Finite Group Databasebeta

← #14300602

Subgroups — #14300602

Order 1920.

808 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic
2C3×D5×C22⋊Q8C211yesmaximal, characteristic
3C2×#91238C212yesmaximal, complemented
4D5⋊#830C211yesmaximal, characteristic
5#11746745C212yesmaximal, complemented
6#10604523—31nomaximal, Hall
7C23×ΓL(1,𝔽16)—21no—
8C2×C3×Q8×D5C2211yescharacteristic
9C22×C3×C4×D5C2211yescharacteristic
10C2×C4⋊ΓL(1,𝔽16)—21no—
11C2×C4⋊ΓL(1,𝔽16)—24no—
12C2×C4⋊ΓL(1,𝔽16)C2212yes—
13C2×D5⋊(C3⋊C8)C2212yes—
14C3×C5×C22⋊Q8C411yescharacteristic, complemented
15C3×#4092C411yescharacteristic, complemented
16C3×#4070C411yescharacteristic, complemented
17C3×#4043C411yescharacteristic, complemented
18C3×#4042—21no—
19C3×D5×C4⋊C4—21no—
20C3×D5×C4⋊C4C2211yescharacteristic
21C3×D5×C22⋊C4—21no—
22C3×#4024—21no—
23#91241—21no—
24#91238—24no—
25#91232—21no—
26#71692—51nomaximal, Hall
27C22×C4⋊AGL(1,𝔽5)—31no—
28D5×C22⋊Q8D311yescharacteristic
29C2×#4009—32no—
30D5⋊(C22⋊C8)—31no—
31#64874—32no—
32C23×C3×D5D411yescharacteristic
33C22×ΓL(1,𝔽16)—24no—
34C22×ΓL(1,𝔽16)—22no—
35C22×ΓL(1,𝔽16)—42no—
36C22×ΓL(1,𝔽16)—42no—
37C2×C3×C5×Q8C2×C411yescharacteristic, complemented
38C22×C3×C4×C5C2×C411yescharacteristic
39C22×C3×Dic5D411yescharacteristic
40C3×Q8×D5—22no—
41C3×Q8×D5—22no—
42C2×C3×C4×D5—22no—
43C2×C3×C4×D5—22no—
44C2×C3×C4×D5—21no—
45C2×C3×C4×D5—21no—
46C2×C3×C4×D5C2×C411yescharacteristic
47C2×C3×C4×D5D411yescharacteristic
48C2×C3×C4×D5D411yescharacteristic
49C2×C3×C4×D5C2311yescharacteristic
50C2×C3×Dic10—21no—
51C2×C3×Dic10C2×C411yescharacteristic, complemented
52C2×C3⋊#170—22no—
53C4⋊ΓL(1,𝔽16)—24no—
54C4⋊ΓL(1,𝔽16)—22no—
55C4⋊ΓL(1,𝔽16)D414yes—
56D5⋊(C3⋊C8)—22no—
57D5⋊(C3⋊C8)—22no—
58C3×C5×C4⋊C4—21no—
59C3×C5×C4⋊C4C2×C411yescharacteristic, complemented
60C3×C5×C22⋊C4—21no—
61C3×C22⋊Dic5—21no—
62C3×#685—21no—
63C3×#685—21no—
64C3×C4⋊Dic5—21no—
65C3×C4⋊Dic5C2×C411yescharacteristic, complemented
66C3×#683—21no—
67C3×#683—21no—
68C3×#683—21no—
69C2×C3×C22⋊Q8—51no—
70C22×C4⋊Dic3—51no—
71C2×#835—52no—
72C2×#830—51no—
73#4529—52no—
74C23×AGL(1,𝔽5)—61no—
75C2×Q8×D5C2×D311yescharacteristic
76C22×C4×D5C2×D311yescharacteristic
77C2×C4⋊AGL(1,𝔽5)—32no—
78C2×C4⋊AGL(1,𝔽5)—64no—
79C2×C4⋊AGL(1,𝔽5)—61no—
80C2×D5⋊C8—32no—
81C5×C22⋊Q8Dic311yescharacteristic, complemented
82#4092Dic311yescharacteristic, complemented
83#4070Dic311yescharacteristic, complemented
84#4043Dic311yescharacteristic, complemented
85#4042—21no—
86D5×C4⋊C4—21no—
87D5×C4⋊C4C2×D311yescharacteristic
88D5×C22⋊C4—21no—
89#4024—21no—
90C22⋊#170—61no—
91#4009—64no—
92#4003—61no—
93#1872—151noSylow
94C23×C3×C5C22⋊C411yescharacteristic
95C22×C3×D5—21no—
96C22×C3×D5—21no—
97C22×C3×D5—21no—
98C22×C3×D5—21no—
99C22×C3×D5C22⋊C411yescharacteristic
100C22×C3×D5SD1612yes—
101C22×C3×D5Q1612yes—
102C22×C3×D5C2×D411yescharacteristic
103C2×ΓL(1,𝔽16)—24no—
104C2×ΓL(1,𝔽16)—48no—
105C2×ΓL(1,𝔽16)—42no—
106C2×ΓL(1,𝔽16)—42no—
107C3×C5×Q8—22no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5—21no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5C22⋊C411yescharacteristic
112C2×C3×C4×C5C22×C411yescharacteristic
113C2×C3×Dic5—22no—
114C2×C3×Dic5—21no—
115C2×C3×Dic5—21no—
116C2×C3×Dic5C22⋊C411yescharacteristic
117C2×C3×Dic5C2×D411yescharacteristic
118C3×C4×D5—21no—
119C3×C4×D5—21no—
120C3×C4×D5—22no—
121C3×C4×D5—22no—
122C3×C4×D5—41no—
123C3×C4×D5—41no—
124C3×C4×D5—41no—
125C3×C4×D5—41no—
126C3×C4×D5C2×D412yes—
127C3×C4×D5#4112yes—
128C3×Dic10—22no—
129C3×Dic10—42no—
130C3⋊#170—42no—
131C22×C3×Q8—51no—
132C23×C3×C4—51no—
133C23×Dic3—101no—
134C3×C22⋊Q8—51no—
135C3×C22⋊Q8—51no—
136C3×C22⋊Q8—51no—
137C3×C22⋊Q8—101no—
138C3×C22⋊Q8—101no—
139C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
140C2×C3×C4⋊C4—51no—
141C2×C3×C4⋊C4—101no—
142C2×C3×C22⋊C4—101no—
143C2×C4⋊Dic3—52no—
144C2×C4⋊Dic3—104no—
145C2×C4⋊Dic3—101no—
146C22×C3⋊C8—52no—
147#835—104no—
148#830—101no—
149#830—101no—
150C23×D5#6711yescharacteristic
151C22×AGL(1,𝔽5)—64no—
152C22×AGL(1,𝔽5)—62no—
153C22×AGL(1,𝔽5)—122no—
154C22×AGL(1,𝔽5)—122no—
155C2×C5×Q8C2×Dic311yescharacteristic, complemented
156C22×C4×C5C2×Dic311yescharacteristic
157C22×Dic5#6711yescharacteristic
158Q8×D5—22no—
159Q8×D5—22no—
160C2×C4×D5—22no—
161C2×C4×D5—22no—
162C2×C4×D5—21no—
163C2×C4×D5—21no—
164C2×C4×D5C2×Dic311yescharacteristic
165C2×C4×D5#6711yescharacteristic
166C2×C4×D5#6711yescharacteristic
167C2×C4×D5C22×D311yescharacteristic
168C2×Dic10—21no—
169C2×Dic10C2×Dic311yescharacteristic, complemented
170C2×#170—62no—
171C4⋊AGL(1,𝔽5)—34no—
172C4⋊AGL(1,𝔽5)—64no—
173C4⋊AGL(1,𝔽5)—62no—
174D5⋊C8—62no—
175D5⋊C8—62no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4C2×Dic311yescharacteristic, complemented
178C5×C22⋊C4—21no—
179C22⋊Dic5—21no—
180#685—21no—
181#685—21no—
182C4⋊Dic5—21no—
183C4⋊Dic5C2×Dic311yescharacteristic, complemented
184#683—21no—
185#683—21no—
186#683—21no—
187C2×C22⋊Q8—51no—
188C22×C4⋊C4—151no—
189C2×#103—152no—
190C2×C22⋊C8—151no—
191#340—152no—
192C22×C3×C5—21no—
193C22×C3×C5—21no—
194C22×C3×C5#10312yescomplemented
195C22×C3×C5C2×C22⋊C411yescharacteristic
196C2×C3×D5—21no—
197C2×C3×D5—21no—
198C2×C3×D5—22no—
199C2×C3×D5—22no—
200C2×C3×D5—22no—
201C2×C3×D5—22no—
202C2×C3×D5#10312yescomplemented
203C2×C3×D5#12011yescharacteristic
204C2×C3×D5#12311yescharacteristic
205C2×C3×D5C2×SD1611yescharacteristic
206C2×C3×D5C2×Q1611yescharacteristic
207C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
208C2×C3×D5#13711yescharacteristic
209ΓL(1,𝔽16)—44no—
210C3×C4×C5—21no—
211C3×C4×C5—41no—
212C3×C4×C5—41no—
213C3×C4×C5C4×D412yes—
214C3×Dic5—21no—
215C3×Dic5—41no—
216C3×Dic5—41no—
217C3×Dic5#12112yes—
218C24×C3—51no—
219C2×C3×Q8—51no—
220C2×C3×Q8—101no—
221C2×C3×Q8—102no—
222C2×C3×Q8—102no—
223C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
224C22×C3×C4—51no—
225C22×C3×C4—51no—
226C22×C3×C4—51no—
227C22×C3×C4—51no—
228C22×C3×C4—51no—
229C22×C3×C4—102no—
230C22×C3×C4—102no—
231C22×C3×C4—101no—
232C22×C3×C4—101no—
233C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
234C22×Dic3—102no—
235C22×Dic3—104no—
236C22×Dic3—202no—
237C22×Dic3—202no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—51no—
240C3×C4⋊C4—101no—
241C3×C4⋊C4—101no—
242C3×C4⋊C4—101no—
243C3×C4⋊C4—101no—
244C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
245C3×C22⋊C4—21no—
246C3×C22⋊C4—101no—
247C3×C22⋊C4—101no—
248C3×C22⋊C4—101no—
249C4⋊Dic3—54no—
250C4⋊Dic3—102no—
251C4⋊Dic3—104no—
252C2×C3⋊C8—102no—
253C2×C3⋊C8—102no—
254C2×C3⋊C8—102no—
255C23×C5#21711yescharacteristic
256C22×D5—21no—
257C22×D5—21no—
258C22×D5—21no—
259C22×D5—21no—
260C22×D5#21512yes—
261C22×D5#21612yes—
262C22×D5#21711yescharacteristic
263C22×D5C2×#6711yescharacteristic
264C2×AGL(1,𝔽5)—64no—
265C2×AGL(1,𝔽5)—128no—
266C2×AGL(1,𝔽5)—122no—
267C2×AGL(1,𝔽5)—122no—
268C5×Q8—22no—
269C2×C4×C5—22no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5#21711yescharacteristic
273C2×C4×C5C22×Dic311yescharacteristic
274C2×Dic5—22no—
275C2×Dic5—21no—
276C2×Dic5—21no—
277C2×Dic5#21711yescharacteristic
278C2×Dic5C2×#6711yescharacteristic
279C4×D5—21no—
280C4×D5—21no—
281C4×D5—22no—
282C4×D5—22no—
283C4×D5—41no—
284C4×D5—41no—
285C4×D5—41no—
286C4×D5—41no—
287C4×D5D3×D412yes—
288C4×D5#23712yes—
289Dic10—22no—
290Dic10—42no—
291#170—122no—
292C22×Q8—51no—
293C23×C4—51no—
294C23×C4—301no—
295C22×C8—152no—
296C22⋊Q8—51no—
297C22⋊Q8—51no—
298C22⋊Q8—51no—
299C22⋊Q8—101no—
300C22⋊Q8—101no—
301C22⋊Q8ΓL(1,𝔽16)11yescharacteristic, complemented
302C2×C4⋊C4—51no—
303C2×C4⋊C4—101no—
304C2×C4⋊C4—152no—
305C2×C4⋊C4—304no—
306C2×C4⋊C4—301no—
307C2×C22⋊C4—101no—
308#103—304no—
309C22⋊C8—301no—
310C22⋊C8—301no—
311C2×C3×C5—22no—
312C2×C3×C5#38611yescharacteristic
313C2×C3×C5C2×#10311yescharacteristic
314C2×C3×C5#41711yescharacteristic
315C3×D5—22no—
316C3×D5#45011yescharacteristic
317C3×D5#45111yescharacteristic
318C3×D5#48311yescharacteristic
319C3×D5#48411yescharacteristic
320C23×C3—51no—
321C23×C3—51no—
322C23×C3—52no—
323C23×C3—52no—
324C23×C3—101no—
325C23×C3—101no—
326C23×C3—101no—
327C23×C3—101no—
328C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
329C3×Q8—22no—
330C3×Q8—102no—
331C3×Q8—202no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—51no—
336C2×C3×C4—52no—
337C2×C3×C4—52no—
338C2×C3×C4—51no—
339C2×C3×C4—102no—
340C2×C3×C4—101no—
341C2×C3×C4—101no—
342C2×C3×C4—102no—
343C2×C3×C4—102no—
344C2×C3×C4—101no—
345C2×C3×C4—101no—
346C2×C3×C4—201no—
347C2×C3×C4—201no—
348C2×C3×C4—201no—
349C2×C3×C4—201no—
350C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
351C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
352C2×Dic3—104no—
353C2×Dic3—202no—
354C2×Dic3—202no—
355C2×Dic3—208no—
356C3⋊C8—202no—
357C22×C5—21no—
358C22×C5—21no—
359C22×C5#83512yescomplemented
360C22×C5C2×#21711yescharacteristic
361C2×D5—21no—
362C2×D5—21no—
363C2×D5—22no—
364C2×D5—22no—
365C2×D5—22no—
366C2×D5—22no—
367C2×D5#83512yescomplemented
368C2×D5#93511yescharacteristic
369C2×D5#93711yescharacteristic
370C2×D5C2×#21511yescharacteristic
371C2×D5C2×#21611yescharacteristic
372C2×D5#94911yescharacteristic
373C2×D5#95111yescharacteristic
374AGL(1,𝔽5)—124no—
375C4×C5—21no—
376C4×C5—41no—
377C4×C5—41no—
378C4×C5D4×Dic312yes—
379Dic5—21no—
380Dic5—41no—
381Dic5—41no—
382Dic5#93912yes—
383C24—51no—
384C2×Q8—51no—
385C2×Q8—101no—
386C2×Q8—102no—
387C2×Q8—102no—
388C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic, complemented
389C22×C4—51no—
390C22×C4—51no—
391C22×C4—51no—
392C22×C4—51no—
393C22×C4—51no—
394C22×C4—102no—
395C22×C4—102no—
396C22×C4—101no—
397C22×C4—101no—
398C22×C4—302no—
399C22×C4—304no—
400C22×C4—602no—
401C22×C4—602no—
402C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
403C2×C8—302no—
404C2×C8—302no—
405C2×C8—302no—
406C4⋊C4—21no—
407C4⋊C4—51no—
408C4⋊C4—101no—
409C4⋊C4—101no—
410C4⋊C4—101no—
411C4⋊C4—101no—
412C4⋊C4—154no—
413C4⋊C4—302no—
414C4⋊C4—304no—
415C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic, complemented
416C22⋊C4—21no—
417C22⋊C4—101no—
418C22⋊C4—101no—
419C22⋊C4—101no—
420C3×C5#187211yescharacteristic, complemented, Hall
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—51no—
424C22×C3—51no—
425C22×C3—51no—
426C22×C3—51no—
427C22×C3—51no—
428C22×C3—51no—
429C22×C3—52no—
430C22×C3—102no—
431C22×C3—102no—
432C22×C3—102no—
433C22×C3—102no—
434C22×C3—101no—
435C22×C3—101no—
436C22×C3#400912yescomplemented
437C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
438C3×C4—21no—
439C3×C4—41no—
440C3×C4—41no—
441C3×C4—52no—
442C3×C4—101no—
443C3×C4—201no—
444C3×C4—201no—
445C3×C4D4×AGL(1,𝔽5)12yes—
446Dic3—204no—
447C2×C5—22no—
448C2×C5#522011yescharacteristic
449C2×C5C2×#83511yescharacteristic
450C2×C5#523211yescharacteristic
451D5—22no—
452D5#504311yescharacteristic
453D5#504411yescharacteristic
454D5#504611yescharacteristic
455D5#505011yescharacteristic
456C23—51no—
457C23—51no—
458C23—52no—
459C23—52no—
460C23—101no—
461C23—101no—
462C23—101no—
463C23—101no—
464C23#676311yescharacteristic
465Q8—22no—
466Q8—102no—
467Q8—202no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—51no—
472C2×C4—52no—
473C2×C4—52no—
474C2×C4—51no—
475C2×C4—102no—
476C2×C4—101no—
477C2×C4—101no—
478C2×C4—102no—
479C2×C4—102no—
480C2×C4—101no—
481C2×C4—101no—
482C2×C4—201no—
483C2×C4—201no—
484C2×C4—201no—
485C2×C4—201no—
486C2×C4—304no—
487C2×C4—602no—
488C2×C4—602no—
489C2×C4—608no—
490C2×C4#676311yescharacteristic
491C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
492C8—602no—
493C2×C3—22no—
494C2×C3—51no—
495C2×C3—51no—
496C2×C3—51no—
497C2×C3—51no—
498C2×C3—102no—
499C2×C3#6577911yescharacteristic
500C2×C3C2×#400911yescharacteristic
501C2×C3#6579311yescharacteristic
502C5#7169211yescharacteristic, complemented, Sylow
503C22—21no—
504C22—21no—
505C22—51no—
506C22—51no—
507C22—51no—
508C22—51no—
509C22—51no—
510C22—51no—
511C22—52no—
512C22—102no—
513C22—102no—
514C22—102no—
515C22—102no—
516C22—101no—
517C22—101no—
518C22#9123812yescomplemented
519C22C2×#676311yescharacteristic
520C4—21no—
521C4—41no—
522C4—41no—
523C4—52no—
524C4—101no—
525C4—201no—
526C4—201no—
527C4—604no—
528C4D4×ΓL(1,𝔽16)12yes—
529C3#1060452311yescharacteristic, complemented, Sylow
530C2—22no—
531C2—51no—
532C2—51no—
533C2—51no—
534C2—51no—
535C2—102no—
536C2#1175243811yescharacteristic
537C2C2×#9123811yescharacteristic
538C2#1175245211yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×C22⋊Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D5D4
cosocleC2×C3×C4×D5C23
hypercenterC22⋊Q8ΓL(1,𝔽16)
hypocenterC3×C5#1872
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#91238C2semidirect
5. #11746745C2semidirect
14. C3×C5×C22⋊Q8C4semidirect
15. C3×#4092C4semidirect
16. C3×#4070C4semidirect
17. C3×#4043C4semidirect
37. C2×C3×C5×Q8C2×C4semidirect
51. C2×C3×Dic10C2×C4semidirect
59. C3×C5×C4⋊C4C2×C4semidirect
65. C3×C4⋊Dic5C2×C4semidirect
81. C5×C22⋊Q8Dic3semidirect
82. #4092Dic3semidirect
83. #4070Dic3semidirect
84. #4043Dic3semidirect
139. C3×C22⋊Q8AGL(1,𝔽5)semidirect
155. C2×C5×Q8C2×Dic3semidirect
169. C2×Dic10C2×Dic3semidirect
177. C5×C4⋊C4C2×Dic3semidirect
183. C4⋊Dic5C2×Dic3semidirect
194. C22×C3×C5#103semidirect
202. C2×C3×D5#103semidirect
223. C2×C3×Q8C2×AGL(1,𝔽5)semidirect
244. C3×C4⋊C4C2×AGL(1,𝔽5)semidirect
301. C22⋊Q8ΓL(1,𝔽16)semidirect
359. C22×C5#835semidirect
367. C2×D5#835semidirect
388. C2×Q8C2×ΓL(1,𝔽16)semidirect
415. C4⋊C4C2×ΓL(1,𝔽16)semidirect
420. C3×C5#1872semidirect
436. C22×C3#4009semidirect
502. C5#71692semidirect
518. C22#91238semidirect
529. C3#10604523semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊ΓL(1,𝔽16)2
C3×D5×C22⋊Q82
C2×#912382
D5⋊#8302
#117467452
#106045233
#716925