Finite Group Databasebeta

← #34257986

Subgroups — #34257986

Order 20224.

660 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772515C211yesmaximal, characteristic, complemented
2#23772507C211yesmaximal, characteristic, complemented
3C4×#14579180C211yesmaximal, characteristic
4#23762273C211yesmaximal, characteristic, complemented
5#23762270C211yesmaximal, characteristic, complemented
6#23762265C211yesmaximal, characteristic, complemented
7#23762246C211yesmaximal, characteristic, complemented
8#23762241C211yesmaximal, characteristic, complemented
9#23762236C211yesmaximal, characteristic, complemented
10#23762233C211yesmaximal, characteristic, complemented
11#334×D79C211yesmaximal, characteristic, complemented
12#23761784C211yesmaximal, characteristic, complemented
13#23761784C211yesmaximal, characteristic, complemented
14#23761691C211yesmaximal, characteristic, complemented
15C79×#1586C211yesmaximal, characteristic, complemented
16C2×#14579180C411yescharacteristic
17C4×#11929623C2211yescharacteristic
18C4×#11929623C2211yescharacteristic
19#19098693C2211yescharacteristic
20#19098690C2211yescharacteristic
21C8×Dic158C2211yescharacteristic
22#19098687C2211yescharacteristic
23C4×Dic316C2211yescharacteristic
24#19098657C411yescharacteristic
25#19098654C411yescharacteristic
26#19098654C411yescharacteristic
27C4×#11929612C2211yescharacteristic
28C4×#11929612C2211yescharacteristic
29D8×Dic79C411yescharacteristic
30#19098641C2211yescharacteristic
31#19098637—21no—
32#19098632—21no—
33D4×C79⋊C8C2211yescharacteristic
34C8×C22⋊D79—21no—
35(C22⋊D79)⋊C8—21no—
36#19098617—21no—
37#19098614C2211yescharacteristic
38#19098614C2211yescharacteristic
39#19098613—21no—
40#19098600C411yescharacteristic
41#19098586—21no—
42C2×Dic79⋊C8—21no—
43C4⋊C8×D79C2211yescharacteristic
44C4⋊C8×D79C2211yescharacteristic
45#19098556C411yescharacteristic
46#19098556C411yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098314C2211yescharacteristic
49#19098313C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098293C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56D4⋊(C79⋊C8)C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098284C2211yescharacteristic
60C4×D8×C79C2211yescharacteristic
61#436×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64D4⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929623C2×C411yescharacteristic
68C2×#11929623C2×C411yescharacteristic
69#14579180—24no—
70C2×Dic316C2×C411yescharacteristic
71C2×#11929612C2×C411yescharacteristic
72C2×#11929612C2×C411yescharacteristic
73C2×#11929608—21no—
74C22×C79⋊C8—21no—
75C2×C8×D79—21no—
76C2×C8×D79—21no—
77C2×C8×D79C2×C411yescharacteristic
78C2×C8×D79C2×C411yescharacteristic
79C4×Dic158C2311yescharacteristic
80C4×Dic158C2311yescharacteristic
81#14579126C2×C411yescharacteristic
82#14579126C2×C411yescharacteristic
83D4×Dic79C2×C411yescharacteristic
84D4×Dic79C2×C411yescharacteristic
85C4×C22⋊D79—21no—
86C4×C22⋊D79—21no—
87(C22⋊D79)⋊C4—21no—
88(C22⋊D79)⋊C4—21no—
89#14579119C2×C411yescharacteristic
90#14579119C2×C411yescharacteristic
91#14579117—21no—
92#14579117—21no—
93C2×C4×Dic79—21no—
94C2×C4×Dic79—21no—
95C42×D79C2311yescharacteristic
96#14579095C2×C411yescharacteristic
97#14579092C2×C411yescharacteristic
98#14579092C2×C411yescharacteristic
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108Dic79⋊C8D411yescharacteristic
109Dic79⋊C8D411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579086C2×C411yescharacteristic
112#14579084C2×C411yescharacteristic
113#14579084C2×C411yescharacteristic
114#14579082C2×C411yescharacteristic
115#14579082C2×C411yescharacteristic
116C22⋊(C79⋊C8)—21no—
117C22⋊(C79⋊C8)—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8D411yescharacteristic
120(C2×D79)⋊C8D411yescharacteristic
121#14579071—21no—
122#14579071—21no—
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C8×Dic79—21no—
127C8×Dic79—21no—
128C8×Dic79C2×C411yescharacteristic
129C8×Dic79C2×C411yescharacteristic
130C2×D8×C79C2×C411yescharacteristic
131C2×M16×C79—21no—
132C22×C8×C79—21no—
133C4×D4×C79C2311yescharacteristic
134C4×D4×C79C2311yescharacteristic
135#107×C79C2×C411yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138#102×C79C2×C411yescharacteristic
139#102×C79C2×C411yescharacteristic
140C22⋊C8×C79—21no—
141C22⋊C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145#11929623—22no—
146#11929623—22no—
147#11929623—22no—
148#11929623M1614yescomplemented
149C2×Dic158C22×C411yescharacteristic
150C2×Dic158C22×C411yescharacteristic
151C2×C22⋊D79—21no—
152C2×C22⋊D79—21no—
153C22×Dic79—21no—
154C22×Dic79—21no—
155C22×Dic79—21no—
156C22×Dic79—21no—
157C2×C4×D79C22×C411yescharacteristic
158C2×C4×D79C2×D411yescharacteristic
159C2×C4×D79#4111yescharacteristic
160Dic316—22no—
161#11929612—22no—
162#11929612—22no—
163#11929608—81no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8—21no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8—81no—
168C2×C79⋊C8C22×C411yescharacteristic
169C2×C79⋊C8C22×C411yescharacteristic
170C2×C79⋊C8C2×D411yescharacteristic
171C2×C79⋊C8C2×D411yescharacteristic
172C8×D79—22no—
173C8×D79—22no—
174C8×D79—81no—
175C8×D79—81no—
176C4⋊Dic79C22×C411yescharacteristic
177C4⋊Dic79C22×C411yescharacteristic
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181Dic79⋊C4—21no—
182C22⋊Dic79—21no—
183C22⋊Dic79—21no—
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79—21no—
189C4×Dic79—21no—
190C4×Dic79—21no—
191C4×Dic79—21no—
192C4×Dic79C22×C411yescharacteristic
193C4×Dic79C2×D411yescharacteristic
194C4×Dic79#4111yescharacteristic
195C2×D4×C79C22×C411yescharacteristic
196C2×D4×C79C22×C411yescharacteristic
197C22×C4×C79—21no—
198C22×C4×C79—21no—
199D8×C79—22no—
200M16×C79—81no—
201C2×C8×C79—21no—
202C2×C8×C79—21no—
203C2×C8×C79—21no—
204C2×C8×C79—81no—
205C2×C8×C79C22×C411yescharacteristic
206C2×C8×C79C22×C411yescharacteristic
207C2×C8×C79C2×D411yescharacteristic
208C2×C8×C79C2×D411yescharacteristic
209C4⋊C4×C79C22×C411yescharacteristic
210C4⋊C4×C79C22×C411yescharacteristic
211C22⋊C4×C79—21no—
212C22⋊C4×C79—21no—
213C42×C79C2411yescharacteristic
214C23×C79—21no—
215C23×C79—21no—
216C22×D79C4×D411yescharacteristic
217D4×C79—21no—
218D4×C79—21no—
219D4×C79—21no—
220D4×C79C2×M1612yes—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79C23×C411yescharacteristic
228C2×C4×C79C22×D411yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C22⋊D79—24no—
231C22⋊D79—42no—
232C2×Dic79—21no—
233C2×Dic79—24no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—42no—
238C2×Dic79—81no—
239C2×Dic79—81no—
240C2×Dic79C4×D411yescharacteristic
241C2×Dic79C4×D411yescharacteristic
242C2×Dic79C4×D411yescharacteristic
243C4×D79—21no—
244C4×D79—21no—
245C4×D79—41no—
246C4×D79C2×M1612yes—
247C4×D79#13112yes—
248Dic158—21no—
249Dic158—21no—
250Dic158—21no—
251Dic158C2×M1612yes—
252C8×C79—21no—
253C8×C79—21no—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79—41no—
257C8×C79—41no—
258C8×C79—41no—
259C79⋊C8—21no—
260C79⋊C8—21no—
261C79⋊C8—21no—
262C79⋊C8—21no—
263C79⋊C8—41no—
264C79⋊C8—41no—
265C79⋊C8—41no—
266C22×C79—22no—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79—41no—
272C22×C79C2×C4×D411yescharacteristic
273C2×D79—21no—
274C2×D79—21no—
275C2×D79#43512yes—
276C4×C79—21no—
277C4×C79—41no—
278C4×C79—41no—
279C4×C79C22×M1611yescharacteristic
280C4×C79C2×#13111yescharacteristic
281C4×C79#57711yescharacteristic
282C4×C79#57711yescharacteristic
283Dic79—22no—
284Dic79—21no—
285Dic79—21no—
286Dic79—41no—
287Dic79—41no—
288Dic79—41no—
289Dic79#43512yes—
290#17882—791nomaximal, Sylow
291C2×C79—42no—
292C2×C79—81no—
293C2×C79C2×#43511yescharacteristic
294C2×C79#295211yescharacteristic
295C2×C79#295711yescharacteristic
296D79—22no—
297#2989—791no—
298#2988—791no—
299C4×#135—791no—
300#1587—791no—
301#1587—791no—
302#1586—791no—
303#1586D7911yescharacteristic, complemented
304#1583—791no—
305#1583—791no—
306#1582—791no—
307#1582—791no—
308C2×#334—791no—
309#1477—791no—
310#1477—791no—
311#1452—791no—
312C79#1788211yescharacteristic, complemented, Sylow
313C2×#135—791no—
314C4×#41—791no—
315C4×#41—791no—
316#446—791no—
317#446—791no—
318C8×Q8—791no—
319C8×Q8—791no—
320C4×Q16—791no—
321C4×Q16—791no—
322C4×SD16—791no—
323C4×SD16—791no—
324C4×SD16—791no—
325C4×SD16—791no—
326C4×D8—791no—
327C4×D8C2×D7911yescharacteristic
328#436—791no—
329#436C2×D7911yescharacteristic
330#435—1581no—
331#435—1581no—
332C8×D4—791no—
333C8×D4—1581no—
334C8×D4—1581no—
335C8×D4C2×D7911yescharacteristic
336#432—1581no—
337#431—791no—
338#431—791no—
339#431—1581no—
340#427—791no—
341#423—1581no—
342C2×C4⋊C8—791no—
343C2×C4⋊C8—791no—
344C2×C4⋊C8—1581no—
345#416—791no—
346#416—791no—
347C2×C4×C8—791no—
348#334—791no—
349#334—791no—
350#334—791no—
351#334C2×D7911yescharacteristic
352Q8⋊C8—791no—
353Q8⋊C8—791no—
354Q8⋊C8—791no—
355Q8⋊C8—791no—
356D4⋊C8—791no—
357D4⋊C8—791no—
358D4⋊C8C2×D7911yescharacteristic
359D4⋊C8C2×D7911yescharacteristic
360#322—791no—
361#322—791no—
362#322—791no—
363#322C2×D7911yescharacteristic
364C2×#41—791no—
365C2×#41—791no—
366#135—1584no—
367C2×Q16—791no—
368C2×SD16—791no—
369C2×SD16—791no—
370C2×D8C4×D7911yescharacteristic
371C2×M16—21no—
372C2×M16—1581no—
373C22×C8—21no—
374C22×C8—791no—
375C22×C8—791no—
376C22×C8—1581no—
377C22×C8—1581no—
378C22×C8—1581no—
379C4×Q8—791no—
380C4×Q8—791no—
381C4×Q8—791no—
382C4×Q8—791no—
383C4×D4—791no—
384C4×D4—791no—
385C4×D4—1581no—
386C4×D4—1581no—
387C4×D4—1581no—
388C4×D4—1581no—
389C4×D4C22×D7911yescharacteristic
390C4×D4C22×D7911yescharacteristic
391#117—791no—
392#117—791no—
393#117—1581no—
394#117—1581no—
395C2×C42—791no—
396C2×C42—1581no—
397C2×C42—1581no—
398#107—791no—
399#107C4×D7911yescharacteristic
400#106—791no—
401#106—791no—
402C4⋊C8—791no—
403C4⋊C8—791no—
404C4⋊C8—791no—
405C4⋊C8—791no—
406C4⋊C8—1581no—
407C4⋊C8—1581no—
408C4⋊C8—1581no—
409C4⋊C8—1581no—
410C4⋊C8—1581no—
411C4⋊C8—1581no—
412C4⋊C8—1581no—
413C4⋊C8C22×D7911yescharacteristic
414C4⋊C8C22×D7911yescharacteristic
415#103—791no—
416#103—791no—
417#103—791no—
418#103—791no—
419#102—791no—
420#102—791no—
421#102C4×D7911yescharacteristic
422#102C4×D7911yescharacteristic
423C22⋊C8—21no—
424C22⋊C8—21no—
425C22⋊C8—791no—
426C22⋊C8—791no—
427C22⋊C8—1581no—
428C22⋊C8—1581no—
429C22⋊C8—1581no—
430#97—1581no—
431#97—1581no—
432C4×C8—791no—
433C4×C8—791no—
434C4×C8—791no—
435C4×C8—791no—
436C4×C8—791no—
437C4×C8—1581no—
438C4×C8—1581no—
439C4×C8C22×D7911yescharacteristic
440C4×C8C22×D7911yescharacteristic
441C4×C8C22×D7911yescharacteristic
442#41—794no—
443#41—1582no—
444#41—1582no—
445#41—1582no—
446C2×Q8—791no—
447C2×Q8—791no—
448C2×D4—1581no—
449C2×D4—1581no—
450C2×D4C2×C4×D7911yescharacteristic
451C2×D4C2×C4×D7911yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—791no—
455C22×C4—791no—
456C22×C4—791no—
457C22×C4—1581no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4—1581no—
461Q16—1582no—
462SD16—1582no—
463SD16—1582no—
464D8—22no—
465M16—81no—
466M16—6321no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—81no—
471C2×C8—791no—
472C2×C8—791no—
473C2×C8—791no—
474C2×C8—791no—
475C2×C8—6321no—
476C2×C8—6321no—
477C2×C8—6321no—
478C2×C8—1582no—
479C2×C8—1582no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8C2×C4×D7911yescharacteristic
484C2×C8C2×C4×D7911yescharacteristic
485C2×C8D4×D7911yescharacteristic
486C2×C8D4×D7911yescharacteristic
487C4⋊C4—791no—
488C4⋊C4—791no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4C2×C4×D7911yescharacteristic
494C4⋊C4C2×C4×D7911yescharacteristic
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—1581no—
498C22⋊C4—1581no—
499C22⋊C4—1581no—
500C22⋊C4—1581no—
501C42—791no—
502C42—791no—
503C42—791no—
504C42—1581no—
505C42—1581no—
506C42—1581no—
507C42—1581no—
508C42—1581no—
509C42—1581no—
510C42C23×D7911yescharacteristic
511C23—21no—
512C23—21no—
513C23—791no—
514Q8—792no—
515Q8—1581no—
516Q8—1581no—
517Q8—1581no—
518D4—21no—
519D4—21no—
520D4—21no—
521D4—3162no—
522D4—1584no—
523D4M16×D7912yes—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—3161no—
531C2×C4—3162no—
532C2×C4—791no—
533C2×C4—792no—
534C2×C4—792no—
535C2×C4—791no—
536C2×C4—791no—
537C2×C4—6321no—
538C2×C4—6321no—
539C2×C4—1581no—
540C2×C4—1581no—
541C2×C4—1581no—
542C2×C4—1584no—
543C2×C4—1581no—
544C2×C4—1581no—
545C2×C4—1581no—
546C2×C4C22×C4×D7911yescharacteristic
547C2×C4C2×D4×D7911yescharacteristic
548C2×C4#41×D7911yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—21no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—3161no—
557C8—3161no—
558C8—3161no—
559C8—1581no—
560C8—1581no—
561C8—1581no—
562C8—1581no—
563C22—22no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—792no—
570C22—1581no—
571C22—1581no—
572C22C4×D4×D7911yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—3161no—
577C4—3161no—
578C4—3161no—
579C4—792no—
580C4—1581no—
581C4—1581no—
582C4—1582no—
583C4C2×M16×D7911yescharacteristic
584C4#131×D7911yescharacteristic
585C4#1909937311yescharacteristic
586C4#1909937311yescharacteristic
587C2—42no—
588C2—81no—
589C2—1582no—
590C2#435×D7911yescharacteristic
591C2#2377228211yescharacteristic
592C2#2377231211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1586C2
socleC22×C79C2×C4×D4
prime radicalC2×#14579180C4
cosocleC42×C79C24
hypercenter#1586D79
hypocenterC79#17882
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
1. #23772515C2semidirect
2. #23772507C2semidirect
4. #23762273C2semidirect
5. #23762270C2semidirect
6. #23762265C2semidirect
7. #23762246C2semidirect
8. #23762241C2semidirect
9. #23762236C2semidirect
10. #23762233C2semidirect
11. #334×D79C2semidirect
12. #23761784C2semidirect
13. #23761784C2semidirect
14. #23761691C2semidirect
15. C79×#1586C2semidirect
148. #11929623M16semidirect
303. #1586D79semidirect
312. C79#17882semidirect

Maximal subgroups

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

subgroupindex
#237725152
#237725072
C4×#145791802
#237622732
#237622702
#237622652
#237622462
#237622412
#237622362
#237622332
#334×D792
#237617842
#237616912
C79×#15862
#1788279