Finite Group Databasebeta

← #34260321

Subgroups — #34260321

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773966C211yesmaximal, characteristic
2#23772563C211yesmaximal, characteristic, complemented
3#23772555C211yesmaximal, characteristic, complemented
4#23762311C211yesmaximal, characteristic, complemented
5#23762307C211yesmaximal, characteristic, complemented
6#23762244C211yesmaximal, characteristic, complemented
7#23762242C211yesmaximal, characteristic, complemented
8#23762238C211yesmaximal, characteristic, complemented
9#23762222C211yesmaximal, characteristic, complemented
10#23762216C211yesmaximal, characteristic, complemented
11#334×D79C211yesmaximal, characteristic, complemented
12#23761812C211yesmaximal, characteristic, complemented
13#23761810C211yesmaximal, characteristic, complemented
14#23761688C211yesmaximal, characteristic, complemented
15C79×#1582C211yesmaximal, characteristic, complemented
16#19099100C2211yescharacteristic
17#19099095C2211yescharacteristic
18#19098963C411yescharacteristic
19#19098944C411yescharacteristic
20#19098919C411yescharacteristic
21#19098916C411yescharacteristic
22#19098803C411yescharacteristic
23#19098798C411yescharacteristic
24#19098793C411yescharacteristic
25#19098789C411yescharacteristic
26#19098693C2211yescharacteristic
27Q8×C79⋊C8C2211yescharacteristic
28#19098687C2211yescharacteristic
29C4×#11929614C2211yescharacteristic
30C4×#11929613C2211yescharacteristic
31C4×#11929609C2211yescharacteristic
32#19098641C2211yescharacteristic
33#19098640C2211yescharacteristic
34#19098637—21no—
35#19098636—21no—
36C8×D316C2211yescharacteristic
37#19098628—21no—
38C8×C22⋊D79—21no—
39#19098625—21no—
40#19098621—21no—
41#19098614C2211yescharacteristic
42#19098589—21no—
43#19098588—21no—
44C4⋊C8×D79C2211yescharacteristic
45C4⋊C8×D79C2211yescharacteristic
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098314C2211yescharacteristic
49#19098313C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098285C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929614C2×C411yescharacteristic
68C2×#11929613C2×C411yescharacteristic
69C2×#11929609C2×C411yescharacteristic
70C2×#11929608—21no—
71C2×#11929607—21no—
72C2×#11929607—21no—
73C2×C8×D79—21no—
74C2×C8×D79—21no—
75C2×C8×D79—21no—
76C2×C8×D79C2×C411yescharacteristic
77C2×C8×D79C2×C411yescharacteristic
78#14579156—21no—
79#14579154—21no—
80#14579152C2×C411yescharacteristic
81#14579152C2×C411yescharacteristic
82#14579150C2×C411yescharacteristic
83#14579148C2×C411yescharacteristic
84#14579146—21no—
85#14579143—21no—
86#14579141C2×C411yescharacteristic
87#14579137C2×C411yescharacteristic
88#14579136C2×C411yescharacteristic
89#14579132C2×C411yescharacteristic
90C4×Dic158C2311yescharacteristic
91C4×D316—21no—
92C4×D316C2311yescharacteristic
93C4×C22⋊D79—21no—
94#14579120—21no—
95#14579118—21no—
96C42×D79C2311yescharacteristic
97#14579094C2×C411yescharacteristic
98#14579094C2×C411yescharacteristic
99#14579093C2×C411yescharacteristic
100C4⋊(C79⋊C8)—21no—
101C4⋊(C79⋊C8)C2311yescharacteristic
102C4⋊(C79⋊C8)C2311yescharacteristic
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8D411yescharacteristic
108Dic79⋊C8D411yescharacteristic
109#14579086C2×C411yescharacteristic
110#14579085C2×C411yescharacteristic
111#14579084C2×C411yescharacteristic
112#14579083C2×C411yescharacteristic
113#14579082C2×C411yescharacteristic
114#14579081C2×C411yescharacteristic
115C22⋊(C79⋊C8)—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8D411yescharacteristic
119(C2×D79)⋊C8D411yescharacteristic
120#14579071—21no—
121#14579071—21no—
122C4×C79⋊C8—21no—
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C8×Dic79—21no—
127C8×Dic79C2×C411yescharacteristic
128C8×Dic79C2×C411yescharacteristic
129C2×SD16×C79C2×C411yescharacteristic
130C22×C8×C79—21no—
131C4×Q8×C79C2311yescharacteristic
132C4×D4×C79C2311yescharacteristic
133#106×C79C2×C411yescharacteristic
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#102×C79C2×C411yescharacteristic
139C22⋊C8×C79—21no—
140#97×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158C22×C411yescharacteristic
145C2×D316—21no—
146C2×D316C22×C411yescharacteristic
147C2×C22⋊D79—21no—
148C2×C4×D79—21no—
149C2×C4×D79C22×C411yescharacteristic
150C2×C4×D79C2×D411yescharacteristic
151C2×C4×D79#4111yescharacteristic
152#11929614—22no—
153#11929613—41no—
154#11929609—22no—
155#11929608—81no—
156#11929607—42no—
157#11929607—81no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8C22×C411yescharacteristic
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C2×D411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C8×D79—22no—
166C8×D79—41no—
167C8×D79—81no—
168C8×D79—81no—
169C8×D79—81no—
170C4⋊Dic79—21no—
171C4⋊Dic79—21no—
172C4⋊Dic79—21no—
173C4⋊Dic79C22×C411yescharacteristic
174C4⋊Dic79C22×C411yescharacteristic
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179C22⋊Dic79—21no—
180C22⋊Dic79—21no—
181C22⋊Dic79—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186(C2×D79)⋊C4—21no—
187(C2×D79)⋊C4—21no—
188C4×Dic79—21no—
189C4×Dic79C22×C411yescharacteristic
190C4×Dic79C2×D411yescharacteristic
191C4×Dic79#4111yescharacteristic
192C2×Q8×C79C22×C411yescharacteristic
193C2×D4×C79C22×C411yescharacteristic
194C22×C4×C79—21no—
195SD16×C79—41no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79—81no—
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C22×C411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C2×C8×C79C2×D411yescharacteristic
204C4⋊C4×C79—21no—
205C4⋊C4×C79C22×C411yescharacteristic
206C4⋊C4×C79C22×C411yescharacteristic
207C22⋊C4×C79—21no—
208C42×C79—21no—
209C42×C79C2411yescharacteristic
210C23×C79—21no—
211C22×D79—21no—
212C22×D79C4×D411yescharacteristic
213Q8×C79—21no—
214Q8×C79—21no—
215D4×C79—21no—
216D4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79C23×C411yescharacteristic
223C2×C4×C79C22×D411yescharacteristic
224C2×C4×C79C2×#4111yescharacteristic
225C22⋊D79—42no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79C4×D411yescharacteristic
230C2×Dic79C4×D411yescharacteristic
231C2×Dic79C4×D411yescharacteristic
232D316—21no—
233D316—21no—
234D316—42no—
235C4×D79—22no—
236C4×D79—21no—
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—21no—
240C4×D79—41no—
241C4×D79—41no—
242Dic158—21no—
243Dic158—21no—
244C8×C79—21no—
245C8×C79—21no—
246C8×C79—21no—
247C8×C79—21no—
248C8×C79—41no—
249C8×C79—41no—
250C8×C79—41no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—21no—
254C79⋊C8—41no—
255C79⋊C8—41no—
256C79⋊C8—41no—
257C79⋊C8#13112yes—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79C2×C4×D411yescharacteristic
262C2×D79—21no—
263C2×D79—21no—
264C2×D79—41no—
265C2×D79—41no—
266C2×D79—41no—
267C2×D79#44412yes—
268C4×C79—21no—
269C4×C79—41no—
270C4×C79—41no—
271C4×C79—41no—
272C4×C79C2×#13111yescharacteristic
273C4×C79#56811yescharacteristic
274C4×C79#57711yescharacteristic
275C4×C79#57711yescharacteristic
276Dic79—21no—
277Dic79—21no—
278Dic79—21no—
279Dic79—41no—
280Dic79—41no—
281Dic79—41no—
282#18093—791nomaximal, Sylow
283C2×C79—81no—
284C2×C79#293511yescharacteristic
285C2×C79#295211yescharacteristic
286C2×C79C2×#44411yescharacteristic
287D79—22no—
288D79—81no—
289#3168—791no—
290#2994—791no—
291#2994—791no—
292#1594—791no—
293#1593—791no—
294#1583—791no—
295#1583—791no—
296#1582—791no—
297#1582D7911yescharacteristic, complemented
298#1580—791no—
299#1579—791no—
300C2×#334—791no—
301#1483—791no—
302#1483—791no—
303#1451—791no—
304C79#1809311yescharacteristic, complemented, Sylow
305#529—791no—
306#529—791no—
307#499—791no—
308#495—791no—
309#488—791no—
310#488—791no—
311#467—791no—
312#466—791no—
313#465—791no—
314#465—791no—
315#446—791no—
316#446C2×D7911yescharacteristic
317C8×Q8—791no—
318C8×Q8—791no—
319C4×Q16—791no—
320C4×SD16—791no—
321C4×SD16C2×D7911yescharacteristic
322C4×D8—791no—
323#436—791no—
324#436—791no—
325#435—1581no—
326#435—1581no—
327C8×D4—791no—
328C8×D4—1581no—
329C8×D4—1581no—
330C8×D4C2×D7911yescharacteristic
331#433—1581no—
332#433—1581no—
333#431—791no—
334#424—1581no—
335#424—1581no—
336C2×C4⋊C8—791no—
337C2×C4⋊C8—791no—
338C4×M16—791no—
339C2×C4×C8—791no—
340#334—791no—
341#334—791no—
342#334—791no—
343#334C2×D7911yescharacteristic
344Q8⋊C8—791no—
345Q8⋊C8—791no—
346Q8⋊C8—791no—
347Q8⋊C8C2×D7911yescharacteristic
348D4⋊C8—791no—
349D4⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8C2×D7911yescharacteristic
352#322—791no—
353#322—791no—
354#322—791no—
355#322C2×D7911yescharacteristic
356C2×Q16—791no—
357C2×SD16—791no—
358C2×SD16C4×D7911yescharacteristic
359C2×D8—791no—
360C2×M16—1581no—
361C2×M16—1581no—
362C2×M16—1581no—
363C22×C8—21no—
364C22×C8—791no—
365C22×C8—791no—
366C22×C8—1581no—
367C22×C8—1581no—
368C22×C8—1581no—
369#126—1581no—
370#126—1581no—
371#125—791no—
372#125—791no—
373#124—791no—
374#124—791no—
375#123—1581no—
376#123—1581no—
377C22⋊Q8—791no—
378C22⋊Q8—791no—
379#121—791no—
380#121—791no—
381C4×Q8—791no—
382C4×Q8C22×D7911yescharacteristic
383C4×D4—791no—
384C4×D4—1581no—
385C4×D4—1581no—
386C4×D4C22×D7911yescharacteristic
387#117—1581no—
388#117—1581no—
389C2×C42—791no—
390#107—791no—
391#107—791no—
392#106—791no—
393#106C4×D7911yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—791no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8—1581no—
402C4⋊C8—1581no—
403C4⋊C8—1581no—
404C4⋊C8C22×D7911yescharacteristic
405C4⋊C8C22×D7911yescharacteristic
406#103—791no—
407#103—791no—
408#103—791no—
409#103C4×D7911yescharacteristic
410#102—791no—
411#102—791no—
412#102—791no—
413#102C4×D7911yescharacteristic
414C22⋊C8—21no—
415C22⋊C8—791no—
416C22⋊C8—791no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420#97—21no—
421#97—1581no—
422#97—1581no—
423C4×C8—791no—
424C4×C8—791no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—791no—
428C4×C8—1581no—
429C4×C8—1581no—
430C4×C8C22×D7911yescharacteristic
431C4×C8C22×D7911yescharacteristic
432C4×C8C22×D7911yescharacteristic
433C2×Q8—791no—
434C2×Q8C2×C4×D7911yescharacteristic
435C2×D4—791no—
436C2×D4—1581no—
437C2×D4—1581no—
438C2×D4C2×C4×D7911yescharacteristic
439C22×C4—21no—
440C22×C4—791no—
441C22×C4—791no—
442C22×C4—791no—
443C22×C4—1581no—
444Q16—1582no—
445SD16—41no—
446SD16—3161no—
447D8—1582no—
448M16—3162no—
449M16—6321no—
450M16—6321no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—81no—
455C2×C8—3161no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—791no—
459C2×C8—791no—
460C2×C8—6321no—
461C2×C8—6321no—
462C2×C8—6321no—
463C2×C8—1582no—
464C2×C8—1581no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8C2×C4×D7911yescharacteristic
468C2×C8C2×C4×D7911yescharacteristic
469C2×C8D4×D7911yescharacteristic
470C2×C8D4×D7911yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—791no—
473C4⋊C4—791no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4C2×C4×D7911yescharacteristic
482C4⋊C4C2×C4×D7911yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—1581no—
485C22⋊C4—1581no—
486C22⋊C4—1581no—
487C22⋊C4—1581no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C22⋊C4—1581no—
492C22⋊C4—1581no—
493C42—21no—
494C42—791no—
495C42—791no—
496C42—791no—
497C42—1581no—
498C42C23×D7911yescharacteristic
499C23—21no—
500C23—791no—
501C23—1581no—
502Q8—21no—
503Q8—21no—
504Q8—1581no—
505Q8—1581no—
506D4—21no—
507D4—21no—
508D4—3162no—
509D4—3162no—
510D4—1581no—
511D4—1581no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—3161no—
518C2×C4—3161no—
519C2×C4—791no—
520C2×C4—791no—
521C2×C4—791no—
522C2×C4—1582no—
523C2×C4—1581no—
524C2×C4—1581no—
525C2×C4—1581no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4C22×C4×D7911yescharacteristic
531C2×C4C2×D4×D7911yescharacteristic
532C2×C4#41×D7911yescharacteristic
533C8—21no—
534C8—21no—
535C8—21no—
536C8—21no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—3161no—
541C8—3161no—
542C8—3161no—
543C8—792no—
544C8—1581no—
545C8—1581no—
546C8—1581no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—3161no—
551C22—3161no—
552C22—3161no—
553C22—792no—
554C22—1581no—
555C22—1581no—
556C22C4×D4×D7911yescharacteristic
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—3161no—
562C4—3161no—
563C4—3161no—
564C4—1581no—
565C4—1581no—
566C4—1581no—
567C4#131×D7911yescharacteristic
568C4#1909932911yescharacteristic
569C4#1909936911yescharacteristic
570C4#1909937011yescharacteristic
571C2—81no—
572C2—6321no—
573C2—1582no—
574C2#2377214111yescharacteristic
575C2#2377227811yescharacteristic
576C2#444×D7911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1582C2
socleC22×C79C2×C4×D4
prime radical#19098798C4
cosocleC42×C79C24
hypercenter#1582D79
hypocenterC79#18093
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1582

Complements

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

NH ≅ G/Ntype
2. #23772563C2semidirect
3. #23772555C2semidirect
4. #23762311C2semidirect
5. #23762307C2semidirect
6. #23762244C2semidirect
7. #23762242C2semidirect
8. #23762238C2semidirect
9. #23762222C2semidirect
10. #23762216C2semidirect
11. #334×D79C2semidirect
12. #23761812C2semidirect
13. #23761810C2semidirect
14. #23761688C2semidirect
15. C79×#1582C2semidirect
297. #1582D79semidirect
304. C79#18093semidirect

Maximal subgroups

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

subgroupindex
#237739662
#237725632
#237725552
#237623112
#237623072
#237622442
#237622422
#237622382
#237622222
#237622162
#334×D792
#237618122
#237618102
#237616882
C79×#15822
#1809379