Finite Group Databasebeta

← #34259463

Subgroups — #34259463

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#23773484C211yesmaximal, characteristic
2#23772563C211yesmaximal, characteristic
3#23772488C211yesmaximal, characteristic, complemented
4#23762295C211yesmaximal, characteristic, complemented
5#23762289C211yesmaximal, characteristic, complemented
6#23762275C211yesmaximal, characteristic, complemented
7#23762268C211yesmaximal, characteristic, complemented
8#23762259C211yesmaximal, characteristic, complemented
9Q16×C79⋊C8C211yesmaximal, characteristic, complemented
10C8×#11929611C211yesmaximal, characteristic, complemented
11#23762182C211yesmaximal, characteristic, complemented
12#23761859C211yesmaximal, characteristic
13Q8⋊C8×D79C211yesmaximal, characteristic, complemented
14#23761685C211yesmaximal, characteristic, complemented
15C79×#1587C211yesmaximal, characteristic, complemented
16#19099100C2211yescharacteristic
17C4×Q8×D79C2211yescharacteristic
18#19098892C411yescharacteristic
19#19098878C411yescharacteristic
20#19098869C411yescharacteristic
21#19098868C411yescharacteristic
22#19098788C411yescharacteristic
23#19098787C411yescharacteristic
24#19098717C411yescharacteristic
25#19098703C411yescharacteristic
26#19098693—21no—
27#19098693C2211yescharacteristic
28#19098690—21no—
29#19098690C2211yescharacteristic
30Q8×C79⋊C8C2211yescharacteristic
31Q8×C79⋊C8C2211yescharacteristic
32C4×#11929614C2211yescharacteristic
33C4×#11929613C2211yescharacteristic
34C4×#11929611C2211yescharacteristic
35#19098640C2211yescharacteristic
36#19098636—21no—
37C8×D316C2211yescharacteristic
38#19098628—21no—
39#19098625—21no—
40#19098619—21no—
41#19098619C2211yescharacteristic
42#19098614—21no—
43#19098614C2211yescharacteristic
44#19098614C2211yescharacteristic
45#19098588—21no—
46C4⋊C8×D79C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098284C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×Q8×D79C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929607—21no—
72C2×C8×D79—21no—
73C2×C8×D79—21no—
74C2×C8×D79—21no—
75#14579156—21no—
76#14579152C2×C411yescharacteristic
77#14579150C2×C411yescharacteristic
78#14579146—21no—
79#14579137C2×C411yescharacteristic
80#14579136C2×C411yescharacteristic
81Q8×Dic79C2×C411yescharacteristic
82C4×Dic158—21no—
83C4×Dic158C2311yescharacteristic
84#14579126—21no—
85#14579126C2×C411yescharacteristic
86C4×D316—21no—
87C4×D316C2311yescharacteristic
88#14579118—21no—
89C4⋊C4×D79—21no—
90C4⋊C4×D79C2×C411yescharacteristic
91C42×D79—21no—
92C42×D79C2311yescharacteristic
93#14579094C2×C411yescharacteristic
94#14579093C2×C411yescharacteristic
95#14579092C2×C411yescharacteristic
96C4⋊(C79⋊C8)—21no—
97C4⋊(C79⋊C8)—21no—
98C4⋊(C79⋊C8)C2311yescharacteristic
99C4⋊(C79⋊C8)C2311yescharacteristic
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
105Dic79⋊C8D411yescharacteristic
106Dic79⋊C8D411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579085C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579083C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8C2×C411yescharacteristic
117(C2×D79)⋊C8C2×C411yescharacteristic
118(C2×D79)⋊C8D411yescharacteristic
119(C2×D79)⋊C8D411yescharacteristic
120#14579071—21no—
121#14579071—21no—
122#14579071—21no—
123C4×C79⋊C8—21no—
124C4×C79⋊C8—21no—
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
128C8×Dic79—21no—
129C2×Q16×C79C2×C411yescharacteristic
130C4×Q8×C79C2311yescharacteristic
131C4×Q8×C79C2311yescharacteristic
132#107×C79C2×C411yescharacteristic
133C4⋊C8×C79—21no—
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#103×C79C2×C411yescharacteristic
139#97×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146Q8×D79—22no—
147Q8×D79—41no—
148C2×D316—21no—
149C2×D316C22×C411yescharacteristic
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79—21no—
153C2×C4×D79C22×C411yescharacteristic
154C2×C4×D79C2×D411yescharacteristic
155C2×C4×D79#4111yescharacteristic
156#11929614—41no—
157#11929613—41no—
158#11929611—22no—
159#11929607—81no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8C22×C411yescharacteristic
164C2×C79⋊C8C22×C411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C8×D79—24no—
168C8×D79—81no—
169C8×D79—81no—
170C4⋊Dic79—21no—
171C4⋊Dic79—21no—
172C4⋊Dic79C22×C411yescharacteristic
173C4⋊Dic79C22×C411yescharacteristic
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179(C2×D79)⋊C4—21no—
180(C2×D79)⋊C4—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184C4×Dic79—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79C22×C411yescharacteristic
188C4×Dic79C2×D411yescharacteristic
189C4×Dic79#4111yescharacteristic
190C2×Q8×C79C22×C411yescharacteristic
191C2×Q8×C79C22×C411yescharacteristic
192Q16×C79—22no—
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79C22×C411yescharacteristic
197C2×C8×C79C22×C411yescharacteristic
198C2×C8×C79C2×D411yescharacteristic
199C2×C8×C79C2×D411yescharacteristic
200C4⋊C4×C79—21no—
201C4⋊C4×C79—21no—
202C4⋊C4×C79C22×C411yescharacteristic
203C4⋊C4×C79C22×C411yescharacteristic
204C42×C79—21no—
205C42×C79—21no—
206C42×C79C2411yescharacteristic
207C22×D79—21no—
208C22×D79C4×D411yescharacteristic
209Q8×C79—21no—
210Q8×C79—21no—
211Q8×C79—21no—
212Q8×C79#13112yes—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79C23×C411yescharacteristic
218C2×C4×C79C22×D411yescharacteristic
219C2×C4×C79C2×#4111yescharacteristic
220C2×Dic79—21no—
221C2×Dic79—21no—
222C2×Dic79—21no—
223C2×Dic79C4×D411yescharacteristic
224C2×Dic79C4×D411yescharacteristic
225C2×Dic79C4×D411yescharacteristic
226D316—21no—
227D316—21no—
228D316—42no—
229C4×D79—22no—
230C4×D79—21no—
231C4×D79—21no—
232C4×D79—21no—
233C4×D79—21no—
234C4×D79—42no—
235C4×D79—42no—
236C4×D79—41no—
237C4×D79—41no—
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×C79C2×C4×D411yescharacteristic
256C2×D79—21no—
257C2×D79—21no—
258C2×D79—41no—
259C2×D79—41no—
260C2×D79—41no—
261C2×D79#44312yes—
262C4×C79—22no—
263C4×C79—21no—
264C4×C79—41no—
265C4×C79—41no—
266C4×C79—41no—
267C4×C79C2×#13111yescharacteristic
268C4×C79#56811yescharacteristic
269C4×C79#57711yescharacteristic
270C4×C79#57811yescharacteristic
271Dic79—21no—
272Dic79—21no—
273Dic79—21no—
274Dic79—41no—
275Dic79—41no—
276Dic79—41no—
277#18017—791nomaximal, Sylow
278C2×C79#293311yescharacteristic
279C2×C79#294711yescharacteristic
280C2×C79C2×#44311yescharacteristic
281D79—22no—
282D79—81no—
283#3108—791no—
284#2994—791no—
285#2984—791no—
286#1591—791no—
287#1590—791no—
288#1588—791no—
289#1587—791no—
290#1587D7911yescharacteristic, complemented
291#1585—791no—
292C8×Q16—791no—
293C8×SD16—791no—
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—1581no—
312#446—1581no—
313#446C2×D7911yescharacteristic
314C8×Q8—791no—
315C8×Q8—791no—
316C8×Q8C2×D7911yescharacteristic
317C4×Q16—791no—
318C4×Q16C2×D7911yescharacteristic
319C4×SD16—791no—
320C4×SD16—791no—
321#436—791no—
322#435—1581no—
323C8×D4—791no—
324C8×D4—1581no—
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⋊C8C2×D7911yescharacteristic
343Q8⋊C8C2×D7911yescharacteristic
344D4⋊C8—791no—
345D4⋊C8—791no—
346#322—791no—
347#322C2×D7911yescharacteristic
348C82—791no—
349C82—791no—
350C22×Q8—791no—
351C2×Q16—791no—
352C2×Q16C4×D7911yescharacteristic
353C2×SD16—791no—
354C2×SD16—791no—
355C2×M16—1581no—
356C22×C8—1581no—
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—1581no—
369C4×Q8—1581no—
370C4×Q8C22×D7911yescharacteristic
371C4×Q8C22×D7911yescharacteristic
372C4×D4—791no—
373C4×D4—1581no—
374#117—1581no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—1581no—
377C2×C42—791no—
378C2×C42—1581no—
379#107—791no—
380#107C4×D7911yescharacteristic
381#106—791no—
382#106—791no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—791no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—791no—
390C4⋊C8—791no—
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#103C4×D7911yescharacteristic
403#103C4×D7911yescharacteristic
404#102—791no—
405#102—791no—
406C22⋊C8—791no—
407C22⋊C8—791no—
408C22⋊C8—791no—
409C22⋊C8—791no—
410C22⋊C8—1581no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—21no—
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—1581no—
430C2×Q8—1582no—
431C2×Q8C2×C4×D7911yescharacteristic
432C2×Q8C2×C4×D7911yescharacteristic
433C2×D4—791no—
434C2×D4—1581no—
435C22×C4—791no—
436C22×C4—791no—
437C22×C4—791no—
438C22×C4—1581no—
439C22×C4—1581no—
440C22×C4—1581no—
441Q16—22no—
442Q16—3161no—
443SD16—3161no—
444SD16—1582no—
445M16—6321no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—791no—
453C2×C8—6321no—
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—21no—
465C4⋊C4—791no—
466C4⋊C4—791no—
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—1581no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C42—21no—
482C42—21no—
483C42—791no—
484C42—791no—
485C42—791no—
486C42—1581no—
487C42—1581no—
488C42—1581no—
489C42C23×D7911yescharacteristic
490C23—791no—
491C23—1581no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—3162no—
496Q8—1581no—
497Q8—1581no—
498Q8#1457916912yes—
499D4—3162no—
500D4—1581no—
501D4—1581no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—3162no—
507C2×C4—3162no—
508C2×C4—3161no—
509C2×C4—3161no—
510C2×C4—791no—
511C2×C4—791no—
512C2×C4—791no—
513C2×C4—1582no—
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—3161no—
539C22—3161no—
540C22—3161no—
541C22—792no—
542C22—1581no—
543C22—1581no—
544C22C4×D4×D7911yescharacteristic
545C4—21no—
546C4—22no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—3161no—
551C4—3161no—
552C4—3161no—
553C4—1581no—
554C4—1581no—
555C4—1581no—
556C4C2×#1457916911yescharacteristic
557C4#1909932811yescharacteristic
558C4#1909937211yescharacteristic
559C4#1909937911yescharacteristic
560C2—6321no—
561C2—1582no—
562C2#2377212611yescharacteristic
563C2#2377224211yescharacteristic
564C2#443×D7911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1587C2
socleC22×C79C2×C4×D4
prime radical#14579136C2×C4
cosocleC42×C79C24
hypercenter#1587D79
hypocenterC79#18017
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1587

Complements

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

NH ≅ G/Ntype
3. #23772488C2semidirect
4. #23762295C2semidirect
5. #23762289C2semidirect
6. #23762275C2semidirect
7. #23762268C2semidirect
8. #23762259C2semidirect
9. Q16×C79⋊C8C2semidirect
10. C8×#11929611C2semidirect
11. #23762182C2semidirect
13. Q8⋊C8×D79C2semidirect
14. #23761685C2semidirect
15. C79×#1587C2semidirect
290. #1587D79semidirect
298. C79#18017semidirect

Maximal subgroups

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

subgroupindex
#237734842
#237725632
#237724882
#237622952
#237622892
#237622752
#237622682
#237622592
Q16×C79⋊C82
C8×#119296112
#237621822
#237618592
Q8⋊C8×D792
#237616852
C79×#15872
#1801779