Finite Group Databasebeta

← #34444723

Subgroups — #34444723

Order 20224.

760 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098999C211yesmaximal, characteristic, complemented
2C22×#14579102C211yesmaximal, characteristic, complemented
3#23775087C211yesmaximal, characteristic, complemented
4#23765664C211yesmaximal, characteristic, complemented
5#23765501C211yesmaximal, characteristic, complemented
6#23765494C211yesmaximal, characteristic, complemented
7#23765455C211yesmaximal, characteristic, complemented
8#23765432C211yesmaximal, characteristic, complemented
9#23765329C211yesmaximal, characteristic, complemented
10#23765321C211yesmaximal, characteristic, complemented
11#23765312C211yesmaximal, characteristic, complemented
12#23765311C211yesmaximal, characteristic, complemented
13#23765274C211yesmaximal, characteristic, complemented
14#23765226C211yesmaximal, characteristic, complemented
15D4×D8×C79C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic, complemented
17C22×D632C2211yescharacteristic, complemented
18#19099209—21no—
19#19098999—24no—
20C2×#14579105C2211yescharacteristic, complemented
21C2×#14579105C2211yescharacteristic, complemented
22C2×#14579105C2211yescharacteristic, complemented
23C2×#14579102—24no—
24C2×#14579102C2211yescharacteristic
25C2×#14579102C2211yescharacteristic
26C2×#14579102C2211yescharacteristic, complemented
27C2×#14579102C2211yescharacteristic, complemented
28C2×#14579102C2211yescharacteristic, complemented
29C2×#14579102C2211yescharacteristic, complemented
30C2×#14579101C2211yescharacteristic, complemented
31C22×C79⋊C16C2211yescharacteristic, complemented
32#19098947C2211yescharacteristic, complemented
33#19098804C2211yescharacteristic, complemented
34#19098799C2211yescharacteristic, complemented
35#19098779—21no—
36#19098716—21no—
37#19098710—21no—
38C4×D632C2211yescharacteristic, complemented
39#19098390C2211yescharacteristic, complemented
40C4⋊(C79⋊C16)C2211yescharacteristic, complemented
41#19098366C2211yescharacteristic, complemented
42#19098364C2211yescharacteristic, complemented
43#19098364C2211yescharacteristic, complemented
44#19098364C2211yescharacteristic, complemented
45#19098363C2211yescharacteristic, complemented
46#19098363C2211yescharacteristic, complemented
47C22⋊(C79⋊C16)C2211yescharacteristic, complemented
48C22⋊(C79⋊C16)C2211yescharacteristic, complemented
49#19098339C2211yescharacteristic, complemented
50C22×D8×C79C2211yescharacteristic
51C22×D8×C79C2211yescharacteristic, complemented
52D42×C79—21no—
53#493×C79C2211yescharacteristic, complemented
54#466×C79C2211yescharacteristic
55#466×C79C2211yescharacteristic, complemented
56#459×C79—21no—
57#447×C79—21no—
58#447×C79—21no—
59C4×D8×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#14579105—22no—
82#14579105—22no—
83#14579105—41no—
84#14579102—22no—
85#14579102—22no—
86#14579102—22no—
87#14579102—22no—
88#14579102—41no—
89#14579102D414yescomplemented
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×D4×C79—21no—
101C22×D4×C79—21no—
102C22×D4×C79—41no—
103C2×D8×C79—22no—
104C2×D8×C79—22no—
105C2×D8×C79—21no—
106C2×D8×C79—21no—
107C2×D8×C79—21no—
108C2×D8×C79—41no—
109C2×D8×C79—41no—
110C2×D8×C79C2311yescharacteristic
111C2×D8×C79C2311yescharacteristic
112C2×D8×C79C2311yescharacteristic
113C22×C8×C79C2311yescharacteristic
114C22×C8×C79C2311yescharacteristic
115#127×C79—21no—
116#121×C79—21no—
117#121×C79—21no—
118#121×C79—41no—
119#120×C79—41no—
120#120×C79—41no—
121C4×D4×C79—21no—
122C4×D4×C79D411yescharacteristic
123#107×C79C2311yescharacteristic
124C4⋊C8×C79D411yescharacteristic
125#102×C79—21no—
126#102×C79—21no—
127#102×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×D412yescomplemented
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×D412yescomplemented
164C24×C79—41no—
165C24×C79—41no—
166C2×D4×C79—21no—
167C2×D4×C79—21no—
168C2×D4×C79—21no—
169C2×D4×C79—42no—
170C2×D4×C79—42no—
171C2×D4×C79—41no—
172C2×D4×C79—41no—
173C2×D4×C79—41no—
174C2×D4×C79—41no—
175C2×D4×C79—41no—
176C2×D4×C79—41no—
177C2×D4×C79—81no—
178C2×D4×C79—81no—
179C2×D4×C79—84no—
180C2×D4×C79D811yescharacteristic
181C22×C4×C79—41no—
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×D411yescharacteristic
184D8×C79—21no—
185D8×C79—21no—
186D8×C79—21no—
187D8×C79—21no—
188D8×C79—21no—
189D8×C79—42no—
190D8×C79—42no—
191D8×C79—81no—
192D8×C79C2×D412yes—
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79C2×D411yescharacteristic
198C2×C8×C79C2×D411yescharacteristic
199C2×C8×C79C2×D411yescharacteristic
200C2×C8×C79C2411yescharacteristic
201C4⋊C4×C79—21no—
202C4⋊C4×C79D811yescharacteristic
203C22⋊C4×C79—41no—
204C22⋊C4×C79—41no—
205C22⋊C4×C79—41no—
206C22⋊C4×C79D811yescharacteristic
207C22⋊C4×C79D811yescharacteristic
208C42×C79C2×D411yescharacteristic
209C23×C79—41no—
210C23×C79—41no—
211C23×C79—41no—
212C23×C79—82no—
213C23×C79—82no—
214C23×C79—82no—
215C23×C79—82no—
216C23×C79—161no—
217C23×C79—161no—
218C23×C79C2×D811yescharacteristic
219C23×C79C2×D811yescharacteristic
220C22×D79—41no—
221C22×D79—41no—
222C22×D79—82no—
223C22×D79—82no—
224C22×D79—161no—
225D4×C79—22no—
226D4×C79—22no—
227D4×C79—41no—
228D4×C79—41no—
229D4×C79—41no—
230D4×C79—41no—
231D4×C79—41no—
232D4×C79—82no—
233D4×C79—82no—
234D4×C79—82no—
235D4×C79—82no—
236D4×C79—82no—
237D4×C79—161no—
238C2×C4×C79—21no—
239C2×C4×C79—21no—
240C2×C4×C79—21no—
241C2×C4×C79—21no—
242C2×C4×C79—41no—
243C2×C4×C79—82no—
244C2×C4×C79C2×D811yescharacteristic
245C2×C4×C79C2×D811yescharacteristic
246C2×C4×C79C2×D811yescharacteristic
247C2×C4×C79C2×D811yescharacteristic
248C2×C4×C79C22×D411yescharacteristic
249C22⋊D79—82no—
250C22⋊D79—161no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253D316—22no—
254D316—41no—
255D316—41no—
256D316—41no—
257D316—82no—
258C4×D79—82no—
259C4×D79—161no—
260Dic158—41no—
261Dic158—41no—
262C8×C79—21no—
263C8×C79—21no—
264C8×C79—21no—
265C8×C79C22×D411yescharacteristic
266C8×C79H(5,𝔽2)11yescharacteristic
267C22×C79—21no—
268C22×C79—21no—
269C22×C79—21no—
270C22×C79—21no—
271C22×C79—21no—
272C22×C79—42no—
273C22×C79—81no—
274C22×C79—81no—
275C22×C79—81no—
276C22×C79—81no—
277C22×C79—81no—
278C22×C79—81no—
279C22×C79—81no—
280C22×C79—81no—
281C22×C79—164no—
282C22×C79—164no—
283C22×C79#50912yescomplemented
284C22×C79C22×D811yescharacteristic
285C2×D79—42no—
286C2×D79—81no—
287C2×D79—81no—
288C2×D79—81no—
289C2×D79—81no—
290C2×D79—81no—
291C2×D79—164no—
292C4×C79—21no—
293C4×C79—21no—
294C4×C79—21no—
295C4×C79—21no—
296C4×C79—81no—
297C4×C79D4211yescharacteristic
298C4×C79#57611yescharacteristic
299Dic79—81no—
300Dic79—81no—
301#33830—791nomaximal, Sylow
302C2×C79—22no—
303C2×C79—41no—
304C2×C79—82no—
305C2×C79—161no—
306C2×C79—161no—
307C2×C79D4×D811yescharacteristic
308C2×C79C2×#50911yescharacteristic
309C2×C79#341511yescharacteristic
310D79—82no—
311D79—161no—
312C2×#509—791no—
313C22×D16—791no—
314#3280—791no—
315D4×D8D7911yescharacteristic, complemented
316#2250—791no—
317#2218—791no—
318#2217—791no—
319#2210—791no—
320#2206—791no—
321#2187—791no—
322#2186—791no—
323#2184—791no—
324#2184—791no—
325#2177—791no—
326#2168—791no—
327C79#3383011yescharacteristic, complemented, Sylow
328C2×#135—791no—
329C22×D8—791no—
330C22×D8C2×D7911yescharacteristic
331C22×D8C2×D7911yescharacteristic, complemented
332#546—1581no—
333D42—21no—
334#509—1584no—
335C2×SD32—791no—
336C2×SD32—791no—
337C2×SD32—791no—
338C2×D16—791no—
339C2×D16—791no—
340C2×D16—791no—
341C2×D16—791no—
342C2×D16—791no—
343C2×D16—791no—
344C2×D16—1584no—
345C2×M32—791no—
346C22×C16—791no—
347#495—791no—
348#493C2×D7911yescharacteristic, complemented
349#467—791no—
350#466—791no—
351#466C2×D7911yescharacteristic
352#466C2×D7911yescharacteristic, complemented
353#463—1581no—
354#459—21no—
355#450—1581no—
356#449—1581no—
357#447—21no—
358#447—21no—
359C4×D8—791no—
360C4×D8C2×D7911yescharacteristic, complemented
361C8×D4C2×D7911yescharacteristic
362#367—791no—
363C4⋊C16—791no—
364#358—791no—
365#357—791no—
366#357—791no—
367#357—791no—
368#357—791no—
369#357—791no—
370C22⋊C16—791no—
371C22⋊C16—791no—
372#346—791no—
373C2×#41—1581no—
374C22×D4—21no—
375C22×D4—21no—
376C22×D4—41no—
377C22×D4—1581no—
378#135—3161no—
379#135—1582no—
380C2×Q16—791no—
381C2×SD16—1581no—
382C2×SD16—1581no—
383C2×D8—22no—
384C2×D8—22no—
385C2×D8—21no—
386C2×D8—21no—
387C2×D8—21no—
388C2×D8—41no—
389C2×D8—41no—
390C2×D8—3161no—
391C2×D8—791no—
392C2×D8—791no—
393C2×D8—1582no—
394C2×D8—1581no—
395C2×D8C22×D7911yescharacteristic
396C2×D8C22×D7911yescharacteristic
397C2×D8C22×D7911yescharacteristic
398C22×C8C22×D7911yescharacteristic
399C22×C8C22×D7911yescharacteristic
400#127—21no—
401#124—1581no—
402#123—3161no—
403C22⋊Q8—1581no—
404#121—21no—
405#121—21no—
406#121—41no—
407#121—3161no—
408#121—1581no—
409#120—41no—
410#120—41no—
411#120—3161no—
412C4×D4—21no—
413C4×D4—1581no—
414C4×D4C22⋊D7911yescharacteristic
415C2×C22⋊C4—3161no—
416SD32—3161no—
417SD32—1582no—
418SD32—1582no—
419D16—3161no—
420D16—794no—
421D16—1582no—
422D16—1582no—
423D16—1582no—
424D16—1582no—
425M32—1582no—
426C2×C16—791no—
427C2×C16—791no—
428C2×C16—791no—
429C2×C16—791no—
430C2×C16—1582no—
431#107—791no—
432#107C22×D7911yescharacteristic
433C4⋊C8C22⋊D7911yescharacteristic
434#103—1581no—
435#102—21no—
436#102—21no—
437#102—21no—
438#102—1581no—
439#102—1581no—
440C22⋊C8C22⋊D7911yescharacteristic
441C22⋊C8C22⋊D7911yescharacteristic
442C4×C8C22×D7911yescharacteristic
443C24—41no—
444C24—41no—
445C24—3161no—
446#41—3162no—
447#41—6321no—
448C2×Q8—1581no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—81no—
461C2×D4—81no—
462C2×D4—84no—
463C2×D4—3162no—
464C2×D4—3161no—
465C2×D4—3161no—
466C2×D4—3161no—
467C2×D4—6321no—
468C2×D4—1581no—
469C2×D4—1581no—
470C2×D4#1192960911yescharacteristic
471C22×C4—41no—
472C22×C4—3161no—
473C22×C4—3161no—
474C22×C4C2×C22⋊D7911yescharacteristic
475C22×C4C2×C22⋊D7911yescharacteristic
476Q16—1581no—
477Q16—1581no—
478SD16—3162no—
479SD16—6321no—
480D8—21no—
481D8—21no—
482D8—21no—
483D8—21no—
484D8—21no—
485D8—42no—
486D8—42no—
487D8—81no—
488D8—3162no—
489D8—792no—
490D8—1581no—
491D8—1581no—
492D8—1581no—
493D8D4×D7912yes—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8C2×C22⋊D7911yescharacteristic
499C2×C8C2×C22⋊D7911yescharacteristic
500C2×C8C2×C22⋊D7911yescharacteristic
501C2×C8C23×D7911yescharacteristic
502C4⋊C4—21no—
503C4⋊C4—3161no—
504C4⋊C4—1581no—
505C4⋊C4#1192960911yescharacteristic
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—3161no—
510C22⋊C4—3161no—
511C22⋊C4—3161no—
512C22⋊C4—3161no—
513C22⋊C4—3161no—
514C22⋊C4#1192960911yescharacteristic
515C22⋊C4#1192960911yescharacteristic
516C42C2×C22⋊D7911yescharacteristic
517C16—792no—
518C16—1581no—
519C16—1581no—
520C16—1581no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—82no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—161no—
529C23—161no—
530C23—3161no—
531C23—3161no—
532C23—6322no—
533C23—6322no—
534C23—12641no—
535C23C2×#1192960911yescharacteristic
536C23C2×#1192960911yescharacteristic
537Q8—3161no—
538Q8—3161no—
539D4—22no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—82no—
547D4—82no—
548D4—82no—
549D4—82no—
550D4—82no—
551D4—161no—
552D4—3161no—
553D4—3161no—
554D4—3161no—
555D4—6322no—
556D4—6322no—
557D4—1582no—
558D4—12641no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—82no—
565C2×C4—3161no—
566C2×C4—3161no—
567C2×C4—6322no—
568C2×C4—12641no—
569C2×C4C2×#1192960911yescharacteristic
570C2×C4C2×#1192960911yescharacteristic
571C2×C4C2×#1192960911yescharacteristic
572C2×C4C2×#1192960911yescharacteristic
573C2×C4C22×C22⋊D7911yescharacteristic
574C8—21no—
575C8—21no—
576C8—21no—
577C8C2×D4×D7911yescharacteristic
578C8#1457920611yescharacteristic
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—164no—
594C22—164no—
595C22—3162no—
596C22—6321no—
597C22—6321no—
598C22—6321no—
599C22—6321no—
600C22—6321no—
601C22—12644no—
602C22#1909899912yescomplemented
603C22C22×#1192960911yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—81no—
609C4—6321no—
610C4—6321no—
611C4D4×C22⋊D7911yescharacteristic
612C4#1909936611yescharacteristic
613C2—22no—
614C2—41no—
615C2—82no—
616C2—161no—
617C2—161no—
618C2—6322no—
619C2—12641no—
620C2D4×#1192960911yescharacteristic
621C2C2×#1909899911yescharacteristic
622C2#2377653911yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingD4×D8×C79C2
socleC22×C79C22×D8
hypercenterD4×D8D79
hypocenterC79#33830
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19098999C2semidirect
2. C22×#14579102C2semidirect
3. #23775087C2semidirect
4. #23765664C2semidirect
5. #23765501C2semidirect
6. #23765494C2semidirect
7. #23765455C2semidirect
8. #23765432C2semidirect
9. #23765329C2semidirect
10. #23765321C2semidirect
11. #23765312C2semidirect
12. #23765311C2semidirect
13. #23765274C2semidirect
14. #23765226C2semidirect
15. D4×D8×C79C2semidirect
16. C2×#14579183C22semidirect
17. C22×D632C22semidirect
20. C2×#14579105C22semidirect
21. C2×#14579105C22semidirect
22. C2×#14579105C22semidirect
26. C2×#14579102C22semidirect
27. C2×#14579102C22semidirect
28. C2×#14579102C22semidirect
29. C2×#14579102C22semidirect
30. C2×#14579101C22semidirect
31. C22×C79⋊C16C22semidirect
32. #19098947C22semidirect
33. #19098804C22semidirect
34. #19098799C22semidirect
38. C4×D632C22semidirect
39. #19098390C22semidirect
40. C4⋊(C79⋊C16)C22semidirect
41. #19098366C22semidirect
42. #19098364C22semidirect
43. #19098364C22semidirect
44. #19098364C22semidirect
45. #19098363C22semidirect
46. #19098363C22semidirect
47. C22⋊(C79⋊C16)C22semidirect
48. C22⋊(C79⋊C16)C22semidirect
49. #19098339C22semidirect
51. C22×D8×C79C22semidirect
53. #493×C79C22semidirect
55. #466×C79C22semidirect
59. C4×D8×C79C22semidirect
89. #14579102D4semidirect
152. D632C2×D4semidirect
163. C79⋊C16C2×D4semidirect
283. C22×C79#509semidirect
315. D4×D8D79semidirect
327. C79#33830semidirect
331. C22×D8C2×D79semidirect
348. #493C2×D79semidirect
352. #466C2×D79semidirect
360. C4×D8C2×D79semidirect
602. C22#19098999semidirect

Maximal subgroups

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

subgroupindex
C2×#190989992
C22×#145791022
#237750872
#237656642
#237655012
#237654942
#237654552
#237654322
#237653292
#237653212
#237653122
#237653112
#237652742
#237652262
D4×D8×C792
#3383079