Finite Group Databasebeta

← #34258536

Subgroups — #34258536

Order 20224.

650 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773429C211yesmaximal, characteristic
2#446×D79C211yesmaximal, characteristic, complemented
3#23772109C211yesmaximal, characteristic
4#23762304C211yesmaximal, characteristic, complemented
5#23762302C211yesmaximal, characteristic, complemented
6#23762300C211yesmaximal, characteristic, complemented
7#23762260C211yesmaximal, characteristic, complemented
8#23762258C211yesmaximal, characteristic, complemented
9#23762256C211yesmaximal, characteristic, complemented
10#23762254C211yesmaximal, characteristic, complemented
11#23762173C211yesmaximal, characteristic, complemented
12#23761827C211yesmaximal, characteristic, complemented
13#23761797C211yesmaximal, characteristic
14#23761691C211yesmaximal, characteristic, complemented
15C79×#1592C211yesmaximal, characteristic, complemented
16C2×#14579169—21no—
17#19099110C2211yescharacteristic
18C4×Q8×D79C2211yescharacteristic
19#19098883C411yescharacteristic
20#19098880C411yescharacteristic
21#19098847C411yescharacteristic
22#19098846C411yescharacteristic
23#19098762C411yescharacteristic
24#19098761C411yescharacteristic
25#19098703C411yescharacteristic
26#19098700C411yescharacteristic
27#19098694C2211yescharacteristic
28#19098693C2211yescharacteristic
29#19098692—21no—
30#19098691—21no—
31#19098690C2211yescharacteristic
32C4×#11929613C2211yescharacteristic
33C4×#11929611C2211yescharacteristic
34C4×#11929612C2211yescharacteristic
35#19098642C2211yescharacteristic
36#19098639—21no—
37D4×C79⋊C8C2211yescharacteristic
38#19098628C2211yescharacteristic
39(C22⋊D79)⋊C8—21no—
40#19098614C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098585C2211yescharacteristic
43C4⋊C8×D79—21no—
44C4⋊C8×D79C2211yescharacteristic
45#19098528—21no—
46#97×D79—21no—
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098315C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098284C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62#436×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929624—21no—
68C2×Q8×D79C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929611C2×C411yescharacteristic
72#14579169—44no—
73C2×#11929607—21no—
74C2×#11929607—21no—
75C22×C79⋊C8—21no—
76C2×C8×D79—21no—
77C2×C8×D79—21no—
78C2×C8×D79D411yescharacteristic
79C2×C8×D79D411yescharacteristic
80#14579160C2×C411yescharacteristic
81#14579157C2×C411yescharacteristic
82#14579147—21no—
83#14579136C2×C411yescharacteristic
84#14579134—21no—
85#14579132C2×C411yescharacteristic
86Q8×Dic79C2×C411yescharacteristic
87C4×Dic158—21no—
88C4×Dic158C2311yescharacteristic
89#14579126—21no—
90#14579126C2×C411yescharacteristic
91C4×D316C2311yescharacteristic
92C4×C22⋊D79—21no—
93C4⋊C4×D79—21no—
94C4⋊C4×D79C2×C411yescharacteristic
95C42×D79—21no—
96C42×D79C2311yescharacteristic
97#14579093C2×C411yescharacteristic
98#14579092C2×C411yescharacteristic
99#14579092C2×C411yescharacteristic
100C4⋊(C79⋊C8)—21no—
101C4⋊(C79⋊C8)C2311yescharacteristic
102C4⋊(C79⋊C8)C2311yescharacteristic
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108Dic79⋊C8C2×C411yescharacteristic
109Dic79⋊C8C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579085C2×C411yescharacteristic
112#14579084C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114#14579082C2×C411yescharacteristic
115#14579081C2×C411yescharacteristic
116C22⋊(C79⋊C8)—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8C2×C411yescharacteristic
121(C2×D79)⋊C8C2×C411yescharacteristic
122#14579072—21no—
123#14579071—21no—
124#14579071—21no—
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
128C8×Dic79D411yescharacteristic
129C8×Dic79D411yescharacteristic
130C2×SD16×C79C2×C411yescharacteristic
131C2×M16×C79—21no—
132C4×Q8×C79C2311yescharacteristic
133C4×D4×C79C2311yescharacteristic
134#106×C79C2×C411yescharacteristic
135C4⋊C8×C79—21no—
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138#103×C79C2×C411yescharacteristic
139#102×C79C2×C411yescharacteristic
140C22⋊C8×C79—21no—
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145#11929624—42no—
146#11929624—42no—
147C2×Dic158—21no—
148C2×Dic158—21no—
149C2×Dic158C22×C411yescharacteristic
150Q8×D79—41no—
151Q8×D79—41no—
152C2×D316—21no—
153C2×D316C22×C411yescharacteristic
154C2×C22⋊D79—21no—
155C2×C22⋊D79—21no—
156C2×C22⋊D79—21no—
157C2×C4×D79—21no—
158C2×C4×D79—21no—
159C2×C4×D79—21no—
160C2×C4×D79C22×C411yescharacteristic
161C2×C4×D79C2×D411yescharacteristic
162C2×C4×D79#4111yescharacteristic
163#11929613—41no—
164#11929612—41no—
165#11929611—41no—
166#11929607—42no—
167#11929607—42no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8—21no—
171C2×C79⋊C8—42no—
172C2×C79⋊C8C22×C411yescharacteristic
173C2×C79⋊C8C22×C411yescharacteristic
174C2×C79⋊C8C2×D411yescharacteristic
175C2×C79⋊C8C2×D411yescharacteristic
176C8×D79—22no—
177C8×D79—22no—
178C8×D79—42no—
179C8×D79—42no—
180C4⋊Dic79—21no—
181C4⋊Dic79C22×C411yescharacteristic
182C4⋊Dic79C22×C411yescharacteristic
183Dic79⋊C4—21no—
184Dic79⋊C4—21no—
185Dic79⋊C4—21no—
186Dic79⋊C4—21no—
187C22⋊Dic79—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190C4×Dic79—21no—
191C4×Dic79—21no—
192C4×Dic79—21no—
193C4×Dic79C22×C411yescharacteristic
194C4×Dic79C2×D411yescharacteristic
195C4×Dic79#4111yescharacteristic
196C2×Q8×C79C22×C411yescharacteristic
197C2×D4×C79C22×C411yescharacteristic
198C22×C4×C79—21no—
199SD16×C79—41no—
200M16×C79—42no—
201C2×C8×C79—21no—
202C2×C8×C79—21no—
203C2×C8×C79—21no—
204C2×C8×C79C22×C411yescharacteristic
205C2×C8×C79C22×C411yescharacteristic
206C2×C8×C79C2×D411yescharacteristic
207C2×C8×C79C2×D411yescharacteristic
208C4⋊C4×C79—21no—
209C4⋊C4×C79C22×C411yescharacteristic
210C4⋊C4×C79C22×C411yescharacteristic
211C22⋊C4×C79—21no—
212C42×C79—21no—
213C42×C79C2411yescharacteristic
214C23×C79—21no—
215C22×D79—21no—
216C22×D79C4×D411yescharacteristic
217Q8×C79—21no—
218Q8×C79—21no—
219D4×C79—21no—
220D4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79C23×C411yescharacteristic
227C2×C4×C79C22×D411yescharacteristic
228C2×C4×C79C2×#4111yescharacteristic
229C22⋊D79—42no—
230C22⋊D79—42no—
231C22⋊D79—42no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79C4×D411yescharacteristic
236C2×Dic79C4×D411yescharacteristic
237C2×Dic79C4×D411yescharacteristic
238D316—21no—
239D316—21no—
240D316—42no—
241C4×D79—21no—
242C4×D79—21no—
243C4×D79—41no—
244C4×D79—42no—
245C4×D79—42no—
246C4×D79—41no—
247C4×D79—41no—
248C4×D79AGL(1,ℤ/8)12yes—
249C4×D79#13712yes—
250Dic158—21no—
251Dic158—21no—
252Dic158—42no—
253Dic158—42no—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79—21no—
257C8×C79—21no—
258C8×C79—41no—
259C8×C79—41no—
260C8×C79—41no—
261C79⋊C8—21no—
262C79⋊C8—21no—
263C79⋊C8—21no—
264C79⋊C8—21no—
265C79⋊C8—41no—
266C79⋊C8—41no—
267C79⋊C8—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79C2×C4×D411yescharacteristic
272C2×D79—21no—
273C2×D79—21no—
274C2×D79—41no—
275C2×D79—41no—
276C2×D79—41no—
277C2×D79#44012yes—
278C4×C79—21no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79—41no—
282C4×C79#56811yescharacteristic
283C4×C79#56811yescharacteristic
284C4×C79C2×AGL(1,ℤ/8)11yescharacteristic
285C4×C79C2×#13711yescharacteristic
286Dic79—21no—
287Dic79—21no—
288Dic79—41no—
289Dic79—41no—
290Dic79—41no—
291Dic79#44012yes—
292#17936—791nomaximal, Sylow
293C2×C79—81no—
294C2×C79#293011yescharacteristic
295C2×C79C2×#44011yescharacteristic
296C2×C79#295511yescharacteristic
297D79—22no—
298D79—81no—
299#3103—791no—
300C2×#446—791no—
301#2931—791no—
302#1592—791no—
303#1592—791no—
304#1592—791no—
305#1592D7911yescharacteristic, complemented
306#1585—791no—
307#1585—791no—
308#1585—791no—
309#1585—791no—
310#1571—791no—
311#1487—791no—
312#1480—791no—
313#1452—791no—
314C79#1793611yescharacteristic, complemented, Sylow
315C2×#131—1581no—
316#532—791no—
317C2×C4×Q8—791no—
318#481—791no—
319#481—791no—
320#475—791no—
321#475—791no—
322#460—791no—
323#460—791no—
324#448—791no—
325#448—791no—
326#446—791no—
327#446—791no—
328#446—791no—
329#446—1581no—
330#446—1581no—
331#446C2×D7911yescharacteristic
332C4×SD16—791no—
333C4×SD16—791no—
334C4×SD16—791no—
335C4×SD16C2×D7911yescharacteristic
336#436—791no—
337#436—1581no—
338#436C2×D7911yescharacteristic
339C8×D4—791no—
340C8×D4—791no—
341C8×D4—1581no—
342#431—791no—
343#431—791no—
344#423—791no—
345C2×C4⋊C8—791no—
346C2×C4⋊C8—1581no—
347#408—1581no—
348C2×#97—1581no—
349C2×C4×C8—791no—
350#335—791no—
351#335—791no—
352#335—791no—
353#335C2×D7911yescharacteristic
354Q8⋊C8—791no—
355Q8⋊C8—791no—
356Q8⋊C8—791no—
357Q8⋊C8C2×D7911yescharacteristic
358D4⋊C8—791no—
359D4⋊C8—791no—
360D4⋊C8—791no—
361D4⋊C8C2×D7911yescharacteristic
362#322—791no—
363#322—791no—
364#322—791no—
365#322C2×D7911yescharacteristic
366C2×#41—1581no—
367C22×Q8—791no—
368C2×SD16—791no—
369C2×SD16—791no—
370C2×SD16—791no—
371C2×SD16C4×D7911yescharacteristic
372#131—3164no—
373C2×M16—21no—
374C2×M16—1581no—
375C2×M16—1581no—
376C22×C8—791no—
377C22×C8—791no—
378C22×C8—1581no—
379C22×C8—1581no—
380C22×C8—1581no—
381C4⋊Q8—791no—
382#127—791no—
383#124—1581no—
384#121—791no—
385#121—791no—
386#121—1581no—
387C4×Q8—791no—
388C4×Q8—791no—
389C4×Q8—791no—
390C4×Q8—1581no—
391C4×Q8—1581no—
392C4×Q8C22×D7911yescharacteristic
393C4×D4—791no—
394C4×D4—1581no—
395C4×D4C22×D7911yescharacteristic
396C2×C4⋊C4—791no—
397C2×C4⋊C4—1581no—
398C2×C42—791no—
399C2×C42—1581no—
400#106—791no—
401#106—791no—
402#106—791no—
403#106C4×D7911yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—791no—
406C4⋊C8—791no—
407C4⋊C8—791no—
408C4⋊C8—791no—
409C4⋊C8—1581no—
410C4⋊C8—1581no—
411C4⋊C8—1581no—
412C4⋊C8—1581no—
413C4⋊C8—1581no—
414C4⋊C8—1581no—
415C4⋊C8C22×D7911yescharacteristic
416C4⋊C8C22×D7911yescharacteristic
417#103—791no—
418#103—791no—
419#103—791no—
420#103C4×D7911yescharacteristic
421#102—791no—
422#102—791no—
423#102—791no—
424#102C4×D7911yescharacteristic
425C22⋊C8—21no—
426C22⋊C8—791no—
427C22⋊C8—791no—
428C22⋊C8—1581no—
429C22⋊C8—1581no—
430C22⋊C8—1581no—
431C22⋊C8—1581no—
432#97—21no—
433#97—1581no—
434#97—1581no—
435#97—1581no—
436C4×C8—791no—
437C4×C8—791no—
438C4×C8—791no—
439C4×C8—791no—
440C4×C8—791no—
441C4×C8C22×D7911yescharacteristic
442C4×C8C22×D7911yescharacteristic
443C4×C8C22×D7911yescharacteristic
444#41—3162no—
445#41—3162no—
446C2×Q8—3161no—
447C2×Q8—3161no—
448C2×Q8—791no—
449C2×Q8—1581no—
450C2×Q8—1581no—
451C2×Q8C2×C4×D7911yescharacteristic
452C2×D4—791no—
453C2×D4—1581no—
454C2×D4—1581no—
455C2×D4—1581no—
456C2×D4—1581no—
457C2×D4C2×C4×D7911yescharacteristic
458C22×C4—21no—
459C22×C4—791no—
460C22×C4—791no—
461C22×C4—791no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1581no—
465SD16—41no—
466SD16—3161no—
467SD16—3161no—
468SD16—3161no—
469M16—42no—
470M16—3162no—
471M16—3162no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—3162no—
476C2×C8—3162no—
477C2×C8—3162no—
478C2×C8—791no—
479C2×C8—791no—
480C2×C8—791no—
481C2×C8—791no—
482C2×C8—1582no—
483C2×C8—1582no—
484C2×C8—1581no—
485C2×C8—1581no—
486C2×C8—1581no—
487C2×C8C2×C4×D7911yescharacteristic
488C2×C8C2×C4×D7911yescharacteristic
489C2×C8D4×D7911yescharacteristic
490C2×C8D4×D7911yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—791no—
493C4⋊C4—791no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4C2×C4×D7911yescharacteristic
500C4⋊C4C2×C4×D7911yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—1581no—
503C22⋊C4—1581no—
504C22⋊C4—1581no—
505C42—21no—
506C42—791no—
507C42—791no—
508C42—791no—
509C42—1581no—
510C42—1581no—
511C42—1581no—
512C42C23×D7911yescharacteristic
513C23—21no—
514C23—791no—
515C23—1581no—
516Q8—21no—
517Q8—21no—
518Q8—3162no—
519Q8—3162no—
520Q8—1581no—
521Q8—1581no—
522D4—21no—
523D4—21no—
524D4—3162no—
525D4—3162no—
526D4—3162no—
527D4—3162no—
528D4—1581no—
529D4—1581no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—3161no—
536C2×C4—3162no—
537C2×C4—3162no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—791no—
541C2×C4—792no—
542C2×C4—792no—
543C2×C4—791no—
544C2×C4—791no—
545C2×C4—1581no—
546C2×C4—1581no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4C22×C4×D7911yescharacteristic
551C2×C4C2×D4×D7911yescharacteristic
552C2×C4#41×D7911yescharacteristic
553C8—21no—
554C8—21no—
555C8—21no—
556C8—21no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—3161no—
561C8—3161no—
562C8—3161no—
563C8—1581no—
564C8—1581no—
565C8—1581no—
566C8—1581no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—3161no—
571C22—3161no—
572C22—3161no—
573C22—792no—
574C22—1581no—
575C22—1581no—
576C22C4×D4×D7911yescharacteristic
577C4—21no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—3161no—
582C4—3161no—
583C4—3161no—
584C4—792no—
585C4—1581no—
586C4—1581no—
587C4#1909932811yescharacteristic
588C4#1909932911yescharacteristic
589C4AGL(1,ℤ/8)×D7911yescharacteristic
590C4#137×D7911yescharacteristic
591C2—81no—
592C2—6321no—
593C2—1582no—
594C2#2377210011yescharacteristic
595C2#440×D7911yescharacteristic
596C2#2377230011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1592C2
socleC22×C79C2×C4×D4
prime radical#19099110C22
cosocleC42×C79C24
hypercenter#1592D79
hypocenterC79#17936
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1592

Complements

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

NH ≅ G/Ntype
2. #446×D79C2semidirect
4. #23762304C2semidirect
5. #23762302C2semidirect
6. #23762300C2semidirect
7. #23762260C2semidirect
8. #23762258C2semidirect
9. #23762256C2semidirect
10. #23762254C2semidirect
11. #23762173C2semidirect
12. #23761827C2semidirect
14. #23761691C2semidirect
15. C79×#1592C2semidirect
305. #1592D79semidirect
314. C79#17936semidirect

Maximal subgroups

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

subgroupindex
#237734292
#446×D792
#237721092
#237623042
#237623022
#237623002
#237622602
#237622582
#237622562
#237622542
#237621732
#237618272
#237617972
#237616912
C79×#15922
#1793679