Finite Group Databasebeta

← #34292686

Subgroups — #34292686

Order 20224.

730 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772198C211yesmaximal, characteristic
2#23772042C211yesmaximal, characteristic, complemented
3#23764292C211yesmaximal, characteristic
4#23764287C211yesmaximal, characteristic
5#23764150C211yesmaximal, characteristic, complemented
6#23764145C211yesmaximal, characteristic, complemented
7#23764114C211yesmaximal, characteristic, complemented
8#23764110C211yesmaximal, characteristic, complemented
9#23764107C211yesmaximal, characteristic, complemented
10#23764103C211yesmaximal, characteristic, complemented
11#23763307C211yesmaximal, characteristic, complemented
12#23763213C211yesmaximal, characteristic, complemented
13#23763210C211yesmaximal, characteristic, complemented
14#23763086C211yesmaximal, characteristic, complemented
15C79×#2922C211yesmaximal, characteristic, complemented
16C2×#14579181C2211yescharacteristic
17C4×#11929624C2211yescharacteristic
18#19098673C412yescomplemented
19#19098668C412yescomplemented
20#19098663C412yescomplemented
21#19098661C412yescomplemented
22#19098622—22no—
23#19098614—22no—
24C2×#14579096C2211yescharacteristic
25C2×#14579096C2211yescharacteristic
26#19098599C2211yescharacteristic
27#19098589C2211yescharacteristic
28#19098589C2211yescharacteristic
29C2×#14579088C2211yescharacteristic
30C2×#14579088C2211yescharacteristic
31C2×#14579087C2211yescharacteristic
32C2×#14579087C2211yescharacteristic
33#19098559C2211yescharacteristic
34#19098557C2211yescharacteristic
35#19098529C2211yescharacteristic
36#19098529C2211yescharacteristic
37C4×#11929608C2211yescharacteristic
38C2×#14579072C2211yescharacteristic
39C2×#14579071C2211yescharacteristic
40C2×#14579071C2211yescharacteristic
41C2×C8×Dic79C2211yescharacteristic
42C2×C8×Dic79C2211yescharacteristic
43#19098409C2211yescharacteristic
44#19098327C2211yescharacteristic
45#19098327C2211yescharacteristic
46#19098324C2211yescharacteristic
47#19098324C2211yescharacteristic
48#19098322C2211yescharacteristic
49#19098322C2211yescharacteristic
50C4×#41×C79C2211yescharacteristic
51#433×C79—22no—
52#431×C79—22no—
53#424×C79C2211yescharacteristic
54#424×C79C2211yescharacteristic
55#408×C79C2211yescharacteristic
56#408×C79C2211yescharacteristic
57C4×M16×C79C2211yescharacteristic
58C2×C4×C8×C79C2211yescharacteristic
59C2×#11929624C2311yescharacteristic
60#14579181—22no—
61#14579181—22no—
62C2×#11929614C2×C411yescharacteristic
63C2×#11929612C2×C411yescharacteristic
64C2×#11929611C2×C411yescharacteristic
65C2×#11929609C2×C411yescharacteristic
66C2×#11929608C2311yescharacteristic
67C2×#11929608C2311yescharacteristic
68C22×C79⋊C8C2311yescharacteristic
69C22×C79⋊C8C2311yescharacteristic
70C4×Dic158C2×C412yes—
71C4×D316C2×C412yes—
72C4×C22⋊D79—22no—
73#14579120C2311yescharacteristic
74#14579118—21no—
75C2×C4×Dic79C2311yescharacteristic
76C2×C4×Dic79C2311yescharacteristic
77C42×D79—21no—
78#14579096—22no—
79#14579096—22no—
80#14579094C2×C412yes—
81#14579092C2×C412yes—
82Dic79⋊C8—22no—
83Dic79⋊C8—22no—
84#14579088—22no—
85#14579088—22no—
86#14579087—22no—
87#14579087—22no—
88#14579085C2×C412yes—
89#14579084C2×C412yes—
90#14579082C2×C412yes—
91#14579081C2×C412yes—
92(C2×D79)⋊C8—22no—
93(C2×D79)⋊C8—22no—
94#14579072—21no—
95#14579072C2×C412yes—
96#14579072C2×C412yes—
97#14579071—22no—
98#14579071—22no—
99C4×C79⋊C8—21no—
100C8×Dic79—22no—
101C8×Dic79—22no—
102C79⋊#95—22no—
103C2×#41×C79C2311yescharacteristic
104C2×M16×C79C2311yescharacteristic
105C2×M16×C79C2311yescharacteristic
106C22×C8×C79C2311yescharacteristic
107C22×C8×C79C2311yescharacteristic
108C4×Q8×C79C2×C412yes—
109C4×D4×C79—22no—
110C4×D4×C79C2×C412yes—
111#117×C79—21no—
112#117×C79C2311yescharacteristic
113C2×C42×C79—21no—
114C2×C42×C79C2311yescharacteristic
115C4⋊C8×C79—22no—
116C4⋊C8×C79—22no—
117C22⋊C8×C79—22no—
118C22⋊C8×C79—22no—
119#97×C79—21no—
120C4×C8×C79—21no—
121C4×C8×C79D412yes—
122C4×C8×C79D412yes—
123#11929624—21no—
124#11929624—21no—
125#11929624—21no—
126#11929624—21no—
127C2×Dic158C22×C411yescharacteristic
128C2×D316C22×C411yescharacteristic
129C2×C22⋊D79—21no—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C2×C4×D79—22no—
133C2×C4×D79—21no—
134#11929614—22no—
135#11929612—22no—
136#11929611—22no—
137#11929609—22no—
138#11929608—21no—
139#11929608—21no—
140#11929608—21no—
141#11929608—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8C22×C412yes—
147C2×C79⋊C8C22×C411yescharacteristic
148C2×C79⋊C8C22×C411yescharacteristic
149C2×C79⋊C8#4112yes—
150C2×C79⋊C8#4112yes—
151C4⋊Dic79C22×C412yes—
152Dic79⋊C4—22no—
153C22⋊Dic79—21no—
154(C2×D79)⋊C4—22no—
155C4×Dic79—22no—
156C4×Dic79—21no—
157C4×Dic79—21no—
158C4×Dic79—21no—
159C4×Dic79#4112yes—
160C4×Dic79#4112yes—
161#41×C79—21no—
162#41×C79—21no—
163#41×C79—21no—
164#41×C79—21no—
165C2×Q8×C79C22×C411yescharacteristic
166C2×D4×C79—21no—
167C2×D4×C79C22×C411yescharacteristic
168C22×C4×C79—22no—
169C22×C4×C79—21no—
170C22×C4×C79#4112yes—
171C22×C4×C79C2411yescharacteristic
172M16×C79—21no—
173M16×C79—21no—
174M16×C79—21no—
175M16×C79—21no—
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79C2×D412yes—
183C2×C8×C79C2×D412yes—
184C4⋊C4×C79—22no—
185C4⋊C4×C79C22×C412yes—
186C22⋊C4×C79—21no—
187C22⋊C4×C79—22no—
188C42×C79—22no—
189C42×C79—21no—
190C42×C79C22×C412yes—
191C42×C79C2×D411yescharacteristic
192C42×C79C2×D411yescharacteristic
193C23×C79—21no—
194C23×C79C2×#4111yescharacteristic
195C22×D79—21no—
196Q8×C79—21no—
197Q8×C79—21no—
198D4×C79—21no—
199D4×C79—21no—
200D4×C79—41no—
201D4×C79—41no—
202C2×C4×C79—22no—
203C2×C4×C79—21no—
204C2×C4×C79—42no—
205C2×C4×C79—44no—
206C2×C4×C79—41no—
207C2×C4×C79—41no—
208C2×C4×C79C4×D412yes—
209C2×C4×C79C4×D412yes—
210C2×C4×C79C23×C411yescharacteristic
211C2×C4×C79C22×D411yescharacteristic
212C2×C4×C79C2×#4111yescharacteristic
213C2×C4×C79C2×#4111yescharacteristic
214C2×C4×C79C2×#4111yescharacteristic
215C2×C4×C79C2×#4111yescharacteristic
216C22⋊D79—41no—
217C22⋊D79—41no—
218C2×Dic79—22no—
219C2×Dic79—21no—
220C2×Dic79—22no—
221C2×Dic79—22no—
222C2×Dic79—42no—
223C2×Dic79—42no—
224D316—21no—
225D316—21no—
226C4×D79—44no—
227C4×D79—41no—
228C4×D79—41no—
229Dic158—21no—
230Dic158—21no—
231C8×C79—22no—
232C8×C79—22no—
233C8×C79—41no—
234C8×C79—41no—
235C79⋊C8—22no—
236C79⋊C8—41no—
237C79⋊C8—41no—
238C79⋊C8#11712yes—
239C79⋊C8#11712yes—
240C22×C79—22no—
241C22×C79—42no—
242C22×C79—41no—
243C22×C79C2×C4×D411yescharacteristic
244C22×C79#52911yescharacteristic
245C22×C79#52911yescharacteristic
246C2×D79—42no—
247C2×D79—41no—
248C4×C79—22no—
249C4×C79—22no—
250C4×C79—42no—
251C4×C79—41no—
252C4×C79C2×#11711yescharacteristic
253C4×C79C4×#4111yescharacteristic
254C4×C79#53211yescharacteristic
255C4×C79#53211yescharacteristic
256Dic79—42no—
257Dic79—42no—
258Dic79—42no—
259Dic79—41no—
260#21296—791nomaximal, Sylow
261C2×C79—21no—
262C2×C79—21no—
263C2×C79—81no—
264C2×C79#229311yescharacteristic
265C2×C79#295512yes—
266D79—81no—
267#2943—791no—
268#2922—791no—
269#2922D7911yescharacteristic, complemented
270#1975—791no—
271#1974—791no—
272#1946—791no—
273#1945—791no—
274#1938—791no—
275#1938—791no—
276#1937—791no—
277#1937—791no—
278#1778—791no—
279#1759—791no—
280#1759—791no—
281#1726—791no—
282C79#2129611yescharacteristic, complemented, Sylow
283C2×#135—791no—
284C4×#41—791no—
285C4×#41C2×D7911yescharacteristic
286#442—792no—
287#441—792no—
288#440—792no—
289#440—792no—
290#433—22no—
291#433—1582no—
292#431—22no—
293#431—1582no—
294C2×#108—791no—
295C2×#108—791no—
296#427—791no—
297#424—791no—
298#424—791no—
299#424C2×D7911yescharacteristic
300#424C2×D7911yescharacteristic
301C2×#104—791no—
302C2×#104—791no—
303C2×#104—791no—
304C2×#104—791no—
305#416—791no—
306#416—791no—
307#408—791no—
308#408—791no—
309#408C2×D7911yescharacteristic
310#408C2×D7911yescharacteristic
311C4×M16—791no—
312C4×M16C2×D7911yescharacteristic
313C2×#97—791no—
314C2×#97—791no—
315C2×#97—791no—
316C2×C4×C8—791no—
317C2×C4×C8—791no—
318C2×C4×C8C2×D7911yescharacteristic
319#376—791no—
320#341—791no—
321#341—791no—
322#339—791no—
323#339—791no—
324#339—791no—
325#339—791no—
326C2×#41—791no—
327C2×#41C22×D7911yescharacteristic
328#135—1582no—
329#135—1582no—
330C2×Q16—791no—
331C2×SD16—791no—
332C2×SD16—791no—
333C2×D8—791no—
334C2×M16—791no—
335C2×M16—791no—
336C2×M16C22×D7911yescharacteristic
337C2×M16C22×D7911yescharacteristic
338C22×C8—791no—
339C22×C8—791no—
340C22×C8C22×D7911yescharacteristic
341C22×C8C22×D7911yescharacteristic
342C4×Q8—792no—
343C4×Q8C4×D7912yes—
344C4×D4—22no—
345C4×D4—792no—
346C4×D4—1582no—
347C4×D4C4×D7912yes—
348#117—21no—
349#117—791no—
350#117—1581no—
351#117C22×D7911yescharacteristic
352C2×C42—21no—
353C2×C42—791no—
354C2×C42—791no—
355C2×C42—1581no—
356C2×C42C22×D7911yescharacteristic
357#108—1582no—
358#108—1582no—
359#107—792no—
360#106—792no—
361C4⋊C8—22no—
362C4⋊C8—22no—
363C4⋊C8—1582no—
364C4⋊C8—1582no—
365#104—1582no—
366#104—1582no—
367#104—1582no—
368#104—1582no—
369#103—792no—
370#103—792no—
371#102—792no—
372#102—792no—
373C22⋊C8—22no—
374C22⋊C8—22no—
375C22⋊C8—1582no—
376C22⋊C8—1582no—
377#97—21no—
378#97—792no—
379#97—792no—
380#97—1582no—
381#97—1582no—
382#97—1581no—
383C4×C8—21no—
384C4×C8—1582no—
385C4×C8—1582no—
386C4×C8—1581no—
387C4×C8C22⋊D7912yes—
388C4×C8C22⋊D7912yes—
389#95—1582no—
390#41—21no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—1581no—
395#41—1581no—
396#41—1581no—
397#41—1581no—
398C2×Q8—791no—
399C2×Q8C2×C4×D7911yescharacteristic
400C2×D4—21no—
401C2×D4—791no—
402C2×D4—1581no—
403C2×D4C2×C4×D7911yescharacteristic
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—1581no—
407C22×C4—1581no—
408C22×C4—1582no—
409C22×C4—1581no—
410C22×C4#1192962412yes—
411C22×C4C23×D7911yescharacteristic
412Q16—1582no—
413SD16—1582no—
414SD16—1582no—
415D8—1582no—
416M16—21no—
417M16—21no—
418M16—21no—
419M16—21no—
420M16—1581no—
421M16—1581no—
422M16—1581no—
423M16—1581no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—792no—
431C2×C8—792no—
432C2×C8—791no—
433C2×C8—791no—
434C2×C8—792no—
435C2×C8—1581no—
436C2×C8—1581no—
437C2×C8—1581no—
438C2×C8—1581no—
439C2×C8C2×C22⋊D7912yes—
440C2×C8C2×C22⋊D7912yes—
441C4⋊C4—22no—
442C4⋊C4—792no—
443C4⋊C4—1582no—
444C4⋊C4C2×C4×D7912yes—
445C22⋊C4—21no—
446C22⋊C4—22no—
447C22⋊C4—1581no—
448C22⋊C4—1582no—
449C42—22no—
450C42—21no—
451C42—792no—
452C42—792no—
453C42—1581no—
454C42—1581no—
455C42—1582no—
456C42—1581no—
457C42C2×C4×D7912yes—
458C42C2×C22⋊D7911yescharacteristic
459C42C2×C22⋊D7911yescharacteristic
460C23—21no—
461C23—1581no—
462C23C2×#1192962411yescharacteristic
463Q8—21no—
464Q8—21no—
465Q8—1581no—
466Q8—1581no—
467D4—21no—
468D4—21no—
469D4—41no—
470D4—41no—
471D4—3161no—
472D4—3161no—
473D4—1581no—
474D4—1581no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—3164no—
482C2×C4—3162no—
483C2×C4—3162no—
484C2×C4—3161no—
485C2×C4—3161no—
486C2×C4—1582no—
487C2×C4—1581no—
488C2×C4—1582no—
489C2×C4—1582no—
490C2×C4C4×C22⋊D7912yes—
491C2×C4C4×C22⋊D7912yes—
492C2×C4C22×C4×D7911yescharacteristic
493C2×C4C22×C22⋊D7911yescharacteristic
494C2×C4#41×D7911yescharacteristic
495C2×C4#41×D7911yescharacteristic
496C2×C4#41×D7911yescharacteristic
497C2×C4#41×D7911yescharacteristic
498C8—22no—
499C8—22no—
500C8—41no—
501C8—41no—
502C8—3161no—
503C8—3161no—
504C8—792no—
505C8—792no—
506C8—1582no—
507C22—22no—
508C22—41no—
509C22—42no—
510C22—3161no—
511C22—3162no—
512C22C2×C4×C22⋊D7911yescharacteristic
513C22#1909909111yescharacteristic
514C22#1909909111yescharacteristic
515C4—22no—
516C4—22no—
517C4—41no—
518C4—42no—
519C4—3162no—
520C4—3162no—
521C4—3161no—
522C4—3162no—
523C4#117×D7911yescharacteristic
524C4#1909903111yescharacteristic
525C4#1909910911yescharacteristic
526C4#1909910911yescharacteristic
527C2—21no—
528C2—21no—
529C2—81no—
530C2—6321no—
531C2#2376585911yescharacteristic
532C2#2377230112yes—
subgroupG/H
centerC2×C4C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#2922C2
socleC22×C79C2×C4×D4
prime radicalC2×#14579181C22
cosocleC22×C4×C79C24
hypercenter#2922D79
hypocenterC79#21296
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2922

Complements

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

NH ≅ G/Ntype
2. #23772042C2semidirect
5. #23764150C2semidirect
6. #23764145C2semidirect
7. #23764114C2semidirect
8. #23764110C2semidirect
9. #23764107C2semidirect
10. #23764103C2semidirect
11. #23763307C2semidirect
12. #23763213C2semidirect
13. #23763210C2semidirect
14. #23763086C2semidirect
15. C79×#2922C2semidirect
18. #19098673C4semidirect
19. #19098668C4semidirect
20. #19098663C4semidirect
21. #19098661C4semidirect
269. #2922D79semidirect
282. C79#21296semidirect

Maximal subgroups

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

subgroupindex
#237721982
#237720422
#237642922
#237642872
#237641502
#237641452
#237641142
#237641102
#237641072
#237641032
#237633072
#237632132
#237632102
#237630862
C79×#29222
#2129679