Finite Group Databasebeta

← #34259347

Subgroups — #34259347

Order 20224.

594 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773469C211yesmaximal, characteristic
2#23772413C211yesmaximal, characteristic
3#23772342C211yesmaximal, characteristic, complemented
4#23762309C211yesmaximal, characteristic, complemented
5#23762295C211yesmaximal, characteristic, complemented
6#23762274C211yesmaximal, characteristic, complemented
7#23762260C211yesmaximal, characteristic, complemented
8#23762238C211yesmaximal, characteristic, complemented
9#23762211C211yesmaximal, characteristic, complemented
10C8×#11929611C211yesmaximal, characteristic, complemented
11#23762184C211yesmaximal, characteristic, complemented
12#23761871C211yesmaximal, characteristic, complemented
13#23761786C211yesmaximal, characteristic
14#23761698C211yesmaximal, characteristic, complemented
15C79×#1584C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17C4×#11929625C2211yescharacteristic
18#19098906C411yescharacteristic
19#19098902C411yescharacteristic
20#19098846C411yescharacteristic
21#19098842C411yescharacteristic
22#19098771C411yescharacteristic
23#19098767C411yescharacteristic
24#19098728C411yescharacteristic
25#19098724C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098694C2211yescharacteristic
28#19098693—21no—
29#19098693C2211yescharacteristic
30#19098692—21no—
31#19098687C2211yescharacteristic
32C4×#11929614C2211yescharacteristic
33C4×#11929613C2211yescharacteristic
34C4×#11929611C2211yescharacteristic
35#19098636—21no—
36#19098633—21no—
37C8×D316C2211yescharacteristic
38#19098628C2211yescharacteristic
39#19098619—21no—
40#19098619C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098588—21no—
44C4⋊C8×D79C2211yescharacteristic
45#19098517—21no—
46C4×#11929607C2211yescharacteristic
47#97×D79—21no—
48#19098315C2211yescharacteristic
49#19098314C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098284C2211yescharacteristic
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×#11929625C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929607D411yescharacteristic
72C2×#11929607D411yescharacteristic
73C2×C8×D79—21no—
74C2×C8×D79—21no—
75C2×C8×D79—21no—
76#14579160C2×C411yescharacteristic
77#14579159C2×C411yescharacteristic
78#14579151—21no—
79#14579141C2×C411yescharacteristic
80#14579139—21no—
81#14579137C2×C411yescharacteristic
82Q8×Dic79C2×C411yescharacteristic
83C4×Dic158—21no—
84C4×Dic158C2311yescharacteristic
85C4×D316—21no—
86C4×D316C2311yescharacteristic
87#14579123—21no—
88#14579123C2×C411yescharacteristic
89#14579119—21no—
90#14579119C2×C411yescharacteristic
91#14579118—21no—
92C42×D79—21no—
93C42×D79C2311yescharacteristic
94#14579094C2×C411yescharacteristic
95#14579093C2×C411yescharacteristic
96#14579092C2×C411yescharacteristic
97C4⋊(C79⋊C8)—21no—
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8C2×C411yescharacteristic
106Dic79⋊C8C2×C411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579085C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579083C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8C2×C411yescharacteristic
118(C2×D79)⋊C8C2×C411yescharacteristic
119#14579072—21no—
120#14579072—21no—
121#14579071—21no—
122#14579071—21no—
123#14579071D411yescharacteristic
124#14579071D411yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
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
144#11929625—22no—
145#11929625—41no—
146C2×Dic158—21no—
147C2×Dic158C22×C411yescharacteristic
148C2×D316—21no—
149C2×D316C22×C411yescharacteristic
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79—21no—
153C2×C4×D79C22×C411yescharacteristic
154C2×C4×D79C2×D411yescharacteristic
155C2×C4×D79#4111yescharacteristic
156#11929614—41no—
157#11929613—22no—
158#11929611—22no—
159#11929607—22no—
160#11929607—41no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8C22×C411yescharacteristic
165C2×C79⋊C8C22×C411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C8×D79—42no—
169C8×D79—42no—
170C8×D79—42no—
171C4⋊Dic79—21no—
172C4⋊Dic79—21no—
173C4⋊Dic79C22×C411yescharacteristic
174C4⋊Dic79C22×C411yescharacteristic
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180(C2×D79)⋊C4—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79C22×C411yescharacteristic
189C4×Dic79C2×D411yescharacteristic
190C4×Dic79#4111yescharacteristic
191C2×Q8×C79C22×C411yescharacteristic
192C2×Q8×C79C22×C411yescharacteristic
193Q16×C79—41no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79C22×C411yescharacteristic
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C2×D411yescharacteristic
200C2×C8×C79C2×D411yescharacteristic
201C4⋊C4×C79—21no—
202C4⋊C4×C79—21no—
203C4⋊C4×C79C22×C411yescharacteristic
204C4⋊C4×C79C22×C411yescharacteristic
205C42×C79—21no—
206C42×C79—21no—
207C42×C79C2411yescharacteristic
208C22×D79—21no—
209C22×D79C4×D411yescharacteristic
210Q8×C79—21no—
211Q8×C79—21no—
212Q8×C79—21no—
213Q8×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79C23×C411yescharacteristic
219C2×C4×C79C22×D411yescharacteristic
220C2×C4×C79C2×#4111yescharacteristic
221C2×Dic79—21no—
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79C4×D411yescharacteristic
225C2×Dic79C4×D411yescharacteristic
226C2×Dic79C4×D411yescharacteristic
227D316—21no—
228D316—42no—
229D316#13112yes—
230C4×D79—21no—
231C4×D79—21no—
232C4×D79—21no—
233C4×D79—21no—
234C4×D79—41no—
235C4×D79—42no—
236C4×D79—41no—
237C4×D79—41no—
238C4×D79—81no—
239Dic158—21no—
240Dic158—21no—
241Dic158—81no—
242C8×C79—21no—
243C8×C79—21no—
244C8×C79—21no—
245C8×C79—41no—
246C8×C79—41no—
247C8×C79—41no—
248C8×C79#13712yes—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C79⋊C8—41no—
256C22×C79C2×C4×D411yescharacteristic
257C2×D79—22no—
258C2×D79—21no—
259C2×D79—21no—
260C2×D79—21no—
261C2×D79—41no—
262C2×D79—41no—
263C4×C79—21no—
264C4×C79—41no—
265C4×C79—41no—
266C4×C79—41no—
267C4×C79—41no—
268C4×C79C2×#13111yescharacteristic
269C4×C79#56811yescharacteristic
270C4×C79C2×#13711yescharacteristic
271C4×C79#57511yescharacteristic
272Dic79—21no—
273Dic79—21no—
274Dic79—21no—
275Dic79—41no—
276Dic79—41no—
277Dic79—41no—
278#18009—791nomaximal, Sylow
279C2×C79#293311yescharacteristic
280C2×C79C4×#13711yescharacteristic
281C2×C79#295711yescharacteristic
282D79—41no—
283D79—42no—
284#3107—791no—
285#2970—791no—
286#2962—791no—
287#1593—791no—
288#1591—791no—
289#1587—791no—
290#1585—791no—
291#1584D7911yescharacteristic, complemented
292#1582—791no—
293SD16⋊C8—791no—
294C8×SD16—791no—
295#1573—791no—
296#1497—791no—
297#1477—791no—
298#1453—791no—
299C79#1800911yescharacteristic, complemented, Sylow
300#533—791no—
301C4×#41—791no—
302#485—791no—
303#485—791no—
304#475—791no—
305#474—791no—
306#462—791no—
307#461—791no—
308#452—791no—
309#452—791no—
310#446—791no—
311#446—791no—
312#446—791no—
313#446—1581no—
314#446—1581no—
315#446C2×D7911yescharacteristic
316C8×Q8—791no—
317C8×Q8C2×D7911yescharacteristic
318C4×Q16—791no—
319C4×Q16C2×D7911yescharacteristic
320C4×SD16—791no—
321C4×SD16—791no—
322#435—1581no—
323#435—1581no—
324C8×D4—791no—
325C8×D4—791no—
326#432—791no—
327#432—1581no—
328#431—791no—
329#431—791no—
330#424—1581no—
331C2×C4⋊C8—791no—
332#405—1581no—
333C4×M16—791no—
334C2×#97—1581no—
335#335—791no—
336#335C2×D7911yescharacteristic
337#334—791no—
338#334—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8—791no—
343Q8⋊C8C2×D7911yescharacteristic
344Q8⋊C8C2×D7911yescharacteristic
345D4⋊C8—791no—
346D4⋊C8—791no—
347#322—791no—
348#322—791no—
349#322C2×D7911yescharacteristic
350C82—791no—
351C2×#41—791no—
352C2×Q16—791no—
353C2×Q16C4×D7911yescharacteristic
354C2×SD16—791no—
355C2×SD16—791no—
356C2×M16—791no—
357C2×M16—791no—
358C22×C8—1581no—
359C22×C8—1581no—
360C22×C8—1581no—
361C4⋊Q8—791no—
362C4⋊Q8—791no—
363#125—1581no—
364C22⋊Q8—791no—
365C22⋊Q8—791no—
366C22⋊Q8—1581no—
367C4×Q8—791no—
368C4×Q8—791no—
369C4×Q8—1581no—
370C4×Q8C22×D7911yescharacteristic
371C4×Q8C22×D7911yescharacteristic
372C4×D4—791no—
373C4×D4—791no—
374C4×D4—1581no—
375C4×D4—1581no—
376#117—791no—
377#117—1581no—
378#117—1581no—
379C2×C42—791no—
380C2×C42—1581no—
381#107—791no—
382#107C4×D7911yescharacteristic
383#106—791no—
384#106—791no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—791no—
390C4⋊C8—791no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8—1581no—
396C4⋊C8—1581no—
397C4⋊C8C22×D7911yescharacteristic
398C4⋊C8C22×D7911yescharacteristic
399#103—791no—
400#103—791no—
401#103—791no—
402#103—791no—
403#103C4×D7911yescharacteristic
404#103C4×D7911yescharacteristic
405#102—791no—
406#102—791no—
407C22⋊C8—791no—
408C22⋊C8—791no—
409C22⋊C8—1581no—
410C22⋊C8—1581no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—21no—
414#97—791no—
415#97—791no—
416#97—1581no—
417#97—1581no—
418#97—1581no—
419#97—1581no—
420C4×C8—21no—
421C4×C8—791no—
422C4×C8—791no—
423C4×C8—791no—
424C4×C8—1581no—
425C4×C8C22×D7911yescharacteristic
426C4×C8C22×D7911yescharacteristic
427C4×C8C22×D7911yescharacteristic
428#41—3161no—
429#41—1582no—
430C2×Q8—791no—
431C2×Q8—1581no—
432C2×Q8C2×C4×D7911yescharacteristic
433C2×Q8C2×C4×D7911yescharacteristic
434C2×D4—791no—
435C2×D4—1581no—
436C22×C4—791no—
437C22×C4—791no—
438C22×C4—791no—
439C22×C4—1581no—
440C22×C4—1581no—
441C22×C4—1581no—
442Q16—41no—
443Q16—3161no—
444SD16—1582no—
445SD16—1582no—
446M16—3161no—
447M16—1582no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—3162no—
452C2×C8—3162no—
453C2×C8—3162no—
454C2×C8—791no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8C2×C4×D7911yescharacteristic
462C2×C8C2×C4×D7911yescharacteristic
463C2×C8D4×D7911yescharacteristic
464C2×C8D4×D7911yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—791no—
468C4⋊C4—791no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4C2×C4×D7911yescharacteristic
477C4⋊C4C2×C4×D7911yescharacteristic
478C22⋊C4—1581no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C22⋊C4—1581no—
482C22⋊C4—1581no—
483C42—21no—
484C42—21no—
485C42—791no—
486C42—791no—
487C42—791no—
488C42—1581no—
489C42—1581no—
490C42—1581no—
491C42C23×D7911yescharacteristic
492C23—791no—
493C23—1581no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—6321no—
499Q8—1581no—
500Q8—1581no—
501D4—3162no—
502D4—792no—
503D4—1581no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—3161no—
509C2×C4—3162no—
510C2×C4—3161no—
511C2×C4—3161no—
512C2×C4—791no—
513C2×C4—791no—
514C2×C4—791no—
515C2×C4—6321no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4—1581no—
520C2×C4—1581no—
521C2×C4—1581no—
522C2×C4—1581no—
523C2×C4C22×C4×D7911yescharacteristic
524C2×C4C2×D4×D7911yescharacteristic
525C2×C4#41×D7911yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—3161no—
533C8—3161no—
534C8—3161no—
535C8—1581no—
536C8—1581no—
537C8—1581no—
538C8—1581no—
539C8#1457919312yes—
540C22—3161no—
541C22—3161no—
542C22—1581no—
543C22—1581no—
544C22—1582no—
545C22—1581no—
546C22C4×D4×D7911yescharacteristic
547C4—21no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—3161no—
553C4—3161no—
554C4—3161no—
555C4—1581no—
556C4—1581no—
557C4—1581no—
558C4#131×D7911yescharacteristic
559C4#1909932911yescharacteristic
560C4C2×#1457919311yescharacteristic
561C4#1909936011yescharacteristic
562C2—3162no—
563C2—3161no—
564C2#2377212311yescharacteristic
565C2C4×#1457919311yescharacteristic
566C2#2377231411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1584C2
socleC22×C79C2×C4×D4
prime radicalC2×#11929625C2×C4
cosocleC42×C79C24
hypercenter#1584D79
hypocenterC79#18009
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
3. #23772342C2semidirect
4. #23762309C2semidirect
5. #23762295C2semidirect
6. #23762274C2semidirect
7. #23762260C2semidirect
8. #23762238C2semidirect
9. #23762211C2semidirect
10. C8×#11929611C2semidirect
11. #23762184C2semidirect
12. #23761871C2semidirect
14. #23761698C2semidirect
15. C79×#1584C2semidirect
291. #1584D79semidirect
299. C79#18009semidirect

Maximal subgroups

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

subgroupindex
#237734692
#237724132
#237723422
#237623092
#237622952
#237622742
#237622602
#237622382
#237622112
C8×#119296112
#237621842
#237618712
#237617862
#237616982
C79×#15842
#1800979