Finite Group Databasebeta

← #34444652

Subgroups — #34444652

Order 20224.

500 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774039C211yesmaximal, characteristic, complemented
2#23765561C211yesmaximal, characteristic, complemented
3#23765554C211yesmaximal, characteristic, complemented
4#23765547C211yesmaximal, characteristic, complemented
5#23765537C211yesmaximal, characteristic, complemented
6#23765511C211yesmaximal, characteristic, complemented
7#23765501C211yesmaximal, characteristic, complemented
8#23765494C211yesmaximal, characteristic, complemented
9#23765484C211yesmaximal, characteristic, complemented
10#23765455C211yesmaximal, characteristic, complemented
11#23765455C211yesmaximal, characteristic, complemented
12#23765437C211yesmaximal, characteristic, complemented
13#23765432C211yesmaximal, characteristic, complemented
14#23765159C211yesmaximal, characteristic, complemented
15C79×#3176C211yesmaximal, characteristic, complemented
16#19099184—21no—
17C2×#14579105C2211yescharacteristic
18C2×#14579105C2211yescharacteristic
19C2×#14579102C2211yescharacteristic
20C2×#14579102C2211yescharacteristic
21C2×#14579101C2211yescharacteristic
22C22×C79⋊C16C2211yescharacteristic
23#19098968C2211yescharacteristic
24#19098947C2211yescharacteristic
25#19098923—21no—
26#19098809—21no—
27#19098804C2211yescharacteristic
28#19098799C2211yescharacteristic
29#19098665—21no—
30C4×Dic316C2211yescharacteristic
31C4×D632C2211yescharacteristic
32#19098390C2211yescharacteristic
33#19098390C2211yescharacteristic
34#19098388C2211yescharacteristic
35#19098388C2211yescharacteristic
36C4⋊(C79⋊C16)C2211yescharacteristic
37C4⋊(C79⋊C16)C2211yescharacteristic
38C4⋊(C79⋊C16)C2211yescharacteristic
39C4⋊(C79⋊C16)C2211yescharacteristic
40#19098366C2211yescharacteristic
41#19098366C2211yescharacteristic
42#19098364C2211yescharacteristic
43#19098364C2211yescharacteristic
44#19098364C2211yescharacteristic
45#19098364C2211yescharacteristic
46#19098363C2211yescharacteristic
47#19098363C2211yescharacteristic
48#540×C79—21no—
49#499×C79C2211yescharacteristic
50#493×C79C2211yescharacteristic
51#490×C79—21no—
52#469×C79—21no—
53#466×C79C2211yescharacteristic
54#466×C79C2211yescharacteristic
55#442×C79—21no—
56C4×D8×C79C2211yescharacteristic
57C4×D8×C79C2211yescharacteristic
58#405×C79C2211yescharacteristic
59C2×Dic316C2311yescharacteristic
60C2×#11929613—21no—
61C2×D632C2311yescharacteristic
62#14579161—21no—
63#14579156—41no—
64#14579149—21no—
65#14579147—41no—
66#14579143—41no—
67#14579140—21no—
68#14579139—41no—
69#14579135—21no—
70C4×Dic158—21no—
71C4×D316—21no—
72#14579105—41no—
73#14579105—41no—
74#14579102—41no—
75#14579102—41no—
76#14579101—41no—
77C2×C79⋊C16—41no—
78C2×C79⋊C16C2311yescharacteristic
79C2×C79⋊C16C2311yescharacteristic
80C2×C79⋊C16C2311yescharacteristic
81C2×C79⋊C16C2311yescharacteristic
82#14579095—21no—
83#14579095C2311yescharacteristic
84#14579095C2311yescharacteristic
85#14579086—21no—
86#14579086—21no—
87#14579083—21no—
88#14579083—21no—
89C2×D8×C79—21no—
90C2×D8×C79C2311yescharacteristic
91C2×D8×C79C2311yescharacteristic
92C2×M16×C79D411yescharacteristic
93C22×C8×C79C2311yescharacteristic
94#127×C79—21no—
95#125×C79—21no—
96#123×C79—41no—
97#123×C79—41no—
98#121×C79—21no—
99#121×C79—21no—
100#121×C79—41no—
101#121×C79—41no—
102C4×D4×C79—21no—
103C4×D4×C79—21no—
104#117×C79D411yescharacteristic
105#107×C79C2311yescharacteristic
106#107×C79C2311yescharacteristic
107#106×C79—21no—
108#102×C79—21no—
109#102×C79—21no—
110#102×C79—21no—
111#102×C79—21no—
112#97×C79D411yescharacteristic
113#97×C79D411yescharacteristic
114C4×C8×C79C2311yescharacteristic
115C4×C8×C79C2311yescharacteristic
116C2×Dic158—21no—
117C2×Dic158—41no—
118C2×D316—21no—
119C2×C22⋊D79—41no—
120C2×C4×D79—41no—
121Dic316—21no—
122Dic316—21no—
123#11929613—42no—
124D632—21no—
125D632—21no—
126C4⋊Dic79—21no—
127C4⋊Dic79—21no—
128C4⋊Dic79—41no—
129Dic79⋊C4—41no—
130Dic79⋊C4—41no—
131C22⋊Dic79—41no—
132(C2×D79)⋊C4—41no—
133(C2×D79)⋊C4—41no—
134(C2×D79)⋊C4—41no—
135(C2×D79)⋊C4—41no—
136C4×Dic79—41no—
137C79⋊C16—21no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141C2×D4×C79—21no—
142C2×D4×C79—21no—
143C2×D4×C79—41no—
144C2×D4×C79—41no—
145C2×D4×C79—41no—
146C2×D4×C79—41no—
147C22×C4×C79—41no—
148C22×C4×C79—41no—
149C22×C4×C79C2×D411yescharacteristic
150D8×C79—21no—
151D8×C79—21no—
152D8×C79—21no—
153D8×C79—21no—
154D8×C79—42no—
155M16×C79—41no—
156C2×C8×C79—21no—
157C2×C8×C79—21no—
158C2×C8×C79C2×D411yescharacteristic
159C2×C8×C79C2×D411yescharacteristic
160C2×C8×C79C2×D411yescharacteristic
161C2×C8×C79C2411yescharacteristic
162C4⋊C4×C79—21no—
163C4⋊C4×C79—21no—
164C4⋊C4×C79—41no—
165C4⋊C4×C79—41no—
166C4⋊C4×C79D811yescharacteristic
167C4⋊C4×C79D811yescharacteristic
168C22⋊C4×C79—41no—
169C22⋊C4×C79—41no—
170C22⋊C4×C79—41no—
171C22⋊C4×C79—41no—
172C22⋊C4×C79D811yescharacteristic
173C22⋊C4×C79D811yescharacteristic
174C42×C79C2×D411yescharacteristic
175C42×C79C2×D411yescharacteristic
176C23×C79—41no—
177C23×C79—41no—
178C23×C79C2×D811yescharacteristic
179C22×D79—41no—
180D4×C79—41no—
181D4×C79—41no—
182D4×C79—41no—
183D4×C79—41no—
184D4×C79—161no—
185D4×C79—161no—
186D4×C79—161no—
187D4×C79—161no—
188C2×C4×C79—21no—
189C2×C4×C79—21no—
190C2×C4×C79—41no—
191C2×C4×C79—41no—
192C2×C4×C79—161no—
193C2×C4×C79—161no—
194C2×C4×C79C2×D811yescharacteristic
195C2×C4×C79C2×D811yescharacteristic
196C2×C4×C79C2×D811yescharacteristic
197C2×C4×C79C2×D811yescharacteristic
198C2×C4×C79C2×D811yescharacteristic
199C2×C4×C79C22×D411yescharacteristic
200C22⋊D79—161no—
201C2×Dic79—41no—
202C2×Dic79—41no—
203C2×Dic79—41no—
204D316—41no—
205D316—41no—
206C4×D79—161no—
207Dic158—41no—
208Dic158—41no—
209Dic158—161no—
210C8×C79—21no—
211C8×C79—21no—
212C8×C79—21no—
213C8×C79H(5,𝔽2)11yescharacteristic
214C8×C792−1+411yescharacteristic
215C22×C79—21no—
216C22×C79—21no—
217C22×C79—21no—
218C22×C79—81no—
219C22×C79—81no—
220C22×C79—81no—
221C22×C79—81no—
222C22×C79—81no—
223C22×C79—81no—
224C22×C79C22×D811yescharacteristic
225C2×D79—81no—
226C2×D79—81no—
227C2×D79—81no—
228C4×C79—21no—
229C4×C79—21no—
230C4×C79—21no—
231C4×C79—21no—
232C4×C79—21no—
233C4×C79—81no—
234C4×C79—81no—
235C4×C79#53711yescharacteristic
236C4×C79#57511yescharacteristic
237Dic79—81no—
238Dic79—81no—
239Dic79—81no—
240#33823—791nomaximal, Sylow
241C2×C79—41no—
242C2×C79—161no—
243C2×C79—161no—
244C2×C79#324511yescharacteristic
245C2×C79#341511yescharacteristic
246C2×C79#341611yescharacteristic
247D79—161no—
248#3176D7911yescharacteristic, complemented
249#3175—791no—
250#2229—791no—
251#2228—791no—
252#2226—791no—
253#2224—791no—
254#2220—791no—
255#2218—791no—
256#2217—791no—
257#2215—791no—
258#2210—791no—
259#2210—791no—
260#2207—791no—
261#2206—791no—
262#2151—791no—
263C79#3382311yescharacteristic, complemented, Sylow
264#542—1581no—
265#540—21no—
266C2×SD32—791no—
267C2×SD32—791no—
268C2×D16—791no—
269C2×D16—791no—
270C2×M32—791no—
271C22×C16—791no—
272#500—791no—
273#499C2×D7911yescharacteristic
274#495—791no—
275#493C2×D7911yescharacteristic
276#490—21no—
277#489—1581no—
278#469—21no—
279#468—1581no—
280#467—791no—
281#466—791no—
282#466C2×D7911yescharacteristic
283#466C2×D7911yescharacteristic
284#442—21no—
285#440—1581no—
286C4×Q16—791no—
287C4×D8—791no—
288C4×D8C2×D7911yescharacteristic
289C4×D8C2×D7911yescharacteristic
290#405C2×D7911yescharacteristic
291#367—791no—
292#367—791no—
293#366—791no—
294#366—791no—
295C4⋊C16—791no—
296C4⋊C16—791no—
297C4⋊C16—791no—
298C4⋊C16—791no—
299#358—791no—
300#358—791no—
301#357—791no—
302#357—791no—
303#357—791no—
304#357—791no—
305#357—791no—
306#357—791no—
307C2×Q16—791no—
308C2×SD16—1581no—
309C2×D8—21no—
310C2×D8—791no—
311C2×D8C22×D7911yescharacteristic
312C2×D8C22×D7911yescharacteristic
313C2×M16C22⋊D7911yescharacteristic
314C22×C8C22×D7911yescharacteristic
315C4⋊Q8—1581no—
316#127—21no—
317#126—3161no—
318#125—21no—
319#124—3161no—
320#124—1581no—
321#123—41no—
322#123—41no—
323#123—3161no—
324C22⋊Q8—3161no—
325C22⋊Q8—1581no—
326#121—21no—
327#121—21no—
328#121—41no—
329#121—41no—
330#121—1581no—
331C4×Q8—1581no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—1581no—
335#117C22⋊D7911yescharacteristic
336SD32—3161no—
337SD32—3161no—
338D16—3161no—
339D16—3161no—
340M32—3161no—
341C2×C16—3161no—
342C2×C16—791no—
343C2×C16—791no—
344C2×C16—791no—
345C2×C16—791no—
346#107—791no—
347#107—791no—
348#107—1581no—
349#107C22×D7911yescharacteristic
350#107C22×D7911yescharacteristic
351#106—21no—
352#103—1581no—
353#103—1581no—
354#102—21no—
355#102—21no—
356#102—21no—
357#102—21no—
358#102—1581no—
359#102—1581no—
360#97C22⋊D7911yescharacteristic
361#97C22⋊D7911yescharacteristic
362C4×C8C22×D7911yescharacteristic
363C4×C8C22×D7911yescharacteristic
364C2×Q8—3161no—
365C2×Q8—1581no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—3161no—
373C2×D4—1581no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—3161no—
377C22×C4C2×C22⋊D7911yescharacteristic
378Q16—1581no—
379Q16—1581no—
380SD16—3162no—
381D8—21no—
382D8—21no—
383D8—21no—
384D8—21no—
385D8—42no—
386D8—1581no—
387D8—1581no—
388M16—41no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8C2×C22⋊D7911yescharacteristic
392C2×C8C2×C22⋊D7911yescharacteristic
393C2×C8C2×C22⋊D7911yescharacteristic
394C2×C8C23×D7911yescharacteristic
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—3161no—
400C4⋊C4—3161no—
401C4⋊C4—3161no—
402C4⋊C4—1581no—
403C4⋊C4—1581no—
404C4⋊C4#1192960911yescharacteristic
405C4⋊C4#1192960911yescharacteristic
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—3161no—
411C22⋊C4—3161no—
412C22⋊C4—3161no—
413C22⋊C4—3161no—
414C22⋊C4—3161no—
415C22⋊C4#1192960911yescharacteristic
416C22⋊C4#1192960911yescharacteristic
417C42—3161no—
418C42C2×C22⋊D7911yescharacteristic
419C42C2×C22⋊D7911yescharacteristic
420C16—1581no—
421C16—1581no—
422C16—1581no—
423C16—1581no—
424C23—41no—
425C23—41no—
426C23—3161no—
427C23C2×#1192960911yescharacteristic
428Q8—3161no—
429Q8—3161no—
430Q8—12641no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—161no—
436D4—161no—
437D4—161no—
438D4—161no—
439D4—3161no—
440D4—3161no—
441D4—12641no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—161no—
447C2×C4—161no—
448C2×C4—3161no—
449C2×C4—3161no—
450C2×C4—3161no—
451C2×C4—12641no—
452C2×C4C2×#1192960911yescharacteristic
453C2×C4C2×#1192960911yescharacteristic
454C2×C4C2×#1192960911yescharacteristic
455C2×C4C2×#1192960911yescharacteristic
456C2×C4C2×#1192960911yescharacteristic
457C2×C4C22×C22⋊D7911yescharacteristic
458C8—21no—
459C8—21no—
460C8—21no—
461C8#1457920711yescharacteristic
462C8#1457920811yescharacteristic
463C22—21no—
464C22—21no—
465C22—21no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—6321no—
473C22—6321no—
474C22—6321no—
475C22C22×#1192960911yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—81no—
482C4—81no—
483C4—6321no—
484C4—6321no—
485C4—6321no—
486C4#1909914111yescharacteristic
487C4#1909936111yescharacteristic
488C2—41no—
489C2—161no—
490C2—161no—
491C2—12641no—
492C2#2377470711yescharacteristic
493C2#2377653911yescharacteristic
494C2#2377654711yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3176C2
socleC22×C79C22×D8
hypercenter#3176D79
hypocenterC79#33823
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3176

Complements

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

NH ≅ G/Ntype
1. #23774039C2semidirect
2. #23765561C2semidirect
3. #23765554C2semidirect
4. #23765547C2semidirect
5. #23765537C2semidirect
6. #23765511C2semidirect
7. #23765501C2semidirect
8. #23765494C2semidirect
9. #23765484C2semidirect
10. #23765455C2semidirect
11. #23765455C2semidirect
12. #23765437C2semidirect
13. #23765432C2semidirect
14. #23765159C2semidirect
15. C79×#3176C2semidirect
248. #3176D79semidirect
263. C79#33823semidirect

Maximal subgroups

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

subgroupindex
#237740392
#237655612
#237655542
#237655472
#237655372
#237655112
#237655012
#237654942
#237654842
#237654552
#237654372
#237654322
#237651592
C79×#31762
#3382379