Finite Group Databasebeta

← #34259294

Subgroups — #34259294

Order 20224.

608 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773420C211yesmaximal, characteristic
2#23772418C211yesmaximal, characteristic, complemented
3#23772404C211yesmaximal, characteristic, complemented
4#23762315C211yesmaximal, characteristic, complemented
5#23762304C211yesmaximal, characteristic, complemented
6#23762280C211yesmaximal, characteristic, complemented
7#23762280C211yesmaximal, characteristic, complemented
8#23762273C211yesmaximal, characteristic, complemented
9#23762266C211yesmaximal, characteristic, complemented
10#23762220C211yesmaximal, characteristic, complemented
11#23762214C211yesmaximal, characteristic, complemented
12#23762180C211yesmaximal, characteristic, complemented
13#23761861C211yesmaximal, characteristic, complemented
14#23761783C211yesmaximal, characteristic, complemented
15C79×#1451C211yesmaximal, characteristic, complemented
16#19099097C2211yescharacteristic
17C4×#11929624C2211yescharacteristic
18#19098905C411yescharacteristic
19#19098888C411yescharacteristic
20#19098871C411yescharacteristic
21#19098864C411yescharacteristic
22#19098786C411yescharacteristic
23#19098779C411yescharacteristic
24#19098727C411yescharacteristic
25#19098710C411yescharacteristic
26#19098693C2211yescharacteristic
27C8×Dic158C2211yescharacteristic
28C8×Dic158C2211yescharacteristic
29C8×Dic158C2211yescharacteristic
30C4×Dic316C2211yescharacteristic
31C4×#11929613C2211yescharacteristic
32C4×#11929613C2211yescharacteristic
33C4×D632C2211yescharacteristic
34#19098642C2211yescharacteristic
35#19098642C2211yescharacteristic
36#19098642C2211yescharacteristic
37#19098639—21no—
38#19098639—21no—
39#19098639—21no—
40#19098632—21no—
41C8×D316C2211yescharacteristic
42#19098625—21no—
43#19098620C2211yescharacteristic
44#19098618—21no—
45#19098616C2211yescharacteristic
46#19098520—21no—
47#19098515—21no—
48#19098316C2211yescharacteristic
49#19098316C2211yescharacteristic
50#19098314C2211yescharacteristic
51#19098314C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098295C2211yescharacteristic
55#19098295C2211yescharacteristic
56#19098292C2211yescharacteristic
57#19098292C2211yescharacteristic
58#19098292C2211yescharacteristic
59#19098292C2211yescharacteristic
60#431×C79C2211yescharacteristic
61C4×M16×C79C2211yescharacteristic
62C2×C4×C8×C79C2211yescharacteristic
63#322×C79C2211yescharacteristic
64#322×C79C2211yescharacteristic
65#322×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929624C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929613C2×C411yescharacteristic
71C2×D632C2×C411yescharacteristic
72C2×#11929607—21no—
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×C8×D79—21no—
76#14579155—21no—
77#14579153C2×C411yescharacteristic
78#14579149C2×C411yescharacteristic
79#14579144—21no—
80#14579140C2×C411yescharacteristic
81#14579135C2×C411yescharacteristic
82C4×Dic158C2×C411yescharacteristic
83C4×Dic158C2311yescharacteristic
84C4×Dic158C2311yescharacteristic
85C4×D316C2×C411yescharacteristic
86C4×D316C2311yescharacteristic
87C4×D316C2311yescharacteristic
88C4×C22⋊D79—21no—
89C4×C22⋊D79—21no—
90C4×C22⋊D79—21no—
91#14579120C2×C411yescharacteristic
92#14579118—21no—
93#14579118—21no—
94C42×D79—21no—
95#14579095C2×C411yescharacteristic
96#14579095C2×C411yescharacteristic
97#14579093C2×C411yescharacteristic
98#14579093C2×C411yescharacteristic
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101C4⋊(C79⋊C8)C2311yescharacteristic
102C4⋊(C79⋊C8)C2311yescharacteristic
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107#14579086C2×C411yescharacteristic
108#14579086C2×C411yescharacteristic
109#14579086C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579083C2×C411yescharacteristic
112#14579083C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114#14579083C2×C411yescharacteristic
115C22⋊(C79⋊C8)—21no—
116C22⋊(C79⋊C8)—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8—21no—
121#14579072—21no—
122#14579071—21no—
123C4×C79⋊C8—21no—
124C8×Dic79—21no—
125C8×Dic79—21no—
126C8×Dic79—21no—
127C2×M16×C79D411yescharacteristic
128C2×M16×C79D411yescharacteristic
129C22×C8×C79—21no—
130C2×C42×C79C2311yescharacteristic
131C4⋊C8×C79C2×C411yescharacteristic
132C4⋊C8×C79C2×C411yescharacteristic
133C22⋊C8×C79C2×C411yescharacteristic
134C22⋊C8×C79C2×C411yescharacteristic
135#97×C79D411yescharacteristic
136#97×C79D411yescharacteristic
137C4×C8×C79—21no—
138C4×C8×C79C2311yescharacteristic
139C4×C8×C79C2311yescharacteristic
140C4×C8×C79C2311yescharacteristic
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929624—22no—
145#11929624—41no—
146C2×Dic158C22×C411yescharacteristic
147C2×Dic158C22×C411yescharacteristic
148C2×D316C22×C411yescharacteristic
149C2×D316C22×C411yescharacteristic
150C2×C22⋊D79—21no—
151C2×C22⋊D79—21no—
152C2×C4×D79—21no—
153C2×C4×D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—21no—
156Dic316—22no—
157#11929613—22no—
158#11929613—41no—
159D632—41no—
160#11929607—42no—
161#11929607—42no—
162#11929607—42no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8—21no—
166C2×C79⋊C8—21no—
167C8×D79—42no—
168C4⋊Dic79C22×C411yescharacteristic
169C4⋊Dic79C22×C411yescharacteristic
170C4⋊Dic79C22×C411yescharacteristic
171C4⋊Dic79C22×C411yescharacteristic
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178C22⋊Dic79—21no—
179C22⋊Dic79—21no—
180(C2×D79)⋊C4—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79—21no—
189C4×Dic79—21no—
190C22×C4×C79C22×C411yescharacteristic
191C22×C4×C79C2×D411yescharacteristic
192C22×C4×C79#4111yescharacteristic
193M16×C79—41no—
194M16×C79—41no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79—42no—
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C22×C411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C2×C8×C79C2×D411yescharacteristic
204C2×C8×C79C2×D411yescharacteristic
205C2×C8×C79C2×D411yescharacteristic
206C42×C79C22×C411yescharacteristic
207C42×C79C2×D411yescharacteristic
208C42×C79#4111yescharacteristic
209C42×C79C2411yescharacteristic
210C23×C79C4×D411yescharacteristic
211C22×D79—21no—
212C22×D79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—41no—
218C2×C4×C79C4×D411yescharacteristic
219C2×C4×C79C4×D411yescharacteristic
220C2×C4×C79C4×D411yescharacteristic
221C2×C4×C79C23×C411yescharacteristic
222C2×C4×C79C22×D411yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C22⋊D79—42no—
225C22⋊D79—81no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232D316—21no—
233D316—21no—
234D316—21no—
235D316—21no—
236C4×D79—42no—
237C4×D79—41no—
238C4×D79—41no—
239C4×D79—41no—
240C4×D79—41no—
241C4×D79—81no—
242Dic158—21no—
243Dic158—21no—
244Dic158—21no—
245Dic158#13112yes—
246C8×C79—22no—
247C8×C79—21no—
248C8×C79—21no—
249C8×C79—21no—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—21no—
253C8×C79—21no—
254C8×C79—21no—
255C8×C79—41no—
256C79⋊C8—41no—
257C79⋊C8—41no—
258C79⋊C8—41no—
259C79⋊C8—41no—
260C22×C79—21no—
261C22×C79—21no—
262C22×C79—21no—
263C22×C79C2×C4×D411yescharacteristic
264C2×D79—41no—
265C2×D79—41no—
266C2×D79—41no—
267C2×D79—41no—
268C2×D79—41no—
269C2×D79—41no—
270C4×C79—21no—
271C4×C79—21no—
272C4×C79—21no—
273C4×C79—21no—
274C4×C79C2×#13111yescharacteristic
275C4×C79#56811yescharacteristic
276C4×C79#57511yescharacteristic
277C4×C79#57511yescharacteristic
278Dic79—22no—
279Dic79—41no—
280Dic79—41no—
281Dic79—41no—
282Dic79—41no—
283Dic79—41no—
284#18005—791nomaximal, Sylow
285C2×C79—41no—
286C2×C79#293311yescharacteristic
287C2×C79#294311yescharacteristic
288C2×C79#295611yescharacteristic
289D79—81no—
290D79—81no—
291#3102—791no—
292#2971—791no—
293#2969—791no—
294#1594—791no—
295#1592—791no—
296#1588—791no—
297#1588—791no—
298#1587—791no—
299#1586—791no—
300#1579—791no—
301SD16⋊C8—791no—
302#1572—791no—
303#1494—791no—
304#1476—791no—
305#1451D7911yescharacteristic, complemented
306C79#1800511yescharacteristic, complemented, Sylow
307#529—791no—
308C4×#41—791no—
309#485—791no—
310#482—791no—
311#479—791no—
312#478—791no—
313#464—791no—
314#463—791no—
315#452—791no—
316#449—791no—
317#446—791no—
318C8×Q8—791no—
319C8×Q8—791no—
320C8×Q8—791no—
321C4×Q16—791no—
322C4×SD16—791no—
323C4×SD16—791no—
324C4×D8—791no—
325#436—791no—
326#436—791no—
327#436—791no—
328#436—1581no—
329#436—1581no—
330#436—1581no—
331#435—1581no—
332C8×D4—791no—
333#433—1581no—
334#432—791no—
335#432—1581no—
336#431—791no—
337#431C2×D7911yescharacteristic
338#405—1581no—
339C4×M16—1581no—
340C4×M16C2×D7911yescharacteristic
341C2×C4×C8C2×D7911yescharacteristic
342#335—791no—
343#335—791no—
344#334—791no—
345#334—791no—
346Q8⋊C8—791no—
347Q8⋊C8—791no—
348Q8⋊C8—791no—
349Q8⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8—791no—
352D4⋊C8—791no—
353D4⋊C8—791no—
354#322C2×D7911yescharacteristic
355#322C2×D7911yescharacteristic
356#322C2×D7911yescharacteristic
357#322C2×D7911yescharacteristic
358C2×#41—791no—
359C2×Q16—791no—
360C2×SD16—791no—
361C2×SD16—791no—
362C2×D8—791no—
363C2×M16—1581no—
364C2×M16—1581no—
365C2×M16—1581no—
366C2×M16D31611yescharacteristic
367C2×M16D31611yescharacteristic
368C22×C8—21no—
369C22×C8—1581no—
370#126—1581no—
371#125—791no—
372#124—791no—
373#123—1581no—
374C22⋊Q8—791no—
375#121—791no—
376C4×Q8—791no—
377C4×Q8—791no—
378C4×Q8—791no—
379C4×D4—791no—
380C4×D4—791no—
381C4×D4—791no—
382C4×D4—1581no—
383C4×D4—1581no—
384C4×D4—1581no—
385#117—791no—
386#117—1581no—
387#117—1581no—
388C2×C42—1581no—
389C2×C42C22×D7911yescharacteristic
390#107—791no—
391#107—791no—
392#106—791no—
393#106—791no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8—1581no—
402C4⋊C8C4×D7911yescharacteristic
403C4⋊C8C4×D7911yescharacteristic
404#103—791no—
405#103—791no—
406#103—791no—
407#103—791no—
408#102—791no—
409#102—791no—
410#102—791no—
411#102—791no—
412C22⋊C8—1581no—
413C22⋊C8—1581no—
414C22⋊C8—1581no—
415C22⋊C8—1581no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8C4×D7911yescharacteristic
419C22⋊C8C4×D7911yescharacteristic
420#97—1581no—
421#97—1581no—
422#97D31611yescharacteristic
423#97D31611yescharacteristic
424C4×C8—21no—
425C4×C8—1581no—
426C4×C8—1581no—
427C4×C8—1581no—
428C4×C8—1581no—
429C4×C8C22×D7911yescharacteristic
430C4×C8C22×D7911yescharacteristic
431C4×C8C22×D7911yescharacteristic
432C4×C8C22×D7911yescharacteristic
433C4×C8C22×D7911yescharacteristic
434C4×C8C22×D7911yescharacteristic
435#41—3161no—
436#41—1582no—
437C2×Q8—791no—
438C2×Q8—791no—
439C2×D4—791no—
440C2×D4—791no—
441C2×D4—1581no—
442C2×D4—1581no—
443C22×C4—1581no—
444C22×C4—1581no—
445C22×C4—1581no—
446C22×C4—1581no—
447C22×C4C2×C4×D7911yescharacteristic
448C22×C4C2×D31611yescharacteristic
449C22×C4#1192962411yescharacteristic
450Q16—1582no—
451SD16—3161no—
452SD16—1582no—
453D8—3161no—
454M16—41no—
455M16—41no—
456M16—3162no—
457M16—3162no—
458M16—3162no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—42no—
462C2×C8—3162no—
463C2×C8—1581no—
464C2×C8—1581no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8C2×C4×D7911yescharacteristic
468C2×C8C2×C4×D7911yescharacteristic
469C2×C8C2×C4×D7911yescharacteristic
470C2×C8C2×C4×D7911yescharacteristic
471C2×C8C2×D31611yescharacteristic
472C2×C8C2×D31611yescharacteristic
473C2×C8C2×D31611yescharacteristic
474C2×C8C2×D31611yescharacteristic
475C4⋊C4—791no—
476C4⋊C4—791no—
477C4⋊C4—791no—
478C4⋊C4—791no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊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—1581no—
494C42—1581no—
495C42—1581no—
496C42—1581no—
497C42C2×C4×D7911yescharacteristic
498C42C2×D31611yescharacteristic
499C42#1192962411yescharacteristic
500C42C23×D7911yescharacteristic
501C23—1581no—
502C23—1581no—
503C23C4×D31611yescharacteristic
504Q8—792no—
505Q8—1581no—
506Q8—1581no—
507Q8—1581no—
508D4—3162no—
509D4—6321no—
510D4—1581no—
511D4—1581no—
512D4—1581no—
513D4—1581no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—3162no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—6321no—
525C2×C4—1581no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4C4×D31611yescharacteristic
532C2×C4C4×D31611yescharacteristic
533C2×C4C4×D31611yescharacteristic
534C2×C4C22×C4×D7911yescharacteristic
535C2×C4C22×D31611yescharacteristic
536C2×C4C2×#1192962411yescharacteristic
537C8—22no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—21no—
543C8—21no—
544C8—21no—
545C8—21no—
546C8—41no—
547C8—3161no—
548C8—3161no—
549C8—3161no—
550C8—3161no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—3161no—
555C22—3161no—
556C22—3161no—
557C22—3161no—
558C22—3161no—
559C22—3161no—
560C22C2×C4×D31611yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—3161no—
566C4—3161no—
567C4—3161no—
568C4—3161no—
569C4—3161no—
570C4—1582no—
571C4#131×D7911yescharacteristic
572C4#1909932811yescharacteristic
573C4#1909936311yescharacteristic
574C4#1909936311yescharacteristic
575C2—41no—
576C2—6321no—
577C2—6321no—
578C2#2377212811yescharacteristic
579C2#2377220211yescharacteristic
580C2#2377231111yescharacteristic
subgroupG/H
centerC2×C4C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1451C2
socleC22×C79C2×C4×D4
prime radical#19098710C4
cosocleC42×C79C24
hypercenter#1451D79
hypocenterC79#18005
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1451

Complements

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

NH ≅ G/Ntype
2. #23772418C2semidirect
3. #23772404C2semidirect
4. #23762315C2semidirect
5. #23762304C2semidirect
6. #23762280C2semidirect
7. #23762280C2semidirect
8. #23762273C2semidirect
9. #23762266C2semidirect
10. #23762220C2semidirect
11. #23762214C2semidirect
12. #23762180C2semidirect
13. #23761861C2semidirect
14. #23761783C2semidirect
15. C79×#1451C2semidirect
305. #1451D79semidirect
306. C79#18005semidirect

Maximal subgroups

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

subgroupindex
#237734202
#237724182
#237724042
#237623152
#237623042
#237622802
#237622732
#237622662
#237622202
#237622142
#237621802
#237618612
#237617832
C79×#14512
#1800579