Finite Group Databasebeta

← #14331132

Subgroups — #14331132

Order 1920.

1040 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×#91836C211yesmaximal, characteristic
2C3×C5×#391C211yesmaximal, characteristic, complemented
3#11751330C211yesmaximal, characteristic, complemented
4#11751090C212yesmaximal, complemented
5#11751041C212yesmaximal, complemented
6#11750770C212yesmaximal, complemented
7#11750769C212yesmaximal, complemented
8#11750766C212yesmaximal
9#11750751C212yesmaximal, complemented
10#10608227—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C22×D60C2211yescharacteristic
13C22×C4×D15C2212yes—
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C42×C5C2211yescharacteristic
16#91836C418yescomplemented
17C2×#6714C2212yes—
18C2×#6714C2212yes—
19C2×#6714C2212yes—
20C2×#6714C2212yes—
21C2×#6714C2212yes—
22C2×C4⋊Dic15C2211yescharacteristic
23C2×#6712C2212yes—
24C2×#6712C2212yes—
25C2×C4×Dic15C2211yescharacteristic
26C2×C4×Dic15C2211yescharacteristic
27C3×C5×#95C2212yes—
28C3×C5×#95C2212yes—
29#91115C2212yes—
30#91115C2212yes—
31#91115C2212yes—
32#91115C2212yes—
33#76496—51nomaximal, Hall
34C2×#4093—31no—
35C5×#391D311yescharacteristic, complemented
36#65519—31no—
37#65278—32no—
38#65229—32no—
39#64958—32no—
40#64957—32no—
41#64954—32no—
42#64939—32no—
43C23×D15C2312yes—
44C2×C3×C5×Q8—41no—
45C2×C3×C5×Q8—41no—
46C2×C3×C5×Q8C2×C414yescomplemented
47C22×C3×C4×C5C2312yes—
48C22×C3×C4×C5C2312yes—
49C22×C3×C4×C5C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15C2312yes—
53C22×Dic15C2312yes—
54C22×Dic15C2312yes—
55C2×D60—41no—
56C2×D60—41no—
57C2×D60C2×C414yes—
58C2×C4×D15—42no—
59C2×C4×D15—42no—
60C2×C4×D15—42no—
61C3×C5×C4⋊C4—41no—
62C3×C42×C5—41no—
63#6714—42no—
64#6714—42no—
65#6714—42no—
66#6714C2×C418yescomplemented
67#6714C2×C418yescomplemented
68C4⋊Dic15—41no—
69#6712—42no—
70#6712—42no—
71C4×Dic15—41no—
72C4×Dic15C2×C414yescomplemented
73C2×#947—51no—
74C3×#391D511yescharacteristic, complemented
75#5229—51no—
76#4989—52no—
77#4940—52no—
78#4669—52no—
79#4668—52no—
80#4665—52no—
81#4650—52no—
82C22×C5×Q8C2×D311yescharacteristic
83C22×D20—31no—
84C22×C4×D5—32no—
85C2×C5×C4⋊C4C2×D311yescharacteristic
86C2×C42×C5C2×D311yescharacteristic
87#4093—38no—
88C2×#685—32no—
89C2×#685—32no—
90C2×#685—32no—
91C2×#685—32no—
92C2×#685—32no—
93C2×C4⋊Dic5—31no—
94C2×#683—32no—
95C2×#683—32no—
96C2×C4×Dic5—31no—
97C2×C4×Dic5—31no—
98C5×#95C2×D312yes—
99C5×#95C2×D312yes—
100#3963—32no—
101#3963—32no—
102#3963—32no—
103#3963—32no—
104#2379—151noSylow
105C23×C3×C5C2411yescharacteristic
106C22×D15—22no—
107C22×D15—22no—
108C22×D15—22no—
109C22×D15—22no—
110C22×D15—22no—
111C22×D15—22no—
112C22×D15C22×C414yes—
113C3×C5×Q8—44no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5C22×C412yes—
127C2×C3×C4×C5C22×C412yes—
128C2×C3×C4×C5C22×C412yes—
129C2×Dic15—22no—
130C2×Dic15—22no—
131C2×Dic15—22no—
132C2×Dic15—22no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15—22no—
137C2×Dic15C22×C414yes—
138D60—44no—
139C4×D15—48no—
140C22×C3×Q8C2×D511yescharacteristic
141C22×D12—51no—
142C22×C4×D3—52no—
143C2×C3×C4⋊C4C2×D511yescharacteristic
144C2×C3×C42C2×D511yescharacteristic
145#947—58no—
146C2×(C2×D3)⋊C4—52no—
147C2×(C2×D3)⋊C4—52no—
148C2×(C2×D3)⋊C4—52no—
149C2×(C2×D3)⋊C4—52no—
150C2×(C2×D3)⋊C4—52no—
151C2×C4⋊Dic3—51no—
152C2×Dic3⋊C4—52no—
153C2×Dic3⋊C4—52no—
154C2×C4×Dic3—51no—
155C2×C4×Dic3—51no—
156C3×#95C2×D512yes—
157C3×#95C2×D512yes—
158C3⋊#95—52no—
159C3⋊#95—52no—
160C3⋊#95—52no—
161C3⋊#95—52no—
162C23×D5—32no—
163C2×C5×Q8—41no—
164C2×C5×Q8—41no—
165C2×C5×Q8C4×D314yescomplemented
166C22×C4×C5C22×D312yes—
167C22×C4×C5C22×D312yes—
168C22×C4×C5C22×D311yescharacteristic
169C22×C4×C5C22×D311yescharacteristic
170C22×C4×C5C22×D311yescharacteristic
171C22×Dic5—32no—
172C22×Dic5—32no—
173C22×Dic5—32no—
174C2×D20—34no—
175C2×D20—121no—
176C2×D20—121no—
177C2×C4×D5—122no—
178C2×C4×D5—122no—
179C2×C4×D5—122no—
180C5×C4⋊C4—41no—
181C42×C5—41no—
182#685—38no—
183#685—38no—
184#685—122no—
185#685—122no—
186#685—122no—
187C4⋊Dic5—121no—
188#683—122no—
189#683—122no—
190C4×Dic5—34no—
191C4×Dic5—121no—
192C2×#124—151no—
193#391—151no—
194#391D1511yescharacteristic, complemented
195#388—152no—
196#388—152no—
197#386—152no—
198#385—152no—
199#385—152no—
200#382—152no—
201C22×C3×C5C23×C411yescharacteristic
202C22×C3×C5H(5,𝔽2)11yescharacteristic
203C22×C3×C5H(5,𝔽2)11yescharacteristic
204C22×C3×C5H(5,𝔽2)12yes—
205C22×C3×C52−1+412yes—
206C2×D15—24no—
207C2×D15—24no—
208C2×D15—24no—
209C2×D15—42no—
210C2×D15—42no—
211C2×D15—42no—
212C2×D15—42no—
213C3×C4×C5—22no—
214C3×C4×C5—22no—
215C3×C4×C5—22no—
216C3×C4×C5—42no—
217C3×C4×C5—42no—
218Dic15—24no—
219Dic15—42no—
220Dic15—42no—
221C23×D3—52no—
222C2×C3×Q8—41no—
223C2×C3×Q8—41no—
224C2×C3×Q8C4×D514yescomplemented
225C22×C3×C4C22×D512yes—
226C22×C3×C4C22×D512yes—
227C22×C3×C4C22×D511yescharacteristic
228C22×C3×C4C22×D511yescharacteristic
229C22×C3×C4C22×D511yescharacteristic
230C22×Dic3—52no—
231C22×Dic3—52no—
232C22×Dic3—52no—
233C2×D12—54no—
234C2×D12—201no—
235C2×D12—201no—
236C2×C4×D3—202no—
237C2×C4×D3—202no—
238C2×C4×D3—202no—
239C3×C4⋊C4—41no—
240C3×C42—41no—
241(C2×D3)⋊C4—58no—
242(C2×D3)⋊C4—58no—
243(C2×D3)⋊C4—202no—
244(C2×D3)⋊C4—202no—
245(C2×D3)⋊C4—202no—
246C4⋊Dic3—201no—
247Dic3⋊C4—202no—
248Dic3⋊C4—202no—
249C4×Dic3—54no—
250C4×Dic3—201no—
251C23×C5C23×D311yescharacteristic
252C22×D5—34no—
253C22×D5—62no—
254C22×D5—62no—
255C22×D5—62no—
256C22×D5—62no—
257C22×D5—62no—
258C22×D5—62no—
259C5×Q8—44no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5C2×C4×D312yes—
273C2×C4×C5C2×C4×D312yes—
274C2×C4×C5C2×C4×D312yes—
275C2×Dic5—34no—
276C2×Dic5—62no—
277C2×Dic5—62no—
278C2×Dic5—62no—
279C2×Dic5—62no—
280C2×Dic5—62no—
281C2×Dic5—62no—
282C2×Dic5—62no—
283C2×Dic5—62no—
284D20—124no—
285C4×D5—128no—
286C22×Q8C2×D1511yescharacteristic
287C22×D4—151no—
288C23×C4—152no—
289#124—158no—
290C2×C4⋊C4—152no—
291C2×C4⋊C4—152no—
292C2×C4⋊C4—151no—
293C2×C4⋊C4C2×D1511yescharacteristic
294C2×C22⋊C4—152no—
295C2×C22⋊C4—152no—
296C2×C22⋊C4—152no—
297C2×C22⋊C4—152no—
298C2×C22⋊C4—152no—
299C2×C42—151no—
300C2×C42—151no—
301C2×C42C2×D1511yescharacteristic
302#95—152no—
303#95—152no—
304#95—152no—
305#95—152no—
306#95C2×D1512yes—
307#95C2×D1512yes—
308C2×C3×C5#51811yescharacteristic
309C2×C3×C5#52012yes—
310C2×C3×C5H(3,𝔽4)11yescharacteristic
311C2×C3×C5#56212yes—
312C2×C3×C5#56311yescharacteristic
313D15—44no—
314C23×C3C23×D511yescharacteristic
315C22×D3—54no—
316C22×D3—102no—
317C22×D3—102no—
318C22×D3—102no—
319C22×D3—102no—
320C22×D3—102no—
321C22×D3—102no—
322C3×Q8—44no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4C2×C4×D512yes—
336C2×C3×C4C2×C4×D512yes—
337C2×C3×C4C2×C4×D512yes—
338C2×Dic3—54no—
339C2×Dic3—102no—
340C2×Dic3—102no—
341C2×Dic3—102no—
342C2×Dic3—102no—
343C2×Dic3—102no—
344C2×Dic3—102no—
345C2×Dic3—102no—
346C2×Dic3—102no—
347D12—204no—
348C4×D3—208no—
349C22×C5C22×C4×D311yescharacteristic
350C22×C5#100411yescharacteristic
351C22×C5#100411yescharacteristic
352C22×C5#100712yes—
353C22×C5#100912yes—
354C2×D5—64no—
355C2×D5—64no—
356C2×D5—64no—
357C2×D5—122no—
358C2×D5—122no—
359C2×D5—122no—
360C2×D5—122no—
361C4×C5—22no—
362C4×C5—22no—
363C4×C5—22no—
364C4×C5—42no—
365C4×C5—42no—
366Dic5—64no—
367Dic5—122no—
368Dic5—122no—
369C24—152no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8C4×D1514yescomplemented
373C2×D4—154no—
374C2×D4—601no—
375C2×D4—601no—
376C22×C4—152no—
377C22×C4—152no—
378C22×C4—152no—
379C22×C4—602no—
380C22×C4—602no—
381C22×C4—602no—
382C22×C4C22×D1512yes—
383C22×C4C22×D1512yes—
384C22×C4C22×D1511yescharacteristic
385C22×C4C22×D1511yescharacteristic
386C22×C4C22×D1511yescharacteristic
387C4⋊C4—41no—
388C4⋊C4—602no—
389C4⋊C4—602no—
390C4⋊C4—601no—
391C22⋊C4—158no—
392C22⋊C4—158no—
393C22⋊C4—602no—
394C22⋊C4—602no—
395C22⋊C4—602no—
396C42—41no—
397C42—154no—
398C42—601no—
399C3×C5#237911yescharacteristic, complemented, Hall
400C22×C3C22×C4×D511yescharacteristic
401C22×C3#414411yescharacteristic
402C22×C3#414411yescharacteristic
403C22×C3#414712yes—
404C22×C3#414912yes—
405C2×D3—104no—
406C2×D3—104no—
407C2×D3—104no—
408C2×D3—202no—
409C2×D3—202no—
410C2×D3—202no—
411C2×D3—202no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—42no—
416C3×C4—42no—
417Dic3—104no—
418Dic3—202no—
419Dic3—202no—
420C2×C5#548611yescharacteristic
421C2×C5#557412yes—
422C2×C5#564712yes—
423C2×C5#568311yescharacteristic
424C2×C5#573511yescharacteristic
425D5—124no—
426C23—154no—
427C23—302no—
428C23—302no—
429C23—302no—
430C23—302no—
431C23—302no—
432C23—302no—
433C23C23×D1511yescharacteristic
434Q8—44no—
435D4—604no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—154no—
449C2×C4—302no—
450C2×C4—302no—
451C2×C4—302no—
452C2×C4—302no—
453C2×C4—302no—
454C2×C4—302no—
455C2×C4—302no—
456C2×C4—302no—
457C2×C4—608no—
458C2×C4C2×C4×D1512yes—
459C2×C4C2×C4×D1512yes—
460C2×C4C2×C4×D1512yes—
461C2×C3#6582011yescharacteristic
462C2×C3#6590812yes—
463C2×C3#6598112yes—
464C2×C3#6601711yescharacteristic
465C2×C3#6606911yescharacteristic
466D3—204no—
467C5#7649611yescharacteristic, complemented, Sylow
468C22—304no—
469C22—304no—
470C22—304no—
471C22—602no—
472C22—602no—
473C22—602no—
474C22—602no—
475C22C22×C4×D1511yescharacteristic
476C22#9209511yescharacteristic
477C22#9209511yescharacteristic
478C22#9209812yes—
479C22#9210012yes—
480C4—22no—
481C4—22no—
482C4—22no—
483C4—42no—
484C4—42no—
485C4—304no—
486C4—602no—
487C4—602no—
488C3#1060822711yescharacteristic, complemented, Sylow
489C2—604no—
490C2#1175624511yescharacteristic
491C2#1175633312yes—
492C2#1175640612yes—
493C2#1175644211yescharacteristic
494C2#1175649411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#391C2
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenter#391D15
hypocenterC3×C5#2379
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C3×C5×#391C2semidirect
3. #11751330C2semidirect
4. #11751090C2semidirect
5. #11751041C2semidirect
6. #11750770C2semidirect
7. #11750769C2semidirect
9. #11750751C2semidirect
16. #91836C4semidirect
35. C5×#391D3semidirect
46. C2×C3×C5×Q8C2×C4semidirect
66. #6714C2×C4semidirect
67. #6714C2×C4semidirect
72. C4×Dic15C2×C4semidirect
74. C3×#391D5semidirect
165. C2×C5×Q8C4×D3semidirect
194. #391D15semidirect
224. C2×C3×Q8C4×D5semidirect
372. C2×Q8C4×D15semidirect
399. C3×C5#2379semidirect
467. C5#76496semidirect
488. C3#10608227semidirect

Maximal subgroups

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

subgroupindex
C2×#918362
C3×C5×#3912
#117513302
#117510902
#117510412
#117507702
#117507692
#117507662
#117507512
#106082273
#764965