Finite Group Databasebeta

← #34328429

Subgroups — #34328429

Order 20224.

756 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098905C211yesmaximal, characteristic
2#23769197C211yesmaximal, characteristic
3#23766016C211yesmaximal, characteristic
4#23764898C211yesmaximal, characteristic, complemented
5#23764893C211yesmaximal, characteristic, complemented
6#23764780C211yesmaximal, characteristic, complemented
7#23764780C211yesmaximal, characteristic, complemented
8#23764143C211yesmaximal, characteristic, complemented
9#23764128C211yesmaximal, characteristic, complemented
10#23764090C211yesmaximal, characteristic, complemented
11#23764076C211yesmaximal, characteristic, complemented
12#23763705C211yesmaximal, characteristic
13#23763595C211yesmaximal, characteristic
14#23763496C211yesmaximal, characteristic
15C2×#433×C79C211yesmaximal, characteristic
16C2×#14579140C2211yescharacteristic
17C2×#14579120C2211yescharacteristic
18C22×C4⋊Dic79C2211yescharacteristic
19#19098905—24no—
20C2×#14579095C2211yescharacteristic
21C2×#14579093C2211yescharacteristic
22C2×#14579093C2211yescharacteristic
23C2×#14579093C2211yescharacteristic
24C2×#14579086C2211yescharacteristic
25C2×#14579086C2211yescharacteristic
26C2×#14579086C2211yescharacteristic
27C2×#14579086C2211yescharacteristic
28#19098501—21no—
29#19098494—21no—
30#19098491—21no—
31#19098491—21no—
32#19098469C2211yescharacteristic
33#19098469C2211yescharacteristic
34#19098461C2211yescharacteristic
35#19098461C2211yescharacteristic
36#19098461C2211yescharacteristic
37#19098461C2211yescharacteristic
38#19098461C2211yescharacteristic
39#19098461C2211yescharacteristic
40#19098459—21no—
41#19098459—21no—
42#19098423C2211yescharacteristic
43#19098421—21no—
44#19098421—21no—
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098326C2211yescharacteristic
51#19098326C2211yescharacteristic
52#19098326C2211yescharacteristic
53C2×#117×C79C2211yescharacteristic
54#433×C79—24no—
55C2×C4⋊C8×C79C2211yescharacteristic
56C2×C4⋊C8×C79C2211yescharacteristic
57C2×C22⋊C8×C79C2211yescharacteristic
58C2×C22⋊C8×C79C2211yescharacteristic
59C2×#97×C79C2211yescharacteristic
60C2×C4×C8×C79C2211yescharacteristic
61C22×Dic158C2311yescharacteristic
62C23×Dic79—21no—
63#14579140—22no—
64#14579140—22no—
65#14579120—22no—
66#14579120—22no—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79—22no—
69C2×C4⋊Dic79—22no—
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
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—
82C2×Dic79⋊C4—21no—
83C2×Dic79⋊C4—21no—
84C2×C22⋊Dic79—21no—
85C2×C22⋊Dic79—21no—
86C2×C22⋊Dic79—21no—
87C2×C4×Dic79—21no—
88#14579095—22no—
89#14579093—22no—
90#14579093—22no—
91#14579093—22no—
92#14579086—22no—
93#14579086—22no—
94#14579086—22no—
95#14579086—22no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C23×C4×C79C2311yescharacteristic
108C22×C8×C79C2311yescharacteristic
109C22×C8×C79C2311yescharacteristic
110C22×C8×C79C2311yescharacteristic
111C22×C8×C79C2311yescharacteristic
112#117×C79—22no—
113#117×C79—22no—
114C2×C4⋊C4×C79D411yescharacteristic
115C2×C4⋊C4×C79D411yescharacteristic
116C2×C22⋊C4×C79D411yescharacteristic
117C2×C22⋊C4×C79D411yescharacteristic
118C2×C42×C79C2311yescharacteristic
119C2×C42×C79C2311yescharacteristic
120C4⋊C8×C79—22no—
121C4⋊C8×C79—22no—
122C22⋊C8×C79—22no—
123C22⋊C8×C79D414yes—
124#97×C79—22no—
125C4×C8×C79—22no—
126C2×Dic158—21no—
127C2×Dic158—21no—
128C2×Dic158—21no—
129C2×Dic158—21no—
130C2×Dic158—21no—
131C2×Dic158—21no—
132C22×Dic79—24no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79—21no—
137C22×Dic79—21no—
138C22×Dic79—21no—
139C22×Dic79—21no—
140C22×Dic79—21no—
141C22×Dic79—81no—
142C22×Dic79—81no—
143C4⋊Dic79—24no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79#4112yes—
157C4⋊Dic79#4112yes—
158Dic79⋊C4—42no—
159Dic79⋊C4—81no—
160Dic79⋊C4—81no—
161Dic79⋊C4—81no—
162Dic79⋊C4—81no—
163Dic79⋊C4—81no—
164Dic79⋊C4—81no—
165C22⋊Dic79—42no—
166C22⋊Dic79—42no—
167C22⋊Dic79—81no—
168C4×Dic79—42no—
169C24×C79C2×D411yescharacteristic
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D412yes—
180C22×C4×C79#4112yes—
181C22×C4×C79C2411yescharacteristic
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79C2×D412yes—
193C2×C8×C79C2×D412yes—
194C4⋊C4×C79—22no—
195C4⋊C4×C79—22no—
196C22⋊C4×C79—22no—
197C22⋊C4×C79—22no—
198C42×C79—21no—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—21no—
202C23×C79—21no—
203C23×C79—21no—
204C23×C79—21no—
205C23×C79—21no—
206C23×C79—21no—
207C23×C79—21no—
208C23×C79#12112yes—
209C23×C79C22×D411yescharacteristic
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—22no—
213C2×C4×C79—22no—
214C2×C4×C79—22no—
215C2×C4×C79—22no—
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—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79#12112yes—
229C2×C4×C79C22×D411yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C792−1+411yescharacteristic
234C2×C4×C792−1+411yescharacteristic
235C2×Dic79—22no—
236C2×Dic79—22no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
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—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—84no—
260Dic158—41no—
261Dic158—41no—
262Dic158—41no—
263Dic158—41no—
264C8×C79—22no—
265C8×C79—22no—
266C8×C79—41no—
267C8×C79—41no—
268C22×C79—22no—
269C22×C79—22no—
270C22×C79—22no—
271C22×C79—41no—
272C22×C79—41no—
273C22×C79—41no—
274C22×C79—41no—
275C22×C79C2×#12111yescharacteristic
276C22×C79#52511yescharacteristic
277C22×C79#53611yescharacteristic
278C22×C79#57711yescharacteristic
279C22×C79#57711yescharacteristic
280C22×C79#57711yescharacteristic
281C22×C79#57811yescharacteristic
282C4×C79—22no—
283C4×C79—41no—
284C4×C79—41no—
285C4×C79—41no—
286C4×C79—41no—
287C4×C79#54711yescharacteristic
288C4×C79#54711yescharacteristic
289C4×C79#55211yescharacteristic
290C4×C79#55211yescharacteristic
291Dic79—42no—
292Dic79—42no—
293Dic79—81no—
294Dic79—81no—
295Dic79—81no—
296Dic79—81no—
297Dic79—81no—
298Dic79—81no—
299#24619—791nomaximal, Sylow
300C2×C79—42no—
301C2×C79#241311yescharacteristic
302C2×C79#306011yescharacteristic
303C2×C79#306311yescharacteristic
304C2×C79#312111yescharacteristic
305C2×C79#312311yescharacteristic
306C2×C79#324211yescharacteristic
307C2×C79#324411yescharacteristic
308C2×#485—791no—
309C2×#433D7911yescharacteristic
310#2625—791no—
311#2316—791no—
312#2088—791no—
313#2087—791no—
314#2066—791no—
315#2066—791no—
316#1944—791no—
317#1941—791no—
318#1935—791no—
319#1933—791no—
320#1872—791no—
321#1853—791no—
322#1825—791no—
323C79#2461911yescharacteristic, complemented, Sylow
324C2×C22⋊Q8—791no—
325C2×#117—791no—
326C2×#117C2×D7911yescharacteristic
327C22×C4⋊C4—791no—
328#485—1584no—
329#433—24no—
330C2×#107—791no—
331C2×#106—791no—
332C2×#106—791no—
333C2×#106—791no—
334C2×C4⋊C8C2×D7911yescharacteristic
335C2×C4⋊C8C2×D7911yescharacteristic
336C2×#103—791no—
337C2×#103—791no—
338C2×#103—791no—
339C2×#103—791no—
340C2×C22⋊C8C2×D7911yescharacteristic
341C2×C22⋊C8C2×D7911yescharacteristic
342C2×#97C2×D7911yescharacteristic
343C2×C4×C8C2×D7911yescharacteristic
344#400—1581no—
345#398—1581no—
346#397—1581no—
347#397—1581no—
348#391—791no—
349#391—791no—
350#389—791no—
351#389—791no—
352#389—791no—
353#389—791no—
354#389—791no—
355#389—791no—
356#389—1581no—
357#389—1581no—
358#380—791no—
359#380—1581no—
360#380—1581no—
361#340—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340—791no—
369C22×Q8—791no—
370C23×C4—1581no—
371C23×C4C22×D7911yescharacteristic
372C22×C8C22×D7911yescharacteristic
373C22×C8C22×D7911yescharacteristic
374C22×C8C22×D7911yescharacteristic
375C22×C8C22×D7911yescharacteristic
376C22⋊Q8—1582no—
377C22⋊Q8—1582no—
378#117—22no—
379#117—22no—
380#117—1582no—
381#117—1582no—
382C2×C4⋊C4—791no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—791no—
387C2×C4⋊C4—791no—
388C2×C4⋊C4—791no—
389C2×C4⋊C4—1581no—
390C2×C4⋊C4—1582no—
391C2×C4⋊C4—1582no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4—1581no—
395C2×C4⋊C4—1581no—
396C2×C4⋊C4—1581no—
397C2×C4⋊C4—1581no—
398C2×C4⋊C4—1581no—
399C2×C4⋊C4D31611yescharacteristic
400C2×C4⋊C4D31611yescharacteristic
401C2×C22⋊C4—1581no—
402C2×C22⋊C4—1581no—
403C2×C22⋊C4—1581no—
404C2×C22⋊C4D31611yescharacteristic
405C2×C22⋊C4D31611yescharacteristic
406C2×C42—1581no—
407C2×C42C22×D7911yescharacteristic
408C2×C42C22×D7911yescharacteristic
409#107—1582no—
410#106—1582no—
411#106—1582no—
412#106—1582no—
413C4⋊C8—22no—
414C4⋊C8—22no—
415#103—1582no—
416#103—1582no—
417#103—1582no—
418#103—1582no—
419C22⋊C8—22no—
420C22⋊C8C22⋊D7914yes—
421#97—22no—
422C4×C8—22no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428#95—1581no—
429#95—1581no—
430#95—1581no—
431#95—1581no—
432#95—1581no—
433#95—1581no—
434C24C2×D31611yescharacteristic
435C2×Q8—1581no—
436C2×Q8—1581no—
437C2×Q8—1581no—
438C2×Q8—1581no—
439C2×Q8—1581no—
440C2×Q8—1581no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—6321no—
446C22×C4—6321no—
447C22×C4—1581no—
448C22×C4—1581no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4—1584no—
456C22×C4C2×C22⋊D7912yes—
457C22×C4C2×D31611yescharacteristic
458C22×C4C2×D31611yescharacteristic
459C22×C4C2×D31611yescharacteristic
460C22×C4C2×D31611yescharacteristic
461C22×C4C2×D31611yescharacteristic
462C22×C4#1192962412yes—
463C22×C4C23×D7911yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8C2×C22⋊D7912yes—
475C2×C8C2×C22⋊D7912yes—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—3162no—
479C4⋊C4—792no—
480C4⋊C4—792no—
481C4⋊C4—6321no—
482C4⋊C4—6321no—
483C4⋊C4—6321no—
484C4⋊C4—6321no—
485C4⋊C4—6321no—
486C4⋊C4—6321no—
487C4⋊C4—1584no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—3162no—
503C22⋊C4—3162no—
504C22⋊C4—6321no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—3162no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23#1457913512yes—
517C23C22×D31611yescharacteristic
518Q8—3161no—
519Q8—3161no—
520Q8—3161no—
521Q8—3161no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
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—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—3161no—
555C2×C4—3161no—
556C2×C4—3161no—
557C2×C4—3161no—
558C2×C4—3161no—
559C2×C4—3161no—
560C2×C4—3161no—
561C2×C4—3161no—
562C2×C4—6324no—
563C2×C4—1582no—
564C2×C4—1582no—
565C2×C4#1457913512yes—
566C2×C4C22×C22⋊D7911yescharacteristic
567C2×C4#41×D7911yescharacteristic
568C2×C4#41×D7911yescharacteristic
569C2×C4C2×#1192962411yescharacteristic
570C2×C4#1457920811yescharacteristic
571C2×C4#1457920811yescharacteristic
572C8—22no—
573C8—22no—
574C8—41no—
575C8—41no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1457913511yescharacteristic
584C22#1909907511yescharacteristic
585C22#1909913911yescharacteristic
586C22#1909937511yescharacteristic
587C22#1909937511yescharacteristic
588C22#1909937511yescharacteristic
589C22#1909938011yescharacteristic
590C4—22no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—3162no—
596C4—3162no—
597C4—6321no—
598C4—6321no—
599C4—6321no—
600C4—6321no—
601C4—6321no—
602C4—6321no—
603C4#1909921211yescharacteristic
604C4#1909921211yescharacteristic
605C4#1909924611yescharacteristic
606C4#1909924611yescharacteristic
607C2—42no—
608C2#2376680111yescharacteristic
609C2#2377306211yescharacteristic
610C2#2377309111yescharacteristic
611C2#2377363111yescharacteristic
612C2#2377365311yescharacteristic
613C2#2377467011yescharacteristic
614C2#2377470011yescharacteristic
subgroupG/H
centerC23C22×D316
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#433×C79C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenterC2×#433D79
hypocenterC79#24619
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
4. #23764898C2semidirect
5. #23764893C2semidirect
6. #23764780C2semidirect
7. #23764780C2semidirect
8. #23764143C2semidirect
9. #23764128C2semidirect
10. #23764090C2semidirect
11. #23764076C2semidirect
323. C79#24619semidirect

Maximal subgroups

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

subgroupindex
C2×#190989052
#237691972
#237660162
#237648982
#237648932
#237647802
#237641432
#237641282
#237640902
#237640762
#237637052
#237635952
#237634962
C2×#433×C792
#2461979