Finite Group Databasebeta

← #34259631

Subgroups — #34259631

Order 20224.

606 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773432C211yesmaximal, characteristic
2#23772490C211yesmaximal, characteristic, complemented
3#23772466C211yesmaximal, characteristic
4#23762301C211yesmaximal, characteristic, complemented
5#23762289C211yesmaximal, characteristic, complemented
6#23762277C211yesmaximal, characteristic, complemented
7#23762256C211yesmaximal, characteristic, complemented
8#23762242C211yesmaximal, characteristic, complemented
9#23762235C211yesmaximal, characteristic, complemented
10#23762211C211yesmaximal, characteristic, complemented
11#23762199C211yesmaximal, characteristic, complemented
12#23761868C211yesmaximal, characteristic
13#23761794C211yesmaximal, characteristic, complemented
14#23761700C211yesmaximal, characteristic, complemented
15C79×#1572C211yesmaximal, characteristic, complemented
16#19099113C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18#19098882C411yescharacteristic
19#19098882C411yescharacteristic
20#19098849C411yescharacteristic
21#19098849C411yescharacteristic
22#19098764C411yescharacteristic
23#19098764C411yescharacteristic
24#19098704C411yescharacteristic
25#19098704C411yescharacteristic
26#19098691—21no—
27#19098691—21no—
28#19098690C2211yescharacteristic
29#19098690C2211yescharacteristic
30#19098687C2211yescharacteristic
31#19098687C2211yescharacteristic
32C4×#11929613C2211yescharacteristic
33C4×#11929613C2211yescharacteristic
34C4×#11929613C2211yescharacteristic
35C4×#11929613C2211yescharacteristic
36#19098642—21no—
37#19098640C2211yescharacteristic
38#19098640C2211yescharacteristic
39C8×D316—21no—
40#19098628C2211yescharacteristic
41#19098628C2211yescharacteristic
42#19098617—21no—
43#19098616C2211yescharacteristic
44#19098616C2211yescharacteristic
45#19098613—21no—
46#19098528—21no—
47#19098528—21no—
48#19098293C2211yescharacteristic
49#19098293C2211yescharacteristic
50#19098293C2211yescharacteristic
51#19098293C2211yescharacteristic
52#19098290C2211yescharacteristic
53#19098290C2211yescharacteristic
54#19098290C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098285C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#432×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62#431×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64#335×C79C2211yescharacteristic
65#334×C79C2211yescharacteristic
66#334×C79C2211yescharacteristic
67C2×#11929624—21no—
68C22×Dic158C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929613C2×C411yescharacteristic
71C2×#11929613C2×C411yescharacteristic
72C2×#11929613C2×C411yescharacteristic
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×C8×D79—21no—
76C2×C8×D79—21no—
77#14579161C2×C411yescharacteristic
78#14579158C2×C411yescharacteristic
79#14579149—21no—
80#14579135—21no—
81#14579135C2×C411yescharacteristic
82#14579135C2×C411yescharacteristic
83C4×Dic158—21no—
84C4×Dic158—21no—
85C4×Dic158C2×C411yescharacteristic
86C4×Dic158C2×C411yescharacteristic
87C4×Dic158C2311yescharacteristic
88C4×Dic158C2311yescharacteristic
89C4×D316—21no—
90C4×D316C2311yescharacteristic
91C4×D316C2311yescharacteristic
92C2×C4⋊Dic79C2×C411yescharacteristic
93C2×Dic79⋊C4—21no—
94C2×C4×Dic79—21no—
95#14579093C2×C411yescharacteristic
96#14579093C2×C411yescharacteristic
97#14579093C2×C411yescharacteristic
98#14579093C2×C411yescharacteristic
99C4⋊(C79⋊C8)—21no—
100C4⋊(C79⋊C8)D411yescharacteristic
101C4⋊(C79⋊C8)D411yescharacteristic
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106#14579086C2×C411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579086C2×C411yescharacteristic
109#14579086C2×C411yescharacteristic
110#14579083C2×C411yescharacteristic
111#14579083C2×C411yescharacteristic
112#14579083C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114C22⋊(C79⋊C8)—21no—
115C22⋊(C79⋊C8)D411yescharacteristic
116C22⋊(C79⋊C8)D411yescharacteristic
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8—21no—
121#14579071—21no—
122#14579071—21no—
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79—21no—
129C2×C42×C79C2311yescharacteristic
130C4⋊C8×C79C2×C411yescharacteristic
131C4⋊C8×C79C2×C411yescharacteristic
132C4⋊C8×C79C2311yescharacteristic
133C4⋊C8×C79C2311yescharacteristic
134C4⋊C8×C79C2311yescharacteristic
135C4⋊C8×C79C2311yescharacteristic
136C22⋊C8×C79—21no—
137C22⋊C8×C79—21no—
138C22⋊C8×C79C2×C411yescharacteristic
139C22⋊C8×C79C2×C411yescharacteristic
140#97×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929624—81no—
145#11929624—81no—
146C2×Dic158—22no—
147C2×Dic158—21no—
148C2×Dic158—21no—
149C2×Dic158—41no—
150C2×Dic158C22×C411yescharacteristic
151C2×Dic158C22×C411yescharacteristic
152C2×D316—21no—
153C2×D316—21no—
154C2×D316C22×C411yescharacteristic
155C2×D316C22×C411yescharacteristic
156C2×C22⋊D79—21no—
157C2×C22⋊D79—21no—
158C22×Dic79—21no—
159C22×Dic79—21no—
160C2×C4×D79—21no—
161C2×C4×D79—21no—
162#11929613—22no—
163#11929613—22no—
164#11929613—41no—
165#11929613—41no—
166#11929607—81no—
167#11929607—81no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8C2×D411yescharacteristic
171C2×C79⋊C8C2×D411yescharacteristic
172C2×C79⋊C8C2×D411yescharacteristic
173C2×C79⋊C8C2×D411yescharacteristic
174C8×D79—81no—
175C8×D79—81no—
176C4⋊Dic79—21no—
177C4⋊Dic79—21no—
178C4⋊Dic79C22×C411yescharacteristic
179C4⋊Dic79C22×C411yescharacteristic
180C4⋊Dic79C22×C411yescharacteristic
181C4⋊Dic79C22×C411yescharacteristic
182Dic79⋊C4—21no—
183Dic79⋊C4—21no—
184Dic79⋊C4—21no—
185Dic79⋊C4—21no—
186(C2×D79)⋊C4—21no—
187(C2×D79)⋊C4—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190C4×Dic79—21no—
191C4×Dic79—21no—
192C4×Dic79—21no—
193C4×Dic79—21no—
194C22×C4×C79C22×C411yescharacteristic
195C22×C4×C79C2×D411yescharacteristic
196C22×C4×C79#4111yescharacteristic
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79—21no—
200C2×C8×C79—21no—
201C2×C8×C79C22×C411yescharacteristic
202C2×C8×C79C22×C411yescharacteristic
203C2×C8×C79C22×C411yescharacteristic
204C2×C8×C79C22×C411yescharacteristic
205C42×C79C22×C411yescharacteristic
206C42×C79C2×D411yescharacteristic
207C42×C79#4111yescharacteristic
208C42×C79C2411yescharacteristic
209C23×C79C4×D411yescharacteristic
210C22×D79—21no—
211C22×D79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—41no—
217C2×C4×C79C4×D411yescharacteristic
218C2×C4×C79C4×D411yescharacteristic
219C2×C4×C79C4×D411yescharacteristic
220C2×C4×C79C23×C411yescharacteristic
221C2×C4×C79C22×D411yescharacteristic
222C2×C4×C79C2×#4111yescharacteristic
223C22⋊D79—81no—
224C22⋊D79—81no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—42no—
232C2×Dic79—42no—
233D316—21no—
234D316—21no—
235D316—21no—
236D316—21no—
237D316—42no—
238D316—81no—
239C4×D79—41no—
240C4×D79—41no—
241C4×D79—41no—
242C4×D79—41no—
243Dic158—21no—
244Dic158—21no—
245Dic158—21no—
246Dic158—42no—
247Dic158—81no—
248Dic158#13112yes—
249C8×C79—21no—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—21no—
253C8×C79—41no—
254C8×C79—41no—
255C8×C79—41no—
256C8×C79—41no—
257C79⋊C8—21no—
258C79⋊C8—21no—
259C79⋊C8—21no—
260C79⋊C8—21no—
261C79⋊C8—41no—
262C79⋊C8—41no—
263C22×C79—21no—
264C22×C79—21no—
265C22×C79—21no—
266C22×C79C2×C4×D411yescharacteristic
267C2×D79—41no—
268C2×D79—41no—
269C2×D79—41no—
270C2×D79—41no—
271C2×D79—41no—
272C2×D79—41no—
273C4×C79—21no—
274C4×C79—21no—
275C4×C79—21no—
276C4×C79—21no—
277C4×C79C2×#13111yescharacteristic
278C4×C79#56811yescharacteristic
279C4×C79#57711yescharacteristic
280C4×C79#57711yescharacteristic
281Dic79—22no—
282Dic79—41no—
283Dic79—41no—
284Dic79—41no—
285Dic79—41no—
286Dic79—41no—
287#18030—791nomaximal, Sylow
288C2×C79—41no—
289C2×C79#293311yescharacteristic
290C2×C79#294911yescharacteristic
291C2×C79#295511yescharacteristic
292D79—81no—
293D79—81no—
294#3103—791no—
295#2984—791no—
296#2979—791no—
297#1592—791no—
298#1590—791no—
299#1588—791no—
300#1585—791no—
301#1583—791no—
302#1582—791no—
303SD16⋊C8—791no—
304C8×SD16—791no—
305#1572D7911yescharacteristic, complemented
306#1496—791no—
307#1479—791no—
308#1454—791no—
309C79#1803011yescharacteristic, complemented, Sylow
310#532—791no—
311C2×C4×Q8—791no—
312#481—791no—
313#481—791no—
314#475—791no—
315#475—791no—
316#460—791no—
317#460—791no—
318#448—791no—
319#448—791no—
320#446—791no—
321#446—791no—
322#446—1581no—
323#446—1581no—
324C8×Q8—791no—
325C8×Q8—791no—
326C4×SD16—791no—
327C4×SD16—791no—
328C4×SD16—791no—
329C4×SD16—791no—
330#436—791no—
331#436—791no—
332#436—1581no—
333C8×D4—791no—
334C8×D4—791no—
335C8×D4—1581no—
336#432—1581no—
337#432C2×D7911yescharacteristic
338#431—791no—
339#431—791no—
340#431—1581no—
341#431C2×D7911yescharacteristic
342#431C2×D7911yescharacteristic
343#408—1581no—
344#408—1581no—
345#335C2×D7911yescharacteristic
346#335C2×D7911yescharacteristic
347#334C2×D7911yescharacteristic
348#334C2×D7911yescharacteristic
349Q8⋊C8—791no—
350Q8⋊C8—791no—
351Q8⋊C8—791no—
352Q8⋊C8—791no—
353D4⋊C8—791no—
354D4⋊C8—791no—
355D4⋊C8—791no—
356D4⋊C8—791no—
357#322—791no—
358#322—791no—
359#322—791no—
360C82—791no—
361C2×#41—1581no—
362C22×Q8—791no—
363C2×SD16—791no—
364C2×SD16—791no—
365C2×SD16—791no—
366C2×SD16—791no—
367C2×M16—1581no—
368C2×M16—1581no—
369C22×C8—1581no—
370C22×C8—1581no—
371C4⋊Q8—791no—
372#127—791no—
373#124—1581no—
374#121—791no—
375#121—791no—
376#121—1581no—
377C4×Q8—791no—
378C4×Q8—791no—
379C4×Q8—791no—
380C4×Q8—791no—
381C4×Q8—1581no—
382C4×Q8—1581no—
383C4×D4—791no—
384C4×D4—791no—
385C4×D4—1581no—
386C2×C4⋊C4—791no—
387C2×C4⋊C4—1581no—
388C2×C42—1581no—
389C2×C42C22×D7911yescharacteristic
390#106—791no—
391#106—791no—
392#106—791no—
393#106—791no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—1581no—
397C4⋊C8—1581no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8C4×D7911yescharacteristic
402C4⋊C8C4×D7911yescharacteristic
403C4⋊C8C22×D7911yescharacteristic
404C4⋊C8C22×D7911yescharacteristic
405C4⋊C8C22×D7911yescharacteristic
406C4⋊C8C22×D7911yescharacteristic
407#103—791no—
408#103—791no—
409#103—791no—
410#103—791no—
411#102—791no—
412#102—791no—
413#102—791no—
414#102—791no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—791no—
418C22⋊C8—791no—
419C22⋊C8—1581no—
420C22⋊C8—1581no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424C22⋊C8C4×D7911yescharacteristic
425C22⋊C8C4×D7911yescharacteristic
426#97—21no—
427#97—1581no—
428#97—1581no—
429C4×C8—21no—
430C4×C8—791no—
431C4×C8—791no—
432C4×C8—791no—
433C4×C8—791no—
434C4×C8—1581no—
435C4×C8—1581no—
436C4×C8C22×D7911yescharacteristic
437C4×C8C22×D7911yescharacteristic
438#41—6321no—
439#41—6321no—
440C2×Q8—3161no—
441C2×Q8—791no—
442C2×Q8—791no—
443C2×Q8—1582no—
444C2×Q8—1581no—
445C2×Q8—1581no—
446C2×D4—791no—
447C2×D4—791no—
448C2×D4—1581no—
449C2×D4—1581no—
450C2×D4—1581no—
451C2×D4—1581no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4C2×C4×D7911yescharacteristic
457C22×C4D4×D7911yescharacteristic
458C22×C4#1192962511yescharacteristic
459SD16—3161no—
460SD16—3161no—
461SD16—1582no—
462SD16—1582no—
463M16—6321no—
464M16—6321no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—791no—
470C2×C8—791no—
471C2×C8—791no—
472C2×C8—791no—
473C2×C8—6321no—
474C2×C8—6321no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8C2×C4×D7911yescharacteristic
478C2×C8C2×C4×D7911yescharacteristic
479C2×C8C2×C4×D7911yescharacteristic
480C2×C8C2×C4×D7911yescharacteristic
481C4⋊C4—791no—
482C4⋊C4—791no—
483C4⋊C4—791no—
484C4⋊C4—791no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C22⋊C4—1581no—
492C22⋊C4—1581no—
493C22⋊C4—1581no—
494C22⋊C4—1581no—
495C42—1581no—
496C42—1581no—
497C42—1581no—
498C42—1581no—
499C42C2×C4×D7911yescharacteristic
500C42D4×D7911yescharacteristic
501C42#1192962511yescharacteristic
502C42C23×D7911yescharacteristic
503C23—1581no—
504C23—1581no—
505C23#1457912311yescharacteristic
506Q8—3162no—
507Q8—792no—
508Q8—6321no—
509Q8—1581no—
510Q8—1581no—
511Q8—1581no—
512D4—3162no—
513D4—6321no—
514D4—6321no—
515D4—6321no—
516D4—1581no—
517D4—1581no—
518D4—1581no—
519D4—1581no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—3162no—
526C2×C4—3162no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4#1457912311yescharacteristic
538C2×C4#1457912311yescharacteristic
539C2×C4#1457912311yescharacteristic
540C2×C4C22×C4×D7911yescharacteristic
541C2×C4C2×D4×D7911yescharacteristic
542C2×C4C2×#1192962511yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8—21no—
547C8—41no—
548C8—41no—
549C8—41no—
550C8—41no—
551C8—3161no—
552C8—3161no—
553C8—1581no—
554C8—1581no—
555C8—1581no—
556C8—1581no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—3161no—
561C22—3161no—
562C22—3161no—
563C22—3161no—
564C22—3161no—
565C22—3161no—
566C22C2×#1457912311yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—3161no—
572C4—3161no—
573C4—3161no—
574C4—3161no—
575C4—3161no—
576C4—1582no—
577C4#131×D7911yescharacteristic
578C4#1909932811yescharacteristic
579C4#1909937011yescharacteristic
580C4#1909937011yescharacteristic
581C2—41no—
582C2—6321no—
583C2—6321no—
584C2#2377212411yescharacteristic
585C2#2377225811yescharacteristic
586C2#2377230211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1572C2
socleC22×C79C2×C4×D4
prime radical#19099113C22
cosocleC42×C79C24
hypercenter#1572D79
hypocenterC79#18030
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
2. #23772490C2semidirect
4. #23762301C2semidirect
5. #23762289C2semidirect
6. #23762277C2semidirect
7. #23762256C2semidirect
8. #23762242C2semidirect
9. #23762235C2semidirect
10. #23762211C2semidirect
11. #23762199C2semidirect
13. #23761794C2semidirect
14. #23761700C2semidirect
15. C79×#1572C2semidirect
305. #1572D79semidirect
309. C79#18030semidirect

Maximal subgroups

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

subgroupindex
#237734322
#237724902
#237724662
#237623012
#237622892
#237622772
#237622562
#237622422
#237622352
#237622112
#237621992
#237618682
#237617942
#237617002
C79×#15722
#1803079