Finite Group Databasebeta

← #34259354

Subgroups — #34259354

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#23773472C211yesmaximal, characteristic
2#23772415C211yesmaximal, characteristic
3#23772347C211yesmaximal, characteristic, complemented
4#23762310C211yesmaximal, characteristic, complemented
5#23762298C211yesmaximal, characteristic, complemented
6#23762273C211yesmaximal, characteristic, complemented
7#23762259C211yesmaximal, characteristic, complemented
8#23762252C211yesmaximal, characteristic, complemented
9#23762238C211yesmaximal, characteristic, complemented
10#23762214C211yesmaximal, characteristic, complemented
11C8×#11929613C211yesmaximal, characteristic, complemented
12#23762185C211yesmaximal, characteristic, complemented
13#23761874C211yesmaximal, characteristic, complemented
14#23761788C211yesmaximal, characteristic
15C79×#1453C211yesmaximal, characteristic, complemented
16#19099120C2211yescharacteristic
17C4×#11929624C2211yescharacteristic
18#19098905C411yescharacteristic
19#19098905C411yescharacteristic
20#19098849C411yescharacteristic
21#19098844C411yescharacteristic
22#19098774C411yescharacteristic
23#19098769C411yescharacteristic
24#19098727C411yescharacteristic
25#19098727C411yescharacteristic
26#19098693C2211yescharacteristic
27#19098693C2211yescharacteristic
28#19098693C2211yescharacteristic
29#19098693C2211yescharacteristic
30#19098691—21no—
31#19098691—21no—
32C8×Dic158C2211yescharacteristic
33C8×Dic158C2211yescharacteristic
34C4×Dic316C2211yescharacteristic
35C4×Dic316C2211yescharacteristic
36C4×#11929613C2211yescharacteristic
37C4×#11929613C2211yescharacteristic
38#19098632—21no—
39#19098632—21no—
40C8×D316C2211yescharacteristic
41C8×D316C2211yescharacteristic
42#19098620C2211yescharacteristic
43#19098618—21no—
44#19098591—21no—
45C2×C4⋊(C79⋊C8)C2211yescharacteristic
46#19098521—21no—
47#97×D79—21no—
48#19098316C2211yescharacteristic
49#19098316C2211yescharacteristic
50#19098314C2211yescharacteristic
51#19098314C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098295C2211yescharacteristic
55#19098295C2211yescharacteristic
56#19098295C2211yescharacteristic
57#19098295C2211yescharacteristic
58#19098292C2211yescharacteristic
59#19098292C2211yescharacteristic
60#431×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62C4×M16×C79C2211yescharacteristic
63#322×C79C2211yescharacteristic
64#322×C79C2211yescharacteristic
65#322×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929624C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×Dic316C2×C411yescharacteristic
70C2×#11929613C2×C411yescharacteristic
71C2×#11929613C2×C411yescharacteristic
72C22×C79⋊C8—21no—
73C2×C8×D79—21no—
74C2×C8×D79—21no—
75#14579161C2×C411yescharacteristic
76#14579161C2×C411yescharacteristic
77#14579153—21no—
78#14579140—21no—
79#14579140C2×C411yescharacteristic
80#14579140C2×C411yescharacteristic
81C4×Dic158—21no—
82C4×Dic158C2×C411yescharacteristic
83C4×Dic158C2311yescharacteristic
84C4×Dic158C2311yescharacteristic
85C4×Dic158C2311yescharacteristic
86C4×D316C2×C411yescharacteristic
87C4×D316C2311yescharacteristic
88C4×C22⋊D79—21no—
89C4×C22⋊D79—21no—
90#14579120—21no—
91#14579120C2×C411yescharacteristic
92#14579118—21no—
93C42×D79—21no—
94#14579095C2×C411yescharacteristic
95#14579095C2×C411yescharacteristic
96#14579093C2×C411yescharacteristic
97#14579093C2×C411yescharacteristic
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101C4⋊(C79⋊C8)C2311yescharacteristic
102C4⋊(C79⋊C8)C2311yescharacteristic
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108Dic79⋊C8—21no—
109#14579086C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579086C2×C411yescharacteristic
112#14579086C2×C411yescharacteristic
113#14579086C2×C411yescharacteristic
114#14579086C2×C411yescharacteristic
115#14579083C2×C411yescharacteristic
116#14579083C2×C411yescharacteristic
117C22⋊(C79⋊C8)—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120#14579072—21no—
121#14579071—21no—
122#14579071—21no—
123#14579071—21no—
124#14579071—21no—
125C8×Dic79—21no—
126C8×Dic79—21no—
127C2×M16×C79D411yescharacteristic
128C2×M16×C79D411yescharacteristic
129C2×C42×C79C2311yescharacteristic
130C4⋊C8×C79—21no—
131C4⋊C8×C79C2×C411yescharacteristic
132C4⋊C8×C79C2×C411yescharacteristic
133C22⋊C8×C79—21no—
134C22⋊C8×C79C2×C411yescharacteristic
135C22⋊C8×C79C2×C411yescharacteristic
136#97×C79D411yescharacteristic
137#97×C79D411yescharacteristic
138C4×C8×C79C2311yescharacteristic
139C4×C8×C79C2311yescharacteristic
140C4×C8×C79C2311yescharacteristic
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929624—22no—
145#11929624—41no—
146C2×Dic158—21no—
147C2×Dic158C22×C411yescharacteristic
148C2×Dic158C22×C411yescharacteristic
149C2×Dic158C22×C411yescharacteristic
150C2×D316C22×C411yescharacteristic
151C2×C22⋊D79—21no—
152C2×C4×D79—21no—
153C2×C4×D79—21no—
154C2×C4×D79—21no—
155Dic316—41no—
156Dic316—41no—
157#11929613—22no—
158#11929613—22no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—42no—
164C8×D79—42no—
165C8×D79—42no—
166C4⋊Dic79—21no—
167C4⋊Dic79—21no—
168C4⋊Dic79—21no—
169C4⋊Dic79C22×C411yescharacteristic
170C4⋊Dic79C22×C411yescharacteristic
171C4⋊Dic79C22×C411yescharacteristic
172C4⋊Dic79C22×C411yescharacteristic
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179C22⋊Dic79—21no—
180C22⋊Dic79—21no—
181C22⋊Dic79—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184C4×Dic79—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79—21no—
189C22×C4×C79C22×C411yescharacteristic
190C22×C4×C79C2×D411yescharacteristic
191C22×C4×C79#4111yescharacteristic
192M16×C79—22no—
193M16×C79—41no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79C22×C411yescharacteristic
197C2×C8×C79C22×C411yescharacteristic
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C2×D411yescharacteristic
201C2×C8×C79C2×D411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C2×C8×C79C2×D411yescharacteristic
204C42×C79C22×C411yescharacteristic
205C42×C79C2×D411yescharacteristic
206C42×C79#4111yescharacteristic
207C42×C79C2411yescharacteristic
208C23×C79C4×D411yescharacteristic
209C22×D79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—41no—
215C2×C4×C79C4×D411yescharacteristic
216C2×C4×C79C4×D411yescharacteristic
217C2×C4×C79C4×D411yescharacteristic
218C2×C4×C79C23×C411yescharacteristic
219C2×C4×C79C22×D411yescharacteristic
220C2×C4×C79C2×#4111yescharacteristic
221C22⋊D79—42no—
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229D316—21no—
230D316#13112yes—
231C4×D79—42no—
232C4×D79—41no—
233C4×D79—41no—
234C4×D79—81no—
235Dic158—21no—
236Dic158—21no—
237Dic158—21no—
238Dic158—21no—
239Dic158—21no—
240Dic158—21no—
241Dic158—81no—
242C8×C79—21no—
243C8×C79—21no—
244C8×C79—21no—
245C8×C79—21no—
246C8×C79—21no—
247C8×C79—21no—
248C8×C79—21no—
249C8×C79—41no—
250C8×C79—41no—
251C8×C79#13712yes—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C79⋊C8—41no—
256C22×C79—21no—
257C22×C79—21no—
258C22×C79—21no—
259C22×C79C2×C4×D411yescharacteristic
260C2×D79—22no—
261C2×D79—41no—
262C2×D79—41no—
263C4×C79—21no—
264C4×C79—21no—
265C4×C79—21no—
266C4×C79—21no—
267C4×C79C2×#13111yescharacteristic
268C4×C79#56811yescharacteristic
269C4×C79C2×#13711yescharacteristic
270C4×C79#57511yescharacteristic
271Dic79—41no—
272Dic79—41no—
273Dic79—41no—
274Dic79—41no—
275Dic79—41no—
276Dic79—41no—
277Dic79—41no—
278#18009—791nomaximal, Sylow
279C2×C79—41no—
280C2×C79#293311yescharacteristic
281C2×C79C4×#13711yescharacteristic
282C2×C79#295711yescharacteristic
283D79—42no—
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#1477—791no—
298#1453D7911yescharacteristic, complemented
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—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#431C2×D7911yescharacteristic
329#431C2×D7911yescharacteristic
330#424—1581no—
331C2×C4⋊C8—791no—
332#405—1581no—
333C4×M16C2×D7911yescharacteristic
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⋊C8—791no—
344Q8⋊C8—791no—
345D4⋊C8—791no—
346D4⋊C8—791no—
347#322C2×D7911yescharacteristic
348#322C2×D7911yescharacteristic
349#322C2×D7911yescharacteristic
350C82C2×D7911yescharacteristic
351C2×#41—791no—
352C2×Q16—791no—
353C2×Q16—791no—
354C2×SD16—791no—
355C2×SD16—791no—
356C2×M16D31611yescharacteristic
357C2×M16D31611yescharacteristic
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—791no—
370C4×Q8—791no—
371C4×Q8—1581no—
372C4×D4—791no—
373C4×D4—791no—
374C4×D4—1581no—
375C4×D4—1581no—
376#117—791no—
377#117—1581no—
378#117—1581no—
379C2×C42—1581no—
380C2×C42C22×D7911yescharacteristic
381#107—791no—
382#107—791no—
383#106—791no—
384#106—791no—
385C4⋊C8—21no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—791no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8—1581no—
396C4⋊C8—1581no—
397C4⋊C8C4×D7911yescharacteristic
398C4⋊C8C4×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—1581no—
409C22⋊C8—1581no—
410C22⋊C8—1581no—
411C22⋊C8C4×D7911yescharacteristic
412C22⋊C8C4×D7911yescharacteristic
413#97—1581no—
414#97—1581no—
415#97—1581no—
416#97—1581no—
417#97—1581no—
418#97D31611yescharacteristic
419#97D31611yescharacteristic
420C4×C8—1581no—
421C4×C8—1581no—
422C4×C8C22×D7911yescharacteristic
423C4×C8C22×D7911yescharacteristic
424C4×C8C22×D7911yescharacteristic
425C4×C8C22×D7911yescharacteristic
426C4×C8C22×D7911yescharacteristic
427C4×C8C22×D7911yescharacteristic
428#41—3161no—
429#41—1582no—
430C2×Q8—791no—
431C2×Q8—791no—
432C2×Q8—791no—
433C2×Q8—1581no—
434C2×D4—791no—
435C2×D4—1581no—
436C22×C4—1581no—
437C22×C4—1581no—
438C22×C4—1581no—
439C22×C4C2×C4×D7911yescharacteristic
440C22×C4C2×D31611yescharacteristic
441C22×C4#1192962411yescharacteristic
442Q16—3161no—
443Q16—3161no—
444SD16—1582no—
445SD16—1582no—
446M16—22no—
447M16—41no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—3162no—
451C2×C8—3162no—
452C2×C8—3162no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8—1581no—
456C2×C8—1581no—
457C2×C8C2×C4×D7911yescharacteristic
458C2×C8C2×C4×D7911yescharacteristic
459C2×C8C2×C4×D7911yescharacteristic
460C2×C8C2×C4×D7911yescharacteristic
461C2×C8C2×D31611yescharacteristic
462C2×C8C2×D31611yescharacteristic
463C2×C8C2×D31611yescharacteristic
464C2×C8C2×D31611yescharacteristic
465C4⋊C4—791no—
466C4⋊C4—791no—
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⋊C4—1581no—
477C4⋊C4—1581no—
478C22⋊C4—1581no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C22⋊C4—1581no—
482C22⋊C4—1581no—
483C42—1581no—
484C42—1581no—
485C42—1581no—
486C42—1581no—
487C42—1581no—
488C42C2×C4×D7911yescharacteristic
489C42C2×D31611yescharacteristic
490C42#1192962411yescharacteristic
491C42C23×D7911yescharacteristic
492C23—1581no—
493C23C4×D31611yescharacteristic
494Q8—6321no—
495Q8—1581no—
496Q8—1581no—
497Q8—1581no—
498Q8—1581no—
499Q8—1581no—
500Q8—1581no—
501D4—3162no—
502D4—792no—
503D4—1581no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—3162no—
510C2×C4—3161no—
511C2×C4—3161no—
512C2×C4—6321no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4—1581no—
520C2×C4C4×D31611yescharacteristic
521C2×C4C4×D31611yescharacteristic
522C2×C4C4×D31611yescharacteristic
523C2×C4C22×C4×D7911yescharacteristic
524C2×C4C22×D31611yescharacteristic
525C2×C4C2×#1192962411yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—21no—
533C8—41no—
534C8—41no—
535C8—3161no—
536C8—3161no—
537C8—3161no—
538C8—3161no—
539C8#1457919412yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—3161no—
544C22—3161no—
545C22—1582no—
546C22C2×C4×D31611yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—3161no—
552C4—3161no—
553C4—3161no—
554C4—3161no—
555C4—3161no—
556C4—3161no—
557C4—3161no—
558C4#131×D7911yescharacteristic
559C4#1909932811yescharacteristic
560C4C2×#1457919411yescharacteristic
561C4#1909936311yescharacteristic
562C2—41no—
563C2—3162no—
564C2#2377212811yescharacteristic
565C2C4×#1457919411yescharacteristic
566C2#2377231811yescharacteristic
subgroupG/H
centerC2×C4C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1453C2
socleC22×C79C2×C4×D4
prime radicalC2×#11929624C2×C4
cosocleC42×C79C24
hypercenter#1453D79
hypocenterC79#18009
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1453

Complements

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

NH ≅ G/Ntype
3. #23772347C2semidirect
4. #23762310C2semidirect
5. #23762298C2semidirect
6. #23762273C2semidirect
7. #23762259C2semidirect
8. #23762252C2semidirect
9. #23762238C2semidirect
10. #23762214C2semidirect
11. C8×#11929613C2semidirect
12. #23762185C2semidirect
13. #23761874C2semidirect
15. C79×#1453C2semidirect
298. #1453D79semidirect
299. C79#18009semidirect

Maximal subgroups

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

subgroupindex
#237734722
#237724152
#237723472
#237623102
#237622982
#237622732
#237622592
#237622522
#237622382
#237622142
C8×#119296132
#237621852
#237618742
#237617882
C79×#14532
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