Finite Group Databasebeta

← #12121328

Subgroups — #12121328

Order 1280.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#474C211yesmaximal, characteristic
2#10614961C211yesmaximal, characteristic, complemented
3#10614064C211yesmaximal, characteristic
4#10602581C211yesmaximal, characteristic, complemented
5#10602402C211yesmaximal, characteristic
6#10601406C211yesmaximal, characteristic, complemented
7#10601405C211yesmaximal, characteristic, complemented
8#10601403C211yesmaximal, characteristic, complemented
9#10601402C211yesmaximal, characteristic, complemented
10#10601260C211yesmaximal, characteristic, complemented
11#10601248C211yesmaximal, characteristic, complemented
12#10600963C211yesmaximal, characteristic
13#10600961C211yesmaximal, characteristic
14#10600945C211yesmaximal, characteristic
15Dic5×#102C211yesmaximal, characteristic
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C4×C5×D4C2211yescharacteristic
18C2×D4×Dic5C2211yescharacteristic
19C2×D4×Dic5C2211yescharacteristic
20C5×#474C418yescomplemented
21C2×C5×#107C2211yescharacteristic
22C2×C5×#107C2211yescharacteristic
23C2×C5×C4⋊C8C2211yescharacteristic
24C2×C5×#102C2211yescharacteristic
25C2×C5×#102C2211yescharacteristic
26#65519—21no—
27#65519—21no—
28C2×#3964C2211yescharacteristic
29C2×#3964C2211yescharacteristic
30#65501—21no—
31C2×#3963—21no—
32C2×#3950C2211yescharacteristic
33C2×#3949C2211yescharacteristic
34C2×#3947C2211yescharacteristic
35C2×C8×Dic5C2211yescharacteristic
36#65274—21no—
37#65274C2211yescharacteristic
38#65274C2211yescharacteristic
39#65274C2211yescharacteristic
40#65274C2211yescharacteristic
41C4⋊C4×Dic5—21no—
42C4⋊C4×Dic5C2211yescharacteristic
43C4⋊C4×Dic5C2211yescharacteristic
44#65239—21no—
45#65239—21no—
46#65225C2211yescharacteristic
47C4×C4⋊Dic5C2211yescharacteristic
48C4×C4⋊Dic5C2211yescharacteristic
49C2×#3935C2211yescharacteristic
50#64771C2211yescharacteristic
51#64771C2211yescharacteristic
52#64770C2211yescharacteristic
53#64770C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#22979—51nomaximal, Sylow
61C22×C5×Q8—21no—
62C22×C5×D4C2311yescharacteristic
63C23×C4×C5—21no—
64C23×Dic5—21no—
65C23×Dic5—21no—
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C5×C4⋊Q8—22no—
69C5×C4⋊Q8C2×C414yescomplemented
70C4×C5×D4—22no—
71C4×C5×D4C2×C414yes—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C5×C22⋊C4—21no—
77C2×C42×C5C2311yescharacteristic
78C2×C22⋊Dic5—21no—
79C2×C22⋊Dic5—21no—
80D4×Dic5—22no—
81D4×Dic5—22no—
82D4×Dic5—41no—
83D4×Dic5—41no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5—21no—
86C2×C4⋊Dic5—21no—
87C2×C4⋊Dic5—21no—
88C2×C4⋊Dic5D411yescharacteristic
89C2×C4⋊Dic5D411yescharacteristic
90C2×C4⋊Dic5C2311yescharacteristic
91C2×C4⋊Dic5C2311yescharacteristic
92C2×C4⋊Dic5C2311yescharacteristic
93C2×C4⋊Dic5C2311yescharacteristic
94C2×C4×Dic5—21no—
95C2×C4×Dic5—21no—
96C2×C4×Dic5—21no—
97C2×C4×Dic5C2311yescharacteristic
98C2×C4×Dic5C2311yescharacteristic
99C22×#168C2311yescharacteristic
100C22×#168C2311yescharacteristic
101C5×#107C2×C414yescomplemented
102C5×#107C2×C414yescomplemented
103C5×C4⋊C8C2×C414yescomplemented
104C5×#102C2×C414yes—
105C5×#102C2×C414yes—
106#3964—22no—
107#3964Q814yescomplemented
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113#3963—21no—
114#3963—21no—
115#3963—21no—
116#3963—21no—
117#3963D411yescharacteristic
118#3963D411yescharacteristic
119#3950—41no—
120#3949—41no—
121#3947—41no—
122C8×Dic5—41no—
123#3935—22no—
124C2×#474D511yescharacteristic
125#2369—51no—
126#2349—51no—
127#1993—51no—
128#1949—51no—
129#1943—51no—
130#1942—51no—
131#1921—51no—
132#1918—51no—
133#1864—51no—
134#1862—51no—
135#1771—51no—
136#1762—51no—
137#1762—51no—
138C4×#102—51no—
139C24×C5—21no—
140C2×C5×Q8—24no—
141C2×C5×Q8—81no—
142C2×C5×Q8—81no—
143C2×C5×D4—21no—
144C2×C5×D4—21no—
145C2×C5×D4—21no—
146C2×C5×D4C22×C412yes—
147C2×C5×D4C2×Q812yes—
148C2×C5×D4#4112yes—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—24no—
153C22×C4×C5—81no—
154C22×C4×C5—81no—
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2411yescharacteristic
158C22×Dic5—21no—
159C22×Dic5—21no—
160C22×Dic5—21no—
161C22×Dic5—21no—
162C22×Dic5—42no—
163C22×Dic5—42no—
164C22×Dic5—42no—
165C22×Dic5—42no—
166C22×Dic5—81no—
167C22×Dic5—81no—
168C22×Dic5C2×D411yescharacteristic
169C22×Dic5C2×D411yescharacteristic
170C22×Dic5C2×D411yescharacteristic
171C22×Dic5C2×D411yescharacteristic
172C2×C5×C8—21no—
173C2×C5×C8—21no—
174C2×C5×C8—21no—
175C2×C5×C8—21no—
176C2×C5×C8C22×C412yes—
177C2×C5×C8C22×C412yes—
178C5×C4⋊C4—24no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4C22×C412yes—
183C5×C4⋊C4C22×C412yes—
184C5×C4⋊C4C22×C412yes—
185C5×C22⋊C4—24no—
186C42×C5C22×C412yes—
187C42×C5#4112yes—
188C22⋊Dic5—42no—
189C22⋊Dic5—42no—
190C4⋊Dic5—21no—
191C4⋊Dic5—22no—
192C4⋊Dic5—21no—
193C4⋊Dic5—21no—
194C4⋊Dic5—21no—
195C4⋊Dic5—21no—
196C4⋊Dic5—21no—
197C4⋊Dic5—21no—
198C4⋊Dic5—41no—
199C4⋊Dic5—81no—
200C4⋊Dic5—81no—
201C4⋊Dic5—81no—
202C4⋊Dic5—81no—
203C4⋊Dic5C2×Q812yes—
204C4×Dic5—21no—
205C4×Dic5—21no—
206C4×Dic5—21no—
207C4×Dic5—21no—
208C4×Dic5—42no—
209C4×Dic5—81no—
210C4×Dic5—81no—
211C2×#168—21no—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168—21no—
215C2×#168—21no—
216C2×#168C2×Q812yes—
217C2×C4⋊Q8C2×D511yescharacteristic
218C2×C4×D4—51no—
219C2×C4×D4—51no—
220C2×C4×D4C2×D511yescharacteristic
221#474Dic518yescomplemented
222C2×#107—51no—
223C2×#107C2×D511yescharacteristic
224C2×#107C2×D511yescharacteristic
225C2×#106—51no—
226C2×C4⋊C8C2×D511yescharacteristic
227C2×#102—51no—
228C2×#102—51no—
229C2×#102C2×D511yescharacteristic
230C2×#102C2×D511yescharacteristic
231C2×#97—51no—
232C2×#97—51no—
233C2×C4×C8—51no—
234#391—101no—
235#391—101no—
236#389—51no—
237#389—51no—
238#389—51no—
239#389—51no—
240#389—101no—
241#385—101no—
242#385—101no—
243#382—51no—
244#380—101no—
245C4×C4⋊C4—51no—
246C4×C4⋊C4—51no—
247C4×C4⋊C4—51no—
248C4×C4⋊C4—51no—
249C4×C4⋊C4—101no—
250C2×#95—101no—
251#340—51no—
252#340—51no—
253#340—51no—
254#340—51no—
255#340—51no—
256#340—51no—
257#340—51no—
258#340—51no—
259#336—51no—
260#336—51no—
261C23×C5—22no—
262C23×C5—22no—
263C23×C5—22no—
264C23×C5—41no—
265C23×C5—41no—
266C23×C5—41no—
267C23×C5—41no—
268C23×C5C22×D411yescharacteristic
269C5×Q8—84no—
270C5×D4—22no—
271C5×D4—22no—
272C5×D4—22no—
273C5×D4C4×Q814yes—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—22no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—84no—
288C2×C4×C5C4×D412yes—
289C2×C4×C5C4×D412yes—
290C2×C4×C5C23×C411yescharacteristic
291C2×C4×C5C22×Q811yescharacteristic
292C2×C4×C5C2×#4111yescharacteristic
293C2×C4×C5C2×#4111yescharacteristic
294C2×C4×C5H(5,𝔽2)11yescharacteristic
295C2×C4×C52−1+411yescharacteristic
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—21no—
299C2×Dic5—21no—
300C2×Dic5—21no—
301C2×Dic5—22no—
302C2×Dic5—21no—
303C2×Dic5—21no—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5—21no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312C2×Dic5—41no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315C2×Dic5—41no—
316C2×Dic5—41no—
317C2×Dic5—41no—
318C2×Dic5—41no—
319C2×Dic5—84no—
320C2×Dic5—84no—
321C2×Dic5#13712yes—
322C5×C8—22no—
323C5×C8—22no—
324#168—22no—
325#168—41no—
326C22×Q8—21no—
327C22×D4C22×D511yescharacteristic
328C23×C4—21no—
329C23×C4—101no—
330C23×C4—101no—
331C22×C8—51no—
332C22×C8—51no—
333C22×C8C22×D511yescharacteristic
334C22×C8C22×D511yescharacteristic
335C4⋊Q8—22no—
336C4⋊Q8C2×Dic514yescomplemented
337C4×D4—22no—
338C4×D4—102no—
339C4×D4—102no—
340C4×D4—201no—
341C4×D4—201no—
342C4×D4C2×Dic514yes—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—51no—
345C2×C4⋊C4—51no—
346C2×C4⋊C4—51no—
347C2×C4⋊C4—51no—
348C2×C4⋊C4—51no—
349C2×C4⋊C4—51no—
350C2×C4⋊C4—101no—
351C2×C4⋊C4—101no—
352C2×C4⋊C4—101no—
353C2×C4⋊C4—101no—
354C2×C4⋊C4C22×D511yescharacteristic
355C2×C4⋊C4C22×D511yescharacteristic
356C2×C4⋊C4C22×D511yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—101no—
359C2×C22⋊C4—101no—
360C2×C42—51no—
361C2×C42—51no—
362C2×C42—101no—
363C2×C42—101no—
364C2×C42—101no—
365C2×C42C22×D511yescharacteristic
366#107—201no—
367#107C2×Dic514yescomplemented
368#107C2×Dic514yescomplemented
369#106—201no—
370C4⋊C8C2×Dic514yescomplemented
371#102—54no—
372#102—102no—
373#102C2×Dic514yes—
374#102C2×Dic514yes—
375#97—102no—
376#97—201no—
377C4×C8—201no—
378#95—51no—
379#95—51no—
380#95—101no—
381#95—101no—
382#95—101no—
383#95—101no—
384#95—101no—
385#95—101no—
386#95—101no—
387#95—101no—
388#95—101no—
389C22×C5—24no—
390C22×C5—42no—
391C22×C5—42no—
392C22×C5—42no—
393C22×C5—42no—
394C22×C5—42no—
395C22×C5—42no—
396C22×C5C2×C4×D411yescharacteristic
397C22×C5#54611yescharacteristic
398C22×C5D4×Q811yescharacteristic
399C22×C5C2×#13711yescharacteristic
400C22×C5#57511yescharacteristic
401C22×C5#57611yescharacteristic
402C22×C5#57611yescharacteristic
403C4×C5—22no—
404C4×C5—22no—
405C4×C5—42no—
406C4×C5—42no—
407C4×C5—42no—
408C4×C5C2×C4×Q811yescharacteristic
409C4×C5#52011yescharacteristic
410C4×C5#55211yescharacteristic
411C4×C5#55411yescharacteristic
412Dic5—22no—
413Dic5—42no—
414Dic5—41no—
415Dic5—41no—
416Dic5—41no—
417Dic5—81no—
418Dic5—81no—
419Dic5—81no—
420C24—21no—
421C2×Q8—24no—
422C2×Q8—81no—
423C2×Q8—81no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4Q8×D512yes—
428C2×D4#71312yes—
429C2×D4C22×Dic512yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—51no—
435C22×C4—51no—
436C22×C4—51no—
437C22×C4—51no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—101no—
441C22×C4—101no—
442C22×C4—101no—
443C22×C4—101no—
444C22×C4—202no—
445C22×C4—202no—
446C22×C4—202no—
447C22×C4—202no—
448C22×C4—401no—
449C22×C4—401no—
450C22×C4D4×D511yescharacteristic
451C22×C4D4×D511yescharacteristic
452C22×C4C23×D511yescharacteristic
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—52no—
458C2×C8—101no—
459C2×C8—101no—
460C2×C8—101no—
461C2×C8—101no—
462C2×C8—101no—
463C2×C8C22×Dic512yes—
464C2×C8C22×Dic512yes—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—52no—
470C4⋊C4—102no—
471C4⋊C4—101no—
472C4⋊C4—101no—
473C4⋊C4—101no—
474C4⋊C4—101no—
475C4⋊C4—101no—
476C4⋊C4—101no—
477C4⋊C4—101no—
478C4⋊C4—201no—
479C4⋊C4—401no—
480C4⋊C4—401no—
481C4⋊C4—401no—
482C4⋊C4—401no—
483C4⋊C4C22×Dic512yes—
484C4⋊C4C22×Dic512yes—
485C4⋊C4C22×Dic512yes—
486C22⋊C4—24no—
487C22⋊C4—202no—
488C22⋊C4—202no—
489C42—101no—
490C42—101no—
491C42—101no—
492C42—101no—
493C42—202no—
494C42—401no—
495C42—401no—
496C42#71112yes—
497C42C22×Dic512yes—
498C2×C5—44no—
499C2×C5#236011yescharacteristic
500C2×C5C4×#13711yescharacteristic
501C2×C5#294811yescharacteristic
502C2×C5#331411yescharacteristic
503C2×C5#331911yescharacteristic
504C2×C5#338511yescharacteristic
505C2×C5#338911yescharacteristic
506C23—22no—
507C23—22no—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C2×D4×D511yescharacteristic
514Q8—84no—
515D4—22no—
516D4—22no—
517D4—22no—
518D4Q8×Dic514yes—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—52no—
533C2×C4—84no—
534C2×C4—101no—
535C2×C4—101no—
536C2×C4—101no—
537C2×C4—101no—
538C2×C4—101no—
539C2×C4—101no—
540C2×C4—101no—
541C2×C4—101no—
542C2×C4—101no—
543C2×C4—101no—
544C2×C4—101no—
545C2×C4—102no—
546C2×C4—201no—
547C2×C4—201no—
548C2×C4—201no—
549C2×C4—201no—
550C2×C4—201no—
551C2×C4—201no—
552C2×C4—201no—
553C2×C4—201no—
554C2×C4—201no—
555C2×C4—201no—
556C2×C4—201no—
557C2×C4—404no—
558C2×C4—404no—
559C2×C4D4×Dic512yes—
560C2×C4D4×Dic512yes—
561C2×C4C2×#71111yescharacteristic
562C2×C4C2×Q8×D511yescharacteristic
563C2×C4C2×#71311yescharacteristic
564C2×C4#414911yescharacteristic
565C2×C4#415011yescharacteristic
566C2×C4C23×Dic511yescharacteristic
567C8—22no—
568C8—22no—
569C8—102no—
570C8—201no—
571C5#2297911yescharacteristic, complemented, Sylow
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22#6596411yescharacteristic
580C22#6596611yescharacteristic
581C22#6609111yescharacteristic
582C22#6609111yescharacteristic
583C22D5×#13711yescharacteristic
584C22#6611111yescharacteristic
585C22C2×D4×Dic511yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—102no—
592C4—202no—
593C4—201no—
594C4—201no—
595C4—201no—
596C4—401no—
597C4—401no—
598C4—401no—
599C4#6596911yescharacteristic
600C4#6597611yescharacteristic
601C4C2×Q8×Dic511yescharacteristic
602C4#6616111yescharacteristic
603C2—44no—
604C2#1061269111yescharacteristic
605C2#1061269311yescharacteristic
606C2#1061271311yescharacteristic
607C2#1061271411yescharacteristic
608C2#1061447811yescharacteristic
609C2#1061555411yescharacteristic
610C2Dic5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#474C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
cosocleC22×C4×C5C24
hypercenterC2×#474D5
hypocenterC5#22979
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
2. #10614961C2semidirect
4. #10602581C2semidirect
6. #10601406C2semidirect
7. #10601405C2semidirect
8. #10601403C2semidirect
9. #10601402C2semidirect
10. #10601260C2semidirect
11. #10601248C2semidirect
20. C5×#474C4semidirect
69. C5×C4⋊Q8C2×C4semidirect
101. C5×#107C2×C4semidirect
102. C5×#107C2×C4semidirect
103. C5×C4⋊C8C2×C4semidirect
107. #3964Q8semidirect
221. #474Dic5semidirect
336. C4⋊Q8C2×Dic5semidirect
367. #107C2×Dic5semidirect
368. #107C2×Dic5semidirect
370. C4⋊C8C2×Dic5semidirect
571. C5#22979semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4742
#106149612
#106140642
#106025812
#106024022
#106014062
#106014052
#106014032
#106014022
#106012602
#106012482
#106009632
#106009612
#106009452
Dic5×#1022
#229795