Finite Group Databasebeta

← #34305941

Subgroups — #34305941

Order 20224.

838 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#106×D79C211yesmaximal, characteristic
2#23766273C211yesmaximal, characteristic
3#23766265C211yesmaximal, characteristic
4#23764143C211yesmaximal, characteristic, complemented
5#23764141C211yesmaximal, characteristic, complemented
6#23763985C211yesmaximal, characteristic, complemented
7#23763983C211yesmaximal, characteristic, complemented
8#23763978C211yesmaximal, characteristic, complemented
9#23763976C211yesmaximal, characteristic, complemented
10#23763489C211yesmaximal, characteristic
11#23763485C211yesmaximal, characteristic
12#23763426C211yesmaximal, characteristic
13#23763300C211yesmaximal, characteristic, complemented
14#23763177C211yesmaximal, characteristic
15C79×#1776C211yesmaximal, characteristic, complemented
16C22×C8×D79C2211yescharacteristic
17C2×#14579118—21no—
18C2×C4⋊C4×D79C2211yescharacteristic
19C2×C4⋊C4×D79C2211yescharacteristic
20C2×#14579095C2211yescharacteristic
21C2×#14579093C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24#106×D79C418yescomplemented
25C2×(C2×D79)⋊C8C2211yescharacteristic
26C2×#14579071C2211yescharacteristic
27#19098461C2211yescharacteristic
28#19098461C2211yescharacteristic
29#19098459C2211yescharacteristic
30#19098459C2211yescharacteristic
31#19098447—21no—
32#19098447—21no—
33#19098445—21no—
34#19098445—21no—
35#19098425—21no—
36#19098422—21no—
37#19098420C2211yescharacteristic
38#19098420C2211yescharacteristic
39C4⋊C4×Dic79C2211yescharacteristic
40C4⋊C4×Dic79C2211yescharacteristic
41C4×(C2×D79)⋊C4—21no—
42C4×(C2×D79)⋊C4C2211yescharacteristic
43#19098326C2211yescharacteristic
44#19098326C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098318C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×#107×C79C2211yescharacteristic
54C2×#106×C79C2211yescharacteristic
55C2×#97×C79C2211yescharacteristic
56C4×C4⋊C4×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×C4×D79—21no—
61C22×C4×D79—21no—
62C22×C4×D79C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C2×C8×D79—22no—
66C2×C8×D79—22no—
67C2×C8×D79C2×C414yes—
68#14579118—44no—
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C4⋊C4×D79—22no—
80C4⋊C4×D79—22no—
81C4⋊C4×D79C2×C414yes—
82C4⋊C4×D79C2×C414yes—
83C2×(C2×D79)⋊C4—21no—
84C2×(C2×D79)⋊C4—21no—
85C2×(C2×D79)⋊C4D411yescharacteristic
86C2×(C2×D79)⋊C4D411yescharacteristic
87C2×C4×Dic79—21no—
88C2×C4×Dic79—21no—
89C2×C4×Dic79C2311yescharacteristic
90#14579095—22no—
91#14579093C2×C414yes—
92#14579092C2×C414yes—
93#14579092C2×C414yes—
94(C2×D79)⋊C8—22no—
95#14579071—22no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95D411yescharacteristic
105C79⋊#95D411yescharacteristic
106C22×C8×C79C2311yescharacteristic
107C22×C8×C79C2311yescharacteristic
108C2×C4⋊C4×C79C2311yescharacteristic
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C42×C79—21no—
113C2×C42×C79—21no—
114C2×C42×C79C2311yescharacteristic
115#107×C79—22no—
116#106×C79C2×C414yes—
117#97×C79—22no—
118#95×C79—21no—
119#95×C79—21no—
120C23×D79C2×D411yescharacteristic
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—21no—
124C22×Dic79—21no—
125C22×Dic79C2×D411yescharacteristic
126C22×Dic79C2×D411yescharacteristic
127C22×Dic79C2×D411yescharacteristic
128C2×C4×D79—24no—
129C2×C4×D79—24no—
130C2×C4×D79—21no—
131C2×C4×D79—21no—
132C2×C4×D79—21no—
133C2×C4×D79—21no—
134C2×C4×D79—42no—
135C2×C4×D79—42no—
136C2×C4×D79—81no—
137C2×C4×D79—81no—
138C2×C4×D79C22×C412yes—
139C2×C4×D79#4112yes—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8C22×C412yes—
146C8×D79—28no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79C22×C412yes—
154C4⋊Dic79C22×C412yes—
155Dic79⋊C4—24no—
156Dic79⋊C4—24no—
157Dic79⋊C4—42no—
158Dic79⋊C4—42no—
159Dic79⋊C4—81no—
160Dic79⋊C4—81no—
161(C2×D79)⋊C4—22no—
162(C2×D79)⋊C4—22no—
163(C2×D79)⋊C4—42no—
164(C2×D79)⋊C4—42no—
165C4×Dic79—21no—
166C4×Dic79—21no—
167C4×Dic79—42no—
168C4×Dic79—81no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2411yescharacteristic
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C2×C8×C79C22×C412yes—
180C2×C8×C79#4112yes—
181C2×C8×C79#4112yes—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79C22×C412yes—
189C4⋊C4×C79C22×C412yes—
190C42×C79—21no—
191C42×C79—21no—
192C42×C79—42no—
193C42×C79—81no—
194C23×C79C22×D411yescharacteristic
195C22×D79—21no—
196C22×D79—21no—
197C22×D79—21no—
198C22×D79—21no—
199C22×D79C4×D412yes—
200C22×D79AGL(1,ℤ/8)12yes—
201C22×D79#13712yes—
202C2×C4×C79—22no—
203C2×C4×C79—22no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—41no—
211C2×C4×C79—41no—
212C2×C4×C79—41no—
213C2×C4×C79—41no—
214C2×C4×C79—41no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79C23×C411yescharacteristic
221C2×C4×C79C2×#4111yescharacteristic
222C2×C4×C79C2×#4111yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C2×C4×C792−1+411yescharacteristic
225C2×C4×C792−1+411yescharacteristic
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—22no—
229C2×Dic79—22no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79C4×D412yes—
247C4×D79—22no—
248C4×D79—22no—
249C4×D79—22no—
250C4×D79—22no—
251C4×D79—48no—
252C4×D79—84no—
253C8×C79—41no—
254C8×C79#11714yescomplemented
255C79⋊C8—22no—
256C79⋊C8—41no—
257C22×C79C2×C4×D411yescharacteristic
258C22×C79#54711yescharacteristic
259C22×C79#54711yescharacteristic
260C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C79C2×#13711yescharacteristic
262C22×C79#57711yescharacteristic
263C22×C79#57711yescharacteristic
264C2×D79—22no—
265C2×D79—22no—
266C2×D79—22no—
267C2×D79—22no—
268C2×D79—22no—
269C2×D79—22no—
270C2×D79#44014yes—
271C4×C79—41no—
272C4×C79—41no—
273C4×C79—42no—
274C4×C79—42no—
275C4×C79—81no—
276C4×C79—81no—
277C4×C79C2×#11711yescharacteristic
278C4×C79#51911yescharacteristic
279C4×C79#55211yescharacteristic
280C4×C79#55211yescharacteristic
281Dic79—22no—
282Dic79—41no—
283Dic79—41no—
284Dic79—42no—
285Dic79—42no—
286Dic79—81no—
287Dic79—81no—
288#22615—791nomaximal, Sylow
289C2×C79#234411yescharacteristic
290C2×C79C2×#44011yescharacteristic
291C2×C79#294911yescharacteristic
292C2×C79#334411yescharacteristic
293C2×C79#334411yescharacteristic
294C2×C79#334511yescharacteristic
295C2×C79#334511yescharacteristic
296D79—24no—
297C22×#106—791no—
298#2345—791no—
299#2344—791no—
300#1944—791no—
301#1944—791no—
302#1920—791no—
303#1920—791no—
304#1919—791no—
305#1919—791no—
306#1824—791no—
307#1823—791no—
308#1808—791no—
309#1776—791no—
310#1776D7911yescharacteristic, complemented
311#1752—791no—
312C79#2261511yescharacteristic, complemented, Sylow
313C23×C8—791no—
314C2×#117—1581no—
315C22×C4⋊C4—791no—
316C22×C4⋊C4—791no—
317C2×#107—791no—
318C2×#107C2×D7911yescharacteristic
319C2×#106—791no—
320C2×#106—791no—
321C2×#106—791no—
322C2×#106—798no—
323C2×#106C2×D7911yescharacteristic
324C2×C22⋊C8—791no—
325C2×#97—791no—
326C2×#97C2×D7911yescharacteristic
327#389—791no—
328#389—791no—
329#389—791no—
330#389—791no—
331#387—1581no—
332#387—1581no—
333#387—1581no—
334#387—1581no—
335#381—1581no—
336#380—791no—
337#380—791no—
338#380—1581no—
339C4×C4⋊C4—791no—
340C4×C4⋊C4—791no—
341C4×C4⋊C4C2×D7911yescharacteristic
342C4×C4⋊C4C2×D7911yescharacteristic
343C4×C22⋊C4—791no—
344C4×C22⋊C4—1581no—
345#340—791no—
346#340—791no—
347#340—791no—
348#340—791no—
349#340—791no—
350#340—791no—
351#340—791no—
352#340—791no—
353#340C2×D7911yescharacteristic
354#340C2×D7911yescharacteristic
355#336—791no—
356#336—791no—
357C23×C4—791no—
358C23×C4—1581no—
359C23×C4—1581no—
360C22×C8—791no—
361C22×C8—794no—
362C22×C8—791no—
363C22×C8—1582no—
364C22×C8—1582no—
365C22×C8C22×D7911yescharacteristic
366C22×C8C22×D7911yescharacteristic
367#117—3164no—
368C2×C4⋊C4—791no—
369C2×C4⋊C4—791no—
370C2×C4⋊C4—794no—
371C2×C4⋊C4—794no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—791no—
374C2×C4⋊C4—1582no—
375C2×C4⋊C4—1582no—
376C2×C4⋊C4—1581no—
377C2×C4⋊C4—1581no—
378C2×C4⋊C4—1581no—
379C2×C4⋊C4—1581no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4C22×D7911yescharacteristic
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C4⋊C4C22×D7911yescharacteristic
385C2×C4⋊C4C22×D7911yescharacteristic
386C2×C22⋊C4—791no—
387C2×C22⋊C4—791no—
388C2×C22⋊C4—1581no—
389C2×C22⋊C4—1581no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—791no—
393C2×C42—1581no—
394C2×C42—1581no—
395C2×C42C22×D7911yescharacteristic
396#107—22no—
397#107—1582no—
398#106—794no—
399#106—794no—
400#106—794no—
401#106C4×D7914yes—
402C22⋊C8—1582no—
403#97—22no—
404#97—1582no—
405#95—21no—
406#95—21no—
407#95—791no—
408#95—791no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412#95—1581no—
413#95—1581no—
414#95—1581no—
415#95—1581no—
416#95—1581no—
417C24—791no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—3162no—
423C22×C4—3162no—
424C22×C4—791no—
425C22×C4—792no—
426C22×C4—792no—
427C22×C4—791no—
428C22×C4—791no—
429C22×C4—6321no—
430C22×C4—6321no—
431C22×C4—1584no—
432C22×C4—1584no—
433C22×C4—1581no—
434C22×C4—1581no—
435C22×C4—1581no—
436C22×C4—1581no—
437C22×C4—1581no—
438C22×C4—1581no—
439C22×C4—1581no—
440C22×C4—1581no—
441C22×C4D4×D7911yescharacteristic
442C22×C4D4×D7911yescharacteristic
443C22×C4C23×D7911yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—792no—
448C2×C8—1581no—
449C2×C8—1581no—
450C2×C8—1581no—
451C2×C8—1581no—
452C2×C8—1581no—
453C2×C8—1588no—
454C2×C8C2×C4×D7912yes—
455C2×C8#1192962412yes—
456C2×C8#1192962412yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—3162no—
464C4⋊C4—3162no—
465C4⋊C4—792no—
466C4⋊C4—792no—
467C4⋊C4—6321no—
468C4⋊C4—6321no—
469C4⋊C4—1584no—
470C4⋊C4—1584no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4C2×C4×D7912yes—
478C4⋊C4C2×C4×D7912yes—
479C22⋊C4—3162no—
480C22⋊C4—3162no—
481C22⋊C4—1582no—
482C22⋊C4—1582no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—81no—
487C42—3162no—
488C42—6321no—
489C42—1581no—
490C42—1581no—
491C23—792no—
492C23—792no—
493C23—792no—
494C23—1581no—
495C23—1581no—
496C23—1581no—
497C23—1581no—
498C23C2×D4×D7911yescharacteristic
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—3168no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—792no—
529C2×C4—6324no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1582no—
533C2×C4—1582no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4—1581no—
539C2×C4—1581no—
540C2×C4—1582no—
541C2×C4—1582no—
542C2×C4—1582no—
543C2×C4—1582no—
544C2×C4C22×C4×D7911yescharacteristic
545C2×C4#41×D7911yescharacteristic
546C2×C4C2×#1192962411yescharacteristic
547C2×C4C2×#1192962411yescharacteristic
548C2×C4#1457920811yescharacteristic
549C2×C4#1457920911yescharacteristic
550C8—41no—
551C8—3161no—
552C8—1582no—
553C8#1457911814yescomplemented
554C22—794no—
555C22—1582no—
556C22—1582no—
557C22—1582no—
558C22—1582no—
559C22—1582no—
560C22—1582no—
561C22C4×D4×D7911yescharacteristic
562C22#1909921411yescharacteristic
563C22#1909921811yescharacteristic
564C22AGL(1,ℤ/8)×D7911yescharacteristic
565C22#137×D7911yescharacteristic
566C22#1909937211yescharacteristic
567C22#1909937311yescharacteristic
568C4—42no—
569C4—42no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—3162no—
575C4—3162no—
576C4—3161no—
577C4—3161no—
578C4—6321no—
579C4—6321no—
580C4—1582no—
581C4C2×#1457911811yescharacteristic
582C4#1909903911yescharacteristic
583C4#1909925211yescharacteristic
584C4#1909925411yescharacteristic
585C2—1584no—
586C2#2376626511yescharacteristic
587C2#440×D7911yescharacteristic
588C2#2377225611yescharacteristic
589C2#2377585611yescharacteristic
590C2#2377586011yescharacteristic
591C2#2377587111yescharacteristic
592C2#2377587511yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1776C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
cosocleC22×C4×C79C24
hypercenter#1776D79
hypocenterC79#22615
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #23764143C2semidirect
5. #23764141C2semidirect
6. #23763985C2semidirect
7. #23763983C2semidirect
8. #23763978C2semidirect
9. #23763976C2semidirect
13. #23763300C2semidirect
15. C79×#1776C2semidirect
24. #106×D79C4semidirect
254. C8×C79#117semidirect
310. #1776D79semidirect
312. C79#22615semidirect
553. C8#14579118semidirect

Maximal subgroups

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

subgroupindex
C2×#106×D792
#237662732
#237662652
#237641432
#237641412
#237639852
#237639832
#237639782
#237639762
#237634892
#237634852
#237634262
#237633002
#237631772
C79×#17762
#2261579