Finite Group Databasebeta

← #34272210

Subgroups — #34272210

Order 20224.

608 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773910C211yesmaximal, characteristic, complemented
2#23773562C211yesmaximal, characteristic
3#41×C79⋊C8C211yesmaximal, characteristic
4#23762802C211yesmaximal, characteristic, complemented
5#23762797C211yesmaximal, characteristic, complemented
6#23762792C211yesmaximal, characteristic, complemented
7#23762787C211yesmaximal, characteristic, complemented
8#23762784C211yesmaximal, characteristic, complemented
9#23762780C211yesmaximal, characteristic, complemented
10#23762777C211yesmaximal, characteristic, complemented
11#23762773C211yesmaximal, characteristic, complemented
12#23762161C211yesmaximal, characteristic, complemented
13#23762111C211yesmaximal, characteristic, complemented
14#23762110C211yesmaximal, characteristic, complemented
15C79×#3116C211yesmaximal, characteristic
16#19099120C2211yescharacteristic
17#19098966C2211yescharacteristic
18#19098966C2211yescharacteristic
19#19098962—21no—
20#19098962—21no—
21#19098959C2211yescharacteristic
22#19098959C2211yescharacteristic
23#19098904—21no—
24#19098858C2211yescharacteristic
25#19098853C2211yescharacteristic
26#19098773C2211yescharacteristic
27#19098768C2211yescharacteristic
28#19098725—21no—
29Q8×C79⋊C8C2211yescharacteristic
30Q8×C79⋊C8C2211yescharacteristic
31#19098668—21no—
32#19098663—21no—
33D4×C79⋊C8—21no—
34D4×C79⋊C8—21no—
35D4×C79⋊C8C2211yescharacteristic
36D4×C79⋊C8C2211yescharacteristic
37#19098620C2211yescharacteristic
38#19098616—21no—
39#19098616C2211yescharacteristic
40#19098599—21no—
41#19098599—21no—
42C2×C4×C79⋊C8—21no—
43C2×C4×C79⋊C8C2211yescharacteristic
44#19098315C2211yescharacteristic
45#19098315C2211yescharacteristic
46#19098313C2211yescharacteristic
47#19098313C2211yescharacteristic
48#19098312C2211yescharacteristic
49#19098312C2211yescharacteristic
50#19098311C2211yescharacteristic
51#19098311C2211yescharacteristic
52#19098310C2211yescharacteristic
53#19098310C2211yescharacteristic
54#19098309C2211yescharacteristic
55#19098309C2211yescharacteristic
56#533×C79C2211yescharacteristic
57C4×#41×C79C2211yescharacteristic
58#485×C79—21no—
59#477×C79C2211yescharacteristic
60#476×C79C2211yescharacteristic
61#462×C79C2211yescharacteristic
62#461×C79C2211yescharacteristic
63#452×C79—21no—
64#441×C79—21no—
65#440×C79—21no—
66#432×C79C2211yescharacteristic
67C2×#11929614—21no—
68C2×#11929612—21no—
69C22×C79⋊C8—21no—
70C22×C79⋊C8—21no—
71C22×C79⋊C8D411yescharacteristic
72C22×C79⋊C8D411yescharacteristic
73#14579161C2311yescharacteristic
74#14579161C2311yescharacteristic
75#14579153—21no—
76#14579140—21no—
77#14579140—21no—
78C4×Dic158—21no—
79C4×Dic158—21no—
80#14579120—21no—
81#14579094—21no—
82#14579094—21no—
83#14579094—21no—
84#14579094—21no—
85#14579094—21no—
86#14579092—21no—
87#14579092—21no—
88#14579092—21no—
89#14579092—21no—
90#14579092—21no—
91C4⋊(C79⋊C8)—21no—
92C4⋊(C79⋊C8)—21no—
93C4⋊(C79⋊C8)C2311yescharacteristic
94C4⋊(C79⋊C8)C2311yescharacteristic
95C4⋊(C79⋊C8)C2311yescharacteristic
96C4⋊(C79⋊C8)C2311yescharacteristic
97#14579085—21no—
98#14579084—21no—
99#14579084—21no—
100#14579082—21no—
101C22⋊(C79⋊C8)—21no—
102C22⋊(C79⋊C8)—21no—
103C22⋊(C79⋊C8)—21no—
104C22⋊(C79⋊C8)—21no—
105#14579072—21no—
106C4×C79⋊C8—21no—
107C4×C79⋊C8—21no—
108C4×C79⋊C8—21no—
109C4×C79⋊C8D411yescharacteristic
110C4×C79⋊C8D411yescharacteristic
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C2×#41×C79D411yescharacteristic
114C2×Q16×C79—21no—
115C2×SD16×C79—21no—
116C4⋊Q8×C79C2311yescharacteristic
117C4⋊Q8×C79C2311yescharacteristic
118#125×C79—21no—
119C22⋊Q8×C79—21no—
120C22⋊Q8×C79—21no—
121C4×Q8×C79—21no—
122C4×Q8×C79—21no—
123C4×Q8×C79D411yescharacteristic
124C4×Q8×C79C2311yescharacteristic
125C4×D4×C79—21no—
126C4×D4×C79—21no—
127C4×D4×C79D411yescharacteristic
128C4×D4×C79C2311yescharacteristic
129#117×C79—21no—
130#117×C79—21no—
131#117×C79D411yescharacteristic
132C2×C42×C79—21no—
133C2×C42×C79C2311yescharacteristic
134#107×C79—21no—
135#106×C79—21no—
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138#103×C79—21no—
139#103×C79—21no—
140#103×C79—21no—
141#102×C79—21no—
142C22⋊C8×C79—21no—
143#97×C79—21no—
144C2×Dic158—21no—
145C2×Dic158—21no—
146#11929614—42no—
147#11929612—42no—
148C2×C79⋊C8—22no—
149C2×C79⋊C8—22no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8—42no—
155C2×C79⋊C8—42no—
156C2×C79⋊C8C2×D411yescharacteristic
157C2×C79⋊C8C2×D411yescharacteristic
158C2×C79⋊C8C2×D411yescharacteristic
159C2×C79⋊C8C2×D411yescharacteristic
160C4⋊Dic79—21no—
161C4⋊Dic79—21no—
162C4⋊Dic79—21no—
163C4⋊Dic79—21no—
164Dic79⋊C4—41no—
165Dic79⋊C4—41no—
166C22⋊Dic79—41no—
167C4×Dic79—41no—
168#41×C79—22no—
169#41×C79—22no—
170C2×Q8×C79—21no—
171C2×Q8×C79—21no—
172C2×Q8×C79C2×D411yescharacteristic
173C2×D4×C79—21no—
174C2×D4×C79C2×D411yescharacteristic
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79#4111yescharacteristic
181Q16×C79—42no—
182SD16×C79—42no—
183C2×C8×C79—21no—
184C2×C8×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—41no—
192C4⋊C4×C79—41no—
193C4⋊C4×C79C2×D411yescharacteristic
194C4⋊C4×C79C2×D411yescharacteristic
195C22⋊C4×C79—21no—
196C22⋊C4×C79—21no—
197C22⋊C4×C79—21no—
198C22⋊C4×C79—41no—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—21no—
202C42×C79—41no—
203C42×C79C2×D411yescharacteristic
204C42×C79C2×D411yescharacteristic
205C42×C79#4111yescharacteristic
206C42×C79C2411yescharacteristic
207C23×C79—21no—
208C23×C79#12111yescharacteristic
209Q8×C79—21no—
210Q8×C79—41no—
211Q8×C79—41no—
212Q8×C79—81no—
213Q8×C79#13512yes—
214D4×C79—24no—
215D4×C79—21no—
216D4×C79#13512yes—
217C2×C4×C79—24no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—41no—
226C2×C4×C79—42no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79#12111yescharacteristic
232C2×C4×C79#12111yescharacteristic
233C2×C4×C79#12111yescharacteristic
234C2×C4×C79C22×D411yescharacteristic
235C2×C4×C79C22×D411yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×Dic79—41no—
238C2×Dic79—41no—
239Dic158—41no—
240Dic158—41no—
241Dic158—81no—
242C8×C79—41no—
243C8×C79—41no—
244C79⋊C8—21no—
245C79⋊C8—21no—
246C79⋊C8—41no—
247C79⋊C8—41no—
248C79⋊C8—41no—
249C79⋊C8—41no—
250C79⋊C8#13712yes—
251C79⋊C8#13712yes—
252C22×C79—21no—
253C22×C79—21no—
254C22×C79—22no—
255C22×C79—21no—
256C22×C79—41no—
257C22×C79—41no—
258C22×C79C2×#12111yescharacteristic
259C4×C79—21no—
260C4×C79—21no—
261C4×C79—22no—
262C4×C79—21no—
263C4×C79—21no—
264C4×C79—41no—
265C4×C79—41no—
266C4×C79—81no—
267C4×C79—81no—
268C4×C79C2×#13511yescharacteristic
269C4×C79C2×#13511yescharacteristic
270C4×C79C2×#13711yescharacteristic
271C4×C79C2×#13711yescharacteristic
272Dic79—81no—
273Dic79—81no—
274#19273—791nomaximal, Sylow
275C2×C79—41no—
276C2×C79—42no—
277C2×C79#303911yescharacteristic
278C2×C79#305011yescharacteristic
279C2×C79#306711yescharacteristic
280#3160—791no—
281#3116—791no—
282#3116D7911yescharacteristic
283C8×#41—791no—
284#1676—791no—
285#1675—791no—
286#1674—791no—
287#1673—791no—
288#1672—791no—
289#1672—791no—
290#1671—791no—
291#1671—791no—
292#1568—791no—
293#1554—791no—
294#1554—791no—
295C79#1927311yescharacteristic, complemented, Sylow
296#533—791no—
297#533C2×D7911yescharacteristic
298C4×#41C2×D7911yescharacteristic
299#500—791no—
300#500—791no—
301#499—1581no—
302#499—1581no—
303#498—791no—
304#498—791no—
305#485—21no—
306#485—1581no—
307#477—791no—
308#477C2×D7911yescharacteristic
309#476—791no—
310#476C2×D7911yescharacteristic
311#462—791no—
312#462C2×D7911yescharacteristic
313#461—791no—
314#461C2×D7911yescharacteristic
315#452—21no—
316#452—1581no—
317C8×Q8—791no—
318C8×Q8—791no—
319#441—21no—
320#441—1581no—
321#440—21no—
322#440—1581no—
323C8×D4—791no—
324C8×D4—791no—
325C8×D4—1581no—
326C8×D4—1581no—
327#432—791no—
328#432C2×D7911yescharacteristic
329#431—791no—
330#431—1581no—
331#427—1581no—
332#427—1581no—
333C2×C4×C8—791no—
334C2×C4×C8—1581no—
335#335—791no—
336#335—791no—
337#334—791no—
338#334—791no—
339#333—791no—
340#333—791no—
341#333—791no—
342#333—791no—
343#332—791no—
344#332—791no—
345#332—791no—
346#332—791no—
347C2×#41C22⋊D7911yescharacteristic
348C2×Q16—21no—
349C2×Q16—1581no—
350C2×SD16—21no—
351C2×SD16—1581no—
352C22×C8—791no—
353C22×C8—791no—
354C22×C8—1581no—
355C22×C8—1581no—
356C4⋊Q8—791no—
357C4⋊Q8—791no—
358C4⋊Q8C22×D7911yescharacteristic
359C4⋊Q8C22×D7911yescharacteristic
360#125—21no—
361#125—1581no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—1581no—
365C22⋊Q8—1581no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—1581no—
369C4×Q8—1581no—
370C4×Q8C22⋊D7911yescharacteristic
371C4×Q8C22×D7911yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C22⋊D7911yescharacteristic
375C4×D4C22×D7911yescharacteristic
376#117—21no—
377#117—21no—
378#117—1581no—
379#117C22⋊D7911yescharacteristic
380C2×C42—21no—
381C2×C42C22×D7911yescharacteristic
382#107—21no—
383#107—1581no—
384#107—1581no—
385#107—1581no—
386#107—1581no—
387#107—1581no—
388#106—21no—
389#106—1581no—
390#106—1581no—
391#106—1581no—
392#106—1581no—
393#106—1581no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8C22×D7911yescharacteristic
401C4⋊C8C22×D7911yescharacteristic
402#103—21no—
403#103—21no—
404#103—21no—
405#103—1581no—
406#103—1581no—
407#103—1581no—
408#102—21no—
409#102—1581no—
410C22⋊C8—21no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413C22⋊C8—1581no—
414C22⋊C8—1581no—
415#97—21no—
416#97—1581no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—791no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8—1581no—
424#41—22no—
425#41—22no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—1581no—
429C2×Q8—1581no—
430C2×Q8C2×C22⋊D7911yescharacteristic
431C2×D4—21no—
432C2×D4C2×C22⋊D7911yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4D4×D7911yescharacteristic
437C22×C4C2×C22⋊D7911yescharacteristic
438C22×C4#1192962311yescharacteristic
439Q16—42no—
440Q16—3162no—
441SD16—42no—
442SD16—3162no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—3162no—
446C2×C8—3162no—
447C2×C8—791no—
448C2×C8—791no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—1581no—
452C2×C8—1581no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8—1582no—
456C2×C8—1582no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—3161no—
466C4⋊C4—3161no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4C2×C22⋊D7911yescharacteristic
472C4⋊C4C2×C22⋊D7911yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—41no—
477C22⋊C4—3161no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—3161no—
483C42D4×D7911yescharacteristic
484C42C2×C22⋊D7911yescharacteristic
485C42#1192962311yescharacteristic
486C42C23×D7911yescharacteristic
487C23—21no—
488C23#1457913311yescharacteristic
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493Q8—3161no—
494Q8—3161no—
495Q8—6321no—
496Q8#1457918112yes—
497D4—24no—
498D4—21no—
499D4#1457918112yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—3161no—
515C2×C4—3161no—
516C2×C4#1457913311yescharacteristic
517C2×C4#1457913311yescharacteristic
518C2×C4#1457913311yescharacteristic
519C2×C4C2×D4×D7911yescharacteristic
520C2×C4C22×C22⋊D7911yescharacteristic
521C2×C4C2×#1192962311yescharacteristic
522C8—41no—
523C8—41no—
524C8—3161no—
525C8—3161no—
526C8—3161no—
527C8—3161no—
528C8—792no—
529C8—792no—
530C8—1581no—
531C8—1581no—
532C22—21no—
533C22—22no—
534C22—21no—
535C22—21no—
536C22—41no—
537C22—41no—
538C22C2×#1457913311yescharacteristic
539C4—21no—
540C4—22no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—41no—
545C4—41no—
546C4—81no—
547C4—81no—
548C4—6321no—
549C4—6321no—
550C4C2×#1457918111yescharacteristic
551C4C2×#1457918111yescharacteristic
552C4#137×D7911yescharacteristic
553C4#137×D7911yescharacteristic
554C2—41no—
555C2—42no—
556C2#2377288511yescharacteristic
557C2#2377297811yescharacteristic
558C2#2377311711yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#3116C2
socleC22×C79C2×#121
prime radicalC2×#41×C79D4
hypercenter#3116D79
hypocenterC79#19273
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
1. #23773910C2semidirect
4. #23762802C2semidirect
5. #23762797C2semidirect
6. #23762792C2semidirect
7. #23762787C2semidirect
8. #23762784C2semidirect
9. #23762780C2semidirect
10. #23762777C2semidirect
11. #23762773C2semidirect
12. #23762161C2semidirect
13. #23762111C2semidirect
14. #23762110C2semidirect
295. C79#19273semidirect

Maximal subgroups

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

subgroupindex
#237739102
#237735622
#41×C79⋊C82
#237628022
#237627972
#237627922
#237627872
#237627842
#237627802
#237627772
#237627732
#237621612
#237621112
#237621102
C79×#31162
#1927379