Finite Group Databasebeta

← #14331126

Subgroups — #14331126

Order 1920.

1040 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91825C211yesmaximal, characteristic
2C3×C5×#386C211yesmaximal, characteristic
3#11751321C211yesmaximal, characteristic
4#11751311C212yesmaximal, complemented
5#11751047C212yesmaximal, complemented
6#11751046C212yesmaximal, complemented
7#11751045C212yesmaximal, complemented
8#11750751C212yesmaximal, complemented
9#11750747C212yesmaximal, complemented
10#10606415—31nomaximal, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×C4×C5C2211yescharacteristic
13C23×Dic15C2211yescharacteristic
14C22×Dic30C2211yescharacteristic
15C2×C3×C5×C22⋊C4C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2212yes—
17C2×#6719C2212yes—
18C2×#6719C2211yescharacteristic
19C2×#6719C2212yes—
20C2×#6719C2212yes—
21#91825C418yescomplemented
22C2×#6712C2212yes—
23C2×#6712C2212yes—
24C2×#6712C2212yes—
25C2×C4×Dic15C2212yes—
26C2×C4×Dic15C2211yescharacteristic
27C3×C5×#95C2212yes—
28#91115C2212yes—
29#91115C2212yes—
30#91115C2212yes—
31#91115C2212yes—
32#91115C2212yes—
33#74684—51nomaximal, Hall
34C2×#4082—31no—
35C5×#386D311yescharacteristic
36#65509—31no—
37#65499—32no—
38#65235—32no—
39#65234—32no—
40#65233—32no—
41#64939—32no—
42#64935—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—42no—
55C22×Dic15C2312yes—
56C22×Dic15C2312yes—
57C22×Dic15C2312yes—
58C22×Dic15C2312yes—
59C2×Dic30—41no—
60C2×Dic30—41no—
61C2×Dic30C2×C414yescomplemented
62C3×C5×C22⋊C4—41no—
63C3×C5×C22⋊C4—42no—
64#6719—41no—
65#6719—42no—
66#6719C2×C418yescomplemented
67#6719C2×C418yescomplemented
68#6712—42no—
69#6712—42no—
70#6712—42no—
71C4×Dic15—42no—
72C4×Dic15C2×C414yescomplemented
73C2×#936—51no—
74C3×#386D511yescharacteristic
75#5220—51no—
76#5210—52no—
77#4946—52no—
78#4945—52no—
79#4944—52no—
80#4650—52no—
81#4646—52no—
82C22×C5×D4C2×D311yescharacteristic
83C23×C4×C5C2×D311yescharacteristic
84C23×Dic5—31no—
85C22×Dic10—31no—
86C2×C5×C22⋊C4C2×D311yescharacteristic
87C2×C5×C22⋊C4C2×D312yes—
88C2×C22⋊Dic5—32no—
89C2×C22⋊Dic5—32no—
90C2×C22⋊Dic5—32no—
91C2×C22⋊Dic5—31no—
92#4082—38no—
93C2×#683—32no—
94C2×#683—32no—
95C2×#683—32no—
96C2×C4×Dic5—31no—
97C2×C4×Dic5—32no—
98C5×#95C2×D312yes—
99#3963—32no—
100#3963—32no—
101#3963—32no—
102#3963—32no—
103#3963—32no—
104#2379—151noSylow
105C23×C3×C5—22no—
106C23×C3×C5—21no—
107C23×C3×C5—22no—
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—
140Dic30—44no—
141C22×C3×D4C2×D511yescharacteristic
142C23×C3×C4C2×D511yescharacteristic
143C23×Dic3—51no—
144C22×Dic6—51no—
145C2×C3×C22⋊C4C2×D511yescharacteristic
146C2×C3×C22⋊C4C2×D512yes—
147C2×#217—52no—
148C2×#217—52no—
149C2×#217—52no—
150C2×#217—51no—
151#936—58no—
152C2×Dic3⋊C4—52no—
153C2×Dic3⋊C4—52no—
154C2×Dic3⋊C4—52no—
155C2×C4×Dic3—51no—
156C2×C4×Dic3—52no—
157C3×#95C2×D512yes—
158C3⋊#95—52no—
159C3⋊#95—52no—
160C3⋊#95—52no—
161C3⋊#95—52no—
162C3⋊#95—52no—
163C24×C5C22×D311yescharacteristic
164C24×C5C22×D311yescharacteristic
165C2×C5×D4—41no—
166C2×C5×D4—41no—
167C2×C5×D4C4×D314yes—
168C22×C4×C5—41no—
169C22×C4×C5—42no—
170C22×C4×C5C22×D312yes—
171C22×C4×C5C22×D312yes—
172C22×C4×C5C22×D311yescharacteristic
173C22×Dic5—32no—
174C22×Dic5—32no—
175C22×Dic5—32no—
176C22×Dic5—32no—
177C22×Dic5—121no—
178C22×Dic5—122no—
179C2×Dic10—34no—
180C2×Dic10—121no—
181C2×Dic10—121no—
182C5×C22⋊C4—41no—
183C5×C22⋊C4—42no—
184C22⋊Dic5—38no—
185C22⋊Dic5—38no—
186C22⋊Dic5—121no—
187C22⋊Dic5—122no—
188#683—122no—
189#683—122no—
190#683—122no—
191C4×Dic5—34no—
192C4×Dic5—122no—
193C2×#124—151no—
194#391—152no—
195#388—152no—
196#388—152no—
197#386—151no—
198#386D1511yescharacteristic
199#385—152no—
200#385—152no—
201#382—152no—
202C22×C3×C5—22no—
203C22×C3×C5—22no—
204C22×C3×C5—24no—
205C22×C3×C5—24no—
206C22×C3×C5—41no—
207C22×C3×C5—41no—
208C22×C3×C5—42no—
209C22×C3×C5—42no—
210C22×C3×C5—41no—
211C22×C3×C5—41no—
212C22×C3×C5C23×C411yescharacteristic
213C22×C3×C5H(5,𝔽2)11yescharacteristic
214C22×C3×C5H(5,𝔽2)11yescharacteristic
215C22×C3×C5H(5,𝔽2)12yes—
216C22×C3×C52−1+412yes—
217C3×C4×C5—22no—
218C3×C4×C5—42no—
219C3×C4×C5—42no—
220Dic15—24no—
221Dic15—24no—
222Dic15—42no—
223Dic15—42no—
224C24×C3C22×D511yescharacteristic
225C24×C3C22×D511yescharacteristic
226C2×C3×D4—41no—
227C2×C3×D4—41no—
228C2×C3×D4C4×D514yes—
229C22×C3×C4—41no—
230C22×C3×C4—42no—
231C22×C3×C4C22×D512yes—
232C22×C3×C4C22×D512yes—
233C22×C3×C4C22×D511yescharacteristic
234C22×Dic3—52no—
235C22×Dic3—52no—
236C22×Dic3—52no—
237C22×Dic3—52no—
238C22×Dic3—201no—
239C22×Dic3—202no—
240C2×Dic6—54no—
241C2×Dic6—201no—
242C2×Dic6—201no—
243C3×C22⋊C4—41no—
244C3×C22⋊C4—42no—
245#217—58no—
246#217—58no—
247#217—201no—
248#217—202no—
249Dic3⋊C4—202no—
250Dic3⋊C4—202no—
251Dic3⋊C4—202no—
252C4×Dic3—54no—
253C4×Dic3—202no—
254C23×C5—22no—
255C23×C5—21no—
256C23×C5—22no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5—22no—
261C23×C5—22no—
262C23×C5C2×C4×D312yes—
263C23×C5C2×C4×D312yes—
264C23×C5C23×D311yescharacteristic
265C5×D4—44no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—44no—
275C2×C4×C5C2×C4×D312yes—
276C2×Dic5—34no—
277C2×Dic5—34no—
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—62no—
288C2×Dic5—124no—
289Dic10—124no—
290C22×Q8—151no—
291C22×D4C2×D1511yescharacteristic
292C23×C4—151no—
293C23×C4C2×D1511yescharacteristic
294#124—158no—
295C2×C4⋊C4—152no—
296C2×C4⋊C4—152no—
297C2×C4⋊C4—152no—
298C2×C22⋊C4—152no—
299C2×C22⋊C4—152no—
300C2×C22⋊C4—152no—
301C2×C22⋊C4—151no—
302C2×C22⋊C4C2×D1512yes—
303C2×C22⋊C4C2×D1511yescharacteristic
304C2×C42—151no—
305C2×C42—152no—
306#95—152no—
307#95—152no—
308#95—152no—
309#95—152no—
310#95—152no—
311#95C2×D1512yes—
312C2×C3×C5—42no—
313C2×C3×C5—42no—
314C2×C3×C5#51811yescharacteristic
315C2×C3×C5#52012yes—
316C2×C3×C5H(3,𝔽4)11yescharacteristic
317C2×C3×C5#56212yes—
318C2×C3×C5#56311yescharacteristic
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3C2×C4×D512yes—
328C23×C3C2×C4×D512yes—
329C23×C3C23×D511yescharacteristic
330C3×D4—44no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—44no—
340C2×C3×C4C2×C4×D512yes—
341C2×Dic3—54no—
342C2×Dic3—54no—
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—102no—
352C2×Dic3—102no—
353C2×Dic3—204no—
354Dic6—204no—
355C22×C5—22no—
356C22×C5—22no—
357C22×C5—24no—
358C22×C5—24no—
359C22×C5—41no—
360C22×C5—41no—
361C22×C5—41no—
362C22×C5—41no—
363C22×C5—42no—
364C22×C5—42no—
365C22×C5C22×C4×D311yescharacteristic
366C22×C5#100411yescharacteristic
367C22×C5#100411yescharacteristic
368C22×C5#100412yes—
369C22×C5#101012yes—
370C4×C5—22no—
371C4×C5—42no—
372C4×C5—42no—
373Dic5—64no—
374Dic5—64no—
375Dic5—122no—
376Dic5—122no—
377C24C22×D1511yescharacteristic
378C24C22×D1511yescharacteristic
379C2×Q8—154no—
380C2×Q8—601no—
381C2×Q8—601no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4C4×D1514yes—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—152no—
388C22×C4—152no—
389C22×C4—152no—
390C22×C4—152no—
391C22×C4—602no—
392C22×C4—601no—
393C22×C4C22×D1512yes—
394C22×C4C22×D1512yes—
395C22×C4C22×D1511yescharacteristic
396C4⋊C4—602no—
397C4⋊C4—602no—
398C4⋊C4—602no—
399C22⋊C4—41no—
400C22⋊C4—42no—
401C22⋊C4—158no—
402C22⋊C4—158no—
403C22⋊C4—601no—
404C22⋊C4—602no—
405C42—154no—
406C42—602no—
407C3×C5#237911yescharacteristic, complemented, Hall
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—24no—
411C22×C3—24no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3C22×C4×D511yescharacteristic
419C22×C3#414411yescharacteristic
420C22×C3#414411yescharacteristic
421C22×C3#414412yes—
422C22×C3#415012yes—
423C3×C4—22no—
424C3×C4—42no—
425C3×C4—42no—
426Dic3—104no—
427Dic3—104no—
428Dic3—202no—
429Dic3—202no—
430C2×C5—42no—
431C2×C5—42no—
432C2×C5#548611yescharacteristic
433C2×C5#554612yes—
434C2×C5#559411yescharacteristic
435C2×C5#562112yes—
436C2×C5#565111yescharacteristic
437C23—22no—
438C23—21no—
439C23—22no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—22no—
445C23C2×C4×D1512yes—
446C23C2×C4×D1512yes—
447C23C23×D1511yescharacteristic
448Q8—604no—
449D4—44no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—44no—
459C2×C4—154no—
460C2×C4—154no—
461C2×C4—302no—
462C2×C4—302no—
463C2×C4—302no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—302no—
467C2×C4—302no—
468C2×C4—302no—
469C2×C4—302no—
470C2×C4—302no—
471C2×C4—604no—
472C2×C4C2×C4×D1512yes—
473C2×C3—42no—
474C2×C3—42no—
475C2×C3#6582011yescharacteristic
476C2×C3#6588012yes—
477C2×C3#6592811yescharacteristic
478C2×C3#6595512yes—
479C2×C3#6598511yescharacteristic
480C5#7468411yescharacteristic, complemented, Sylow
481C22—24no—
482C22—24no—
483C22—22no—
484C22—22no—
485C22—41no—
486C22—41no—
487C22—42no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22C22×C4×D1511yescharacteristic
492C22#9209512yes—
493C22#9209511yescharacteristic
494C22#9209511yescharacteristic
495C22#9210112yes—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—304no—
500C4—304no—
501C4—602no—
502C4—602no—
503C3#1060641511yescharacteristic, complemented, Sylow
504C2—42no—
505C2—42no—
506C2#1175624511yescharacteristic
507C2#1175630512yes—
508C2#1175635311yescharacteristic
509C2#1175638012yes—
510C2#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#2379
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. #11751047C2semidirect
6. #11751046C2semidirect
7. #11751045C2semidirect
8. #11750751C2semidirect
9. #11750747C2semidirect
21. #91825C4semidirect
61. C2×Dic30C2×C4semidirect
66. #6719C2×C4semidirect
67. #6719C2×C4semidirect
72. C4×Dic15C2×C4semidirect
407. C3×C5#2379semidirect
480. C5#74684semidirect
503. C3#10606415semidirect

Maximal subgroups

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

subgroupindex
C2×#918252
C3×C5×#3862
#117513212
#117513112
#117510472
#117510462
#117510452
#117507512
#117507472
#106064153
#746845