Finite Group Databasebeta

← #34259344

Subgroups — #34259344

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#23773471C211yesmaximal, characteristic
2#23772414C211yesmaximal, characteristic
3#23772343C211yesmaximal, characteristic, complemented
4#23762306C211yesmaximal, characteristic, complemented
5#23762294C211yesmaximal, characteristic, complemented
6#23762274C211yesmaximal, characteristic, complemented
7#23762254C211yesmaximal, characteristic, complemented
8#23762253C211yesmaximal, characteristic, complemented
9#23762233C211yesmaximal, characteristic, complemented
10#23762209C211yesmaximal, characteristic, complemented
11#23762203C211yesmaximal, characteristic, complemented
12#23762183C211yesmaximal, characteristic, complemented
13#23761874C211yesmaximal, characteristic, complemented
14#23761695C211yesmaximal, characteristic, complemented
15C79×#1477C211yesmaximal, characteristic
16#19099120C2211yescharacteristic
17#19098904C411yescharacteristic
18#19098904C411yescharacteristic
19#19098848C411yescharacteristic
20#19098843C411yescharacteristic
21#19098773C411yescharacteristic
22#19098768C411yescharacteristic
23#19098725C411yescharacteristic
24#19098725C411yescharacteristic
25#19098694C2211yescharacteristic
26#19098694C2211yescharacteristic
27#19098692—21no—
28#19098692—21no—
29#19098690C2211yescharacteristic
30#19098690C2211yescharacteristic
31#19098687C2211yescharacteristic
32#19098687C2211yescharacteristic
33C4×#11929614C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929612C2211yescharacteristic
36C4×#11929612C2211yescharacteristic
37#19098635—21no—
38#19098635—21no—
39D4×C79⋊C8C2211yescharacteristic
40D4×C79⋊C8C2211yescharacteristic
41#19098620—21no—
42#19098620C2211yescharacteristic
43#19098616C2211yescharacteristic
44#19098589—21no—
45#19098518—21no—
46C4×#11929608C2211yescharacteristic
47C2×#14579072—21no—
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098313C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098295C2211yescharacteristic
55#19098295C2211yescharacteristic
56#19098285C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C4×#41×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62C2×C4⋊C8×C79C2211yescharacteristic
63Q8⋊C8×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66D4⋊C8×C79C2211yescharacteristic
67C2×#11929614C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929608D411yescharacteristic
72C2×#11929608D411yescharacteristic
73C22×C79⋊C8—21no—
74C22×C79⋊C8—21no—
75#14579161C2×C411yescharacteristic
76#14579161C2×C411yescharacteristic
77#14579153—21no—
78#14579140—21no—
79#14579140C2×C411yescharacteristic
80#14579140C2×C411yescharacteristic
81C4×Dic158—21no—
82C4×Dic158C2311yescharacteristic
83C4×Dic158C2311yescharacteristic
84#14579120—21no—
85#14579094C2×C411yescharacteristic
86#14579094C2×C411yescharacteristic
87#14579092C2×C411yescharacteristic
88#14579092C2×C411yescharacteristic
89C4⋊(C79⋊C8)—21no—
90C4⋊(C79⋊C8)—21no—
91C4⋊(C79⋊C8)C2×C411yescharacteristic
92C4⋊(C79⋊C8)C2×C411yescharacteristic
93C4⋊(C79⋊C8)C2311yescharacteristic
94C4⋊(C79⋊C8)C2311yescharacteristic
95Dic79⋊C8—21no—
96Dic79⋊C8—21no—
97Dic79⋊C8—21no—
98Dic79⋊C8—21no—
99#14579085C2×C411yescharacteristic
100#14579085C2×C411yescharacteristic
101#14579084C2×C411yescharacteristic
102#14579084C2×C411yescharacteristic
103#14579084C2×C411yescharacteristic
104#14579084C2×C411yescharacteristic
105#14579082C2×C411yescharacteristic
106#14579082C2×C411yescharacteristic
107C22⋊(C79⋊C8)—21no—
108C22⋊(C79⋊C8)—21no—
109C22⋊(C79⋊C8)—21no—
110C22⋊(C79⋊C8)C2×C411yescharacteristic
111C22⋊(C79⋊C8)C2×C411yescharacteristic
112#14579072—21no—
113#14579072—21no—
114#14579072—21no—
115#14579072D411yescharacteristic
116#14579072D411yescharacteristic
117#14579071—21no—
118#14579071—21no—
119C4×C79⋊C8C2311yescharacteristic
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C8×Dic79—21no—
124C8×Dic79—21no—
125C2×#41×C79C2×C411yescharacteristic
126C22×C8×C79—21no—
127C4×Q8×C79C2×C411yescharacteristic
128C4×Q8×C79C2311yescharacteristic
129C4×D4×C79—21no—
130C4×D4×C79—21no—
131C4×D4×C79C2×C411yescharacteristic
132C4×D4×C79C2311yescharacteristic
133#117×C79—21no—
134#117×C79C2×C411yescharacteristic
135C2×C42×C79—21no—
136C2×C42×C79C2311yescharacteristic
137C4⋊C8×C79—21no—
138C4⋊C8×C79—21no—
139C4⋊C8×C79C2311yescharacteristic
140C4⋊C8×C79C2311yescharacteristic
141C22⋊C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×Dic158C22×C411yescharacteristic
147#11929614—41no—
148#11929614—41no—
149#11929612—22no—
150#11929612—22no—
151#11929608—22no—
152#11929608—41no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8—42no—
156C2×C79⋊C8—42no—
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
165C4⋊Dic79—21no—
166C4⋊Dic79—21no—
167C4⋊Dic79—21no—
168C4⋊Dic79C22×C411yescharacteristic
169C4⋊Dic79C22×C411yescharacteristic
170Dic79⋊C4—21no—
171Dic79⋊C4—21no—
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174C22⋊Dic79—21no—
175C22⋊Dic79—21no—
176C4×Dic79—21no—
177C4×Dic79—21no—
178#41×C79—22no—
179#41×C79—41no—
180C2×Q8×C79C22×C411yescharacteristic
181C2×D4×C79—21no—
182C2×D4×C79C22×C411yescharacteristic
183C22×C4×C79—21no—
184C22×C4×C79—21no—
185C22×C4×C79—21no—
186C22×C4×C79C22×C411yescharacteristic
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79#4111yescharacteristic
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79—21no—
193C2×C8×C79—42no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79C22×C411yescharacteristic
197C4⋊C4×C79C22×C411yescharacteristic
198C22⋊C4×C79—21no—
199C22⋊C4×C79—21no—
200C22⋊C4×C79—21no—
201C42×C79—21no—
202C42×C79—21no—
203C42×C79—21no—
204C42×C79C22×C411yescharacteristic
205C42×C79C2×D411yescharacteristic
206C42×C79#4111yescharacteristic
207C42×C79C2411yescharacteristic
208C23×C79—21no—
209C23×C79C4×D411yescharacteristic
210Q8×C79—21no—
211Q8×C79—21no—
212D4×C79—21no—
213D4×C79—42no—
214D4×C79#13112yes—
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×C79—42no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—81no—
227C2×C4×C79C4×D411yescharacteristic
228C2×C4×C79C4×D411yescharacteristic
229C2×C4×C79C4×D411yescharacteristic
230C2×C4×C79C23×C411yescharacteristic
231C2×C4×C79C22×D411yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237Dic158—21no—
238Dic158—21no—
239Dic158—21no—
240Dic158—21no—
241Dic158—81no—
242C8×C79—41no—
243C8×C79—41no—
244C8×C79—41no—
245C8×C79—41no—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C79⋊C8#13712yes—
256C22×C79—21no—
257C22×C79—22no—
258C22×C79—21no—
259C22×C79—21no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79C2×C4×D411yescharacteristic
263C4×C79—21no—
264C4×C79—21no—
265C4×C79—21no—
266C4×C79—21no—
267C4×C79—41no—
268C4×C79—41no—
269C4×C79—41no—
270C4×C79C2×#13111yescharacteristic
271C4×C79#56811yescharacteristic
272C4×C79C2×#13711yescharacteristic
273C4×C79#57511yescharacteristic
274Dic79—41no—
275Dic79—41no—
276Dic79—41no—
277Dic79—41no—
278#18009—791nomaximal, Sylow
279C2×C79—41no—
280C2×C79—42no—
281C2×C79#293311yescharacteristic
282C2×C79C4×#13711yescharacteristic
283C2×C79#295711yescharacteristic
284#3107—791no—
285#2970—791no—
286#2962—791no—
287#1593—791no—
288#1591—791no—
289#1587—791no—
290#1585—791no—
291#1584—791no—
292#1582—791no—
293SD16⋊C8—791no—
294C8×SD16—791no—
295#1573—791no—
296#1497—791no—
297#1477D7911yescharacteristic
298#1453—791no—
299C79#1800911yescharacteristic, complemented, Sylow
300#533—791no—
301C4×#41C2×D7911yescharacteristic
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—791no—
314#446—1581no—
315#446—1581no—
316C8×Q8—791no—
317C8×Q8—791no—
318C4×Q16—791no—
319C4×Q16—791no—
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#431C2×D7911yescharacteristic
330#424—1581no—
331C2×C4⋊C8C2×D7911yescharacteristic
332#405—1581no—
333C4×M16—791no—
334C2×#97—1581no—
335#335—791no—
336#335—791no—
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⋊C8C2×D7911yescharacteristic
346D4⋊C8C2×D7911yescharacteristic
347#322—791no—
348#322—791no—
349#322—791no—
350C82—791no—
351C2×#41C4×D7911yescharacteristic
352C2×Q16—791no—
353C2×Q16—791no—
354C2×SD16—791no—
355C2×SD16—791no—
356C2×M16—791no—
357C2×M16—791no—
358C22×C8—21no—
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×Q8C4×D7911yescharacteristic
371C4×Q8C22×D7911yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C4×D7911yescharacteristic
375C4×D4C22×D7911yescharacteristic
376#117—21no—
377#117—1581no—
378#117C4×D7911yescharacteristic
379C2×C42—21no—
380C2×C42C22×D7911yescharacteristic
381#107—791no—
382#107—791no—
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#103—791no—
404#103—791no—
405#102—791no—
406#102—791no—
407C22⋊C8—21no—
408C22⋊C8—791no—
409C22⋊C8—791no—
410C22⋊C8—1581no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—791no—
414#97—791no—
415#97—1581no—
416#97—1581no—
417#97—1581no—
418#97—1581no—
419#97—1581no—
420C4×C8—791no—
421C4×C8—791no—
422C4×C8—791no—
423C4×C8—791no—
424C4×C8—1581no—
425C4×C8—1581no—
426C4×C8C22×D7911yescharacteristic
427C4×C8C22×D7911yescharacteristic
428#41—22no—
429#41—41no—
430C2×Q8—791no—
431C2×Q8—791no—
432C2×Q8—1581no—
433C2×Q8C2×C4×D7911yescharacteristic
434C2×D4—21no—
435C2×D4C2×C4×D7911yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4C2×C4×D7911yescharacteristic
440C22×C4D4×D7911yescharacteristic
441C22×C4#1192962311yescharacteristic
442Q16—3161no—
443Q16—3161no—
444SD16—1582no—
445SD16—1582no—
446M16—3161no—
447M16—1582no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—42no—
453C2×C8—3162no—
454C2×C8—3162no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—791no—
459C2×C8—791no—
460C2×C8—791no—
461C2×C8—791no—
462C2×C8—791no—
463C2×C8—1581no—
464C2×C8—1581no—
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—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—1581no—
482C22⋊C4—1581no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—1581no—
487C42—1581no—
488C42C2×C4×D7911yescharacteristic
489C42D4×D7911yescharacteristic
490C42#1192962311yescharacteristic
491C42C23×D7911yescharacteristic
492C23—21no—
493C23(C22⋊D79)⋊C411yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—6321no—
497Q8—1581no—
498Q8—1581no—
499Q8—1581no—
500Q8—1581no—
501D4—21no—
502D4—42no—
503D4#1457916712yes—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4—1581no—
520C2×C4(C22⋊D79)⋊C411yescharacteristic
521C2×C4(C22⋊D79)⋊C411yescharacteristic
522C2×C4(C22⋊D79)⋊C411yescharacteristic
523C2×C4C22×C4×D7911yescharacteristic
524C2×C4C2×D4×D7911yescharacteristic
525C2×C4C2×#1192962311yescharacteristic
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—3161no—
531C8—3161no—
532C8—792no—
533C8—1581no—
534C8—1581no—
535C8—1581no—
536C8—1581no—
537C8—1581no—
538C8—1581no—
539C8—1581no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—41no—
545C22—41no—
546C22C2×(C22⋊D79)⋊C411yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—3161no—
555C4—3161no—
556C4—3161no—
557C4—3161no—
558C4C2×#1457916711yescharacteristic
559C4#1909932911yescharacteristic
560C4#137×D7911yescharacteristic
561C4#1909935911yescharacteristic
562C2—41no—
563C2—42no—
564C2#2377212111yescharacteristic
565C2#2377222411yescharacteristic
566C2#2377231311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1477C2
socleC22×C79C2×C4×D4
prime radicalC2×#41×C79C2×C4
cosocleC42×C79C24
hypercenter#1477D79
hypocenterC79#18009
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
3. #23772343C2semidirect
4. #23762306C2semidirect
5. #23762294C2semidirect
6. #23762274C2semidirect
7. #23762254C2semidirect
8. #23762253C2semidirect
9. #23762233C2semidirect
10. #23762209C2semidirect
11. #23762203C2semidirect
12. #23762183C2semidirect
13. #23761874C2semidirect
14. #23761695C2semidirect
299. C79#18009semidirect

Maximal subgroups

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

subgroupindex
#237734712
#237724142
#237723432
#237623062
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