Finite Group Databasebeta

← #12121395

Subgroups — #12121395

Order 1280.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3076C211yesmaximal, characteristic
2#10615061C211yesmaximal, characteristic, complemented
3#10614243C211yesmaximal, characteristic
4#10602633C211yesmaximal, characteristic, complemented
5#10602436C211yesmaximal, characteristic
6#10601456C211yesmaximal, characteristic, complemented
7#10601455C211yesmaximal, characteristic, complemented
8#10601454C211yesmaximal, characteristic, complemented
9#10601454C211yesmaximal, characteristic, complemented
10#10601267C211yesmaximal, characteristic, complemented
11#10601235C211yesmaximal, characteristic, complemented
12#10601085C211yesmaximal, characteristic
13#10601085C211yesmaximal, characteristic
14#10601083C211yesmaximal, characteristic
15#10601082C211yesmaximal, characteristic
16C5×#533C2211yescharacteristic
17C4×C5×#41C2211yescharacteristic
18C2×#4094C2211yescharacteristic
19C2×#4094C2211yescharacteristic
20C5×#479C411yescharacteristic, complemented
21C5×#479C411yescharacteristic, complemented
22C5×#478C411yescharacteristic, complemented
23C5×#478C411yescharacteristic, complemented
24C5×#477C411yescharacteristic, complemented
25C5×#476C411yescharacteristic, complemented
26C5×#475C411yescharacteristic, complemented
27C5×#474C411yescharacteristic, complemented
28C5×#427C2211yescharacteristic
29C5×#427C2211yescharacteristic
30C2×C5×C4⋊C8C2211yescharacteristic
31C5×#416C2211yescharacteristic
32C5×#416C2211yescharacteristic
33C2×#3969C2211yescharacteristic
34C2×#3969C2211yescharacteristic
35C2×#3951C2211yescharacteristic
36C2×#3951C2211yescharacteristic
37C2×#3947C2211yescharacteristic
38C2×C8×Dic5C2211yescharacteristic
39#65313—21no—
40#65313—21no—
41#65313—21no—
42Q8×#168—21no—
43#65288—21no—
44#65288—21no—
45#65288C2211yescharacteristic
46#65288C2211yescharacteristic
47#65288C2211yescharacteristic
48#65288C2211yescharacteristic
49#65286C2211yescharacteristic
50#65286C2211yescharacteristic
51#65221C2211yescharacteristic
52#65213—21no—
53#65213C2211yescharacteristic
54C2×C4⋊#168—21no—
55C2×C4⋊#168C2211yescharacteristic
56C4×#681C2211yescharacteristic
57#64772C2211yescharacteristic
58#64772C2211yescharacteristic
59#64770C2211yescharacteristic
60#64770C2211yescharacteristic
61#64752C2211yescharacteristic
62#64752C2211yescharacteristic
63#64752C2211yescharacteristic
64#64752C2211yescharacteristic
65#64752C2211yescharacteristic
66#64752C2211yescharacteristic
67#22984—51nomaximal, Sylow
68C2×C5×#41C2311yescharacteristic
69C22×C5×C8C2311yescharacteristic
70C22×C5×C8C2311yescharacteristic
71C5×C4⋊Q8C2×C411yescharacteristic
72C5×C4⋊Q8C2×C411yescharacteristic
73C5×#125C2×C411yescharacteristic
74C5×#125C2×C411yescharacteristic
75C5×C22⋊Q8—21no—
76C5×C22⋊Q8—21no—
77C4×C5×Q8—21no—
78C4×C5×Q8—21no—
79C4×C5×Q8C2×C411yescharacteristic
80C4×C5×Q8C2×C411yescharacteristic
81C4×C5×D4—21no—
82C4×C5×D4—21no—
83C4×C5×D4C2×C411yescharacteristic
84C4×C5×D4C2×C411yescharacteristic
85C5×#117—21no—
86C5×#117C2311yescharacteristic
87C5×#117C2311yescharacteristic
88C5×#117C2311yescharacteristic
89C2×C42×C5—21no—
90C2×C42×C5C2311yescharacteristic
91#4094—22no—
92#4094—22no—
93#4094—41no—
94#4094—41no—
95C2×C4×Dic5C2311yescharacteristic
96C2×C4×Dic5C2311yescharacteristic
97C2×#681—21no—
98C2×#681—21no—
99C2×#681C2311yescharacteristic
100C2×#681C2311yescharacteristic
101C2×#681C2311yescharacteristic
102C2×#681C2311yescharacteristic
103C22×#168—21no—
104C22×#168—21no—
105C22×#168C2311yescharacteristic
106C22×#168C2311yescharacteristic
107C5×#107C2×C411yescharacteristic
108C5×#107C2×C411yescharacteristic
109C5×#107C2×C411yescharacteristic
110C5×#107C2×C411yescharacteristic
111C5×#106C2×C411yescharacteristic
112C5×#106C2×C411yescharacteristic
113C5×#106C2×C411yescharacteristic
114C5×#106C2×C411yescharacteristic
115C5×C4⋊C8C2×C411yescharacteristic
116C5×C4⋊C8C2×C411yescharacteristic
117C5×C4⋊C8C2×C411yescharacteristic
118C5×C4⋊C8C2×C411yescharacteristic
119C5×#103C2×C411yescharacteristic
120C5×#103C2×C411yescharacteristic
121C5×#103C2×C411yescharacteristic
122C5×#103C2×C411yescharacteristic
123C5×#102C2×C411yescharacteristic
124C5×#102C2×C411yescharacteristic
125C5×#102C2×C411yescharacteristic
126C5×#102C2×C411yescharacteristic
127#3969—22no—
128#3969Q814yescomplemented
129#3963—21no—
130#3963—21no—
131#3951—41no—
132#3951—41no—
133#3947—41no—
134C8×Dic5—41no—
135C4⋊#168—21no—
136C4⋊#168—21no—
137C4⋊#168—21no—
138C4⋊#168—21no—
139C4⋊#168—21no—
140C4⋊#168—21no—
141C4⋊#168—21no—
142C4⋊#168—21no—
143C4⋊#168—21no—
144C4⋊#168—21no—
145C4⋊#168D411yescharacteristic
146C4⋊#168D411yescharacteristic
147C4⋊#168D411yescharacteristic
148C4⋊#168D411yescharacteristic
149#3935—21no—
150#3935—21no—
151#3935—21no—
152C4×#168—21no—
153C4×#168—21no—
154C4×#168—21no—
155#3076D511yescharacteristic
156#2962—51no—
157#2906—51no—
158#1995—51no—
159#1950—51no—
160#1946—51no—
161#1945—51no—
162#1922—51no—
163#1922—51no—
164#1866—51no—
165#1866—51no—
166#1768—51no—
167#1765—51no—
168#1764—51no—
169#1759—51no—
170C5×#41—21no—
171C5×#41—21no—
172C5×#41C2×Q812yes—
173C5×#41#4112yes—
174C2×C5×Q8—21no—
175C2×C5×Q8—21no—
176C2×C5×Q8C22×C411yescharacteristic
177C2×C5×D4—21no—
178C2×C5×D4C22×C411yescharacteristic
179C22×C4×C5—21no—
180C22×C4×C5—21no—
181C22×C4×C5—21no—
182C22×C4×C5#4111yescharacteristic
183C22×C4×C5#4111yescharacteristic
184C22×C4×C5C2411yescharacteristic
185C22×Dic5—21no—
186C22×Dic5—21no—
187C2×C5×C8—21no—
188C2×C5×C8—21no—
189C2×C5×C8—21no—
190C2×C5×C8—21no—
191C2×C5×C8C22×C411yescharacteristic
192C2×C5×C8C22×C411yescharacteristic
193C2×C5×C8C22×C411yescharacteristic
194C2×C5×C8C22×C411yescharacteristic
195C5×C4⋊C4—21no—
196C5×C4⋊C4—21no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4—21no—
199C5×C4⋊C4—21no—
200C5×C4⋊C4—21no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4C22×C411yescharacteristic
204C5×C4⋊C4C22×C411yescharacteristic
205C5×C4⋊C4C22×C411yescharacteristic
206C5×C4⋊C4C22×C411yescharacteristic
207C5×C4⋊C4C22×C411yescharacteristic
208C5×C4⋊C4C22×C411yescharacteristic
209C5×C22⋊C4—21no—
210C5×C22⋊C4—21no—
211C5×C22⋊C4—21no—
212C5×C22⋊C4—21no—
213C5×C22⋊C4—21no—
214C42×C5—21no—
215C42×C5—21no—
216C42×C5—21no—
217C42×C5—21no—
218C42×C5—21no—
219C42×C5C22×C411yescharacteristic
220C42×C5C22×C411yescharacteristic
221C42×C5C2×D411yescharacteristic
222C42×C5C2×D411yescharacteristic
223C4×Dic5—21no—
224C4×Dic5—21no—
225C4×Dic5—21no—
226C4×Dic5C2×Q812yes—
227#681—21no—
228#681—21no—
229#681—21no—
230#681—21no—
231#681—21no—
232#681—21no—
233#681—21no—
234#681—42no—
235#681—42no—
236#681C2×Q812yes—
237C2×#168—21no—
238C2×#168—21no—
239C2×#168—21no—
240C2×#168—21no—
241C2×#168—21no—
242C2×#168—21no—
243C2×#168—21no—
244C2×#168—21no—
245C2×#168—42no—
246C2×#168—42no—
247C2×#168C2×D411yescharacteristic
248C2×#168C2×D411yescharacteristic
249C2×#168C2×D411yescharacteristic
250C2×#168C2×D411yescharacteristic
251C2×#131—51no—
252C2×#131—51no—
253#533C2×D511yescharacteristic
254C4×#41C2×D511yescharacteristic
255#479Dic511yescharacteristic, complemented
256#479Dic511yescharacteristic, complemented
257#478Dic511yescharacteristic, complemented
258#478Dic511yescharacteristic, complemented
259#477Dic511yescharacteristic, complemented
260#476Dic511yescharacteristic, complemented
261#475Dic511yescharacteristic, complemented
262#474Dic511yescharacteristic, complemented
263#446—101no—
264#446—101no—
265#446—101no—
266C8×Q8—101no—
267C2×#108—51no—
268C2×#108—51no—
269#427C2×D511yescharacteristic
270#427C2×D511yescharacteristic
271#424—51no—
272#424—51no—
273#424—51no—
274#424—51no—
275#424—101no—
276#424—101no—
277#423—51no—
278#423—101no—
279C2×C4⋊C8—51no—
280C2×C4⋊C8—101no—
281C2×C4⋊C8C2×D511yescharacteristic
282C2×#104—51no—
283C2×#104—51no—
284#416C2×D511yescharacteristic
285#416C2×D511yescharacteristic
286#405—51no—
287#405—51no—
288C4×M16—51no—
289C2×#97—51no—
290C2×C4×C8—51no—
291#376—51no—
292#341—51no—
293#341—51no—
294#339—51no—
295#339—51no—
296#339—51no—
297#339—51no—
298#339—51no—
299#339—51no—
300#336—51no—
301#336—51no—
302C23×C5—21no—
303C23×C5C2×#4111yescharacteristic
304C5×Q8—21no—
305C5×Q8—42no—
306C5×Q8—81no—
307C5×Q8C4×Q812yes—
308C5×D4—22no—
309C5×D4—22no—
310C5×D4—21no—
311C5×D4C4×Q812yes—
312C2×C4×C5—21no—
313C2×C4×C5—22no—
314C2×C4×C5—22no—
315C2×C4×C5—21no—
316C2×C4×C5—21no—
317C2×C4×C5—21no—
318C2×C4×C5—21no—
319C2×C4×C5—21no—
320C2×C4×C5—21no—
321C2×C4×C5—41no—
322C2×C4×C5—41no—
323C2×C4×C5—42no—
324C2×C4×C5—81no—
325C2×C4×C5C4×D411yescharacteristic
326C2×C4×C5C4×D411yescharacteristic
327C2×C4×C5C4×D411yescharacteristic
328C2×C4×C5C4×D411yescharacteristic
329C2×C4×C5C23×C411yescharacteristic
330C2×C4×C5C22×D411yescharacteristic
331C2×C4×C5C22×Q811yescharacteristic
332C2×C4×C5C2×#4111yescharacteristic
333C2×C4×C5H(5,𝔽2)11yescharacteristic
334C2×C4×C52−1+411yescharacteristic
335C2×Dic5—22no—
336C2×Dic5—41no—
337C2×Dic5—41no—
338C2×Dic5—41no—
339C2×Dic5—41no—
340C2×Dic5—41no—
341C5×C8—21no—
342C5×C8—21no—
343C5×C8—21no—
344C5×C8—21no—
345#168—21no—
346#168—21no—
347#168—21no—
348#168—21no—
349#168—22no—
350#168—41no—
351#168—41no—
352#168—41no—
353C2×#41C22×D511yescharacteristic
354#131—102no—
355#131—102no—
356#131—201no—
357#131—201no—
358C2×M16—51no—
359C2×M16—51no—
360C2×M16—51no—
361C2×M16—51no—
362C2×M16—101no—
363C2×M16—101no—
364C22×C8—51no—
365C22×C8—51no—
366C22×C8—101no—
367C22×C8—101no—
368C22×C8C22×D511yescharacteristic
369C22×C8C22×D511yescharacteristic
370C4⋊Q8C2×Dic511yescharacteristic
371C4⋊Q8C2×Dic511yescharacteristic
372#125C2×Dic511yescharacteristic
373#125C2×Dic511yescharacteristic
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8C2×Dic511yescharacteristic
379C4×Q8C2×Dic511yescharacteristic
380C4×D4—21no—
381C4×D4—21no—
382C4×D4C2×Dic511yescharacteristic
383C4×D4C2×Dic511yescharacteristic
384#117—21no—
385#117C22×D511yescharacteristic
386#117C22×D511yescharacteristic
387#117C22×D511yescharacteristic
388C2×C42—21no—
389C2×C42—51no—
390C2×C42—51no—
391C2×C42C22×D511yescharacteristic
392#108—201no—
393#108—201no—
394#107C2×Dic511yescharacteristic
395#107C2×Dic511yescharacteristic
396#107C2×Dic511yescharacteristic
397#107C2×Dic511yescharacteristic
398#106C2×Dic511yescharacteristic
399#106C2×Dic511yescharacteristic
400#106C2×Dic511yescharacteristic
401#106C2×Dic511yescharacteristic
402C4⋊C8—51no—
403C4⋊C8—51no—
404C4⋊C8—51no—
405C4⋊C8—51no—
406C4⋊C8—101no—
407C4⋊C8—101no—
408C4⋊C8—101no—
409C4⋊C8—101no—
410C4⋊C8—101no—
411C4⋊C8—101no—
412C4⋊C8—101no—
413C4⋊C8—101no—
414C4⋊C8—101no—
415C4⋊C8—101no—
416C4⋊C8C2×Dic511yescharacteristic
417C4⋊C8C2×Dic511yescharacteristic
418C4⋊C8C2×Dic511yescharacteristic
419C4⋊C8C2×Dic511yescharacteristic
420#104—54no—
421#104—102no—
422#103C2×Dic511yescharacteristic
423#103C2×Dic511yescharacteristic
424#103C2×Dic511yescharacteristic
425#103C2×Dic511yescharacteristic
426#102C2×Dic511yescharacteristic
427#102C2×Dic511yescharacteristic
428#102C2×Dic511yescharacteristic
429#102C2×Dic511yescharacteristic
430#97—101no—
431#97—101no—
432#97—101no—
433#97—201no—
434C4×C8—101no—
435C4×C8—101no—
436C4×C8—101no—
437C4×C8—201no—
438#95—101no—
439#95—101no—
440C22×C5—21no—
441C22×C5—21no—
442C22×C5—22no—
443C22×C5—41no—
444C22×C5—41no—
445C22×C5C2×C4×D411yescharacteristic
446C22×C5#55211yescharacteristic
447C22×C5#55411yescharacteristic
448C4×C5—22no—
449C4×C5—21no—
450C4×C5—21no—
451C4×C5—21no—
452C4×C5—21no—
453C4×C5—41no—
454C4×C5—41no—
455C4×C5—41no—
456C4×C5—41no—
457C4×C5—41no—
458C4×C5—41no—
459C4×C5C2×C4×Q811yescharacteristic
460C4×C5#52011yescharacteristic
461C4×C5#54611yescharacteristic
462C4×C5D4×Q811yescharacteristic
463Dic5—42no—
464Dic5—81no—
465#41—21no—
466#41—21no—
467#41Q8×D512yes—
468#41#71312yes—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8C22×Dic511yescharacteristic
472C2×D4—21no—
473C2×D4C22×Dic511yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—101no—
478C22×C4—101no—
479C22×C4#71111yescharacteristic
480C22×C4#71111yescharacteristic
481C22×C4C23×D511yescharacteristic
482M16—52no—
483M16—101no—
484M16—101no—
485M16—101no—
486M16—101no—
487M16—101no—
488M16—101no—
489M16—101no—
490M16—202no—
491M16—202no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—51no—
497C2×C8—51no—
498C2×C8—51no—
499C2×C8—51no—
500C2×C8—101no—
501C2×C8—101no—
502C2×C8—101no—
503C2×C8—101no—
504C2×C8—101no—
505C2×C8—101no—
506C2×C8—101no—
507C2×C8—101no—
508C2×C8—202no—
509C2×C8—202no—
510C2×C8C22×Dic511yescharacteristic
511C2×C8C22×Dic511yescharacteristic
512C2×C8C22×Dic511yescharacteristic
513C2×C8C22×Dic511yescharacteristic
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4C22×Dic511yescharacteristic
523C4⋊C4C22×Dic511yescharacteristic
524C4⋊C4C22×Dic511yescharacteristic
525C4⋊C4C22×Dic511yescharacteristic
526C4⋊C4C22×Dic511yescharacteristic
527C4⋊C4C22×Dic511yescharacteristic
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—52no—
539C42—101no—
540C42—101no—
541C42—101no—
542C42D4×D511yescharacteristic
543C42D4×D511yescharacteristic
544C42C22×Dic511yescharacteristic
545C42C22×Dic511yescharacteristic
546C2×C5—21no—
547C2×C5—21no—
548C2×C5—42no—
549C2×C5#236011yescharacteristic
550C2×C5#295511yescharacteristic
551C2×C5#295711yescharacteristic
552C23—21no—
553C23C2×#71111yescharacteristic
554Q8—21no—
555Q8—42no—
556Q8—81no—
557Q8Q8×Dic512yes—
558D4—22no—
559D4—22no—
560D4—21no—
561D4Q8×Dic512yes—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—81no—
575C2×C4—102no—
576C2×C4—201no—
577C2×C4—201no—
578C2×C4—201no—
579C2×C4—201no—
580C2×C4—201no—
581C2×C4D4×Dic511yescharacteristic
582C2×C4D4×Dic511yescharacteristic
583C2×C4D4×Dic511yescharacteristic
584C2×C4D4×Dic511yescharacteristic
585C2×C4C2×D4×D511yescharacteristic
586C2×C4C2×Q8×D511yescharacteristic
587C2×C4C2×#71311yescharacteristic
588C2×C4#414911yescharacteristic
589C2×C4#415011yescharacteristic
590C2×C4C23×Dic511yescharacteristic
591C8—21no—
592C8—21no—
593C8—21no—
594C8—21no—
595C8—101no—
596C8—101no—
597C8—101no—
598C8—101no—
599C8—102no—
600C8—201no—
601C8—201no—
602C8—201no—
603C5#2298411yescharacteristic, complemented, Sylow
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22#6596911yescharacteristic
610C22#6597611yescharacteristic
611C22C2×D4×Dic511yescharacteristic
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—22no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—202no—
624C4—401no—
625C4#6596411yescharacteristic
626C4#6596611yescharacteristic
627C4C2×Q8×Dic511yescharacteristic
628C4#6616111yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#1061447811yescharacteristic
633C2#1061564111yescharacteristic
634C2#1061573011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#3076C2
socleC22×C5C2×C4×D4
prime radicalC2×C5×#41C23
cosocleC22×C4×C5C24
hypercenter#3076D5
hypocenterC5#22984
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
2. #10615061C2semidirect
4. #10602633C2semidirect
6. #10601456C2semidirect
7. #10601455C2semidirect
8. #10601454C2semidirect
9. #10601454C2semidirect
10. #10601267C2semidirect
11. #10601235C2semidirect
20. C5×#479C4semidirect
21. C5×#479C4semidirect
22. C5×#478C4semidirect
23. C5×#478C4semidirect
24. C5×#477C4semidirect
25. C5×#476C4semidirect
26. C5×#475C4semidirect
27. C5×#474C4semidirect
128. #3969Q8semidirect
255. #479Dic5semidirect
256. #479Dic5semidirect
257. #478Dic5semidirect
258. #478Dic5semidirect
259. #477Dic5semidirect
260. #476Dic5semidirect
261. #475Dic5semidirect
262. #474Dic5semidirect
603. C5#22984semidirect

Maximal subgroups

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

subgroupindex
C5×#30762
#106150612
#106142432
#106026332
#106024362
#106014562
#106014552
#106014542
#106012672
#106012352
#106010852
#106010832
#106010822
#229845