Finite Group Databasebeta

← #34259697

Subgroups — #34259697

Order 20224.

656 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772415C211yesmaximal, characteristic, complemented
2C4×#14579181C211yesmaximal, characteristic
3#23762295C212yesmaximal, complemented
4#23762236C212yesmaximal, complemented
5#23762234C212yesmaximal, complemented
6C8×#11929609C212yesmaximal, complemented
7#23762161C211yesmaximal, characteristic, complemented
8#23761834C211yesmaximal, characteristic, complemented
9#23761832C211yesmaximal, characteristic, complemented
10M16×C79⋊C8C211yesmaximal, characteristic, complemented
11C79×#2970C211yesmaximal, characteristic, complemented
12C2×#14579181C411yescharacteristic
13C4×#11929624C2211yescharacteristic
14#19098693C2212yes—
15C4×#11929614C411yescharacteristic
16C4×#11929614C2211yescharacteristic
17C4×#11929611C411yescharacteristic
18C4×#11929611C2211yescharacteristic
19C4×#11929612C411yescharacteristic
20C4×#11929612C2211yescharacteristic
21C4×#11929609C411yescharacteristic
22C4×#11929609C2211yescharacteristic
23#19098632—22no—
24C8×D316C2212yes—
25#19098620C2211yescharacteristic
26#19098618—21no—
27#19098616C2211yescharacteristic
28#19098599C411yescharacteristic
29#19098559C411yescharacteristic
30#19098557C411yescharacteristic
31#97×D79—21no—
32C2×C4×C79⋊C8C2211yescharacteristic
33#19098315C2212yes—
34#19098313C2212yes—
35Q8⋊(C79⋊C8)C2212yes—
36#19098293C2212yes—
37D4⋊(C79⋊C8)C2212yes—
38#19098290C2212yes—
39#19098284C2212yes—
40C8×C79⋊C8C2212yes—
41C4×#41×C79C2211yescharacteristic
42#446×C79C2212yes—
43#435×C79—22no—
44C8×D4×C79C2212yes—
45#432×C79—21no—
46#432×C79C2211yescharacteristic
47C4×M16×C79C2211yescharacteristic
48C2×#97×C79—21no—
49C2×#11929624C2×C411yescharacteristic
50#14579181—24no—
51C2×#11929614C2×C411yescharacteristic
52C2×#11929612C2×C411yescharacteristic
53C2×#11929611C2×C411yescharacteristic
54C2×#11929609C2×C411yescharacteristic
55C22×C79⋊C8C2×C411yescharacteristic
56C22×C79⋊C8C2×C411yescharacteristic
57C2×C8×D79—22no—
58C4×Dic158C2×C411yescharacteristic
59C4×Dic158C2311yescharacteristic
60C4×D316C2×C411yescharacteristic
61C4×D316C2311yescharacteristic
62C4×C22⋊D79—21no—
63C4×C22⋊D79—21no—
64#14579120C2×C411yescharacteristic
65#14579118—21no—
66C42×D79—21no—
67#14579094C2×C411yescharacteristic
68#14579094C2×C411yescharacteristic
69#14579092C2×C411yescharacteristic
70#14579092C2×C411yescharacteristic
71C4⋊(C79⋊C8)—21no—
72C4⋊(C79⋊C8)C2312yes—
73Dic79⋊C8—22no—
74#14579085C2×C411yescharacteristic
75#14579085C2×C411yescharacteristic
76#14579084C2×C411yescharacteristic
77#14579084C2×C411yescharacteristic
78#14579082C2×C411yescharacteristic
79#14579082C2×C411yescharacteristic
80#14579081C2×C411yescharacteristic
81#14579081C2×C411yescharacteristic
82C22⋊(C79⋊C8)—21no—
83C22⋊(C79⋊C8)—21no—
84(C2×D79)⋊C8—22no—
85#14579072—21no—
86#14579071—22no—
87C4×C79⋊C8C2×C411yescharacteristic
88C4×C79⋊C8C2×C411yescharacteristic
89C4×C79⋊C8C2312yes—
90C4×C79⋊C8C2311yescharacteristic
91C4×C79⋊C8C2311yescharacteristic
92C2×#41×C79C2×C411yescharacteristic
93C2×M16×C79D412yes—
94C22×C8×C79—22no—
95C4×Q8×C79C2×C411yescharacteristic
96C4×Q8×C79C2311yescharacteristic
97C4×D4×C79—21no—
98C4×D4×C79—21no—
99C4×D4×C79C2×C411yescharacteristic
100C4×D4×C79C2311yescharacteristic
101#117×C79—21no—
102#117×C79C2×C411yescharacteristic
103C2×C42×C79—21no—
104C2×C42×C79C2311yescharacteristic
105C4⋊C8×C79—22no—
106C4⋊C8×C79C2312yes—
107C22⋊C8×C79—21no—
108C22⋊C8×C79—22no—
109#97×C79—22no—
110#97×C79—21no—
111#97×C79D411yescharacteristic
112#97×C79D411yescharacteristic
113C4×C8×C79C2312yes—
114#11929624—22no—
115#11929624—41no—
116C2×Dic158C22×C411yescharacteristic
117C2×D316C22×C411yescharacteristic
118C2×C22⋊D79—21no—
119C2×C4×D79—21no—
120C2×C4×D79—21no—
121C2×C4×D79—21no—
122#11929614—22no—
123#11929612—22no—
124#11929611—22no—
125#11929609M1614yescomplemented
126C2×C79⋊C8—22no—
127C2×C79⋊C8—22no—
128C2×C79⋊C8—22no—
129C2×C79⋊C8—41no—
130C2×C79⋊C8C22×C411yescharacteristic
131C2×C79⋊C8C22×C411yescharacteristic
132C2×C79⋊C8C22×C411yescharacteristic
133C2×C79⋊C8C22×C411yescharacteristic
134C8×D79—44no—
135C4⋊Dic79C22×C411yescharacteristic
136C4⋊Dic79C22×C411yescharacteristic
137Dic79⋊C4—21no—
138Dic79⋊C4—21no—
139C22⋊Dic79—21no—
140(C2×D79)⋊C4—21no—
141(C2×D79)⋊C4—21no—
142C4×Dic79—21no—
143C4×Dic79—21no—
144C4×Dic79—21no—
145#41×C79—22no—
146#41×C79—41no—
147C2×Q8×C79C22×C411yescharacteristic
148C2×D4×C79—21no—
149C2×D4×C79C22×C411yescharacteristic
150C22×C4×C79—21no—
151C22×C4×C79—21no—
152C22×C4×C79—21no—
153C22×C4×C79C22×C411yescharacteristic
154C22×C4×C79C2×D411yescharacteristic
155C22×C4×C79#4111yescharacteristic
156M16×C79—24no—
157C2×C8×C79—22no—
158C2×C8×C79—44no—
159C2×C8×C79C2×D412yes—
160C2×C8×C79C2×D412yes—
161C4⋊C4×C79—21no—
162C4⋊C4×C79—21no—
163C4⋊C4×C79C22×C411yescharacteristic
164C4⋊C4×C79C22×C411yescharacteristic
165C22⋊C4×C79—21no—
166C22⋊C4×C79—21no—
167C22⋊C4×C79—21no—
168C42×C79—21no—
169C42×C79—21no—
170C42×C79—21no—
171C42×C79C22×C411yescharacteristic
172C42×C79C2×D411yescharacteristic
173C42×C79#4111yescharacteristic
174C42×C79C2411yescharacteristic
175C23×C79—21no—
176C23×C79C4×D411yescharacteristic
177C22×D79—21no—
178Q8×C79—21no—
179Q8×C79—21no—
180D4×C79—21no—
181D4×C79—42no—
182D4×C79C2×M1612yes—
183C2×C4×C79—21no—
184C2×C4×C79—21no—
185C2×C4×C79—22no—
186C2×C4×C79—21no—
187C2×C4×C79—21no—
188C2×C4×C79—21no—
189C2×C4×C79—41no—
190C2×C4×C79—42no—
191C2×C4×C79—42no—
192C2×C4×C79—81no—
193C2×C4×C79C4×D411yescharacteristic
194C2×C4×C79C4×D411yescharacteristic
195C2×C4×C79C4×D411yescharacteristic
196C2×C4×C79C23×C411yescharacteristic
197C2×C4×C79C22×D411yescharacteristic
198C2×C4×C79C2×#4111yescharacteristic
199C22⋊D79—42no—
200C2×Dic79—21no—
201C2×Dic79—21no—
202C2×Dic79—21no—
203D316—21no—
204D316C2×M1612yes—
205C4×D79—42no—
206C4×D79—42no—
207C4×D79—81no—
208Dic158—21no—
209Dic158—21no—
210C8×C79—22no—
211C8×C79—42no—
212C8×C79#13514yescomplemented
213C79⋊C8—21no—
214C79⋊C8—21no—
215C79⋊C8—21no—
216C79⋊C8—42no—
217C79⋊C8—42no—
218C79⋊C8C2×M1612yes—
219C22×C79—21no—
220C22×C79—22no—
221C22×C79—21no—
222C22×C79—21no—
223C22×C79—41no—
224C22×C79—41no—
225C22×C79C2×C4×D411yescharacteristic
226C2×D79—22no—
227C2×D79—41no—
228C2×D79—41no—
229C4×C79—21no—
230C4×C79—21no—
231C4×C79—21no—
232C4×C79—21no—
233C4×C79—41no—
234C4×C79—41no—
235C4×C79—41no—
236C4×C79C22×M1611yescharacteristic
237C4×C79#56811yescharacteristic
238C4×C79C2×#13512yes—
239Dic79—41no—
240Dic79—41no—
241Dic79—41no—
242#18035—791nomaximal, Sylow
243C2×C79—41no—
244C2×C79—42no—
245C2×C79D4×M1611yescharacteristic
246C2×C79C4×#13511yescharacteristic
247C2×C79#295711yescharacteristic
248D79—42no—
249#2970—791no—
250#2970D7911yescharacteristic, complemented
251C4×#135—791no—
252#1591—792no—
253#1582—792no—
254#1582—792no—
255C8×D8—792no—
256#1568—791no—
257#1488—791no—
258#1488—791no—
259C8×M16—791no—
260C79#1803511yescharacteristic, complemented, Sylow
261C2×#135—791no—
262C4×#41—791no—
263C4×#41C2×D7911yescharacteristic
264#446—792no—
265#446C2×D7912yes—
266C4×Q16—791no—
267C4×Q16—791no—
268C4×SD16—791no—
269C4×SD16—791no—
270C4×SD16—791no—
271C4×SD16—791no—
272C4×D8—791no—
273C4×D8—791no—
274#435—22no—
275#435—1582no—
276C8×D4—792no—
277C8×D4C2×D7912yes—
278#432—21no—
279#432—791no—
280#432—1581no—
281#432C2×D7911yescharacteristic
282#431—791no—
283#427—791no—
284#416—791no—
285#416—791no—
286C4×M16C2×D7911yescharacteristic
287C2×#97—21no—
288C2×#97—1581no—
289C2×C4×C8—791no—
290#335—792no—
291#334—792no—
292Q8⋊C8—792no—
293Q8⋊C8—792no—
294D4⋊C8—792no—
295D4⋊C8—792no—
296#322—792no—
297C82—792no—
298C2×#41—791no—
299C2×#41C4×D7911yescharacteristic
300#135—1584no—
301C2×Q16—791no—
302C2×SD16—791no—
303C2×SD16—791no—
304C2×D8—791no—
305C2×M16C22⋊D7912yes—
306C22×C8—22no—
307C22×C8—791no—
308C22×C8—791no—
309C22×C8—1582no—
310C4×Q8—791no—
311C4×Q8—791no—
312C4×Q8C4×D7911yescharacteristic
313C4×Q8C22×D7911yescharacteristic
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—791no—
317C4×D4—791no—
318C4×D4—1581no—
319C4×D4—1581no—
320C4×D4C4×D7911yescharacteristic
321C4×D4C22×D7911yescharacteristic
322#117—21no—
323#117—791no—
324#117—1581no—
325#117C4×D7911yescharacteristic
326C2×C42—21no—
327C2×C42—1581no—
328C2×C42C22×D7911yescharacteristic
329#107—791no—
330#107—791no—
331#106—791no—
332#106—791no—
333C4⋊C8—22no—
334C4⋊C8—792no—
335C4⋊C8—1582no—
336C4⋊C8—1581no—
337C4⋊C8C22×D7912yes—
338#103—791no—
339#103—791no—
340#103—791no—
341#103—791no—
342#102—791no—
343#102—791no—
344#102—791no—
345#102—791no—
346C22⋊C8—21no—
347C22⋊C8—22no—
348C22⋊C8—1581no—
349C22⋊C8—1581no—
350C22⋊C8—1582no—
351#97—22no—
352#97—21no—
353#97—1582no—
354#97—1581no—
355#97C22⋊D7911yescharacteristic
356#97C22⋊D7911yescharacteristic
357C4×C8—791no—
358C4×C8—791no—
359C4×C8—791no—
360C4×C8—791no—
361C4×C8—792no—
362C4×C8C22×D7912yes—
363#41—22no—
364#41—41no—
365#41—3161no—
366#41—1582no—
367C2×Q8—791no—
368C2×Q8C2×C4×D7911yescharacteristic
369C2×D4—21no—
370C2×D4—791no—
371C2×D4—1581no—
372C2×D4C2×C4×D7911yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—1581no—
377C22×C4—1581no—
378C22×C4—1581no—
379C22×C4C2×C4×D7911yescharacteristic
380C22×C4C2×C22⋊D7911yescharacteristic
381C22×C4#1192962411yescharacteristic
382Q16—1582no—
383SD16—1582no—
384SD16—1582no—
385D8—794no—
386M16—24no—
387C2×C8—22no—
388C2×C8—44no—
389C2×C8—3164no—
390C2×C8—3161no—
391C2×C8—791no—
392C2×C8—791no—
393C2×C8—791no—
394C2×C8—791no—
395C2×C8—1582no—
396C2×C8—1582no—
397C2×C8—1582no—
398C2×C8C2×C22⋊D7912yes—
399C2×C8C2×C22⋊D7912yes—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—791no—
403C4⋊C4—791no—
404C4⋊C4—1581no—
405C4⋊C4—1581no—
406C4⋊C4C2×C4×D7911yescharacteristic
407C4⋊C4C2×C4×D7911yescharacteristic
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—1581no—
412C22⋊C4—1581no—
413C22⋊C4—1581no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—1581no—
418C42—1581no—
419C42—1581no—
420C42C2×C4×D7911yescharacteristic
421C42C2×C22⋊D7911yescharacteristic
422C42#1192962411yescharacteristic
423C42C23×D7911yescharacteristic
424C23—21no—
425C23—1581no—
426C23C4×C22⋊D7911yescharacteristic
427Q8—21no—
428Q8—21no—
429Q8—1581no—
430Q8—1581no—
431D4—21no—
432D4—42no—
433D4—3162no—
434D4—792no—
435D4—1581no—
436D4M16×D7912yes—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—3162no—
448C2×C4—3162no—
449C2×C4—6321no—
450C2×C4—1581no—
451C2×C4—1581no—
452C2×C4—1581no—
453C2×C4C4×C22⋊D7911yescharacteristic
454C2×C4C4×C22⋊D7911yescharacteristic
455C2×C4C4×C22⋊D7911yescharacteristic
456C2×C4C22×C4×D7911yescharacteristic
457C2×C4C22×C22⋊D7911yescharacteristic
458C2×C4C2×#1192962411yescharacteristic
459C8—22no—
460C8—42no—
461C8—3162no—
462C8—3162no—
463C8—792no—
464C8—1581no—
465C8—1581no—
466C8—1581no—
467C8#1457918114yescomplemented
468C22—21no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—3161no—
475C22—3161no—
476C22—1582no—
477C22C2×C4×C22⋊D7911yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—3161no—
486C4—3161no—
487C4—3161no—
488C4C2×M16×D7911yescharacteristic
489C4#1909932911yescharacteristic
490C4C2×#1457918112yes—
491C2—41no—
492C2—42no—
493C2—3162no—
494C2M16×C22⋊D7911yescharacteristic
495C2C4×#1457918111yescharacteristic
496C2#2377231511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#2970C2
socleC22×C79C2×C4×D4
prime radicalC2×#14579181C4
cosocleC42×C79C24
hypercenter#2970D79
hypocenterC79#18035
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2970

Complements

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

NH ≅ G/Ntype
1. #23772415C2semidirect
3. #23762295C2semidirect
4. #23762236C2semidirect
5. #23762234C2semidirect
6. C8×#11929609C2semidirect
7. #23762161C2semidirect
8. #23761834C2semidirect
9. #23761832C2semidirect
10. M16×C79⋊C8C2semidirect
11. C79×#2970C2semidirect
125. #11929609M16semidirect
212. C8×C79#135semidirect
250. #2970D79semidirect
260. C79#18035semidirect
467. C8#14579181semidirect

Maximal subgroups

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

subgroupindex
#237724152
C4×#145791812
#237622952
#237622362
#237622342
C8×#119296092
#237621612
#237618342
#237618322
M16×C79⋊C82
C79×#29702
#1803579