Finite Group Databasebeta

← #12127995

Subgroups — #12127995

Order 1280.

1178 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
1C5×#2297C211yesmaximal, characteristic
2C2×#65521C211yesmaximal, characteristic
3C2×#65521C211yesmaximal, characteristic
4C2×#65284C211yesmaximal, characteristic, complemented
5C2×#65217C211yesmaximal, characteristic, complemented
6C2×#65213C211yesmaximal, characteristic, complemented
7#10602374C211yesmaximal, characteristic, complemented
8#10600476C212yesmaximal, complemented
9#10600475C212yesmaximal, complemented
10#10600446C212yesmaximal, complemented
11#10600445C212yesmaximal, complemented
12C22×C5×#41C2211yescharacteristic
13C2×C5×#117C2211yescharacteristic
14C22×C42×C5C2211yescharacteristic
15C22×C4⋊Dic5C2211yescharacteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C22×#683—21no—
18C22×#681C2211yescharacteristic
19C22×#681C2211yescharacteristic
20C5×#391C2211yescharacteristic
21C5×#391C2211yescharacteristic
22C5×#390C2211yescharacteristic
23C5×#390C2211yescharacteristic
24C5×#386—21no—
25C5×#386—21no—
26C4×C5×C22⋊C4—21no—
27C4×C5×C22⋊C4—21no—
28#65521—24no—
29#65521—24no—
30C2×#3967C2212yes—
31C2×#3967C2212yes—
32C2×#3964C2212yes—
33C2×#3964C2212yes—
34C2×#3963—21no—
35#65284—24no—
36C2×#3940C2212yes—
37C2×#3939C2212yes—
38#65217—21no—
39#65217—21no—
40#65217C2212yes—
41#65217C2212yes—
42#65213—24no—
43C2×C4⋊#168C2212yes—
44C2×C4⋊#168C2212yes—
45#64749C2212yes—
46#64749C2212yes—
47#23617—51nomaximal, Sylow
48C2×C5×#41—22no—
49C2×C5×#41—22no—
50C2×C5×#41—22no—
51C2×C5×#41—22no—
52C2×C5×#41—22no—
53C2×C5×#41Q814yes—
54C22×C5×Q8C2311yescharacteristic
55C22×C5×D4—21no—
56C22×C5×D4C2311yescharacteristic
57C23×C4×C5—21no—
58C23×C4×C5D411yescharacteristic
59C23×C4×C5D411yescharacteristic
60C23×C4×C5C2311yescharacteristic
61C23×Dic5—21no—
62C23×Dic5—21no—
63C5×#117—22no—
64C5×#117—22no—
65C2×C5×C4⋊C4C2311yescharacteristic
66C2×C5×C4⋊C4C2311yescharacteristic
67C2×C5×C22⋊C4—21no—
68C2×C5×C22⋊C4—21no—
69C2×C5×C22⋊C4—21no—
70C2×C5×C22⋊C4—21no—
71C2×C5×C22⋊C4—21no—
72C2×C42×C5—22no—
73C2×C42×C5—22no—
74C2×C42×C5—21no—
75C2×C42×C5D411yescharacteristic
76C2×C42×C5D411yescharacteristic
77C2×C42×C5C2311yescharacteristic
78C2×C42×C5C2311yescharacteristic
79C2×C4⋊Dic5—21no—
80C2×C4⋊Dic5—22no—
81C2×C4⋊Dic5—22no—
82C2×C4⋊Dic5—22no—
83C2×C4⋊Dic5—21no—
84C2×C4⋊Dic5D414yes—
85C2×C4⋊Dic5C2312yes—
86C2×C4⋊Dic5C2312yes—
87C2×#683—22no—
88C2×#683—22no—
89C2×#683—44no—
90C2×#681—22no—
91C2×#681—22no—
92C2×#681—22no—
93C2×#681—22no—
94C2×#681—22no—
95C2×#681D414yes—
96C22×#168C2312yes—
97C22×#168C2312yes—
98C5×#95—21no—
99C5×#95—21no—
100C5×#95—21no—
101C5×#95—21no—
102#3967—24no—
103#3967—24no—
104#3964—24no—
105#3964—24no—
106#3963—22no—
107#3963—22no—
108#3940—24no—
109#3939—24no—
110C4⋊#168—24no—
111C4⋊#168—24no—
112C2×#428—51no—
113C2×#423—51no—
114C2×#417—51no—
115C2×#417—51no—
116#2297D511yescharacteristic
117C2×#389—51no—
118#2068—52no—
119#2067—52no—
120#2066—52no—
121#2065—52no—
122#1736—51no—
123C24×C5—21no—
124C24×C5C2×D411yescharacteristic
125C5×#41—28no—
126C5×#41—28no—
127C5×#41—44no—
128C5×#41—44no—
129C2×C5×Q8—21no—
130C2×C5×Q8—21no—
131C2×C5×Q8—21no—
132C2×C5×Q8—21no—
133C2×C5×Q8—21no—
134C2×C5×Q8C2×Q812yes—
135C2×C5×D4—24no—
136C2×C5×D4—21no—
137C2×C5×D4—21no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4—21no—
141C2×C5×D4—42no—
142C2×C5×D4—42no—
143C2×C5×D4C2×Q812yes—
144C22×C4×C5—22no—
145C22×C4×C5—22no—
146C22×C4×C5—21no—
147C22×C4×C5—24no—
148C22×C4×C5—22no—
149C22×C4×C5—22no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—41no—
153C22×C4×C5—41no—
154C22×C4×C5—42no—
155C22×C4×C5—42no—
156C22×C4×C5C2×D412yes—
157C22×C4×C5C2×D412yes—
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×Q812yes—
164C22×C4×C5#4112yes—
165C22×C4×C5#4112yes—
166C22×C4×C5#4112yes—
167C22×C4×C5C2411yescharacteristic
168C22×Dic5—24no—
169C22×Dic5—22no—
170C22×Dic5—22no—
171C22×Dic5—42no—
172C22×Dic5—42no—
173C22×Dic5—42no—
174C22×Dic5—42no—
175C22×Dic5—42no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C22⋊C4—42no—
181C5×C22⋊C4—42no—
182C5×C22⋊C4—42no—
183C5×C22⋊C4—42no—
184C5×C22⋊C4—42no—
185C42×C5—22no—
186C42×C5—22no—
187C42×C5—21no—
188C42×C5—21no—
189C42×C5—21no—
190C42×C5—21no—
191C42×C5—42no—
192C4⋊Dic5—24no—
193C4⋊Dic5—22no—
194C4⋊Dic5—22no—
195C4⋊Dic5—22no—
196C4⋊Dic5—42no—
197C4⋊Dic5C2×D414yes—
198#683—44no—
199#683—44no—
200#681—28no—
201#681—44no—
202C2×#168—22no—
203C2×#168—22no—
204C2×#168—22no—
205C2×#168—22no—
206C2×#168—22no—
207C2×#168C2×D414yes—
208C22×#41C2×D511yescharacteristic
209C22×M16—51no—
210C22×M16—51no—
211C2×#117C2×D511yescharacteristic
212C22×C4⋊C4—51no—
213C22×C4⋊C4—51no—
214C22×C4⋊C4—101no—
215C22×C42C2×D511yescharacteristic
216#428—104no—
217C2×#107—52no—
218C2×#106—52no—
219#423—104no—
220C2×C4⋊C8—52no—
221C2×C4⋊C8—52no—
222#417—104no—
223#417—104no—
224C2×#103—52no—
225C2×#103—52no—
226C2×#102—52no—
227C2×#102—52no—
228#391C2×D511yescharacteristic
229#391C2×D511yescharacteristic
230#390C2×D511yescharacteristic
231#390C2×D511yescharacteristic
232#389—52no—
233#389—52no—
234#389—101no—
235#389—101no—
236#386—21no—
237#386—21no—
238C4×C22⋊C4—21no—
239C4×C22⋊C4—21no—
240C2×#95—101no—
241#340—52no—
242#340—52no—
243C23×C5—21no—
244C23×C5—21no—
245C23×C5—21no—
246C23×C5—21no—
247C23×C5—22no—
248C23×C5—41no—
249C23×C5—41no—
250C23×C5—41no—
251C23×C5—41no—
252C23×C5—41no—
253C23×C5—41no—
254C23×C5#12112yes—
255C23×C5#12112yes—
256C23×C5C22⋊Q812yes—
257C23×C5C22×D411yescharacteristic
258C5×Q8—22no—
259C5×Q8—22no—
260C5×Q8—41no—
261C5×Q8—41no—
262C5×D4—22no—
263C5×D4—22no—
264C5×D4—48no—
265C5×D4—41no—
266C5×D4—41no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—22no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—44no—
279C2×C4×C5—44no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—48no—
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#12114yes—
290C2×C4×C5#12112yes—
291C2×C4×C5#12112yes—
292C2×C4×C5C22⋊Q814yes—
293C2×C4×C5C22⋊Q814yes—
294C2×C4×C5C22⋊Q812yes—
295C2×C4×C5#12614yes—
296C2×C4×C5AGL(1,ℤ/8)12yes—
297C2×C4×C5#13712yes—
298C2×C4×C5C22×D411yescharacteristic
299C2×C4×C5C22×D411yescharacteristic
300C2×C4×C5C22×Q811yescharacteristic
301C2×C4×C5C2×#4111yescharacteristic
302C2×C4×C5C2×#4111yescharacteristic
303C2×C4×C5C2×#4111yescharacteristic
304C2×Dic5—24no—
305C2×Dic5—48no—
306C2×Dic5—42no—
307C2×Dic5—42no—
308C2×Dic5—42no—
309C2×Dic5—42no—
310C2×Dic5—42no—
311C2×Dic5—84no—
312#168—24no—
313#168—42no—
314C2×#41—22no—
315C2×#41—22no—
316C2×#41—22no—
317C2×#41—22no—
318C2×#41—22no—
319C2×#41Dic1014yes—
320C22×Q8C22×D511yescharacteristic
321C22×D4—21no—
322C22×D4C22×D511yescharacteristic
323C23×C4—21no—
324C23×C4—101no—
325C23×C4—101no—
326C23×C4C22⋊D511yescharacteristic
327C23×C4C22⋊D511yescharacteristic
328C23×C4C22×D511yescharacteristic
329C2×M16—54no—
330C2×M16—102no—
331C2×M16—102no—
332C2×M16—102no—
333C2×M16—102no—
334C2×M16—102no—
335C22×C8—52no—
336C22×C8—52no—
337#117—22no—
338#117—22no—
339C2×C4⋊C4—54no—
340C2×C4⋊C4—52no—
341C2×C4⋊C4—52no—
342C2×C4⋊C4—102no—
343C2×C4⋊C4—101no—
344C2×C4⋊C4—102no—
345C2×C4⋊C4—102no—
346C2×C4⋊C4—101no—
347C2×C4⋊C4—102no—
348C2×C4⋊C4—102no—
349C2×C4⋊C4—204no—
350C2×C4⋊C4C22×D511yescharacteristic
351C2×C4⋊C4C22×D511yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42C22⋊D511yescharacteristic
361C2×C42C22⋊D511yescharacteristic
362C2×C42C22×D511yescharacteristic
363C2×C42C22×D511yescharacteristic
364#107—104no—
365#106—104no—
366C4⋊C8—104no—
367C4⋊C8—104no—
368#103—104no—
369#103—104no—
370#102—104no—
371#102—104no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—102no—
377#95—102no—
378C22×C5—22no—
379C22×C5—22no—
380C22×C5—22no—
381C22×C5—22no—
382C22×C5—22no—
383C22×C5—22no—
384C22×C5—42no—
385C22×C5—42no—
386C22×C5—42no—
387C22×C5—81no—
388C22×C5—81no—
389C22×C5—81no—
390C22×C5—81no—
391C22×C5#39614yes—
392C22×C5C2×#12111yescharacteristic
393C22×C5C2×#12111yescharacteristic
394C22×C5C2×C22⋊Q811yescharacteristic
395C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
396C22×C5C2×#13711yescharacteristic
397C22×C5#57511yescharacteristic
398C22×C5#57511yescharacteristic
399C4×C5—22no—
400C4×C5—22no—
401C4×C5—42no—
402C4×C5—41no—
403C4×C5—41no—
404C4×C5—81no—
405C4×C5—81no—
406C4×C5#39614yes—
407C4×C5C2×#12111yescharacteristic
408C4×C5C2×C22⋊Q811yescharacteristic
409C4×C5C2×C22⋊Q811yescharacteristic
410C4×C5C2×#12611yescharacteristic
411Dic5—44no—
412Dic5—82no—
413C24—21no—
414C24C2×C22⋊D511yescharacteristic
415#41—28no—
416#41—28no—
417#41—44no—
418#41—44no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8C2×Dic1012yes—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4C2×Dic1012yes—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—102no—
447C22×C4—104no—
448C22×C4—102no—
449C22×C4—202no—
450C22×C4—202no—
451C22×C4—202no—
452C22×C4—202no—
453C22×C4—202no—
454C22×C4C2×Dic1012yes—
455C22×C4#70912yes—
456C22×C4D4×D512yes—
457C22×C4D4×D512yes—
458C22×C4#71112yes—
459C22×C4#71112yes—
460C22×C4C2×C22⋊D511yescharacteristic
461C22×C4C2×C22⋊D511yescharacteristic
462C22×C4C2×C22⋊D511yescharacteristic
463C22×C4C2×C22⋊D511yescharacteristic
464C22×C4C2×C22⋊D511yescharacteristic
465C22×C4C23×D511yescharacteristic
466M16—108no—
467M16—204no—
468C2×C8—54no—
469C2×C8—102no—
470C2×C8—102no—
471C2×C8—102no—
472C2×C8—102no—
473C2×C8—102no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—54no—
479C4⋊C4—102no—
480C4⋊C4—104no—
481C4⋊C4—102no—
482C4⋊C4—102no—
483C4⋊C4—204no—
484C4⋊C4—202no—
485C4⋊C4—204no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—42no—
498C2×C5—24no—
499C2×C5—82no—
500C2×C5C2×#39611yescharacteristic
501C2×C5#303611yescharacteristic
502C2×C5#303711yescharacteristic
503C2×C5#305511yescharacteristic
504C2×C5#305611yescharacteristic
505C2×C5#307911yescharacteristic
506C2×C5#308011yescharacteristic
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23#407012yes—
519C23#408412yes—
520C23#408512yes—
521C23C22×C22⋊D511yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—48no—
529D4—41no—
530D4—41no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—48no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—104no—
554C2×C4—208no—
555C2×C4—202no—
556C2×C4—202no—
557C2×C4—202no—
558C2×C4—202no—
559C2×C4—202no—
560C2×C4—404no—
561C2×C4#402414yes—
562C2×C4#402414yes—
563C2×C4#402514yes—
564C2×C4#403014yes—
565C2×C4#407012yes—
566C2×C4#408412yes—
567C2×C4#408512yes—
568C2×C4#409512yes—
569C2×C4#409712yes—
570C2×C4C22×Dic1011yescharacteristic
571C2×C4C2×#70911yescharacteristic
572C2×C4C2×D4×D511yescharacteristic
573C2×C4C2×D4×D511yescharacteristic
574C2×C4C2×#71111yescharacteristic
575C2×C4C2×#71111yescharacteristic
576C8—104no—
577C8—202no—
578C5#2361711yescharacteristic, complemented, Sylow
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22#6523714yes—
593C22C2×#407011yescharacteristic
594C22C2×#408411yescharacteristic
595C22C2×#408511yescharacteristic
596C22C2×#409511yescharacteristic
597C22#6615611yescharacteristic
598C22#6615611yescharacteristic
599C22C2×#409711yescharacteristic
600C4—22no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—204no—
608C4—402no—
609C4#6523714yes—
610C4C2×#402411yescharacteristic
611C4C2×#402411yescharacteristic
612C4C2×#402511yescharacteristic
613C4C2×#403011yescharacteristic
614C2—24no—
615C2—82no—
616C2C2×#6523711yescharacteristic
617C2#1061423911yescharacteristic
618C2#1061424011yescharacteristic
619C2#1061465511yescharacteristic
620C2#1061465611yescharacteristic
621C2#1061502811yescharacteristic
622C2#1061504511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2297C2
socleC23×C5C22×D4
prime radicalC22×C5×#41C22
hypercenter#2297D5
hypocenterC5#23617
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
4. C2×#65284C2semidirect
5. C2×#65217C2semidirect
6. C2×#65213C2semidirect
7. #10602374C2semidirect
8. #10600476C2semidirect
9. #10600475C2semidirect
10. #10600446C2semidirect
11. #10600445C2semidirect
578. C5#23617semidirect

Maximal subgroups

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

subgroupindex
C5×#22972
C2×#655212
C2×#652842
C2×#652172
C2×#652132
#106023742
#106004762
#106004752
#106004462
#106004452
#236175