Finite Group Databasebeta

← #34259425

Subgroups — #34259425

Order 20224.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773447C211yesmaximal, characteristic
2#23772562C211yesmaximal, characteristic, complemented
3#23762288C211yesmaximal, characteristic, complemented
4#23762283C211yesmaximal, characteristic, complemented
5#23762276C211yesmaximal, characteristic, complemented
6#23762264C211yesmaximal, characteristic, complemented
7#23762262C211yesmaximal, characteristic, complemented
8#23762257C211yesmaximal, characteristic, complemented
9C8×#11929612C211yesmaximal, characteristic, complemented
10C8×#11929609C211yesmaximal, characteristic, complemented
11#23762186C211yesmaximal, characteristic, complemented
12#23761858C211yesmaximal, characteristic, complemented
13C2×D4⋊(C79⋊C8)C211yesmaximal, characteristic, complemented
14#23761684C211yesmaximal, characteristic, complemented
15C79×#2974C211yesmaximal, characteristic, complemented
16#19099097C2211yescharacteristic
17#19098898C411yescharacteristic
18#19098881C411yescharacteristic
19#19098863C411yescharacteristic
20#19098863C411yescharacteristic
21#19098778C411yescharacteristic
22#19098778C411yescharacteristic
23#19098714C411yescharacteristic
24#19098697C411yescharacteristic
25#19098693C2211yescharacteristic, complemented
26C8×Dic158C2211yescharacteristic, complemented
27C4×#11929612C2211yescharacteristic
28C4×#11929612C2211yescharacteristic
29C4×#11929609C2211yescharacteristic
30C4×#11929609C2211yescharacteristic
31#19098642C2211yescharacteristic, complemented
32#19098631—21no—
33C8×D316C2211yescharacteristic, complemented
34(C22⋊D79)⋊C8—21no—
35#19098623—21no—
36#19098616C2211yescharacteristic
37#19098588—21no—
38C2×C4⋊(C79⋊C8)C2211yescharacteristic
39C2×C4×C79⋊C8C2211yescharacteristic
40#19098315C2211yescharacteristic, complemented
41#19098315C2211yescharacteristic, complemented
42#19098313C2211yescharacteristic, complemented
43#19098313C2211yescharacteristic, complemented
44#19098293C2211yescharacteristic, complemented
45#19098293C2211yescharacteristic, complemented
46D4⋊(C79⋊C8)C2211yescharacteristic
47D4⋊(C79⋊C8)C2211yescharacteristic
48D4⋊(C79⋊C8)C2211yescharacteristic
49D4⋊(C79⋊C8)C2211yescharacteristic
50#19098290C2211yescharacteristic, complemented
51#19098290C2211yescharacteristic, complemented
52#19098285C2211yescharacteristic, complemented
53#19098285C2211yescharacteristic, complemented
54C8×C79⋊C8C2211yescharacteristic, complemented
55C8×C79⋊C8C2211yescharacteristic, complemented
56C2×C4×D4×C79C2211yescharacteristic
57#436×C79C2211yescharacteristic
58#436×C79C2211yescharacteristic
59#435×C79—21no—
60#435×C79—21no—
61C8×D4×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#432×C79C2211yescharacteristic
64#431×C79C2211yescharacteristic
65#407×C79—21no—
66C2×C22⋊C8×C79—21no—
67C2×#11929612C2×C411yescharacteristic
68C2×#11929612C2×C411yescharacteristic
69C2×#11929609C2×C411yescharacteristic
70C2×#11929609C2×C411yescharacteristic
71C2×#11929607—21no—
72C22×C79⋊C8—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×Dic158C2311yescharacteristic
82C4×D316C2311yescharacteristic
83C4×C22⋊D79—21no—
84#14579118—21no—
85#14579094C2×C411yescharacteristic
86#14579094C2×C411yescharacteristic
87#14579092C2×C411yescharacteristic
88#14579092C2×C411yescharacteristic
89C4⋊(C79⋊C8)—21no—
90C4⋊(C79⋊C8)C2×C411yescharacteristic
91C4⋊(C79⋊C8)C2×C411yescharacteristic
92C4⋊(C79⋊C8)C2311yescharacteristic
93C4⋊(C79⋊C8)C2311yescharacteristic
94Dic79⋊C8—21no—
95Dic79⋊C8—21no—
96#14579084C2×C411yescharacteristic
97#14579084C2×C411yescharacteristic
98#14579082C2×C411yescharacteristic
99#14579082C2×C411yescharacteristic
100#14579082C2×C411yescharacteristic
101#14579082C2×C411yescharacteristic
102#14579081C2×C411yescharacteristic
103#14579081C2×C411yescharacteristic
104C22⋊(C79⋊C8)C2×C411yescharacteristic
105C22⋊(C79⋊C8)C2×C411yescharacteristic
106(C2×D79)⋊C8—21no—
107(C2×D79)⋊C8—21no—
108#14579071—21no—
109C4×C79⋊C8—21no—
110C4×C79⋊C8C2311yescharacteristic
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C4×C79⋊C8C2311yescharacteristic
114C8×Dic79—21no—
115C22×D4×C79C2×C411yescharacteristic
116C23×C4×C79—21no—
117C2×M16×C79—21no—
118C2×M16×C79—21no—
119C22×C8×C79—21no—
120C22×C8×C79—21no—
121C4×D4×C79—21no—
122C4×D4×C79—21no—
123C4×D4×C79C2×C411yescharacteristic
124C4×D4×C79C2×C411yescharacteristic
125C4×D4×C79C2311yescharacteristic
126C4×D4×C79C2311yescharacteristic
127C2×C4⋊C4×C79C2×C411yescharacteristic
128C2×C22⋊C4×C79—21no—
129C2×C42×C79C2311yescharacteristic
130C4⋊C8×C79D411yescharacteristic
131C4⋊C8×C79D411yescharacteristic
132C4⋊C8×C79C2311yescharacteristic
133C4⋊C8×C79C2311yescharacteristic
134C22⋊C8×C79—21no—
135C22⋊C8×C79—21no—
136C22⋊C8×C79—21no—
137C22⋊C8×C79—21no—
138C22⋊C8×C79—21no—
139C22⋊C8×C79D411yescharacteristic
140C22⋊C8×C79D411yescharacteristic
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158C22×C411yescharacteristic
145C2×D316C22×C411yescharacteristic
146C2×C22⋊D79—21no—
147C2×C4×D79—21no—
148#11929612—22no—
149#11929612—41no—
150#11929609—22no—
151#11929609—41no—
152#11929607—81no—
153C2×C79⋊C8—24no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8—21no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8—81no—
159C2×C79⋊C8C22×C411yescharacteristic
160C2×C79⋊C8C22×C411yescharacteristic
161C2×C79⋊C8C22×C411yescharacteristic
162C2×C79⋊C8C22×C411yescharacteristic
163C8×D79—81no—
164C4⋊Dic79C22×C411yescharacteristic
165C4⋊Dic79C22×C411yescharacteristic
166Dic79⋊C4—21no—
167Dic79⋊C4—21no—
168Dic79⋊C4—21no—
169Dic79⋊C4—21no—
170C22⋊Dic79—21no—
171(C2×D79)⋊C4—21no—
172(C2×D79)⋊C4—21no—
173(C2×D79)⋊C4—21no—
174(C2×D79)⋊C4—21no—
175C4×Dic79—21no—
176C24×C79—21no—
177C2×D4×C79—22no—
178C2×D4×C79—21no—
179C2×D4×C79—41no—
180C2×D4×C79C22×C411yescharacteristic
181C2×D4×C79C22×C411yescharacteristic
182C22×C4×C79—21no—
183C22×C4×C79—21no—
184C22×C4×C79—21no—
185C22×C4×C79—21no—
186C22×C4×C79—21no—
187C22×C4×C79—81no—
188C22×C4×C79—81no—
189C22×C4×C79C22×C411yescharacteristic
190C22×C4×C79C2×D411yescharacteristic
191C22×C4×C79#4111yescharacteristic
192M16×C79—81no—
193M16×C79—81no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—81no—
197C2×C8×C79—81no—
198C2×C8×C79C2×D411yescharacteristic
199C2×C8×C79C2×D411yescharacteristic
200C2×C8×C79C2×D411yescharacteristic
201C2×C8×C79C2×D411yescharacteristic
202C4⋊C4×C79—21no—
203C4⋊C4×C79C22×C411yescharacteristic
204C4⋊C4×C79C22×C411yescharacteristic
205C22⋊C4×C79—21no—
206C22⋊C4×C79—21no—
207C22⋊C4×C79—21no—
208C22⋊C4×C79—21no—
209C42×C79C22×C411yescharacteristic
210C42×C79C2×D411yescharacteristic
211C42×C79#4111yescharacteristic
212C42×C79C2411yescharacteristic
213C23×C79—21no—
214C23×C79—21no—
215C23×C79—42no—
216C23×C79—42no—
217C23×C79—81no—
218C23×C79C4×D411yescharacteristic
219C22×D79—21no—
220D4×C79—21no—
221D4×C79—21no—
222D4×C79—21no—
223D4×C79—42no—
224D4×C79#13112yes—
225C2×C4×C79—22no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79—81no—
237C2×C4×C79—81no—
238C2×C4×C79—81no—
239C2×C4×C79C4×D411yescharacteristic
240C2×C4×C79C4×D411yescharacteristic
241C2×C4×C79C4×D411yescharacteristic
242C2×C4×C79C23×C411yescharacteristic
243C2×C4×C79C22×D411yescharacteristic
244C2×C4×C79C2×#4111yescharacteristic
245C22⋊D79—42no—
246C2×Dic79—21no—
247C2×Dic79—21no—
248C2×Dic79—21no—
249D316—21no—
250D316—21no—
251C4×D79—41no—
252C4×D79—41no—
253Dic158—21no—
254Dic158—21no—
255C8×C79—21no—
256C8×C79—21no—
257C8×C79—21no—
258C8×C79—21no—
259C8×C79—41no—
260C8×C79—41no—
261C79⋊C8—22no—
262C79⋊C8—22no—
263C79⋊C8—21no—
264C79⋊C8—21no—
265C79⋊C8—21no—
266C79⋊C8—21no—
267C79⋊C8—41no—
268C79⋊C8—41no—
269C22×C79—21no—
270C22×C79—21no—
271C22×C79—22no—
272C22×C79—41no—
273C22×C79—41no—
274C22×C79—41no—
275C22×C79—41no—
276C22×C79—41no—
277C22×C79—84no—
278C22×C79#44312yes—
279C22×C79C2×C4×D411yescharacteristic
280C2×D79—41no—
281C2×D79—41no—
282C2×D79—41no—
283C4×C79—21no—
284C4×C79—21no—
285C4×C79—21no—
286C4×C79—21no—
287C4×C79—41no—
288C4×C79—41no—
289C4×C79C2×#13111yescharacteristic
290C4×C79#56811yescharacteristic
291C4×C79#57611yescharacteristic
292C4×C79#57711yescharacteristic
293Dic79—41no—
294Dic79—41no—
295Dic79—41no—
296#18014—791nomaximal, Sylow
297C2×C79—22no—
298C2×C79—42no—
299C2×C79—81no—
300C2×C79#293311yescharacteristic
301C2×C79#294611yescharacteristic
302C2×C79C2×#44311yescharacteristic
303D79—81no—
304#3105—791no—
305#2994—791no—
306#2974D7911yescharacteristic, complemented
307#1590—791no—
308#1589—791no—
309#1588—791no—
310#1586—791no—
311#1586—791no—
312#1585—791no—
313C8×SD16—791no—
314C8×D8—791no—
315#1573—791no—
316#1494—791no—
317C2×D4⋊C8—791no—
318#1450—791no—
319C79#1801411yescharacteristic, complemented, Sylow
320#529—791no—
321C2×C4×D4C2×D7911yescharacteristic
322#484—791no—
323#481—791no—
324#478—791no—
325#478—791no—
326#463—791no—
327#463—791no—
328#450—791no—
329#447—791no—
330#446—791no—
331C8×Q8—791no—
332C4×SD16—791no—
333C4×SD16—791no—
334C4×D8—791no—
335C4×D8—791no—
336#436—791no—
337#436C2×D7911yescharacteristic
338#436C2×D7911yescharacteristic
339#435—21no—
340#435—21no—
341#435—1581no—
342C8×D4—791no—
343C8×D4—1581no—
344C8×D4C2×D7911yescharacteristic
345C8×D4C2×D7911yescharacteristic
346#433—1581no—
347#432C2×D7911yescharacteristic
348#431—791no—
349#431C2×D7911yescharacteristic
350#424—1581no—
351C2×C4⋊C8—791no—
352#407—21no—
353C2×C22⋊C8—21no—
354C2×C4×C8—791no—
355#335—791no—
356#335—791no—
357#334—791no—
358#334—791no—
359Q8⋊C8—791no—
360Q8⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8—791no—
364D4⋊C8—791no—
365D4⋊C8—791no—
366D4⋊C8—791no—
367#322—791no—
368#322—791no—
369C82—791no—
370C82—791no—
371C22×D4C4×D7911yescharacteristic
372C23×C4—21no—
373C2×SD16—791no—
374C2×SD16—791no—
375C2×D8—791no—
376C2×D8—791no—
377C2×M16—21no—
378C2×M16—21no—
379C2×M16—1581no—
380C22×C8—21no—
381C22×C8—21no—
382C22×C8—1581no—
383C22×C8—1581no—
384C22×C8—1581no—
385#126—1581no—
386#125—791no—
387#124—791no—
388#123—1581no—
389C22⋊Q8—791no—
390#121—791no—
391C4×Q8—791no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—791no—
395C4×D4—1581no—
396C4×D4C4×D7911yescharacteristic
397C4×D4C4×D7911yescharacteristic
398C4×D4C22×D7911yescharacteristic
399C4×D4C22×D7911yescharacteristic
400#117—1581no—
401C2×C4⋊C4C4×D7911yescharacteristic
402C2×C22⋊C4—21no—
403C2×C42C22×D7911yescharacteristic
404#107—791no—
405#107—791no—
406#106—791no—
407#106—791no—
408C4⋊C8—791no—
409C4⋊C8—791no—
410C4⋊C8—791no—
411C4⋊C8—791no—
412C4⋊C8—1581no—
413C4⋊C8—1581no—
414C4⋊C8—1581no—
415C4⋊C8C22⋊D7911yescharacteristic
416C4⋊C8C22⋊D7911yescharacteristic
417C4⋊C8C22×D7911yescharacteristic
418C4⋊C8C22×D7911yescharacteristic
419#103—791no—
420#103—791no—
421#102—791no—
422#102—791no—
423#102—791no—
424#102—791no—
425#102—791no—
426#102—791no—
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—21no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—791no—
433C22⋊C8—791no—
434C22⋊C8—1581no—
435C22⋊C8—1581no—
436C22⋊C8C22⋊D7911yescharacteristic
437C22⋊C8C22⋊D7911yescharacteristic
438#97—21no—
439#97—1581no—
440C4×C8—791no—
441C4×C8—791no—
442C4×C8—791no—
443C4×C8—791no—
444C4×C8—1581no—
445C4×C8—1581no—
446C4×C8C22×D7911yescharacteristic
447C4×C8C22×D7911yescharacteristic
448C24—21no—
449C2×Q8—791no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—41no—
453C2×D4—791no—
454C2×D4—1581no—
455C2×D4C2×C4×D7911yescharacteristic
456C2×D4C2×C4×D7911yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—1581no—
465C22×C4C2×C4×D7911yescharacteristic
466C22×C4C2×C22⋊D7911yescharacteristic
467C22×C4#1192962411yescharacteristic
468SD16—3161no—
469SD16—1582no—
470D8—3161no—
471D8—1582no—
472M16—81no—
473M16—81no—
474M16—6321no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—81no—
478C2×C8—81no—
479C2×C8—791no—
480C2×C8—791no—
481C2×C8—791no—
482C2×C8—791no—
483C2×C8—6321no—
484C2×C8—6321no—
485C2×C8—1581no—
486C2×C8—1581no—
487C2×C8—1584no—
488C2×C8—1581no—
489C2×C8—1581no—
490C2×C8C2×C22⋊D7911yescharacteristic
491C2×C8C2×C22⋊D7911yescharacteristic
492C2×C8C2×C22⋊D7911yescharacteristic
493C2×C8C2×C22⋊D7911yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—791no—
496C4⋊C4—791no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4C2×C4×D7911yescharacteristic
502C4⋊C4C2×C4×D7911yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—1581no—
508C22⋊C4—1581no—
509C22⋊C4—1581no—
510C22⋊C4—1581no—
511C22⋊C4—1581no—
512C42—1581no—
513C42C2×C4×D7911yescharacteristic
514C42C2×C22⋊D7911yescharacteristic
515C42#1192962411yescharacteristic
516C42C23×D7911yescharacteristic
517C23—21no—
518C23—21no—
519C23—42no—
520C23—42no—
521C23—81no—
522C23—1581no—
523C23C4×C22⋊D7911yescharacteristic
524Q8—1581no—
525Q8—1581no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—42no—
530D4—3162no—
531D4—1581no—
532D4—1581no—
533D4#1457916912yes—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—1581no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4C4×C22⋊D7911yescharacteristic
554C2×C4C4×C22⋊D7911yescharacteristic
555C2×C4C4×C22⋊D7911yescharacteristic
556C2×C4C22×C4×D7911yescharacteristic
557C2×C4C22×C22⋊D7911yescharacteristic
558C2×C4C2×#1192962411yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—41no—
565C8—3161no—
566C8—3161no—
567C8—1582no—
568C8—1582no—
569C8—1581no—
570C8—1581no—
571C8—1581no—
572C8—1581no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—84no—
582C22—3161no—
583C22—3161no—
584C22—3161no—
585C22#1909867812yes—
586C22C2×C4×C22⋊D7911yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—3161no—
594C4—3161no—
595C4—3161no—
596C4C2×#1457916911yescharacteristic
597C4#1909932911yescharacteristic
598C4#1909936611yescharacteristic
599C4#1909937411yescharacteristic
600C2—22no—
601C2—42no—
602C2—81no—
603C2—6321no—
604C2#2377212511yescharacteristic
605C2#2377223211yescharacteristic
606C2C2×#1909867811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#2974C2
socleC22×C79C2×C4×D4
prime radical#19098697C4
cosocleC42×C79C24
hypercenter#2974D79
hypocenterC79#18014
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2974

Complements

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

NH ≅ G/Ntype
2. #23772562C2semidirect
3. #23762288C2semidirect
4. #23762283C2semidirect
5. #23762276C2semidirect
6. #23762264C2semidirect
7. #23762262C2semidirect
8. #23762257C2semidirect
9. C8×#11929612C2semidirect
10. C8×#11929609C2semidirect
11. #23762186C2semidirect
12. #23761858C2semidirect
13. C2×D4⋊(C79⋊C8)C2semidirect
14. #23761684C2semidirect
15. C79×#2974C2semidirect
25. #19098693C22semidirect
26. C8×Dic158C22semidirect
31. #19098642C22semidirect
33. C8×D316C22semidirect
40. #19098315C22semidirect
41. #19098315C22semidirect
42. #19098313C22semidirect
43. #19098313C22semidirect
44. #19098293C22semidirect
45. #19098293C22semidirect
50. #19098290C22semidirect
51. #19098290C22semidirect
52. #19098285C22semidirect
53. #19098285C22semidirect
54. C8×C79⋊C8C22semidirect
55. C8×C79⋊C8C22semidirect
306. #2974D79semidirect
319. C79#18014semidirect

Maximal subgroups

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

subgroupindex
#237734472
#237725622
#237622882
#237622832
#237622762
#237622642
#237622622
#237622572
C8×#119296122
C8×#119296092
#237621862
#237618582
C2×D4⋊(C79⋊C8)2
#237616842
C79×#29742
#1801479