Finite Group Databasebeta

← #34273777

Subgroups — #34273777

Order 20224.

594 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#23774555C211yesmaximal, characteristic, complemented
2#23773562C211yesmaximal, characteristic
3#23772511C211yesmaximal, characteristic
4#23762856C211yesmaximal, characteristic, complemented
5#23762849C211yesmaximal, characteristic, complemented
6#23762828C211yesmaximal, characteristic, complemented
7#23762821C211yesmaximal, characteristic, complemented
8#23762792C211yesmaximal, characteristic, complemented
9#23762787C211yesmaximal, characteristic, complemented
10#23762780C211yesmaximal, characteristic, complemented
11#23762773C211yesmaximal, characteristic, complemented
12#23762178C211yesmaximal, characteristic, complemented
13#23762100C211yesmaximal, characteristic, complemented
14#23762055C211yesmaximal, characteristic, complemented
15C79×#3107C211yesmaximal, characteristic
16#19099120C2211yescharacteristic
17#19098966C2211yescharacteristic
18#19098959C2211yescharacteristic
19#19098913C2211yescharacteristic
20#19098913C2211yescharacteristic
21#19098904—21no—
22#19098901—21no—
23#19098901—21no—
24#19098870—21no—
25#19098870—21no—
26#19098858C2211yescharacteristic
27#19098853C2211yescharacteristic
28#19098773C2211yescharacteristic
29#19098768C2211yescharacteristic
30#19098725—21no—
31Q8×C79⋊C8C2211yescharacteristic
32Q8×C79⋊C8C2211yescharacteristic
33#19098668—21no—
34#19098663—21no—
35D4×C79⋊C8—21no—
36D4×C79⋊C8—21no—
37D4×C79⋊C8C2211yescharacteristic
38D4×C79⋊C8C2211yescharacteristic
39#19098620C2211yescharacteristic
40#19098616—21no—
41#19098616C2211yescharacteristic
42C2×C4⋊(C79⋊C8)—21no—
43C2×C4⋊(C79⋊C8)C2211yescharacteristic
44#19098315C2211yescharacteristic
45#19098315C2211yescharacteristic
46#19098313C2211yescharacteristic
47#19098313C2211yescharacteristic
48#19098311C2211yescharacteristic
49#19098311C2211yescharacteristic
50#19098310C2211yescharacteristic
51#19098310C2211yescharacteristic
52#19098309C2211yescharacteristic
53#19098309C2211yescharacteristic
54#19098309C2211yescharacteristic
55#19098309C2211yescharacteristic
56#533×C79C2211yescharacteristic
57C4×#41×C79C2211yescharacteristic
58#485×C79—21no—
59#475×C79C2211yescharacteristic
60#474×C79C2211yescharacteristic
61#462×C79C2211yescharacteristic
62#461×C79C2211yescharacteristic
63#452×C79—21no—
64C4×Q16×C79—21no—
65C4×SD16×C79—21no—
66#431×C79C2211yescharacteristic
67C2×#11929614—21no—
68C2×#11929612—21no—
69C22×C79⋊C8—21no—
70C22×C79⋊C8—21no—
71C22×C79⋊C8—21no—
72#14579161C2311yescharacteristic
73#14579161C2311yescharacteristic
74#14579153—21no—
75#14579140—21no—
76#14579140—21no—
77C4×Dic158—21no—
78C4×Dic158—21no—
79#14579120—21no—
80#14579094—21no—
81#14579094—21no—
82#14579094—21no—
83#14579092—21no—
84#14579092—21no—
85#14579092—21no—
86C4⋊(C79⋊C8)—21no—
87C4⋊(C79⋊C8)—21no—
88C4⋊(C79⋊C8)—21no—
89C4⋊(C79⋊C8)D411yescharacteristic
90C4⋊(C79⋊C8)D411yescharacteristic
91C4⋊(C79⋊C8)C2311yescharacteristic
92C4⋊(C79⋊C8)C2311yescharacteristic
93C4⋊(C79⋊C8)C2311yescharacteristic
94C4⋊(C79⋊C8)C2311yescharacteristic
95#14579085—21no—
96#14579085—21no—
97#14579085—21no—
98#14579085—21no—
99#14579085—21no—
100#14579084—21no—
101#14579084—21no—
102#14579082—21no—
103C22⋊(C79⋊C8)—21no—
104C22⋊(C79⋊C8)—21no—
105C22⋊(C79⋊C8)—21no—
106C22⋊(C79⋊C8)D411yescharacteristic
107C22⋊(C79⋊C8)D411yescharacteristic
108#14579072—21no—
109C4×C79⋊C8—21no—
110C4×C79⋊C8—21no—
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—
143C4×C8×C79—21no—
144C2×Dic158—21no—
145C2×Dic158—21no—
146#11929614—42no—
147#11929612—42no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—81no—
153C2×C79⋊C8—81no—
154C2×C79⋊C8—81no—
155C2×C79⋊C8C2×D411yescharacteristic
156C2×C79⋊C8C2×D411yescharacteristic
157C2×C79⋊C8C2×D411yescharacteristic
158C2×C79⋊C8C2×D411yescharacteristic
159C4⋊Dic79—21no—
160C4⋊Dic79—21no—
161C4⋊Dic79—21no—
162C4⋊Dic79—21no—
163Dic79⋊C4—41no—
164Dic79⋊C4—41no—
165C22⋊Dic79—41no—
166C4×Dic79—41no—
167#41×C79—22no—
168#41×C79—22no—
169C2×Q8×C79—21no—
170C2×Q8×C79—21no—
171C2×Q8×C79C2×D411yescharacteristic
172C2×D4×C79—21no—
173C2×D4×C79C2×D411yescharacteristic
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79#4111yescharacteristic
180Q16×C79—42no—
181SD16×C79—42no—
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C4⋊C4×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—41no—
191C4⋊C4×C79—41no—
192C4⋊C4×C79C2×D411yescharacteristic
193C4⋊C4×C79C2×D411yescharacteristic
194C22⋊C4×C79—21no—
195C22⋊C4×C79—21no—
196C22⋊C4×C79—21no—
197C22⋊C4×C79—41no—
198C42×C79—21no—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—41no—
202C42×C79C2×D411yescharacteristic
203C42×C79C2×D411yescharacteristic
204C42×C79#4111yescharacteristic
205C42×C79C2411yescharacteristic
206C23×C79—21no—
207C23×C79#12111yescharacteristic
208Q8×C79—21no—
209Q8×C79—41no—
210Q8×C79—41no—
211Q8×C79—41no—
212Q8×C79—41no—
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—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—81no—
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—21no—
247C79⋊C8—21no—
248C79⋊C8—41no—
249C79⋊C8—41no—
250C79⋊C8—41no—
251C79⋊C8—41no—
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×C79#57711yescharacteristic
271C4×C79#57811yescharacteristic
272Dic79—81no—
273Dic79—81no—
274#19418—791nomaximal, Sylow
275C2×C79—41no—
276C2×C79—42no—
277C2×C79#304611yescharacteristic
278C2×C79#305011yescharacteristic
279C2×C79#306911yescharacteristic
280#3231—791no—
281#3116—791no—
282#3107D7911yescharacteristic
283#2988—791no—
284#1684—791no—
285#1683—791no—
286#1680—791no—
287#1679—791no—
288#1674—791no—
289#1673—791no—
290#1672—791no—
291#1671—791no—
292#1572—791no—
293#1552—791no—
294#1542—791no—
295C79#1941811yescharacteristic, complemented, Sylow
296#533—791no—
297#533C2×D7911yescharacteristic
298C4×#41C2×D7911yescharacteristic
299#500—791no—
300#498—791no—
301#487—791no—
302#487—791no—
303#485—21no—
304#485—1581no—
305#485—1581no—
306#485—1581no—
307#479—1581no—
308#479—1581no—
309#477—791no—
310#476—791no—
311#475C2×D7911yescharacteristic
312#474C2×D7911yescharacteristic
313#462—791no—
314#462C2×D7911yescharacteristic
315#461—791no—
316#461C2×D7911yescharacteristic
317#452—21no—
318#452—1581no—
319C8×Q8—791no—
320C8×Q8—791no—
321#441—1581no—
322#440—1581no—
323C4×Q16—21no—
324C4×SD16—21no—
325C8×D4—791no—
326C8×D4—791no—
327C8×D4—1581no—
328C8×D4—1581no—
329#432—791no—
330#431—791no—
331#431—1581no—
332#431C2×D7911yescharacteristic
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#332—791no—
342#332—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—1581no—
353C22×C8—1581no—
354C22×C8—1581no—
355C4⋊Q8—791no—
356C4⋊Q8—791no—
357C4⋊Q8C22×D7911yescharacteristic
358C4⋊Q8C22×D7911yescharacteristic
359#125—21no—
360#125—1581no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—1581no—
364C22⋊Q8—1581no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—1581no—
368C4×Q8—1581no—
369C4×Q8C22⋊D7911yescharacteristic
370C4×Q8C22×D7911yescharacteristic
371C4×D4—21no—
372C4×D4—21no—
373C4×D4C22⋊D7911yescharacteristic
374C4×D4C22×D7911yescharacteristic
375#117—21no—
376#117—21no—
377#117—1581no—
378#117C22⋊D7911yescharacteristic
379C2×C42—21no—
380C2×C42C22×D7911yescharacteristic
381#107—21no—
382#107—1581no—
383#107—1581no—
384#107—1581no—
385#106—21no—
386#106—1581no—
387#106—1581no—
388#106—1581no—
389C4⋊C8—791no—
390C4⋊C8—791no—
391C4⋊C8—791no—
392C4⋊C8—791no—
393C4⋊C8—791no—
394C4⋊C8—791no—
395C4⋊C8—1581no—
396C4⋊C8—1581no—
397C4⋊C8—1581no—
398C4⋊C8C22×D7911yescharacteristic
399C4⋊C8C22×D7911yescharacteristic
400#103—21no—
401#103—21no—
402#103—21no—
403#103—1581no—
404#103—1581no—
405#103—1581no—
406#103—1581no—
407#103—1581no—
408#103—1581no—
409#103—1581no—
410#102—21no—
411#102—1581no—
412C22⋊C8—21no—
413C22⋊C8—791no—
414C22⋊C8—791no—
415C22⋊C8—1581no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418#97—1581no—
419C4×C8—21no—
420C4×C8—791no—
421C4×C8—791no—
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—791no—
446C2×C8—791no—
447C2×C8—791no—
448C2×C8—791no—
449C2×C8—6321no—
450C2×C8—6321no—
451C2×C8—6321no—
452C2×C8—1581no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8—1581no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—3161no—
465C4⋊C4—3161no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4C2×C22⋊D7911yescharacteristic
471C4⋊C4C2×C22⋊D7911yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—41no—
476C22⋊C4—3161no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—41no—
481C42—3161no—
482C42D4×D7911yescharacteristic
483C42C2×C22⋊D7911yescharacteristic
484C42#1192962311yescharacteristic
485C42C23×D7911yescharacteristic
486C23—21no—
487C23#1457913311yescharacteristic
488Q8—21no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
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—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—81no—
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—1581no—
529C8—1581no—
530C8—1581no—
531C8—1581no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—21no—
536C22—41no—
537C22—41no—
538C22C2×#1457913311yescharacteristic
539C4—21no—
540C4—21no—
541C4—22no—
542C4—21no—
543C4—21no—
544C4—41no—
545C4—41no—
546C4—81no—
547C4—81no—
548C4—6321no—
549C4—6321no—
550C4C2×#1457918111yescharacteristic
551C4C2×#1457918111yescharacteristic
552C4#1909937011yescharacteristic
553C4#1909937711yescharacteristic
554C2—41no—
555C2—42no—
556C2#2377294811yescharacteristic
557C2#2377297811yescharacteristic
558C2#2377313411yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#3107C2
socleC22×C79C2×#121
prime radicalC2×#41×C79D4
hypercenter#3107D79
hypocenterC79#19418
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
1. #23774555C2semidirect
4. #23762856C2semidirect
5. #23762849C2semidirect
6. #23762828C2semidirect
7. #23762821C2semidirect
8. #23762792C2semidirect
9. #23762787C2semidirect
10. #23762780C2semidirect
11. #23762773C2semidirect
12. #23762178C2semidirect
13. #23762100C2semidirect
14. #23762055C2semidirect
295. C79#19418semidirect

Maximal subgroups

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

subgroupindex
#237745552
#237735622
#237725112
#237628562
#237628492
#237628282
#237628212
#237627922
#237627872
#237627802
#237627732
#237621782
#237621002
#237620552
C79×#31072
#1941879