Finite Group Databasebeta

← #12101482

Subgroups — #12101482

Order 1280.

974 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65170C211yesmaximal, characteristic
2D5×#389C211yesmaximal, characteristic
3C4×#4038C211yesmaximal, characteristic
4C5×#1921C211yesmaximal, characteristic, complemented
5#10601402C211yesmaximal, characteristic, complemented
6#10600395C211yesmaximal, characteristic, complemented
7#10600254C211yesmaximal, characteristic, complemented
8#10599660C211yesmaximal, characteristic
9#10599660C211yesmaximal, characteristic
10#10599660C211yesmaximal, characteristic
11#10599610C211yesmaximal, characteristic, complemented
12#10599609C211yesmaximal, characteristic, complemented
13#10599580C211yesmaximal, characteristic, complemented
14#10599453C211yesmaximal, characteristic
15#10599416C211yesmaximal, characteristic, complemented
16C22×#676C2211yescharacteristic
17C22×#676C2211yescharacteristic
18C2×#4038C2211yescharacteristic
19C2×#4038C2211yescharacteristic
20C2×D5×C4⋊C4—21no—
21C2×D5×C4⋊C4C2211yescharacteristic
22C2×D5×C4⋊C4C2211yescharacteristic
23C2×C42×D5C2211yescharacteristic
24C2×C5×#107C2211yescharacteristic
25C5×#389C2211yescharacteristic
26C4×C5×C4⋊C4C2211yescharacteristic
27C2×#3949C2211yescharacteristic
28#65274C2211yescharacteristic
29C4⋊C4×Dic5C2211yescharacteristic
30C4⋊C4×Dic5C2211yescharacteristic
31#65262C2211yescharacteristic
32#65261—21no—
33C2×#3940C2211yescharacteristic
34C2×#3939C2211yescharacteristic
35C4×#685—21no—
36C4×C4⋊Dic5C2211yescharacteristic
37C42×Dic5—21no—
38#65217C2211yescharacteristic
39#65170C418yescomplemented
40#64955—21no—
41D5×#95—21no—
42#64937—21no—
43#64936—21no—
44C5×#340C2211yescharacteristic
45C5×#340C2211yescharacteristic
46C5×#340C2211yescharacteristic
47C5×#336C2211yescharacteristic
48#64771C2211yescharacteristic
49#64771C2211yescharacteristic
50#64771C2211yescharacteristic
51#64770C2211yescharacteristic
52#64770C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64744C2211yescharacteristic
60#21034—51nomaximal, Sylow
61C22×C4×D5—21no—
62C22×C4×D5—21no—
63C22×C4×D5D411yescharacteristic
64C22×C4×D5D411yescharacteristic
65C22×C4×D5C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C4⋊C4C2311yescharacteristic
73C2×C42×C5—21no—
74C2×C42×C5C2311yescharacteristic
75C2×#685—21no—
76C2×#685—21no—
77C2×C4⋊Dic5—21no—
78C2×C4⋊Dic5C2311yescharacteristic
79C2×C4⋊Dic5C2311yescharacteristic
80C2×C4⋊Dic5C2311yescharacteristic
81C2×C4⋊Dic5C2311yescharacteristic
82C2×#683—21no—
83C2×#683—21no—
84C2×#683—21no—
85C2×#683—21no—
86C2×C4×Dic5—21no—
87C2×C4×Dic5—21no—
88C2×C4×Dic5—21no—
89C2×C4×Dic5—21no—
90C2×C4×Dic5—21no—
91C2×C4×Dic5D411yescharacteristic
92C2×C4×Dic5D411yescharacteristic
93C2×C4×Dic5C2311yescharacteristic
94C22×#168C2311yescharacteristic
95C22×#168C2311yescharacteristic
96C2×#676—22no—
97C2×#676—22no—
98C2×#676—22no—
99C2×#676—22no—
100C2×#676—22no—
101C2×#676C2×C414yes—
102#4038—22no—
103#4038—22no—
104#4038C2×C414yes—
105#4038Q814yes—
106D5×C4⋊C4—22no—
107D5×C4⋊C4—22no—
108D5×C4⋊C4—22no—
109D5×C4⋊C4—44no—
110D5×C4⋊C4C2×C414yes—
111C42×D5—22no—
112C42×D5—22no—
113C5×#107C2×C414yescomplemented
114C5×#95—21no—
115C5×#95—21no—
116#3963—21no—
117#3963—21no—
118#3963—21no—
119#3963—21no—
120#3963—21no—
121#3963—21no—
122#3949C2×C414yescomplemented
123#3940C2×C414yescomplemented
124#3939C2×C414yescomplemented
125C2×#428—51no—
126C2×#389—51no—
127C4×#117—51no—
128#1921—51no—
129#1921D511yescharacteristic, complemented
130#1920—51no—
131#1920—51no—
132#1919—51no—
133#1919—51no—
134#1918—51no—
135#1918—51no—
136#1736—51no—
137#1736—51no—
138#1736—51no—
139#1728—51no—
140C23×D5C2×D411yescharacteristic
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2411yescharacteristic
148C22×Dic5—21no—
149C22×Dic5—21no—
150C22×Dic5—21no—
151C22×Dic5—21no—
152C22×Dic5C2×D411yescharacteristic
153C22×Dic5C2×D411yescharacteristic
154C22×Dic5C2×D411yescharacteristic
155C2×C4×D5—24no—
156C2×C4×D5—22no—
157C2×C4×D5—22no—
158C2×C4×D5—22no—
159C2×C4×D5—22no—
160C2×C4×D5—41no—
161C2×C4×D5—41no—
162C2×C4×D5—42no—
163C2×C4×D5—42no—
164C2×C4×D5—42no—
165C2×C4×D5—42no—
166C2×C4×D5—42no—
167C2×C4×D5C22×C412yes—
168C2×C4×D5C2×Q812yes—
169C2×C4×D5#4112yes—
170C2×C4×D5#4112yes—
171C2×C4×D5#4112yes—
172C2×C4×D5#4112yes—
173C2×C5×C8—21no—
174C2×C5×C8—21no—
175C2×C5×C8—21no—
176C2×C5×C8—21no—
177C2×C5×C8—21no—
178C2×C5×C8C22×C412yes—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—21no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—42no—
185C5×C4⋊C4C22×C412yes—
186C5×C4⋊C4C22×C412yes—
187C5×C4⋊C4C2×Q812yes—
188C42×C5—21no—
189C42×C5—21no—
190C42×C5—42no—
191#685—24no—
192#685—24no—
193C4⋊Dic5—21no—
194C4⋊Dic5—21no—
195C4⋊Dic5—21no—
196C4⋊Dic5—21no—
197C4⋊Dic5—21no—
198C4⋊Dic5—42no—
199C4⋊Dic5C22×C412yes—
200C4⋊Dic5C22×C412yes—
201C4⋊Dic5C2×Q812yes—
202#683—24no—
203#683—42no—
204#683—42no—
205#683—42no—
206C4×Dic5—21no—
207C4×Dic5—21no—
208C4×Dic5—24no—
209C4×Dic5—24no—
210C4×Dic5—22no—
211C4×Dic5—22no—
212C4×Dic5—42no—
213C4×Dic5—42no—
214C4×Dic5—42no—
215C2×#168—21no—
216C2×#168—21no—
217C2×#168—21no—
218C2×#168—21no—
219C2×#168—21no—
220C2×#168C22×C412yes—
221#676—28no—
222#676—44no—
223C22×M16—51no—
224C22×M16—51no—
225C2×#117—51no—
226C2×#117—51no—
227C22×C4⋊C4—51no—
228C22×C4⋊C4—51no—
229C22×C4⋊C4—101no—
230C22×C42—51no—
231#428—58no—
232C2×#107—51no—
233C2×#107C2×D511yescharacteristic
234C2×#106—51no—
235C2×#106—51no—
236#389—51no—
237#389—51no—
238#389—51no—
239#389—101no—
240#389—101no—
241#389C2×D511yescharacteristic
242C4×C4⋊C4—51no—
243C4×C4⋊C4—51no—
244C4×C4⋊C4—51no—
245C4×C4⋊C4C2×D511yescharacteristic
246C4×C22⋊C4—101no—
247C4×C22⋊C4—101no—
248#376—101no—
249C2×#95—101no—
250C43—101no—
251#340—51no—
252#340—51no—
253#340—51no—
254#340—51no—
255#340—51no—
256#340—51no—
257#340—51no—
258#340—51no—
259#340—51no—
260#340C2×D511yescharacteristic
261#340C2×D511yescharacteristic
262#340C2×D511yescharacteristic
263#336—51no—
264#336—51no—
265#336—51no—
266#336C2×D511yescharacteristic
267C23×C5C22×D411yescharacteristic
268C22×D5—21no—
269C22×D5—21no—
270C22×D5—21no—
271C22×D5—21no—
272C22×D5C4×D412yes—
273C22×D5C22⋊Q812yes—
274C22×D5#12312yes—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5#13712yes—
293C2×C4×C5C23×C411yescharacteristic
294C2×C4×C5C22×Q811yescharacteristic
295C2×C4×C5C2×#4111yescharacteristic
296C2×C4×C5C2×#4111yescharacteristic
297C2×C4×C5C2×#4111yescharacteristic
298C2×C4×C5C2×#4111yescharacteristic
299C2×Dic5—21no—
300C2×Dic5—21no—
301C2×Dic5—22no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312C2×Dic5—41no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315C2×Dic5—41no—
316C2×Dic5C4×D412yes—
317C2×Dic5C22⋊Q812yes—
318C2×Dic5#12312yes—
319C2×Dic5AGL(1,ℤ/8)12yes—
320C4×D5—44no—
321C4×D5—44no—
322C4×D5—48no—
323C4×D5—84no—
324C4×D5#11714yes—
325C4×D5C4×Q814yes—
326C4×D5#12514yes—
327C4×D5#12614yes—
328C5×C8—22no—
329C5×C8—41no—
330#168—22no—
331#168—41no—
332C23×C4—51no—
333C23×C4—51no—
334C23×C4—51no—
335C23×C4—101no—
336C23×C4—101no—
337C2×M16—54no—
338C2×M16—102no—
339C2×M16—102no—
340C2×M16—102no—
341C2×M16—102no—
342C2×M16—102no—
343C22×C8—51no—
344C22×C8—51no—
345C22×C8C22×D511yescharacteristic
346C22×C8C22×D511yescharacteristic
347#117—54no—
348#117—54no—
349#117—102no—
350#117—102no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—51no—
353C2×C4⋊C4—51no—
354C2×C4⋊C4—51no—
355C2×C4⋊C4—54no—
356C2×C4⋊C4—51no—
357C2×C4⋊C4—102no—
358C2×C4⋊C4—102no—
359C2×C4⋊C4—102no—
360C2×C4⋊C4—101no—
361C2×C4⋊C4—101no—
362C2×C4⋊C4—101no—
363C2×C4⋊C4—101no—
364C2×C4⋊C4—101no—
365C2×C4⋊C4—204no—
366C2×C4⋊C4C22×D511yescharacteristic
367C2×C4⋊C4C22×D511yescharacteristic
368C2×C4⋊C4C22×D511yescharacteristic
369C2×C4⋊C4C22×D511yescharacteristic
370C2×C22⋊C4—101no—
371C2×C22⋊C4—101no—
372C2×C42—21no—
373C2×C42—51no—
374C2×C42—51no—
375C2×C42—51no—
376C2×C42—102no—
377C2×C42—102no—
378C2×C42—101no—
379C2×C42—101no—
380C2×C42—101no—
381C2×C42—101no—
382C2×C42—101no—
383C2×C42C22×D511yescharacteristic
384#107—54no—
385#107C4×D514yescomplemented
386#106—54no—
387#106—54no—
388#95—21no—
389#95—21no—
390#95—101no—
391#95—101no—
392#95—101no—
393#95—101no—
394#95—101no—
395#95—101no—
396C22×C5C2×C4×D411yescharacteristic
397C22×C5C2×C22⋊Q811yescharacteristic
398C22×C5C2×#12311yescharacteristic
399C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
400C22×C5C2×#13711yescharacteristic
401C22×C5#57511yescharacteristic
402C22×C5#57511yescharacteristic
403C2×D5—22no—
404C2×D5—22no—
405C2×D5—22no—
406C2×D5—22no—
407C2×D5—22no—
408C2×D5—22no—
409C2×D5#38714yes—
410C4×C5—22no—
411C4×C5—42no—
412C4×C5—41no—
413C4×C5—42no—
414C4×C5—42no—
415C4×C5—81no—
416C4×C5C2×#11711yescharacteristic
417C4×C5C2×C4×Q811yescharacteristic
418C4×C5C2×#12511yescharacteristic
419C4×C5C2×#12611yescharacteristic
420Dic5—22no—
421Dic5—42no—
422Dic5—41no—
423Dic5—42no—
424Dic5—42no—
425Dic5—81no—
426Dic5#38714yes—
427C24—51no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—51no—
433C22×C4—52no—
434C22×C4—52no—
435C22×C4—52no—
436C22×C4—52no—
437C22×C4—52no—
438C22×C4—52no—
439C22×C4—51no—
440C22×C4—51no—
441C22×C4—102no—
442C22×C4—102no—
443C22×C4—102no—
444C22×C4—104no—
445C22×C4—101no—
446C22×C4—101no—
447C22×C4—101no—
448C22×C4—102no—
449C22×C4—101no—
450C22×C4—201no—
451C22×C4—201no—
452C22×C4—202no—
453C22×C4—202no—
454C22×C4—202no—
455C22×C4—202no—
456C22×C4—202no—
457C22×C4D4×D511yescharacteristic
458C22×C4D4×D511yescharacteristic
459C22×C4C23×D511yescharacteristic
460M16—108no—
461M16—204no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—52no—
468C2×C8—101no—
469C2×C8—101no—
470C2×C8—101no—
471C2×C8—101no—
472C2×C8—101no—
473C2×C8C2×C4×D512yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—52no—
481C4⋊C4—52no—
482C4⋊C4—52no—
483C4⋊C4—104no—
484C4⋊C4—101no—
485C4⋊C4—101no—
486C4⋊C4—101no—
487C4⋊C4—101no—
488C4⋊C4—101no—
489C4⋊C4—202no—
490C4⋊C4—202no—
491C4⋊C4—202no—
492C4⋊C4—202no—
493C4⋊C4C2×C4×D512yes—
494C4⋊C4C2×C4×D512yes—
495C4⋊C4Q8×D512yes—
496C22⋊C4—104no—
497C22⋊C4—104no—
498C42—21no—
499C42—21no—
500C42—42no—
501C42—102no—
502C42—102no—
503C42—104no—
504C42—104no—
505C42—101no—
506C42—101no—
507C42—202no—
508C42—202no—
509C42—202no—
510C2×C5C2×#38711yescharacteristic
511C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
512C2×C5C4×#13711yescharacteristic
513C2×C5#307911yescharacteristic
514C2×C5#308011yescharacteristic
515C2×C5#309411yescharacteristic
516C2×C5#309511yescharacteristic
517D5—24no—
518C23—52no—
519C23—52no—
520C23—52no—
521C23—101no—
522C23—101no—
523C23—101no—
524C23—101no—
525C23C2×D4×D511yescharacteristic
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—52no—
544C2×C4—52no—
545C2×C4—52no—
546C2×C4—54no—
547C2×C4—54no—
548C2×C4—54no—
549C2×C4—54no—
550C2×C4—52no—
551C2×C4—101no—
552C2×C4—101no—
553C2×C4—101no—
554C2×C4—101no—
555C2×C4—102no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—101no—
559C2×C4—204no—
560C2×C4—204no—
561C2×C4—208no—
562C2×C4—201no—
563C2×C4—201no—
564C2×C4—201no—
565C2×C4—201no—
566C2×C4—201no—
567C2×C4—201no—
568C2×C4—201no—
569C2×C4—201no—
570C2×C4—201no—
571C2×C4—404no—
572C2×C4#406412yes—
573C2×C4C22×C4×D511yescharacteristic
574C2×C4C2×Q8×D511yescharacteristic
575C2×C4D5×#4111yescharacteristic
576C2×C4D5×#4111yescharacteristic
577C2×C4D5×#4111yescharacteristic
578C2×C4D5×#4111yescharacteristic
579C8—22no—
580C8—41no—
581C8—102no—
582C8—201no—
583C5#2103411yescharacteristic, complemented, Sylow
584C22—54no—
585C22—102no—
586C22—102no—
587C22—102no—
588C22—102no—
589C22—102no—
590C22—102no—
591C22C4×D4×D511yescharacteristic
592C22D5×C22⋊Q811yescharacteristic
593C22D5×#12311yescharacteristic
594C22C2×#406411yescharacteristic
595C22#6610211yescharacteristic
596C22D5×AGL(1,ℤ/8)11yescharacteristic
597C22#6610711yescharacteristic
598C4—22no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—54no—
604C4—81no—
605C4—102no—
606C4—202no—
607C4—202no—
608C4—201no—
609C4—202no—
610C4—401no—
611C4D5×#11711yescharacteristic
612C4C4×Q8×D511yescharacteristic
613C4D5×#12511yescharacteristic
614C4D5×#12611yescharacteristic
615C2—104no—
616C2D5×#38711yescharacteristic
617C2C4×#406411yescharacteristic
618C2#1061159611yescharacteristic
619C2#1061284011yescharacteristic
620C2#1061289811yescharacteristic
621C2#1061310411yescharacteristic
622C2#1061320011yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1921C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1921D5
hypocenterC5#21034
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
4. C5×#1921C2semidirect
5. #10601402C2semidirect
6. #10600395C2semidirect
7. #10600254C2semidirect
11. #10599610C2semidirect
12. #10599609C2semidirect
13. #10599580C2semidirect
15. #10599416C2semidirect
39. #65170C4semidirect
113. C5×#107C2×C4semidirect
122. #3949C2×C4semidirect
123. #3940C2×C4semidirect
124. #3939C2×C4semidirect
129. #1921D5semidirect
385. #107C4×D5semidirect
583. C5#21034semidirect

Maximal subgroups

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

subgroupindex
C2×#651702
D5×#3892
C4×#40382
C5×#19212
#106014022
#106003952
#106002542
#105996602
#105996102
#105996092
#105995802
#105994532
#105994162
#210345