Finite Group Databasebeta

← #34258908

Subgroups — #34258908

Order 20224.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774000C211yesmaximal, characteristic
2#23772545C211yesmaximal, characteristic
3#23772495C211yesmaximal, characteristic, complemented
4#23762306C211yesmaximal, characteristic, complemented
5#23762305C211yesmaximal, characteristic, complemented
6#23762294C211yesmaximal, characteristic, complemented
7#23762287C211yesmaximal, characteristic, complemented
8#23762216C211yesmaximal, characteristic, complemented
9#23762215C211yesmaximal, characteristic, complemented
10#23762204C211yesmaximal, characteristic, complemented
11#23762197C211yesmaximal, characteristic, complemented
12#23762165C211yesmaximal, characteristic, complemented
13#23761861C211yesmaximal, characteristic
14C8×#11929608C211yesmaximal, characteristic, complemented
15C79×#1493C211yesmaximal, characteristic, complemented
16#19099097C2211yescharacteristic
17#19098969C411yescharacteristic
18#19098955C411yescharacteristic
19#19098931C411yescharacteristic
20#19098925C411yescharacteristic
21#19098819C411yescharacteristic
22#19098819C411yescharacteristic
23#19098811C411yescharacteristic
24#19098811C411yescharacteristic
25#19098694—21no—
26#19098694—21no—
27#19098694C2211yescharacteristic
28#19098694C2211yescharacteristic
29#19098690C2211yescharacteristic
30Q8×C79⋊C8C2211yescharacteristic
31Q8×C79⋊C8C2211yescharacteristic
32#19098687C2211yescharacteristic
33C4×#11929614C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929611C2211yescharacteristic
36C4×#11929611C2211yescharacteristic
37#19098640C2211yescharacteristic
38#19098639—21no—
39#19098632—21no—
40#19098628C2211yescharacteristic
41#19098624—21no—
42#19098624—21no—
43#19098623—21no—
44#19098620C2211yescharacteristic
45#19098616C2211yescharacteristic
46#19098588—21no—
47C4×#11929608C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098313C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098292C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098285C2211yescharacteristic
57#19098285C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#533×C79C2211yescharacteristic
61#423×C79C2211yescharacteristic
62C2×C4×C8×C79C2211yescharacteristic
63Q8⋊C8×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66Q8⋊C8×C79C2211yescharacteristic
67C2×#11929614C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929611C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929608C2×C411yescharacteristic
72C2×#11929608C2×C411yescharacteristic
73C2×#11929607—21no—
74C2×C8×D79—21no—
75#14579155—21no—
76#14579153C2×C411yescharacteristic
77#14579149C2×C411yescharacteristic
78#14579144—21no—
79#14579140C2×C411yescharacteristic
80#14579135C2×C411yescharacteristic
81C4×Dic158C2311yescharacteristic
82C4×D316C2311yescharacteristic
83C4×C22⋊D79—21no—
84#14579118—21no—
85#14579094C2×C411yescharacteristic
86#14579094C2×C411yescharacteristic
87#14579092C2×C411yescharacteristic
88#14579092C2×C411yescharacteristic
89C4⋊(C79⋊C8)—21no—
90C4⋊(C79⋊C8)—21no—
91C4⋊(C79⋊C8)—21no—
92C4⋊(C79⋊C8)—21no—
93C4⋊(C79⋊C8)D411yescharacteristic
94C4⋊(C79⋊C8)D411yescharacteristic
95C4⋊(C79⋊C8)C2311yescharacteristic
96C4⋊(C79⋊C8)C2311yescharacteristic
97Dic79⋊C8—21no—
98Dic79⋊C8—21no—
99#14579085C2×C411yescharacteristic
100#14579085C2×C411yescharacteristic
101#14579085C2×C411yescharacteristic
102#14579085C2×C411yescharacteristic
103#14579084C2×C411yescharacteristic
104#14579084C2×C411yescharacteristic
105#14579081C2×C411yescharacteristic
106#14579081C2×C411yescharacteristic
107C22⋊(C79⋊C8)—21no—
108C22⋊(C79⋊C8)D411yescharacteristic
109C22⋊(C79⋊C8)D411yescharacteristic
110(C2×D79)⋊C8—21no—
111(C2×D79)⋊C8—21no—
112#14579072—21no—
113#14579072—21no—
114#14579072—21no—
115#14579072C2×C411yescharacteristic
116#14579072C2×C411yescharacteristic
117#14579071—21no—
118C4×C79⋊C8—21no—
119C4×C79⋊C8—21no—
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C8×Dic79—21no—
125C2×M16×C79—21no—
126C22×C8×C79—21no—
127C4⋊Q8×C79C2×C411yescharacteristic
128C4⋊Q8×C79C2×C411yescharacteristic
129#125×C79—21no—
130C22⋊Q8×C79—21no—
131C22⋊Q8×C79C2×C411yescharacteristic
132C22⋊Q8×C79C2×C411yescharacteristic
133C4×Q8×C79—21no—
134C4×Q8×C79C2311yescharacteristic
135C4×Q8×C79C2311yescharacteristic
136#117×C79—21no—
137C2×C42×C79C2311yescharacteristic
138C4⋊C8×C79—21no—
139C4⋊C8×C79C2311yescharacteristic
140C4⋊C8×C79C2311yescharacteristic
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158C22×C411yescharacteristic
145C2×D316C22×C411yescharacteristic
146C2×C22⋊D79—21no—
147C2×C4×D79—21no—
148#11929614—22no—
149#11929614—22no—
150#11929611—22no—
151#11929611—22no—
152#11929608—22no—
153#11929608—22no—
154#11929607—81no—
155C2×C79⋊C8—21no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8C22×C411yescharacteristic
158C2×C79⋊C8C22×C411yescharacteristic
159C2×C79⋊C8C22×C411yescharacteristic
160C2×C79⋊C8C22×C411yescharacteristic
161C2×C79⋊C8C2×D411yescharacteristic
162C2×C79⋊C8C2×D411yescharacteristic
163C2×C79⋊C8C2×D411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C8×D79—81no—
166C4⋊Dic79C22×C411yescharacteristic
167C4⋊Dic79C22×C411yescharacteristic
168Dic79⋊C4—21no—
169Dic79⋊C4—21no—
170Dic79⋊C4—21no—
171Dic79⋊C4—21no—
172C22⋊Dic79—21no—
173(C2×D79)⋊C4—21no—
174(C2×D79)⋊C4—21no—
175(C2×D79)⋊C4—21no—
176(C2×D79)⋊C4—21no—
177C4×Dic79—21no—
178C2×Q8×C79—21no—
179C2×Q8×C79C22×C411yescharacteristic
180C2×Q8×C79C22×C411yescharacteristic
181C22×C4×C79C22×C411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79#4111yescharacteristic
184M16×C79—81no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79—42no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79C22×C411yescharacteristic
198C4⋊C4×C79C22×C411yescharacteristic
199C22⋊C4×C79—21no—
200C22⋊C4×C79—21no—
201C42×C79—21no—
202C42×C79—21no—
203C42×C79C22×C411yescharacteristic
204C42×C79C2×D411yescharacteristic
205C42×C79#4111yescharacteristic
206C42×C79C2411yescharacteristic
207C23×C79C4×D411yescharacteristic
208C22×D79—21no—
209Q8×C79—21no—
210Q8×C79—21no—
211Q8×C79—21no—
212Q8×C79—21no—
213Q8×C79—81no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—41no—
223C2×C4×C79C4×D411yescharacteristic
224C2×C4×C79C4×D411yescharacteristic
225C2×C4×C79C4×D411yescharacteristic
226C2×C4×C79C23×C411yescharacteristic
227C2×C4×C79C22×D411yescharacteristic
228C2×C4×C79C2×#4111yescharacteristic
229C22⋊D79—81no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233D316—21no—
234D316—21no—
235C4×D79—41no—
236C4×D79—41no—
237Dic158—21no—
238Dic158—21no—
239C8×C79—22no—
240C8×C79—41no—
241C8×C79—41no—
242C8×C79—41no—
243C79⋊C8—21no—
244C79⋊C8—21no—
245C79⋊C8—21no—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—41no—
250C79⋊C8—41no—
251C79⋊C8#13112yes—
252C79⋊C8#13112yes—
253C22×C79—21no—
254C22×C79—21no—
255C22×C79—21no—
256C22×C79C2×C4×D411yescharacteristic
257C2×D79—41no—
258C2×D79—41no—
259C2×D79—41no—
260C4×C79—21no—
261C4×C79—21no—
262C4×C79—21no—
263C4×C79—21no—
264C4×C79—41no—
265C4×C79—41no—
266C4×C79—41no—
267C4×C79—41no—
268C4×C79C2×#13111yescharacteristic
269C4×C79C2×#13111yescharacteristic
270C4×C79#57711yescharacteristic
271C4×C79#57811yescharacteristic
272Dic79—41no—
273Dic79—41no—
274Dic79—41no—
275#17969—791nomaximal, Sylow
276C2×C79—41no—
277C2×C79#293111yescharacteristic
278C2×C79#294711yescharacteristic
279C2×C79#295611yescharacteristic
280D79—81no—
281#3172—791no—
282#2993—791no—
283#2985—791no—
284#1593—791no—
285#1592—791no—
286#1591—791no—
287#1590—791no—
288#1579—791no—
289SD16⋊C8—791no—
290C8×Q16—791no—
291C8×SD16—791no—
292#1569—791no—
293#1494—791no—
294#1493D7911yescharacteristic, complemented
295C8×M16—791no—
296C79#1796911yescharacteristic, complemented, Sylow
297#533C2×D7911yescharacteristic
298#529—791no—
299#501—791no—
300#497—791no—
301#491—791no—
302#489—791no—
303#470—791no—
304#470—791no—
305#468—791no—
306#468—791no—
307#446—791no—
308#446—791no—
309#446—791no—
310#446—1581no—
311#446—1581no—
312C8×Q8—791no—
313C8×Q8—791no—
314C8×Q8—791no—
315C4×Q16—791no—
316C4×Q16—791no—
317C4×SD16—791no—
318C4×SD16—791no—
319#436—791no—
320#436—1581no—
321#435—1581no—
322C8×D4—791no—
323#433—1581no—
324#433—1581no—
325#433—1581no—
326#432—791no—
327#431—791no—
328#424—1581no—
329#423C2×D7911yescharacteristic
330C4×M16—791no—
331C2×C4×C8C2×D7911yescharacteristic
332#335—791no—
333#335—791no—
334#334—791no—
335#334—791no—
336Q8⋊C8—791no—
337Q8⋊C8—791no—
338Q8⋊C8C2×D7911yescharacteristic
339Q8⋊C8C2×D7911yescharacteristic
340Q8⋊C8C2×D7911yescharacteristic
341Q8⋊C8C2×D7911yescharacteristic
342D4⋊C8—791no—
343D4⋊C8—791no—
344#322—791no—
345#322—791no—
346C82—791no—
347C82—791no—
348C2×Q16—791no—
349C2×Q16—791no—
350C2×SD16—791no—
351C2×SD16—791no—
352C2×M16—21no—
353C2×M16—791no—
354C2×M16—791no—
355C2×M16—1581no—
356C22×C8—21no—
357C22×C8—1581no—
358C4⋊Q8C4×D7911yescharacteristic
359C4⋊Q8C4×D7911yescharacteristic
360#126—1581no—
361#125—21no—
362#125—791no—
363#124—791no—
364#123—1581no—
365C22⋊Q8—21no—
366C22⋊Q8—791no—
367C22⋊Q8C4×D7911yescharacteristic
368C22⋊Q8C4×D7911yescharacteristic
369#121—791no—
370C4×Q8—21no—
371C4×Q8—791no—
372C4×Q8C22×D7911yescharacteristic
373C4×Q8C22×D7911yescharacteristic
374C4×D4—791no—
375C4×D4—1581no—
376#117—21no—
377#117—1581no—
378C2×C42C22×D7911yescharacteristic
379#107—791no—
380#107—791no—
381#106—791no—
382#106—791no—
383C4⋊C8—21no—
384C4⋊C8—791no—
385C4⋊C8—791no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—1581no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8C22×D7911yescharacteristic
395C4⋊C8C22×D7911yescharacteristic
396#103—791no—
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103—791no—
402#102—791no—
403#102—791no—
404C22⋊C8—791no—
405C22⋊C8—791no—
406C22⋊C8—1581no—
407C22⋊C8—1581no—
408C22⋊C8—1581no—
409#97—791no—
410#97—791no—
411#97—1581no—
412#97—1581no—
413#97—1581no—
414#97—1581no—
415C4×C8—21no—
416C4×C8—791no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—1581no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8C22×D7911yescharacteristic
424C4×C8C22×D7911yescharacteristic
425C2×Q8—21no—
426C2×Q8—791no—
427C2×Q8C2×C4×D7911yescharacteristic
428C2×Q8C2×C4×D7911yescharacteristic
429C2×D4—791no—
430C2×D4—1581no—
431C22×C4—1581no—
432C22×C4C2×C4×D7911yescharacteristic
433C22×C4D4×D7911yescharacteristic
434C22×C4#1192962311yescharacteristic
435Q16—1582no—
436Q16—1582no—
437SD16—1582no—
438SD16—1582no—
439M16—81no—
440M16—6321no—
441M16—1582no—
442M16—1582no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—42no—
448C2×C8—791no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—791no—
453C2×C8—791no—
454C2×C8—791no—
455C2×C8—791no—
456C2×C8—6321no—
457C2×C8—1581no—
458C2×C8—1581no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—791no—
467C4⋊C4—791no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4C2×C4×D7911yescharacteristic
473C4⋊C4C2×C4×D7911yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—1581no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C42—21no—
482C42—21no—
483C42—1581no—
484C42C2×C4×D7911yescharacteristic
485C42D4×D7911yescharacteristic
486C42#1192962311yescharacteristic
487C42C23×D7911yescharacteristic
488C23—1581no—
489C23(C22⋊D79)⋊C411yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—81no—
495Q8—1581no—
496Q8—1581no—
497D4—6321no—
498D4—1581no—
499D4—1581no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—3161no—
510C2×C4—3161no—
511C2×C4—1581no—
512C2×C4—1581no—
513C2×C4—1581no—
514C2×C4(C22⋊D79)⋊C411yescharacteristic
515C2×C4(C22⋊D79)⋊C411yescharacteristic
516C2×C4(C22⋊D79)⋊C411yescharacteristic
517C2×C4C22×C4×D7911yescharacteristic
518C2×C4C2×D4×D7911yescharacteristic
519C2×C4C2×#1192962311yescharacteristic
520C8—22no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—3161no—
525C8—3161no—
526C8—792no—
527C8—792no—
528C8—1581no—
529C8—1581no—
530C8—1581no—
531C8—1581no—
532C8—1581no—
533C8—1581no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—3161no—
538C22—3161no—
539C22—3161no—
540C22C2×(C22⋊D79)⋊C411yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—3161no—
550C4—3161no—
551C4—3161no—
552C4#131×D7911yescharacteristic
553C4#131×D7911yescharacteristic
554C4#1909936911yescharacteristic
555C4#1909937611yescharacteristic
556C2—41no—
557C2—6321no—
558C2#2377210811yescharacteristic
559C2#2377223711yescharacteristic
560C2#2377230611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1493C2
socleC22×C79C2×C4×D4
prime radical#14579135C2×C4
cosocleC42×C79C24
hypercenter#1493D79
hypocenterC79#17969
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
3. #23772495C2semidirect
4. #23762306C2semidirect
5. #23762305C2semidirect
6. #23762294C2semidirect
7. #23762287C2semidirect
8. #23762216C2semidirect
9. #23762215C2semidirect
10. #23762204C2semidirect
11. #23762197C2semidirect
12. #23762165C2semidirect
14. C8×#11929608C2semidirect
15. C79×#1493C2semidirect
294. #1493D79semidirect
296. C79#17969semidirect

Maximal subgroups

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

subgroupindex
#237740002
#237725452
#237724952
#237623062
#237623052
#237622942
#237622872
#237622162
#237622152
#237622042
#237621972
#237621652
#237618612
C8×#119296082
C79×#14932
#1796979