Finite Group Databasebeta

← #99×ΓL(1,𝔽16) #14295381

Subgroups — #99×ΓL(1,𝔽16) #14295381

Order 1920.

824 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×D4×ΓL(1,𝔽16)C211yesmaximal, characteristic
2D5⋊#834C211yesmaximal, characteristic
3C22⋊C4×ΓL(1,𝔽16)C212yesmaximal, complemented
4C3×D5×#99C211yesmaximal, characteristic
5#11746737C212yesmaximal, complemented
6AGL(1,𝔽5)×#99—31nomaximal, Hall
7C23×ΓL(1,𝔽16)C411yescharacteristic, complemented
8C23×ΓL(1,𝔽16)C2211yescharacteristic
9C2×C3×D4×D5C2211yescharacteristic
10C2×#6763C411yescharacteristic, complemented
11C2×#6763C2211yescharacteristic
12C2×#6763C2212yes—
13D4×ΓL(1,𝔽16)—22no—
14D4×ΓL(1,𝔽16)—22no—
15C2×C4⋊ΓL(1,𝔽16)C411yescharacteristic, complemented
16C2×C4×ΓL(1,𝔽16)—22no—
17C2×C4×ΓL(1,𝔽16)C411yescharacteristic, complemented
18C3×D5×C22⋊C4C2212yes—
19#91240C412yescomplemented
20#91235—22no—
21#91228C412yescomplemented
22C3×C5×#99C411yescharacteristic, complemented
23C3×#3966C411yescharacteristic, complemented
24C3×#3938C412yescomplemented
25Dic3×#99—51nomaximal, Hall
26C2×D4×AGL(1,𝔽5)—31no—
27D5⋊#99—31no—
28C22⋊C4×AGL(1,𝔽5)—32no—
29D5×#99D311yescharacteristic
30#64866—32no—
31C23×C3×D5C2×C411yescharacteristic
32C23×C3×D5C2311yescharacteristic
33C22×ΓL(1,𝔽16)—22no—
34C22×ΓL(1,𝔽16)—21no—
35C22×ΓL(1,𝔽16)—21no—
36C22×ΓL(1,𝔽16)—42no—
37C22×ΓL(1,𝔽16)—42no—
38C22×ΓL(1,𝔽16)C2×C411yescharacteristic
39C22×ΓL(1,𝔽16)C2×C411yescharacteristic
40C22×ΓL(1,𝔽16)C2×C411yescharacteristic
41C22×ΓL(1,𝔽16)C2×C411yescharacteristic
42C22×ΓL(1,𝔽16)D412yes—
43C22×ΓL(1,𝔽16)D412yes—
44C2×C3×C5×D4C2×C411yescharacteristic
45C2×C3×C22⋊D5C2×C411yescharacteristic
46C2×C3×C22⋊D5C2×C411yescharacteristic
47C3×D4×D5—22no—
48C3×D4×D5—22no—
49C2×C3×D20C2×C411yescharacteristic
50C2×C3×C4×D5—22no—
51C2×C3×C4×D5C2×C411yescharacteristic
52#6763—22no—
53#6763—22no—
54#6763—22no—
55#6763C2×C414yescomplemented
56C4⋊ΓL(1,𝔽16)—22no—
57C4×ΓL(1,𝔽16)—21no—
58C4×ΓL(1,𝔽16)—21no—
59C4×ΓL(1,𝔽16)—44no—
60C3×C5×C22⋊C4C2×C412yescomplemented
61C3×C22⋊Dic5C2×C412yescomplemented
62C3×#685—22no—
63C2×D4×Dic3—51no—
64C2×C3×#99—51no—
65C2×#834—51no—
66Dic3×C22⋊C4—52no—
67#4527—52no—
68C23×AGL(1,𝔽5)—31no—
69C23×AGL(1,𝔽5)—31no—
70C2×D4×D5C2×D311yescharacteristic
71C2×C22⋊AGL(1,𝔽5)—32no—
72C2×C22⋊AGL(1,𝔽5)—31no—
73C2×C22⋊AGL(1,𝔽5)—31no—
74D4×AGL(1,𝔽5)—62no—
75D4×AGL(1,𝔽5)—62no—
76C2×C4⋊AGL(1,𝔽5)—31no—
77C2×C4×AGL(1,𝔽5)—31no—
78C2×C4×AGL(1,𝔽5)—62no—
79D5×C22⋊C4C2×D312yes—
80#4011—32no—
81(C2×AGL(1,𝔽5))⋊C4—62no—
82#3999—32no—
83C5×#99Dic311yescharacteristic, complemented
84#3966Dic311yescharacteristic, complemented
85#3938Dic312yescomplemented
86C4×#99—151noSylow
87C23×C3×C5C4211yescharacteristic, complemented
88C23×C3×C5C22×C411yescharacteristic
89C22×C3×D5—21no—
90C22×C3×D5—21no—
91C22×C3×D5—41no—
92C22×C3×D5—41no—
93C22×C3×D5—41no—
94C22×C3×D5—41no—
95C22×C3×D5C4211yescharacteristic, complemented
96C22×C3×D5C22×C411yescharacteristic
97C22×C3×D5C22×C411yescharacteristic
98C22×C3×D5C2×D412yes—
99C22×C3×D5#4112yes—
100C2×ΓL(1,𝔽16)—24no—
101C2×ΓL(1,𝔽16)—22no—
102C2×ΓL(1,𝔽16)—21no—
103C2×ΓL(1,𝔽16)—21no—
104C2×ΓL(1,𝔽16)—24no—
105C2×ΓL(1,𝔽16)—41no—
106C2×ΓL(1,𝔽16)—42no—
107C2×ΓL(1,𝔽16)—42no—
108C2×ΓL(1,𝔽16)—42no—
109C2×ΓL(1,𝔽16)—42no—
110C2×ΓL(1,𝔽16)—42no—
111C2×ΓL(1,𝔽16)—41no—
112C2×ΓL(1,𝔽16)—41no—
113C2×ΓL(1,𝔽16)C22⋊C411yescharacteristic
114C2×ΓL(1,𝔽16)C22⋊C412yes—
115C2×ΓL(1,𝔽16)C22⋊C411yescharacteristic
116C3×C5×D4—22no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5C4211yescharacteristic, complemented
119C3×C22⋊D5—22no—
120C3×C22⋊D5—22no—
121C2×C3×Dic5—22no—
122C2×C3×Dic5C4211yescharacteristic, complemented
123C3×D20—22no—
124C3×C4×D5—21no—
125C3×C4×D5—21no—
126C3×C4×D5—42no—
127C3×C4×D5—42no—
128C22×C3×D4—51no—
129C23×Dic3—51no—
130C23×Dic3—51no—
131C2×C3×C22⋊C4—52no—
132C2×#217—52no—
133C2×#217—51no—
134C2×#217—51no—
135D4×Dic3—102no—
136D4×Dic3—102no—
137C2×C4⋊Dic3—51no—
138C2×C4×Dic3—51no—
139C2×C4×Dic3—102no—
140C3×#99—51no—
141C3×#99—52no—
142C3×#99AGL(1,𝔽5)11yescharacteristic, complemented
143#834—52no—
144#834—52no—
145C3⋊#95—102no—
146C23×D5C4×D311yescharacteristic
147C23×D5C22×D311yescharacteristic
148C22×AGL(1,𝔽5)—31no—
149C22×AGL(1,𝔽5)—32no—
150C22×AGL(1,𝔽5)—31no—
151C22×AGL(1,𝔽5)—32no—
152C22×AGL(1,𝔽5)—31no—
153C22×AGL(1,𝔽5)—31no—
154C22×AGL(1,𝔽5)—61no—
155C22×AGL(1,𝔽5)—61no—
156C22×AGL(1,𝔽5)—62no—
157C22×AGL(1,𝔽5)—122no—
158C22×AGL(1,𝔽5)—122no—
159C2×C5×D4C2×Dic311yescharacteristic
160C2×C22⋊D5C4×D311yescharacteristic
161C2×C22⋊D5C2×Dic311yescharacteristic
162D4×D5—22no—
163D4×D5—22no—
164C2×D20C4×D311yescharacteristic
165C2×C4×D5—22no—
166C2×C4×D5C4×D311yescharacteristic
167C22⋊AGL(1,𝔽5)—34no—
168C22⋊AGL(1,𝔽5)—62no—
169C22⋊AGL(1,𝔽5)—62no—
170C22⋊AGL(1,𝔽5)—62no—
171C4⋊AGL(1,𝔽5)—62no—
172C4×AGL(1,𝔽5)—61no—
173C4×AGL(1,𝔽5)—61no—
174C4×AGL(1,𝔽5)—124no—
175C5×C22⋊C4C2×Dic312yescomplemented
176C22⋊Dic5C2×Dic312yescomplemented
177#685—22no—
178C2×C4×D4—151no—
179C2×#99—51no—
180C2×#99—151no—
181C4×C22⋊C4—152no—
182#342—152no—
183C22×C3×C5—21no—
184C22×C3×C5—41no—
185C22×C3×C5—41no—
186C22×C3×C5C2×C4211yescharacteristic
187C22×C3×C5C4×D412yes—
188C2×C3×D5—22no—
189C2×C3×D5—21no—
190C2×C3×D5—22no—
191C2×C3×D5—21no—
192C2×C3×D5—21no—
193C2×C3×D5—41no—
194C2×C3×D5—42no—
195C2×C3×D5—42no—
196C2×C3×D5—41no—
197C2×C3×D5—41no—
198C2×C3×D5—41no—
199C2×C3×D5C2×C22⋊C411yescharacteristic
200C2×C3×D5#11711yescharacteristic
201C2×C3×D5C4×D412yes—
202ΓL(1,𝔽16)—21no—
203ΓL(1,𝔽16)—22no—
204ΓL(1,𝔽16)—44no—
205ΓL(1,𝔽16)—41no—
206ΓL(1,𝔽16)—41no—
207ΓL(1,𝔽16)#9912yesdirect product factor
208C3×C4×C5—21no—
209C3×C4×C5—42no—
210C3×Dic5—21no—
211C3×Dic5—42no—
212C24×C3—51no—
213C24×C3—51no—
214C2×C3×D4—51no—
215C2×C3×D4—51no—
216C2×C3×D4—51no—
217C2×C3×D4—102no—
218C2×C3×D4—102no—
219C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
220C22×C3×C4—51no—
221C22×C3×C4—102no—
222C22×Dic3—51no—
223C22×Dic3—52no—
224C22×Dic3—52no—
225C22×Dic3—51no—
226C22×Dic3—51no—
227C22×Dic3—51no—
228C22×Dic3—101no—
229C22×Dic3—101no—
230C22×Dic3—102no—
231C22×Dic3—202no—
232C22×Dic3—202no—
233C3×C22⋊C4—52no—
234C3×C22⋊C4—102no—
235C3×C22⋊C4C2×AGL(1,𝔽5)12yescomplemented
236#217—54no—
237#217—102no—
238#217—102no—
239#217—102no—
240C4⋊Dic3—102no—
241C4×Dic3—101no—
242C4×Dic3—101no—
243C4×Dic3—204no—
244C23×C5C4×Dic311yescharacteristic, complemented
245C23×C5C22×Dic311yescharacteristic
246C22×D5—21no—
247C22×D5—21no—
248C22×D5—41no—
249C22×D5—41no—
250C22×D5—41no—
251C22×D5—41no—
252C22×D5C4×Dic311yescharacteristic, complemented
253C22×D5C2×C4×D311yescharacteristic
254C22×D5C2×C4×D311yescharacteristic
255C22×D5D3×D412yes—
256C22×D5#23712yes—
257C2×AGL(1,𝔽5)—31no—
258C2×AGL(1,𝔽5)—31no—
259C2×AGL(1,𝔽5)—32no—
260C2×AGL(1,𝔽5)—64no—
261C2×AGL(1,𝔽5)—62no—
262C2×AGL(1,𝔽5)—61no—
263C2×AGL(1,𝔽5)—61no—
264C2×AGL(1,𝔽5)—64no—
265C2×AGL(1,𝔽5)—122no—
266C2×AGL(1,𝔽5)—121no—
267C2×AGL(1,𝔽5)—122no—
268C2×AGL(1,𝔽5)—122no—
269C2×AGL(1,𝔽5)—122no—
270C2×AGL(1,𝔽5)—122no—
271C2×AGL(1,𝔽5)—121no—
272C2×AGL(1,𝔽5)—121no—
273C5×D4—22no—
274C2×C4×C5—22no—
275C2×C4×C5C4×Dic311yescharacteristic, complemented
276C22⋊D5—22no—
277C22⋊D5—22no—
278C2×Dic5—22no—
279C2×Dic5C4×Dic311yescharacteristic, complemented
280D20—22no—
281C4×D5—21no—
282C4×D5—21no—
283C4×D5—42no—
284C4×D5—42no—
285C22×D4—51no—
286C23×C4—151no—
287C23×C4—151no—
288C4×D4—302no—
289C4×D4—302no—
290C2×C4⋊C4—151no—
291C2×C22⋊C4—52no—
292C2×C22⋊C4—152no—
293C2×C22⋊C4—151no—
294C2×C22⋊C4—151no—
295C2×C42—151no—
296C2×C42—302no—
297#99—51no—
298#99—52no—
299#99—152no—
300#99—152no—
301#99ΓL(1,𝔽16)11yescharacteristic, direct product factor
302#95—302no—
303C2×C3×C5—22no—
304C2×C3×C5—21no—
305C2×C3×C5—41no—
306C2×C3×C5C4×C22⋊C411yescharacteristic
307C3×D5—22no—
308C3×D5—21no—
309C3×D5—41no—
310C3×D5C2×#9911yescharacteristic
311C3×D5#41011yescharacteristic
312C23×C3—51no—
313C23×C3—52no—
314C23×C3—51no—
315C23×C3—52no—
316C23×C3—51no—
317C23×C3—101no—
318C23×C3—101no—
319C23×C3—201no—
320C23×C3—201no—
321C23×C3—201no—
322C23×C3—201no—
323C23×C3C4×AGL(1,𝔽5)11yescharacteristic, complemented
324C23×C3C22×AGL(1,𝔽5)11yescharacteristic
325C3×D4—22no—
326C3×D4—102no—
327C3×D4—102no—
328C3×D4—102no—
329C2×C3×C4—22no—
330C2×C3×C4—51no—
331C2×C3×C4—101no—
332C2×C3×C4—101no—
333C2×C3×C4—102no—
334C2×C3×C4—202no—
335C2×C3×C4—202no—
336C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic, complemented
337C2×Dic3—51no—
338C2×Dic3—51no—
339C2×Dic3—52no—
340C2×Dic3—104no—
341C2×Dic3—102no—
342C2×Dic3—101no—
343C2×Dic3—101no—
344C2×Dic3—104no—
345C2×Dic3—202no—
346C2×Dic3—201no—
347C2×Dic3—202no—
348C2×Dic3—202no—
349C2×Dic3—202no—
350C2×Dic3—202no—
351C2×Dic3—201no—
352C2×Dic3—201no—
353C22×C5—21no—
354C22×C5—41no—
355C22×C5—41no—
356C22×C5C2×C4×Dic311yescharacteristic
357C22×C5D4×Dic312yes—
358C2×D5—22no—
359C2×D5—21no—
360C2×D5—22no—
361C2×D5—21no—
362C2×D5—21no—
363C2×D5—41no—
364C2×D5—42no—
365C2×D5—42no—
366C2×D5—41no—
367C2×D5—41no—
368C2×D5—41no—
369C2×D5#87711yescharacteristic
370C2×D5D3×C22⋊C411yescharacteristic
371C2×D5#67⋊C412yes—
372AGL(1,𝔽5)—32no—
373AGL(1,𝔽5)—62no—
374AGL(1,𝔽5)—61no—
375AGL(1,𝔽5)—124no—
376AGL(1,𝔽5)—121no—
377AGL(1,𝔽5)—121no—
378C4×C5—21no—
379C4×C5—42no—
380Dic5—21no—
381Dic5—42no—
382C24—51no—
383C24—51no—
384C2×D4—51no—
385C2×D4—51no—
386C2×D4—51no—
387C2×D4—102no—
388C2×D4—102no—
389C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
390C22×C4—51no—
391C22×C4—102no—
392C22×C4—151no—
393C22×C4—152no—
394C22×C4—152no—
395C22×C4—151no—
396C22×C4—151no—
397C22×C4—151no—
398C22×C4—301no—
399C22×C4—301no—
400C22×C4—302no—
401C22×C4—602no—
402C22×C4—602no—
403C4⋊C4—302no—
404C22⋊C4—52no—
405C22⋊C4—102no—
406C22⋊C4—154no—
407C22⋊C4—302no—
408C22⋊C4—302no—
409C22⋊C4—302no—
410C22⋊C4C2×ΓL(1,𝔽16)12yescomplemented
411C42—301no—
412C42—301no—
413C42—604no—
414C3×C5C4×#9911yescharacteristic, complemented, Hall
415C22×C3—21no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—51no—
419C22×C3—51no—
420C22×C3—52no—
421C22×C3—102no—
422C22×C3—101no—
423C22×C3—102no—
424C22×C3—101no—
425C22×C3—101no—
426C22×C3—202no—
427C22×C3—201no—
428C22×C3—201no—
429C22×C3—201no—
430C22×C3—201no—
431C22×C3—202no—
432C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
433C22×C3D4×AGL(1,𝔽5)12yes—
434C3×C4—21no—
435C3×C4—42no—
436C3×C4—101no—
437C3×C4—202no—
438Dic3—52no—
439Dic3—102no—
440Dic3—101no—
441Dic3—201no—
442Dic3—201no—
443Dic3—204no—
444C2×C5—22no—
445C2×C5—21no—
446C2×C5—41no—
447C2×C5Dic3×C22⋊C411yescharacteristic
448D5—22no—
449D5—21no—
450D5—41no—
451D5#474011yescharacteristic
452D5D3×#9911yescharacteristic
453C23—51no—
454C23—52no—
455C23—51no—
456C23—52no—
457C23—51no—
458C23—101no—
459C23—101no—
460C23—201no—
461C23—201no—
462C23—201no—
463C23—201no—
464C23C4×ΓL(1,𝔽16)11yescharacteristic, complemented
465C23C22×ΓL(1,𝔽16)11yescharacteristic
466D4—22no—
467D4—102no—
468D4—102no—
469D4—102no—
470C2×C4—22no—
471C2×C4—51no—
472C2×C4—101no—
473C2×C4—101no—
474C2×C4—102no—
475C2×C4—151no—
476C2×C4—151no—
477C2×C4—152no—
478C2×C4—202no—
479C2×C4—202no—
480C2×C4—304no—
481C2×C4—302no—
482C2×C4—301no—
483C2×C4—304no—
484C2×C4—301no—
485C2×C4—602no—
486C2×C4—602no—
487C2×C4—601no—
488C2×C4—602no—
489C2×C4—602no—
490C2×C4—602no—
491C2×C4—601no—
492C2×C4—601no—
493C2×C4C4×ΓL(1,𝔽16)11yescharacteristic, complemented
494C2×C3—22no—
495C2×C3—21no—
496C2×C3—41no—
497C2×C3—51no—
498C2×C3—51no—
499C2×C3—102no—
500C2×C3—101no—
501C2×C3—201no—
502C2×C3C22⋊C4×AGL(1,𝔽5)11yescharacteristic
503C5Dic3×#9911yescharacteristic, complemented, Sylow
504C22—21no—
505C22—41no—
506C22—41no—
507C22—51no—
508C22—51no—
509C22—52no—
510C22—102no—
511C22—102no—
512C22—101no—
513C22—101no—
514C22—101no—
515C22—202no—
516C22—201no—
517C22—201no—
518C22—201no—
519C22—201no—
520C22—202no—
521C22C2×C4×ΓL(1,𝔽16)11yescharacteristic
522C22D4×ΓL(1,𝔽16)12yes—
523C4—21no—
524C4—42no—
525C4—101no—
526C4—152no—
527C4—202no—
528C4—301no—
529C4—302no—
530C4—601no—
531C4—601no—
532C4—604no—
533C3AGL(1,𝔽5)×#9911yescharacteristic, complemented, Sylow
534C2—22no—
535C2—21no—
536C2—41no—
537C2—51no—
538C2—51no—
539C2—102no—
540C2—101no—
541C2—201no—
542C2C22⋊C4×ΓL(1,𝔽16)11yescharacteristic
subgroupG/H
centerC2C22⋊C4×ΓL(1,𝔽16)
commutatorC22×C3×C5C2×C42
FrattiniC23C22×ΓL(1,𝔽16)
FittingC3×C5×#99C4
socleC2×C3×C5C4×C22⋊C4
prime radicalC2×C3×D4×D5C22
cosocleC23×C3×D5C23
hypercenter#99ΓL(1,𝔽16)
hypocenterC3×C5C4×#99
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
3. C22⋊C4×ΓL(1,𝔽16)C2semidirect
5. #11746737C2semidirect
7. C23×ΓL(1,𝔽16)C4semidirect
10. C2×#6763C4semidirect
15. C2×C4⋊ΓL(1,𝔽16)C4semidirect
17. C2×C4×ΓL(1,𝔽16)C4semidirect
19. #91240C4semidirect
21. #91228C4semidirect
22. C3×C5×#99C4semidirect
23. C3×#3966C4semidirect
24. C3×#3938C4semidirect
55. #6763C2×C4semidirect
60. C3×C5×C22⋊C4C2×C4semidirect
61. C3×C22⋊Dic5C2×C4semidirect
83. C5×#99Dic3semidirect
84. #3966Dic3semidirect
85. #3938Dic3semidirect
87. C23×C3×C5C42semidirect
95. C22×C3×D5C42semidirect
118. C2×C3×C4×C5C42semidirect
122. C2×C3×Dic5C42semidirect
142. C3×#99AGL(1,𝔽5)semidirect
175. C5×C22⋊C4C2×Dic3semidirect
176. C22⋊Dic5C2×Dic3semidirect
207. ΓL(1,𝔽16)#99direct (N × H)
235. C3×C22⋊C4C2×AGL(1,𝔽5)semidirect
244. C23×C5C4×Dic3semidirect
252. C22×D5C4×Dic3semidirect
275. C2×C4×C5C4×Dic3semidirect
279. C2×Dic5C4×Dic3semidirect
301. #99ΓL(1,𝔽16)direct (N × H)
323. C23×C3C4×AGL(1,𝔽5)semidirect
336. C2×C3×C4C4×AGL(1,𝔽5)semidirect
410. C22⋊C4C2×ΓL(1,𝔽16)semidirect
414. C3×C5C4×#99semidirect
464. C23C4×ΓL(1,𝔽16)semidirect
493. C2×C4C4×ΓL(1,𝔽16)semidirect
503. C5Dic3×#99semidirect
533. C3AGL(1,𝔽5)×#99semidirect

Maximal subgroups

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

subgroupindex
C2×D4×ΓL(1,𝔽16)2
D5⋊#8342
C22⋊C4×ΓL(1,𝔽16)2
C3×D5×#992
#117467372
AGL(1,𝔽5)×#993
Dic3×#995