Finite Group Databasebeta

← #14322638

Subgroups — #14322638

Order 1920.

1344 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×(C2×D3)⋊C4C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4#11752028C212yesmaximal, complemented
5#11752025C212yesmaximal
6#11751856C214yesmaximal, complemented
7#11751849C214yesmaximal, complemented
8#10617070—31nomaximal, Hall
9C23×D3×D5C2211yescharacteristic
10C22×C3×C4×D5C2211yescharacteristic
11C22×D5×Dic3C2211yescharacteristic
12C2×#6763C2212yes—
13C2×C4×ΓL(1,𝔽16)C2212yescomplemented
14C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
15C2×C3×C4×AGL(1,𝔽5)C2212yescomplemented
16C2×Dic3⋊AGL(1,𝔽5)C2214yescomplemented
17C2×(C2×D3)⋊AGL(1,𝔽5)C2214yes—
18C2×(C2×D3)⋊AGL(1,𝔽5)C2214yes—
19C2×#6714C411yescharacteristic, complemented
20C2×C5×(C2×D3)⋊C4C411yescharacteristic, complemented
21C2×#6667C411yescharacteristic, complemented
22C2×(C2×D3)⋊Dic5C411yescharacteristic, complemented
23D5×(C2×D3)⋊C4C412yescomplemented
24D5×(C2×D3)⋊C4C412yescomplemented
25D5×(C2×D3)⋊C4C2212yes—
26D5×(C2×D3)⋊C4C2212yes—
27#91235C2212yescomplemented
28C3×(C2×AGL(1,𝔽5))⋊C4C2212yescomplemented
29(C2×AGL(1,𝔽5))⋊Dic3C2214yescomplemented
30#73567—51nomaximal, Hall
31C2×D5×C22⋊C4—31no—
32#65800—32no—
33#65765—32no—
34C4×C22⋊AGL(1,𝔽5)—31no—
35C4×C22⋊AGL(1,𝔽5)D311yescharacteristic, complemented
36#65700—34no—
37#65700—34no—
38C23×D15C2×C411yescharacteristic
39C23×C5×D3C2×C411yescharacteristic
40C23×C3×D5C2311yescharacteristic
41C22×D3×D5—24no—
42C22×D3×D5—42no—
43C22×D3×D5—42no—
44C22×D3×D5C2×C412yes—
45C22×D3×D5C2312yes—
46C22×ΓL(1,𝔽16)C2314yes—
47C22×C3×AGL(1,𝔽5)C2314yes—
48C22×C3×C4×C5C2×C411yescharacteristic
49C22×Dic15C2×C411yescharacteristic
50C22×C5×Dic3C2×C411yescharacteristic
51C22×C3×Dic5C2×C411yescharacteristic
52C2×C3×C4×D5—42no—
53C2×C3×C4×D5C2×C411yescharacteristic
54C2×C3×C4×D5C2×C411yescharacteristic
55C2×C3×C4×D5C2311yescharacteristic
56C2×C3×C4×D5C2311yescharacteristic
57C2×D5×Dic3—42no—
58C2×D5×Dic3C2×C412yes—
59C2×D5×Dic3C2312yes—
60#6763—24no—
61C4×ΓL(1,𝔽16)—42no—
62C3×C22⋊AGL(1,𝔽5)—24no—
63C3×C4×AGL(1,𝔽5)—42no—
64Dic3⋊AGL(1,𝔽5)—44no—
65(C2×D3)⋊AGL(1,𝔽5)—44no—
66(C2×D3)⋊AGL(1,𝔽5)—44no—
67#6714C2×C412yes—
68#6714C2×C412yes—
69C5×(C2×D3)⋊C4C2×C412yes—
70C5×(C2×D3)⋊C4C2×C412yes—
71#6667C2×C412yes—
72#6667C2×C412yes—
73(C2×D3)⋊Dic5C2×C412yes—
74(C2×D3)⋊Dic5C2×C412yes—
75C22×(C2×D3)⋊C4—51no—
76C3×C4×C22⋊C4—51no—
77C4×#217—51no—
78#4955—52no—
79#4942—52no—
80#4940—54no—
81#4669—54no—
82C24×D5—31no—
83C22×C4×D5—31no—
84C22×C4×D5C2×D311yescharacteristic
85C2×C22⋊AGL(1,𝔽5)—32no—
86C2×C22⋊AGL(1,𝔽5)—34no—
87C2×C22⋊AGL(1,𝔽5)—34no—
88C2×C22⋊AGL(1,𝔽5)C2×D312yes—
89C2×C4⋊AGL(1,𝔽5)—34no—
90C2×C4×AGL(1,𝔽5)—32no—
91C2×C4×AGL(1,𝔽5)C2×D312yescomplemented
92C2×C5×C22⋊C4—31no—
93C2×C22⋊Dic5—31no—
94C2×#685—31no—
95C2×#685—31no—
96D5×C22⋊C4—32no—
97D5×C22⋊C4—32no—
98D5×C22⋊C4—32no—
99D5×C22⋊C4—32no—
100(C2×AGL(1,𝔽5))⋊C4—34no—
101(C2×AGL(1,𝔽5))⋊C4—32no—
102(C2×AGL(1,𝔽5))⋊C4C2×D312yescomplemented
103#2312—151noSylow
104C23×C3×C5C22×C411yescharacteristic
105C22×D15—22no—
106C22×D15—41no—
107C22×D15—41no—
108C22×D15C22×C412yes—
109C22×C5×D3—22no—
110C22×C5×D3—41no—
111C22×C5×D3—41no—
112C22×C5×D3C22×C412yes—
113C22×C3×D5—22no—
114C22×C3×D5—22no—
115C22×C3×D5C22×C411yescharacteristic
116C22×C3×D5#4114yes—
117C22×C3×D5C2411yescharacteristic
118C2×D3×D5—24no—
119C2×D3×D5—24no—
120C2×D3×D5—24no—
121C2×D3×D5—24no—
122C2×D3×D5—42no—
123C2×D3×D5—42no—
124C2×D3×D5—416no—
125C2×ΓL(1,𝔽16)—24no—
126C2×ΓL(1,𝔽16)—24no—
127C2×ΓL(1,𝔽16)—24no—
128C2×C3×AGL(1,𝔽5)—24no—
129C2×C3×AGL(1,𝔽5)—24no—
130C2×C3×AGL(1,𝔽5)—24no—
131C2×C3×C4×C5—41no—
132C2×C3×C4×C5C22×C411yescharacteristic
133C2×C3×C4×C5C22×C411yescharacteristic
134C2×Dic15—41no—
135C2×Dic15C22×C412yes—
136C2×C5×Dic3—41no—
137C2×C5×Dic3C22×C412yes—
138C2×C3×Dic5—41no—
139C2×C3×Dic5C22×C411yescharacteristic
140C2×C3×Dic5C22×C411yescharacteristic
141C3×C4×D5—22no—
142C3×C4×D5—22no—
143C3×C4×D5—41no—
144C3×C4×D5—41no—
145D5×Dic3—24no—
146D5×Dic3—42no—
147C24×D3—51no—
148C23×C3×C4—51no—
149C23×Dic3—51no—
150C2×C3×C22⋊C4—52no—
151C2×C3×C42—52no—
152C2×#217—52no—
153C2×(C2×D3)⋊C4—51no—
154C2×(C2×D3)⋊C4—51no—
155C2×(C2×D3)⋊C4—51no—
156C2×(C2×D3)⋊C4—52no—
157C2×(C2×D3)⋊C4—52no—
158C2×(C2×D3)⋊C4—52no—
159C2×(C2×D3)⋊C4—52no—
160C2×(C2×D3)⋊C4—54no—
161C2×(C2×D3)⋊C4—54no—
162C2×(C2×D3)⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
163C2×Dic3⋊C4—54no—
164C2×C4×Dic3—52no—
165C3×#95—52no—
166C3⋊#95—54no—
167C3⋊#95—52no—
168C24×C5—31no—
169C23×D5—31no—
170C23×D5—32no—
171C23×D5—32no—
172C23×D5—64no—
173C23×D5—122no—
174C23×D5—122no—
175C23×D5C22×D311yescharacteristic
176C22×AGL(1,𝔽5)—34no—
177C22×AGL(1,𝔽5)C22×D314yes—
178C22×C4×C5—31no—
179C22×C4×C5C4×D311yescharacteristic
180C22×Dic5—31no—
181C22×Dic5C4×D311yescharacteristic
182C2×C4×D5—32no—
183C2×C4×D5—32no—
184C2×C4×D5—42no—
185C2×C4×D5—122no—
186C2×C4×D5C4×D311yescharacteristic
187C2×C4×D5C4×D311yescharacteristic
188C2×C4×D5C22×D311yescharacteristic
189C2×C4×D5C22×D311yescharacteristic
190C22⋊AGL(1,𝔽5)—24no—
191C22⋊AGL(1,𝔽5)—64no—
192C22⋊AGL(1,𝔽5)—124no—
193C22⋊AGL(1,𝔽5)—124no—
194C4⋊AGL(1,𝔽5)—124no—
195C4×AGL(1,𝔽5)—42no—
196C4×AGL(1,𝔽5)—122no—
197C5×C22⋊C4—32no—
198C5×C22⋊C4—32no—
199C22⋊Dic5—32no—
200C22⋊Dic5—32no—
201#685—32no—
202#685—32no—
203#685—32no—
204#685—32no—
205C22×C22⋊C4—151no—
206#388—154no—
207#388—154no—
208#380—152no—
209#379—152no—
210C4×C22⋊C4—51no—
211C4×C22⋊C4—151no—
212C22×C3×C5—21no—
213C22×C3×C5—21no—
214C22×C3×C5#11712yes—
215C22×C3×C5C23×C411yescharacteristic
216C2×D15—22no—
217C2×D15—22no—
218C2×D15—22no—
219C2×D15—44no—
220C2×C5×D3—22no—
221C2×C5×D3—22no—
222C2×C5×D3—22no—
223C2×C5×D3—44no—
224C2×C3×D5—21no—
225C2×C3×D5—21no—
226C2×C3×D5—28no—
227C2×C3×D5#11712yes—
228C2×C3×D5C2×#4112yes—
229C2×C3×D5H(5,𝔽2)12yes—
230C2×C3×D5H(5,𝔽2)12yes—
231D3×D5—48no—
232D3×D5—48no—
233ΓL(1,𝔽16)—44no—
234C3×AGL(1,𝔽5)—44no—
235C3×C4×C5—21no—
236C3×C4×C5—21no—
237Dic15—22no—
238C3×Dic5—21no—
239C3×Dic5—21no—
240C5×Dic3—22no—
241C24×C3—51no—
242C23×D3—51no—
243C23×D3—52no—
244C23×D3—52no—
245C23×D3—104no—
246C23×D3—202no—
247C23×D3—202no—
248C23×D3C2×AGL(1,𝔽5)11yescharacteristic
249C22×C3×C4—51no—
250C22×C3×C4—51no—
251C22×C3×C4—51no—
252C22×C3×C4—51no—
253C22×C3×C4—51no—
254C22×C3×C4—54no—
255C22×C3×C4—202no—
256C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
257C22×Dic3—51no—
258C22×Dic3—52no—
259C22×Dic3—52no—
260C22×Dic3—54no—
261C22×Dic3—202no—
262C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
263C3×C22⋊C4—104no—
264C3×C42—202no—
265#217—104no—
266(C2×D3)⋊C4—52no—
267(C2×D3)⋊C4—52no—
268(C2×D3)⋊C4—52no—
269(C2×D3)⋊C4—52no—
270(C2×D3)⋊C4—52no—
271(C2×D3)⋊C4—52no—
272(C2×D3)⋊C4—204no—
273(C2×D3)⋊C4—204no—
274(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
275(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
276Dic3⋊C4—204no—
277C4×Dic3—202no—
278C23×C5—32no—
279C23×C5—62no—
280C23×C5—121no—
281C23×C5—121no—
282C23×C5C2×C4×D311yescharacteristic
283C22×D5—22no—
284C22×D5—22no—
285C22×D5—32no—
286C22×D5—62no—
287C22×D5—64no—
288C22×D5—64no—
289C22×D5—64no—
290C22×D5—64no—
291C22×D5—122no—
292C22×D5—122no—
293C22×D5—1216no—
294C22×D5—121no—
295C22×D5—121no—
296C22×D5C2×C4×D311yescharacteristic
297C22×D5#23514yes—
298C22×D5C23×D311yescharacteristic
299C2×AGL(1,𝔽5)—24no—
300C2×AGL(1,𝔽5)—24no—
301C2×AGL(1,𝔽5)—24no—
302C2×AGL(1,𝔽5)—64no—
303C2×AGL(1,𝔽5)—64no—
304C2×AGL(1,𝔽5)—64no—
305C2×C4×C5—32no—
306C2×C4×C5—41no—
307C2×C4×C5—121no—
308C2×C4×C5C2×C4×D311yescharacteristic
309C2×C4×C5C2×C4×D311yescharacteristic
310C2×Dic5—32no—
311C2×Dic5—41no—
312C2×Dic5—121no—
313C2×Dic5C2×C4×D311yescharacteristic
314C2×Dic5C2×C4×D311yescharacteristic
315C4×D5—22no—
316C4×D5—22no—
317C4×D5—41no—
318C4×D5—41no—
319C4×D5—64no—
320C4×D5—122no—
321C25—151no—
322C23×C4—51no—
323C23×C4—151no—
324C2×C4⋊C4—154no—
325C2×C22⋊C4—31no—
326C2×C22⋊C4—52no—
327C2×C22⋊C4—152no—
328C2×C22⋊C4—152no—
329C2×C22⋊C4—152no—
330C2×C22⋊C4—152no—
331C2×C22⋊C4—151no—
332C2×C22⋊C4—151no—
333C2×C22⋊C4—151no—
334C2×C22⋊C4—152no—
335C2×C22⋊C4—154no—
336C2×C22⋊C4—154no—
337C2×C42—52no—
338C2×C42—152no—
339#95—52no—
340#95—152no—
341#95—154no—
342C2×C3×C5—22no—
343C2×C3×C5C2×#11711yescharacteristic
344C2×C3×C5#51811yescharacteristic
345C2×C3×C5#51811yescharacteristic
346D15—42no—
347C3×D5—22no—
348C3×D5#53814yes—
349C5×D3—42no—
350C23×C3—51no—
351C23×C3—54no—
352C23×C3—51no—
353C23×C3—102no—
354C23×C3—102no—
355C23×C3C22×AGL(1,𝔽5)11yescharacteristic
356C22×D3—22no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—52no—
360C22×D3—104no—
361C22×D3—104no—
362C22×D3—104no—
363C22×D3—104no—
364C22×D3—102no—
365C22×D3—202no—
366C22×D3—202no—
367C22×D3—2016no—
368C22×D3—201no—
369C22×D3—201no—
370C22×D3C22×AGL(1,𝔽5)12yes—
371C2×C3×C4—41no—
372C2×C3×C4—51no—
373C2×C3×C4—51no—
374C2×C3×C4—102no—
375C2×C3×C4—102no—
376C2×C3×C4—104no—
377C2×C3×C4—104no—
378C2×C3×C4—104no—
379C2×C3×C4—201no—
380C2×C3×C4—201no—
381C2×C3×C4—201no—
382C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
383C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
384C2×Dic3—41no—
385C2×Dic3—52no—
386C2×Dic3—104no—
387C2×Dic3—104no—
388C2×Dic3—104no—
389C2×Dic3—104no—
390C2×Dic3—202no—
391C2×Dic3—201no—
392C2×Dic3C22×AGL(1,𝔽5)12yes—
393C22×C5—21no—
394C22×C5—21no—
395C22×C5—62no—
396C22×C5—62no—
397C22×C5—62no—
398C22×C5—124no—
399C22×C5#87212yes—
400C22×C5C22×C4×D311yescharacteristic
401C2×D5—21no—
402C2×D5—21no—
403C2×D5—28no—
404C2×D5—62no—
405C2×D5—62no—
406C2×D5—62no—
407C2×D5—128no—
408C2×D5—124no—
409C2×D5—128no—
410C2×D5#87212yes—
411C2×D5C2×#23512yes—
412C2×D5#100412yes—
413C2×D5#100912yes—
414AGL(1,𝔽5)—44no—
415AGL(1,𝔽5)—124no—
416C4×C5—21no—
417C4×C5—21no—
418C4×C5—62no—
419Dic5—21no—
420Dic5—21no—
421Dic5—62no—
422C24—31no—
423C24—51no—
424C24—152no—
425C24—152no—
426C24—151no—
427C24—304no—
428C24—602no—
429C24—602no—
430C22×C4—31no—
431C22×C4—51no—
432C22×C4—51no—
433C22×C4—51no—
434C22×C4—51no—
435C22×C4—51no—
436C22×C4—54no—
437C22×C4—152no—
438C22×C4—152no—
439C22×C4—151no—
440C22×C4—154no—
441C22×C4—202no—
442C22×C4—602no—
443C22×C4D3×AGL(1,𝔽5)11yescharacteristic
444C4⋊C4—604no—
445C22⋊C4—32no—
446C22⋊C4—32no—
447C22⋊C4—104no—
448C22⋊C4—152no—
449C22⋊C4—152no—
450C22⋊C4—152no—
451C22⋊C4—152no—
452C22⋊C4—152no—
453C22⋊C4—152no—
454C22⋊C4—304no—
455C22⋊C4—604no—
456C22⋊C4—604no—
457C42—202no—
458C42—602no—
459C3×C5#231211yescharacteristic, complemented, Hall
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—52no—
463C22×C3—52no—
464C22×C3—52no—
465C22×C3—52no—
466C22×C3—108no—
467C22×C3—101no—
468C22×C3—101no—
469C22×C3#413012yes—
470C22×C3C23×AGL(1,𝔽5)11yescharacteristic
471C2×D3—22no—
472C2×D3—22no—
473C2×D3—22no—
474C2×D3—44no—
475C2×D3—102no—
476C2×D3—102no—
477C2×D3—102no—
478C2×D3—208no—
479C2×D3—208no—
480C2×D3—204no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—101no—
484C3×C4—101no—
485C3×C4—204no—
486Dic3—22no—
487Dic3—102no—
488Dic3—204no—
489C2×C5—22no—
490C2×C5—122no—
491C2×C5C2×#87211yescharacteristic
492C2×C5#548611yescharacteristic
493C2×C5#552111yescharacteristic
494D5—22no—
495D5—122no—
496D5#556014yes—
497C23—32no—
498C23—51no—
499C23—54no—
500C23—51no—
501C23—62no—
502C23—102no—
503C23—102no—
504C23—121no—
505C23—121no—
506C23—152no—
507C23—304no—
508C23—304no—
509C23—304no—
510C23—304no—
511C23—302no—
512C23—601no—
513C23—601no—
514C23—602no—
515C23—602no—
516C23—6016no—
517C23C2×D3×AGL(1,𝔽5)11yescharacteristic
518C2×C4—32no—
519C2×C4—41no—
520C2×C4—51no—
521C2×C4—51no—
522C2×C4—102no—
523C2×C4—102no—
524C2×C4—104no—
525C2×C4—104no—
526C2×C4—104no—
527C2×C4—121no—
528C2×C4—152no—
529C2×C4—201no—
530C2×C4—201no—
531C2×C4—201no—
532C2×C4—304no—
533C2×C4—304no—
534C2×C4—304no—
535C2×C4—304no—
536C2×C4—602no—
537C2×C4—601no—
538C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
539C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
540C2×C3—22no—
541C2×C3—54no—
542C2×C3—102no—
543C2×C3C2×#413011yescharacteristic
544C2×C3#6625811yescharacteristic
545C2×C3#6625811yescharacteristic
546D3—42no—
547D3—202no—
548C5#7356711yescharacteristic, complemented, Sylow
549C22—21no—
550C22—21no—
551C22—52no—
552C22—52no—
553C22—52no—
554C22—52no—
555C22—62no—
556C22—62no—
557C22—62no—
558C22—108no—
559C22—101no—
560C22—101no—
561C22—124no—
562C22—302no—
563C22—302no—
564C22—302no—
565C22—604no—
566C22—608no—
567C22—608no—
568C22#9190912yes—
569C22C22×D3×AGL(1,𝔽5)11yescharacteristic
570C4—21no—
571C4—21no—
572C4—62no—
573C4—101no—
574C4—101no—
575C4—204no—
576C4—302no—
577C4—604no—
578C3#1061707011yescharacteristic, complemented, Sylow
579C2—22no—
580C2—54no—
581C2—102no—
582C2—122no—
583C2—602no—
584C2C2×#9190911yescharacteristic
585C2#1175683711yescharacteristic
586C2#1175687111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C4×C5C2×C4
prime radicalC23×D3×D5C22
cosocleC22×C3×D5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2312
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C4×#6763C2semidirect
3. C3×C4×C22⋊AGL(1,𝔽5)C2semidirect
4. #11752028C2semidirect
6. #11751856C2semidirect
7. #11751849C2semidirect
13. C2×C4×ΓL(1,𝔽16)C22semidirect
15. C2×C3×C4×AGL(1,𝔽5)C22semidirect
16. C2×Dic3⋊AGL(1,𝔽5)C22semidirect
19. C2×#6714C4semidirect
20. C2×C5×(C2×D3)⋊C4C4semidirect
21. C2×#6667C4semidirect
22. C2×(C2×D3)⋊Dic5C4semidirect
23. D5×(C2×D3)⋊C4C4semidirect
24. D5×(C2×D3)⋊C4C4semidirect
27. #91235C22semidirect
28. C3×(C2×AGL(1,𝔽5))⋊C4C22semidirect
29. (C2×AGL(1,𝔽5))⋊Dic3C22semidirect
35. C4×C22⋊AGL(1,𝔽5)D3semidirect
91. C2×C4×AGL(1,𝔽5)C2×D3semidirect
102. (C2×AGL(1,𝔽5))⋊C4C2×D3semidirect
162. C2×(C2×D3)⋊C4AGL(1,𝔽5)semidirect
459. C3×C5#2312semidirect
548. C5#73567semidirect
578. C3#10617070semidirect

Maximal subgroups

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

subgroupindex
C2×D5×(C2×D3)⋊C42
C4×#67632
C3×C4×C22⋊AGL(1,𝔽5)2
#117520282
#117520252
#117518562
#117518492
#106170703
#735675