Finite Group Databasebeta

← #14299458

Subgroups — #14299458

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91989C211yesmaximal, characteristic
2#11752424C211yesmaximal, characteristic
3C2×#91223C211yesmaximal, characteristic, complemented
4C2×#91222C211yesmaximal, characteristic, complemented
5C3×D5×C22⋊C8C211yesmaximal, characteristic
6#11746764C211yesmaximal, characteristic, complemented
7#11746764C211yesmaximal, characteristic, complemented
8#10603912—31nomaximal, Hall
9C22×C3×C4×D5C2211yescharacteristic
10C2×#6763—21no—
11C2×#6763—21no—
12#91989—22no—
13#91989—22no—
14C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
15C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
16C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
17C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
18C2×C4×ΓL(1,𝔽16)—21no—
19C2×C3×C8×D5C2211yescharacteristic
20C2×C3×C8×D5C2211yescharacteristic
21#91235—21no—
22#91223—22no—
23#91222—22no—
24C3×C5×C22⋊C8C411yescharacteristic, complemented
25C3×C22⋊#168C411yescharacteristic, complemented
26C3×#3952C411yescharacteristic, complemented
27C3×#3952C411yescharacteristic, complemented
28#69929—51nomaximal, Hall
29C2×#4130—31no—
30#65765—31no—
31C2×#3994—31no—
32C2×#3993—31no—
33D5×C22⋊C8D311yescharacteristic
34#64893—31no—
35#64893—31no—
36C23×C3×D5D411yescharacteristic
37C22×ΓL(1,𝔽16)—21no—
38C22×ΓL(1,𝔽16)—21no—
39C22×ΓL(1,𝔽16)—21no—
40C22×ΓL(1,𝔽16)—21no—
41C22×C3×C4×C5C2×C411yescharacteristic
42C22×C3×Dic5Q811yescharacteristic
43C2×C3×C4×D5—21no—
44C2×C3×C4×D5—21no—
45C2×C3×C4×D5—21no—
46C2×C3×C4×D5—21no—
47C2×C3×C4×D5C2×C411yescharacteristic
48C2×C3×C4×D5D411yescharacteristic
49C2×C3×C4×D5Q811yescharacteristic
50C2×C3×C4×D5C2311yescharacteristic
51#6763—24no—
52#6763—42no—
53C4⋊ΓL(1,𝔽16)—21no—
54C4⋊ΓL(1,𝔽16)—21no—
55C4⋊ΓL(1,𝔽16)—21no—
56C4⋊ΓL(1,𝔽16)—21no—
57C4⋊ΓL(1,𝔽16)—21no—
58C4⋊ΓL(1,𝔽16)—21no—
59C4⋊ΓL(1,𝔽16)D412yes—
60C4⋊ΓL(1,𝔽16)D412yes—
61C4×ΓL(1,𝔽16)—24no—
62C2×C3×C5×C8C2×C411yescharacteristic
63C2×C3×C5×C8C2×C411yescharacteristic
64C2×C3×#168C2×C411yescharacteristic
65C2×C3×#168C2×C411yescharacteristic
66C3×C8×D5—21no—
67C3×C8×D5—21no—
68C3×C8×D5—21no—
69C3×C8×D5—21no—
70C2×#926—51no—
71C2×C3×C22⋊C8—51no—
72#5212—51no—
73C2×#818—51no—
74C2×#817—51no—
75#4551—51no—
76#4551—51no—
77C22×C4×D5C2×D311yescharacteristic
78C2×C22⋊AGL(1,𝔽5)—61no—
79C2×C22⋊AGL(1,𝔽5)—61no—
80#4130—62no—
81#4130—62no—
82C2×C4⋊AGL(1,𝔽5)—31no—
83C2×C4⋊AGL(1,𝔽5)—31no—
84C2×C4⋊AGL(1,𝔽5)—31no—
85C2×C4⋊AGL(1,𝔽5)—31no—
86C2×C4×AGL(1,𝔽5)—61no—
87C2×C8×D5C2×D311yescharacteristic
88C2×C8×D5C2×D311yescharacteristic
89(C2×AGL(1,𝔽5))⋊C4—61no—
90#3994—62no—
91#3993—62no—
92C5×C22⋊C8Dic311yescharacteristic, complemented
93C22⋊#168Dic311yescharacteristic, complemented
94#3952Dic311yescharacteristic, complemented
95#3952Dic311yescharacteristic, complemented
96#1854—151noSylow
97C23×C3×C5C4⋊C411yescharacteristic
98C22×C3×D5—21no—
99C22×C3×D5—21no—
100C22×C3×D5—21no—
101C22×C3×D5—21no—
102C22×C3×D5—21no—
103C22×C3×D5—21no—
104C22×C3×D5C4⋊C411yescharacteristic
105C22×C3×D5C2×D411yescharacteristic
106C2×ΓL(1,𝔽16)—22no—
107C2×ΓL(1,𝔽16)—22no—
108C2×ΓL(1,𝔽16)—41no—
109C2×ΓL(1,𝔽16)—41no—
110C2×ΓL(1,𝔽16)—41no—
111C2×ΓL(1,𝔽16)—41no—
112C2×ΓL(1,𝔽16)—41no—
113C2×ΓL(1,𝔽16)—41no—
114C2×ΓL(1,𝔽16)—41no—
115C2×ΓL(1,𝔽16)—41no—
116C2×ΓL(1,𝔽16)—41no—
117C2×ΓL(1,𝔽16)—41no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5C4⋊C411yescharacteristic
121C2×C3×C4×C5C22×C411yescharacteristic
122C2×C3×Dic5—21no—
123C2×C3×Dic5—21no—
124C2×C3×Dic5C4⋊C411yescharacteristic
125C2×C3×Dic5C2×Q811yescharacteristic
126C3×C4×D5—21no—
127C3×C4×D5—21no—
128C3×C4×D5—21no—
129C3×C4×D5—21no—
130C3×C4×D5—21no—
131C3×C4×D5—21no—
132C3×C4×D5C2×D411yescharacteristic
133C3×C4×D5C2×D411yescharacteristic
134C3×C4×D5#4111yescharacteristic
135C3×C4×D5#4111yescharacteristic
136C3×C5×C8—21no—
137C3×C5×C8—21no—
138C3×#168—21no—
139C3×#168—21no—
140C23×C3×C4—51no—
141C22×C3×C8—51no—
142C22×C3×C8—51no—
143C2×#217—101no—
144C2×#217—101no—
145#926—102no—
146#926—102no—
147C2×C4⋊Dic3—51no—
148C2×C4⋊Dic3—51no—
149C2×C4⋊Dic3—51no—
150C2×C4⋊Dic3—51no—
151C2×C4×Dic3—101no—
152C3×C22⋊C8—51no—
153C3×C22⋊C8—51no—
154C3×C22⋊C8—51no—
155C3×C22⋊C8AGL(1,𝔽5)11yescharacteristic, complemented
156C3⋊#95—101no—
157#818—102no—
158#817—102no—
159C23×D5D1211yescharacteristic
160C22×AGL(1,𝔽5)—61no—
161C22×AGL(1,𝔽5)—61no—
162C22×AGL(1,𝔽5)—61no—
163C22×AGL(1,𝔽5)—61no—
164C22×C4×C5C2×Dic311yescharacteristic
165C22×Dic5Dic611yescharacteristic
166C2×C4×D5—21no—
167C2×C4×D5—21no—
168C2×C4×D5—21no—
169C2×C4×D5—21no—
170C2×C4×D5Dic611yescharacteristic
171C2×C4×D5D1211yescharacteristic
172C2×C4×D5C2×Dic311yescharacteristic
173C2×C4×D5C22×D311yescharacteristic
174C22⋊AGL(1,𝔽5)—64no—
175C22⋊AGL(1,𝔽5)—122no—
176C4⋊AGL(1,𝔽5)—32no—
177C4⋊AGL(1,𝔽5)—32no—
178C4⋊AGL(1,𝔽5)—61no—
179C4⋊AGL(1,𝔽5)—61no—
180C4⋊AGL(1,𝔽5)—61no—
181C4⋊AGL(1,𝔽5)—61no—
182C4⋊AGL(1,𝔽5)—61no—
183C4⋊AGL(1,𝔽5)—61no—
184C4×AGL(1,𝔽5)—64no—
185C2×C5×C8C2×Dic311yescharacteristic
186C2×C5×C8C2×Dic311yescharacteristic
187C2×#168C2×Dic311yescharacteristic
188C2×#168C2×Dic311yescharacteristic
189C8×D5—21no—
190C8×D5—21no—
191C8×D5—21no—
192C8×D5—21no—
193C2×#117—151no—
194C2×#107—151no—
195C2×#106—151no—
196C2×C22⋊C8—51no—
197#380—151no—
198#340—151no—
199#340—151no—
200C22×C3×C5—21no—
201C22×C3×C5—21no—
202C22×C3×C5—21no—
203C22×C3×C5C2×C4⋊C411yescharacteristic
204C2×C3×D5—21no—
205C2×C3×D5—21no—
206C2×C3×D5—21no—
207C2×C3×D5—41no—
208C2×C3×D5—41no—
209C2×C3×D5—41no—
210C2×C3×D5—41no—
211C2×C3×D5#12011yescharacteristic
212C2×C3×D5#12311yescharacteristic
213C2×C3×D5#13511yescharacteristic
214C2×C3×D5#13511yescharacteristic
215C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
216C2×C3×D5#13711yescharacteristic
217ΓL(1,𝔽16)—42no—
218ΓL(1,𝔽16)—42no—
219ΓL(1,𝔽16)—81no—
220ΓL(1,𝔽16)—81no—
221C3×C4×C5—21no—
222C3×C4×C5C4×D411yescharacteristic
223C3×C4×C5C4×D411yescharacteristic
224C3×Dic5—21no—
225C3×Dic5C22⋊Q811yescharacteristic
226C3×Dic5C22⋊Q811yescharacteristic
227C24×C3—51no—
228C22×C3×C4—51no—
229C22×C3×C4—51no—
230C22×C3×C4—51no—
231C22×C3×C4—51no—
232C22×C3×C4—51no—
233C22×C3×C4—101no—
234C22×C3×C4—101no—
235C22×C3×C4—101no—
236C22×C3×C4—101no—
237C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
238C22×Dic3—101no—
239C22×Dic3—101no—
240C22×Dic3—101no—
241C22×Dic3—101no—
242C2×C3×C8—51no—
243C2×C3×C8—51no—
244C2×C3×C8—101no—
245C2×C3×C8—101no—
246C2×C3×C8—101no—
247C2×C3×C8—101no—
248C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
249C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
250#217—104no—
251#217—202no—
252C4⋊Dic3—52no—
253C4⋊Dic3—52no—
254C4⋊Dic3—101no—
255C4⋊Dic3—101no—
256C4⋊Dic3—101no—
257C4⋊Dic3—101no—
258C4⋊Dic3—101no—
259C4⋊Dic3—101no—
260C4×Dic3—104no—
261C23×C5C4⋊Dic311yescharacteristic
262C22×D5—21no—
263C22×D5—21no—
264C22×D5—21no—
265C22×D5—21no—
266C22×D5—21no—
267C22×D5—21no—
268C22×D5C4⋊Dic311yescharacteristic
269C22×D5C2×D1211yescharacteristic
270C2×AGL(1,𝔽5)—62no—
271C2×AGL(1,𝔽5)—62no—
272C2×AGL(1,𝔽5)—121no—
273C2×AGL(1,𝔽5)—121no—
274C2×AGL(1,𝔽5)—121no—
275C2×AGL(1,𝔽5)—121no—
276C2×AGL(1,𝔽5)—121no—
277C2×AGL(1,𝔽5)—121no—
278C2×AGL(1,𝔽5)—121no—
279C2×AGL(1,𝔽5)—121no—
280C2×AGL(1,𝔽5)—121no—
281C2×AGL(1,𝔽5)—121no—
282C2×C4×C5—21no—
283C2×C4×C5—21no—
284C2×C4×C5C4⋊Dic311yescharacteristic
285C2×C4×C5C22×Dic311yescharacteristic
286C2×Dic5—21no—
287C2×Dic5—21no—
288C2×Dic5C4⋊Dic311yescharacteristic
289C2×Dic5C2×Dic611yescharacteristic
290C4×D5—21no—
291C4×D5—21no—
292C4×D5—21no—
293C4×D5—21no—
294C4×D5—21no—
295C4×D5—21no—
296C4×D5D3×D411yescharacteristic
297C4×D5D3×D411yescharacteristic
298C4×D5#23711yescharacteristic
299C4×D5#23711yescharacteristic
300C5×C8—21no—
301C5×C8—21no—
302#168—21no—
303#168—21no—
304C23×C4—51no—
305C22×C8—51no—
306C22×C8—51no—
307#117—302no—
308#117—302no—
309C2×C4⋊C4—151no—
310C2×C4⋊C4—151no—
311C2×C4⋊C4—151no—
312C2×C4⋊C4—151no—
313C2×C22⋊C4—301no—
314C2×C22⋊C4—301no—
315C2×C42—301no—
316#107—302no—
317#106—302no—
318C22⋊C8—51no—
319C22⋊C8—51no—
320C22⋊C8—51no—
321C22⋊C8ΓL(1,𝔽16)11yescharacteristic, complemented
322#95—301no—
323C2×C3×C5—41no—
324C2×C3×C5#38511yescharacteristic
325C2×C3×C5#42711yescharacteristic
326C2×C3×C5#42811yescharacteristic
327C3×D5—41no—
328C3×D5#44911yescharacteristic
329C3×D5#45211yescharacteristic
330C3×D5#48211yescharacteristic
331C3×D5#48511yescharacteristic
332C23×C3—51no—
333C23×C3—51no—
334C23×C3—101no—
335C23×C3—101no—
336C23×C3—101no—
337C23×C3—101no—
338C23×C3—101no—
339C23×C3—101no—
340C23×C3C4⋊AGL(1,𝔽5)11yescharacteristic
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—51no—
344C2×C3×C4—51no—
345C2×C3×C4—51no—
346C2×C3×C4—51no—
347C2×C3×C4—51no—
348C2×C3×C4—51no—
349C2×C3×C4—101no—
350C2×C3×C4—101no—
351C2×C3×C4—101no—
352C2×C3×C4—101no—
353C2×C3×C4—101no—
354C2×C3×C4—101no—
355C2×C3×C4—101no—
356C2×C3×C4—101no—
357C2×C3×C4C4⋊AGL(1,𝔽5)11yescharacteristic
358C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
359C2×Dic3—102no—
360C2×Dic3—102no—
361C2×Dic3—201no—
362C2×Dic3—201no—
363C2×Dic3—201no—
364C2×Dic3—201no—
365C2×Dic3—201no—
366C2×Dic3—201no—
367C2×Dic3—201no—
368C2×Dic3—201no—
369C2×Dic3—201no—
370C2×Dic3—201no—
371C3×C8—21no—
372C3×C8—21no—
373C3×C8—101no—
374C3×C8—101no—
375C22×C5—21no—
376C22×C5—21no—
377C22×C5—21no—
378C22×C5C2×C4⋊Dic311yescharacteristic
379C2×D5—21no—
380C2×D5—21no—
381C2×D5—21no—
382C2×D5—41no—
383C2×D5—41no—
384C2×D5—41no—
385C2×D5—41no—
386C2×D5#88211yescharacteristic
387C2×D5#88611yescharacteristic
388C2×D5#90411yescharacteristic
389C2×D5#90411yescharacteristic
390C2×D5#90811yescharacteristic
391C2×D5#90911yescharacteristic
392AGL(1,𝔽5)—122no—
393AGL(1,𝔽5)—122no—
394AGL(1,𝔽5)—241no—
395AGL(1,𝔽5)—241no—
396C4×C5—21no—
397C4×C5D4×Dic311yescharacteristic
398C4×C5D4×Dic311yescharacteristic
399Dic5—21no—
400Dic5#87811yescharacteristic
401Dic5#87811yescharacteristic
402C24—51no—
403C22×C4—51no—
404C22×C4—51no—
405C22×C4—51no—
406C22×C4—51no—
407C22×C4—51no—
408C22×C4—101no—
409C22×C4—101no—
410C22×C4—101no—
411C22×C4—101no—
412C22×C4—301no—
413C22×C4—301no—
414C22×C4—301no—
415C22×C4—301no—
416C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
417C2×C8—51no—
418C2×C8—51no—
419C2×C8—101no—
420C2×C8—101no—
421C2×C8—101no—
422C2×C8—101no—
423C2×C8C2×ΓL(1,𝔽16)11yescharacteristic
424C2×C8C2×ΓL(1,𝔽16)11yescharacteristic
425C4⋊C4—152no—
426C4⋊C4—152no—
427C4⋊C4—301no—
428C4⋊C4—301no—
429C4⋊C4—301no—
430C4⋊C4—301no—
431C4⋊C4—301no—
432C4⋊C4—301no—
433C22⋊C4—304no—
434C22⋊C4—602no—
435C42—304no—
436C3×C5#185411yescharacteristic, complemented, Hall
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—51no—
441C22×C3—51no—
442C22×C3—51no—
443C22×C3—51no—
444C22×C3—51no—
445C22×C3—51no—
446C22×C3—101no—
447C22×C3—101no—
448C22×C3—101no—
449C22×C3—201no—
450C22×C3—201no—
451C22×C3—201no—
452C22×C3—201no—
453C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
454C3×C4—21no—
455C3×C4—51no—
456C3×C4—51no—
457C3×C4—101no—
458C3×C4D4×AGL(1,𝔽5)11yescharacteristic
459C3×C4D4×AGL(1,𝔽5)11yescharacteristic
460Dic3—202no—
461Dic3—202no—
462Dic3—401no—
463Dic3—401no—
464C2×C5—41no—
465C2×C5#495011yescharacteristic
466C2×C5#510611yescharacteristic
467C2×C5#512111yescharacteristic
468D5—41no—
469D5#472311yescharacteristic
470D5#473311yescharacteristic
471D5#473411yescharacteristic
472D5#473711yescharacteristic
473C23—51no—
474C23—51no—
475C23—101no—
476C23—101no—
477C23—101no—
478C23—101no—
479C23—101no—
480C23—101no—
481C23C4⋊ΓL(1,𝔽16)11yescharacteristic
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—51no—
485C2×C4—51no—
486C2×C4—51no—
487C2×C4—51no—
488C2×C4—51no—
489C2×C4—51no—
490C2×C4—101no—
491C2×C4—101no—
492C2×C4—101no—
493C2×C4—101no—
494C2×C4—101no—
495C2×C4—101no—
496C2×C4—101no—
497C2×C4—101no—
498C2×C4—302no—
499C2×C4—302no—
500C2×C4—601no—
501C2×C4—601no—
502C2×C4—601no—
503C2×C4—601no—
504C2×C4—601no—
505C2×C4—601no—
506C2×C4—601no—
507C2×C4—601no—
508C2×C4—601no—
509C2×C4—601no—
510C2×C4C4⋊ΓL(1,𝔽16)11yescharacteristic
511C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
512C8—21no—
513C8—21no—
514C8—101no—
515C8—101no—
516C2×C3—41no—
517C2×C3—51no—
518C2×C3—51no—
519C2×C3—51no—
520C2×C3—51no—
521C2×C3—201no—
522C2×C3#6569911yescharacteristic
523C2×C3#6572111yescharacteristic
524C2×C3#6572711yescharacteristic
525C5#6992911yescharacteristic, complemented, Sylow
526C22—21no—
527C22—21no—
528C22—21no—
529C22—51no—
530C22—51no—
531C22—51no—
532C22—51no—
533C22—51no—
534C22—51no—
535C22—101no—
536C22—101no—
537C22—101no—
538C22—201no—
539C22—201no—
540C22—201no—
541C22—201no—
542C22C2×C4⋊ΓL(1,𝔽16)11yescharacteristic
543C4—21no—
544C4—51no—
545C4—51no—
546C4—101no—
547C4—602no—
548C4—602no—
549C4—1201no—
550C4—1201no—
551C4D4×ΓL(1,𝔽16)11yescharacteristic
552C4D4×ΓL(1,𝔽16)11yescharacteristic
553C3#1060391211yescharacteristic, complemented, Sylow
554C2—41no—
555C2—51no—
556C2—51no—
557C2—51no—
558C2—51no—
559C2—201no—
560C2#1175235811yescharacteristic
561C2#1175238011yescharacteristic
562C2#1175238611yescharacteristic
subgroupG/H
centerC22C2×C4⋊ΓL(1,𝔽16)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×C22⋊C8C4
socleC22×C3×C5C2×C4⋊C4
prime radicalC23×C3×D5D4
cosocleC2×C3×C4×D5C23
hypercenterC22⋊C8ΓL(1,𝔽16)
hypocenterC3×C5#1854
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×#91223C2semidirect
4. C2×#91222C2semidirect
6. #11746764C2semidirect
7. #11746764C2semidirect
24. C3×C5×C22⋊C8C4semidirect
25. C3×C22⋊#168C4semidirect
26. C3×#3952C4semidirect
27. C3×#3952C4semidirect
92. C5×C22⋊C8Dic3semidirect
93. C22⋊#168Dic3semidirect
94. #3952Dic3semidirect
95. #3952Dic3semidirect
155. C3×C22⋊C8AGL(1,𝔽5)semidirect
321. C22⋊C8ΓL(1,𝔽16)semidirect
436. C3×C5#1854semidirect
525. C5#69929semidirect
553. C3#10603912semidirect

Maximal subgroups

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

subgroupindex
C2×#919892
#117524242
C2×#912232
C2×#912222
C3×D5×C22⋊C82
#117467642
#106039123
#699295