Finite Group Databasebeta

← #34258344

Subgroups — #34258344

Order 20224.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773480C211yesmaximal, characteristic
2#23772551C211yesmaximal, characteristic
3#23762278C211yesmaximal, characteristic, complemented
4#23762271C211yesmaximal, characteristic, complemented
5#23762250C211yesmaximal, characteristic, complemented
6#23762243C211yesmaximal, characteristic, complemented
7#23762217C211yesmaximal, characteristic, complemented
8#23762210C211yesmaximal, characteristic, complemented
9C8×#11929614C211yesmaximal, characteristic, complemented
10C8×#11929611C211yesmaximal, characteristic, complemented
11#23762188C211yesmaximal, characteristic, complemented
12#23761829C211yesmaximal, characteristic, complemented
13#23761806C211yesmaximal, characteristic
14#23761700C211yesmaximal, characteristic, complemented
15C79×#2982C211yesmaximal, characteristic, complemented
16#19099097C2211yescharacteristic
17#19098893C411yescharacteristic
18#19098876C411yescharacteristic
19#19098870C411yescharacteristic
20#19098870C411yescharacteristic
21#19098785C411yescharacteristic
22#19098785C411yescharacteristic
23#19098719C411yescharacteristic
24#19098702C411yescharacteristic
25#19098693C2211yescharacteristic
26C8×Dic158C2211yescharacteristic
27C4×#11929614C2211yescharacteristic
28C4×#11929614C2211yescharacteristic
29C4×#11929611C2211yescharacteristic
30C4×#11929611C2211yescharacteristic
31#19098642C2211yescharacteristic
32#19098638—21no—
33#19098637—21no—
34C8×D316C2211yescharacteristic
35#19098623—21no—
36#19098616C2211yescharacteristic
37#19098616C2211yescharacteristic
38#19098588—21no—
39#19098587C2211yescharacteristic
40#19098315C2211yescharacteristic
41#19098315C2211yescharacteristic
42#19098313C2211yescharacteristic
43#19098313C2211yescharacteristic
44Q8⋊(C79⋊C8)C2211yescharacteristic
45Q8⋊(C79⋊C8)C2211yescharacteristic
46Q8⋊(C79⋊C8)C2211yescharacteristic
47Q8⋊(C79⋊C8)C2211yescharacteristic
48#19098293C2211yescharacteristic
49#19098293C2211yescharacteristic
50#19098290C2211yescharacteristic
51#19098290C2211yescharacteristic
52#19098285C2211yescharacteristic
53#19098285C2211yescharacteristic
54#19098284C2211yescharacteristic
55C8×C79⋊C8C2211yescharacteristic
56C2×C4×Q8×C79C2211yescharacteristic
57C8×Q8×C79—21no—
58C8×Q8×C79—21no—
59C8×Q8×C79C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C8×Q8×C79C2211yescharacteristic
63#431×C79—21no—
64#431×C79—21no—
65#431×C79C2211yescharacteristic
66#431×C79C2211yescharacteristic
67C2×#11929614C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929611C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×#11929607—21no—
73C2×C8×D79—21no—
74#14579155—21no—
75#14579153C2×C411yescharacteristic
76#14579149C2×C411yescharacteristic
77#14579144—21no—
78#14579140C2×C411yescharacteristic
79#14579135C2×C411yescharacteristic
80C4×Dic158C2311yescharacteristic
81C4×D316C2311yescharacteristic
82C4×C22⋊D79—21no—
83#14579118—21no—
84#14579094C2×C411yescharacteristic
85#14579094C2×C411yescharacteristic
86#14579092C2×C411yescharacteristic
87#14579092C2×C411yescharacteristic
88C4⋊(C79⋊C8)—21no—
89C4⋊(C79⋊C8)—21no—
90C4⋊(C79⋊C8)C2×C411yescharacteristic
91C4⋊(C79⋊C8)C2×C411yescharacteristic
92C4⋊(C79⋊C8)C2311yescharacteristic
93C4⋊(C79⋊C8)C2311yescharacteristic
94Dic79⋊C8—21no—
95Dic79⋊C8—21no—
96#14579085C2×C411yescharacteristic
97#14579085C2×C411yescharacteristic
98#14579085C2×C411yescharacteristic
99#14579085C2×C411yescharacteristic
100#14579084C2×C411yescharacteristic
101#14579084C2×C411yescharacteristic
102#14579081C2×C411yescharacteristic
103#14579081C2×C411yescharacteristic
104C22⋊(C79⋊C8)—21no—
105C22⋊(C79⋊C8)C2×C411yescharacteristic
106C22⋊(C79⋊C8)C2×C411yescharacteristic
107(C2×D79)⋊C8—21no—
108(C2×D79)⋊C8—21no—
109#14579071—21no—
110C4×C79⋊C8C2311yescharacteristic
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C4×C79⋊C8C2311yescharacteristic
114C8×Dic79—21no—
115C22×Q8×C79C2×C411yescharacteristic
116C4×Q8×C79—21no—
117C4×Q8×C79—21no—
118C4×Q8×C79C2×C411yescharacteristic
119C4×Q8×C79C2×C411yescharacteristic
120C4×Q8×C79C2311yescharacteristic
121C4×Q8×C79C2311yescharacteristic
122C2×C4⋊C4×C79—21no—
123C2×C4⋊C4×C79C2×C411yescharacteristic
124C2×C42×C79—21no—
125C2×C42×C79C2311yescharacteristic
126C4⋊C8×C79—21no—
127C4⋊C8×C79—21no—
128C4⋊C8×C79—21no—
129C4⋊C8×C79—21no—
130C4⋊C8×C79D411yescharacteristic
131C4⋊C8×C79D411yescharacteristic
132C4⋊C8×C79C2311yescharacteristic
133C4⋊C8×C79C2311yescharacteristic
134C22⋊C8×C79—21no—
135C22⋊C8×C79D411yescharacteristic
136C22⋊C8×C79D411yescharacteristic
137C4×C8×C79—21no—
138C4×C8×C79—21no—
139C4×C8×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158C22×C411yescharacteristic
145C2×D316C22×C411yescharacteristic
146C2×C22⋊D79—21no—
147C2×C4×D79—21no—
148#11929614—22no—
149#11929614—22no—
150#11929611—22no—
151#11929611—22no—
152#11929608—81no—
153#11929607—81no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8—21no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8C22×C411yescharacteristic
159C2×C79⋊C8C22×C411yescharacteristic
160C2×C79⋊C8C22×C411yescharacteristic
161C2×C79⋊C8C22×C411yescharacteristic
162C8×D79—81no—
163C4⋊Dic79C22×C411yescharacteristic
164C4⋊Dic79C22×C411yescharacteristic
165Dic79⋊C4—21no—
166Dic79⋊C4—21no—
167Dic79⋊C4—21no—
168Dic79⋊C4—21no—
169C22⋊Dic79—21no—
170(C2×D79)⋊C4—21no—
171(C2×D79)⋊C4—21no—
172(C2×D79)⋊C4—21no—
173(C2×D79)⋊C4—21no—
174C4×Dic79—21no—
175C2×Q8×C79—22no—
176C2×Q8×C79—22no—
177C2×Q8×C79—21no—
178C2×Q8×C79C22×C411yescharacteristic
179C2×Q8×C79C22×C411yescharacteristic
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79C22×C411yescharacteristic
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79#4111yescharacteristic
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79C2×D411yescharacteristic
188C2×C8×C79C2×D411yescharacteristic
189C2×C8×C79C2×D411yescharacteristic
190C2×C8×C79C2×D411yescharacteristic
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79C22×C411yescharacteristic
197C4⋊C4×C79C22×C411yescharacteristic
198C42×C79—21no—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—21no—
202C42×C79C22×C411yescharacteristic
203C42×C79C2×D411yescharacteristic
204C42×C79#4111yescharacteristic
205C42×C79C2411yescharacteristic
206C23×C79C4×D411yescharacteristic
207C22×D79—21no—
208Q8×C79—24no—
209Q8×C79—21no—
210Q8×C79—21no—
211Q8×C79#13112yes—
212Q8×C79#13112yes—
213C2×C4×C79—24no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—41no—
223C2×C4×C79—81no—
224C2×C4×C79C4×D411yescharacteristic
225C2×C4×C79C4×D411yescharacteristic
226C2×C4×C79C4×D411yescharacteristic
227C2×C4×C79C23×C411yescharacteristic
228C2×C4×C79C22×D411yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C22⋊D79—81no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234D316—21no—
235D316—21no—
236C4×D79—41no—
237C4×D79—41no—
238Dic158—21no—
239Dic158—21no—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—21no—
244C8×C79—41no—
245C8×C79—41no—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—41no—
251C79⋊C8—41no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C22×C79—21no—
255C22×C79—21no—
256C22×C79—21no—
257C22×C79C2×C4×D411yescharacteristic
258C2×D79—41no—
259C2×D79—41no—
260C2×D79—41no—
261C4×C79—22no—
262C4×C79—22no—
263C4×C79—21no—
264C4×C79—21no—
265C4×C79—21no—
266C4×C79—21no—
267C4×C79—41no—
268C4×C79—41no—
269C4×C79C2×#13111yescharacteristic
270C4×C79C2×#13111yescharacteristic
271C4×C79#57711yescharacteristic
272C4×C79#57811yescharacteristic
273Dic79—41no—
274Dic79—41no—
275Dic79—41no—
276#17919—791nomaximal, Sylow
277C2×C79—41no—
278C2×C79#292911yescharacteristic
279C2×C79#294711yescharacteristic
280C2×C79#295711yescharacteristic
281D79—81no—
282#3108—791no—
283#2993—791no—
284#2982D7911yescharacteristic, complemented
285#1588—791no—
286#1587—791no—
287#1584—791no—
288#1583—791no—
289#1579—791no—
290SD16⋊C8—791no—
291C8×Q16—791no—
292C8×SD16—791no—
293#1574—791no—
294#1487—791no—
295#1482—791no—
296#1454—791no—
297C79#1791911yescharacteristic, complemented, Sylow
298#529—791no—
299C2×C4×Q8C2×D7911yescharacteristic
300#483—791no—
301#480—791no—
302#479—791no—
303#479—791no—
304#464—791no—
305#464—791no—
306#451—791no—
307#448—791no—
308#446—791no—
309C8×Q8—21no—
310C8×Q8—21no—
311C8×Q8—791no—
312C8×Q8C2×D7911yescharacteristic
313C8×Q8C2×D7911yescharacteristic
314C8×Q8C2×D7911yescharacteristic
315C8×Q8C2×D7911yescharacteristic
316C4×Q16—791no—
317C4×Q16—791no—
318C4×SD16—791no—
319C4×SD16—791no—
320#436—791no—
321#436—1581no—
322#435—1581no—
323C8×D4—791no—
324#433—1581no—
325#431—21no—
326#431—21no—
327#431—791no—
328#431—791no—
329#431C2×D7911yescharacteristic
330#431C2×D7911yescharacteristic
331#424—1581no—
332#423—791no—
333#335—791no—
334#335—791no—
335#334—791no—
336#334—791no—
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8—791no—
343D4⋊C8—791no—
344D4⋊C8—791no—
345#322—791no—
346#322—791no—
347#322—791no—
348C82—791no—
349C22×Q8C4×D7911yescharacteristic
350C2×Q16—791no—
351C2×Q16—791no—
352C2×SD16—791no—
353C2×SD16—791no—
354C2×M16—1581no—
355C2×M16—1581no—
356C22×C8—1581no—
357#126—1581no—
358#125—791no—
359#124—791no—
360#123—1581no—
361C22⋊Q8—791no—
362#121—791no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—791no—
366C4×Q8C4×D7911yescharacteristic
367C4×Q8C4×D7911yescharacteristic
368C4×Q8C22×D7911yescharacteristic
369C4×Q8C22×D7911yescharacteristic
370C4×D4—791no—
371C4×D4—1581no—
372#117—1581no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C4×D7911yescharacteristic
375C2×C42—21no—
376C2×C42C22×D7911yescharacteristic
377#107—791no—
378#107—791no—
379#106—791no—
380#106—791no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—791no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8C22⋊D7911yescharacteristic
394C4⋊C8C22⋊D7911yescharacteristic
395C4⋊C8C22×D7911yescharacteristic
396C4⋊C8C22×D7911yescharacteristic
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103—791no—
402#103—791no—
403#102—791no—
404#102—791no—
405C22⋊C8—21no—
406C22⋊C8—791no—
407C22⋊C8—791no—
408C22⋊C8—1581no—
409C22⋊C8—1581no—
410C22⋊C8—1581no—
411C22⋊C8C22⋊D7911yescharacteristic
412C22⋊C8C22⋊D7911yescharacteristic
413#97—1581no—
414C4×C8—21no—
415C4×C8—21no—
416C4×C8—21no—
417C4×C8—21no—
418C4×C8—21no—
419C4×C8—791no—
420C4×C8—791no—
421C4×C8—791no—
422C4×C8—791no—
423C4×C8—1581no—
424C4×C8C22×D7911yescharacteristic
425C4×C8C22×D7911yescharacteristic
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—791no—
430C2×Q8C2×C4×D7911yescharacteristic
431C2×Q8C2×C4×D7911yescharacteristic
432C2×D4—791no—
433C2×D4—1581no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—1581no—
437C22×C4C2×C4×D7911yescharacteristic
438C22×C4C2×C22⋊D7911yescharacteristic
439C22×C4#1192962411yescharacteristic
440Q16—1582no—
441Q16—1582no—
442SD16—1582no—
443SD16—1582no—
444M16—6321no—
445M16—6321no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—791no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—6321no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8—1581no—
456C2×C8—1581no—
457C2×C8C2×C22⋊D7911yescharacteristic
458C2×C8C2×C22⋊D7911yescharacteristic
459C2×C8C2×C22⋊D7911yescharacteristic
460C2×C8C2×C22⋊D7911yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—791no—
467C4⋊C4—791no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4C2×C4×D7911yescharacteristic
473C4⋊C4C2×C4×D7911yescharacteristic
474C22⋊C4—1581no—
475C22⋊C4—1581no—
476C22⋊C4—1581no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—1581no—
484C42C2×C4×D7911yescharacteristic
485C42C2×C22⋊D7911yescharacteristic
486C42#1192962411yescharacteristic
487C42C23×D7911yescharacteristic
488C23—1581no—
489C23C4×C22⋊D7911yescharacteristic
490Q8—24no—
491Q8—21no—
492Q8—21no—
493Q8—1581no—
494Q8—1581no—
495Q8#1457916712yes—
496Q8#1457916712yes—
497D4—6321no—
498D4—1581no—
499D4—1581no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—81no—
511C2×C4—3161no—
512C2×C4—3161no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4C4×C22⋊D7911yescharacteristic
517C2×C4C4×C22⋊D7911yescharacteristic
518C2×C4C4×C22⋊D7911yescharacteristic
519C2×C4C22×C4×D7911yescharacteristic
520C2×C4C22×C22⋊D7911yescharacteristic
521C2×C4C2×#1192962411yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—3161no—
529C8—3161no—
530C8—3161no—
531C8—3161no—
532C8—1581no—
533C8—1581no—
534C8—1581no—
535C8—1581no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—3161no—
540C22—3161no—
541C22—3161no—
542C22C2×C4×C22⋊D7911yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—22no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—3161no—
552C4—3161no—
553C4—3161no—
554C4C2×#1457916711yescharacteristic
555C4C2×#1457916711yescharacteristic
556C4#1909937111yescharacteristic
557C4#1909937811yescharacteristic
558C2—41no—
559C2—6321no—
560C2#2377209411yescharacteristic
561C2#2377224311yescharacteristic
562C2#2377231511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#2982C2
socleC22×C79C2×C4×D4
prime radical#14579135C2×C4
cosocleC42×C79C24
hypercenter#2982D79
hypocenterC79#17919
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2982

Complements

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

NH ≅ G/Ntype
3. #23762278C2semidirect
4. #23762271C2semidirect
5. #23762250C2semidirect
6. #23762243C2semidirect
7. #23762217C2semidirect
8. #23762210C2semidirect
9. C8×#11929614C2semidirect
10. C8×#11929611C2semidirect
11. #23762188C2semidirect
12. #23761829C2semidirect
14. #23761700C2semidirect
15. C79×#2982C2semidirect
284. #2982D79semidirect
297. C79#17919semidirect

Maximal subgroups

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

subgroupindex
#237734802
#237725512
#237622782
#237622712
#237622502
#237622432
#237622172
#237622102
C8×#119296142
C8×#119296112
#237621882
#237618292
#237618062
#237617002
C79×#29822
#1791979