Finite Group Databasebeta

← #14296839

Subgroups — #14296839

Order 1920.

1760 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D4×D5C211yesmaximal, characteristic
2#11752459C212yesmaximal, complemented
3C2×#91240C212yesmaximal, complemented
4C2×#91228C212yesmaximal, complemented
5#10604941—31nomaximal, Hall
6C24×C3×D5C411yescharacteristic, complemented
7C24×C3×D5C2211yescharacteristic
8C22×C3×C5×D4C411yescharacteristic, complemented
9C22×C3×C22⋊D5C411yescharacteristic, complemented
10C22×C3×C22⋊D5C2211yescharacteristic
11C2×C3×D4×D5—24no—
12C2×C3×D4×D5—24no—
13C2×C3×D4×D5—24no—
14C22×C3×D20C2211yescharacteristic
15C22×C3×C4×D5C411yescharacteristic, complemented
16C2×#6763—24no—
17C2×#6763C2214yescomplemented
18#91240—24no—
19#91228—24no—
20#72130—51nomaximal, Hall
21C22×D4×D5D311yescharacteristic
22#65800—32no—
23C2×#4011—32no—
24C2×#3999—32no—
25C24×C3×C5C2×C411yescharacteristic
26C24×C3×C5C2×C411yescharacteristic
27C23×C3×D5—22no—
28C23×C3×D5—22no—
29C23×C3×D5—22no—
30C23×C3×D5—22no—
31C23×C3×D5—22no—
32C23×C3×D5—44no—
33C23×C3×D5—44no—
34C23×C3×D5C2×C411yescharacteristic
35C23×C3×D5C2×C411yescharacteristic
36C23×C3×D5D414yes—
37C23×C3×D5C2311yescharacteristic
38C22×ΓL(1,𝔽16)—24no—
39C2×C3×C5×D4—22no—
40C2×C3×C5×D4—22no—
41C2×C3×C5×D4—22no—
42C22×C3×C4×C5C2×C411yescharacteristic
43C2×C3×C22⋊D5—22no—
44C2×C3×C22⋊D5—22no—
45C2×C3×C22⋊D5—22no—
46C2×C3×C22⋊D5—22no—
47C2×C3×C22⋊D5—22no—
48C2×C3×C22⋊D5D414yes—
49C22×C3×Dic5C2×C411yescharacteristic
50C3×D4×D5—232no—
51C2×C3×D20—22no—
52C2×C3×D20—22no—
53C2×C3×D20D414yes—
54C2×C3×C4×D5—22no—
55C2×C3×C4×D5—22no—
56C2×C3×C4×D5—22no—
57#6763—24no—
58#6763—24no—
59#6763—48no—
60C23×C3×D4—51no—
61#5247—52no—
62C2×#834—52no—
63C2×#834—52no—
64C24×D5Dic311yescharacteristic, complemented
65C24×D5C2×D311yescharacteristic
66C22×C5×D4Dic311yescharacteristic, complemented
67C22×C22⋊D5Dic311yescharacteristic, complemented
68C22×C22⋊D5C2×D311yescharacteristic
69C2×D4×D5—24no—
70C2×D4×D5—24no—
71C2×D4×D5—24no—
72C22×D20C2×D311yescharacteristic
73C22×C4×D5Dic311yescharacteristic, complemented
74C2×C22⋊AGL(1,𝔽5)—34no—
75C2×C22⋊AGL(1,𝔽5)—64no—
76#4011—64no—
77#3999—64no—
78#1781—151noSylow
79C23×C3×C5—21no—
80C23×C3×C5—21no—
81C23×C3×C5—21no—
82C23×C3×C5—21no—
83C23×C3×C5—42no—
84C23×C3×C5—42no—
85C23×C3×C5C22⋊C412yes—
86C23×C3×C5C22⋊C412yescomplemented
87C23×C3×C5C22×C411yescharacteristic
88C22×C3×D5—21no—
89C22×C3×D5—21no—
90C22×C3×D5—24no—
91C22×C3×D5—28no—
92C22×C3×D5—21no—
93C22×C3×D5—21no—
94C22×C3×D5—28no—
95C22×C3×D5—28no—
96C22×C3×D5—44no—
97C22×C3×D5—42no—
98C22×C3×D5—48no—
99C22×C3×D5—48no—
100C22×C3×D5—48no—
101C22×C3×D5—48no—
102C22×C3×D5—44no—
103C22×C3×D5—44no—
104C22×C3×D5—42no—
105C22×C3×D5C22⋊C412yes—
106C22×C3×D5C22⋊C412yescomplemented
107C22×C3×D5C2×D412yes—
108C22×C3×D5C2×D412yes—
109C22×C3×D5C2×D412yes—
110C2×ΓL(1,𝔽16)—48no—
111C2×ΓL(1,𝔽16)—44no—
112C3×C5×D4—28no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5C22⋊C412yescomplemented
116C3×C22⋊D5—28no—
117C3×C22⋊D5—28no—
118C2×C3×Dic5—21no—
119C2×C3×Dic5—21no—
120C2×C3×Dic5C22⋊C412yescomplemented
121C3×D20—28no—
122C3×C4×D5—28no—
123C25×C3—51no—
124C25×C3—51no—
125C22×C3×D4—51no—
126C22×C3×D4—51no—
127C22×C3×D4—51no—
128C22×C3×D4—104no—
129C22×C3×D4—104no—
130C22×C3×D4—104no—
131C22×C3×D4AGL(1,𝔽5)11yescharacteristic, complemented
132C23×C3×C4—51no—
133C2×#217—54no—
134C2×#217—104no—
135#834—104no—
136#834—104no—
137C24×C5C2×Dic311yescharacteristic
138C24×C5C2×Dic311yescharacteristic
139C23×D5—22no—
140C23×D5—22no—
141C23×D5—22no—
142C23×D5—22no—
143C23×D5—22no—
144C23×D5—44no—
145C23×D5—44no—
146C23×D5C2×Dic311yescharacteristic
147C23×D5C2×Dic311yescharacteristic
148C23×D5#6714yes—
149C23×D5C22×D311yescharacteristic
150C22×AGL(1,𝔽5)—64no—
151C2×C5×D4—22no—
152C2×C5×D4—22no—
153C2×C5×D4—22no—
154C22×C4×C5C2×Dic311yescharacteristic
155C2×C22⋊D5—22no—
156C2×C22⋊D5—22no—
157C2×C22⋊D5—22no—
158C2×C22⋊D5—22no—
159C2×C22⋊D5—22no—
160C2×C22⋊D5#6714yes—
161C22×Dic5C2×Dic311yescharacteristic
162D4×D5—232no—
163C2×D20—22no—
164C2×D20—22no—
165C2×D20#6714yes—
166C2×C4×D5—22no—
167C2×C4×D5—22no—
168C2×C4×D5—22no—
169C22⋊AGL(1,𝔽5)—64no—
170C22⋊AGL(1,𝔽5)—64no—
171C22⋊AGL(1,𝔽5)—128no—
172C23×D4—51no—
173C2×#99—152no—
174C2×#99—152no—
175#379—152no—
176C22×C3×C5—22no—
177C22×C3×C5—22no—
178C22×C3×C5—22no—
179C22×C3×C5—24no—
180C22×C3×C5—42no—
181C22×C3×C5—42no—
182C22×C3×C5—42no—
183C22×C3×C5—42no—
184C22×C3×C5—42no—
185C22×C3×C5C2×C22⋊C411yescharacteristic
186C22×C3×C5C2×C22⋊C411yescharacteristic
187C22×C3×C5C2×C22⋊C411yescharacteristic
188C2×C3×D5—24no—
189C2×C3×D5—24no—
190C2×C3×D5—22no—
191C2×C3×D5—24no—
192C2×C3×D5—22no—
193C2×C3×D5—22no—
194C2×C3×D5—42no—
195C2×C3×D5—42no—
196C2×C3×D5—416no—
197C2×C3×D5—416no—
198C2×C3×D5—42no—
199C2×C3×D5—42no—
200C2×C3×D5—42no—
201C2×C3×D5#9914yescomplemented
202C2×C3×D5#12014yes—
203ΓL(1,𝔽16)—84no—
204C3×C4×C5—22no—
205C3×Dic5—22no—
206C24×C3—51no—
207C24×C3—51no—
208C24×C3—54no—
209C24×C3—51no—
210C24×C3—102no—
211C24×C3—102no—
212C24×C3—102no—
213C24×C3—102no—
214C24×C3—102no—
215C24×C3—204no—
216C24×C3—204no—
217C24×C3C2×AGL(1,𝔽5)11yescharacteristic
218C24×C3C2×AGL(1,𝔽5)11yescharacteristic
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C2×C3×D4—54no—
223C2×C3×D4—54no—
224C2×C3×D4—102no—
225C2×C3×D4—102no—
226C2×C3×D4—102no—
227C2×C3×D4—1032no—
228C2×C3×D4—102no—
229C2×C3×D4—102no—
230C2×C3×D4—102no—
231C2×C3×D4—102no—
232C22×C3×C4—51no—
233C22×C3×C4—102no—
234C22×C3×C4—102no—
235C22×C3×C4—102no—
236C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
237C22×Dic3—104no—
238#217—104no—
239#217—104no—
240#217—208no—
241C23×C5—21no—
242C23×C5—21no—
243C23×C5—21no—
244C23×C5—21no—
245C23×C5—42no—
246C23×C5—42no—
247C23×C5#21712yes—
248C23×C5#21712yescomplemented
249C23×C5C22×Dic311yescharacteristic
250C22×D5—21no—
251C22×D5—21no—
252C22×D5—24no—
253C22×D5—28no—
254C22×D5—21no—
255C22×D5—21no—
256C22×D5—28no—
257C22×D5—28no—
258C22×D5—44no—
259C22×D5—42no—
260C22×D5—48no—
261C22×D5—48no—
262C22×D5—48no—
263C22×D5—48no—
264C22×D5—44no—
265C22×D5—44no—
266C22×D5—42no—
267C22×D5#21712yes—
268C22×D5#21712yescomplemented
269C22×D5C2×#6712yes—
270C22×D5C2×#6712yes—
271C22×D5C2×#6712yes—
272C2×AGL(1,𝔽5)—128no—
273C2×AGL(1,𝔽5)—124no—
274C5×D4—28no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5#21712yescomplemented
278C22⋊D5—28no—
279C22⋊D5—28no—
280C2×Dic5—21no—
281C2×Dic5—21no—
282C2×Dic5#21712yescomplemented
283D20—28no—
284C4×D5—28no—
285C25—51no—
286C25—51no—
287C22×D4—51no—
288C22×D4—51no—
289C22×D4—51no—
290C22×D4—104no—
291C22×D4—104no—
292C22×D4—104no—
293C22×D4ΓL(1,𝔽16)11yescharacteristic, complemented
294C23×C4—51no—
295C2×C22⋊C4—154no—
296C2×C22⋊C4—304no—
297#99—304no—
298#99—304no—
299C2×C3×C5—21no—
300C2×C3×C5—21no—
301C2×C3×C5—42no—
302C2×C3×C5—42no—
303C2×C3×C5#37911yescharacteristic
304C2×C3×C5C2×#9912yes—
305C3×D5—24no—
306C3×D5—24no—
307C3×D5—42no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—52no—
315C23×C3—52no—
316C23×C3—52no—
317C23×C3—52no—
318C23×C3—52no—
319C23×C3—104no—
320C23×C3—108no—
321C23×C3—108no—
322C23×C3—108no—
323C23×C3—101no—
324C23×C3—101no—
325C23×C3—101no—
326C23×C3—101no—
327C23×C3—202no—
328C23×C3—204no—
329C23×C3—202no—
330C23×C3—204no—
331C23×C3—208no—
332C23×C3—208no—
333C23×C3—208no—
334C23×C3—208no—
335C23×C3—204no—
336C23×C3C22⋊AGL(1,𝔽5)12yes—
337C23×C3C22⋊AGL(1,𝔽5)12yescomplemented
338C23×C3C22×AGL(1,𝔽5)11yescharacteristic
339C3×D4—28no—
340C3×D4—108no—
341C3×D4—108no—
342C3×D4—108no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—52no—
346C2×C3×C4—108no—
347C2×C3×C4—101no—
348C2×C3×C4—101no—
349C2×C3×C4C22⋊AGL(1,𝔽5)12yescomplemented
350C2×Dic3—204no—
351C2×Dic3—208no—
352C22×C5—22no—
353C22×C5—24no—
354C22×C5—22no—
355C22×C5—22no—
356C22×C5—42no—
357C22×C5—42no—
358C22×C5—42no—
359C22×C5—42no—
360C22×C5—42no—
361C22×C5C2×#21711yescharacteristic
362C22×C5C2×#21711yescharacteristic
363C22×C5C2×#21711yescharacteristic
364C2×D5—24no—
365C2×D5—24no—
366C2×D5—24no—
367C2×D5—22no—
368C2×D5—22no—
369C2×D5—22no—
370C2×D5—42no—
371C2×D5—42no—
372C2×D5—416no—
373C2×D5—416no—
374C2×D5—42no—
375C2×D5—42no—
376C2×D5—42no—
377C2×D5#83414yescomplemented
378C2×D5#95314yes—
379AGL(1,𝔽5)—244no—
380C4×C5—22no—
381Dic5—22no—
382C24—51no—
383C24—51no—
384C24—54no—
385C24—51no—
386C24—102no—
387C24—102no—
388C24—102no—
389C24—102no—
390C24—102no—
391C24—204no—
392C24—204no—
393C24C2×ΓL(1,𝔽16)11yescharacteristic
394C24C2×ΓL(1,𝔽16)11yescharacteristic
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—54no—
399C2×D4—54no—
400C2×D4—102no—
401C2×D4—102no—
402C2×D4—102no—
403C2×D4—1032no—
404C2×D4—102no—
405C2×D4—102no—
406C2×D4—102no—
407C2×D4—102no—
408C22×C4—51no—
409C22×C4—102no—
410C22×C4—102no—
411C22×C4—102no—
412C22×C4—304no—
413C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
414C22⋊C4—304no—
415C22⋊C4—304no—
416C22⋊C4—608no—
417C3×C5#178111yescharacteristic, complemented, Hall
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—24no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—54no—
428C22×C3—54no—
429C22×C3—104no—
430C22×C3—104no—
431C22×C3—102no—
432C22×C3—104no—
433C22×C3—102no—
434C22×C3—102no—
435C22×C3—202no—
436C22×C3—202no—
437C22×C3—202no—
438C22×C3—202no—
439C22×C3—2016no—
440C22×C3—2016no—
441C22×C3—202no—
442C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
443C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
444C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
445C3×C4—22no—
446C3×C4—102no—
447Dic3—404no—
448C2×C5—21no—
449C2×C5—21no—
450C2×C5—42no—
451C2×C5—42no—
452C2×C5C2×#83412yes—
453C2×C5#524711yescharacteristic
454D5—24no—
455D5—24no—
456D5—42no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—42no—
462C23—42no—
463C23—52no—
464C23—52no—
465C23—52no—
466C23—52no—
467C23—52no—
468C23—104no—
469C23—108no—
470C23—108no—
471C23—108no—
472C23—101no—
473C23—101no—
474C23—101no—
475C23—101no—
476C23—202no—
477C23—204no—
478C23—202no—
479C23—204no—
480C23—208no—
481C23—208no—
482C23—208no—
483C23—208no—
484C23—204no—
485C23#676312yes—
486C23#676312yescomplemented
487C23C22×ΓL(1,𝔽16)11yescharacteristic
488D4—28no—
489D4—108no—
490D4—108no—
491D4—108no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—52no—
495C2×C4—108no—
496C2×C4—101no—
497C2×C4—101no—
498C2×C4—604no—
499C2×C4—608no—
500C2×C4#676312yescomplemented
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—42no—
504C2×C3—42no—
505C2×C3—104no—
506C2×C3—104no—
507C2×C3—202no—
508C2×C3D5⋊#9912yes—
509C2×C3#6580011yescharacteristic
510C5#7213011yescharacteristic, complemented, Sylow
511C22—22no—
512C22—22no—
513C22—22no—
514C22—24no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—54no—
521C22—54no—
522C22—104no—
523C22—104no—
524C22—102no—
525C22—104no—
526C22—102no—
527C22—102no—
528C22—202no—
529C22—202no—
530C22—202no—
531C22—202no—
532C22—2016no—
533C22—2016no—
534C22—202no—
535C22C2×#676311yescharacteristic
536C22C2×#676311yescharacteristic
537C22C2×#676311yescharacteristic
538C4—22no—
539C4—102no—
540C4—1204no—
541C3#1060494111yescharacteristic, complemented, Sylow
542C2—21no—
543C2—21no—
544C2—42no—
545C2—42no—
546C2—104no—
547C2—104no—
548C2—202no—
549C2D5⋊#83412yes—
550C2#1175245911yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC23×C3×C5C22×C4
FrattiniC23C22×ΓL(1,𝔽16)
FittingC22×C3×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D4×D5C2
cosocleC23×C3×D5C23
hypercenterC22×D4ΓL(1,𝔽16)
hypocenterC3×C5#1781
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. #11752459C2semidirect
3. C2×#91240C2semidirect
4. C2×#91228C2semidirect
6. C24×C3×D5C4semidirect
8. C22×C3×C5×D4C4semidirect
9. C22×C3×C22⋊D5C4semidirect
15. C22×C3×C4×D5C4semidirect
17. C2×#6763C22semidirect
64. C24×D5Dic3semidirect
66. C22×C5×D4Dic3semidirect
67. C22×C22⋊D5Dic3semidirect
73. C22×C4×D5Dic3semidirect
86. C23×C3×C5C22⋊C4semidirect
106. C22×C3×D5C22⋊C4semidirect
115. C2×C3×C4×C5C22⋊C4semidirect
120. C2×C3×Dic5C22⋊C4semidirect
131. C22×C3×D4AGL(1,𝔽5)semidirect
201. C2×C3×D5#99semidirect
248. C23×C5#217semidirect
268. C22×D5#217semidirect
277. C2×C4×C5#217semidirect
282. C2×Dic5#217semidirect
293. C22×D4ΓL(1,𝔽16)semidirect
337. C23×C3C22⋊AGL(1,𝔽5)semidirect
349. C2×C3×C4C22⋊AGL(1,𝔽5)semidirect
377. C2×D5#834semidirect
417. C3×C5#1781semidirect
486. C23#6763semidirect
500. C2×C4#6763semidirect
510. C5#72130semidirect
541. C3#10604941semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D4×D52
#117524592
C2×#912402
C2×#912282
#106049413
#721305