Finite Group Databasebeta

← #14328004

Subgroups — #14328004

Order 1920.

1120 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×C4×#1220C211yesmaximal, characteristic
2C3×C4×C5×C22⋊C4C211yesmaximal, characteristic, complemented
3#11751312C211yesmaximal, characteristic, complemented
4#11751058C212yesmaximal
5#11751046C212yesmaximal, complemented
6#11751041C212yesmaximal, complemented
7C4×#6712C212yesmaximal, complemented
8#11750767C211yesmaximal, characteristic, complemented
9#11750765C211yesmaximal, characteristic, complemented
10#11750751C211yesmaximal, characteristic, complemented
11#11750750C211yesmaximal, characteristic, complemented
12#10605194—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×#1220C2211yescharacteristic
15C22×C4×D15C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2212yes—
17C2×C3×C42×C5C2212yes—
18C2×#6719C2212yes—
19C2×#6719C2211yescharacteristic
20C4×#1220C418yescomplemented
21C2×#6714C2212yes—
22C2×#6714C2212yes—
23C2×#6714C2211yescharacteristic
24C2×#6712C2212yes—
25C2×#6712C2212yes—
26C2×#6712C2211yescharacteristic
27C2×C4×Dic15C2212yes—
28C2×C4×Dic15C2212yes—
29C2×C4×Dic15C2211yescharacteristic
30C3×C5×#95C2211yescharacteristic
31C3×C5×#95C2211yescharacteristic
32#91115C2211yescharacteristic
33#91115C2211yescharacteristic
34#91115C2212yes—
35#91115C2212yes—
36#91115C2212yes—
37#73463—51nomaximal, Hall
38C2×C4×C22⋊D5—31no—
39C4×C5×C22⋊C4D311yescharacteristic, complemented
40#65500—31no—
41#65246—32no—
42#65234—32no—
43#65229—32no—
44C4×#683—32no—
45#64955—31no—
46#64953—31no—
47#64939—31no—
48#64938—31no—
49C24×C3×C5C2311yescharacteristic
50C23×D15C2311yescharacteristic
51C22×C3×C4×C5—42no—
52C22×C3×C4×C5C2×C414yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2312yes—
57C2×#1220—22no—
58C2×#1220—22no—
59C2×#1220C2×C414yes—
60C22×Dic15C2312yes—
61C22×Dic15C2312yes—
62C22×Dic15C2311yescharacteristic
63C22×Dic15C2311yescharacteristic
64C22×Dic15C2311yescharacteristic
65C2×C4×D15—42no—
66C2×C4×D15C2×C414yes—
67C3×C5×C22⋊C4—24no—
68C3×C42×C5—42no—
69#6719—42no—
70#6719C2×C414yes—
71#6714—24no—
72#6714—42no—
73#6714C2×C414yes—
74#6712—24no—
75#6712—24no—
76#6712C2×C414yes—
77C4×Dic15—24no—
78C4×Dic15—42no—
79C4×Dic15C2×C414yes—
80C2×C4×#67—51no—
81C3×C4×C22⋊C4D511yescharacteristic, complemented
82#5211—51no—
83#4957—52no—
84#4945—52no—
85#4940—52no—
86C4×Dic3⋊C4—52no—
87#4666—51no—
88#4664—51no—
89#4650—51no—
90#4649—51no—
91C23×C4×C5C2×D311yescharacteristic
92C22×C22⋊D5—31no—
93C22×C4×D5—31no—
94C2×C5×C22⋊C4C2×D312yes—
95C2×C42×C5C2×D312yes—
96C2×C22⋊Dic5—31no—
97C2×C22⋊Dic5—32no—
98C4×C22⋊D5—38no—
99C2×#685—31no—
100C2×#685—32no—
101C2×#685—32no—
102C2×#683—31no—
103C2×#683—32no—
104C2×#683—32no—
105C2×C4×Dic5—31no—
106C2×C4×Dic5—32no—
107C2×C4×Dic5—32no—
108C5×#95C2×D311yescharacteristic
109C5×#95C2×D311yescharacteristic
110#3963—31no—
111#3963—31no—
112#3963—32no—
113#3963—32no—
114#3963—32no—
115#2357—151noSylow
116C23×C3×C5—22no—
117C23×C3×C5—21no—
118C23×C3×C5—21no—
119C23×C3×C5—22no—
120C23×C3×C5C22×C412yes—
121C23×C3×C5C2411yescharacteristic
122C22×D15—22no—
123C22×D15—21no—
124C22×D15—21no—
125C22×D15—22no—
126C22×D15C22×C412yes—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—44no—
135C2×C3×C4×C5C22×C412yes—
136C2×C3×C4×C5C22×C412yes—
137C2×C3×C4×C5#4114yes—
138#1220—28no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141C2×Dic15—22no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15—22no—
147C2×Dic15C22×C412yes—
148C2×Dic15C22×C412yes—
149C2×Dic15C22×C412yes—
150C2×Dic15#4112yes—
151C2×Dic15#4112yes—
152C4×D15—44no—
153C23×C3×C4C2×D511yescharacteristic
154C22×#67—51no—
155C22×C4×D3—51no—
156C2×C3×C22⋊C4C2×D512yes—
157C2×C3×C42C2×D512yes—
158C2×#217—51no—
159C2×#217—52no—
160C4×#67—58no—
161C2×(C2×D3)⋊C4—51no—
162C2×(C2×D3)⋊C4—52no—
163C2×(C2×D3)⋊C4—52no—
164C2×Dic3⋊C4—51no—
165C2×Dic3⋊C4—52no—
166C2×Dic3⋊C4—52no—
167C2×C4×Dic3—51no—
168C2×C4×Dic3—52no—
169C2×C4×Dic3—52no—
170C3×#95C2×D511yescharacteristic
171C3×#95C2×D511yescharacteristic
172C3⋊#95—51no—
173C3⋊#95—51no—
174C3⋊#95—52no—
175C3⋊#95—52no—
176C3⋊#95—52no—
177C24×C5C22×D311yescharacteristic
178C23×D5—31no—
179C22×C4×C5—42no—
180C22×C4×C5C4×D314yes—
181C22×C4×C5C22×D312yes—
182C22×C4×C5C22×D311yescharacteristic
183C22×C4×C5C22×D311yescharacteristic
184C22×C4×C5C22×D312yes—
185C2×C22⋊D5—34no—
186C2×C22⋊D5—62no—
187C2×C22⋊D5—62no—
188C22×Dic5—31no—
189C22×Dic5—31no—
190C22×Dic5—31no—
191C22×Dic5—32no—
192C22×Dic5—32no—
193C2×C4×D5—34no—
194C2×C4×D5—122no—
195C5×C22⋊C4—24no—
196C42×C5—42no—
197C22⋊Dic5—34no—
198C22⋊Dic5—122no—
199#685—34no—
200#685—64no—
201#685—122no—
202#683—34no—
203#683—64no—
204#683—64no—
205C4×Dic5—34no—
206C4×Dic5—64no—
207C4×Dic5—122no—
208C2×C4×D4—151no—
209#390—152no—
210#388—152no—
211#388—152no—
212#382—151no—
213#381—151no—
214#381—151no—
215C4×C4⋊C4—152no—
216C4×C4⋊C4—151no—
217C4×C22⋊C4—151no—
218C4×C22⋊C4D1511yescharacteristic, complemented
219C22×C3×C5—22no—
220C22×C3×C5—22no—
221C22×C3×C5—22no—
222C22×C3×C5—42no—
223C22×C3×C5—42no—
224C22×C3×C5C23×C411yescharacteristic
225C22×C3×C5C2×#4112yes—
226C22×C3×C5C2×#4111yescharacteristic
227C22×C3×C5C2×#4111yescharacteristic
228C22×C3×C5H(5,𝔽2)12yes—
229C2×D15—22no—
230C2×D15—22no—
231C2×D15—22no—
232C2×D15—42no—
233C2×D15—42no—
234C3×C4×C5—22no—
235C3×C4×C5—22no—
236C3×C4×C5—24no—
237C3×C4×C5—42no—
238Dic15—22no—
239Dic15—22no—
240Dic15—42no—
241Dic15—42no—
242Dic15#11714yes—
243C24×C3C22×D511yescharacteristic
244C23×D3—51no—
245C22×C3×C4—42no—
246C22×C3×C4C4×D514yes—
247C22×C3×C4C22×D512yes—
248C22×C3×C4C22×D511yescharacteristic
249C22×C3×C4C22×D511yescharacteristic
250C22×C3×C4C22×D512yes—
251C2×#67—54no—
252C2×#67—102no—
253C2×#67—102no—
254C22×Dic3—51no—
255C22×Dic3—51no—
256C22×Dic3—51no—
257C22×Dic3—52no—
258C22×Dic3—52no—
259C2×C4×D3—54no—
260C2×C4×D3—202no—
261C3×C22⋊C4—24no—
262C3×C42—42no—
263#217—54no—
264#217—202no—
265(C2×D3)⋊C4—54no—
266(C2×D3)⋊C4—104no—
267(C2×D3)⋊C4—202no—
268Dic3⋊C4—54no—
269Dic3⋊C4—104no—
270Dic3⋊C4—104no—
271C4×Dic3—54no—
272C4×Dic3—104no—
273C4×Dic3—202no—
274C23×C5—22no—
275C23×C5—22no—
276C23×C5—21no—
277C23×C5—21no—
278C23×C5C2×C4×D312yes—
279C23×C5C23×D311yescharacteristic
280C22×D5—32no—
281C22×D5—62no—
282C22×D5—62no—
283C22×D5—61no—
284C22×D5—61no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—44no—
293C2×C4×C5C2×C4×D312yes—
294C2×C4×C5C2×C4×D312yes—
295C2×C4×C5#23514yes—
296C22⋊D5—68no—
297C2×Dic5—32no—
298C2×Dic5—32no—
299C2×Dic5—32no—
300C2×Dic5—32no—
301C2×Dic5—32no—
302C2×Dic5—62no—
303C2×Dic5—62no—
304C2×Dic5—62no—
305C2×Dic5—62no—
306C2×Dic5—62no—
307C2×Dic5—62no—
308C2×Dic5—62no—
309C2×Dic5—62no—
310C4×D5—124no—
311C22×D4—151no—
312C23×C4—151no—
313C23×C4C2×D1511yescharacteristic
314C4×D4—158no—
315C2×C4⋊C4—151no—
316C2×C4⋊C4—152no—
317C2×C4⋊C4—152no—
318C2×C22⋊C4—151no—
319C2×C22⋊C4—151no—
320C2×C22⋊C4—152no—
321C2×C22⋊C4—152no—
322C2×C22⋊C4—152no—
323C2×C22⋊C4C2×D1512yes—
324C2×C42—151no—
325C2×C42—152no—
326C2×C42—152no—
327C2×C42C2×D1512yes—
328#95—151no—
329#95—151no—
330#95—152no—
331#95—152no—
332#95—152no—
333#95C2×D1511yescharacteristic
334#95C2×D1511yescharacteristic
335C2×C3×C5—42no—
336C2×C3×C5C2×#11711yescharacteristic
337C2×C3×C5C4×#4111yescharacteristic
338C2×C3×C5#51811yescharacteristic
339C2×C3×C5#55312yes—
340C2×C3×C5#55912yes—
341D15—42no—
342C23×C3—22no—
343C23×C3—22no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3C2×C4×D512yes—
347C23×C3C23×D511yescharacteristic
348C22×D3—52no—
349C22×D3—102no—
350C22×D3—102no—
351C22×D3—101no—
352C22×D3—101no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—44no—
361C2×C3×C4C2×C4×D512yes—
362C2×C3×C4C2×C4×D512yes—
363C2×C3×C4#70914yes—
364#67—108no—
365C2×Dic3—52no—
366C2×Dic3—52no—
367C2×Dic3—52no—
368C2×Dic3—52no—
369C2×Dic3—52no—
370C2×Dic3—102no—
371C2×Dic3—102no—
372C2×Dic3—102no—
373C2×Dic3—102no—
374C2×Dic3—102no—
375C2×Dic3—102no—
376C2×Dic3—102no—
377C2×Dic3—102no—
378C4×D3—204no—
379C22×C5—22no—
380C22×C5—22no—
381C22×C5—22no—
382C22×C5—42no—
383C22×C5—42no—
384C22×C5C22×C4×D311yescharacteristic
385C22×C5C2×#23512yes—
386C22×C5#100412yes—
387C22×C5D3×#4111yescharacteristic
388C22×C5D3×#4111yescharacteristic
389C2×D5—62no—
390C2×D5—62no—
391C2×D5—62no—
392C2×D5—122no—
393C2×D5—122no—
394C4×C5—22no—
395C4×C5—22no—
396C4×C5—24no—
397C4×C5—42no—
398Dic5—34no—
399Dic5—62no—
400Dic5—62no—
401Dic5—122no—
402Dic5—122no—
403C24—151no—
404C24C22×D1511yescharacteristic
405C2×D4—154no—
406C2×D4—302no—
407C2×D4—302no—
408C22×C4—42no—
409C22×C4—151no—
410C22×C4—151no—
411C22×C4—151no—
412C22×C4—154no—
413C22×C4—152no—
414C22×C4—152no—
415C22×C4—602no—
416C22×C4C4×D1514yes—
417C22×C4C22×D1512yes—
418C22×C4C22×D1511yescharacteristic
419C22×C4C22×D1511yescharacteristic
420C22×C4C22×D1512yes—
421C4⋊C4—154no—
422C4⋊C4—304no—
423C4⋊C4—304no—
424C22⋊C4—24no—
425C22⋊C4—154no—
426C22⋊C4—154no—
427C22⋊C4—304no—
428C22⋊C4—602no—
429C22⋊C4—602no—
430C42—42no—
431C42—154no—
432C42—304no—
433C42—602no—
434C3×C5#235711yescharacteristic, complemented, Hall
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3C22×C4×D511yescharacteristic
441C22×C3C2×#70912yes—
442C22×C3#414412yes—
443C22×C3D5×#4111yescharacteristic
444C22×C3D5×#4111yescharacteristic
445C2×D3—102no—
446C2×D3—102no—
447C2×D3—102no—
448C2×D3—202no—
449C2×D3—202no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—24no—
453C3×C4—42no—
454Dic3—54no—
455Dic3—102no—
456Dic3—102no—
457Dic3—202no—
458Dic3—202no—
459C2×C5—42no—
460C2×C5C4×#23511yescharacteristic
461C2×C5#548611yescharacteristic
462C2×C5D3×#11711yescharacteristic
463C2×C5#554012yes—
464C2×C5#555912yes—
465D5—122no—
466C23—22no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—152no—
471C23—302no—
472C23—302no—
473C23—301no—
474C23—301no—
475C23C2×C4×D1512yes—
476C23C23×D1511yescharacteristic
477D4—308no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—44no—
486C2×C4—152no—
487C2×C4—152no—
488C2×C4—152no—
489C2×C4—152no—
490C2×C4—152no—
491C2×C4—302no—
492C2×C4—302no—
493C2×C4—302no—
494C2×C4—302no—
495C2×C4—302no—
496C2×C4—302no—
497C2×C4—302no—
498C2×C4—302no—
499C2×C4—604no—
500C2×C4C2×C4×D1512yes—
501C2×C4C2×C4×D1512yes—
502C2×C4#681714yes—
503C2×C3—42no—
504C2×C3C4×#70911yescharacteristic
505C2×C3#6582011yescharacteristic
506C2×C3D5×#11711yescharacteristic
507C2×C3#6587412yes—
508C2×C3#6589312yes—
509D3—202no—
510C5#7346311yescharacteristic, complemented, Sylow
511C22—22no—
512C22—22no—
513C22—22no—
514C22—42no—
515C22—42no—
516C22—302no—
517C22—302no—
518C22—302no—
519C22—602no—
520C22—602no—
521C22C22×C4×D1511yescharacteristic
522C22C2×#681712yes—
523C22#9209512yes—
524C22#41×D1511yescharacteristic
525C22#41×D1511yescharacteristic
526C4—24no—
527C4—22no—
528C4—22no—
529C4—42no—
530C4—154no—
531C4—302no—
532C4—302no—
533C4—602no—
534C4—602no—
535C3#1060519411yescharacteristic, complemented, Sylow
536C2—42no—
537C2—602no—
538C2C4×#681711yescharacteristic
539C2#1175624511yescharacteristic
540C2D15×#11711yescharacteristic
541C2#1175629912yes—
542C2#1175631812yes—
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C4×C5×C22⋊C4C2
socle, cosocleC23×C3×C5C24
prime radicalC22×#1220C22
hypercenterC4×C22⋊C4D15
hypocenterC3×C5#2357
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×C5×C22⋊C4C2semidirect
3. #11751312C2semidirect
5. #11751046C2semidirect
6. #11751041C2semidirect
7. C4×#6712C2semidirect
8. #11750767C2semidirect
9. #11750765C2semidirect
10. #11750751C2semidirect
11. #11750750C2semidirect
20. C4×#1220C4semidirect
39. C4×C5×C22⋊C4D3semidirect
81. C3×C4×C22⋊C4D5semidirect
218. C4×C22⋊C4D15semidirect
434. C3×C5#2357semidirect
510. C5#73463semidirect
535. C3#10605194semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#12202
C3×C4×C5×C22⋊C42
#117513122
#117510582
#117510462
#117510412
C4×#67122
#117507672
#117507652
#117507512
#117507502
#106051943
#734635