Finite Group Databasebeta

← #34296261

Subgroups — #34296261

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×#19098916C211yesmaximal, characteristic
2#23766137C211yesmaximal, characteristic, complemented
3#23766112C211yesmaximal, characteristic
4#23764367C211yesmaximal, characteristic, complemented
5#23764215C211yesmaximal, characteristic
6#23764099C211yesmaximal, characteristic
7#23764088C211yesmaximal, characteristic, complemented
8#23764080C211yesmaximal, characteristic
9#23764077C211yesmaximal, characteristic, complemented
10#23763526C211yesmaximal, characteristic, complemented
11#23763231C211yesmaximal, characteristic, complemented
12C4×#14579084C211yesmaximal, characteristic, complemented
13C4×#14579082C211yesmaximal, characteristic
14#23763088C211yesmaximal, characteristic, complemented
15C79×#1762C211yesmaximal, characteristic
16C2×#14579152C2211yescharacteristic
17C2×#14579148C2211yescharacteristic
18C2×C4×Dic158C2211yescharacteristic
19#19098916C418yescomplemented
20C2×#14579086C2211yescharacteristic
21C2×#14579086C2211yescharacteristic
22C2×#14579084C2211yescharacteristic
23C2×#14579084C2211yescharacteristic
24C2×#14579082C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×#14579071C2211yescharacteristic
27C2×#14579071C2211yescharacteristic
28C2×C4×C79⋊C8C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30C2×C8×Dic79C2211yescharacteristic
31#19098468C2211yescharacteristic
32#19098466C2211yescharacteristic
33#19098462C2211yescharacteristic
34#19098461—21no—
35#19098460—21no—
36#19098459C2211yescharacteristic
37#19098451—21no—
38#19098451—21no—
39#19098447—21no—
40#19098446—21no—
41#19098436—21no—
42#19098436—21no—
43#19098432C2211yescharacteristic
44#19098432C2211yescharacteristic
45C4×C4⋊Dic79C2211yescharacteristic
46#19098409C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098325C2211yescharacteristic
53C2×C4×D4×C79C2211yescharacteristic
54C2×#102×C79C2211yescharacteristic
55C2×#102×C79C2211yescharacteristic
56C2×#97×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×Dic158C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63#14579152—22no—
64#14579152C2×C414yescomplemented
65#14579148—22no—
66#14579148C2×C414yes—
67C4×Dic158—22no—
68C4×Dic158—22no—
69C2×C4⋊Dic79—21no—
70C2×C4⋊Dic79—21no—
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×Dic79⋊C4—21no—
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×C22⋊Dic79—21no—
82C2×C22⋊Dic79—21no—
83C2×C22⋊Dic79—21no—
84C2×C22⋊Dic79—21no—
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79C2311yescharacteristic
88C2×C4×Dic79C2311yescharacteristic
89#14579086—22no—
90#14579086C2×C414yescomplemented
91#14579084—22no—
92#14579084C2×C414yescomplemented
93#14579082—22no—
94#14579082C2×C414yes—
95#14579071—22no—
96#14579071—22no—
97C4×C79⋊C8—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×D4×C79C2311yescharacteristic
110C23×C4×C79—21no—
111C22×C8×C79C2311yescharacteristic
112C22×C8×C79C2311yescharacteristic
113C4×D4×C79—22no—
114C4×D4×C79—22no—
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C22⋊C4×C79—21no—
119C2×C42×C79—21no—
120C2×C42×C79C2311yescharacteristic
121#102×C79—22no—
122#102×C79C2×C414yes—
123#97×C79—22no—
124#95×C79—21no—
125C2×Dic158—21no—
126C2×Dic158—21no—
127C2×Dic158—21no—
128C2×Dic158—21no—
129C2×Dic158—21no—
130C2×Dic158C22×C412yes—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79C2×D411yescharacteristic
136C22×Dic79C2×D411yescharacteristic
137C22×Dic79C2×D411yescharacteristic
138C22×Dic79C2×D411yescharacteristic
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8C22×C412yes—
144C2×C79⋊C8#4112yes—
145C4⋊Dic79—24no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—81no—
152C4⋊Dic79C22×C412yes—
153Dic79⋊C4—24no—
154Dic79⋊C4—42no—
155Dic79⋊C4—81no—
156Dic79⋊C4—81no—
157Dic79⋊C4—81no—
158Dic79⋊C4—81no—
159Dic79⋊C4—81no—
160C22⋊Dic79—24no—
161C22⋊Dic79—24no—
162C22⋊Dic79—81no—
163C22⋊Dic79—81no—
164C4×Dic79—21no—
165C4×Dic79—21no—
166C4×Dic79—42no—
167C4×Dic79—81no—
168C4×Dic79C22×C412yes—
169C4×Dic79#4112yes—
170C24×C79—21no—
171C2×D4×C79—21no—
172C2×D4×C79—21no—
173C2×D4×C79—21no—
174C2×D4×C79—21no—
175C2×D4×C79—21no—
176C2×D4×C79C22×C412yes—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—42no—
181C22×C4×C79—81no—
182C22×C4×C79—81no—
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2411yescharacteristic
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79C22×C412yes—
191C2×C8×C79#4112yes—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79C22×C412yes—
198C22⋊C4×C79—42no—
199C42×C79—81no—
200C42×C79#4112yes—
201C42×C79#4112yes—
202C23×C79—22no—
203C23×C79—41no—
204C23×C79—41no—
205C23×C79—41no—
206C23×C79—41no—
207C23×C79—41no—
208C23×C79—41no—
209C23×C79C22×D411yescharacteristic
210D4×C79—22no—
211D4×C79—22no—
212D4×C79—41no—
213D4×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—41no—
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#12412yes—
229C2×C4×C79#12412yes—
230C2×C4×C79C23×C411yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—22no—
241C2×Dic79—22no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—21no—
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—41no—
258C2×Dic79C4×D412yes—
259C2×Dic79C4×D412yes—
260Dic158—22no—
261Dic158—22no—
262Dic158—41no—
263Dic158—41no—
264C8×C79—22no—
265C8×C79—22no—
266C79⋊C8—22no—
267C79⋊C8—22no—
268C22×C79—42no—
269C22×C79—42no—
270C22×C79—42no—
271C22×C79—81no—
272C22×C79—81no—
273C22×C79—81no—
274C22×C79—81no—
275C22×C79C2×C4×D411yescharacteristic
276C22×C79C2×#12411yescharacteristic
277C22×C79#53211yescharacteristic
278C22×C79#57611yescharacteristic
279C22×C79#57711yescharacteristic
280C22×C79#57711yescharacteristic
281C22×C79#57811yescharacteristic
282C4×C79—22no—
283C4×C79—22no—
284C4×C79—42no—
285C4×C79—81no—
286C4×C79—81no—
287C4×C79C4×#4111yescharacteristic
288C4×C79C4×#4111yescharacteristic
289C4×C79#52911yescharacteristic
290C4×C79#52911yescharacteristic
291Dic79—22no—
292Dic79—22no—
293Dic79—41no—
294Dic79—41no—
295Dic79—42no—
296Dic79—42no—
297Dic79—81no—
298Dic79—81no—
299#21643—791nomaximal, Sylow
300C2×C79—82no—
301C2×C79C4×#12411yescharacteristic
302C2×C79#295111yescharacteristic
303C2×C79#295211yescharacteristic
304C2×C79#313611yescharacteristic
305C2×C79#313711yescharacteristic
306C2×C79#317811yescharacteristic
307C2×C79#317911yescharacteristic
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#1762D7911yescharacteristic
320C4×#103—791no—
321C4×#102—791no—
322#1726—791no—
323C79#2164311yescharacteristic, complemented, Sylow
324C2×#125—791no—
325C2×#124—791no—
326C2×C4×Q8—791no—
327C2×C4×D4C2×D7911yescharacteristic
328#488—798no—
329C2×#103—791no—
330C2×#103—791no—
331C2×#103—791no—
332C2×#103—791no—
333C2×#102—791no—
334C2×#102—791no—
335C2×#102C2×D7911yescharacteristic
336C2×#102C2×D7911yescharacteristic
337C2×#97—791no—
338C2×#97—791no—
339C2×#97C2×D7911yescharacteristic
340C2×C4×C8—791no—
341C2×C4×C8—791no—
342C2×C4×C8—791no—
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×Q8—791no—
369C22×D4C22×D7911yescharacteristic
370C23×C4—21no—
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—1582no—
380C4×Q8—1582no—
381C4×D4—22no—
382C4×D4—22no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—1581no—
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—21no—
399C2×C22⋊C4—1581no—
400C2×C22⋊C4—1581no—
401C2×C22⋊C4—1581no—
402C2×C22⋊C4—1581no—
403C2×C42—21no—
404C2×C42—791no—
405C2×C42—791no—
406C2×C42—1581no—
407C2×C42—1581no—
408C2×C42C22×D7911yescharacteristic
409#103—794no—
410#103—794no—
411#103—1582no—
412#103—1582no—
413#102—22no—
414#102—794no—
415#102—1582no—
416#102C4×D7914yes—
417#97—22no—
418#97—1582no—
419#97—1582no—
420C4×C8—794no—
421C4×C8—1582no—
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—21no—
434C2×Q8—792no—
435C2×Q8—1581no—
436C2×Q8—1581no—
437C2×Q8—1581no—
438C2×Q8—1581no—
439C2×Q8—1581no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4C2×C4×D7912yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—791no—
453C22×C4—791no—
454C22×C4—791no—
455C22×C4—791no—
456C22×C4—1581no—
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—3162no—
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—42no—
497C22⋊C4—6321no—
498C22⋊C4—6321no—
499C22⋊C4—1584no—
500C22⋊C4—1584no—
501C42—81no—
502C42—3162no—
503C42—792no—
504C42—792no—
505C42—6321no—
506C42—1581no—
507C42—1581no—
508C42#1192962312yes—
509C42#1192962412yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D7911yescharacteristic
518Q8—3161no—
519Q8—3161no—
520Q8—1582no—
521Q8—1582no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4—3161no—
541C2×C4—3161no—
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—1582no—
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—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C4×D4×D7911yescharacteristic
584C22C2×#1457914711yescharacteristic
585C22#1909911011yescharacteristic
586C22#1909936711yescharacteristic
587C22#1909937311yescharacteristic
588C22#1909937311yescharacteristic
589C22#1909937911yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—3162no—
596C4—3161no—
597C4—3161no—
598C4—3162no—
599C4—6321no—
600C4—6321no—
601C4—1582no—
602C4—1582no—
603C4C4×#1192962311yescharacteristic
604C4C4×#1192962411yescharacteristic
605C4#1909909111yescharacteristic
606C4#1909909311yescharacteristic
607C2—82no—
608C2C4×#1457914711yescharacteristic
609C2#2377227211yescharacteristic
610C2#2377228211yescharacteristic
611C2#2377375011yescharacteristic
612C2#2377376811yescharacteristic
613C2#2377406611yescharacteristic
614C2#2377408911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1762C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenter#1762D79
hypocenterC79#21643
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #23766137C2semidirect
4. #23764367C2semidirect
7. #23764088C2semidirect
9. #23764077C2semidirect
10. #23763526C2semidirect
11. #23763231C2semidirect
12. C4×#14579084C2semidirect
14. #23763088C2semidirect
19. #19098916C4semidirect
64. #14579152C2×C4semidirect
90. #14579086C2×C4semidirect
92. #14579084C2×C4semidirect
99. C8×Dic79C2×C4semidirect
323. C79#21643semidirect

Maximal subgroups

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

subgroupindex
C2×#190989162
#237661372
#237661122
#237643672
#237642152
#237640992
#237640882
#237640802
#237640772
#237635262
#237632312
C4×#145790842
C4×#145790822
#237630882
C79×#17622
#2164379