Finite Group Databasebeta

← #34288817

Subgroups — #34288817

Order 20224.

886 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×Dic79C211yesmaximal, characteristic
2#23765802C211yesmaximal, characteristic
3#23765795C211yesmaximal, characteristic
4#23763567C211yesmaximal, characteristic, complemented
5#23763565C211yesmaximal, characteristic, complemented
6#23763532C211yesmaximal, characteristic, complemented
7#23763532C211yesmaximal, characteristic, complemented
8#23763433C211yesmaximal, characteristic
9#23763431C211yesmaximal, characteristic
10#23763431C211yesmaximal, characteristic
11#107×Dic79C211yesmaximal, characteristic, complemented
12#23763292C211yesmaximal, characteristic, complemented
13#106×Dic79C211yesmaximal, characteristic, complemented
14#23763288C211yesmaximal, characteristic, complemented
15C79×#1809C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×#14579117—21no—
18C22×C4×Dic79C2211yescharacteristic
19C2×#14579095C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579093C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25M16×Dic79C418yescomplemented
26C2×#14579071C2211yescharacteristic
27C2×#14579071C2211yescharacteristic
28C2×C8×Dic79C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30#19098432C2211yescharacteristic
31#19098432C2211yescharacteristic
32#19098430—21no—
33#19098428C2211yescharacteristic
34#19098428C2211yescharacteristic
35#19098427—21no—
36#19098424—21no—
37#19098423C2211yescharacteristic
38#19098423C2211yescharacteristic
39#19098421—21no—
40C4⋊C4×Dic79C2211yescharacteristic
41C4⋊C4×Dic79C2211yescharacteristic
42C4⋊C4×Dic79C2211yescharacteristic
43C4⋊C4×Dic79C2211yescharacteristic
44#19098414—21no—
45C22⋊C4×Dic79—21no—
46#19098408—21no—
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098325C2211yescharacteristic
53C22×M16×C79C2211yescharacteristic
54C2×#107×C79C2211yescharacteristic
55C2×#106×C79C2211yescharacteristic
56#380×C79C2211yescharacteristic
57#380×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C23×Dic79D411yescharacteristic
61C23×Dic79D411yescharacteristic
62C2×#11929608—22no—
63C2×#11929608—22no—
64C2×#11929608C2×C414yes—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579117—44no—
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×Dic79⋊C4—21no—
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×C22⋊Dic79—21no—
77C2×C22⋊Dic79—21no—
78C2×C4×Dic79—21no—
79C2×C4×Dic79—21no—
80C2×C4×Dic79—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79C2×C414yes—
83C2×C4×Dic79D411yescharacteristic
84C2×C4×Dic79D411yescharacteristic
85C2×C4×Dic79Q814yes—
86C2×C4×Dic79C2311yescharacteristic
87C2×C4×Dic79C2311yescharacteristic
88#14579095—22no—
89#14579094—22no—
90#14579094—22no—
91#14579093—22no—
92#14579092—22no—
93#14579092—22no—
94#14579071C2×C414yes—
95#14579071C2×C414yes—
96C8×Dic79C2×C414yes—
97C8×Dic79C2×C414yes—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C23×C4×C79C2311yescharacteristic
109C2×M16×C79—22no—
110C2×M16×C79—22no—
111C2×M16×C79C2×C414yes—
112C22×C8×C79C2311yescharacteristic
113C22×C8×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C22⋊C4×C79—21no—
119C2×C22⋊C4×C79—21no—
120#107×C79—22no—
121#106×C79—22no—
122#95×C79—21no—
123#95×C79—21no—
124C22×Dic79—22no—
125C22×Dic79—22no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—41no—
131C22×Dic79—41no—
132C22×Dic79C4⋊C414yes—
133C22×Dic79C4⋊C414yes—
134C22×Dic79C2×D411yescharacteristic
135C22×Dic79C2×D411yescharacteristic
136C22×Dic79C2×D411yescharacteristic
137C22×Dic79C2×D411yescharacteristic
138#11929608—28no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8C22×C412yes—
142C2×C79⋊C8C22×C412yes—
143C2×C79⋊C8#4112yes—
144C2×C79⋊C8#4112yes—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153Dic79⋊C4—42no—
154Dic79⋊C4—42no—
155Dic79⋊C4—42no—
156Dic79⋊C4—42no—
157C22⋊Dic79—42no—
158C22⋊Dic79—42no—
159C4×Dic79—42no—
160C4×Dic79—42no—
161C4×Dic79—42no—
162C4×Dic79—42no—
163C4×Dic79C4⋊C414yes—
164C4×Dic79C4⋊C414yes—
165C4×Dic79C22×C412yes—
166C4×Dic79C22×C412yes—
167C4×Dic79C2×Q812yes—
168C4×Dic79C2×Q812yes—
169C24×C79C2×D411yescharacteristic
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79C22×C412yes—
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2×Q812yes—
181C22×C4×C79C2411yescharacteristic
182M16×C79—28no—
183C2×C8×C79—21no—
184C2×C8×C79—21no—
185C2×C8×C79C22×C412yes—
186C2×C8×C79C22×C412yes—
187C2×C8×C79#4112yes—
188C2×C8×C79#4112yes—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C22⋊C4×C79—42no—
198C22⋊C4×C79—42no—
199C23×C79—21no—
200C23×C79—21no—
201C23×C79—21no—
202C23×C79—21no—
203C23×C79—21no—
204C23×C79—21no—
205C23×C79C2×C4⋊C412yes—
206C23×C79C22×D411yescharacteristic
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—22no—
210C2×C4×C79—22no—
211C2×C4×C79—22no—
212C2×C4×C79—22no—
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×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79C2×C4⋊C412yes—
226C2×C4×C79C23×C411yescharacteristic
227C2×C4×C79C22×Q811yescharacteristic
228C2×C4×C79C2×#4111yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—28no—
239C2×Dic79—44no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79C2×C4⋊C412yes—
253C2×Dic79C2×C4⋊C412yes—
254C2×Dic79C2×C4⋊C412yes—
255C2×Dic79C2×C4⋊C412yes—
256C2×Dic79AGL(1,ℤ/8)12yes—
257C2×Dic79#13712yes—
258C8×C79—22no—
259C8×C79#11714yescomplemented
260C79⋊C8—22no—
261C79⋊C8#11714yescomplemented
262C22×C79—22no—
263C22×C79—22no—
264C22×C79—22no—
265C22×C79—41no—
266C22×C79—41no—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79C22×C4⋊C411yescharacteristic
270C22×C79#53211yescharacteristic
271C22×C79#53211yescharacteristic
272C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C79C2×#13711yescharacteristic
274C22×C79#57511yescharacteristic
275C22×C79#57511yescharacteristic
276C4×C79—22no—
277C4×C79—81no—
278C4×C79—81no—
279C4×C79—81no—
280C4×C79—81no—
281C4×C79C2×#11711yescharacteristic
282C4×C79C2×#11711yescharacteristic
283C4×C79#53311yescharacteristic
284C4×C79#53311yescharacteristic
285Dic79—22no—
286Dic79—22no—
287Dic79—22no—
288Dic79—81no—
289Dic79—81no—
290Dic79—81no—
291Dic79—81no—
292Dic79#42814yes—
293#20819—791nomaximal, Sylow
294C2×C79—42no—
295C2×C79#228511yescharacteristic
296C2×C79C2×#42811yescharacteristic
297C2×C79#291111yescharacteristic
298C2×C79#317111yescharacteristic
299C2×C79#317111yescharacteristic
300C2×C79#317211yescharacteristic
301C2×C79#317211yescharacteristic
302C2×C4×M16—791no—
303#2285—791no—
304#2283—791no—
305#1847—791no—
306#1847—791no—
307#1836—791no—
308#1836—791no—
309#1809—791no—
310#1809—791no—
311#1809—791no—
312#1809D7911yescharacteristic
313C4×#107—791no—
314C4×#107—791no—
315C4×#106—791no—
316C4×#106—791no—
317C79#2081911yescharacteristic, complemented, Sylow
318C22×M16—791no—
319C22×M16C2×D7911yescharacteristic
320C2×#117—1581no—
321C22×C42—791no—
322C2×#107—791no—
323C2×#107—791no—
324C2×#107—791no—
325C2×#107C2×D7911yescharacteristic
326C2×#106—791no—
327C2×#106—791no—
328C2×#106—791no—
329C2×#106C2×D7911yescharacteristic
330C4×M16—798no—
331C2×#97—791no—
332C2×#97—791no—
333C2×C4×C8—791no—
334C2×C4×C8—791no—
335#384—791no—
336#384—791no—
337#383—1581no—
338#382—791no—
339#382—791no—
340#382—1581no—
341#381—1581no—
342#380—791no—
343#380—791no—
344#380—1581no—
345#380C2×D7911yescharacteristic
346#380C2×D7911yescharacteristic
347C4×C4⋊C4—791no—
348C4×C4⋊C4—791no—
349C4×C4⋊C4—791no—
350C4×C4⋊C4—791no—
351C4×C4⋊C4—1581no—
352C4×C22⋊C4—1581no—
353#376—1581no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340C2×D7911yescharacteristic
361#340C2×D7911yescharacteristic
362C23×C4—791no—
363C23×C4—791no—
364C23×C4C22×D7911yescharacteristic
365C2×M16—22no—
366C2×M16—22no—
367C2×M16—794no—
368C2×M16—1582no—
369C2×M16—1582no—
370C2×M16C4×D7914yes—
371C22×C8—791no—
372C22×C8—791no—
373C22×C8C22×D7911yescharacteristic
374C22×C8C22×D7911yescharacteristic
375#117—3164no—
376C2×C4⋊C4—791no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—791no—
379C2×C4⋊C4—791no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4—1581no—
383C2×C4⋊C4—1581no—
384C2×C4⋊C4C22×D7911yescharacteristic
385C2×C4⋊C4C22×D7911yescharacteristic
386C2×C4⋊C4C22×D7911yescharacteristic
387C2×C4⋊C4C22×D7911yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1581no—
391C2×C22⋊C4—1581no—
392C2×C42—794no—
393C2×C42—791no—
394C2×C42—791no—
395C2×C42—794no—
396C2×C42—791no—
397C2×C42—791no—
398C2×C42—1581no—
399C2×C42—1581no—
400C2×C42—1581no—
401C2×C42—1581no—
402#107—22no—
403#107—1582no—
404#107—1582no—
405#107—1582no—
406#106—22no—
407#106—1582no—
408#106—1582no—
409#106—1582no—
410#97—794no—
411#97—794no—
412C4×C8—794no—
413C4×C8—794no—
414#95—21no—
415#95—21no—
416#95—1581no—
417#95—1581no—
418#95—1581no—
419#95—1581no—
420#95—1581no—
421#95—1581no—
422#95—1581no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426C24D4×D7911yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—3161no—
436C22×C4—3161no—
437C22×C4—794no—
438C22×C4—791no—
439C22×C4—794no—
440C22×C4—791no—
441C22×C4—791no—
442C22×C4—791no—
443C22×C4—1581no—
444C22×C4—1581no—
445C22×C4—1581no—
446C22×C4—1581no—
447C22×C4—1582no—
448C22×C4—1582no—
449C22×C4C2×C4×D7912yes—
450C22×C4D4×D7911yescharacteristic
451C22×C4Q8×D7912yes—
452C22×C4C23×D7911yescharacteristic
453M16—28no—
454M16—1588no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—792no—
458C2×C8—792no—
459C2×C8—792no—
460C2×C8—792no—
461C2×C8—1581no—
462C2×C8—1581no—
463C2×C8C2×C4×D7912yes—
464C2×C8C2×C4×D7912yes—
465C2×C8#1192962312yes—
466C2×C8#1192962312yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—3162no—
476C4⋊C4—3162no—
477C4⋊C4—3162no—
478C4⋊C4—3162no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—3162no—
490C22⋊C4—3162no—
491C42—3162no—
492C42—3162no—
493C42—3162no—
494C42—3162no—
495C42—792no—
496C42—792no—
497C42—794no—
498C42—794no—
499C42—792no—
500C42—792no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23C4⋊C4×D7912yes—
508C23C2×D4×D7911yescharacteristic
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3164no—
540C2×C4—792no—
541C2×C4—792no—
542C2×C4—792no—
543C2×C4—792no—
544C2×C4—792no—
545C2×C4—792no—
546C2×C4—1588no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4C4⋊C4×D7912yes—
554C2×C4C22×C4×D7911yescharacteristic
555C2×C4C2×Q8×D7911yescharacteristic
556C2×C4#41×D7911yescharacteristic
557C2×C4#41×D7911yescharacteristic
558C2×C4C2×#1192962311yescharacteristic
559C2×C4C2×#1192962311yescharacteristic
560C8—22no—
561C8—794no—
562C8—1582no—
563C8#1457911714yescomplemented
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22C2×C4⋊C4×D7911yescharacteristic
572C22#1909910811yescharacteristic
573C22#1909910811yescharacteristic
574C22AGL(1,ℤ/8)×D7911yescharacteristic
575C22#137×D7911yescharacteristic
576C22#1909935911yescharacteristic
577C22#1909936211yescharacteristic
578C4—22no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—794no—
584C4—6321no—
585C4—6321no—
586C4—6321no—
587C4—6321no—
588C4—1582no—
589C4—1582no—
590C4—1582no—
591C4#117×D7911yescharacteristic
592C4C2×#1457911711yescharacteristic
593C4#1909911511yescharacteristic
594C4#1909911511yescharacteristic
595C2—42no—
596C2#2376580211yescharacteristic
597C2#428×D7911yescharacteristic
598C2#2377198711yescharacteristic
599C2#2377398811yescharacteristic
600C2#2377399011yescharacteristic
601C2#2377399911yescharacteristic
602C2#2377400111yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1809C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenter#1809D79
hypocenterC79#20819
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #23763567C2semidirect
5. #23763565C2semidirect
6. #23763532C2semidirect
7. #23763532C2semidirect
11. #107×Dic79C2semidirect
12. #23763292C2semidirect
13. #106×Dic79C2semidirect
14. #23763288C2semidirect
25. M16×Dic79C4semidirect
259. C8×C79#117semidirect
261. C79⋊C8#117semidirect
317. C79#20819semidirect
563. C8#14579117semidirect

Maximal subgroups

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

subgroupindex
C2×M16×Dic792
#237658022
#237657952
#237635672
#237635652
#237635322
#237634332
#237634312
#107×Dic792
#237632922
#106×Dic792
#237632882
C79×#18092
#2081979