Finite Group Databasebeta

← #34103509

Subgroups — #34103509

Order 20096.

1246 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D157C211yesmaximal, characteristic, complemented
2C2×#19097745C211yesmaximal, characteristic, complemented
3C2×#19097743C211yesmaximal, characteristic, complemented
4C2×#19097741C211yesmaximal, characteristic, complemented
5C2×#19097741C211yesmaximal, characteristic, complemented
6C22×#14578931C211yesmaximal, characteristic, complemented
7#23757191C211yesmaximal, characteristic, complemented
8#23757189C211yesmaximal, characteristic, complemented
9#23757183C211yesmaximal, characteristic, complemented
10#23757182C211yesmaximal, characteristic, complemented
11#23757151C211yesmaximal, characteristic, complemented
12#23757150C211yesmaximal, characteristic, complemented
13#23757147C211yesmaximal, characteristic, complemented
14#23757128C211yesmaximal, characteristic, complemented
15#386×C157C211yesmaximal, characteristic, complemented
16C24×D157C2211yescharacteristic
17C24×D157C2211yescharacteristic
18C22×D628C2211yescharacteristic
19C22×D628C2211yescharacteristic
20C22×D628C2211yescharacteristic, complemented
21C22×D628C2211yescharacteristic, complemented
22C22×C22⋊D157C2211yescharacteristic
23C22×C22⋊D157C2211yescharacteristic
24C22×C22⋊D157C2211yescharacteristic, complemented
25C22×C22⋊D157C2211yescharacteristic, complemented
26C2×D4×D157—24no—
27C2×D4×D157—24no—
28C2×D4×D157—24no—
29C23×Dic157C2211yescharacteristic, complemented
30C22×C4×D157C2211yescharacteristic
31#19097745—24no—
32#19097743—24no—
33#19097741—24no—
34#19097741—24no—
35C2×C4⋊Dic157C2211yescharacteristic, complemented
36C2×#14578933C2211yescharacteristic, complemented
37C2×C22⋊Dic157C2211yescharacteristic, complemented
38C2×#14578931—24no—
39C2×#14578931C2211yescharacteristic
40C2×#14578931C2211yescharacteristic
41C2×#14578931C2211yescharacteristic
42C2×#14578931C2211yescharacteristic, complemented
43C2×#14578931C2211yescharacteristic, complemented
44C2×#14578931C2211yescharacteristic, complemented
45C2×#14578931C2211yescharacteristic, complemented
46C2×#14578931C2211yescharacteristic, complemented
47C2×#14578931C2211yescharacteristic, complemented
48C2×#14578931C2211yescharacteristic, complemented
49C2×#14578931C2211yescharacteristic, complemented
50#19097678C2211yescharacteristic, complemented
51#19097678C2211yescharacteristic, complemented
52C22×D4×C157C2211yescharacteristic
53C23×C4×C157C2211yescharacteristic, complemented
54C2×C22⋊C4×C157C2211yescharacteristic, complemented
55C2×C22⋊C4×C157C2211yescharacteristic, complemented
56C2×C22⋊C4×C157C2211yescharacteristic, complemented
57#95×C157C2211yescharacteristic, complemented
58#95×C157C2211yescharacteristic, complemented
59C23×D157—22no—
60C23×D157—22no—
61C23×D157—22no—
62C23×D157—22no—
63C23×D157—22no—
64C23×D157—22no—
65C23×D157—22no—
66C23×D157—22no—
67C23×D157—22no—
68C23×D157—41no—
69C23×D157—41no—
70C23×D157—41no—
71C23×D157—41no—
72C23×D157D414yescomplemented
73C23×D157C2311yescharacteristic
74C23×D157C2311yescharacteristic
75C23×D157C2311yescharacteristic
76C23×D157C2311yescharacteristic
77C2×D628—22no—
78C2×D628—22no—
79C2×D628—22no—
80C2×D628—22no—
81C2×D628—22no—
82C2×D628—22no—
83C2×D628—22no—
84C2×D628—22no—
85C2×D628—22no—
86C2×D628—41no—
87C2×D628—41no—
88C2×D628—41no—
89C2×C22⋊D157—22no—
90C2×C22⋊D157—22no—
91C2×C22⋊D157—22no—
92C2×C22⋊D157—22no—
93C2×C22⋊D157—22no—
94C2×C22⋊D157—22no—
95C2×C22⋊D157—22no—
96C2×C22⋊D157—22no—
97C2×C22⋊D157—22no—
98C2×C22⋊D157—22no—
99C2×C22⋊D157—41no—
100C2×C22⋊D157D414yescomplemented
101D4×D157—232no—
102C22×Dic157—22no—
103C22×Dic157—22no—
104C22×Dic157—41no—
105C22×Dic157C2311yescharacteristic
106C22×Dic157C2311yescharacteristic
107C22×Dic157C2311yescharacteristic
108C22×Dic157C2311yescharacteristic
109C2×C4×D157—22no—
110C2×C4×D157—22no—
111C2×C4×D157—22no—
112C4⋊Dic157—22no—
113#14578933—22no—
114C22⋊Dic157—22no—
115#14578931—22no—
116#14578931—22no—
117#14578931—22no—
118#14578931—22no—
119#14578931—22no—
120#14578931—22no—
121#14578931—22no—
122#14578931—22no—
123#14578931—22no—
124#14578931—41no—
125#14578931D414yescomplemented
126C24×C157C2311yescharacteristic
127C24×C157C2311yescharacteristic
128C2×D4×C157—22no—
129C2×D4×C157—22no—
130C2×D4×C157D414yescomplemented
131C22×C4×C157—22no—
132C22×C4×C157—22no—
133C22×C4×C157—41no—
134C22×C4×C157C2311yescharacteristic
135C22×C4×C157C2311yescharacteristic
136C22×C4×C157C2311yescharacteristic
137C22×C4×C157C2311yescharacteristic
138C22×C4×C157C2311yescharacteristic
139C22⋊C4×C157—22no—
140C22⋊C4×C157—22no—
141C22⋊C4×C157—41no—
142C22×D157—21no—
143C22×D157—21no—
144C22×D157—21no—
145C22×D157—21no—
146C22×D157—28no—
147C22×D157—28no—
148C22×D157—21no—
149C22×D157—21no—
150C22×D157—21no—
151C22×D157—21no—
152C22×D157—21no—
153C22×D157—21no—
154C22×D157—21no—
155C22×D157—21no—
156C22×D157—21no—
157C22×D157—21no—
158C22×D157—21no—
159C22×D157—21no—
160C22×D157—28no—
161C22×D157—21no—
162C22×D157—21no—
163C22×D157—21no—
164C22×D157—21no—
165C22×D157—21no—
166C22×D157—21no—
167C22×D157—21no—
168C22×D157—44no—
169C22×D157—44no—
170C22×D157—44no—
171C22×D157—44no—
172C22×D157—44no—
173C22×D157—44no—
174C22×D157—44no—
175C22×D157—44no—
176C22×D157—44no—
177C22×D157—44no—
178C22×D157—44no—
179C22×D157C2×D412yes—
180C22×D157C2×D412yes—
181C22×D157C2×D412yes—
182C22×D157C2×D412yes—
183C22×D157#4112yes—
184D628—28no—
185D628—44no—
186D628—44no—
187D628—44no—
188C22⋊D157—28no—
189C22⋊D157—28no—
190C22⋊D157—44no—
191C22⋊D157—44no—
192C2×Dic157—21no—
193C2×Dic157—21no—
194C2×Dic157—21no—
195C2×Dic157—21no—
196C2×Dic157—21no—
197C2×Dic157—21no—
198C2×Dic157—21no—
199C2×Dic157—21no—
200C2×Dic157—21no—
201C2×Dic157—21no—
202C2×Dic157—44no—
203C2×Dic157C2×D412yes—
204C2×Dic157C2×D412yescomplemented
205C4×D157—28no—
206C23×C157—21no—
207C23×C157—21no—
208C23×C157—21no—
209C23×C157—21no—
210C23×C157—21no—
211C23×C157—21no—
212C23×C157—21no—
213C23×C157—21no—
214C23×C157—21no—
215C23×C157—21no—
216C23×C157C2×D412yes—
217C23×C157C2×D412yes—
218C23×C157C2×D412yes—
219C23×C157C2×D412yes—
220C23×C157C2411yescharacteristic
221D4×C157—28no—
222C2×C4×C157—21no—
223C2×C4×C157—21no—
224C2×C4×C157—21no—
225C2×C4×C157—21no—
226C2×C4×C157—21no—
227C2×C4×C157—21no—
228C2×C4×C157—21no—
229C2×C4×C157—21no—
230C2×C4×C157—21no—
231C2×C4×C157—21no—
232C2×C4×C157—21no—
233C2×C4×C157—21no—
234C2×C4×C157—21no—
235C2×C4×C157—44no—
236C2×C4×C157C2×D412yes—
237C2×C4×C157C2×D412yescomplemented
238C22×C157—22no—
239C22×C157—22no—
240C22×C157—22no—
241C22×C157—22no—
242C22×C157—22no—
243C22×C157—22no—
244C22×C157—22no—
245C22×C157—22no—
246C22×C157—22no—
247C22×C157—41no—
248C22×C157—41no—
249C22×C157—41no—
250C22×C157—41no—
251C22×C157#12014yescomplemented
252C22×C157C22×D411yescharacteristic
253C22×C157C22×D411yescharacteristic
254C22×C157C22×D411yescharacteristic
255C22×C157C22×D411yescharacteristic
256C22×C157C2×#4111yescharacteristic
257C22×C157H(5,𝔽2)11yescharacteristic
258C22×C157H(5,𝔽2)11yescharacteristic
259C2×D157—22no—
260C2×D157—22no—
261C2×D157—22no—
262C2×D157—22no—
263C2×D157—22no—
264C2×D157—22no—
265C2×D157—22no—
266C2×D157—22no—
267C2×D157—22no—
268C2×D157—22no—
269C2×D157—22no—
270C2×D157—22no—
271C2×D157—416no—
272C2×D157—416no—
273C2×D157—41no—
274C2×D157—41no—
275C2×D157—41no—
276C2×D157—41no—
277C2×D157—41no—
278C2×D157—41no—
279C2×D157—41no—
280C2×D157—41no—
281C2×D157—41no—
282C2×D157—41no—
283C2×D157—41no—
284C2×D157—41no—
285C2×D157—41no—
286C2×D157—41no—
287C2×D157—41no—
288C2×D157#12114yescomplemented
289C4×C157—22no—
290C4×C157—22no—
291C4×C157—41no—
292C4×C157—41no—
293C4×C157—41no—
294Dic157—22no—
295Dic157—22no—
296Dic157—41no—
297Dic157—41no—
298C2×C157—24no—
299C2×C157—42no—
300C2×C157C2×#12011yescharacteristic
301C2×C157C2×#12111yescharacteristic
302C2×C157#53411yescharacteristic
303C2×C157D4211yescharacteristic
304C2×C157D4211yescharacteristic
305C2×C157#54611yescharacteristic
306C2×C157#54611yescharacteristic
307D157—24no—
308D157—42no—
309D157—42no—
310D157—81no—
311C157#241011yescharacteristic, complemented, Sylow
312#2410—1571nomaximal, Sylow
313C23×D4—1571no—
314C2×#121—1571no—
315C2×#121—1571no—
316C2×#120—1571no—
317C2×#120—1571no—
318C22×C22⋊C4—1571no—
319#394—1571no—
320#394—1571no—
321#392—1571no—
322#392—1571no—
323#386—1571no—
324#386—1571no—
325#386D15711yescharacteristic, complemented
326#385—1571no—
327#379—1571no—
328C25—1571no—
329C25—1571no—
330C22×D4—3144no—
331C22×D4—3144no—
332C22×D4—3144no—
333C22×D4—1571no—
334C22×D4—1571no—
335C22×D4—1571no—
336C22×D4—1571no—
337C22×D4—1571no—
338C22×D4—1571no—
339C22×D4—1571no—
340C22×D4—1571no—
341C22×D4C2×D15711yescharacteristic
342C23×C4—1571no—
343C23×C4—1571no—
344C23×C4C2×D15711yescharacteristic, complemented
345#121—3144no—
346#121—3144no—
347#120—3144no—
348#120—3144no—
349C2×C4⋊C4—1571no—
350C2×C4⋊C4—1571no—
351C2×C22⋊C4—3144no—
352C2×C22⋊C4—1571no—
353C2×C22⋊C4—1571no—
354C2×C22⋊C4—1571no—
355C2×C22⋊C4—1571no—
356C2×C22⋊C4—1571no—
357C2×C22⋊C4—1571no—
358C2×C22⋊C4—1571no—
359C2×C22⋊C4—1571no—
360C2×C22⋊C4—1571no—
361C2×C22⋊C4—1571no—
362C2×C22⋊C4—1571no—
363C2×C22⋊C4—1571no—
364C2×C22⋊C4C2×D15711yescharacteristic, complemented
365C2×C22⋊C4C2×D15711yescharacteristic, complemented
366C2×C22⋊C4C2×D15711yescharacteristic, complemented
367#95—1571no—
368#95—1571no—
369#95C2×D15711yescharacteristic, complemented
370#95C2×D15711yescharacteristic, complemented
371C24—3142no—
372C24—3142no—
373C24—3142no—
374C24—3142no—
375C24—3142no—
376C24—3142no—
377C24—3142no—
378C24—3142no—
379C24—3142no—
380C24—6281no—
381C24—6281no—
382C24—6281no—
383C24—6281no—
384C24—1571no—
385C24—1571no—
386C24—1571no—
387C24—1574no—
388C24—1571no—
389C24C22×D15711yescharacteristic
390C24C22×D15711yescharacteristic
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—3142no—
394C2×D4—3142no—
395C2×D4—3142no—
396C2×D4—3142no—
397C2×D4—3142no—
398C2×D4—3142no—
399C2×D4—3142no—
400C2×D4—31432no—
401C2×D4—3142no—
402C2×D4—3142no—
403C2×D4—3142no—
404C2×D4—3142no—
405C2×D4—3142no—
406C2×D4—3142no—
407C2×D4—3142no—
408C2×D4—3142no—
409C2×D4—3142no—
410C2×D4—3142no—
411C2×D4—3142no—
412C2×D4—3142no—
413C2×D4—6281no—
414C2×D4—6281no—
415C2×D4—6281no—
416C2×D4—6281no—
417C2×D4—1574no—
418C2×D4D62814yescomplemented
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—41no—
422C22×C4—3142no—
423C22×C4—3142no—
424C22×C4—3142no—
425C22×C4—3142no—
426C22×C4—3142no—
427C22×C4—6281no—
428C22×C4—1571no—
429C22×C4—1571no—
430C22×C4—1571no—
431C22×C4—1571no—
432C22×C4C22×D15711yescharacteristic
433C22×C4C22×D15711yescharacteristic
434C22×C4C22×D15711yescharacteristic
435C22×C4C22×D15711yescharacteristic
436C22×C4C22×D15711yescharacteristic
437C4⋊C4—3142no—
438C4⋊C4—3142no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—41no—
442C22⋊C4—3142no—
443C22⋊C4—3142no—
444C22⋊C4—3142no—
445C22⋊C4—3142no—
446C22⋊C4—3142no—
447C22⋊C4—3142no—
448C22⋊C4—3142no—
449C22⋊C4—3142no—
450C22⋊C4—3142no—
451C22⋊C4—3142no—
452C22⋊C4—6281no—
453C22⋊C4—1574no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—3148no—
465C23—3148no—
466C23—3141no—
467C23—3141no—
468C23—3141no—
469C23—3141no—
470C23—3148no—
471C23—3141no—
472C23—3141no—
473C23—3141no—
474C23—3141no—
475C23—3141no—
476C23—3141no—
477C23—3141no—
478C23—3141no—
479C23—3141no—
480C23—3141no—
481C23—3141no—
482C23—3141no—
483C23—3141no—
484C23—3141no—
485C23—3141no—
486C23—3141no—
487C23—3141no—
488C23—3141no—
489C23—3141no—
490C23—6284no—
491C23—6284no—
492C23—6284no—
493C23—6284no—
494C23—6284no—
495C23—6284no—
496C23—6284no—
497C23—6284no—
498C23—6284no—
499C23—6284no—
500C23—6284no—
501C23—1572no—
502C23—1572no—
503C23—1572no—
504C23—1572no—
505C23—1572no—
506C23D4×D15712yes—
507C23D4×D15712yes—
508C23C2×D62812yes—
509C23C2×D62812yes—
510C23C23×D15711yescharacteristic
511D4—28no—
512D4—3148no—
513D4—3148no—
514D4—3148no—
515D4—6284no—
516D4—6284no—
517D4—6284no—
518D4—6284no—
519D4—6284no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—44no—
534C2×C4—3148no—
535C2×C4—3141no—
536C2×C4—3141no—
537C2×C4—3141no—
538C2×C4—3141no—
539C2×C4—3141no—
540C2×C4—3141no—
541C2×C4—3141no—
542C2×C4—3141no—
543C2×C4—3141no—
544C2×C4—3141no—
545C2×C4—6284no—
546C2×C4—1572no—
547C2×C4—1572no—
548C2×C4D4×D15712yescomplemented
549C2×C4C2×D62812yes—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—3142no—
564C22—3142no—
565C22—3142no—
566C22—3142no—
567C22—3142no—
568C22—3142no—
569C22—3142no—
570C22—3142no—
571C22—3142no—
572C22—3142no—
573C22—3142no—
574C22—3142no—
575C22—62816no—
576C22—62816no—
577C22—6281no—
578C22—6281no—
579C22—6281no—
580C22—6281no—
581C22—6281no—
582C22—6281no—
583C22—6281no—
584C22—6281no—
585C22—6281no—
586C22—6281no—
587C22—6281no—
588C22—6281no—
589C22—6281no—
590C22—6281no—
591C22—6281no—
592C22—1574no—
593C22#1909774114yescomplemented
594C22C2×D4×D15711yescharacteristic
595C22C2×D4×D15711yescharacteristic
596C22C2×D4×D15711yescharacteristic
597C22C22×D62811yescharacteristic
598C22#41×D15711yescharacteristic
599C22#1909781611yescharacteristic
600C22#1909781711yescharacteristic
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—3142no—
607C4—3142no—
608C4—6281no—
609C4—6281no—
610C2—24no—
611C2—42no—
612C2—3144no—
613C2—6282no—
614C2—6282no—
615C2—12561no—
616C2C2×#1909774111yescharacteristic
617C2#121×D15711yescharacteristic
618C2#2375783111yescharacteristic
619C2#2375790511yescharacteristic
620C2D4×D62811yescharacteristic
621C2#2375791511yescharacteristic
622C2#2375791811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#386×C157C2
hypercenter#386D157
hypocenterC157#2410
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×D4×D157C2semidirect
2. C2×#19097745C2semidirect
3. C2×#19097743C2semidirect
4. C2×#19097741C2semidirect
5. C2×#19097741C2semidirect
6. C22×#14578931C2semidirect
7. #23757191C2semidirect
8. #23757189C2semidirect
9. #23757183C2semidirect
10. #23757182C2semidirect
11. #23757151C2semidirect
12. #23757150C2semidirect
13. #23757147C2semidirect
14. #23757128C2semidirect
15. #386×C157C2semidirect
20. C22×D628C22semidirect
21. C22×D628C22semidirect
24. C22×C22⋊D157C22semidirect
25. C22×C22⋊D157C22semidirect
29. C23×Dic157C22semidirect
35. C2×C4⋊Dic157C22semidirect
36. C2×#14578933C22semidirect
37. C2×C22⋊Dic157C22semidirect
42. C2×#14578931C22semidirect
43. C2×#14578931C22semidirect
44. C2×#14578931C22semidirect
45. C2×#14578931C22semidirect
46. C2×#14578931C22semidirect
47. C2×#14578931C22semidirect
48. C2×#14578931C22semidirect
49. C2×#14578931C22semidirect
50. #19097678C22semidirect
51. #19097678C22semidirect
53. C23×C4×C157C22semidirect
54. C2×C22⋊C4×C157C22semidirect
55. C2×C22⋊C4×C157C22semidirect
56. C2×C22⋊C4×C157C22semidirect
57. #95×C157C22semidirect
58. #95×C157C22semidirect
72. C23×D157D4semidirect
100. C2×C22⋊D157D4semidirect
125. #14578931D4semidirect
130. C2×D4×C157D4semidirect
204. C2×Dic157C2×D4semidirect
237. C2×C4×C157C2×D4semidirect
251. C22×C157#120semidirect
288. C2×D157#121semidirect
311. C157#2410semidirect
325. #386D157semidirect
344. C23×C4C2×D157semidirect
364. C2×C22⋊C4C2×D157semidirect
365. C2×C22⋊C4C2×D157semidirect
366. C2×C22⋊C4C2×D157semidirect
369. #95C2×D157semidirect
370. #95C2×D157semidirect
418. C2×D4D628semidirect
548. C2×C4D4×D157semidirect
593. C22#19097741semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1572
C2×#190977452
C2×#190977432
C2×#190977412
C22×#145789312
#237571912
#237571892
#237571832
#237571822
#237571512
#237571502
#237571472
#237571282
#386×C1572
#2410157