Finite Group Databasebeta

← #14323029

Subgroups — #14323029

Order 1920.

1136 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×C3×C4×C22⋊D5C211yesmaximal, characteristic
2C2×#91115C211yesmaximal, characteristic, complemented
3C4×C5×#217C211yesmaximal, characteristic, complemented
4#11749515C212yesmaximal, complemented
5#11749500C212yesmaximal
6#11748770C212yesmaximal, complemented
7#11748738C211yesmaximal, characteristic, complemented
8#11748737C211yesmaximal, characteristic, complemented
9#11748612C211yesmaximal, characteristic, complemented
10#11748590C212yesmaximal, complemented
11#11748580C211yesmaximal, characteristic, complemented
12#10605299—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C23×Dic15C2212yes—
15C22×C3×C22⋊D5C2211yescharacteristic
16C22×C3×C4×D5C2211yescharacteristic
17C2×C5×#217C2212yes—
18C2×C4×C5×Dic3C2212yes—
19C2×C3×C22⋊Dic5C2211yescharacteristic
20C3×C4×C22⋊D5C418yescomplemented
21C2×C3×#685C2211yescharacteristic
22C2×C3×#683C2211yescharacteristic
23C2×C3×C4×Dic5C2211yescharacteristic
24C2×#6668C2212yes—
25C2×#6668C2212yes—
26C2×#6665C2212yes—
27C2×#6665C2212yes—
28#91115C2211yescharacteristic
29#91115C2211yescharacteristic
30#91115C2212yes—
31C5×C3⋊#95C2211yescharacteristic
32C5×C3⋊#95C2211yescharacteristic
33#90994C2212yes—
34#90994C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#82324—51nomaximal, Hall
38C2×C4×C22⋊D5D311yescharacteristic
39C4×C5×C22⋊C4—31no—
40C2×#3963—31no—
41#65247—32no—
42#65235—32no—
43#65229—32no—
44#65222—32no—
45#64956—31no—
46#64953—31no—
47#64940—31no—
48#64939—31no—
49C24×C3×C5C2311yescharacteristic
50C23×C3×D5C2311yescharacteristic
51C22×C3×C4×C5—42no—
52C22×C3×C4×C5C2×C414yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15—24no—
56C22×Dic15—42no—
57C22×Dic15—42no—
58C22×Dic15C2312yes—
59C22×Dic15C2312yes—
60C22×C5×Dic3C2312yes—
61C22×C5×Dic3C2312yes—
62C2×C3×C22⋊D5—24no—
63C2×C3×C22⋊D5C2×C414yes—
64C22×C3×Dic5C2311yescharacteristic
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2311yescharacteristic
67C2×C3×C4×D5—42no—
68C2×C3×C4×D5C2×C414yescomplemented
69C5×#217—24no—
70C4×C5×Dic3—42no—
71C3×C22⋊Dic5C2×C414yes—
72C3×#685C2×C414yescomplemented
73C3×#683C2×C414yescomplemented
74C3×C4×Dic5C2×C414yescomplemented
75#6668—42no—
76#6668—42no—
77#6665—42no—
78#6665—42no—
79C2×C3×C4×D4—51no—
80#5220—52no—
81#5212—51no—
82#5211—51no—
83C4×#217D511yescharacteristic, complemented
84C2×C3⋊#95—51no—
85#4980—52no—
86#4951—52no—
87#4950—52no—
88#4937—51no—
89#4936—51no—
90C23×C4×C5C2×D311yescharacteristic
91C22×C22⋊D5C2×D311yescharacteristic
92C23×Dic5—32no—
93C22×C4×D5C2×D311yescharacteristic
94C2×C5×C22⋊C4—32no—
95C2×C42×C5—32no—
96C2×C22⋊Dic5C2×D311yescharacteristic
97C4×C22⋊D5Dic318yescomplemented
98C2×#685—32no—
99C2×#685—32no—
100C2×#685C2×D311yescharacteristic
101C2×#683—32no—
102C2×#683—32no—
103C2×#683C2×D311yescharacteristic
104C2×C4×Dic5C2×D311yescharacteristic
105C5×#95—31no—
106C5×#95—31no—
107#3963—31no—
108#3963—31no—
109#3963—32no—
110#3963—32no—
111#3963—32no—
112#3963—32no—
113#3963—32no—
114#2318—151noSylow
115C23×C3×C5—21no—
116C23×C3×C5—22no—
117C23×C3×C5—21no—
118C23×C3×C5C22×C412yes—
119C23×C3×C5#4114yes—
120C23×C3×C5C2411yescharacteristic
121C22×C3×D5—22no—
122C22×C3×D5—21no—
123C22×C3×D5—22no—
124C22×C3×D5—21no—
125C22×C3×D5C22×C412yes—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—44no—
129C2×C3×C4×C5C22×C412yes—
130C2×C3×C4×C5C22×C412yes—
131C2×Dic15—22no—
132C2×Dic15—22no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15—22no—
137C2×Dic15—48no—
138C2×C5×Dic3—22no—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C3×C22⋊D5—28no—
145C2×C3×Dic5—22no—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—21no—
149C2×C3×Dic5C22×C412yes—
150C2×C3×Dic5C22×C412yes—
151C2×C3×Dic5C22×C412yes—
152C3×C4×D5—44no—
153C22×C3×D4—51no—
154C23×C3×C4—51no—
155C23×C3×C4C2×D511yescharacteristic
156C23×Dic3—52no—
157C3×C4×D4—58no—
158C2×C3×C4⋊C4—51no—
159C2×C3×C22⋊C4—51no—
160C2×C3×C22⋊C4—51no—
161C2×C3×C42—51no—
162C2×#217—52no—
163C2×#217—52no—
164C2×#217C2×D512yes—
165C2×C4⋊Dic3—52no—
166C2×C4⋊Dic3—52no—
167C2×C4×Dic3C2×D512yes—
168C3⋊#95—51no—
169C3⋊#95—51no—
170C3⋊#95—52no—
171C3⋊#95—52no—
172C3⋊#95—52no—
173C3⋊#95—52no—
174C3⋊#95—52no—
175C3⋊#95C2×D511yescharacteristic
176C3⋊#95C2×D511yescharacteristic
177C24×C5C22×D311yescharacteristic
178C23×D5C22×D311yescharacteristic
179C22×C4×C5—32no—
180C22×C4×C5—32no—
181C22×C4×C5—42no—
182C22×C4×C5C2×Dic314yes—
183C22×C4×C5C22×D311yescharacteristic
184C22×C4×C5C22×D311yescharacteristic
185C2×C22⋊D5—24no—
186C2×C22⋊D5C2×Dic314yes—
187C22×Dic5—32no—
188C22×Dic5—32no—
189C22×Dic5—64no—
190C22×Dic5—122no—
191C22×Dic5—122no—
192C22×Dic5C22×D311yescharacteristic
193C22×Dic5C22×D311yescharacteristic
194C22×Dic5C22×D311yescharacteristic
195C2×C4×D5—42no—
196C2×C4×D5C2×Dic314yescomplemented
197C5×C22⋊C4—64no—
198C42×C5—122no—
199C22⋊Dic5C2×Dic314yes—
200#685—122no—
201#685—122no—
202#685C2×Dic314yescomplemented
203#683—122no—
204#683—122no—
205#683C2×Dic314yescomplemented
206C4×Dic5C2×Dic314yescomplemented
207C2×C4×D4—51no—
208#388—152no—
209#387—152no—
210#386—152no—
211#385—152no—
212#383—151no—
213#382—151no—
214#381—151no—
215#380—151no—
216C4×C22⋊C4—31no—
217C2×#95—151no—
218C22×C3×C5—22no—
219C22×C3×C5—22no—
220C22×C3×C5—24no—
221C22×C3×C5—24no—
222C22×C3×C5#11714yes—
223C22×C3×C5C23×C411yescharacteristic
224C22×C3×C5C2×#4112yes—
225C22×C3×C5H(5,𝔽2)12yes—
226C22×C3×C5H(5,𝔽2)11yescharacteristic
227C22×C3×C52−1+411yescharacteristic
228C2×C3×D5—22no—
229C2×C3×D5—22no—
230C2×C3×D5—22no—
231C2×C3×D5—42no—
232C2×C3×D5—42no—
233C3×C4×C5—22no—
234C3×C4×C5—22no—
235Dic15—42no—
236Dic15—42no—
237C3×Dic5—22no—
238C3×Dic5—22no—
239C3×Dic5—22no—
240C5×Dic3—42no—
241C5×Dic3—42no—
242C24×C3—51no—
243C24×C3C22×D511yescharacteristic
244C2×C3×D4—54no—
245C2×C3×D4—104no—
246C22×C3×C4—42no—
247C22×C3×C4—51no—
248C22×C3×C4—51no—
249C22×C3×C4—51no—
250C22×C3×C4—54no—
251C22×C3×C4—202no—
252C22×C3×C4C4×D514yes—
253C22×C3×C4C22×D511yescharacteristic
254C22×C3×C4C22×D511yescharacteristic
255C22×Dic3—52no—
256C22×Dic3—52no—
257C22×Dic3—104no—
258C22×Dic3—202no—
259C22×Dic3—202no—
260C22×Dic3C22×D512yes—
261C22×Dic3C22×D512yes—
262C3×C4⋊C4—54no—
263C3×C22⋊C4—54no—
264C3×C22⋊C4—54no—
265C3×C42—54no—
266#217—24no—
267#217—202no—
268#217—202no—
269C4⋊Dic3—202no—
270C4⋊Dic3—202no—
271C4×Dic3—42no—
272C23×C5—22no—
273C23×C5—21no—
274C23×C5—21no—
275C23×C5#23514yes—
276C23×C5C22×Dic312yes—
277C23×C5C23×D311yescharacteristic
278C22×D5—22no—
279C22×D5—22no—
280C22×D5—21no—
281C22×D5—21no—
282C22×D5C22×Dic312yes—
283C2×C4×C5—22no—
284C2×C4×C5—22no—
285C2×C4×C5—44no—
286C2×C4×C5—62no—
287C2×C4×C5—62no—
288C2×C4×C5—62no—
289C2×C4×C5—62no—
290C2×C4×C5—62no—
291C2×C4×C5—62no—
292C2×C4×C5C22×Dic312yes—
293C2×C4×C5C22×Dic312yes—
294C22⋊D5—28no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—21no—
298C2×Dic5—21no—
299C2×Dic5—62no—
300C2×Dic5—62no—
301C2×Dic5—62no—
302C2×Dic5—62no—
303C2×Dic5—62no—
304C2×Dic5—62no—
305C2×Dic5—128no—
306C2×Dic5C22×Dic312yes—
307C2×Dic5C22×Dic312yes—
308C2×Dic5C22×Dic312yes—
309C4×D5—44no—
310C22×D4—51no—
311C23×C4—51no—
312C23×C4—152no—
313C23×C4D3×D511yescharacteristic
314C4×D4—58no—
315C2×C4⋊C4—51no—
316C2×C4⋊C4—152no—
317C2×C4⋊C4—152no—
318C2×C22⋊C4—32no—
319C2×C22⋊C4—51no—
320C2×C22⋊C4—51no—
321C2×C22⋊C4—152no—
322C2×C22⋊C4—152no—
323C2×C42—32no—
324C2×C42—51no—
325#95—31no—
326#95—31no—
327#95—152no—
328#95—152no—
329#95—152no—
330#95—152no—
331#95—151no—
332#95—151no—
333#95—152no—
334C2×C3×C5—24no—
335C2×C3×C5C2×#11711yescharacteristic
336C2×C3×C5#51811yescharacteristic
337C2×C3×C5#52011yescharacteristic
338C2×C3×C5#53812yes—
339C2×C3×C5#53912yes—
340C3×D5—42no—
341C23×C3—22no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—52no—
345C23×C3—102no—
346C23×C3—102no—
347C23×C3—101no—
348C23×C3—101no—
349C23×C3C2×C4×D512yes—
350C23×C3#70914yes—
351C23×C3C23×D511yescharacteristic
352C3×D4—108no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—44no—
356C2×C3×C4—52no—
357C2×C3×C4—52no—
358C2×C3×C4—52no—
359C2×C3×C4—102no—
360C2×C3×C4—102no—
361C2×C3×C4—101no—
362C2×C3×C4—101no—
363C2×C3×C4—204no—
364C2×C3×C4C2×C4×D512yes—
365C2×C3×C4C2×C4×D512yes—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—102no—
373C2×Dic3—102no—
374C2×Dic3—102no—
375C2×Dic3—102no—
376C2×Dic3—102no—
377C2×Dic3—102no—
378C2×Dic3—208no—
379C22×C5—22no—
380C22×C5—24no—
381C22×C5—24no—
382C22×C5—22no—
383C22×C5#92614yes—
384C22×C5C2×#23512yes—
385C22×C5#100412yes—
386C22×C5#100911yescharacteristic
387C22×C5#101011yescharacteristic
388C22×C5C23×Dic311yescharacteristic
389C2×D5—22no—
390C2×D5—22no—
391C2×D5—22no—
392C2×D5—42no—
393C2×D5—42no—
394C4×C5—22no—
395C4×C5—22no—
396C4×C5—122no—
397C4×C5—122no—
398Dic5—22no—
399Dic5—22no—
400Dic5—22no—
401Dic5—122no—
402Dic5—122no—
403C24—51no—
404C24C2×D3×D511yescharacteristic
405C2×D4—54no—
406C2×D4—104no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—42no—
410C22×C4—51no—
411C22×C4—51no—
412C22×C4—51no—
413C22×C4—54no—
414C22×C4—152no—
415C22×C4—152no—
416C22×C4—202no—
417C22×C4—304no—
418C22×C4—602no—
419C22×C4—602no—
420C22×C4D5×Dic314yes—
421C22×C4C2×D3×D511yescharacteristic
422C22×C4C2×D3×D511yescharacteristic
423C4⋊C4—54no—
424C4⋊C4—602no—
425C4⋊C4—602no—
426C22⋊C4—54no—
427C22⋊C4—54no—
428C22⋊C4—64no—
429C22⋊C4—602no—
430C22⋊C4—602no—
431C42—54no—
432C42—122no—
433C3×C5#231811yescharacteristic, complemented, Hall
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—24no—
437C22×C3—24no—
438C22×C3—102no—
439C22×C3—102no—
440C22×C3—102no—
441C22×C3—202no—
442C22×C3—202no—
443C22×C3#401814yes—
444C22×C3C22×C4×D511yescharacteristic
445C22×C3C2×#70912yes—
446C22×C3#414412yes—
447C22×C3#414911yescharacteristic
448C22×C3#415011yescharacteristic
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—102no—
452C3×C4—102no—
453C3×C4—102no—
454Dic3—42no—
455Dic3—42no—
456Dic3—202no—
457Dic3—202no—
458C2×C5—24no—
459C2×C5#556012yes—
460C2×C5#556412yes—
461C2×C5C2×#92611yescharacteristic
462C2×C5#579811yescharacteristic
463C2×C5#582711yescharacteristic
464D5—42no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—52no—
469C23—101no—
470C23—102no—
471C23—101no—
472C23—102no—
473C23#677114yes—
474C23C2×D5×Dic312yes—
475C23C22×D3×D511yescharacteristic
476D4—108no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—44no—
480C2×C4—52no—
481C2×C4—52no—
482C2×C4—52no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—101no—
490C2×C4—102no—
491C2×C4—101no—
492C2×C4—102no—
493C2×C4—204no—
494C2×C4—302no—
495C2×C4—302no—
496C2×C4—302no—
497C2×C4—302no—
498C2×C4—302no—
499C2×C4—302no—
500C2×C4—608no—
501C2×C4C2×D5×Dic312yes—
502C2×C4C2×D5×Dic312yes—
503C2×C3—24no—
504C2×C3—202no—
505C2×C3C2×#401811yescharacteristic
506C2×C3#6582011yescharacteristic
507C2×C3#6585711yescharacteristic
508C2×C3#6589412yes—
509C2×C3#6589812yes—
510C5#8232411yescharacteristic, complemented, Sylow
511C22—22no—
512C22—24no—
513C22—24no—
514C22—22no—
515C22—102no—
516C22—102no—
517C22—102no—
518C22—202no—
519C22—202no—
520C22#9135714yes—
521C22#9200111yescharacteristic
522C22C2×#677112yes—
523C22#9201911yescharacteristic
524C22C22×D5×Dic311yescharacteristic
525C22#9204912yes—
526C4—22no—
527C4—22no—
528C4—102no—
529C4—102no—
530C4—102no—
531C4—122no—
532C4—122no—
533C4—602no—
534C4—602no—
535C3#1060529911yescharacteristic, complemented, Sylow
536C2—24no—
537C2—202no—
538C2C2×#9135711yescharacteristic
539C2#1175450911yescharacteristic
540C2#1175459612yes—
541C2#1175459912yes—
542C2#1175540611yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×C22⋊D5C22
hypercenterC23×C4D3×D5
hypocenterC3×C5#2318
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×#91115C2semidirect
3. C4×C5×#217C2semidirect
4. #11749515C2semidirect
6. #11748770C2semidirect
7. #11748738C2semidirect
8. #11748737C2semidirect
9. #11748612C2semidirect
10. #11748590C2semidirect
11. #11748580C2semidirect
20. C3×C4×C22⋊D5C4semidirect
68. C2×C3×C4×D5C2×C4semidirect
72. C3×#685C2×C4semidirect
73. C3×#683C2×C4semidirect
74. C3×C4×Dic5C2×C4semidirect
83. C4×#217D5semidirect
97. C4×C22⋊D5Dic3semidirect
196. C2×C4×D5C2×Dic3semidirect
202. #685C2×Dic3semidirect
205. #683C2×Dic3semidirect
206. C4×Dic5C2×Dic3semidirect
433. C3×C5#2318semidirect
510. C5#82324semidirect
535. C3#10605299semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊D52
C2×#911152
C4×C5×#2172
#117495152
#117495002
#117487702
#117487382
#117487372
#117486122
#117485902
#117485802
#106052993
#823245