Finite Group Databasebeta

← #34445300

Subgroups — #34445300

Order 20224.

610 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775769C211yesmaximal, characteristic, complemented
2#23765641C211yesmaximal, characteristic, complemented
3#23765588C211yesmaximal, characteristic, complemented
4#23765578C211yesmaximal, characteristic, complemented
5#23765557C211yesmaximal, characteristic, complemented
6#23765511C211yesmaximal, characteristic, complemented
7#23765494C211yesmaximal, characteristic, complemented
8#23765437C211yesmaximal, characteristic, complemented
9#23765329C211yesmaximal, characteristic, complemented
10#23765329C211yesmaximal, characteristic, complemented
11#23765264C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765116C211yesmaximal, characteristic, complemented
14C2×#19098364C211yesmaximal, characteristic, complemented
15D4×D8×C79C211yesmaximal, characteristic
16#19099253—21no—
17C2×#14579105C2211yescharacteristic
18C2×#14579105C2211yescharacteristic, complemented
19C2×#14579105C2211yescharacteristic, complemented
20C2×#14579101C2211yescharacteristic, complemented
21C22×C79⋊C16C2211yescharacteristic, complemented
22#19098965C2211yescharacteristic, complemented
23#19098905—21no—
24#19098894—21no—
25#19098871—21no—
26#19098804C2211yescharacteristic, complemented
27#19098804C2211yescharacteristic, complemented
28C4×Dic316C2211yescharacteristic, complemented
29#19098601C2211yescharacteristic, complemented
30C2×#14579095C2211yescharacteristic, complemented
31#19098390C2211yescharacteristic, complemented
32#19098388C2211yescharacteristic, complemented
33#19098388C2211yescharacteristic, complemented
34C4⋊(C79⋊C16)C2211yescharacteristic, complemented
35#19098366C2211yescharacteristic, complemented
36#19098364C2211yescharacteristic
37#19098364C2211yescharacteristic
38#19098364C2211yescharacteristic
39#19098364C2211yescharacteristic, complemented
40#19098364C2211yescharacteristic, complemented
41#19098364C2211yescharacteristic, complemented
42#19098364C2211yescharacteristic, complemented
43#19098364C2211yescharacteristic, complemented
44#19098364C2211yescharacteristic, complemented
45C22⋊(C79⋊C16)C2211yescharacteristic, complemented
46C22⋊(C79⋊C16)C2211yescharacteristic, complemented
47#19098339C2211yescharacteristic, complemented
48C22×D8×C79C2211yescharacteristic
49C22×D8×C79C2211yescharacteristic
50D42×C79—21no—
51#493×C79C2211yescharacteristic
52#466×C79C2211yescharacteristic
53#466×C79C2211yescharacteristic
54#459×C79—21no—
55#447×C79—21no—
56#447×C79—21no—
57C4×D8×C79C2211yescharacteristic
58C8×D4×C79C2211yescharacteristic
59C2×Dic316C2311yescharacteristic
60#14579154—41no—
61#14579153—21no—
62#14579151—41no—
63#14579142—41no—
64#14579140—21no—
65#14579140—21no—
66C4×Dic158—21no—
67#14579120—21no—
68#14579117—41no—
69C2×C4⋊Dic79—21no—
70#14579105—22no—
71#14579105—41no—
72#14579105—41no—
73#14579101—41no—
74C2×C79⋊C16—41no—
75C2×C79⋊C16C2311yescharacteristic
76C2×C79⋊C16C2311yescharacteristic
77C2×C79⋊C16C2311yescharacteristic
78C2×C79⋊C16C2311yescharacteristic
79#14579095—21no—
80#14579095C2311yescharacteristic
81#14579095C2311yescharacteristic
82#14579095C2311yescharacteristic
83#14579093—21no—
84#14579093—21no—
85#14579086—21no—
86#14579086—21no—
87#14579086—21no—
88C22×D4×C79—21no—
89C22×D4×C79—21no—
90C22×D4×C79—41no—
91C2×D8×C79—22no—
92C2×D8×C79—22no—
93C2×D8×C79—21no—
94C2×D8×C79—21no—
95C2×D8×C79—21no—
96C2×D8×C79—41no—
97C2×D8×C79—41no—
98C2×D8×C79C2311yescharacteristic
99C2×D8×C79C2311yescharacteristic
100C2×D8×C79C2311yescharacteristic
101C22×C8×C79C2311yescharacteristic
102C22×C8×C79C2311yescharacteristic
103#127×C79—21no—
104#121×C79—21no—
105#121×C79—21no—
106#121×C79—41no—
107#120×C79—41no—
108#120×C79—41no—
109C4×D4×C79—21no—
110C4×D4×C79D411yescharacteristic
111#107×C79C2311yescharacteristic
112C4⋊C8×C79D411yescharacteristic
113#102×C79—21no—
114#102×C79—21no—
115#102×C79—21no—
116C22⋊C8×C79D411yescharacteristic
117C22⋊C8×C79D411yescharacteristic
118C4×C8×C79C2311yescharacteristic
119C2×Dic158—21no—
120C22×Dic79—41no—
121Dic316—21no—
122Dic316—21no—
123C4⋊Dic79—21no—
124C4⋊Dic79—21no—
125C4⋊Dic79—21no—
126C4⋊Dic79—41no—
127Dic79⋊C4—41no—
128Dic79⋊C4—41no—
129Dic79⋊C4—41no—
130Dic79⋊C4—41no—
131Dic79⋊C4—41no—
132C22⋊Dic79—41no—
133C22⋊Dic79—41no—
134C22⋊Dic79—41no—
135C4×Dic79—41no—
136C4×Dic79—41no—
137C79⋊C16—21no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141C24×C79—41no—
142C24×C79—41no—
143C2×D4×C79—21no—
144C2×D4×C79—21no—
145C2×D4×C79—21no—
146C2×D4×C79—42no—
147C2×D4×C79—42no—
148C2×D4×C79—41no—
149C2×D4×C79—41no—
150C2×D4×C79—41no—
151C2×D4×C79—41no—
152C2×D4×C79—41no—
153C2×D4×C79—41no—
154C2×D4×C79—81no—
155C2×D4×C79—81no—
156C2×D4×C79—84no—
157C2×D4×C79D811yescharacteristic
158C22×C4×C79—41no—
159C22×C4×C79C2×D411yescharacteristic
160C22×C4×C79C2×D411yescharacteristic
161D8×C79—21no—
162D8×C79—21no—
163D8×C79—21no—
164D8×C79—21no—
165D8×C79—21no—
166D8×C79—42no—
167D8×C79—42no—
168D8×C79—81no—
169D8×C79#4112yes—
170C2×C8×C79—21no—
171C2×C8×C79—21no—
172C2×C8×C79—21no—
173C2×C8×C79—21no—
174C2×C8×C79C2×D411yescharacteristic
175C2×C8×C79C2×D411yescharacteristic
176C2×C8×C79C2×D411yescharacteristic
177C2×C8×C79C2411yescharacteristic
178C4⋊C4×C79—21no—
179C4⋊C4×C79D811yescharacteristic
180C22⋊C4×C79—41no—
181C22⋊C4×C79—41no—
182C22⋊C4×C79—41no—
183C22⋊C4×C79D811yescharacteristic
184C22⋊C4×C79D811yescharacteristic
185C42×C79C2×D411yescharacteristic
186C23×C79—41no—
187C23×C79—41no—
188C23×C79—41no—
189C23×C79—82no—
190C23×C79—82no—
191C23×C79—82no—
192C23×C79—82no—
193C23×C79—161no—
194C23×C79—161no—
195C23×C79C2×D811yescharacteristic
196C23×C79C2×D811yescharacteristic
197D4×C79—22no—
198D4×C79—22no—
199D4×C79—41no—
200D4×C79—41no—
201D4×C79—41no—
202D4×C79—41no—
203D4×C79—41no—
204D4×C79—82no—
205D4×C79—82no—
206D4×C79—82no—
207D4×C79—82no—
208D4×C79—82no—
209D4×C79—161no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—41no—
215C2×C4×C79—82no—
216C2×C4×C79C2×D811yescharacteristic
217C2×C4×C79C2×D811yescharacteristic
218C2×C4×C79C2×D811yescharacteristic
219C2×C4×C79C2×D811yescharacteristic
220C2×C4×C79C22×D411yescharacteristic
221C2×Dic79—41no—
222C2×Dic79—41no—
223C2×Dic79—41no—
224C2×Dic79—41no—
225C2×Dic79—82no—
226Dic158—41no—
227Dic158—41no—
228C8×C79—21no—
229C8×C79—21no—
230C8×C79—21no—
231C8×C79C2×#4111yescharacteristic
232C8×C79H(5,𝔽2)11yescharacteristic
233C22×C79—21no—
234C22×C79—21no—
235C22×C79—21no—
236C22×C79—21no—
237C22×C79—21no—
238C22×C79—42no—
239C22×C79—81no—
240C22×C79—81no—
241C22×C79—81no—
242C22×C79—81no—
243C22×C79—81no—
244C22×C79—81no—
245C22×C79—81no—
246C22×C79—81no—
247C22×C79—164no—
248C22×C79—164no—
249C22×C79#50912yes—
250C22×C79C22×D811yescharacteristic
251C4×C79—21no—
252C4×C79—21no—
253C4×C79—21no—
254C4×C79—21no—
255C4×C79—81no—
256C4×C79#54611yescharacteristic
257C4×C79#57711yescharacteristic
258Dic79—81no—
259Dic79—81no—
260Dic79—81no—
261Dic79—81no—
262#33877—791nomaximal, Sylow
263C2×C79—22no—
264C2×C79—41no—
265C2×C79—82no—
266C2×C79—161no—
267C2×C79—161no—
268C2×C79#329411yescharacteristic
269C2×C79C2×#50911yescharacteristic
270C2×C79#341611yescharacteristic
271#3336—791no—
272D4×D8D7911yescharacteristic
273#2244—791no—
274#2234—791no—
275#2232—791no—
276#2228—791no—
277#2220—791no—
278#2217—791no—
279#2207—791no—
280#2187—791no—
281#2187—791no—
282#2175—791no—
283#2168—791no—
284#2140—791no—
285C2×#357—791no—
286C79#3387711yescharacteristic, complemented, Sylow
287C22×D8C2×D7911yescharacteristic
288C22×D8C2×D7911yescharacteristic
289#552—1581no—
290D42—21no—
291C2×SD32—791no—
292C2×SD32—791no—
293C2×SD32—791no—
294C2×M32—791no—
295C22×C16—791no—
296#499—791no—
297#493C2×D7911yescharacteristic
298#485—1581no—
299#483—1581no—
300#479—1581no—
301#467—791no—
302#467—791no—
303#466C2×D7911yescharacteristic
304#466C2×D7911yescharacteristic
305#459—21no—
306#447—21no—
307#447—21no—
308C4×Q16—791no—
309C4×D8C2×D7911yescharacteristic
310C8×D4C2×D7911yescharacteristic
311#427—791no—
312C2×#107—791no—
313#367—791no—
314#366—791no—
315#366—791no—
316C4⋊C16—791no—
317#358—791no—
318#357—791no—
319#357—791no—
320#357—791no—
321#357—791no—
322#357—791no—
323#357—791no—
324#357—791no—
325#357—791no—
326#357—791no—
327C22⋊C16—791no—
328C22⋊C16—791no—
329#346—791no—
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—41no—
333C2×Q16—791no—
334C2×D8—22no—
335C2×D8—22no—
336C2×D8—21no—
337C2×D8—21no—
338C2×D8—21no—
339C2×D8—41no—
340C2×D8—41no—
341C2×D8C22×D7911yescharacteristic
342C2×D8C22×D7911yescharacteristic
343C2×D8C22×D7911yescharacteristic
344C22×C8C22×D7911yescharacteristic
345C22×C8C22×D7911yescharacteristic
346#127—21no—
347#126—3161no—
348#125—3161no—
349#125—1581no—
350#123—3161no—
351C22⋊Q8—1581no—
352C22⋊Q8—1581no—
353#121—21no—
354#121—21no—
355#121—41no—
356#120—41no—
357#120—41no—
358C4×Q8—1581no—
359C4×D4—21no—
360C4×D4C22⋊D7911yescharacteristic
361#117—3161no—
362#117—1581no—
363C2×C4⋊C4—1581no—
364SD32—3161no—
365SD32—3161no—
366SD32—1582no—
367M32—3161no—
368C2×C16—3161no—
369C2×C16—791no—
370C2×C16—791no—
371C2×C16—791no—
372C2×C16—791no—
373#107—791no—
374#107—791no—
375#107—791no—
376#107—1581no—
377#107C22×D7911yescharacteristic
378#106—1581no—
379#106—1581no—
380C4⋊C8C22⋊D7911yescharacteristic
381#103—1581no—
382#103—1581no—
383#103—1581no—
384#102—21no—
385#102—21no—
386#102—21no—
387C22⋊C8C22⋊D7911yescharacteristic
388C22⋊C8C22⋊D7911yescharacteristic
389C4×C8C22×D7911yescharacteristic
390C24—41no—
391C24—41no—
392C2×Q8—1581no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4—84no—
407C2×D4#1192960911yescharacteristic
408C22×C4—41no—
409C22×C4—3161no—
410C22×C4C2×C22⋊D7911yescharacteristic
411C22×C4C2×C22⋊D7911yescharacteristic
412Q16—1581no—
413Q16—1581no—
414D8—21no—
415D8—21no—
416D8—21no—
417D8—21no—
418D8—21no—
419D8—42no—
420D8—42no—
421D8—81no—
422D8#1192962312yes—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8C2×C22⋊D7911yescharacteristic
428C2×C8C2×C22⋊D7911yescharacteristic
429C2×C8C2×C22⋊D7911yescharacteristic
430C2×C8C23×D7911yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—3161no—
433C4⋊C4—3161no—
434C4⋊C4—3161no—
435C4⋊C4—3161no—
436C4⋊C4—3161no—
437C4⋊C4—3161no—
438C4⋊C4—1581no—
439C4⋊C4—1581no—
440C4⋊C4—1581no—
441C4⋊C4#1192960911yescharacteristic
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—3161no—
446C22⋊C4—3161no—
447C22⋊C4—3161no—
448C22⋊C4#1192960911yescharacteristic
449C22⋊C4#1192960911yescharacteristic
450C42—3161no—
451C42—3161no—
452C42C2×C22⋊D7911yescharacteristic
453C16—1581no—
454C16—1581no—
455C16—1581no—
456C16—1581no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—161no—
465C23—161no—
466C23C2×#1192960911yescharacteristic
467C23C2×#1192960911yescharacteristic
468Q8—3161no—
469Q8—3161no—
470D4—22no—
471D4—22no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—82no—
478D4—82no—
479D4—82no—
480D4—82no—
481D4—82no—
482D4—161no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—82no—
489C2×C4—3161no—
490C2×C4—3161no—
491C2×C4—3161no—
492C2×C4—3161no—
493C2×C4—6322no—
494C2×C4C2×#1192960911yescharacteristic
495C2×C4C2×#1192960911yescharacteristic
496C2×C4C2×#1192960911yescharacteristic
497C2×C4C2×#1192960911yescharacteristic
498C2×C4C22×C22⋊D7911yescharacteristic
499C8—21no—
500C8—21no—
501C8—21no—
502C8C2×#1192962311yescharacteristic
503C8#1457920611yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—164no—
519C22—164no—
520C22#1909899912yes—
521C22C22×#1192960911yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—81no—
527C4—6321no—
528C4—6321no—
529C4—6321no—
530C4—6321no—
531C4#1909920711yescharacteristic
532C4#1909937411yescharacteristic
533C2—22no—
534C2—41no—
535C2—82no—
536C2—161no—
537C2—161no—
538C2#2377521311yescharacteristic
539C2C2×#1909899911yescharacteristic
540C2#2377654711yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
Fitting, prime radicalD4×D8×C79C2
socleC22×C79C22×D8
hypercenterD4×D8D79
hypocenterC79#33877
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. #23775769C2semidirect
2. #23765641C2semidirect
3. #23765588C2semidirect
4. #23765578C2semidirect
5. #23765557C2semidirect
6. #23765511C2semidirect
7. #23765494C2semidirect
8. #23765437C2semidirect
9. #23765329C2semidirect
10. #23765329C2semidirect
11. #23765264C2semidirect
12. #23765226C2semidirect
13. #23765116C2semidirect
14. C2×#19098364C2semidirect
18. C2×#14579105C22semidirect
19. C2×#14579105C22semidirect
20. C2×#14579101C22semidirect
21. C22×C79⋊C16C22semidirect
22. #19098965C22semidirect
26. #19098804C22semidirect
27. #19098804C22semidirect
28. C4×Dic316C22semidirect
29. #19098601C22semidirect
30. C2×#14579095C22semidirect
31. #19098390C22semidirect
32. #19098388C22semidirect
33. #19098388C22semidirect
34. C4⋊(C79⋊C16)C22semidirect
35. #19098366C22semidirect
39. #19098364C22semidirect
40. #19098364C22semidirect
41. #19098364C22semidirect
42. #19098364C22semidirect
43. #19098364C22semidirect
44. #19098364C22semidirect
45. C22⋊(C79⋊C16)C22semidirect
46. C22⋊(C79⋊C16)C22semidirect
47. #19098339C22semidirect
286. C79#33877semidirect

Maximal subgroups

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

subgroupindex
#237757692
#237656412
#237655882
#237655782
#237655572
#237655112
#237654942
#237654372
#237653292
#237652642
#237652262
#237651162
C2×#190983642
D4×D8×C792
#3387779