Finite Group Databasebeta

← #34296271

Subgroups — #34296271

Order 20224.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098919C211yesmaximal, characteristic
2#23766137C211yesmaximal, characteristic, complemented
3#23766107C211yesmaximal, characteristic
4#23764367C211yesmaximal, characteristic, complemented
5#23764213C211yesmaximal, characteristic
6#23764098C211yesmaximal, characteristic
7#23764087C211yesmaximal, characteristic, complemented
8#23764081C211yesmaximal, characteristic
9#23764078C211yesmaximal, characteristic, complemented
10#23763526C211yesmaximal, characteristic, complemented
11#23763230C211yesmaximal, characteristic, complemented
12#23763228C211yesmaximal, characteristic
13C4×#14579083C211yesmaximal, characteristic
14#23763086C211yesmaximal, characteristic, complemented
15C4×#103×C79C211yesmaximal, characteristic, complemented
16C2×#14579152C2211yescharacteristic
17C2×#14579150C2211yescharacteristic
18C2×C4×D316C2211yescharacteristic
19#19098919C418yescomplemented
20C2×#14579085C2211yescharacteristic
21C2×#14579085C2211yescharacteristic
22C2×#14579083C2211yescharacteristic
23C2×#14579083C2211yescharacteristic
24C2×#14579081C2211yescharacteristic
25C2×#14579081C2211yescharacteristic
26C2×#14579072C2211yescharacteristic
27C2×#14579071C2211yescharacteristic
28C2×#14579071C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30C2×C8×Dic79C2211yescharacteristic
31#19098470C2211yescharacteristic
32#19098464C2211yescharacteristic
33#19098462C2211yescharacteristic
34#19098461—21no—
35#19098460—21no—
36#19098459C2211yescharacteristic
37#19098454—21no—
38#19098454—21no—
39#19098447—21no—
40#19098446—21no—
41#19098434—21no—
42#19098434—21no—
43#19098432C2211yescharacteristic
44#19098432C2211yescharacteristic
45C4×C4⋊Dic79C2211yescharacteristic
46#19098409C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098325C2211yescharacteristic
53C2×C4×Q8×C79C2211yescharacteristic
54C2×#103×C79C2211yescharacteristic
55C2×#103×C79C2211yescharacteristic
56C2×C4×C8×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×D316C2311yescharacteristic
61C22×C4×D79—21no—
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64#14579152—22no—
65#14579152C2×C414yescomplemented
66#14579150—22no—
67#14579150C2×C414yes—
68C4×D316—22no—
69C4×D316—22no—
70C2×C4⋊Dic79—21no—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×(C2×D79)⋊C4—21no—
84C2×(C2×D79)⋊C4—21no—
85C2×(C2×D79)⋊C4—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79C2311yescharacteristic
88C2×C4×Dic79C2311yescharacteristic
89#14579085—22no—
90#14579085C2×C414yescomplemented
91#14579083—22no—
92#14579083C2×C414yes—
93#14579081—22no—
94#14579081C2×C414yes—
95#14579072—22no—
96#14579071—22no—
97#14579071—22no—
98C8×Dic79—22no—
99C8×Dic79C2×C414yescomplemented
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95D411yescharacteristic
106C79⋊#95D411yescharacteristic
107C79⋊#95D411yescharacteristic
108C79⋊#95D411yescharacteristic
109C22×Q8×C79C2311yescharacteristic
110C22×C8×C79C2311yescharacteristic
111C22×C8×C79C2311yescharacteristic
112C4×Q8×C79—22no—
113C4×Q8×C79—22no—
114C2×C4⋊C4×C79—21no—
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C42×C79—21no—
119C2×C42×C79—21no—
120C2×C42×C79C2311yescharacteristic
121#103×C79—22no—
122#103×C79C2×C414yescomplemented
123C4×C8×C79—22no—
124#95×C79—21no—
125C23×D79—21no—
126C2×D316—21no—
127C2×D316—21no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316—21no—
131C2×D316C22×C412yes—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79C2×D411yescharacteristic
136C22×Dic79C2×D411yescharacteristic
137C22×Dic79C2×D411yescharacteristic
138C22×Dic79C2×D411yescharacteristic
139C2×C4×D79—42no—
140C2×C4×D79—81no—
141C2×C4×D79—81no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8C22×C412yes—
147C2×C79⋊C8#4112yes—
148C4⋊Dic79—24no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—81no—
155C4⋊Dic79C22×C412yes—
156Dic79⋊C4—24no—
157Dic79⋊C4—81no—
158Dic79⋊C4—81no—
159Dic79⋊C4—81no—
160Dic79⋊C4—81no—
161Dic79⋊C4—81no—
162(C2×D79)⋊C4—24no—
163(C2×D79)⋊C4—24no—
164(C2×D79)⋊C4—42no—
165(C2×D79)⋊C4—81no—
166(C2×D79)⋊C4—81no—
167C4×Dic79—21no—
168C4×Dic79—21no—
169C4×Dic79—81no—
170C4×Dic79C22×C412yes—
171C4×Dic79#4112yes—
172C2×Q8×C79—21no—
173C2×Q8×C79—21no—
174C2×Q8×C79—21no—
175C2×Q8×C79—21no—
176C2×Q8×C79—21no—
177C2×Q8×C79C22×C412yes—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2411yescharacteristic
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79C22×C412yes—
190C2×C8×C79#4112yes—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—42no—
197C4⋊C4×C79C22×C412yes—
198C42×C79—42no—
199C42×C79—81no—
200C42×C79#4112yes—
201C42×C79#4112yes—
202C23×C79C22×D411yescharacteristic
203C22×D79—22no—
204C22×D79—41no—
205C22×D79—41no—
206C22×D79—41no—
207C22×D79—41no—
208C22×D79—41no—
209C22×D79—41no—
210Q8×C79—22no—
211Q8×C79—22no—
212Q8×C79—41no—
213Q8×C79—41no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—22no—
219C2×C4×C79—22no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79#12412yes—
231C2×C4×C79#12412yes—
232C2×C4×C79C23×C411yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—22no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—21no—
248C2×Dic79—21no—
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×Dic79C4×D412yes—
258C2×Dic79C4×D412yes—
259D316—22no—
260D316—22no—
261D316—41no—
262D316—41no—
263C4×D79—84no—
264C8×C79—22no—
265C8×C79—22no—
266C79⋊C8—22no—
267C79⋊C8—22no—
268C22×C79C2×C4×D411yescharacteristic
269C22×C79C2×#12411yescharacteristic
270C22×C79#53211yescharacteristic
271C22×C79#57611yescharacteristic
272C22×C79#57711yescharacteristic
273C22×C79#57711yescharacteristic
274C22×C79#57811yescharacteristic
275C2×D79—42no—
276C2×D79—42no—
277C2×D79—42no—
278C2×D79—81no—
279C2×D79—81no—
280C2×D79—81no—
281C2×D79—81no—
282C4×C79—22no—
283C4×C79—22no—
284C4×C79—42no—
285C4×C79—42no—
286C4×C79—81no—
287C4×C79—81no—
288C4×C79C4×#4111yescharacteristic
289C4×C79C4×#4111yescharacteristic
290C4×C79#52911yescharacteristic
291C4×C79#52911yescharacteristic
292Dic79—22no—
293Dic79—22no—
294Dic79—41no—
295Dic79—41no—
296Dic79—42no—
297Dic79—81no—
298Dic79—81no—
299#21643—791nomaximal, Sylow
300C2×C79C4×#12411yescharacteristic
301C2×C79#295111yescharacteristic
302C2×C79#295211yescharacteristic
303C2×C79#313611yescharacteristic
304C2×C79#313711yescharacteristic
305C2×C79#317811yescharacteristic
306C2×C79#317911yescharacteristic
307D79—82no—
308C2×#488—791no—
309#2330—791no—
310#2326—791no—
311#1992—791no—
312#1959—791no—
313#1936—791no—
314#1935—791no—
315#1934—791no—
316#1933—791no—
317#1834—791no—
318#1763—791no—
319#1762—791no—
320C4×#103D7911yescharacteristic, complemented
321C4×#102—791no—
322#1726—791no—
323C79#2164311yescharacteristic, complemented, Sylow
324C2×#125—791no—
325C2×#124—791no—
326C2×C4×Q8C2×D7911yescharacteristic
327C2×C4×D4—791no—
328#488—798no—
329C2×#103—791no—
330C2×#103—791no—
331C2×#103C2×D7911yescharacteristic
332C2×#103C2×D7911yescharacteristic
333C2×#102—791no—
334C2×#102—791no—
335C2×#102—791no—
336C2×#102—791no—
337C2×#97—791no—
338C2×#97—791no—
339C2×#97—791no—
340C2×C4×C8—791no—
341C2×C4×C8—791no—
342C2×C4×C8C2×D7911yescharacteristic
343#391—791no—
344#390—791no—
345#389—791no—
346#389—791no—
347#389—1581no—
348#389—1581no—
349#388—1581no—
350#388—1581no—
351#387—1581no—
352#387—1581no—
353#385—1581no—
354#385—1581no—
355#384—791no—
356#384—791no—
357C4×C4⋊C4—791no—
358C4×C4⋊C4C2×D7911yescharacteristic
359#376—791no—
360#340—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340C2×D7911yescharacteristic
367#340C2×D7911yescharacteristic
368C22×Q8C22×D7911yescharacteristic
369C22×D4—791no—
370C23×C4—1581no—
371C22×C8—791no—
372C22×C8—791no—
373C22×C8C22×D7911yescharacteristic
374C22×C8C22×D7911yescharacteristic
375#125—794no—
376#125—1582no—
377#124—794no—
378#124—1582no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×D4—1582no—
382C4×D4—1582no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—791no—
387C2×C4⋊C4—1581no—
388C2×C4⋊C4—1581no—
389C2×C4⋊C4—1581no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4—1581no—
395C2×C4⋊C4C22×D7911yescharacteristic
396C2×C4⋊C4C22×D7911yescharacteristic
397C2×C4⋊C4C22×D7911yescharacteristic
398C2×C22⋊C4—1581no—
399C2×C22⋊C4—1581no—
400C2×C22⋊C4—1581no—
401C2×C22⋊C4—1581no—
402C2×C22⋊C4—1581no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—791no—
406C2×C42—791no—
407C2×C42—1581no—
408C2×C42C22×D7911yescharacteristic
409#103—22no—
410#103—794no—
411#103—1582no—
412#103C4×D7914yescomplemented
413#102—794no—
414#102—794no—
415#102—1582no—
416#102—1582no—
417#97—1582no—
418#97—1582no—
419#97—1582no—
420C4×C8—22no—
421C4×C8—794no—
422C4×C8—1582no—
423#95—21no—
424#95—791no—
425#95—791no—
426#95—791no—
427#95—791no—
428#95—1581no—
429#95—1581no—
430#95—1581no—
431#95—1581no—
432#95—1581no—
433C24—1581no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C2×C4×D7912yes—
440C2×D4—792no—
441C2×D4—1581no—
442C2×D4—1581no—
443C2×D4—1581no—
444C2×D4—1581no—
445C2×D4—1581no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—3162no—
451C22×C4—791no—
452C22×C4—791no—
453C22×C4—791no—
454C22×C4—791no—
455C22×C4—6321no—
456C22×C4—6321no—
457C22×C4—1581no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4D4×D7911yescharacteristic
461C22×C4D4×D7911yescharacteristic
462C22×C4C23×D7911yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—792no—
468C2×C8—792no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8—1581no—
472C2×C8—1581no—
473C2×C8C2×C4×D7912yes—
474C2×C8#1192962412yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—792no—
482C4⋊C4—6321no—
483C4⋊C4—6321no—
484C4⋊C4—6321no—
485C4⋊C4—6321no—
486C4⋊C4—6321no—
487C4⋊C4—6321no—
488C4⋊C4—1584no—
489C4⋊C4—1584no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4C2×C4×D7912yes—
496C22⋊C4—3162no—
497C22⋊C4—6321no—
498C22⋊C4—6321no—
499C22⋊C4—1584no—
500C22⋊C4—1584no—
501C42—42no—
502C42—81no—
503C42—792no—
504C42—792no—
505C42—6321no—
506C42—1581no—
507C42—1581no—
508C42#1192962312yes—
509C42#1192962412yes—
510C23—3161no—
511C23—3161no—
512C23—3161no—
513C23—3161no—
514C23—3161no—
515C23—3161no—
516C23—1582no—
517C23C2×D4×D7911yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—3161no—
523D4—3161no—
524D4—1582no—
525D4—1582no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
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—792no—
551C2×C4—792no—
552C2×C4—6324no—
553C2×C4—1582no—
554C2×C4—1581no—
555C2×C4—1581no—
556C2×C4—1581no—
557C2×C4—1581no—
558C2×C4—1581no—
559C2×C4—1581no—
560C2×C4—1581no—
561C2×C4—1581no—
562C2×C4—1581no—
563C2×C4—1581no—
564C2×C4#1457914712yes—
565C2×C4#1457914712yes—
566C2×C4C22×C4×D7911yescharacteristic
567C2×C4#41×D7911yescharacteristic
568C2×C4#41×D7911yescharacteristic
569C2×C4C2×#1192962311yescharacteristic
570C2×C4C2×#1192962411yescharacteristic
571C2×C4C2×#1192962411yescharacteristic
572C8—22no—
573C8—22no—
574C8—1582no—
575C8—1582no—
576C22—3162no—
577C22—3162no—
578C22—3162no—
579C22—6321no—
580C22—6321no—
581C22—6321no—
582C22—6321no—
583C22C4×D4×D7911yescharacteristic
584C22C2×#1457914711yescharacteristic
585C22#1909911011yescharacteristic
586C22#1909936511yescharacteristic
587C22#1909936911yescharacteristic
588C22#1909937011yescharacteristic
589C22#1909937611yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—81no—
595C4—81no—
596C4—3162no—
597C4—3161no—
598C4—3161no—
599C4—6321no—
600C4—6321no—
601C4—1582no—
602C4—1582no—
603C4C4×#1192962311yescharacteristic
604C4C4×#1192962411yescharacteristic
605C4#1909909111yescharacteristic
606C4#1909909311yescharacteristic
607C2—6322no—
608C2C4×#1457914711yescharacteristic
609C2#2377227511yescharacteristic
610C2#2377227811yescharacteristic
611C2#2377375811yescharacteristic
612C2#2377376311yescharacteristic
613C2#2377407211yescharacteristic
614C2#2377407911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC4×#103×C79C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenterC4×#103D79
hypocenterC79#21643
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. #23766137C2semidirect
4. #23764367C2semidirect
7. #23764087C2semidirect
9. #23764078C2semidirect
10. #23763526C2semidirect
11. #23763230C2semidirect
14. #23763086C2semidirect
15. C4×#103×C79C2semidirect
19. #19098919C4semidirect
65. #14579152C2×C4semidirect
90. #14579085C2×C4semidirect
99. C8×Dic79C2×C4semidirect
122. #103×C79C2×C4semidirect
320. C4×#103D79semidirect
323. C79#21643semidirect
412. #103C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190989192
#237661372
#237661072
#237643672
#237642132
#237640982
#237640872
#237640812
#237640782
#237635262
#237632302
#237632282
C4×#145790832
#237630862
C4×#103×C792
#2164379