Finite Group Databasebeta

← #34184578

Subgroups — #34184578

Order 20224.

686 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771849C211yesmaximal, characteristic
2#23763344C211yesmaximal, characteristic
3#23761920C211yesmaximal, characteristic, complemented
4#23761764C211yesmaximal, characteristic, complemented
5#23761277C211yesmaximal, characteristic, complemented
6#23761247C211yesmaximal, characteristic, complemented
7C2×#452×C79C211yesmaximal, characteristic
8C2×#14579168—21no—
9C2×#14579088—21no—
10C2×#14579088—21no—
11#19098530—22no—
12#19098530—21no—
13#19098527—21no—
14#19098527C2211yescharacteristic
15#19098527C2211yescharacteristic
16C2×C22⋊(C79⋊C8)—21no—
17C2×C22⋊(C79⋊C8)C2211yescharacteristic
18C4×C22⋊Dic79C2211yescharacteristic
19#19098300—22no—
20#19098289—21no—
21#19098289—21no—
22C22×#41×C79C2211yescharacteristic
23C22×Q16×C79C411yescharacteristic, complemented
24C22×SD16×C79C411yescharacteristic, complemented
25C2×C22⋊Q8×C79C2211yescharacteristic
26#452×C79—21no—
27#452×C79—21no—
28#452×C79—21no—
29#452×C79—21no—
30C2×#103×C79C411yescharacteristic, complemented
31C2×#102×C79C411yescharacteristic, complemented
32C2×C22⋊C8×C79C2211yescharacteristic
33#14579168—42no—
34#14579168—42no—
35C2×#11929608—22no—
36C2×#11929608—21no—
37C2×#11929608—21no—
38C22×C79⋊C8—22no—
39C22×C79⋊C8—21no—
40C2×C22⋊Dic79—21no—
41C2×C4×Dic79—21no—
42#14579088—42no—
43#14579088—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×C79D411yescharacteristic
59C2×#41×C79D411yescharacteristic
60C2×#41×C79D411yescharacteristic
61C2×#41×C79D411yescharacteristic
62C22×Q8×C79C2×C411yescharacteristic
63C22×Q8×C79C2×C411yescharacteristic
64C22×D4×C79—21no—
65C22×D4×C79C2×C411yescharacteristic
66C23×C4×C79—21no—
67C23×C4×C79C2311yescharacteristic
68C2×Q16×C79—21no—
69C2×Q16×C79—21no—
70C2×Q16×C79—42no—
71C2×SD16×C79—21no—
72C2×SD16×C79—21no—
73C2×SD16×C79—42no—
74C22×C8×C79C2×C411yescharacteristic
75C22×C8×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—
93#11929608—41no—
94#11929608—41no—
95C2×C79⋊C8—21no—
96C2×C79⋊C8—22no—
97C2×C79⋊C8—21no—
98C2×C79⋊C8—44no—
99C2×C79⋊C8—41no—
100C2×C79⋊C8—41no—
101C2×C79⋊C8—41no—
102C22⋊Dic79—22no—
103C22⋊Dic79—41no—
104C4×Dic79—41no—
105C4×Dic79—41no—
106C24×C79—21no—
107C24×C79#4111yescharacteristic
108#41×C79—22no—
109#41×C79—22no—
110#41×C79—22no—
111#41×C79—22no—
112#41×C79—41no—
113#41×C79—41no—
114#41×C79—41no—
115#41×C79—41no—
116#41×C79—44no—
117#41×C79—44no—
118C2×Q8×C79—21no—
119C2×Q8×C79—22no—
120C2×Q8×C79—21no—
121C2×Q8×C79—21no—
122C2×Q8×C79—41no—
123C2×Q8×C79—41no—
124C2×Q8×C79—41no—
125C2×Q8×C79C22⋊C411yescharacteristic
126C2×Q8×C79C22⋊C411yescharacteristic
127C2×D4×C79—21no—
128C2×D4×C79—22no—
129C2×D4×C79—21no—
130C2×D4×C79—21no—
131C2×D4×C79—21no—
132C2×D4×C79—21no—
133C2×D4×C79—41no—
134C2×D4×C79—44no—
135C2×D4×C79C22⋊C411yescharacteristic
136C2×D4×C79C22⋊C411yescharacteristic
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—21no—
144C22×C4×C79—41no—
145C22×C4×C79—41no—
146C22×C4×C79—44no—
147C22×C4×C79C22×C411yescharacteristic
148C22×C4×C79C2×D411yescharacteristic
149C22×C4×C79C2×D411yescharacteristic
150C22×C4×C79C2×D411yescharacteristic
151C22×C4×C79C2×D411yescharacteristic
152C22×C4×C79#4111yescharacteristic
153Q16×C79—42no—
154Q16×C79—42no—
155SD16×C79—42no—
156SD16×C79—42no—
157C2×C8×C79—21no—
158C2×C8×C79—21no—
159C2×C8×C79—41no—
160C2×C8×C79—41no—
161C4⋊C4×C79—21no—
162C4⋊C4×C79—21no—
163C4⋊C4×C79—41no—
164C4⋊C4×C79—41no—
165C22⋊C4×C79—41no—
166C22⋊C4×C79—41no—
167C23×C79—21no—
168C23×C79—21no—
169C23×C79—21no—
170C23×C79—21no—
171C23×C79—41no—
172C23×C79—41no—
173C23×C79—42no—
174C23×C79—41no—
175C23×C79—41no—
176C23×C79—81no—
177C23×C79C4×D411yescharacteristic
178C23×C79#12111yescharacteristic
179C23×C79#12311yescharacteristic
180Q8×C79—21no—
181Q8×C79—21no—
182Q8×C79—21no—
183Q8×C79—42no—
184Q8×C79—41no—
185Q8×C79—41no—
186Q8×C79—81no—
187Q8×C79#9912yes—
188D4×C79—24no—
189D4×C79—21no—
190D4×C79—21no—
191D4×C79—21no—
192D4×C79—42no—
193D4×C79—42no—
194D4×C79—42no—
195D4×C79—42no—
196D4×C79#9912yes—
197C2×C4×C79—24no—
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—21no—
206C2×C4×C79—41no—
207C2×C4×C79—41no—
208C2×C4×C79—41no—
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—41no—
216C2×C4×C79—41no—
217C2×C4×C79—81no—
218C2×C4×C79C2×C22⋊C411yescharacteristic
219C2×C4×C79C4×D411yescharacteristic
220C2×C4×C79#12011yescharacteristic
221C2×C4×C79#12111yescharacteristic
222C2×Dic79—42no—
223C2×Dic79—81no—
224C2×Dic79—81no—
225C8×C79—41no—
226C8×C79—41no—
227C79⋊C8—41no—
228C79⋊C8—42no—
229C79⋊C8—41no—
230C79⋊C8—81no—
231C22×C79—21no—
232C22×C79—21no—
233C22×C79—21no—
234C22×C79—21no—
235C22×C79—21no—
236C22×C79—22no—
237C22×C79—21no—
238C22×C79—21no—
239C22×C79—21no—
240C22×C79—41no—
241C22×C79—41no—
242C22×C79—41no—
243C22×C79—41no—
244C22×C79—41no—
245C22×C79—81no—
246C22×C79—81no—
247C22×C79—82no—
248C22×C79—82no—
249C22×C79#38611yescharacteristic
250C22×C79#44311yescharacteristic
251C22×C79#44411yescharacteristic
252C4×C79—21no—
253C4×C79—22no—
254C4×C79—21no—
255C4×C79—21no—
256C4×C79—41no—
257C4×C79—81no—
258C4×C79—81no—
259C4×C79C2×#9911yescharacteristic
260C4×C79C2×#9911yescharacteristic
261Dic79—82no—
262#9239—791nomaximal, Sylow
263C2×C79—21no—
264C2×C79—21no—
265C2×C79—41no—
266C2×C79—41no—
267C2×C79—42no—
268C2×C79—81no—
269C2×C79#187611yescharacteristic
270C2×C79#189811yescharacteristic
271C2×C79#190211yescharacteristic
272C2×#452D7911yescharacteristic
273#2880—791no—
274#1789—791no—
275#1508—791no—
276#1472—791no—
277#1293—791no—
278#1282—791no—
279C79#923911yescharacteristic, complemented, Sylow
280C22×#41C2×D7911yescharacteristic
281C22×Q16Dic7911yescharacteristic, complemented
282C22×SD16Dic7911yescharacteristic, complemented
283C2×#131—1581no—
284C2×C22⋊Q8C2×D7911yescharacteristic
285#452—21no—
286#452—21no—
287#452—21no—
288#452—21no—
289C2×#104—1581no—
290C2×#104—1581no—
291C2×#103Dic7911yescharacteristic, complemented
292C2×#102Dic7911yescharacteristic, complemented
293#408—1582no—
294#408—1581no—
295#407—791no—
296#407—791no—
297#407—1581no—
298C2×C22⋊C8—791no—
299C2×C22⋊C8—1581no—
300C2×C22⋊C8C2×D7911yescharacteristic
301C4×C22⋊C4—791no—
302#328—1582no—
303#324—1581no—
304#324—1581no—
305C2×#41—21no—
306C2×#41—21no—
307C2×#41—42no—
308C2×#41—42no—
309C2×#41C22⋊D7911yescharacteristic
310C2×#41C22⋊D7911yescharacteristic
311C2×#41C22⋊D7911yescharacteristic
312C2×#41C22⋊D7911yescharacteristic
313C22×Q8C2×Dic7911yescharacteristic
314C22×Q8C2×Dic7911yescharacteristic
315C22×D4—21no—
316C22×D4C2×Dic7911yescharacteristic
317C23×C4—21no—
318C23×C4C22×D7911yescharacteristic
319C2×Q16—21no—
320C2×Q16—21no—
321C2×Q16—42no—
322C2×SD16—21no—
323C2×SD16—21no—
324C2×SD16—42no—
325#131—3162no—
326#131—3162no—
327C2×M16—1581no—
328C2×M16—1582no—
329C2×M16—1581no—
330C22×C8—1582no—
331C22×C8—1581no—
332C22×C8C2×Dic7911yescharacteristic
333C22×C8C2×Dic7911yescharacteristic
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4C2×Dic7911yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—1581no—
342C2×C42—1581no—
343#104—3162no—
344#104—3162no—
345#103—21no—
346#103—21no—
347#102—21no—
348#102—21no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—791no—
352C22⋊C8—791no—
353C22⋊C8—792no—
354C22⋊C8—1581no—
355C22⋊C8—1581no—
356C22⋊C8—1582no—
357C22⋊C8—1582no—
358C22⋊C8—1581no—
359C22⋊C8—1581no—
360#95—1581no—
361C24—21no—
362C24#1192962311yescharacteristic
363#41—22no—
364#41—22no—
365#41—22no—
366#41—22no—
367#41—41no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—44no—
372#41—44no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8C22⋊Dic7911yescharacteristic
381C2×Q8C22⋊Dic7911yescharacteristic
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—44no—
390C2×D4C22⋊Dic7911yescharacteristic
391C2×D4C22⋊Dic7911yescharacteristic
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—44no—
402C22×C4—3161no—
403C22×C4C22×Dic7911yescharacteristic
404C22×C4D4×D7911yescharacteristic
405C22×C4D4×D7911yescharacteristic
406C22×C4C2×C22⋊D7911yescharacteristic
407C22×C4C2×C22⋊D7911yescharacteristic
408C22×C4#1192962311yescharacteristic
409Q16—42no—
410Q16—42no—
411SD16—42no—
412SD16—42no—
413M16—3164no—
414M16—3161no—
415M16—3161no—
416M16—3161no—
417M16—3161no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—3164no—
423C2×C8—3161no—
424C2×C8—3161no—
425C2×C8—3161no—
426C2×C8—1581no—
427C2×C8—1581no—
428C2×C8—1582no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—3161no—
436C22⋊C4—1582no—
437C42—3161no—
438C42—3161no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—81no—
449C23D4×Dic7911yescharacteristic
450C23#1457913311yescharacteristic
451C23#1457914211yescharacteristic
452Q8—21no—
453Q8—21no—
454Q8—21no—
455Q8—42no—
456Q8—41no—
457Q8—41no—
458Q8—81no—
459Q8#1457907612yes—
460D4—24no—
461D4—21no—
462D4—21no—
463D4—21no—
464D4—42no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4#1457907612yes—
469C2×C4—24no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—81no—
490C2×C4—3162no—
491C2×C4—6321no—
492C2×C4—6321no—
493C2×C4C2×C22⋊Dic7911yescharacteristic
494C2×C4D4×Dic7911yescharacteristic
495C2×C4#1457912911yescharacteristic
496C2×C4#1457913311yescharacteristic
497C8—41no—
498C8—41no—
499C8—3161no—
500C8—3162no—
501C8—3161no—
502C8—6321no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—22no—
510C22—21no—
511C22—21no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—81no—
518C22—81no—
519C22—82no—
520C22—82no—
521C22#1909844311yescharacteristic
522C22#1909868011yescharacteristic
523C22#1909868511yescharacteristic
524C4—21no—
525C4—21no—
526C4—22no—
527C4—21no—
528C4—41no—
529C4—81no—
530C4—81no—
531C4—6322no—
532C4C2×#1457907611yescharacteristic
533C4C2×#1457907611yescharacteristic
534C2—21no—
535C2—21no—
536C2—41no—
537C2—41no—
538C2—42no—
539C2—81no—
540C2#2376372911yescharacteristic
541C2#2376385411yescharacteristic
542C2#2376387511yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#452×C79C2
socleC22×C79#386
prime radicalC22×#41×C79C22
cosocleC23×C4×C79C23
hypercenterC2×#452D79
hypocenterC79#9239
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#452

Complements

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

NH ≅ G/Ntype
3. #23761920C2semidirect
4. #23761764C2semidirect
5. #23761277C2semidirect
6. #23761247C2semidirect
23. C22×Q16×C79C4semidirect
24. C22×SD16×C79C4semidirect
30. C2×#103×C79C4semidirect
31. C2×#102×C79C4semidirect
279. C79#9239semidirect
281. C22×Q16Dic79semidirect
282. C22×SD16Dic79semidirect
291. C2×#103Dic79semidirect
292. C2×#102Dic79semidirect

Maximal subgroups

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

subgroupindex
#237718492
#237633442
#237619202
#237617642
#237612772
#237612472
C2×#452×C792
#923979