Finite Group Databasebeta

← #34180193

Subgroups — #34180193

Order 20224.

756 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764349C211yesmaximal, characteristic, complemented
2#23763420C211yesmaximal, characteristic, complemented
3#23762085C211yesmaximal, characteristic, complemented
4#23762077C211yesmaximal, characteristic, complemented
5#23761895C211yesmaximal, characteristic, complemented
6#23761787C211yesmaximal, characteristic, complemented
7C79×#3511C211yesmaximal, characteristic
8#19098616C2211yescharacteristic
9#19098587C2211yescharacteristic
10C2×C4⋊(C79⋊C8)C2211yescharacteristic
11C2×#14579088—21no—
12C4×C4⋊Dic79C2211yescharacteristic
13#19098325—22no—
14#19098322—21no—
15#19098309—21no—
16#19098309—22no—
17#19098307—21no—
18#19098297—22no—
19Q8⋊(C79⋊C8)—21no—
20D4⋊(C79⋊C8)—21no—
21#556×C79C412yescomplemented
22#555×C79—21no—
23#554×C79—21no—
24#553×C79—22no—
25#548×C79—22no—
26#547×C79C412yescomplemented
27#542×C79—22no—
28#537×C79—21no—
29#533×C79—21no—
30#532×C79—21no—
31#532×C79C2211yescharacteristic
32C2×C4⋊Q8×C79C2211yescharacteristic
33#520×C79—21no—
34C4×#41×C79C2211yescharacteristic
35C2×#11929608—21no—
36C22×C79⋊C8—21no—
37C2×C4⋊Dic79—21no—
38C2×C4×Dic79—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44#14579088—42no—
45C22⋊(C79⋊C8)—21no—
46C4×C79⋊C8—21no—
47C79⋊#95—21no—
48C2×#41×C79—22no—
49C2×#41×C79—21no—
50C2×#41×C79D411yescharacteristic
51C22×Q8×C79—21no—
52C4⋊Q8×C79—21no—
53C4⋊Q8×C79—21no—
54C4⋊Q8×C79C2×C412yes—
55C4⋊Q8×C79D412yes—
56C4⋊Q8×C79D411yescharacteristic
57#127×C79D411yescharacteristic
58#126×C79—24no—
59#125×C79—22no—
60#124×C79—24no—
61#124×C79—22no—
62#124×C79—21no—
63#123×C79—24no—
64C22⋊Q8×C79—22no—
65C22⋊Q8×C79—24no—
66C22⋊Q8×C79—22no—
67C22⋊Q8×C79—22no—
68#121×C79—22no—
69#121×C79—22no—
70#121×C79—24no—
71#121×C79—22no—
72#121×C79—21no—
73#121×C79D412yes—
74C4×Q8×C79—22no—
75C4×Q8×C79—22no—
76C4×Q8×C79C2×C411yescharacteristic
77C4×Q8×C79D411yescharacteristic
78C4×D4×C79—22no—
79C4×D4×C79—24no—
80C4×D4×C79—22no—
81C4×D4×C79—21no—
82C4×D4×C79—21no—
83C4×D4×C79—21no—
84C4×D4×C79C2×C411yescharacteristic
85C4×D4×C79C2×C412yes—
86C4×D4×C79D411yescharacteristic
87#117×C79—22no—
88#117×C79—22no—
89#117×C79—21no—
90#117×C79D411yescharacteristic
91C2×C4⋊C4×C79—21no—
92C2×C4⋊C4×C79—22no—
93C2×C4⋊C4×C79D412yes—
94C2×C42×C79—21no—
95C2×C42×C79C2311yescharacteristic
96C22×Dic79—41no—
97#11929608—41no—
98#11929608—41no—
99C2×C79⋊C8—42no—
100C2×C79⋊C8—41no—
101C2×C79⋊C8—41no—
102C2×C79⋊C8—41no—
103C4⋊Dic79—22no—
104C4⋊Dic79—41no—
105C4×Dic79—41no—
106C4×Dic79—41no—
107#41×C79—21no—
108#41×C79—21no—
109#41×C79—21no—
110#41×C79—21no—
111#41×C79—41no—
112#41×C79—41no—
113#41×C79—41no—
114#41×C79—41no—
115#41×C79—84no—
116C2×Q8×C79—22no—
117C2×Q8×C79—21no—
118C2×Q8×C79—22no—
119C2×Q8×C79—82no—
120C2×Q8×C79C22⋊C411yescharacteristic
121C2×D4×C79—21no—
122C2×D4×C79—22no—
123C2×D4×C79—24no—
124C2×D4×C79—21no—
125C2×D4×C79—21no—
126C2×D4×C79C22⋊C411yescharacteristic
127C2×D4×C79C22⋊C412yes—
128C22×C4×C79—24no—
129C22×C4×C79—22no—
130C22×C4×C79—21no—
131C22×C4×C79—22no—
132C22×C4×C79—21no—
133C22×C4×C79—21no—
134C22×C4×C79C2×D411yescharacteristic
135C22×C4×C79C2×D411yescharacteristic
136C22×C4×C79C2×D411yescharacteristic
137C4⋊C4×C79—24no—
138C4⋊C4×C79—24no—
139C4⋊C4×C79—22no—
140C4⋊C4×C79—21no—
141C4⋊C4×C79—22no—
142C4⋊C4×C79—22no—
143C4⋊C4×C79—21no—
144C4⋊C4×C79—21no—
145C4⋊C4×C79C22⋊C411yescharacteristic
146C4⋊C4×C79C22⋊C414yes—
147C4⋊C4×C79C22⋊C411yescharacteristic
148C4⋊C4×C79C22⋊C412yes—
149C22⋊C4×C79—24no—
150C22⋊C4×C79—24no—
151C22⋊C4×C79—24no—
152C22⋊C4×C79—21no—
153C22⋊C4×C79—21no—
154C22⋊C4×C79—22no—
155C22⋊C4×C79—21no—
156C22⋊C4×C79—22no—
157C42×C79—24no—
158C42×C79—21no—
159C42×C79—21no—
160C42×C79—21no—
161C42×C79C22×C411yescharacteristic
162C42×C79C2×D411yescharacteristic
163C42×C79C2×D411yescharacteristic
164C42×C79C2×D411yescharacteristic
165C23×C79—21no—
166C23×C79—22no—
167C23×C79#12011yescharacteristic
168Q8×C79—21no—
169Q8×C79—21no—
170Q8×C79—44no—
171Q8×C79—41no—
172Q8×C79—41no—
173Q8×C79—82no—
174D4×C79—21no—
175D4×C79—21no—
176D4×C79—42no—
177D4×C79—41no—
178D4×C79—42no—
179D4×C79—41no—
180D4×C79—42no—
181D4×C79—41no—
182D4×C79—41no—
183D4×C79—41no—
184D4×C79—41no—
185D4×C79—84no—
186C2×C4×C79—21no—
187C2×C4×C79—21no—
188C2×C4×C79—21no—
189C2×C4×C79—21no—
190C2×C4×C79—22no—
191C2×C4×C79—21no—
192C2×C4×C79—24no—
193C2×C4×C79—41no—
194C2×C4×C79—42no—
195C2×C4×C79—42no—
196C2×C4×C79—42no—
197C2×C4×C79—41no—
198C2×C4×C79—41no—
199C2×C4×C79—41no—
200C2×C4×C79—41no—
201C2×C4×C79—82no—
202C2×C4×C79—84no—
203C2×C4×C79C2×C22⋊C411yescharacteristic
204C2×C4×C79C2×C22⋊C411yescharacteristic
205C2×C4×C79C2×C22⋊C411yescharacteristic
206C2×C4×C79#12011yescharacteristic
207C2×C4×C79#12011yescharacteristic
208C2×C4×C79#12011yescharacteristic
209C2×C4×C79#13511yescharacteristic
210C2×C4×C79#13511yescharacteristic
211C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
212C2×C4×C79#13711yescharacteristic
213C2×Dic79—42no—
214C2×Dic79—81no—
215C2×Dic79—81no—
216C79⋊C8—81no—
217C79⋊C8—81no—
218C79⋊C8—81no—
219C22×C79—21no—
220C22×C79—21no—
221C22×C79—41no—
222C22×C79—42no—
223C22×C79—41no—
224C22×C79—42no—
225C22×C79—41no—
226C22×C79—42no—
227C22×C79#37911yescharacteristic
228C22×C79#44911yescharacteristic
229C22×C79#45211yescharacteristic
230C4×C79—21no—
231C4×C79—21no—
232C4×C79—21no—
233C4×C79—41no—
234C4×C79—41no—
235C4×C79—44no—
236C4×C79—41no—
237C4×C79—42no—
238C4×C79#41611yescharacteristic
239C4×C79#41711yescharacteristic
240C4×C79#44911yescharacteristic
241C4×C79#45211yescharacteristic
242C4×C79#45411yescharacteristic
243C4×C79#45611yescharacteristic
244Dic79—82no—
245#8422—791nomaximal, Sylow
246C2×C79—21no—
247C2×C79—21no—
248C2×C79—81no—
249C2×C79—82no—
250C2×C79#178811yescharacteristic
251C2×C79#179311yescharacteristic
252C2×C79#179511yescharacteristic
253#3511D7911yescharacteristic
254#1987—791no—
255#1806—791no—
256#1549—791no—
257#1547—791no—
258#1502—791no—
259#1477—791no—
260C79#842211yescharacteristic, complemented, Sylow
261#556Dic7912yescomplemented
262#555—21no—
263#554—21no—
264#553—22no—
265#548—22no—
266#547Dic7912yescomplemented
267#542—22no—
268#537—21no—
269#533—21no—
270#532—21no—
271#532C2×D7911yescharacteristic
272C2×C4⋊Q8C2×D7911yescharacteristic
273#520—21no—
274C4×#41C2×D7911yescharacteristic
275#431—791no—
276#423—791no—
277C2×C4⋊C8—791no—
278C2×#104—1581no—
279C4×C4⋊C4—791no—
280#340—1582no—
281#339—1581no—
282#332—1581no—
283#332—1582no—
284#331—1581no—
285#327—1582no—
286Q8⋊C8—1581no—
287D4⋊C8—1581no—
288C2×#41—22no—
289C2×#41—21no—
290C2×#41C22⋊D7911yescharacteristic
291C22×Q8—21no—
292C2×M16—1581no—
293C22×C8—1581no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8C2×Dic7912yes—
297C4⋊Q8C22⋊D7912yes—
298C4⋊Q8C22⋊D7911yescharacteristic
299#127C22⋊D7911yescharacteristic
300#126—24no—
301#125—22no—
302#124—24no—
303#124—22no—
304#124—21no—
305#123—24no—
306C22⋊Q8—22no—
307C22⋊Q8—24no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310#121—22no—
311#121—22no—
312#121—24no—
313#121—22no—
314#121—21no—
315#121C22⋊D7912yes—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8C2×Dic7911yescharacteristic
319C4×Q8C22⋊D7911yescharacteristic
320C4×D4—22no—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4C2×Dic7911yescharacteristic
327C4×D4C2×Dic7912yes—
328C4×D4C22⋊D7911yescharacteristic
329#117—22no—
330#117—22no—
331#117—21no—
332#117C22⋊D7911yescharacteristic
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—1581no—
336C2×C4⋊C4C22⋊D7912yes—
337C2×C42—21no—
338C2×C42—1581no—
339C2×C42C22×D7911yescharacteristic
340C4⋊C8—1581no—
341C4⋊C8—1581no—
342C4⋊C8—1581no—
343C4⋊C8—1581no—
344C4⋊C8—1581no—
345#104—3162no—
346C22⋊C8—1581no—
347C4×C8—1581no—
348#95—1581no—
349#41—21no—
350#41—21no—
351#41—21no—
352#41—21no—
353#41—41no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—84no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—82no—
362C2×Q8C22⋊Dic7911yescharacteristic
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4C22⋊Dic7911yescharacteristic
369C2×D4C22⋊Dic7912yes—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—3161no—
377C22×C4C2×C22⋊D7911yescharacteristic
378C22×C4C2×C22⋊D7911yescharacteristic
379C22×C4C2×C22⋊D7911yescharacteristic
380M16—3161no—
381M16—3161no—
382C2×C8—3162no—
383C2×C8—3161no—
384C2×C8—3161no—
385C2×C8—3161no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—3161no—
395C4⋊C4—1582no—
396C4⋊C4C22⋊Dic7911yescharacteristic
397C4⋊C4C22⋊Dic7914yes—
398C4⋊C4C22⋊Dic7911yescharacteristic
399C4⋊C4C22⋊Dic7912yes—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—22no—
406C22⋊C4—21no—
407C22⋊C4—22no—
408C42—24no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—3161no—
413C42—3161no—
414C42C22×Dic7911yescharacteristic
415C42C2×C22⋊D7911yescharacteristic
416C42C2×C22⋊D7911yescharacteristic
417C42C2×C22⋊D7911yescharacteristic
418C23—21no—
419C23—22no—
420C23#1457913011yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—44no—
424Q8—41no—
425Q8—41no—
426Q8—82no—
427D4—21no—
428D4—21no—
429D4—42no—
430D4—41no—
431D4—42no—
432D4—41no—
433D4—42no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—84no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—84no—
456C2×C4—3162no—
457C2×C4—6321no—
458C2×C4—6321no—
459C2×C4C2×C22⋊Dic7911yescharacteristic
460C2×C4C2×C22⋊Dic7911yescharacteristic
461C2×C4C2×C22⋊Dic7911yescharacteristic
462C2×C4#1457913011yescharacteristic
463C2×C4#1457913011yescharacteristic
464C2×C4#1457913011yescharacteristic
465C2×C4#1457918111yescharacteristic
466C2×C4#1457918111yescharacteristic
467C2×C4#1457918611yescharacteristic
468C2×C4#1457919111yescharacteristic
469C8—6321no—
470C8—6321no—
471C8—6321no—
472C22—21no—
473C22—21no—
474C22—41no—
475C22—42no—
476C22—41no—
477C22—42no—
478C22—41no—
479C22—42no—
480C22#1909841911yescharacteristic
481C22#1909870811yescharacteristic
482C22#1909872511yescharacteristic
483C4—21no—
484C4—21no—
485C4—21no—
486C4—41no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—44no—
491C4—6322no—
492C4#1909855911yescharacteristic
493C4#1909856411yescharacteristic
494C4#1909870811yescharacteristic
495C4#1909872511yescharacteristic
496C4#1909873611yescharacteristic
497C4#1909874611yescharacteristic
498C2—21no—
499C2—21no—
500C2—81no—
501C2—82no—
502C2#2376334011yescharacteristic
503C2#2376335911yescharacteristic
504C2#2376337011yescharacteristic
subgroupG/H
centerC22#19098419
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3511C2
socleC22×C79#379
cosocleC2×C42×C79C23
hypercenter#3511D79
hypocenterC79#8422
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #23764349C2semidirect
2. #23763420C2semidirect
3. #23762085C2semidirect
4. #23762077C2semidirect
5. #23761895C2semidirect
6. #23761787C2semidirect
21. #556×C79C4semidirect
26. #547×C79C4semidirect
260. C79#8422semidirect
261. #556Dic79semidirect
266. #547Dic79semidirect

Maximal subgroups

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

subgroupindex
#237643492
#237634202
#237620852
#237620772
#237618952
#237617872
C79×#35112
#842279