Finite Group Databasebeta

← #34259601

Subgroups — #34259601

Order 20224.

616 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773409C211yesmaximal, characteristic
2#23772501C211yesmaximal, characteristic, complemented
3#23772437C211yesmaximal, 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#23761872C211yesmaximal, characteristic, complemented
13#23761790C211yesmaximal, characteristic
14#23761700C211yesmaximal, characteristic, complemented
15C79×#1572C211yesmaximal, characteristic, complemented
16#19099120C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18#19098894C411yescharacteristic
19#19098894C411yescharacteristic
20#19098854C411yescharacteristic
21#19098854C411yescharacteristic
22#19098769C411yescharacteristic
23#19098769C411yescharacteristic
24#19098716C411yescharacteristic
25#19098716C411yescharacteristic
26#19098693—21no—
27#19098690C2211yescharacteristic, complemented
28#19098690C2211yescharacteristic, complemented
29C8×Dic158—21no—
30#19098687C2211yescharacteristic, complemented
31#19098687C2211yescharacteristic, complemented
32C4×#11929613C2211yescharacteristic
33C4×#11929613C2211yescharacteristic
34C4×#11929613C2211yescharacteristic
35C4×#11929613C2211yescharacteristic
36#19098640C2211yescharacteristic
37#19098640C2211yescharacteristic
38#19098636—21no—
39#19098636—21no—
40#19098628C2211yescharacteristic
41#19098628C2211yescharacteristic
42#19098621—21no—
43#19098621—21no—
44#19098616C2211yescharacteristic
45#19098616C2211yescharacteristic
46#19098525—21no—
47C22⋊C8×D79—21no—
48#19098293C2211yescharacteristic, complemented
49#19098293C2211yescharacteristic, complemented
50#19098293C2211yescharacteristic, complemented
51#19098293C2211yescharacteristic, complemented
52#19098290C2211yescharacteristic
53#19098290C2211yescharacteristic
54#19098290C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098285C2211yescharacteristic, complemented
57#19098285C2211yescharacteristic, complemented
58#19098284C2211yescharacteristic, complemented
59C8×C79⋊C8C2211yescharacteristic, complemented
60#432×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62#431×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic, complemented
64#335×C79C2211yescharacteristic, complemented
65#334×C79C2211yescharacteristic, complemented
66#334×C79C2211yescharacteristic, complemented
67C22×D316C2×C411yescharacteristic
68C22×C4×D79—21no—
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#14579161C2×C411yescharacteristic
79#14579153—21no—
80#14579140—21no—
81#14579140C2×C411yescharacteristic
82#14579140C2×C411yescharacteristic
83C4×Dic158—21no—
84C4×Dic158C2311yescharacteristic
85C4×Dic158C2311yescharacteristic
86C4×D316—21no—
87C4×D316—21no—
88C4×D316C2×C411yescharacteristic
89C4×D316C2×C411yescharacteristic
90C4×D316C2311yescharacteristic
91C4×D316C2311yescharacteristic
92#14579120—21no—
93C2×C4⋊Dic79C2×C411yescharacteristic
94C2×(C2×D79)⋊C4—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
144C23×D79—21no—
145C2×Dic158—21no—
146C2×Dic158C22×C411yescharacteristic
147C2×Dic158C22×C411yescharacteristic
148C2×D316—22no—
149C2×D316—21no—
150C2×D316—41no—
151C2×D316C22×C411yescharacteristic
152C2×D316C22×C411yescharacteristic
153C22×Dic79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—21no—
156C2×C4×D79—21no—
157C2×C4×D79—21no—
158C2×C4×D79—81no—
159C2×C4×D79—81no—
160#11929613—22no—
161#11929613—22no—
162#11929613—41no—
163#11929613—41no—
164#11929607—81no—
165#11929607—81no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8C2×D411yescharacteristic
169C2×C79⋊C8C2×D411yescharacteristic
170C2×C79⋊C8C2×D411yescharacteristic
171C2×C79⋊C8C2×D411yescharacteristic
172C8×D79—81no—
173C8×D79—81no—
174C4⋊Dic79—21no—
175C4⋊Dic79—21no—
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—
186C22⋊Dic79—21no—
187C22⋊Dic79—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191(C2×D79)⋊C4—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—
212C22×D79—42no—
213C22×D79—42no—
214C22×D79—81no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—41no—
220C2×C4×C79C4×D411yescharacteristic
221C2×C4×C79C4×D411yescharacteristic
222C2×C4×C79C4×D411yescharacteristic
223C2×C4×C79C23×C411yescharacteristic
224C2×C4×C79C22×D411yescharacteristic
225C2×C4×C79C2×#4111yescharacteristic
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—81no—
233D316—21no—
234D316—21no—
235D316—21no—
236D316—42no—
237D316#13112yes—
238C4×D79—41no—
239C4×D79—41no—
240C4×D79—41no—
241C4×D79—41no—
242C4×D79—81no—
243C4×D79—81no—
244Dic158—21no—
245Dic158—21no—
246Dic158—21no—
247Dic158—21no—
248Dic158—42no—
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—22no—
268C2×D79—41no—
269C2×D79—41no—
270C2×D79—41no—
271C2×D79—41no—
272C2×D79—41no—
273C2×D79—84no—
274C4×C79—21no—
275C4×C79—21no—
276C4×C79—21no—
277C4×C79—21no—
278C4×C79C2×#13111yescharacteristic
279C4×C79#56811yescharacteristic
280C4×C79#57711yescharacteristic
281C4×C79#57711yescharacteristic
282Dic79—41no—
283Dic79—41no—
284Dic79—41no—
285Dic79—41no—
286Dic79—41no—
287Dic79—41no—
288#18028—791nomaximal, Sylow
289C2×C79—41no—
290C2×C79#293311yescharacteristic
291C2×C79#294911yescharacteristic
292C2×C79#295511yescharacteristic
293D79—42no—
294D79—81no—
295#3101—791no—
296#2986—791no—
297#2974—791no—
298#1592—791no—
299#1590—791no—
300#1588—791no—
301#1585—791no—
302#1583—791no—
303#1582—791no—
304SD16⋊C8—791no—
305C8×SD16—791no—
306#1572D7911yescharacteristic, complemented
307#1497—791no—
308#1478—791no—
309#1454—791no—
310C79#1802811yescharacteristic, complemented, Sylow
311#533—791no—
312C2×C4×D4—791no—
313#483—791no—
314#483—791no—
315#476—791no—
316#476—791no—
317#461—791no—
318#461—791no—
319#450—791no—
320#450—791no—
321#446—791no—
322#446—791no—
323#446—1581no—
324C8×Q8—791no—
325C8×Q8—791no—
326C8×Q8—1581no—
327C4×SD16—791no—
328C4×SD16—791no—
329C4×SD16—791no—
330C4×SD16—791no—
331#436—791no—
332#436—791no—
333#435—1581no—
334#435—1581no—
335C8×D4—791no—
336C8×D4—791no—
337#433—1581no—
338#433—1581no—
339#432C2×D7911yescharacteristic
340#431—791no—
341#431—791no—
342#431C2×D7911yescharacteristic
343#431C2×D7911yescharacteristic
344#407—1581no—
345C2×C22⋊C8—1581no—
346#335C2×D7911yescharacteristic, complemented
347#335C2×D7911yescharacteristic, complemented
348#334C2×D7911yescharacteristic, complemented
349#334C2×D7911yescharacteristic, complemented
350Q8⋊C8—791no—
351Q8⋊C8—791no—
352Q8⋊C8—791no—
353Q8⋊C8—791no—
354D4⋊C8—791no—
355D4⋊C8—791no—
356D4⋊C8—791no—
357D4⋊C8—791no—
358#322—791no—
359#322—791no—
360#322—791no—
361C82—791no—
362C22×D4—791no—
363C23×C4—1581no—
364C2×SD16—791no—
365C2×SD16—791no—
366C2×SD16—791no—
367C2×SD16—791no—
368C2×M16—1581no—
369C2×M16—1581no—
370C22×C8—1581no—
371C22×C8—1581no—
372C4⋊Q8—791no—
373C4⋊Q8—791no—
374#125—1581no—
375C22⋊Q8—791no—
376C22⋊Q8—791no—
377C22⋊Q8—1581no—
378C4×Q8—791no—
379C4×Q8—791no—
380C4×Q8—1581no—
381C4×D4—791no—
382C4×D4—791no—
383C4×D4—791no—
384C4×D4—791no—
385C4×D4—1581no—
386C4×D4—1581no—
387#117—1581no—
388C2×C4⋊C4—791no—
389C2×C22⋊C4—1581no—
390C2×C42C22×D7911yescharacteristic
391#106—791no—
392#106—791no—
393#106—791no—
394#106—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—1581no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8—1581no—
402C4⋊C8C4×D7911yescharacteristic
403C4⋊C8C4×D7911yescharacteristic
404C4⋊C8C22×D7911yescharacteristic
405C4⋊C8C22×D7911yescharacteristic
406C4⋊C8C22×D7911yescharacteristic
407C4⋊C8C22×D7911yescharacteristic
408#103—791no—
409#103—791no—
410#103—791no—
411#103—791no—
412#102—791no—
413#102—791no—
414#102—791no—
415#102—791no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8—791no—
419C22⋊C8—791no—
420C22⋊C8—1581no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424C22⋊C8—1581no—
425C22⋊C8C4×D7911yescharacteristic
426C22⋊C8C4×D7911yescharacteristic
427#97—21no—
428#97—1581no—
429#97—1581no—
430C4×C8—21no—
431C4×C8—791no—
432C4×C8—791no—
433C4×C8—791no—
434C4×C8—791no—
435C4×C8—1581no—
436C4×C8—1581no—
437C4×C8C22×D7911yescharacteristic
438C4×C8C22×D7911yescharacteristic
439C24—1581no—
440C2×Q8—791no—
441C2×Q8—791no—
442C2×Q8—1581no—
443C2×D4—3161no—
444C2×D4—791no—
445C2×D4—791no—
446C2×D4—1582no—
447C2×D4—1581no—
448C22×C4—6321no—
449C22×C4—6321no—
450C22×C4—1581no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4C2×C4×D7911yescharacteristic
456C22×C4D4×D7911yescharacteristic
457C22×C4#1192962511yescharacteristic
458SD16—3161no—
459SD16—3161no—
460SD16—1582no—
461SD16—1582no—
462M16—6321no—
463M16—6321no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—791no—
469C2×C8—791no—
470C2×C8—791no—
471C2×C8—791no—
472C2×C8—6321no—
473C2×C8—6321no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8C2×C4×D7911yescharacteristic
477C2×C8C2×C4×D7911yescharacteristic
478C2×C8C2×C4×D7911yescharacteristic
479C2×C8C2×C4×D7911yescharacteristic
480C4⋊C4—791no—
481C4⋊C4—791no—
482C4⋊C4—791no—
483C4⋊C4—791no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C22⋊C4—1581no—
493C22⋊C4—1581no—
494C22⋊C4—1581no—
495C22⋊C4—1581no—
496C22⋊C4—1581no—
497C22⋊C4—1581no—
498C42—1581no—
499C42—1581no—
500C42C2×C4×D7911yescharacteristic
501C42D4×D7911yescharacteristic
502C42#1192962511yescharacteristic
503C42C23×D7911yescharacteristic
504C23—3162no—
505C23—3162no—
506C23—6321no—
507C23—1581no—
508C23—1581no—
509C23#1457912311yescharacteristic
510Q8—3162no—
511Q8—1581no—
512Q8—1581no—
513Q8—1581no—
514Q8—1581no—
515D4—3162no—
516D4—792no—
517D4—1581no—
518D4—1581no—
519D4—1581no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—6321no—
530C2×C4—6321no—
531C2×C4—6321no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4#1457912311yescharacteristic
539C2×C4#1457912311yescharacteristic
540C2×C4#1457912311yescharacteristic
541C2×C4C22×C4×D7911yescharacteristic
542C2×C4C2×D4×D7911yescharacteristic
543C2×C4C2×#1192962511yescharacteristic
544C8—21no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—3161no—
553C8—3161no—
554C8—1581no—
555C8—1581no—
556C8—1581no—
557C8—1581no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—3161no—
562C22—3161no—
563C22—3161no—
564C22—3161no—
565C22—3161no—
566C22—6324no—
567C22—1582no—
568C22C2×#1457912311yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—3161no—
574C4—3161no—
575C4—3161no—
576C4—3161no—
577C4—3161no—
578C4—3161no—
579C4#131×D7911yescharacteristic
580C4#1909932811yescharacteristic
581C4#1909937011yescharacteristic
582C4#1909937011yescharacteristic
583C2—41no—
584C2—3162no—
585C2—6321no—
586C2#2377212411yescharacteristic
587C2#2377225811yescharacteristic
588C2#2377230211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1572C2
socleC22×C79C2×C4×D4
prime radicalC22×D316C2×C4
cosocleC42×C79C24
hypercenter#1572D79
hypocenterC79#18028
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. #23772501C2semidirect
4. #23762301C2semidirect
5. #23762289C2semidirect
6. #23762277C2semidirect
7. #23762256C2semidirect
8. #23762242C2semidirect
9. #23762235C2semidirect
10. #23762211C2semidirect
11. #23762199C2semidirect
12. #23761872C2semidirect
14. #23761700C2semidirect
15. C79×#1572C2semidirect
27. #19098690C22semidirect
28. #19098690C22semidirect
30. #19098687C22semidirect
31. #19098687C22semidirect
48. #19098293C22semidirect
49. #19098293C22semidirect
50. #19098293C22semidirect
51. #19098293C22semidirect
56. #19098285C22semidirect
57. #19098285C22semidirect
58. #19098284C22semidirect
59. C8×C79⋊C8C22semidirect
63. #335×C79C22semidirect
64. #335×C79C22semidirect
65. #334×C79C22semidirect
66. #334×C79C22semidirect
306. #1572D79semidirect
310. C79#18028semidirect
346. #335C2×D79semidirect
347. #335C2×D79semidirect
348. #334C2×D79semidirect
349. #334C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734092
#237725012
#237724372
#237623012
#237622892
#237622772
#237622562
#237622422
#237622352
#237622112
#237621992
#237618722
#237617902
#237617002
C79×#15722
#1802879