Finite Group Databasebeta

← #12259472

Subgroups — #12259472

Order 1280.

2470 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×Dic5C211yesmaximal, characteristic
2C5×#2339C211yesmaximal, characteristic
3#10615819C211yesmaximal, characteristic
4D4×#683C212yesmaximal, complemented
5#10614048C212yesmaximal, complemented
6#10608220C212yesmaximal, complemented
7#10608214C212yesmaximal, complemented
8#10606404C214yesmaximal, complemented
9#10606394C214yesmaximal, complemented
10#10605790C212yesmaximal, complemented
11#10605782C212yesmaximal, complemented
12#10605612C211yesmaximal, characteristic, complemented
13#10605595C211yesmaximal, characteristic, complemented
14#10605190C211yesmaximal, characteristic, complemented
15#10605168C211yesmaximal, characteristic, complemented
16#10605066C212yesmaximal, complemented
17C2×#64939C212yesmaximal, complemented
18C22×C5×#41C2211yescharacteristic
19C2×C4×C5×D4C2212yes—
20#41×Dic5C4116yescomplemented
21C2×Q8×Dic5C2211yescharacteristic
22C2×Q8×Dic5C2211yescharacteristic
23C2×D4×Dic5C2212yes—
24C2×D4×Dic5C2211yescharacteristic
25C2×D4×Dic5C2211yescharacteristic
26C2×D4×Dic5C2212yes—
27C2×D4×Dic5C2212yes—
28C2×#4072C2212yes—
29C2×#4072C2211yescharacteristic
30C22×#683C2214yes—
31C22×C4×Dic5C2211yescharacteristic
32C22×C4×Dic5C2212yes—
33C2×#4023C2212yes—
34C5×#391C2212yes—
35C5×#386C2214yes—
36C5×#381C2212yes—
37C5×#380C2211yescharacteristic
38C4×C5×C22⋊C4C2211yescharacteristic
39C2×C5×#95C2212yes—
40#65519C2212yes—
41#65509C2214yes—
42#65501C2211yescharacteristic
43#65500C2212yes—
44#65499C2214yes—
45C4×C22⋊Dic5C2214yes—
46C4×C22⋊Dic5C2211yescharacteristic
47C2×#3963C2212yes—
48C2×#3963C2214yes—
49#65265C2212yes—
50C4⋊C4×Dic5C2212yes—
51C4⋊C4×Dic5C2212yes—
52#65262C2212yes—
53#65235C2214yes—
54#65234C2214yes—
55#65233C2214yes—
56#65232C2212yes—
57#65231C2212yes—
58#65231C2214yes—
59#65231C2212yes—
60C22⋊C4×Dic5C2214yes—
61C22⋊C4×Dic5C2214yes—
62C4×C4⋊Dic5C2212yes—
63#65221C2212yes—
64C4×#683C2212yes—
65#65217C2212yes—
66#64943C2214yes—
67#64943C2214yes—
68#64942C2214yes—
69#64942C2214yes—
70#64940C2211yescharacteristic
71#64940C2214yes—
72#64940C2211yescharacteristic
73#64939C2212yes—
74#64939C2212yes—
75#64939C2212yes—
76#64939C2214yes—
77#64939C2214yes—
78#64939C2211yescharacteristic
79#64939C2211yescharacteristic
80#64938C2212yes—
81#64938C2214yes—
82#64937C2212yes—
83#64937C2212yes—
84#34191—51nomaximal, Sylow
85C2×C5×#41—28no—
86C2×C5×#41C2×C418yes—
87C22×C5×Q8C2311yescharacteristic
88C22×C5×D4C2312yes—
89C22×C5×D4C2311yescharacteristic
90C23×C4×C5C2312yes—
91C23×C4×C5C2311yescharacteristic
92C23×C4×C5C2314yes—
93C23×Dic5C2311yescharacteristic
94C23×Dic5C2312yes—
95C23×Dic5C2314yes—
96C23×Dic5C2312yes—
97C23×Dic5C2311yescharacteristic
98C4×C5×D4—28no—
99C2×C5×C4⋊C4C2312yes—
100C2×C5×C22⋊C4C2314yes—
101C2×C5×C22⋊C4C2312yes—
102C2×C5×C22⋊C4C2312yes—
103C2×C42×C5C2312yes—
104C2×C22⋊Dic5C2314yes—
105C2×C22⋊Dic5C2312yes—
106C2×C22⋊Dic5C2312yes—
107C2×C22⋊Dic5C2312yes—
108C2×C22⋊Dic5C2312yes—
109C2×C22⋊Dic5C2312yes—
110Q8×Dic5C2×C418yescomplemented
111Q8×Dic5C2×C418yescomplemented
112D4×Dic5—28no—
113D4×Dic5C2×C418yes—
114D4×Dic5C2×C418yes—
115D4×Dic5C2×C4116yescomplemented
116D4×Dic5C2×C4116yescomplemented
117#4072C2×C418yescomplemented
118#4072C2×C4116yescomplemented
119C2×C4⋊Dic5C2312yes—
120C2×C4⋊Dic5C2312yes—
121C2×C4⋊Dic5C2312yes—
122C2×C4⋊Dic5C2311yescharacteristic
123C2×C4⋊Dic5C2311yescharacteristic
124C2×#683—48no—
125C2×#683C2314yes—
126C2×#683C2312yes—
127C2×#683C2312yes—
128C2×#683C2314yes—
129C2×#683C2314yes—
130C2×#683C2314yes—
131C2×C4×Dic5C2×C418yescomplemented
132C2×C4×Dic5C2×C4116yescomplemented
133C2×C4×Dic5C2312yes—
134C2×C4×Dic5C2314yes—
135C2×C4×Dic5C2314yes—
136C2×C4×Dic5C2314yes—
137C2×C4×Dic5C2311yescharacteristic
138C2×C4×Dic5C2312yes—
139C2×C4×Dic5C2311yescharacteristic
140C2×C4×Dic5C2312yes—
141C2×C4×Dic5C2312yes—
142C2×C4×Dic5C2311yescharacteristic
143C2×C4×Dic5C2311yescharacteristic
144#4023—44no—
145C5×#95C2311yescharacteristic
146C5×#95C2312yes—
147C5×#95C2314yes—
148C5×#95C2311yescharacteristic
149C5×#95C2314yes—
150#3963C2311yescharacteristic
151#3963C2312yes—
152#3963C2314yes—
153#3963C2314yes—
154#3963C2314yes—
155#3963C2311yescharacteristic
156#3963C2312yes—
157#3963C2314yes—
158#3963C2314yes—
159#3963C2314yes—
160#3963C2314yes—
161#3963C2312yes—
162#3963C2312yes—
163#3963C2312yes—
164#3963C2312yes—
165#3963C2312yes—
166C2×C4×#41—51no—
167#2355—52no—
168#2352—54no—
169D4×C4⋊C4—52no—
170#2346—51no—
171#2346—51no—
172#2343—52no—
173#2343—52no—
174#2339—51no—
175#2339D511yescharacteristic
176#2320—52no—
177#2318—54no—
178#2285—51no—
179#2283—52no—
180#2283—51no—
181C2×#382—52no—
182#2275—52no—
183C24×C5C2411yescharacteristic
184C24×C5C2412yes—
185C5×#41—232no—
186C2×C5×Q8—24no—
187C2×C5×Q8C22×C414yes—
188C2×C5×D4—28no—
189C2×C5×D4C22×C418yes—
190C2×C5×D4C22×C414yes—
191C2×C5×D4#4118yes—
192C22×C4×C5—28no—
193C22×C4×C5—24no—
194C22×C4×C5—28no—
195C22×C4×C5—44no—
196C22×C4×C5—44no—
197C22×C4×C5C22×C418yes—
198C22×C4×C5C22×C414yes—
199C22×C4×C5C2412yes—
200C22×C4×C5C2412yes—
201C22×C4×C5C2414yes—
202C22×C4×C5C2411yescharacteristic
203C22×C4×C5C2412yes—
204C22×C4×C5C2411yescharacteristic
205C22×Dic5—22no—
206C22×Dic5—24no—
207C22×Dic5—24no—
208C22×Dic5—22no—
209C22×Dic5—28no—
210C22×Dic5—44no—
211C22×Dic5—42no—
212C22×Dic5—44no—
213C22×Dic5—44no—
214C22×Dic5C22×C418yes—
215C22×Dic5C22×C414yes—
216C22×Dic5C22×C418yes—
217C22×Dic5C22×C414yes—
218C22×Dic5C2412yes—
219C22×Dic5C2414yes—
220C22×Dic5C2412yes—
221C22×Dic5C2411yescharacteristic
222C22×Dic5C2412yes—
223C22×Dic5C2412yes—
224C22×Dic5C2411yescharacteristic
225C22×Dic5C2411yescharacteristic
226C22×Dic5C2411yescharacteristic
227C5×C4⋊C4—24no—
228C5×C22⋊C4—28no—
229C5×C22⋊C4—42no—
230C5×C22⋊C4—42no—
231C42×C5—24no—
232C22⋊Dic5—28no—
233C22⋊Dic5—42no—
234C22⋊Dic5—42no—
235C22⋊Dic5C22×C418yes—
236C22⋊Dic5C22×C418yes—
237C22⋊Dic5C22×C418yes—
238C4⋊Dic5—24no—
239C4⋊Dic5C22×C414yes—
240C4⋊Dic5C22×C414yes—
241C4⋊Dic5C22×C418yes—
242C4⋊Dic5C22×C418yes—
243#683—28no—
244#683—24no—
245#683—24no—
246#683—44no—
247#683—44no—
248#683—44no—
249C4×Dic5—24no—
250C4×Dic5—44no—
251C4×Dic5—44no—
252C4×Dic5—44no—
253C4×Dic5C22×C414yes—
254C4×Dic5C22×C418yes—
255C4×Dic5C22×C414yes—
256C4×Dic5C22×C414yes—
257C4×Dic5C22×C414yes—
258C4×Dic5C22×C418yes—
259C4×Dic5C22×C418yes—
260C22×#41C2×D511yescharacteristic
261C4×#41—516no—
262C2×C4×Q8—51no—
263C2×C4×Q8—51no—
264C2×C4×D4—52no—
265C2×C4×D4—51no—
266C2×C4×D4—52no—
267C2×C4×D4—51no—
268C2×C4×D4—52no—
269C2×C4×D4C2×D512yes—
270C2×#117—52no—
271C2×#117—51no—
272C2×#117—52no—
273C22×C4⋊C4—54no—
274C22×C42—52no—
275C22×C42—51no—
276#391—52no—
277#391C2×D512yes—
278#389—52no—
279#389—52no—
280#388—54no—
281#388—54no—
282#387—54no—
283#387—54no—
284#387—54no—
285#387—54no—
286#386—54no—
287#386C2×D514yes—
288#385—54no—
289#385—54no—
290#384—52no—
291#383—54no—
292#383—51no—
293#383—51no—
294#382—54no—
295#382—52no—
296#382—52no—
297#382—54no—
298#382—51no—
299#382—52no—
300#382—51no—
301#381—52no—
302#381—52no—
303#381C2×D512yes—
304#380—51no—
305#380—52no—
306#380—54no—
307#380—52no—
308#380C2×D511yescharacteristic
309C4×C4⋊C4—52no—
310C4×C4⋊C4—52no—
311C4×C4⋊C4—52no—
312C4×C4⋊C4—52no—
313C4×C4⋊C4—52no—
314C4×C4⋊C4—54no—
315C4×C22⋊C4—51no—
316C4×C22⋊C4—54no—
317C4×C22⋊C4—54no—
318C4×C22⋊C4—54no—
319C4×C22⋊C4C2×D511yescharacteristic
320#376—52no—
321#376—52no—
322#376—52no—
323C2×#95—54no—
324C2×#95—52no—
325C2×#95C2×D512yes—
326C23×C5—22no—
327C23×C5—22no—
328C23×C5—24no—
329C23×C5—22no—
330C23×C5—21no—
331C23×C5—21no—
332C23×C5—22no—
333C23×C5C23×C412yes—
334C23×C5C23×C414yes—
335C23×C5C2×#4118yes—
336C23×C5C2511yescharacteristic
337C5×Q8—28no—
338C5×D4—216no—
339C5×D4#117116yescomplemented
340C2×C4×C5—216no—
341C2×C4×C5—22no—
342C2×C4×C5—24no—
343C2×C4×C5—28no—
344C2×C4×C5—22no—
345C2×C4×C5—22no—
346C2×C4×C5—24no—
347C2×C4×C5—24no—
348C2×C4×C5—22no—
349C2×C4×C5—22no—
350C2×C4×C5—22no—
351C2×C4×C5—24no—
352C2×C4×C5—22no—
353C2×C4×C5—88no—
354C2×C4×C5C23×C412yes—
355C2×C4×C5C23×C414yes—
356C2×C4×C5C23×C412yes—
357C2×C4×C5C2×#4114yes—
358C2×Dic5—21no—
359C2×Dic5—22no—
360C2×Dic5—21no—
361C2×Dic5—22no—
362C2×Dic5—24no—
363C2×Dic5—22no—
364C2×Dic5—22no—
365C2×Dic5—216no—
366C2×Dic5—28no—
367C2×Dic5—22no—
368C2×Dic5—22no—
369C2×Dic5—24no—
370C2×Dic5—24no—
371C2×Dic5—22no—
372C2×Dic5—22no—
373C2×Dic5—22no—
374C2×Dic5—24no—
375C2×Dic5—22no—
376C2×Dic5—48no—
377C2×Dic5—44no—
378C2×Dic5—88no—
379C2×Dic5C23×C412yes—
380C2×Dic5C23×C412yes—
381C2×Dic5C23×C414yes—
382C2×Dic5C23×C412yes—
383C2×Dic5C23×C414yes—
384C2×Dic5C23×C412yes—
385C2×Dic5H(5,𝔽2)12yes—
386C2×Dic52−1+412yes—
387C2×#41—28no—
388C2×#41C4×D518yes—
389C22×Q8C22×D511yescharacteristic
390C22×D4C22×D512yes—
391C22×D4C22×D511yescharacteristic
392C23×C4—52no—
393C23×C4—51no—
394C23×C4—52no—
395C23×C4—51no—
396C23×C4—54no—
397C23×C4C22×D512yes—
398C23×C4C22×D511yescharacteristic
399C23×C4C22×D514yes—
400C4×Q8—58no—
401C4×Q8—58no—
402C4×D4—28no—
403C4×D4—58no—
404C4×D4—58no—
405C4×D4—516no—
406C4×D4—516no—
407C4×D4—108no—
408#117—516no—
409#117—58no—
410#117—204no—
411C2×C4⋊C4—51no—
412C2×C4⋊C4—52no—
413C2×C4⋊C4—52no—
414C2×C4⋊C4—51no—
415C2×C4⋊C4—52no—
416C2×C4⋊C4—54no—
417C2×C4⋊C4—52no—
418C2×C4⋊C4—52no—
419C2×C4⋊C4—54no—
420C2×C4⋊C4—54no—
421C2×C4⋊C4—54no—
422C2×C4⋊C4—208no—
423C2×C4⋊C4C22×D512yes—
424C2×C22⋊C4—52no—
425C2×C22⋊C4—52no—
426C2×C22⋊C4—52no—
427C2×C22⋊C4—54no—
428C2×C22⋊C4—52no—
429C2×C22⋊C4—52no—
430C2×C22⋊C4C22×D514yes—
431C2×C22⋊C4C22×D512yes—
432C2×C22⋊C4C22×D512yes—
433C2×C42—516no—
434C2×C42—58no—
435C2×C42—52no—
436C2×C42—51no—
437C2×C42—51no—
438C2×C42—52no—
439C2×C42—51no—
440C2×C42—52no—
441C2×C42—51no—
442C2×C42—52no—
443C2×C42—54no—
444C2×C42—54no—
445C2×C42—54no—
446C2×C42C22×D512yes—
447#95—52no—
448#95—51no—
449#95—52no—
450#95—54no—
451#95—54no—
452#95—54no—
453#95—54no—
454#95—51no—
455#95—52no—
456#95—52no—
457#95—52no—
458#95—54no—
459#95—54no—
460#95—52no—
461#95—52no—
462#95—54no—
463#95C22×D511yescharacteristic
464#95C22×D512yes—
465#95C22×D514yes—
466#95C22×D511yescharacteristic
467#95C22×D514yes—
468C22×C5—24no—
469C22×C5—24no—
470C22×C5—28no—
471C22×C5—28no—
472C22×C5—22no—
473C22×C5—22no—
474C22×C5—22no—
475C22×C5—42no—
476C22×C5—42no—
477C22×C5C2×#11718yes—
478C22×C5C24×C411yescharacteristic
479C22×C5C22×#4112yes—
480C22×C5C2×H(5,𝔽2)11yescharacteristic
481C22×C5C2×2−1+411yescharacteristic
482C22×C5#58512yes—
483C4×C5—22no—
484C4×C5—24no—
485C4×C5—44no—
486C4×C5—42no—
487C4×C5—42no—
488C4×C5C2×#11714yes—
489Dic5—22no—
490Dic5—24no—
491Dic5—22no—
492Dic5—24no—
493Dic5—22no—
494Dic5—44no—
495Dic5—42no—
496Dic5—42no—
497Dic5#52014yes—
498C24C23×D511yescharacteristic
499C24C23×D512yes—
500#41—232no—
501C2×Q8—24no—
502C2×Q8C2×C4×D514yes—
503C2×D4—28no—
504C2×D4C2×C4×D518yes—
505C2×D4C2×C4×D514yes—
506C2×D4#71118yes—
507C22×C4—28no—
508C22×C4—24no—
509C22×C4—28no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—51no—
513C22×C4—54no—
514C22×C4—58no—
515C22×C4—51no—
516C22×C4—52no—
517C22×C4—51no—
518C22×C4—51no—
519C22×C4—52no—
520C22×C4—54no—
521C22×C4—58no—
522C22×C4—52no—
523C22×C4—52no—
524C22×C4—54no—
525C22×C4—104no—
526C22×C4—102no—
527C22×C4—104no—
528C22×C4—102no—
529C22×C4—108no—
530C22×C4—204no—
531C22×C4—202no—
532C22×C4—204no—
533C22×C4—204no—
534C22×C4C2×C4×D518yes—
535C22×C4C2×C4×D514yes—
536C22×C4C23×D512yes—
537C22×C4C23×D512yes—
538C22×C4C23×D514yes—
539C22×C4C23×D511yescharacteristic
540C22×C4C23×D512yes—
541C22×C4C23×D511yescharacteristic
542C4⋊C4—24no—
543C4⋊C4—58no—
544C4⋊C4—58no—
545C4⋊C4—54no—
546C4⋊C4—54no—
547C4⋊C4—104no—
548C4⋊C4—108no—
549C4⋊C4—104no—
550C4⋊C4—104no—
551C4⋊C4—204no—
552C4⋊C4—204no—
553C4⋊C4—204no—
554C22⋊C4—28no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—58no—
558C22⋊C4—58no—
559C22⋊C4—58no—
560C22⋊C4—108no—
561C22⋊C4—202no—
562C22⋊C4—202no—
563C42—24no—
564C42—54no—
565C42—58no—
566C42—54no—
567C42—58no—
568C42—54no—
569C42—58no—
570C42—54no—
571C42—104no—
572C42—204no—
573C42—204no—
574C42—204no—
575C2×C5—28no—
576C2×C5—42no—
577C2×C5C22×#11711yescharacteristic
578C2×C5C2×#52011yescharacteristic
579C2×C5#342811yescharacteristic
580C2×C5#351412yes—
581C2×C5#351512yes—
582C23—24no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—21no—
588C23—21no—
589C23C22×C4×D512yes—
590C23C22×C4×D514yes—
591C23C2×#71118yes—
592C23C24×D511yescharacteristic
593Q8—28no—
594D4—216no—
595D4#4023116yescomplemented
596C2×C4—216no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—52no—
610C2×C4—52no—
611C2×C4—52no—
612C2×C4—52no—
613C2×C4—54no—
614C2×C4—52no—
615C2×C4—52no—
616C2×C4—54no—
617C2×C4—88no—
618C2×C4—1016no—
619C2×C4—108no—
620C2×C4—104no—
621C2×C4—102no—
622C2×C4—102no—
623C2×C4—101no—
624C2×C4—102no—
625C2×C4—101no—
626C2×C4—102no—
627C2×C4—102no—
628C2×C4—102no—
629C2×C4—104no—
630C2×C4—102no—
631C2×C4—104no—
632C2×C4—102no—
633C2×C4—104no—
634C2×C4—102no—
635C2×C4—102no—
636C2×C4—208no—
637C2×C4—204no—
638C2×C4—408no—
639C2×C4C22×C4×D512yes—
640C2×C4C22×C4×D514yes—
641C2×C4C22×C4×D512yes—
642C2×C4C2×#71114yes—
643C5#3419111yescharacteristic, complemented, Sylow
644C22—24no—
645C22—28no—
646C22—24no—
647C22—28no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—42no—
652C22—42no—
653C22C2×#402318yes—
654C22C23×C4×D511yescharacteristic
655C22C22×#71112yes—
656C22#6628312yes—
657C22D5×H(5,𝔽2)11yescharacteristic
658C22D5×2−1+411yescharacteristic
659C4—22no—
660C4—24no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—54no—
665C4—102no—
666C4—104no—
667C4—102no—
668C4—102no—
669C4—104no—
670C4—202no—
671C4—204no—
672C4—202no—
673C4C2×#402314yes—
674C2—28no—
675C2—42no—
676C2C22×#402311yescharacteristic
677C2#1061773511yescharacteristic
678C2D5×#52011yescharacteristic
679C2#1061853812yes—
680C2#1061858312yes—
subgroupG/H
center, FrattiniC23C24×D5
commutatorC22×C5C24×C4
FittingC5×#2339C2
socle, cosocleC23×C5C25
prime radicalC22×C5×#41C22
hypercenter#2339D5
hypocenterC5#34191
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #2339

Complements

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

NH ≅ G/Ntype
4. D4×#683C2semidirect
5. #10614048C2semidirect
6. #10608220C2semidirect
7. #10608214C2semidirect
8. #10606404C2semidirect
9. #10606394C2semidirect
10. #10605790C2semidirect
11. #10605782C2semidirect
12. #10605612C2semidirect
13. #10605595C2semidirect
14. #10605190C2semidirect
15. #10605168C2semidirect
16. #10605066C2semidirect
17. C2×#64939C2semidirect
20. #41×Dic5C4semidirect
110. Q8×Dic5C2×C4semidirect
111. Q8×Dic5C2×C4semidirect
115. D4×Dic5C2×C4semidirect
116. D4×Dic5C2×C4semidirect
117. #4072C2×C4semidirect
118. #4072C2×C4semidirect
131. C2×C4×Dic5C2×C4semidirect
132. C2×C4×Dic5C2×C4semidirect
339. C5×D4#117semidirect
595. D4#4023semidirect
643. C5#34191semidirect

Maximal subgroups

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

subgroupindex
C2×#41×Dic52
C5×#23392
#106158192
D4×#6832
#106140482
#106082202
#106082142
#106064042
#106063942
#106057902
#106057822
#106056122
#106055952
#106051902
#106051682
#106050662
C2×#649392
#341915