Finite Group Databasebeta

← #34295663

Subgroups — #34295663

Order 20224.

768 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773315C211yesmaximal, characteristic
2#23772099C211yesmaximal, characteristic
3#23771818C211yesmaximal, characteristic, complemented
4#23764047C211yesmaximal, characteristic, complemented
5#23764047C211yesmaximal, characteristic, complemented
6#23763998C211yesmaximal, characteristic, complemented
7#23763998C211yesmaximal, characteristic, complemented
8#23763702C211yesmaximal, characteristic
9#23763497C211yesmaximal, characteristic, complemented
10#23763287C211yesmaximal, characteristic, complemented
11#23763282C211yesmaximal, characteristic, complemented
12#23763243C211yesmaximal, characteristic, complemented
13#23763236C211yesmaximal, characteristic, complemented
14#23763186C211yesmaximal, characteristic, complemented
15C79×#1855C211yesmaximal, characteristic, complemented
16C2×#14579169C2211yescharacteristic
17C22×#11929608C2211yescharacteristic
18#19099074C2211yescharacteristic
19C2×#14579120C2211yescharacteristic
20#19098905C411yescharacteristic, complemented
21#19098905C411yescharacteristic, complemented
22#19098899C411yescharacteristic, complemented
23#19098894C411yescharacteristic, complemented
24#19098888C411yescharacteristic, complemented
25#19098888C411yescharacteristic, complemented
26#19098882C411yescharacteristic, complemented
27#19098877C411yescharacteristic, complemented
28#19098638—21no—
29#19098638—21no—
30#19098637—21no—
31#19098637—21no—
32#19098601C2211yescharacteristic
33#19098601C2211yescharacteristic
34#19098591C2211yescharacteristic
35C2×#14579087C2211yescharacteristic
36C2×#14579087C2211yescharacteristic
37#19098560C2211yescharacteristic
38#19098560C2211yescharacteristic
39#19098531C2211yescharacteristic
40#19098521C2211yescharacteristic
41#19098521C2211yescharacteristic
42#19098521C2211yescharacteristic
43#19098521C2211yescharacteristic
44#19098521C2211yescharacteristic
45#19098518—21no—
46#19098518—21no—
47M16×Dic79—21no—
48M16×Dic79—21no—
49C2×#14579071C2211yescharacteristic
50C2×C8×Dic79C2211yescharacteristic
51C4×C22⋊Dic79C2211yescharacteristic
52#19098324C2211yescharacteristic
53#19098324C2211yescharacteristic
54#19098324C2211yescharacteristic
55#19098324C2211yescharacteristic
56#19098324C2211yescharacteristic
57#19098324C2211yescharacteristic
58#19098317C2211yescharacteristic
59#19098317C2211yescharacteristic
60C22×M16×C79C2211yescharacteristic
61C2×#108×C79C2211yescharacteristic
62C2×#108×C79C2211yescharacteristic
63#407×C79C2211yescharacteristic
64C2×C22⋊C8×C79C2211yescharacteristic
65#341×C79C2211yescharacteristic
66#341×C79C2211yescharacteristic
67C2×#11929624C2311yescharacteristic
68#14579169—22no—
69#14579169—22no—
70C2×#11929608—22no—
71C2×#11929608—22no—
72C2×#11929608D411yescharacteristic
73C2×#11929608D411yescharacteristic
74C2×#11929608D411yescharacteristic
75C2×#11929608D411yescharacteristic
76C2×#11929607—21no—
77C22×C79⋊C8C2311yescharacteristic
78C22×C79⋊C8C2311yescharacteristic
79C2×C8×D79—21no—
80#14579144—21no—
81#14579140C2×C411yescharacteristic
82#14579140C2×C411yescharacteristic
83#14579135C2×C411yescharacteristic
84#14579135C2×C411yescharacteristic
85#14579130—21no—
86C4×C22⋊D79—21no—
87C4×C22⋊D79—21no—
88#14579120—21no—
89#14579120—21no—
90#14579120C2×C411yescharacteristic
91#14579120C2×C411yescharacteristic
92#14579120C2311yescharacteristic
93#14579120C2311yescharacteristic
94#14579120C2311yescharacteristic
95C2×C4⋊Dic79C2×C411yescharacteristic
96C2×C4⋊Dic79C2×C411yescharacteristic
97C2×C22⋊Dic79—21no—
98C2×C22⋊Dic79—21no—
99C2×C4×Dic79—21no—
100C2×C4×Dic79C2311yescharacteristic
101C2×C4×Dic79C2311yescharacteristic
102#14579095C2×C411yescharacteristic
103#14579095C2×C411yescharacteristic
104#14579095C2×C411yescharacteristic
105#14579095C2×C411yescharacteristic
106#14579093C2×C411yescharacteristic
107#14579093C2×C411yescharacteristic
108#14579093C2×C411yescharacteristic
109#14579093C2×C411yescharacteristic
110Dic79⋊C8—21no—
111Dic79⋊C8—21no—
112#14579087—22no—
113#14579087—22no—
114#14579086C2×C411yescharacteristic
115#14579086C2×C411yescharacteristic
116#14579086C2×C411yescharacteristic
117#14579086C2×C411yescharacteristic
118#14579083C2×C411yescharacteristic
119#14579083C2×C411yescharacteristic
120#14579083C2×C411yescharacteristic
121#14579083C2×C411yescharacteristic
122(C2×D79)⋊C8—21no—
123(C2×D79)⋊C8—21no—
124#14579071—21no—
125#14579071—21no—
126#14579071—21no—
127#14579071—21no—
128#14579071—21no—
129#14579071—22no—
130C8×Dic79—21no—
131C8×Dic79—21no—
132C8×Dic79—21no—
133C8×Dic79—21no—
134C8×Dic79—21no—
135C8×Dic79—22no—
136C79⋊#95—21no—
137C23×C4×C79C2311yescharacteristic
138C2×M16×C79—21no—
139C2×M16×C79—22no—
140C2×M16×C79—22no—
141C2×M16×C79C2311yescharacteristic
142C2×M16×C79C2311yescharacteristic
143C2×M16×C79C2311yescharacteristic
144C2×M16×C79C2311yescharacteristic
145C22×C8×C79—21no—
146C22×C8×C79C2311yescharacteristic
147C22×C8×C79C2311yescharacteristic
148#108×C79—22no—
149#108×C79—22no—
150C22⋊C8×C79—21no—
151C22⋊C8×C79—21no—
152C22⋊C8×C79C2×C411yescharacteristic
153C22⋊C8×C79C2×C411yescharacteristic
154C22⋊C8×C79C2×C411yescharacteristic
155C22⋊C8×C79C2×C411yescharacteristic
156#11929624—21no—
157#11929624—21no—
158#11929624—21no—
159#11929624—21no—
160C2×Dic158C22×C411yescharacteristic
161C2×D316C22×C411yescharacteristic
162C2×C22⋊D79—21no—
163C2×C22⋊D79—21no—
164C2×C22⋊D79—21no—
165C22×Dic79—21no—
166C22×Dic79—21no—
167C22×Dic79—21no—
168C22×Dic79—21no—
169C2×C4×D79—21no—
170#11929608—22no—
171#11929608—22no—
172#11929608—22no—
173#11929608—22no—
174#11929607—42no—
175C2×C79⋊C8—21no—
176C2×C79⋊C8—21no—
177C2×C79⋊C8C2×D411yescharacteristic
178C2×C79⋊C8C2×D411yescharacteristic
179C2×C79⋊C8C2×D411yescharacteristic
180C2×C79⋊C8C2×D411yescharacteristic
181C2×C79⋊C8#4112yes—
182C2×C79⋊C8#4112yes—
183C8×D79—42no—
184C4⋊Dic79—21no—
185C4⋊Dic79—21no—
186C4⋊Dic79C22×C411yescharacteristic
187C4⋊Dic79C22×C411yescharacteristic
188C4⋊Dic79C22×C411yescharacteristic
189C4⋊Dic79C22×C411yescharacteristic
190C4⋊Dic79C22×C411yescharacteristic
191C4⋊Dic79C22×C411yescharacteristic
192Dic79⋊C4—21no—
193Dic79⋊C4—21no—
194C22⋊Dic79—21no—
195C22⋊Dic79—21no—
196C22⋊Dic79—21no—
197C22⋊Dic79—21no—
198C22⋊Dic79—21no—
199C22⋊Dic79—21no—
200C22⋊Dic79—21no—
201C22⋊Dic79—42no—
202(C2×D79)⋊C4—21no—
203(C2×D79)⋊C4—21no—
204C4×Dic79—21no—
205C4×Dic79—21no—
206C4×Dic79—21no—
207C4×Dic79—21no—
208C4×Dic79—21no—
209C4×Dic79—21no—
210C4×Dic79—21no—
211C4×Dic79—21no—
212C4×Dic79—21no—
213C24×C79#4111yescharacteristic
214C22×C4×C79—21no—
215C22×C4×C79—21no—
216C22×C4×C79—21no—
217C22×C4×C79—21no—
218C22×C4×C79C22×C411yescharacteristic
219C22×C4×C79C22×C411yescharacteristic
220C22×C4×C79C2×D411yescharacteristic
221C22×C4×C79C2×D411yescharacteristic
222C22×C4×C79#4111yescharacteristic
223C22×C4×C79C2411yescharacteristic
224M16×C79—24no—
225M16×C79—21no—
226M16×C79—21no—
227M16×C79—21no—
228M16×C79—21no—
229M16×C79—21no—
230M16×C79—21no—
231M16×C79#4112yes—
232M16×C79#4112yes—
233C2×C8×C79—24no—
234C2×C8×C79—21no—
235C2×C8×C79—21no—
236C2×C8×C79—21no—
237C2×C8×C79—21no—
238C2×C8×C79—21no—
239C2×C8×C79—21no—
240C2×C8×C79—21no—
241C2×C8×C79—21no—
242C2×C8×C79C22×C411yescharacteristic
243C2×C8×C79C22×C411yescharacteristic
244C2×C8×C79C22×C411yescharacteristic
245C2×C8×C79C22×C411yescharacteristic
246C23×C79—21no—
247C23×C79—21no—
248C23×C79—41no—
249C23×C79—41no—
250C23×C79C4×D411yescharacteristic
251C23×C79C4×D411yescharacteristic
252C23×C79C2×#4111yescharacteristic
253C22×D79—21no—
254C2×C4×C79—21no—
255C2×C4×C79—21no—
256C2×C4×C79—21no—
257C2×C4×C79—21no—
258C2×C4×C79—21no—
259C2×C4×C79—21no—
260C2×C4×C79—21no—
261C2×C4×C79—21no—
262C2×C4×C79—21no—
263C2×C4×C79—21no—
264C2×C4×C79C4×D411yescharacteristic
265C2×C4×C79C4×D411yescharacteristic
266C2×C4×C79C23×C411yescharacteristic
267C2×C4×C79C22×D411yescharacteristic
268C2×C4×C79C2×#4111yescharacteristic
269C2×C4×C79C2×#4111yescharacteristic
270C2×C4×C79C2×#4111yescharacteristic
271C2×C4×C79C2×#4111yescharacteristic
272C22⋊D79—41no—
273C22⋊D79—41no—
274C22⋊D79—81no—
275C22⋊D79—81no—
276C2×Dic79—22no—
277C2×Dic79—21no—
278C2×Dic79—21no—
279C2×Dic79—21no—
280C2×Dic79—21no—
281C2×Dic79—21no—
282C2×Dic79—21no—
283C2×Dic79—21no—
284C2×Dic79—41no—
285C2×Dic79—41no—
286C2×Dic79—41no—
287C2×Dic79—41no—
288C2×Dic79—41no—
289C2×Dic79—81no—
290C2×Dic79—81no—
291D316—21no—
292D316—21no—
293C4×D79—41no—
294C4×D79—41no—
295Dic158—21no—
296Dic158—21no—
297C8×C79—21no—
298C8×C79—21no—
299C8×C79—21no—
300C8×C79—21no—
301C8×C79—22no—
302C8×C79—22no—
303C8×C79—41no—
304C8×C79—41no—
305C79⋊C8—21no—
306C79⋊C8—21no—
307C79⋊C8#12312yes—
308C79⋊C8#12312yes—
309C22×C79—21no—
310C22×C79—21no—
311C22×C79—21no—
312C22×C79—21no—
313C22×C79—21no—
314C22×C79—21no—
315C22×C79—21no—
316C22×C79—21no—
317C22×C79—81no—
318C22×C79—81no—
319C22×C79C2×C4×D411yescharacteristic
320C22×C79#52911yescharacteristic
321C22×C79#52911yescharacteristic
322C2×D79—41no—
323C2×D79—41no—
324C2×D79—41no—
325C4×C79—21no—
326C4×C79—21no—
327C4×C79C4×#4111yescharacteristic
328C4×C79C4×#4111yescharacteristic
329C4×C79C2×#12311yescharacteristic
330C4×C79#52511yescharacteristic
331Dic79—42no—
332Dic79—41no—
333Dic79—41no—
334Dic79—41no—
335Dic79—41no—
336Dic79—41no—
337Dic79—41no—
338Dic79—41no—
339Dic79—81no—
340#21594—791nomaximal, Sylow
341C2×C79—21no—
342C2×C79—21no—
343C2×C79—41no—
344C2×C79—41no—
345C2×C79C4×#12311yescharacteristic
346C2×C79#295611yescharacteristic
347C2×C79#295711yescharacteristic
348D79—81no—
349#3091—791no—
350#2930—791no—
351#2873—791no—
352#1929—791no—
353#1929—791no—
354#1922—791no—
355#1922—791no—
356#1871—791no—
357#1855D7911yescharacteristic, complemented
358#1826—791no—
359#1773—791no—
360#1772—791no—
361#1765—791no—
362#1764—791no—
363#1753—791no—
364C79#2159411yescharacteristic, complemented, Sylow
365C2×#131—791no—
366C22×M16—791no—
367C22×M16C2×D7911yescharacteristic
368#525—791no—
369C2×#117—791no—
370#485—791no—
371#485—791no—
372#484—791no—
373#483—791no—
374#482—791no—
375#482—791no—
376#481—791no—
377#480—791no—
378#436—1581no—
379#436—1581no—
380#435—1581no—
381#435—1581no—
382C2×#108C2×D7911yescharacteristic
383C2×#108C2×D7911yescharacteristic
384#427—791no—
385#427—791no—
386#424—791no—
387C2×#104—791no—
388C2×#104—791no—
389#416—791no—
390#416—791no—
391#408—791no—
392#407C2×D7911yescharacteristic
393C2×C22⋊C8C2×D7911yescharacteristic
394#405—791no—
395#405—791no—
396#405—791no—
397#405—791no—
398#405—791no—
399#405—1581no—
400#405—1581no—
401C4×M16—1581no—
402C4×M16—1581no—
403C2×#97—791no—
404C2×C4×C8—791no—
405C4×C22⋊C4—791no—
406#341C2×D7911yescharacteristic
407#341C2×D7911yescharacteristic
408#339—791no—
409#339—791no—
410#339—791no—
411#339—791no—
412#339—791no—
413#339—791no—
414#336—791no—
415#336—791no—
416C2×#41—791no—
417C23×C4C22×D7911yescharacteristic
418#131—1582no—
419#131—1582no—
420C2×M16—21no—
421C2×M16—22no—
422C2×M16—22no—
423C2×M16—791no—
424C2×M16—791no—
425C2×M16—791no—
426C2×M16—791no—
427C2×M16—1581no—
428C2×M16—1582no—
429C2×M16—1582no—
430C2×M16C22×D7911yescharacteristic
431C2×M16C22×D7911yescharacteristic
432C2×M16C22×D7911yescharacteristic
433C2×M16C22×D7911yescharacteristic
434C22×C8—21no—
435C22×C8—791no—
436C22×C8—791no—
437C22×C8—1581no—
438C22×C8C22×D7911yescharacteristic
439C22×C8C22×D7911yescharacteristic
440#123—1581no—
441C22⋊Q8—791no—
442C22⋊Q8—791no—
443#121—791no—
444#121—791no—
445#120—1581no—
446C4×D4—1581no—
447C4×D4—1581no—
448#117—791no—
449#117—791no—
450#117—791no—
451#117—791no—
452#117—791no—
453#117—1581no—
454#117—1581no—
455C2×C4⋊C4—791no—
456C2×C4⋊C4—791no—
457C2×C22⋊C4—1581no—
458C2×C22⋊C4—1581no—
459C2×C42—791no—
460C2×C42—791no—
461C2×C42—1581no—
462#108—22no—
463#108—22no—
464#107—791no—
465#107—791no—
466#107—791no—
467#107—791no—
468#106—791no—
469#106—791no—
470#106—791no—
471#106—791no—
472C4⋊C8—1581no—
473C4⋊C8—1581no—
474#104—1582no—
475#104—1582no—
476#103—791no—
477#103—791no—
478#103—791no—
479#103—791no—
480#102—791no—
481#102—791no—
482#102—791no—
483#102—791no—
484C22⋊C8—21no—
485C22⋊C8—21no—
486C22⋊C8—1581no—
487C22⋊C8—1581no—
488C22⋊C8C4×D7911yescharacteristic
489C22⋊C8C4×D7911yescharacteristic
490C22⋊C8C4×D7911yescharacteristic
491C22⋊C8C4×D7911yescharacteristic
492#97—1582no—
493#97—1581no—
494#97—1581no—
495#97—1581no—
496#97—1581no—
497#97—1581no—
498C4×C8—1582no—
499C4×C8—1581no—
500C4×C8—1581no—
501C4×C8—1581no—
502C4×C8—1581no—
503C4×C8—1581no—
504#95—1581no—
505C24#1192962511yescharacteristic
506#41—1581no—
507#41—1581no—
508#41—1581no—
509#41—1581no—
510C2×Q8—791no—
511C2×D4—791no—
512C2×D4—1581no—
513C2×D4—1581no—
514C2×D4—1581no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—1581no—
520C22×C4—1581no—
521C22×C4—1581no—
522C22×C4—1581no—
523C22×C4—1581no—
524C22×C4C2×C4×D7911yescharacteristic
525C22×C4C2×C4×D7911yescharacteristic
526C22×C4D4×D7911yescharacteristic
527C22×C4D4×D7911yescharacteristic
528C22×C4#1192962511yescharacteristic
529C22×C4C23×D7911yescharacteristic
530M16—24no—
531M16—21no—
532M16—21no—
533M16—21no—
534M16—21no—
535M16—21no—
536M16—21no—
537M16—3162no—
538M16—1582no—
539M16—1582no—
540M16—1582no—
541M16—1582no—
542M16#1192962312yes—
543M16#1192962312yes—
544C2×C8—24no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—3162no—
554C2×C8—792no—
555C2×C8—792no—
556C2×C8—791no—
557C2×C8—791no—
558C2×C8—791no—
559C2×C8—791no—
560C2×C8—1581no—
561C2×C8—1581no—
562C2×C8C2×C4×D7911yescharacteristic
563C2×C8C2×C4×D7911yescharacteristic
564C2×C8C2×C4×D7911yescharacteristic
565C2×C8C2×C4×D7911yescharacteristic
566C4⋊C4—791no—
567C4⋊C4—791no—
568C4⋊C4—791no—
569C4⋊C4—791no—
570C4⋊C4—791no—
571C4⋊C4—791no—
572C4⋊C4—1581no—
573C4⋊C4—1581no—
574C4⋊C4—1581no—
575C4⋊C4—1581no—
576C22⋊C4—3162no—
577C22⋊C4—1581no—
578C22⋊C4—1581no—
579C22⋊C4—1581no—
580C22⋊C4—1581no—
581C22⋊C4—1581no—
582C22⋊C4—1581no—
583C22⋊C4—1581no—
584C22⋊C4—1581no—
585C22⋊C4—1581no—
586C42—1581no—
587C42—1581no—
588C42—1581no—
589C42—1581no—
590C42—1581no—
591C42—1581no—
592C42—1581no—
593C42—1581no—
594C42—1581no—
595C23—21no—
596C23—21no—
597C23—41no—
598C23—41no—
599C23—1581no—
600C23#1457912311yescharacteristic
601C23#1457912311yescharacteristic
602C23C2×#1192962511yescharacteristic
603Q8—1581no—
604Q8—1581no—
605D4—3161no—
606D4—3161no—
607D4—6321no—
608D4—6321no—
609D4—1581no—
610D4—1581no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—3161no—
622C2×C4—3161no—
623C2×C4—3161no—
624C2×C4—3161no—
625C2×C4—3161no—
626C2×C4—3161no—
627C2×C4—3161no—
628C2×C4—6321no—
629C2×C4—6321no—
630C2×C4—1582no—
631C2×C4—1581no—
632C2×C4—1581no—
633C2×C4—1581no—
634C2×C4—1581no—
635C2×C4—1581no—
636C2×C4—1581no—
637C2×C4—1581no—
638C2×C4#1457912311yescharacteristic
639C2×C4#1457912311yescharacteristic
640C2×C4C22×C4×D7911yescharacteristic
641C2×C4C2×D4×D7911yescharacteristic
642C2×C4#41×D7911yescharacteristic
643C2×C4#41×D7911yescharacteristic
644C2×C4C2×#1192962311yescharacteristic
645C2×C4C2×#1192962311yescharacteristic
646C8—21no—
647C8—21no—
648C8—21no—
649C8—21no—
650C8—22no—
651C8—22no—
652C8—41no—
653C8—41no—
654C8—792no—
655C8—792no—
656C8—1581no—
657C8—1581no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—81no—
667C22—81no—
668C22—3161no—
669C22—3161no—
670C22—3161no—
671C22C2×#1457912311yescharacteristic
672C22#1909909911yescharacteristic
673C22#1909909911yescharacteristic
674C4—21no—
675C4—21no—
676C4—3162no—
677C4—3161no—
678C4—3161no—
679C4—3161no—
680C4—3161no—
681C4—3161no—
682C4—3161no—
683C4—3161no—
684C4—6321no—
685C4C4×#1192962311yescharacteristic
686C4C4×#1192962311yescharacteristic
687C4#123×D7911yescharacteristic
688C4#1909906911yescharacteristic
689C2—21no—
690C2—21no—
691C2—41no—
692C2—41no—
693C2—6321no—
694C2#2376591911yescharacteristic
695C2#2377230911yescharacteristic
696C2#2377231611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1855C2
socleC22×C79C2×C4×D4
prime radical#19099074C22
cosocleC22×C4×C79C24
hypercenter#1855D79
hypocenterC79#21594
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1855

Complements

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

NH ≅ G/Ntype
3. #23771818C2semidirect
4. #23764047C2semidirect
5. #23764047C2semidirect
6. #23763998C2semidirect
7. #23763998C2semidirect
9. #23763497C2semidirect
10. #23763287C2semidirect
11. #23763282C2semidirect
12. #23763243C2semidirect
13. #23763236C2semidirect
14. #23763186C2semidirect
15. C79×#1855C2semidirect
20. #19098905C4semidirect
21. #19098905C4semidirect
22. #19098899C4semidirect
23. #19098894C4semidirect
24. #19098888C4semidirect
25. #19098888C4semidirect
26. #19098882C4semidirect
27. #19098877C4semidirect
357. #1855D79semidirect
364. C79#21594semidirect

Maximal subgroups

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

subgroupindex
#237733152
#237720992
#237718182
#237640472
#237639982
#237637022
#237634972
#237632872
#237632822
#237632432
#237632362
#237631862
C79×#18552
#2159479