Finite Group Databasebeta

← #34347062

Subgroups — #34347062

Order 20224.

826 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×#19098604C211yesmaximal, characteristic
2#23767098C211yesmaximal, characteristic
3#23767087C211yesmaximal, characteristic
4#23764884C211yesmaximal, characteristic, complemented
5#23764874C211yesmaximal, characteristic, complemented
6#23764730C211yesmaximal, characteristic, complemented
7#23764725C211yesmaximal, characteristic, complemented
8#23764139C211yesmaximal, characteristic, complemented
9#23764128C211yesmaximal, characteristic, complemented
10#23763596C211yesmaximal, characteristic
11#23763586C211yesmaximal, characteristic
12#23763567C211yesmaximal, characteristic, complemented
13#23763434C211yesmaximal, characteristic
14#23763183C211yesmaximal, characteristic
15C2×#500×C79C211yesmaximal, characteristic, complemented
16C22×#11929607C2211yescharacteristic
17C2×#14579137—21no—
18C2×#14579119C2211yescharacteristic
19C2×C4⋊C4×D79C2211yescharacteristic
20#19098604—24no—
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579093C2211yescharacteristic
24C2×#14579093C2211yescharacteristic
25C2×#14579092C2211yescharacteristic
26C2×#14579092C2211yescharacteristic
27C2×(C2×D79)⋊C8C2211yescharacteristic
28C2×#14579071C2211yescharacteristic
29#19098496—21no—
30#19098485—21no—
31#19098482—21no—
32#19098474—21no—
33#19098470—21no—
34#19098462C2211yescharacteristic
35#19098462C2211yescharacteristic
36#19098461C2211yescharacteristic
37#19098461C2211yescharacteristic
38#19098444—21no—
39#19098434—21no—
40#19098432C2211yescharacteristic
41#19098428C2211yescharacteristic
42#19098420C2211yescharacteristic
43#19098420C2211yescharacteristic
44C4×(C2×D79)⋊C4C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098318C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×C4⋊Q8×C79C2211yescharacteristic
54C2×C4⋊Q8×C79C2211yescharacteristic
55#500×C79—24no—
56C2×#107×C79C2211yescharacteristic
57C2×#107×C79C2211yescharacteristic
58C2×#107×C79C2211yescharacteristic
59C2×#107×C79C2211yescharacteristic
60C2×C4×C8×C79C2211yescharacteristic
61C22×C4×D79—21no—
62C22×C4×D79C2311yescharacteristic
63C2×#11929607—22no—
64C2×#11929607—22no—
65C2×#11929607Q814yescomplemented
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68#14579137—44no—
69#14579119—22no—
70#14579119—41no—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79—21no—
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×Dic79⋊C4—21no—
82C4⋊C4×D79—22no—
83C4⋊C4×D79—41no—
84C2×(C2×D79)⋊C4—21no—
85C2×(C2×D79)⋊C4—21no—
86C2×(C2×D79)⋊C4—21no—
87C2×(C2×D79)⋊C4D411yescharacteristic
88C2×(C2×D79)⋊C4D411yescharacteristic
89C2×C4×Dic79—21no—
90C2×C4×Dic79C2311yescharacteristic
91#14579094—22no—
92#14579094—41no—
93#14579093—22no—
94#14579093—22no—
95#14579092—22no—
96#14579092—41no—
97(C2×D79)⋊C8Q814yescomplemented
98#14579071—22no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95D411yescharacteristic
108C79⋊#95D411yescharacteristic
109C22×Q8×C79—21no—
110C22×Q8×C79—21no—
111C22×C8×C79C2311yescharacteristic
112C22×C8×C79C2311yescharacteristic
113C4⋊Q8×C79—22no—
114C4⋊Q8×C79—22no—
115C4⋊Q8×C79—41no—
116C4⋊Q8×C79—41no—
117C2×C4⋊C4×C79—21no—
118C2×C4⋊C4×C79—21no—
119C2×C4⋊C4×C79C2311yescharacteristic
120C2×C4⋊C4×C79C2311yescharacteristic
121C2×C4⋊C4×C79C2311yescharacteristic
122C2×C4⋊C4×C79C2311yescharacteristic
123C2×C42×C79C2311yescharacteristic
124#107×C79—22no—
125#107×C79—22no—
126#107×C79—22no—
127#107×C79D414yes—
128C4×C8×C79—22no—
129C23×D79C2×D411yescharacteristic
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79C2×D411yescharacteristic
135C22×Dic79C2×D411yescharacteristic
136C22×Dic79C2×D411yescharacteristic
137C2×C4×D79—21no—
138C2×C4×D79—21no—
139C2×C4×D79—21no—
140C2×C4×D79—21no—
141C2×C4×D79—42no—
142C2×C4×D79—42no—
143C2×C4×D79—81no—
144C2×C4×D79C2×Q812yes—
145C2×C4×D79C2×Q812yes—
146#11929607—28no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8C2×Q812yes—
152C2×C79⋊C8C2×Q812yes—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79—21no—
161C4⋊Dic79—42no—
162C4⋊Dic79—81no—
163Dic79⋊C4—42no—
164Dic79⋊C4—42no—
165Dic79⋊C4—42no—
166Dic79⋊C4—42no—
167Dic79⋊C4—81no—
168(C2×D79)⋊C4—22no—
169(C2×D79)⋊C4—22no—
170(C2×D79)⋊C4—42no—
171(C2×D79)⋊C4—42no—
172(C2×D79)⋊C4—81no—
173C4×Dic79—21no—
174C4×Dic79—21no—
175C4×Dic79—42no—
176C2×Q8×C79—42no—
177C2×Q8×C79—42no—
178C2×Q8×C79—42no—
179C2×Q8×C79—42no—
180C2×Q8×C79—81no—
181C2×Q8×C79—81no—
182C22×C4×C79—21no—
183C22×C4×C79—21no—
184C22×C4×C79—21no—
185C22×C4×C79—21no—
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79C2411yescharacteristic
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79C2×D412yes—
192C2×C8×C79C2×Q812yes—
193C2×C8×C79C2×Q812yes—
194C2×C8×C79#4112yes—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79—21no—
200C4⋊C4×C79—21no—
201C4⋊C4×C79—42no—
202C4⋊C4×C79—42no—
203C4⋊C4×C79C2×D412yes—
204C4⋊C4×C79C2×D412yes—
205C42×C79—21no—
206C42×C79—21no—
207C23×C79C22×D411yescharacteristic
208C22×D79—21no—
209C22×D79—21no—
210C22×D79—21no—
211C22×D79—21no—
212C22×D79—21no—
213C22×D79—21no—
214C22×D79C4⋊Q812yes—
215Q8×C79—84no—
216Q8×C79—84no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—22no—
222C2×C4×C79—22no—
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×C79—41no—
233C2×C4×C79AGL(1,ℤ/8)12yes—
234C2×C4×C79#13712yes—
235C2×C4×C79C22×D411yescharacteristic
236C2×C4×C79C22×Q811yescharacteristic
237C2×C4×C79C22×Q811yescharacteristic
238C2×C4×C79C2×#4111yescharacteristic
239C2×C4×C79H(5,𝔽2)11yescharacteristic
240C2×C4×C792−1+411yescharacteristic
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—21no—
248C2×Dic79—21no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79C4⋊Q812yes—
262C4×D79—22no—
263C4×D79—22no—
264C4×D79—22no—
265C4×D79—22no—
266C4×D79—84no—
267C8×C79—22no—
268C8×C79C22⋊Q814yescomplemented
269C79⋊C8—22no—
270C79⋊C8—22no—
271C22×C79C2×C4⋊Q811yescharacteristic
272C22×C79D4211yescharacteristic
273C22×C79#54711yescharacteristic
274C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C79C2×#13711yescharacteristic
276C22×C79#57611yescharacteristic
277C22×C79#57811yescharacteristic
278C2×D79—22no—
279C2×D79—22no—
280C2×D79—22no—
281C2×D79—41no—
282C2×D79—41no—
283C2×D79—41no—
284C2×D79—41no—
285C4×C79—22no—
286C4×C79—22no—
287C4×C79—42no—
288C4×C79—42no—
289C4×C79—81no—
290C4×C79—81no—
291C4×C79C2×C22⋊Q811yescharacteristic
292C4×C79#54411yescharacteristic
293C4×C79D4×Q811yescharacteristic
294C4×C79#55411yescharacteristic
295Dic79—22no—
296Dic79—41no—
297Dic79—41no—
298Dic79—81no—
299Dic79—81no—
300Dic79—81no—
301Dic79—81no—
302#26074—791nomaximal, Sylow
303C2×C79#244011yescharacteristic
304C2×C79#316411yescharacteristic
305C2×C79#316511yescharacteristic
306C2×C79#326411yescharacteristic
307C2×C79#327011yescharacteristic
308C2×C79#335311yescharacteristic
309C2×C79#335711yescharacteristic
310D79—42no—
311C2×#500D7911yescharacteristic, complemented
312C2×#428—791no—
313#2440—791no—
314#2439—791no—
315#2085—791no—
316#2083—791no—
317#2058—791no—
318#2057—791no—
319#1943—791no—
320#1941—791no—
321#1854—791no—
322#1852—791no—
323#1847—791no—
324#1809—791no—
325#1753—791no—
326C79#2607411yescharacteristic, complemented, Sylow
327C22×M16—791no—
328C2×C4⋊Q8C2×D7911yescharacteristic
329C2×C4⋊Q8C2×D7911yescharacteristic
330C2×C22⋊Q8—1581no—
331C2×#117—791no—
332C22×C4⋊C4—791no—
333#500—24no—
334#428—1584no—
335C2×#107—791no—
336C2×#107—791no—
337C2×#107C2×D7911yescharacteristic
338C2×#107C2×D7911yescharacteristic
339C2×#107C2×D7911yescharacteristic
340C2×#107C2×D7911yescharacteristic
341C2×#106—791no—
342C2×#106—791no—
343C2×#106—791no—
344C2×#106—791no—
345C2×C22⋊C8—791no—
346C2×#97—791no—
347C2×C4×C8C2×D7911yescharacteristic
348#398—1581no—
349#396—1581no—
350#395—1581no—
351#393—1581no—
352#391—1581no—
353#389—791no—
354#389—791no—
355#389—791no—
356#389—791no—
357#387—1581no—
358#385—1581no—
359#384—791no—
360#382—791no—
361#380—791no—
362#380—791no—
363C4×C22⋊C4—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340—791no—
369#340—791no—
370#336—791no—
371#336—791no—
372C22×Q8—21no—
373C22×Q8—21no—
374C23×C4—791no—
375C23×C4—1581no—
376C2×M16—794no—
377C2×M16—1582no—
378C2×M16—1582no—
379C22×C8—791no—
380C22×C8—791no—
381C22×C8C22×D7911yescharacteristic
382C22×C8C22×D7911yescharacteristic
383C4⋊Q8—22no—
384C4⋊Q8—22no—
385C4⋊Q8—41no—
386C4⋊Q8—41no—
387C22⋊Q8—3164no—
388#117—3161no—
389#117—1582no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—3161no—
393C2×C4⋊C4—791no—
394C2×C4⋊C4—791no—
395C2×C4⋊C4—791no—
396C2×C4⋊C4—791no—
397C2×C4⋊C4—1581no—
398C2×C4⋊C4—1581no—
399C2×C4⋊C4—1581no—
400C2×C4⋊C4—1581no—
401C2×C4⋊C4—1581no—
402C2×C4⋊C4—1582no—
403C2×C4⋊C4—1581no—
404C2×C4⋊C4—1581no—
405C2×C4⋊C4C22×D7911yescharacteristic
406C2×C4⋊C4C22×D7911yescharacteristic
407C2×C4⋊C4C22×D7911yescharacteristic
408C2×C4⋊C4C22×D7911yescharacteristic
409C2×C22⋊C4—791no—
410C2×C22⋊C4—791no—
411C2×C22⋊C4—1581no—
412C2×C22⋊C4—1581no—
413C2×C22⋊C4—1581no—
414C2×C42—791no—
415C2×C42—1581no—
416C2×C42C22×D7911yescharacteristic
417#107—22no—
418#107—22no—
419#107—22no—
420#107—3161no—
421#107—1582no—
422#107C22⋊D7914yes—
423#106—3161no—
424#106—1582no—
425#106—1582no—
426#106—1582no—
427C22⋊C8—794no—
428#97—1582no—
429C4×C8—22no—
430#95—791no—
431#95—791no—
432#95—1581no—
433#95—1581no—
434#95—1581no—
435#95—1581no—
436#95—1581no—
437#95—1581no—
438#95—1581no—
439#95—1581no—
440C24—791no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—81no—
446C2×Q8—81no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—3162no—
452C22×C4—3162no—
453C22×C4—791no—
454C22×C4—792no—
455C22×C4—792no—
456C22×C4—791no—
457C22×C4—791no—
458C22×C4—6321no—
459C22×C4—1581no—
460C22×C4—1581no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4—1581no—
467C22×C4D4×D7911yescharacteristic
468C22×C4D4×D7911yescharacteristic
469C22×C4C23×D7911yescharacteristic
470M16—1588no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—792no—
474C2×C8—792no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8C2×C22⋊D7912yes—
480C2×C8Q8×D7912yes—
481C2×C8Q8×D7912yes—
482C2×C8#1192962512yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—3162no—
492C4⋊C4—3162no—
493C4⋊C4—3162no—
494C4⋊C4—3162no—
495C4⋊C4—3162no—
496C4⋊C4—6321no—
497C4⋊C4—6321no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4—1581no—
502C4⋊C4—1581no—
503C4⋊C4—1581no—
504C4⋊C4—1581no—
505C4⋊C4—1581no—
506C4⋊C4C2×C22⋊D7912yes—
507C4⋊C4C2×C22⋊D7912yes—
508C22⋊C4—3162no—
509C22⋊C4—3162no—
510C22⋊C4—6321no—
511C22⋊C4—1582no—
512C22⋊C4—1582no—
513C42—21no—
514C42—21no—
515C42—3162no—
516C42—1581no—
517C42—1581no—
518C23—792no—
519C23—1581no—
520C23—1581no—
521C23—1581no—
522C23—1581no—
523C23—1581no—
524C23—1581no—
525C23C2×D4×D7911yescharacteristic
526Q8—84no—
527Q8—84no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
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—3161no—
556C2×C4—792no—
557C2×C4—6324no—
558C2×C4—1581no—
559C2×C4—1581no—
560C2×C4—1581no—
561C2×C4—1581no—
562C2×C4—1581no—
563C2×C4—1581no—
564C2×C4—1581no—
565C2×C4—1581no—
566C2×C4—1582no—
567C2×C4—1582no—
568C2×C4—1582no—
569C2×C4—1582no—
570C2×C4#1457918512yes—
571C2×C4#1457919312yes—
572C2×C4C22×C22⋊D7911yescharacteristic
573C2×C4C2×Q8×D7911yescharacteristic
574C2×C4C2×Q8×D7911yescharacteristic
575C2×C4C2×#1192962511yescharacteristic
576C2×C4#1457920611yescharacteristic
577C2×C4#1457920911yescharacteristic
578C8—22no—
579C8—1582no—
580C8—1582no—
581C8#1457913714yescomplemented
582C22—3161no—
583C22—3161no—
584C22—3161no—
585C22—3161no—
586C22—1582no—
587C22—1582no—
588C22—1582no—
589C22C4⋊Q8×D7911yescharacteristic
590C22D4×C22⋊D7911yescharacteristic
591C22#1909921611yescharacteristic
592C22C2×#1457918511yescharacteristic
593C22C2×#1457919311yescharacteristic
594C22#1909936411yescharacteristic
595C22#1909937611yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—3161no—
603C4—3161no—
604C4—6321no—
605C4—6321no—
606C4—6321no—
607C4—6321no—
608C4—1582no—
609C4C2×#1457913711yescharacteristic
610C4#1909919011yescharacteristic
611C4Q8×C22⋊D7911yescharacteristic
612C4#1909927411yescharacteristic
613C2—3162no—
614C2#2376709811yescharacteristic
615C2#2377393311yescharacteristic
616C2#2377394311yescharacteristic
617C2#2377488011yescharacteristic
618C2#2377497011yescharacteristic
619C2#2377598311yescharacteristic
620C2#2377602511yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#500×C79C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
hypercenterC2×#500D79
hypocenterC79#26074
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
4. #23764884C2semidirect
5. #23764874C2semidirect
6. #23764730C2semidirect
7. #23764725C2semidirect
8. #23764139C2semidirect
9. #23764128C2semidirect
12. #23763567C2semidirect
15. C2×#500×C79C2semidirect
65. C2×#11929607Q8semidirect
97. (C2×D79)⋊C8Q8semidirect
268. C8×C79C22⋊Q8semidirect
311. C2×#500D79semidirect
326. C79#26074semidirect
581. C8#14579137semidirect

Maximal subgroups

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

subgroupindex
C2×#190986042
#237670982
#237670872
#237648842
#237648742
#237647302
#237647252
#237641392
#237641282
#237635962
#237635862
#237635672
#237634342
#237631832
C2×#500×C792
#2607479