Finite Group Databasebeta

← #34271099

Subgroups — #34271099

Order 20224.

696 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773793C211yesmaximal, characteristic, complemented
2C2×#19098853C211yesmaximal, characteristic, complemented
3D4×#11929608C211yesmaximal, characteristic, complemented
4#23762973C212yesmaximal, complemented
5#23762931C212yesmaximal, complemented
6#23762849C212yesmaximal, complemented
7#23762807C212yesmaximal, complemented
8#23762153C211yesmaximal, characteristic, complemented
9#23762080C211yesmaximal, characteristic, complemented
10#23762077C211yesmaximal, characteristic, complemented
11C79×#3100C211yesmaximal, characteristic
12C22×#11929608—21no—
13C2×#14579161C2211yescharacteristic
14#19098922—22no—
15#19098913C2212yescomplemented
16#19098909C2212yescomplemented
17#19098853—21no—
18#19098853—21no—
19#19098853C2212yes—
20#19098853C2212yes—
21#19098641C2212yes—
22#19098635—22no—
23D4×C79⋊C8C2212yes—
24C2×#14579092—21no—
25#19098587C2211yescharacteristic
26C2×C4⋊(C79⋊C8)C2211yescharacteristic
27#19098564—21no—
28#19098564—21no—
29C2×#14579082—21no—
30#19098527—21no—
31C4×#11929608C2211yescharacteristic
32#19098315C2212yescomplemented
33#19098313C2212yescomplemented
34#19098311C2212yescomplemented
35#19098311C2212yescomplemented
36#19098309C2212yescomplemented
37#19098307C2212yescomplemented
38#532×C79C2211yescharacteristic
39C2×C4×D4×C79C2211yescharacteristic
40#480×C79—21no—
41#474×C79C2211yescharacteristic
42#474×C79C2211yescharacteristic
43#459×C79C2211yescharacteristic
44#459×C79C2211yescharacteristic
45#447×C79—21no—
46C4×D8×C79—21no—
47C4×D8×C79—21no—
48#431×C79C2211yescharacteristic
49C22×Dic158—21no—
50C2×#11929608—21no—
51C2×#11929608—22no—
52C2×#11929608—44no—
53C2×#11929608D411yescharacteristic
54C2×#11929608D411yescharacteristic
55C22×C79⋊C8—21no—
56C22×C79⋊C8—22no—
57#14579161—21no—
58#14579161—21no—
59#14579161—21no—
60#14579161C2312yes—
61C2×C4⋊Dic79—21no—
62C2×C4⋊Dic79—21no—
63#14579092—22no—
64#14579092—22no—
65C4⋊(C79⋊C8)—21no—
66C4⋊(C79⋊C8)—21no—
67C4⋊(C79⋊C8)C2312yes—
68C4⋊(C79⋊C8)C2312yes—
69#14579085—22no—
70#14579085—22no—
71#14579082—22no—
72#14579082—22no—
73#14579082—22no—
74#14579082—22no—
75C22⋊(C79⋊C8)—22no—
76C22⋊(C79⋊C8)—22no—
77#14579072D412yes—
78C4×C79⋊C8C2312yes—
79C2×#41×C79—21no—
80C22×D4×C79D411yescharacteristic
81C23×C4×C79—21no—
82C2×D8×C79—21no—
83C2×D8×C79—21no—
84C4⋊Q8×C79C2311yescharacteristic
85#127×C79C2311yescharacteristic
86#124×C79—21no—
87#121×C79—21no—
88#121×C79—21no—
89C4×D4×C79—21no—
90C4×D4×C79—21no—
91C4×D4×C79—21no—
92C4×D4×C79—21no—
93C4×D4×C79D411yescharacteristic
94C4×D4×C79D411yescharacteristic
95C4×D4×C79C2311yescharacteristic
96C4×D4×C79C2311yescharacteristic
97C2×C4⋊C4×C79D411yescharacteristic
98C2×C22⋊C4×C79—21no—
99C2×C42×C79C2311yescharacteristic
100#107×C79—21no—
101#107×C79—21no—
102C4⋊C8×C79C2311yescharacteristic
103C4⋊C8×C79C2311yescharacteristic
104#102×C79—21no—
105#102×C79—21no—
106#102×C79—21no—
107#102×C79—21no—
108C22⋊C8×C79—21no—
109C4×C8×C79—21no—
110C2×Dic158—22no—
111C2×Dic158—41no—
112C2×Dic158—82no—
113C22×Dic79—41no—
114#11929608—22no—
115#11929608—22no—
116#11929608—44no—
117#11929608—42no—
118C2×C79⋊C8—22no—
119C2×C79⋊C8—22no—
120C2×C79⋊C8—42no—
121C2×C79⋊C8—44no—
122C2×C79⋊C8C2×D412yes—
123C2×C79⋊C8C2×D412yes—
124C4⋊Dic79—22no—
125C4⋊Dic79—22no—
126C4⋊Dic79—41no—
127C4⋊Dic79—41no—
128C24×C79—21no—
129#41×C79—42no—
130#41×C79—42no—
131C2×Q8×C79—21no—
132C2×D4×C79—22no—
133C2×D4×C79—21no—
134C2×D4×C79—21no—
135C2×D4×C79—21no—
136C2×D4×C79—21no—
137C2×D4×C79—41no—
138C2×D4×C79—41no—
139C2×D4×C79C2×D411yescharacteristic
140C2×D4×C79C2×D411yescharacteristic
141C22×C4×C79—21no—
142C22×C4×C79—21no—
143C22×C4×C79—21no—
144C22×C4×C79—21no—
145C22×C4×C79—21no—
146C22×C4×C79—42no—
147C22×C4×C79—42no—
148C22×C4×C79—41no—
149C22×C4×C79C2×D411yescharacteristic
150C22×C4×C79C2×D411yescharacteristic
151C22×C4×C79#4111yescharacteristic
152D8×C79—42no—
153D8×C79—42no—
154C2×C8×C79—21no—
155C2×C8×C79—21no—
156C4⋊C4×C79—21no—
157C4⋊C4×C79—21no—
158C4⋊C4×C79—21no—
159C4⋊C4×C79C2×D411yescharacteristic
160C4⋊C4×C79C2×D411yescharacteristic
161C22⋊C4×C79—21no—
162C22⋊C4×C79—21no—
163C22⋊C4×C79—21no—
164C22⋊C4×C79—21no—
165C22⋊C4×C79—41no—
166C22⋊C4×C79—41no—
167C42×C79C2×D411yescharacteristic
168C42×C79C2×D411yescharacteristic
169C42×C79#4111yescharacteristic
170C42×C79C2411yescharacteristic
171C23×C79—21no—
172C23×C79—21no—
173C23×C79—42no—
174C23×C79—42no—
175C23×C79—41no—
176C23×C79—81no—
177C23×C79#12111yescharacteristic
178Q8×C79—41no—
179Q8×C79—41no—
180D4×C79—21no—
181D4×C79—21no—
182D4×C79—21no—
183D4×C79—42no—
184D4×C79—41no—
185D4×C79—41no—
186D4×C79—41no—
187D4×C79—41no—
188D4×C79—81no—
189D4×C79—82no—
190D4×C79#13712yes—
191C2×C4×C79—21no—
192C2×C4×C79—21no—
193C2×C4×C79—21no—
194C2×C4×C79—21no—
195C2×C4×C79—41no—
196C2×C4×C79—42no—
197C2×C4×C79—41no—
198C2×C4×C79—41no—
199C2×C4×C79—41no—
200C2×C4×C79—41no—
201C2×C4×C79—42no—
202C2×C4×C79—42no—
203C2×C4×C79—41no—
204C2×C4×C79—82no—
205C2×C4×C79#12111yescharacteristic
206C2×C4×C79#12111yescharacteristic
207C2×C4×C79#12111yescharacteristic
208C2×C4×C79#13512yes—
209C2×C4×C79AGL(1,ℤ/8)12yes—
210C2×C4×C79C22×D411yescharacteristic
211C2×C4×C79C22×D411yescharacteristic
212C2×C4×C79C2×#4111yescharacteristic
213C2×Dic79—42no—
214C2×Dic79—82no—
215Dic158—82no—
216Dic158—82no—
217C8×C79—41no—
218C8×C79—41no—
219C79⋊C8—22no—
220C79⋊C8—22no—
221C79⋊C8—42no—
222C79⋊C8—42no—
223C22×C79—21no—
224C22×C79—21no—
225C22×C79—22no—
226C22×C79—41no—
227C22×C79—41no—
228C22×C79—41no—
229C22×C79—41no—
230C22×C79—41no—
231C22×C79—84no—
232C22×C79—81no—
233C22×C79—81no—
234C22×C79—81no—
235C22×C79#46312yes—
236C22×C79C2×#12111yescharacteristic
237C4×C79—21no—
238C4×C79—21no—
239C4×C79—21no—
240C4×C79—41no—
241C4×C79—41no—
242C4×C79—81no—
243C4×C79#46312yes—
244C4×C79C2×#13511yescharacteristic
245C4×C79C2×AGL(1,ℤ/8)11yescharacteristic
246C4×C79C2×#13711yescharacteristic
247C4×C79#57511yescharacteristic
248Dic79—82no—
249#19168—791nomaximal, Sylow
250C2×C79—22no—
251C2×C79—42no—
252C2×C79—81no—
253C2×C79—161no—
254C2×C79C2×#46311yescharacteristic
255C2×C79#305611yescharacteristic
256C2×C79#306811yescharacteristic
257#3141—791no—
258#3100D7911yescharacteristic
259C2×#476—791no—
260D4×M16—791no—
261#1703—792no—
262#1697—792no—
263#1683—792no—
264#1677—792no—
265#1565—791no—
266#1548—791no—
267#1547—791no—
268C79#1916811yescharacteristic, complemented, Sylow
269C22×M16—1581no—
270#532C2×D7911yescharacteristic
271C2×C4⋊Q8—791no—
272C2×C4×D4C2×D7911yescharacteristic
273#489—1582no—
274#487—792no—
275#486—792no—
276#480—21no—
277#476—792no—
278#476—792no—
279#476—1581no—
280#476—1581no—
281#474C2×D7911yescharacteristic
282#474C2×D7911yescharacteristic
283#459C2×D7911yescharacteristic
284#459C2×D7911yescharacteristic
285#447—21no—
286C4×D8—21no—
287C4×D8—21no—
288#436—792no—
289#435—1582no—
290C8×D4—792no—
291#431C2×D7911yescharacteristic
292C2×#106—1581no—
293#423—791no—
294C2×C4⋊C8—791no—
295#417—1581no—
296#417—1581no—
297C2×#102—1581no—
298#407—1581no—
299C4×M16—791no—
300#335—792no—
301#334—792no—
302#333—792no—
303#333—792no—
304#332—792no—
305#331—792no—
306C2×#41—21no—
307C22×Q8—1581no—
308C22×D4C22⋊D7911yescharacteristic
309C23×C4—21no—
310C2×D8—21no—
311C2×D8—21no—
312C2×M16—3164no—
313C2×M16—791no—
314C2×M16—791no—
315C2×M16—1582no—
316C2×M16—1581no—
317C22×C8—1582no—
318C22×C8—1581no—
319C4⋊Q8—792no—
320C4⋊Q8—1581no—
321C4⋊Q8—1581no—
322C4⋊Q8—1581no—
323C4⋊Q8C22×D7911yescharacteristic
324#127C22×D7911yescharacteristic
325#124—21no—
326#121—21no—
327#121—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4C22⋊D7911yescharacteristic
333C4×D4C22⋊D7911yescharacteristic
334C4×D4C22×D7911yescharacteristic
335C4×D4C22×D7911yescharacteristic
336C2×C4⋊C4—1581no—
337C2×C4⋊C4—1581no—
338C2×C4⋊C4C22⋊D7911yescharacteristic
339C2×C22⋊C4—21no—
340C2×C42C22×D7911yescharacteristic
341#107—21no—
342#107—21no—
343#106—1582no—
344#106—1582no—
345C4⋊C8—792no—
346C4⋊C8—792no—
347C4⋊C8—1581no—
348C4⋊C8—1581no—
349C4⋊C8C22×D7911yescharacteristic
350C4⋊C8C22×D7911yescharacteristic
351#103—1582no—
352#103—1582no—
353#102—21no—
354#102—21no—
355#102—21no—
356#102—21no—
357#102—1582no—
358#102—1582no—
359#102—1582no—
360#102—1582no—
361C22⋊C8—21no—
362C22⋊C8—1582no—
363C22⋊C8—1582no—
364#97—792no—
365C4×C8—21no—
366C4×C8—792no—
367C24—21no—
368#41—42no—
369#41—42no—
370C2×Q8—21no—
371C2×Q8—3161no—
372C2×Q8—6322no—
373C2×Q8—1582no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4C2×C22⋊D7911yescharacteristic
382C2×D4C2×C22⋊D7911yescharacteristic
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—3161no—
392C22×C4D4×D7911yescharacteristic
393C22×C4C2×C22⋊D7911yescharacteristic
394C22×C4#1192962311yescharacteristic
395D8—42no—
396D8—42no—
397M16—3162no—
398M16—3164no—
399M16—1582no—
400M16—1582no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—3162no—
404C2×C8—3164no—
405C2×C8—792no—
406C2×C8—792no—
407C2×C8—1582no—
408C2×C8—1582no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—3161no—
413C4⋊C4—3161no—
414C4⋊C4—1582no—
415C4⋊C4—1582no—
416C4⋊C4C2×C22⋊D7911yescharacteristic
417C4⋊C4C2×C22⋊D7911yescharacteristic
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C42D4×D7911yescharacteristic
425C42C2×C22⋊D7911yescharacteristic
426C42#1192962311yescharacteristic
427C42C23×D7911yescharacteristic
428C23—21no—
429C23—21no—
430C23—42no—
431C23—42no—
432C23—41no—
433C23—81no—
434C23#1457913311yescharacteristic
435Q8—41no—
436Q8—41no—
437Q8—6322no—
438Q8—6322no—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—42no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—41no—
447D4—81no—
448D4—82no—
449D4#1457919112yes—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—82no—
464C2×C4—3162no—
465C2×C4—6322no—
466C2×C4#1457913311yescharacteristic
467C2×C4#1457913311yescharacteristic
468C2×C4#1457913311yescharacteristic
469C2×C4#1457918012yes—
470C2×C4#1457918512yes—
471C2×C4C2×D4×D7911yescharacteristic
472C2×C4C22×C22⋊D7911yescharacteristic
473C2×C4C2×#1192962311yescharacteristic
474C8—41no—
475C8—41no—
476C8—3162no—
477C8—3162no—
478C8—1582no—
479C8—1582no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—84no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22#1909877612yes—
493C22C2×#1457913311yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—41no—
498C4—41no—
499C4—81no—
500C4—6322no—
501C4#1909877612yes—
502C4C2×#1457918011yescharacteristic
503C4C2×#1457918511yescharacteristic
504C4C2×#1457919111yescharacteristic
505C4#1909936111yescharacteristic
506C2—22no—
507C2—42no—
508C2—81no—
509C2—161no—
510C2C2×#1909877611yescharacteristic
511C2#2377303211yescharacteristic
512C2#2377313011yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
Fitting, prime radicalC79×#3100C2
socleC22×C79C2×#121
hypercenter#3100D79
hypocenterC79#19168
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #23773793C2semidirect
2. C2×#19098853C2semidirect
3. D4×#11929608C2semidirect
4. #23762973C2semidirect
5. #23762931C2semidirect
6. #23762849C2semidirect
7. #23762807C2semidirect
8. #23762153C2semidirect
9. #23762080C2semidirect
10. #23762077C2semidirect
15. #19098913C22semidirect
16. #19098909C22semidirect
32. #19098315C22semidirect
33. #19098313C22semidirect
34. #19098311C22semidirect
35. #19098311C22semidirect
36. #19098309C22semidirect
37. #19098307C22semidirect
268. C79#19168semidirect

Maximal subgroups

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

subgroupindex
#237737932
C2×#190988532
D4×#119296082
#237629732
#237629312
#237628492
#237628072
#237621532
#237620802
#237620772
C79×#31002
#1916879