Finite Group Databasebeta

← #34259685

Subgroups — #34259685

Order 20224.

598 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#23773422C211yesmaximal, characteristic
2#23772563C211yesmaximal, characteristic, complemented
3#23772528C211yesmaximal, characteristic, complemented
4#23762282C211yesmaximal, characteristic, complemented
5#23762279C211yesmaximal, characteristic, complemented
6#23762277C211yesmaximal, characteristic, complemented
7#23762248C211yesmaximal, characteristic, complemented
8#23762227C211yesmaximal, characteristic, complemented
9#23762214C211yesmaximal, characteristic, complemented
10#23762204C211yesmaximal, characteristic, complemented
11#23762173C211yesmaximal, characteristic, complemented
12#23761864C211yesmaximal, characteristic, complemented
13#23761784C211yesmaximal, characteristic, complemented
14#23761701C211yesmaximal, characteristic, complemented
15C79×#1592C211yesmaximal, characteristic, complemented
16#19099100C2211yescharacteristic
17C4×#11929623C2211yescharacteristic
18#19098902C411yescharacteristic
19#19098889C411yescharacteristic
20#19098873C411yescharacteristic
21#19098865C411yescharacteristic
22#19098788C411yescharacteristic
23#19098780C411yescharacteristic
24#19098722C411yescharacteristic
25#19098709C411yescharacteristic
26Q8×C79⋊C8C2211yescharacteristic
27C8×Dic158C2211yescharacteristic
28#19098687C2211yescharacteristic
29C4×#11929614C2211yescharacteristic
30C4×#11929613C2211yescharacteristic
31C4×#11929609C2211yescharacteristic
32#19098641C2211yescharacteristic
33#19098640C2211yescharacteristic
34#19098639—21no—
35#19098638—21no—
36#19098636—21no—
37C8×D316C2211yescharacteristic
38#19098628—21no—
39#19098625—21no—
40#19098614C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098613—21no—
44#19098588—21no—
45#19098586—21no—
46#19098585C2211yescharacteristic
47C4⋊C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098315C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62#436×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929623C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929609C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×#11929607—21no—
73C2×#11929607—21no—
74C2×C8×D79—21no—
75C2×C8×D79—21no—
76#14579156—21no—
77#14579152C2×C411yescharacteristic
78#14579150C2×C411yescharacteristic
79#14579146—21no—
80#14579137C2×C411yescharacteristic
81#14579136C2×C411yescharacteristic
82C4×Dic158C2311yescharacteristic
83#14579126C2×C411yescharacteristic
84C4×D316—21no—
85C4×D316C2311yescharacteristic
86D4×Dic79C2×C411yescharacteristic
87C4×C22⋊D79—21no—
88(C22⋊D79)⋊C4—21no—
89#14579119C2×C411yescharacteristic
90#14579118—21no—
91#14579117—21no—
92C2×C4×Dic79—21no—
93C42×D79C2311yescharacteristic
94#14579094C2×C411yescharacteristic
95#14579094C2×C411yescharacteristic
96#14579093C2×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⋊C8C2×C411yescharacteristic
106Dic79⋊C8C2×C411yescharacteristic
107Dic79⋊C8D411yescharacteristic
108Dic79⋊C8D411yescharacteristic
109#14579086C2×C411yescharacteristic
110#14579085C2×C411yescharacteristic
111#14579084C2×C411yescharacteristic
112#14579083C2×C411yescharacteristic
113#14579082C2×C411yescharacteristic
114#14579081C2×C411yescharacteristic
115C22⋊(C79⋊C8)—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8C2×C411yescharacteristic
120(C2×D79)⋊C8C2×C411yescharacteristic
121(C2×D79)⋊C8D411yescharacteristic
122(C2×D79)⋊C8D411yescharacteristic
123C4×C79⋊C8—21no—
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79—21no—
129C2×SD16×C79C2×C411yescharacteristic
130C2×M16×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—
140#97×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929623—22no—
145#11929623—41no—
146C2×Dic158C22×C411yescharacteristic
147C2×D316—21no—
148C2×D316C22×C411yescharacteristic
149C2×C22⋊D79—21no—
150C22×Dic79—21no—
151C22×Dic79—21no—
152C2×C4×D79—21no—
153C2×C4×D79C22×C411yescharacteristic
154C2×C4×D79C2×D411yescharacteristic
155C2×C4×D79#4111yescharacteristic
156#11929614—22no—
157#11929613—22no—
158#11929609—41no—
159#11929608—81no—
160#11929607—81no—
161#11929607—81no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8C22×C411yescharacteristic
166C2×C79⋊C8C22×C411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C2×C79⋊C8C2×D411yescharacteristic
169C8×D79—81no—
170C8×D79—81no—
171C4⋊Dic79—21no—
172C4⋊Dic79C22×C411yescharacteristic
173C4⋊Dic79C22×C411yescharacteristic
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178C22⋊Dic79—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—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79C22×C411yescharacteristic
189C4×Dic79C2×D411yescharacteristic
190C4×Dic79#4111yescharacteristic
191C2×Q8×C79C22×C411yescharacteristic
192C2×D4×C79C22×C411yescharacteristic
193C22×C4×C79—21no—
194SD16×C79—41no—
195M16×C79—81no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C2×D411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C4⋊C4×C79—21no—
204C4⋊C4×C79C22×C411yescharacteristic
205C4⋊C4×C79C22×C411yescharacteristic
206C22⋊C4×C79—21no—
207C42×C79—21no—
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C22×D79—21no—
211C22×D79C4×D411yescharacteristic
212Q8×C79—21no—
213Q8×C79—21no—
214D4×C79—21no—
215D4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79C23×C411yescharacteristic
222C2×C4×C79C22×D411yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C22⋊D79—42no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—42no—
229C2×Dic79—42no—
230C2×Dic79C4×D411yescharacteristic
231C2×Dic79C4×D411yescharacteristic
232C2×Dic79C4×D411yescharacteristic
233D316—21no—
234D316—21no—
235D316—42no—
236C4×D79—21no—
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—21no—
240C4×D79—41no—
241C4×D79—41no—
242C4×D79—41no—
243Dic158—21no—
244Dic158#13112yes—
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×C79—41no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79C2×C4×D411yescharacteristic
263C2×D79—21no—
264C2×D79—21no—
265C2×D79—21no—
266C2×D79—41no—
267C2×D79—41no—
268C2×D79—41no—
269C4×C79—21no—
270C4×C79—41no—
271C4×C79—41no—
272C4×C79—41no—
273C4×C79C2×#13111yescharacteristic
274C4×C79#56811yescharacteristic
275C4×C79#57611yescharacteristic
276C4×C79#57811yescharacteristic
277Dic79—22no—
278Dic79—21no—
279Dic79—21no—
280Dic79—21no—
281Dic79—41no—
282Dic79—41no—
283#18033—791nomaximal, Sylow
284C2×C79—81no—
285C2×C79#293311yescharacteristic
286C2×C79#295111yescharacteristic
287C2×C79#295511yescharacteristic
288D79—41no—
289D79—81no—
290#3102—791no—
291#2994—791no—
292#2991—791no—
293#1592D7911yescharacteristic, complemented
294#1589—791no—
295#1588—791no—
296#1588—791no—
297#1584—791no—
298#1581—791no—
299SD16⋊C8—791no—
300C8×Q16—791no—
301#1571—791no—
302#1495—791no—
303#1477—791no—
304#1454—791no—
305C79#1803311yescharacteristic, 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×Q8—791no—
320C4×Q16—791no—
321C4×SD16—791no—
322C4×SD16C2×D7911yescharacteristic
323C4×D8—791no—
324#436—791no—
325#436—791no—
326#436—1581no—
327#436—1581no—
328#436C2×D7911yescharacteristic
329#435—1581no—
330C8×D4—791no—
331C8×D4—1581no—
332#433—1581no—
333#431—791no—
334#431—791no—
335#431—791no—
336#431—1581no—
337#424—1581no—
338#423—791no—
339#423—1581no—
340C2×C4⋊C8—791no—
341#335—791no—
342#335—791no—
343#335—791no—
344#335C2×D7911yescharacteristic
345Q8⋊C8—791no—
346Q8⋊C8—791no—
347Q8⋊C8—791no—
348Q8⋊C8C2×D7911yescharacteristic
349D4⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8—791no—
352D4⋊C8C2×D7911yescharacteristic
353#322—791no—
354#322—791no—
355#322C2×D7911yescharacteristic
356C82—791no—
357C2×#41—791no—
358C2×Q16—791no—
359C2×SD16—791no—
360C2×SD16C4×D7911yescharacteristic
361C2×D8—791no—
362C2×M16—21no—
363C2×M16—1581no—
364C2×M16—1581no—
365C2×M16—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×Q8—791no—
376C4×Q8C22×D7911yescharacteristic
377C4×D4—791no—
378C4×D4—791no—
379C4×D4—1581no—
380C4×D4—1581no—
381C4×D4—1581no—
382C4×D4C22×D7911yescharacteristic
383#117—791no—
384#117—1581no—
385#117—1581no—
386C2×C42—791no—
387C2×C42—1581no—
388#107—791no—
389#107—791no—
390#106—791no—
391#106C4×D7911yescharacteristic
392C4⋊C8—21no—
393C4⋊C8—791no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—791no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8—1581no—
402C4⋊C8—1581no—
403C4⋊C8—1581no—
404C4⋊C8—1581no—
405C4⋊C8C22×D7911yescharacteristic
406C4⋊C8C22×D7911yescharacteristic
407#103—791no—
408#103—791no—
409#103—791no—
410#103C4×D7911yescharacteristic
411#102—791no—
412#102—791no—
413#102—791no—
414#102C4×D7911yescharacteristic
415C22⋊C8—21no—
416C22⋊C8—791no—
417C22⋊C8—791no—
418C22⋊C8—791no—
419C22⋊C8—791no—
420C22⋊C8—1581no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424#97—21no—
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—3161no—
435#41—1582no—
436C2×Q8—791no—
437C2×Q8C2×C4×D7911yescharacteristic
438C2×D4—791no—
439C2×D4—1581no—
440C2×D4—1581no—
441C2×D4C2×C4×D7911yescharacteristic
442C22×C4—21no—
443C22×C4—791no—
444C22×C4—791no—
445C22×C4—791no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4—1581no—
449Q16—1582no—
450SD16—41no—
451SD16—1582no—
452D8—3161no—
453M16—81no—
454M16—6321no—
455M16—6321no—
456M16—6321no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—791no—
461C2×C8—791no—
462C2×C8—791no—
463C2×C8—791no—
464C2×C8—6321no—
465C2×C8—6321no—
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—791no—
475C4⋊C4—791no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4C2×C4×D7911yescharacteristic
482C4⋊C4C2×C4×D7911yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—1581no—
485C22⋊C4—1581no—
486C22⋊C4—1581no—
487C22⋊C4—1581no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C42—21no—
492C42—791no—
493C42—791no—
494C42—791no—
495C42—1581no—
496C42—1581no—
497C42—1581no—
498C42C23×D7911yescharacteristic
499C23—21no—
500C23—791no—
501C23—1581no—
502Q8—21no—
503Q8—21no—
504Q8—792no—
505Q8—1581no—
506D4—21no—
507D4—21no—
508D4—3162no—
509D4—3162no—
510D4—1581no—
511D4—1581no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—3161no—
518C2×C4—3162no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3162no—
522C2×C4—791no—
523C2×C4—791no—
524C2×C4—791no—
525C2×C4—1581no—
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—41no—
550C22—41no—
551C22—41no—
552C22—3161no—
553C22—3161no—
554C22—3161no—
555C22—1581no—
556C22—1581no—
557C22—1581no—
558C22C4×D4×D7911yescharacteristic
559C4—21no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—3161no—
564C4—3161no—
565C4—1581no—
566C4—1581no—
567C4—1581no—
568C4—1582no—
569C4#131×D7911yescharacteristic
570C4#1909932911yescharacteristic
571C4#1909936711yescharacteristic
572C4#1909937911yescharacteristic
573C2—81no—
574C2—3161no—
575C2—6321no—
576C2#2377212911yescharacteristic
577C2#2377227211yescharacteristic
578C2#2377230011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1592C2
socleC22×C79C2×C4×D4
prime radical#19098709C4
cosocleC42×C79C24
hypercenter#1592D79
hypocenterC79#18033
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1592

Complements

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

NH ≅ G/Ntype
2. #23772563C2semidirect
3. #23772528C2semidirect
4. #23762282C2semidirect
5. #23762279C2semidirect
6. #23762277C2semidirect
7. #23762248C2semidirect
8. #23762227C2semidirect
9. #23762214C2semidirect
10. #23762204C2semidirect
11. #23762173C2semidirect
12. #23761864C2semidirect
13. #23761784C2semidirect
14. #23761701C2semidirect
15. C79×#1592C2semidirect
293. #1592D79semidirect
305. C79#18033semidirect

Maximal subgroups

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

subgroupindex
#237734222
#237725632
#237725282
#237622822
#237622792
#237622772
#237622482
#237622272
#237622142
#237622042
#237621732
#237618642
#237617842
#237617012
C79×#15922
#1803379