Finite Group Databasebeta

← #34258850

Subgroups — #34258850

Order 20224.

662 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773991C211yesmaximal, characteristic
2#23772374C211yesmaximal, characteristic, complemented
3#23772109C211yesmaximal, characteristic, complemented
4#23762315C211yesmaximal, characteristic, complemented
5#23762312C211yesmaximal, characteristic, complemented
6#23762300C211yesmaximal, characteristic, complemented
7SD16×C79⋊C8C211yesmaximal, characteristic, complemented
8#23762197C211yesmaximal, characteristic, complemented
9#23762194C211yesmaximal, characteristic, complemented
10#23762192C211yesmaximal, characteristic, complemented
11#23762166C211yesmaximal, characteristic, complemented
12#23761867C211yesmaximal, characteristic, complemented
13#23761864C211yesmaximal, characteristic, complemented
14#23761679C211yesmaximal, characteristic, complemented
15C79×#1592C211yesmaximal, characteristic, complemented
16C2×#14579169—21no—
17#19099110C2211yescharacteristic
18#19099100C2211yescharacteristic
19#19098970C411yescharacteristic
20#19098949C411yescharacteristic
21#19098926C411yescharacteristic
22#19098924C411yescharacteristic
23#19098817C411yescharacteristic
24#19098815C411yescharacteristic
25#19098812C411yescharacteristic
26#19098806C411yescharacteristic
27Q8×C79⋊C8C2211yescharacteristic
28C4×#11929611C2211yescharacteristic
29C4×D632C2211yescharacteristic
30C4×#11929609C2211yescharacteristic
31#19098642C2211yescharacteristic
32#19098642C2211yescharacteristic
33#19098639—21no—
34#19098636—21no—
35D4×C79⋊C8C2211yescharacteristic
36#19098628—21no—
37#19098628C2211yescharacteristic
38#19098628C2211yescharacteristic
39(C22⋊D79)⋊C8—21no—
40#19098623—21no—
41#19098619C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098585C2211yescharacteristic
44#19098528—21no—
45#19098520—21no—
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098314C2211yescharacteristic
50#19098313C2211yescharacteristic
51Q8⋊(C79⋊C8)C2211yescharacteristic
52#19098292C2211yescharacteristic
53#19098292C2211yescharacteristic
54D4⋊(C79⋊C8)C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
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×#11929624—21no—
68C2×#11929611C2×C411yescharacteristic
69C2×D632C2×C411yescharacteristic
70C2×#11929609C2×C411yescharacteristic
71#14579169—44no—
72C2×#11929607—21no—
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×#11929607C2×C411yescharacteristic
76C2×#11929607C2×C411yescharacteristic
77C22×C79⋊C8—21no—
78C2×C8×D79—21no—
79C2×C8×D79—21no—
80C2×C8×D79D411yescharacteristic
81C2×C8×D79D411yescharacteristic
82#14579160C2×C411yescharacteristic
83#14579157C2×C411yescharacteristic
84#14579156—21no—
85#14579152C2×C411yescharacteristic
86#14579150C2×C411yescharacteristic
87#14579147—21no—
88#14579146—21no—
89#14579137C2×C411yescharacteristic
90#14579136C2×C411yescharacteristic
91#14579136C2×C411yescharacteristic
92#14579134—21no—
93#14579132C2×C411yescharacteristic
94C4×D316—21no—
95C4×D316C2311yescharacteristic
96C4×D316C2311yescharacteristic
97C4×C22⋊D79—21no—
98#14579118—21no—
99C42×D79C2311yescharacteristic
100#14579095C2×C411yescharacteristic
101#14579094C2×C411yescharacteristic
102#14579092C2×C411yescharacteristic
103C4⋊(C79⋊C8)—21no—
104C4⋊(C79⋊C8)C2311yescharacteristic
105C4⋊(C79⋊C8)C2311yescharacteristic
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108#14579085C2×C411yescharacteristic
109#14579083C2×C411yescharacteristic
110#14579083C2×C411yescharacteristic
111#14579082C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113#14579081C2×C411yescharacteristic
114C22⋊(C79⋊C8)—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8—21no—
121#14579071—21no—
122#14579071C2×C411yescharacteristic
123#14579071C2×C411yescharacteristic
124C4×C79⋊C8—21no—
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
128C8×Dic79D411yescharacteristic
129C8×Dic79D411yescharacteristic
130C2×SD16×C79C2×C411yescharacteristic
131C2×M16×C79—21no—
132C4×Q8×C79C2311yescharacteristic
133C4×D4×C79C2311yescharacteristic
134#106×C79C2×C411yescharacteristic
135C4⋊C8×C79—21no—
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138#103×C79C2×C411yescharacteristic
139#102×C79C2×C411yescharacteristic
140C22⋊C8×C79—21no—
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145#11929624—42no—
146#11929624—42no—
147C2×Dic158—21no—
148C2×D316—21no—
149C2×D316—21no—
150C2×D316C22×C411yescharacteristic
151C2×D316C22×C411yescharacteristic
152C2×C22⋊D79—21no—
153C2×C22⋊D79—21no—
154C2×C22⋊D79—21no—
155C2×C4×D79—21no—
156C2×C4×D79—21no—
157C2×C4×D79C22×C411yescharacteristic
158C2×C4×D79C2×D411yescharacteristic
159C2×C4×D79#4111yescharacteristic
160#11929611—22no—
161D632—22no—
162#11929609—22no—
163#11929607—22no—
164#11929607—22no—
165#11929607—42no—
166#11929607—42no—
167#11929607—42no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8—21no—
171C2×C79⋊C8—42no—
172C2×C79⋊C8C22×C411yescharacteristic
173C2×C79⋊C8C22×C411yescharacteristic
174C2×C79⋊C8C2×D411yescharacteristic
175C2×C79⋊C8C2×D411yescharacteristic
176C8×D79—22no—
177C8×D79—41no—
178C8×D79—42no—
179C8×D79—42no—
180C4⋊Dic79—21no—
181C4⋊Dic79C22×C411yescharacteristic
182C4⋊Dic79C22×C411yescharacteristic
183Dic79⋊C4—21no—
184Dic79⋊C4—21no—
185Dic79⋊C4—21no—
186C22⋊Dic79—21no—
187(C2×D79)⋊C4—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191(C2×D79)⋊C4—21no—
192(C2×D79)⋊C4—21no—
193(C2×D79)⋊C4—21no—
194C4×Dic79C22×C411yescharacteristic
195C4×Dic79C2×D411yescharacteristic
196C4×Dic79#4111yescharacteristic
197C2×Q8×C79C22×C411yescharacteristic
198C2×D4×C79C22×C411yescharacteristic
199C22×C4×C79—21no—
200SD16×C79—22no—
201M16×C79—42no—
202C2×C8×C79—21no—
203C2×C8×C79—21no—
204C2×C8×C79—21no—
205C2×C8×C79C22×C411yescharacteristic
206C2×C8×C79C22×C411yescharacteristic
207C2×C8×C79C2×D411yescharacteristic
208C2×C8×C79C2×D411yescharacteristic
209C4⋊C4×C79—21no—
210C4⋊C4×C79C22×C411yescharacteristic
211C4⋊C4×C79C22×C411yescharacteristic
212C22⋊C4×C79—21no—
213C42×C79—21no—
214C42×C79C2411yescharacteristic
215C23×C79—21no—
216C22×D79—21no—
217C22×D79—21no—
218C22×D79C4×D411yescharacteristic
219Q8×C79—21no—
220Q8×C79—21no—
221D4×C79—21no—
222D4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79C23×C411yescharacteristic
229C2×C4×C79C22×D411yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C22⋊D79—42no—
232C22⋊D79—42no—
233C22⋊D79—42no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79C4×D411yescharacteristic
237C2×Dic79C4×D411yescharacteristic
238C2×Dic79C4×D411yescharacteristic
239D316—21no—
240D316—21no—
241D316—21no—
242D316—21no—
243D316—42no—
244D316—42no—
245C4×D79—21no—
246C4×D79—21no—
247C4×D79—21no—
248C4×D79—21no—
249C4×D79—41no—
250C4×D79—41no—
251C4×D79—41no—
252C4×D79—41no—
253C4×D79—41no—
254Dic158—42no—
255C8×C79—21no—
256C8×C79—21no—
257C8×C79—21no—
258C8×C79—41no—
259C8×C79—41no—
260C8×C79—41no—
261C8×C79#13112yescomplemented
262C79⋊C8—21no—
263C79⋊C8—21no—
264C79⋊C8—41no—
265C79⋊C8—41no—
266C79⋊C8—41no—
267C79⋊C8#13112yescomplemented
268C79⋊C8AGL(1,ℤ/8)12yes—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79—41no—
272C22×C79C2×C4×D411yescharacteristic
273C2×D79—21no—
274C2×D79—21no—
275C2×D79—21no—
276C2×D79—41no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—41no—
280C2×D79—41no—
281C2×D79—41no—
282C4×C79—21no—
283C4×C79—41no—
284C4×C79—41no—
285C4×C79—41no—
286C4×C79C2×#13111yescharacteristic
287C4×C79C2×#13111yescharacteristic
288C4×C79C2×AGL(1,ℤ/8)11yescharacteristic
289C4×C79#57511yescharacteristic
290Dic79—21no—
291Dic79—21no—
292Dic79—21no—
293Dic79—41no—
294Dic79—41no—
295#17964—791nomaximal, Sylow
296C2×C79—81no—
297C2×C79#293111yescharacteristic
298C2×C79C4×AGL(1,ℤ/8)11yescharacteristic
299C2×C79#295511yescharacteristic
300D79—41no—
301D79—81no—
302D79—81no—
303#3171—791no—
304#2966—791no—
305#2931—791no—
306#1594—791no—
307#1594—791no—
308#1592—791no—
309#1592D7911yescharacteristic, complemented
310C8×SD16—791no—
311C8×SD16—791no—
312C8×D8—791no—
313C8×D8—791no—
314#1569—791no—
315#1496—791no—
316#1495—791no—
317C8×M16—791no—
318C79#1796411yescharacteristic, complemented, Sylow
319C2×#131—1581no—
320#532—791no—
321#529—791no—
322#501—791no—
323#496—791no—
324#490—791no—
325#489—791no—
326#469—791no—
327#469—791no—
328#468—791no—
329#468—791no—
330#446C2×D7911yescharacteristic
331C8×Q8—791no—
332C4×SD16—791no—
333C4×SD16C2×D7911yescharacteristic
334C4×D8—791no—
335C4×D8—791no—
336#436—791no—
337#436—791no—
338#436—1581no—
339#436C2×D7911yescharacteristic
340#435—1581no—
341C8×D4—791no—
342C8×D4—791no—
343C8×D4—791no—
344C8×D4—1581no—
345C8×D4—1581no—
346#433—1581no—
347#432—791no—
348#431—791no—
349#423—791no—
350#408—1581no—
351#405—1581no—
352C4×M16—791no—
353C2×C4×C8—791no—
354#335—791no—
355#335C2×D7911yescharacteristic
356#334—791no—
357#334—791no—
358Q8⋊C8—791no—
359Q8⋊C8C2×D7911yescharacteristic
360D4⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8—791no—
364D4⋊C8—791no—
365D4⋊C8C2×D7911yescharacteristic
366#322—791no—
367#322C2×D7911yescharacteristic
368C82—791no—
369C82—791no—
370C2×#41—1581no—
371C2×SD16—791no—
372C2×SD16C4×D7911yescharacteristic
373C2×D8—791no—
374C2×D8—791no—
375#131—3164no—
376C2×M16—21no—
377C2×M16—791no—
378C2×M16—791no—
379C2×M16—1581no—
380C2×M16—1581no—
381C2×M16—1581no—
382C22×C8—791no—
383C22×C8—791no—
384C22×C8—1581no—
385C22×C8—1581no—
386C22×C8—1581no—
387C4⋊Q8—791no—
388#127—791no—
389#126—1581no—
390#125—791no—
391#124—791no—
392#124—1581no—
393#123—1581no—
394C22⋊Q8—791no—
395#121—791no—
396#121—791no—
397#121—791no—
398#121—1581no—
399C4×Q8C22×D7911yescharacteristic
400C4×D4—791no—
401C4×D4—791no—
402C4×D4—1581no—
403C4×D4—1581no—
404C4×D4C22×D7911yescharacteristic
405#117—1581no—
406C2×C42—791no—
407#107—791no—
408#107—791no—
409#106—791no—
410#106C4×D7911yescharacteristic
411C4⋊C8—21no—
412C4⋊C8—791no—
413C4⋊C8—791no—
414C4⋊C8—1581no—
415C4⋊C8—1581no—
416C4⋊C8—1581no—
417C4⋊C8C22×D7911yescharacteristic
418C4⋊C8C22×D7911yescharacteristic
419#103—791no—
420#103C4×D7911yescharacteristic
421#102—791no—
422#102—791no—
423#102—791no—
424#102—791no—
425#102—791no—
426#102C4×D7911yescharacteristic
427C22⋊C8—21no—
428C22⋊C8—1581no—
429C22⋊C8—1581no—
430C22⋊C8—1581no—
431C22⋊C8—1581no—
432C22⋊C8—1581no—
433C22⋊C8—1581no—
434C22⋊C8—1581no—
435#97—21no—
436#97—791no—
437#97—791no—
438#97—1581no—
439C4×C8—791no—
440C4×C8—791no—
441C4×C8—791no—
442C4×C8—791no—
443C4×C8—791no—
444C4×C8—1581no—
445C4×C8C22×D7911yescharacteristic
446C4×C8C22×D7911yescharacteristic
447C4×C8C22×D7911yescharacteristic
448#41—3162no—
449#41—3162no—
450C2×Q8—1581no—
451C2×Q8C2×C4×D7911yescharacteristic
452C2×D4—791no—
453C2×D4—791no—
454C2×D4—1581no—
455C2×D4—1581no—
456C2×D4—1581no—
457C2×D4—1581no—
458C2×D4—1581no—
459C2×D4C2×C4×D7911yescharacteristic
460C22×C4—21no—
461C22×C4—791no—
462C22×C4—791no—
463C22×C4—791no—
464C22×C4—1581no—
465C22×C4—1581no—
466SD16—22no—
467SD16—1582no—
468D8—1582no—
469D8—1582no—
470M16—42no—
471M16—3162no—
472M16—3162no—
473M16—3162no—
474M16—1582no—
475M16—1582no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—3161no—
480C2×C8—3162no—
481C2×C8—3162no—
482C2×C8—3162no—
483C2×C8—791no—
484C2×C8—791no—
485C2×C8—791no—
486C2×C8—791no—
487C2×C8—1582no—
488C2×C8—1581no—
489C2×C8—1581no—
490C2×C8—1581no—
491C2×C8C2×C4×D7911yescharacteristic
492C2×C8C2×C4×D7911yescharacteristic
493C2×C8D4×D7911yescharacteristic
494C2×C8D4×D7911yescharacteristic
495C4⋊C4—21no—
496C4⋊C4—791no—
497C4⋊C4—791no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4—1581no—
502C4⋊C4C2×C4×D7911yescharacteristic
503C4⋊C4C2×C4×D7911yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C22⋊C4—1581no—
508C22⋊C4—1581no—
509C22⋊C4—1581no—
510C22⋊C4—1581no—
511C22⋊C4—1581no—
512C22⋊C4—1581no—
513C42—21no—
514C42—791no—
515C42—791no—
516C42—791no—
517C42C23×D7911yescharacteristic
518C23—21no—
519C23—791no—
520C23—1581no—
521C23—1581no—
522Q8—21no—
523Q8—21no—
524Q8—3162no—
525D4—21no—
526D4—21no—
527D4—3162no—
528D4—3162no—
529D4—3162no—
530D4—3162no—
531D4—3162no—
532D4—1581no—
533D4—1581no—
534D4—1581no—
535D4—1581no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—791no—
547C2×C4—791no—
548C2×C4—791no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4—1581no—
554C2×C4—1581no—
555C2×C4C22×C4×D7911yescharacteristic
556C2×C4C2×D4×D7911yescharacteristic
557C2×C4#41×D7911yescharacteristic
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—3161no—
565C8—3161no—
566C8—3161no—
567C8—792no—
568C8—792no—
569C8—1581no—
570C8—1581no—
571C8#1457916912yescomplemented
572C22—41no—
573C22—41no—
574C22—41no—
575C22—3161no—
576C22—3161no—
577C22—3161no—
578C22—3161no—
579C22—3161no—
580C22—3161no—
581C22—1581no—
582C22—1581no—
583C22—1581no—
584C22C4×D4×D7911yescharacteristic
585C4—21no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—3161no—
590C4—3161no—
591C4—1581no—
592C4—1581no—
593C4—1581no—
594C4#131×D7911yescharacteristic
595C4C2×#1457916911yescharacteristic
596C4AGL(1,ℤ/8)×D7911yescharacteristic
597C4#1909936211yescharacteristic
598C2—81no—
599C2—3161no—
600C2—6321no—
601C2—6321no—
602C2#2377210911yescharacteristic
603C2#2377220511yescharacteristic
604C2#2377230011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1592C2
socleC22×C79C2×C4×D4
prime radical#23773991C2
cosocleC42×C79C24
hypercenter#1592D79
hypocenterC79#17964
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. #23772374C2semidirect
3. #23772109C2semidirect
4. #23762315C2semidirect
5. #23762312C2semidirect
6. #23762300C2semidirect
7. SD16×C79⋊C8C2semidirect
8. #23762197C2semidirect
9. #23762194C2semidirect
10. #23762192C2semidirect
11. #23762166C2semidirect
12. #23761867C2semidirect
13. #23761864C2semidirect
14. #23761679C2semidirect
15. C79×#1592C2semidirect
261. C8×C79#131semidirect
267. C79⋊C8#131semidirect
309. #1592D79semidirect
318. C79#17964semidirect
571. C8#14579169semidirect

Maximal subgroups

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

subgroupindex
#237739912
#237723742
#237721092
#237623152
#237623122
#237623002
SD16×C79⋊C82
#237621972
#237621942
#237621922
#237621662
#237618672
#237618642
#237616792
C79×#15922
#1796479