Finite Group Databasebeta

← #12970859

Subgroups — #12970859

Order 1280.

2098 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#542C211yesmaximal, characteristic, complemented
2#10617636C211yesmaximal, characteristic, complemented
3#10615195C218yesmaximal, complemented
4#10615194C218yesmaximal, complemented
5#10614930C212yesmaximal, complemented
6#10614674C212yesmaximal, complemented
7#10614466C212yesmaximal, complemented
8#10614355C212yesmaximal, complemented
9#10614172C212yesmaximal, complemented
10#10614038C212yesmaximal, complemented
11#10613940C211yesmaximal, characteristic, complemented
12C22×#709C2212yes—
13C5×#542—24no—
14C5×#542C2218yescomplemented
15C2×C5×C4⋊Q8C2211yescharacteristic, complemented
16C2×C5×#126—21no—
17C2×C5×#124—21no—
18C2×C5×#124C2211yescharacteristic, complemented
19C2×C5×#123—21no—
20C2×C5×C22⋊Q8—21no—
21C2×C5×C22⋊Q8C2211yescharacteristic
22C2×C5×#121C2211yescharacteristic
23C2×C4×C5×Q8C2211yescharacteristic, complemented
24C2×C4×C5×D4C2211yescharacteristic, complemented
25C2×C5×#117C2211yescharacteristic
26C2×#4088C2214yescomplemented
27C2×#4078C2214yescomplemented
28#65860C2214yescomplemented
29#65859C2214yescomplemented
30#65858—24no—
31#65844—21no—
32#65844—21no—
33#65827—21no—
34#65827—21no—
35#65811C2212yes—
36#65811C2212yes—
37#65496C2212yes—
38#65382C2214yescomplemented
39#65378C2218yescomplemented
40#65377C2214yescomplemented
41#65349C2214yescomplemented
42#65348C2218yescomplemented
43#65347C2214yescomplemented
44#65339C2214yescomplemented
45#65336C2218yescomplemented
46#65335C2214yescomplemented
47#65329C2214yescomplemented
48#65326C2218yes—
49#65325C2214yescomplemented
50#65320C2214yescomplemented
51#65317C2218yescomplemented
52#65316C2214yescomplemented
53#65308C2214yescomplemented
54#65305C2218yescomplemented
55#65304C2214yescomplemented
56#65289C2218yescomplemented
57#65255C2214yescomplemented
58#65255C2214yescomplemented
59#65213C2212yescomplemented
60C4×#681C2212yescomplemented
61#60810—51nomaximal, Sylow
62C22×C5×Q8—21no—
63C22×C5×Q8C2311yescharacteristic
64C22×C5×D4—21no—
65C22×C5×D4C2311yescharacteristic
66C23×C4×C5—21no—
67C23×C4×C5C2311yescharacteristic
68C22×C22⋊D5—22no—
69C2×#709—22no—
70C2×#709—22no—
71C2×#709—48no—
72C2×#709C2314yes—
73C2×#709C2314yes—
74C22×D20C2312yes—
75C22×C4×D5—22no—
76C22×Dic10C2312yes—
77C5×C4⋊Q8—22no—
78C5×C4⋊Q8C2314yes—
79C5×#126—28no—
80C5×#124—28no—
81C5×#124—22no—
82C5×#124C2314yes—
83C5×#123—28no—
84C5×C22⋊Q8—22no—
85C5×C22⋊Q8—28no—
86C5×C22⋊Q8C2314yes—
87C5×#121—22no—
88C5×#121C2314yes—
89C4×C5×Q8—22no—
90C4×C5×Q8C2314yes—
91C4×C5×D4—22no—
92C4×C5×D4C2314yes—
93C5×#117D414yes—
94C5×#117C2314yes—
95C2×C5×C4⋊C4—21no—
96C2×C5×C4⋊C4—21no—
97C2×C5×C4⋊C4—21no—
98C2×C5×C4⋊C4D411yescharacteristic
99C2×C5×C4⋊C4D411yescharacteristic
100C2×C5×C4⋊C4C2311yescharacteristic
101C2×C5×C4⋊C4C2311yescharacteristic
102C2×C5×C22⋊C4—21no—
103C2×C5×C22⋊C4—21no—
104C2×C5×C22⋊C4—21no—
105C2×C5×C22⋊C4—21no—
106C2×C5×C22⋊C4—21no—
107C2×C5×C22⋊C4D411yescharacteristic
108C2×C5×C22⋊C4D411yescharacteristic
109C2×C42×C5—21no—
110C2×C42×C5C2311yescharacteristic
111C2×C42×C5C2311yescharacteristic
112C2×#689C2318yes—
113#4088—48no—
114C2×#688C2318yes—
115C2×#687C2318yes—
116#4078—48no—
117C2×#686C2318yes—
118#4076—22no—
119#4076—22no—
120#4076—22no—
121#4076—22no—
122C2×#681C2314yes—
123#4043—24no—
124#4043—24no—
125#4041—24no—
126#4041—24no—
127#4032—24no—
128#4032—24no—
129#4028—24no—
130#4028—24no—
131#4021C2314yes—
132#4021C2314yes—
133#4020C2312yes—
134#4020C2312yes—
135C4×D20—22no—
136C4×D20—22no—
137C4×D20—22no—
138C4×D20—22no—
139#4015C2312yes—
140#4015C2312yes—
141C22⋊#168C2314yes—
142C22⋊#168C2314yes—
143#3942C2318yes—
144#3942C2318yes—
145#3941C2318yes—
146#3941C2318yes—
147C4⋊#168C2314yes—
148C4⋊#168C2314yes—
149#3935C2314yes—
150C4×#168C2314yes—
151C2×#542D511yescharacteristic, complemented
152#3449—51no—
153#3225—58no—
154#3224—58no—
155#3153—52no—
156#3141—52no—
157#3037—52no—
158#3036—52no—
159#3007—52no—
160#3006—52no—
161#2921—51no—
162C24×C5—21no—
163C24×C5C2×D411yescharacteristic
164C23×D5—22no—
165C2×C5×Q8—21no—
166C2×C5×Q8—24no—
167C2×C5×Q8—24no—
168C2×C5×Q8—82no—
169C2×C5×Q8C2412yes—
170C2×C5×D4—24no—
171C2×C5×D4—21no—
172C2×C5×D4—24no—
173C2×C5×D4—82no—
174C2×C5×D4C2412yes—
175C22×C4×C5—24no—
176C22×C4×C5—21no—
177C22×C4×C5—21no—
178C22×C4×C5—21no—
179C22×C4×C5—24no—
180C22×C4×C5—82no—
181C22×C4×C5C2×D411yescharacteristic
182C22×C4×C5C2×D412yes—
183C22×C4×C5C2×D411yescharacteristic
184C22×C4×C5C2×D411yescharacteristic
185C22×C4×C5C2×D411yescharacteristic
186C22×C4×C5C2×D411yescharacteristic
187C22×C4×C5C2412yes—
188C22×C4×C5C2411yescharacteristic
189C2×C22⋊D5—24no—
190C2×C22⋊D5—24no—
191C2×C22⋊D5—48no—
192C22×Dic5—22no—
193#709—28no—
194#709—28no—
195#709—44no—
196#709—44no—
197#709—48no—
198#709—48no—
199C2×D20—22no—
200C2×D20—22no—
201C2×D20—22no—
202C2×D20—22no—
203C2×D20—22no—
204C2×D20—44no—
205C2×D20C2414yes—
206C2×C4×D5—24no—
207C2×C4×D5—24no—
208C2×C4×D5—48no—
209C2×Dic10—22no—
210C2×Dic10—44no—
211C2×Dic10C2414yes—
212C5×C4⋊C4—24no—
213C5×C4⋊C4—24no—
214C5×C4⋊C4—24no—
215C5×C4⋊C4—21no—
216C5×C4⋊C4—21no—
217C5×C4⋊C4C2×D414yes—
218C5×C4⋊C4C2×D414yes—
219C5×C4⋊C4C2412yes—
220C5×C4⋊C4C2412yes—
221C5×C22⋊C4—24no—
222C5×C22⋊C4—24no—
223C5×C22⋊C4—24no—
224C5×C22⋊C4—24no—
225C5×C22⋊C4—24no—
226C5×C22⋊C4C2×D414yes—
227C5×C22⋊C4C2×D414yes—
228C42×C5—24no—
229C42×C5C2×D412yes—
230C42×C5C2×D412yes—
231C42×C5C2412yes—
232C42×C5C2412yes—
233#689—48no—
234#688—48no—
235#687—48no—
236#686—48no—
237#685—24no—
238#685—24no—
239#685—24no—
240#685—24no—
241C4⋊Dic5—22no—
242C4⋊Dic5—22no—
243C4⋊Dic5—22no—
244C4⋊Dic5—22no—
245#681—28no—
246C2×#168C2418yes—
247C22×#41—52no—
248C2×#137—54no—
249C2×AGL(1,ℤ/8)—54no—
250#542—24no—
251#542C2×D518yescomplemented
252#537—101no—
253#537—101no—
254#536—54no—
255#535—54no—
256#534—101no—
257#534—101no—
258#532—52no—
259#532—52no—
260C2×C4⋊Q8C2×D511yescharacteristic, complemented
261C2×#126—21no—
262C2×#124—21no—
263C2×#124C2×D511yescharacteristic, complemented
264#525—104no—
265C2×#123—21no—
266C2×C22⋊Q8—21no—
267C2×C22⋊Q8C2×D511yescharacteristic
268C2×#121C2×D511yescharacteristic
269C2×C4×Q8C2×D511yescharacteristic, complemented
270C2×C4×D4C2×D511yescharacteristic, complemented
271C2×#117C2×D511yescharacteristic
272#497—54no—
273#496—54no—
274#495—58no—
275#489—58no—
276#487—54no—
277#486—54no—
278#464—58no—
279#463—58no—
280#462—54no—
281#461—54no—
282#460—54no—
283#459—54no—
284#452—58no—
285#451—54no—
286#450—54no—
287#449—58no—
288#448—54no—
289#447—54no—
290#433—58no—
291#423—52no—
292#417—54no—
293#417—54no—
294#407—52no—
295C4×M16—52no—
296C23×C5—22no—
297C23×C5—22no—
298C23×C5—41no—
299C23×C5—41no—
300C23×C5—42no—
301C23×C5—42no—
302C23×C5—42no—
303C23×C5C22×D412yes—
304C23×C5C22×D411yescharacteristic
305C22×D5—24no—
306C22×D5—44no—
307C22×D5—82no—
308C22×D5—82no—
309C5×Q8—24no—
310C5×Q8—42no—
311C5×Q8—84no—
312C5×D4—24no—
313C5×D4—42no—
314C5×D4—84no—
315C2×C4×C5—22no—
316C2×C4×C5—24no—
317C2×C4×C5—24no—
318C2×C4×C5—22no—
319C2×C4×C5—22no—
320C2×C4×C5—22no—
321C2×C4×C5—41no—
322C2×C4×C5—41no—
323C2×C4×C5—41no—
324C2×C4×C5—41no—
325C2×C4×C5—42no—
326C2×C4×C5—42no—
327C2×C4×C5—42no—
328C2×C4×C5—84no—
329C2×C4×C5C22×D412yes—
330C2×C4×C5C22×D412yes—
331C2×C4×C5C22×D412yes—
332C2×C4×C5C22×D412yes—
333C2×C4×C5C22×D412yes—
334C2×C4×C5C22×D411yescharacteristic
335C2×C4×C5H(5,𝔽2)14yes—
336C2×C4×C5C2511yescharacteristic
337C22⋊D5—48no—
338C22⋊D5—48no—
339C2×Dic5—24no—
340C2×Dic5—44no—
341D20—24no—
342D20—24no—
343D20—44no—
344D20—82no—
345D20—82no—
346D20—82no—
347D20—82no—
348C4×D5—48no—
349C4×D5—48no—
350Dic10—24no—
351Dic10—24no—
352Dic10—44no—
353#168—28no—
354C2×#41—54no—
355C2×#41—54no—
356C2×#41—102no—
357C2×#41—102no—
358C2×#41—208no—
359C22×Q8—21no—
360C22×Q8—52no—
361C22×Q8C22×D511yescharacteristic
362C22×D4—21no—
363C22×D4—52no—
364C22×D4—102no—
365C22×D4C22×D511yescharacteristic
366C23×C4—21no—
367C23×C4—102no—
368C23×C4C22×D511yescharacteristic
369#137—208no—
370AGL(1,ℤ/8)—208no—
371C2×Q16—58no—
372C2×SD16—58no—
373C2×SD16—58no—
374C2×D8—58no—
375C2×M16—54no—
376C4⋊Q8—22no—
377C4⋊Q8—52no—
378C4⋊Q8—52no—
379C4⋊Q8C22×D514yes—
380#127—52no—
381#127—52no—
382#126—28no—
383#124—28no—
384#124—22no—
385#124—54no—
386#124—54no—
387#124C22×D514yes—
388#123—28no—
389#123—104no—
390#123—104no—
391C22⋊Q8—22no—
392C22⋊Q8—28no—
393C22⋊Q8—104no—
394C22⋊Q8—104no—
395C22⋊Q8C22×D514yes—
396#121—22no—
397#121—102no—
398#121—102no—
399#121—102no—
400#121—102no—
401#121—104no—
402#121—104no—
403#121C22×D514yes—
404#120—104no—
405#120—104no—
406C4×Q8—22no—
407C4×Q8C22×D514yes—
408C4×D4—22no—
409C4×D4—102no—
410C4×D4—102no—
411C4×D4—102no—
412C4×D4—102no—
413C4×D4C22×D514yes—
414#117C22⋊D514yes—
415#117C22×D514yes—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4C22⋊D511yescharacteristic
420C2×C4⋊C4C22⋊D511yescharacteristic
421C2×C4⋊C4C22×D511yescharacteristic
422C2×C4⋊C4C22×D511yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4C22⋊D511yescharacteristic
429C2×C22⋊C4C22⋊D511yescharacteristic
430C2×C42—21no—
431C2×C42C22×D511yescharacteristic
432C2×C42C22×D511yescharacteristic
433C4⋊C8—54no—
434C4⋊C8—54no—
435#103—58no—
436#103—58no—
437#102—58no—
438#102—58no—
439C22⋊C8—54no—
440C22⋊C8—54no—
441#97—54no—
442C4×C8—54no—
443C22×C5—21no—
444C22×C5—21no—
445C22×C5—22no—
446C22×C5—22no—
447C22×C5—22no—
448C22×C5—42no—
449C22×C5—42no—
450C22×C5—42no—
451C22×C5—82no—
452C22×C5—84no—
453C22×C5#53512yes—
454C22×C5C23×D411yescharacteristic
455C2×D5—44no—
456C2×D5—44no—
457C2×D5—44no—
458C2×D5—88no—
459C4×C5—22no—
460C4×C5—22no—
461C4×C5—22no—
462C4×C5—22no—
463C4×C5—22no—
464C4×C5—42no—
465C4×C5—42no—
466C4×C5—42no—
467C4×C5#53512yes—
468C4×C5C2×H(5,𝔽2)12yes—
469Dic5—44no—
470C24—21no—
471C24—102no—
472C24C2×C22⋊D511yescharacteristic
473#41—108no—
474#41—108no—
475#41—204no—
476#41—204no—
477#41—208no—
478#41—208no—
479C2×Q8—21no—
480C2×Q8—24no—
481C2×Q8—24no—
482C2×Q8—54no—
483C2×Q8—82no—
484C2×Q8—102no—
485C2×Q8—204no—
486C2×Q8C23×D512yes—
487C2×D4—24no—
488C2×D4—24no—
489C2×D4—21no—
490C2×D4—54no—
491C2×D4—82no—
492C2×D4—104no—
493C2×D4—104no—
494C2×D4—102no—
495C2×D4—102no—
496C2×D4—102no—
497C2×D4—102no—
498C2×D4—102no—
499C2×D4—204no—
500C2×D4—208no—
501C2×D4C23×D512yes—
502C22×C4—24no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—24no—
507C22×C4—82no—
508C22×C4—102no—
509C22×C4—104no—
510C22×C4—104no—
511C22×C4—208no—
512C22×C4C2×C22⋊D511yescharacteristic
513C22×C4C2×C22⋊D512yes—
514C22×C4C2×C22⋊D511yescharacteristic
515C22×C4C2×C22⋊D511yescharacteristic
516C22×C4C2×C22⋊D511yescharacteristic
517C22×C4C2×C22⋊D511yescharacteristic
518C22×C4C23×D512yes—
519C22×C4C23×D511yescharacteristic
520Q16—208no—
521SD16—208no—
522SD16—208no—
523D8—208no—
524M16—108no—
525C2×C8—58no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—102no—
532C4⋊C4—102no—
533C4⋊C4—102no—
534C4⋊C4—102no—
535C4⋊C4C2×C22⋊D514yes—
536C4⋊C4C2×C22⋊D514yes—
537C4⋊C4C23×D512yes—
538C4⋊C4C23×D512yes—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—104no—
545C22⋊C4—104no—
546C22⋊C4—104no—
547C22⋊C4—104no—
548C22⋊C4C2×C22⋊D514yes—
549C22⋊C4C2×C22⋊D514yes—
550C42—24no—
551C42C2×C22⋊D512yes—
552C42C2×C22⋊D512yes—
553C42C23×D512yes—
554C42C23×D512yes—
555C2×C5—22no—
556C2×C5—42no—
557C2×C5—82no—
558C2×C5C2×#53511yescharacteristic
559C2×C5#358511yescharacteristic
560C2×C5#358611yescharacteristic
561D5—84no—
562C23—22no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—104no—
570C23—204no—
571C23—402no—
572C23—402no—
573C23C22×C22⋊D512yes—
574C23C22×C22⋊D511yescharacteristic
575Q8—24no—
576Q8—42no—
577Q8—84no—
578Q8—104no—
579Q8—104no—
580Q8—204no—
581D4—24no—
582D4—42no—
583D4—84no—
584D4—104no—
585D4—104no—
586D4—204no—
587D4—208no—
588D4—208no—
589D4—402no—
590D4—402no—
591D4—402no—
592D4—402no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—84no—
607C2×C4—104no—
608C2×C4—208no—
609C2×C4—204no—
610C2×C4—208no—
611C2×C4#414914yes—
612C2×C4C22×C22⋊D512yes—
613C2×C4C22×C22⋊D512yes—
614C2×C4C22×C22⋊D512yes—
615C2×C4C22×C22⋊D512yes—
616C2×C4C22×C22⋊D512yes—
617C2×C4C22×C22⋊D511yescharacteristic
618C2×C4C24×D511yescharacteristic
619C8—108no—
620C5#6081011yescharacteristic, complemented, Sylow
621C22—21no—
622C22—21no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—82no—
630C22—84no—
631C22—204no—
632C22—204no—
633C22—204no—
634C22—408no—
635C22#6616412yes—
636C22C23×C22⋊D511yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—204no—
646C4#6616412yes—
647C4C2×#414912yes—
648C2—22no—
649C2—42no—
650C2—82no—
651C2—404no—
652C2C2×#6616411yescharacteristic
653C2#1061917811yescharacteristic
654C2#1061918811yescharacteristic
subgroupG/H
centerC22C23×C22⋊D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#542C2
socleC22×C5C23×D4
hypercenterC2×#542D5
hypocenterC5#60810
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. C2×C5×#542C2semidirect
2. #10617636C2semidirect
3. #10615195C2semidirect
4. #10615194C2semidirect
5. #10614930C2semidirect
6. #10614674C2semidirect
7. #10614466C2semidirect
8. #10614355C2semidirect
9. #10614172C2semidirect
10. #10614038C2semidirect
11. #10613940C2semidirect
14. C5×#542C22semidirect
15. C2×C5×C4⋊Q8C22semidirect
18. C2×C5×#124C22semidirect
23. C2×C4×C5×Q8C22semidirect
24. C2×C4×C5×D4C22semidirect
26. C2×#4088C22semidirect
27. C2×#4078C22semidirect
28. #65860C22semidirect
29. #65859C22semidirect
38. #65382C22semidirect
39. #65378C22semidirect
40. #65377C22semidirect
41. #65349C22semidirect
42. #65348C22semidirect
43. #65347C22semidirect
44. #65339C22semidirect
45. #65336C22semidirect
46. #65335C22semidirect
47. #65329C22semidirect
49. #65325C22semidirect
50. #65320C22semidirect
51. #65317C22semidirect
52. #65316C22semidirect
53. #65308C22semidirect
54. #65305C22semidirect
55. #65304C22semidirect
56. #65289C22semidirect
57. #65255C22semidirect
58. #65255C22semidirect
59. #65213C22semidirect
60. C4×#681C22semidirect
151. C2×#542D5semidirect
251. #542C2×D5semidirect
260. C2×C4⋊Q8C2×D5semidirect
263. C2×#124C2×D5semidirect
269. C2×C4×Q8C2×D5semidirect
270. C2×C4×D4C2×D5semidirect
620. C5#60810semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5422
#106176362
#106151952
#106151942
#106149302
#106146742
#106144662
#106143552
#106141722
#106140382
#106139402
#608105