Finite Group Databasebeta

← #34259455

Subgroups — #34259455

Order 20224.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773478C211yesmaximal, characteristic
2#23772555C211yesmaximal, characteristic
3#23772488C211yesmaximal, characteristic, complemented
4#23762298C211yesmaximal, characteristic, complemented
5#23762293C211yesmaximal, characteristic, complemented
6#23762279C211yesmaximal, characteristic, complemented
7#23762271C211yesmaximal, characteristic, complemented
8#23762270C211yesmaximal, characteristic, complemented
9#23762257C211yesmaximal, characteristic, complemented
10#23762204C211yesmaximal, characteristic, complemented
11#23762182C211yesmaximal, characteristic, complemented
12#23761857C211yesmaximal, characteristic
13Q8⋊C8×D79C211yesmaximal, characteristic, complemented
14#23761682C211yesmaximal, characteristic, complemented
15C8×SD16×C79C211yesmaximal, characteristic, complemented
16#19099095C2211yescharacteristic
17C4×Q8×D79C2211yescharacteristic
18#19098895C411yescharacteristic
19#19098875C411yescharacteristic
20#19098873C411yescharacteristic
21#19098872C411yescharacteristic
22#19098784C411yescharacteristic
23#19098783C411yescharacteristic
24#19098720C411yescharacteristic
25#19098700C411yescharacteristic
26#19098693—21no—
27#19098693C2211yescharacteristic
28#19098693C2211yescharacteristic
29#19098690—21no—
30#19098690C2211yescharacteristic
31Q8×C79⋊C8C2211yescharacteristic
32#19098687C2211yescharacteristic
33C4×Dic316C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929612C2211yescharacteristic
36#19098641C2211yescharacteristic
37#19098637—21no—
38C8×C22⋊D79—21no—
39#19098621—21no—
40#19098619—21no—
41#19098619C2211yescharacteristic
42#19098614—21no—
43#19098614C2211yescharacteristic
44#19098614C2211yescharacteristic
45#19098589—21no—
46C4⋊C8×D79C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098293C2211yescharacteristic
56D4⋊(C79⋊C8)C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×Q8×D79C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×#11929614C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×C8×D79—21no—
73C2×C8×D79—21no—
74#14579154—21no—
75#14579152C2×C411yescharacteristic
76#14579148C2×C411yescharacteristic
77#14579143—21no—
78#14579141C2×C411yescharacteristic
79#14579132C2×C411yescharacteristic
80Q8×Dic79C2×C411yescharacteristic
81C4×Dic158—21no—
82C4×Dic158C2311yescharacteristic
83C4×Dic158C2311yescharacteristic
84#14579126—21no—
85#14579126C2×C411yescharacteristic
86C4×C22⋊D79—21no—
87#14579120—21no—
88C4⋊C4×D79—21no—
89C4⋊C4×D79C2×C411yescharacteristic
90C42×D79—21no—
91C42×D79C2311yescharacteristic
92#14579095C2×C411yescharacteristic
93#14579094C2×C411yescharacteristic
94#14579092C2×C411yescharacteristic
95C4⋊(C79⋊C8)—21no—
96C4⋊(C79⋊C8)C2311yescharacteristic
97C4⋊(C79⋊C8)C2311yescharacteristic
98Dic79⋊C8—21no—
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
105Dic79⋊C8D411yescharacteristic
106Dic79⋊C8D411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579086C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579084C2×C411yescharacteristic
112#14579082C2×C411yescharacteristic
113C22⋊(C79⋊C8)—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8C2×C411yescharacteristic
116(C2×D79)⋊C8C2×C411yescharacteristic
117(C2×D79)⋊C8D411yescharacteristic
118(C2×D79)⋊C8D411yescharacteristic
119#14579071—21no—
120#14579071—21no—
121#14579071—21no—
122#14579071—21no—
123C4×C79⋊C8—21no—
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79—21no—
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×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×Dic158C22×C411yescharacteristic
147Q8×D79—22no—
148Q8×D79—41no—
149C2×C22⋊D79—21no—
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79C22×C411yescharacteristic
153C2×C4×D79C2×D411yescharacteristic
154C2×C4×D79#4111yescharacteristic
155Dic316—41no—
156#11929614—22no—
157#11929612—41no—
158#11929608—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—24no—
167C8×D79—81no—
168C4⋊Dic79—21no—
169C4⋊Dic79—21no—
170C4⋊Dic79—21no—
171C4⋊Dic79C22×C411yescharacteristic
172C4⋊Dic79C22×C411yescharacteristic
173Dic79⋊C4—21no—
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—
182C4×Dic79—21no—
183C4×Dic79—21no—
184C4×Dic79—21no—
185C4×Dic79—21no—
186C4×Dic79C22×C411yescharacteristic
187C4×Dic79C2×D411yescharacteristic
188C4×Dic79#4111yescharacteristic
189C2×Q8×C79C22×C411yescharacteristic
190C2×D4×C79C22×C411yescharacteristic
191C22×C4×C79—21no—
192SD16×C79—22no—
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—81no—
197C2×C8×C79C22×C411yescharacteristic
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C2×D411yescharacteristic
200C2×C8×C79C2×D411yescharacteristic
201C4⋊C4×C79—21no—
202C4⋊C4×C79C22×C411yescharacteristic
203C4⋊C4×C79C22×C411yescharacteristic
204C22⋊C4×C79—21no—
205C42×C79—21no—
206C42×C79C2411yescharacteristic
207C23×C79—21no—
208C22×D79C4×D411yescharacteristic
209Q8×C79—21no—
210Q8×C79#13112yes—
211D4×C79—21no—
212D4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79C23×C411yescharacteristic
219C2×C4×C79C22×D411yescharacteristic
220C2×C4×C79C2×#4111yescharacteristic
221C22⋊D79—42no—
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79C4×D411yescharacteristic
227C2×Dic79C4×D411yescharacteristic
228C2×Dic79C4×D411yescharacteristic
229C4×D79—22no—
230C4×D79—21no—
231C4×D79—21no—
232C4×D79—21no—
233C4×D79—21no—
234C4×D79—42no—
235C4×D79—42no—
236Dic158—21no—
237Dic158—21no—
238Dic158—21no—
239Dic158—21no—
240Dic158—42no—
241C8×C79—22no—
242C8×C79—21no—
243C8×C79—21no—
244C8×C79—21no—
245C8×C79—21no—
246C8×C79—41no—
247C8×C79—41no—
248C79⋊C8—22no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C22×C79—41no—
256C22×C79—41no—
257C22×C79—41no—
258C22×C79C2×C4×D411yescharacteristic
259C2×D79—21no—
260C2×D79—21no—
261C2×D79#44312yes—
262C4×C79—22no—
263C4×C79—21no—
264C4×C79—41no—
265C4×C79—41no—
266C4×C79C2×#13111yescharacteristic
267C4×C79#56811yescharacteristic
268C4×C79#57711yescharacteristic
269C4×C79#57811yescharacteristic
270Dic79—21no—
271Dic79—21no—
272Dic79—21no—
273Dic79—41no—
274Dic79—41no—
275Dic79—41no—
276Dic79—41no—
277#18017—791nomaximal, Sylow
278C2×C79—81no—
279C2×C79#293311yescharacteristic
280C2×C79#294711yescharacteristic
281C2×C79C2×#44311yescharacteristic
282D79—22no—
283#3108—791no—
284#2994—791no—
285#2984—791no—
286#1591—791no—
287#1590—791no—
288#1588—791no—
289#1587—791no—
290#1587—791no—
291#1585—791no—
292C8×Q16—791no—
293C8×SD16D7911yescharacteristic, complemented
294#1573—791no—
295#1494—791no—
296C2×Q8⋊C8—791no—
297#1450—791no—
298C79#1801711yescharacteristic, complemented, Sylow
299#529—791no—
300C2×C4×Q8—791no—
301#483—791no—
302#480—791no—
303#479—791no—
304#479—791no—
305#464—791no—
306#464—791no—
307#451—791no—
308#448—791no—
309#446—791no—
310#446—791no—
311#446—791no—
312#446—1581no—
313#446—1581no—
314C8×Q8—791no—
315C8×Q8—791no—
316C8×Q8C2×D7911yescharacteristic
317C4×Q16—791no—
318C4×Q16—791no—
319C4×SD16—791no—
320C4×SD16C2×D7911yescharacteristic
321#436—791no—
322#435—1581no—
323C8×D4—1581no—
324C8×D4C2×D7911yescharacteristic
325#433—1581no—
326#432—791no—
327#432—1581no—
328#431—791no—
329#431—791no—
330#431—1581no—
331#424—1581no—
332C2×C4⋊C8—791no—
333C2×C4×C8—791no—
334#335—791no—
335#335—791no—
336#334—791no—
337#334C2×D7911yescharacteristic
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8—791no—
343Q8⋊C8C2×D7911yescharacteristic
344D4⋊C8—791no—
345D4⋊C8C2×D7911yescharacteristic
346#322—791no—
347#322—791no—
348C82—791no—
349C82C2×D7911yescharacteristic
350C22×Q8—791no—
351C2×Q16—791no—
352C2×Q16—791no—
353C2×SD16—791no—
354C2×SD16C4×D7911yescharacteristic
355C2×M16—1581no—
356C22×C8—21no—
357C22×C8—1581no—
358C22×C8—1581no—
359#126—1581no—
360#125—791no—
361#124—791no—
362#123—1581no—
363C22⋊Q8—791no—
364#121—791no—
365C4×Q8—791no—
366C4×Q8—791no—
367C4×Q8—791no—
368C4×Q8—791no—
369C4×Q8—1581no—
370C4×Q8—1581no—
371C4×Q8C22×D7911yescharacteristic
372C4×D4—1581no—
373C4×D4C22×D7911yescharacteristic
374#117—1581no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—1581no—
377C2×C42—791no—
378C2×C42—1581no—
379#107—791no—
380#107—791no—
381#106—791no—
382#106C4×D7911yescharacteristic
383C4⋊C8—21no—
384C4⋊C8—791no—
385C4⋊C8—791no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—791no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8—1581no—
396C4⋊C8C22×D7911yescharacteristic
397C4⋊C8C22×D7911yescharacteristic
398#103—791no—
399#103—791no—
400#103—791no—
401#103—791no—
402#103—791no—
403#103C4×D7911yescharacteristic
404#102—791no—
405#102C4×D7911yescharacteristic
406C22⋊C8—21no—
407C22⋊C8—791no—
408C22⋊C8—791no—
409C22⋊C8—791no—
410C22⋊C8—791no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—1581no—
414#97—1581no—
415#97—1581no—
416#97—1581no—
417C4×C8—21no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—791no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8—1581no—
424C4×C8C22×D7911yescharacteristic
425C4×C8C22×D7911yescharacteristic
426C4×C8C22×D7911yescharacteristic
427C2×Q8—3161no—
428C2×Q8—791no—
429C2×Q8—791no—
430C2×Q8—1581no—
431C2×Q8—1582no—
432C2×Q8C2×C4×D7911yescharacteristic
433C2×D4—1581no—
434C2×D4C2×C4×D7911yescharacteristic
435C22×C4—21no—
436C22×C4—791no—
437C22×C4—791no—
438C22×C4—791no—
439C22×C4—1581no—
440C22×C4—1581no—
441Q16—3161no—
442Q16—1582no—
443SD16—22no—
444SD16—3161no—
445M16—6321no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—81no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—791no—
453C2×C8—791no—
454C2×C8—6321no—
455C2×C8—1584no—
456C2×C8—1581no—
457C2×C8—1581no—
458C2×C8—1581no—
459C2×C8C2×C4×D7911yescharacteristic
460C2×C8C2×C4×D7911yescharacteristic
461C2×C8D4×D7911yescharacteristic
462C2×C8D4×D7911yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—791no—
465C4⋊C4—791no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4C2×C4×D7911yescharacteristic
475C4⋊C4C2×C4×D7911yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C42—21no—
482C42—791no—
483C42—791no—
484C42—791no—
485C42—1581no—
486C42—1581no—
487C42—1581no—
488C42—1581no—
489C42C23×D7911yescharacteristic
490C23—21no—
491C23—791no—
492Q8—21no—
493Q8—3162no—
494Q8—1581no—
495Q8—1581no—
496Q8—1581no—
497Q8—1581no—
498Q8#1457916712yes—
499D4—21no—
500D4—21no—
501D4—3162no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—3162no—
508C2×C4—3162no—
509C2×C4—791no—
510C2×C4—791no—
511C2×C4—791no—
512C2×C4—1582no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4—1581no—
520C2×C4—1581no—
521C2×C4C22×C4×D7911yescharacteristic
522C2×C4C2×D4×D7911yescharacteristic
523C2×C4#41×D7911yescharacteristic
524C8—22no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—3161no—
532C8—3161no—
533C8—1582no—
534C8—1581no—
535C8—1581no—
536C8—1581no—
537C8—1581no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—792no—
542C22—1581no—
543C22—1581no—
544C22C4×D4×D7911yescharacteristic
545C4—21no—
546C4—22no—
547C4—41no—
548C4—41no—
549C4—3161no—
550C4—3161no—
551C4—3161no—
552C4—3161no—
553C4—1581no—
554C4—1581no—
555C4—1581no—
556C4C2×#1457916711yescharacteristic
557C4#1909932811yescharacteristic
558C4#1909936911yescharacteristic
559C4#1909937711yescharacteristic
560C2—81no—
561C2—1582no—
562C2#2377212011yescharacteristic
563C2#2377223611yescharacteristic
564C2#443×D7911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC8×SD16×C79C2
socleC22×C79C2×C4×D4
prime radical#14579132C2×C4
cosocleC42×C79C24
hypercenterC8×SD16D79
hypocenterC79#18017
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
3. #23772488C2semidirect
4. #23762298C2semidirect
5. #23762293C2semidirect
6. #23762279C2semidirect
7. #23762271C2semidirect
8. #23762270C2semidirect
9. #23762257C2semidirect
10. #23762204C2semidirect
11. #23762182C2semidirect
13. Q8⋊C8×D79C2semidirect
14. #23761682C2semidirect
15. C8×SD16×C79C2semidirect
293. C8×SD16D79semidirect
298. C79#18017semidirect

Maximal subgroups

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

subgroupindex
#237734782
#237725552
#237724882
#237622982
#237622932
#237622792
#237622712
#237622702
#237622572
#237622042
#237621822
#237618572
Q8⋊C8×D792
#237616822
C8×SD16×C792
#1801779