Finite Group Databasebeta

← #34296463

Subgroups — #34296463

Order 20224.

738 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773743C211yesmaximal, characteristic
2#23772339C211yesmaximal, characteristic, complemented
3#23772115C211yesmaximal, characteristic
4#23764375C211yesmaximal, characteristic, complemented
5#23764226C211yesmaximal, characteristic
6#23764111C211yesmaximal, characteristic, complemented
7#23764107C211yesmaximal, characteristic, complemented
8#23764103C211yesmaximal, characteristic, complemented
9#23763539C211yesmaximal, characteristic, complemented
10#23763245C211yesmaximal, characteristic, complemented
11#23763241C211yesmaximal, characteristic, complemented
12#23763240C211yesmaximal, characteristic, complemented
13#23763236C211yesmaximal, characteristic, complemented
14#97×Dic79C211yesmaximal, characteristic, complemented
15C79×#1938C211yesmaximal, characteristic, complemented
16C2×#14579169C2211yescharacteristic
17C2×#14579168C2211yescharacteristic
18#19099116C2211yescharacteristic
19#19099109C2211yescharacteristic
20#19098930C412yescomplemented
21#19098926C412yescomplemented
22#19098924C412yescomplemented
23#19098921C412yescomplemented
24#19098692—22no—
25#19098690—22no—
26#19098637—22no—
27#19098631—22no—
28#19098589C2211yescharacteristic
29#19098586C2211yescharacteristic
30#19098586C2211yescharacteristic
31C2×#14579089C2211yescharacteristic
32C2×#14579088C2211yescharacteristic
33C2×#14579087C2211yescharacteristic
34#19098560C2211yescharacteristic
35#19098559C2211yescharacteristic
36#19098557C2211yescharacteristic
37#19098529C2211yescharacteristic
38#19098521C2211yescharacteristic
39#19098519C2211yescharacteristic
40C2×#14579072C2211yescharacteristic
41C2×#14579071C2211yescharacteristic
42C2×#14579071C2211yescharacteristic
43C2×C8×Dic79C2211yescharacteristic
44C2×C8×Dic79C2211yescharacteristic
45C42×Dic79C2211yescharacteristic
46#19098328C2211yescharacteristic
47#19098327C2211yescharacteristic
48#19098327C2211yescharacteristic
49#19098324C2211yescharacteristic
50#19098323C2211yescharacteristic
51#19098322C2211yescharacteristic
52#424×C79C2211yescharacteristic
53C2×#104×C79C2211yescharacteristic
54#416×C79C2211yescharacteristic
55#408×C79C2211yescharacteristic
56C2×#97×C79C2211yescharacteristic
57#341×C79C2211yescharacteristic
58#339×C79C2211yescharacteristic
59C2×#11929624C2311yescharacteristic
60#14579169—22no—
61#14579169—22no—
62#14579168—22no—
63#14579168—22no—
64C2×#11929608—21no—
65C2×#11929608C2311yescharacteristic
66C2×#11929608C2311yescharacteristic
67C2×#11929607—21no—
68C22×C79⋊C8—21no—
69C22×C79⋊C8C2311yescharacteristic
70C22×C79⋊C8C2311yescharacteristic
71C2×C8×D79—21no—
72#14579160C2×C412yes—
73#14579159C2×C411yescharacteristic
74#14579157C2×C411yescharacteristic
75#14579152C2×C412yes—
76#14579150C2×C411yescharacteristic
77#14579148C2×C411yescharacteristic
78#14579138—22no—
79#14579133—22no—
80C4×Dic158—22no—
81C4×C22⋊D79—22no—
82#14579120C2311yescharacteristic
83C2×C4×Dic79—22no—
84C2×C4×Dic79C2311yescharacteristic
85C2×C4×Dic79C2311yescharacteristic
86Dic79⋊C8—22no—
87Dic79⋊C8—22no—
88Dic79⋊C8—22no—
89#14579089—22no—
90#14579088—22no—
91#14579087—22no—
92#14579086C2×C412yes—
93#14579085C2×C412yes—
94#14579084C2×C412yes—
95#14579083C2×C412yes—
96#14579082C2×C412yes—
97#14579081C2×C412yes—
98(C2×D79)⋊C8—22no—
99#14579072—21no—
100#14579072—22no—
101#14579071—21no—
102#14579071C2×C412yes—
103#14579071C2×C412yes—
104#14579071D412yes—
105#14579071D412yes—
106C4×C79⋊C8—21no—
107C8×Dic79—21no—
108C8×Dic79—22no—
109C8×Dic79—22no—
110C2×#41×C79C2311yescharacteristic
111C2×M16×C79C2311yescharacteristic
112C2×M16×C79C2311yescharacteristic
113C22×C8×C79C2311yescharacteristic
114C22×C8×C79C2311yescharacteristic
115#117×C79C2311yescharacteristic
116C2×C42×C79C2311yescharacteristic
117C4⋊C8×C79—22no—
118#104×C79—22no—
119#103×C79C2×C412yes—
120#102×C79C2×C412yes—
121C22⋊C8×C79—22no—
122#97×C79—22no—
123#11929624—21no—
124#11929624—21no—
125#11929624—21no—
126#11929624—21no—
127C2×Dic158—22no—
128C2×Dic158C22×C411yescharacteristic
129C2×D316C22×C411yescharacteristic
130C2×C22⋊D79—21no—
131C2×C22⋊D79—22no—
132C22×Dic79—21no—
133C22×Dic79#4112yes—
134C2×C4×D79—21no—
135#11929608—21no—
136#11929608—21no—
137#11929608—21no—
138#11929608—21no—
139#11929608—42no—
140#11929607—42no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—42no—
146C2×C79⋊C8C22×C412yes—
147C2×C79⋊C8C2×D411yescharacteristic
148C2×C79⋊C8C2×D411yescharacteristic
149C2×C79⋊C8#4112yes—
150C2×C79⋊C8#4112yes—
151C8×D79—42no—
152C4⋊Dic79—22no—
153C4⋊Dic79C22×C412yes—
154Dic79⋊C4—22no—
155Dic79⋊C4—22no—
156Dic79⋊C4—22no—
157C22⋊Dic79—22no—
158C22⋊Dic79—21no—
159(C2×D79)⋊C4—22no—
160C4×Dic79—21no—
161C4×Dic79—21no—
162C4×Dic79—44no—
163C4×Dic79C22×C411yescharacteristic
164C4×Dic79C22×C411yescharacteristic
165C4×Dic79C2×D412yes—
166C4×Dic79#4112yes—
167#41×C79—21no—
168#41×C79—21no—
169#41×C79—21no—
170#41×C79—21no—
171C2×Q8×C79C22×C411yescharacteristic
172C2×D4×C79—21no—
173C2×D4×C79C22×C411yescharacteristic
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79C2411yescharacteristic
177M16×C79—21no—
178M16×C79—21no—
179M16×C79—21no—
180M16×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79C22×C412yes—
188C2×C8×C79C2×D412yes—
189C4⋊C4×C79C22×C412yes—
190C22⋊C4×C79—21no—
191C42×C79—21no—
192C42×C79—21no—
193C42×C79#4112yes—
194C23×C79—21no—
195C23×C79C2×#4111yescharacteristic
196C22×D79—21no—
197Q8×C79—21no—
198Q8×C79—21no—
199D4×C79—21no—
200D4×C79—21no—
201D4×C79—41no—
202D4×C79—41no—
203C2×C4×C79—21no—
204C2×C4×C79—22no—
205C2×C4×C79—22no—
206C2×C4×C79—42no—
207C2×C4×C79—41no—
208C2×C4×C79—41no—
209C2×C4×C79C23×C411yescharacteristic
210C2×C4×C79C22×D411yescharacteristic
211C2×C4×C79C2×#4111yescharacteristic
212C2×C4×C79C2×#4111yescharacteristic
213C2×C4×C79C2×#4111yescharacteristic
214C2×C4×C79C2×#4111yescharacteristic
215C22⋊D79—41no—
216C22⋊D79—41no—
217C22⋊D79—82no—
218C2×Dic79—24no—
219C2×Dic79—22no—
220C2×Dic79—22no—
221C2×Dic79—21no—
222C2×Dic79—42no—
223C2×Dic79C4×D412yes—
224C2×Dic79C4×D412yes—
225D316—21no—
226D316—21no—
227C4×D79—41no—
228C4×D79—41no—
229Dic158—21no—
230Dic158—21no—
231Dic158—82no—
232C8×C79—22no—
233C8×C79—22no—
234C8×C79—41no—
235C8×C79—41no—
236C79⋊C8—22no—
237C79⋊C8—41no—
238C79⋊C8—41no—
239C79⋊C8#12412yes—
240C79⋊C8#12412yes—
241C22×C79—22no—
242C22×C79—42no—
243C22×C79—41no—
244C22×C79C2×C4×D411yescharacteristic
245C22×C79#52911yescharacteristic
246C22×C79#52911yescharacteristic
247C2×D79—42no—
248C2×D79—41no—
249C4×C79—42no—
250C4×C79—42no—
251C4×C79—41no—
252C4×C79C4×#4111yescharacteristic
253C4×C79C4×#4111yescharacteristic
254C4×C79C2×#12411yescharacteristic
255C4×C79#53211yescharacteristic
256Dic79—24no—
257Dic79—22no—
258Dic79—22no—
259Dic79—42no—
260Dic79—41no—
261#21662—791nomaximal, Sylow
262C2×C79—21no—
263C2×C79—21no—
264C2×C79—81no—
265C2×C79C4×#12411yescharacteristic
266C2×C79#295612yes—
267D79—81no—
268#3135—791no—
269#2961—791no—
270#2932—791no—
271#1994—791no—
272#1962—791no—
273#1938—791no—
274#1938D7911yescharacteristic, complemented
275#1937—791no—
276#1937—791no—
277#1838—791no—
278#1765—791no—
279#1765—791no—
280#1764—791no—
281#1764—791no—
282C4×#97—791no—
283C79#2166211yescharacteristic, complemented, Sylow
284C2×#131—791no—
285C2×#131—791no—
286#533—791no—
287#532—791no—
288#491—792no—
289#490—792no—
290#489—792no—
291#489—792no—
292#446—1582no—
293#446—1582no—
294#435—1582no—
295#435—1582no—
296#424—791no—
297#424C2×D7911yescharacteristic
298#423—791no—
299#423—791no—
300C2×#104—791no—
301C2×#104—791no—
302C2×#104—791no—
303C2×#104C2×D7911yescharacteristic
304#416—791no—
305#416—791no—
306#416—791no—
307#416C2×D7911yescharacteristic
308#408—791no—
309#408C2×D7911yescharacteristic
310#405—791no—
311#405—791no—
312C2×#97—791no—
313C2×#97—791no—
314C2×#97—791no—
315C2×#97C2×D7911yescharacteristic
316C2×C4×C8—791no—
317C2×C4×C8—791no—
318C43—791no—
319#341—791no—
320#341—791no—
321#341—791no—
322#341C2×D7911yescharacteristic
323#339—791no—
324#339—791no—
325#339—791no—
326#339C2×D7911yescharacteristic
327C2×#41—791no—
328C2×#41C22×D7911yescharacteristic
329#131—1582no—
330#131—1582no—
331#131—1582no—
332#131—1582no—
333C2×M16—791no—
334C2×M16—791no—
335C2×M16—1581no—
336C2×M16—1581no—
337C2×M16C22×D7911yescharacteristic
338C2×M16C22×D7911yescharacteristic
339C22×C8—791no—
340C22×C8—791no—
341C22×C8—1581no—
342C22×C8—1581no—
343C22×C8C22×D7911yescharacteristic
344C22×C8C22×D7911yescharacteristic
345C4⋊Q8—792no—
346C4⋊Q8—791no—
347#127—791no—
348#125—792no—
349#124—791no—
350#124—791no—
351C22⋊Q8—1582no—
352#121—1582no—
353C4×Q8—1582no—
354C4×D4—1582no—
355#117—791no—
356#117C22×D7911yescharacteristic
357C2×C42—791no—
358C2×C42—791no—
359C2×C42—1582no—
360C2×C42C22×D7911yescharacteristic
361C4⋊C8—22no—
362C4⋊C8—1582no—
363C4⋊C8—1582no—
364C4⋊C8—1582no—
365#104—22no—
366#104—1582no—
367#104—1582no—
368#104—1582no—
369#103—792no—
370#103—792no—
371#103—792no—
372#103C4×D7912yes—
373#102—792no—
374#102—792no—
375#102—792no—
376#102C4×D7912yes—
377C22⋊C8—22no—
378C22⋊C8—1582no—
379#97—22no—
380#97—792no—
381#97—792no—
382#97—792no—
383#97—792no—
384#97—1582no—
385#97—1581no—
386#97—1581no—
387C4×C8—1581no—
388C4×C8—1582no—
389C4×C8—1582no—
390C4×C8—1581no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—21no—
395#41—1581no—
396#41—1581no—
397#41—1581no—
398#41—1581no—
399C2×Q8—791no—
400C2×Q8—1582no—
401C2×Q8C2×C4×D7911yescharacteristic
402C2×D4—21no—
403C2×D4—791no—
404C2×D4—1582no—
405C2×D4—1581no—
406C2×D4C2×C4×D7911yescharacteristic
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—792no—
410C22×C4—1581no—
411C22×C4—1581no—
412C22×C4C23×D7911yescharacteristic
413M16—21no—
414M16—21no—
415M16—21no—
416M16—21no—
417M16—3162no—
418M16—3162no—
419M16—1581no—
420M16—1581no—
421M16—1581no—
422M16—1581no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—3162no—
430C2×C8—3162no—
431C2×C8—792no—
432C2×C8—792no—
433C2×C8—791no—
434C2×C8—791no—
435C2×C8—792no—
436C2×C8—1581no—
437C2×C8—1581no—
438C2×C8—1581no—
439C2×C8—1581no—
440C2×C8C2×C4×D7912yes—
441C2×C8D4×D7912yes—
442C4⋊C4—792no—
443C4⋊C4—1582no—
444C4⋊C4—1582no—
445C4⋊C4—1582no—
446C4⋊C4—1582no—
447C4⋊C4C2×C4×D7912yes—
448C22⋊C4—21no—
449C22⋊C4—1582no—
450C22⋊C4—1582no—
451C22⋊C4—1581no—
452C42—21no—
453C42—21no—
454C42—3164no—
455C42—791no—
456C42—791no—
457C42—792no—
458C42—792no—
459C42—1581no—
460C42—1581no—
461C42#1192962312yes—
462C23—21no—
463C23—1581no—
464C23#41×D7911yescharacteristic
465Q8—21no—
466Q8—21no—
467Q8—6322no—
468Q8—1581no—
469Q8—1581no—
470D4—21no—
471D4—21no—
472D4—41no—
473D4—41no—
474D4—3161no—
475D4—3161no—
476D4—6322no—
477D4—1581no—
478D4—1581no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—3161no—
486C2×C4—3161no—
487C2×C4—3162no—
488C2×C4—792no—
489C2×C4—792no—
490C2×C4—1582no—
491C2×C4—1581no—
492C2×C4—1584no—
493C2×C4—1582no—
494C2×C4C22×C4×D7911yescharacteristic
495C2×C4C2×D4×D7911yescharacteristic
496C2×C4#41×D7911yescharacteristic
497C2×C4#41×D7911yescharacteristic
498C2×C4#41×D7911yescharacteristic
499C2×C4C2×#1192962311yescharacteristic
500C8—22no—
501C8—22no—
502C8—41no—
503C8—41no—
504C8—3161no—
505C8—3161no—
506C8—792no—
507C8—792no—
508C8—1582no—
509C22—22no—
510C22—42no—
511C22—41no—
512C22—3162no—
513C22—3161no—
514C22C4×D4×D7911yescharacteristic
515C22#1909909011yescharacteristic
516C22#1909909611yescharacteristic
517C4—42no—
518C4—42no—
519C4—41no—
520C4—3162no—
521C4—3161no—
522C4—1584no—
523C4—1582no—
524C4—1582no—
525C4C4×#1192962311yescharacteristic
526C4#1909903111yescharacteristic
527C4#124×D7911yescharacteristic
528C4#1909910811yescharacteristic
529C2—21no—
530C2—21no—
531C2—81no—
532C2—6321no—
533C2#2376594111yescharacteristic
534C2#2377230812yes—
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1938C2
socleC22×C79C2×C4×D4
prime radical#19099109C22
cosocleC22×C4×C79C24
hypercenter#1938D79
hypocenterC79#21662
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1938

Complements

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

NH ≅ G/Ntype
2. #23772339C2semidirect
4. #23764375C2semidirect
6. #23764111C2semidirect
7. #23764107C2semidirect
8. #23764103C2semidirect
9. #23763539C2semidirect
10. #23763245C2semidirect
11. #23763241C2semidirect
12. #23763240C2semidirect
13. #23763236C2semidirect
14. #97×Dic79C2semidirect
15. C79×#1938C2semidirect
20. #19098930C4semidirect
21. #19098926C4semidirect
22. #19098924C4semidirect
23. #19098921C4semidirect
274. #1938D79semidirect
283. C79#21662semidirect

Maximal subgroups

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

subgroupindex
#237737432
#237723392
#237721152
#237643752
#237642262
#237641112
#237641072
#237641032
#237635392
#237632452
#237632412
#237632402
#237632362
#97×Dic792
C79×#19382
#2166279