Finite Group Databasebeta

← #34296039

Subgroups — #34296039

Order 20224.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098853C211yesmaximal, characteristic
2#23766148C211yesmaximal, characteristic, complemented
3#23764352C211yesmaximal, characteristic, complemented
4#23764168C211yesmaximal, characteristic, complemented
5#23763985C211yesmaximal, characteristic, complemented
6#23763978C211yesmaximal, characteristic, complemented
7#23763661C211yesmaximal, characteristic
8#23763647C211yesmaximal, characteristic
9#23763410C211yesmaximal, characteristic
10#23763298C211yesmaximal, characteristic, complemented
11C4×#14579092C211yesmaximal, characteristic, complemented
12#23763257C211yesmaximal, characteristic, complemented
13#23763225C211yesmaximal, characteristic
14C4×#14579082C211yesmaximal, characteristic
15C79×#2275C211yesmaximal, characteristic
16C2×#14579161C2211yescharacteristic
17#19098853C418yescomplemented
18C2×#14579094C2211yescharacteristic
19C2×#14579092C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×C4⋊(C79⋊C8)C2211yescharacteristic
23C2×C4⋊(C79⋊C8)C2211yescharacteristic
24C2×#14579082C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×#14579072C2211yescharacteristic
29C2×C4×C79⋊C8C2211yescharacteristic
30#19098467—21no—
31#19098467—21no—
32#19098461C2211yescharacteristic
33#19098461C2211yescharacteristic
34#19098446—21no—
35#19098446—21no—
36#19098429C2211yescharacteristic
37C4×C4⋊Dic79C2211yescharacteristic
38C4×C4⋊Dic79C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098318C2211yescharacteristic
48#19098318C2211yescharacteristic
49C2×C4×D4×C79C2211yescharacteristic
50C2×C4×D4×C79C2211yescharacteristic
51C2×C4×D4×C79C2211yescharacteristic
52C4×C4⋊C4×C79C2211yescharacteristic
53C4×C4⋊C4×C79C2211yescharacteristic
54C4×C4⋊C4×C79C2211yescharacteristic
55C4×C22⋊C4×C79—21no—
56C4×C22⋊C4×C79—21no—
57C4×C22⋊C4×C79—21no—
58#376×C79C2211yescharacteristic
59C2×#95×C79—21no—
60C22×Dic158—21no—
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65#14579161—22no—
66#14579161C2×C414yescomplemented
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×Dic79⋊C4—21no—
73C2×Dic79⋊C4—21no—
74C2×C4×Dic79—21no—
75C2×C4×Dic79—21no—
76#14579094—22no—
77#14579092—22no—
78#14579092C2×C414yescomplemented
79#14579092C2×C414yescomplemented
80C4⋊(C79⋊C8)—22no—
81C4⋊(C79⋊C8)C2×C414yescomplemented
82#14579082—22no—
83#14579082—22no—
84#14579082C2×C414yes—
85#14579082C2×C414yes—
86#14579072—22no—
87C4×C79⋊C8—22no—
88C79⋊#95—21no—
89C79⋊#95—21no—
90C79⋊#95—21no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C22×D4×C79C2311yescharacteristic
95C23×C4×C79—21no—
96C23×C4×C79—21no—
97C23×C4×C79—21no—
98C4×D4×C79—22no—
99C4×D4×C79—22no—
100C4×D4×C79—22no—
101C4×D4×C79—22no—
102C4×D4×C79C2×C414yes—
103C4×D4×C79Q814yescomplemented
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C4⋊C4×C79C2311yescharacteristic
106C2×C4⋊C4×C79C2311yescharacteristic
107C2×C22⋊C4×C79—21no—
108C2×C22⋊C4×C79—21no—
109C2×C22⋊C4×C79—21no—
110C2×C42×C79—21no—
111C2×C42×C79—21no—
112C2×C42×C79—21no—
113C2×C42×C79C2311yescharacteristic
114C2×C42×C79C2311yescharacteristic
115C2×C42×C79C2311yescharacteristic
116#95×C79—21no—
117#95×C79—21no—
118#95×C79—21no—
119#95×C79D411yescharacteristic
120#95×C79D411yescharacteristic
121#95×C79D411yescharacteristic
122#95×C79D411yescharacteristic
123C2×Dic158—24no—
124C2×Dic158—81no—
125C2×Dic158—81no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8C22×C412yes—
141C2×C79⋊C8C22×C412yes—
142C4⋊Dic79—24no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79C22×C412yes—
150C4⋊Dic79C22×C412yes—
151Dic79⋊C4—81no—
152Dic79⋊C4—81no—
153C4×Dic79—81no—
154C4×Dic79—81no—
155C24×C79—21no—
156C2×D4×C79—21no—
157C2×D4×C79—21no—
158C2×D4×C79—21no—
159C2×D4×C79C22×C412yes—
160C2×D4×C79C2×Q812yes—
161C2×D4×C79#4112yes—
162C22×C4×C79—21no—
163C22×C4×C79—21no—
164C22×C4×C79—21no—
165C22×C4×C79—24no—
166C22×C4×C79—24no—
167C22×C4×C79—24no—
168C22×C4×C79—81no—
169C22×C4×C79—81no—
170C22×C4×C79—81no—
171C22×C4×C79—81no—
172C22×C4×C79—81no—
173C22×C4×C79—81no—
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2411yescharacteristic
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79C22×C412yes—
185C4⋊C4×C79C2×Q812yes—
186C4⋊C4×C79#4112yes—
187C22⋊C4×C79—24no—
188C22⋊C4×C79—24no—
189C22⋊C4×C79—24no—
190C42×C79—21no—
191C42×C79—21no—
192C42×C79—81no—
193C42×C79—81no—
194C42×C79—81no—
195C42×C79C22×C412yes—
196C42×C79C2×Q812yes—
197C42×C79#4112yes—
198C42×C79#4112yes—
199C23×C79—22no—
200C23×C79—22no—
201C23×C79—22no—
202C23×C79—41no—
203C23×C79—41no—
204C23×C79—41no—
205C23×C79—41no—
206C23×C79C22×D411yescharacteristic
207D4×C79—22no—
208D4×C79—22no—
209D4×C79—22no—
210D4×C79C4×Q814yes—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—22no—
220C2×C4×C79—22no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—22no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79—84no—
235C2×C4×C79—84no—
236C2×C4×C79—84no—
237C2×C4×C79C4×D412yes—
238C2×C4×C79C4×D412yes—
239C2×C4×C79C22⋊Q812yes—
240C2×C4×C79C22⋊Q812yes—
241C2×C4×C79C23×C411yescharacteristic
242C2×C4×C79C22×Q811yescharacteristic
243C2×C4×C79C2×#4111yescharacteristic
244C2×C4×C79C2×#4111yescharacteristic
245C2×C4×C79C2×#4111yescharacteristic
246C2×C4×C79C2×#4111yescharacteristic
247C2×Dic79—22no—
248C2×Dic79—22no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259Dic158—84no—
260C79⋊C8—22no—
261C79⋊C8—22no—
262C79⋊C8—41no—
263C79⋊C8—41no—
264C22×C79—24no—
265C22×C79—42no—
266C22×C79—42no—
267C22×C79—42no—
268C22×C79—42no—
269C22×C79—42no—
270C22×C79—42no—
271C22×C79C2×C4×D411yescharacteristic
272C22×C79C2×C22⋊Q811yescharacteristic
273C22×C79#52511yescharacteristic
274C22×C79#57611yescharacteristic
275C22×C79#57711yescharacteristic
276C22×C79#57711yescharacteristic
277C22×C79#57711yescharacteristic
278C4×C79—22no—
279C4×C79—22no—
280C4×C79—22no—
281C4×C79—22no—
282C4×C79—42no—
283C4×C79—41no—
284C4×C79—42no—
285C4×C79—41no—
286C4×C79—42no—
287C4×C79C2×C4×Q811yescharacteristic
288C4×C79C4×#4111yescharacteristic
289C4×C79#52911yescharacteristic
290C4×C79#53311yescharacteristic
291Dic79—42no—
292Dic79—42no—
293Dic79—81no—
294Dic79—81no—
295#21623—791nomaximal, Sylow
296C2×C79—44no—
297C2×C79C4×C22⋊Q811yescharacteristic
298C2×C79#294611yescharacteristic
299C2×C79#294911yescharacteristic
300C2×C79#308111yescharacteristic
301C2×C79#308411yescharacteristic
302C2×C79#311811yescharacteristic
303C2×C79#312511yescharacteristic
304C2×#476—791no—
305#2331—791no—
306#2275D7911yescharacteristic
307#1988—791no—
308#1949—791no—
309#1920—791no—
310#1919—791no—
311#1864—791no—
312#1862—791no—
313#1804—791no—
314#1776—791no—
315C4×#106—791no—
316#1768—791no—
317#1762—791no—
318C4×#102—791no—
319C79#2162311yescharacteristic, complemented, Sylow
320C2×C4⋊Q8—791no—
321C2×C4×D4C2×D7911yescharacteristic
322C2×C4×D4C2×D7911yescharacteristic
323C2×C4×D4C2×D7911yescharacteristic
324#476—798no—
325C2×#107—791no—
326C2×#106—791no—
327C2×#106—791no—
328C2×#106—791no—
329C2×C4⋊C8—791no—
330C2×C4⋊C8—791no—
331C2×#102—791no—
332C2×#102—791no—
333C2×#102—791no—
334C2×#102—791no—
335C2×#97—791no—
336C2×C4×C8—791no—
337#391—1581no—
338#391—1581no—
339#389—791no—
340#389—791no—
341#387—1581no—
342#387—1581no—
343#382—791no—
344C4×C4⋊C4—791no—
345C4×C4⋊C4—791no—
346C4×C4⋊C4C2×D7911yescharacteristic
347C4×C4⋊C4C2×D7911yescharacteristic
348C4×C4⋊C4C2×D7911yescharacteristic
349C4×C22⋊C4—21no—
350C4×C22⋊C4—21no—
351C4×C22⋊C4—21no—
352#376C2×D7911yescharacteristic
353C2×#95—21no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#340—791no—
362#336—791no—
363#336—791no—
364C22×Q8—1581no—
365C22×D4C22×D7911yescharacteristic
366C23×C4—21no—
367C23×C4—21no—
368C23×C4—21no—
369C22×C8—791no—
370C22×C8—791no—
371C22×C8—791no—
372C22×C8—791no—
373C4⋊Q8—794no—
374C4⋊Q8—1582no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4Dic15814yescomplemented
380C4×D4C4×D7914yes—
381C2×C4⋊C4—791no—
382C2×C4⋊C4—791no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—1581no—
386C2×C4⋊C4—1581no—
387C2×C4⋊C4—1581no—
388C2×C4⋊C4C22×D7911yescharacteristic
389C2×C4⋊C4C22×D7911yescharacteristic
390C2×C4⋊C4C22×D7911yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—1581no—
398C2×C42—1581no—
399C2×C42C22×D7911yescharacteristic
400C2×C42C22×D7911yescharacteristic
401C2×C42C22×D7911yescharacteristic
402#107—1582no—
403#106—794no—
404#106—794no—
405#106—1582no—
406C4⋊C8—794no—
407C4⋊C8—1582no—
408#102—794no—
409#102—794no—
410#102—1582no—
411#102—1582no—
412#97—1582no—
413C4×C8—1582no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—1581no—
418#95—1581no—
419#95—1581no—
420#95—1581no—
421#95—1581no—
422#95—1581no—
423#95C22⋊D7911yescharacteristic
424#95C22⋊D7911yescharacteristic
425#95C22⋊D7911yescharacteristic
426#95C22⋊D7911yescharacteristic
427C24—21no—
428C2×Q8—6321no—
429C2×Q8—6321no—
430C2×Q8—1584no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×C4×D7912yes—
435C2×D4C2×Dic15812yes—
436C2×D4#1192962412yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4C2×C22⋊D7911yescharacteristic
454C22×C4C2×C22⋊D7911yescharacteristic
455C22×C4C2×C22⋊D7911yescharacteristic
456C22×C4C2×C22⋊D7911yescharacteristic
457C22×C4C2×C22⋊D7911yescharacteristic
458C22×C4C2×C22⋊D7911yescharacteristic
459C22×C4C23×D7911yescharacteristic
460C2×C8—792no—
461C2×C8—792no—
462C2×C8—1581no—
463C2×C8—1581no—
464C2×C8—1581no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8—1581no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—792no—
476C4⋊C4—792no—
477C4⋊C4—6321no—
478C4⋊C4—6321no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1584no—
486C4⋊C4C2×C4×D7912yes—
487C4⋊C4C2×Dic15812yes—
488C4⋊C4#1192962412yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C42—21no—
493C42—21no—
494C42—81no—
495C42—81no—
496C42—81no—
497C42—6321no—
498C42—6321no—
499C42C2×C4×D7912yes—
500C42C2×Dic15812yes—
501C42#1192962412yes—
502C42#1192962412yes—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C22×C22⋊D7911yescharacteristic
511Q8—6324no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C4×Dic15814yes—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—3161no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4—1582no—
553C2×C4—1582no—
554C2×C4C4×C22⋊D7912yes—
555C2×C4C4×C22⋊D7912yes—
556C2×C4#1457914012yes—
557C2×C4#1457914012yes—
558C2×C4C22×C4×D7911yescharacteristic
559C2×C4C22×Dic15811yescharacteristic
560C2×C4C2×#1192962411yescharacteristic
561C2×C4C2×#1192962411yescharacteristic
562C2×C4C2×#1192962411yescharacteristic
563C2×C4C2×#1192962411yescharacteristic
564C8—3161no—
565C8—3161no—
566C8—1582no—
567C8—1582no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22C2×C4×C22⋊D7911yescharacteristic
576C22C2×#1457914011yescharacteristic
577C22#1909907411yescharacteristic
578C22#1909936611yescharacteristic
579C22#1909937111yescharacteristic
580C22#1909937411yescharacteristic
581C22#1909937411yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—42no—
587C4—42no—
588C4—41no—
589C4—42no—
590C4—41no—
591C4—3162no—
592C4—3162no—
593C4—6321no—
594C4—6321no—
595C4C2×C4×Dic15811yescharacteristic
596C4C4×#1192962411yescharacteristic
597C4#1909909711yescharacteristic
598C4#1909912011yescharacteristic
599C2—44no—
600C2C4×#1457914011yescharacteristic
601C2#2377222811yescharacteristic
602C2#2377225711yescharacteristic
603C2#2377323111yescharacteristic
604C2#2377326211yescharacteristic
605C2#2377358411yescharacteristic
606C2#2377367711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#2275C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenter#2275D79
hypocenterC79#21623
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ #376 ◁ #2275

Complements

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

NH ≅ G/Ntype
2. #23766148C2semidirect
3. #23764352C2semidirect
4. #23764168C2semidirect
5. #23763985C2semidirect
6. #23763978C2semidirect
10. #23763298C2semidirect
11. C4×#14579092C2semidirect
12. #23763257C2semidirect
17. #19098853C4semidirect
66. #14579161C2×C4semidirect
78. #14579092C2×C4semidirect
79. #14579092C2×C4semidirect
81. C4⋊(C79⋊C8)C2×C4semidirect
103. C4×D4×C79Q8semidirect
319. C79#21623semidirect
379. C4×D4Dic158semidirect

Maximal subgroups

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

subgroupindex
C2×#190988532
#237661482
#237643522
#237641682
#237639852
#237639782
#237636612
#237636472
#237634102
#237632982
C4×#145790922
#237632572
#237632252
C4×#145790822
C79×#22752
#2162379