Finite Group Databasebeta

← #34258911

Subgroups — #34258911

Order 20224.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774002C211yesmaximal, characteristic
2#23772550C211yesmaximal, characteristic
3#23772491C211yesmaximal, characteristic, complemented
4#23762306C211yesmaximal, characteristic, complemented
5#23762299C211yesmaximal, characteristic, complemented
6#23762296C211yesmaximal, characteristic, complemented
7#23762291C211yesmaximal, characteristic, complemented
8#23762220C211yesmaximal, characteristic, complemented
9C8×#11929614C211yesmaximal, characteristic, complemented
10C8×#11929612C211yesmaximal, characteristic, complemented
11#23762166C211yesmaximal, characteristic, complemented
12#23761857C211yesmaximal, characteristic
13#23761853C211yesmaximal, characteristic, complemented
14#23761679C211yesmaximal, characteristic, complemented
15C79×SD16⋊C8C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17#19099095C2211yescharacteristic
18#19098970C411yescharacteristic
19#19098954C411yescharacteristic
20#19098930C411yescharacteristic
21#19098921C411yescharacteristic
22#19098822C411yescharacteristic
23#19098820C411yescharacteristic
24#19098808C411yescharacteristic
25#19098806C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098693—21no—
28#19098692—21no—
29#19098690C2211yescharacteristic
30#19098690C2211yescharacteristic
31C8×Dic158C2211yescharacteristic
32C8×Dic158C2211yescharacteristic
33C4×Dic316C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929612C2211yescharacteristic
36#19098641C2211yescharacteristic
37#19098638—21no—
38#19098634—21no—
39#19098623—21no—
40#19098622—21no—
41#19098621—21no—
42#19098619C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098589—21no—
45#19098585C2211yescharacteristic
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098314C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098293C2211yescharacteristic
56D4⋊(C79⋊C8)C2211yescharacteristic
57#19098285C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×Dic316C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929608—21no—
71C2×#11929607—21no—
72C2×#11929607C2×C411yescharacteristic
73C2×#11929607C2×C411yescharacteristic
74C2×C8×D79—21no—
75#14579160C2×C411yescharacteristic
76#14579159C2×C411yescharacteristic
77#14579154—21no—
78#14579152C2×C411yescharacteristic
79#14579151—21no—
80#14579148C2×C411yescharacteristic
81#14579143—21no—
82#14579141C2×C411yescharacteristic
83#14579141C2×C411yescharacteristic
84#14579139—21no—
85#14579137C2×C411yescharacteristic
86#14579132C2×C411yescharacteristic
87C4×Dic158—21no—
88C4×Dic158C2311yescharacteristic
89C4×Dic158C2311yescharacteristic
90C4×C22⋊D79—21no—
91#14579120—21no—
92#14579118—21no—
93C42×D79C2311yescharacteristic
94#14579095C2×C411yescharacteristic
95#14579094C2×C411yescharacteristic
96#14579092C2×C411yescharacteristic
97C4⋊(C79⋊C8)—21no—
98C4⋊(C79⋊C8)C2311yescharacteristic
99C4⋊(C79⋊C8)C2311yescharacteristic
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8D411yescharacteristic
106Dic79⋊C8D411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579086C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579084C2×C411yescharacteristic
112#14579082C2×C411yescharacteristic
113C22⋊(C79⋊C8)—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8D411yescharacteristic
116(C2×D79)⋊C8D411yescharacteristic
117#14579072—21no—
118#14579071—21no—
119#14579071—21no—
120#14579071—21no—
121#14579071C2×C411yescharacteristic
122#14579071C2×C411yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C8×Dic79—21no—
127C8×Dic79—21no—
128C8×Dic79—21no—
129C2×SD16×C79C2×C411yescharacteristic
130C22×C8×C79—21no—
131C4×Q8×C79C2311yescharacteristic
132C4×D4×C79C2311yescharacteristic
133#106×C79C2×C411yescharacteristic
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#102×C79C2×C411yescharacteristic
139C22⋊C8×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×Dic158C22×C411yescharacteristic
147C2×C22⋊D79—21no—
148C2×C4×D79C22×C411yescharacteristic
149C2×C4×D79C2×D411yescharacteristic
150C2×C4×D79#4111yescharacteristic
151Dic316—22no—
152#11929614—22no—
153#11929612—22no—
154#11929608—81no—
155#11929607—22no—
156#11929607—22no—
157#11929607—81no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8C22×C411yescharacteristic
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C2×D411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C8×D79—42no—
166C4⋊Dic79—21no—
167C4⋊Dic79—21no—
168C4⋊Dic79—21no—
169C4⋊Dic79C22×C411yescharacteristic
170C4⋊Dic79C22×C411yescharacteristic
171Dic79⋊C4—21no—
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178C22⋊Dic79—21no—
179C22⋊Dic79—21no—
180C22⋊Dic79—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184C4×Dic79—21no—
185C4×Dic79—21no—
186C4×Dic79C22×C411yescharacteristic
187C4×Dic79C2×D411yescharacteristic
188C4×Dic79#4111yescharacteristic
189C2×Q8×C79C22×C411yescharacteristic
190C2×D4×C79C22×C411yescharacteristic
191C22×C4×C79—21no—
192SD16×C79—22no—
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—81no—
197C2×C8×C79C22×C411yescharacteristic
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C2×D411yescharacteristic
200C2×C8×C79C2×D411yescharacteristic
201C4⋊C4×C79—21no—
202C4⋊C4×C79C22×C411yescharacteristic
203C4⋊C4×C79C22×C411yescharacteristic
204C22⋊C4×C79—21no—
205C42×C79—21no—
206C42×C79C2411yescharacteristic
207C23×C79—21no—
208C22×D79C4×D411yescharacteristic
209Q8×C79—21no—
210Q8×C79—21no—
211D4×C79—21no—
212D4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79C23×C411yescharacteristic
219C2×C4×C79C22×D411yescharacteristic
220C2×C4×C79C2×#4111yescharacteristic
221C22⋊D79—81no—
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79C4×D411yescharacteristic
227C2×Dic79C4×D411yescharacteristic
228C2×Dic79C4×D411yescharacteristic
229C4×D79—21no—
230C4×D79—21no—
231C4×D79—21no—
232C4×D79—21no—
233C4×D79—41no—
234Dic158—21no—
235Dic158—21no—
236Dic158—21no—
237Dic158—21no—
238Dic158—81no—
239C8×C79—21no—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—41no—
243C8×C79—41no—
244C8×C79—41no—
245C8×C79#13112yes—
246C79⋊C8—22no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—41no—
251C79⋊C8—41no—
252C79⋊C8#13112yes—
253C22×C79—41no—
254C22×C79—41no—
255C22×C79—41no—
256C22×C79C2×C4×D411yescharacteristic
257C2×D79—21no—
258C2×D79—21no—
259C2×D79—21no—
260C4×C79—21no—
261C4×C79—41no—
262C4×C79—41no—
263C4×C79—41no—
264C4×C79C2×#13111yescharacteristic
265C4×C79C2×#13111yescharacteristic
266C4×C79#57711yescharacteristic
267C4×C79#57811yescharacteristic
268Dic79—21no—
269Dic79—21no—
270Dic79—21no—
271Dic79—41no—
272Dic79—41no—
273Dic79—41no—
274Dic79—41no—
275#17969—791nomaximal, Sylow
276C2×C79—81no—
277C2×C79#293111yescharacteristic
278C2×C79#294711yescharacteristic
279C2×C79#295611yescharacteristic
280D79—41no—
281#3172—791no—
282#2993—791no—
283#2985—791no—
284#1593—791no—
285#1592—791no—
286#1591—791no—
287#1590—791no—
288#1579—791no—
289SD16⋊C8D7911yescharacteristic, complemented
290C8×Q16—791no—
291C8×SD16—791no—
292#1569—791no—
293#1494—791no—
294#1493—791no—
295C8×M16—791no—
296C79#1796911yescharacteristic, complemented, Sylow
297#533—791no—
298#529—791no—
299#501—791no—
300#497—791no—
301#491—791no—
302#489—791no—
303#470—791no—
304#470—791no—
305#468—791no—
306#468—791no—
307#446—791no—
308#446—791no—
309#446—791no—
310#446—1581no—
311#446—1581no—
312C8×Q8—791no—
313C8×Q8—791no—
314C8×Q8C2×D7911yescharacteristic
315C4×Q16—791no—
316C4×Q16—791no—
317C4×SD16—791no—
318C4×SD16C2×D7911yescharacteristic
319#436—791no—
320#436—1581no—
321#435—1581no—
322C8×D4C2×D7911yescharacteristic
323#433—1581no—
324#433—1581no—
325#433—1581no—
326#432—791no—
327#431—791no—
328#424—1581no—
329#423—791no—
330C4×M16—791no—
331C2×C4×C8—791no—
332#335—791no—
333#335C2×D7911yescharacteristic
334#334—791no—
335#334—791no—
336Q8⋊C8—791no—
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8C2×D7911yescharacteristic
342D4⋊C8—791no—
343D4⋊C8C2×D7911yescharacteristic
344#322—791no—
345#322C2×D7911yescharacteristic
346C82—791no—
347C82—791no—
348C2×Q16—791no—
349C2×Q16—791no—
350C2×SD16—791no—
351C2×SD16C4×D7911yescharacteristic
352C2×M16—791no—
353C2×M16—791no—
354C2×M16—1581no—
355C2×M16—1581no—
356C22×C8—21no—
357C22×C8—1581no—
358C4⋊Q8—791no—
359C4⋊Q8—791no—
360#126—1581no—
361#125—791no—
362#125—1581no—
363#124—791no—
364#123—1581no—
365C22⋊Q8—791no—
366C22⋊Q8—791no—
367C22⋊Q8—791no—
368C22⋊Q8—1581no—
369#121—791no—
370C4×Q8—791no—
371C4×Q8—791no—
372C4×Q8—1581no—
373C4×Q8C22×D7911yescharacteristic
374C4×D4—1581no—
375C4×D4C22×D7911yescharacteristic
376#117—1581no—
377#117—1581no—
378C2×C42—791no—
379#107—791no—
380#107—791no—
381#106—791no—
382#106C4×D7911yescharacteristic
383C4⋊C8—21no—
384C4⋊C8—791no—
385C4⋊C8—791no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—1581no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8C22×D7911yescharacteristic
395C4⋊C8C22×D7911yescharacteristic
396#103—791no—
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103C4×D7911yescharacteristic
402#102—791no—
403#102C4×D7911yescharacteristic
404C22⋊C8—21no—
405C22⋊C8—791no—
406C22⋊C8—791no—
407C22⋊C8—1581no—
408C22⋊C8—1581no—
409#97—791no—
410#97—791no—
411#97—1581no—
412#97—1581no—
413#97—1581no—
414#97—1581no—
415C4×C8—21no—
416C4×C8—791no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—1581no—
420C4×C8—1581no—
421C4×C8—1581no—
422C4×C8C22×D7911yescharacteristic
423C4×C8C22×D7911yescharacteristic
424C4×C8C22×D7911yescharacteristic
425C2×Q8—791no—
426C2×Q8—791no—
427C2×Q8—1581no—
428C2×Q8C2×C4×D7911yescharacteristic
429C2×D4—1581no—
430C2×D4C2×C4×D7911yescharacteristic
431C22×C4—21no—
432C22×C4—791no—
433C22×C4—791no—
434C22×C4—791no—
435Q16—1582no—
436Q16—1582no—
437SD16—22no—
438SD16—1582no—
439M16—6321no—
440M16—6321no—
441M16—1582no—
442M16—1582no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—81no—
447C2×C8—3162no—
448C2×C8—791no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—1581no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8C2×C4×D7911yescharacteristic
456C2×C8C2×C4×D7911yescharacteristic
457C2×C8D4×D7911yescharacteristic
458C2×C8D4×D7911yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—791no—
461C4⋊C4—791no—
462C4⋊C4—1581no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4C2×C4×D7911yescharacteristic
473C4⋊C4C2×C4×D7911yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—1581no—
476C22⋊C4—1581no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C42—21no—
482C42—791no—
483C42—791no—
484C42—791no—
485C42—1581no—
486C42—1581no—
487C42C23×D7911yescharacteristic
488C23—21no—
489C23—791no—
490Q8—21no—
491Q8—21no—
492Q8—6321no—
493Q8—1581no—
494Q8—1581no—
495Q8—1581no—
496Q8—1581no—
497D4—21no—
498D4—21no—
499D4—6321no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—3161no—
506C2×C4—791no—
507C2×C4—791no—
508C2×C4—791no—
509C2×C4—1581no—
510C2×C4—1581no—
511C2×C4—1581no—
512C2×C4—1581no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4—1581no—
517C2×C4C22×C4×D7911yescharacteristic
518C2×C4C2×D4×D7911yescharacteristic
519C2×C4#41×D7911yescharacteristic
520C8—21no—
521C8—21no—
522C8—21no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—3161no—
527C8—3161no—
528C8—792no—
529C8—1582no—
530C8—1581no—
531C8—1581no—
532C8—1581no—
533C8#1457916712yes—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—1581no—
538C22—1581no—
539C22—1581no—
540C22C4×D4×D7911yescharacteristic
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—3161no—
546C4—3161no—
547C4—3161no—
548C4—3161no—
549C4—1581no—
550C4—1581no—
551C4—1581no—
552C4#131×D7911yescharacteristic
553C4C2×#1457916711yescharacteristic
554C4#1909937311yescharacteristic
555C4#1909937911yescharacteristic
556C2—81no—
557C2—3161no—
558C2#2377210711yescharacteristic
559C2#2377224011yescharacteristic
560C2#2377230811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×SD16⋊C8C2
socleC22×C79C2×C4×D4
prime radical#14579132C2×C4
cosocleC42×C79C24
hypercenterSD16⋊C8D79
hypocenterC79#17969
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
3. #23772491C2semidirect
4. #23762306C2semidirect
5. #23762299C2semidirect
6. #23762296C2semidirect
7. #23762291C2semidirect
8. #23762220C2semidirect
9. C8×#11929614C2semidirect
10. C8×#11929612C2semidirect
11. #23762166C2semidirect
13. #23761853C2semidirect
14. #23761679C2semidirect
15. C79×SD16⋊C8C2semidirect
289. SD16⋊C8D79semidirect
296. C79#17969semidirect

Maximal subgroups

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

subgroupindex
#237740022
#237725502
#237724912
#237623062
#237622992
#237622962
#237622912
#237622202
C8×#119296142
C8×#119296122
#237621662
#237618572
#237618532
#237616792
C79×SD16⋊C82
#1796979