Finite Group Databasebeta

← #34258381

Subgroups — #34258381

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#23773469C211yesmaximal, characteristic
2#23772510C211yesmaximal, characteristic
3#23772496C211yesmaximal, characteristic, complemented
4#23762271C211yesmaximal, characteristic, complemented
5#23762255C211yesmaximal, characteristic, complemented
6#23762238C211yesmaximal, characteristic, complemented
7#23762219C211yesmaximal, characteristic, complemented
8#23762211C211yesmaximal, characteristic, complemented
9#23762204C211yesmaximal, characteristic, complemented
10#23762197C211yesmaximal, characteristic, complemented
11#23762184C211yesmaximal, characteristic, complemented
12#23761833C211yesmaximal, characteristic
13#23761801C211yesmaximal, characteristic, complemented
14#23761701C211yesmaximal, 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#19098693C2211yescharacteristic
27#19098691—21no—
28#19098690—21no—
29Q8×C79⋊C8C2211yescharacteristic
30Q8×C79⋊C8C2211yescharacteristic
31#19098687C2211yescharacteristic
32C4×#11929614C2211yescharacteristic
33C4×#11929613C2211yescharacteristic
34C4×#11929611C2211yescharacteristic
35C8×D316—21no—
36C8×D316C2211yescharacteristic
37#19098628—21no—
38#19098628C2211yescharacteristic
39#19098625—21no—
40#19098623—21no—
41#19098619C2211yescharacteristic
42#19098614—21no—
43#19098614C2211yescharacteristic
44#19098614C2211yescharacteristic
45#19098614C2211yescharacteristic
46C4⋊C8×D79—21no—
47C4⋊C8×D79C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098314C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
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×C8×D79—21no—
72C2×C8×D79—21no—
73C2×C8×D79—21no—
74#14579160C2×C411yescharacteristic
75#14579159C2×C411yescharacteristic
76#14579151—21no—
77#14579141C2×C411yescharacteristic
78#14579139—21no—
79#14579137C2×C411yescharacteristic
80Q8×Dic79C2×C411yescharacteristic
81C4×Dic158—21no—
82C4×Dic158C2311yescharacteristic
83C4×D316—21no—
84C4×D316C2311yescharacteristic
85#14579123—21no—
86#14579123C2×C411yescharacteristic
87#14579119—21no—
88#14579119C2×C411yescharacteristic
89#14579118—21no—
90C42×D79—21no—
91C42×D79C2311yescharacteristic
92#14579094C2×C411yescharacteristic
93#14579093C2×C411yescharacteristic
94#14579092C2×C411yescharacteristic
95C4⋊(C79⋊C8)—21no—
96C4⋊(C79⋊C8)—21no—
97C4⋊(C79⋊C8)C2311yescharacteristic
98C4⋊(C79⋊C8)C2311yescharacteristic
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
105Dic79⋊C8D411yescharacteristic
106Dic79⋊C8D411yescharacteristic
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(C2×D79)⋊C8D411yescharacteristic
120(C2×D79)⋊C8D411yescharacteristic
121#14579071—21no—
122#14579071—21no—
123C4×C79⋊C8—21no—
124C4×C79⋊C8—21no—
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—22no—
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—22no—
157#11929613—22no—
158#11929611—22no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C22×C411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C8×D79—81no—
167C8×D79—81no—
168C8×D79—81no—
169C4⋊Dic79—21no—
170C4⋊Dic79—21no—
171C4⋊Dic79C22×C411yescharacteristic
172C4⋊Dic79C22×C411yescharacteristic
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178(C2×D79)⋊C4—21no—
179(C2×D79)⋊C4—21no—
180(C2×D79)⋊C4—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183C4×Dic79—21no—
184C4×Dic79—21no—
185C4×Dic79—21no—
186C4×Dic79C22×C411yescharacteristic
187C4×Dic79C2×D411yescharacteristic
188C4×Dic79#4111yescharacteristic
189C2×Q8×C79C22×C411yescharacteristic
190C2×Q8×C79C22×C411yescharacteristic
191Q16×C79—22no—
192C2×C8×C79—21no—
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79C22×C411yescharacteristic
196C2×C8×C79C22×C411yescharacteristic
197C2×C8×C79C2×D411yescharacteristic
198C2×C8×C79C2×D411yescharacteristic
199C4⋊C4×C79—21no—
200C4⋊C4×C79—21no—
201C4⋊C4×C79C22×C411yescharacteristic
202C4⋊C4×C79C22×C411yescharacteristic
203C42×C79—21no—
204C42×C79—21no—
205C42×C79C2411yescharacteristic
206C22×D79—21no—
207C22×D79C4×D411yescharacteristic
208Q8×C79—21no—
209Q8×C79—21no—
210Q8×C79—21no—
211Q8×C79#13112yes—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79C23×C411yescharacteristic
217C2×C4×C79C22×D411yescharacteristic
218C2×C4×C79C2×#4111yescharacteristic
219C2×Dic79—21no—
220C2×Dic79—21no—
221C2×Dic79—21no—
222C2×Dic79C4×D411yescharacteristic
223C2×Dic79C4×D411yescharacteristic
224C2×Dic79C4×D411yescharacteristic
225D316—24no—
226D316—21no—
227D316#13112yes—
228C4×D79—24no—
229C4×D79—21no—
230C4×D79—21no—
231C4×D79—21no—
232C4×D79—21no—
233C4×D79—41no—
234C4×D79—41no—
235C4×D79—41no—
236C4×D79—81no—
237Dic158—21no—
238Dic158—21no—
239Dic158—81no—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—21no—
244C8×C79—41no—
245C8×C79—41no—
246C8×C79—41no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—41no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C22×C79C2×C4×D411yescharacteristic
255C2×D79—22no—
256C2×D79—21no—
257C2×D79—21no—
258C2×D79—21no—
259C2×D79—41no—
260C2×D79—41no—
261C4×C79—22no—
262C4×C79—21no—
263C4×C79—41no—
264C4×C79—41no—
265C4×C79—41no—
266C4×C79C2×#13111yescharacteristic
267C4×C79C2×#13111yescharacteristic
268C4×C79#57711yescharacteristic
269C4×C79#57811yescharacteristic
270Dic79—21no—
271Dic79—21no—
272Dic79—21no—
273Dic79—41no—
274Dic79—41no—
275Dic79—41no—
276#17921—791nomaximal, Sylow
277C2×C79#292911yescharacteristic
278C2×C79#294711yescharacteristic
279C2×C79#295711yescharacteristic
280D79—41no—
281D79—42no—
282#3107—791no—
283#2988—791no—
284#2985—791no—
285#1587—791no—
286#1585—791no—
287#1584D7911yescharacteristic, complemented
288#1582—791no—
289#1579—791no—
290SD16⋊C8—791no—
291C8×Q16—791no—
292C8×SD16—791no—
293#1573—791no—
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—1581no—
310#446—1581no—
311#446C2×D7911yescharacteristic
312C8×Q8—791no—
313C8×Q8—791no—
314C8×Q8—791no—
315C8×Q8C2×D7911yescharacteristic
316C4×Q16—791no—
317C4×Q16C2×D7911yescharacteristic
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#432—791no—
327#431—791no—
328#431—791no—
329#431—791no—
330#431—1581no—
331C2×C4⋊C8—791no—
332C2×C4⋊C8—1581no—
333#335—791no—
334#335C2×D7911yescharacteristic
335#334—791no—
336#334—791no—
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8C2×D7911yescharacteristic
342Q8⋊C8C2×D7911yescharacteristic
343D4⋊C8—791no—
344D4⋊C8—791no—
345#322—791no—
346#322—791no—
347#322C2×D7911yescharacteristic
348C82—791no—
349C2×#41—791no—
350C2×Q16—791no—
351C2×Q16C4×D7911yescharacteristic
352C2×SD16—791no—
353C2×SD16—791no—
354C22×C8—1581no—
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—1581no—
366C4×Q8C22×D7911yescharacteristic
367C4×Q8C22×D7911yescharacteristic
368C4×D4—791no—
369C4×D4—791no—
370C4×D4—1581no—
371C4×D4—1581no—
372#117—791no—
373#117—1581no—
374#117—1581no—
375C2×C42—791no—
376C2×C42—1581no—
377#107—791no—
378#107C4×D7911yescharacteristic
379#106—791no—
380#106—791no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—791no—
384C4⋊C8—791no—
385C4⋊C8—791no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8C22×D7911yescharacteristic
396C4⋊C8C22×D7911yescharacteristic
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103C4×D7911yescharacteristic
402#103C4×D7911yescharacteristic
403#102—791no—
404#102—791no—
405C22⋊C8—791no—
406C22⋊C8—791no—
407C22⋊C8—791no—
408C22⋊C8—791no—
409C22⋊C8—1581no—
410C22⋊C8—1581no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—21no—
414#97—1581no—
415#97—1581no—
416C4×C8—21no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—1581no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8C22×D7911yescharacteristic
424C4×C8C22×D7911yescharacteristic
425C4×C8C22×D7911yescharacteristic
426#41—1582no—
427#41—1582no—
428C2×Q8—791no—
429C2×Q8—1581no—
430C2×Q8C2×C4×D7911yescharacteristic
431C2×Q8C2×C4×D7911yescharacteristic
432C2×D4—791no—
433C2×D4—1581no—
434C22×C4—791no—
435C22×C4—791no—
436C22×C4—791no—
437C22×C4—1581no—
438C22×C4—1581no—
439C22×C4—1581no—
440Q16—22no—
441Q16—1582no—
442SD16—1582no—
443SD16—1582no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—791no—
448C2×C8—791no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—6321no—
452C2×C8—6321no—
453C2×C8—6321no—
454C2×C8—1581no—
455C2×C8—1581no—
456C2×C8—1581no—
457C2×C8C2×C4×D7911yescharacteristic
458C2×C8C2×C4×D7911yescharacteristic
459C2×C8D4×D7911yescharacteristic
460C2×C8D4×D7911yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
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⋊C4C2×C4×D7911yescharacteristic
473C4⋊C4C2×C4×D7911yescharacteristic
474C22⋊C4—1581no—
475C22⋊C4—1581no—
476C22⋊C4—1581no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C42—21no—
480C42—21no—
481C42—791no—
482C42—791no—
483C42—791no—
484C42—1581no—
485C42—1581no—
486C42—1581no—
487C42C23×D7911yescharacteristic
488C23—791no—
489C23—1581no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—6321no—
494Q8—1581no—
495Q8—1581no—
496Q8#1457916912yes—
497D4—792no—
498D4—1584no—
499D4—1581no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—3161no—
505C2×C4—3161no—
506C2×C4—3161no—
507C2×C4—791no—
508C2×C4—791no—
509C2×C4—791no—
510C2×C4—6321no—
511C2×C4—1584no—
512C2×C4—1581no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4C22×C4×D7911yescharacteristic
520C2×C4C2×D4×D7911yescharacteristic
521C2×C4#41×D7911yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—3161no—
530C8—3161no—
531C8—3161no—
532C8—1581no—
533C8—1581no—
534C8—1581no—
535C8—1581no—
536C22—3161no—
537C22—3161no—
538C22—1581no—
539C22—1581no—
540C22—1582no—
541C22—1581no—
542C22C4×D4×D7911yescharacteristic
543C4—21no—
544C4—22no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—3161no—
549C4—3161no—
550C4—3161no—
551C4—1581no—
552C4—1581no—
553C4—1581no—
554C4#131×D7911yescharacteristic
555C4C2×#1457916911yescharacteristic
556C4#1909937011yescharacteristic
557C4#1909937611yescharacteristic
558C2—3162no—
559C2—3161no—
560C2#2377209111yescharacteristic
561C2#2377223811yescharacteristic
562C2#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#17921
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. #23772496C2semidirect
4. #23762271C2semidirect
5. #23762255C2semidirect
6. #23762238C2semidirect
7. #23762219C2semidirect
8. #23762211C2semidirect
9. #23762204C2semidirect
10. #23762197C2semidirect
11. #23762184C2semidirect
13. #23761801C2semidirect
14. #23761701C2semidirect
15. C79×#1584C2semidirect
287. #1584D79semidirect
297. C79#17921semidirect

Maximal subgroups

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

subgroupindex
#237734692
#237725102
#237724962
#237622712
#237622552
#237622382
#237622192
#237622112
#237622042
#237621972
#237621842
#237618332
#237618012
#237617012
C79×#15842
#1792179