Finite Group Databasebeta

← #34445360

Subgroups — #34445360

Order 20224.

570 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775793C211yesmaximal, characteristic, complemented
2#23765636C211yesmaximal, characteristic, complemented
3#23765600C211yesmaximal, characteristic, complemented
4#23765578C211yesmaximal, characteristic, complemented
5#23765537C211yesmaximal, characteristic, complemented
6#23765511C211yesmaximal, characteristic, complemented
7#23765489C211yesmaximal, characteristic, complemented
8#23765459C211yesmaximal, characteristic, complemented
9#23765340C211yesmaximal, characteristic, complemented
10#23765329C211yesmaximal, characteristic, complemented
11#23765264C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765112C211yesmaximal, characteristic, complemented
14C2×#19098364C211yesmaximal, characteristic, complemented
15C79×#3280C211yesmaximal, characteristic
16#19099270—21no—
17C2×#14579107C2211yescharacteristic, complemented
18C2×#14579105C2211yescharacteristic
19C2×#14579105C2211yescharacteristic
20C2×#14579101C2211yescharacteristic, complemented
21C22×C79⋊C16C2211yescharacteristic
22#19098968C2211yescharacteristic, complemented
23#19098899—21no—
24#19098899—21no—
25#19098859—21no—
26#19098804C2211yescharacteristic
27#19098804C2211yescharacteristic, complemented
28C4×Dic316C2211yescharacteristic, complemented
29C2×#14579095C2211yescharacteristic
30C2×#14579095C2211yescharacteristic, complemented
31#19098388C2211yescharacteristic, complemented
32#19098388C2211yescharacteristic, complemented
33#19098388C2211yescharacteristic, complemented
34C4⋊(C79⋊C16)C2211yescharacteristic, complemented
35#19098367C2211yescharacteristic, complemented
36#19098367C2211yescharacteristic, complemented
37#19098367C2211yescharacteristic, complemented
38#19098366C2211yescharacteristic, complemented
39#19098364C2211yescharacteristic
40#19098364C2211yescharacteristic
41#19098364C2211yescharacteristic
42#19098364C2211yescharacteristic
43#19098364C2211yescharacteristic
44#19098364C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic, complemented
47#19098339C2211yescharacteristic, complemented
48C2×#135×C79C2211yescharacteristic
49C22×D8×C79C2211yescharacteristic
50#546×C79—21no—
51#495×C79C2211yescharacteristic
52#467×C79C2211yescharacteristic
53#466×C79C2211yescharacteristic
54#463×C79—21no—
55#450×C79—21no—
56#449×C79—21no—
57C4×D8×C79C2211yescharacteristic
58C8×D4×C79C2211yescharacteristic
59C2×Dic316C2311yescharacteristic
60#14579161—21no—
61#14579152—41no—
62#14579140—21no—
63#14579140—21no—
64#14579138—41no—
65#14579138—41no—
66C4×Dic158—21no—
67D4×Dic79—41no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79—21no—
70#14579107—41no—
71#14579105—22no—
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#14579095—21no—
81#14579095—21no—
82#14579095C2311yescharacteristic
83#14579095C2311yescharacteristic
84#14579095C2311yescharacteristic
85#14579086—21no—
86#14579086—21no—
87#14579086—21no—
88C2×#41×C79—21no—
89C22×D4×C79—21no—
90#135×C79—22no—
91#135×C79—41no—
92C2×Q16×C79C2311yescharacteristic
93C2×SD16×C79—21no—
94C2×SD16×C79—21no—
95C2×D8×C79—22no—
96C2×D8×C79—21no—
97C2×D8×C79—41no—
98C2×D8×C79C2311yescharacteristic
99C2×D8×C79C2311yescharacteristic
100C22×C8×C79C2311yescharacteristic
101C22×C8×C79C2311yescharacteristic
102#124×C79—21no—
103#123×C79—41no—
104C22⋊Q8×C79—21no—
105#121×C79—21no—
106#121×C79—41no—
107#120×C79—41no—
108C4×D4×C79—21no—
109C4×D4×C79D411yescharacteristic
110C2×C22⋊C4×C79—41no—
111#107×C79C2311yescharacteristic
112C4⋊C8×C79D411yescharacteristic
113#103×C79—21no—
114#102×C79—21no—
115#102×C79—21no—
116C22⋊C8×C79D411yescharacteristic
117C22⋊C8×C79D411yescharacteristic
118C4×C8×C79C2311yescharacteristic
119C2×Dic158—21no—
120C2×Dic158—41no—
121C22×Dic79—41no—
122C22×Dic79—41no—
123Dic316—21no—
124Dic316—21no—
125C4⋊Dic79—21no—
126C4⋊Dic79—21no—
127C4⋊Dic79—21no—
128C4⋊Dic79—41no—
129C4⋊Dic79—41no—
130C4⋊Dic79—41no—
131Dic79⋊C4—41no—
132Dic79⋊C4—41no—
133Dic79⋊C4—41no—
134C22⋊Dic79—41no—
135C22⋊Dic79—41no—
136C4×Dic79—41no—
137C79⋊C16—21no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141C24×C79—41no—
142#41×C79—42no—
143#41×C79—81no—
144C2×Q8×C79—21no—
145C2×D4×C79—21no—
146C2×D4×C79—21no—
147C2×D4×C79—42no—
148C2×D4×C79—41no—
149C2×D4×C79—41no—
150C2×D4×C79—41no—
151C2×D4×C79—81no—
152C2×D4×C79D811yescharacteristic
153C22×C4×C79—41no—
154C22×C4×C79—41no—
155C22×C4×C79C2×D411yescharacteristic
156C22×C4×C79C2×D411yescharacteristic
157Q16×C79—21no—
158Q16×C79—21no—
159SD16×C79—42no—
160SD16×C79—81no—
161D8×C79—21no—
162D8×C79—21no—
163D8×C79—21no—
164D8×C79—42no—
165D8×C79#4112yes—
166C2×C8×C79—21no—
167C2×C8×C79—21no—
168C2×C8×C79—21no—
169C2×C8×C79—21no—
170C2×C8×C79C2×D411yescharacteristic
171C2×C8×C79C2×D411yescharacteristic
172C2×C8×C79C2×D411yescharacteristic
173C2×C8×C79C2411yescharacteristic
174C4⋊C4×C79—21no—
175C4⋊C4×C79—41no—
176C4⋊C4×C79D811yescharacteristic
177C22⋊C4×C79—41no—
178C22⋊C4×C79—41no—
179C22⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C22⋊C4×C79—41no—
182C22⋊C4×C79D811yescharacteristic
183C22⋊C4×C79D811yescharacteristic
184C42×C79C2×D411yescharacteristic
185C23×C79—41no—
186C23×C79—41no—
187C23×C79—82no—
188C23×C79—82no—
189C23×C79—161no—
190C23×C79C2×D811yescharacteristic
191C23×C79C2×D811yescharacteristic
192Q8×C79—41no—
193Q8×C79—41no—
194D4×C79—22no—
195D4×C79—22no—
196D4×C79—41no—
197D4×C79—41no—
198D4×C79—41no—
199D4×C79—82no—
200D4×C79—82no—
201D4×C79—161no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—41no—
207C2×C4×C79—41no—
208C2×C4×C79—82no—
209C2×C4×C79—161no—
210C2×C4×C79C2×D811yescharacteristic
211C2×C4×C79C2×D811yescharacteristic
212C2×C4×C79C2×D811yescharacteristic
213C2×C4×C79C2×D811yescharacteristic
214C2×C4×C79C22×D411yescharacteristic
215C2×Dic79—41no—
216C2×Dic79—41no—
217C2×Dic79—41no—
218C2×Dic79—41no—
219C2×Dic79—161no—
220C2×Dic79—161no—
221Dic158—41no—
222Dic158—41no—
223Dic158—161no—
224C8×C79—21no—
225C8×C79—21no—
226C8×C79—21no—
227C8×C79C2×#4111yescharacteristic
228C8×C79H(5,𝔽2)11yescharacteristic
229C22×C79—21no—
230C22×C79—21no—
231C22×C79—21no—
232C22×C79—21no—
233C22×C79—21no—
234C22×C79—42no—
235C22×C79—81no—
236C22×C79—81no—
237C22×C79—81no—
238C22×C79—81no—
239C22×C79—81no—
240C22×C79—164no—
241C22×C79#51012yes—
242C22×C79C22×D811yescharacteristic
243C4×C79—21no—
244C4×C79—21no—
245C4×C79—21no—
246C4×C79—21no—
247C4×C79—81no—
248C4×C79—81no—
249C4×C79#54611yescharacteristic
250C4×C79#57711yescharacteristic
251Dic79—81no—
252Dic79—81no—
253Dic79—81no—
254Dic79—81no—
255#33881—791nomaximal, Sylow
256C2×C79—22no—
257C2×C79—41no—
258C2×C79—82no—
259C2×C79—161no—
260C2×C79#329411yescharacteristic
261C2×C79C2×#51011yescharacteristic
262C2×C79#341511yescharacteristic
263#3338—791no—
264#3280D7911yescharacteristic
265#2243—791no—
266#2236—791no—
267#2232—791no—
268#2224—791no—
269#2220—791no—
270#2216—791no—
271#2211—791no—
272#2189—791no—
273#2187—791no—
274#2175—791no—
275#2168—791no—
276#2139—791no—
277C2×#357—791no—
278C79#3388111yescharacteristic, complemented, Sylow
279C2×#135C2×D7911yescharacteristic
280C22×D8C2×D7911yescharacteristic
281#554—1581no—
282#546—21no—
283C2×Q32—791no—
284C2×SD32—791no—
285C2×SD32—791no—
286C2×M32—791no—
287C22×C16—791no—
288#500—791no—
289#495C2×D7911yescharacteristic
290#484—1581no—
291#484—1581no—
292#477—1581no—
293#467—791no—
294#467—791no—
295#467C2×D7911yescharacteristic
296#466C2×D7911yescharacteristic
297#463—21no—
298#450—21no—
299#449—21no—
300C4×Q16—791no—
301C4×D8C2×D7911yescharacteristic
302C8×D4C2×D7911yescharacteristic
303C2×#107—791no—
304C2×#107—791no—
305#366—791no—
306#366—791no—
307#366—791no—
308C4⋊C16—791no—
309#358—791no—
310#358—791no—
311#358—791no—
312#358—791no—
313#357—791no—
314#357—791no—
315#357—791no—
316#357—791no—
317#357—791no—
318#357—791no—
319C22⋊C16—791no—
320C22⋊C16—791no—
321#346—791no—
322C2×#41—21no—
323C22×D4—21no—
324#135—22no—
325#135—41no—
326C2×Q16—791no—
327C2×Q16C22×D7911yescharacteristic
328C2×SD16—21no—
329C2×SD16—21no—
330C2×D8—22no—
331C2×D8—21no—
332C2×D8—41no—
333C2×D8C22×D7911yescharacteristic
334C2×D8C22×D7911yescharacteristic
335C22×C8C22×D7911yescharacteristic
336C22×C8C22×D7911yescharacteristic
337C4⋊Q8—1581no—
338#125—3161no—
339#124—21no—
340#123—41no—
341C22⋊Q8—21no—
342C22⋊Q8—3161no—
343C22⋊Q8—3161no—
344C22⋊Q8—1581no—
345C22⋊Q8—1581no—
346#121—21no—
347#121—41no—
348#120—41no—
349C4×Q8—1581no—
350C4×D4—21no—
351C4×D4—3161no—
352C4×D4C22⋊D7911yescharacteristic
353C2×C4⋊C4—1581no—
354C2×C4⋊C4—1581no—
355C2×C22⋊C4—41no—
356Q32—3161no—
357SD32—3161no—
358SD32—1582no—
359M32—3161no—
360C2×C16—3161no—
361C2×C16—791no—
362C2×C16—791no—
363C2×C16—791no—
364C2×C16—791no—
365#107—791no—
366#107—791no—
367#107—791no—
368#107—1581no—
369#107—1581no—
370#107—1581no—
371#107C22×D7911yescharacteristic
372C4⋊C8C22⋊D7911yescharacteristic
373#103—21no—
374#103—1581no—
375#103—1581no—
376#103—1581no—
377#102—21no—
378#102—21no—
379C22⋊C8C22⋊D7911yescharacteristic
380C22⋊C8C22⋊D7911yescharacteristic
381C4×C8C22×D7911yescharacteristic
382C24—41no—
383#41—42no—
384#41—81no—
385C2×Q8—21no—
386C2×Q8—3161no—
387C2×Q8—1581no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—81no—
395C2×D4#1192960911yescharacteristic
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—3161no—
399C22×C4—3161no—
400C22×C4C2×C22⋊D7911yescharacteristic
401C22×C4C2×C22⋊D7911yescharacteristic
402Q16—21no—
403Q16—21no—
404Q16—1581no—
405Q16—1581no—
406SD16—42no—
407SD16—81no—
408D8—21no—
409D8—21no—
410D8—21no—
411D8—42no—
412D8#1192962312yes—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8C2×C22⋊D7911yescharacteristic
418C2×C8C2×C22⋊D7911yescharacteristic
419C2×C8C2×C22⋊D7911yescharacteristic
420C2×C8C23×D7911yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—3161no—
424C4⋊C4—3161no—
425C4⋊C4—3161no—
426C4⋊C4—3161no—
427C4⋊C4—3161no—
428C4⋊C4—3161no—
429C4⋊C4—1581no—
430C4⋊C4—1581no—
431C4⋊C4—1581no—
432C4⋊C4#1192960911yescharacteristic
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—3161no—
439C22⋊C4—3161no—
440C22⋊C4#1192960911yescharacteristic
441C22⋊C4#1192960911yescharacteristic
442C42—3161no—
443C42C2×C22⋊D7911yescharacteristic
444C16—1581no—
445C16—1581no—
446C16—1581no—
447C16—1581no—
448C23—41no—
449C23—41no—
450C23—82no—
451C23—82no—
452C23—161no—
453C23C2×#1192960911yescharacteristic
454C23C2×#1192960911yescharacteristic
455Q8—41no—
456Q8—41no—
457Q8—3161no—
458Q8—3161no—
459Q8—12641no—
460D4—22no—
461D4—22no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—82no—
466D4—82no—
467D4—161no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—82no—
475C2×C4—161no—
476C2×C4—3161no—
477C2×C4—3161no—
478C2×C4—3161no—
479C2×C4—3161no—
480C2×C4—12641no—
481C2×C4—12641no—
482C2×C4C2×#1192960911yescharacteristic
483C2×C4C2×#1192960911yescharacteristic
484C2×C4C2×#1192960911yescharacteristic
485C2×C4C2×#1192960911yescharacteristic
486C2×C4C22×C22⋊D7911yescharacteristic
487C8—21no—
488C8—21no—
489C8—21no—
490C8C2×#1192962311yescharacteristic
491C8#1457920611yescharacteristic
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—42no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—164no—
504C22#1909900312yes—
505C22C22×#1192960911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—81no—
512C4—6321no—
513C4—6321no—
514C4—6321no—
515C4—6321no—
516C4#1909920711yescharacteristic
517C4#1909937411yescharacteristic
518C2—22no—
519C2—41no—
520C2—82no—
521C2—161no—
522C2#2377521311yescharacteristic
523C2C2×#1909900311yescharacteristic
524C2#2377653911yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
Fitting, prime radicalC79×#3280C2
socleC22×C79C22×D8
hypercenter#3280D79
hypocenterC79#33881
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #23775793C2semidirect
2. #23765636C2semidirect
3. #23765600C2semidirect
4. #23765578C2semidirect
5. #23765537C2semidirect
6. #23765511C2semidirect
7. #23765489C2semidirect
8. #23765459C2semidirect
9. #23765340C2semidirect
10. #23765329C2semidirect
11. #23765264C2semidirect
12. #23765226C2semidirect
13. #23765112C2semidirect
14. C2×#19098364C2semidirect
17. C2×#14579107C22semidirect
20. C2×#14579101C22semidirect
22. #19098968C22semidirect
27. #19098804C22semidirect
28. C4×Dic316C22semidirect
30. C2×#14579095C22semidirect
31. #19098388C22semidirect
32. #19098388C22semidirect
33. #19098388C22semidirect
34. C4⋊(C79⋊C16)C22semidirect
35. #19098367C22semidirect
36. #19098367C22semidirect
37. #19098367C22semidirect
38. #19098366C22semidirect
46. C22⋊(C79⋊C16)C22semidirect
47. #19098339C22semidirect
278. C79#33881semidirect

Maximal subgroups

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

subgroupindex
#237757932
#237656362
#237656002
#237655782
#237655372
#237655112
#237654892
#237654592
#237653402
#237653292
#237652642
#237652262
#237651122
C2×#190983642
C79×#32802
#3388179