Finite Group Databasebeta

← #34292642

Subgroups — #34292642

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
1C2×#19098664C211yesmaximal, characteristic
2#23766045C211yesmaximal, characteristic, complemented
3#23766037C211yesmaximal, characteristic
4#23764256C211yesmaximal, characteristic
5#23764138C211yesmaximal, characteristic, complemented
6#23764131C211yesmaximal, characteristic, complemented
7#23764094C211yesmaximal, characteristic
8#23764087C211yesmaximal, characteristic, complemented
9#23764086C211yesmaximal, characteristic
10#23764079C211yesmaximal, characteristic, complemented
11#23763299C211yesmaximal, characteristic, complemented
12#103×Dic79C211yesmaximal, characteristic, complemented
13#102×Dic79C211yesmaximal, characteristic
14#23763088C211yesmaximal, characteristic, complemented
15C79×#1973C211yesmaximal, characteristic
16C2×Q8×Dic79C2211yescharacteristic
17C2×D4×Dic79C2211yescharacteristic
18#19098664C418yescomplemented
19C2×#14579095C2211yescharacteristic
20C2×#14579093C2211yescharacteristic
21C2×#14579085C2211yescharacteristic
22C2×#14579085C2211yescharacteristic
23C2×#14579082C2211yescharacteristic
24C2×#14579082C2211yescharacteristic
25C2×#14579071C2211yescharacteristic
26C2×#14579071C2211yescharacteristic
27C2×C4×C79⋊C8C2211yescharacteristic
28C2×C8×Dic79C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30#19098470C2211yescharacteristic
31#19098466C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098461C2211yescharacteristic
34#19098459C2211yescharacteristic
35#19098459C2211yescharacteristic
36#19098457—21no—
37#19098451—21no—
38#19098448—21no—
39#19098446—21no—
40#19098430—21no—
41#19098427—21no—
42#19098424—21no—
43#19098421—21no—
44C4⋊C4×Dic79C2211yescharacteristic
45C4⋊C4×Dic79C2211yescharacteristic
46#19098409C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098325C2211yescharacteristic
53C22×SD16×C79C2211yescharacteristic
54C2×#107×C79C2211yescharacteristic
55C2×#103×C79C2211yescharacteristic
56C2×#102×C79C2211yescharacteristic
57C2×#97×C79C2211yescharacteristic
58#391×C79C2211yescharacteristic
59#390×C79C2211yescharacteristic
60C23×Dic79—21no—
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63Q8×Dic79—22no—
64Q8×Dic79C2×C414yes—
65D4×Dic79—22no—
66D4×Dic79C2×C414yes—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×Dic79⋊C4—21no—
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×C22⋊Dic79—21no—
77C2×C22⋊Dic79—21no—
78C2×C22⋊Dic79—21no—
79C2×C4×Dic79—21no—
80C2×C4×Dic79—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79C2311yescharacteristic
83C2×C4×Dic79C2311yescharacteristic
84#14579095C2×C414yes—
85#14579093—22no—
86#14579085—22no—
87#14579085C2×C414yes—
88#14579082—22no—
89#14579082C2×C414yes—
90#14579071—22no—
91#14579071C2×C414yes—
92C4×C79⋊C8—22no—
93C8×Dic79—22no—
94C8×Dic79—22no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95D411yescharacteristic
104C79⋊#95D411yescharacteristic
105C79⋊#95D411yescharacteristic
106C79⋊#95D411yescharacteristic
107C22×Q8×C79C2311yescharacteristic
108C22×D4×C79C2311yescharacteristic
109C2×SD16×C79—22no—
110C2×SD16×C79—22no—
111C2×SD16×C79C2×C414yes—
112C22×C8×C79C2311yescharacteristic
113C22×C8×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C22⋊C4×C79—21no—
117C2×C22⋊C4×C79—21no—
118C2×C42×C79C2311yescharacteristic
119#107×C79—22no—
120#103×C79—22no—
121#102×C79—22no—
122#97×C79—22no—
123#95×C79—21no—
124#95×C79—21no—
125C22×Dic79—24no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—81no—
131C22×Dic79—81no—
132C22×Dic79C2×D411yescharacteristic
133C22×Dic79C2×D411yescharacteristic
134C22×Dic79C2×D411yescharacteristic
135C22×Dic79C2×D411yescharacteristic
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8C22×C412yes—
141C2×C79⋊C8#4112yes—
142C4⋊Dic79—24no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79C22×C412yes—
150C4⋊Dic79C22×C412yes—
151Dic79⋊C4—81no—
152Dic79⋊C4—81no—
153Dic79⋊C4—81no—
154Dic79⋊C4—81no—
155C22⋊Dic79—24no—
156C22⋊Dic79—81no—
157C22⋊Dic79—81no—
158C4×Dic79—24no—
159C4×Dic79—21no—
160C4×Dic79—21no—
161C4×Dic79—81no—
162C4×Dic79—81no—
163C4×Dic79C22×C412yes—
164C4×Dic79#4112yes—
165C24×C79—21no—
166C2×Q8×C79—21no—
167C2×Q8×C79—21no—
168C2×Q8×C79—21no—
169C2×Q8×C79—21no—
170C2×Q8×C79—21no—
171C2×Q8×C79C22×C412yes—
172C2×D4×C79—21no—
173C2×D4×C79—21no—
174C2×D4×C79—21no—
175C2×D4×C79—21no—
176C2×D4×C79—21no—
177C2×D4×C79C22×C412yes—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2411yescharacteristic
184SD16×C79—28no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79C22×C412yes—
188C2×C8×C79#4112yes—
189C2×C8×C79#4112yes—
190C2×C8×C79#4112yes—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C22⋊C4×C79—81no—
196C22⋊C4×C79—81no—
197C42×C79—21no—
198C42×C79—21no—
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—22no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
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—41no—
229C2×C4×C79—41no—
230C2×C4×C79C23×C411yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—22no—
241C2×Dic79—22no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—21no—
248C2×Dic79—41no—
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—84no—
259C2×Dic79C4×D412yes—
260C2×Dic79C4×D412yes—
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—41no—
280C4×C79—41no—
281C4×C79—42no—
282C4×C79—81no—
283C4×C79—81no—
284C4×C79C2×#11711yescharacteristic
285C4×C79C4×#4111yescharacteristic
286C4×C79#52911yescharacteristic
287C4×C79#52911yescharacteristic
288Dic79—22no—
289Dic79—22no—
290Dic79—41no—
291Dic79—41no—
292Dic79—42no—
293Dic79—42no—
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×#440—791no—
306#2320—791no—
307#2319—791no—
308#1973D7911yescharacteristic
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×Q8—791no—
323C2×C4×D4—791no—
324#440—798no—
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#391C2×D7911yescharacteristic
342#390—791no—
343#390C2×D7911yescharacteristic
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—1581no—
368C2×SD16—22no—
369C2×SD16—22no—
370C2×SD16C4×D7914yes—
371C22×C8—791no—
372C22×C8—791no—
373C22×C8C22×D7911yescharacteristic
374C22×C8C22×D7911yescharacteristic
375C4×Q8—794no—
376C4×Q8—1582no—
377C4×D4—794no—
378C4×D4—1582no—
379C2×C4⋊C4—791no—
380C2×C4⋊C4—791no—
381C2×C4⋊C4—791no—
382C2×C4⋊C4—791no—
383C2×C4⋊C4—1581no—
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—21no—
392C2×C22⋊C4—1581no—
393C2×C22⋊C4—1581no—
394C2×C22⋊C4—1581no—
395C2×C42—791no—
396C2×C42—791no—
397C2×C42—1581no—
398C2×C42—1581no—
399C2×C42—1581no—
400C2×C42C22×D7911yescharacteristic
401#107—22no—
402#107—794no—
403#106—1582no—
404#103—22no—
405#103—794no—
406#103—1582no—
407#102—22no—
408#102—794no—
409#102—1582no—
410#97—22no—
411#97—794no—
412#97—1582no—
413C4×C8—1582no—
414C4×C8—1582no—
415C4×C8—1582no—
416#95—21no—
417#95—21no—
418#95—791no—
419#95—791no—
420#95—791no—
421#95—791no—
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×Q8C2×C4×D7912yes—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D7912yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—791no—
447C22×C4—791no—
448C22×C4—791no—
449C22×C4—791no—
450C22×C4—6321no—
451C22×C4—6321no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4—1584no—
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×C8C2×C4×D7912yes—
470C2×C8#1192962312yes—
471C2×C8#1192962312yes—
472C2×C8#1192962312yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—792no—
478C4⋊C4—792no—
479C4⋊C4—6321no—
480C4⋊C4—6321no—
481C4⋊C4—6321no—
482C4⋊C4—6321no—
483C4⋊C4—1584no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—6321no—
493C22⋊C4—6321no—
494C22⋊C4—1584no—
495C42—21no—
496C42—21no—
497C42—792no—
498C42—792no—
499C42—6321no—
500C42—6321no—
501C42—1584no—
502C42—1581no—
503C42—1581no—
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—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
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—792no—
546C2×C4—792no—
547C2×C4—6324no—
548C2×C4—1582no—
549C2×C4—1582no—
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×C4—1581no—
559C2×C4—1581no—
560C2×C4C22×C4×D7911yescharacteristic
561C2×C4#41×D7911yescharacteristic
562C2×C4#41×D7911yescharacteristic
563C2×C4C2×#1192962311yescharacteristic
564C2×C4C2×#1192962311yescharacteristic
565C2×C4C2×#1192962311yescharacteristic
566C8—22no—
567C8—1582no—
568C8—1582no—
569C8#1457911714yes—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C4×D4×D7911yescharacteristic
578C22#1909910811yescharacteristic
579C22#1909911111yescharacteristic
580C22#1909936711yescharacteristic
581C22#1909937211yescharacteristic
582C22#1909937311yescharacteristic
583C22#1909937911yescharacteristic
584C4—41no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—3162no—
590C4—3162no—
591C4—3161no—
592C4—3161no—
593C4—6321no—
594C4—6321no—
595C4—1582no—
596C4—1582no—
597C4C2×#1457911711yescharacteristic
598C4C4×#1192962311yescharacteristic
599C4#1909909211yescharacteristic
600C4#1909909411yescharacteristic
601C2—82no—
602C2#2376586211yescharacteristic
603C2#2377223111yescharacteristic
604C2#2377224011yescharacteristic
605C2#2377402311yescharacteristic
606C2#2377403411yescharacteristic
607C2#2377406511yescharacteristic
608C2#2377408011yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1973C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenter#1973D79
hypocenterC79#21291
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1973

Complements

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

NH ≅ G/Ntype
2. #23766045C2semidirect
5. #23764138C2semidirect
6. #23764131C2semidirect
8. #23764087C2semidirect
10. #23764079C2semidirect
11. #23763299C2semidirect
12. #103×Dic79C2semidirect
14. #23763088C2semidirect
18. #19098664C4semidirect
320. C79#21291semidirect

Maximal subgroups

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

subgroupindex
C2×#190986642
#237660452
#237660372
#237642562
#237641382
#237641312
#237640942
#237640872
#237640862
#237640792
#237632992
#103×Dic792
#102×Dic792
#237630882
C79×#19732
#2129179