Finite Group Databasebeta

← #34443392

Subgroups — #34443392

Order 20224.

558 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774050C211yesmaximal, characteristic, complemented
2#23765585C211yesmaximal, characteristic, complemented
3#23765580C211yesmaximal, characteristic, complemented
4#23765564C211yesmaximal, characteristic, complemented
5#23765557C211yesmaximal, characteristic, complemented
6#23765440C211yesmaximal, characteristic, complemented
7#23765435C211yesmaximal, characteristic, complemented
8#23765323C211yesmaximal, characteristic, complemented
9#23765323C211yesmaximal, characteristic, complemented
10#23765274C211yesmaximal, characteristic, complemented
11#23765262C211yesmaximal, characteristic, complemented
12C4×#14579104C211yesmaximal, characteristic, complemented
13C4×#14579102C211yesmaximal, characteristic, complemented
14#23765207C211yesmaximal, characteristic, complemented
15C79×#2951C211yesmaximal, characteristic, complemented
16#19099168—21no—
17C2×#14579104C2211yescharacteristic
18C2×#14579104C2211yescharacteristic
19C2×#14579102C2211yescharacteristic
20C2×#14579102C2211yescharacteristic
21#19098965C2211yescharacteristic
22#19098940C2211yescharacteristic
23#19098928—21no—
24#19098816—21no—
25#19098799C2211yescharacteristic
26#19098799C2211yescharacteristic
27#19098676—21no—
28C4×D632C2211yescharacteristic
29C4×D632C2211yescharacteristic
30#19098390C2211yescharacteristic
31#19098390C2211yescharacteristic
32#19098388C2211yescharacteristic
33#19098388C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35C4⋊(C79⋊C16)C2211yescharacteristic
36#19098367C2211yescharacteristic
37#19098367C2211yescharacteristic
38#19098364C2211yescharacteristic
39#19098364C2211yescharacteristic
40#19098363C2211yescharacteristic
41#19098363C2211yescharacteristic
42#19098363C2211yescharacteristic
43#19098363C2211yescharacteristic
44C22⋊(C79⋊C16)C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46#19098339C2211yescharacteristic
47C4×C79⋊C16C2211yescharacteristic
48C2×#135×C79C2211yescharacteristic
49#520×C79—21no—
50#440×C79—21no—
51#440×C79—21no—
52C4×Q16×C79C2211yescharacteristic
53C4×Q16×C79C2211yescharacteristic
54C4×D8×C79C2211yescharacteristic
55C4×D8×C79C2211yescharacteristic
56C2×#107×C79C2211yescharacteristic
57#417×C79—21no—
58#405×C79C2211yescharacteristic
59C2×D632—21no—
60C2×D632C2311yescharacteristic
61C2×D632C2311yescharacteristic
62#14579158—21no—
63#14579153—21no—
64#14579146—41no—
65#14579146—41no—
66#14579135—21no—
67#14579135—21no—
68#14579134—41no—
69#14579134—41no—
70C4×D316—21no—
71C4×D316—21no—
72#14579104—22no—
73#14579104—22no—
74#14579102—22no—
75#14579102—22no—
76C2×C79⋊C16C2311yescharacteristic
77C2×C79⋊C16C2311yescharacteristic
78C2×C79⋊C16C2311yescharacteristic
79C2×C79⋊C16C2311yescharacteristic
80#14579095C2311yescharacteristic
81#14579095C2311yescharacteristic
82#14579093—21no—
83#14579083—21no—
84#14579083—21no—
85#14579083—21no—
86#14579083—21no—
87C2×#41×C79—21no—
88#135×C79—22no—
89#135×C79—22no—
90C2×Q16×C79C2311yescharacteristic
91C2×SD16×C79—21no—
92C2×D8×C79C2311yescharacteristic
93C2×M16×C79D411yescharacteristic
94C22×C8×C79C2311yescharacteristic
95C4×Q8×C79—21no—
96C4×Q8×C79—21no—
97C4×D4×C79—21no—
98C4×D4×C79—21no—
99C4×D4×C79—41no—
100C4×D4×C79—41no—
101#117×C79—41no—
102#117×C79D411yescharacteristic
103C2×C4⋊C4×C79—21no—
104C2×C4⋊C4×C79—41no—
105#107×C79—21no—
106#107×C79C2311yescharacteristic
107#107×C79C2311yescharacteristic
108#103×C79—21no—
109#103×C79—21no—
110#102×C79—21no—
111#102×C79—21no—
112#97×C79D411yescharacteristic
113#97×C79D411yescharacteristic
114C4×C8×C79C2311yescharacteristic
115C4×C8×C79C2311yescharacteristic
116C2×D316—21no—
117C2×D316—21no—
118C2×D316—41no—
119C2×D316—41no—
120C2×C22⋊D79—41no—
121C2×C22⋊D79—41no—
122C2×C4×D79—41no—
123C2×C4×D79—41no—
124D632—21no—
125D632—21no—
126D632—21no—
127D632—21no—
128D632—42no—
129C4⋊Dic79—21no—
130C4⋊Dic79—21no—
131Dic79⋊C4—41no—
132Dic79⋊C4—41no—
133(C2×D79)⋊C4—41no—
134(C2×D79)⋊C4—41no—
135(C2×D79)⋊C4—41no—
136(C2×D79)⋊C4—41no—
137C79⋊C16—21no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141#41×C79—24no—
142#41×C79—42no—
143C2×Q8×C79—21no—
144C2×D4×C79—21no—
145C2×D4×C79—41no—
146C22×C4×C79—41no—
147C22×C4×C79—41no—
148C22×C4×C79—41no—
149C22×C4×C79—41no—
150C22×C4×C79C2×D411yescharacteristic
151Q16×C79—21no—
152Q16×C79#4112yes—
153SD16×C79—24no—
154D8×C79—21no—
155D8×C79#4112yes—
156M16×C79—41no—
157C2×C8×C79—21no—
158C2×C8×C79—21no—
159C2×C8×C79C2×D411yescharacteristic
160C2×C8×C79C2×D411yescharacteristic
161C2×C8×C79C2×D411yescharacteristic
162C2×C8×C79C2411yescharacteristic
163C4⋊C4×C79—21no—
164C4⋊C4×C79—21no—
165C4⋊C4×C79—41no—
166C4⋊C4×C79—41no—
167C4⋊C4×C79—41no—
168C4⋊C4×C79D811yescharacteristic
169C4⋊C4×C79D811yescharacteristic
170C22⋊C4×C79—41no—
171C22⋊C4×C79—41no—
172C22⋊C4×C79D811yescharacteristic
173C22⋊C4×C79D811yescharacteristic
174C42×C79—41no—
175C42×C79—41no—
176C42×C79C2×D411yescharacteristic
177C42×C79C2×D411yescharacteristic
178C23×C79—41no—
179C23×C79C2×D811yescharacteristic
180C22×D79—41no—
181C22×D79—41no—
182Q8×C79—22no—
183Q8×C79—41no—
184D4×C79—22no—
185D4×C79—44no—
186D4×C79—41no—
187C2×C4×C79—21no—
188C2×C4×C79—21no—
189C2×C4×C79—44no—
190C2×C4×C79—41no—
191C2×C4×C79—41no—
192C2×C4×C79—41no—
193C2×C4×C79—161no—
194C2×C4×C79—161no—
195C2×C4×C79—161no—
196C2×C4×C79C2×D811yescharacteristic
197C2×C4×C79C2×D811yescharacteristic
198C2×C4×C79C2×D811yescharacteristic
199C2×C4×C79C2×D811yescharacteristic
200C2×C4×C79C2×D811yescharacteristic
201C2×C4×C79C22×D411yescharacteristic
202C22⋊D79—161no—
203C22⋊D79—161no—
204C2×Dic79—41no—
205C2×Dic79—41no—
206D316—41no—
207D316—41no—
208D316—41no—
209D316—41no—
210D316—161no—
211D316—161no—
212C4×D79—161no—
213C4×D79—161no—
214C8×C79—21no—
215C8×C79—21no—
216C8×C79—21no—
217C8×C79C2×#4111yescharacteristic
218C8×C79C2×#4111yescharacteristic
219C22×C79—21no—
220C22×C79—21no—
221C22×C79—21no—
222C22×C79—42no—
223C22×C79—81no—
224C22×C79—81no—
225C22×C79C22×D811yescharacteristic
226C2×D79—81no—
227C2×D79—81no—
228C2×D79—81no—
229C2×D79—81no—
230C2×D79—81no—
231C2×D79—81no—
232C4×C79—21no—
233C4×C79—21no—
234C4×C79—21no—
235C4×C79—21no—
236C4×C79—21no—
237C4×C79—42no—
238C4×C79—81no—
239C4×C79—81no—
240C4×C79#52511yescharacteristic
241C4×C79#57511yescharacteristic
242Dic79—81no—
243Dic79—81no—
244#33679—791nomaximal, Sylow
245C2×C79—41no—
246C2×C79—82no—
247C2×C79#310011yescharacteristic
248C2×C79#341511yescharacteristic
249C2×C79#341611yescharacteristic
250D79—161no—
251D79—161no—
252#3176—791no—
253#2951D7911yescharacteristic, complemented
254#2233—791no—
255#2232—791no—
256#2229—791no—
257#2228—791no—
258#2207—791no—
259#2206—791no—
260#2186—791no—
261#2186—791no—
262#2177—791no—
263#2175—791no—
264C4×SD32—791no—
265C4×D16—791no—
266#2164—791no—
267C79#3367911yescharacteristic, complemented, Sylow
268C2×#135C2×D7911yescharacteristic
269#540—1581no—
270#520—21no—
271C2×SD32—791no—
272C2×SD32—791no—
273C2×D16—791no—
274C2×D16—791no—
275#499—791no—
276#493—791no—
277#490—1581no—
278#469—1581no—
279#466—791no—
280#466—791no—
281#442—1581no—
282#440—21no—
283#440—21no—
284C4×Q16C2×D7911yescharacteristic
285C4×Q16C2×D7911yescharacteristic
286C4×D8—791no—
287C4×D8—791no—
288C4×D8C2×D7911yescharacteristic
289C4×D8C2×D7911yescharacteristic
290C2×#107C2×D7911yescharacteristic
291#417—21no—
292#405C2×D7911yescharacteristic
293#367—791no—
294#367—791no—
295#366—791no—
296#366—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—
307C22⋊C16—791no—
308C22⋊C16—791no—
309#346—791no—
310C4×C16—791no—
311C2×#41—21no—
312#135—22no—
313#135—22no—
314C2×Q16C22×D7911yescharacteristic
315C2×SD16—21no—
316C2×D8—791no—
317C2×D8—791no—
318C2×D8—1581no—
319C2×D8C22×D7911yescharacteristic
320C2×M16C22⋊D7911yescharacteristic
321C22×C8C22×D7911yescharacteristic
322#127—1581no—
323#125—1581no—
324#123—3161no—
325#123—3161no—
326#121—3161no—
327#121—3161no—
328#121—1581no—
329#121—1581no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—41no—
335C4×D4—41no—
336C4×D4—1581no—
337C4×D4—1581no—
338#117—41no—
339#117C22⋊D7911yescharacteristic
340C2×C4⋊C4—21no—
341C2×C4⋊C4—41no—
342SD32—1582no—
343SD32—1582no—
344D16—1582no—
345D16—1582no—
346C2×C16—791no—
347C2×C16—791no—
348C2×C16—791no—
349C2×C16—791no—
350#107—21no—
351#107—791no—
352#107—791no—
353#107C22×D7911yescharacteristic
354#107C22×D7911yescharacteristic
355#106—1581no—
356#103—21no—
357#103—21no—
358#102—21no—
359#102—21no—
360#102—1581no—
361#102—1581no—
362#102—1581no—
363#102—1581no—
364#97C22⋊D7911yescharacteristic
365#97C22⋊D7911yescharacteristic
366C4×C8C22×D7911yescharacteristic
367C4×C8C22×D7911yescharacteristic
368#41—24no—
369#41—42no—
370C2×Q8—21no—
371C2×D4—21no—
372C2×D4—41no—
373C2×D4—3161no—
374C2×D4—3161no—
375C2×D4—3161no—
376C2×D4—3161no—
377C2×D4—1581no—
378C2×D4—1581no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—3161no—
384C22×C4—3161no—
385C22×C4C2×C22⋊D7911yescharacteristic
386Q16—21no—
387Q16#1192962412yes—
388SD16—24no—
389D8—21no—
390D8—3162no—
391D8—1581no—
392D8—1581no—
393D8—1581no—
394D8—1581no—
395D8#1192962412yes—
396M16—41no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8C2×C22⋊D7911yescharacteristic
400C2×C8C2×C22⋊D7911yescharacteristic
401C2×C8C2×C22⋊D7911yescharacteristic
402C2×C8C23×D7911yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—3161no—
409C4⋊C4—3161no—
410C4⋊C4—1581no—
411C4⋊C4—1581no—
412C4⋊C4#1192960911yescharacteristic
413C4⋊C4#1192960911yescharacteristic
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—3161no—
417C22⋊C4—3161no—
418C22⋊C4—3161no—
419C22⋊C4—3161no—
420C22⋊C4#1192960911yescharacteristic
421C22⋊C4#1192960911yescharacteristic
422C42—41no—
423C42—41no—
424C42C2×C22⋊D7911yescharacteristic
425C42C2×C22⋊D7911yescharacteristic
426C16—1581no—
427C16—1581no—
428C16—1581no—
429C16—1581no—
430C23—41no—
431C23—3161no—
432C23—3161no—
433C23C2×#1192960911yescharacteristic
434Q8—22no—
435Q8—41no—
436D4—22no—
437D4—44no—
438D4—41no—
439D4—3161no—
440D4—3161no—
441D4—3161no—
442D4—3161no—
443D4—12641no—
444D4—12641no—
445D4—12641no—
446D4—12641no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—161no—
454C2×C4—161no—
455C2×C4—161no—
456C2×C4—3161no—
457C2×C4—3161no—
458C2×C4—12641no—
459C2×C4—12641no—
460C2×C4C2×#1192960911yescharacteristic
461C2×C4C2×#1192960911yescharacteristic
462C2×C4C2×#1192960911yescharacteristic
463C2×C4C2×#1192960911yescharacteristic
464C2×C4C2×#1192960911yescharacteristic
465C2×C4C22×C22⋊D7911yescharacteristic
466C8—21no—
467C8—21no—
468C8—21no—
469C8C2×#1192962411yescharacteristic
470C8C2×#1192962411yescharacteristic
471C22—21no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—81no—
476C22—81no—
477C22—6321no—
478C22—6321no—
479C22—6321no—
480C22—6321no—
481C22—6321no—
482C22—6321no—
483C22C22×#1192960911yescharacteristic
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—42no—
490C4—81no—
491C4—81no—
492C4—6321no—
493C4—6321no—
494C4#1909907411yescharacteristic
495C4#1909936111yescharacteristic
496C2—41no—
497C2—82no—
498C2—12641no—
499C2—12641no—
500C2#2377340011yescharacteristic
501C2#2377653911yescharacteristic
502C2#2377654411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#2951C2
socleC22×C79C22×D8
hypercenter#2951D79
hypocenterC79#33679
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #23774050C2semidirect
2. #23765585C2semidirect
3. #23765580C2semidirect
4. #23765564C2semidirect
5. #23765557C2semidirect
6. #23765440C2semidirect
7. #23765435C2semidirect
8. #23765323C2semidirect
9. #23765323C2semidirect
10. #23765274C2semidirect
11. #23765262C2semidirect
12. C4×#14579104C2semidirect
13. C4×#14579102C2semidirect
14. #23765207C2semidirect
15. C79×#2951C2semidirect
253. #2951D79semidirect
267. C79#33679semidirect

Maximal subgroups

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

subgroupindex
#237740502
#237655852
#237655802
#237655642
#237655572
#237654402
#237654352
#237653232
#237652742
#237652622
C4×#145791042
C4×#145791022
#237652072
C79×#29512
#3367979