Finite Group Databasebeta

← #34259460

Subgroups — #34259460

Order 20224.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773482C211yesmaximal, characteristic
2#23772556C211yesmaximal, characteristic
3#23772490C211yesmaximal, characteristic, complemented
4#23762296C211yesmaximal, characteristic, complemented
5#23762289C211yesmaximal, characteristic, complemented
6#23762277C211yesmaximal, characteristic, complemented
7#23762270C211yesmaximal, characteristic, complemented
8#23762270C211yesmaximal, characteristic, complemented
9#23762256C211yesmaximal, characteristic, complemented
10#23762206C211yesmaximal, characteristic, complemented
11#23762199C211yesmaximal, characteristic, complemented
12#23761858C211yesmaximal, characteristic
13C2×#19098293C211yesmaximal, characteristic, complemented
14#23761684C211yesmaximal, characteristic, complemented
15C79×#1573C211yesmaximal, characteristic, complemented
16#19099097C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18#19098894C411yescharacteristic
19#19098877C411yescharacteristic
20#19098871C411yescharacteristic
21#19098871C411yescharacteristic
22#19098786C411yescharacteristic
23#19098786C411yescharacteristic
24#19098721C411yescharacteristic
25#19098704C411yescharacteristic
26#19098691—21no—
27#19098691—21no—
28#19098690C2211yescharacteristic
29#19098690C2211yescharacteristic
30#19098690C2211yescharacteristic
31#19098687C2211yescharacteristic
32#19098687C2211yescharacteristic
33#19098687C2211yescharacteristic
34C4×Dic316C2211yescharacteristic
35C4×Dic316C2211yescharacteristic
36C4×#11929613C2211yescharacteristic
37C4×#11929613C2211yescharacteristic
38#19098640C2211yescharacteristic
39#19098634—21no—
40#19098628C2211yescharacteristic
41C8×C22⋊D79—21no—
42#19098623—21no—
43#19098617—21no—
44#19098616C2211yescharacteristic
45#19098613—21no—
46#19098588—21no—
47C2×C4×C79⋊C8C2211yescharacteristic
48#19098293C2211yescharacteristic
49#19098293C2211yescharacteristic
50#19098293C2211yescharacteristic
51#19098293C2211yescharacteristic
52#19098293C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098290C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098285C2211yescharacteristic
57#19098285C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
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
67C22×Dic158C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×Dic316C2×C411yescharacteristic
70C2×#11929613C2×C411yescharacteristic
71C2×#11929613C2×C411yescharacteristic
72C2×#11929607—21no—
73C22×C79⋊C8—21no—
74C2×C8×D79—21no—
75#14579155—21no—
76#14579153C2×C411yescharacteristic
77#14579149C2×C411yescharacteristic
78#14579144—21no—
79#14579140C2×C411yescharacteristic
80#14579135C2×C411yescharacteristic
81C4×Dic158—21no—
82C4×Dic158—21no—
83C4×Dic158C2×C411yescharacteristic
84C4×Dic158C2×C411yescharacteristic
85C4×Dic158C2311yescharacteristic
86C4×Dic158C2311yescharacteristic
87C4×Dic158C2311yescharacteristic
88C4×D316C2311yescharacteristic
89C4×C22⋊D79—21no—
90#14579118—21no—
91C2×C4⋊Dic79C2×C411yescharacteristic
92C2×Dic79⋊C4—21no—
93C2×C4×Dic79—21no—
94#14579095C2×C411yescharacteristic
95#14579095C2×C411yescharacteristic
96#14579093C2×C411yescharacteristic
97#14579093C2×C411yescharacteristic
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)D411yescharacteristic
115C22⋊(C79⋊C8)D411yescharacteristic
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118#14579071—21no—
119#14579071—21no—
120#14579071—21no—
121C4×C79⋊C8—21no—
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
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
144C2×Dic158—22no—
145C2×Dic158—21no—
146C2×Dic158—41no—
147C2×Dic158C22×C411yescharacteristic
148C2×Dic158C22×C411yescharacteristic
149C2×Dic158C22×C411yescharacteristic
150C2×D316C22×C411yescharacteristic
151C2×C22⋊D79—21no—
152C22×Dic79—21no—
153C22×Dic79—21no—
154C2×C4×D79—21no—
155Dic316—22no—
156Dic316—41no—
157#11929613—22no—
158#11929613—41no—
159#11929607—81no—
160C2×C79⋊C8—24no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8C2×D411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C8×D79—81no—
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—
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181C22⋊Dic79—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79—21no—
189C4×Dic79—21no—
190C4×Dic79—21no—
191C22×C4×C79C22×C411yescharacteristic
192C22×C4×C79C2×D411yescharacteristic
193C22×C4×C79#4111yescharacteristic
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—81no—
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C22×C411yescharacteristic
202C2×C8×C79C22×C411yescharacteristic
203C42×C79C22×C411yescharacteristic
204C42×C79C2×D411yescharacteristic
205C42×C79#4111yescharacteristic
206C42×C79C2411yescharacteristic
207C23×C79C4×D411yescharacteristic
208C22×D79—21no—
209C2×C4×C79—22no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79C4×D411yescharacteristic
215C2×C4×C79C4×D411yescharacteristic
216C2×C4×C79C4×D411yescharacteristic
217C2×C4×C79C23×C411yescharacteristic
218C2×C4×C79C22×D411yescharacteristic
219C2×C4×C79C2×#4111yescharacteristic
220C22⋊D79—42no—
221C2×Dic79—21no—
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—42no—
229C2×Dic79—42no—
230D316—21no—
231D316—21no—
232C4×D79—41no—
233C4×D79—41no—
234Dic158—21no—
235Dic158—21no—
236Dic158—21no—
237Dic158—21no—
238Dic158—21no—
239Dic158—42no—
240Dic158#13112yes—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—21no—
244C8×C79—21no—
245C8×C79—41no—
246C8×C79—41no—
247C8×C79—41no—
248C8×C79—41no—
249C79⋊C8—22no—
250C79⋊C8—22no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C22×C79—21no—
256C22×C79—21no—
257C22×C79#44312yes—
258C22×C79C2×C4×D411yescharacteristic
259C2×D79—41no—
260C2×D79—41no—
261C2×D79—41no—
262C4×C79—21no—
263C4×C79—21no—
264C4×C79—21no—
265C4×C79—21no—
266C4×C79C2×#13111yescharacteristic
267C4×C79#56811yescharacteristic
268C4×C79#57711yescharacteristic
269C4×C79#57811yescharacteristic
270Dic79—22no—
271Dic79—41no—
272Dic79—41no—
273Dic79—41no—
274Dic79—41no—
275Dic79—41no—
276Dic79—41no—
277#18017—791nomaximal, Sylow
278C2×C79—22no—
279C2×C79#293311yescharacteristic
280C2×C79#294711yescharacteristic
281C2×C79C2×#44311yescharacteristic
282D79—81no—
283#3108—791no—
284#2994—791no—
285#2984—791no—
286#1591—791no—
287#1590—791no—
288#1588—791no—
289#1587—791no—
290#1587—791no—
291#1585—791no—
292C8×Q16—791no—
293C8×SD16—791no—
294#1573D7911yescharacteristic, complemented
295#1494—791no—
296C2×Q8⋊C8—791no—
297#1450—791no—
298C79#1801711yescharacteristic, complemented, Sylow
299#529—791no—
300C2×C4×Q8—791no—
301#483—791no—
302#480—791no—
303#479—791no—
304#479—791no—
305#464—791no—
306#464—791no—
307#451—791no—
308#448—791no—
309#446—791no—
310#446—791no—
311#446—791no—
312#446—1581no—
313#446—1581no—
314C8×Q8—791no—
315C8×Q8—791no—
316C8×Q8—791no—
317C4×Q16—791no—
318C4×Q16—791no—
319C4×SD16—791no—
320C4×SD16—791no—
321#436—791no—
322#435—1581no—
323C8×D4—791no—
324C8×D4—1581no—
325#433—1581no—
326#432—1581no—
327#432C2×D7911yescharacteristic
328#431—791no—
329#431—1581no—
330#431C2×D7911yescharacteristic
331#424—1581no—
332C2×C4⋊C8C2×D7911yescharacteristic
333C2×C4×C8—791no—
334#335C2×D7911yescharacteristic
335#335C2×D7911yescharacteristic
336#334C2×D7911yescharacteristic
337#334C2×D7911yescharacteristic
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8—791no—
343Q8⋊C8—791no—
344D4⋊C8—791no—
345D4⋊C8—791no—
346#322—791no—
347#322—791no—
348C82—791no—
349C82—791no—
350C22×Q8—791no—
351C2×Q16—791no—
352C2×Q16—791no—
353C2×SD16—791no—
354C2×SD16—791no—
355C2×M16—1581no—
356C22×C8—21no—
357C22×C8—1581no—
358C22×C8—1581no—
359#126—1581no—
360#125—791no—
361#124—791no—
362#123—1581no—
363C22⋊Q8—791no—
364#121—791no—
365C4×Q8—791no—
366C4×Q8—791no—
367C4×Q8—791no—
368C4×Q8—791no—
369C4×Q8—791no—
370C4×Q8—1581no—
371C4×Q8—1581no—
372C4×D4—791no—
373C4×D4—1581no—
374#117—1581no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—1581no—
377C2×C42—1581no—
378C2×C42C22×D7911yescharacteristic
379#107—791no—
380#107—791no—
381#106—791no—
382#106—791no—
383C4⋊C8—21no—
384C4⋊C8—791no—
385C4⋊C8—791no—
386C4⋊C8—1581no—
387C4⋊C8—1581no—
388C4⋊C8—1581no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8C4×D7911yescharacteristic
393C4⋊C8C4×D7911yescharacteristic
394C4⋊C8C22×D7911yescharacteristic
395C4⋊C8C22×D7911yescharacteristic
396C4⋊C8C22×D7911yescharacteristic
397C4⋊C8C22×D7911yescharacteristic
398#103—791no—
399#103—791no—
400#103—791no—
401#103—791no—
402#103—791no—
403#103—791no—
404#102—791no—
405#102—791no—
406C22⋊C8—21no—
407C22⋊C8—791no—
408C22⋊C8—791no—
409C22⋊C8—1581no—
410C22⋊C8—1581no—
411C22⋊C8C4×D7911yescharacteristic
412C22⋊C8C4×D7911yescharacteristic
413#97—21no—
414#97—1581no—
415#97—1581no—
416#97—1581no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—791no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8—1581no—
424C4×C8—1581no—
425C4×C8C22×D7911yescharacteristic
426C4×C8C22×D7911yescharacteristic
427C2×Q8—3161no—
428C2×Q8—791no—
429C2×Q8—791no—
430C2×Q8—791no—
431C2×Q8—1582no—
432C2×Q8—1581no—
433C2×D4—791no—
434C2×D4—1581no—
435C22×C4—1581no—
436C22×C4—1581no—
437C22×C4—1581no—
438C22×C4C2×C4×D7911yescharacteristic
439C22×C4D4×D7911yescharacteristic
440C22×C4#1192962511yescharacteristic
441Q16—3161no—
442Q16—1582no—
443SD16—3161no—
444SD16—1582no—
445M16—6321no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—81no—
451C2×C8—791no—
452C2×C8—791no—
453C2×C8—791no—
454C2×C8—791no—
455C2×C8—6321no—
456C2×C8—1584no—
457C2×C8—1581no—
458C2×C8—1581no—
459C2×C8C2×C4×D7911yescharacteristic
460C2×C8C2×C4×D7911yescharacteristic
461C2×C8C2×C4×D7911yescharacteristic
462C2×C8C2×C4×D7911yescharacteristic
463C4⋊C4—791no—
464C4⋊C4—791no—
465C4⋊C4—791no—
466C4⋊C4—791no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C22⋊C4—1581no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C42—1581no—
482C42—1581no—
483C42—1581no—
484C42—1581no—
485C42—1581no—
486C42C2×C4×D7911yescharacteristic
487C42D4×D7911yescharacteristic
488C42#1192962511yescharacteristic
489C42C23×D7911yescharacteristic
490C23—1581no—
491C23#1457912311yescharacteristic
492Q8—3162no—
493Q8—792no—
494Q8—1581no—
495Q8—1581no—
496Q8—1581no—
497Q8—1581no—
498Q8—1581no—
499D4—3162no—
500D4—1581no—
501D4—1581no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—3162no—
508C2×C4—3162no—
509C2×C4—3161no—
510C2×C4—3161no—
511C2×C4—1581no—
512C2×C4—1581no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4#1457912311yescharacteristic
519C2×C4#1457912311yescharacteristic
520C2×C4#1457912311yescharacteristic
521C2×C4C22×C4×D7911yescharacteristic
522C2×C4C2×D4×D7911yescharacteristic
523C2×C4C2×#1192962511yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—1582no—
533C8—1582no—
534C8—1581no—
535C8—1581no—
536C8—1581no—
537C8—1581no—
538C22—21no—
539C22—21no—
540C22—3161no—
541C22—3161no—
542C22—3161no—
543C22#1909867912yes—
544C22C2×#1457912311yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—3161no—
550C4—3161no—
551C4—3161no—
552C4—3161no—
553C4—3161no—
554C4—3161no—
555C4—1582no—
556C4#131×D7911yescharacteristic
557C4#1909932811yescharacteristic
558C4#1909937311yescharacteristic
559C4#1909937911yescharacteristic
560C2—22no—
561C2—6321no—
562C2#2377212411yescharacteristic
563C2#2377224111yescharacteristic
564C2C2×#1909867911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1573C2
socleC22×C79C2×C4×D4
prime radical#14579135C2×C4
cosocleC42×C79C24
hypercenter#1573D79
hypocenterC79#18017
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. #23772490C2semidirect
4. #23762296C2semidirect
5. #23762289C2semidirect
6. #23762277C2semidirect
7. #23762270C2semidirect
8. #23762270C2semidirect
9. #23762256C2semidirect
10. #23762206C2semidirect
11. #23762199C2semidirect
13. C2×#19098293C2semidirect
14. #23761684C2semidirect
15. C79×#1573C2semidirect
294. #1573D79semidirect
298. C79#18017semidirect

Maximal subgroups

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

subgroupindex
#237734822
#237725562
#237724902
#237622962
#237622892
#237622772
#237622702
#237622562
#237622062
#237621992
#237618582
C2×#190982932
#237616842
C79×#15732
#1801779