Finite Group Databasebeta

← #34259682

Subgroups — #34259682

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773420C211yesmaximal, characteristic
2#23772562C211yesmaximal, characteristic, complemented
3#23772529C211yesmaximal, characteristic, complemented
4#23762304C211yesmaximal, characteristic, complemented
5#23762286C211yesmaximal, characteristic, complemented
6#23762280C211yesmaximal, characteristic, complemented
7#23762280C211yesmaximal, characteristic, complemented
8#23762252C211yesmaximal, characteristic, complemented
9#23762231C211yesmaximal, characteristic, complemented
10#23762214C211yesmaximal, characteristic, complemented
11C8×Dic316C211yesmaximal, characteristic, complemented
12#23762175C211yesmaximal, characteristic, complemented
13#23761866C211yesmaximal, characteristic, complemented
14#23761788C211yesmaximal, characteristic, complemented
15C79×#1454C211yesmaximal, 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#19098638—21no—
38#19098638—21no—
39#19098631—21no—
40C8×D316C2211yescharacteristic
41(C22⋊D79)⋊C8—21no—
42#19098623—21no—
43#19098615—21no—
44#19098588—21no—
45#19098587C2211yescharacteristic
46#19098585—21no—
47C2×C4⋊(C79⋊C8)C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098316C2211yescharacteristic
50#19098316C2211yescharacteristic
51#19098316C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098295C2211yescharacteristic
55#19098295C2211yescharacteristic
56#19098292C2211yescharacteristic
57#19098292C2211yescharacteristic
58#19098292C2211yescharacteristic
59#19098292C2211yescharacteristic
60#431×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62#431×C79C2211yescharacteristic
63#322×C79C2211yescharacteristic
64#322×C79C2211yescharacteristic
65#322×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929624C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929613C2×C411yescharacteristic
71C2×D632C2×C411yescharacteristic
72C2×#11929608—21no—
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×#11929607—21no—
76C22×C79⋊C8—21no—
77C2×C8×D79—21no—
78#14579155—21no—
79#14579153C2×C411yescharacteristic
80#14579149C2×C411yescharacteristic
81#14579144—21no—
82#14579140C2×C411yescharacteristic
83#14579135C2×C411yescharacteristic
84C4×Dic158C2×C411yescharacteristic
85C4×Dic158C2311yescharacteristic
86C4×Dic158C2311yescharacteristic
87C4×D316C2×C411yescharacteristic
88C4×D316C2311yescharacteristic
89C4×D316C2311yescharacteristic
90C4×C22⋊D79—21no—
91C4×C22⋊D79—21no—
92C4×C22⋊D79—21no—
93#14579120C2×C411yescharacteristic
94#14579118—21no—
95#14579118—21no—
96C42×D79—21no—
97#14579095C2×C411yescharacteristic
98#14579095C2×C411yescharacteristic
99#14579093C2×C411yescharacteristic
100#14579093C2×C411yescharacteristic
101C4⋊(C79⋊C8)—21no—
102C4⋊(C79⋊C8)—21no—
103C4⋊(C79⋊C8)C2311yescharacteristic
104C4⋊(C79⋊C8)C2311yescharacteristic
105C4⋊(C79⋊C8)C2311yescharacteristic
106C4⋊(C79⋊C8)C2311yescharacteristic
107Dic79⋊C8—21no—
108Dic79⋊C8—21no—
109Dic79⋊C8—21no—
110Dic79⋊C8—21no—
111#14579086C2×C411yescharacteristic
112#14579086C2×C411yescharacteristic
113#14579086C2×C411yescharacteristic
114#14579086C2×C411yescharacteristic
115#14579083C2×C411yescharacteristic
116#14579083C2×C411yescharacteristic
117#14579083C2×C411yescharacteristic
118#14579083C2×C411yescharacteristic
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8—21no—
121(C2×D79)⋊C8—21no—
122(C2×D79)⋊C8—21no—
123#14579071—21no—
124C8×Dic79—21no—
125C8×Dic79—21no—
126C8×Dic79—21no—
127C2×C42×C79C2311yescharacteristic
128C4⋊C8×C79—21no—
129C4⋊C8×C79C2×C411yescharacteristic
130C4⋊C8×C79C2×C411yescharacteristic
131C4⋊C8×C79D411yescharacteristic
132C4⋊C8×C79D411yescharacteristic
133C22⋊C8×C79—21no—
134C22⋊C8×C79C2×C411yescharacteristic
135C22⋊C8×C79C2×C411yescharacteristic
136C22⋊C8×C79D411yescharacteristic
137C22⋊C8×C79D411yescharacteristic
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#11929608—81no—
161#11929607—81no—
162#11929607—81no—
163#11929607—81no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8—21no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8—81no—
169C8×D79—81no—
170C4⋊Dic79C22×C411yescharacteristic
171C4⋊Dic79C22×C411yescharacteristic
172C4⋊Dic79C22×C411yescharacteristic
173C4⋊Dic79C22×C411yescharacteristic
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—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×Dic79—21no—
190C4×Dic79—21no—
191C4×Dic79—21no—
192C22×C4×C79C22×C411yescharacteristic
193C22×C4×C79C2×D411yescharacteristic
194C22×C4×C79#4111yescharacteristic
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79C22×C411yescharacteristic
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C2×D411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C2×C8×C79C2×D411yescharacteristic
204C2×C8×C79C2×D411yescharacteristic
205C42×C79C22×C411yescharacteristic
206C42×C79C2×D411yescharacteristic
207C42×C79#4111yescharacteristic
208C42×C79C2411yescharacteristic
209C23×C79C4×D411yescharacteristic
210C22×D79—21no—
211C22×D79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—41no—
217C2×C4×C79C4×D411yescharacteristic
218C2×C4×C79C4×D411yescharacteristic
219C2×C4×C79C4×D411yescharacteristic
220C2×C4×C79C23×C411yescharacteristic
221C2×C4×C79C22×D411yescharacteristic
222C2×C4×C79C2×#4111yescharacteristic
223C22⋊D79—42no—
224C22⋊D79—42no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231D316—21no—
232D316—21no—
233D316—21no—
234D316—21no—
235C4×D79—42no—
236C4×D79—42no—
237C4×D79—41no—
238C4×D79—41no—
239C4×D79—41no—
240C4×D79—41no—
241Dic158—21no—
242Dic158—21no—
243Dic158—21no—
244Dic158#13112yes—
245C8×C79—21no—
246C8×C79—21no—
247C8×C79—21no—
248C8×C79—21no—
249C8×C79—21no—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—21no—
253C8×C79—41no—
254C8×C79—41no—
255C79⋊C8—41no—
256C79⋊C8—41no—
257C79⋊C8—41no—
258C79⋊C8—41no—
259C22×C79—21no—
260C22×C79—21no—
261C22×C79—21no—
262C22×C79C2×C4×D411yescharacteristic
263C2×D79—41no—
264C2×D79—41no—
265C2×D79—41no—
266C2×D79—41no—
267C2×D79—41no—
268C2×D79—41no—
269C4×C79—21no—
270C4×C79—21no—
271C4×C79—21no—
272C4×C79—21no—
273C4×C79C2×#13111yescharacteristic
274C4×C79#56811yescharacteristic
275C4×C79#57611yescharacteristic
276C4×C79#57811yescharacteristic
277Dic79—22no—
278Dic79—41no—
279Dic79—41no—
280Dic79—41no—
281Dic79—41no—
282Dic79—41no—
283#18033—791nomaximal, Sylow
284C2×C79—41no—
285C2×C79#293311yescharacteristic
286C2×C79#295111yescharacteristic
287C2×C79#295511yescharacteristic
288D79—81no—
289D79—81no—
290#3102—791no—
291#2994—791no—
292#2991—791no—
293#1592—791no—
294#1589—791no—
295#1588—791no—
296#1588—791no—
297#1584—791no—
298#1581—791no—
299SD16⋊C8—791no—
300C8×Q16—791no—
301#1571—791no—
302#1495—791no—
303#1477—791no—
304#1454D7911yescharacteristic, complemented
305C79#1803311yescharacteristic, complemented, Sylow
306#529—791no—
307C4×#41—791no—
308#485—791no—
309#482—791no—
310#479—791no—
311#478—791no—
312#464—791no—
313#463—791no—
314#452—791no—
315#449—791no—
316#446—791no—
317C8×Q8—791no—
318C8×Q8—791no—
319C8×Q8—791no—
320C4×Q16—791no—
321C4×SD16—791no—
322C4×SD16—791no—
323C4×D8—791no—
324#436—791no—
325#436—791no—
326#436—791no—
327#436—1581no—
328#436—1581no—
329#435—1581no—
330C8×D4—791no—
331C8×D4—1581no—
332#433—1581no—
333#431—1581no—
334#431C2×D7911yescharacteristic
335#431C2×D7911yescharacteristic
336#431C2×D7911yescharacteristic
337#424—1581no—
338#423—791no—
339#423—1581no—
340C2×C4⋊C8—791no—
341#335—791no—
342#335—791no—
343#335—791no—
344#335—791no—
345Q8⋊C8—791no—
346Q8⋊C8—791no—
347Q8⋊C8—791no—
348Q8⋊C8—791no—
349D4⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8—791no—
352D4⋊C8—791no—
353#322C2×D7911yescharacteristic
354#322C2×D7911yescharacteristic
355#322C2×D7911yescharacteristic
356C82C2×D7911yescharacteristic
357C2×#41—791no—
358C2×Q16—791no—
359C2×SD16—791no—
360C2×SD16—791no—
361C2×D8—791no—
362C2×M16—1581no—
363C2×M16—1581no—
364C2×M16—1581no—
365C2×M16—1581no—
366C22×C8—1581no—
367C22×C8—1581no—
368#126—1581no—
369#125—791no—
370#124—791no—
371#123—1581no—
372C22⋊Q8—791no—
373#121—791no—
374C4×Q8—791no—
375C4×Q8—791no—
376C4×Q8—791no—
377C4×D4—791no—
378C4×D4—791no—
379C4×D4—791no—
380C4×D4—1581no—
381C4×D4—1581no—
382C4×D4—1581no—
383#117—791no—
384#117—1581no—
385#117—1581no—
386C2×C42—1581no—
387C2×C42C22×D7911yescharacteristic
388#107—791no—
389#107—791no—
390#106—791no—
391#106—791no—
392C4⋊C8—21no—
393C4⋊C8—791no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—1581no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8—1581no—
402C4⋊C8—1581no—
403C4⋊C8C4×D7911yescharacteristic
404C4⋊C8C4×D7911yescharacteristic
405C4⋊C8D31611yescharacteristic
406C4⋊C8D31611yescharacteristic
407#103—791no—
408#103—791no—
409#103—791no—
410#103—791no—
411#102—791no—
412#102—791no—
413#102—791no—
414#102—791no—
415C22⋊C8—21no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420C22⋊C8C4×D7911yescharacteristic
421C22⋊C8C4×D7911yescharacteristic
422C22⋊C8D31611yescharacteristic
423C22⋊C8D31611yescharacteristic
424#97—1581no—
425C4×C8—1581no—
426C4×C8—1581no—
427C4×C8—1581no—
428C4×C8C22×D7911yescharacteristic
429C4×C8C22×D7911yescharacteristic
430C4×C8C22×D7911yescharacteristic
431C4×C8C22×D7911yescharacteristic
432C4×C8C22×D7911yescharacteristic
433C4×C8C22×D7911yescharacteristic
434#41—3161no—
435#41—1582no—
436C2×Q8—791no—
437C2×Q8—791no—
438C2×D4—791no—
439C2×D4—791no—
440C2×D4—1581no—
441C2×D4—1581no—
442C22×C4—1581no—
443C22×C4—1581no—
444C22×C4—1581no—
445C22×C4—1581no—
446C22×C4C2×C4×D7911yescharacteristic
447C22×C4C2×D31611yescharacteristic
448C22×C4#1192962411yescharacteristic
449Q16—1582no—
450SD16—3161no—
451SD16—1582no—
452D8—3161no—
453M16—6321no—
454M16—6321no—
455M16—6321no—
456M16—6321no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—6321no—
460C2×C8—6321no—
461C2×C8—1581no—
462C2×C8—1581no—
463C2×C8—1581no—
464C2×C8—1581no—
465C2×C8C2×C4×D7911yescharacteristic
466C2×C8C2×C4×D7911yescharacteristic
467C2×C8C2×C4×D7911yescharacteristic
468C2×C8C2×C4×D7911yescharacteristic
469C2×C8C2×D31611yescharacteristic
470C2×C8C2×D31611yescharacteristic
471C2×C8C2×D31611yescharacteristic
472C2×C8C2×D31611yescharacteristic
473C4⋊C4—791no—
474C4⋊C4—791no—
475C4⋊C4—791no—
476C4⋊C4—791no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C22⋊C4—1581no—
484C22⋊C4—1581no—
485C22⋊C4—1581no—
486C22⋊C4—1581no—
487C22⋊C4—1581no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C42—1581no—
492C42—1581no—
493C42—1581no—
494C42—1581no—
495C42C2×C4×D7911yescharacteristic
496C42C2×D31611yescharacteristic
497C42#1192962411yescharacteristic
498C42C23×D7911yescharacteristic
499C23—1581no—
500C23—1581no—
501C23C4×D31611yescharacteristic
502Q8—792no—
503Q8—1581no—
504Q8—1581no—
505Q8—1581no—
506D4—3162no—
507D4—3162no—
508D4—1581no—
509D4—1581no—
510D4—1581no—
511D4—1581no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—3162no—
518C2×C4—3162no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—1581no—
524C2×C4—1581no—
525C2×C4—1581no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4C4×D31611yescharacteristic
530C2×C4C4×D31611yescharacteristic
531C2×C4C4×D31611yescharacteristic
532C2×C4C22×C4×D7911yescharacteristic
533C2×C4C22×D31611yescharacteristic
534C2×C4C2×#1192962411yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—21no—
543C8—41no—
544C8—41no—
545C8—3161no—
546C8—3161no—
547C8—3161no—
548C8—3161no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—3161no—
553C22—3161no—
554C22—3161no—
555C22—3161no—
556C22—3161no—
557C22—3161no—
558C22C2×C4×D31611yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—3161no—
564C4—3161no—
565C4—3161no—
566C4—3161no—
567C4—3161no—
568C4—1582no—
569C4#131×D7911yescharacteristic
570C4#1909932811yescharacteristic
571C4#1909936811yescharacteristic
572C4#1909938011yescharacteristic
573C2—41no—
574C2—6321no—
575C2—6321no—
576C2#2377212811yescharacteristic
577C2#2377227711yescharacteristic
578C2#2377230411yescharacteristic
subgroupG/H
centerC2×C4C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1454C2
socleC22×C79C2×C4×D4
prime radical#19098710C4
cosocleC42×C79C24
hypercenter#1454D79
hypocenterC79#18033
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
2. #23772562C2semidirect
3. #23772529C2semidirect
4. #23762304C2semidirect
5. #23762286C2semidirect
6. #23762280C2semidirect
7. #23762280C2semidirect
8. #23762252C2semidirect
9. #23762231C2semidirect
10. #23762214C2semidirect
11. C8×Dic316C2semidirect
12. #23762175C2semidirect
13. #23761866C2semidirect
14. #23761788C2semidirect
15. C79×#1454C2semidirect
304. #1454D79semidirect
305. C79#18033semidirect

Maximal subgroups

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

subgroupindex
#237734202
#237725622
#237725292
#237623042
#237622862
#237622802
#237622522
#237622312
#237622142
C8×Dic3162
#237621752
#237618662
#237617882
C79×#14542
#1803379