Finite Group Databasebeta

← #34447415

Subgroups — #34447415

Order 20224.

2070 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099176C211yesmaximal, characteristic
2#23766203C214yesmaximal, complemented
3#23766174C214yesmaximal
4#23766119C211yesmaximal, characteristic, complemented
5#23766083C212yesmaximal
6#23766047C212yesmaximal, complemented
7#23766008C211yesmaximal, characteristic
8#23765967C211yesmaximal, characteristic, complemented
9#125×Dic79C211yesmaximal, characteristic, complemented
10#23765959C214yesmaximal, complemented
11#23765950C214yesmaximal
12#23765930C212yesmaximal
13C4×#14579126C212yesmaximal, complemented
14#23765739C211yesmaximal, characteristic
15C79×#2330C211yesmaximal, characteristic, complemented
16#19099176C4116yescomplemented
17C2×#14579161C2211yescharacteristic
18C2×#14579156C2214yes—
19C2×#14579152C2214yes—
20C2×#14579141C2212yes—
21C2×C4×Dic158C2212yes—
22C2×#14579126C2212yes—
23C2×#14579126C2212yes—
24C2×#14579118C2211yescharacteristic
25C2×#14579118C2211yescharacteristic
26C2×C42×D79C2211yescharacteristic
27#19098469C2212yes—
28#19098468C2212yes—
29#19098462C2212yes—
30#19098461C2214yes—
31#19098459C2214yes—
32#19098455C2214yes—
33#19098454C2214yes—
34#19098453C2214yes—
35#19098450C2214yes—
36#19098449C2214yes—
37#19098448C2214yes—
38#19098448C2214yes—
39#19098448C2214yes—
40#19098446C2214yes—
41#19098444C2214yes—
42#19098444C2214yes—
43#19098434C2214yes—
44#19098430C2214yes—
45#19098429C2211yescharacteristic
46#19098428C2214yes—
47#19098420C2212yes—
48C4⋊C4×Dic79C2212yes—
49C4⋊C4×Dic79C2214yes—
50C4⋊C4×Dic79C2212yes—
51C4⋊C4×Dic79C2214yes—
52C4⋊C4×Dic79C2214yes—
53C4×Dic79⋊C4C2214yes—
54#19098414C2214yes—
55#19098414C2212yes—
56#19098414C2214yes—
57#19098414C2212yes—
58#19098410C2212yes—
59#19098410C2214yes—
60#19098410C2212yes—
61#19098409C2211yescharacteristic
62C42×Dic79C2214yes—
63C42×Dic79C2211yescharacteristic
64C42×Dic79C2211yescharacteristic
65C2×#125×C79C2211yescharacteristic
66#389×C79C2212yes—
67#389×C79C2214yes—
68#387×C79C2214yes—
69#384×C79C2211yescharacteristic
70C4×C4⋊C4×C79C2212yes—
71#376×C79C2211yescharacteristic
72C22×Dic158C2312yes—
73C22×C4×D79C2311yescharacteristic
74C22×C4×D79C2311yescharacteristic
75C22×C4×D79C2311yescharacteristic
76#14579161C2×C418yescomplemented
77#14579156C2×C4132yes—
78#14579152C2×C4132yescomplemented
79#14579141C2×C4116yes—
80C4×Dic158—28no—
81#14579126—28no—
82#14579126C2×C4116yescomplemented
83#14579118—24no—
84#14579118C2×C418yes—
85C2×C4⋊Dic79C2312yes—
86C2×C4⋊Dic79C2314yes—
87C2×C4⋊Dic79C2314yes—
88C2×C4⋊Dic79C2314yes—
89C2×Dic79⋊C4C2314yes—
90C2×Dic79⋊C4C2312yes—
91C2×Dic79⋊C4C2314yes—
92C2×Dic79⋊C4C2312yes—
93C2×Dic79⋊C4C2314yes—
94C2×(C2×D79)⋊C4C2312yes—
95C2×(C2×D79)⋊C4C2314yes—
96C2×(C2×D79)⋊C4C2312yes—
97C2×(C2×D79)⋊C4C2314yes—
98C2×C4×Dic79C2312yes—
99C2×C4×Dic79C2311yescharacteristic
100C2×C4×Dic79C2314yes—
101C2×C4×Dic79C2312yes—
102C2×C4×Dic79C2314yes—
103C2×C4×Dic79C2314yes—
104C2×C4×Dic79C2311yescharacteristic
105C2×C4×Dic79C2311yescharacteristic
106C2×C4×Dic79C2312yes—
107C2×C4×Dic79C2312yes—
108C2×C4×Dic79C2314yes—
109C2×C4×Dic79C2311yescharacteristic
110C2×C4×Dic79C2314yes—
111C2×C4×Dic79C2311yescharacteristic
112C42×D79—24no—
113C79⋊#95C2312yes—
114C79⋊#95C2312yes—
115C79⋊#95C2314yes—
116C79⋊#95C2314yes—
117C79⋊#95C2314yes—
118C79⋊#95C2314yes—
119C79⋊#95C2314yes—
120C79⋊#95C2314yes—
121C79⋊#95C2314yes—
122C79⋊#95C2314yes—
123C79⋊#95C2314yes—
124#125×C79C2×C418yescomplemented
125C2×C4⋊C4×C79C2312yes—
126C2×C4⋊C4×C79C2314yes—
127C2×C4⋊C4×C79C2314yes—
128C2×C4⋊C4×C79C2314yes—
129C2×C4⋊C4×C79C2312yes—
130C2×C42×C79C2311yescharacteristic
131C2×C42×C79C2314yes—
132C2×C42×C79C2311yescharacteristic
133C2×C42×C79C2311yescharacteristic
134#95×C79C2312yes—
135#95×C79C2312yes—
136#95×C79C2314yes—
137#95×C79C2314yes—
138C23×D79C2411yescharacteristic
139C2×Dic158—24no—
140C2×Dic158—24no—
141C2×Dic158C22×C418yes—
142C22×Dic79C2412yes—
143C22×Dic79C2414yes—
144C22×Dic79C2414yes—
145C22×Dic79C2412yes—
146C22×Dic79C2411yescharacteristic
147C22×Dic79C2411yescharacteristic
148C22×Dic79C2411yescharacteristic
149C2×C4×D79—22no—
150C2×C4×D79—22no—
151C2×C4×D79—24no—
152C2×C4×D79—22no—
153C2×C4×D79—22no—
154C2×C4×D79—42no—
155C2×C4×D79C22×C414yes—
156C4⋊Dic79—24no—
157C4⋊Dic79—44no—
158C4⋊Dic79C22×C4116yes—
159C4⋊Dic79C22×C4116yes—
160Dic79⋊C4—28no—
161Dic79⋊C4—24no—
162Dic79⋊C4—28no—
163Dic79⋊C4C22×C418yes—
164Dic79⋊C4C22×C4116yes—
165(C2×D79)⋊C4—24no—
166(C2×D79)⋊C4—44no—
167(C2×D79)⋊C4C22×C418yes—
168(C2×D79)⋊C4C22×C4116yes—
169C4×Dic79—22no—
170C4×Dic79—22no—
171C4×Dic79—28no—
172C4×Dic79—24no—
173C4×Dic79—28no—
174C4×Dic79—24no—
175C4×Dic79—22no—
176C4×Dic79—28no—
177C4×Dic79—42no—
178C4×Dic79—42no—
179C4×Dic79—44no—
180C4×Dic79C22×C414yes—
181C4×Dic79C22×C4116yes—
182C4×Dic79#4114yes—
183C22×C4×C79C2411yescharacteristic
184C22×C4×C79C2411yescharacteristic
185C22×C4×C79C2414yes—
186C22×C4×C79C2412yes—
187C22×C4×C79C2414yes—
188C22×C4×C79C2412yes—
189C22×C4×C79C2411yescharacteristic
190C4⋊C4×C79—24no—
191C4⋊C4×C79—44no—
192C4⋊C4×C79—44no—
193C4⋊C4×C79C22×C418yes—
194C4⋊C4×C79C22×C4116yes—
195C42×C79—22no—
196C42×C79—44no—
197C42×C79C22×C414yes—
198C42×C79#4114yes—
199C23×C79C2511yescharacteristic
200C22×D79—21no—
201C22×D79—21no—
202C22×D79—22no—
203C22×D79—22no—
204C22×D79C23×C412yes—
205C2×C4×C79—22no—
206C2×C4×C79—24no—
207C2×C4×C79—22no—
208C2×C4×C79—22no—
209C2×C4×C79—22no—
210C2×C4×C79—24no—
211C2×C4×C79—24no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—22no—
215C2×C4×C79—24no—
216C2×C4×C79—24no—
217C2×C4×C79—22no—
218C2×C4×C79C23×C418yes—
219C2×C4×C79C23×C414yes—
220C2×C4×C79C23×C412yes—
221C2×C4×C79C2×#4112yes—
222C2×C4×C79C2×#4114yes—
223C2×C4×C79C2×#4112yes—
224C2×Dic79—21no—
225C2×Dic79—24no—
226C2×Dic79—24no—
227C2×Dic79—21no—
228C2×Dic79—22no—
229C2×Dic79—22no—
230C2×Dic79—22no—
231C2×Dic79—22no—
232C2×Dic79—24no—
233C2×Dic79—22no—
234C2×Dic79—24no—
235C2×Dic79—24no—
236C2×Dic79—22no—
237C2×Dic79C23×C414yes—
238C2×Dic79C23×C418yes—
239C2×Dic79C23×C412yes—
240C2×Dic79C2×#4112yes—
241C2×Dic79C2×#4112yes—
242C2×Dic792−1+414yes—
243C4×D79—28no—
244C4×D79—44no—
245C4×D79—44no—
246Dic158—216no—
247C22×C79C24×C411yescharacteristic
248C22×C79C22×#4111yescharacteristic
249C22×C79C22×#4111yescharacteristic
250C22×C79C2×2−1+412yes—
251C22×C79#58512yes—
252C2×D79—22no—
253C2×D79—24no—
254C2×D79—41no—
255C2×D79—41no—
256C2×D79—42no—
257C4×C79—24no—
258C4×C79—28no—
259C4×C79—24no—
260C4×C79—41no—
261C4×C79—44no—
262C4×C79—41no—
263C4×C79C4×#4114yes—
264Dic79—28no—
265Dic79—22no—
266Dic79—24no—
267Dic79—22no—
268Dic79—42no—
269Dic79—44no—
270Dic79#54114yescomplemented
271C4×#541—791nomaximal, Sylow
272C2×C79C2×C4×#4111yescharacteristic
273C2×C79C4×2−1+412yes—
274C2×C79C2×#54111yescharacteristic
275C2×C79#345511yescharacteristic
276C2×C79#346112yes—
277D79—42no—
278C2×#541—791no—
279#2336—794no—
280#2334—794no—
281#2330D7911yescharacteristic, complemented
282#2327—791no—
283#2323—792no—
284#2320—792no—
285#2314—791no—
286C4×C4⋊Q8—791no—
287C4×#125—794no—
288C4×#125—791no—
289C4×#126—794no—
290C4×C22⋊Q8—792no—
291C42×Q8—792no—
292C4×#117—791no—
293C79C4×#54111yescharacteristic, complemented, Sylow
294#541—7916no—
295C2×C4⋊Q8—791no—
296C2×#126—794no—
297C2×#125—794no—
298C2×#125C2×D7911yescharacteristic
299C2×C22⋊Q8—792no—
300C2×C4×Q8—792no—
301C2×C4×Q8—792no—
302C2×C4×Q8—792no—
303C2×#117—791no—
304C2×#117—791no—
305C22×C42—791no—
306#391—792no—
307#391—792no—
308#389—792no—
309#389—794no—
310#389—794no—
311#389C2×D7914yes—
312#389C2×D7912yes—
313#388—794no—
314#388—794no—
315#388—794no—
316#387—794no—
317#387—794no—
318#387—794no—
319#387—794no—
320#387—794no—
321#387—794no—
322#387—794no—
323#387—794no—
324#387C2×D7914yes—
325#385—794no—
326#384C2×D7911yescharacteristic
327#383—794no—
328#382—794no—
329#382—791no—
330#380—792no—
331C4×C4⋊C4—794no—
332C4×C4⋊C4—794no—
333C4×C4⋊C4—794no—
334C4×C4⋊C4—792no—
335C4×C4⋊C4—792no—
336C4×C4⋊C4—794no—
337C4×C4⋊C4—794no—
338C4×C4⋊C4—792no—
339C4×C4⋊C4—794no—
340C4×C4⋊C4—792no—
341C4×C4⋊C4C2×D7912yes—
342C4×C22⋊C4—792no—
343C4×C22⋊C4—794no—
344C4×C22⋊C4—792no—
345#376—791no—
346#376C2×D7911yescharacteristic
347C43—794no—
348C43—791no—
349C43—791no—
350C22×Q8—792no—
351C23×C4—791no—
352C23×C4—791no—
353C23×C4—791no—
354C4⋊Q8—798no—
355#126—7932no—
356#125—7932no—
357#125C4×D7918yescomplemented
358C22⋊Q8—7916no—
359C4×Q8—7916no—
360C4×Q8—1588no—
361C4×Q8—1588no—
362#117—798no—
363#117—1584no—
364C2×C4⋊C4—794no—
365C2×C4⋊C4—792no—
366C2×C4⋊C4—794no—
367C2×C4⋊C4—794no—
368C2×C4⋊C4—794no—
369C2×C4⋊C4—792no—
370C2×C4⋊C4—792no—
371C2×C4⋊C4—794no—
372C2×C4⋊C4—794no—
373C2×C4⋊C4C22×D7912yes—
374C2×C4⋊C4C22×D7914yes—
375C2×C4⋊C4C22×D7914yes—
376C2×C4⋊C4C22×D7912yes—
377C2×C4⋊C4C22×D7914yes—
378C2×C22⋊C4—794no—
379C2×C22⋊C4—792no—
380C2×C22⋊C4—794no—
381C2×C22⋊C4—792no—
382C2×C42—792no—
383C2×C42—792no—
384C2×C42—794no—
385C2×C42—791no—
386C2×C42—791no—
387C2×C42—794no—
388C2×C42—791no—
389C2×C42—794no—
390C2×C42—794no—
391C2×C42—792no—
392C2×C42—794no—
393C2×C42—792no—
394C2×C42—791no—
395C2×C42—791no—
396C2×C42—1584no—
397C2×C42C22×D7911yescharacteristic
398C2×C42C22×D7914yes—
399C2×C42C22×D7911yescharacteristic
400C2×C42C22×D7911yescharacteristic
401#95—792no—
402#95—792no—
403#95—794no—
404#95—794no—
405#95—794no—
406#95—794no—
407#95—794no—
408#95—794no—
409#95—794no—
410#95—794no—
411#95—794no—
412#95C22×D7912yes—
413#95C22×D7912yes—
414#95C22×D7914yes—
415#95C22×D7914yes—
416C24—791no—
417C2×Q8—798no—
418C2×Q8—1584no—
419C2×Q8—1584no—
420C22×C4—3162no—
421C22×C4—791no—
422C22×C4—792no—
423C22×C4—791no—
424C22×C4—794no—
425C22×C4—791no—
426C22×C4—794no—
427C22×C4—794no—
428C22×C4—792no—
429C22×C4—1582no—
430C22×C4—1582no—
431C22×C4—1584no—
432C22×C4—1582no—
433C22×C4—1582no—
434C22×C4C23×D7911yescharacteristic
435C22×C4C23×D7911yescharacteristic
436C22×C4C23×D7914yes—
437C22×C4C23×D7912yes—
438C22×C4C23×D7914yes—
439C22×C4C23×D7912yes—
440C22×C4C23×D7911yescharacteristic
441C4⋊C4—24no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—3164no—
445C4⋊C4—798no—
446C4⋊C4—7916no—
447C4⋊C4—7916no—
448C4⋊C4—7916no—
449C4⋊C4—1588no—
450C4⋊C4—1584no—
451C4⋊C4—1588no—
452C4⋊C4—1584no—
453C4⋊C4C2×C4×D7918yes—
454C4⋊C4C2×C4×D79116yes—
455C22⋊C4—3164no—
456C22⋊C4—798no—
457C22⋊C4—7916no—
458C22⋊C4—1584no—
459C42—22no—
460C42—44no—
461C42—3162no—
462C42—3162no—
463C42—3164no—
464C42—794no—
465C42—794no—
466C42—7916no—
467C42—1588no—
468C42—1584no—
469C42—1584no—
470C42—1582no—
471C42—1582no—
472C42—1588no—
473C42—1582no—
474C42—1588no—
475C42C2×C4×D7914yes—
476C42#1192962514yes—
477C23—792no—
478C23—1582no—
479C23—1582no—
480C23—1581no—
481C23—1581no—
482C23C24×D7911yescharacteristic
483Q8—15816no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—3164no—
498C2×C4—3164no—
499C2×C4—794no—
500C2×C4—792no—
501C2×C4—794no—
502C2×C4—798no—
503C2×C4—792no—
504C2×C4—792no—
505C2×C4—1584no—
506C2×C4—1582no—
507C2×C4—1584no—
508C2×C4—1582no—
509C2×C4—1582no—
510C2×C4—1584no—
511C2×C4—1582no—
512C2×C4—1582no—
513C2×C4—1582no—
514C2×C4—1581no—
515C2×C4—1584no—
516C2×C4—1584no—
517C2×C4—1581no—
518C2×C4—1588no—
519C2×C4C22×C4×D7918yes—
520C2×C4C22×C4×D7914yes—
521C2×C4C22×C4×D7912yes—
522C2×C4#41×D7912yes—
523C2×C4C2×#1192962514yes—
524C2×C4C2×#1192962512yes—
525C22—3161no—
526C22—3162no—
527C22—3161no—
528C22—1582no—
529C22—1584no—
530C22C23×C4×D7911yescharacteristic
531C22C2×#41×D7911yescharacteristic
532C22C22×#1192962511yescharacteristic
533C222−1+4×D7912yes—
534C22#1909939812yes—
535C4—24no—
536C4—28no—
537C4—24no—
538C4—41no—
539C4—44no—
540C4—41no—
541C4—3164no—
542C4—3162no—
543C4—794no—
544C4—1582no—
545C4—1582no—
546C4—1588no—
547C4—1584no—
548C4C4×#1192962514yes—
549C2—3162no—
550C2C2×C4×#1192962511yescharacteristic
551C2#2377661112yes—
552C2#541×D7911yescharacteristic
553C2#2377678411yescharacteristic
554C2#2377683412yes—
subgroupG/H
center, FrattiniC23C24×D79
commutatorC22×C79C24×C4
FittingC79×#2330C2
socle, cosocleC23×C79C25
prime radicalC23×D79C24
hypercenter#2330D79
hypocenterC79C4×#541
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #23766203C2semidirect
4. #23766119C2semidirect
6. #23766047C2semidirect
8. #23765967C2semidirect
9. #125×Dic79C2semidirect
10. #23765959C2semidirect
13. C4×#14579126C2semidirect
15. C79×#2330C2semidirect
16. #19099176C4semidirect
76. #14579161C2×C4semidirect
78. #14579152C2×C4semidirect
82. #14579126C2×C4semidirect
124. #125×C79C2×C4semidirect
270. Dic79#541semidirect
281. #2330D79semidirect
293. C79C4×#541semidirect
357. #125C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190991762
#237662032
#237661742
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