Finite Group Databasebeta

← #34291339

Subgroups — #34291339

Order 20224.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098661C211yesmaximal, characteristic
2#23765762C211yesmaximal, characteristic
3#23764038C211yesmaximal, characteristic, complemented
4#23764031C211yesmaximal, characteristic, complemented
5#23763989C211yesmaximal, characteristic, complemented
6#23763968C211yesmaximal, characteristic, complemented
7#23763737C211yesmaximal, characteristic
8#23763737C211yesmaximal, characteristic
9#23763660C211yesmaximal, characteristic
10#23763646C211yesmaximal, characteristic
11#23763298C211yesmaximal, characteristic, complemented
12C4×#14579085C211yesmaximal, characteristic, complemented
13C4×#14579081C211yesmaximal, characteristic
14#23763152C211yesmaximal, characteristic, complemented
15C79×#2320C211yesmaximal, characteristic, complemented
16C22×#11929611C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18C2×C4×D316C2211yescharacteristic
19C2×C4×D316C2211yescharacteristic
20#19098661C418yescomplemented
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579085C2211yescharacteristic
25C2×#14579085C2211yescharacteristic
26C2×#14579085C2211yescharacteristic
27C2×#14579081C2211yescharacteristic
28C2×#14579081C2211yescharacteristic
29C2×#14579081C2211yescharacteristic
30C2×#14579072C2211yescharacteristic
31C2×C4×C79⋊C8C2211yescharacteristic
32C4×C4⋊Dic79C2211yescharacteristic
33C4×C4⋊Dic79C2211yescharacteristic
34C4×C4⋊Dic79C2211yescharacteristic
35#19098410—21no—
36#19098410—21no—
37#19098410—21no—
38#95×D79—21no—
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098318C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48#19098318C2211yescharacteristic
49C2×C4×Q8×C79C2211yescharacteristic
50#391×C79C2211yescharacteristic
51#391×C79C2211yescharacteristic
52#389×C79C2211yescharacteristic
53#389×C79C2211yescharacteristic
54#387×C79—21no—
55#387×C79—21no—
56#383×C79—21no—
57#382×C79—21no—
58C4×C4⋊C4×C79C2211yescharacteristic
59#376×C79C2211yescharacteristic
60C22×D316C2311yescharacteristic
61C22×C4×D79—21no—
62C22×C4×D79—21no—
63C22×C4×D79—21no—
64C2×#11929611—22no—
65C2×#11929611—22no—
66C2×#11929611C2×C414yes—
67C22×C79⋊C8C2311yescharacteristic
68C22×C79⋊C8C2311yescharacteristic
69C22×C79⋊C8C2311yescharacteristic
70C22×C79⋊C8C2311yescharacteristic
71C4×D316—22no—
72C4×D316—22no—
73C4×D316—22no—
74C4×D316—22no—
75C4×D316—22no—
76C4×D316C2×C414yes—
77C2×C4⋊Dic79C2311yescharacteristic
78C2×C4⋊Dic79C2311yescharacteristic
79C2×C4⋊Dic79C2311yescharacteristic
80C2×(C2×D79)⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79—21no—
86#14579094—22no—
87#14579094C2×C414yescomplemented
88#14579092—22no—
89#14579085—22no—
90#14579085—22no—
91#14579085C2×C414yescomplemented
92#14579081—22no—
93#14579081—22no—
94#14579081C2×C414yes—
95#14579072C2×C414yescomplemented
96C4×C79⋊C8—22no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C22×Q8×C79C2311yescharacteristic
101C4×Q8×C79—22no—
102C4×Q8×C79C2×C414yescomplemented
103C2×C4⋊C4×C79—21no—
104C2×C4⋊C4×C79—21no—
105C2×C4⋊C4×C79—21no—
106C2×C4⋊C4×C79C2311yescharacteristic
107C2×C4⋊C4×C79C2311yescharacteristic
108C2×C4⋊C4×C79C2311yescharacteristic
109C2×C42×C79—21no—
110C2×C42×C79—21no—
111C2×C42×C79C2311yescharacteristic
112C2×C42×C79C2311yescharacteristic
113C2×C42×C79C2311yescharacteristic
114#95×C79—21no—
115#95×C79—21no—
116#95×C79—21no—
117#95×C79—21no—
118#95×C79—21no—
119#95×C79—21no—
120#95×C79—21no—
121#95×C79D411yescharacteristic
122#95×C79D411yescharacteristic
123#95×C79D411yescharacteristic
124#95×C79D411yescharacteristic
125C23×D79—21no—
126C2×D316—21no—
127C2×D316—21no—
128C2×D316—21no—
129C2×D316C22×C412yes—
130C2×D316#4112yes—
131C2×D316#4112yes—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C2×C4×D79—24no—
136C2×C4×D79—24no—
137C2×C4×D79—24no—
138C2×C4×D79—81no—
139C2×C4×D79—81no—
140C2×C4×D79—81no—
141C2×C4×D79—81no—
142C2×C4×D79—81no—
143C2×C4×D79—81no—
144#11929611—28no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8C22×C412yes—
156C2×C79⋊C8C22×C412yes—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79C22×C412yes—
161C4⋊Dic79#4112yes—
162C4⋊Dic79#4112yes—
163(C2×D79)⋊C4—24no—
164(C2×D79)⋊C4—24no—
165(C2×D79)⋊C4—24no—
166C4×Dic79—81no—
167C4×Dic79—81no—
168C4×Dic79—81no—
169C2×Q8×C79—21no—
170C2×Q8×C79—21no—
171C2×Q8×C79—21no—
172C2×Q8×C79—21no—
173C2×Q8×C79—21no—
174C2×Q8×C79C22×C412yes—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2411yescharacteristic
186C4⋊C4×C79—24no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—81no—
193C4⋊C4×C79—81no—
194C4⋊C4×C79C22×C412yes—
195C42×C79—24no—
196C42×C79—21no—
197C42×C79—21no—
198C42×C79—21no—
199C42×C79—21no—
200C42×C79—81no—
201C42×C79C22×C412yes—
202C42×C79#4112yes—
203C23×C79C22×D411yescharacteristic
204C22×D79—22no—
205C22×D79—22no—
206C22×D79—22no—
207C22×D79—41no—
208C22×D79—41no—
209C22×D79—41no—
210C22×D79—41no—
211Q8×C79—22no—
212Q8×C79—22no—
213Q8×C79—41no—
214Q8×C79—41no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—22no—
220C2×C4×C79—22no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
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—21no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79—41no—
237C2×C4×C79—41no—
238C2×C4×C79—41no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—41no—
242C2×C4×C79—41no—
243C2×C4×C79C4×D412yes—
244C2×C4×C79C4×D412yes—
245C2×C4×C79C23×C411yescharacteristic
246C2×C4×C79C2×#4111yescharacteristic
247C2×C4×C79C2×#4111yescharacteristic
248C2×C4×C79C2×#4111yescharacteristic
249C2×C4×C79C2×#4111yescharacteristic
250C2×C4×C79C2×#4111yescharacteristic
251C2×Dic79—22no—
252C2×Dic79—22no—
253C2×Dic79—22no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260D316—22no—
261D316—22no—
262D316—22no—
263D316#11714yes—
264C4×D79—84no—
265C4×D79—84no—
266C4×D79—84no—
267C79⋊C8—22no—
268C79⋊C8—22no—
269C79⋊C8—41no—
270C79⋊C8—41no—
271C22×C79C2×C4×D411yescharacteristic
272C22×C79#52511yescharacteristic
273C22×C79#52511yescharacteristic
274C22×C79#57611yescharacteristic
275C22×C79#57711yescharacteristic
276C22×C79#57711yescharacteristic
277C22×C79#57811yescharacteristic
278C2×D79—24no—
279C2×D79—42no—
280C2×D79—42no—
281C2×D79—42no—
282C2×D79—42no—
283C2×D79—42no—
284C2×D79—42no—
285C4×C79—22no—
286C4×C79—22no—
287C4×C79—41no—
288C4×C79—41no—
289C4×C79—41no—
290C4×C79—41no—
291C4×C79—42no—
292C4×C79—42no—
293C4×C79—81no—
294C4×C79—81no—
295C4×C79C2×#11711yescharacteristic
296C4×C79C4×#4111yescharacteristic
297C4×C79#52911yescharacteristic
298C4×C79#52911yescharacteristic
299Dic79—42no—
300Dic79—42no—
301Dic79—42no—
302#21125—791nomaximal, Sylow
303C2×C79#229011yescharacteristic
304C2×C79#294611yescharacteristic
305C2×C79#294711yescharacteristic
306C2×C79#311811yescharacteristic
307C2×C79#312011yescharacteristic
308C2×C79#312311yescharacteristic
309C2×C79#312411yescharacteristic
310D79—44no—
311C2×#440—791no—
312#2320D7911yescharacteristic, complemented
313#2275—791no—
314#1928—791no—
315#1927—791no—
316#1921—791no—
317#1918—791no—
318#1877—791no—
319#1877—791no—
320#1864—791no—
321#1862—791no—
322#1776—791no—
323C4×#103—791no—
324C4×#102—791no—
325#1746—791no—
326C79#2112511yescharacteristic, complemented, Sylow
327C22×SD16—791no—
328C2×C4×Q8C2×D7911yescharacteristic
329C2×C4×D4—791no—
330C2×C4×D4—791no—
331C2×C4×D4—791no—
332#440—798no—
333C2×#107—791no—
334C2×#107—791no—
335C2×#106—791no—
336C2×#103—791no—
337C2×#103—791no—
338C2×#103—791no—
339C2×#102—791no—
340C2×#102—791no—
341C2×#102—791no—
342C2×#97—791no—
343C2×C4×C8—791no—
344#391C2×D7911yescharacteristic
345#391C2×D7911yescharacteristic
346#389C2×D7911yescharacteristic
347#389C2×D7911yescharacteristic
348#387—21no—
349#387—21no—
350#383—21no—
351#382—21no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—791no—
354C4×C4⋊C4—791no—
355C4×C4⋊C4C2×D7911yescharacteristic
356C4×C22⋊C4—1581no—
357C4×C22⋊C4—1581no—
358C4×C22⋊C4—1581no—
359#376C2×D7911yescharacteristic
360C2×#95—1581no—
361#340—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#336—791no—
368#336—791no—
369#336—791no—
370#336—791no—
371C22×Q8C22×D7911yescharacteristic
372C22×D4—791no—
373C23×C4—1581no—
374C23×C4—1581no—
375C23×C4—1581no—
376C2×SD16—794no—
377C2×SD16—1582no—
378C2×SD16—1582no—
379C22×C8—791no—
380C22×C8—791no—
381C22×C8—791no—
382C22×C8—791no—
383C4×Q8—22no—
384C4×Q8C4×D7914yescomplemented
385C4×D4—794no—
386C4×D4—1582no—
387C4×D4—1582no—
388C4×D4—1582no—
389C4×D4—1582no—
390C4×D4—1582no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—791no—
395C2×C4⋊C4—791no—
396C2×C4⋊C4—791no—
397C2×C4⋊C4C22×D7911yescharacteristic
398C2×C4⋊C4C22×D7911yescharacteristic
399C2×C4⋊C4C22×D7911yescharacteristic
400C2×C22⋊C4—1581no—
401C2×C22⋊C4—1581no—
402C2×C22⋊C4—1581no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—1581no—
406C2×C42—1581no—
407C2×C42—1581no—
408C2×C42C22×D7911yescharacteristic
409C2×C42C22×D7911yescharacteristic
410C2×C42C22×D7911yescharacteristic
411#107—794no—
412#107—1582no—
413#106—1582no—
414#103—794no—
415#103—1582no—
416#103—1582no—
417#102—794no—
418#102—1582no—
419#102—1582no—
420#97—794no—
421C4×C8—1582no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—1581no—
430#95—1581no—
431#95—1581no—
432#95C22⋊D7911yescharacteristic
433#95C22⋊D7911yescharacteristic
434#95C22⋊D7911yescharacteristic
435#95C22⋊D7911yescharacteristic
436C24—1581no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D7912yes—
443C2×D4—792no—
444C2×D4—792no—
445C2×D4—792no—
446C2×D4—1581no—
447C2×D4—1581no—
448C2×D4—1581no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—6321no—
454C22×C4—6321no—
455C22×C4—6321no—
456C22×C4—6321no—
457C22×C4—6321no—
458C22×C4—6321no—
459C22×C4—1584no—
460C22×C4—1584no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1584no—
465C22×C4C2×C22⋊D7911yescharacteristic
466C22×C4C2×C22⋊D7911yescharacteristic
467C22×C4C2×C22⋊D7911yescharacteristic
468C22×C4C2×C22⋊D7911yescharacteristic
469C22×C4C2×C22⋊D7911yescharacteristic
470C22×C4C2×C22⋊D7911yescharacteristic
471C22×C4C23×D7911yescharacteristic
472SD16—1588no—
473C2×C8—792no—
474C2×C8—792no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C4⋊C4—21no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—792no—
494C4⋊C4—792no—
495C4⋊C4—792no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4C2×C4×D7912yes—
500C22⋊C4—1584no—
501C22⋊C4—1584no—
502C22⋊C4—1584no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—24no—
508C42—81no—
509C42—6321no—
510C42—6321no—
511C42—6321no—
512C42C2×C4×D7912yes—
513C42#1192962412yes—
514C23—3161no—
515C23—3161no—
516C23—3161no—
517C23—3161no—
518C23—1582no—
519C23—1582no—
520C23—1582no—
521C23C22×C22⋊D7911yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—794no—
527D4—1582no—
528D4—1582no—
529D4—1582no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—3161no—
559C2×C4—3161no—
560C2×C4—3161no—
561C2×C4—3161no—
562C2×C4—3161no—
563C2×C4—3161no—
564C2×C4—6324no—
565C2×C4—6324no—
566C2×C4—6324no—
567C2×C4—1582no—
568C2×C4—1582no—
569C2×C4—1582no—
570C2×C4C4×C22⋊D7912yes—
571C2×C4C4×C22⋊D7912yes—
572C2×C4C22×C4×D7911yescharacteristic
573C2×C4#41×D7911yescharacteristic
574C2×C4#41×D7911yescharacteristic
575C2×C4#41×D7911yescharacteristic
576C2×C4#41×D7911yescharacteristic
577C2×C4C2×#1192962411yescharacteristic
578C8—3161no—
579C8—3161no—
580C8—1582no—
581C8—1582no—
582C22—3162no—
583C22—3162no—
584C22—3162no—
585C22—3162no—
586C22—3162no—
587C22—3162no—
588C22—1584no—
589C22C2×C4×C22⋊D7911yescharacteristic
590C22#1909907011yescharacteristic
591C22#1909907011yescharacteristic
592C22#1909936611yescharacteristic
593C22#1909937111yescharacteristic
594C22#1909937111yescharacteristic
595C22#1909937811yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—3162no—
607C4—3162no—
608C4—3162no—
609C4#117×D7911yescharacteristic
610C4#1909903111yescharacteristic
611C4#1909909111yescharacteristic
612C4#1909909111yescharacteristic
613C2—3164no—
614C2#2376583711yescharacteristic
615C2#2377222811yescharacteristic
616C2#2377224311yescharacteristic
617C2#2377357811yescharacteristic
618C2#2377360811yescharacteristic
619C2#2377364711yescharacteristic
620C2#2377365611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#2320C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenter#2320D79
hypocenterC79#21125
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ #376 ◁ #2320

Complements

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

NH ≅ G/Ntype
3. #23764038C2semidirect
4. #23764031C2semidirect
5. #23763989C2semidirect
6. #23763968C2semidirect
11. #23763298C2semidirect
12. C4×#14579085C2semidirect
14. #23763152C2semidirect
15. C79×#2320C2semidirect
20. #19098661C4semidirect
87. #14579094C2×C4semidirect
91. #14579085C2×C4semidirect
95. #14579072C2×C4semidirect
102. C4×Q8×C79C2×C4semidirect
312. #2320D79semidirect
326. C79#21125semidirect
384. C4×Q8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190986612
#237657622
#237640382
#237640312
#237639892
#237639682
#237637372
#237636602
#237636462
#237632982
C4×#145790852
C4×#145790812
#237631522
C79×#23202
#2112579