Finite Group Databasebeta

← #34461460

Subgroups — #34461460

Order 20224.

2042 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771710C211yesmaximal, characteristic, complemented
2#23771708C212yesmaximal, complemented
3#23769372C214yesmaximal, complemented
4#23768623C214yesmaximal, complemented
5#23768622C214yesmaximal, complemented
6#23767127C212yesmaximal, complemented
7#23767126C212yesmaximal, complemented
8#23767122C212yesmaximal, complemented
9#23767120C212yesmaximal, complemented
10#23766011C214yesmaximal
11#380×D79C212yesmaximal
12C2×C4×(C2×D79)⋊C4C211yesmaximal, characteristic
13C79×#2851C211yesmaximal, characteristic, complemented
14C23×C4×D79C2211yescharacteristic
15C2×#14579153C2211yescharacteristic, complemented
16C2×#14579152C2212yescomplemented
17C2×C4⋊C4×D79C2212yes—
18C2×C4⋊C4×D79C2212yes—
19C22×(C2×D79)⋊C4C2212yes—
20C2×C22⋊C4×D79C2214yes—
21C22×C4×Dic79C2211yescharacteristic
22#19098504C2214yescomplemented
23#19098504C2214yescomplemented
24#19098503C2214yescomplemented
25#19098501C2214yescomplemented
26#19098497C2214yescomplemented
27#19098496C2214yescomplemented
28#19098494C2214yescomplemented
29#19098493C2214yescomplemented
30#19098462C2212yescomplemented
31#19098461C2212yescomplemented
32#19098459C2212yescomplemented
33#19098450C2214yescomplemented
34#19098447C2214yescomplemented
35#19098446C2214yescomplemented
36#19098438C2214yes—
37#19098437C2214yes—
38#19098437C2214yes—
39#19098436C2214yes—
40#19098435C2214yes—
41#19098434C2214yes—
42#19098434C2214yes—
43#19098429C2211yescharacteristic, complemented
44#19098428C2212yescomplemented
45#19098423C2212yes—
46#19098422C2214yes—
47#19098422C2214yes—
48#19098422C2214yes—
49#19098421C2214yes—
50#19098420C2212yes—
51#19098420C2212yes—
52#19098420C2214yes—
53#19098420C2214yes—
54C4×(C2×D79)⋊C4C2212yes—
55C4×(C2×D79)⋊C4C2212yes—
56C4×(C2×D79)⋊C4C2212yes—
57C4×(C2×D79)⋊C4C2212yes—
58C2×C79⋊#95C2212yes—
59#95×D79C2214yes—
60C2×#125×C79C2211yescharacteristic, complemented
61C22×C42×C79C2211yescharacteristic
62#389×C79C2212yescomplemented
63#387×C79C2214yescomplemented
64#385×C79C2214yes—
65#382×C79C2211yescharacteristic, complemented
66#380×C79C2212yes—
67C24×D79C2311yescharacteristic
68C23×Dic79C2312yes—
69C23×Dic79C2311yescharacteristic
70C22×C4×D79—42no—
71C22×C4×D79—42no—
72C22×C4×D79Q818yes—
73C22×C4×D79C2312yes—
74C22×C4×D79C2314yes—
75C22×C4×D79C2314yes—
76C22×C4×D79C2312yes—
77#14579153—42no—
78#14579152—44no—
79C2×C4⋊Dic79C2312yes—
80C2×C4⋊Dic79C2312yes—
81C2×C4⋊Dic79C2314yes—
82C2×C4⋊Dic79C2314yes—
83C2×Dic79⋊C4C2314yes—
84C2×Dic79⋊C4C2314yes—
85C2×Dic79⋊C4C2314yes—
86C2×Dic79⋊C4C2314yes—
87C2×Dic79⋊C4C2314yes—
88C2×Dic79⋊C4C2314yes—
89C4⋊C4×D79—44no—
90C4⋊C4×D79—44no—
91C2×C22⋊Dic79C2314yes—
92C2×(C2×D79)⋊C4—28no—
93C2×(C2×D79)⋊C4C2312yes—
94C2×(C2×D79)⋊C4C2312yes—
95C2×(C2×D79)⋊C4C2312yes—
96C2×(C2×D79)⋊C4C2312yes—
97C2×(C2×D79)⋊C4C2314yes—
98C2×(C2×D79)⋊C4C2314yes—
99C22⋊C4×D79—216no—
100C2×C4×Dic79—42no—
101C2×C4×Dic79C2312yes—
102C2×C4×Dic79C2311yescharacteristic
103C2×C4×Dic79C2311yescharacteristic
104C79⋊#95C2314yes—
105C79⋊#95C2314yes—
106C79⋊#95C2314yes—
107C79⋊#95C2312yes—
108C79⋊#95C2312yes—
109C79⋊#95C2312yes—
110C79⋊#95C2312yes—
111C79⋊#95C2314yes—
112C79⋊#95C2314yes—
113C79⋊#95C2314yes—
114C79⋊#95C2314yes—
115C79⋊#95C2314yes—
116C79⋊#95C2314yes—
117C23×C4×C79C2311yescharacteristic
118C23×C4×C79C2312yes—
119#125×C79—42no—
120C2×C4⋊C4×C79C2312yes—
121C2×C4⋊C4×C79C2312yes—
122C2×C4⋊C4×C79C2314yes—
123C2×C4⋊C4×C79C2314yes—
124C2×C22⋊C4×C79C2314yes—
125C2×C42×C79—42no—
126C2×C42×C79C2311yescharacteristic
127C2×C42×C79C2311yescharacteristic
128C2×C42×C79C2312yes—
129#95×C79C2314yes—
130#95×C79C2314yes—
131#95×C79C2314yes—
132C23×D79—42no—
133C23×D79—41no—
134C23×D79—41no—
135C23×D79C2×Q814yes—
136C23×D79#4118yes—
137C23×D79C2412yes—
138C22×Dic79—24no—
139C22×Dic79—42no—
140C22×Dic79—41no—
141C22×Dic79—41no—
142C22×Dic79—42no—
143C22×Dic79C2×Q814yes—
144C22×Dic79C2412yes—
145C22×Dic79C2412yes—
146C22×Dic79C2414yes—
147C22×Dic79C2414yes—
148C22×Dic79C2412yes—
149C2×C4×D79—28no—
150C2×C4×D79—28no—
151C2×C4×D79—24no—
152C2×C4×D79—24no—
153C2×C4×D79—42no—
154C2×C4×D79—42no—
155C2×C4×D79—44no—
156C2×C4×D79—44no—
157C2×C4×D79—44no—
158C2×C4×D79—44no—
159C2×C4×D79—416no—
160C2×C4×D79C2×Q818yes—
161C2×C4×D79C2×Q818yes—
162C4⋊Dic79—24no—
163C4⋊Dic79—24no—
164C4⋊Dic79—44no—
165C4⋊Dic79—44no—
166Dic79⋊C4—44no—
167Dic79⋊C4—44no—
168Dic79⋊C4—44no—
169Dic79⋊C4—44no—
170Dic79⋊C4—44no—
171Dic79⋊C4—44no—
172C22⋊Dic79—28no—
173(C2×D79)⋊C4—28no—
174(C2×D79)⋊C4—28no—
175(C2×D79)⋊C4—24no—
176(C2×D79)⋊C4—24no—
177(C2×D79)⋊C4—24no—
178(C2×D79)⋊C4—24no—
179C4×Dic79—24no—
180C4×Dic79—41no—
181C4×Dic79—41no—
182C24×C79C2411yescharacteristic
183C22×C4×C79—22no—
184C22×C4×C79—22no—
185C22×C4×C79—24no—
186C22×C4×C79—42no—
187C22×C4×C79—42no—
188C22×C4×C79C2×Q814yes—
189C22×C4×C79C2414yes—
190C22×C4×C79C2412yes—
191C22×C4×C79C2412yes—
192C22×C4×C79C2414yes—
193C22×C4×C79C2411yescharacteristic
194C22×C4×C79C2411yescharacteristic
195C4⋊C4×C79—24no—
196C4⋊C4×C79—24no—
197C4⋊C4×C79—44no—
198C4⋊C4×C79—44no—
199C22⋊C4×C79—28no—
200C42×C79—22no—
201C42×C79—22no—
202C42×C79—42no—
203C23×C79—21no—
204C23×C79—21no—
205C23×C79—22no—
206C23×C79C22×Q812yes—
207C23×C79C2×#4114yes—
208C23×C79C2511yescharacteristic
209C22×D79—22no—
210C22×D79—22no—
211C22×D79—22no—
212C22×D79—22no—
213C22×D79—44no—
214C22×D79—44no—
215C22×D79—48no—
216C22×D79—48no—
217C22×D79#125116yescomplemented
218C22×D79C22×Q814yes—
219C22×D79C2×#4114yes—
220C22×D79C2×#4114yes—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—28no—
224C2×C4×C79—22no—
225C2×C4×C79—22no—
226C2×C4×C79—24no—
227C2×C4×C79—24no—
228C2×C4×C79—24no—
229C2×C4×C79—22no—
230C2×C4×C79—24no—
231C2×C4×C79—22no—
232C2×C4×C79—22no—
233C2×C4×C79—24no—
234C2×C4×C79—24no—
235C2×C4×C79—22no—
236C2×C4×C79—48no—
237C2×C4×C79C22×Q812yes—
238C2×C4×C79C22×Q812yes—
239C2×C4×C79H(5,𝔽2)12yes—
240C2×C4×C79H(5,𝔽2)12yes—
241C2×Dic79—22no—
242C2×Dic79—22no—
243C2×Dic79—22no—
244C2×Dic79—22no—
245C2×Dic79—24no—
246C2×Dic79—24no—
247C2×Dic79—22no—
248C2×Dic79—22no—
249C2×Dic79—24no—
250C2×Dic79—24no—
251C2×Dic79—24no—
252C2×Dic79—22no—
253C2×Dic79—24no—
254C2×Dic79—22no—
255C2×Dic79—44no—
256C2×Dic79—48no—
257C2×Dic79C22×Q814yes—
258C4×D79—216no—
259C4×D79—48no—
260C4×D79—88no—
261C4×D79—88no—
262C22×C79—22no—
263C22×C79—22no—
264C22×C79—24no—
265C22×C79—24no—
266C22×C79C2×#12514yes—
267C22×C79C23×Q811yescharacteristic
268C22×C79C22×#4112yes—
269C22×C79C2×H(5,𝔽2)11yescharacteristic
270C22×C79C2×H(5,𝔽2)11yescharacteristic
271C22×C79#58512yes—
272C2×D79—24no—
273C2×D79—24no—
274C2×D79—24no—
275C2×D79—24no—
276C2×D79—24no—
277C2×D79—24no—
278C2×D79—416no—
279C2×D79C2×#12518yes—
280C4×C79—22no—
281C4×C79—42no—
282C4×C79—42no—
283C4×C79—44no—
284C4×C79—44no—
285C4×C79#54314yes—
286Dic79—24no—
287Dic79—44no—
288Dic79—44no—
289Dic79—42no—
290Dic79—42no—
291#35020—791nomaximal, Sylow
292C2×C79—24no—
293C2×C79C22×#12511yescharacteristic
294C2×C79C2×#54311yescharacteristic
295C2×C79#345811yescharacteristic
296C2×C79#350512yes—
297C2×C79#350512yes—
298D79—28no—
299#2851—791no—
300#2851—792no—
301#2851D7911yescharacteristic, complemented
302#2646—794no—
303#2569—794no—
304#2569—794no—
305#2442—792no—
306#2442—792no—
307#2442—792no—
308#2442—792no—
309#2315—794no—
310C2×#380—792no—
311C2×C4×C22⋊C4—791no—
312C79#3502011yescharacteristic, complemented, Sylow
313C24×C4—791no—
314C2×#125—792no—
315C2×#125—791no—
316C2×#125C2×D7911yescharacteristic, complemented
317C22×C4⋊C4—792no—
318C22×C4⋊C4—792no—
319C22×C22⋊C4—792no—
320C22×C22⋊C4—794no—
321C22×C42—791no—
322C22×C42C2×D7911yescharacteristic
323#400—794no—
324#400—794no—
325#400—794no—
326#400—794no—
327#398—794no—
328#398—794no—
329#398—794no—
330#398—794no—
331#389—792no—
332#389—792no—
333#389—792no—
334#389C2×D7912yescomplemented
335#387—794no—
336#387—794no—
337#387—794no—
338#387C2×D7914yescomplemented
339#385—794no—
340#385—794no—
341#385—794no—
342#385—794no—
343#385—794no—
344#385—794no—
345#385—794no—
346#385C2×D7914yes—
347#382—792no—
348#382—791no—
349#382C2×D7911yescharacteristic, complemented
350#380—792no—
351#380—794no—
352#380—794no—
353#380—792no—
354#380—792no—
355#380—794no—
356#380—794no—
357#380—794no—
358#380—794no—
359#380C2×D7912yes—
360C4×C22⋊C4—792no—
361C4×C22⋊C4—792no—
362C4×C22⋊C4—792no—
363C4×C22⋊C4—792no—
364C2×#95—792no—
365C2×#95—794no—
366C25—791no—
367C23×C4—3162no—
368C23×C4—3162no—
369C23×C4—791no—
370C23×C4—798no—
371C23×C4—792no—
372C23×C4—792no—
373C23×C4—794no—
374C23×C4—794no—
375C23×C4—792no—
376C23×C4C22×D7911yescharacteristic
377C23×C4C22×D7912yes—
378#125—42no—
379#125—3162no—
380#125—3164no—
381C2×C4⋊C4—3164no—
382C2×C4⋊C4—3164no—
383C2×C4⋊C4—794no—
384C2×C4⋊C4—794no—
385C2×C4⋊C4—794no—
386C2×C4⋊C4—794no—
387C2×C4⋊C4—794no—
388C2×C4⋊C4—794no—
389C2×C4⋊C4—794no—
390C2×C4⋊C4—794no—
391C2×C4⋊C4—792no—
392C2×C4⋊C4—792no—
393C2×C4⋊C4C22×D7912yes—
394C2×C4⋊C4C22×D7912yes—
395C2×C4⋊C4C22×D7914yes—
396C2×C4⋊C4C22×D7914yes—
397C2×C22⋊C4—794no—
398C2×C22⋊C4—792no—
399C2×C22⋊C4—792no—
400C2×C22⋊C4—792no—
401C2×C22⋊C4—792no—
402C2×C22⋊C4—794no—
403C2×C22⋊C4—794no—
404C2×C22⋊C4—15816no—
405C2×C22⋊C4—1588no—
406C2×C22⋊C4C22×D7914yes—
407C2×C42—42no—
408C2×C42—3162no—
409C2×C42—791no—
410C2×C42—791no—
411C2×C42—792no—
412C2×C42C22×D7911yescharacteristic
413C2×C42C22×D7911yescharacteristic
414C2×C42C22×D7912yes—
415#95—792no—
416#95—792no—
417#95—792no—
418#95—792no—
419#95—794no—
420#95—794no—
421#95—794no—
422#95—794no—
423#95—794no—
424#95—794no—
425#95—794no—
426#95—794no—
427#95—794no—
428#95C22×D7914yes—
429#95C22×D7914yes—
430#95C22×D7914yes—
431C24—3161no—
432C24—3161no—
433C24—3162no—
434C24—794no—
435C24—798no—
436C24—792no—
437C24C23×D7911yescharacteristic
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—3162no—
444C22×C4—3162no—
445C22×C4—31616no—
446C22×C4—3164no—
447C22×C4—3164no—
448C22×C4—3164no—
449C22×C4—3164no—
450C22×C4—3162no—
451C22×C4—3161no—
452C22×C4—3161no—
453C22×C4—3162no—
454C22×C4—794no—
455C22×C4—798no—
456C22×C4—792no—
457C22×C4—798no—
458C22×C4—792no—
459C22×C4—792no—
460C22×C4—794no—
461C22×C4—794no—
462C22×C4—1588no—
463C22×C4—1588no—
464C22×C4—1584no—
465C22×C4—1584no—
466C22×C4—1584no—
467C22×C4Q8×D7914yes—
468C22×C4C23×D7912yes—
469C22×C4C23×D7912yes—
470C22×C4C23×D7914yes—
471C22×C4C23×D7914yes—
472C22×C4C23×D7911yescharacteristic
473C22×C4C23×D7911yescharacteristic
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—3164no—
479C4⋊C4—3164no—
480C4⋊C4—3164no—
481C4⋊C4—3164no—
482C4⋊C4—3164no—
483C4⋊C4—3164no—
484C4⋊C4—3164no—
485C4⋊C4—3164no—
486C4⋊C4—1584no—
487C4⋊C4—1584no—
488C22⋊C4—28no—
489C22⋊C4—1588no—
490C22⋊C4—1588no—
491C22⋊C4—1584no—
492C22⋊C4—1584no—
493C22⋊C4—1584no—
494C22⋊C4—1584no—
495C22⋊C4—1588no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—3161no—
500C42—3161no—
501C42—1584no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—3164no—
506C23—3164no—
507C23—3168no—
508C23—3168no—
509C23—7916no—
510C23—794no—
511C23—794no—
512C23—794no—
513C23—1582no—
514C23—1582no—
515C23—1582no—
516C23—1582no—
517C23C2×Q8×D7912yes—
518C23#41×D7914yes—
519C23C24×D7911yescharacteristic
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—3168no—
537C2×C4—3168no—
538C2×C4—3164no—
539C2×C4—794no—
540C2×C4—6328no—
541C2×C4—6328no—
542C2×C4—1584no—
543C2×C4—1582no—
544C2×C4—1582no—
545C2×C4—1584no—
546C2×C4—1584no—
547C2×C4—1582no—
548C2×C4—1582no—
549C2×C4—1584no—
550C2×C4—1584no—
551C2×C4—1582no—
552C2×C4—1582no—
553C2×C4—1584no—
554C2×C4—1582no—
555C2×C4—1582no—
556C2×C4—15816no—
557C2×C4C2×Q8×D7912yes—
558C2×C4C2×Q8×D7912yes—
559C2×C4#1457920612yes—
560C2×C4#1457920712yes—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—31616no—
566C22—798no—
567C22—1584no—
568C22—1584no—
569C22—1584no—
570C22—1584no—
571C22—1584no—
572C22—1584no—
573C22#125×D7914yes—
574C22C22×Q8×D7911yescharacteristic
575C22C2×#41×D7912yes—
576C22C2×#1457920611yescharacteristic
577C22C2×#1457920711yescharacteristic
578C22#1909939912yes—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—3164no—
585C4—3162no—
586C4—3162no—
587C4—3164no—
588C4—1584no—
589C4#1909918814yes—
590C2—24no—
591C2—1588no—
592C2C2×#125×D7911yescharacteristic
593C2C2×#1909918811yescharacteristic
594C2#2377680311yescharacteristic
595C2#2377722812yes—
596C2#2377723512yes—
subgroupG/H
center, FrattiniC23C24×D79
commutator, socle, cosocleC23×C79C25
FittingC79×#2851C2
prime radicalC24×D79C23
hypercenter#2851D79
hypocenterC79#35020
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. #23771710C2semidirect
2. #23771708C2semidirect
3. #23769372C2semidirect
4. #23768623C2semidirect
5. #23768622C2semidirect
6. #23767127C2semidirect
7. #23767126C2semidirect
8. #23767122C2semidirect
9. #23767120C2semidirect
13. C79×#2851C2semidirect
15. C2×#14579153C22semidirect
16. C2×#14579152C22semidirect
22. #19098504C22semidirect
23. #19098504C22semidirect
24. #19098503C22semidirect
25. #19098501C22semidirect
26. #19098497C22semidirect
27. #19098496C22semidirect
28. #19098494C22semidirect
29. #19098493C22semidirect
30. #19098462C22semidirect
31. #19098461C22semidirect
32. #19098459C22semidirect
33. #19098450C22semidirect
34. #19098447C22semidirect
35. #19098446C22semidirect
43. #19098429C22semidirect
44. #19098428C22semidirect
60. C2×#125×C79C22semidirect
62. #389×C79C22semidirect
63. #387×C79C22semidirect
65. #382×C79C22semidirect
217. C22×D79#125semidirect
301. #2851D79semidirect
312. C79#35020semidirect
316. C2×#125C2×D79semidirect
334. #389C2×D79semidirect
338. #387C2×D79semidirect
349. #382C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237717102
#237717082
#237693722
#237686232
#237686222
#237671272
#237671262
#237671222
#237671202
#237660112
#380×D792
C2×C4×(C2×D79)⋊C42
C79×#28512
#3502079