Finite Group Databasebeta

← #12121100

Subgroups — #12121100

Order 1280.

926 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#683C211yesmaximal, characteristic
2C2×#65048C211yesmaximal, characteristic
3#10605383C211yesmaximal, characteristic, complemented
4C4×C5×#102C211yesmaximal, characteristic
5#10602399C211yesmaximal, characteristic
6#10600961C211yesmaximal, characteristic
7Dic5×#102C211yesmaximal, characteristic
8Dic5×#102C211yesmaximal, characteristic
9C8×#683C211yesmaximal, characteristic, complemented
10#10600563C211yesmaximal, characteristic, complemented
11#10600251C211yesmaximal, characteristic, complemented
12#10599652C211yesmaximal, characteristic, complemented
13#10599607C211yesmaximal, characteristic, complemented
14#10599596C211yesmaximal, characteristic, complemented
15#10599588C211yesmaximal, characteristic, complemented
16C2×C4×C5×D4C2211yescharacteristic
17C2×D4×Dic5C2211yescharacteristic
18C2×D4×Dic5C2211yescharacteristic
19C22×#683—21no—
20C2×#4034C2211yescharacteristic
21C2×C4×Dic10—21no—
22C2×C5×#102C2211yescharacteristic
23C2×C5×#102C2211yescharacteristic
24C2×C4×C5×C8C2211yescharacteristic
25C4×C5×C4⋊C4C2211yescharacteristic
26C2×#3964C2211yescharacteristic
27C2×#3964C2211yescharacteristic
28#65499—21no—
29C2×#3950C2211yescharacteristic
30C2×#3950C2211yescharacteristic
31C2×#3946C2211yescharacteristic
32C2×C8×Dic5C2211yescharacteristic
33C2×C8×Dic5C2211yescharacteristic
34#65274C2211yescharacteristic
35#65265C2211yescharacteristic
36C4⋊C4×Dic5C2211yescharacteristic
37C4⋊C4×Dic5C2211yescharacteristic
38#65262C2211yescharacteristic
39C2×#3939C2211yescharacteristic
40C2×#3939C2211yescharacteristic
41#65235—21no—
42#65231—21no—
43#65222—21no—
44C4×#683C2211yescharacteristic
45#65217C2211yescharacteristic
46#65217C2211yescharacteristic
47#65048C418yescomplemented
48#64944—21no—
49#64935—21no—
50C5×#340C2211yescharacteristic
51C5×#340C2211yescharacteristic
52#64771C2211yescharacteristic
53#64771C2211yescharacteristic
54#64770C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64744C2211yescharacteristic
60#22962—51nomaximal, Sylow
61C22×C5×D4C2311yescharacteristic
62C23×C4×C5—21no—
63C23×Dic5—21no—
64C23×Dic5—21no—
65C22×Dic10—21no—
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C4×C5×D4—22no—
69C4×C5×D4—41no—
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C4⋊C4C2311yescharacteristic
73C2×C5×C22⋊C4—21no—
74C2×C42×C5—21no—
75C2×C42×C5C2311yescharacteristic
76C2×C22⋊Dic5—21no—
77C2×C22⋊Dic5—21no—
78D4×Dic5—22no—
79D4×Dic5—22no—
80D4×Dic5—41no—
81D4×Dic5C2×C414yes—
82C2×C4⋊Dic5—21no—
83C2×C4⋊Dic5C2311yescharacteristic
84C2×C4⋊Dic5C2311yescharacteristic
85C2×C4⋊Dic5C2311yescharacteristic
86C2×C4⋊Dic5C2311yescharacteristic
87C2×#683—21no—
88C2×#683—21no—
89C2×#683—21no—
90C2×#683—21no—
91C2×#683—21no—
92C2×#683—21no—
93C2×#683—44no—
94C2×#683D411yescharacteristic
95C2×#683D411yescharacteristic
96C2×C4×Dic5—21no—
97C2×C4×Dic5—21no—
98C2×C4×Dic5C2311yescharacteristic
99C2×C4×Dic5C2311yescharacteristic
100C22×#168C2311yescharacteristic
101C22×#168C2311yescharacteristic
102#4034—22no—
103#4034C2×C414yescomplemented
104C4×Dic10—44no—
105C5×#102C2×C414yes—
106C5×#102Q814yescomplemented
107C4×C5×C8—41no—
108C5×#95—21no—
109#3964—22no—
110#3964C2×C414yes—
111#3963—21no—
112#3963—21no—
113#3963—21no—
114#3963—21no—
115#3963—21no—
116#3963—21no—
117#3963D411yescharacteristic
118#3963D411yescharacteristic
119#3950—41no—
120#3950C2×C414yescomplemented
121#3946C2×C414yescomplemented
122C8×Dic5—22no—
123C8×Dic5—41no—
124#3939—41no—
125#3939C2×C414yescomplemented
126C2×#474—51no—
127#2365—51no—
128D4×C4⋊C4—51no—
129#1990—51no—
130#1947—51no—
131#1942—51no—
132#1942—51no—
133#1918—51no—
134#1918—51no—
135#1862—51no—
136#1862—51no—
137C8×C4⋊C4—51no—
138C4×#102—51no—
139C4×#102—51no—
140C4×#102D511yescharacteristic
141C24×C5—21no—
142C2×C5×D4—21no—
143C2×C5×D4—21no—
144C2×C5×D4—21no—
145C2×C5×D4C22×C412yes—
146C2×C5×D4C2×Q812yes—
147C2×C5×D4#4112yes—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—42no—
152C22×C4×C5—42no—
153C22×C4×C5—81no—
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2411yescharacteristic
157C22×Dic5—21no—
158C22×Dic5—21no—
159C22×Dic5—24no—
160C22×Dic5—21no—
161C22×Dic5—21no—
162C22×Dic5—42no—
163C22×Dic5—42no—
164C22×Dic5—42no—
165C22×Dic5—42no—
166C22×Dic5—81no—
167C22×Dic5C2×D411yescharacteristic
168C22×Dic5C2×D411yescharacteristic
169C22×Dic5C2×D411yescharacteristic
170C22×Dic5C2×D411yescharacteristic
171C2×Dic10—24no—
172C2×Dic10—42no—
173C2×Dic10—42no—
174C2×C5×C8—21no—
175C2×C5×C8—21no—
176C2×C5×C8—21no—
177C2×C5×C8—21no—
178C2×C5×C8C22×C412yes—
179C2×C5×C8C2×Q812yes—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—21no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4C22×C412yes—
185C5×C4⋊C4C2×Q812yes—
186C5×C22⋊C4—42no—
187C42×C5—21no—
188C42×C5—21no—
189C42×C5—42no—
190C22⋊Dic5—24no—
191C22⋊Dic5—42no—
192C4⋊Dic5—21no—
193C4⋊Dic5—21no—
194C4⋊Dic5—21no—
195C4⋊Dic5—21no—
196C4⋊Dic5—21no—
197C4⋊Dic5—21no—
198C4⋊Dic5—42no—
199C4⋊Dic5C22×C412yes—
200C4⋊Dic5C22×C412yes—
201#683—24no—
202#683—22no—
203#683—42no—
204#683—42no—
205#683—42no—
206#683—42no—
207#683—81no—
208#683D814yes—
209C4×Dic5—21no—
210C4×Dic5—21no—
211C4×Dic5—42no—
212C4×Dic5—42no—
213C4×Dic5C22×C412yes—
214C4×Dic5#4112yes—
215C2×#168—21no—
216C2×#168—21no—
217C2×#168—21no—
218C2×#168—21no—
219C2×#168—21no—
220C2×#168C22×C412yes—
221C2×C4⋊Q8—51no—
222C2×C4×Q8—101no—
223C2×C4×D4—51no—
224C2×C4×D4—51no—
225C2×C4×D4C2×D511yescharacteristic
226C22×C4⋊C4—101no—
227#474—58no—
228C2×#107—51no—
229C2×#107—51no—
230C2×#107—51no—
231C2×#107—51no—
232C2×C4⋊C8—51no—
233C2×#102—51no—
234C2×#102—51no—
235C2×#102C2×D511yescharacteristic
236C2×#102C2×D511yescharacteristic
237C2×C4×C8—51no—
238C2×C4×C8—51no—
239C2×C4×C8C2×D511yescharacteristic
240#391—101no—
241#389—51no—
242#389—51no—
243#389—51no—
244#389—51no—
245#387—101no—
246#387—101no—
247#385—101no—
248#385—101no—
249#384—51no—
250#380—101no—
251C4×C4⋊C4—51no—
252C4×C4⋊C4—51no—
253C4×C4⋊C4—51no—
254C4×C4⋊C4C2×D511yescharacteristic
255#340—51no—
256#340—51no—
257#340—51no—
258#340—51no—
259#340—51no—
260#340—51no—
261#340C2×D511yescharacteristic
262#340C2×D511yescharacteristic
263#336—51no—
264#336—51no—
265C23×C5—22no—
266C23×C5—22no—
267C23×C5—22no—
268C23×C5—41no—
269C23×C5—41no—
270C23×C5—41no—
271C23×C5—41no—
272C23×C5C22×D411yescharacteristic
273C5×D4—22no—
274C5×D4—22no—
275C5×D4—22no—
276C5×D4C4×Q814yescomplemented
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—22no—
280C2×C4×C5—21no—
281C2×C4×C5—22no—
282C2×C4×C5—21no—
283C2×C4×C5—21no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5—84no—
292C2×C4×C5C2×D812yes—
293C2×C4×C5C23×C411yescharacteristic
294C2×C4×C5C22×Q811yescharacteristic
295C2×C4×C5C2×#4111yescharacteristic
296C2×C4×C5C2×#4111yescharacteristic
297C2×C4×C5H(5,𝔽2)11yescharacteristic
298C2×C4×C52−1+411yescharacteristic
299C2×Dic5—21no—
300C2×Dic5—21no—
301C2×Dic5—21no—
302C2×Dic5—21no—
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5—21no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312C2×Dic5—48no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315C2×Dic5—41no—
316C2×Dic5—41no—
317C2×Dic5—41no—
318C2×Dic5—41no—
319C2×Dic5—84no—
320C2×Dic5C4×D412yes—
321C2×Dic5C4×D412yes—
322C2×Dic5C2×D812yes—
323C2×Dic5C2×D812yes—
324C2×Dic5#13512yes—
325Dic10—48no—
326C5×C8—22no—
327C5×C8—22no—
328#168—22no—
329#168—41no—
330C22×Q8—101no—
331C22×D4C22×D511yescharacteristic
332C23×C4—21no—
333C23×C4—101no—
334C23×C4—101no—
335C22×C8—51no—
336C22×C8—51no—
337C22×C8C22×D511yescharacteristic
338C22×C8C22×D511yescharacteristic
339C4⋊Q8—54no—
340C4⋊Q8—102no—
341C4×Q8—204no—
342C4×D4—22no—
343C4×D4—41no—
344C4×D4—54no—
345C4×D4—102no—
346C4×D4—102no—
347C4×D4—201no—
348C2×C4⋊C4—51no—
349C2×C4⋊C4—51no—
350C2×C4⋊C4—51no—
351C2×C4⋊C4—51no—
352C2×C4⋊C4—51no—
353C2×C4⋊C4—51no—
354C2×C4⋊C4—101no—
355C2×C4⋊C4—101no—
356C2×C4⋊C4—101no—
357C2×C4⋊C4—101no—
358C2×C4⋊C4—101no—
359C2×C4⋊C4—101no—
360C2×C4⋊C4—101no—
361C2×C4⋊C4—204no—
362C2×C4⋊C4C22×D511yescharacteristic
363C2×C4⋊C4C22×D511yescharacteristic
364C2×C4⋊C4C22×D511yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—101no—
367C2×C22⋊C4—101no—
368C2×C42—21no—
369C2×C42—51no—
370C2×C42—51no—
371C2×C42—101no—
372C2×C42—101no—
373C2×C42C22×D511yescharacteristic
374#107—54no—
375#107—54no—
376#107—201no—
377#107—201no—
378C4⋊C8—54no—
379#102—54no—
380#102—102no—
381#102Dic1014yescomplemented
382#102C4×D514yes—
383C4×C8—41no—
384C4×C8—102no—
385C4×C8—201no—
386#95—21no—
387#95—51no—
388#95—51no—
389#95—101no—
390#95—101no—
391#95—101no—
392#95—101no—
393#95—101no—
394#95—101no—
395C22×C5—24no—
396C22×C5—42no—
397C22×C5—42no—
398C22×C5—42no—
399C22×C5—42no—
400C22×C5—42no—
401C22×C5—42no—
402C22×C5C2×C4×D411yescharacteristic
403C22×C5#54611yescharacteristic
404C22×C5D4×Q811yescharacteristic
405C22×C5C22×D811yescharacteristic
406C22×C5C2×#13511yescharacteristic
407C22×C5#57711yescharacteristic
408C22×C5#57811yescharacteristic
409C4×C5—22no—
410C4×C5—42no—
411C4×C5—42no—
412C4×C5—41no—
413C4×C5—81no—
414C4×C5C2×C4×Q811yescharacteristic
415C4×C5#52011yescharacteristic
416C4×C5#55211yescharacteristic
417C4×C5#55411yescharacteristic
418Dic5—22no—
419Dic5—22no—
420Dic5—42no—
421Dic5—42no—
422Dic5—41no—
423Dic5—81no—
424Dic5—81no—
425Dic5C4×D814yes—
426C24—21no—
427C2×Q8—104no—
428C2×Q8—202no—
429C2×Q8—202no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4C2×Dic1012yes—
434C2×D4C2×C4×D512yes—
435C2×D4#70912yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—51no—
442C22×C4—51no—
443C22×C4—51no—
444C22×C4—51no—
445C22×C4—81no—
446C22×C4—104no—
447C22×C4—101no—
448C22×C4—101no—
449C22×C4—101no—
450C22×C4—101no—
451C22×C4—202no—
452C22×C4—202no—
453C22×C4—202no—
454C22×C4—202no—
455C22×C4—401no—
456C22×C4D4×D511yescharacteristic
457C22×C4D4×D511yescharacteristic
458C22×C4C23×D511yescharacteristic
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—52no—
464C2×C8—101no—
465C2×C8—101no—
466C2×C8—101no—
467C2×C8—101no—
468C2×C8—101no—
469C2×C8C2×Dic1012yes—
470C2×C8C2×C4×D512yes—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—54no—
476C4⋊C4—52no—
477C4⋊C4—52no—
478C4⋊C4—102no—
479C4⋊C4—104no—
480C4⋊C4—101no—
481C4⋊C4—101no—
482C4⋊C4—101no—
483C4⋊C4—101no—
484C4⋊C4—101no—
485C4⋊C4—101no—
486C4⋊C4—202no—
487C4⋊C4—202no—
488C4⋊C4—202no—
489C4⋊C4—202no—
490C4⋊C4—202no—
491C4⋊C4—401no—
492C4⋊C4C2×Dic1012yes—
493C4⋊C4C2×C4×D512yes—
494C22⋊C4—42no—
495C22⋊C4—104no—
496C22⋊C4—202no—
497C42—21no—
498C42—21no—
499C42—42no—
500C42—52no—
501C42—52no—
502C42—101no—
503C42—101no—
504C42—202no—
505C42—202no—
506C2×C5—44no—
507C2×C5#236011yescharacteristic
508C2×C5C2×C4×D811yescharacteristic
509C2×C5#294711yescharacteristic
510C2×C5#329411yescharacteristic
511C2×C5#330211yescharacteristic
512C2×C5Q8×D811yescharacteristic
513C2×C5#338111yescharacteristic
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D511yescharacteristic
522Q8—208no—
523D4—22no—
524D4—22no—
525D4—22no—
526D4C4×Dic1014yescomplemented
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—52no—
542C2×C4—52no—
543C2×C4—52no—
544C2×C4—52no—
545C2×C4—52no—
546C2×C4—84no—
547C2×C4—101no—
548C2×C4—101no—
549C2×C4—101no—
550C2×C4—101no—
551C2×C4—101no—
552C2×C4—101no—
553C2×C4—101no—
554C2×C4—102no—
555C2×C4—102no—
556C2×C4—208no—
557C2×C4—201no—
558C2×C4—201no—
559C2×C4—201no—
560C2×C4—201no—
561C2×C4—201no—
562C2×C4—201no—
563C2×C4—201no—
564C2×C4—201no—
565C2×C4—201no—
566C2×C4—201no—
567C2×C4—404no—
568C2×C4D5×D812yes—
569C2×C4C22×Dic1011yescharacteristic
570C2×C4C22×C4×D511yescharacteristic
571C2×C4C2×#70911yescharacteristic
572C2×C4#414411yescharacteristic
573C2×C4D5×#4111yescharacteristic
574C2×C4#415011yescharacteristic
575C8—22no—
576C8—22no—
577C8—102no—
578C8—201no—
579C5#2296211yescharacteristic, complemented, Sylow
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22#6582411yescharacteristic
588C22D4×Dic1011yescharacteristic
589C22C4×D4×D511yescharacteristic
590C22C2×D5×D811yescharacteristic
591C22#6609611yescharacteristic
592C22D5×#13511yescharacteristic
593C22#6610511yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—54no—
599C4—81no—
600C4—102no—
601C4—102no—
602C4—202no—
603C4—202no—
604C4—201no—
605C4—401no—
606C4—401no—
607C4C2×C4×Dic1011yescharacteristic
608C4#6586511yescharacteristic
609C4#6587311yescharacteristic
610C4#6588011yescharacteristic
611C2—44no—
612C2#1060519711yescharacteristic
613C2#1061048811yescharacteristic
614C2#1061058011yescharacteristic
615C2D8×Dic1011yescharacteristic
616C2C4×D5×D811yescharacteristic
617C2#1061136411yescharacteristic
618C2#1061137911yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC4×C5×#102C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
cosocleC22×C4×C5C24
hypercenterC4×#102D5
hypocenterC5#22962
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. #10605383C2semidirect
9. C8×#683C2semidirect
10. #10600563C2semidirect
11. #10600251C2semidirect
12. #10599652C2semidirect
13. #10599607C2semidirect
14. #10599596C2semidirect
15. #10599588C2semidirect
47. #65048C4semidirect
103. #4034C2×C4semidirect
106. C5×#102Q8semidirect
120. #3950C2×C4semidirect
121. #3946C2×C4semidirect
125. #3939C2×C4semidirect
276. C5×D4C4×Q8semidirect
381. #102Dic10semidirect
526. D4C4×Dic10semidirect
579. C5#22962semidirect

Maximal subgroups

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

subgroupindex
D4×#6832
C2×#650482
#106053832
C4×C5×#1022
#106023992
#106009612
Dic5×#1022
C8×#6832
#106005632
#106002512
#105996522
#105996072
#105995962
#105995882
#229625