Finite Group Databasebeta

← #12116972

Subgroups — #12116972

Order 1280.

838 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×#106C211yesmaximal, characteristic
2#10605499C211yesmaximal, characteristic
3#10605498C211yesmaximal, characteristic
4C5×#1776C211yesmaximal, characteristic, complemented
5#10601423C211yesmaximal, characteristic
6#10601407C211yesmaximal, characteristic, complemented
7#10601346C211yesmaximal, characteristic, complemented
8#10600624C211yesmaximal, characteristic
9#10600613C211yesmaximal, characteristic, complemented
10#10600253C211yesmaximal, characteristic, complemented
11#10600252C211yesmaximal, characteristic, complemented
12#10599665C211yesmaximal, characteristic
13#10599664C211yesmaximal, characteristic
14#10599582C211yesmaximal, characteristic, complemented
15#10599581C211yesmaximal, characteristic, complemented
16C22×C8×D5C2211yescharacteristic
17C2×D5×C4⋊C4C2211yescharacteristic
18C2×D5×C4⋊C4C2211yescharacteristic
19C2×#4018—21no—
20C2×C5×#107C2211yescharacteristic
21C2×C5×#106C2211yescharacteristic
22C2×C5×#97C2211yescharacteristic
23C4×C5×C4⋊C4C2211yescharacteristic
24C4×C5×C4⋊C4C2211yescharacteristic
25C2×#3952C2211yescharacteristic
26C2×#3950C2211yescharacteristic
27C2×#3949C2211yescharacteristic
28C2×#3947C2211yescharacteristic
29#65276C2211yescharacteristic
30#65276C2211yescharacteristic
31C4⋊C4×Dic5C2211yescharacteristic
32C4⋊C4×Dic5C2211yescharacteristic
33#65262C2211yescharacteristic
34#65262C2211yescharacteristic
35C2×#3940C2211yescharacteristic
36C2×#3940C2211yescharacteristic
37C4×#685—21no—
38C4×#685C2211yescharacteristic
39#65222—21no—
40#65222—21no—
41#65217C2211yescharacteristic
42#65217C2211yescharacteristic
43D5×#106C418yescomplemented
44#64956—21no—
45#64953—21no—
46#64941—21no—
47#64941—21no—
48C5×#340C2211yescharacteristic
49C5×#340C2211yescharacteristic
50#64771C2211yescharacteristic
51#64771C2211yescharacteristic
52#64770C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64744C2211yescharacteristic
60#22615—51nomaximal, Sylow
61C22×C4×D5—21no—
62C22×C4×D5—21no—
63C22×C4×D5C2311yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C2×C5×C4⋊C4C2311yescharacteristic
67C2×C5×C4⋊C4C2311yescharacteristic
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C42×C5—21no—
71C2×C42×C5—21no—
72C2×C42×C5C2311yescharacteristic
73C2×#685—21no—
74C2×#685—21no—
75C2×#685D411yescharacteristic
76C2×#685D411yescharacteristic
77C2×C4⋊Dic5C2311yescharacteristic
78C2×C4⋊Dic5C2311yescharacteristic
79C2×C4⋊Dic5C2311yescharacteristic
80C2×C4⋊Dic5C2311yescharacteristic
81C2×#683—21no—
82C2×#683—21no—
83C2×#683—21no—
84C2×#683—21no—
85C2×#683—21no—
86C2×#683—21no—
87C2×C4×Dic5—21no—
88C2×C4×Dic5—21no—
89C2×C4×Dic5C2311yescharacteristic
90C22×#168C2311yescharacteristic
91C22×#168C2311yescharacteristic
92C2×C8×D5—22no—
93C2×C8×D5—22no—
94C2×C8×D5C2×C414yes—
95D5×C4⋊C4—22no—
96D5×C4⋊C4—22no—
97D5×C4⋊C4C2×C414yes—
98D5×C4⋊C4C2×C414yes—
99#4018—44no—
100C5×#107—22no—
101C5×#106C2×C414yes—
102C5×#97—22no—
103C5×#95—21no—
104C5×#95—21no—
105#3963—21no—
106#3963—21no—
107#3963—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113#3963D411yescharacteristic
114#3963D411yescharacteristic
115#3952—22no—
116#3950—22no—
117#3949C2×C414yes—
118#3947—22no—
119#3940C2×C414yes—
120#3940C2×C414yes—
121C22×#106—51no—
122#2345—51no—
123#2344—51no—
124#1944—51no—
125#1944—51no—
126#1920—51no—
127#1920—51no—
128#1919—51no—
129#1919—51no—
130#1824—51no—
131#1823—51no—
132#1808—51no—
133#1776—51no—
134#1776D511yescharacteristic, complemented
135#1752—51no—
136C23×D5C2×D411yescharacteristic
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5C2×D411yescharacteristic
142C22×C4×C5C2×D411yescharacteristic
143C22×C4×C5C2411yescharacteristic
144C22×Dic5—21no—
145C22×Dic5—21no—
146C22×Dic5—21no—
147C22×Dic5—21no—
148C22×Dic5C2×D411yescharacteristic
149C22×Dic5C2×D411yescharacteristic
150C22×Dic5C2×D411yescharacteristic
151C2×C4×D5—21no—
152C2×C4×D5—21no—
153C2×C4×D5—21no—
154C2×C4×D5—21no—
155C2×C4×D5—24no—
156C2×C4×D5—24no—
157C2×C4×D5—42no—
158C2×C4×D5—42no—
159C2×C4×D5—81no—
160C2×C4×D5—81no—
161C2×C4×D5C22×C412yes—
162C2×C4×D5#4112yes—
163C2×C5×C8—21no—
164C2×C5×C8—21no—
165C2×C5×C8—21no—
166C2×C5×C8C22×C412yes—
167C2×C5×C8#4112yes—
168C2×C5×C8#4112yes—
169C5×C4⋊C4—21no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4C22×C412yes—
176C5×C4⋊C4C22×C412yes—
177C42×C5—21no—
178C42×C5—21no—
179C42×C5—42no—
180C42×C5—81no—
181#685—22no—
182#685—22no—
183#685—42no—
184#685—42no—
185C4⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5—21no—
189C4⋊Dic5—21no—
190C4⋊Dic5—21no—
191C4⋊Dic5C22×C412yes—
192C4⋊Dic5C22×C412yes—
193#683—24no—
194#683—24no—
195#683—42no—
196#683—42no—
197#683—81no—
198#683—81no—
199C4×Dic5—21no—
200C4×Dic5—21no—
201C4×Dic5—42no—
202C4×Dic5—81no—
203C2×#168—21no—
204C2×#168—21no—
205C2×#168—21no—
206C2×#168—21no—
207C2×#168—21no—
208C2×#168C22×C412yes—
209C8×D5—28no—
210C23×C8—51no—
211C2×#117—101no—
212C22×C4⋊C4—51no—
213C22×C4⋊C4—51no—
214C2×#107—51no—
215C2×#107C2×D511yescharacteristic
216C2×#106—51no—
217C2×#106—51no—
218C2×#106—51no—
219C2×#106—58no—
220C2×#106C2×D511yescharacteristic
221C2×C22⋊C8—51no—
222C2×#97—51no—
223C2×#97C2×D511yescharacteristic
224#389—51no—
225#389—51no—
226#389—51no—
227#389—51no—
228#387—101no—
229#387—101no—
230#387—101no—
231#387—101no—
232#381—101no—
233#380—51no—
234#380—51no—
235#380—101no—
236C4×C4⋊C4—51no—
237C4×C4⋊C4—51no—
238C4×C4⋊C4C2×D511yescharacteristic
239C4×C4⋊C4C2×D511yescharacteristic
240C4×C22⋊C4—51no—
241C4×C22⋊C4—101no—
242#340—51no—
243#340—51no—
244#340—51no—
245#340—51no—
246#340—51no—
247#340—51no—
248#340—51no—
249#340—51no—
250#340C2×D511yescharacteristic
251#340C2×D511yescharacteristic
252#336—51no—
253#336—51no—
254C23×C5C22×D411yescharacteristic
255C22×D5—21no—
256C22×D5—21no—
257C22×D5—21no—
258C22×D5—21no—
259C22×D5C4×D412yes—
260C22×D5AGL(1,ℤ/8)12yes—
261C22×D5#13712yes—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—41no—
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—41no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5C23×C411yescharacteristic
281C2×C4×C5C2×#4111yescharacteristic
282C2×C4×C5C2×#4111yescharacteristic
283C2×C4×C5C2×#4111yescharacteristic
284C2×C4×C52−1+411yescharacteristic
285C2×C4×C52−1+411yescharacteristic
286C2×Dic5—21no—
287C2×Dic5—21no—
288C2×Dic5—22no—
289C2×Dic5—22no—
290C2×Dic5—21no—
291C2×Dic5—21no—
292C2×Dic5—21no—
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—21no—
296C2×Dic5—41no—
297C2×Dic5—41no—
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×Dic5C4×D412yes—
307C4×D5—22no—
308C4×D5—22no—
309C4×D5—22no—
310C4×D5—22no—
311C4×D5—48no—
312C4×D5—84no—
313C5×C8—41no—
314C5×C8#11714yescomplemented
315#168—22no—
316#168—41no—
317C23×C4—51no—
318C23×C4—101no—
319C23×C4—101no—
320C22×C8—51no—
321C22×C8—54no—
322C22×C8—51no—
323C22×C8—102no—
324C22×C8—102no—
325C22×C8C22×D511yescharacteristic
326C22×C8C22×D511yescharacteristic
327#117—204no—
328C2×C4⋊C4—51no—
329C2×C4⋊C4—51no—
330C2×C4⋊C4—54no—
331C2×C4⋊C4—54no—
332C2×C4⋊C4—51no—
333C2×C4⋊C4—51no—
334C2×C4⋊C4—102no—
335C2×C4⋊C4—102no—
336C2×C4⋊C4—101no—
337C2×C4⋊C4—101no—
338C2×C4⋊C4—101no—
339C2×C4⋊C4—101no—
340C2×C4⋊C4—101no—
341C2×C4⋊C4—101no—
342C2×C4⋊C4C22×D511yescharacteristic
343C2×C4⋊C4C22×D511yescharacteristic
344C2×C4⋊C4C22×D511yescharacteristic
345C2×C4⋊C4C22×D511yescharacteristic
346C2×C22⋊C4—51no—
347C2×C22⋊C4—51no—
348C2×C22⋊C4—101no—
349C2×C22⋊C4—101no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—51no—
353C2×C42—101no—
354C2×C42—101no—
355C2×C42C22×D511yescharacteristic
356#107—22no—
357#107—102no—
358#106—54no—
359#106—54no—
360#106—54no—
361#106C4×D514yes—
362C22⋊C8—102no—
363#97—22no—
364#97—102no—
365#95—21no—
366#95—21no—
367#95—51no—
368#95—51no—
369#95—101no—
370#95—101no—
371#95—101no—
372#95—101no—
373#95—101no—
374#95—101no—
375#95—101no—
376#95—101no—
377C22×C5C2×C4×D411yescharacteristic
378C22×C5#54711yescharacteristic
379C22×C5#54711yescharacteristic
380C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
381C22×C5C2×#13711yescharacteristic
382C22×C5#57711yescharacteristic
383C22×C5#57711yescharacteristic
384C2×D5—22no—
385C2×D5—22no—
386C2×D5—22no—
387C2×D5—22no—
388C2×D5—22no—
389C2×D5—22no—
390C2×D5#44014yes—
391C4×C5—41no—
392C4×C5—41no—
393C4×C5—42no—
394C4×C5—42no—
395C4×C5—81no—
396C4×C5—81no—
397C4×C5C2×#11711yescharacteristic
398C4×C5#51911yescharacteristic
399C4×C5#55211yescharacteristic
400C4×C5#55211yescharacteristic
401Dic5—22no—
402Dic5—41no—
403Dic5—41no—
404Dic5—42no—
405Dic5—42no—
406Dic5—81no—
407Dic5—81no—
408C24—51no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—51no—
414C22×C4—52no—
415C22×C4—52no—
416C22×C4—51no—
417C22×C4—51no—
418C22×C4—104no—
419C22×C4—104no—
420C22×C4—101no—
421C22×C4—101no—
422C22×C4—101no—
423C22×C4—101no—
424C22×C4—101no—
425C22×C4—101no—
426C22×C4—101no—
427C22×C4—101no—
428C22×C4—202no—
429C22×C4—202no—
430C22×C4—401no—
431C22×C4—401no—
432C22×C4D4×D511yescharacteristic
433C22×C4D4×D511yescharacteristic
434C22×C4C23×D511yescharacteristic
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—52no—
439C2×C8—101no—
440C2×C8—101no—
441C2×C8—108no—
442C2×C8—101no—
443C2×C8—101no—
444C2×C8—101no—
445C2×C8C2×C4×D512yes—
446C2×C8#70912yes—
447C2×C8#70912yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—52no—
455C4⋊C4—52no—
456C4⋊C4—104no—
457C4⋊C4—104no—
458C4⋊C4—101no—
459C4⋊C4—101no—
460C4⋊C4—101no—
461C4⋊C4—101no—
462C4⋊C4—101no—
463C4⋊C4—101no—
464C4⋊C4—202no—
465C4⋊C4—202no—
466C4⋊C4—401no—
467C4⋊C4—401no—
468C4⋊C4C2×C4×D512yes—
469C4⋊C4C2×C4×D512yes—
470C22⋊C4—102no—
471C22⋊C4—102no—
472C22⋊C4—202no—
473C22⋊C4—202no—
474C42—21no—
475C42—21no—
476C42—42no—
477C42—81no—
478C42—101no—
479C42—101no—
480C42—202no—
481C42—401no—
482C2×C5#234411yescharacteristic
483C2×C5C2×#44011yescharacteristic
484C2×C5#294911yescharacteristic
485C2×C5#334411yescharacteristic
486C2×C5#334411yescharacteristic
487C2×C5#334511yescharacteristic
488C2×C5#334511yescharacteristic
489D5—24no—
490C23—52no—
491C23—52no—
492C23—52no—
493C23—101no—
494C23—101no—
495C23—101no—
496C23—101no—
497C23C2×D4×D511yescharacteristic
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—52no—
517C2×C4—101no—
518C2×C4—101no—
519C2×C4—102no—
520C2×C4—102no—
521C2×C4—102no—
522C2×C4—102no—
523C2×C4—101no—
524C2×C4—101no—
525C2×C4—101no—
526C2×C4—101no—
527C2×C4—102no—
528C2×C4—102no—
529C2×C4—101no—
530C2×C4—101no—
531C2×C4—208no—
532C2×C4—201no—
533C2×C4—201no—
534C2×C4—201no—
535C2×C4—201no—
536C2×C4—201no—
537C2×C4—201no—
538C2×C4—201no—
539C2×C4—201no—
540C2×C4—201no—
541C2×C4—201no—
542C2×C4—404no—
543C2×C4C22×C4×D511yescharacteristic
544C2×C4C2×#70911yescharacteristic
545C2×C4C2×#70911yescharacteristic
546C2×C4#414711yescharacteristic
547C2×C4D5×#4111yescharacteristic
548C2×C4#415011yescharacteristic
549C8—41no—
550C8—102no—
551C8—201no—
552C8#401814yescomplemented
553C5#2261511yescharacteristic, complemented, Sylow
554C22—54no—
555C22—102no—
556C22—102no—
557C22—102no—
558C22—102no—
559C22—102no—
560C22—102no—
561C22#6584111yescharacteristic
562C22C4×D4×D511yescharacteristic
563C22#6588511yescharacteristic
564C22D5×AGL(1,ℤ/8)11yescharacteristic
565C22#6610911yescharacteristic
566C22D5×#13711yescharacteristic
567C22#6611211yescharacteristic
568C4—41no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—102no—
575C4—202no—
576C4—202no—
577C4—201no—
578C4—201no—
579C4—401no—
580C4—401no—
581C4C2×#401811yescharacteristic
582C4#6586311yescharacteristic
583C4#6590611yescharacteristic
584C4#6591511yescharacteristic
585C2—104no—
586C2#1060549811yescharacteristic
587C2#1061095511yescharacteristic
588C2#1061095611yescharacteristic
589C2D5×#44011yescharacteristic
590C2#1061150811yescharacteristic
591C2#1061157611yescharacteristic
592C2#1061161511yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1776C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1776D5
hypocenterC5#22615
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. C5×#1776C2semidirect
6. #10601407C2semidirect
7. #10601346C2semidirect
9. #10600613C2semidirect
10. #10600253C2semidirect
11. #10600252C2semidirect
14. #10599582C2semidirect
15. #10599581C2semidirect
43. D5×#106C4semidirect
134. #1776D5semidirect
314. C5×C8#117semidirect
552. C8#4018semidirect
553. C5#22615semidirect

Maximal subgroups

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

subgroupindex
C2×D5×#1062
#106054992
#106054982
C5×#17762
#106014232
#106014072
#106013462
#106006242
#106006132
#106002532
#106002522
#105996652
#105996642
#105995822
#105995812
#226155