Finite Group Databasebeta

← #34259324

Subgroups — #34259324

Order 20224.

662 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773461C211yesmaximal, characteristic
2#23772415C211yesmaximal, characteristic, complemented
3#23772146C211yesmaximal, characteristic, complemented
4#23762304C211yesmaximal, characteristic, complemented
5#23762292C211yesmaximal, characteristic, complemented
6#23762266C211yesmaximal, characteristic, complemented
7#23762259C211yesmaximal, characteristic, complemented
8#23762238C211yesmaximal, characteristic, complemented
9#23762231C211yesmaximal, characteristic, complemented
10#23762225C211yesmaximal, characteristic, complemented
11C8×D632C211yesmaximal, characteristic, complemented
12#23762185C211yesmaximal, characteristic, complemented
13#23761870C211yesmaximal, characteristic, complemented
14#23761788C211yesmaximal, characteristic, complemented
15C79×#1453C211yesmaximal, characteristic, complemented
16C2×#14579169—21no—
17#19099113C2211yescharacteristic
18C4×#11929624C2211yescharacteristic
19#19098888C411yescharacteristic
20#19098888C411yescharacteristic
21#19098859C411yescharacteristic
22#19098854C411yescharacteristic
23#19098764C411yescharacteristic
24#19098759C411yescharacteristic
25#19098710C411yescharacteristic
26#19098710C411yescharacteristic
27#19098693C2211yescharacteristic
28#19098693C2211yescharacteristic
29C4×#11929613C2211yescharacteristic
30C4×#11929613C2211yescharacteristic
31C4×D632C2211yescharacteristic
32C4×D632C2211yescharacteristic
33#19098642C2211yescharacteristic
34#19098642C2211yescharacteristic
35#19098636—21no—
36#19098636—21no—
37#19098632—21no—
38#19098632—21no—
39C8×D316C2211yescharacteristic
40C8×D316C2211yescharacteristic
41C8×D316C2211yescharacteristic
42C8×D316C2211yescharacteristic
43#19098620C2211yescharacteristic
44#19098618—21no—
45C2×C4⋊(C79⋊C8)C2211yescharacteristic
46#19098531—21no—
47#97×D79—21no—
48#19098316C2211yescharacteristic
49#19098316C2211yescharacteristic
50#19098314C2211yescharacteristic
51#19098314C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098292C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098292C2211yescharacteristic
57#19098292C2211yescharacteristic
58#19098292C2211yescharacteristic
59#19098292C2211yescharacteristic
60#431×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62C4×M16×C79C2211yescharacteristic
63#322×C79C2211yescharacteristic
64#322×C79C2211yescharacteristic
65#322×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929624—21no—
68C2×#11929624C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929613C2×C411yescharacteristic
71C2×D632C2×C411yescharacteristic
72C2×D632C2×C411yescharacteristic
73#14579169—44no—
74C2×#11929607—21no—
75C2×#11929607—21no—
76C22×C79⋊C8—21no—
77C2×C8×D79—21no—
78C2×C8×D79—21no—
79C2×C8×D79—21no—
80C2×C8×D79—21no—
81#14579161C2×C411yescharacteristic
82#14579158C2×C411yescharacteristic
83#14579149—21no—
84#14579135—21no—
85#14579135C2×C411yescharacteristic
86#14579135C2×C411yescharacteristic
87C4×Dic158C2×C411yescharacteristic
88C4×Dic158C2311yescharacteristic
89C4×D316—21no—
90C4×D316C2×C411yescharacteristic
91C4×D316C2311yescharacteristic
92C4×D316C2311yescharacteristic
93C4×D316C2311yescharacteristic
94C4×C22⋊D79—21no—
95C4×C22⋊D79—21no—
96#14579120C2×C411yescharacteristic
97#14579118—21no—
98C42×D79—21no—
99#14579095C2×C411yescharacteristic
100#14579095C2×C411yescharacteristic
101#14579093C2×C411yescharacteristic
102#14579093C2×C411yescharacteristic
103C4⋊(C79⋊C8)—21no—
104C4⋊(C79⋊C8)C2311yescharacteristic
105C4⋊(C79⋊C8)C2311yescharacteristic
106C4⋊(C79⋊C8)C2311yescharacteristic
107C4⋊(C79⋊C8)C2311yescharacteristic
108Dic79⋊C8—21no—
109Dic79⋊C8—21no—
110#14579086C2×C411yescharacteristic
111#14579086C2×C411yescharacteristic
112#14579083C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114#14579083C2×C411yescharacteristic
115#14579083C2×C411yescharacteristic
116#14579083C2×C411yescharacteristic
117#14579083C2×C411yescharacteristic
118C22⋊(C79⋊C8)—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8—21no—
121(C2×D79)⋊C8—21no—
122(C2×D79)⋊C8—21no—
123(C2×D79)⋊C8—21no—
124(C2×D79)⋊C8—21no—
125#14579072—21no—
126#14579071—21no—
127#14579071—21no—
128C2×M16×C79D411yescharacteristic
129C2×M16×C79D411yescharacteristic
130C2×C42×C79C2311yescharacteristic
131C4⋊C8×C79—21no—
132C4⋊C8×C79C2×C411yescharacteristic
133C4⋊C8×C79C2×C411yescharacteristic
134C22⋊C8×C79—21no—
135C22⋊C8×C79C2×C411yescharacteristic
136C22⋊C8×C79C2×C411yescharacteristic
137#97×C79D411yescharacteristic
138#97×C79D411yescharacteristic
139C4×C8×C79C2311yescharacteristic
140C4×C8×C79C2311yescharacteristic
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145#11929624—22no—
146#11929624—41no—
147#11929624—42no—
148#11929624—42no—
149C2×Dic158—21no—
150C2×Dic158C22×C411yescharacteristic
151C2×D316—21no—
152C2×D316—21no—
153C2×D316C22×C411yescharacteristic
154C2×D316C22×C411yescharacteristic
155C2×D316C22×C411yescharacteristic
156C2×C22⋊D79—21no—
157C2×C22⋊D79—21no—
158C2×C22⋊D79—21no—
159C2×C4×D79—21no—
160C2×C4×D79—21no—
161C2×C4×D79—21no—
162C2×C4×D79—21no—
163C2×C4×D79—21no—
164#11929613—41no—
165#11929613—41no—
166D632—22no—
167D632—22no—
168#11929607—42no—
169#11929607—42no—
170C2×C79⋊C8—21no—
171C2×C79⋊C8—21no—
172C2×C79⋊C8—21no—
173C2×C79⋊C8—21no—
174C2×C79⋊C8—42no—
175C8×D79—42no—
176C8×D79—42no—
177C8×D79—42no—
178C8×D79—42no—
179C4⋊Dic79—21no—
180C4⋊Dic79C22×C411yescharacteristic
181C4⋊Dic79C22×C411yescharacteristic
182C4⋊Dic79C22×C411yescharacteristic
183C4⋊Dic79C22×C411yescharacteristic
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—
193C4×Dic79—21no—
194C4×Dic79—21no—
195C4×Dic79—21no—
196C22×C4×C79C22×C411yescharacteristic
197C22×C4×C79C2×D411yescharacteristic
198C22×C4×C79#4111yescharacteristic
199M16×C79—22no—
200M16×C79—41no—
201C2×C8×C79—21no—
202C2×C8×C79—21no—
203C2×C8×C79C22×C411yescharacteristic
204C2×C8×C79C22×C411yescharacteristic
205C2×C8×C79C22×C411yescharacteristic
206C2×C8×C79C22×C411yescharacteristic
207C2×C8×C79C2×D411yescharacteristic
208C2×C8×C79C2×D411yescharacteristic
209C2×C8×C79C2×D411yescharacteristic
210C2×C8×C79C2×D411yescharacteristic
211C42×C79C22×C411yescharacteristic
212C42×C79C2×D411yescharacteristic
213C42×C79#4111yescharacteristic
214C42×C79C2411yescharacteristic
215C23×C79C4×D411yescharacteristic
216C22×D79—21no—
217C22×D79—21no—
218C22×D79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—41no—
224C2×C4×C79C4×D411yescharacteristic
225C2×C4×C79C4×D411yescharacteristic
226C2×C4×C79C4×D411yescharacteristic
227C2×C4×C79C23×C411yescharacteristic
228C2×C4×C79C22×D411yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C22⋊D79—42no—
231C22⋊D79—42no—
232C22⋊D79—42no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—21no—
238D316—21no—
239D316—21no—
240D316—21no—
241D316—21no—
242D316—21no—
243D316—42no—
244D316—81no—
245D316#13112yescomplemented
246C4×D79—42no—
247C4×D79—41no—
248C4×D79—41no—
249C4×D79—41no—
250C4×D79—41no—
251C4×D79—41no—
252C4×D79—41no—
253C4×D79—81no—
254Dic158—21no—
255Dic158—21no—
256Dic158—42no—
257C8×C79—21no—
258C8×C79—21no—
259C8×C79—21no—
260C8×C79—21no—
261C8×C79—21no—
262C8×C79—21no—
263C8×C79—21no—
264C8×C79—41no—
265C8×C79—41no—
266C8×C79AGL(1,ℤ/8)12yes—
267C79⋊C8—41no—
268C79⋊C8—41no—
269C79⋊C8—41no—
270C79⋊C8—41no—
271C22×C79—21no—
272C22×C79—21no—
273C22×C79—21no—
274C22×C79C2×C4×D411yescharacteristic
275C2×D79—22no—
276C2×D79—41no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—41no—
280C2×D79—41no—
281C2×D79—41no—
282C2×D79—41no—
283C2×D79—41no—
284C4×C79—21no—
285C4×C79—21no—
286C4×C79—21no—
287C4×C79—21no—
288C4×C79C2×#13111yescharacteristic
289C4×C79#56811yescharacteristic
290C4×C79C2×AGL(1,ℤ/8)11yescharacteristic
291C4×C79#57511yescharacteristic
292Dic79—41no—
293Dic79—41no—
294Dic79—41no—
295Dic79—41no—
296Dic79—41no—
297#18007—791nomaximal, Sylow
298C2×C79—41no—
299C2×C79#293311yescharacteristic
300C2×C79C4×AGL(1,ℤ/8)11yescharacteristic
301C2×C79#295711yescharacteristic
302D79—42no—
303D79—81no—
304D79—81no—
305#3106—791no—
306#2970—791no—
307#2935—791no—
308#1592—791no—
309#1590—791no—
310#1586—791no—
311#1585—791no—
312#1582—791no—
313#1581—791no—
314#1580—791no—
315C8×D8—791no—
316#1573—791no—
317#1496—791no—
318#1477—791no—
319#1453D7911yescharacteristic, complemented
320C79#1800711yescharacteristic, complemented, Sylow
321C2×#131—1581no—
322#532—791no—
323C4×#41—791no—
324#482—791no—
325#482—791no—
326#477—791no—
327#476—791no—
328#460—791no—
329#459—791no—
330#449—791no—
331#449—791no—
332#446—791no—
333#446—791no—
334C4×SD16—791no—
335C4×SD16—791no—
336C4×D8—791no—
337C4×D8—791no—
338#436—791no—
339#436—791no—
340#435—1581no—
341#435—1581no—
342#435—1581no—
343#435—1581no—
344C8×D4—791no—
345C8×D4—791no—
346C8×D4—791no—
347C8×D4—791no—
348#432—791no—
349#432—1581no—
350#431C2×D7911yescharacteristic
351#431C2×D7911yescharacteristic
352C2×C4⋊C8—791no—
353#408—1581no—
354C4×M16C2×D7911yescharacteristic
355C2×#97—1581no—
356#335—791no—
357#335—791no—
358#334—791no—
359#334—791no—
360Q8⋊C8—791no—
361Q8⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8—791no—
364D4⋊C8—791no—
365D4⋊C8—791no—
366D4⋊C8—791no—
367D4⋊C8—791no—
368#322C2×D7911yescharacteristic
369#322C2×D7911yescharacteristic
370#322C2×D7911yescharacteristic
371C82C2×D7911yescharacteristic
372C2×#41—791no—
373C2×#41—1581no—
374C2×SD16—791no—
375C2×SD16—791no—
376C2×D8—791no—
377C2×D8—791no—
378#131—3164no—
379C2×M16—1581no—
380C2×M16—1581no—
381C2×M16D31611yescharacteristic
382C2×M16D31611yescharacteristic
383C22×C8—1581no—
384C22×C8—1581no—
385C22×C8—1581no—
386C22×C8—1581no—
387C22×C8—1581no—
388C4⋊Q8—791no—
389#127—791no—
390#124—1581no—
391#121—791no—
392#121—791no—
393#121—1581no—
394C4×Q8—791no—
395C4×Q8—791no—
396C4×D4—791no—
397C4×D4—791no—
398C4×D4—791no—
399C4×D4—791no—
400C4×D4—1581no—
401C4×D4—1581no—
402C4×D4—1581no—
403#117—791no—
404#117—1581no—
405C2×C42—1581no—
406C2×C42C22×D7911yescharacteristic
407#107—791no—
408#107—791no—
409#106—791no—
410#106—791no—
411C4⋊C8—21no—
412C4⋊C8—791no—
413C4⋊C8—791no—
414C4⋊C8—791no—
415C4⋊C8—791no—
416C4⋊C8—1581no—
417C4⋊C8—1581no—
418C4⋊C8—1581no—
419C4⋊C8C4×D7911yescharacteristic
420C4⋊C8C4×D7911yescharacteristic
421#103—791no—
422#103—791no—
423#102—791no—
424#102—791no—
425#102—791no—
426#102—791no—
427#102—791no—
428#102—791no—
429C22⋊C8—21no—
430C22⋊C8—1581no—
431C22⋊C8—1581no—
432C22⋊C8—1581no—
433C22⋊C8—1581no—
434C22⋊C8—1581no—
435C22⋊C8—1581no—
436C22⋊C8—1581no—
437C22⋊C8C4×D7911yescharacteristic
438C22⋊C8C4×D7911yescharacteristic
439#97—1581no—
440#97—1581no—
441#97—1581no—
442#97D31611yescharacteristic
443#97D31611yescharacteristic
444C4×C8C22×D7911yescharacteristic
445C4×C8C22×D7911yescharacteristic
446C4×C8C22×D7911yescharacteristic
447C4×C8C22×D7911yescharacteristic
448C4×C8C22×D7911yescharacteristic
449C4×C8C22×D7911yescharacteristic
450#41—3162no—
451#41—3162no—
452#41—3161no—
453#41—1582no—
454C2×Q8—791no—
455C2×Q8—1581no—
456C2×D4—791no—
457C2×D4—791no—
458C2×D4—791no—
459C2×D4—1581no—
460C2×D4—1581no—
461C2×D4—1581no—
462C2×D4—1581no—
463C2×D4—1581no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4—1581no—
467C22×C4—1581no—
468C22×C4—1581no—
469C22×C4C2×C4×D7911yescharacteristic
470C22×C4C2×D31611yescharacteristic
471C22×C4#1192962411yescharacteristic
472SD16—3161no—
473SD16—3161no—
474D8—1582no—
475D8—1582no—
476M16—22no—
477M16—41no—
478M16—3162no—
479M16—3162no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—3162no—
483C2×C8—3162no—
484C2×C8—3162no—
485C2×C8—3162no—
486C2×C8—3162no—
487C2×C8—1581no—
488C2×C8—1581no—
489C2×C8—1581no—
490C2×C8—1581no—
491C2×C8C2×C4×D7911yescharacteristic
492C2×C8C2×C4×D7911yescharacteristic
493C2×C8C2×C4×D7911yescharacteristic
494C2×C8C2×C4×D7911yescharacteristic
495C2×C8C2×D31611yescharacteristic
496C2×C8C2×D31611yescharacteristic
497C2×C8C2×D31611yescharacteristic
498C2×C8C2×D31611yescharacteristic
499C4⋊C4—791no—
500C4⋊C4—791no—
501C4⋊C4—791no—
502C4⋊C4—791no—
503C4⋊C4—1581no—
504C4⋊C4—1581no—
505C4⋊C4—1581no—
506C22⋊C4—1581no—
507C22⋊C4—1581no—
508C22⋊C4—1581no—
509C22⋊C4—1581no—
510C22⋊C4—1581no—
511C22⋊C4—1581no—
512C22⋊C4—1581no—
513C42—1581no—
514C42—1581no—
515C42—1581no—
516C42C2×C4×D7911yescharacteristic
517C42C2×D31611yescharacteristic
518C42#1192962411yescharacteristic
519C42C23×D7911yescharacteristic
520C23—1581no—
521C23—1581no—
522C23—1581no—
523C23C4×D31611yescharacteristic
524Q8—3162no—
525Q8—1581no—
526Q8—1581no—
527D4—3162no—
528D4—3162no—
529D4—3162no—
530D4—3162no—
531D4—792no—
532D4—6321no—
533D4—1581no—
534D4—1581no—
535D4—1581no—
536D4—1581no—
537D4—1581no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—3162no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—3161no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—6321no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4—1581no—
554C2×C4—1581no—
555C2×C4—1581no—
556C2×C4C4×D31611yescharacteristic
557C2×C4C4×D31611yescharacteristic
558C2×C4C4×D31611yescharacteristic
559C2×C4C22×C4×D7911yescharacteristic
560C2×C4C22×D31611yescharacteristic
561C2×C4C2×#1192962411yescharacteristic
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—41no—
570C8—41no—
571C8—3161no—
572C8—3161no—
573C8—3161no—
574C8—3161no—
575C8#1457918912yes—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—3161no—
580C22—3161no—
581C22—3161no—
582C22—3161no—
583C22—3161no—
584C22—3161no—
585C22—3161no—
586C22—3161no—
587C22—1582no—
588C22C2×C4×D31611yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—3161no—
594C4—3161no—
595C4—3161no—
596C4—3161no—
597C4—3161no—
598C4#131×D7911yescharacteristic
599C4#1909932811yescharacteristic
600C4C2×#1457918911yescharacteristic
601C4#1909936311yescharacteristic
602C2—41no—
603C2—3162no—
604C2—6321no—
605C2—6321no—
606C2#2377212811yescharacteristic
607C2C4×#1457918911yescharacteristic
608C2#2377231811yescharacteristic
subgroupG/H
centerC2×C4C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1453C2
socleC22×C79C2×C4×D4
prime radical#23773461C2
cosocleC42×C79C24
hypercenter#1453D79
hypocenterC79#18007
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1453

Complements

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

NH ≅ G/Ntype
2. #23772415C2semidirect
3. #23772146C2semidirect
4. #23762304C2semidirect
5. #23762292C2semidirect
6. #23762266C2semidirect
7. #23762259C2semidirect
8. #23762238C2semidirect
9. #23762231C2semidirect
10. #23762225C2semidirect
11. C8×D632C2semidirect
12. #23762185C2semidirect
13. #23761870C2semidirect
14. #23761788C2semidirect
15. C79×#1453C2semidirect
245. D316#131semidirect
319. #1453D79semidirect
320. C79#18007semidirect

Maximal subgroups

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

subgroupindex
#237734612
#237724152
#237721462
#237623042
#237622922
#237622662
#237622592
#237622382
#237622312
#237622252
C8×D6322
#237621852
#237618702
#237617882
C79×#14532
#1800779