Finite Group Databasebeta

← #34257977

Subgroups — #34257977

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#23772516C211yesmaximal, characteristic, complemented
2#23772510C211yesmaximal, characteristic, complemented
3C4×#14579184C211yesmaximal, characteristic
4#23762278C211yesmaximal, characteristic, complemented
5#23762275C211yesmaximal, characteristic, complemented
6#23762260C211yesmaximal, characteristic, complemented
7#23762255C211yesmaximal, characteristic, complemented
8#23762251C211yesmaximal, characteristic, complemented
9#23762231C211yesmaximal, characteristic, complemented
10#23762228C211yesmaximal, characteristic, complemented
11#335×D79C211yesmaximal, characteristic, complemented
12#23761786C211yesmaximal, characteristic, complemented
13#23761786C211yesmaximal, characteristic, complemented
14#23761691C211yesmaximal, characteristic, complemented
15C79×#1584C211yesmaximal, characteristic, complemented
16C2×#14579184C411yescharacteristic
17C4×#11929625C2211yescharacteristic
18C4×#11929625C2211yescharacteristic
19#19098694C2211yescharacteristic
20Q8×C79⋊C8C2211yescharacteristic
21Q16×Dic79C411yescharacteristic
22C4×#11929611C2211yescharacteristic
23C4×#11929611C2211yescharacteristic
24#19098648C411yescharacteristic
25#19098648C411yescharacteristic
26C4×D632C2211yescharacteristic
27#19098645C411yescharacteristic
28#19098642C2211yescharacteristic
29#19098640C2211yescharacteristic
30#19098636—21no—
31#19098633—21no—
32C8×D316—21no—
33C8×D316C2211yescharacteristic
34#19098628—21no—
35#19098628C2211yescharacteristic
36#19098618—21no—
37#19098614—21no—
38#19098614C2211yescharacteristic
39#19098614C2211yescharacteristic
40#19098600C411yescharacteristic
41#19098585—21no—
42C4⋊C8×D79—21no—
43C4⋊C8×D79C2211yescharacteristic
44C4⋊C8×D79C2211yescharacteristic
45#19098558C411yescharacteristic
46#19098558C411yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098315C2211yescharacteristic
51Q8⋊(C79⋊C8)C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098292C2211yescharacteristic
54#19098292C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098284C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929625C2×C411yescharacteristic
68C2×#11929625C2×C411yescharacteristic
69#14579184—24no—
70C2×#11929611C2×C411yescharacteristic
71C2×#11929611C2×C411yescharacteristic
72C2×D632C2×C411yescharacteristic
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×C8×D79—21no—
76C2×C8×D79—21no—
77C2×C8×D79—21no—
78C2×C8×D79—21no—
79C2×C8×D79C2×C411yescharacteristic
80C2×C8×D79C2×C411yescharacteristic
81Q8×Dic79C2×C411yescharacteristic
82Q8×Dic79C2×C411yescharacteristic
83C4×D316—21no—
84C4×D316—21no—
85C4×D316C2311yescharacteristic
86C4×D316C2311yescharacteristic
87#14579123—21no—
88#14579123—21no—
89#14579123C2×C411yescharacteristic
90#14579123C2×C411yescharacteristic
91#14579119—21no—
92#14579119—21no—
93#14579119C2×C411yescharacteristic
94#14579119C2×C411yescharacteristic
95C42×D79—21no—
96C42×D79—21no—
97C42×D79C2311yescharacteristic
98#14579095C2×C411yescharacteristic
99#14579092C2×C411yescharacteristic
100#14579092C2×C411yescharacteristic
101C4⋊(C79⋊C8)—21no—
102C4⋊(C79⋊C8)—21no—
103C4⋊(C79⋊C8)C2311yescharacteristic
104C4⋊(C79⋊C8)C2311yescharacteristic
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108Dic79⋊C8D411yescharacteristic
109Dic79⋊C8D411yescharacteristic
110#14579085C2×C411yescharacteristic
111#14579085C2×C411yescharacteristic
112#14579083C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114#14579081C2×C411yescharacteristic
115#14579081C2×C411yescharacteristic
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8—21no—
121(C2×D79)⋊C8D411yescharacteristic
122(C2×D79)⋊C8D411yescharacteristic
123#14579072—21no—
124C4×C79⋊C8—21no—
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
128C8×Dic79C2×C411yescharacteristic
129C8×Dic79C2×C411yescharacteristic
130C2×Q16×C79C2×C411yescharacteristic
131C4×Q8×C79C2311yescharacteristic
132C4×Q8×C79C2311yescharacteristic
133#107×C79C2×C411yescharacteristic
134C4⋊C8×C79—21no—
135C4⋊C8×C79—21no—
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138#103×C79C2×C411yescharacteristic
139#103×C79C2×C411yescharacteristic
140#97×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145#11929625—22no—
146#11929625—22no—
147#11929625—22no—
148#11929625M1614yescomplemented
149C2×D316—21no—
150C2×D316—21no—
151C2×D316C22×C411yescharacteristic
152C2×D316C22×C411yescharacteristic
153C2×C4×D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—21no—
156C2×C4×D79—21no—
157C2×C4×D79—21no—
158C2×C4×D79—21no—
159C2×C4×D79C22×C411yescharacteristic
160C2×C4×D79C2×D411yescharacteristic
161C2×C4×D79#4111yescharacteristic
162#11929611—22no—
163#11929611—22no—
164D632—22no—
165#11929607—81no—
166#11929607—81no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8C22×C411yescharacteristic
171C2×C79⋊C8C22×C411yescharacteristic
172C2×C79⋊C8C2×D411yescharacteristic
173C2×C79⋊C8C2×D411yescharacteristic
174C8×D79—22no—
175C8×D79—22no—
176C8×D79—81no—
177C8×D79—81no—
178C8×D79—81no—
179C8×D79—81no—
180C4⋊Dic79—21no—
181C4⋊Dic79—21no—
182C4⋊Dic79C22×C411yescharacteristic
183C4⋊Dic79C22×C411yescharacteristic
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186(C2×D79)⋊C4—21no—
187(C2×D79)⋊C4—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190C4×Dic79—21no—
191C4×Dic79—21no—
192C4×Dic79—21no—
193C4×Dic79—21no—
194C4×Dic79C22×C411yescharacteristic
195C4×Dic79C2×D411yescharacteristic
196C4×Dic79#4111yescharacteristic
197C2×Q8×C79C22×C411yescharacteristic
198C2×Q8×C79C22×C411yescharacteristic
199Q16×C79—22no—
200C2×C8×C79—21no—
201C2×C8×C79—21no—
202C2×C8×C79—21no—
203C2×C8×C79C22×C411yescharacteristic
204C2×C8×C79C22×C411yescharacteristic
205C2×C8×C79C2×D411yescharacteristic
206C2×C8×C79C2×D411yescharacteristic
207C4⋊C4×C79—21no—
208C4⋊C4×C79—21no—
209C4⋊C4×C79C22×C411yescharacteristic
210C4⋊C4×C79C22×C411yescharacteristic
211C42×C79—21no—
212C42×C79—21no—
213C42×C79C2411yescharacteristic
214C22×D79—21no—
215C22×D79—21no—
216C22×D79C4×D411yescharacteristic
217Q8×C79—21no—
218Q8×C79—21no—
219Q8×C79—21no—
220Q8×C79C2×M1612yes—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79C23×C411yescharacteristic
226C2×C4×C79C22×D411yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79C4×D411yescharacteristic
231C2×Dic79C4×D411yescharacteristic
232C2×Dic79C4×D411yescharacteristic
233D316—24no—
234D316—21no—
235D316—21no—
236D316—21no—
237D316—42no—
238D316C2×M1612yes—
239C4×D79—21no—
240C4×D79—21no—
241C4×D79—24no—
242C4×D79—41no—
243C4×D79—41no—
244C4×D79—41no—
245C4×D79—41no—
246C4×D79—41no—
247C4×D79—42no—
248C4×D79—81no—
249C4×D79—81no—
250C4×D79C2×M1612yes—
251C4×D79#13112yes—
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×C79C2×C4×D411yescharacteristic
267C2×D79—21no—
268C2×D79—21no—
269C2×D79—22no—
270C2×D79—41no—
271C2×D79—41no—
272C2×D79—41no—
273C2×D79—41no—
274C2×D79—41no—
275C2×D79#43512yes—
276C4×C79—22no—
277C4×C79—21no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79C22×M1611yescharacteristic
282C4×C79C2×#13111yescharacteristic
283C4×C79#57611yescharacteristic
284C4×C79#57811yescharacteristic
285Dic79—21no—
286Dic79—21no—
287Dic79—41no—
288Dic79—41no—
289Dic79#43512yes—
290#17881—791nomaximal, Sylow
291C2×C79C2×#43511yescharacteristic
292C2×C79#295111yescharacteristic
293C2×C79#295711yescharacteristic
294D79—22no—
295D79—42no—
296D79—81no—
297#2989—791no—
298#2988—791no—
299C4×#135—791no—
300#1588—791no—
301#1588—791no—
302#1585—791no—
303#1585—791no—
304#1584—791no—
305#1584D7911yescharacteristic, complemented
306#1581—791no—
307#1581—791no—
308C2×#335—791no—
309#1477—791no—
310#1477—791no—
311#1452—791no—
312C79#1788111yescharacteristic, complemented, Sylow
313C2×#135—791no—
314C4×#41—791no—
315C4×#41—791no—
316#446—791no—
317#446C2×D7911yescharacteristic
318C8×Q8—791no—
319C8×Q8C2×D7911yescharacteristic
320C4×Q16—791no—
321C4×Q16C2×D7911yescharacteristic
322C4×SD16—791no—
323C4×SD16—791no—
324C4×SD16—791no—
325C4×SD16—791no—
326C4×D8—791no—
327C4×D8—791no—
328#436—791no—
329#436—791no—
330#435—1581no—
331#435—1581no—
332C8×D4—791no—
333C8×D4—791no—
334C8×D4—1581no—
335C8×D4—1581no—
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#335—791no—
349#335—791no—
350#335—791no—
351#335C2×D7911yescharacteristic
352Q8⋊C8—791no—
353Q8⋊C8—791no—
354Q8⋊C8C2×D7911yescharacteristic
355Q8⋊C8C2×D7911yescharacteristic
356D4⋊C8—791no—
357D4⋊C8—791no—
358D4⋊C8—791no—
359D4⋊C8—791no—
360#322—791no—
361#322—791no—
362#322—791no—
363#322C2×D7911yescharacteristic
364C2×#41—791no—
365C2×#41—791no—
366#135—1584no—
367C2×Q16C4×D7911yescharacteristic
368C2×SD16—791no—
369C2×SD16—791no—
370C2×D8—791no—
371C2×M16—1581no—
372C2×M16—1581no—
373C22×C8—791no—
374C22×C8—791no—
375C22×C8—1581no—
376C22×C8—1581no—
377C22×C8—1581no—
378C22×C8—1581no—
379C4×Q8—791no—
380C4×Q8—791no—
381C4×Q8C22×D7911yescharacteristic
382C4×Q8C22×D7911yescharacteristic
383C4×D4—791no—
384C4×D4—791no—
385C4×D4—791no—
386C4×D4—791no—
387C4×D4—1581no—
388C4×D4—1581no—
389C4×D4—1581no—
390C4×D4—1581no—
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—21no—
403C4⋊C8—21no—
404C4⋊C8—791no—
405C4⋊C8—791no—
406C4⋊C8—791no—
407C4⋊C8—791no—
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#103C4×D7911yescharacteristic
418#103C4×D7911yescharacteristic
419#102—791no—
420#102—791no—
421#102—791no—
422#102—791no—
423C22⋊C8—791no—
424C22⋊C8—791no—
425C22⋊C8—1581no—
426C22⋊C8—1581no—
427C22⋊C8—1581no—
428C22⋊C8—1581no—
429C22⋊C8—1581no—
430#97—21no—
431#97—1581no—
432C4×C8—21no—
433C4×C8—791no—
434C4×C8—791no—
435C4×C8—791no—
436C4×C8—791no—
437C4×C8—791no—
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×Q8C2×C4×D7911yescharacteristic
447C2×Q8C2×C4×D7911yescharacteristic
448C2×D4—791no—
449C2×D4—791no—
450C2×D4—1581no—
451C2×D4—1581no—
452C22×C4—791no—
453C22×C4—791no—
454C22×C4—791no—
455C22×C4—1581no—
456C22×C4—1581no—
457C22×C4—1581no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4—1581no—
461Q16—22no—
462SD16—1582no—
463SD16—1582no—
464D8—1582no—
465M16—6321no—
466M16—6321no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—791no—
471C2×C8—791no—
472C2×C8—791no—
473C2×C8—791no—
474C2×C8—6321no—
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—21no—
488C4⋊C4—21no—
489C4⋊C4—791no—
490C4⋊C4—791no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4C2×C4×D7911yescharacteristic
494C4⋊C4C2×C4×D7911yescharacteristic
495C22⋊C4—1581no—
496C22⋊C4—1581no—
497C22⋊C4—1581no—
498C22⋊C4—1581no—
499C22⋊C4—1581no—
500C22⋊C4—1581no—
501C42—21no—
502C42—21no—
503C42—791no—
504C42—791no—
505C42—791no—
506C42—1581no—
507C42—1581no—
508C42—1581no—
509C42—1581no—
510C42C23×D7911yescharacteristic
511C23—791no—
512C23—1581no—
513C23—1581no—
514Q8—21no—
515Q8—21no—
516Q8—21no—
517Q8M16×D7912yes—
518D4—3162no—
519D4—792no—
520D4—1584no—
521D4—1581no—
522D4—1581no—
523D4—1581no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—3161no—
529C2×C4—3162no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—791no—
535C2×C4—792no—
536C2×C4—792no—
537C2×C4—791no—
538C2×C4—791no—
539C2×C4—6321no—
540C2×C4—6321no—
541C2×C4—1584no—
542C2×C4—1581no—
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—3161no—
564C22—3161no—
565C22—3161no—
566C22—3161no—
567C22—3161no—
568C22—792no—
569C22—1582no—
570C22—1581no—
571C22—1581no—
572C22C4×D4×D7911yescharacteristic
573C4—21no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—3161no—
579C4—3161no—
580C4—792no—
581C4—1581no—
582C4—1581no—
583C4C2×M16×D7911yescharacteristic
584C4#131×D7911yescharacteristic
585C4#1909936511yescharacteristic
586C4#1909937611yescharacteristic
587C2—3162no—
588C2—6321no—
589C2—1582no—
590C2#435×D7911yescharacteristic
591C2#2377227511yescharacteristic
592C2#2377231411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1584C2
socleC22×C79C2×C4×D4
prime radicalC2×#14579184C4
cosocleC42×C79C24
hypercenter#1584D79
hypocenterC79#17881
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
1. #23772516C2semidirect
2. #23772510C2semidirect
4. #23762278C2semidirect
5. #23762275C2semidirect
6. #23762260C2semidirect
7. #23762255C2semidirect
8. #23762251C2semidirect
9. #23762231C2semidirect
10. #23762228C2semidirect
11. #335×D79C2semidirect
12. #23761786C2semidirect
13. #23761786C2semidirect
14. #23761691C2semidirect
15. C79×#1584C2semidirect
148. #11929625M16semidirect
305. #1584D79semidirect
312. C79#17881semidirect

Maximal subgroups

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

subgroupindex
#237725162
#237725102
C4×#145791842
#237622782
#237622752
#237622602
#237622552
#237622512
#237622312
#237622282
#335×D792
#237617862
#237616912
C79×#15842
#1788179