Finite Group Databasebeta

← #34258955

Subgroups — #34258955

Order 20224.

592 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773966C211yesmaximal, characteristic
2#23772550C211yesmaximal, characteristic, complemented
3#23772548C211yesmaximal, characteristic, complemented
4#23762303C211yesmaximal, characteristic, complemented
5#23762301C211yesmaximal, characteristic, complemented
6#23762294C211yesmaximal, characteristic, complemented
7#23762282C211yesmaximal, characteristic, complemented
8#23762214C211yesmaximal, characteristic, complemented
9C8×#11929614C211yesmaximal, characteristic, complemented
10C8×#11929609C211yesmaximal, characteristic, complemented
11#23762157C211yesmaximal, characteristic, complemented
12#23761864C211yesmaximal, characteristic, complemented
13#23761862C211yesmaximal, characteristic, complemented
14#23761685C211yesmaximal, characteristic, complemented
15C79×SD16⋊C8C211yesmaximal, characteristic, complemented
16#19099100C2211yescharacteristic
17#19099095C2211yescharacteristic
18#19098963C411yescharacteristic
19#19098944C411yescharacteristic
20#19098919C411yescharacteristic
21#19098916C411yescharacteristic
22#19098803C411yescharacteristic
23#19098798C411yescharacteristic
24#19098793C411yescharacteristic
25#19098789C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098690C2211yescharacteristic
28C8×Dic158C2211yescharacteristic
29C4×#11929614C2211yescharacteristic
30C4×#11929613C2211yescharacteristic
31C4×#11929609C2211yescharacteristic
32#19098642—21no—
33#19098641C2211yescharacteristic
34#19098640C2211yescharacteristic
35#19098638—21no—
36#19098634—21no—
37#19098633—21no—
38C8×D316C2211yescharacteristic
39#19098622—21no—
40#19098621—21no—
41#19098614C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098589—21no—
44#19098588—21no—
45#19098585C2211yescharacteristic
46#19098585C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098315C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098284C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#335×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×#11929607—21no—
74C2×C8×D79—21no—
75C2×C8×D79C2×C411yescharacteristic
76C2×C8×D79C2×C411yescharacteristic
77#14579156—21no—
78#14579154—21no—
79#14579152C2×C411yescharacteristic
80#14579152C2×C411yescharacteristic
81#14579150C2×C411yescharacteristic
82#14579148C2×C411yescharacteristic
83#14579146—21no—
84#14579143—21no—
85#14579141C2×C411yescharacteristic
86#14579137C2×C411yescharacteristic
87#14579136C2×C411yescharacteristic
88#14579132C2×C411yescharacteristic
89C4×Dic158C2311yescharacteristic
90C4×D316—21no—
91C4×D316C2311yescharacteristic
92C4×C22⋊D79—21no—
93#14579120—21no—
94#14579118—21no—
95C42×D79C2311yescharacteristic
96#14579094C2×C411yescharacteristic
97#14579094C2×C411yescharacteristic
98#14579093C2×C411yescharacteristic
99C4⋊(C79⋊C8)—21no—
100C4⋊(C79⋊C8)C2311yescharacteristic
101C4⋊(C79⋊C8)C2311yescharacteristic
102Dic79⋊C8—21no—
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)⋊C8—21no—
119(C2×D79)⋊C8D411yescharacteristic
120(C2×D79)⋊C8D411yescharacteristic
121#14579072—21no—
122#14579071—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—
140C4×C8×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—22no—
154#11929609—22no—
155#11929608—81no—
156#11929607—81no—
157#11929607—81no—
158#11929607—81no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C22×C411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C8×D79—22no—
167C8×D79—22no—
168C8×D79—81no—
169C4⋊Dic79—21no—
170C4⋊Dic79—21no—
171C4⋊Dic79—21no—
172C4⋊Dic79C22×C411yescharacteristic
173C4⋊Dic79C22×C411yescharacteristic
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178C22⋊Dic79—21no—
179C22⋊Dic79—21no—
180C22⋊Dic79—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—
186(C2×D79)⋊C4—21no—
187C4×Dic79—21no—
188C4×Dic79C22×C411yescharacteristic
189C4×Dic79C2×D411yescharacteristic
190C4×Dic79#4111yescharacteristic
191C2×Q8×C79C22×C411yescharacteristic
192C2×D4×C79C22×C411yescharacteristic
193C22×C4×C79—21no—
194SD16×C79—22no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—81no—
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C2×D411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C4⋊C4×C79—21no—
204C4⋊C4×C79C22×C411yescharacteristic
205C4⋊C4×C79C22×C411yescharacteristic
206C22⋊C4×C79—21no—
207C42×C79—21no—
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C22×D79—21no—
211C22×D79C4×D411yescharacteristic
212Q8×C79—21no—
213Q8×C79—21no—
214D4×C79—21no—
215D4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79C23×C411yescharacteristic
222C2×C4×C79C22×D411yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C22⋊D79—81no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79C4×D411yescharacteristic
229C2×Dic79C4×D411yescharacteristic
230C2×Dic79C4×D411yescharacteristic
231D316—21no—
232D316—21no—
233D316—81no—
234C4×D79—21no—
235C4×D79—21no—
236C4×D79—21no—
237C4×D79—21no—
238C4×D79—41no—
239C4×D79—41no—
240C4×D79—41no—
241Dic158—21no—
242Dic158—21no—
243C8×C79—21no—
244C8×C79—21no—
245C8×C79—21no—
246C8×C79—41no—
247C8×C79—41no—
248C8×C79—41no—
249C8×C79#13112yes—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C79⋊C8—41no—
256C79⋊C8#13112yes—
257C22×C79—41no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79C2×C4×D411yescharacteristic
261C2×D79—21no—
262C2×D79—21no—
263C2×D79—21no—
264C2×D79—41no—
265C2×D79—41no—
266C2×D79—41no—
267C4×C79—21no—
268C4×C79—41no—
269C4×C79—41no—
270C4×C79—41no—
271C4×C79C2×#13111yescharacteristic
272C4×C79C2×#13111yescharacteristic
273C4×C79#57611yescharacteristic
274C4×C79#57811yescharacteristic
275Dic79—21no—
276Dic79—21no—
277Dic79—21no—
278Dic79—41no—
279Dic79—41no—
280Dic79—41no—
281#17973—791nomaximal, Sylow
282C2×C79—81no—
283C2×C79#293111yescharacteristic
284C2×C79#295111yescharacteristic
285C2×C79#295711yescharacteristic
286D79—41no—
287D79—81no—
288#3168—791no—
289#2993—791no—
290#2993—791no—
291#1592—791no—
292#1592—791no—
293#1591—791no—
294#1589—791no—
295SD16⋊C8—791no—
296SD16⋊C8D7911yescharacteristic, complemented
297C8×Q16—791no—
298C8×D8—791no—
299#1567—791no—
300#1495—791no—
301#1495—791no—
302#1450—791no—
303C79#1797311yescharacteristic, complemented, Sylow
304#529—791no—
305#529—791no—
306#499—791no—
307#495—791no—
308#488—791no—
309#488—791no—
310#467—791no—
311#466—791no—
312#465—791no—
313#465—791no—
314#446—791no—
315#446—791no—
316C8×Q8—791no—
317C8×Q8C2×D7911yescharacteristic
318C4×Q16—791no—
319C4×SD16—791no—
320C4×SD16C2×D7911yescharacteristic
321C4×D8—791no—
322#436—791no—
323#436—791no—
324#436—1581no—
325#436—1581no—
326#435—1581no—
327#435—1581no—
328C8×D4—791no—
329C8×D4C2×D7911yescharacteristic
330#433—1581no—
331#433—1581no—
332#431—791no—
333#431—791no—
334#424—1581no—
335#424—1581no—
336#423—791no—
337#423—791no—
338C2×C4×C8—791no—
339#335—791no—
340#335—791no—
341#335—791no—
342#335C2×D7911yescharacteristic
343Q8⋊C8—791no—
344Q8⋊C8—791no—
345Q8⋊C8—791no—
346Q8⋊C8C2×D7911yescharacteristic
347D4⋊C8—791no—
348D4⋊C8—791no—
349D4⋊C8—791no—
350D4⋊C8C2×D7911yescharacteristic
351#322—791no—
352#322C2×D7911yescharacteristic
353C82—791no—
354C82—791no—
355C2×Q16—791no—
356C2×SD16—791no—
357C2×SD16C4×D7911yescharacteristic
358C2×D8—791no—
359C2×M16—1581no—
360C2×M16—1581no—
361C2×M16—1581no—
362C2×M16—1581no—
363C22×C8—21no—
364C22×C8—791no—
365C22×C8—791no—
366C22×C8—1581no—
367#126—1581no—
368#126—1581no—
369#125—791no—
370#125—791no—
371#124—791no—
372#124—791no—
373#123—1581no—
374#123—1581no—
375C22⋊Q8—791no—
376C22⋊Q8—791no—
377#121—791no—
378#121—791no—
379C4×Q8—791no—
380C4×Q8C22×D7911yescharacteristic
381C4×D4—791no—
382C4×D4—1581no—
383C4×D4—1581no—
384C4×D4C22×D7911yescharacteristic
385#117—1581no—
386#117—1581no—
387C2×C42—791no—
388#107—791no—
389#107—791no—
390#106—791no—
391#106C4×D7911yescharacteristic
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⋊C8C22×D7911yescharacteristic
404C4⋊C8C22×D7911yescharacteristic
405#103—791no—
406#103—791no—
407#103—791no—
408#103C4×D7911yescharacteristic
409#102—791no—
410#102—791no—
411#102—791no—
412#102C4×D7911yescharacteristic
413C22⋊C8—21no—
414C22⋊C8—791no—
415C22⋊C8—791no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420#97—1581no—
421#97—1581no—
422C4×C8—21no—
423C4×C8—791no—
424C4×C8—791no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—791no—
428C4×C8—1581no—
429C4×C8C22×D7911yescharacteristic
430C4×C8C22×D7911yescharacteristic
431C4×C8C22×D7911yescharacteristic
432C2×Q8—791no—
433C2×Q8C2×C4×D7911yescharacteristic
434C2×D4—791no—
435C2×D4—1581no—
436C2×D4—1581no—
437C2×D4C2×C4×D7911yescharacteristic
438C22×C4—21no—
439C22×C4—791no—
440C22×C4—791no—
441C22×C4—791no—
442C22×C4—1581no—
443Q16—1582no—
444SD16—22no—
445SD16—1582no—
446D8—1582no—
447M16—6321no—
448M16—6321no—
449M16—6321no—
450M16—6321no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—81no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—791no—
459C2×C8—6321no—
460C2×C8—1582no—
461C2×C8—1581no—
462C2×C8—1581no—
463C2×C8—1581no—
464C2×C8—1582no—
465C2×C8C2×C4×D7911yescharacteristic
466C2×C8C2×C4×D7911yescharacteristic
467C2×C8D4×D7911yescharacteristic
468C2×C8D4×D7911yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—791no—
471C4⋊C4—791no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4C2×C4×D7911yescharacteristic
480C4⋊C4C2×C4×D7911yescharacteristic
481C22⋊C4—21no—
482C22⋊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—21no—
492C42—791no—
493C42—791no—
494C42—791no—
495C42—1581no—
496C42C23×D7911yescharacteristic
497C23—21no—
498C23—791no—
499C23—1581no—
500Q8—21no—
501Q8—21no—
502Q8—1581no—
503Q8—1581no—
504D4—21no—
505D4—21no—
506D4—6321no—
507D4—6321no—
508D4—1581no—
509D4—1581no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—3161no—
516C2×C4—3161no—
517C2×C4—3161no—
518C2×C4—791no—
519C2×C4—791no—
520C2×C4—791no—
521C2×C4—1581no—
522C2×C4—1581no—
523C2×C4—1581no—
524C2×C4—1581no—
525C2×C4—1581no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4C22×C4×D7911yescharacteristic
529C2×C4C2×D4×D7911yescharacteristic
530C2×C4#41×D7911yescharacteristic
531C8—21no—
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—3161no—
538C8—3161no—
539C8—3161no—
540C8—792no—
541C8—1581no—
542C8—1581no—
543C8—1581no—
544C8#1457916712yes—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—3161no—
549C22—3161no—
550C22—3161no—
551C22—1581no—
552C22—1581no—
553C22—1581no—
554C22C4×D4×D7911yescharacteristic
555C4—21no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—3161no—
560C4—3161no—
561C4—3161no—
562C4—1581no—
563C4—1581no—
564C4—1581no—
565C4#131×D7911yescharacteristic
566C4C2×#1457916711yescharacteristic
567C4#1909936711yescharacteristic
568C4#1909937911yescharacteristic
569C2—81no—
570C2—3161no—
571C2—6321no—
572C2#2377210711yescharacteristic
573C2#2377227211yescharacteristic
574C2#2377231411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×SD16⋊C8C2
socleC22×C79C2×C4×D4
prime radical#19098798C4
cosocleC42×C79C24
hypercenterSD16⋊C8D79
hypocenterC79#17973
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #23772550C2semidirect
3. #23772548C2semidirect
4. #23762303C2semidirect
5. #23762301C2semidirect
6. #23762294C2semidirect
7. #23762282C2semidirect
8. #23762214C2semidirect
9. C8×#11929614C2semidirect
10. C8×#11929609C2semidirect
11. #23762157C2semidirect
12. #23761864C2semidirect
13. #23761862C2semidirect
14. #23761685C2semidirect
15. C79×SD16⋊C8C2semidirect
296. SD16⋊C8D79semidirect
303. C79#17973semidirect

Maximal subgroups

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

subgroupindex
#237739662
#237725502
#237725482
#237623032
#237623012
#237622942
#237622822
#237622142
C8×#119296142
C8×#119296092
#237621572
#237618642
#237618622
#237616852
C79×SD16⋊C82
#1797379