Finite Group Databasebeta

← #34340429

Subgroups — #34340429

Order 20224.

648 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773953C211yesmaximal, characteristic, complemented
2C2×#19098863C211yesmaximal, characteristic
3#23772349C211yesmaximal, characteristic, complemented
4#23764913C211yesmaximal, characteristic, complemented
5#23764908C211yesmaximal, characteristic, complemented
6#23764760C211yesmaximal, characteristic
7#23764760C211yesmaximal, characteristic
8#23764392C211yesmaximal, characteristic, complemented
9#23764182C211yesmaximal, characteristic, complemented
10#23764177C211yesmaximal, characteristic, complemented
11#23764173C211yesmaximal, characteristic, complemented
12#23764065C211yesmaximal, characteristic
13#23764065C211yesmaximal, characteristic
14C4×#14579078C211yesmaximal, characteristic
15C79×#3077C211yesmaximal, characteristic
16C2×#14579153C2211yescharacteristic
17#19098969—21no—
18#19098969—21no—
19#19098962—21no—
20#19098962—21no—
21#19098863—21no—
22#19098863—21no—
23#19098863—21no—
24#19098863—21no—
25#19098694—21no—
26#19098694—21no—
27#19098694—21no—
28Q8×C79⋊C8—21no—
29C2×#14579096C2211yescharacteristic
30C2×#14579096C2211yescharacteristic
31#19098606C2211yescharacteristic
32#19098606C2211yescharacteristic
33#19098606C2211yescharacteristic
34#19098606C2211yescharacteristic
35C2×#14579094C2211yescharacteristic
36C2×#14579092C2211yescharacteristic
37#19098590C2211yescharacteristic
38#19098590C2211yescharacteristic
39C2×C4⋊(C79⋊C8)C2211yescharacteristic
40C2×#14579082C2211yescharacteristic
41C2×#14579082C2211yescharacteristic
42C2×#14579078C2211yescharacteristic
43C2×#14579078C2211yescharacteristic
44#19098519C2211yescharacteristic
45#19098519C2211yescharacteristic
46C4×#11929608C2211yescharacteristic
47C4×#11929608C2211yescharacteristic
48#19098332C2211yescharacteristic
49#19098332C2211yescharacteristic
50#19098331C2211yescharacteristic
51#19098331C2211yescharacteristic
52#19098331C2211yescharacteristic
53#19098331C2211yescharacteristic
54#19098327C2211yescharacteristic
55#19098327C2211yescharacteristic
56#533×C79C2211yescharacteristic
57C2×C4×D4×C79C2211yescharacteristic
58#478×C79—21no—
59#478×C79—21no—
60#476×C79—21no—
61#474×C79—21no—
62#428×C79C2211yescharacteristic
63#428×C79C2211yescharacteristic
64#423×C79C2211yescharacteristic
65#418×C79C2211yescharacteristic
66#418×C79C2211yescharacteristic
67C2×#11929608C2311yescharacteristic
68C2×#11929608C2311yescharacteristic
69C2×#11929608C2311yescharacteristic
70C2×#11929608C2311yescharacteristic
71C22×C79⋊C8C2311yescharacteristic
72C22×C79⋊C8C2311yescharacteristic
73#14579153—21no—
74#14579153—21no—
75#14579153—21no—
76#14579153—21no—
77C2×C4⋊Dic79C2311yescharacteristic
78C2×C4⋊Dic79C2311yescharacteristic
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81#14579096—22no—
82#14579096—22no—
83#14579094—21no—
84#14579094—21no—
85#14579094—21no—
86#14579094—21no—
87#14579094—21no—
88#14579094—21no—
89#14579092—21no—
90#14579092—21no—
91#14579092—21no—
92#14579092—21no—
93#14579092—21no—
94#14579092—21no—
95C4⋊(C79⋊C8)—21no—
96C4⋊(C79⋊C8)—21no—
97C4⋊(C79⋊C8)—21no—
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)D411yescharacteristic
100C4⋊(C79⋊C8)D411yescharacteristic
101C4⋊(C79⋊C8)D411yescharacteristic
102C4⋊(C79⋊C8)D411yescharacteristic
103#14579082—21no—
104#14579082—21no—
105#14579082—21no—
106#14579082—21no—
107#14579078—22no—
108#14579078Q814yescomplemented
109#14579072—21no—
110#14579072—21no—
111#14579072—21no—
112#14579072—21no—
113C4×C79⋊C8—21no—
114C4×C79⋊C8—21no—
115C4×C79⋊C8—21no—
116C4×C79⋊C8—21no—
117C22×D4×C79C2311yescharacteristic
118C23×C4×C79—21no—
119C2×M16×C79C2311yescharacteristic
120C2×M16×C79C2311yescharacteristic
121C4⋊Q8×C79D411yescharacteristic
122C4⋊Q8×C79D411yescharacteristic
123#125×C79D411yescharacteristic
124#125×C79D411yescharacteristic
125C22⋊Q8×C79—21no—
126C22⋊Q8×C79—21no—
127C4×Q8×C79—21no—
128C4×Q8×C79—21no—
129C4×D4×C79—21no—
130C4×D4×C79—21no—
131C4×D4×C79—21no—
132C4×D4×C79—21no—
133#117×C79C2311yescharacteristic
134#117×C79C2311yescharacteristic
135C2×C4⋊C4×C79C2311yescharacteristic
136C2×C22⋊C4×C79—21no—
137C2×C42×C79C2311yescharacteristic
138#107×C79—21no—
139#107×C79—21no—
140#106×C79—21no—
141#106×C79—21no—
142C4⋊C8×C79—21no—
143C4⋊C8×C79—21no—
144#102×C79—21no—
145#102×C79—21no—
146#102×C79—21no—
147#102×C79—21no—
148C22×Dic79—21no—
149C22×Dic79—21no—
150#11929608—21no—
151#11929608—21no—
152#11929608—21no—
153#11929608—21no—
154#11929608—21no—
155#11929608C2×Q812yes—
156#11929608C2×Q812yes—
157#11929608#4112yes—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8—21no—
166C2×C79⋊C8C2×D411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C2×C79⋊C8C2×D411yescharacteristic
169C2×C79⋊C8C2×D411yescharacteristic
170C4⋊Dic79—21no—
171C4⋊Dic79—21no—
172C4⋊Dic79—21no—
173C4⋊Dic79—21no—
174Dic79⋊C4—41no—
175Dic79⋊C4—41no—
176Dic79⋊C4—41no—
177Dic79⋊C4—41no—
178C24×C79—21no—
179C2×Q8×C79—21no—
180C2×Q8×C79—21no—
181C2×D4×C79—21no—
182C2×D4×C79—21no—
183C2×D4×C79—21no—
184C2×D4×C79—21no—
185C2×D4×C79—21no—
186C2×D4×C79C2×Q812yes—
187C22×C4×C79—21no—
188C22×C4×C79—41no—
189C22×C4×C79—41no—
190C22×C4×C79—82no—
191C22×C4×C79#4111yescharacteristic
192C22×C4×C79#4111yescharacteristic
193C22×C4×C79C2411yescharacteristic
194M16×C79—21no—
195M16×C79—21no—
196M16×C79—21no—
197M16×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79—21no—
200C4⋊C4×C79—21no—
201C4⋊C4×C79—21no—
202C4⋊C4×C79—21no—
203C4⋊C4×C79—21no—
204C4⋊C4×C79—21no—
205C4⋊C4×C79—21no—
206C4⋊C4×C79—21no—
207C4⋊C4×C79—21no—
208C4⋊C4×C79C2×D411yescharacteristic
209C4⋊C4×C79C2×D411yescharacteristic
210C4⋊C4×C79C2×D411yescharacteristic
211C4⋊C4×C79C2×D411yescharacteristic
212C22⋊C4×C79—21no—
213C22⋊C4×C79—21no—
214C22⋊C4×C79—41no—
215C22⋊C4×C79—41no—
216C42×C79—21no—
217C42×C79—21no—
218C42×C79C2×D411yescharacteristic
219C42×C79C2×D411yescharacteristic
220C42×C79C2×D411yescharacteristic
221C42×C79C2×D411yescharacteristic
222C23×C79—22no—
223C23×C79—41no—
224C23×C79—41no—
225C23×C79—81no—
226C23×C79—81no—
227C23×C79C2×#4111yescharacteristic
228Q8×C79—42no—
229Q8×C79—81no—
230D4×C79—22no—
231D4×C79—22no—
232D4×C79—41no—
233D4×C79—41no—
234C2×C4×C79—21no—
235C2×C4×C79—21no—
236C2×C4×C79—21no—
237C2×C4×C79—21no—
238C2×C4×C79—41no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—82no—
242C2×C4×C79—82no—
243C2×C4×C79—81no—
244C2×C4×C79#12111yescharacteristic
245C2×C4×C79#12111yescharacteristic
246C2×C4×C79#12111yescharacteristic
247C2×C4×C79#12111yescharacteristic
248C2×C4×C79C22×D411yescharacteristic
249C2×C4×C79C22×D411yescharacteristic
250C2×C4×C79C22×Q811yescharacteristic
251C2×C4×C79C2×#4111yescharacteristic
252C2×C4×C792−1+411yescharacteristic
253C2×C4×C792−1+411yescharacteristic
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—81no—
257C2×Dic79—81no—
258C8×C79—41no—
259C8×C79—41no—
260C79⋊C8—21no—
261C79⋊C8—21no—
262C79⋊C8—21no—
263C79⋊C8—21no—
264C79⋊C8—22no—
265C79⋊C8—22no—
266C79⋊C8—22no—
267C79⋊C8—41no—
268C22×C79—21no—
269C22×C79—21no—
270C22×C79—42no—
271C22×C79—82no—
272C22×C79—82no—
273C22×C79—81no—
274C22×C79—81no—
275C22×C79C2×#12111yescharacteristic
276C22×C79#53311yescharacteristic
277C22×C79#54111yescharacteristic
278C4×C79—21no—
279C4×C79—21no—
280C4×C79—21no—
281C4×C79—21no—
282C4×C79—41no—
283C4×C79—41no—
284C4×C79—41no—
285C4×C79—41no—
286C4×C79—81no—
287C4×C79#54711yescharacteristic
288C4×C79#54811yescharacteristic
289C4×C79D4×Q811yescharacteristic
290C4×C79D4×Q811yescharacteristic
291Dic79—81no—
292Dic79—81no—
293#25493—791nomaximal, Sylow
294C2×C79—21no—
295C2×C79—21no—
296C2×C79—82no—
297C2×C79#242911yescharacteristic
298C2×C79#306711yescharacteristic
299C2×C79#306811yescharacteristic
300#3166—791no—
301#3077D7911yescharacteristic
302C2×#478—791no—
303#2962—791no—
304#2090—791no—
305#2089—791no—
306#2063—791no—
307#2063—791no—
308#1998—791no—
309#1952—791no—
310#1951—791no—
311#1950—791no—
312#1931—791no—
313#1931—791no—
314C4×#100—791no—
315C79#2549311yescharacteristic, complemented, Sylow
316#533C2×D7911yescharacteristic
317C2×#125—791no—
318C2×C4×D4C2×D7911yescharacteristic
319#501—1581no—
320#501—1581no—
321#499—1581no—
322#499—1581no—
323#478—21no—
324#478—21no—
325#478—1581no—
326#478—1581no—
327#478—1581no—
328#478—1581no—
329#476—21no—
330#474—21no—
331#446—1581no—
332#446—1581no—
333#446—1581no—
334C8×Q8—1581no—
335C2×#108—791no—
336C2×#108—791no—
337#428—791no—
338#428—791no—
339#428—791no—
340#428—791no—
341#428C2×D7911yescharacteristic
342#428C2×D7911yescharacteristic
343C2×#107—791no—
344C2×#106—791no—
345#424—791no—
346#424—791no—
347#423C2×D7911yescharacteristic
348C2×C4⋊C8—791no—
349#418C2×D7911yescharacteristic
350#418C2×D7911yescharacteristic
351C2×#102—791no—
352C2×#102—791no—
353C2×#100—791no—
354C2×#100—791no—
355#405—791no—
356#405—791no—
357C4×M16—791no—
358C4×M16—791no—
359#343—791no—
360#343—791no—
361#343—791no—
362#343—791no—
363#343—791no—
364#343—791no—
365#341—791no—
366#341—791no—
367C22×D4C22×D7911yescharacteristic
368C23×C4—21no—
369C2×M16—791no—
370C2×M16—791no—
371C2×M16—791no—
372C2×M16—791no—
373C2×M16C22×D7911yescharacteristic
374C2×M16C22×D7911yescharacteristic
375C22×C8—791no—
376C22×C8—791no—
377C4⋊Q8C22⋊D7911yescharacteristic
378C4⋊Q8C22⋊D7911yescharacteristic
379#125—1581no—
380#125—1581no—
381#125—1581no—
382#125—1581no—
383#125C22⋊D7911yescharacteristic
384#125C22⋊D7911yescharacteristic
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393#117C22×D7911yescharacteristic
394#117C22×D7911yescharacteristic
395C2×C4⋊C4—791no—
396C2×C4⋊C4—791no—
397C2×C4⋊C4—1581no—
398C2×C4⋊C4—1581no—
399C2×C4⋊C4C22×D7911yescharacteristic
400C2×C22⋊C4—21no—
401C2×C42C22×D7911yescharacteristic
402#108—1582no—
403#108—1582no—
404#107—21no—
405#107—21no—
406#107—1581no—
407#107—1581no—
408#107—1581no—
409#107—1581no—
410#107—1581no—
411#107—1581no—
412#106—21no—
413#106—21no—
414#106—1581no—
415#106—1581no—
416#106—1581no—
417#106—1581no—
418#106—1581no—
419#106—1581no—
420C4⋊C8—21no—
421C4⋊C8—21no—
422C4⋊C8—791no—
423C4⋊C8—791no—
424C4⋊C8—791no—
425C4⋊C8—791no—
426C4⋊C8—1581no—
427C4⋊C8—1581no—
428C4⋊C8—1581no—
429C4⋊C8—1581no—
430#102—21no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—1581no—
435#102—1581no—
436#102—1581no—
437#102—1581no—
438#100—794no—
439#100—1582no—
440#97—1581no—
441#97—1581no—
442#97—1581no—
443#97—1581no—
444C4×C8—1581no—
445C4×C8—1581no—
446C4×C8—1581no—
447C4×C8—1581no—
448C24—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4Q8×D7912yes—
457C22×C4—21no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—82no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4#1192962311yescharacteristic
464C22×C4#1192962311yescharacteristic
465C22×C4C23×D7911yescharacteristic
466M16—21no—
467M16—21no—
468M16—21no—
469M16—21no—
470M16—792no—
471M16—792no—
472M16—792no—
473M16—1581no—
474M16—1581no—
475M16—1581no—
476M16—1581no—
477M16—1581no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—791no—
481C2×C8—791no—
482C2×C8—791no—
483C2×C8—791no—
484C2×C8—1581no—
485C2×C8—1581no—
486C2×C8—1581no—
487C2×C8—1581no—
488C2×C8—1581no—
489C2×C8—1581no—
490C2×C8—1581no—
491C2×C8—1581no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—3161no—
501C4⋊C4—3161no—
502C4⋊C4—3161no—
503C4⋊C4—3161no—
504C4⋊C4—1581no—
505C4⋊C4—1581no—
506C4⋊C4—1581no—
507C4⋊C4—1581no—
508C4⋊C4C2×C22⋊D7911yescharacteristic
509C4⋊C4C2×C22⋊D7911yescharacteristic
510C4⋊C4C2×C22⋊D7911yescharacteristic
511C4⋊C4C2×C22⋊D7911yescharacteristic
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C42—21no—
517C42—21no—
518C42D4×D7911yescharacteristic
519C42D4×D7911yescharacteristic
520C42C2×C22⋊D7911yescharacteristic
521C42C2×C22⋊D7911yescharacteristic
522C23—22no—
523C23—41no—
524C23—41no—
525C23—81no—
526C23—81no—
527C23C2×#1192962311yescharacteristic
528Q8—42no—
529Q8—81no—
530D4—22no—
531D4—22no—
532D4—41no—
533D4—41no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—6321no—
547C2×C4—6321no—
548C2×C4#1457913311yescharacteristic
549C2×C4#1457913311yescharacteristic
550C2×C4#1457913311yescharacteristic
551C2×C4#1457913311yescharacteristic
552C2×C4C2×D4×D7911yescharacteristic
553C2×C4C22×C22⋊D7911yescharacteristic
554C2×C4C2×Q8×D7911yescharacteristic
555C2×C4#41×D7911yescharacteristic
556C2×C4#1457920811yescharacteristic
557C2×C4#1457920911yescharacteristic
558C8—41no—
559C8—41no—
560C8—3161no—
561C8—1581no—
562C8—1581no—
563C8—1581no—
564C8—1581no—
565C8—1582no—
566C8—1582no—
567C8—1582no—
568C22—21no—
569C22—21no—
570C22—42no—
571C22—82no—
572C22—82no—
573C22—81no—
574C22—81no—
575C22C2×#1457913311yescharacteristic
576C22#1909911511yescharacteristic
577C22#1909917311yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—6321no—
588C4—6321no—
589C4#1909921511yescharacteristic
590C4#1909922011yescharacteristic
591C4#1909922611yescharacteristic
592C4Q8×C22⋊D7911yescharacteristic
593C2—21no—
594C2—21no—
595C2—82no—
596C2#2376700111yescharacteristic
597C2#2377311711yescharacteristic
598C2#2377312511yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#3077C2
socleC22×C79C2×#121
prime radicalC22×D4×C79C23
hypercenter#3077D79
hypocenterC79#25493
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #23773953C2semidirect
3. #23772349C2semidirect
4. #23764913C2semidirect
5. #23764908C2semidirect
8. #23764392C2semidirect
9. #23764182C2semidirect
10. #23764177C2semidirect
11. #23764173C2semidirect
108. #14579078Q8semidirect
315. C79#25493semidirect

Maximal subgroups

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

subgroupindex
#237739532
C2×#190988632
#237723492
#237649132
#237649082
#237647602
#237643922
#237641822
#237641772
#237641732
#237640652
C4×#145790782
C79×#30772
#2549379