
239C3×Dic5—22no—
240C3×Dic5—42no—
241C5×Dic3—22no—
242C5×Dic3—42no—
243C24×C3—51no—
244C23×D3C22×D511yescharacteristic
245C22×C3×C4—51no—
246C22×C3×C4—54no—
247C22×C3×C4—52no—
248C22×C3×C4—202no—
249C22×C3×C4C22×D512yes—
250C22×C3×C4C22×D511yescharacteristic
251C2×#67—54no—
252C2×#67—102no—
253C2×#67—102no—
254C22×Dic3—51no—
255C22×Dic3—51no—
256C22×Dic3—52no—
257C22×Dic3C22×D512yes—
258C22×Dic3C22×D511yescharacteristic
259C2×C4×D3—42no—
260C2×C4×D3C4×D514yes—
261C3×C22⋊C4—202no—
262#217—54no—
263#217—104no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—54no—
266(C2×D3)⋊C4—104no—
267Dic3⋊C4—54no—
268Dic3⋊C4—104no—
269Dic3⋊C4—104no—
270C4×Dic3—54no—
271C4×Dic3—104no—
272C4×Dic3—202no—
273C4×Dic3—202no—
274C23×C5—32no—
275C23×C5—62no—
276C23×C5—62no—
277C23×C5—61no—
278C23×C5—61no—
279C23×C5C23×D311yescharacteristic
280C22×D5—22no—
281C22×D5—22no—
282C22×D5—21no—
283C22×D5—21no—
284C22×D5C2×C4×D312yes—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—32no—
290C2×C4×C5—62no—
291C2×C4×C5—62no—
292C2×C4×C5—62no—
293C2×C4×C5—62no—
294C2×C4×C5—124no—
295C2×C4×C5C2×C4×D312yes—
296C22⋊D5—68no—
297C2×Dic5—22no—
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—32no—
302C2×Dic5—32no—
303C2×Dic5—32no—
304C2×Dic5—32no—
305C2×Dic5—34no—
306C2×Dic5—62no—
307C2×Dic5—62no—
308C2×Dic5—62no—
309C2×Dic5C2×C4×D312yes—
310C4×D5—44no—
311C22×D4—151no—
312C23×C4—31no—
313C23×C4—51no—
314C4×D4—158no—
315C2×C4⋊C4—151no—
316C2×C4⋊C4—152no—
317C2×C4⋊C4—152no—
318C2×C22⋊C4—32no—
319C2×C22⋊C4—52no—
320C2×C22⋊C4—151no—
321C2×C22⋊C4—151no—
322C2×C22⋊C4—152no—
323C2×C22⋊C4—152no—
324C2×C42—151no—
325C2×C42—152no—
326C2×C42—152no—
327C2×C42—152no—
328#95—31no—
329#95—32no—
330#95—51no—
331#95—52no—
332#95—152no—
333#95—151no—
334#95D3×D511yescharacteristic
335C2×C3×C5C2×#11711yescharacteristic
336C2×C3×C5C4×#4111yescharacteristic
337C2×C3×C5#51811yescharacteristic
338C2×C3×C5#55312yes—
339C2×C3×C5#55912yes—
340C3×D5—42no—
341C5×D3—42no—
342C23×C3—52no—
343C23×C3—102no—
344C23×C3—102no—
345C23×C3—101no—
346C23×C3—101no—
347C23×C3C23×D511yescharacteristic
348C22×D3—22no—
349C22×D3—22no—
350C22×D3—21no—
351C22×D3—21no—
352C22×D3C2×C4×D512yes—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—52no—
358C2×C3×C4—102no—
359C2×C3×C4—102no—
360C2×C3×C4—102no—
361C2×C3×C4—102no—
362C2×C3×C4—204no—
363C2×C3×C4C2×C4×D512yes—
364#67—108no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—52no—
370C2×Dic3—52no—
371C2×Dic3—52no—
372C2×Dic3—52no—
373C2×Dic3—54no—
374C2×Dic3—102no—
375C2×Dic3—102no—
376C2×Dic3—102no—
377C2×Dic3C2×C4×D512yes—
378C4×D3—44no—
379C22×C5—62no—
380C22×C5—62no—
381C22×C5—62no—
382C22×C5—122no—
383C22×C5—122no—
384C22×C5C22×C4×D311yescharacteristic
385C22×C5#100412yes—
386C22×C5D3×#4112yes—
387C22×C5D3×#4111yescharacteristic
388C22×C5D3×#4111yescharacteristic
389C2×D5—22no—
390C2×D5—22no—
391C2×D5—22no—
392C2×D5—42no—
393C2×D5—42no—
394C4×C5—22no—
395C4×C5—42no—
396C4×C5—62no—
397C4×C5—122no—
398Dic5—22no—
399Dic5—34no—
400Dic5—42no—
401Dic5—64no—
402Dic5—62no—
403C24—31no—
404C24—51no—
405C2×D4—154no—
406C2×D4—302no—
407C2×D4—302no—
408C22×C4—31no—
409C22×C4—34no—
410C22×C4—32no—
411C22×C4—51no—
412C22×C4—54no—
413C22×C4—52no—
414C22×C4—122no—
415C22×C4—151no—
416C22×C4—151no—
417C22×C4—152no—
418C22×C4—202no—
419C22×C4C2×D3×D512yes—
420C22×C4C2×D3×D511yescharacteristic
421C4⋊C4—154no—
422C4⋊C4—304no—
423C4⋊C4—304no—
424C22⋊C4—122no—
425C22⋊C4—154no—
426C22⋊C4—154no—
427C22⋊C4—202no—
428C22⋊C4—304no—
429C22⋊C4—304no—
430C42—154no—
431C42—304no—
432C42—602no—
433C42—602no—
434C3×C5#235711yescharacteristic, complemented, Hall
435C22×C3—102no—
436C22×C3—102no—
437C22×C3—102no—
438C22×C3—202no—
439C22×C3—202no—
440C22×C3C22×C4×D511yescharacteristic
441C22×C3#414412yes—
442C22×C3D5×#4112yes—
443C22×C3D5×#4111yescharacteristic
444C22×C3D5×#4111yescharacteristic
445C2×D3—22no—
446C2×D3—22no—
447C2×D3—22no—
448C2×D3—42no—
449C2×D3—42no—
450C3×C4—22no—
451C3×C4—42no—
452C3×C4—102no—
453C3×C4—202no—
454Dic3—22no—
455Dic3—42no—
456Dic3—54no—
457Dic3—102no—
458Dic3—104no—
459C2×C5—122no—
460C2×C5#548611yescharacteristic
461C2×C5D3×#11711yescharacteristic
462C2×C5#552211yescharacteristic
463C2×C5#566112yes—
464C2×C5#566412yes—
465D5—42no—
466C23—32no—
467C23—52no—
468C23—62no—
469C23—61no—
470C23—62no—
471C23—61no—
472C23—102no—
473C23—101no—
474C23—102no—
475C23—101no—
476C23C22×D3×D511yescharacteristic
477D4—308no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—32no—
483C2×C4—52no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—102no—
489C2×C4—102no—
490C2×C4—102no—
491C2×C4—102no—
492C2×C4—124no—
493C2×C4—152no—
494C2×C4—152no—
495C2×C4—152no—
496C2×C4—152no—
497C2×C4—154no—
498C2×C4—204no—
499C2×C4—302no—
500C2×C4—302no—
501C2×C4—302no—
502C2×C4C4×D3×D512yes—
503C2×C3—202no—
504C2×C3#6582011yescharacteristic
505C2×C3D5×#11711yescharacteristic
506C2×C3#6585611yescharacteristic
507C2×C3#6599512yes—
508C2×C3#6599812yes—
509D3—42no—
510C5#7406111yescharacteristic, complemented, Sylow
511C22—62no—
512C22—62no—
513C22—62no—
514C22—102no—
515C22—102no—
516C22—102no—
517C22—122no—
518C22—122no—
519C22—202no—
520C22—202no—
521C22C2×C4×D3×D511yescharacteristic
522C22#9201612yes—
523C22#9202411yescharacteristic
524C22#9202512yes—
525C22#9203411yescharacteristic
526C4—22no—
527C4—42no—
528C4—62no—
529C4—102no—
530C4—122no—
531C4—154no—
532C4—202no—
533C4—304no—
534C4—302no—
535C3#1060579211yescharacteristic, complemented, Sylow
536C2—122no—
537C2—202no—
538C2#1175266711yescharacteristic
539C2#1175306511yescharacteristic
540C2#1175315212yes—
541C2#1175378812yes—
542C2#1175383511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC3×C5×#95C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1201C22
hypercenter#95D3×D5
hypocenterC3×C5#2357
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
2. #11750750C2semidirect
3. C5×#4664C2semidirect
4. C3×#64953C2semidirect
6. #11748867C2semidirect
7. #11748720C2semidirect
8. #11748719C2semidirect
9. #11748707C2semidirect
10. #11748702C2semidirect
11. #11748663C2semidirect
29. C4×#1201C4semidirect
45. #64953D3semidirect
87. #4664D5semidirect
434. C3×C5#2357semidirect
510. C5#74061semidirect
535. C3#10605792semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#12012
#117507502
C5×#46642
C3×#649532
#117489152
#117488672
#117487202
#117487192
#117487072
#117487022
#117486632
#106057923
#740615