Finite Group Databasebeta

← #34258233

Subgroups — #34258233

Order 20224.

626 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773424C211yesmaximal, characteristic
2#23772374C211yesmaximal, characteristic, complemented
3C8×#11929625C211yesmaximal, characteristic, complemented
4#23762278C211yesmaximal, characteristic, complemented
5#23762266C211yesmaximal, characteristic, complemented
6#23762255C211yesmaximal, characteristic, complemented
7#23762223C211yesmaximal, characteristic, complemented
8#23762219C211yesmaximal, characteristic, complemented
9#23762215C211yesmaximal, characteristic, complemented
10#23762215C211yesmaximal, characteristic, complemented
11#23762177C211yesmaximal, characteristic, complemented
12#23761833C211yesmaximal, characteristic, complemented
13#23761807C211yesmaximal, characteristic, complemented
14#23761691C211yesmaximal, characteristic, complemented
15C79×#1587C211yesmaximal, characteristic, complemented
16#19099100C2211yescharacteristic
17C4×#11929625C2211yescharacteristic
18#19098903C411yescharacteristic
19#19098890C411yescharacteristic
20#19098872C411yescharacteristic
21#19098866C411yescharacteristic
22#19098787C411yescharacteristic
23#19098781C411yescharacteristic
24#19098724C411yescharacteristic
25#19098711C411yescharacteristic
26Q8×C79⋊C8C2211yescharacteristic
27Q8×C79⋊C8C2211yescharacteristic
28C4×#11929611C2211yescharacteristic
29C4×#11929611C2211yescharacteristic
30C4×D632C2211yescharacteristic
31#19098642C2211yescharacteristic
32#19098636—21no—
33C8×D316—21no—
34C8×D316C2211yescharacteristic
35#19098628—21no—
36#19098628—21no—
37#19098628C2211yescharacteristic
38#19098628C2211yescharacteristic
39#19098623—21no—
40#19098619C2211yescharacteristic
41#19098615—21no—
42#19098615C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098614C2211yescharacteristic
45#19098520—21no—
46C4×C8×D79—21no—
47C4×C8×D79C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098314C2211yescharacteristic
51Q8⋊(C79⋊C8)C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098292C2211yescharacteristic
54#19098292C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098284C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929625C2×C411yescharacteristic
68C2×#11929611C2×C411yescharacteristic
69C2×#11929611C2×C411yescharacteristic
70C2×D632C2×C411yescharacteristic
71C2×#11929607—21no—
72C2×C8×D79—21no—
73C2×C8×D79—21no—
74C2×C8×D79—21no—
75C2×C8×D79D411yescharacteristic
76C2×C8×D79D411yescharacteristic
77#14579156—21no—
78#14579152C2×C411yescharacteristic
79#14579150C2×C411yescharacteristic
80#14579146—21no—
81#14579137C2×C411yescharacteristic
82#14579136C2×C411yescharacteristic
83Q8×Dic79C2×C411yescharacteristic
84C4×D316—21no—
85C4×D316—21no—
86C4×D316C2311yescharacteristic
87C4×D316C2311yescharacteristic
88#14579123—21no—
89#14579123C2×C411yescharacteristic
90#14579119—21no—
91#14579119C2×C411yescharacteristic
92#14579118—21no—
93C42×D79—21no—
94C42×D79C2311yescharacteristic
95#14579095C2×C411yescharacteristic
96#14579092C2×C411yescharacteristic
97#14579092C2×C411yescharacteristic
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)—21no—
100C4⋊(C79⋊C8)C2311yescharacteristic
101C4⋊(C79⋊C8)C2311yescharacteristic
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
105#14579085C2×C411yescharacteristic
106#14579085C2×C411yescharacteristic
107#14579083C2×C411yescharacteristic
108#14579083C2×C411yescharacteristic
109#14579081C2×C411yescharacteristic
110#14579081C2×C411yescharacteristic
111(C2×D79)⋊C8—21no—
112(C2×D79)⋊C8—21no—
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8C2×C411yescharacteristic
119(C2×D79)⋊C8C2×C411yescharacteristic
120#14579071—21no—
121C4×C79⋊C8—21no—
122C4×C79⋊C8—21no—
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
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
139#97×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929625—22no—
145#11929625—22no—
146C2×D316—21no—
147C2×D316—21no—
148C2×D316C22×C411yescharacteristic
149C2×D316C22×C411yescharacteristic
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79—21no—
153C2×C4×D79—21no—
154C2×C4×D79C22×C411yescharacteristic
155C2×C4×D79C2×D411yescharacteristic
156C2×C4×D79#4111yescharacteristic
157#11929611—22no—
158#11929611—22no—
159D632—22no—
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—41no—
169C8×D79—41no—
170C8×D79—42no—
171C8×D79—42no—
172C8×D79—42no—
173C4⋊Dic79—21no—
174C4⋊Dic79—21no—
175C4⋊Dic79C22×C411yescharacteristic
176C4⋊Dic79C22×C411yescharacteristic
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179(C2×D79)⋊C4—21no—
180(C2×D79)⋊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—
186(C2×D79)⋊C4—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×D79—21no—
211C22×D79C4×D411yescharacteristic
212Q8×C79—21no—
213Q8×C79—21no—
214Q8×C79—21no—
215Q8×C79#13112yes—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79C23×C411yescharacteristic
221C2×C4×C79C22×D411yescharacteristic
222C2×C4×C79C2×#4111yescharacteristic
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79C4×D411yescharacteristic
226C2×Dic79C4×D411yescharacteristic
227C2×Dic79C4×D411yescharacteristic
228D316—24no—
229D316—21no—
230D316—21no—
231D316—21no—
232D316—42no—
233D316#13112yes—
234C4×D79—24no—
235C4×D79—21no—
236C4×D79—21no—
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—41no—
240C4×D79—42no—
241C4×D79—41no—
242C4×D79—41no—
243C4×D79—41no—
244C4×D79—41no—
245C8×C79—21no—
246C8×C79—21no—
247C8×C79—21no—
248C8×C79—21no—
249C8×C79—41no—
250C8×C79—41no—
251C8×C79—41no—
252C79⋊C8—21no—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C79⋊C8—21no—
256C79⋊C8—41no—
257C79⋊C8—41no—
258C79⋊C8—41no—
259C22×C79C2×C4×D411yescharacteristic
260C2×D79—22no—
261C2×D79—21no—
262C2×D79—21no—
263C2×D79—21no—
264C2×D79—41no—
265C2×D79—41no—
266C2×D79—41no—
267C2×D79—41no—
268C2×D79—41no—
269C4×C79—22no—
270C4×C79—21no—
271C4×C79—41no—
272C4×C79—41no—
273C4×C79—41no—
274C4×C79C2×#13111yescharacteristic
275C4×C79C2×#13111yescharacteristic
276C4×C79#57511yescharacteristic
277C4×C79#57511yescharacteristic
278Dic79—21no—
279Dic79—21no—
280Dic79—21no—
281Dic79—41no—
282Dic79—41no—
283#17909—791nomaximal, Sylow
284C2×C79#292911yescharacteristic
285C2×C79#294311yescharacteristic
286C2×C79#295511yescharacteristic
287D79—41no—
288D79—42no—
289D79—81no—
290#3102—791no—
291#2966—791no—
292C8×#41—791no—
293#1588—791no—
294#1587D7911yescharacteristic, complemented
295#1586—791no—
296#1585—791no—
297#1580—791no—
298#1579—791no—
299SD16⋊C8—791no—
300SD16⋊C8—791no—
301#1572—791no—
302#1488—791no—
303#1482—791no—
304#1452—791no—
305C79#1790911yescharacteristic, complemented, Sylow
306#529—791no—
307C4×#41—791no—
308#485—791no—
309#482—791no—
310#479—791no—
311#478—791no—
312#464—791no—
313#463—791no—
314#452—791no—
315#449—791no—
316#446C2×D7911yescharacteristic
317C8×Q8—791no—
318C8×Q8—791no—
319C8×Q8C2×D7911yescharacteristic
320C4×Q16C2×D7911yescharacteristic
321C4×SD16—791no—
322C4×SD16—791no—
323C4×D8—791no—
324#436—791no—
325#435—1581no—
326C8×D4—791no—
327C8×D4—791no—
328C8×D4—791no—
329C8×D4—1581no—
330C8×D4—1581no—
331C8×D4—1581no—
332#433—1581no—
333#432—791no—
334#431—791no—
335#431—791no—
336#431—791no—
337#431—1581no—
338#405—1581no—
339C2×C4×C8—791no—
340C2×C4×C8—1581no—
341#335—791no—
342#335—791no—
343#334—791no—
344#334C2×D7911yescharacteristic
345Q8⋊C8—791no—
346Q8⋊C8—791no—
347Q8⋊C8C2×D7911yescharacteristic
348Q8⋊C8C2×D7911yescharacteristic
349D4⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8—791no—
352D4⋊C8—791no—
353#322—791no—
354#322—791no—
355#322—791no—
356#322C2×D7911yescharacteristic
357C2×#41—791no—
358C2×Q16C4×D7911yescharacteristic
359C2×SD16—791no—
360C2×SD16—791no—
361C2×D8—791no—
362C2×M16—1581no—
363C22×C8—791no—
364C22×C8—791no—
365C22×C8—1581no—
366C22×C8—1581no—
367C22×C8—1581no—
368#126—1581no—
369#125—791no—
370#124—791no—
371#123—1581no—
372C22⋊Q8—791no—
373#121—791no—
374C4×Q8—791no—
375C4×Q8C22×D7911yescharacteristic
376C4×Q8C22×D7911yescharacteristic
377C4×D4—791no—
378C4×D4—791no—
379C4×D4—791no—
380C4×D4—1581no—
381C4×D4—1581no—
382C4×D4—1581no—
383#117—791no—
384#117—1581no—
385#117—1581no—
386C2×C42—791no—
387C2×C42—1581no—
388#107—791no—
389#107C4×D7911yescharacteristic
390#106—791no—
391#106—791no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8C22×D7911yescharacteristic
402C4⋊C8C22×D7911yescharacteristic
403#103—791no—
404#103—791no—
405#103C4×D7911yescharacteristic
406#103C4×D7911yescharacteristic
407#102—791no—
408#102—791no—
409#102—791no—
410#102—791no—
411C22⋊C8—791no—
412C22⋊C8—791no—
413C22⋊C8—1581no—
414C22⋊C8—1581no—
415C22⋊C8—1581no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420#97—21no—
421#97—1581no—
422C4×C8—21no—
423C4×C8—791no—
424C4×C8—791no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—791no—
428C4×C8—1581no—
429C4×C8—1581no—
430C4×C8—1581no—
431C4×C8C22×D7911yescharacteristic
432C4×C8C22×D7911yescharacteristic
433C4×C8C22×D7911yescharacteristic
434#41—1582no—
435#41—1582no—
436C2×Q8C2×C4×D7911yescharacteristic
437C2×Q8C2×C4×D7911yescharacteristic
438C2×D4—791no—
439C2×D4—791no—
440C2×D4—1581no—
441C2×D4—1581no—
442C22×C4—791no—
443C22×C4—791no—
444C22×C4—791no—
445C22×C4—1581no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4—1581no—
449Q16—22no—
450SD16—1582no—
451SD16—1582no—
452D8—1582no—
453M16—3162no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—3162no—
458C2×C8—3161no—
459C2×C8—3161no—
460C2×C8—3162no—
461C2×C8—3162no—
462C2×C8—791no—
463C2×C8—791no—
464C2×C8—791no—
465C2×C8—791no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8C2×C4×D7911yescharacteristic
470C2×C8C2×C4×D7911yescharacteristic
471C2×C8D4×D7911yescharacteristic
472C2×C8D4×D7911yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—791no—
476C4⋊C4—791no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4C2×C4×D7911yescharacteristic
482C4⋊C4C2×C4×D7911yescharacteristic
483C22⋊C4—1581no—
484C22⋊C4—1581no—
485C22⋊C4—1581no—
486C22⋊C4—1581no—
487C22⋊C4—1581no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C42—21no—
492C42—21no—
493C42—791no—
494C42—791no—
495C42—791no—
496C42—1581no—
497C42—1581no—
498C42C23×D7911yescharacteristic
499C23—791no—
500C23—1581no—
501C23—1581no—
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8#1457916912yes—
506D4—3162no—
507D4—792no—
508D4—1584no—
509D4—1581no—
510D4—1581no—
511D4—1581no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—3161no—
517C2×C4—3162no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—791no—
523C2×C4—791no—
524C2×C4—791no—
525C2×C4—1584no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4C22×C4×D7911yescharacteristic
533C2×C4C2×D4×D7911yescharacteristic
534C2×C4#41×D7911yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—3161no—
543C8—3161no—
544C8—3161no—
545C8—1581no—
546C8—1581no—
547C8—1581no—
548C8—1581no—
549C22—3161no—
550C22—3161no—
551C22—3161no—
552C22—3161no—
553C22—3161no—
554C22—1581no—
555C22—1581no—
556C22—1582no—
557C22—1581no—
558C22C4×D4×D7911yescharacteristic
559C4—21no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—3161no—
565C4—3161no—
566C4—1581no—
567C4—1581no—
568C4—1581no—
569C4#131×D7911yescharacteristic
570C4C2×#1457916911yescharacteristic
571C4#1909936211yescharacteristic
572C4#1909936211yescharacteristic
573C2—3162no—
574C2—3161no—
575C2—6321no—
576C2#2377209111yescharacteristic
577C2#2377219911yescharacteristic
578C2#2377229811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1587C2
socleC22×C79C2×C4×D4
prime radical#19098711C4
cosocleC42×C79C24
hypercenter#1587D79
hypocenterC79#17909
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1587

Complements

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

NH ≅ G/Ntype
2. #23772374C2semidirect
3. C8×#11929625C2semidirect
4. #23762278C2semidirect
5. #23762266C2semidirect
6. #23762255C2semidirect
7. #23762223C2semidirect
8. #23762219C2semidirect
9. #23762215C2semidirect
10. #23762215C2semidirect
11. #23762177C2semidirect
12. #23761833C2semidirect
13. #23761807C2semidirect
14. #23761691C2semidirect
15. C79×#1587C2semidirect
294. #1587D79semidirect
305. C79#17909semidirect

Maximal subgroups

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

subgroupindex
#237734242
#237723742
C8×#119296252
#237622782
#237622662
#237622552
#237622232
#237622192
#237622152
#237621772
#237618332
#237618072
#237616912
C79×#15872
#1790979