Finite Group Databasebeta

← #34258304

Subgroups — #34258304

Order 20224.

624 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773469C211yesmaximal, characteristic
2C8×#11929625C211yesmaximal, characteristic
3#23772329C211yesmaximal, characteristic, complemented
4#23762251C211yesmaximal, characteristic, complemented
5#23762247C211yesmaximal, characteristic, complemented
6#23762239C211yesmaximal, characteristic, complemented
7#23762235C211yesmaximal, characteristic, complemented
8#23762217C211yesmaximal, characteristic, complemented
9#23762214C211yesmaximal, characteristic, complemented
10#23762210C211yesmaximal, characteristic, complemented
11#23762187C211yesmaximal, characteristic, complemented
12#23761838C211yesmaximal, characteristic
13#23761819C211yesmaximal, characteristic, complemented
14#23761688C211yesmaximal, characteristic, complemented
15C79×#1579C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17C4×#11929625C2211yescharacteristic
18#19098906C411yescharacteristic
19#19098902C411yescharacteristic
20#19098846C411yescharacteristic
21#19098842C411yescharacteristic
22#19098771C411yescharacteristic
23#19098767C411yescharacteristic
24#19098728C411yescharacteristic
25#19098724C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098692—21no—
28#19098690C2211yescharacteristic
29Q8×C79⋊C8C2211yescharacteristic
30C8×Dic158—21no—
31C8×Dic158C2211yescharacteristic
32C4×#11929614C2211yescharacteristic
33C4×#11929613C2211yescharacteristic
34C4×#11929611C2211yescharacteristic
35C8×D316—21no—
36C8×D316C2211yescharacteristic
37#19098628—21no—
38#19098628C2211yescharacteristic
39#19098615—21no—
40#19098615C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098588—21no—
43#19098585C2211yescharacteristic
44#19098517—21no—
45C4×#11929607C2211yescharacteristic
46C4×C8×D79—21no—
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098285C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929625C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929607—21no—
72C2×#11929607—21no—
73C2×C8×D79—21no—
74C2×C8×D79—21no—
75C2×C8×D79D411yescharacteristic
76C2×C8×D79D411yescharacteristic
77#14579160C2×C411yescharacteristic
78#14579159C2×C411yescharacteristic
79#14579151—21no—
80#14579141C2×C411yescharacteristic
81#14579139—21no—
82#14579137C2×C411yescharacteristic
83Q8×Dic79C2×C411yescharacteristic
84C4×Dic158—21no—
85C4×Dic158C2311yescharacteristic
86C4×D316—21no—
87C4×D316C2311yescharacteristic
88#14579123—21no—
89#14579123C2×C411yescharacteristic
90#14579119—21no—
91#14579119C2×C411yescharacteristic
92#14579118—21no—
93C42×D79—21no—
94C42×D79C2311yescharacteristic
95#14579094C2×C411yescharacteristic
96#14579093C2×C411yescharacteristic
97#14579092C2×C411yescharacteristic
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)—21no—
100C4⋊(C79⋊C8)C2311yescharacteristic
101C4⋊(C79⋊C8)C2311yescharacteristic
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8C2×C411yescharacteristic
106Dic79⋊C8C2×C411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579085C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579083C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8C2×C411yescharacteristic
117(C2×D79)⋊C8C2×C411yescharacteristic
118#14579072—21no—
119#14579071—21no—
120#14579071—21no—
121C4×C79⋊C8—21no—
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C8×Dic79—21no—
126C8×Dic79—21no—
127C8×Dic79D411yescharacteristic
128C8×Dic79D411yescharacteristic
129C2×Q16×C79C2×C411yescharacteristic
130C4×Q8×C79C2311yescharacteristic
131C4×Q8×C79C2311yescharacteristic
132#107×C79C2×C411yescharacteristic
133C4⋊C8×C79—21no—
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#103×C79C2×C411yescharacteristic
139C4×C8×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929625—22no—
145#11929625—22no—
146C2×Dic158—21no—
147C2×Dic158C22×C411yescharacteristic
148C2×D316—21no—
149C2×D316C22×C411yescharacteristic
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79—21no—
153C2×C4×D79C22×C411yescharacteristic
154C2×C4×D79C2×D411yescharacteristic
155C2×C4×D79#4111yescharacteristic
156#11929614—22no—
157#11929613—22no—
158#11929611—22no—
159#11929607—42no—
160#11929607—42no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8C22×C411yescharacteristic
165C2×C79⋊C8C22×C411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C8×D79—22no—
169C8×D79—41no—
170C8×D79—42no—
171C8×D79—42no—
172C4⋊Dic79—21no—
173C4⋊Dic79—21no—
174C4⋊Dic79C22×C411yescharacteristic
175C4⋊Dic79C22×C411yescharacteristic
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79—21no—
189C4×Dic79C22×C411yescharacteristic
190C4×Dic79C2×D411yescharacteristic
191C4×Dic79#4111yescharacteristic
192C2×Q8×C79C22×C411yescharacteristic
193C2×Q8×C79C22×C411yescharacteristic
194Q16×C79—22no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C2×D411yescharacteristic
201C2×C8×C79C2×D411yescharacteristic
202C4⋊C4×C79—21no—
203C4⋊C4×C79—21no—
204C4⋊C4×C79C22×C411yescharacteristic
205C4⋊C4×C79C22×C411yescharacteristic
206C42×C79—21no—
207C42×C79—21no—
208C42×C79C2411yescharacteristic
209C22×D79—21no—
210C22×D79C4×D411yescharacteristic
211Q8×C79—21no—
212Q8×C79—21no—
213Q8×C79—21no—
214Q8×C79#13112yes—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79C23×C411yescharacteristic
220C2×C4×C79C22×D411yescharacteristic
221C2×C4×C79C2×#4111yescharacteristic
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79C4×D411yescharacteristic
226C2×Dic79C4×D411yescharacteristic
227C2×Dic79C4×D411yescharacteristic
228D316—24no—
229D316—21no—
230D316#13112yes—
231C4×D79—24no—
232C4×D79—21no—
233C4×D79—21no—
234C4×D79—21no—
235C4×D79—21no—
236C4×D79—22no—
237C4×D79—41no—
238C4×D79—41no—
239C4×D79—42no—
240Dic158—21no—
241Dic158—21no—
242Dic158—42no—
243C8×C79—21no—
244C8×C79—21no—
245C8×C79—21no—
246C8×C79—21no—
247C8×C79—41no—
248C8×C79—41no—
249C8×C79—41no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C79⋊C8—41no—
256C79⋊C8#13712yes—
257C22×C79C2×C4×D411yescharacteristic
258C2×D79—22no—
259C2×D79—21no—
260C2×D79—21no—
261C2×D79—21no—
262C2×D79—41no—
263C2×D79—41no—
264C4×C79—22no—
265C4×C79—21no—
266C4×C79—41no—
267C4×C79—41no—
268C4×C79—41no—
269C4×C79C2×#13111yescharacteristic
270C4×C79C2×#13111yescharacteristic
271C4×C79C2×#13711yescharacteristic
272C4×C79#57511yescharacteristic
273Dic79—21no—
274Dic79—21no—
275Dic79—41no—
276Dic79—41no—
277Dic79—41no—
278Dic79#44412yes—
279#17916—791nomaximal, Sylow
280C2×C79#292911yescharacteristic
281C2×C79C4×#13711yescharacteristic
282C2×C79C2×#44411yescharacteristic
283D79—41no—
284D79—42no—
285#3107—791no—
286C8×#41—791no—
287#2960—791no—
288#1584—791no—
289#1584—791no—
290#1582—791no—
291#1582—791no—
292#1579—791no—
293#1579D7911yescharacteristic, complemented
294SD16⋊C8—791no—
295SD16⋊C8—791no—
296#1574—791no—
297#1489—791no—
298#1485—791no—
299#1451—791no—
300C79#1791611yescharacteristic, complemented, Sylow
301#533—791no—
302C4×#41—791no—
303#485—791no—
304#485—791no—
305#475—791no—
306#474—791no—
307#462—791no—
308#461—791no—
309#452—791no—
310#452—791no—
311#446—791no—
312#446—791no—
313#446—1581no—
314C8×Q8—791no—
315C8×Q8—791no—
316C8×Q8—1581no—
317C8×Q8C2×D7911yescharacteristic
318C8×Q8C2×D7911yescharacteristic
319C4×Q16—791no—
320C4×Q16C2×D7911yescharacteristic
321C4×SD16—791no—
322C4×SD16—791no—
323C8×D4—791no—
324C8×D4—791no—
325C8×D4—1581no—
326C8×D4—1581no—
327#431—791no—
328#431—791no—
329#431—1581no—
330#424—1581no—
331#423—791no—
332#405—1581no—
333C4×M16—791no—
334C2×C4×C8—791no—
335C2×C4×C8—1581no—
336#335—791no—
337#335—791no—
338#334—791no—
339#334C2×D7911yescharacteristic
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8—791no—
343Q8⋊C8—791no—
344Q8⋊C8C2×D7911yescharacteristic
345Q8⋊C8C2×D7911yescharacteristic
346D4⋊C8—791no—
347D4⋊C8—791no—
348#322—791no—
349#322—791no—
350#322—791no—
351#322C2×D7911yescharacteristic
352C2×#41—791no—
353C2×Q16—791no—
354C2×Q16C4×D7911yescharacteristic
355C2×SD16—791no—
356C2×SD16—791no—
357C2×M16—1581no—
358C2×M16—1581no—
359C22×C8—791no—
360C22×C8—791no—
361C22×C8—1581no—
362C22×C8—1581no—
363C4⋊Q8—791no—
364C4⋊Q8—791no—
365#125—1581no—
366C22⋊Q8—791no—
367C22⋊Q8—791no—
368C22⋊Q8—1581no—
369C4×Q8—791no—
370C4×Q8—791no—
371C4×Q8—1581no—
372C4×Q8C22×D7911yescharacteristic
373C4×Q8C22×D7911yescharacteristic
374C4×D4—791no—
375C4×D4—791no—
376C4×D4—1581no—
377C4×D4—1581no—
378#117—791no—
379#117—1581no—
380#117—1581no—
381C2×C42—791no—
382C2×C42—1581no—
383#107—791no—
384#107C4×D7911yescharacteristic
385#106—791no—
386#106—791no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—791no—
390C4⋊C8—791no—
391C4⋊C8—791no—
392C4⋊C8—791no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8—1581no—
396C4⋊C8—1581no—
397C4⋊C8—1581no—
398C4⋊C8C22×D7911yescharacteristic
399C4⋊C8C22×D7911yescharacteristic
400#103—791no—
401#103—791no—
402#103—791no—
403#103—791no—
404#103C4×D7911yescharacteristic
405#103C4×D7911yescharacteristic
406#102—791no—
407#102—791no—
408C22⋊C8—791no—
409C22⋊C8—791no—
410C22⋊C8—1581no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—1581no—
414#97—1581no—
415#97—1581no—
416C4×C8—21no—
417C4×C8—21no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—791no—
421C4×C8—791no—
422C4×C8—791no—
423C4×C8—1581no—
424C4×C8—1581no—
425C4×C8—1581no—
426C4×C8C22×D7911yescharacteristic
427C4×C8C22×D7911yescharacteristic
428C4×C8C22×D7911yescharacteristic
429#41—1582no—
430#41—1582no—
431C2×Q8—791no—
432C2×Q8—1581no—
433C2×Q8C2×C4×D7911yescharacteristic
434C2×Q8C2×C4×D7911yescharacteristic
435C2×D4—791no—
436C2×D4—1581no—
437C22×C4—791no—
438C22×C4—791no—
439C22×C4—791no—
440C22×C4—1581no—
441C22×C4—1581no—
442C22×C4—1581no—
443Q16—22no—
444Q16—1582no—
445SD16—1582no—
446SD16—1582no—
447M16—3162no—
448M16—3162no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—3162no—
453C2×C8—3161no—
454C2×C8—3162no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—791no—
459C2×C8—1582no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1581no—
463C2×C8C2×C4×D7911yescharacteristic
464C2×C8C2×C4×D7911yescharacteristic
465C2×C8D4×D7911yescharacteristic
466C2×C8D4×D7911yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—791no—
470C4⋊C4—791no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4C2×C4×D7911yescharacteristic
479C4⋊C4C2×C4×D7911yescharacteristic
480C22⋊C4—1581no—
481C22⋊C4—1581no—
482C22⋊C4—1581no—
483C22⋊C4—1581no—
484C22⋊C4—1581no—
485C42—21no—
486C42—21no—
487C42—791no—
488C42—791no—
489C42—791no—
490C42—1581no—
491C42—1581no—
492C42—1581no—
493C42C23×D7911yescharacteristic
494C23—791no—
495C23—1581no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—3162no—
500Q8—1581no—
501Q8—1581no—
502Q8#1457916712yes—
503D4—792no—
504D4—1584no—
505D4—1581no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—3162no—
511C2×C4—3161no—
512C2×C4—3161no—
513C2×C4—791no—
514C2×C4—791no—
515C2×C4—791no—
516C2×C4—1582no—
517C2×C4—1584no—
518C2×C4—1581no—
519C2×C4—1581no—
520C2×C4—1581no—
521C2×C4—1581no—
522C2×C4—1581no—
523C2×C4—1581no—
524C2×C4—1581no—
525C2×C4C22×C4×D7911yescharacteristic
526C2×C4C2×D4×D7911yescharacteristic
527C2×C4#41×D7911yescharacteristic
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—3161no—
536C8—3161no—
537C8—3161no—
538C8—792no—
539C8—1581no—
540C8—1581no—
541C8—1581no—
542C22—3161no—
543C22—3161no—
544C22—1581no—
545C22—1581no—
546C22—1582no—
547C22—1581no—
548C22C4×D4×D7911yescharacteristic
549C4—21no—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—3161no—
555C4—3161no—
556C4—3161no—
557C4—792no—
558C4—1581no—
559C4—1581no—
560C4#131×D7911yescharacteristic
561C4C2×#1457916711yescharacteristic
562C4#137×D7911yescharacteristic
563C4#1909936211yescharacteristic
564C2—3162no—
565C2—3161no—
566C2#2377208911yescharacteristic
567C2#2377221611yescharacteristic
568C2#444×D7911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1579C2
socleC22×C79C2×C4×D4
prime radicalC2×#11929625C2×C4
cosocleC42×C79C24
hypercenter#1579D79
hypocenterC79#17916
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1579

Complements

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

NH ≅ G/Ntype
3. #23772329C2semidirect
4. #23762251C2semidirect
5. #23762247C2semidirect
6. #23762239C2semidirect
7. #23762235C2semidirect
8. #23762217C2semidirect
9. #23762214C2semidirect
10. #23762210C2semidirect
11. #23762187C2semidirect
13. #23761819C2semidirect
14. #23761688C2semidirect
15. C79×#1579C2semidirect
293. #1579D79semidirect
300. C79#17916semidirect

Maximal subgroups

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

subgroupindex
#237734692
C8×#119296252
#237723292
#237622512
#237622472
#237622392
#237622352
#237622172
#237622142
#237622102
#237621872
#237618382
#237618192
#237616882
C79×#15792
#1791679