Finite Group Databasebeta

← #34258372

Subgroups — #34258372

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773472C211yesmaximal, characteristic
2#23772509C211yesmaximal, characteristic
3#23772497C211yesmaximal, characteristic, complemented
4#23762270C211yesmaximal, characteristic, complemented
5#23762256C211yesmaximal, characteristic, complemented
6#23762249C211yesmaximal, characteristic, complemented
7#23762235C211yesmaximal, characteristic, complemented
8#23762218C211yesmaximal, characteristic, complemented
9#23762211C211yesmaximal, characteristic, complemented
10#23762206C211yesmaximal, characteristic, complemented
11#23762199C211yesmaximal, characteristic, complemented
12#23761832C211yesmaximal, characteristic
13#23761802C211yesmaximal, characteristic, complemented
14#23761700C211yesmaximal, characteristic, complemented
15C79×#1573C211yesmaximal, characteristic, complemented
16#19099120C2211yescharacteristic
17C4×#11929624C2211yescharacteristic
18#19098905C411yescharacteristic
19#19098905C411yescharacteristic
20#19098849C411yescharacteristic
21#19098844C411yescharacteristic
22#19098774C411yescharacteristic
23#19098769C411yescharacteristic
24#19098727C411yescharacteristic
25#19098727C411yescharacteristic
26#19098691—21no—
27#19098691—21no—
28#19098690C2211yescharacteristic
29#19098690C2211yescharacteristic
30#19098687C2211yescharacteristic
31#19098687C2211yescharacteristic
32#19098687C2211yescharacteristic
33#19098687C2211yescharacteristic
34C4×Dic316C2211yescharacteristic
35C4×Dic316C2211yescharacteristic
36C4×#11929613C2211yescharacteristic
37C4×#11929613C2211yescharacteristic
38#19098628C2211yescharacteristic
39#19098628C2211yescharacteristic
40C8×C22⋊D79—21no—
41C8×C22⋊D79—21no—
42#19098621—21no—
43#19098621—21no—
44#19098616C2211yescharacteristic
45#19098616C2211yescharacteristic
46#19098615—21no—
47C4⋊C8×D79—21no—
48#19098293C2211yescharacteristic
49#19098293C2211yescharacteristic
50#19098293C2211yescharacteristic
51#19098293C2211yescharacteristic
52#19098293C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098290C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098285C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#432×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62C2×C4⋊C8×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64#335×C79C2211yescharacteristic
65#334×C79C2211yescharacteristic
66#334×C79C2211yescharacteristic
67C2×#11929624C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×Dic316C2×C411yescharacteristic
70C2×#11929613C2×C411yescharacteristic
71C2×#11929613C2×C411yescharacteristic
72C2×C8×D79—21no—
73C2×C8×D79—21no—
74#14579161C2×C411yescharacteristic
75#14579161C2×C411yescharacteristic
76#14579153—21no—
77#14579140—21no—
78#14579140C2×C411yescharacteristic
79#14579140C2×C411yescharacteristic
80C4×Dic158—21no—
81C4×Dic158C2×C411yescharacteristic
82C4×Dic158C2311yescharacteristic
83C4×Dic158C2311yescharacteristic
84C4×Dic158C2311yescharacteristic
85C4×D316C2×C411yescharacteristic
86C4×D316C2311yescharacteristic
87C4×C22⋊D79—21no—
88C4×C22⋊D79—21no—
89#14579120—21no—
90#14579120C2×C411yescharacteristic
91#14579118—21no—
92C42×D79—21no—
93#14579095C2×C411yescharacteristic
94#14579095C2×C411yescharacteristic
95#14579093C2×C411yescharacteristic
96#14579093C2×C411yescharacteristic
97C4⋊(C79⋊C8)—21no—
98C4⋊(C79⋊C8)D411yescharacteristic
99C4⋊(C79⋊C8)D411yescharacteristic
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106#14579086C2×C411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579086C2×C411yescharacteristic
109#14579086C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579086C2×C411yescharacteristic
112#14579083C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114C22⋊(C79⋊C8)—21no—
115C22⋊(C79⋊C8)D411yescharacteristic
116C22⋊(C79⋊C8)D411yescharacteristic
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119#14579071—21no—
120#14579071—21no—
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C8×Dic79—21no—
126C8×Dic79—21no—
127C8×Dic79—21no—
128C8×Dic79—21no—
129C22×C8×C79—21no—
130C2×C42×C79C2311yescharacteristic
131C4⋊C8×C79—21no—
132C4⋊C8×C79C2×C411yescharacteristic
133C4⋊C8×C79C2×C411yescharacteristic
134C4⋊C8×C79C2311yescharacteristic
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138C22⋊C8×C79—21no—
139C22⋊C8×C79C2×C411yescharacteristic
140C22⋊C8×C79C2×C411yescharacteristic
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929624—22no—
145#11929624—22no—
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—22no—
156Dic316—22no—
157#11929613—22no—
158#11929613—22no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8C2×D411yescharacteristic
162C2×C79⋊C8C2×D411yescharacteristic
163C2×C79⋊C8C2×D411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C8×D79—81no—
166C8×D79—81no—
167C4⋊Dic79—21no—
168C4⋊Dic79—21no—
169C4⋊Dic79—21no—
170C4⋊Dic79C22×C411yescharacteristic
171C4⋊Dic79C22×C411yescharacteristic
172C4⋊Dic79C22×C411yescharacteristic
173C4⋊Dic79C22×C411yescharacteristic
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180C22⋊Dic79—21no—
181C22⋊Dic79—21no—
182C22⋊Dic79—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79—21no—
189C4×Dic79—21no—
190C22×C4×C79C22×C411yescharacteristic
191C22×C4×C79C2×D411yescharacteristic
192C22×C4×C79#4111yescharacteristic
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79—81no—
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C22×C411yescharacteristic
202C42×C79C22×C411yescharacteristic
203C42×C79C2×D411yescharacteristic
204C42×C79#4111yescharacteristic
205C42×C79C2411yescharacteristic
206C23×C79C4×D411yescharacteristic
207C22×D79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—41no—
213C2×C4×C79C4×D411yescharacteristic
214C2×C4×C79C4×D411yescharacteristic
215C2×C4×C79C4×D411yescharacteristic
216C2×C4×C79C23×C411yescharacteristic
217C2×C4×C79C22×D411yescharacteristic
218C2×C4×C79C2×#4111yescharacteristic
219C22⋊D79—24no—
220C2×Dic79—21no—
221C2×Dic79—21no—
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227D316—21no—
228D316#13112yes—
229C4×D79—24no—
230C4×D79—41no—
231C4×D79—41no—
232C4×D79—81no—
233Dic158—21no—
234Dic158—21no—
235Dic158—21no—
236Dic158—21no—
237Dic158—21no—
238Dic158—81no—
239Dic158#13112yes—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—21no—
244C8×C79—41no—
245C8×C79—41no—
246C8×C79—41no—
247C8×C79—41no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C22×C79—21no—
255C22×C79—21no—
256C22×C79—21no—
257C22×C79C2×C4×D411yescharacteristic
258C2×D79—22no—
259C2×D79—41no—
260C2×D79—41no—
261C4×C79—21no—
262C4×C79—21no—
263C4×C79—21no—
264C4×C79—21no—
265C4×C79C2×#13111yescharacteristic
266C4×C79C2×#13111yescharacteristic
267C4×C79#57711yescharacteristic
268C4×C79#57811yescharacteristic
269Dic79—22no—
270Dic79—41no—
271Dic79—41no—
272Dic79—41no—
273Dic79—41no—
274Dic79—41no—
275Dic79—41no—
276#17921—791nomaximal, Sylow
277C2×C79—41no—
278C2×C79#292911yescharacteristic
279C2×C79#294711yescharacteristic
280C2×C79#295711yescharacteristic
281D79—42no—
282#3107—791no—
283#2988—791no—
284#2985—791no—
285#1587—791no—
286#1585—791no—
287#1584—791no—
288#1582—791no—
289#1579—791no—
290SD16⋊C8—791no—
291C8×Q16—791no—
292C8×SD16—791no—
293#1573D7911yescharacteristic, complemented
294#1488—791no—
295#1481—791no—
296#1454—791no—
297C79#1792111yescharacteristic, complemented, Sylow
298#533—791no—
299C4×#41—791no—
300#485—791no—
301#485—791no—
302#475—791no—
303#474—791no—
304#462—791no—
305#461—791no—
306#452—791no—
307#452—791no—
308#446—791no—
309#446—791no—
310#446—1581no—
311#446—1581no—
312C8×Q8—791no—
313C8×Q8—791no—
314C8×Q8—791no—
315C8×Q8—791no—
316C4×Q16—791no—
317C4×Q16—791no—
318C4×SD16—791no—
319C4×SD16—791no—
320C8×D4—791no—
321C8×D4—791no—
322C8×D4—1581no—
323C8×D4—1581no—
324#433—1581no—
325#433—1581no—
326#432C2×D7911yescharacteristic
327#431—791no—
328#431—791no—
329#431—1581no—
330#431C2×D7911yescharacteristic
331C2×C4⋊C8—1581no—
332C2×C4⋊C8C2×D7911yescharacteristic
333#335C2×D7911yescharacteristic
334#335C2×D7911yescharacteristic
335#334C2×D7911yescharacteristic
336#334C2×D7911yescharacteristic
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8—791no—
343D4⋊C8—791no—
344D4⋊C8—791no—
345#322—791no—
346#322—791no—
347#322—791no—
348C82—791no—
349C2×#41—791no—
350C2×Q16—791no—
351C2×Q16—791no—
352C2×SD16—791no—
353C2×SD16—791no—
354C22×C8—21no—
355C22×C8—1581no—
356C22×C8—1581no—
357C4⋊Q8—791no—
358C4⋊Q8—791no—
359#125—1581no—
360C22⋊Q8—791no—
361C22⋊Q8—791no—
362C22⋊Q8—1581no—
363C4×Q8—791no—
364C4×Q8—791no—
365C4×Q8—791no—
366C4×Q8—791no—
367C4×Q8—1581no—
368C4×D4—791no—
369C4×D4—791no—
370C4×D4—1581no—
371C4×D4—1581no—
372#117—791no—
373#117—1581no—
374#117—1581no—
375C2×C42—1581no—
376C2×C42C22×D7911yescharacteristic
377#107—791no—
378#107—791no—
379#106—791no—
380#106—791no—
381C4⋊C8—21no—
382C4⋊C8—791no—
383C4⋊C8—791no—
384C4⋊C8—1581no—
385C4⋊C8—1581no—
386C4⋊C8—1581no—
387C4⋊C8—1581no—
388C4⋊C8—1581no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8C4×D7911yescharacteristic
392C4⋊C8C4×D7911yescharacteristic
393C4⋊C8C22×D7911yescharacteristic
394C4⋊C8C22×D7911yescharacteristic
395C4⋊C8C22×D7911yescharacteristic
396C4⋊C8C22×D7911yescharacteristic
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103—791no—
402#103—791no—
403#102—791no—
404#102—791no—
405C22⋊C8—21no—
406C22⋊C8—791no—
407C22⋊C8—791no—
408C22⋊C8—1581no—
409C22⋊C8—1581no—
410C22⋊C8—1581no—
411C22⋊C8C4×D7911yescharacteristic
412C22⋊C8C4×D7911yescharacteristic
413#97—21no—
414#97—1581no—
415#97—1581no—
416C4×C8—791no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—1581no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8—1581no—
424C4×C8C22×D7911yescharacteristic
425C4×C8C22×D7911yescharacteristic
426#41—1582no—
427#41—1582no—
428C2×Q8—791no—
429C2×Q8—791no—
430C2×Q8—791no—
431C2×Q8—1581no—
432C2×D4—791no—
433C2×D4—1581no—
434C22×C4—1581no—
435C22×C4—1581no—
436C22×C4—1581no—
437C22×C4C2×C4×D7911yescharacteristic
438C22×C4D4×D7911yescharacteristic
439C22×C4#1192962511yescharacteristic
440Q16—1582no—
441Q16—1582no—
442SD16—1582no—
443SD16—1582no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—81no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—791no—
453C2×C8—6321no—
454C2×C8—6321no—
455C2×C8—1581no—
456C2×C8—1581no—
457C2×C8C2×C4×D7911yescharacteristic
458C2×C8C2×C4×D7911yescharacteristic
459C2×C8C2×C4×D7911yescharacteristic
460C2×C8C2×C4×D7911yescharacteristic
461C4⋊C4—791no—
462C4⋊C4—791no—
463C4⋊C4—791no—
464C4⋊C4—791no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C22⋊C4—1581no—
475C22⋊C4—1581no—
476C22⋊C4—1581no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C42—1581no—
480C42—1581no—
481C42—1581no—
482C42—1581no—
483C42—1581no—
484C42C2×C4×D7911yescharacteristic
485C42D4×D7911yescharacteristic
486C42#1192962511yescharacteristic
487C42C23×D7911yescharacteristic
488C23—1581no—
489C23#1457912311yescharacteristic
490Q8—792no—
491Q8—6321no—
492Q8—1581no—
493Q8—1581no—
494Q8—1581no—
495Q8—1581no—
496Q8—1581no—
497D4—792no—
498D4—1584no—
499D4—1581no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—3161no—
506C2×C4—3161no—
507C2×C4—6321no—
508C2×C4—1584no—
509C2×C4—1581no—
510C2×C4—1581no—
511C2×C4—1581no—
512C2×C4—1581no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4#1457912311yescharacteristic
517C2×C4#1457912311yescharacteristic
518C2×C4#1457912311yescharacteristic
519C2×C4C22×C4×D7911yescharacteristic
520C2×C4C2×D4×D7911yescharacteristic
521C2×C4C2×#1192962511yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—3161no—
531C8—3161no—
532C8—1581no—
533C8—1581no—
534C8—1581no—
535C8—1581no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—3161no—
540C22—3161no—
541C22—1582no—
542C22C2×#1457912311yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—3161no—
548C4—3161no—
549C4—3161no—
550C4—3161no—
551C4—3161no—
552C4—3161no—
553C4—1582no—
554C4#131×D7911yescharacteristic
555C4#131×D7911yescharacteristic
556C4#1909937311yescharacteristic
557C4#1909937911yescharacteristic
558C2—41no—
559C2—3162no—
560C2#2377209311yescharacteristic
561C2#2377224111yescharacteristic
562C2#2377231611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1573C2
socleC22×C79C2×C4×D4
prime radicalC2×#11929624C2×C4
cosocleC42×C79C24
hypercenter#1573D79
hypocenterC79#17921
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1573

Complements

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

NH ≅ G/Ntype
3. #23772497C2semidirect
4. #23762270C2semidirect
5. #23762256C2semidirect
6. #23762249C2semidirect
7. #23762235C2semidirect
8. #23762218C2semidirect
9. #23762211C2semidirect
10. #23762206C2semidirect
11. #23762199C2semidirect
13. #23761802C2semidirect
14. #23761700C2semidirect
15. C79×#1573C2semidirect
293. #1573D79semidirect
297. C79#17921semidirect

Maximal subgroups

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

subgroupindex
#237734722
#237725092
#237724972
#237622702
#237622562
#237622492
#237622352
#237622182
#237622112
#237622062
#237621992
#237618322
#237618022
#237617002
C79×#15732
#1792179