Finite Group Databasebeta

← #34310294

Subgroups — #34310294

Order 20224.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098841C211yesmaximal, characteristic
2#23766487C211yesmaximal, characteristic, complemented
3#23766305C211yesmaximal, characteristic
4#23764371C211yesmaximal, characteristic, complemented
5#23764171C211yesmaximal, characteristic, complemented
6#23764137C211yesmaximal, characteristic, complemented
7#23764133C211yesmaximal, characteristic, complemented
8#23763992C211yesmaximal, characteristic, complemented
9#23763970C211yesmaximal, characteristic, complemented
10#23763661C211yesmaximal, characteristic
11#23763649C211yesmaximal, characteristic
12#23763274C211yesmaximal, characteristic, complemented
13#23763227C211yesmaximal, characteristic
14#23763227C211yesmaximal, characteristic
15C4×#102×C79C211yesmaximal, characteristic
16C2×#14579160C2211yescharacteristic
17C2×D4×Dic79C2211yescharacteristic
18C2×D4×Dic79C2211yescharacteristic
19#19098841C418yescomplemented
20C2×#14579095C2211yescharacteristic
21C2×#14579095C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×Dic79⋊C8C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579071C2211yescharacteristic
28C2×#14579071C2211yescharacteristic
29#19098469—21no—
30#19098467—21no—
31#19098462C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098461C2211yescharacteristic
34#19098460—21no—
35#19098459C2211yescharacteristic
36#19098436—21no—
37#19098435—21no—
38#19098428C2211yescharacteristic
39#19098421—21no—
40C4⋊C4×Dic79C2211yescharacteristic
41C4⋊C4×Dic79C2211yescharacteristic
42C4×Dic79⋊C4—21no—
43C4×Dic79⋊C4C2211yescharacteristic
44C2×C79⋊#95—21no—
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098318C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×C4×D4×C79C2211yescharacteristic
54C2×#102×C79C2211yescharacteristic
55C2×#102×C79C2211yescharacteristic
56C2×C4×C8×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×Dic158—21no—
61C23×Dic79—21no—
62C23×Dic79—21no—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65#14579160—22no—
66#14579160C2×C414yescomplemented
67D4×Dic79—22no—
68D4×Dic79—22no—
69D4×Dic79—41no—
70D4×Dic79C2×C414yes—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4D411yescharacteristic
81C2×Dic79⋊C4D411yescharacteristic
82C2×C22⋊Dic79—21no—
83C2×C22⋊Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79—21no—
86C2×C4×Dic79C2311yescharacteristic
87C2×C4×Dic79C2311yescharacteristic
88#14579095—41no—
89#14579095C2×C414yescomplemented
90#14579094C2×C414yescomplemented
91#14579092—41no—
92Dic79⋊C8C2×C414yescomplemented
93#14579082—22no—
94#14579082C2×C414yes—
95#14579071—22no—
96#14579071—41no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95D411yescharacteristic
106C79⋊#95D411yescharacteristic
107C22×D4×C79C2311yescharacteristic
108C23×C4×C79—21no—
109C22×C8×C79C2311yescharacteristic
110C22×C8×C79C2311yescharacteristic
111C4×D4×C79—22no—
112C4×D4×C79—41no—
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C22⋊C4×C79—21no—
117C2×C42×C79—21no—
118C2×C42×C79C2311yescharacteristic
119#102×C79C2×C414yes—
120#102×C79Q814yescomplemented
121C4×C8×C79—41no—
122#95×C79—21no—
123C2×Dic158—24no—
124C2×Dic158—81no—
125C2×Dic158—81no—
126C22×Dic79—24no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—21no—
131C22×Dic79—42no—
132C22×Dic79—42no—
133C22×Dic79—81no—
134C22×Dic79—81no—
135C22×Dic79—81no—
136C22×Dic79C2×D411yescharacteristic
137C22×Dic79C2×D411yescharacteristic
138C22×Dic79C2×D411yescharacteristic
139C22×Dic79C2×D411yescharacteristic
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8C22×C412yes—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—81no—
153C4⋊Dic79C22×C412yes—
154C4⋊Dic79C22×C412yes—
155Dic79⋊C4—24no—
156Dic79⋊C4—22no—
157Dic79⋊C4—41no—
158Dic79⋊C4—81no—
159Dic79⋊C4—81no—
160Dic79⋊C4—81no—
161C22⋊Dic79—24no—
162C22⋊Dic79—42no—
163C4×Dic79—21no—
164C4×Dic79—21no—
165C4×Dic79—42no—
166C4×Dic79—81no—
167C4×Dic79C22×C412yes—
168C4×Dic79#4112yes—
169C24×C79—21no—
170C2×D4×C79—21no—
171C2×D4×C79—21no—
172C2×D4×C79—21no—
173C2×D4×C79C22×C412yes—
174C2×D4×C79C2×Q812yes—
175C2×D4×C79#4112yes—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—42no—
180C22×C4×C79—42no—
181C22×C4×C79—81no—
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×C79C2×Q812yes—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79C22×C412yes—
196C4⋊C4×C79C2×Q812yes—
197C22⋊C4×C79—42no—
198C42×C79—21no—
199C42×C79—21no—
200C42×C79—81no—
201C23×C79—22no—
202C23×C79—22no—
203C23×C79—22no—
204C23×C79—41no—
205C23×C79—41no—
206C23×C79—41no—
207C23×C79—41no—
208C23×C79C22×D411yescharacteristic
209D4×C79—22no—
210D4×C79—22no—
211D4×C79—22no—
212D4×C79C4×Q814yes—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—22no—
216C2×C4×C79—21no—
217C2×C4×C79—22no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
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—84no—
228C2×C4×C79#13712yes—
229C2×C4×C79C23×C411yescharacteristic
230C2×C4×C79C22×Q811yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79H(5,𝔽2)11yescharacteristic
234C2×C4×C792−1+411yescharacteristic
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—22no—
240C2×Dic79—22no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—41no—
248C2×Dic79—41no—
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—84no—
258C2×Dic79—84no—
259C2×Dic79C4×D412yes—
260C2×Dic79C4×D412yes—
261Dic158—84no—
262C8×C79—22no—
263C8×C79—22no—
264C79⋊C8—22no—
265C79⋊C8—41no—
266C22×C79—24no—
267C22×C79—42no—
268C22×C79—42no—
269C22×C79—42no—
270C22×C79—42no—
271C22×C79—42no—
272C22×C79—42no—
273C22×C79C2×C4×D411yescharacteristic
274C22×C79#54611yescharacteristic
275C22×C79D4×Q811yescharacteristic
276C22×C79C2×#13711yescharacteristic
277C22×C79#57511yescharacteristic
278C22×C79#57611yescharacteristic
279C22×C79#57611yescharacteristic
280C4×C79—22no—
281C4×C79—42no—
282C4×C79—42no—
283C4×C79—41no—
284C4×C79—81no—
285C4×C79C2×C4×Q811yescharacteristic
286C4×C79#52011yescharacteristic
287C4×C79#55211yescharacteristic
288C4×C79#55411yescharacteristic
289Dic79—22no—
290Dic79—22no—
291Dic79—42no—
292Dic79—42no—
293Dic79—41no—
294Dic79—41no—
295Dic79—81no—
296Dic79—81no—
297#22979—791nomaximal, Sylow
298C2×C79—44no—
299C2×C79#236011yescharacteristic
300C2×C79C4×#13711yescharacteristic
301C2×C79#294811yescharacteristic
302C2×C79#331411yescharacteristic
303C2×C79#331911yescharacteristic
304C2×C79#338511yescharacteristic
305C2×C79#338911yescharacteristic
306C2×#474—791no—
307#2369—791no—
308#2349—791no—
309#1993—791no—
310#1949—791no—
311#1943—791no—
312#1942—791no—
313#1921—791no—
314#1918—791no—
315#1864—791no—
316#1862—791no—
317#1771—791no—
318#1762—791no—
319#1762—791no—
320C4×#102D7911yescharacteristic
321C79#2297911yescharacteristic, complemented, Sylow
322C2×C4⋊Q8—791no—
323C2×C4×D4—791no—
324C2×C4×D4—791no—
325C2×C4×D4C2×D7911yescharacteristic
326#474—798no—
327C2×#107—791no—
328C2×#107—791no—
329C2×#107—791no—
330C2×#106—791no—
331C2×C4⋊C8—791no—
332C2×#102—791no—
333C2×#102—791no—
334C2×#102C2×D7911yescharacteristic
335C2×#102C2×D7911yescharacteristic
336C2×#97—791no—
337C2×#97—791no—
338C2×C4×C8C2×D7911yescharacteristic
339#391—1581no—
340#391—1581no—
341#389—791no—
342#389—791no—
343#389—791no—
344#389—791no—
345#389—1581no—
346#385—1581no—
347#385—1581no—
348#382—791no—
349#380—1581no—
350C4×C4⋊C4—791no—
351C4×C4⋊C4—791no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—1581no—
354C4×C4⋊C4C2×D7911yescharacteristic
355C2×#95—1581no—
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340C2×D7911yescharacteristic
363#340C2×D7911yescharacteristic
364#336—791no—
365#336—791no—
366C22×Q8—1581no—
367C22×D4C22×D7911yescharacteristic
368C23×C4—21no—
369C23×C4—1581no—
370C23×C4—1581no—
371C22×C8—791no—
372C22×C8—791no—
373C22×C8C22×D7911yescharacteristic
374C22×C8C22×D7911yescharacteristic
375C4⋊Q8—794no—
376C4⋊Q8—1582no—
377C4×D4—22no—
378C4×D4—41no—
379C4×D4—3161no—
380C4×D4—794no—
381C4×D4—1582no—
382C4×D4—1582no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—791no—
387C2×C4⋊C4—791no—
388C2×C4⋊C4—791no—
389C2×C4⋊C4—1581no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4C22×D7911yescharacteristic
395C2×C4⋊C4C22×D7911yescharacteristic
396C2×C4⋊C4C22×D7911yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1581no—
399C2×C22⋊C4—1581no—
400C2×C42—21no—
401C2×C42—791no—
402C2×C42—791no—
403C2×C42—1581no—
404C2×C42—1581no—
405C2×C42C22×D7911yescharacteristic
406#107—3161no—
407#107—794no—
408#107—794no—
409#106—3161no—
410C4⋊C8—794no—
411#102—794no—
412#102—1582no—
413#102Dic15814yescomplemented
414#102C4×D7914yes—
415#97—3161no—
416#97—1582no—
417C4×C8—41no—
418#95—21no—
419#95—791no—
420#95—791no—
421#95—1581no—
422#95—1581no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428#95—1581no—
429C24—21no—
430C2×Q8—6321no—
431C2×Q8—6321no—
432C2×Q8—1584no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4C2×C4×D7912yes—
437C2×D4C2×Dic15812yes—
438C2×D4#1192962412yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—81no—
445C22×C4—3162no—
446C22×C4—3162no—
447C22×C4—791no—
448C22×C4—791no—
449C22×C4—791no—
450C22×C4—791no—
451C22×C4—6321no—
452C22×C4—6321no—
453C22×C4—6321no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4—1581no—
457C22×C4—1581no—
458C22×C4—1584no—
459C22×C4D4×D7911yescharacteristic
460C22×C4D4×D7911yescharacteristic
461C22×C4C23×D7911yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—792no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8—1581no—
472C2×C8C2×C4×D7912yes—
473C2×C8C2×Dic15812yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—3161no—
479C4⋊C4—792no—
480C4⋊C4—792no—
481C4⋊C4—6321no—
482C4⋊C4—6321no—
483C4⋊C4—6321no—
484C4⋊C4—6321no—
485C4⋊C4—1584no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1582no—
493C4⋊C4C2×C4×D7912yes—
494C4⋊C4C2×Dic15812yes—
495C22⋊C4—42no—
496C22⋊C4—3162no—
497C22⋊C4—1584no—
498C42—21no—
499C42—21no—
500C42—81no—
501C42—3162no—
502C42—792no—
503C42—792no—
504C42—6321no—
505C42—1581no—
506C42—1581no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×D4×D7911yescharacteristic
515Q8—6324no—
516D4—22no—
517D4—22no—
518D4—22no—
519D4C4×Dic15814yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—84no—
535C2×C4—3161no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—792no—
546C2×C4—792no—
547C2×C4—6324no—
548C2×C4—6324no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4—1582no—
552C2×C4—1582no—
553C2×C4—1581no—
554C2×C4—1581no—
555C2×C4—1581no—
556C2×C4—1581no—
557C2×C4—1581no—
558C2×C4—1581no—
559C2×C4—1581no—
560C2×C4—1581no—
561C2×C4#1457919012yes—
562C2×C4C22×C4×D7911yescharacteristic
563C2×C4C22×Dic15811yescharacteristic
564C2×C4#41×D7911yescharacteristic
565C2×C4C2×#1192962411yescharacteristic
566C2×C4#1457920611yescharacteristic
567C2×C4#1457920811yescharacteristic
568C8—22no—
569C8—22no—
570C8—3161no—
571C8—1582no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C4×D4×D7911yescharacteristic
580C22#1909920311yescharacteristic
581C22D4×Dic15811yescharacteristic
582C22C2×#1457919011yescharacteristic
583C22#1909936011yescharacteristic
584C22#1909936411yescharacteristic
585C22#1909936511yescharacteristic
586C4—22no—
587C4—41no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—3161no—
592C4—3162no—
593C4—3162no—
594C4—3161no—
595C4—6321no—
596C4—6321no—
597C4—1582no—
598C4—1582no—
599C4C2×C4×Dic15811yescharacteristic
600C4#1909904211yescharacteristic
601C4#1909924611yescharacteristic
602C4#1909927211yescharacteristic
603C2—44no—
604C2#2376640611yescharacteristic
605C2C4×#1457919011yescharacteristic
606C2#2377224711yescharacteristic
607C2#2377547711yescharacteristic
608C2#2377555211yescharacteristic
609C2#2377630811yescharacteristic
610C2#2377635611yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC4×#102×C79C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenterC4×#102D79
hypocenterC79#22979
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #23766487C2semidirect
4. #23764371C2semidirect
5. #23764171C2semidirect
6. #23764137C2semidirect
7. #23764133C2semidirect
8. #23763992C2semidirect
9. #23763970C2semidirect
12. #23763274C2semidirect
19. #19098841C4semidirect
66. #14579160C2×C4semidirect
89. #14579095C2×C4semidirect
90. #14579094C2×C4semidirect
92. Dic79⋊C8C2×C4semidirect
120. #102×C79Q8semidirect
321. C79#22979semidirect
413. #102Dic158semidirect

Maximal subgroups

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

subgroupindex
C2×#190988412
#237664872
#237663052
#237643712
#237641712
#237641372
#237641332
#237639922
#237639702
#237636612
#237636492
#237632742
#237632272
C4×#102×C792
#2297979