Finite Group Databasebeta

← #14330691

Subgroups — #14330691

Order 1920.

1024 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91824C211yesmaximal, characteristic
2C3×C5×#386C211yesmaximal, characteristic
3#11751322C211yesmaximal, characteristic
4#11751311C212yesmaximal, complemented
5#11751046C212yesmaximal, complemented
6#11751045C212yesmaximal, complemented
7#11751043C212yesmaximal, complemented
8#11750753C212yesmaximal, complemented
9#11750748C212yesmaximal, complemented
10#10606413—31nomaximal, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×C4×C5C2211yescharacteristic
13C23×Dic15C2211yescharacteristic
14C2×C3×C5×C22⋊C4C2212yes—
15C2×C3×C5×C22⋊C4C2211yescharacteristic
16C2×#6719C2212yes—
17C2×#6719C2212yes—
18C2×#6719C2212yes—
19C2×#6719C2211yescharacteristic
20#91824C418yescomplemented
21C2×C4⋊Dic15C2211yescharacteristic
22C2×#6712C2212yes—
23C2×#6712C2212yes—
24C2×#6712C2212yes—
25C2×#6712C2212yes—
26C2×C4×Dic15C2212yes—
27C2×C4×Dic15C2211yescharacteristic
28C3×C5×#95C2212yes—
29#91115C2212yes—
30#91115C2212yes—
31#91115C2212yes—
32#91115C2212yes—
33#74682—51nomaximal, Hall
34C2×#4081—31no—
35C5×#386D311yescharacteristic
36#65510—31no—
37#65499—32no—
38#65234—32no—
39#65233—32no—
40#65231—32no—
41#64941—32no—
42#64936—32no—
43C24×C3×C5C2311yescharacteristic
44C24×C3×C5C2311yescharacteristic
45C2×C3×C5×D4—41no—
46C2×C3×C5×D4—41no—
47C2×C3×C5×D4C2×C414yes—
48C22×C3×C4×C5—41no—
49C22×C3×C4×C5—42no—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2312yes—
52C22×C3×C4×C5C2311yescharacteristic
53C22×Dic15—41no—
54C22×Dic15—41no—
55C22×Dic15C2×C414yescomplemented
56C22×Dic15C2312yes—
57C22×Dic15C2312yes—
58C22×Dic15C2312yes—
59C22×Dic15C2312yes—
60C3×C5×C22⋊C4—42no—
61C3×C5×C22⋊C4—41no—
62#6719—42no—
63#6719—42no—
64#6719C2×C418yes—
65#6719C2×C414yescomplemented
66C4⋊Dic15—41no—
67#6712—42no—
68#6712—42no—
69#6712—42no—
70#6712C2×C418yescomplemented
71C4×Dic15—42no—
72C4×Dic15—41no—
73C2×#935—51no—
74C3×#386D511yescharacteristic
75#5221—51no—
76#5210—52no—
77#4945—52no—
78#4944—52no—
79#4942—52no—
80#4652—52no—
81#4647—52no—
82C22×C5×D4C2×D311yescharacteristic
83C23×C4×C5C2×D311yescharacteristic
84C23×Dic5—31no—
85C2×C5×C22⋊C4C2×D312yes—
86C2×C5×C22⋊C4C2×D311yescharacteristic
87C2×C22⋊Dic5—31no—
88C2×C22⋊Dic5—32no—
89C2×C22⋊Dic5—32no—
90C2×C22⋊Dic5—32no—
91#4081—38no—
92C2×C4⋊Dic5—31no—
93C2×#683—32no—
94C2×#683—32no—
95C2×#683—32no—
96C2×#683—32no—
97C2×C4×Dic5—32no—
98C2×C4×Dic5—31no—
99C5×#95C2×D312yes—
100#3963—32no—
101#3963—32no—
102#3963—32no—
103#3963—32no—
104#2376—151noSylow
105C23×C3×C5—22no—
106C23×C3×C5—22no—
107C23×C3×C5—21no—
108C23×C3×C5—21no—
109C23×C3×C5—21no—
110C23×C3×C5—21no—
111C23×C3×C5—22no—
112C23×C3×C5—22no—
113C23×C3×C5C22×C412yes—
114C23×C3×C5C22×C412yes—
115C23×C3×C5C2411yescharacteristic
116C3×C5×D4—44no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—22no—
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—44no—
126C2×C3×C4×C5C22×C412yes—
127C2×Dic15—22no—
128C2×Dic15—22no—
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×Dic15—44no—
138C2×Dic15C22×C414yes—
139C2×Dic15C22×C414yes—
140C22×C3×D4C2×D511yescharacteristic
141C23×C3×C4C2×D511yescharacteristic
142C23×Dic3—51no—
143C2×C3×C22⋊C4C2×D512yes—
144C2×C3×C22⋊C4C2×D511yescharacteristic
145C2×#217—51no—
146C2×#217—52no—
147C2×#217—52no—
148C2×#217—52no—
149#935—58no—
150C2×C4⋊Dic3—51no—
151C2×Dic3⋊C4—52no—
152C2×Dic3⋊C4—52no—
153C2×Dic3⋊C4—52no—
154C2×Dic3⋊C4—52no—
155C2×C4×Dic3—52no—
156C2×C4×Dic3—51no—
157C3×#95C2×D512yes—
158C3⋊#95—52no—
159C3⋊#95—52no—
160C3⋊#95—52no—
161C3⋊#95—52no—
162C24×C5C22×D311yescharacteristic
163C24×C5C22×D311yescharacteristic
164C2×C5×D4—41no—
165C2×C5×D4—41no—
166C2×C5×D4C4×D314yes—
167C22×C4×C5—41no—
168C22×C4×C5—42no—
169C22×C4×C5C22×D312yes—
170C22×C4×C5C22×D312yes—
171C22×C4×C5C22×D311yescharacteristic
172C22×Dic5—32no—
173C22×Dic5—32no—
174C22×Dic5—34no—
175C22×Dic5—32no—
176C22×Dic5—32no—
177C22×Dic5—121no—
178C22×Dic5—121no—
179C5×C22⋊C4—42no—
180C5×C22⋊C4—41no—
181C22⋊Dic5—38no—
182C22⋊Dic5—34no—
183C22⋊Dic5—122no—
184C22⋊Dic5—122no—
185C4⋊Dic5—121no—
186#683—38no—
187#683—122no—
188#683—122no—
189#683—122no—
190C4×Dic5—122no—
191C4×Dic5—121no—
192C2×#123—151no—
193#390—151no—
194#389—152no—
195#388—152no—
196#388—152no—
197#387—152no—
198#386D1511yescharacteristic
199#385—152no—
200#380—152no—
201C22×C3×C5—22no—
202C22×C3×C5—22no—
203C22×C3×C5—24no—
204C22×C3×C5—24no—
205C22×C3×C5—41no—
206C22×C3×C5—41no—
207C22×C3×C5—42no—
208C22×C3×C5—42no—
209C22×C3×C5—41no—
210C22×C3×C5—41no—
211C22×C3×C5C23×C411yescharacteristic
212C22×C3×C5H(5,𝔽2)11yescharacteristic
213C22×C3×C5H(5,𝔽2)11yescharacteristic
214C22×C3×C5H(5,𝔽2)12yes—
215C22×C3×C52−1+412yes—
216C3×C4×C5—22no—
217C3×C4×C5—42no—
218C3×C4×C5—42no—
219Dic15—24no—
220Dic15—24no—
221Dic15—42no—
222Dic15—42no—
223C24×C3C22×D511yescharacteristic
224C24×C3C22×D511yescharacteristic
225C2×C3×D4—41no—
226C2×C3×D4—41no—
227C2×C3×D4C4×D514yes—
228C22×C3×C4—41no—
229C22×C3×C4—42no—
230C22×C3×C4C22×D512yes—
231C22×C3×C4C22×D512yes—
232C22×C3×C4C22×D511yescharacteristic
233C22×Dic3—52no—
234C22×Dic3—52no—
235C22×Dic3—54no—
236C22×Dic3—52no—
237C22×Dic3—52no—
238C22×Dic3—201no—
239C22×Dic3—201no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—41no—
242#217—58no—
243#217—54no—
244#217—202no—
245#217—202no—
246C4⋊Dic3—201no—
247Dic3⋊C4—58no—
248Dic3⋊C4—202no—
249Dic3⋊C4—202no—
250Dic3⋊C4—202no—
251C4×Dic3—202no—
252C4×Dic3—201no—
253C23×C5—22no—
254C23×C5—22no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—22no—
260C23×C5—22no—
261C23×C5C2×C4×D312yes—
262C23×C5C2×C4×D312yes—
263C23×C5C23×D311yescharacteristic
264C5×D4—44no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—44no—
274C2×C4×C5C2×C4×D312yes—
275C2×Dic5—34no—
276C2×Dic5—34no—
277C2×Dic5—62no—
278C2×Dic5—62no—
279C2×Dic5—62no—
280C2×Dic5—62no—
281C2×Dic5—62no—
282C2×Dic5—62no—
283C2×Dic5—62no—
284C2×Dic5—62no—
285C2×Dic5—62no—
286C2×Dic5—62no—
287C2×Dic5—124no—
288C22×D4C2×D1511yescharacteristic
289C23×C4—151no—
290C23×C4C2×D1511yescharacteristic
291#123—158no—
292C2×C4⋊C4—152no—
293C2×C4⋊C4—152no—
294C2×C4⋊C4—152no—
295C2×C4⋊C4—152no—
296C2×C4⋊C4—151no—
297C2×C22⋊C4—152no—
298C2×C22⋊C4—151no—
299C2×C22⋊C4—152no—
300C2×C22⋊C4—152no—
301C2×C22⋊C4C2×D1512yes—
302C2×C22⋊C4C2×D1511yescharacteristic
303C2×C42—151no—
304C2×C42—152no—
305#95—152no—
306#95—152no—
307#95—152no—
308#95—152no—
309#95C2×D1512yes—
310C2×C3×C5—42no—
311C2×C3×C5—42no—
312C2×C3×C5#51811yescharacteristic
313C2×C3×C5#52012yes—
314C2×C3×C5#56011yescharacteristic
315C2×C3×C5#56212yes—
316C2×C3×C5#56311yescharacteristic
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3C2×C4×D512yes—
326C23×C3C2×C4×D512yes—
327C23×C3C23×D511yescharacteristic
328C3×D4—44no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—44no—
338C2×C3×C4C2×C4×D512yes—
339C2×Dic3—54no—
340C2×Dic3—54no—
341C2×Dic3—102no—
342C2×Dic3—102no—
343C2×Dic3—102no—
344C2×Dic3—102no—
345C2×Dic3—102no—
346C2×Dic3—102no—
347C2×Dic3—102no—
348C2×Dic3—102no—
349C2×Dic3—102no—
350C2×Dic3—102no—
351C2×Dic3—204no—
352C22×C5—22no—
353C22×C5—22no—
354C22×C5—24no—
355C22×C5—24no—
356C22×C5—41no—
357C22×C5—41no—
358C22×C5—41no—
359C22×C5—41no—
360C22×C5—42no—
361C22×C5—42no—
362C22×C5C22×C4×D311yescharacteristic
363C22×C5#100411yescharacteristic
364C22×C5#100411yescharacteristic
365C22×C5#100412yes—
366C22×C5#101012yes—
367C4×C5—22no—
368C4×C5—42no—
369C4×C5—42no—
370Dic5—64no—
371Dic5—64no—
372Dic5—122no—
373Dic5—122no—
374C24C22×D1511yescharacteristic
375C24C22×D1511yescharacteristic
376C2×D4—41no—
377C2×D4—41no—
378C2×D4C4×D1514yes—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—152no—
382C22×C4—152no—
383C22×C4—154no—
384C22×C4—152no—
385C22×C4—152no—
386C22×C4—601no—
387C22×C4—601no—
388C22×C4C22×D1512yes—
389C22×C4C22×D1512yes—
390C22×C4C22×D1511yescharacteristic
391C4⋊C4—158no—
392C4⋊C4—602no—
393C4⋊C4—602no—
394C4⋊C4—602no—
395C4⋊C4—601no—
396C22⋊C4—42no—
397C22⋊C4—41no—
398C22⋊C4—154no—
399C22⋊C4—158no—
400C22⋊C4—602no—
401C22⋊C4—602no—
402C42—601no—
403C42—602no—
404C3×C5#237611yescharacteristic, complemented, Hall
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3C22×C4×D511yescharacteristic
416C22×C3#414411yescharacteristic
417C22×C3#414411yescharacteristic
418C22×C3#414412yes—
419C22×C3#415012yes—
420C3×C4—22no—
421C3×C4—42no—
422C3×C4—42no—
423Dic3—104no—
424Dic3—104no—
425Dic3—202no—
426Dic3—202no—
427C2×C5—42no—
428C2×C5—42no—
429C2×C5#548611yescharacteristic
430C2×C5#554612yes—
431C2×C5#559511yescharacteristic
432C2×C5#562112yes—
433C2×C5#565111yescharacteristic
434C23—22no—
435C23—21no—
436C23—22no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23C2×C4×D1512yes—
443C23C2×C4×D1512yes—
444C23C23×D1511yescharacteristic
445D4—44no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—44no—
455C2×C4—154no—
456C2×C4—154no—
457C2×C4—302no—
458C2×C4—302no—
459C2×C4—302no—
460C2×C4—302no—
461C2×C4—302no—
462C2×C4—302no—
463C2×C4—302no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—302no—
467C2×C4—604no—
468C2×C4C2×C4×D1512yes—
469C2×C3—42no—
470C2×C3—42no—
471C2×C3#6582011yescharacteristic
472C2×C3#6588012yes—
473C2×C3#6592911yescharacteristic
474C2×C3#6595512yes—
475C2×C3#6598511yescharacteristic
476C5#7468211yescharacteristic, complemented, Sylow
477C22—24no—
478C22—24no—
479C22—22no—
480C22—22no—
481C22—41no—
482C22—41no—
483C22—42no—
484C22—42no—
485C22—41no—
486C22—41no—
487C22C22×C4×D1511yescharacteristic
488C22#9209512yes—
489C22#9209511yescharacteristic
490C22#9209511yescharacteristic
491C22#9210112yes—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—304no—
496C4—304no—
497C4—602no—
498C4—602no—
499C3#1060641311yescharacteristic, complemented, Sylow
500C2—42no—
501C2—42no—
502C2#1175624511yescharacteristic
503C2#1175630512yes—
504C2#1175635411yescharacteristic
505C2#1175638012yes—
506C2#1175641011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#386C2
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×C5×D4C22
hypercenter#386D15
hypocenterC3×C5#2376
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
4. #11751311C2semidirect
5. #11751046C2semidirect
6. #11751045C2semidirect
7. #11751043C2semidirect
8. #11750753C2semidirect
9. #11750748C2semidirect
20. #91824C4semidirect
55. C22×Dic15C2×C4semidirect
65. #6719C2×C4semidirect
70. #6712C2×C4semidirect
404. C3×C5#2376semidirect
476. C5#74682semidirect
499. C3#10606413semidirect

Maximal subgroups

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

subgroupindex
C2×#918242
C3×C5×#3862
#117513222
#117513112
#117510462
#117510452
#117510432
#117507532
#117507482
#106064133
#746825