Finite Group Databasebeta

← #34185320

Subgroups — #34185320

Order 20224.

678 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771849C211yesmaximal, characteristic
2#23763340C211yesmaximal, characteristic
3#23761972C211yesmaximal, characteristic, complemented
4#23761764C211yesmaximal, characteristic, complemented
5#23761305C211yesmaximal, characteristic, complemented
6#23761287C211yesmaximal, characteristic, complemented
7C79×#3007C211yesmaximal, characteristic
8C2×#14579168—21no—
9C2×#14579085—21no—
10C2×#14579082—21no—
11#19098530—22no—
12#19098530—21no—
13#19098527—21no—
14#19098527C2211yescharacteristic
15C2×C22⋊(C79⋊C8)—21no—
16C2×C22⋊(C79⋊C8)C2211yescharacteristic
17C2×C22⋊(C79⋊C8)C2211yescharacteristic
18#19098423C2211yescharacteristic
19#19098300—21no—
20#19098300—21no—
21#19098289—22no—
22C22×#41×C79C2211yescharacteristic
23C2×#137×C79C411yescharacteristic, complemented
24C2×#137×C79C411yescharacteristic, complemented
25C2×C22⋊Q8×C79C2211yescharacteristic
26#452×C79—21no—
27#452×C79—21no—
28#451×C79—21no—
29#450×C79—21no—
30#417×C79C411yescharacteristic, complemented
31#417×C79C411yescharacteristic, complemented
32#407×C79C2211yescharacteristic
33#14579168—42no—
34#14579168—42no—
35C2×#11929608—22no—
36C2×#11929608—21no—
37C22×C79⋊C8—22no—
38C22×C79⋊C8—21no—
39C22×C79⋊C8—21no—
40C2×C4⋊Dic79—21no—
41C2×C22⋊Dic79—21no—
42#14579085—42no—
43#14579082—42no—
44C22⋊(C79⋊C8)—22no—
45C22⋊(C79⋊C8)—22no—
46C22⋊(C79⋊C8)—21no—
47C22⋊(C79⋊C8)—21no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)—21no—
50C22⋊(C79⋊C8)D411yescharacteristic
51C22⋊(C79⋊C8)D411yescharacteristic
52C22⋊(C79⋊C8)D412yes—
53C79⋊#95—21no—
54C2×#41×C79—21no—
55C2×#41×C79—21no—
56C2×#41×C79—42no—
57C2×#41×C79—42no—
58C2×#41×C79C2×C411yescharacteristic
59C2×#41×C79C2×C411yescharacteristic
60C2×#41×C79D411yescharacteristic
61C2×#41×C79D411yescharacteristic
62C22×Q8×C79C2×C411yescharacteristic
63C22×Q8×C79D411yescharacteristic
64C22×D4×C79—21no—
65C22×D4×C79D411yescharacteristic
66C23×C4×C79—21no—
67C23×C4×C79C2311yescharacteristic
68#137×C79—42no—
69#137×C79—42no—
70C2×Q16×C79—21no—
71C2×Q16×C79—21no—
72C2×SD16×C79—21no—
73C2×SD16×C79—21no—
74C2×M16×C79C2×C411yescharacteristic
75C2×M16×C79C2×C411yescharacteristic
76C22⋊Q8×C79—21no—
77C22⋊Q8×C79—21no—
78C22⋊Q8×C79—21no—
79C22⋊Q8×C79—21no—
80C2×C4⋊C4×C79—21no—
81C2×C4⋊C4×C79C2×C411yescharacteristic
82C2×C22⋊C4×C79—21no—
83#103×C79—21no—
84#103×C79—21no—
85#102×C79—21no—
86#102×C79—21no—
87C22⋊C8×C79—21no—
88C22⋊C8×C79—21no—
89C22×Dic79—41no—
90#11929608—44no—
91#11929608—41no—
92#11929608—41no—
93C2×C79⋊C8—21no—
94C2×C79⋊C8—22no—
95C2×C79⋊C8—21no—
96C2×C79⋊C8—44no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8—41no—
99C2×C79⋊C8—41no—
100C2×C79⋊C8—41no—
101C2×C79⋊C8—41no—
102C4⋊Dic79—41no—
103C4⋊Dic79—41no—
104C22⋊Dic79—81no—
105C24×C79—21no—
106C24×C79#4111yescharacteristic
107#41×C79—22no—
108#41×C79—22no—
109#41×C79—22no—
110#41×C79—22no—
111#41×C79—41no—
112#41×C79—41no—
113#41×C79—41no—
114#41×C79—41no—
115#41×C79—44no—
116#41×C79—44no—
117C2×Q8×C79—21no—
118C2×Q8×C79—22no—
119C2×Q8×C79—21no—
120C2×Q8×C79—21no—
121C2×Q8×C79—41no—
122C2×Q8×C79—41no—
123C2×Q8×C79—41no—
124C2×Q8×C79C22⋊C411yescharacteristic
125C2×Q8×C79C22⋊C411yescharacteristic
126C2×D4×C79—21no—
127C2×D4×C79—22no—
128C2×D4×C79—21no—
129C2×D4×C79—21no—
130C2×D4×C79—21no—
131C2×D4×C79—21no—
132C2×D4×C79—41no—
133C2×D4×C79—44no—
134C2×D4×C79C22⋊C411yescharacteristic
135C2×D4×C79C22⋊C411yescharacteristic
136C22×C4×C79—21no—
137C22×C4×C79—21no—
138C22×C4×C79—21no—
139C22×C4×C79—21no—
140C22×C4×C79—21no—
141C22×C4×C79—21no—
142C22×C4×C79—21no—
143C22×C4×C79—41no—
144C22×C4×C79—41no—
145C22×C4×C79—44no—
146C22×C4×C79C22×C411yescharacteristic
147C22×C4×C79C2×D411yescharacteristic
148C22×C4×C79C2×D411yescharacteristic
149C22×C4×C79C2×D411yescharacteristic
150C22×C4×C79C2×D411yescharacteristic
151C22×C4×C79#4111yescharacteristic
152Q16×C79—42no—
153Q16×C79—42no—
154SD16×C79—42no—
155SD16×C79—42no—
156M16×C79—41no—
157M16×C79—41no—
158C2×C8×C79—21no—
159C2×C8×C79—21no—
160C4⋊C4×C79—21no—
161C4⋊C4×C79—21no—
162C4⋊C4×C79—41no—
163C4⋊C4×C79—41no—
164C22⋊C4×C79—41no—
165C22⋊C4×C79—41no—
166C23×C79—21no—
167C23×C79—21no—
168C23×C79—21no—
169C23×C79—21no—
170C23×C79—41no—
171C23×C79—41no—
172C23×C79—42no—
173C23×C79—41no—
174C23×C79—41no—
175C23×C79—81no—
176C23×C79C4×D411yescharacteristic
177C23×C79#12111yescharacteristic
178C23×C79#12311yescharacteristic
179Q8×C79—21no—
180Q8×C79—21no—
181Q8×C79—21no—
182Q8×C79—42no—
183Q8×C79—41no—
184Q8×C79—41no—
185Q8×C79—81no—
186Q8×C79#9912yes—
187D4×C79—24no—
188D4×C79—21no—
189D4×C79—21no—
190D4×C79—21no—
191D4×C79—42no—
192D4×C79—42no—
193D4×C79—42no—
194D4×C79—42no—
195D4×C79#9912yes—
196C2×C4×C79—24no—
197C2×C4×C79—21no—
198C2×C4×C79—21no—
199C2×C4×C79—21no—
200C2×C4×C79—21no—
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—41no—
206C2×C4×C79—41no—
207C2×C4×C79—41no—
208C2×C4×C79—42no—
209C2×C4×C79—42no—
210C2×C4×C79—42no—
211C2×C4×C79—42no—
212C2×C4×C79—42no—
213C2×C4×C79—41no—
214C2×C4×C79—41no—
215C2×C4×C79—81no—
216C2×C4×C79C2×C22⋊C411yescharacteristic
217C2×C4×C79C4×D411yescharacteristic
218C2×C4×C79#12011yescharacteristic
219C2×C4×C79#12111yescharacteristic
220C2×Dic79—81no—
221C2×Dic79—81no—
222C2×Dic79—81no—
223C8×C79—41no—
224C8×C79—41no—
225C79⋊C8—41no—
226C79⋊C8—42no—
227C79⋊C8—41no—
228C79⋊C8—81no—
229C22×C79—21no—
230C22×C79—21no—
231C22×C79—21no—
232C22×C79—21no—
233C22×C79—21no—
234C22×C79—21no—
235C22×C79—22no—
236C22×C79—21no—
237C22×C79—21no—
238C22×C79—41no—
239C22×C79—41no—
240C22×C79—41no—
241C22×C79—41no—
242C22×C79—41no—
243C22×C79—81no—
244C22×C79—81no—
245C22×C79—82no—
246C22×C79—82no—
247C22×C79#38611yescharacteristic
248C22×C79#47111yescharacteristic
249C22×C79#47311yescharacteristic
250C4×C79—21no—
251C4×C79—22no—
252C4×C79—21no—
253C4×C79—21no—
254C4×C79—41no—
255C4×C79—81no—
256C4×C79—81no—
257C4×C79C2×#9911yescharacteristic
258C4×C79C2×#9911yescharacteristic
259Dic79—161no—
260#9345—791nomaximal, Sylow
261C2×C79—21no—
262C2×C79—21no—
263C2×C79—41no—
264C2×C79—41no—
265C2×C79—42no—
266C2×C79—81no—
267C2×C79#188111yescharacteristic
268C2×C79#189811yescharacteristic
269C2×C79#190811yescharacteristic
270#3007D7911yescharacteristic
271#2880—791no—
272#1788—791no—
273#1520—791no—
274#1472—791no—
275#1305—791no—
276#1298—791no—
277C79#934511yescharacteristic, complemented, Sylow
278C22×#41C2×D7911yescharacteristic
279C2×#137Dic7911yescharacteristic, complemented
280C2×#137Dic7911yescharacteristic, complemented
281C2×#131—1581no—
282C2×C22⋊Q8C2×D7911yescharacteristic
283#452—21no—
284#452—21no—
285#451—21no—
286#450—21no—
287#417Dic7911yescharacteristic, complemented
288#417Dic7911yescharacteristic, complemented
289C2×#103—1581no—
290C2×#102—1581no—
291#408—1582no—
292#408—1581no—
293#407—791no—
294#407—1581no—
295#407C2×D7911yescharacteristic
296C2×C22⋊C8—791no—
297C2×C22⋊C8—791no—
298C2×C22⋊C8—1581no—
299#380—791no—
300#328—1581no—
301#328—1581no—
302#324—1582no—
303C2×#41—21no—
304C2×#41—21no—
305C2×#41—42no—
306C2×#41—42no—
307C2×#41C2×Dic7911yescharacteristic
308C2×#41C2×Dic7911yescharacteristic
309C2×#41C22⋊D7911yescharacteristic
310C2×#41C22⋊D7911yescharacteristic
311C22×Q8C2×Dic7911yescharacteristic
312C22×Q8C22⋊D7911yescharacteristic
313C22×D4—21no—
314C22×D4C22⋊D7911yescharacteristic
315C23×C4—21no—
316C23×C4C22×D7911yescharacteristic
317#137—42no—
318#137—42no—
319C2×Q16—21no—
320C2×Q16—21no—
321C2×SD16—21no—
322C2×SD16—21no—
323#131—3162no—
324#131—3162no—
325C2×M16—1582no—
326C2×M16—1581no—
327C2×M16C2×Dic7911yescharacteristic
328C2×M16C2×Dic7911yescharacteristic
329C22×C8—1581no—
330C22×C8—1582no—
331C22×C8—1581no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—1581no—
338C2×C4⋊C4C2×Dic7911yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—1581no—
341#103—21no—
342#103—21no—
343#103—3162no—
344#102—21no—
345#102—21no—
346#102—3162no—
347C22⋊C8—21no—
348C22⋊C8—21no—
349C22⋊C8—792no—
350C22⋊C8—791no—
351C22⋊C8—791no—
352C22⋊C8—1581no—
353C22⋊C8—1581no—
354C22⋊C8—1582no—
355C22⋊C8—1582no—
356C22⋊C8—1581no—
357C22⋊C8—1581no—
358#95—1581no—
359C24—21no—
360C24#1192962311yescharacteristic
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—44no—
370#41—44no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8C22⋊Dic7911yescharacteristic
379C2×Q8C22⋊Dic7911yescharacteristic
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—44no—
388C2×D4C22⋊Dic7911yescharacteristic
389C2×D4C22⋊Dic7911yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—44no—
400C22×C4—3161no—
401C22×C4C22×Dic7911yescharacteristic
402C22×C4D4×D7911yescharacteristic
403C22×C4D4×D7911yescharacteristic
404C22×C4C2×C22⋊D7911yescharacteristic
405C22×C4C2×C22⋊D7911yescharacteristic
406C22×C4#1192962311yescharacteristic
407Q16—42no—
408Q16—42no—
409SD16—42no—
410SD16—42no—
411M16—41no—
412M16—41no—
413M16—3164no—
414M16—3161no—
415M16—3161no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—3164no—
419C2×C8—3161no—
420C2×C8—3161no—
421C2×C8—3161no—
422C2×C8—3161no—
423C2×C8—3161no—
424C2×C8—1581no—
425C2×C8—1582no—
426C2×C8—1581no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—3161no—
432C4⋊C4—3161no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—6321no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—81no—
446C23D4×Dic7911yescharacteristic
447C23#1457913311yescharacteristic
448C23#1457914211yescharacteristic
449Q8—21no—
450Q8—21no—
451Q8—21no—
452Q8—42no—
453Q8—41no—
454Q8—41no—
455Q8—81no—
456Q8#1457907612yes—
457D4—24no—
458D4—21no—
459D4—21no—
460D4—21no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4#1457907612yes—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—81no—
486C2×C4—6321no—
487C2×C4—6321no—
488C2×C4—6321no—
489C2×C4C2×C22⋊Dic7911yescharacteristic
490C2×C4D4×Dic7911yescharacteristic
491C2×C4#1457912911yescharacteristic
492C2×C4#1457913311yescharacteristic
493C8—41no—
494C8—41no—
495C8—3161no—
496C8—3162no—
497C8—3161no—
498C8—6321no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22—82no—
517C22#1909844311yescharacteristic
518C22#1909882811yescharacteristic
519C22#1909883611yescharacteristic
520C4—21no—
521C4—21no—
522C4—22no—
523C4—21no—
524C4—41no—
525C4—81no—
526C4—81no—
527C4—12641no—
528C4C2×#1457907611yescharacteristic
529C4C2×#1457907611yescharacteristic
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—42no—
535C2—81no—
536C2#2376376411yescharacteristic
537C2#2376385411yescharacteristic
538C2#2376391711yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#3007C2
socleC22×C79#386
prime radicalC22×#41×C79C22
cosocleC23×C4×C79C23
hypercenter#3007D79
hypocenterC79#9345
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3007

Complements

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

NH ≅ G/Ntype
3. #23761972C2semidirect
4. #23761764C2semidirect
5. #23761305C2semidirect
6. #23761287C2semidirect
23. C2×#137×C79C4semidirect
24. C2×#137×C79C4semidirect
30. #417×C79C4semidirect
31. #417×C79C4semidirect
277. C79#9345semidirect
279. C2×#137Dic79semidirect
280. C2×#137Dic79semidirect
287. #417Dic79semidirect
288. #417Dic79semidirect

Maximal subgroups

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

subgroupindex
#237718492
#237633402
#237619722
#237617642
#237613052
#237612872
C79×#30072
#934579