Finite Group Databasebeta

← #34391380

Subgroups — #34391380

Order 20224.

724 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23765877C211yesmaximal, characteristic, complemented
2#23764901C211yesmaximal, characteristic, complemented
3#23764778C211yesmaximal, characteristic, complemented
4#23764774C211yesmaximal, characteristic
5#23764717C211yesmaximal, characteristic
6#23764253C211yesmaximal, characteristic
7#23764125C211yesmaximal, characteristic, complemented
8#23764086C211yesmaximal, characteristic
9#23764075C211yesmaximal, characteristic, complemented
10#23764024C211yesmaximal, characteristic
11#23763975C211yesmaximal, characteristic, complemented
12#23763758C211yesmaximal, characteristic
13#23763667C211yesmaximal, characteristic, complemented
14C4×C4⋊(C79⋊C8)C211yesmaximal, characteristic, complemented
15C79×#2860C211yesmaximal, characteristic
16C22×#11929612C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18C2×#14579092C2211yescharacteristic
19C2×#14579092C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
21C2×C4⋊(C79⋊C8)C2211yescharacteristic
22C2×C4⋊(C79⋊C8)C2211yescharacteristic
23C2×#14579084C2211yescharacteristic
24C2×#14579084C2211yescharacteristic
25C2×#14579084C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×#14579082C2211yescharacteristic
29C2×C4×C79⋊C8C2211yescharacteristic
30C2×C4×C79⋊C8C2211yescharacteristic
31#19098469C2211yescharacteristic
32#19098469C2211yescharacteristic
33#19098461C2211yescharacteristic
34#19098461C2211yescharacteristic
35#19098447—21no—
36#19098447—21no—
37C4×Dic79⋊C4—21no—
38C4×C4⋊Dic79C2211yescharacteristic
39#19098409—21no—
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×#126×C79—21no—
49#390×C79C2211yescharacteristic
50#390×C79C2211yescharacteristic
51#390×C79C2211yescharacteristic
52#389×C79C2211yescharacteristic
53#389×C79C2211yescharacteristic
54#389×C79C2211yescharacteristic
55#388×C79—21no—
56#388×C79—21no—
57#388×C79—21no—
58C43×C79C2211yescharacteristic
59C22×Dic158C2311yescharacteristic
60C2×#11929612—22no—
61C2×#11929612—22no—
62C2×#11929612—41no—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C4×Dic158—22no—
68C4×Dic158—22no—
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×Dic79⋊C4—21no—
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×C4×Dic79—21no—
76C2×C4×Dic79—21no—
77#14579092—22no—
78#14579092—22no—
79#14579092—41no—
80C4⋊(C79⋊C8)—22no—
81C4⋊(C79⋊C8)—22no—
82#14579084—22no—
83#14579084—22no—
84#14579084—41no—
85#14579082—22no—
86#14579082—22no—
87#14579082—41no—
88C4×C79⋊C8—41no—
89C4×C79⋊C8—41no—
90C79⋊#95—21no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C22×D4×C79C2311yescharacteristic
98#126×C79—44no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79—21no—
102C2×C4⋊C4×C79C2311yescharacteristic
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C22⋊C4×C79—21no—
106C2×C22⋊C4×C79—21no—
107C2×C22⋊C4×C79—21no—
108C2×C22⋊C4×C79—21no—
109C2×C22⋊C4×C79—21no—
110C2×C22⋊C4×C79—21no—
111C2×C42×C79D411yescharacteristic
112C2×C42×C79D411yescharacteristic
113C2×C42×C79D411yescharacteristic
114C2×C42×C79D411yescharacteristic
115C2×C42×C79C2311yescharacteristic
116C2×C42×C79C2311yescharacteristic
117C2×C42×C79C2311yescharacteristic
118#95×C79—21no—
119#95×C79—21no—
120#95×C79—21no—
121C2×Dic158—21no—
122C2×Dic158—21no—
123C2×Dic158—21no—
124C2×Dic158—21no—
125C2×Dic158—21no—
126C2×Dic158#4112yes—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—21no—
131#11929612—44no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8#4112yes—
143C2×C79⋊C8#4112yes—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79#4112yes—
150Dic79⋊C4—24no—
151Dic79⋊C4—42no—
152Dic79⋊C4—42no—
153C4×Dic79—24no—
154C4×Dic79—42no—
155C24×C79—21no—
156C2×D4×C79—21no—
157C2×D4×C79—21no—
158C2×D4×C79—21no—
159C2×D4×C79—21no—
160C2×D4×C79—21no—
161C2×D4×C79—21no—
162C22×C4×C79—21no—
163C22×C4×C79—21no—
164C22×C4×C79—21no—
165C22×C4×C79C2×D411yescharacteristic
166C22×C4×C79C2×D411yescharacteristic
167C22×C4×C79C2×D411yescharacteristic
168C22×C4×C79C2×D411yescharacteristic
169C22×C4×C79C2×D411yescharacteristic
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2411yescharacteristic
172C4⋊C4×C79—21no—
173C4⋊C4×C79—21no—
174C4⋊C4×C79—21no—
175C4⋊C4×C79—21no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—21no—
178C4⋊C4×C79—42no—
179C4⋊C4×C79—42no—
180C4⋊C4×C79—42no—
181C22⋊C4×C79—42no—
182C22⋊C4×C79—42no—
183C22⋊C4×C79—42no—
184C22⋊C4×C79—42no—
185C22⋊C4×C79—42no—
186C22⋊C4×C79—42no—
187C42×C79—22no—
188C42×C79—21no—
189C42×C79—21no—
190C42×C79—21no—
191C42×C79—21no—
192C42×C79—21no—
193C42×C79—21no—
194C42×C79—41no—
195C42×C79—41no—
196C42×C79SD1614yescomplemented
197C23×C79—41no—
198C23×C79—41no—
199C23×C79—41no—
200C23×C79—41no—
201C23×C79—41no—
202C23×C79—41no—
203C23×C79—41no—
204C23×C79C22×D411yescharacteristic
205D4×C79—41no—
206D4×C79—41no—
207D4×C79—41no—
208D4×C79—41no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79C2×SD1612yes—
233C2×C4×C79C2×SD1612yes—
234C2×C4×C79C2×SD1612yes—
235C2×C4×C79#13512yes—
236C2×C4×C79C2×#4111yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×C4×C79C2×#4111yescharacteristic
239C2×C4×C79C2×#4111yescharacteristic
240C2×C4×C79H(5,𝔽2)11yescharacteristic
241C2×C4×C792−1+411yescharacteristic
242C2×Dic79—22no—
243C2×Dic79—22no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254Dic158—22no—
255Dic158—22no—
256Dic158—41no—
257Dic158—41no—
258C79⋊C8—22no—
259C79⋊C8—22no—
260C79⋊C8—41no—
261C79⋊C8—41no—
262C22×C79—81no—
263C22×C79—81no—
264C22×C79—81no—
265C22×C79—81no—
266C22×C79—81no—
267C22×C79—81no—
268C22×C79—81no—
269C22×C79#52511yescharacteristic
270C22×C79#53211yescharacteristic
271C22×C79#53711yescharacteristic
272C22×C79C22×SD1611yescharacteristic
273C22×C79C2×#13511yescharacteristic
274C22×C79#57511yescharacteristic
275C22×C79#57511yescharacteristic
276C4×C79—22no—
277C4×C79—22no—
278C4×C79—22no—
279C4×C79—22no—
280C4×C79—41no—
281C4×C79—41no—
282C4×C79—81no—
283C4×C79—81no—
284C4×C79—81no—
285C4×C79#55111yescharacteristic
286C4×C79#55211yescharacteristic
287C4×C79#55311yescharacteristic
288C4×C79#55611yescharacteristic
289Dic79—42no—
290Dic79—42no—
291Dic79—81no—
292Dic79—81no—
293#29434—791nomaximal, Sylow
294C2×C79—161no—
295C2×C79#260111yescharacteristic
296C2×C79#310211yescharacteristic
297C2×C79#310311yescharacteristic
298C2×C79#316711yescharacteristic
299C2×C79#318311yescharacteristic
300C2×C79#324611yescharacteristic
301C2×C79#325011yescharacteristic
302#2860D7911yescharacteristic
303#2295—791no—
304#2088—791no—
305#2066—791no—
306#2065—791no—
307#2055—791no—
308#1968—791no—
309#1941—791no—
310#1934—791no—
311#1933—791no—
312#1926—791no—
313#1919—791no—
314#1880—791no—
315#1865—791no—
316C4×C4⋊C8—791no—
317C79#2943411yescharacteristic, complemented, Sylow
318C22×SD16—791no—
319C2×#126—21no—
320C2×C4×Q8—791no—
321C2×#106—791no—
322C2×#106—791no—
323C2×#106—791no—
324C2×C4⋊C8—791no—
325C2×C4⋊C8—791no—
326C2×#103—791no—
327C2×#103—791no—
328C2×#103—791no—
329C2×#102—791no—
330C2×#102—791no—
331C2×#102—791no—
332C2×C4×C8—791no—
333C2×C4×C8—791no—
334#391—791no—
335#391—791no—
336#390C2×D7911yescharacteristic
337#390C2×D7911yescharacteristic
338#390C2×D7911yescharacteristic
339#389—791no—
340#389—791no—
341#389C2×D7911yescharacteristic
342#389C2×D7911yescharacteristic
343#389C2×D7911yescharacteristic
344#388—21no—
345#388—21no—
346#388—21no—
347#387—1581no—
348#387—1581no—
349C4×C4⋊C4—791no—
350C4×C4⋊C4—1581no—
351#376—1581no—
352C43C2×D7911yescharacteristic
353#340—791no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358#340—791no—
359#336—791no—
360#336—791no—
361C22×Q8—791no—
362C22×D4C22×D7911yescharacteristic
363C2×SD16—3161no—
364C2×SD16—1582no—
365C2×SD16—1582no—
366C22×C8—791no—
367C22×C8—791no—
368C22×C8—791no—
369C22×C8—791no—
370#126—44no—
371C4×Q8—1582no—
372C4×Q8—1582no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—791no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—791no—
379C2×C4⋊C4—1581no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4C22×D7911yescharacteristic
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C4⋊C4C22×D7911yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—1581no—
392C2×C42—1581no—
393C2×C42C22⋊D7911yescharacteristic
394C2×C42C22⋊D7911yescharacteristic
395C2×C42C22⋊D7911yescharacteristic
396C2×C42C22⋊D7911yescharacteristic
397C2×C42C22×D7911yescharacteristic
398C2×C42C22×D7911yescharacteristic
399C2×C42C22×D7911yescharacteristic
400#106—3161no—
401#106—1582no—
402#106—1582no—
403C4⋊C8—1582no—
404C4⋊C8—1582no—
405#103—3161no—
406#103—1582no—
407#103—1582no—
408#102—3161no—
409#102—1582no—
410#102—1582no—
411C4×C8—3161no—
412C4×C8—3161no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—1581no—
417#95—1581no—
418#95—1581no—
419#95—1581no—
420#95—1581no—
421#95—1581no—
422#95—1581no—
423C24—21no—
424C2×Q8—792no—
425C2×Q8—1581no—
426C2×Q8—1581no—
427C2×Q8—1581no—
428C2×Q8—1581no—
429C2×Q8—1581no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—1581no—
440C22×C4—1581no—
441C22×C4—1581no—
442C22×C4—1581no—
443C22×C4C2×C22⋊D7911yescharacteristic
444C22×C4C2×C22⋊D7911yescharacteristic
445C22×C4C2×C22⋊D7911yescharacteristic
446C22×C4C2×C22⋊D7911yescharacteristic
447C22×C4C2×C22⋊D7911yescharacteristic
448C22×C4C2×C22⋊D7911yescharacteristic
449C22×C4C23×D7911yescharacteristic
450SD16—3164no—
451C2×C8—792no—
452C2×C8—792no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8—1581no—
456C2×C8—1581no—
457C2×C8—1581no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1581no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—3162no—
473C4⋊C4—3162no—
474C4⋊C4—792no—
475C4⋊C4—1584no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—3162no—
497C42—1584no—
498C42#1192961214yescomplemented
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C22×C22⋊D7911yescharacteristic
507Q8—3161no—
508Q8—3161no—
509Q8—1582no—
510Q8—1582no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—3161no—
548C2×C4—1582no—
549C2×C4—1582no—
550C2×C4C2×#1192961212yes—
551C2×C4C2×#1192961212yes—
552C2×C4C2×#1192961212yes—
553C2×C4#1457918112yes—
554C2×C4#41×D7911yescharacteristic
555C2×C4#41×D7911yescharacteristic
556C2×C4#41×D7911yescharacteristic
557C2×C4#41×D7911yescharacteristic
558C2×C4#1457920711yescharacteristic
559C2×C4#1457920811yescharacteristic
560C8—3161no—
561C8—3161no—
562C8—1582no—
563C8—1582no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22#1909907011yescharacteristic
572C22#1909910911yescharacteristic
573C22#1909914111yescharacteristic
574C22C22×#1192961211yescharacteristic
575C22C2×#1457918111yescharacteristic
576C22#1909936111yescharacteristic
577C22#1909936111yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—3162no—
588C4—3162no—
589C4—6321no—
590C4—6321no—
591C4#1909923811yescharacteristic
592C4#1909924911yescharacteristic
593C4#1909926411yescharacteristic
594C4#1909928011yescharacteristic
595C2—161no—
596C2#2376899911yescharacteristic
597C2#2377341211yescharacteristic
598C2#2377342711yescharacteristic
599C2#2377395511yescharacteristic
600C2#2377412311yescharacteristic
601C2#2377471611yescharacteristic
602C2#2377474711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2860C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
hypercenter#2860D79
hypocenterC79#29434
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C43 ◁ #2860

Complements

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

NH ≅ G/Ntype
1. #23765877C2semidirect
2. #23764901C2semidirect
3. #23764778C2semidirect
7. #23764125C2semidirect
9. #23764075C2semidirect
11. #23763975C2semidirect
13. #23763667C2semidirect
14. C4×C4⋊(C79⋊C8)C2semidirect
196. C42×C79SD16semidirect
317. C79#29434semidirect
498. C42#11929612semidirect

Maximal subgroups

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

subgroupindex
#237658772
#237649012
#237647782
#237647742
#237647172
#237642532
#237641252
#237640862
#237640752
#237640242
#237639752
#237637582
#237636672
C4×C4⋊(C79⋊C8)2
C79×#28602
#2943479