Finite Group Databasebeta

← #34287801

Subgroups — #34287801

Order 20224.

1462 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099044C211yesmaximal, characteristic
2C2×#19098589C211yesmaximal, characteristic, complemented
3C2×#19098562C211yesmaximal, characteristic
4C2×#19098560C211yesmaximal, characteristic
5#23766024C211yesmaximal, characteristic, complemented
6#23763414C212yesmaximal, complemented
7#23763409C212yesmaximal, complemented
8#23763404C212yesmaximal, complemented
9#23763399C212yesmaximal, complemented
10#23763119C211yesmaximal, characteristic, complemented
11C2×#340×C79C211yesmaximal, characteristic, complemented
12C22×#11929624C2211yescharacteristic
13C22×#11929608C2211yescharacteristic
14#19099044—24no—
15#19099044C418yescomplemented
16C2×#14579126C2212yes—
17C2×#14579123C2212yes—
18C2×C4×C22⋊D79—22no—
19C2×#14579120C2211yescharacteristic
20C2×#14579120C2211yescharacteristic
21C2×#14579119—21no—
22C2×C4⋊C4×D79—21no—
23#19098589—24no—
24C2×Dic79⋊C8C2212yes—
25C2×Dic79⋊C8C2212yes—
26#19098562—24no—
27#19098560—24no—
28C2×#14579086C2211yescharacteristic
29C2×#14579086C2211yescharacteristic
30C2×#14579084C2212yes—
31C2×#14579083C2211yescharacteristic
32C2×#14579083C2211yescharacteristic
33C2×#14579081C2212yes—
34#19098459—21no—
35#19098459—21no—
36#19098436—22no—
37#19098432C2212yes—
38#19098428C2212yes—
39#19098325C2212yes—
40#19098325C2212yes—
41C23×C8×C79C2211yescharacteristic
42C22×C4⋊C4×C79C2211yescharacteristic
43C22×C4⋊C4×C79C2211yescharacteristic
44#340×C79C2212yes—
45#340×C79C2212yes—
46C2×#11929624—22no—
47C2×#11929624—22no—
48C2×#11929624—22no—
49C2×#11929624C2×C414yes—
50C2×#11929624D414yes—
51C2×#11929624Q814yes—
52C22×Dic158C2311yescharacteristic
53C22×D316C2311yescharacteristic
54C22×C22⋊D79—21no—
55C22×C4×D79—21no—
56C22×C4×D79—22no—
57C2×#11929608—22no—
58C2×#11929608—22no—
59C2×#11929608—22no—
60C22×C79⋊C8C2312yes—
61#14579126—24no—
62#14579126C2×C418yescomplemented
63#14579123—24no—
64#14579123C2×C418yescomplemented
65C4×C22⋊D79—216no—
66#14579120—22no—
67#14579120—22no—
68#14579120C2×C414yes—
69#14579120D414yes—
70#14579119—28no—
71C2×C4⋊Dic79D411yescharacteristic
72C2×C4⋊Dic79D411yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×Dic79⋊C4—22no—
76C2×Dic79⋊C4—22no—
77C2×Dic79⋊C4—22no—
78C4⋊C4×D79—28no—
79C2×C22⋊Dic79—21no—
80C2×C22⋊Dic79D412yes—
81C2×(C2×D79)⋊C4—22no—
82C2×C4×Dic79—22no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79C2312yes—
85Dic79⋊C8—24no—
86Dic79⋊C8—24no—
87#14579086—22no—
88#14579086—22no—
89#14579084—24no—
90#14579083—22no—
91#14579083—22no—
92#14579081—24no—
93C79⋊#95—22no—
94C79⋊#95—22no—
95C23×C4×C79—21no—
96C23×C4×C79—21no—
97C23×C4×C79C2311yescharacteristic
98C22×C8×C79—22no—
99C22×C8×C79—22no—
100C22×C8×C79—22no—
101C22×C8×C79C2311yescharacteristic
102C22×C8×C79C2311yescharacteristic
103C2×C4⋊C4×C79—21no—
104C2×C4⋊C4×C79—22no—
105C2×C4⋊C4×C79—22no—
106C2×C4⋊C4×C79—22no—
107C2×C4⋊C4×C79—21no—
108C2×C4⋊C4×C79C2×C414yes—
109C2×C4⋊C4×C79C2312yes—
110C2×C4⋊C4×C79C2312yes—
111C23×D79—21no—
112#11929624—28no—
113#11929624—28no—
114#11929624—28no—
115#11929624C4⋊C4116yescomplemented
116C2×Dic158—21no—
117C2×Dic158—21no—
118C2×Dic158—21no—
119C2×Dic158C22×C412yes—
120C2×Dic158C2×D412yes—
121C2×Dic158C2×Q812yes—
122C2×D316—21no—
123C2×D316—21no—
124C2×D316—21no—
125C2×D316C22×C412yes—
126C2×D316C2×D412yes—
127C2×D316C2×Q812yes—
128C2×C22⋊D79—24no—
129C2×C22⋊D79—24no—
130C2×C22⋊D79—24no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79C2×D412yes—
135C22×Dic79C2×D412yes—
136C2×C4×D79—28no—
137C2×C4×D79—24no—
138C2×C4×D79—24no—
139C2×C4×D79—24no—
140C2×C4×D79—82no—
141C2×C4×D79—82no—
142#11929608—44no—
143C2×C79⋊C8—22no—
144C2×C79⋊C8—22no—
145C2×C79⋊C8—22no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79C22⋊C414yes—
151C4⋊Dic79C22⋊C414yes—
152Dic79⋊C4—28no—
153Dic79⋊C4—82no—
154Dic79⋊C4—82no—
155C22⋊Dic79—42no—
156C22⋊Dic79C22⋊C418yes—
157(C2×D79)⋊C4—28no—
158C4×Dic79—28no—
159C4×Dic79—42no—
160C4×Dic79C22×C414yes—
161C4×Dic79C2×D414yes—
162C24×C79C2×D411yescharacteristic
163C22×C4×C79—22no—
164C22×C4×C79—24no—
165C22×C4×C79—22no—
166C22×C4×C79—42no—
167C22×C4×C79—81no—
168C22×C4×C79—81no—
169C22×C4×C79—81no—
170C22×C4×C79—81no—
171C22×C4×C79C22×C412yes—
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D412yes—
174C22×C4×C79C2×D412yes—
175C22×C4×C79C2×Q812yes—
176C22×C4×C79#4112yes—
177C22×C4×C79#4112yes—
178C22×C4×C79C2411yescharacteristic
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79—21no—
185C2×C8×C79—44no—
186C4⋊C4×C79—24no—
187C4⋊C4×C79—22no—
188C4⋊C4×C79—22no—
189C4⋊C4×C79—22no—
190C4⋊C4×C79—42no—
191C4⋊C4×C79C22×C414yes—
192C23×C79—21no—
193C23×C79—21no—
194C23×C79—21no—
195C23×C79—21no—
196C23×C79C2×C22⋊C412yes—
197C23×C79#12112yes—
198C23×C79C22⋊Q812yes—
199C23×C79C22×D411yescharacteristic
200C22×D79—22no—
201C22×D79—22no—
202C22×D79—22no—
203C22×D79—41no—
204C22×D79—41no—
205C22×D79—41no—
206C22×D79—41no—
207C2×C4×C79—24no—
208C2×C4×C79—42no—
209C2×C4×C79—42no—
210C2×C4×C79—42no—
211C2×C4×C79—42no—
212C2×C4×C79—42no—
213C2×C4×C79—84no—
214C2×C4×C79—84no—
215C2×C4×C79C2×C22⋊C412yes—
216C2×C4×C79C2×C4⋊C414yes—
217C2×C4×C79#11714yes—
218C2×C4×C79#12114yes—
219C2×C4×C79#12112yes—
220C2×C4×C79C22⋊Q814yes—
221C2×C4×C79C22⋊Q812yes—
222C2×C4×C79C23×C411yescharacteristic
223C2×C4×C79C22×D411yescharacteristic
224C2×C4×C79C22×D411yescharacteristic
225C2×C4×C79C22×Q811yescharacteristic
226C2×C4×C79C2×#4111yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C22⋊D79—216no—
229C2×Dic79—22no—
230C2×Dic79—22no—
231C2×Dic79—22no—
232C2×Dic79—22no—
233C2×Dic79—22no—
234C2×Dic79—22no—
235C2×Dic79—22no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79C2×C22⋊C414yes—
243C2×Dic79C2×C22⋊C414yes—
244D316—22no—
245D316—22no—
246D316—22no—
247D316C2×C4⋊C414yes—
248C4×D79—216no—
249C4×D79—88no—
250Dic158—22no—
251Dic158—22no—
252Dic158—22no—
253Dic158C2×C4⋊C414yes—
254C8×C79—41no—
255C8×C79—41no—
256C79⋊C8—42no—
257C22×C79—22no—
258C22×C79—22no—
259C22×C79—22no—
260C22×C79—22no—
261C22×C79—22no—
262C22×C79—22no—
263C22×C79#38014yes—
264C22×C79C22×C22⋊C411yescharacteristic
265C22×C79C2×#12111yescharacteristic
266C22×C79C2×C22⋊Q811yescharacteristic
267C22×C79#57611yescharacteristic
268C22×C79#57711yescharacteristic
269C22×C79#57711yescharacteristic
270C22×C79#57811yescharacteristic
271C2×D79—24no—
272C2×D79—42no—
273C2×D79—42no—
274C2×D79—42no—
275C2×D79—42no—
276C2×D79—42no—
277C2×D79—42no—
278C4×C79—44no—
279C4×C79—82no—
280C4×C79#38014yes—
281C4×C79C22×C4⋊C411yescharacteristic
282C4×C79C2×#11711yescharacteristic
283C4×C79C2×#12111yescharacteristic
284C4×C79C2×C22⋊Q811yescharacteristic
285Dic79—24no—
286Dic79—24no—
287Dic79—24no—
288Dic79—81no—
289Dic79—81no—
290#20652—791nomaximal, Sylow
291C2×C79—24no—
292C2×C79C2×#38011yescharacteristic
293C2×C79#289711yescharacteristic
294C2×C79#289811yescharacteristic
295C2×C79#304511yescharacteristic
296C2×C79#304611yescharacteristic
297C2×C79#308311yescharacteristic
298C2×C79#308411yescharacteristic
299D79—44no—
300C2×#520—791no—
301C2×#424—791no—
302C2×#417—791no—
303C2×#416—791no—
304#2317—791no—
305#1805—792no—
306#1804—792no—
307#1803—792no—
308#1802—792no—
309#1736—791no—
310C2×#340D7911yescharacteristic, complemented
311C79#2065211yescharacteristic, complemented, Sylow
312C22×#41—791no—
313C22×M16—791no—
314C23×C8C2×D7911yescharacteristic
315#520—798no—
316#520—1584no—
317C2×C4×Q8—792no—
318C2×C4×D4—792no—
319C2×C4×D4—1582no—
320C2×#117—791no—
321C2×#117—791no—
322C2×#117—1581no—
323C22×C4⋊C4—1581no—
324C22×C4⋊C4C2×D7911yescharacteristic
325C22×C4⋊C4C2×D7911yescharacteristic
326#424—1584no—
327C2×C4⋊C8—792no—
328C2×C4⋊C8—792no—
329#417—1584no—
330#416—1584no—
331C2×#103—791no—
332C2×#103—791no—
333C2×#103—792no—
334C2×#102—791no—
335C2×#102—791no—
336C2×#102—792no—
337#389—1581no—
338#389—1581no—
339#385—1582no—
340#384—792no—
341#382—792no—
342#340—792no—
343#340—792no—
344#340C2×D7912yes—
345#340C2×D7912yes—
346C2×#41—794no—
347C2×#41—794no—
348C2×#41—794no—
349C2×#41—1582no—
350C2×#41—1582no—
351C2×#41—1582no—
352C22×Q8—791no—
353C22×D4—791no—
354C22×D4—1581no—
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—1582no—
358C23×C4—1581no—
359C23×C4C22×D7911yescharacteristic
360C2×M16—1582no—
361C2×M16—1582no—
362C2×M16—1582no—
363C22×C8—22no—
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—792no—
367C22×C8C22×D7911yescharacteristic
368C22×C8C22×D7911yescharacteristic
369C4×Q8—798no—
370C4×Q8—1584no—
371C4×D4—798no—
372C4×D4—1584no—
373C4×D4—15816no—
374#117—794no—
375#117—794no—
376#117—1582no—
377#117—1582no—
378#117—1588no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—791no—
387C2×C4⋊C4—791no—
388C2×C4⋊C4—1588no—
389C2×C4⋊C4—1582no—
390C2×C4⋊C4—1582no—
391C2×C4⋊C4—1582no—
392C2×C4⋊C4C4×D7914yes—
393C2×C4⋊C4C22×D7912yes—
394C2×C4⋊C4C22×D7912yes—
395C2×C22⋊C4—792no—
396C2×C22⋊C4—1581no—
397C2×C22⋊C4—1582no—
398C2×C42—792no—
399C2×C42—1581no—
400C2×C42—1582no—
401C4⋊C8—1584no—
402C4⋊C8—1584no—
403#103—1584no—
404#103—1582no—
405#103—1582no—
406#102—1584no—
407#102—1582no—
408#102—1582no—
409#95—1582no—
410#95—1582no—
411C24—1581no—
412C24D4×D7911yescharacteristic
413#41—7916no—
414#41—1588no—
415#41—1588no—
416#41—1588no—
417C2×Q8—792no—
418C2×Q8—792no—
419C2×Q8—792no—
420C2×Q8—1581no—
421C2×Q8—1581no—
422C2×Q8—1581no—
423C2×D4—792no—
424C2×D4—792no—
425C2×D4—792no—
426C2×D4—1584no—
427C2×D4—1584no—
428C2×D4—1584no—
429C2×D4—1581no—
430C2×D4—1581no—
431C2×D4—1581no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—792no—
441C22×C4—792no—
442C22×C4—6322no—
443C22×C4—6322no—
444C22×C4—1588no—
445C22×C4—1581no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4—1584no—
449C22×C4—1584no—
450C22×C4—1584no—
451C22×C4C2×C4×D7912yes—
452C22×C4D4×D7911yescharacteristic
453C22×C4D4×D7912yes—
454C22×C4D4×D7912yes—
455C22×C4Q8×D7912yes—
456C22×C4#1192962412yes—
457C22×C4#1192962412yes—
458C22×C4C23×D7911yescharacteristic
459M16—3164no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—44no—
467C2×C8—1582no—
468C2×C8—1582no—
469C2×C8—1582no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—42no—
475C4⋊C4—794no—
476C4⋊C4—794no—
477C4⋊C4—6322no—
478C4⋊C4—6322no—
479C4⋊C4—1588no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4C2×C4×D7914yes—
485C22⋊C4—3162no—
486C22⋊C4—798no—
487C22⋊C4—1588no—
488C42—3162no—
489C42—794no—
490C42—794no—
491C42—1588no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—3161no—
497C23—3161no—
498C23—3161no—
499C23—3161no—
500C23—1582no—
501C23—1582no—
502C23—1582no—
503C23C22⋊C4×D7912yes—
504C23#1457913412yes—
505C23#1457913912yes—
506C23C2×D4×D7911yescharacteristic
507Q8—794no—
508Q8—1582no—
509Q8—1582no—
510Q8—1582no—
511D4—794no—
512D4—15816no—
513D4—1582no—
514D4—1582no—
515D4—1582no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—794no—
531C2×C4—794no—
532C2×C4—6328no—
533C2×C4—1582no—
534C2×C4—1582no—
535C2×C4—1582no—
536C2×C4—1582no—
537C2×C4—1582no—
538C2×C4—1582no—
539C2×C4—1582no—
540C2×C4—15816no—
541C2×C4C22⋊C4×D7912yes—
542C2×C4C4⋊C4×D7914yes—
543C2×C4#1457911814yes—
544C2×C4#1457913412yes—
545C2×C4#1457913414yes—
546C2×C4#1457913912yes—
547C2×C4#1457913914yes—
548C2×C4C22×C4×D7911yescharacteristic
549C2×C4C2×D4×D7911yescharacteristic
550C2×C4C2×D4×D7911yescharacteristic
551C2×C4C2×Q8×D7911yescharacteristic
552C2×C4C2×#1192962411yescharacteristic
553C2×C4C2×#1192962411yescharacteristic
554C8—41no—
555C8—41no—
556C8—3162no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—3162no—
564C22—3162no—
565C22—3162no—
566C22—3162no—
567C22—3162no—
568C22—3162no—
569C22—1584no—
570C22#1909842214yes—
571C22C2×C22⋊C4×D7911yescharacteristic
572C22C2×#1457913411yescharacteristic
573C22C2×#1457913911yescharacteristic
574C22#1909936411yescharacteristic
575C22#1909936911yescharacteristic
576C22#1909936911yescharacteristic
577C22#1909937611yescharacteristic
578C4—44no—
579C4—82no—
580C4—6321no—
581C4—6321no—
582C4—1584no—
583C4—1584no—
584C4—1584no—
585C4#1909842214yes—
586C4C2×C4⋊C4×D7911yescharacteristic
587C4C2×#1457911811yescharacteristic
588C4C2×#1457913411yescharacteristic
589C4C2×#1457913911yescharacteristic
590C2—24no—
591C2—3164no—
592C2C2×#1909842211yescharacteristic
593C2#2377192111yescharacteristic
594C2#2377192811yescharacteristic
595C2#2377294411yescharacteristic
596C2#2377295511yescharacteristic
597C2#2377325611yescharacteristic
598C2#2377326711yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#340×C79C2
socleC23×C79C22×D4
prime radicalC22×#11929624C22
cosocleC22×C4×C79C24
hypercenterC2×#340D79
hypocenterC79#20652
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C2×#19098589C2semidirect
5. #23766024C2semidirect
6. #23763414C2semidirect
7. #23763409C2semidirect
8. #23763404C2semidirect
9. #23763399C2semidirect
10. #23763119C2semidirect
11. C2×#340×C79C2semidirect
15. #19099044C4semidirect
62. #14579126C2×C4semidirect
64. #14579123C2×C4semidirect
115. #11929624C4⋊C4semidirect
310. C2×#340D79semidirect
311. C79#20652semidirect

Maximal subgroups

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

subgroupindex
C2×#190990442
C2×#190985892
C2×#190985622
C2×#190985602
#237660242
#237634142
#237634092
#237634042
#237633992
#237631192
C2×#340×C792
#2065279