Finite Group Databasebeta

← #12133231

Subgroups — #12133231

Order 1280.

1350 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
1C2×C5×#547C211yesmaximal, characteristic
2C2×#65383C211yesmaximal, characteristic, complemented
3#10614107C211yesmaximal, characteristic, complemented
4C2×C4×#687C211yesmaximal, characteristic, complemented
5#10602580C211yesmaximal, characteristic, complemented
6#10602478C212yesmaximal, complemented
7#10602473C212yesmaximal, complemented
8#10602407C211yesmaximal, characteristic, complemented
9#10602397C211yesmaximal, characteristic, complemented
10#10602352C212yesmaximal, complemented
11#10602349C212yesmaximal, complemented
12C22×C5×#41—21no—
13C5×#547—24no—
14C5×#547—24no—
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C5×#123—22no—
17C2×C5×C22⋊Q8—21no—
18C2×C5×#121C2212yes—
19C2×C4×C5×D4C2211yescharacteristic
20C2×C4×C5×D4C2211yescharacteristic
21C22×C5×C4⋊C4C2212yes—
22C22×#687C2211yescharacteristic
23C2×#4070C2212yes—
24C2×C4×Dic10C2211yescharacteristic
25#65510—21no—
26C2×#3964C2211yescharacteristic
27C2×#3964C2212yes—
28#65501C2212yes—
29C2×C22⋊#168C2212yes—
30C2×C22⋊#168C2212yes—
31#65383—24no—
32C4×#687—24no—
33#65262—21no—
34C2×#3942C2211yescharacteristic
35C2×#3942C2212yes—
36C2×#3940C2211yescharacteristic
37C2×#3940C2212yes—
38#65237—22no—
39#65224C2211yescharacteristic
40C2×C4⋊#168C2211yescharacteristic
41C2×C4×#168C2211yescharacteristic
42#64749C2212yes—
43#64749C2212yes—
44#64749C2212yes—
45#24120—51nomaximal, Sylow
46C2×C5×#41—28no—
47C2×C5×#41—44no—
48C2×C5×#41—44no—
49C22×C5×Q8—21no—
50C22×C5×D4—22no—
51C22×C5×D4D411yescharacteristic
52C22×C5×D4C2311yescharacteristic
53C23×C4×C5—22no—
54C23×C4×C5—21no—
55C23×C4×C5C2312yes—
56C22×Dic10C2311yescharacteristic
57C5×C4⋊Q8—22no—
58C5×C4⋊Q8D414yes—
59C5×#123—216no—
60C5×C22⋊Q8—28no—
61C5×#121—24no—
62C5×#121—24no—
63C4×C5×D4—22no—
64C4×C5×D4—22no—
65C4×C5×D4—22no—
66C4×C5×D4D414yes—
67C2×C5×C4⋊C4—22no—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4—24no—
70C2×C5×C4⋊C4—24no—
71C2×C5×C4⋊C4D411yescharacteristic
72C2×C5×C4⋊C4C2311yescharacteristic
73C2×C5×C4⋊C4C2312yes—
74C2×C5×C22⋊C4—21no—
75C2×C5×C22⋊C4—22no—
76C2×C5×C22⋊C4D412yes—
77C2×C42×C5C2311yescharacteristic
78C2×C22⋊Dic5—22no—
79C2×C22⋊Dic5—22no—
80C2×#687—22no—
81C2×#687—22no—
82C2×#687—22no—
83C2×C4⋊Dic5—21no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5C2311yescharacteristic
86C2×C4⋊Dic5C2312yes—
87#4070—24no—
88#4070—42no—
89C2×#683—21no—
90C2×#683—22no—
91C2×C4×Dic5—21no—
92C22×#168C2311yescharacteristic
93C22×#168C2312yes—
94C22×#168C2311yescharacteristic
95C4×Dic10—22no—
96C4×Dic10—22no—
97#3964—22no—
98#3964—42no—
99#3963—22no—
100C22⋊#168—24no—
101C22⋊#168—42no—
102#3942—22no—
103#3942—24no—
104#3940—22no—
105#3940—24no—
106C4⋊#168—22no—
107C4×#168—22no—
108C2×#547D511yescharacteristic
109C2×#475—51no—
110C2×C4×SD16—51no—
111#2538—51no—
112#2034—52no—
113#2000—52no—
114#1990—51no—
115#1954—51no—
116#1872—52no—
117#1853—52no—
118#1802—51no—
119C24×C5—21no—
120C24×C5C2×D412yes—
121C5×#41—432no—
122C2×C5×Q8—24no—
123C2×C5×Q8—42no—
124C2×C5×Q8—42no—
125C2×C5×D4—28no—
126C2×C5×D4—22no—
127C2×C5×D4—22no—
128C2×C5×D4—21no—
129C2×C5×D4—21no—
130C2×C5×D4—21no—
131C2×C5×D4—21no—
132C2×C5×D4—21no—
133C2×C5×D4—44no—
134C2×C5×D4—44no—
135C2×C5×D4SD1614yescomplemented
136C2×C5×D4C2×D412yes—
137C22×C4×C5—28no—
138C22×C4×C5—21no—
139C22×C4×C5—22no—
140C22×C4×C5—24no—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—22no—
145C22×C4×C5—22no—
146C22×C4×C5—22no—
147C22×C4×C5—44no—
148C22×C4×C5—44no—
149C22×C4×C5—42no—
150C22×C4×C5—42no—
151C22×C4×C5C2×D412yes—
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2411yescharacteristic
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—22no—
158C2×Dic10—21no—
159C2×Dic10—21no—
160C2×Dic10—21no—
161C2×Dic10—21no—
162C2×Dic10—21no—
163C2×Dic10#4112yescomplemented
164C5×C4⋊C4—28no—
165C5×C4⋊C4—24no—
166C5×C4⋊C4—22no—
167C5×C4⋊C4—21no—
168C5×C4⋊C4—22no—
169C5×C4⋊C4C2×D412yes—
170C5×C4⋊C4C2×D414yes—
171C5×C22⋊C4—28no—
172C5×C22⋊C4—24no—
173C5×C22⋊C4—24no—
174C42×C5C2×D412yes—
175C42×C5C2×D412yes—
176C22⋊Dic5—44no—
177C22⋊Dic5—82no—
178#687—28no—
179C4⋊Dic5—21no—
180C4⋊Dic5—21no—
181C4⋊Dic5—22no—
182C4⋊Dic5—22no—
183C4⋊Dic5—42no—
184C4⋊Dic5—81no—
185#683—44no—
186#683—42no—
187C4×Dic5—42no—
188C2×#168—21no—
189C2×#168—21no—
190C2×#168—21no—
191C2×#168—22no—
192C2×#168—22no—
193C2×#168—21no—
194C2×#168—22no—
195C2×#168—21no—
196C2×#168#4112yescomplemented
197C22×#41—21no—
198C22×SD16—51no—
199#547—24no—
200#547—24no—
201C2×C4⋊Q8C2×D511yescharacteristic
202C2×#123—22no—
203C2×C22⋊Q8—21no—
204C2×C22⋊Q8—52no—
205C2×#121C2×D512yes—
206C2×C4×Q8—51no—
207C2×C4×D4C2×D511yescharacteristic
208C2×C4×D4C2×D511yescharacteristic
209C22×C4⋊C4C2×D512yes—
210#475—104no—
211C4×SD16—104no—
212C2×#106—52no—
213C2×#106—51no—
214C2×C4⋊C8—51no—
215C2×#103—52no—
216C2×#103—51no—
217C2×#102—51no—
218C2×#102—52no—
219C2×C22⋊C8—52no—
220C2×C22⋊C8—52no—
221C2×C4×C8—51no—
222#396—102no—
223#390—101no—
224#389—101no—
225#384—51no—
226#380—52no—
227#340—52no—
228#340—52no—
229#340—52no—
230C23×C5—22no—
231C23×C5—22no—
232C23×C5—22no—
233C23×C5—22no—
234C23×C5—22no—
235C23×C5—22no—
236C23×C5—22no—
237C23×C5—41no—
238C23×C5—41no—
239C23×C5—41no—
240C23×C5—41no—
241C23×C5—41no—
242C23×C5—41no—
243C23×C5C2×SD1614yes—
244C23×C5C22×D411yescharacteristic
245C5×Q8—48no—
246C5×D4—22no—
247C5×D4—22no—
248C5×D4—28no—
249C5×D4—416no—
250C5×D4—41no—
251C5×D4—41no—
252C2×C4×C5—21no—
253C2×C4×C5—21no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—24no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—416no—
260C2×C4×C5—41no—
261C2×C4×C5—42no—
262C2×C4×C5—41no—
263C2×C4×C5—42no—
264C2×C4×C5—42no—
265C2×C4×C5—42no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—48no—
269C2×C4×C5#12012yes—
270C2×C4×C5#12012yes—
271C2×C4×C5C2×SD1612yes—
272C2×C4×C5AGL(1,ℤ/8)12yes—
273C2×C4×C5C22×D411yescharacteristic
274C2×C4×C5C22×D411yescharacteristic
275C2×C4×C5C2×#4111yescharacteristic
276C2×C4×C5C2×#4111yescharacteristic
277C2×C4×C5H(5,𝔽2)11yescharacteristic
278C2×C4×C52−1+411yescharacteristic
279C2×Dic5—22no—
280C2×Dic5—41no—
281C2×Dic5—41no—
282C2×Dic5—41no—
283C2×Dic5—42no—
284C2×Dic5—42no—
285C2×Dic5—42no—
286C2×Dic5—41no—
287C2×Dic5—41no—
288Dic10—22no—
289Dic10—22no—
290Dic10—41no—
291Dic10—41no—
292#168—22no—
293#168—41no—
294#168—42no—
295C2×#41—28no—
296C2×#41—44no—
297C2×#41—44no—
298C22×Q8—21no—
299C22×Q8—51no—
300C22×D4—22no—
301C22×D4C22⋊D511yescharacteristic
302C22×D4C22×D511yescharacteristic
303C23×C4—22no—
304C23×C4—21no—
305C23×C4C22×D512yes—
306C2×SD16—102no—
307C2×SD16—102no—
308C2×SD16—102no—
309C22×C8—51no—
310C22×C8—51no—
311C22×C8—52no—
312C4⋊Q8—22no—
313C4⋊Q8C22⋊D514yes—
314#123—216no—
315C22⋊Q8—28no—
316C22⋊Q8—104no—
317C22⋊Q8—202no—
318#121—24no—
319#121—24no—
320C4×Q8—102no—
321C4×Q8—102no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4C22⋊D514yes—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—51no—
331C2×C4⋊C4—52no—
332C2×C4⋊C4—101no—
333C2×C4⋊C4—102no—
334C2×C4⋊C4—101no—
335C2×C4⋊C4—101no—
336C2×C4⋊C4C22⋊D511yescharacteristic
337C2×C4⋊C4C22×D511yescharacteristic
338C2×C4⋊C4C22×D512yes—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—102no—
342C2×C22⋊C4—102no—
343C2×C22⋊C4C22⋊D512yes—
344C2×C42—101no—
345C2×C42C22×D511yescharacteristic
346#106—102no—
347#106—104no—
348C4⋊C8—102no—
349#103—102no—
350#103—104no—
351#102—102no—
352#102—202no—
353C22⋊C8—104no—
354C22⋊C8—202no—
355C4×C8—102no—
356#95—102no—
357C22×C5—24no—
358C22×C5—24no—
359C22×C5—24no—
360C22×C5—42no—
361C22×C5—42no—
362C22×C5—42no—
363C22×C5—42no—
364C22×C5—42no—
365C22×C5—42no—
366C22×C5—42no—
367C22×C5—81no—
368C22×C5—81no—
369C22×C5—81no—
370C22×C5—81no—
371C22×C5C2×#12011yescharacteristic
372C22×C5#54611yescharacteristic
373C22×C5#54711yescharacteristic
374C22×C5C22×SD1611yescharacteristic
375C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
376C22×C5#57711yescharacteristic
377C22×C5#57811yescharacteristic
378C4×C5—22no—
379C4×C5—42no—
380C4×C5—42no—
381C4×C5—44no—
382C4×C5#45014yes—
383C4×C5#52511yescharacteristic
384C4×C5#53711yescharacteristic
385C4×C5#54811yescharacteristic
386C4×C5#55011yescharacteristic
387Dic5—42no—
388Dic5—82no—
389Dic5—81no—
390C24—21no—
391C24C2×C22⋊D512yes—
392#41—432no—
393C2×Q8—24no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—52no—
397C2×Q8—101no—
398C2×Q8—101no—
399C2×Q8—101no—
400C2×Q8—101no—
401C2×Q8—101no—
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4#68714yescomplemented
413C2×D4C2×C22⋊D512yes—
414C22×C4—28no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—101no—
429C22×C4—101no—
430C22×C4—102no—
431C22×C4C2×C22⋊D512yes—
432C22×C4C2×C22⋊D511yescharacteristic
433C22×C4C2×C22⋊D511yescharacteristic
434C22×C4C23×D511yescharacteristic
435SD16—108no—
436C2×C8—52no—
437C2×C8—101no—
438C2×C8—101no—
439C2×C8—102no—
440C2×C8—101no—
441C2×C8—102no—
442C2×C8—101no—
443C2×C8—101no—
444C2×C8—102no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—102no—
451C4⋊C4—102no—
452C4⋊C4—101no—
453C4⋊C4—101no—
454C4⋊C4—204no—
455C4⋊C4—202no—
456C4⋊C4—202no—
457C4⋊C4—401no—
458C4⋊C4C2×C22⋊D512yes—
459C4⋊C4C2×C22⋊D514yes—
460C22⋊C4—28no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—204no—
464C22⋊C4—402no—
465C42—202no—
466C42C2×C22⋊D512yes—
467C42C2×C22⋊D512yes—
468C2×C5—44no—
469C2×C5—82no—
470C2×C5#240711yescharacteristic
471C2×C5C2×#45011yescharacteristic
472C2×C5#301311yescharacteristic
473C2×C5#329811yescharacteristic
474C2×C5#333211yescharacteristic
475C2×C5#333511yescharacteristic
476C2×C5#335311yescharacteristic
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C2×#68714yes—
491C23C22×C22⋊D511yescharacteristic
492Q8—48no—
493Q8—102no—
494Q8—102no—
495Q8—201no—
496Q8—201no—
497D4—22no—
498D4—22no—
499D4—28no—
500D4—416no—
501D4—41no—
502D4—41no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—416no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—48no—
520C2×C4—102no—
521C2×C4—201no—
522C2×C4—201no—
523C2×C4—202no—
524C2×C4—202no—
525C2×C4—201no—
526C2×C4—202no—
527C2×C4—201no—
528C2×C4—201no—
529C2×C4#407812yes—
530C2×C4C2×#68712yes—
531C2×C4#409912yes—
532C2×C4#409912yes—
533C2×C4D5×#4111yescharacteristic
534C2×C4D5×#4111yescharacteristic
535C2×C4#414911yescharacteristic
536C2×C4#415011yescharacteristic
537C2×C4C22×C22⋊D511yescharacteristic
538C2×C4C22×C22⋊D511yescharacteristic
539C8—102no—
540C8—201no—
541C8—202no—
542C5#2412011yescharacteristic, complemented, Sylow
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#407811yescharacteristic
558C22C22×#68711yescharacteristic
559C22#6615811yescharacteristic
560C22#6616311yescharacteristic
561C22#6617111yescharacteristic
562C22#6617211yescharacteristic
563C22C2×#409911yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—44no—
568C4—202no—
569C4—401no—
570C4—402no—
571C4#6550714yes—
572C4#6616211yescharacteristic
573C4#6616511yescharacteristic
574C4#6616711yescharacteristic
575C4#6616811yescharacteristic
576C2—44no—
577C2—82no—
578C2#1061524811yescharacteristic
579C2#1061525111yescharacteristic
580C2#1061526211yescharacteristic
581C2#1061526411yescharacteristic
582C2C2×#6550711yescharacteristic
583C2#1061582811yescharacteristic
584C2#1061584511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
Fitting, prime radicalC2×C5×#547C2
socleC23×C5C22×D4
hypercenterC2×#547D5
hypocenterC5#24120
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. C2×#65383C2semidirect
3. #10614107C2semidirect
4. C2×C4×#687C2semidirect
5. #10602580C2semidirect
6. #10602478C2semidirect
7. #10602473C2semidirect
8. #10602407C2semidirect
9. #10602397C2semidirect
10. #10602352C2semidirect
11. #10602349C2semidirect
135. C2×C5×D4SD16semidirect
163. C2×Dic10#41semidirect
196. C2×#168#41semidirect
412. C2×D4#687semidirect
542. C5#24120semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5472
C2×#653832
#106141072
C2×C4×#6872
#106025802
#106024782
#106024732
#106024072
#106023972
#106023522
#106023492
#241205