Finite Group Databasebeta

← #34292649

Subgroups — #34292649

Order 20224.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766051C211yesmaximal, characteristic, complemented
2#23766043C211yesmaximal, characteristic
3#23764283C211yesmaximal, characteristic
4#23764256C211yesmaximal, characteristic
5#23764139C211yesmaximal, characteristic, complemented
6#23764134C211yesmaximal, characteristic, complemented
7#23764099C211yesmaximal, characteristic
8#23764092C211yesmaximal, characteristic, complemented
9#23764085C211yesmaximal, characteristic
10#23764078C211yesmaximal, characteristic, complemented
11#23763297C211yesmaximal, characteristic, complemented
12C4×#14579085C211yesmaximal, characteristic, complemented
13C4×#14579082C211yesmaximal, characteristic
14#23763088C211yesmaximal, characteristic, complemented
15C2×#440×C79C211yesmaximal, characteristic
16C2×#14579095C2211yescharacteristic
17C2×#14579093C2211yescharacteristic
18C2×#14579085C2211yescharacteristic
19C2×#14579085C2211yescharacteristic
20C2×#14579082C2211yescharacteristic
21C2×#14579082C2211yescharacteristic
22C2×#14579071C2211yescharacteristic
23C2×#14579071C2211yescharacteristic
24C2×C4×C79⋊C8C2211yescharacteristic
25C2×C8×Dic79C2211yescharacteristic
26C2×C8×Dic79C2211yescharacteristic
27#19098470C2211yescharacteristic
28#19098470C2211yescharacteristic
29#19098466C2211yescharacteristic
30#19098466C2211yescharacteristic
31#19098462C2211yescharacteristic
32#19098462C2211yescharacteristic
33#19098462C2211yescharacteristic
34#19098462C2211yescharacteristic
35#19098456—21no—
36#19098456—21no—
37#19098449—21no—
38#19098449—21no—
39#19098431—21no—
40#19098429—21no—
41#19098426—21no—
42#19098423—21no—
43C4×C4⋊Dic79C2211yescharacteristic
44C4×C4⋊Dic79C2211yescharacteristic
45#19098409C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52C22×SD16×C79C2211yescharacteristic
53C2×C4×Q8×C79C2211yescharacteristic
54C2×C4×D4×C79C2211yescharacteristic
55#440×C79C418yescomplemented
56C2×#107×C79C2211yescharacteristic
57C2×#103×C79C2211yescharacteristic
58C2×#102×C79C2211yescharacteristic
59C2×#97×C79C2211yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C2×C4⋊Dic79—21no—
63C2×C4⋊Dic79—21no—
64C2×C4⋊Dic79—21no—
65C2×C4⋊Dic79—21no—
66C2×C4⋊Dic79C2311yescharacteristic
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C22⋊Dic79—21no—
71C2×C22⋊Dic79—21no—
72C2×C22⋊Dic79—21no—
73C2×C22⋊Dic79—21no—
74C2×C4×Dic79—21no—
75C2×C4×Dic79—21no—
76C2×C4×Dic79C2311yescharacteristic
77C2×C4×Dic79C2311yescharacteristic
78#14579095—22no—
79#14579093—22no—
80#14579085—22no—
81#14579085—22no—
82#14579082—22no—
83#14579082—22no—
84#14579071—22no—
85#14579071—22no—
86C4×C79⋊C8—22no—
87C8×Dic79—22no—
88C8×Dic79—22no—
89C79⋊#95—21no—
90C79⋊#95—21no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95D411yescharacteristic
100C79⋊#95D411yescharacteristic
101C79⋊#95D411yescharacteristic
102C79⋊#95D411yescharacteristic
103C22×Q8×C79C2311yescharacteristic
104C22×D4×C79C2311yescharacteristic
105C23×C4×C79—21no—
106C2×SD16×C79—22no—
107C2×SD16×C79—22no—
108C2×SD16×C79C2×C414yes—
109C22×C8×C79C2311yescharacteristic
110C22×C8×C79C2311yescharacteristic
111C4×Q8×C79—22no—
112C4×Q8×C79C2×C414yes—
113C4×D4×C79—22no—
114C4×D4×C79C2×C414yes—
115C2×C4⋊C4×C79—21no—
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C22⋊C4×C79—21no—
119C2×C42×C79—21no—
120C2×C42×C79C2311yescharacteristic
121#107×C79C2×C414yes—
122#103×C79C2×C414yes—
123#102×C79C2×C414yes—
124#97×C79C2×C414yes—
125C22×Dic79—21no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79C2×D411yescharacteristic
130C22×Dic79C2×D411yescharacteristic
131C22×Dic79C2×D411yescharacteristic
132C22×Dic79C2×D411yescharacteristic
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8#4112yes—
138C2×C79⋊C8#4112yes—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—81no—
148C4⋊Dic79—81no—
149C4⋊Dic79—81no—
150C4⋊Dic79—81no—
151C22⋊Dic79—81no—
152C22⋊Dic79—81no—
153C22⋊Dic79—81no—
154C22⋊Dic79—81no—
155C4×Dic79—21no—
156C4×Dic79—21no—
157C4×Dic79—21no—
158C4×Dic79—21no—
159C4×Dic79—81no—
160C4×Dic79—81no—
161C24×C79—21no—
162C2×Q8×C79—21no—
163C2×Q8×C79—21no—
164C2×Q8×C79—21no—
165C2×Q8×C79—21no—
166C2×Q8×C79—21no—
167C2×Q8×C79C22×C412yes—
168C2×D4×C79—21no—
169C2×D4×C79—21no—
170C2×D4×C79—21no—
171C2×D4×C79—21no—
172C2×D4×C79—21no—
173C2×D4×C79C22×C412yes—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—24no—
178C22×C4×C79—81no—
179C22×C4×C79—81no—
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2411yescharacteristic
183SD16×C79—28no—
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79C22×C412yes—
187C2×C8×C79C22×C412yes—
188C2×C8×C79#4112yes—
189C2×C8×C79#4112yes—
190C4⋊C4×C79—24no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79C22×C412yes—
194C4⋊C4×C79C22×C412yes—
195C22⋊C4×C79—24no—
196C42×C79—24no—
197C42×C79C22×C412yes—
198C42×C79#4112yes—
199C23×C79—22no—
200C23×C79—41no—
201C23×C79—41no—
202C23×C79—41no—
203C23×C79—41no—
204C23×C79—41no—
205C23×C79—41no—
206C23×C79C22×D411yescharacteristic
207Q8×C79—22no—
208Q8×C79—22no—
209Q8×C79—41no—
210Q8×C79—41no—
211D4×C79—22no—
212D4×C79—22no—
213D4×C79—41no—
214D4×C79—41no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—22no—
220C2×C4×C79—22no—
221C2×C4×C79—22no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—84no—
229C2×C4×C79C4×D412yes—
230C2×C4×C79C4×D412yes—
231C2×C4×C79C23×C411yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—21no—
248C2×Dic79—21no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C8×C79—22no—
262C8×C79#11714yes—
263C79⋊C8—22no—
264C79⋊C8—22no—
265C22×C79—42no—
266C22×C79—42no—
267C22×C79—42no—
268C22×C79—81no—
269C22×C79—81no—
270C22×C79—81no—
271C22×C79—81no—
272C22×C79C2×C4×D411yescharacteristic
273C22×C79#53211yescharacteristic
274C22×C79#53211yescharacteristic
275C22×C79#57611yescharacteristic
276C22×C79#57711yescharacteristic
277C22×C79#57711yescharacteristic
278C22×C79#57811yescharacteristic
279C4×C79—22no—
280C4×C79—22no—
281C4×C79—42no—
282C4×C79—42no—
283C4×C79—42no—
284C4×C79C2×#11711yescharacteristic
285C4×C79C4×#4111yescharacteristic
286C4×C79#52911yescharacteristic
287C4×C79#52911yescharacteristic
288Dic79—41no—
289Dic79—41no—
290Dic79—41no—
291Dic79—41no—
292Dic79—81no—
293Dic79—81no—
294Dic79—81no—
295Dic79—81no—
296#21291—791nomaximal, Sylow
297C2×C79—82no—
298C2×C79#229311yescharacteristic
299C2×C79#294611yescharacteristic
300C2×C79#294711yescharacteristic
301C2×C79#317411yescharacteristic
302C2×C79#317511yescharacteristic
303C2×C79#317811yescharacteristic
304C2×C79#317911yescharacteristic
305C2×#440D7911yescharacteristic
306#2320—791no—
307#2319—791no—
308#1973—791no—
309#1968—791no—
310#1943—791no—
311#1942—791no—
312#1936—791no—
313#1935—791no—
314#1934—791no—
315#1933—791no—
316#1776—791no—
317C4×#103—791no—
318C4×#102—791no—
319#1726—791no—
320C79#2129111yescharacteristic, complemented, Sylow
321C22×SD16C2×D7911yescharacteristic
322C2×C4×Q8C2×D7911yescharacteristic
323C2×C4×D4C2×D7911yescharacteristic
324#440Dic7918yescomplemented
325C2×#107—791no—
326C2×#107C2×D7911yescharacteristic
327C2×#106—791no—
328C2×#103—791no—
329C2×#103—791no—
330C2×#103C2×D7911yescharacteristic
331C2×#102—791no—
332C2×#102—791no—
333C2×#102C2×D7911yescharacteristic
334C2×#97—791no—
335C2×#97—791no—
336C2×#97C2×D7911yescharacteristic
337C2×C4×C8—791no—
338C2×C4×C8—791no—
339C2×C4×C8—791no—
340#391—791no—
341#391—791no—
342#390—791no—
343#390—791no—
344#389—791no—
345#389—791no—
346#389—791no—
347#389—791no—
348#388—1581no—
349#388—1581no—
350#387—1581no—
351#387—1581no—
352#383—1581no—
353#382—1581no—
354#381—1581no—
355#380—1581no—
356C4×C4⋊C4—791no—
357C4×C4⋊C4—791no—
358#376—791no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365C22×Q8C22×D7911yescharacteristic
366C22×D4C22×D7911yescharacteristic
367C23×C4—21no—
368C2×SD16—22no—
369C2×SD16—22no—
370C2×SD16C2×Dic7914yes—
371C22×C8—791no—
372C22×C8—791no—
373C22×C8C22×D7911yescharacteristic
374C22×C8C22×D7911yescharacteristic
375C4×Q8—22no—
376C4×Q8C2×Dic7914yes—
377C4×D4—22no—
378C4×D4C2×Dic7914yes—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—791no—
381C2×C4⋊C4—791no—
382C2×C4⋊C4—791no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—1581no—
385C2×C4⋊C4—1581no—
386C2×C4⋊C4—1581no—
387C2×C4⋊C4—1581no—
388C2×C4⋊C4C22×D7911yescharacteristic
389C2×C4⋊C4C22×D7911yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1581no—
392C2×C22⋊C4—1581no—
393C2×C22⋊C4—1581no—
394C2×C22⋊C4—1581no—
395C2×C42—21no—
396C2×C42—791no—
397C2×C42—791no—
398C2×C42—1581no—
399C2×C42—1581no—
400C2×C42C22×D7911yescharacteristic
401#107—1582no—
402#107C2×Dic7914yes—
403#106—1582no—
404#103—1582no—
405#103—1582no—
406#103C2×Dic7914yes—
407#102—1582no—
408#102—1582no—
409#102C2×Dic7914yes—
410#97—1582no—
411#97—1582no—
412#97C2×Dic7914yes—
413C4×C8—1582no—
414C4×C8—1582no—
415C4×C8—1582no—
416#95—791no—
417#95—791no—
418#95—791no—
419#95—791no—
420#95—1581no—
421#95—1581no—
422#95—1581no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428#95—1581no—
429#95—1581no—
430C24—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C22×Dic7912yes—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C22×Dic7912yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—791no—
450C22×C4—791no—
451C22×C4—791no—
452C22×C4—791no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4—1581no—
457C22×C4D4×D7911yescharacteristic
458C22×C4D4×D7911yescharacteristic
459C22×C4C23×D7911yescharacteristic
460SD16—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—792no—
464C2×C8—792no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8C22×Dic7912yes—
470C2×C8C22×Dic7912yes—
471C2×C8#1192962412yes—
472C2×C8#1192962412yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—6321no—
477C4⋊C4—6321no—
478C4⋊C4—6321no—
479C4⋊C4—6321no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4C22×Dic7912yes—
489C4⋊C4C22×Dic7912yes—
490C22⋊C4—24no—
491C22⋊C4—6321no—
492C22⋊C4—6321no—
493C22⋊C4—6321no—
494C22⋊C4—6321no—
495C42—24no—
496C42—6321no—
497C42—6321no—
498C42—1581no—
499C42—1581no—
500C42—1581no—
501C42—1581no—
502C42C22×Dic7912yes—
503C42#1192962312yes—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D7911yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—84no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—1581no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4—1581no—
554C2×C4—1581no—
555C2×C4—1581no—
556C2×C4—1581no—
557C2×C4—1581no—
558C2×C4D4×Dic7912yes—
559C2×C4D4×Dic7912yes—
560C2×C4C23×Dic7911yescharacteristic
561C2×C4#41×D7911yescharacteristic
562C2×C4#41×D7911yescharacteristic
563C2×C4C2×#1192962311yescharacteristic
564C2×C4C2×#1192962411yescharacteristic
565C2×C4C2×#1192962411yescharacteristic
566C8—22no—
567C8—1582no—
568C8—1582no—
569C8#1457912014yes—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C2×D4×Dic7911yescharacteristic
578C22#1909911011yescharacteristic
579C22#1909911011yescharacteristic
580C22#1909936711yescharacteristic
581C22#1909937211yescharacteristic
582C22#1909937311yescharacteristic
583C22#1909937911yescharacteristic
584C4—22no—
585C4—22no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—3161no—
590C4—3161no—
591C4—3161no—
592C4—3161no—
593C4—6321no—
594C4—6321no—
595C4—6321no—
596C4—6321no—
597C4C2×#1457912011yescharacteristic
598C4#41×Dic7911yescharacteristic
599C4#1909909511yescharacteristic
600C4#1909909511yescharacteristic
601C2—82no—
602C2#2376586511yescharacteristic
603C2#2377223311yescharacteristic
604C2#2377224411yescharacteristic
605C2#2377402611yescharacteristic
606C2#2377403711yescharacteristic
607C2#2377406611yescharacteristic
608C2#2377408111yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#440×C79C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenterC2×#440D79
hypocenterC79#21291
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #23766051C2semidirect
5. #23764139C2semidirect
6. #23764134C2semidirect
8. #23764092C2semidirect
10. #23764078C2semidirect
11. #23763297C2semidirect
12. C4×#14579085C2semidirect
14. #23763088C2semidirect
55. #440×C79C4semidirect
320. C79#21291semidirect
324. #440Dic79semidirect

Maximal subgroups

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

subgroupindex
#237660512
#237660432
#237642832
#237642562
#237641392
#237641342
#237640992
#237640922
#237640852
#237640782
#237632972
C4×#145790852
C4×#145790822
#237630882
C2×#440×C792
#2129179