Finite Group Databasebeta

← #34310287

Subgroups — #34310287

Order 20224.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098840C211yesmaximal, characteristic
2#23766311C211yesmaximal, characteristic
3#23764374C211yesmaximal, characteristic, complemented
4#23764168C211yesmaximal, characteristic, complemented
5#23764135C211yesmaximal, characteristic, complemented
6#23764130C211yesmaximal, characteristic, complemented
7#23763992C211yesmaximal, characteristic, complemented
8#23763971C211yesmaximal, characteristic, complemented
9#23763660C211yesmaximal, characteristic
10#23763646C211yesmaximal, characteristic
11#23763275C211yesmaximal, characteristic, complemented
12#23763228C211yesmaximal, characteristic
13#23763228C211yesmaximal, characteristic
14C4×#14579081C211yesmaximal, characteristic
15C79×#2369C211yesmaximal, characteristic, complemented
16C2×C4×D316C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18C2×C4×D316C2211yescharacteristic
19#19098840C418yescomplemented
20C2×#14579094C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×#14579081C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×#14579081C2211yescharacteristic
28C2×#14579081C2211yescharacteristic
29C2×#14579072C2211yescharacteristic
30C2×#14579072C2211yescharacteristic
31C2×C4×C79⋊C8C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098461C2211yescharacteristic
34#19098434—21no—
35#19098434—21no—
36#19098429C2211yescharacteristic
37#19098422—21no—
38#95×D79—21no—
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098318C2211yescharacteristic
48#19098318C2211yescharacteristic
49C2×C4⋊Q8×C79C2211yescharacteristic
50#391×C79—21no—
51#391×C79—21no—
52#389×C79—21no—
53#389×C79C2211yescharacteristic
54#389×C79C2211yescharacteristic
55C4×C4⋊C4×C79—21no—
56C4×C4⋊C4×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C4⋊C4×C79C2211yescharacteristic
59C4×C4⋊C4×C79C2211yescharacteristic
60C22×D316C2311yescharacteristic
61C22×C4×D79—21no—
62C22×C4×D79—21no—
63C22×C4×D79—21no—
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68C4×D316—22no—
69C4×D316—22no—
70C4×D316—22no—
71C4×D316—41no—
72C4×D316—41no—
73C4×D316C2×C414yes—
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×(C2×D79)⋊C4—21no—
80C2×(C2×D79)⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82#14579094—41no—
83#14579094C2×C414yescomplemented
84#14579094C2×C414yescomplemented
85#14579092—41no—
86C4⋊(C79⋊C8)C2×C414yescomplemented
87#14579081—22no—
88#14579081C2×C414yes—
89#14579081C2×C414yes—
90#14579081Q814yescomplemented
91#14579072—22no—
92#14579072—41no—
93C4×C79⋊C8—41no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C22×Q8×C79—21no—
99C4⋊Q8×C79—22no—
100C4⋊Q8×C79C2×C414yescomplemented
101C2×C4⋊C4×C79—21no—
102C2×C4⋊C4×C79—21no—
103C2×C4⋊C4×C79—21no—
104C2×C4⋊C4×C79D411yescharacteristic
105C2×C4⋊C4×C79D411yescharacteristic
106C2×C4⋊C4×C79C2311yescharacteristic
107C2×C4⋊C4×C79C2311yescharacteristic
108C2×C4⋊C4×C79C2311yescharacteristic
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C42×C79—21no—
111C2×C42×C79—21no—
112C2×C42×C79—21no—
113C2×C42×C79C2311yescharacteristic
114C2×C42×C79C2311yescharacteristic
115C2×C42×C79C2311yescharacteristic
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
123C23×D79—21no—
124C2×D316—21no—
125C2×D316—21no—
126C2×D316—21no—
127C2×D316C22×C412yes—
128C2×D316C2×Q812yes—
129C2×D316#4112yes—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C2×C4×D79—24no—
134C2×C4×D79—42no—
135C2×C4×D79—42no—
136C2×C4×D79—42no—
137C2×C4×D79—42no—
138C2×C4×D79—81no—
139C2×C4×D79—81no—
140C2×C4×D79—81no—
141C2×C4×D79—81no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
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⋊C8C22×C412yes—
152C2×C79⋊C8C22×C412yes—
153C2×C79⋊C8C2×Q812yes—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79C22×C412yes—
160Dic79⋊C4—81no—
161Dic79⋊C4—81no—
162(C2×D79)⋊C4—24no—
163(C2×D79)⋊C4—42no—
164(C2×D79)⋊C4—42no—
165C2×Q8×C79—24no—
166C2×Q8×C79—81no—
167C2×Q8×C79—81no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2411yescharacteristic
179C4⋊C4×C79—24no—
180C4⋊C4×C79—22no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—41no—
187C4⋊C4×C79—81no—
188C4⋊C4×C79—81no—
189C4⋊C4×C79C22×C412yes—
190C4⋊C4×C79C22×C412yes—
191C4⋊C4×C79C2×Q812yes—
192C42×C79—21no—
193C42×C79—21no—
194C42×C79—21no—
195C42×C79—21no—
196C42×C79—42no—
197C42×C79—81no—
198C42×C79—81no—
199C42×C79C22×C412yes—
200C42×C79#4112yes—
201C23×C79C22×D411yescharacteristic
202C22×D79—22no—
203C22×D79—22no—
204C22×D79—22no—
205C22×D79—41no—
206C22×D79—41no—
207C22×D79—41no—
208C22×D79—41no—
209Q8×C79—84no—
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—22no—
217C2×C4×C79—22no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—22no—
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—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79—41no—
237C2×C4×C79C4×D412yes—
238C2×C4×C79C4×D412yes—
239C2×C4×C79#13712yes—
240C2×C4×C79C23×C411yescharacteristic
241C2×C4×C79C22×Q811yescharacteristic
242C2×C4×C79C2×#4111yescharacteristic
243C2×C4×C79C2×#4111yescharacteristic
244C2×C4×C79H(5,𝔽2)11yescharacteristic
245C2×C4×C792−1+411yescharacteristic
246C2×Dic79—22no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255D316—22no—
256D316—22no—
257D316—22no—
258D316C4×Q814yes—
259C4×D79—84no—
260C4×D79—84no—
261C4×D79—84no—
262C79⋊C8—22no—
263C79⋊C8—22no—
264C79⋊C8—22no—
265C79⋊C8—41no—
266C22×C79C2×C4×D411yescharacteristic
267C22×C79#54611yescharacteristic
268C22×C79D4×Q811yescharacteristic
269C22×C79C2×#13711yescharacteristic
270C22×C79#57511yescharacteristic
271C22×C79#57611yescharacteristic
272C22×C79#57611yescharacteristic
273C2×D79—24no—
274C2×D79—42no—
275C2×D79—42no—
276C2×D79—42no—
277C2×D79—42no—
278C2×D79—42no—
279C2×D79—42no—
280C4×C79—22no—
281C4×C79—22no—
282C4×C79—22no—
283C4×C79—42no—
284C4×C79—42no—
285C4×C79—42no—
286C4×C79—41no—
287C4×C79—41no—
288C4×C79—41no—
289C4×C79—81no—
290C4×C79C2×C4×Q811yescharacteristic
291C4×C79#52011yescharacteristic
292C4×C79#55211yescharacteristic
293C4×C79#55411yescharacteristic
294Dic79—42no—
295Dic79—81no—
296Dic79—81no—
297#22979—791nomaximal, Sylow
298C2×C79#236011yescharacteristic
299C2×C79C4×#13711yescharacteristic
300C2×C79#294811yescharacteristic
301C2×C79#331411yescharacteristic
302C2×C79#331911yescharacteristic
303C2×C79#338511yescharacteristic
304C2×C79#338911yescharacteristic
305D79—44no—
306C2×#474—791no—
307#2369D7911yescharacteristic, complemented
308#2349—791no—
309#1993—791no—
310#1949—791no—
311#1943—791no—
312#1942—791no—
313#1921—791no—
314#1918—791no—
315#1864—791no—
316#1862—791no—
317#1771—791no—
318#1762—791no—
319#1762—791no—
320C4×#102—791no—
321C79#2297911yescharacteristic, complemented, Sylow
322C2×C4⋊Q8C2×D7911yescharacteristic
323C2×C4×D4—791no—
324C2×C4×D4—791no—
325C2×C4×D4—791no—
326#474—798no—
327C2×#107—791no—
328C2×#107—791no—
329C2×#107—791no—
330C2×#106—791no—
331C2×C4⋊C8—791no—
332C2×#102—791no—
333C2×#102—791no—
334C2×#102—791no—
335C2×#102—791no—
336C2×#97—791no—
337C2×#97—791no—
338C2×C4×C8—791no—
339#391—21no—
340#391—21no—
341#389—21no—
342#389—791no—
343#389—791no—
344#389C2×D7911yescharacteristic
345#389C2×D7911yescharacteristic
346#385—1581no—
347#385—1581no—
348#382—791no—
349#380—1581no—
350C4×C4⋊C4—21no—
351C4×C4⋊C4C2×D7911yescharacteristic
352C4×C4⋊C4C2×D7911yescharacteristic
353C4×C4⋊C4C2×D7911yescharacteristic
354C4×C4⋊C4C2×D7911yescharacteristic
355C2×#95—1581no—
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364#336—791no—
365#336—791no—
366C22×Q8—21no—
367C22×D4—791no—
368C23×C4—1581no—
369C23×C4—1581no—
370C23×C4—1581no—
371C22×C8—791no—
372C22×C8—791no—
373C22×C8—791no—
374C22×C8—791no—
375C4⋊Q8—22no—
376C4⋊Q8C4×D7914yescomplemented
377C4×D4—3161no—
378C4×D4—3161no—
379C4×D4—794no—
380C4×D4—1582no—
381C4×D4—1582no—
382C4×D4—1582no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—791no—
387C2×C4⋊C4—791no—
388C2×C4⋊C4—791no—
389C2×C4⋊C4—1581no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4C22⋊D7911yescharacteristic
392C2×C4⋊C4C22⋊D7911yescharacteristic
393C2×C4⋊C4C22×D7911yescharacteristic
394C2×C4⋊C4C22×D7911yescharacteristic
395C2×C4⋊C4C22×D7911yescharacteristic
396C2×C4⋊C4C22×D7911yescharacteristic
397C2×C22⋊C4—1581no—
398C2×C22⋊C4—1581no—
399C2×C22⋊C4—1581no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42C22×D7911yescharacteristic
404C2×C42C22×D7911yescharacteristic
405C2×C42C22×D7911yescharacteristic
406#107—3161no—
407#107—794no—
408#107—794no—
409#106—3161no—
410C4⋊C8—794no—
411#102—794no—
412#102—794no—
413#102—794no—
414#102—1582no—
415#97—3161no—
416#97—1582no—
417C4×C8—3161no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95C22⋊D7911yescharacteristic
428#95C22⋊D7911yescharacteristic
429C24—1581no—
430C2×Q8—24no—
431C2×Q8—81no—
432C2×Q8—81no—
433C2×D4—792no—
434C2×D4—792no—
435C2×D4—792no—
436C2×D4—1581no—
437C2×D4—1581no—
438C2×D4—1581no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—3162no—
444C22×C4—3162no—
445C22×C4—3162no—
446C22×C4—3162no—
447C22×C4—6321no—
448C22×C4—6321no—
449C22×C4—6321no—
450C22×C4—6321no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1584no—
455C22×C4C2×C22⋊D7911yescharacteristic
456C22×C4C2×C22⋊D7911yescharacteristic
457C22×C4C2×C22⋊D7911yescharacteristic
458C22×C4C2×C22⋊D7911yescharacteristic
459C22×C4C2×C22⋊D7911yescharacteristic
460C22×C4C2×C22⋊D7911yescharacteristic
461C22×C4C23×D7911yescharacteristic
462C2×C8—792no—
463C2×C8—792no—
464C2×C8—792no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8—1581no—
472C2×C8—1581no—
473C2×C8—1581no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—792no—
485C4⋊C4—6321no—
486C4⋊C4—6321no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4C2×C4×D7912yes—
493C4⋊C4C2×C4×D7912yes—
494C4⋊C4Q8×D7912yes—
495C22⋊C4—3162no—
496C22⋊C4—3162no—
497C22⋊C4—1584no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—42no—
503C42—81no—
504C42—81no—
505C42C2×C4×D7912yes—
506C42#1192962412yes—
507C23—3161no—
508C23—3161no—
509C23—3161no—
510C23—3161no—
511C23—1582no—
512C23—1582no—
513C23—1582no—
514C23C22×C22⋊D7911yescharacteristic
515Q8—84no—
516D4—794no—
517D4—1582no—
518D4—1582no—
519D4—1582no—
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—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—3161no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—3161no—
555C2×C4—6324no—
556C2×C4—6324no—
557C2×C4—6324no—
558C2×C4—1582no—
559C2×C4C4×C22⋊D7912yes—
560C2×C4C4×C22⋊D7912yes—
561C2×C4#1457919212yes—
562C2×C4C22×C4×D7911yescharacteristic
563C2×C4C2×Q8×D7911yescharacteristic
564C2×C4#41×D7911yescharacteristic
565C2×C4C2×#1192962411yescharacteristic
566C2×C4#1457920711yescharacteristic
567C2×C4#1457920911yescharacteristic
568C8—3161no—
569C8—1582no—
570C8—1582no—
571C8—1582no—
572C22—3162no—
573C22—3162no—
574C22—3162no—
575C22—3162no—
576C22—3162no—
577C22—3162no—
578C22—1584no—
579C22C2×C4×C22⋊D7911yescharacteristic
580C22#1909920111yescharacteristic
581C22Q8×C22⋊D7911yescharacteristic
582C22C2×#1457919211yescharacteristic
583C22#1909936111yescharacteristic
584C22#1909936611yescharacteristic
585C22#1909936611yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—81no—
596C4—3162no—
597C4—6321no—
598C4—6321no—
599C4C4×Q8×D7911yescharacteristic
600C4#1909904511yescharacteristic
601C4#1909925411yescharacteristic
602C4#1909927311yescharacteristic
603C2—3164no—
604C2#2376640111yescharacteristic
605C2C4×#1457919211yescharacteristic
606C2#2377225011yescharacteristic
607C2#2377547511yescharacteristic
608C2#2377555011yescharacteristic
609C2#2377630611yescharacteristic
610C2#2377635411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#2369C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenter#2369D79
hypocenterC79#22979
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2369

Complements

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

NH ≅ G/Ntype
3. #23764374C2semidirect
4. #23764168C2semidirect
5. #23764135C2semidirect
6. #23764130C2semidirect
7. #23763992C2semidirect
8. #23763971C2semidirect
11. #23763275C2semidirect
15. C79×#2369C2semidirect
19. #19098840C4semidirect
83. #14579094C2×C4semidirect
84. #14579094C2×C4semidirect
86. C4⋊(C79⋊C8)C2×C4semidirect
90. #14579081Q8semidirect
100. C4⋊Q8×C79C2×C4semidirect
307. #2369D79semidirect
321. C79#22979semidirect
376. C4⋊Q8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190988402
#237663112
#237643742
#237641682
#237641352
#237641302
#237639922
#237639712
#237636602
#237636462
#237632752
#237632282
C4×#145790812
C79×#23692
#2297979