Finite Group Databasebeta

← #12111487

Subgroups — #12111487

Order 1280.

732 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2319C211yesmaximal, characteristic
2C2×#65298C211yesmaximal, characteristic
3#10605121C211yesmaximal, characteristic, complemented
4#10602397C211yesmaximal, characteristic
5#10600438C211yesmaximal, characteristic
6#10600437C211yesmaximal, characteristic
7#10600430C211yesmaximal, characteristic
8#10600430C211yesmaximal, characteristic
9#10600413C211yesmaximal, characteristic, complemented
10#10600411C211yesmaximal, characteristic, complemented
11#10600395C211yesmaximal, characteristic, complemented
12#10600393C211yesmaximal, characteristic, complemented
13#10600284C211yesmaximal, characteristic, complemented
14#10600260C211yesmaximal, characteristic, complemented
15#10600229C211yesmaximal, characteristic, complemented
16C2×C4×C5×D4C2211yescharacteristic
17C2×#4016C2211yescharacteristic
18C5×#390C2211yescharacteristic
19C5×#390C2211yescharacteristic
20C5×#389C2211yescharacteristic
21C5×#389C2211yescharacteristic
22C5×#388—21no—
23C5×#388—21no—
24C5×#381—21no—
25C5×#380—21no—
26C4×C5×C4⋊C4C2211yescharacteristic
27C5×#376C2211yescharacteristic
28C2×#3964C2211yescharacteristic
29C2×#3964C2211yescharacteristic
30C2×#3964C2211yescharacteristic
31C2×#3964C2211yescharacteristic
32#65298C418yescomplemented
33C2×#3940C2211yescharacteristic
34C2×#3940C2211yescharacteristic
35C2×#3940C2211yescharacteristic
36C2×#3939C2211yescharacteristic
37#65224C2211yescharacteristic
38C4×C4⋊Dic5C2211yescharacteristic
39C4×C4⋊Dic5C2211yescharacteristic
40#65217C2211yescharacteristic
41#65217C2211yescharacteristic
42C2×C4⋊#168C2211yescharacteristic
43C2×C4⋊#168C2211yescharacteristic
44C2×#3935C2211yescharacteristic
45C2×C4×#168C2211yescharacteristic
46#64944—21no—
47#64944—21no—
48#64936—21no—
49#64936—21no—
50#64749C2211yescharacteristic
51#64749C2211yescharacteristic
52#64749C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64744C2211yescharacteristic
59#64744C2211yescharacteristic
60#22122—51nomaximal, Sylow
61C22×C5×D4C2311yescharacteristic
62C23×C4×C5—21no—
63C4×C5×D4—22no—
64C4×C5×D4C2×C414yes—
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4C2311yescharacteristic
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C22⋊C4—21no—
71C2×C5×C22⋊C4—21no—
72C2×C5×C22⋊C4—21no—
73C2×C5×C22⋊C4—21no—
74C2×C5×C22⋊C4—21no—
75C2×C42×C5—21no—
76C2×C42×C5C2311yescharacteristic
77C2×C42×C5C2311yescharacteristic
78C2×C42×C5C2311yescharacteristic
79C2×C4⋊Dic5—21no—
80C2×C4⋊Dic5—21no—
81C2×C4⋊Dic5C2311yescharacteristic
82C2×C4⋊Dic5C2311yescharacteristic
83C2×C4⋊Dic5C2311yescharacteristic
84C2×C4⋊Dic5C2311yescharacteristic
85C2×#683—21no—
86C2×#683—21no—
87C2×#683—21no—
88C2×#683—21no—
89C2×C4×Dic5—21no—
90C2×C4×Dic5—21no—
91C22×#168C2311yescharacteristic
92C22×#168C2311yescharacteristic
93C22×#168C2311yescharacteristic
94C22×#168C2311yescharacteristic
95#4016—22no—
96#4016C2×C414yes—
97C5×#95—21no—
98C5×#95—21no—
99C5×#95—21no—
100C5×#95D411yescharacteristic
101C5×#95D411yescharacteristic
102C5×#95D411yescharacteristic
103C5×#95D411yescharacteristic
104#3964—41no—
105#3964—41no—
106#3964C2×C414yes—
107#3964C2×C414yes—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3940—41no—
113#3940—41no—
114#3940C2×C414yes—
115#3939C2×C414yes—
116C4⋊#168—22no—
117C4⋊#168C2×C414yes—
118#3935—41no—
119C4×#168—41no—
120C2×#478—51no—
121#2330—51no—
122#2319D511yescharacteristic
123#1989—51no—
124#1948—51no—
125#1943—51no—
126#1941—51no—
127#1936—51no—
128#1934—51no—
129#1926—51no—
130#1926—51no—
131#1920—51no—
132#1918—51no—
133#1802—51no—
134#1768—51no—
135C24×C5—21no—
136C2×C5×D4—21no—
137C2×C5×D4—21no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4—21no—
141C2×C5×D4C22×C412yes—
142C22×C4×C5—21no—
143C22×C4×C5—24no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—81no—
147C22×C4×C5—81no—
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2411yescharacteristic
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—21no—
158C22×Dic5—21no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—21no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4—81no—
165C5×C4⋊C4—81no—
166C5×C4⋊C4C22×C412yes—
167C5×C22⋊C4—24no—
168C5×C22⋊C4—81no—
169C5×C22⋊C4—81no—
170C5×C22⋊C4—81no—
171C5×C22⋊C4—81no—
172C42×C5—21no—
173C42×C5—21no—
174C42×C5—21no—
175C42×C5—21no—
176C42×C5—81no—
177C42×C5C22×C412yes—
178C42×C5#4112yes—
179C4⋊Dic5—21no—
180C4⋊Dic5—21no—
181C4⋊Dic5—21no—
182C4⋊Dic5—21no—
183C4⋊Dic5—21no—
184C4⋊Dic5—21no—
185C4⋊Dic5—81no—
186C4⋊Dic5—81no—
187C4⋊Dic5C22×C412yes—
188C4⋊Dic5C22×C412yes—
189#683—24no—
190#683—24no—
191#683—81no—
192#683—81no—
193C4×Dic5—81no—
194C4×Dic5—81no—
195C2×#168—21no—
196C2×#168—21no—
197C2×#168—21no—
198C2×#168—21no—
199C2×#168—21no—
200C2×#168—21no—
201C2×#168—21no—
202C2×#168—21no—
203C2×#168—21no—
204C2×#168—21no—
205C2×#168C22×C412yes—
206C2×#168C22×C412yes—
207C2×#125—51no—
208C2×C4×D4C2×D511yescharacteristic
209#478—58no—
210C2×#107—51no—
211C2×#106—51no—
212C2×#106—51no—
213C2×#106—51no—
214C2×C4⋊C8—51no—
215C2×C4⋊C8—51no—
216C2×#102—51no—
217C2×#102—51no—
218C2×#102—51no—
219C2×#102—51no—
220C2×#97—51no—
221C2×C4×C8—51no—
222#390C2×D511yescharacteristic
223#390C2×D511yescharacteristic
224#389—51no—
225#389—51no—
226#389—101no—
227#389—101no—
228#389C2×D511yescharacteristic
229#389C2×D511yescharacteristic
230#388—21no—
231#388—21no—
232#387—101no—
233#387—101no—
234#384—51no—
235#381—21no—
236#380—21no—
237C4×C4⋊C4—51no—
238C4×C4⋊C4—51no—
239C4×C4⋊C4C2×D511yescharacteristic
240#376C2×D511yescharacteristic
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×D4—22no—
260C5×D4—22no—
261C5×D4—41no—
262C5×D4—41no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—22no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—41no—
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—41no—
288C2×C4×C5—84no—
289C2×C4×C5C4×D412yes—
290C2×C4×C5C4×D412yes—
291C2×C4×C5C23×C411yescharacteristic
292C2×C4×C5C2×#4111yescharacteristic
293C2×C4×C5H(5,𝔽2)11yescharacteristic
294C2×C4×C52−1+411yescharacteristic
295C2×C4×C52−1+411yescharacteristic
296C2×C4×C52−1+411yescharacteristic
297C2×Dic5—22no—
298C2×Dic5—22no—
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—
309#168—22no—
310#168—22no—
311#168—41no—
312#168—41no—
313C22×D4C22×D511yescharacteristic
314C23×C4—21no—
315C22×C8—51no—
316C22×C8—51no—
317C22×C8—51no—
318C22×C8—51no—
319#125—54no—
320#125—102no—
321C4×D4—22no—
322C4×D4C4×D514yes—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—51no—
326C2×C4⋊C4—51no—
327C2×C4⋊C4—51no—
328C2×C4⋊C4—51no—
329C2×C4⋊C4—101no—
330C2×C4⋊C4—101no—
331C2×C4⋊C4—101no—
332C2×C4⋊C4—101no—
333C2×C4⋊C4—101no—
334C2×C4⋊C4—101no—
335C2×C4⋊C4C22×D511yescharacteristic
336C2×C4⋊C4C22×D511yescharacteristic
337C2×C4⋊C4C22×D511yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C42—21no—
344C2×C42—101no—
345C2×C42—101no—
346C2×C42C22×D511yescharacteristic
347C2×C42C22×D511yescharacteristic
348C2×C42C22×D511yescharacteristic
349#107—54no—
350#106—54no—
351#106—201no—
352#106—201no—
353C4⋊C8—54no—
354C4⋊C8—102no—
355#102—54no—
356#102—54no—
357#102—201no—
358#102—201no—
359#97—201no—
360C4×C8—201no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—101no—
365#95—101no—
366#95—101no—
367#95—101no—
368#95C22⋊D511yescharacteristic
369#95C22⋊D511yescharacteristic
370#95C22⋊D511yescharacteristic
371#95C22⋊D511yescharacteristic
372C22×C5—42no—
373C22×C5—42no—
374C22×C5—42no—
375C22×C5—81no—
376C22×C5—81no—
377C22×C5—81no—
378C22×C5—81no—
379C22×C5C2×C4×D411yescharacteristic
380C22×C5#53611yescharacteristic
381C22×C5#53711yescharacteristic
382C22×C5#57611yescharacteristic
383C22×C5#57711yescharacteristic
384C22×C5#57711yescharacteristic
385C22×C5#57711yescharacteristic
386C4×C5—22no—
387C4×C5—22no—
388C4×C5—42no—
389C4×C5—41no—
390C4×C5—41no—
391C4×C5—41no—
392C4×C5—41no—
393C4×C5—81no—
394C4×C5—81no—
395C4×C5#51911yescharacteristic
396C4×C5#52011yescharacteristic
397C4×C5#54111yescharacteristic
398C4×C5#54211yescharacteristic
399Dic5—42no—
400Dic5—42no—
401Dic5—81no—
402Dic5—81no—
403C24—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4C2×C4×D512yes—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—101no—
417C22×C4—101no—
418C22×C4—101no—
419C22×C4—101no—
420C22×C4C2×C22⋊D511yescharacteristic
421C22×C4C2×C22⋊D511yescharacteristic
422C22×C4C2×C22⋊D511yescharacteristic
423C22×C4C2×C22⋊D511yescharacteristic
424C22×C4C2×C22⋊D511yescharacteristic
425C22×C4C2×C22⋊D511yescharacteristic
426C22×C4C23×D511yescharacteristic
427C2×C8—52no—
428C2×C8—52no—
429C2×C8—101no—
430C2×C8—101no—
431C2×C8—101no—
432C2×C8—101no—
433C2×C8—101no—
434C2×C8—101no—
435C2×C8—101no—
436C2×C8—101no—
437C2×C8—101no—
438C2×C8—101no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—52no—
445C4⋊C4—52no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—101no—
449C4⋊C4—101no—
450C4⋊C4—104no—
451C4⋊C4—104no—
452C4⋊C4—101no—
453C4⋊C4—101no—
454C4⋊C4—101no—
455C4⋊C4—101no—
456C4⋊C4—401no—
457C4⋊C4—401no—
458C4⋊C4—401no—
459C4⋊C4—401no—
460C4⋊C4C2×C4×D512yes—
461C22⋊C4—24no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—81no—
471C42—401no—
472C42—401no—
473C42C2×C4×D512yes—
474C42#70912yes—
475C2×C5—82no—
476C2×C5#232311yescharacteristic
477C2×C5#294611yescharacteristic
478C2×C5#295211yescharacteristic
479C2×C5#324111yescharacteristic
480C2×C5#324411yescharacteristic
481C2×C5#325711yescharacteristic
482C2×C5#326011yescharacteristic
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C22×C22⋊D511yescharacteristic
491D4—22no—
492D4—22no—
493D4—41no—
494D4—41no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—84no—
521C2×C4—102no—
522C2×C4—102no—
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×C4C4×C22⋊D512yes—
534C2×C4C4×C22⋊D512yes—
535C2×C4C22×C4×D511yescharacteristic
536C2×C4C2×#70911yescharacteristic
537C2×C4#414911yescharacteristic
538C2×C4#415011yescharacteristic
539C2×C4#415011yescharacteristic
540C2×C4#415011yescharacteristic
541C8—102no—
542C8—102no—
543C8—201no—
544C8—201no—
545C5#2212211yescharacteristic, complemented, Sylow
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×C4×C22⋊D511yescharacteristic
554C22#6615911yescharacteristic
555C22#6616711yescharacteristic
556C22#6616911yescharacteristic
557C22#6617011yescharacteristic
558C22#6617011yescharacteristic
559C22#6617111yescharacteristic
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—202no—
570C4—202no—
571C4—401no—
572C4—401no—
573C4#6585011yescharacteristic
574C4#6585211yescharacteristic
575C4#6585711yescharacteristic
576C4#6586811yescharacteristic
577C2—82no—
578C2#1061396611yescharacteristic
579C2#1061426811yescharacteristic
580C2#1061428911yescharacteristic
581C2#1061516111yescharacteristic
582C2#1061517211yescharacteristic
583C2#1061518611yescharacteristic
584C2#1061519611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2319C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
cosocleC22×C4×C5C24
hypercenter#2319D5
hypocenterC5#22122
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
3. #10605121C2semidirect
9. #10600413C2semidirect
10. #10600411C2semidirect
11. #10600395C2semidirect
12. #10600393C2semidirect
13. #10600284C2semidirect
14. #10600260C2semidirect
15. #10600229C2semidirect
32. #65298C4semidirect
545. C5#22122semidirect

Maximal subgroups

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

subgroupindex
C5×#23192
C2×#652982
#106051212
#106023972
#106004382
#106004372
#106004302
#106004132
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