Finite Group Databasebeta

← #34351199

Subgroups — #34351199

Order 20224.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098870C211yesmaximal, characteristic, complemented
2C2×#19098606C211yesmaximal, characteristic
3#23767124C211yesmaximal, characteristic
4#23764948C211yesmaximal, characteristic, complemented
5#23764896C211yesmaximal, characteristic, complemented
6#23764795C211yesmaximal, characteristic, complemented
7#23764738C211yesmaximal, characteristic, complemented
8#23764168C211yesmaximal, characteristic, complemented
9#23763978C211yesmaximal, characteristic, complemented
10#23763834C211yesmaximal, characteristic
11#23763708C211yesmaximal, characteristic
12#23763668C211yesmaximal, characteristic, complemented
13#23763599C211yesmaximal, characteristic
14#23763585C211yesmaximal, characteristic
15C79×#2362C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×#14579153C2211yescharacteristic
18C22×C4⋊Dic79C2211yescharacteristic
19#19098870—24no—
20#19098606—24no—
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579094C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×#14579092C2211yescharacteristic
26C2×#14579092C2211yescharacteristic
27C2×C4⋊(C79⋊C8)C2211yescharacteristic
28C2×C4⋊(C79⋊C8)C2211yescharacteristic
29C2×#14579085C2211yescharacteristic
30C2×#14579085C2211yescharacteristic
31C2×#14579085C2211yescharacteristic
32C2×#14579085C2211yescharacteristic
33C2×C22⋊(C79⋊C8)C2211yescharacteristic
34#19098505—21no—
35#19098497—21no—
36#19098461C2211yescharacteristic
37#19098461C2211yescharacteristic
38#19098435—21no—
39#19098429C2211yescharacteristic
40#19098423C2211yescharacteristic
41#19098421—21no—
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098318C2211yescharacteristic
49#19098318C2211yescharacteristic
50C2×C22⋊Q8×C79C2211yescharacteristic
51C2×C4×Q8×C79C2211yescharacteristic
52C2×#117×C79C2211yescharacteristic
53#391×C79C2211yescharacteristic
54#389×C79C2211yescharacteristic
55#388×C79—21no—
56#388×C79—21no—
57#387×C79—21no—
58C4×C4⋊C4×C79—21no—
59C4×C4⋊C4×C79C2211yescharacteristic
60C4×C22⋊C4×C79C2211yescharacteristic
61C23×Dic79—21no—
62C2×#11929608—22no—
63C2×#11929608—22no—
64C2×#11929608—22no—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68C22×C79⋊C8C2311yescharacteristic
69#14579153—22no—
70#14579153—41no—
71C2×C4⋊Dic79—22no—
72C2×C4⋊Dic79—21no—
73C2×C4⋊Dic79—41no—
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×Dic79⋊C4—21no—
82C2×C22⋊Dic79—21no—
83#14579094—22no—
84#14579094—22no—
85#14579094—22no—
86#14579092—22no—
87#14579092—41no—
88#14579092D414yes—
89C4⋊(C79⋊C8)—22no—
90C4⋊(C79⋊C8)—22no—
91#14579085—22no—
92#14579085—22no—
93#14579085—22no—
94#14579085—41no—
95C22⋊(C79⋊C8)—22no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C22×Q8×C79C2311yescharacteristic
102C23×C4×C79C2311yescharacteristic
103C22⋊Q8×C79—22no—
104C22⋊Q8×C79Q814yes—
105C4×Q8×C79—22no—
106C4×Q8×C79—41no—
107#117×C79—22no—
108#117×C79—41no—
109C2×C4⋊C4×C79—21no—
110C2×C4⋊C4×C79—21no—
111C2×C4⋊C4×C79—21no—
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C22⋊C4×C79—21no—
116C2×C22⋊C4×C79—21no—
117C2×C22⋊C4×C79D411yescharacteristic
118C2×C22⋊C4×C79D411yescharacteristic
119C2×C42×C79—21no—
120C2×C42×C79—21no—
121C2×C42×C79—21no—
122C2×C42×C79C2311yescharacteristic
123C2×C42×C79C2311yescharacteristic
124#95×C79—21no—
125#95×C79—21no—
126#95×C79—21no—
127#95×C79—21no—
128#95×C79D411yescharacteristic
129#95×C79D411yescharacteristic
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—42no—
135C22×Dic79—42no—
136C22×Dic79—81no—
137#11929608—28no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8C2×D412yes—
149C2×C79⋊C8#4112yes—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79—42no—
158C4⋊Dic79C2×D412yes—
159Dic79⋊C4—42no—
160Dic79⋊C4—42no—
161Dic79⋊C4—81no—
162Dic79⋊C4—81no—
163C22⋊Dic79—42no—
164C24×C79C2×D411yescharacteristic
165C2×Q8×C79—21no—
166C2×Q8×C79—21no—
167C2×Q8×C79—21no—
168C2×Q8×C79—21no—
169C2×Q8×C79—21no—
170C2×Q8×C79C2×Q812yes—
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×C79—21no—
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2×Q812yes—
185C22×C4×C79#4112yes—
186C22×C4×C79C2411yescharacteristic
187C4⋊C4×C79—24no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—42no—
193C4⋊C4×C79—42no—
194C4⋊C4×C79C2×D412yes—
195C4⋊C4×C79C2×Q812yes—
196C22⋊C4×C79—24no—
197C22⋊C4×C79—41no—
198C22⋊C4×C79—41no—
199C22⋊C4×C79—42no—
200C42×C79—21no—
201C42×C79—21no—
202C42×C79—21no—
203C42×C79—21no—
204C42×C79—42no—
205C42×C79—42no—
206C42×C79—42no—
207C23×C79—21no—
208C23×C79—21no—
209C23×C79—21no—
210C23×C79—21no—
211C23×C79—21no—
212C23×C79—21no—
213C23×C79C22⋊Q812yes—
214C23×C79C22×D411yescharacteristic
215Q8×C79—22no—
216Q8×C79—22no—
217Q8×C79—41no—
218Q8×C79—41no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—22no—
222C2×C4×C79—22no—
223C2×C4×C79—22no—
224C2×C4×C79—22no—
225C2×C4×C79—22no—
226C2×C4×C79—22no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—22no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—21no—
234C2×C4×C79—21no—
235C2×C4×C79—21no—
236C2×C4×C79—21no—
237C2×C4×C79—21no—
238C2×C4×C79—21no—
239C2×C4×C79—21no—
240C2×C4×C79—41no—
241C2×C4×C79—41no—
242C2×C4×C79—41no—
243C2×C4×C79—41no—
244C2×C4×C79—41no—
245C2×C4×C79—41no—
246C2×C4×C79—41no—
247C2×C4×C79—41no—
248C2×C4×C79—41no—
249C2×C4×C79C22⋊Q812yes—
250C2×C4×C79C22×D411yescharacteristic
251C2×C4×C79C22×Q811yescharacteristic
252C2×C4×C79C2×#4111yescharacteristic
253C2×C4×C79C2×#4111yescharacteristic
254C2×C4×C79H(5,𝔽2)11yescharacteristic
255C2×C4×C792−1+411yescharacteristic
256C2×Dic79—22no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79—41no—
265C2×Dic79—41no—
266C2×Dic79—41no—
267C2×Dic79—41no—
268C2×Dic79—84no—
269C79⋊C8—22no—
270C79⋊C8—22no—
271C79⋊C8—41no—
272C79⋊C8—41no—
273C22×C79—22no—
274C22×C79—22no—
275C22×C79—22no—
276C22×C79—41no—
277C22×C79—41no—
278C22×C79—41no—
279C22×C79—41no—
280C22×C79C2×C22⋊Q811yescharacteristic
281C22×C79D4211yescharacteristic
282C22×C79#54711yescharacteristic
283C22×C79#57511yescharacteristic
284C22×C79#57511yescharacteristic
285C22×C79#57711yescharacteristic
286C22×C79#57811yescharacteristic
287C4×C79—22no—
288C4×C79—42no—
289C4×C79—41no—
290C4×C79—41no—
291C4×C79—42no—
292C4×C79—41no—
293C4×C79—42no—
294C4×C79—41no—
295C4×C79—81no—
296C4×C79#52511yescharacteristic
297C4×C79#53911yescharacteristic
298C4×C79D4×Q811yescharacteristic
299C4×C79#55411yescharacteristic
300Dic79—42no—
301Dic79—81no—
302Dic79—81no—
303Dic79—81no—
304#26367—791nomaximal, Sylow
305C2×C79—42no—
306C2×C79#244911yescharacteristic
307C2×C79#308011yescharacteristic
308C2×C79#308211yescharacteristic
309C2×C79#326911yescharacteristic
310C2×C79#327511yescharacteristic
311C2×C79#334911yescharacteristic
312C2×C79#335111yescharacteristic
313C2×#479—791no—
314C2×#428—791no—
315#2442—791no—
316#2362D7911yescharacteristic
317#2097—791no—
318#2087—791no—
319#2068—791no—
320#2059—791no—
321#1949—791no—
322#1919—791no—
323#1894—791no—
324#1872—791no—
325#1865—791no—
326#1854—791no—
327#1851—791no—
328C79#2636711yescharacteristic, complemented, Sylow
329C22×M16—791no—
330C2×#125—791no—
331C2×C22⋊Q8C2×D7911yescharacteristic
332C2×C4×Q8C2×D7911yescharacteristic
333C2×#117C2×D7911yescharacteristic
334C22×C4⋊C4—791no—
335#479—1584no—
336#428—1584no—
337C2×#107—791no—
338C2×#107—791no—
339C2×#107—791no—
340C2×#106—791no—
341C2×#106—791no—
342C2×#106—791no—
343C2×C4⋊C8—791no—
344C2×C4⋊C8—791no—
345C2×#103—791no—
346C2×#103—791no—
347C2×#103—791no—
348C2×#103—791no—
349C2×C22⋊C8—791no—
350#400—1581no—
351#398—1581no—
352#391C2×D7911yescharacteristic
353#389—791no—
354#389—791no—
355#389C2×D7911yescharacteristic
356#388—21no—
357#388—21no—
358#387—21no—
359#385—1581no—
360#382—791no—
361#380—791no—
362#380—1581no—
363C4×C4⋊C4—21no—
364C4×C4⋊C4C2×D7911yescharacteristic
365C4×C22⋊C4C2×D7911yescharacteristic
366#340—791no—
367#340—791no—
368#340—791no—
369#340—791no—
370#340—791no—
371#340—791no—
372#336—791no—
373#336—791no—
374C22×Q8C22×D7911yescharacteristic
375C23×C4—1581no—
376C23×C4C22×D7911yescharacteristic
377C2×M16—1582no—
378C2×M16—1582no—
379C2×M16—1582no—
380C22×C8—791no—
381C22×C8—791no—
382C22×C8—791no—
383C22×C8—791no—
384#125—3161no—
385#125—1582no—
386C22⋊Q8—22no—
387C22⋊Q8Dic15814yes—
388C4×Q8—22no—
389C4×Q8—41no—
390#117—22no—
391#117—41no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—3161no—
396C2×C4⋊C4—791no—
397C2×C4⋊C4—791no—
398C2×C4⋊C4—791no—
399C2×C4⋊C4—791no—
400C2×C4⋊C4—1581no—
401C2×C4⋊C4—1581no—
402C2×C4⋊C4—1581no—
403C2×C4⋊C4—1582no—
404C2×C4⋊C4—1581no—
405C2×C4⋊C4—1581no—
406C2×C4⋊C4C22×D7911yescharacteristic
407C2×C4⋊C4C22×D7911yescharacteristic
408C2×C4⋊C4C22×D7911yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1581no—
412C2×C22⋊C4C22⋊D7911yescharacteristic
413C2×C22⋊C4C22⋊D7911yescharacteristic
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42C22×D7911yescharacteristic
418C2×C42C22×D7911yescharacteristic
419#107—1582no—
420#107—1582no—
421#107—1582no—
422#106—3161no—
423#106—794no—
424#106—1582no—
425C4⋊C8—1582no—
426C4⋊C8—1582no—
427#103—3161no—
428#103—1582no—
429#103—1582no—
430#103—1582no—
431C22⋊C8—1582no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—1581no—
437#95—1581no—
438#95—1581no—
439#95—1581no—
440#95—1581no—
441#95C22⋊D7911yescharacteristic
442#95C22⋊D7911yescharacteristic
443C24C2×C22⋊D7911yescharacteristic
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8C2×Dic15812yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—3162no—
459C22×C4—3162no—
460C22×C4—6321no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4C2×C22⋊D7911yescharacteristic
466C22×C4C2×C22⋊D7911yescharacteristic
467C22×C4C2×C22⋊D7911yescharacteristic
468C22×C4C2×C22⋊D7911yescharacteristic
469C22×C4C2×C22⋊D7911yescharacteristic
470C22×C4C2×Dic15812yes—
471C22×C4#1192962412yes—
472C22×C4C23×D7911yescharacteristic
473M16—1588no—
474C2×C8—792no—
475C2×C8—792no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C2×C8—1581no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—3162no—
494C4⋊C4—3162no—
495C4⋊C4—3162no—
496C4⋊C4—792no—
497C4⋊C4—6321no—
498C4⋊C4—6321no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4—1581no—
502C4⋊C4—1581no—
503C4⋊C4—1581no—
504C4⋊C4—1581no—
505C4⋊C4—1581no—
506C4⋊C4D4×D7912yes—
507C4⋊C4C2×Dic15812yes—
508C22⋊C4—24no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—42no—
512C22⋊C4—3162no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—42no—
519C42—42no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1457914012yes—
527C23C22×C22⋊D7911yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—3161no—
563C2×C4—3161no—
564C2×C4—3161no—
565C2×C4—3161no—
566C2×C4—3161no—
567C2×C4—3161no—
568C2×C4—3161no—
569C2×C4—3161no—
570C2×C4—3161no—
571C2×C4—3161no—
572C2×C4—3161no—
573C2×C4—6324no—
574C2×C4—1582no—
575C2×C4#1457914012yes—
576C2×C4C2×D4×D7911yescharacteristic
577C2×C4C22×Dic15811yescharacteristic
578C2×C4#41×D7911yescharacteristic
579C2×C4C2×#1192962411yescharacteristic
580C2×C4#1457920611yescharacteristic
581C2×C4#1457920811yescharacteristic
582C8—3161no—
583C8—3161no—
584C8—1582no—
585C8—1582no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1457914011yescharacteristic
594C22D4×C22⋊D7911yescharacteristic
595C22#1909921211yescharacteristic
596C22#1909936111yescharacteristic
597C22#1909936111yescharacteristic
598C22#1909937111yescharacteristic
599C22#1909937811yescharacteristic
600C4—22no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—3162no—
610C4—6321no—
611C4—6321no—
612C4—6321no—
613C4#1909907211yescharacteristic
614C4#1909915511yescharacteristic
615C4D4×Dic15811yescharacteristic
616C4#1909927211yescharacteristic
617C2—42no—
618C2#2376720311yescharacteristic
619C2#2377322211yescharacteristic
620C2#2377324211yescharacteristic
621C2#2377495111yescharacteristic
622C2#2377503211yescharacteristic
623C2#2377592411yescharacteristic
624C2#2377596011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2362C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2362D79
hypocenterC79#26367
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
1. C2×#19098870C2semidirect
4. #23764948C2semidirect
5. #23764896C2semidirect
6. #23764795C2semidirect
7. #23764738C2semidirect
8. #23764168C2semidirect
9. #23763978C2semidirect
12. #23763668C2semidirect
328. C79#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#190988702
C2×#190986062
#237671242
#237649482
#237648962
#237647952
#237647382
#237641682
#237639782
#237638342
#237637082
#237636682
#237635992
#237635852
C79×#23622
#2636779