Finite Group Databasebeta

← #12021626

Subgroups — #12021626

Order 1280.

1730 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×C22⋊C4C211yesmaximal, characteristic
2C2×#65501C211yesmaximal, characteristic
3C2×#65240C212yesmaximal
4C2×#65225C211yesmaximal, characteristic, complemented
5#10602920C211yesmaximal, characteristic
6#10602339C212yesmaximal
7#10602328C211yesmaximal, characteristic, complemented
8#10602327C211yesmaximal, characteristic, complemented
9C4×#3963C211yesmaximal, characteristic, complemented
10#10600212C212yesmaximal, complemented
11#10600200C212yesmaximal, complemented
12C24×C4×C5C2211yescharacteristic
13C22×C5×C22⋊C4C2211yescharacteristic
14C22×C5×C22⋊C4C2211yescharacteristic
15C22×C42×C5C2211yescharacteristic
16C22×C42×C5C2211yescharacteristic
17C22×C22⋊Dic5C2212yes—
18C22×C22⋊Dic5C2212yes—
19C22×C4⋊Dic5C2212yes—
20C22×C4⋊Dic5C2212yes—
21C22×C4⋊Dic5C2212yes—
22C22×C4×Dic5C2212yes—
23C4×C5×C22⋊C4C418yescomplemented
24C2×C5×#95C2211yescharacteristic
25C2×C5×#95C2211yescharacteristic
26#65501—24no—
27C2×#3963C2212yes—
28C2×#3963C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2211yescharacteristic
34C2×#3963C2211yescharacteristic
35C2×#3963C2211yescharacteristic
36C2×#3963C2211yescharacteristic
37#65240—28no—
38#65225—24no—
39#13915—51nomaximal, Sylow
40C25×C5C2311yescharacteristic
41C23×C4×C5—22no—
42C23×C4×C5—24no—
43C23×C4×C5—22no—
44C23×C4×C5—42no—
45C23×C4×C5C2×C414yes—
46C23×C4×C5C2311yescharacteristic
47C23×C4×C5C2311yescharacteristic
48C23×C4×C5C2311yescharacteristic
49C23×C4×C5C2311yescharacteristic
50C23×C4×C5C2311yescharacteristic
51C23×C4×C5C2311yescharacteristic
52C23×Dic5C2312yes—
53C23×Dic5C2312yes—
54C23×Dic5C2312yes—
55C23×Dic5C2312yes—
56C2×C5×C22⋊C4—24no—
57C2×C5×C22⋊C4C2×C414yes—
58C2×C5×C22⋊C4C2×C414yes—
59C2×C5×C22⋊C4D414yes—
60C2×C5×C22⋊C4Q814yes—
61C2×C42×C5—22no—
62C2×C42×C5—22no—
63C2×C42×C5—42no—
64C2×C42×C5C2×C414yes—
65C2×C42×C5C2×C414yes—
66C2×C22⋊Dic5—24no—
67C2×C22⋊Dic5—24no—
68C2×C22⋊Dic5—24no—
69C2×C22⋊Dic5—24no—
70C2×C22⋊Dic5—42no—
71C2×C22⋊Dic5—42no—
72C2×C4⋊Dic5—24no—
73C2×C4⋊Dic5—24no—
74C2×C4⋊Dic5—24no—
75C2×C4⋊Dic5—24no—
76C2×C4⋊Dic5—24no—
77C2×C4⋊Dic5—42no—
78C2×C4⋊Dic5—42no—
79C2×C4⋊Dic5—42no—
80C2×C4⋊Dic5—42no—
81C2×C4×Dic5—24no—
82C2×C4×Dic5—24no—
83C2×C4×Dic5—42no—
84C5×#95C2×C414yes—
85C5×#95C2×C414yes—
86#3963—22no—
87#3963—24no—
88#3963—24no—
89#3963—22no—
90#3963—24no—
91#3963—24no—
92#3963—22no—
93#3963—24no—
94#3963—24no—
95#3963D414yes—
96C2×#388—52no—
97C2×#382—51no—
98C2×#380—51no—
99C2×C4×C22⋊C4D511yescharacteristic
100#1444—52no—
101#1439—52no—
102#1436—52no—
103C4⋊#95—51no—
104#1434—51no—
105C22⋊#95—51no—
106C4×#95—51no—
107C24×C5—21no—
108C24×C5—22no—
109C24×C5—22no—
110C24×C5—22no—
111C24×C5—21no—
112C24×C5—21no—
113C24×C5—22no—
114C24×C5—21no—
115C24×C5C22×C412yes—
116C24×C5C2×D412yes—
117C24×C5C2×Q812yes—
118C24×C5C2411yescharacteristic
119C22×C4×C5—21no—
120C22×C4×C5—21no—
121C22×C4×C5—21no—
122C22×C4×C5—22no—
123C22×C4×C5—21no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—21no—
130C22×C4×C5—21no—
131C22×C4×C5—21no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—21no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—28no—
138C22×C4×C5—28no—
139C22×C4×C5—44no—
140C22×C4×C5—48no—
141C22×C4×C5—48no—
142C22×C4×C5C22×C412yes—
143C22×C4×C5C22×C412yes—
144C22×C4×C5C22×C412yes—
145C22×C4×C5C22×C412yes—
146C22×C4×C5C22×C412yes—
147C22×C4×C5C22×C412yes—
148C22×C4×C5C2×D412yes—
149C22×C4×C5C2×D412yes—
150C22×C4×C5C2×D412yes—
151C22×C4×C5C2×Q812yes—
152C22×C4×C5C2×Q812yes—
153C22×C4×C5#4114yes—
154C22×C4×C5#4114yes—
155C22×C4×C5#4112yes—
156C22×Dic5—22no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—22no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—22no—
169C22×Dic5—22no—
170C22×Dic5—22no—
171C22×Dic5—22no—
172C22×Dic5—22no—
173C22×Dic5—22no—
174C22×Dic5—22no—
175C22×Dic5—22no—
176C22×Dic5—22no—
177C22×Dic5—22no—
178C22×Dic5—22no—
179C22×Dic5—22no—
180C22×Dic5—22no—
181C22×Dic5—22no—
182C22×Dic5C2×D414yes—
183C22×Dic5#4114yes—
184C5×C22⋊C4—28no—
185C5×C22⋊C4C4⋊C4116yescomplemented
186C42×C5—28no—
187C42×C5—44no—
188C22⋊Dic5—48no—
189C22⋊Dic5—48no—
190C4⋊Dic5—48no—
191C4⋊Dic5—48no—
192C4⋊Dic5—48no—
193C4×Dic5—48no—
194C24×C4C2×D511yescharacteristic
195C22×C4⋊C4—52no—
196C22×C4⋊C4—52no—
197C22×C4⋊C4—52no—
198C22×C22⋊C4—52no—
199C22×C22⋊C4—52no—
200C22×C22⋊C4C2×D511yescharacteristic
201C22×C22⋊C4C2×D511yescharacteristic
202C22×C42—52no—
203C22×C42C2×D511yescharacteristic
204C22×C42C2×D511yescharacteristic
205#388—108no—
206#382—104no—
207#380—104no—
208C4×C22⋊C4Dic518yescomplemented
209C2×#95—51no—
210C2×#95—51no—
211C2×#95—52no—
212C2×#95—51no—
213C2×#95—52no—
214C2×#95—51no—
215C2×#95—52no—
216C2×#95—52no—
217C2×#95—52no—
218C2×#95—52no—
219C2×#95C2×D511yescharacteristic
220C2×#95C2×D511yescharacteristic
221C23×C5—22no—
222C23×C5—22no—
223C23×C5—22no—
224C23×C5—22no—
225C23×C5—22no—
226C23×C5—22no—
227C23×C5—24no—
228C23×C5—24no—
229C23×C5—22no—
230C23×C5—24no—
231C23×C5—22no—
232C23×C5—22no—
233C23×C5—24no—
234C23×C5—22no—
235C23×C5—42no—
236C23×C5—42no—
237C23×C5—42no—
238C23×C5—42no—
239C23×C5—42no—
240C23×C5—42no—
241C23×C5—42no—
242C23×C5—42no—
243C23×C5C2×C4⋊C414yes—
244C23×C5C23×C411yescharacteristic
245C23×C5C22×D411yescharacteristic
246C23×C5C22×D411yescharacteristic
247C23×C5C22×Q811yescharacteristic
248C23×C5C2×#4112yes—
249C23×C5C2×#4112yes—
250C23×C5C2×#4111yescharacteristic
251C23×C5C2×#4112yes—
252C23×C5H(5,𝔽2)12yes—
253C23×C5H(5,𝔽2)11yescharacteristic
254C23×C52−1+411yescharacteristic
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—24no—
266C2×C4×C5—24no—
267C2×C4×C5—22no—
268C2×C4×C5—24no—
269C2×C4×C5—24no—
270C2×C4×C5—24no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—24no—
274C2×C4×C5—24no—
275C2×C4×C5—24no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—416no—
281C2×C4×C5—42no—
282C2×C4×C5—42no—
283C2×C4×C5C2×C4⋊C414yes—
284C2×C4×C5C2×C4⋊C414yes—
285C2×C4×C5#11714yes—
286C2×C4×C5#11714yes—
287C2×C4×C5C4×D414yes—
288C2×Dic5—24no—
289C2×Dic5—24no—
290C2×Dic5—24no—
291C2×Dic5—24no—
292C2×Dic5—24no—
293C2×Dic5—24no—
294C2×Dic5—42no—
295C2×Dic5—42no—
296C2×Dic5—42no—
297C2×Dic5—42no—
298C2×Dic5—42no—
299C2×Dic5—42no—
300C2×Dic5—42no—
301C2×Dic5—42no—
302C2×Dic5—42no—
303C2×Dic5—42no—
304C2×Dic5—42no—
305C2×Dic5—42no—
306C2×Dic5—42no—
307C2×Dic5—42no—
308C2×Dic5—42no—
309C2×Dic5—42no—
310C2×Dic5—42no—
311C2×Dic5—42no—
312C2×Dic5—42no—
313C2×Dic5—42no—
314C2×Dic5—42no—
315C2×Dic5—42no—
316C25C22×D511yescharacteristic
317C23×C4—24no—
318C23×C4—22no—
319C23×C4—22no—
320C23×C4—42no—
321C23×C4—52no—
322C23×C4—52no—
323C23×C4—52no—
324C23×C4—52no—
325C23×C4C2×Dic514yes—
326C23×C4C22×D511yescharacteristic
327C23×C4C22×D511yescharacteristic
328C23×C4C22×D511yescharacteristic
329C23×C4C22×D511yescharacteristic
330C23×C4C22×D511yescharacteristic
331C23×C4C22×D511yescharacteristic
332C2×C4⋊C4—104no—
333C2×C4⋊C4—104no—
334C2×C4⋊C4—104no—
335C2×C4⋊C4—104no—
336C2×C4⋊C4—104no—
337C2×C4⋊C4—202no—
338C2×C4⋊C4—202no—
339C2×C4⋊C4—202no—
340C2×C4⋊C4—202no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—104no—
343C2×C22⋊C4—104no—
344C2×C22⋊C4—104no—
345C2×C22⋊C4—104no—
346C2×C22⋊C4—202no—
347C2×C22⋊C4—202no—
348C2×C22⋊C4Dic1014yes—
349C2×C22⋊C4D2014yes—
350C2×C22⋊C4C2×Dic514yes—
351C2×C22⋊C4C2×Dic514yes—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—42no—
355C2×C42—104no—
356C2×C42—104no—
357C2×C42—202no—
358C2×C42C2×Dic514yes—
359C2×C42C2×Dic514yes—
360#95—54no—
361#95—104no—
362#95—104no—
363#95—104no—
364#95—104no—
365#95—104no—
366#95—104no—
367#95—102no—
368#95—102no—
369#95—102no—
370#95C2×Dic514yes—
371#95C2×Dic514yes—
372C22×C5—24no—
373C22×C5—24no—
374C22×C5—24no—
375C22×C5—42no—
376C22×C5—44no—
377C22×C5—42no—
378C22×C5—42no—
379C22×C5—42no—
380C22×C5—44no—
381C22×C5—44no—
382C22×C5—44no—
383C22×C5C22×C4⋊C411yescharacteristic
384C22×C5C2×#11711yescharacteristic
385C22×C5C2×#11711yescharacteristic
386C22×C5C2×C4×D411yescharacteristic
387C22×C5C4×#4111yescharacteristic
388C22×C5#51811yescharacteristic
389C22×C5#52011yescharacteristic
390C22×C5#52912yes—
391C22×C5#53212yes—
392C22×C5#53212yes—
393C22×C5#53312yes—
394C22×C5#53812yes—
395C22×C5#53912yes—
396C22×C5D4211yescharacteristic
397C22×C5#54612yes—
398C22×C5#54612yes—
399C22×C5#54711yescharacteristic
400C22×C5D4×Q811yescharacteristic
401C22×C5#55112yes—
402C22×C5#55212yes—
403C22×C5#55412yes—
404C22×C5#55411yescharacteristic
405C22×C5#55512yes—
406C4×C5—24no—
407C4×C5—24no—
408C4×C5—24no—
409C4×C5—24no—
410C4×C5—24no—
411C4×C5—42no—
412Dic5—44no—
413Dic5—44no—
414Dic5—82no—
415Dic5—82no—
416C24—22no—
417C24—22no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24—21no—
423C24—21no—
424C24C2×Dic1012yes—
425C24C2×D2012yes—
426C24C22×Dic512yes—
427C24C23×D511yescharacteristic
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—54no—
452C22×C4—54no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—102no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—102no—
462C22×C4—102no—
463C22×C4—102no—
464C22×C4—102no—
465C22×C4—102no—
466C22×C4—102no—
467C22×C4—102no—
468C22×C4—102no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4—102no—
472C22×C4—102no—
473C22×C4—102no—
474C22×C4—102no—
475C22×C4—102no—
476C22×C4—102no—
477C22×C4—102no—
478C22×C4—102no—
479C22×C4C2×Dic1012yes—
480C22×C4C2×Dic1012yes—
481C22×C4C2×D2012yes—
482C22×C4C2×D2012yes—
483C22×C4#70914yes—
484C22×C4#70914yes—
485C22×C4D4×D512yes—
486C22×C4#71112yes—
487C22×C4C22×Dic512yes—
488C22×C4C22×Dic512yes—
489C22×C4C22×Dic512yes—
490C22×C4C22×Dic512yes—
491C22×C4C22×Dic512yes—
492C22×C4C22×Dic512yes—
493C4⋊C4—208no—
494C4⋊C4—208no—
495C4⋊C4—208no—
496C22⋊C4—28no—
497C22⋊C4—208no—
498C22⋊C4—208no—
499C22⋊C4C4⋊Dic5116yescomplemented
500C42—28no—
501C42—44no—
502C42—208no—
503C2×C5—44no—
504C2×C5#228511yescharacteristic
505C2×C5#229311yescharacteristic
506C2×C5#231811yescharacteristic
507C2×C5#234211yescharacteristic
508C2×C5#234611yescharacteristic
509C2×C5D4×C4⋊C411yescharacteristic
510C2×C5#234911yescharacteristic
511C2×C5#255312yes—
512C2×C5#255512yes—
513C2×C5#255612yes—
514C2×C5#257412yes—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C2×C4⋊Dic514yes—
538C23C22×Dic1011yescharacteristic
539C23C22×D2011yescharacteristic
540C23C2×#70912yes—
541C23C2×#70912yes—
542C23C2×D4×D511yescharacteristic
543C23C2×#71111yescharacteristic
544C23#414412yes—
545C23D5×#4112yes—
546C23#414911yescharacteristic
547C23#415011yescharacteristic
548C23C23×Dic511yescharacteristic
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—104no—
578C2×C4—104no—
579C2×C4—104no—
580C2×C4—104no—
581C2×C4—104no—
582C2×C4—104no—
583C2×C4—202no—
584C2×C4—202no—
585C2×C4—202no—
586C2×C4—202no—
587C2×C4—202no—
588C2×C4—202no—
589C2×C4—202no—
590C2×C4—202no—
591C2×C4—202no—
592C2×C4—202no—
593C2×C4—202no—
594C2×C4—202no—
595C2×C4—202no—
596C2×C4—202no—
597C2×C4—202no—
598C2×C4—202no—
599C2×C4—202no—
600C2×C4—202no—
601C2×C4—202no—
602C2×C4—202no—
603C2×C4—202no—
604C2×C4—202no—
605C2×C4C2×C4⋊Dic514yes—
606C2×C4C2×C4⋊Dic514yes—
607C2×C4#407214yes—
608C2×C4#407214yes—
609C2×C4D4×Dic514yes—
610C5#1391511yescharacteristic, complemented, Sylow
611C22—24no—
612C22—24no—
613C22—24no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22#6580412yes—
623C22#6581112yes—
624C22#6582412yes—
625C22#6582912yes—
626C22#6586312yes—
627C22#6586612yes—
628C22D4×Dic1011yescharacteristic
629C22#6587312yes—
630C22#6587711yescharacteristic
631C22#6588212yes—
632C22D4×D2011yescharacteristic
633C22#6588812yes—
634C22#6588911yescharacteristic
635C22#6589112yes—
636C22#6589412yes—
637C22#6589812yes—
638C22C22×C4⋊Dic511yescharacteristic
639C22C2×#407211yescharacteristic
640C22C2×#407211yescharacteristic
641C22C2×D4×Dic511yescharacteristic
642C22#6613211yescharacteristic
643C22#41×Dic511yescharacteristic
644C22#6616111yescharacteristic
645C4—24no—
646C4—24no—
647C4—24no—
648C4—24no—
649C4—24no—
650C4—42no—
651C4—204no—
652C4—204no—
653C4—402no—
654C4—402no—
655C2—44no—
656C2#1060526712yes—
657C2#1060527312yes—
658C2#1060533012yes—
659C2#1060533112yes—
660C2#1061379311yescharacteristic
661C2#1061382811yescharacteristic
662C2#1061397011yescharacteristic
663C2#1061405011yescharacteristic
664C2#1061405511yescharacteristic
665C2D4×C4⋊Dic511yescharacteristic
666C2#1061406411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C4×C5×C22⋊C4C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×C4×C22⋊C4D5
hypocenterC5#13915
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#65225C2semidirect
7. #10602328C2semidirect
8. #10602327C2semidirect
9. C4×#3963C2semidirect
10. #10600212C2semidirect
11. #10600200C2semidirect
23. C4×C5×C22⋊C4C4semidirect
185. C5×C22⋊C4C4⋊C4semidirect
208. C4×C22⋊C4Dic5semidirect
499. C22⋊C4C4⋊Dic5semidirect
610. C5#13915semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×C22⋊C42
C2×#655012
C2×#652402
C2×#652252
#106029202
#106023392
#106023282
#106023272
C4×#39632
#106002122
#106002002
#139155