Finite Group Databasebeta

← #34444758

Subgroups — #34444758

Order 20224.

686 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099004C211yesmaximal, characteristic, complemented
2C22×#14579104C211yesmaximal, characteristic, complemented
3#23775087C211yesmaximal, characteristic
4#23765669C211yesmaximal, characteristic, complemented
5#23765515C211yesmaximal, characteristic, complemented
6#23765489C211yesmaximal, characteristic, complemented
7#23765462C211yesmaximal, characteristic, complemented
8#23765442C211yesmaximal, characteristic, complemented
9#23765338C211yesmaximal, characteristic, complemented
10#23765334C211yesmaximal, characteristic, complemented
11#23765328C211yesmaximal, characteristic, complemented
12#23765317C211yesmaximal, characteristic, complemented
13#23765262C211yesmaximal, characteristic, complemented
14#23765226C211yesmaximal, characteristic, complemented
15D4×Q16×C79C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic
17C22×D632C2211yescharacteristic
18#19099209—21no—
19#19099004—24no—
20C2×#14579107C2211yescharacteristic, complemented
21C2×#14579107C2211yescharacteristic, complemented
22C2×#14579107C2211yescharacteristic, complemented
23C2×#14579104—24no—
24C2×#14579104C2211yescharacteristic
25C2×#14579104C2211yescharacteristic
26C2×#14579104C2211yescharacteristic
27C2×#14579104C2211yescharacteristic
28C2×#14579104C2211yescharacteristic
29C2×#14579104C2211yescharacteristic
30C2×#14579101C2211yescharacteristic, complemented
31C22×C79⋊C16C2211yescharacteristic
32#19098947C2211yescharacteristic
33#19098804C2211yescharacteristic
34#19098799C2211yescharacteristic
35#19098779—21no—
36#19098716—21no—
37#19098710—21no—
38C4×D632C2211yescharacteristic
39#19098388C2211yescharacteristic, complemented
40C4⋊(C79⋊C16)C2211yescharacteristic, complemented
41#19098367C2211yescharacteristic, complemented
42#19098367C2211yescharacteristic, complemented
43#19098367C2211yescharacteristic, complemented
44#19098366C2211yescharacteristic, complemented
45#19098363C2211yescharacteristic
46#19098363C2211yescharacteristic
47C22⋊(C79⋊C16)C2211yescharacteristic
48C22⋊(C79⋊C16)C2211yescharacteristic, complemented
49#19098339C2211yescharacteristic, complemented
50C22×Q16×C79C2211yescharacteristic
51C22×Q16×C79C2211yescharacteristic, complemented
52D4×Q8×C79—21no—
53#494×C79C2211yescharacteristic, complemented
54#467×C79C2211yescharacteristic
55#467×C79C2211yescharacteristic, complemented
56#462×C79—21no—
57#451×C79—21no—
58#451×C79—21no—
59C4×Q16×C79C2211yescharacteristic, complemented
60C8×D4×C79C2211yescharacteristic
61C2×#11929624—21no—
62C22×D316—21no—
63#14579183—22no—
64#14579183—41no—
65C2×Dic316C2311yescharacteristic
66C2×#11929613—21no—
67C2×#11929613—21no—
68C2×D632—22no—
69C2×D632—21no—
70C2×D632—41no—
71C2×D632C2311yescharacteristic
72C2×D632C2311yescharacteristic
73#14579149—21no—
74#14579146—41no—
75#14579140—21no—
76#14579135—21no—
77#14579134—41no—
78#14579129—41no—
79C4×D316—21no—
80C22⋊C4×D79—41no—
81#14579107—22no—
82#14579107—22no—
83#14579107—41no—
84#14579104—22no—
85#14579104—22no—
86#14579104—22no—
87#14579104—22no—
88#14579104—41no—
89#14579104D414yescomplemented
90#14579101—22no—
91C2×C79⋊C16—22no—
92C2×C79⋊C16C2311yescharacteristic
93C2×C79⋊C16C2311yescharacteristic
94C2×C79⋊C16C2311yescharacteristic
95C2×C79⋊C16C2311yescharacteristic
96#14579095C2311yescharacteristic
97#14579086—21no—
98#14579083—21no—
99#14579083—21no—
100C22×Q8×C79—21no—
101C22×Q8×C79—21no—
102C2×Q16×C79—22no—
103C2×Q16×C79—22no—
104C2×Q16×C79—21no—
105C2×Q16×C79—21no—
106C2×Q16×C79—21no—
107C2×Q16×C79—41no—
108C2×Q16×C79—41no—
109C2×Q16×C79C2311yescharacteristic
110C2×Q16×C79C2311yescharacteristic
111C2×Q16×C79C2311yescharacteristic
112C22×C8×C79C2311yescharacteristic
113C22×C8×C79C2311yescharacteristic
114C4⋊Q8×C79—21no—
115C4⋊Q8×C79—41no—
116C22⋊Q8×C79—21no—
117C22⋊Q8×C79—21no—
118C22⋊Q8×C79—41no—
119C22⋊Q8×C79—41no—
120C4×Q8×C79—21no—
121C4×D4×C79—41no—
122C4×D4×C79D411yescharacteristic
123#107×C79C2311yescharacteristic
124C4⋊C8×C79D411yescharacteristic
125#103×C79—21no—
126#103×C79—21no—
127#103×C79—21no—
128C22⋊C8×C79D411yescharacteristic
129C22⋊C8×C79D411yescharacteristic
130C4×C8×C79C2311yescharacteristic
131C23×D79—41no—
132#11929624—42no—
133#11929624—81no—
134C2×Dic158—21no—
135C2×D316—21no—
136C2×D316—21no—
137C2×D316—42no—
138C2×D316—41no—
139C2×D316—81no—
140C2×C22⋊D79—41no—
141C2×C22⋊D79—41no—
142C2×C4×D79—41no—
143C2×C4×D79—41no—
144Dic316—21no—
145Dic316—21no—
146#11929613—42no—
147#11929613—81no—
148D632—21no—
149D632—21no—
150D632—21no—
151D632—42no—
152D632C2×D412yes—
153C4⋊Dic79—21no—
154Dic79⋊C4—41no—
155C22⋊Dic79—41no—
156(C2×D79)⋊C4—41no—
157(C2×D79)⋊C4—41no—
158(C2×D79)⋊C4—41no—
159(C2×D79)⋊C4—41no—
160C79⋊C16—21no—
161C79⋊C16—21no—
162C79⋊C16—21no—
163C79⋊C16C2×D412yes—
164C2×Q8×C79—21no—
165C2×Q8×C79—21no—
166C2×Q8×C79—21no—
167C2×Q8×C79—42no—
168C2×Q8×C79—42no—
169C2×Q8×C79—41no—
170C2×Q8×C79—41no—
171C2×Q8×C79—81no—
172C2×Q8×C79—81no—
173C2×D4×C79D811yescharacteristic
174C22×C4×C79—41no—
175C22×C4×C79—41no—
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178Q16×C79—21no—
179Q16×C79—21no—
180Q16×C79—21no—
181Q16×C79—21no—
182Q16×C79—21no—
183Q16×C79—42no—
184Q16×C79—42no—
185Q16×C79—81no—
186Q16×C79C2×D412yes—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79C2×D411yescharacteristic
192C2×C8×C79C2×D411yescharacteristic
193C2×C8×C79C2×D411yescharacteristic
194C2×C8×C79C2411yescharacteristic
195C4⋊C4×C79—21no—
196C4⋊C4×C79—41no—
197C4⋊C4×C79—41no—
198C4⋊C4×C79—41no—
199C4⋊C4×C79—41no—
200C4⋊C4×C79—41no—
201C4⋊C4×C79D811yescharacteristic
202C22⋊C4×C79—41no—
203C22⋊C4×C79—41no—
204C22⋊C4×C79D811yescharacteristic
205C22⋊C4×C79D811yescharacteristic
206C42×C79—41no—
207C42×C79C2×D411yescharacteristic
208C23×C79C2×D811yescharacteristic
209C23×C79C2×D811yescharacteristic
210C22×D79—41no—
211C22×D79—41no—
212C22×D79—82no—
213C22×D79—82no—
214C22×D79—161no—
215Q8×C79—22no—
216Q8×C79—41no—
217Q8×C79—41no—
218Q8×C79—41no—
219Q8×C79—41no—
220Q8×C79—41no—
221Q8×C79—82no—
222Q8×C79—82no—
223D4×C79—22no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—82no—
233C2×C4×C79—82no—
234C2×C4×C79C2×D811yescharacteristic
235C2×C4×C79C2×D811yescharacteristic
236C2×C4×C79C2×D811yescharacteristic
237C2×C4×C79C2×D811yescharacteristic
238C2×C4×C79C22×D411yescharacteristic
239C22⋊D79—82no—
240C22⋊D79—161no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243D316—22no—
244D316—41no—
245D316—41no—
246D316—41no—
247D316—82no—
248C4×D79—82no—
249C4×D79—161no—
250Dic158—41no—
251Dic158—41no—
252C8×C79—21no—
253C8×C79—21no—
254C8×C79—21no—
255C8×C79C22×D411yescharacteristic
256C8×C79H(5,𝔽2)11yescharacteristic
257C22×C79—21no—
258C22×C79—21no—
259C22×C79—21no—
260C22×C79—21no—
261C22×C79—21no—
262C22×C79#51012yescomplemented
263C22×C79C22×D811yescharacteristic
264C2×D79—42no—
265C2×D79—81no—
266C2×D79—81no—
267C2×D79—81no—
268C2×D79—81no—
269C2×D79—81no—
270C2×D79—164no—
271C4×C79—21no—
272C4×C79—21no—
273C4×C79—21no—
274C4×C79—21no—
275C4×C79—42no—
276C4×C79—81no—
277C4×C79—81no—
278C4×C79—81no—
279C4×C79D4211yescharacteristic
280C4×C79#57611yescharacteristic
281Dic79—81no—
282Dic79—81no—
283#33832—791nomaximal, Sylow
284C2×C79—22no—
285C2×C79—41no—
286C2×C79D4×D811yescharacteristic
287C2×C79C2×#51011yescharacteristic
288C2×C79#341611yescharacteristic
289D79—82no—
290D79—161no—
291C2×#510—791no—
292C22×SD32—791no—
293D4×Q16D7911yescharacteristic, complemented
294#3280—791no—
295#2251—791no—
296#2220—791no—
297#2216—791no—
298#2211—791no—
299#2208—791no—
300#2189—791no—
301#2188—791no—
302#2187—791no—
303#2185—791no—
304#2175—791no—
305#2168—791no—
306C79#3383211yescharacteristic, complemented, Sylow
307C2×#135—791no—
308C22×Q16C2×D7911yescharacteristic
309C22×Q16C2×D7911yescharacteristic, complemented
310C22×D8—791no—
311D4×Q8—21no—
312#546—1581no—
313#510—1584no—
314C2×Q32—791no—
315C2×Q32—791no—
316C2×Q32—791no—
317C2×SD32—791no—
318C2×SD32—791no—
319C2×SD32—791no—
320C2×SD32—791no—
321C2×SD32—791no—
322C2×SD32—791no—
323C2×SD32—1584no—
324C2×M32—791no—
325C22×C16—791no—
326#495—791no—
327#494C2×D7911yescharacteristic, complemented
328#467—791no—
329#467C2×D7911yescharacteristic
330#467C2×D7911yescharacteristic, complemented
331#466—791no—
332#463—1581no—
333#462—21no—
334#451—21no—
335#451—21no—
336#450—1581no—
337#449—1581no—
338C4×Q16C2×D7911yescharacteristic, complemented
339C4×D8—791no—
340C8×D4C2×D7911yescharacteristic
341#366—791no—
342C4⋊C16—791no—
343#358—791no—
344#358—791no—
345#358—791no—
346#358—791no—
347#357—791no—
348#357—791no—
349C22⋊C16—791no—
350C22⋊C16—791no—
351#346—791no—
352C2×#41—1581no—
353C22×Q8—21no—
354C22×Q8—21no—
355C22×D4—1581no—
356#135—3161no—
357#135—1582no—
358C2×Q16—22no—
359C2×Q16—22no—
360C2×Q16—21no—
361C2×Q16—21no—
362C2×Q16—21no—
363C2×Q16—41no—
364C2×Q16—41no—
365C2×Q16—791no—
366C2×Q16C22×D7911yescharacteristic
367C2×Q16C22×D7911yescharacteristic
368C2×Q16C22×D7911yescharacteristic
369C2×SD16—1581no—
370C2×SD16—1581no—
371C2×D8—3161no—
372C2×D8—791no—
373C2×D8—791no—
374C2×D8—1582no—
375C2×D8—1581no—
376C22×C8C22×D7911yescharacteristic
377C22×C8C22×D7911yescharacteristic
378C4⋊Q8—21no—
379C4⋊Q8—41no—
380#124—1581no—
381#123—3161no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—41no—
385C22⋊Q8—41no—
386C22⋊Q8—1581no—
387#121—3161no—
388#121—1581no—
389#120—3161no—
390C4×Q8—21no—
391C4×D4—41no—
392C4×D4—1581no—
393C4×D4C22⋊D7911yescharacteristic
394C2×C22⋊C4—3161no—
395Q32—3161no—
396Q32—1582no—
397Q32—1582no—
398SD32—3161no—
399SD32—794no—
400SD32—1582no—
401SD32—1582no—
402SD32—1582no—
403SD32—1582no—
404M32—1582no—
405C2×C16—791no—
406C2×C16—791no—
407C2×C16—791no—
408C2×C16—791no—
409C2×C16—1582no—
410#107—791no—
411#107C22×D7911yescharacteristic
412C4⋊C8C22⋊D7911yescharacteristic
413#103—21no—
414#103—21no—
415#103—21no—
416#103—1581no—
417#102—1581no—
418#102—1581no—
419C22⋊C8C22⋊D7911yescharacteristic
420C22⋊C8C22⋊D7911yescharacteristic
421C4×C8C22×D7911yescharacteristic
422C24—3161no—
423#41—3162no—
424#41—6321no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—41no—
431C2×Q8—41no—
432C2×Q8—81no—
433C2×Q8—81no—
434C2×Q8—1581no—
435C2×D4—3162no—
436C2×D4—3161no—
437C2×D4—3161no—
438C2×D4—3161no—
439C2×D4—6321no—
440C2×D4—1581no—
441C2×D4—1581no—
442C2×D4#1192960911yescharacteristic
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—3161no—
446C22×C4—3161no—
447C22×C4C2×C22⋊D7911yescharacteristic
448C22×C4C2×C22⋊D7911yescharacteristic
449Q16—21no—
450Q16—21no—
451Q16—21no—
452Q16—21no—
453Q16—21no—
454Q16—42no—
455Q16—42no—
456Q16—81no—
457Q16—1581no—
458Q16—1581no—
459Q16D4×D7912yes—
460SD16—3162no—
461SD16—6321no—
462D8—3162no—
463D8—792no—
464D8—1581no—
465D8—1581no—
466D8—1581no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8C2×C22⋊D7911yescharacteristic
472C2×C8C2×C22⋊D7911yescharacteristic
473C2×C8C2×C22⋊D7911yescharacteristic
474C2×C8C23×D7911yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—3161no—
482C4⋊C4—1581no—
483C4⋊C4#1192960911yescharacteristic
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—3161no—
487C22⋊C4—3161no—
488C22⋊C4—3161no—
489C22⋊C4—3161no—
490C22⋊C4—3161no—
491C22⋊C4#1192960911yescharacteristic
492C22⋊C4#1192960911yescharacteristic
493C42—41no—
494C42C2×C22⋊D7911yescharacteristic
495C16—792no—
496C16—1581no—
497C16—1581no—
498C16—1581no—
499C23—3161no—
500C23—3161no—
501C23—6322no—
502C23—6322no—
503C23—12641no—
504C23C2×#1192960911yescharacteristic
505C23C2×#1192960911yescharacteristic
506Q8—22no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—82no—
513Q8—82no—
514Q8—3161no—
515Q8—3161no—
516D4—22no—
517D4—3161no—
518D4—3161no—
519D4—3161no—
520D4—6322no—
521D4—6322no—
522D4—1582no—
523D4—12641no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—6322no—
537C2×C4—12641no—
538C2×C4C2×#1192960911yescharacteristic
539C2×C4C2×#1192960911yescharacteristic
540C2×C4C2×#1192960911yescharacteristic
541C2×C4C2×#1192960911yescharacteristic
542C2×C4C22×C22⋊D7911yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8C2×D4×D7911yescharacteristic
547C8#1457920611yescharacteristic
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—3162no—
554C22—6321no—
555C22—6321no—
556C22—6321no—
557C22—6321no—
558C22—6321no—
559C22—12644no—
560C22#1909900412yescomplemented
561C22C22×#1192960911yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—6321no—
571C4—6321no—
572C4D4×C22⋊D7911yescharacteristic
573C4#1909936611yescharacteristic
574C2—22no—
575C2—41no—
576C2—6322no—
577C2—12641no—
578C2D4×#1192960911yescharacteristic
579C2C2×#1909900411yescharacteristic
580C2#2377654411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingD4×Q16×C79C2
socleC22×C79C22×D8
prime radical#23775087C2
hypercenterD4×Q16D79
hypocenterC79#33832
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#19099004C2semidirect
2. C22×#14579104C2semidirect
4. #23765669C2semidirect
5. #23765515C2semidirect
6. #23765489C2semidirect
7. #23765462C2semidirect
8. #23765442C2semidirect
9. #23765338C2semidirect
10. #23765334C2semidirect
11. #23765328C2semidirect
12. #23765317C2semidirect
13. #23765262C2semidirect
14. #23765226C2semidirect
15. D4×Q16×C79C2semidirect
20. C2×#14579107C22semidirect
21. C2×#14579107C22semidirect
22. C2×#14579107C22semidirect
30. C2×#14579101C22semidirect
39. #19098388C22semidirect
40. C4⋊(C79⋊C16)C22semidirect
41. #19098367C22semidirect
42. #19098367C22semidirect
43. #19098367C22semidirect
44. #19098366C22semidirect
48. C22⋊(C79⋊C16)C22semidirect
49. #19098339C22semidirect
51. C22×Q16×C79C22semidirect
53. #494×C79C22semidirect
55. #467×C79C22semidirect
59. C4×Q16×C79C22semidirect
89. #14579104D4semidirect
262. C22×C79#510semidirect
293. D4×Q16D79semidirect
306. C79#33832semidirect
309. C22×Q16C2×D79semidirect
327. #494C2×D79semidirect
330. #467C2×D79semidirect
338. C4×Q16C2×D79semidirect
560. C22#19099004semidirect

Maximal subgroups

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

subgroupindex
C2×#190990042
C22×#145791042
#237750872
#237656692
#237655152
#237654892
#237654622
#237654422
#237653382
#237653342
#237653282
#237653172
#237652622
#237652262
D4×Q16×C792
#3383279