Finite Group Databasebeta

← #12111506

Subgroups — #12111506

Order 1280.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#476C211yesmaximal, characteristic
2#10614959C211yesmaximal, characteristic, complemented
3#10614056C211yesmaximal, characteristic
4#10602573C211yesmaximal, characteristic, complemented
5#10602398C211yesmaximal, characteristic
6#10601355C211yesmaximal, characteristic, complemented
7#10601353C211yesmaximal, characteristic, complemented
8#10601351C211yesmaximal, characteristic, complemented
9#10601350C211yesmaximal, characteristic, complemented
10#10601248C211yesmaximal, characteristic, complemented
11#10601235C211yesmaximal, characteristic, complemented
12#10600980C211yesmaximal, characteristic
13#10600978C211yesmaximal, characteristic
14#10600949C211yesmaximal, characteristic
15#10600948C211yesmaximal, characteristic
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C4×C5×D4C2211yescharacteristic
18C5×#476C418yescomplemented
19C2×C5×#106C2211yescharacteristic
20C2×C5×#106C2211yescharacteristic
21C2×C5×C4⋊C8C2211yescharacteristic
22C2×C5×#102C2211yescharacteristic
23C2×C5×#102C2211yescharacteristic
24#65519—21no—
25#65519—21no—
26#65510C2211yescharacteristic
27#65510C2211yescharacteristic
28C2×#3964C2211yescharacteristic
29C2×#3964C2211yescharacteristic
30#65501—21no—
31#65500—21no—
32C2×#3950C2211yescharacteristic
33C2×#3949C2211yescharacteristic
34C2×#3947C2211yescharacteristic
35C2×C8×Dic5C2211yescharacteristic
36#65274C2211yescharacteristic
37#65274C2211yescharacteristic
38#65274C2211yescharacteristic
39#65274C2211yescharacteristic
40#65270—21no—
41#65270—21no—
42C4⋊C4×Dic5C2211yescharacteristic
43C4⋊C4×Dic5C2211yescharacteristic
44#65240—21no—
45#65240—21no—
46#65225C2211yescharacteristic
47C4×C4⋊Dic5C2211yescharacteristic
48#65221C2211yescharacteristic
49C2×C4⋊#168C2211yescharacteristic
50#64771C2211yescharacteristic
51#64771C2211yescharacteristic
52#64770C2211yescharacteristic
53#64770C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#22123—51nomaximal, Sylow
61C22×C5×Q8—21no—
62C22×C5×D4C2311yescharacteristic
63C23×C4×C5—21no—
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C5×C4⋊Q8—22no—
67C5×C4⋊Q8C2×C414yes—
68C4×C5×D4—22no—
69C4×C5×D4C2×C414yes—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C4⋊C4C2311yescharacteristic
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C22⋊C4—21no—
75C2×C42×C5C2311yescharacteristic
76C2×C22⋊Dic5—21no—
77C2×C22⋊Dic5—21no—
78C2×C22⋊Dic5—21no—
79C2×C22⋊Dic5—21no—
80C2×C4⋊Dic5—21no—
81C2×C4⋊Dic5—21no—
82C2×C4⋊Dic5—21no—
83C2×C4⋊Dic5—21no—
84C2×C4⋊Dic5C2311yescharacteristic
85C2×C4⋊Dic5C2311yescharacteristic
86C2×C4⋊Dic5C2311yescharacteristic
87C2×C4⋊Dic5C2311yescharacteristic
88C2×C4×Dic5—21no—
89C2×C4×Dic5—21no—
90C2×C4×Dic5—21no—
91C2×C4×Dic5C2311yescharacteristic
92C2×C4×Dic5C2311yescharacteristic
93C22×#168C2311yescharacteristic
94C22×#168C2311yescharacteristic
95C5×#106C2×C414yes—
96C5×#106C2×C414yes—
97C5×C4⋊C8C2×C414yes—
98C5×#102C2×C414yes—
99C5×#102C2×C414yes—
100#3964—41no—
101#3964—41no—
102#3963—21no—
103#3963—21no—
104#3963—21no—
105#3963—21no—
106#3963—21no—
107#3963—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963D411yescharacteristic
112#3963D411yescharacteristic
113#3963D411yescharacteristic
114#3963D411yescharacteristic
115#3950—41no—
116#3949—41no—
117#3947—41no—
118C8×Dic5—41no—
119C4⋊#168—22no—
120C2×#476D511yescharacteristic
121#2331—51no—
122#2319—51no—
123#1988—51no—
124#1949—51no—
125#1944—51no—
126#1941—51no—
127#1936—51no—
128#1934—51no—
129#1926—51no—
130#1925—51no—
131#1920—51no—
132#1919—51no—
133#1804—51no—
134#1768—51no—
135C24×C5—21no—
136C2×C5×Q8—24no—
137C2×C5×Q8—81no—
138C2×C5×Q8—81no—
139C2×C5×D4—21no—
140C2×C5×D4—21no—
141C2×C5×D4—21no—
142C2×C5×D4—21no—
143C2×C5×D4—21no—
144C2×C5×D4C22×C412yes—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—24no—
149C22×C4×C5—81no—
150C22×C4×C5—81no—
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2411yescharacteristic
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—21no—
158C22×Dic5C2×D411yescharacteristic
159C22×Dic5C2×D411yescharacteristic
160C22×Dic5C2×D411yescharacteristic
161C22×Dic5C2×D411yescharacteristic
162C2×C5×C8—21no—
163C2×C5×C8—21no—
164C2×C5×C8—21no—
165C2×C5×C8—21no—
166C2×C5×C8C22×C412yes—
167C2×C5×C8C22×C412yes—
168C5×C4⋊C4—24no—
169C5×C4⋊C4—21no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4C22×C412yes—
173C5×C4⋊C4C22×C412yes—
174C5×C4⋊C4C22×C412yes—
175C5×C22⋊C4—24no—
176C42×C5C22×C412yes—
177C42×C5#4112yes—
178C22⋊Dic5—81no—
179C22⋊Dic5—81no—
180C22⋊Dic5—81no—
181C22⋊Dic5—81no—
182C4⋊Dic5—21no—
183C4⋊Dic5—21no—
184C4⋊Dic5—21no—
185C4⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5—21no—
189C4⋊Dic5—21no—
190C4⋊Dic5—81no—
191C4⋊Dic5—81no—
192C4⋊Dic5—81no—
193C4⋊Dic5—81no—
194C4×Dic5—21no—
195C4×Dic5—21no—
196C4×Dic5—21no—
197C4×Dic5—21no—
198C4×Dic5—81no—
199C4×Dic5—81no—
200C4×Dic5—81no—
201C2×#168—21no—
202C2×#168—21no—
203C2×#168—21no—
204C2×#168—21no—
205C2×#168—21no—
206C2×#168—21no—
207C2×C4⋊Q8C2×D511yescharacteristic
208C2×C4×D4C2×D511yescharacteristic
209#476Dic518yescomplemented
210C2×#107—51no—
211C2×#106—51no—
212C2×#106C2×D511yescharacteristic
213C2×#106C2×D511yescharacteristic
214C2×C4⋊C8—51no—
215C2×C4⋊C8C2×D511yescharacteristic
216C2×#102—51no—
217C2×#102—51no—
218C2×#102C2×D511yescharacteristic
219C2×#102C2×D511yescharacteristic
220C2×#97—51no—
221C2×C4×C8—51no—
222#391—101no—
223#391—101no—
224#390—51no—
225#390—51no—
226#389—51no—
227#389—51no—
228#389—51no—
229#389—51no—
230#388—101no—
231#388—101no—
232#387—101no—
233#387—101no—
234#382—51no—
235#381—101no—
236#380—101no—
237C4×C4⋊C4—51no—
238C4×C4⋊C4—51no—
239C4×C4⋊C4—51no—
240#376—51no—
241#340—51no—
242#340—51no—
243#340—51no—
244#340—51no—
245#340—51no—
246#340—51no—
247#340—51no—
248#340—51no—
249#336—51no—
250#336—51no—
251C23×C5—22no—
252C23×C5—41no—
253C23×C5—41no—
254C23×C5—41no—
255C23×C5—41no—
256C23×C5—41no—
257C23×C5—41no—
258C23×C5C22×D411yescharacteristic
259C5×Q8—84no—
260C5×D4—22no—
261C5×D4—22no—
262C5×D4—41no—
263C5×D4—41no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—41no—
272C2×C4×C5—41no—
273C2×C4×C5—41no—
274C2×C4×C5—41no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—84no—
278C2×C4×C5C4×D412yes—
279C2×C4×C5C4×D412yes—
280C2×C4×C5C23×C411yescharacteristic
281C2×C4×C5C2×#4111yescharacteristic
282C2×C4×C5H(5,𝔽2)11yescharacteristic
283C2×C4×C52−1+411yescharacteristic
284C2×C4×C52−1+411yescharacteristic
285C2×C4×C52−1+411yescharacteristic
286C2×Dic5—21no—
287C2×Dic5—21no—
288C2×Dic5—21no—
289C2×Dic5—21no—
290C2×Dic5—21no—
291C2×Dic5—21no—
292C2×Dic5—21no—
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—21no—
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—41no—
299C2×Dic5—41no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C5×C8—22no—
311C5×C8—22no—
312#168—41no—
313#168—41no—
314C22×Q8—21no—
315C22×D4C22×D511yescharacteristic
316C23×C4—21no—
317C22×C8—51no—
318C22×C8—51no—
319C22×C8C22×D511yescharacteristic
320C22×C8C22×D511yescharacteristic
321C4⋊Q8—22no—
322C4⋊Q8C2×Dic514yes—
323C4×D4—22no—
324C4×D4C2×Dic514yes—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—51no—
327C2×C4⋊C4—51no—
328C2×C4⋊C4—51no—
329C2×C4⋊C4—51no—
330C2×C4⋊C4—101no—
331C2×C4⋊C4—101no—
332C2×C4⋊C4—101no—
333C2×C4⋊C4—101no—
334C2×C4⋊C4C22×D511yescharacteristic
335C2×C4⋊C4C22×D511yescharacteristic
336C2×C4⋊C4C22×D511yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—101no—
339C2×C22⋊C4—101no—
340C2×C22⋊C4—101no—
341C2×C22⋊C4—101no—
342C2×C42—51no—
343C2×C42—51no—
344C2×C42—101no—
345C2×C42—101no—
346C2×C42—101no—
347C2×C42C22×D511yescharacteristic
348#107—201no—
349#106—201no—
350#106C2×Dic514yes—
351#106C2×Dic514yes—
352C4⋊C8—102no—
353C4⋊C8C2×Dic514yes—
354#102—201no—
355#102—201no—
356#102C2×Dic514yes—
357#102C2×Dic514yes—
358#97—201no—
359C4×C8—201no—
360#95—51no—
361#95—51no—
362#95—51no—
363#95—51no—
364#95—101no—
365#95—101no—
366#95—101no—
367#95—101no—
368#95—101no—
369#95—101no—
370#95—101no—
371#95—101no—
372#95—101no—
373C22×C5—42no—
374C22×C5—42no—
375C22×C5—42no—
376C22×C5—81no—
377C22×C5—81no—
378C22×C5—81no—
379C22×C5—81no—
380C22×C5C2×C4×D411yescharacteristic
381C22×C5#53611yescharacteristic
382C22×C5#53711yescharacteristic
383C22×C5#57611yescharacteristic
384C22×C5#57711yescharacteristic
385C22×C5#57711yescharacteristic
386C22×C5#57711yescharacteristic
387C4×C5—22no—
388C4×C5—22no—
389C4×C5—42no—
390C4×C5—42no—
391C4×C5—42no—
392C4×C5#51911yescharacteristic
393C4×C5#52011yescharacteristic
394C4×C5#54111yescharacteristic
395C4×C5#54211yescharacteristic
396Dic5—41no—
397Dic5—41no—
398Dic5—41no—
399Dic5—41no—
400Dic5—81no—
401Dic5—81no—
402Dic5—81no—
403Dic5—81no—
404C24—21no—
405C2×Q8—24no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C22×Dic512yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—51no—
419C22×C4—51no—
420C22×C4—51no—
421C22×C4—51no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—101no—
425C22×C4—101no—
426C22×C4—101no—
427C22×C4—101no—
428C22×C4D4×D511yescharacteristic
429C22×C4D4×D511yescharacteristic
430C22×C4C23×D511yescharacteristic
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—101no—
436C2×C8—101no—
437C2×C8—101no—
438C2×C8—101no—
439C2×C8—101no—
440C2×C8—101no—
441C2×C8C22×Dic512yes—
442C2×C8C22×Dic512yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—24no—
447C4⋊C4—101no—
448C4⋊C4—101no—
449C4⋊C4—101no—
450C4⋊C4—101no—
451C4⋊C4—101no—
452C4⋊C4—101no—
453C4⋊C4—101no—
454C4⋊C4—101no—
455C4⋊C4—401no—
456C4⋊C4—401no—
457C4⋊C4—401no—
458C4⋊C4—401no—
459C4⋊C4C22×Dic512yes—
460C4⋊C4C22×Dic512yes—
461C4⋊C4C22×Dic512yes—
462C22⋊C4—24no—
463C22⋊C4—401no—
464C22⋊C4—401no—
465C22⋊C4—401no—
466C22⋊C4—401no—
467C42—101no—
468C42—101no—
469C42—101no—
470C42—101no—
471C42—401no—
472C42—401no—
473C42—401no—
474C42#71112yes—
475C42C22×Dic512yes—
476C2×C5—82no—
477C2×C5#232311yescharacteristic
478C2×C5#294611yescharacteristic
479C2×C5#294911yescharacteristic
480C2×C5#324111yescharacteristic
481C2×C5#324411yescharacteristic
482C2×C5#325711yescharacteristic
483C2×C5#326311yescharacteristic
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×D4×D511yescharacteristic
492Q8—84no—
493D4—22no—
494D4—22no—
495D4—41no—
496D4—41no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—84no—
511C2×C4—101no—
512C2×C4—101no—
513C2×C4—101no—
514C2×C4—101no—
515C2×C4—101no—
516C2×C4—101no—
517C2×C4—101no—
518C2×C4—101no—
519C2×C4—101no—
520C2×C4—101no—
521C2×C4—101no—
522C2×C4—101no—
523C2×C4—201no—
524C2×C4—201no—
525C2×C4—201no—
526C2×C4—201no—
527C2×C4—201no—
528C2×C4—201no—
529C2×C4—201no—
530C2×C4—201no—
531C2×C4—201no—
532C2×C4—201no—
533C2×C4—201no—
534C2×C4—201no—
535C2×C4D4×Dic512yes—
536C2×C4D4×Dic512yes—
537C2×C4C2×#71111yescharacteristic
538C2×C4#414711yescharacteristic
539C2×C4#414711yescharacteristic
540C2×C4#414911yescharacteristic
541C2×C4#415011yescharacteristic
542C2×C4C23×Dic511yescharacteristic
543C8—22no—
544C8—22no—
545C8—201no—
546C8—201no—
547C5#2212311yescharacteristic, complemented, Sylow
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22#6596211yescharacteristic
556C22#6596311yescharacteristic
557C22#6609611yescharacteristic
558C22#6609611yescharacteristic
559C22#6610811yescharacteristic
560C22#6610911yescharacteristic
561C22C2×D4×Dic511yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—201no—
568C4—201no—
569C4—201no—
570C4—201no—
571C4—401no—
572C4—401no—
573C4—401no—
574C4—401no—
575C4#6595911yescharacteristic
576C4#6597711yescharacteristic
577C4#6614611yescharacteristic
578C4#6616111yescharacteristic
579C2—82no—
580C2#1061278111yescharacteristic
581C2#1061279911yescharacteristic
582C2#1061280011yescharacteristic
583C2#1061280511yescharacteristic
584C2#1061447511yescharacteristic
585C2#1061557411yescharacteristic
586C2#1061566111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#476C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
cosocleC22×C4×C5C24
hypercenterC2×#476D5
hypocenterC5#22123
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
2. #10614959C2semidirect
4. #10602573C2semidirect
6. #10601355C2semidirect
7. #10601353C2semidirect
8. #10601351C2semidirect
9. #10601350C2semidirect
10. #10601248C2semidirect
11. #10601235C2semidirect
18. C5×#476C4semidirect
209. #476Dic5semidirect
547. C5#22123semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4762
#106149592
#106140562
#106025732
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