Finite Group Databasebeta

← #12121092

Subgroups — #12121092

Order 1280.

788 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10614222C211yesmaximal, characteristic
2C2×#65313C211yesmaximal, characteristic, complemented
3C2×#65128C211yesmaximal, characteristic
4C5×#1985C211yesmaximal, characteristic, complemented
5#10601261C211yesmaximal, characteristic, complemented
6#10600623C211yesmaximal, characteristic
7#10600603C211yesmaximal, characteristic, complemented
8#10599460C211yesmaximal, characteristic
9#10599459C211yesmaximal, characteristic
10#10599449C211yesmaximal, characteristic, complemented
11#10599449C211yesmaximal, characteristic, complemented
12#10599443C211yesmaximal, characteristic
13#10599441C211yesmaximal, characteristic
14#10599411C211yesmaximal, characteristic, complemented
15#10599410C211yesmaximal, characteristic, complemented
16C2×C4×C5×Q8C2211yescharacteristic
17C22×#676C2211yescharacteristic
18C22×C8×D5C2211yescharacteristic
19C2×#4018C2211yescharacteristic
20C2×C5×C4⋊C8C2211yescharacteristic
21C2×C5×#97C2211yescharacteristic
22#65520C411yescharacteristic
23#65518C411yescharacteristic
24C2×#3952C2211yescharacteristic
25C2×#3952C2211yescharacteristic
26C2×#3947C2211yescharacteristic
27C2×#3946C2211yescharacteristic
28C2×#3946C2211yescharacteristic
29C2×#3946C2211yescharacteristic
30C2×#3946C2211yescharacteristic
31C2×C8×Dic5C2211yescharacteristic
32#65313—24no—
33#65283C411yescharacteristic
34#65283C411yescharacteristic
35#65281C411yescharacteristic
36#65273C411yescharacteristic
37#65273C411yescharacteristic
38#65271C411yescharacteristic
39C4×#685C2211yescharacteristic
40C4×#685C2211yescharacteristic
41C4×C4⋊Dic5C2211yescharacteristic
42C4×#683C2211yescharacteristic
43C4×#683C2211yescharacteristic
44C2×C4⋊#168C2211yescharacteristic
45C2×C4⋊#168C2211yescharacteristic
46C2×C4⋊#168C2211yescharacteristic
47C2×#3935C2211yescharacteristic
48C2×#3935C2211yescharacteristic
49C2×C4×#168C2211yescharacteristic
50#65128—24no—
51C5×#336C2211yescharacteristic
52C5×#336C2211yescharacteristic
53C5×#336C2211yescharacteristic
54C5×#336C2211yescharacteristic
55#64770C2211yescharacteristic
56#64770C2211yescharacteristic
57#64770C2211yescharacteristic
58#64770C2211yescharacteristic
59#64770C2211yescharacteristic
60#64770C2211yescharacteristic
61#22961—51nomaximal, Sylow
62C22×C5×Q8C2×C411yescharacteristic
63C22×C4×D5C2311yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C4×C5×Q8—22no—
69C4×C5×Q8—22no—
70C2×C5×C4⋊C4C2×C411yescharacteristic
71C2×C5×C4⋊C4C2×C411yescharacteristic
72C2×C5×C4⋊C4C2×C411yescharacteristic
73C2×C42×C5C2311yescharacteristic
74C2×C42×C5C2311yescharacteristic
75C2×C42×C5C2311yescharacteristic
76C2×#685C2×C411yescharacteristic
77C2×#685C2×C411yescharacteristic
78C2×#685C2×C411yescharacteristic
79C2×#685C2×C411yescharacteristic
80C2×#685C2×C411yescharacteristic
81C2×#685C2×C411yescharacteristic
82C2×C4⋊Dic5C2×C411yescharacteristic
83C2×C4⋊Dic5C2×C411yescharacteristic
84C2×#683C2×C411yescharacteristic
85C2×#683C2×C411yescharacteristic
86C2×#683C2×C411yescharacteristic
87C2×#683C2×C411yescharacteristic
88C2×#683C2×C411yescharacteristic
89C2×#683C2×C411yescharacteristic
90C2×C4×Dic5C2311yescharacteristic
91C2×C4×Dic5C2311yescharacteristic
92C2×C4×Dic5C2311yescharacteristic
93C22×#168C2311yescharacteristic
94C22×#168C2311yescharacteristic
95C22×#168C2311yescharacteristic
96C22×#168C2311yescharacteristic
97C2×#676—22no—
98C2×#676—22no—
99C2×#676—41no—
100C2×C8×D5—22no—
101C2×C8×D5—22no—
102C2×C8×D5—41no—
103#4018—22no—
104#4018—22no—
105C5×C4⋊C8—22no—
106C5×#97—22no—
107#3963C2×C411yescharacteristic
108#3963C2×C411yescharacteristic
109#3963C2×C411yescharacteristic
110#3963C2×C411yescharacteristic
111#3963C2×C411yescharacteristic
112#3963C2×C411yescharacteristic
113#3963C2×C411yescharacteristic
114#3963C2×C411yescharacteristic
115#3963C2×C411yescharacteristic
116#3963C2×C411yescharacteristic
117#3952—22no—
118#3952Q814yescomplemented
119#3947—41no—
120#3946—22no—
121#3946—22no—
122#3946—41no—
123#3946—41no—
124C8×Dic5—41no—
125C4⋊#168—22no—
126C4⋊#168—22no—
127C4⋊#168D414yes—
128#3935—22no—
129#3935—22no—
130C4×#168—22no—
131C2×#446—51no—
132C2×#424—51no—
133#2396—51no—
134#1985D511yescharacteristic, complemented
135#1940—51no—
136#1939—51no—
137#1939—51no—
138#1917—51no—
139#1917—51no—
140#1851—51no—
141#1850—51no—
142#1771—51no—
143#1753—51no—
144#1752—51no—
145#1750—51no—
146C23×D5C22×C411yescharacteristic
147C2×C5×Q8—22no—
148C2×C5×Q8—41no—
149C2×C5×Q8—41no—
150C22×C4×C5C22×C411yescharacteristic
151C22×C4×C5C22×C411yescharacteristic
152C22×C4×C5C22×C411yescharacteristic
153C22×C4×C5C22×C411yescharacteristic
154C22×C4×C5C22×C411yescharacteristic
155C22×C4×C5C22×C411yescharacteristic
156C22×C4×C5C2411yescharacteristic
157C22×Dic5C22×C411yescharacteristic
158C22×Dic5C22×C411yescharacteristic
159C22×Dic5C22×C411yescharacteristic
160C22×Dic5C22×C411yescharacteristic
161C22×Dic5C22×C411yescharacteristic
162C22×Dic5C22×C411yescharacteristic
163C22×Dic5C22×C411yescharacteristic
164C2×C4×D5—21no—
165C2×C4×D5—21no—
166C2×C4×D5—21no—
167C2×C4×D5—21no—
168C2×C4×D5C2×Q812yes—
169C2×C4×D5#4112yes—
170C2×C5×C8—21no—
171C2×C5×C8—21no—
172C2×C5×C8—21no—
173C2×C5×C8—21no—
174C2×C5×C8—21no—
175C2×C5×C8—21no—
176C2×C5×C8—21no—
177C2×C5×C8—21no—
178C2×C5×C8—21no—
179C2×C5×C8—21no—
180C2×C5×C8—21no—
181C2×C5×C8C2×Q812yes—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—22no—
185C42×C5—21no—
186C42×C5—21no—
187C42×C5—21no—
188C42×C5—21no—
189C42×C5C2×D412yes—
190C42×C5#4112yes—
191#685—22no—
192#685—22no—
193#685—22no—
194#685—41no—
195#685—41no—
196#685—41no—
197C4⋊Dic5—22no—
198C4⋊Dic5—41no—
199#683—22no—
200#683—22no—
201#683—22no—
202#683—41no—
203#683—41no—
204#683—41no—
205C4×Dic5—21no—
206C4×Dic5—21no—
207C4×Dic5—21no—
208C4×Dic5—21no—
209C4×Dic5—21no—
210C4×Dic5—21no—
211C2×#168—21no—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168—21no—
215C2×#168—21no—
216C2×#168—21no—
217C2×#168—21no—
218C2×#168—21no—
219C2×#168—21no—
220C2×#168C2×D412yes—
221C2×#168C2×D412yes—
222C2×#168C2×Q812yes—
223#676—44no—
224C8×D5—44no—
225C22×M16—51no—
226C23×C8—51no—
227C2×C4×Q8C2×D511yescharacteristic
228C2×#117—51no—
229#446—104no—
230#424—104no—
231C2×C4⋊C8—51no—
232C2×C4⋊C8—51no—
233C2×C4⋊C8—51no—
234C2×C4⋊C8—51no—
235C2×C4⋊C8—51no—
236C2×C4⋊C8—51no—
237C2×C4⋊C8—51no—
238C2×C4⋊C8C2×D511yescharacteristic
239C2×C22⋊C8—51no—
240C2×C22⋊C8—51no—
241C2×#97—51no—
242C2×#97—51no—
243C2×#97—51no—
244C2×#97C2×D511yescharacteristic
245C2×C4×C8—51no—
246C2×C4×C8—51no—
247#400—51no—
248#399—51no—
249#398—51no—
250#398—51no—
251#396—51no—
252#396—51no—
253#395—51no—
254#393—51no—
255C4×C4⋊C4—51no—
256C4×C4⋊C4—51no—
257C4×C4⋊C4—51no—
258C4×C22⋊C4—51no—
259C4×C22⋊C4—51no—
260#336—51no—
261#336—51no—
262#336—51no—
263#336—51no—
264#336—51no—
265#336—51no—
266#336C2×D511yescharacteristic
267#336C2×D511yescharacteristic
268#336C2×D511yescharacteristic
269#336C2×D511yescharacteristic
270C23×C5C23×C411yescharacteristic
271C22×D5—21no—
272C22×D5—21no—
273C22×D5—21no—
274C22×D5—21no—
275C22×D5—21no—
276C22×D5—21no—
277C22×D5C4×Q812yes—
278C5×Q8—44no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—21no—
284C2×C4×C5—21no—
285C2×C4×C5—21no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—21no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5C4×D412yes—
296C2×C4×C5C4×D412yes—
297C2×C4×C5C22×D411yescharacteristic
298C2×C4×C5C22×Q811yescharacteristic
299C2×C4×C5C2×#4111yescharacteristic
300C2×C4×C5C2×#4111yescharacteristic
301C2×C4×C5H(5,𝔽2)11yescharacteristic
302C2×C4×C52−1+411yescharacteristic
303C2×Dic5—21no—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5—21no—
308C2×Dic5—21no—
309C2×Dic5—21no—
310C2×Dic5—21no—
311C2×Dic5—21no—
312C2×Dic5—21no—
313C2×Dic5—21no—
314C2×Dic5—21no—
315C2×Dic5—21no—
316C2×Dic5—21no—
317C2×Dic5—21no—
318C2×Dic5—21no—
319C2×Dic5—21no—
320C2×Dic5—21no—
321C2×Dic5—21no—
322C2×Dic5C4×Q812yes—
323C2×Dic5#13112yes—
324C4×D5—22no—
325C4×D5—22no—
326C4×D5—22no—
327C4×D5—22no—
328C5×C8—22no—
329C5×C8—41no—
330C5×C8—41no—
331C5×C8—41no—
332#168—22no—
333#168—22no—
334#168—22no—
335#168—41no—
336C22×Q8C4×D511yescharacteristic
337C23×C4—51no—
338C2×M16—102no—
339C2×M16—102no—
340C2×M16—201no—
341C22×C8—51no—
342C22×C8—51no—
343C22×C8—51no—
344C22×C8—51no—
345C22×C8—102no—
346C22×C8—102no—
347C22×C8—201no—
348C22×C8C22×D511yescharacteristic
349C22×C8C22×D511yescharacteristic
350C22×C8C22×D511yescharacteristic
351C22×C8C22×D511yescharacteristic
352C4×Q8—22no—
353C4×Q8—22no—
354#117—102no—
355#117—102no—
356C2×C4⋊C4—51no—
357C2×C4⋊C4—51no—
358C2×C4⋊C4—51no—
359C2×C4⋊C4—51no—
360C2×C4⋊C4—51no—
361C2×C4⋊C4—51no—
362C2×C4⋊C4—51no—
363C2×C4⋊C4—51no—
364C2×C4⋊C4C4×D511yescharacteristic
365C2×C4⋊C4C4×D511yescharacteristic
366C2×C4⋊C4C4×D511yescharacteristic
367C2×C22⋊C4—51no—
368C2×C22⋊C4—51no—
369C2×C22⋊C4—51no—
370C2×C22⋊C4—51no—
371C2×C22⋊C4—51no—
372C2×C22⋊C4—51no—
373C2×C42—51no—
374C2×C42—51no—
375C2×C42—51no—
376C2×C42C22×D511yescharacteristic
377C2×C42C22×D511yescharacteristic
378C2×C42C22×D511yescharacteristic
379C4⋊C8—22no—
380C4⋊C8—54no—
381C4⋊C8—102no—
382C4⋊C8—102no—
383C4⋊C8—102no—
384C4⋊C8—102no—
385C4⋊C8—201no—
386C4⋊C8—201no—
387C22⋊C8—54no—
388C22⋊C8—102no—
389#97—22no—
390#97—102no—
391#97—102no—
392#97—201no—
393C4×C8—102no—
394C4×C8—201no—
395#95—51no—
396#95—51no—
397#95—51no—
398#95—51no—
399#95—51no—
400#95—51no—
401#95—51no—
402#95—51no—
403#95—51no—
404#95—51no—
405C22×C5C2×C4×D411yescharacteristic
406C22×C5C2×C4×Q811yescharacteristic
407C22×C5#52011yescharacteristic
408C22×C5C2×#13111yescharacteristic
409C22×C5#56811yescharacteristic
410C22×C5#56811yescharacteristic
411C22×C5#56811yescharacteristic
412C2×D5—22no—
413C2×D5—22no—
414C2×D5—22no—
415C2×D5—41no—
416C2×D5—41no—
417C2×D5—41no—
418C2×D5—41no—
419C4×C5—22no—
420C4×C5—22no—
421C4×C5—41no—
422C4×C5—41no—
423C4×C5—41no—
424C4×C5—41no—
425C4×C5#54611yescharacteristic
426C4×C5D4×Q811yescharacteristic
427C4×C5#55211yescharacteristic
428C4×C5#55411yescharacteristic
429Dic5—22no—
430Dic5—22no—
431Dic5—41no—
432Dic5—41no—
433Dic5—41no—
434Dic5—41no—
435Dic5—41no—
436C24—51no—
437C2×Q8—22no—
438C2×Q8—41no—
439C2×Q8—41no—
440C22×C4—51no—
441C22×C4—51no—
442C22×C4—52no—
443C22×C4—52no—
444C22×C4—51no—
445C22×C4—51no—
446C22×C4—51no—
447C22×C4—51no—
448C22×C4—51no—
449C22×C4—101no—
450C22×C4—101no—
451C22×C4—101no—
452C22×C4—101no—
453C22×C4C2×C4×D511yescharacteristic
454C22×C4C2×C4×D511yescharacteristic
455C22×C4C2×C4×D511yescharacteristic
456C22×C4C2×C4×D511yescharacteristic
457C22×C4C2×C4×D511yescharacteristic
458C22×C4C2×C4×D511yescharacteristic
459C22×C4C23×D511yescharacteristic
460M16—204no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—52no—
473C2×C8—52no—
474C2×C8—52no—
475C2×C8—101no—
476C2×C8—101no—
477C2×C8—101no—
478C2×C8—101no—
479C2×C8—101no—
480C2×C8—101no—
481C2×C8—101no—
482C2×C8—101no—
483C2×C8—101no—
484C2×C8—204no—
485C2×C8Q8×D512yes—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—102no—
490C4⋊C4—102no—
491C4⋊C4—102no—
492C4⋊C4—102no—
493C4⋊C4—201no—
494C4⋊C4—201no—
495C4⋊C4—201no—
496C4⋊C4—201no—
497C22⋊C4—102no—
498C22⋊C4—102no—
499C22⋊C4—102no—
500C22⋊C4—201no—
501C22⋊C4—201no—
502C22⋊C4—201no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—101no—
508C42—101no—
509C42—101no—
510C42—101no—
511C42—101no—
512C42—101no—
513C42D4×D512yes—
514C42#71112yes—
515C2×C5#236011yescharacteristic
516C2×C5#293011yescharacteristic
517C2×C5#293511yescharacteristic
518C2×C5#296111yescharacteristic
519C2×C5#296211yescharacteristic
520C2×C5#299211yescharacteristic
521C2×C5#299511yescharacteristic
522D5—42no—
523C23—52no—
524C23—101no—
525C23—101no—
526C23—101no—
527C23—101no—
528C23—101no—
529C23—101no—
530C23C22×C4×D511yescharacteristic
531Q8—44no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—52no—
549C2×C4—52no—
550C2×C4—101no—
551C2×C4—101no—
552C2×C4—101no—
553C2×C4—101no—
554C2×C4—102no—
555C2×C4—102no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—101no—
559C2×C4—101no—
560C2×C4—101no—
561C2×C4—101no—
562C2×C4—101no—
563C2×C4—101no—
564C2×C4—101no—
565C2×C4—101no—
566C2×C4—101no—
567C2×C4—101no—
568C2×C4—101no—
569C2×C4—101no—
570C2×C4—101no—
571C2×C4—101no—
572C2×C4—101no—
573C2×C4#402712yes—
574C2×C4#402712yes—
575C2×C4C2×D4×D511yescharacteristic
576C2×C4C2×#71111yescharacteristic
577C2×C4C2×Q8×D511yescharacteristic
578C2×C4#414711yescharacteristic
579C2×C4D5×#4111yescharacteristic
580C2×C4#414911yescharacteristic
581C8—22no—
582C8—41no—
583C8—41no—
584C8—41no—
585C8—102no—
586C8—102no—
587C8—102no—
588C8—201no—
589C5#2296111yescharacteristic, complemented, Sylow
590C22—102no—
591C22—102no—
592C22—102no—
593C22—201no—
594C22—201no—
595C22—201no—
596C22—201no—
597C22C2×#402711yescharacteristic
598C22C4×Q8×D511yescharacteristic
599C22#6590811yescharacteristic
600C22#6607911yescharacteristic
601C22#6607911yescharacteristic
602C22D5×#13111yescharacteristic
603C22#6608411yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—102no—
611C4—102no—
612C4—201no—
613C4—201no—
614C4—201no—
615C4—201no—
616C4—201no—
617C4#6596911yescharacteristic
618C4#6597311yescharacteristic
619C4#6601011yescharacteristic
620C4#6606111yescharacteristic
621C2—202no—
622C2#1060834611yescharacteristic
623C2#1061035811yescharacteristic
624C2#1061039111yescharacteristic
625C2#1061167711yescharacteristic
626C2#1061168011yescharacteristic
627C2#1061169011yescharacteristic
628C2#1061169211yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D5
commutator, socleC23×C5C23×C4
FittingC5×#1985C2
prime radicalC23×D5C22×C4
cosocleC22×C4×C5C24
hypercenter#1985D5
hypocenterC5#22961
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. C2×#65313C2semidirect
4. C5×#1985C2semidirect
5. #10601261C2semidirect
7. #10600603C2semidirect
10. #10599449C2semidirect
11. #10599449C2semidirect
14. #10599411C2semidirect
15. #10599410C2semidirect
118. #3952Q8semidirect
134. #1985D5semidirect
589. C5#22961semidirect

Maximal subgroups

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

subgroupindex
#106142222
C2×#653132
C2×#651282
C5×#19852
#106012612
#106006232
#106006032
#105994602
#105994592
#105994492
#105994432
#105994412
#105994112
#105994102
#229615