Finite Group Databasebeta

← #12133618

Subgroups — #12133618

Order 1280.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65804C211yesmaximal, characteristic, complemented
2C4×C5×#121C211yesmaximal, characteristic
3C2×#65308C211yesmaximal, characteristic, complemented
4C2×#65297C211yesmaximal, characteristic, complemented
5#10602478C212yesmaximal
6#10602323C211yesmaximal, characteristic, complemented
7#10600453C212yesmaximal, complemented
8#10600425C212yesmaximal, complemented
9#10600319C212yesmaximal, complemented
10#10600282C211yesmaximal, characteristic, complemented
11#10600279C211yesmaximal, characteristic, complemented
12C2×C5×#121C2211yescharacteristic
13C2×C4×C5×D4—21no—
14C2×C4×C5×D4C2211yescharacteristic
15C22×C42×C5C2211yescharacteristic
16C22×#687C2212yes—
17C2×#4072—21no—
18C2×#4070—21no—
19C2×#4070C2212yes—
20#65804—24no—
21#65804—24no—
22C2×#4016—21no—
23C2×#4015C2211yescharacteristic
24C2×#4015C2211yescharacteristic
25C2×C4×Dic10—21no—
26C2×C4×Dic10C2212yes—
27C5×#390C2211yescharacteristic
28C5×#389C2211yescharacteristic
29C5×#388—21no—
30C5×#386—21no—
31C5×#380C2211yescharacteristic
32C4×C5×C4⋊C4C2211yescharacteristic
33C4×C5×C22⋊C4—21no—
34C2×#3964C2212yes—
35C2×C22⋊#168C2212yes—
36#65308—24no—
37#65297—24no—
38C2×#3942C2212yes—
39C2×#3942C2212yes—
40C2×#3942C2212yes—
41C2×#3940C2212yes—
42C2×C4⋊#168C2211yescharacteristic
43C2×C4⋊#168C2211yescharacteristic
44#64749C2212yes—
45#64749C2212yes—
46#64744C2212yes—
47#24151—51nomaximal, Sylow
48C22×C5×D4—21no—
49C22×C5×D4C2311yescharacteristic
50C23×C4×C5—21no—
51C23×C4×C5D411yescharacteristic
52C23×C4×C5D411yescharacteristic
53C23×C4×C5C2311yescharacteristic
54C22×Dic10—21no—
55C22×Dic10C2312yes—
56C5×#121—22no—
57C5×#121—22no—
58C4×C5×D4—22no—
59C4×C5×D4—22no—
60C4×C5×D4—44no—
61C2×C5×C4⋊C4—21no—
62C2×C5×C4⋊C4C2311yescharacteristic
63C2×C5×C4⋊C4C2311yescharacteristic
64C2×C5×C4⋊C4C2311yescharacteristic
65C2×C5×C22⋊C4—21no—
66C2×C5×C22⋊C4—21no—
67C2×C5×C22⋊C4—21no—
68C2×C5×C22⋊C4—21no—
69C2×C5×C22⋊C4—21no—
70C2×C42×C5—22no—
71C2×C42×C5—22no—
72C2×C42×C5—21no—
73C2×C42×C5D411yescharacteristic
74C2×C42×C5D411yescharacteristic
75C2×C42×C5C2311yescharacteristic
76C2×C42×C5C2311yescharacteristic
77C2×C22⋊Dic5—22no—
78C2×#687—24no—
79C2×#687—24no—
80C2×#687—42no—
81#4072—28no—
82C2×C4⋊Dic5—21no—
83C2×C4⋊Dic5—21no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5C2312yes—
86#4070—28no—
87#4070—24no—
88#4070—24no—
89C2×#683—22no—
90C2×#683—22no—
91C2×C4×Dic5—22no—
92C22×#168C2312yes—
93C22×#168C2312yes—
94#4016—28no—
95#4015—22no—
96#4015—22no—
97#4015D414yescomplemented
98#4015D414yescomplemented
99C4×Dic10—28no—
100C4×Dic10—24no—
101C4×Dic10—24no—
102C5×#95—21no—
103C5×#95—21no—
104C5×#95—21no—
105#3964—24no—
106C22⋊#168—42no—
107#3942—24no—
108#3942—24no—
109#3942—42no—
110#3940—24no—
111C4⋊#168—22no—
112C4⋊#168—22no—
113C2×#533—51no—
114C2×#476—51no—
115C2×#461—51no—
116C4×#121D511yescharacteristic
117#2035—52no—
118#2000—52no—
119#1987—51no—
120#1981—51no—
121#1880—52no—
122#1865—52no—
123#1797—51no—
124C24×C5—21no—
125C24×C5C2×D411yescharacteristic
126C2×C5×D4—24no—
127C2×C5×D4—21no—
128C2×C5×D4—21no—
129C2×C5×D4—21no—
130C2×C5×D4—21no—
131C2×C5×D4—21no—
132C2×C5×D4—42no—
133C2×C5×D4—42no—
134C2×C5×D4#4112yes—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—21no—
138C22×C4×C5—22no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—41no—
148C22×C4×C5—41no—
149C22×C4×C5—42no—
150C22×C4×C5—42no—
151C22×C4×C5—42no—
152C22×C4×C5SD1614yescomplemented
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2411yescharacteristic
159C22×Dic5—22no—
160C22×Dic5—22no—
161C2×Dic10—22no—
162C2×Dic10—22no—
163C2×Dic10—22no—
164C2×Dic10—22no—
165C2×Dic10—24no—
166C2×Dic10—22no—
167C2×Dic10—42no—
168C2×Dic10—42no—
169C2×Dic10C2×D414yes—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—42no—
176C5×C4⋊C4#4112yes—
177C5×C22⋊C4—24no—
178C5×C22⋊C4—42no—
179C5×C22⋊C4—42no—
180C5×C22⋊C4—42no—
181C5×C22⋊C4—42no—
182C42×C5—22no—
183C42×C5—22no—
184C42×C5—21no—
185C42×C5—21no—
186C42×C5—42no—
187C42×C5C2×D412yes—
188C42×C5C2×D412yes—
189C22⋊Dic5—28no—
190#687—48no—
191C4⋊Dic5—24no—
192C4⋊Dic5—24no—
193C4⋊Dic5—24no—
194C4⋊Dic5—22no—
195C4⋊Dic5C2×D414yes—
196#683—28no—
197#683—28no—
198C4×Dic5—28no—
199C2×#168—22no—
200C2×#168—22no—
201C2×#168—22no—
202C2×#168—22no—
203C2×#168—22no—
204C2×#168—22no—
205C22×SD16—52no—
206#533—104no—
207#533—104no—
208C2×C4⋊Q8—51no—
209C2×C4⋊Q8—51no—
210C2×#125—101no—
211C2×C22⋊Q8—52no—
212C2×C22⋊Q8—101no—
213C2×#121C2×D511yescharacteristic
214C2×C4×Q8—52no—
215C2×C4×Q8—101no—
216C2×C4×D4—21no—
217C2×C4×D4C2×D511yescharacteristic
218C2×#117—101no—
219C22×C42C2×D511yescharacteristic
220#476—104no—
221#461—104no—
222C2×#106—52no—
223C2×C4⋊C8—51no—
224C2×C4⋊C8—51no—
225C2×#103—52no—
226C2×#103—52no—
227C2×#103—52no—
228C2×#102—52no—
229C2×C22⋊C8—52no—
230#390C2×D511yescharacteristic
231#389C2×D511yescharacteristic
232#388—21no—
233#386—21no—
234#380C2×D511yescharacteristic
235C4×C4⋊C4C2×D511yescharacteristic
236C4×C22⋊C4—21no—
237#340—52no—
238#340—52no—
239#336—52no—
240C23×C5—21no—
241C23×C5—21no—
242C23×C5—21no—
243C23×C5—21no—
244C23×C5—21no—
245C23×C5—21no—
246C23×C5—22no—
247C23×C5—41no—
248C23×C5—41no—
249C23×C5—41no—
250C23×C5—41no—
251C23×C5—41no—
252C23×C5—41no—
253C23×C5C2×SD1612yes—
254C23×C5C22×D411yescharacteristic
255C5×D4—22no—
256C5×D4—22no—
257C5×D4—48no—
258C5×D4—41no—
259C5×D4—41no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—28no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—22no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—44no—
273C2×C4×C5—41no—
274C2×C4×C5—41no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—84no—
286C2×C4×C5#12012yes—
287C2×C4×C5#12012yes—
288C2×C4×C5C2×SD1612yes—
289C2×C4×C5C2×SD1612yes—
290C2×C4×C5#13712yes—
291C2×C4×C5C22×D411yescharacteristic
292C2×C4×C5C22×D411yescharacteristic
293C2×C4×C5C2×#4111yescharacteristic
294C2×C4×C5C2×#4111yescharacteristic
295C2×C4×C5H(5,𝔽2)11yescharacteristic
296C2×C4×C52−1+411yescharacteristic
297C2×Dic5—24no—
298C2×Dic5—24no—
299C2×Dic5—42no—
300C2×Dic5—42no—
301C2×Dic5—42no—
302C2×Dic5—42no—
303Dic10—24no—
304Dic10—24no—
305Dic10—48no—
306Dic10—42no—
307Dic10—42no—
308#168—42no—
309#168—42no—
310C22×Q8—52no—
311C22×Q8—101no—
312C22×D4—21no—
313C22×D4C22×D511yescharacteristic
314C23×C4—21no—
315C23×C4C22⋊D511yescharacteristic
316C23×C4C22⋊D511yescharacteristic
317C23×C4C22×D511yescharacteristic
318C2×SD16—104no—
319C2×SD16—104no—
320C2×SD16—202no—
321C22×C8—52no—
322C22×C8—52no—
323C4⋊Q8—54no—
324C4⋊Q8—54no—
325C4⋊Q8—102no—
326C4⋊Q8—102no—
327#125—108no—
328C22⋊Q8—104no—
329C22⋊Q8—104no—
330C22⋊Q8—108no—
331#121—22no—
332#121—22no—
333C4×Q8—104no—
334C4×Q8—104no—
335C4×Q8—108no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—44no—
339#117—108no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—52no—
342C2×C4⋊C4—101no—
343C2×C4⋊C4—101no—
344C2×C4⋊C4—102no—
345C2×C4⋊C4—102no—
346C2×C4⋊C4—101no—
347C2×C4⋊C4C22×D511yescharacteristic
348C2×C4⋊C4C22×D511yescharacteristic
349C2×C4⋊C4C22×D511yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—102no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—102no—
360C2×C42C22⋊D511yescharacteristic
361C2×C42C22⋊D511yescharacteristic
362C2×C42C22×D511yescharacteristic
363C2×C42C22×D511yescharacteristic
364#106—104no—
365C4⋊C8—102no—
366C4⋊C8—102no—
367#103—104no—
368#103—104no—
369#103—202no—
370#102—104no—
371C22⋊C8—202no—
372#95—21no—
373#95—21no—
374#95—21no—
375C22×C5—22no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5—41no—
379C22×C5—41no—
380C22×C5—41no—
381C22×C5—41no—
382C22×C5—42no—
383C22×C5—42no—
384C22×C5—42no—
385C22×C5—81no—
386C22×C5—81no—
387C22×C5—81no—
388C22×C5—81no—
389C22×C5C2×#12011yescharacteristic
390C22×C5#52511yescharacteristic
391C22×C5#53711yescharacteristic
392C22×C5C22×SD1611yescharacteristic
393C22×C5C2×#13711yescharacteristic
394C22×C5#57511yescharacteristic
395C22×C5#57511yescharacteristic
396C4×C5—22no—
397C4×C5—22no—
398C4×C5—42no—
399C4×C5—41no—
400C4×C5—41no—
401C4×C5—81no—
402C4×C5—81no—
403C4×C5#44814yes—
404C4×C5#54611yescharacteristic
405C4×C5#54711yescharacteristic
406C4×C5#54811yescharacteristic
407C4×C5#55011yescharacteristic
408Dic5—44no—
409Dic5—44no—
410C24—21no—
411C24C2×C22⋊D511yescharacteristic
412C2×Q8—54no—
413C2×Q8—102no—
414C2×Q8—104no—
415C2×Q8—102no—
416C2×Q8—102no—
417C2×Q8—102no—
418C2×Q8—102no—
419C2×Q8—202no—
420C2×Q8—202no—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4#70912yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—102no—
448C22×C4—102no—
449C22×C4#68714yescomplemented
450C22×C4C2×C22⋊D511yescharacteristic
451C22×C4C2×C22⋊D511yescharacteristic
452C22×C4C2×C22⋊D511yescharacteristic
453C22×C4C2×C22⋊D511yescharacteristic
454C22×C4C2×C22⋊D511yescharacteristic
455C22×C4C23×D511yescharacteristic
456SD16—208no—
457C2×C8—102no—
458C2×C8—102no—
459C2×C8—102no—
460C2×C8—102no—
461C2×C8—102no—
462C2×C8—102no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—54no—
470C4⋊C4—108no—
471C4⋊C4—108no—
472C4⋊C4—102no—
473C4⋊C4—104no—
474C4⋊C4—104no—
475C4⋊C4—104no—
476C4⋊C4#70912yes—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—108no—
483C42—21no—
484C42—21no—
485C42—22no—
486C42—22no—
487C42—42no—
488C42—108no—
489C42D4×D512yes—
490C42D4×D512yes—
491C2×C5—42no—
492C2×C5—82no—
493C2×C5#240711yescharacteristic
494C2×C5C2×#44811yescharacteristic
495C2×C5#300511yescharacteristic
496C2×C5#310111yescharacteristic
497C2×C5#311611yescharacteristic
498C2×C5#324611yescharacteristic
499C2×C5#325211yescharacteristic
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C2×#68712yes—
514C23C22×C22⋊D511yescharacteristic
515Q8—104no—
516Q8—104no—
517Q8—208no—
518Q8—202no—
519Q8—202no—
520D4—22no—
521D4—22no—
522D4—48no—
523D4—41no—
524D4—41no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4—104no—
552C2×C4—104no—
553C2×C4—202no—
554C2×C4—202no—
555C2×C4—202no—
556C2×C4—202no—
557C2×C4C2×#68712yes—
558C2×C4C2×#68712yes—
559C2×C4#408312yes—
560C2×C4#408312yes—
561C2×C4#409712yes—
562C2×C4C2×#70911yescharacteristic
563C2×C4C2×#70911yescharacteristic
564C2×C4C2×D4×D511yescharacteristic
565C2×C4C2×D4×D511yescharacteristic
566C2×C4#414911yescharacteristic
567C2×C4#415011yescharacteristic
568C8—202no—
569C8—202no—
570C5#2415111yescharacteristic, complemented, Sylow
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22#6612511yescharacteristic
586C22C22×#68711yescharacteristic
587C22C2×#408311yescharacteristic
588C22#6615611yescharacteristic
589C22#6615611yescharacteristic
590C22C2×#409711yescharacteristic
591C22#6616711yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—204no—
600C4—204no—
601C4#6532914yes—
602C4#6588411yescharacteristic
603C4#6588511yescharacteristic
604C4#6588611yescharacteristic
605C4#6588711yescharacteristic
606C2—42no—
607C2—82no—
608C2#1061404511yescharacteristic
609C2#1061413311yescharacteristic
610C2#1061426611yescharacteristic
611C2C2×#6532911yescharacteristic
612C2#1061465911yescharacteristic
613C2#1061488711yescharacteristic
614C2#1061505111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC4×C5×#121C2
socleC23×C5C22×D4
prime radicalC2×C5×#121C22
hypercenterC4×#121D5
hypocenterC5#24151
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#65804C2semidirect
3. C2×#65308C2semidirect
4. C2×#65297C2semidirect
6. #10602323C2semidirect
7. #10600453C2semidirect
8. #10600425C2semidirect
9. #10600319C2semidirect
10. #10600282C2semidirect
11. #10600279C2semidirect
97. #4015D4semidirect
98. #4015D4semidirect
152. C22×C4×C5SD16semidirect
449. C22×C4#687semidirect
570. C5#24151semidirect

Maximal subgroups

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

subgroupindex
C2×#658042
C4×C5×#1212
C2×#653082
C2×#652972
#106024782
#106023232
#106004532
#106004252
#106003192
#106002822
#106002792
#241515