Finite Group Databasebeta

← #34259666

Subgroups — #34259666

Order 20224.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773437C211yesmaximal, characteristic
2#23772498C211yesmaximal, characteristic
3#23772488C211yesmaximal, characteristic
4#23762307C211yesmaximal, characteristic, complemented
5#23762296C211yesmaximal, characteristic, complemented
6#23762268C211yesmaximal, characteristic, complemented
7#23762268C211yesmaximal, characteristic, complemented
8#23762252C211yesmaximal, characteristic, complemented
9#23762217C211yesmaximal, characteristic, complemented
10Q16×C79⋊C8C211yesmaximal, characteristic, complemented
11#23762179C211yesmaximal, characteristic
12#23761871C211yesmaximal, characteristic
13#23761793C211yesmaximal, characteristic
14#23761701C211yesmaximal, characteristic
15C79×#1584C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17C4×Q8×D79C2211yescharacteristic
18#19098900C411yescharacteristic
19#19098897C411yescharacteristic
20#19098857C411yescharacteristic
21#19098856C411yescharacteristic
22#19098772C411yescharacteristic
23#19098771C411yescharacteristic
24#19098720C411yescharacteristic
25#19098717C411yescharacteristic
26#19098693—21no—
27#19098693C2211yescharacteristic
28#19098691—21no—
29#19098690—21no—
30#19098690C2211yescharacteristic
31#19098690C2211yescharacteristic
32Q8×C79⋊C8C2211yescharacteristic
33Q8×C79⋊C8C2211yescharacteristic
34C8×Dic158C2211yescharacteristic
35#19098687—21no—
36C4×Dic316C2211yescharacteristic
37C4×#11929614C2211yescharacteristic
38C4×#11929614C2211yescharacteristic
39#19098625—21no—
40#19098623—21no—
41#19098619—21no—
42#19098619C2211yescharacteristic
43#19098615C2211yescharacteristic
44#19098614—21no—
45#19098614C2211yescharacteristic
46#19098614C2211yescharacteristic
47#19098614C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098313C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54Q8⋊(C79⋊C8)C2211yescharacteristic
55#19098293C2211yescharacteristic
56#19098293C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×Q8×D79C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×#11929614C2×C411yescharacteristic
70C2×#11929614C2×C411yescharacteristic
71#14579160C2×C411yescharacteristic
72#14579159C2×C411yescharacteristic
73#14579151—21no—
74#14579141C2×C411yescharacteristic
75#14579139—21no—
76#14579137C2×C411yescharacteristic
77Q8×Dic79C2×C411yescharacteristic
78C4×Dic158—21no—
79C4×Dic158—21no—
80C4×Dic158C2311yescharacteristic
81C4×Dic158C2311yescharacteristic
82#14579126—21no—
83#14579126C2×C411yescharacteristic
84#14579118—21no—
85C4⋊C4×D79—21no—
86C4⋊C4×D79C2×C411yescharacteristic
87C42×D79—21no—
88C42×D79C2311yescharacteristic
89#14579095C2×C411yescharacteristic
90#14579094C2×C411yescharacteristic
91#14579094C2×C411yescharacteristic
92C4⋊(C79⋊C8)—21no—
93C4⋊(C79⋊C8)—21no—
94C4⋊(C79⋊C8)C2311yescharacteristic
95C4⋊(C79⋊C8)C2311yescharacteristic
96Dic79⋊C8—21no—
97Dic79⋊C8—21no—
98Dic79⋊C8—21no—
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101Dic79⋊C8C2×C411yescharacteristic
102Dic79⋊C8C2×C411yescharacteristic
103Dic79⋊C8D411yescharacteristic
104Dic79⋊C8D411yescharacteristic
105#14579086C2×C411yescharacteristic
106#14579086C2×C411yescharacteristic
107#14579085C2×C411yescharacteristic
108#14579085C2×C411yescharacteristic
109#14579084C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111(C2×D79)⋊C8—21no—
112(C2×D79)⋊C8—21no—
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8C2×C411yescharacteristic
115(C2×D79)⋊C8C2×C411yescharacteristic
116(C2×D79)⋊C8D411yescharacteristic
117(C2×D79)⋊C8D411yescharacteristic
118#14579071—21no—
119#14579071—21no—
120#14579071—21no—
121#14579071—21no—
122C4×C79⋊C8—21no—
123C4×C79⋊C8—21no—
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79—21no—
129C2×Q16×C79C2×C411yescharacteristic
130C4×Q8×C79C2311yescharacteristic
131C4×Q8×C79C2311yescharacteristic
132#107×C79C2×C411yescharacteristic
133C4⋊C8×C79—21no—
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#103×C79C2×C411yescharacteristic
139#97×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158—21no—
146C2×Dic158C22×C411yescharacteristic
147C2×Dic158C22×C411yescharacteristic
148Q8×D79—22no—
149Q8×D79—41no—
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79C22×C411yescharacteristic
153C2×C4×D79C2×D411yescharacteristic
154C2×C4×D79#4111yescharacteristic
155Dic316—41no—
156#11929614—22no—
157#11929614—41no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8C22×C411yescharacteristic
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C2×D411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C4⋊Dic79—21no—
166C4⋊Dic79—21no—
167C4⋊Dic79C22×C411yescharacteristic
168C4⋊Dic79C22×C411yescharacteristic
169Dic79⋊C4—21no—
170Dic79⋊C4—21no—
171Dic79⋊C4—21no—
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177(C2×D79)⋊C4—21no—
178(C2×D79)⋊C4—21no—
179C4×Dic79—21no—
180C4×Dic79—21no—
181C4×Dic79—21no—
182C4×Dic79—21no—
183C4×Dic79C22×C411yescharacteristic
184C4×Dic79C2×D411yescharacteristic
185C4×Dic79#4111yescharacteristic
186C2×Q8×C79C22×C411yescharacteristic
187C2×Q8×C79C22×C411yescharacteristic
188Q16×C79—22no—
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79C22×C411yescharacteristic
193C2×C8×C79C22×C411yescharacteristic
194C2×C8×C79C2×D411yescharacteristic
195C2×C8×C79C2×D411yescharacteristic
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79C22×C411yescharacteristic
199C4⋊C4×C79C22×C411yescharacteristic
200C42×C79—21no—
201C42×C79—21no—
202C42×C79C2411yescharacteristic
203C22×D79C4×D411yescharacteristic
204Q8×C79—21no—
205Q8×C79—21no—
206Q8×C79—21no—
207Q8×C79#13112yes—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79C23×C411yescharacteristic
213C2×C4×C79C22×D411yescharacteristic
214C2×C4×C79C2×#4111yescharacteristic
215C2×Dic79—21no—
216C2×Dic79—21no—
217C2×Dic79—21no—
218C2×Dic79—21no—
219C2×Dic79C4×D411yescharacteristic
220C2×Dic79C4×D411yescharacteristic
221C2×Dic79C4×D411yescharacteristic
222C4×D79—21no—
223C4×D79—21no—
224C4×D79—21no—
225C4×D79—21no—
226C4×D79—41no—
227C4×D79—42no—
228C4×D79—42no—
229Dic158—21no—
230Dic158—21no—
231Dic158—21no—
232Dic158—21no—
233Dic158—42no—
234Dic158—42no—
235C8×C79—21no—
236C8×C79—21no—
237C8×C79—21no—
238C8×C79—21no—
239C8×C79—41no—
240C8×C79—41no—
241C8×C79—41no—
242C79⋊C8—21no—
243C79⋊C8—21no—
244C79⋊C8—21no—
245C79⋊C8—21no—
246C79⋊C8—41no—
247C79⋊C8—41no—
248C79⋊C8—41no—
249C22×C79C2×C4×D411yescharacteristic
250C2×D79—21no—
251C2×D79—21no—
252C2×D79—21no—
253C4×C79—22no—
254C4×C79—21no—
255C4×C79—41no—
256C4×C79—41no—
257C4×C79—41no—
258C4×C79C2×#13111yescharacteristic
259C4×C79#56811yescharacteristic
260C4×C79#57811yescharacteristic
261C4×C79#57811yescharacteristic
262Dic79—21no—
263Dic79—21no—
264Dic79—21no—
265Dic79—41no—
266Dic79—41no—
267Dic79—41no—
268Dic79—41no—
269#18032—791nomaximal, Sylow
270C2×C79#293311yescharacteristic
271C2×C79#295011yescharacteristic
272C2×C79#295511yescharacteristic
273D79—41no—
274#3104—791no—
275#2986—791no—
276#2984—791no—
277#1593—791no—
278#1591—791no—
279#1587—791no—
280#1587—791no—
281#1584—791no—
282#1584D7911yescharacteristic, complemented
283#1579—791no—
284C8×Q16—791no—
285#1572—791no—
286#1497—791no—
287#1479—791no—
288#1454—791no—
289C79#1803211yescharacteristic, complemented, Sylow
290#533—791no—
291C2×C4×Q8—791no—
292#484—791no—
293#484—791no—
294#477—791no—
295#477—791no—
296#462—791no—
297#462—791no—
298#451—791no—
299#451—791no—
300#446—791no—
301#446—791no—
302#446—791no—
303#446—1581no—
304#446—1581no—
305#446—1581no—
306#446C2×D7911yescharacteristic
307C8×Q8—791no—
308C8×Q8—791no—
309C8×Q8—791no—
310C8×Q8—1581no—
311C8×Q8C2×D7911yescharacteristic
312C4×Q16—791no—
313C4×Q16—791no—
314C4×Q16—791no—
315C4×Q16C2×D7911yescharacteristic
316#433—1581no—
317#433—1581no—
318#432—791no—
319#432—1581no—
320#431—791no—
321#431—791no—
322#431—791no—
323#431—791no—
324#431—1581no—
325#335—791no—
326#335C2×D7911yescharacteristic
327#334—791no—
328#334—791no—
329Q8⋊C8—791no—
330Q8⋊C8—791no—
331Q8⋊C8—791no—
332Q8⋊C8—791no—
333Q8⋊C8—791no—
334Q8⋊C8—791no—
335Q8⋊C8C2×D7911yescharacteristic
336Q8⋊C8C2×D7911yescharacteristic
337#322—791no—
338#322—791no—
339#322C2×D7911yescharacteristic
340C82—791no—
341C22×Q8—791no—
342C2×Q16—791no—
343C2×Q16—791no—
344C2×Q16—791no—
345C2×Q16C4×D7911yescharacteristic
346C4⋊Q8—791no—
347C4⋊Q8—791no—
348#125—1581no—
349C22⋊Q8—791no—
350C22⋊Q8—791no—
351C22⋊Q8—1581no—
352C4×Q8—791no—
353C4×Q8—791no—
354C4×Q8—791no—
355C4×Q8—791no—
356C4×Q8—1581no—
357C4×Q8—1581no—
358C4×Q8—1581no—
359C4×Q8C22×D7911yescharacteristic
360C4×Q8C22×D7911yescharacteristic
361#117—1581no—
362C2×C4⋊C4—791no—
363C2×C4⋊C4—1581no—
364C2×C42—791no—
365C2×C42—1581no—
366#107—791no—
367#107—791no—
368#107—791no—
369#107C4×D7911yescharacteristic
370C4⋊C8—21no—
371C4⋊C8—21no—
372C4⋊C8—791no—
373C4⋊C8—791no—
374C4⋊C8—791no—
375C4⋊C8—791no—
376C4⋊C8—791no—
377C4⋊C8—791no—
378C4⋊C8—1581no—
379C4⋊C8—1581no—
380C4⋊C8—1581no—
381C4⋊C8—1581no—
382C4⋊C8—1581no—
383C4⋊C8—1581no—
384C4⋊C8—1581no—
385C4⋊C8C22×D7911yescharacteristic
386C4⋊C8C22×D7911yescharacteristic
387#103—791no—
388#103—791no—
389#103—791no—
390#103—791no—
391#103—791no—
392#103—791no—
393#103C4×D7911yescharacteristic
394#103C4×D7911yescharacteristic
395C22⋊C8—791no—
396C22⋊C8—791no—
397C22⋊C8—791no—
398C22⋊C8—791no—
399C22⋊C8—1581no—
400C22⋊C8—1581no—
401C22⋊C8—1581no—
402#97—21no—
403#97—1581no—
404#97—1581no—
405#97—1581no—
406#97—1581no—
407C4×C8—21no—
408C4×C8—791no—
409C4×C8—791no—
410C4×C8—791no—
411C4×C8—1581no—
412C4×C8—1581no—
413C4×C8—1581no—
414C4×C8—1581no—
415C4×C8C22×D7911yescharacteristic
416C4×C8C22×D7911yescharacteristic
417C4×C8C22×D7911yescharacteristic
418C2×Q8—3161no—
419C2×Q8—791no—
420C2×Q8—791no—
421C2×Q8—1581no—
422C2×Q8—1581no—
423C2×Q8—1582no—
424C2×Q8C2×C4×D7911yescharacteristic
425C2×Q8C2×C4×D7911yescharacteristic
426C22×C4—791no—
427C22×C4—791no—
428C22×C4—791no—
429C22×C4—1581no—
430C22×C4—1581no—
431Q16—22no—
432Q16—3161no—
433Q16—3161no—
434Q16—1582no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—791no—
439C2×C8—791no—
440C2×C8—791no—
441C2×C8—791no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8C2×C4×D7911yescharacteristic
446C2×C8C2×C4×D7911yescharacteristic
447C2×C8D4×D7911yescharacteristic
448C2×C8D4×D7911yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—791no—
452C4⋊C4—791no—
453C4⋊C4—1581no—
454C4⋊C4—1581no—
455C4⋊C4—1581no—
456C4⋊C4—1581no—
457C4⋊C4—1581no—
458C4⋊C4—1581no—
459C4⋊C4—1581no—
460C4⋊C4—1581no—
461C4⋊C4—1581no—
462C4⋊C4—1581no—
463C4⋊C4C2×C4×D7911yescharacteristic
464C4⋊C4C2×C4×D7911yescharacteristic
465C22⋊C4—1581no—
466C22⋊C4—1581no—
467C42—21no—
468C42—21no—
469C42—791no—
470C42—791no—
471C42—791no—
472C42—1581no—
473C42—1581no—
474C42—1581no—
475C42—1581no—
476C42C23×D7911yescharacteristic
477C23—791no—
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—3162no—
482Q8—3162no—
483Q8—1581no—
484Q8—1581no—
485Q8—1581no—
486Q8—1581no—
487Q8#1457916912yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—3162no—
493C2×C4—3161no—
494C2×C4—3162no—
495C2×C4—791no—
496C2×C4—791no—
497C2×C4—791no—
498C2×C4—1581no—
499C2×C4—1581no—
500C2×C4—1581no—
501C2×C4—1581no—
502C2×C4—1581no—
503C2×C4—1581no—
504C2×C4—1581no—
505C2×C4—1581no—
506C2×C4C22×C4×D7911yescharacteristic
507C2×C4C2×D4×D7911yescharacteristic
508C2×C4#41×D7911yescharacteristic
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—3161no—
517C8—3161no—
518C8—3161no—
519C8—1581no—
520C8—1581no—
521C8—1581no—
522C8—1581no—
523C22—1581no—
524C22—1581no—
525C22—1581no—
526C22C4×D4×D7911yescharacteristic
527C4—21no—
528C4—22no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—3161no—
533C4—3161no—
534C4—3161no—
535C4—3161no—
536C4—1581no—
537C4—1581no—
538C4—1581no—
539C4C2×#1457916911yescharacteristic
540C4#1909932811yescharacteristic
541C4#1909937611yescharacteristic
542C4#1909937711yescharacteristic
543C2—3161no—
544C2#2377212611yescharacteristic
545C2#2377226311yescharacteristic
546C2#2377230011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1584C2
socleC22×C79C2×C4×D4
prime radicalC22×D79C4×D4
cosocleC42×C79C24
hypercenter#1584D79
hypocenterC79#18032
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #23762307C2semidirect
5. #23762296C2semidirect
6. #23762268C2semidirect
7. #23762268C2semidirect
8. #23762252C2semidirect
9. #23762217C2semidirect
10. Q16×C79⋊C8C2semidirect
15. C79×#1584C2semidirect
282. #1584D79semidirect
289. C79#18032semidirect

Maximal subgroups

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

subgroupindex
#237734372
#237724982
#237724882
#237623072
#237622962
#237622682
#237622522
#237622172
Q16×C79⋊C82
#237621792
#237618712
#237617932
#237617012
C79×#15842
#1803279