Finite Group Databasebeta

← #14326446

Subgroups — #14326446

Order 1920.

1064 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91761C211yesmaximal, characteristic
2C5×#4932C211yesmaximal, characteristic, complemented
3C3×#65225C211yesmaximal, characteristic, complemented
4#11748898C212yesmaximal
5#11748888C211yesmaximal, characteristic
6#11748881C212yesmaximal
7#11748847C212yesmaximal, complemented
8#11748674C212yesmaximal, complemented
9#11748663C211yesmaximal, characteristic, complemented
10#11748621C211yesmaximal, characteristic, complemented
11#11748606C211yesmaximal, characteristic
12#10605147—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×#1200C2212yes—
15C2×C3×C42×C5C2211yescharacteristic
16C2×#6714C2211yescharacteristic
17C2×#6714C2211yescharacteristic
18C2×#6712C2211yescharacteristic
19C2×#6712C2211yescharacteristic
20C2×C4×Dic15C2211yescharacteristic
21#91761C418yescomplemented
22C2×C4×C5×Dic3C2212yes—
23C2×C3×C4⋊Dic5C2212yes—
24C2×#6669C2212yes—
25C2×#6667C2212yes—
26C2×Dic3×Dic5C2212yes—
27C5×C3⋊#95C2211yescharacteristic
28C5×C3⋊#95C2211yescharacteristic
29C5×C3⋊#95C2212yes—
30C3×#3963C2211yescharacteristic
31C3×#3963C2212yes—
32C3×#3963C2211yescharacteristic
33#90994C2212yes—
34#90994C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#74227—51nomaximal, Hall
38C2×#4018—31no—
39C5×#376—31no—
40C4×#685—32no—
41#65225D311yescharacteristic, complemented
42#65222—32no—
43#64957—32no—
44#64953—31no—
45D5×#95—31no—
46#64944—32no—
47#64940—31no—
48#64938—31no—
49C23×D15C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C22×Dic15C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C2×C4×D15—24no—
57C2×C4×D15C2×C414yes—
58C22×C5×Dic3C2312yes—
59C22×C5×Dic3C2312yes—
60C22×C3×Dic5C2312yes—
61C22×C3×Dic5C2312yes—
62C2×#1200—24no—
63C2×#1200—24no—
64C2×#1200—42no—
65C3×C42×C5C2×C414yescomplemented
66#6714C2×C414yes—
67#6714C2×C414yes—
68#6712C2×C414yescomplemented
69#6712C2×C414yescomplemented
70C4×Dic15C2×C414yescomplemented
71C4×C5×Dic3—24no—
72C3×C4⋊Dic5—42no—
73#6669—24no—
74#6667—24no—
75Dic3×Dic5—42no—
76C2×#872—51no—
77C3×#382—51no—
78#4989—52no—
79#4978—52no—
80#4932D511yescharacteristic, complemented
81#4666—52no—
82#4664—51no—
83D3×#95—51no—
84#4652—52no—
85#4651—51no—
86#4649—51no—
87C22×C4×D5—31no—
88C22×C4×D5—32no—
89C2×C42×C5—32no—
90C2×C42×C5C2×D311yescharacteristic
91C2×#685—31no—
92C2×#685—31no—
93C2×#685—32no—
94C2×C4⋊Dic5C2×D312yes—
95C2×#683—31no—
96C2×#683—31no—
97C2×#683—32no—
98C2×C4×Dic5—31no—
99C2×C4×Dic5—32no—
100#4018—38no—
101C5×#95—31no—
102C5×#95—31no—
103C5×#95—32no—
104#3963—32no—
105#3963—32no—
106#3963—32no—
107#3963—32no—
108#3963C2×D311yescharacteristic
109#3963C2×D311yescharacteristic
110#3963C2×D312yes—
111#2343—151noSylow
112C23×C3×C5C2411yescharacteristic
113C22×D15—22no—
114C22×D15—21no—
115C22×D15—21no—
116C22×D15C22×C412yes—
117C22×D15#4114yes—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5—21no—
122C2×C3×C4×C5C22×C412yes—
123C2×C3×C4×C5C22×C412yes—
124C2×C3×C4×C5C22×C412yes—
125C2×Dic15—22no—
126C2×Dic15—22no—
127C2×Dic15—21no—
128C2×Dic15—21no—
129C2×Dic15C22×C412yes—
130C2×Dic15C22×C412yes—
131C2×Dic15C22×C412yes—
132C4×D15—28no—
133C2×C5×Dic3—22no—
134C2×C5×Dic3—22no—
135C2×C5×Dic3—22no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3—22no—
138C2×C5×Dic3#4114yes—
139C2×C3×Dic5—22no—
140C2×C3×Dic5—22no—
141C2×C3×Dic5—22no—
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—22no—
145#1200—48no—
146C22×C4×D3—51no—
147C22×C4×D3—52no—
148C2×C3×C4⋊C4—52no—
149C2×C3×C42C2×D511yescharacteristic
150C2×(C2×D3)⋊C4—51no—
151C2×(C2×D3)⋊C4—51no—
152C2×(C2×D3)⋊C4—52no—
153C2×Dic3⋊C4—51no—
154C2×Dic3⋊C4—51no—
155C2×Dic3⋊C4—52no—
156C2×C4×Dic3—51no—
157C2×C4×Dic3—52no—
158C2×C4×Dic3C2×D512yes—
159#872—58no—
160C3×#95—51no—
161C3×#95—51no—
162C3×#95—52no—
163C3⋊#95—52no—
164C3⋊#95—52no—
165C3⋊#95—52no—
166C3⋊#95—52no—
167C3⋊#95C2×D511yescharacteristic
168C3⋊#95C2×D511yescharacteristic
169C3⋊#95C2×D512yes—
170C23×D5—31no—
171C22×C4×C5—32no—
172C22×C4×C5—32no—
173C22×C4×C5C22×D311yescharacteristic
174C22×C4×C5C22×D311yescharacteristic
175C22×C4×C5C22×D311yescharacteristic
176C22×Dic5—31no—
177C22×Dic5—31no—
178C22×Dic5—31no—
179C22×Dic5C22×D312yes—
180C22×Dic5C22×D312yes—
181C2×C4×D5—34no—
182C2×C4×D5—64no—
183C2×C4×D5—64no—
184C2×C4×D5—64no—
185C2×C4×D5—122no—
186C42×C5—64no—
187C42×C5C4×D314yescomplemented
188#685—34no—
189#685—34no—
190#685—64no—
191C4⋊Dic5—42no—
192#683—34no—
193#683—34no—
194#683—64no—
195C4×Dic5—34no—
196C4×Dic5—122no—
197C2×#117—151no—
198#387—152no—
199#387—152no—
200#385—152no—
201#383—151no—
202#382—51no—
203#381—151no—
204C4×C4⋊C4—151no—
205C4×C22⋊C4—152no—
206#376—31no—
207C2×#95—151no—
208C22×C3×C5C23×C411yescharacteristic
209C22×C3×C5C2×#4112yes—
210C22×C3×C5C2×#4112yes—
211C22×C3×C5H(5,𝔽2)11yescharacteristic
212C22×C3×C52−1+411yescharacteristic
213C2×D15—22no—
214C2×D15—22no—
215C2×D15—24no—
216C2×D15—24no—
217C2×D15#11714yes—
218C3×C4×C5—22no—
219C3×C4×C5—22no—
220C3×C4×C5—22no—
221Dic15—22no—
222Dic15—22no—
223Dic15—22no—
224C3×Dic5—42no—
225C3×Dic5—42no—
226C5×Dic3—24no—
227C5×Dic3—42no—
228C23×D3—51no—
229C22×C3×C4—52no—
230C22×C3×C4—52no—
231C22×C3×C4C22×D511yescharacteristic
232C22×C3×C4C22×D511yescharacteristic
233C22×C3×C4C22×D511yescharacteristic
234C22×Dic3—51no—
235C22×Dic3—51no—
236C22×Dic3—51no—
237C22×Dic3C22×D512yes—
238C22×Dic3C22×D512yes—
239C2×C4×D3—54no—
240C2×C4×D3—104no—
241C2×C4×D3—104no—
242C2×C4×D3—104no—
243C2×C4×D3—202no—
244C3×C4⋊C4—202no—
245C3×C42C4×D514yescomplemented
246(C2×D3)⋊C4—54no—
247(C2×D3)⋊C4—54no—
248(C2×D3)⋊C4—104no—
249Dic3⋊C4—54no—
250Dic3⋊C4—54no—
251Dic3⋊C4—104no—
252C4×Dic3—24no—
253C4×Dic3—54no—
254C4×Dic3—202no—
255C23×C5C23×D311yescharacteristic
256C22×D5—32no—
257C22×D5—34no—
258C22×D5—61no—
259C22×D5—61no—
260C22×D5—62no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—34no—
266C2×C4×C5—62no—
267C2×C4×C5—62no—
268C2×C4×C5—62no—
269C2×C4×C5—62no—
270C2×C4×C5—62no—
271C2×C4×C5C2×C4×D312yes—
272C2×C4×C5C2×C4×D312yes—
273C2×C4×C5C2×C4×D312yes—
274C2×Dic5—22no—
275C2×Dic5—22no—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—32no—
281C2×Dic5—32no—
282C2×Dic5—32no—
283C2×Dic5—61no—
284C2×Dic5—61no—
285C2×Dic5—62no—
286C2×Dic5—62no—
287C4×D5—68no—
288C4×D5—128no—
289C23×C4—152no—
290C23×C4—151no—
291#117—158no—
292C2×C4⋊C4—52no—
293C2×C4⋊C4—151no—
294C2×C4⋊C4—151no—
295C2×C4⋊C4—152no—
296C2×C22⋊C4—151no—
297C2×C22⋊C4—151no—
298C2×C22⋊C4—152no—
299C2×C42—32no—
300C2×C42—151no—
301C2×C42—152no—
302C2×C42D3×D511yescharacteristic
303#95—31no—
304#95—31no—
305#95—32no—
306#95—51no—
307#95—52no—
308#95—51no—
309#95—152no—
310#95—152no—
311#95—152no—
312#95—152no—
313C2×C3×C5C2×#11711yescharacteristic
314C2×C3×C5C4×#4111yescharacteristic
315C2×C3×C5#52011yescharacteristic
316C2×C3×C5#55112yes—
317C2×C3×C5#55212yes—
318D15—24no—
319C23×C3C23×D511yescharacteristic
320C22×D3—52no—
321C22×D3—54no—
322C22×D3—102no—
323C22×D3—101no—
324C22×D3—101no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—102no—
330C2×C3×C4—102no—
331C2×C3×C4—102no—
332C2×C3×C4—102no—
333C2×C3×C4—102no—
334C2×C3×C4—102no—
335C2×C3×C4C2×C4×D512yes—
336C2×C3×C4C2×C4×D512yes—
337C2×C3×C4C2×C4×D512yes—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—52no—
344C2×Dic3—52no—
345C2×Dic3—52no—
346C2×Dic3—102no—
347C2×Dic3—102no—
348C2×Dic3—101no—
349C2×Dic3—101no—
350C2×Dic3#70914yes—
351C4×D3—108no—
352C4×D3—208no—
353C22×C5C22×C4×D311yescharacteristic
354C22×C5#100411yescharacteristic
355C22×C5#100711yescharacteristic
356C22×C5D3×#4112yes—
357C22×C5D3×#4112yes—
358C2×D5—34no—
359C2×D5—64no—
360C2×D5—64no—
361C2×D5—62no—
362C2×D5—62no—
363C4×C5—22no—
364C4×C5—22no—
365C4×C5—22no—
366C4×C5—64no—
367C4×C5—122no—
368Dic5—42no—
369Dic5—42no—
370Dic5—62no—
371Dic5—62no—
372Dic5—62no—
373C24—151no—
374C22×C4—32no—
375C22×C4—32no—
376C22×C4—52no—
377C22×C4—52no—
378C22×C4—151no—
379C22×C4—151no—
380C22×C4—151no—
381C22×C4—154no—
382C22×C4—304no—
383C22×C4—304no—
384C22×C4—304no—
385C22×C4—602no—
386C22×C4C2×D3×D511yescharacteristic
387C22×C4C2×D3×D511yescharacteristic
388C22×C4C2×D3×D511yescharacteristic
389C4⋊C4—154no—
390C4⋊C4—154no—
391C4⋊C4—202no—
392C4⋊C4—304no—
393C22⋊C4—154no—
394C22⋊C4—154no—
395C22⋊C4—304no—
396C42—64no—
397C42—154no—
398C42—602no—
399C42#120014yescomplemented
400C3×C5#234311yescharacteristic, complemented, Hall
401C22×C3C22×C4×D511yescharacteristic
402C22×C3C2×#70912yes—
403C22×C3D5×#4112yes—
404C22×C3#414911yescharacteristic
405C22×C3#415011yescharacteristic
406C2×D3—54no—
407C2×D3—102no—
408C2×D3—102no—
409C2×D3—104no—
410C2×D3—104no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—202no—
415C3×C4—202no—
416Dic3—24no—
417Dic3—42no—
418Dic3—102no—
419Dic3—102no—
420Dic3—102no—
421C2×C5#550411yescharacteristic
422C2×C5D3×#11711yescharacteristic
423C2×C5#552211yescharacteristic
424C2×C5#567912yes—
425C2×C5#569312yes—
426D5—64no—
427C23—154no—
428C23—152no—
429C23—301no—
430C23—302no—
431C23—301no—
432C23C22×D3×D511yescharacteristic
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—34no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—102no—
444C2×C4—102no—
445C2×C4—102no—
446C2×C4—102no—
447C2×C4—102no—
448C2×C4—102no—
449C2×C4—152no—
450C2×C4—152no—
451C2×C4—152no—
452C2×C4—308no—
453C2×C4—301no—
454C2×C4—302no—
455C2×C4—302no—
456C2×C4—301no—
457C2×C4—608no—
458C2×C4C2×#120012yes—
459C2×C4C2×#120012yes—
460C2×C4C2×#120012yes—
461C2×C3C4×#70911yescharacteristic
462C2×C3D5×#11711yescharacteristic
463C2×C3#6585711yescharacteristic
464C2×C3#6586112yes—
465C2×C3#6586312yes—
466D3—104no—
467C5#7422711yescharacteristic, complemented, Sylow
468C22—154no—
469C22—302no—
470C22—304no—
471C22—304no—
472C22—302no—
473C22#9200411yescharacteristic
474C22D3×#70912yes—
475C22#9201612yes—
476C22#9201711yescharacteristic
477C22C22×#120011yescharacteristic
478C4—22no—
479C4—22no—
480C4—22no—
481C4—64no—
482C4—122no—
483C4—202no—
484C4—202no—
485C4—302no—
486C4—302no—
487C4—302no—
488C3#1060514711yescharacteristic, complemented, Sylow
489C2—304no—
490C2#1175261512yes—
491C2#1175276612yes—
492C2#1175443611yescharacteristic
493C2#1175446111yescharacteristic
494C2#1175449611yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#2343
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. C5×#4932C2semidirect
3. C3×#65225C2semidirect
7. #11748847C2semidirect
8. #11748674C2semidirect
9. #11748663C2semidirect
10. #11748621C2semidirect
21. #91761C4semidirect
41. #65225D3semidirect
65. C3×C42×C5C2×C4semidirect
68. #6712C2×C4semidirect
69. #6712C2×C4semidirect
70. C4×Dic15C2×C4semidirect
80. #4932D5semidirect
187. C42×C5C4×D3semidirect
245. C3×C42C4×D5semidirect
399. C42#1200semidirect
400. C3×C5#2343semidirect
467. C5#74227semidirect
488. C3#10605147semidirect

Maximal subgroups

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

subgroupindex
C2×#917612
C5×#49322
C3×#652252
#117488982
#117488882
#117488812
#117488472
#117486742
#117486632
#117486212
#117486062
#106051473
#742275