Finite Group Databasebeta

← #34259633

Subgroups — #34259633

Order 20224.

606 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773431C211yesmaximal, characteristic
2#23772466C211yesmaximal, characteristic
3#23762300C211yesmaximal, characteristic, complemented
4#23762290C211yesmaximal, characteristic, complemented
5#23762278C211yesmaximal, characteristic, complemented
6#23762255C211yesmaximal, characteristic, complemented
7#23762243C211yesmaximal, characteristic, complemented
8#23762234C211yesmaximal, characteristic, complemented
9#23762210C211yesmaximal, characteristic, complemented
10C8×#11929611C211yesmaximal, characteristic, complemented
11#23762181C211yesmaximal, characteristic, complemented
12#23761868C211yesmaximal, characteristic
13#23761795C211yesmaximal, characteristic, complemented
14#23761700C211yesmaximal, characteristic, complemented
15C79×#2984C211yesmaximal, characteristic, complemented
16#19099113C2211yescharacteristic
17#19098881C411yescharacteristic
18#19098881C411yescharacteristic
19#19098848C411yescharacteristic
20#19098848C411yescharacteristic
21#19098763C411yescharacteristic
22#19098763C411yescharacteristic
23#19098702C411yescharacteristic
24#19098702C411yescharacteristic
25C4×#11929611C2211yescharacteristic
26C4×#11929611C2211yescharacteristic
27C4×#11929611C2211yescharacteristic
28C4×#11929611C2211yescharacteristic
29#19098642C2211yescharacteristic
30#19098642C2211yescharacteristic
31#19098640—21no—
32C8×D316C2211yescharacteristic
33C8×D316C2211yescharacteristic
34#19098628—21no—
35#19098616C2211yescharacteristic
36#19098616C2211yescharacteristic
37#19098616C2211yescharacteristic
38#19098528—21no—
39#19098528—21no—
40#19098315C2211yescharacteristic
41#19098315C2211yescharacteristic
42#19098313C2211yescharacteristic
43#19098313C2211yescharacteristic
44Q8⋊(C79⋊C8)C2211yescharacteristic
45Q8⋊(C79⋊C8)C2211yescharacteristic
46Q8⋊(C79⋊C8)C2211yescharacteristic
47Q8⋊(C79⋊C8)C2211yescharacteristic
48#19098290C2211yescharacteristic
49#19098290C2211yescharacteristic
50#19098290C2211yescharacteristic
51#19098290C2211yescharacteristic
52#19098285C2211yescharacteristic
53#19098285C2211yescharacteristic
54#19098284C2211yescharacteristic
55C8×C79⋊C8C2211yescharacteristic
56C2×C4×Q8×C79C2211yescharacteristic
57#446×C79—21no—
58#446×C79—21no—
59#446×C79C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C8×Q8×C79C2211yescharacteristic
63#432×C79—21no—
64#432×C79C2211yescharacteristic
65#431×C79—21no—
66#431×C79C2211yescharacteristic
67C2×#11929624—21no—
68C2×#11929611C2×C411yescharacteristic
69C2×#11929611C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929611C2×C411yescharacteristic
72C2×#11929607—21no—
73C2×#11929607—21no—
74C2×C8×D79—21no—
75C2×C8×D79—21no—
76#14579161C2×C411yescharacteristic
77#14579158C2×C411yescharacteristic
78#14579149—21no—
79#14579135—21no—
80#14579135C2×C411yescharacteristic
81#14579135C2×C411yescharacteristic
82C4×D316—21no—
83C4×D316C2311yescharacteristic
84C4×D316C2311yescharacteristic
85#14579092C2×C411yescharacteristic
86#14579092C2×C411yescharacteristic
87#14579092C2×C411yescharacteristic
88#14579092C2×C411yescharacteristic
89C4⋊(C79⋊C8)—21no—
90C4⋊(C79⋊C8)C2×C411yescharacteristic
91C4⋊(C79⋊C8)C2×C411yescharacteristic
92C4⋊(C79⋊C8)C2311yescharacteristic
93C4⋊(C79⋊C8)C2311yescharacteristic
94#14579085C2×C411yescharacteristic
95#14579085C2×C411yescharacteristic
96#14579085C2×C411yescharacteristic
97#14579085C2×C411yescharacteristic
98#14579081C2×C411yescharacteristic
99#14579081C2×C411yescharacteristic
100#14579081C2×C411yescharacteristic
101#14579081C2×C411yescharacteristic
102C22⋊(C79⋊C8)—21no—
103C22⋊(C79⋊C8)—21no—
104C22⋊(C79⋊C8)C2×C411yescharacteristic
105C22⋊(C79⋊C8)C2×C411yescharacteristic
106(C2×D79)⋊C8—21no—
107(C2×D79)⋊C8—21no—
108(C2×D79)⋊C8—21no—
109(C2×D79)⋊C8—21no—
110C4×C79⋊C8—21no—
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C4×C79⋊C8C2311yescharacteristic
114C4×C79⋊C8C2311yescharacteristic
115C22×Q8×C79C2×C411yescharacteristic
116C4×Q8×C79—21no—
117C4×Q8×C79—21no—
118C4×Q8×C79C2×C411yescharacteristic
119C4×Q8×C79C2×C411yescharacteristic
120C4×Q8×C79C2311yescharacteristic
121C4×Q8×C79C2311yescharacteristic
122C2×C4⋊C4×C79—21no—
123C2×C4⋊C4×C79C2×C411yescharacteristic
124C2×C42×C79—21no—
125C2×C42×C79C2311yescharacteristic
126C4⋊C8×C79—21no—
127C4⋊C8×C79—21no—
128C4⋊C8×C79—21no—
129C4⋊C8×C79—21no—
130C4⋊C8×C79D411yescharacteristic
131C4⋊C8×C79D411yescharacteristic
132C4⋊C8×C79C2311yescharacteristic
133C4⋊C8×C79C2311yescharacteristic
134C22⋊C8×C79—21no—
135C22⋊C8×C79D411yescharacteristic
136C22⋊C8×C79D411yescharacteristic
137#97×C79—21no—
138#97×C79—21no—
139#97×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929624—81no—
145#11929624—81no—
146C2×Dic158—21no—
147C2×D316—21no—
148C2×D316—21no—
149C2×D316C22×C411yescharacteristic
150C2×D316C22×C411yescharacteristic
151C2×C22⋊D79—21no—
152C2×C22⋊D79—21no—
153C2×C4×D79—21no—
154C2×C4×D79—21no—
155#11929611—22no—
156#11929611—22no—
157#11929611—41no—
158#11929611—41no—
159#11929607—81no—
160#11929607—81no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8C22×C411yescharacteristic
166C2×C79⋊C8C22×C411yescharacteristic
167C2×C79⋊C8C22×C411yescharacteristic
168C2×C79⋊C8C22×C411yescharacteristic
169C8×D79—81no—
170C8×D79—81no—
171C4⋊Dic79—21no—
172C4⋊Dic79C22×C411yescharacteristic
173C4⋊Dic79C22×C411yescharacteristic
174(C2×D79)⋊C4—21no—
175(C2×D79)⋊C4—21no—
176(C2×D79)⋊C4—21no—
177(C2×D79)⋊C4—21no—
178C2×Q8×C79—21no—
179C2×Q8×C79—22no—
180C2×Q8×C79—41no—
181C2×Q8×C79C22×C411yescharacteristic
182C2×Q8×C79C22×C411yescharacteristic
183C22×C4×C79—21no—
184C22×C4×C79—21no—
185C22×C4×C79C22×C411yescharacteristic
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79#4111yescharacteristic
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79C2×D411yescharacteristic
191C2×C8×C79C2×D411yescharacteristic
192C2×C8×C79C2×D411yescharacteristic
193C2×C8×C79C2×D411yescharacteristic
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79C22×C411yescharacteristic
200C4⋊C4×C79C22×C411yescharacteristic
201C42×C79—21no—
202C42×C79—21no—
203C42×C79—21no—
204C42×C79—21no—
205C42×C79C22×C411yescharacteristic
206C42×C79C2×D411yescharacteristic
207C42×C79#4111yescharacteristic
208C42×C79C2411yescharacteristic
209C23×C79C4×D411yescharacteristic
210C22×D79—21no—
211C22×D79—21no—
212Q8×C79—21no—
213Q8×C79—21no—
214Q8×C79—21no—
215Q8×C79—42no—
216Q8×C79#13112yes—
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—41no—
226C2×C4×C79—42no—
227C2×C4×C79—42no—
228C2×C4×C79C4×D411yescharacteristic
229C2×C4×C79C4×D411yescharacteristic
230C2×C4×C79C4×D411yescharacteristic
231C2×C4×C79C23×C411yescharacteristic
232C2×C4×C79C22×D411yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C22⋊D79—81no—
235C22⋊D79—81no—
236C2×Dic79—21no—
237C2×Dic79—21no—
238D316—21no—
239D316—21no—
240D316—21no—
241D316—21no—
242D316—42no—
243D316—81no—
244C4×D79—41no—
245C4×D79—41no—
246C4×D79—41no—
247C4×D79—41no—
248Dic158—81no—
249C8×C79—21no—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—21no—
253C8×C79—41no—
254C8×C79—41no—
255C79⋊C8—21no—
256C79⋊C8—21no—
257C79⋊C8—21no—
258C79⋊C8—21no—
259C79⋊C8—41no—
260C79⋊C8—41no—
261C79⋊C8—41no—
262C79⋊C8—41no—
263C22×C79—21no—
264C22×C79—21no—
265C22×C79—21no—
266C22×C79C2×C4×D411yescharacteristic
267C2×D79—41no—
268C2×D79—41no—
269C2×D79—41no—
270C2×D79—41no—
271C2×D79—41no—
272C2×D79—41no—
273C4×C79—21no—
274C4×C79—22no—
275C4×C79—21no—
276C4×C79—21no—
277C4×C79—21no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79C2×#13111yescharacteristic
282C4×C79#56811yescharacteristic
283C4×C79#57711yescharacteristic
284C4×C79#57711yescharacteristic
285Dic79—41no—
286Dic79—41no—
287#18030—791nomaximal, Sylow
288C2×C79—41no—
289C2×C79#293311yescharacteristic
290C2×C79#294911yescharacteristic
291C2×C79#295511yescharacteristic
292D79—81no—
293D79—81no—
294#3103—791no—
295#2984D7911yescharacteristic, complemented
296#2979—791no—
297#1592—791no—
298#1590—791no—
299#1588—791no—
300#1585—791no—
301#1583—791no—
302#1582—791no—
303SD16⋊C8—791no—
304C8×SD16—791no—
305#1572—791no—
306#1496—791no—
307#1479—791no—
308#1454—791no—
309C79#1803011yescharacteristic, complemented, Sylow
310#532—791no—
311C2×C4×Q8C2×D7911yescharacteristic
312#481—791no—
313#481—791no—
314#475—791no—
315#475—791no—
316#460—791no—
317#460—791no—
318#448—791no—
319#448—791no—
320#446—21no—
321#446—21no—
322#446C2×D7911yescharacteristic
323#446C2×D7911yescharacteristic
324C8×Q8C2×D7911yescharacteristic
325C8×Q8C2×D7911yescharacteristic
326C4×SD16—791no—
327C4×SD16—791no—
328C4×SD16—791no—
329C4×SD16—791no—
330#436—791no—
331#436—791no—
332#436—1581no—
333C8×D4—791no—
334C8×D4—791no—
335C8×D4—1581no—
336#432—21no—
337#432C2×D7911yescharacteristic
338#431—21no—
339#431—791no—
340#431—791no—
341#431—791no—
342#431C2×D7911yescharacteristic
343#408—1581no—
344#408—1581no—
345#335—791no—
346#335—791no—
347#334—791no—
348#334—791no—
349Q8⋊C8—791no—
350Q8⋊C8—791no—
351Q8⋊C8—791no—
352Q8⋊C8—791no—
353D4⋊C8—791no—
354D4⋊C8—791no—
355D4⋊C8—791no—
356D4⋊C8—791no—
357#322—791no—
358#322—791no—
359#322—791no—
360C82—791no—
361C2×#41—1581no—
362C22×Q8C4×D7911yescharacteristic
363C2×SD16—791no—
364C2×SD16—791no—
365C2×SD16—791no—
366C2×SD16—791no—
367C2×M16—1581no—
368C2×M16—1581no—
369C22×C8—1581no—
370C22×C8—1581no—
371C4⋊Q8—791no—
372#127—791no—
373#124—1581no—
374#121—791no—
375#121—791no—
376#121—1581no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8C4×D7911yescharacteristic
380C4×Q8C4×D7911yescharacteristic
381C4×Q8C22×D7911yescharacteristic
382C4×Q8C22×D7911yescharacteristic
383C4×D4—791no—
384C4×D4—791no—
385C4×D4—1581no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4C4×D7911yescharacteristic
388C2×C42—21no—
389C2×C42C22×D7911yescharacteristic
390#106—791no—
391#106—791no—
392#106—791no—
393#106—791no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—21no—
397C4⋊C8—21no—
398C4⋊C8—791no—
399C4⋊C8—791no—
400C4⋊C8—791no—
401C4⋊C8—791no—
402C4⋊C8—1581no—
403C4⋊C8C22⋊D7911yescharacteristic
404C4⋊C8C22⋊D7911yescharacteristic
405C4⋊C8C22×D7911yescharacteristic
406C4⋊C8C22×D7911yescharacteristic
407#103—791no—
408#103—791no—
409#103—791no—
410#103—791no—
411#102—791no—
412#102—791no—
413#102—791no—
414#102—791no—
415C22⋊C8—21no—
416C22⋊C8—791no—
417C22⋊C8—791no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420C22⋊C8—1581no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424C22⋊C8C22⋊D7911yescharacteristic
425C22⋊C8C22⋊D7911yescharacteristic
426#97—21no—
427#97—21no—
428#97—21no—
429C4×C8—21no—
430C4×C8—21no—
431C4×C8—791no—
432C4×C8—791no—
433C4×C8—791no—
434C4×C8—791no—
435C4×C8—1581no—
436C4×C8C22×D7911yescharacteristic
437C4×C8C22×D7911yescharacteristic
438#41—6321no—
439#41—6321no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—1581no—
444C2×Q8C2×C4×D7911yescharacteristic
445C2×Q8C2×C4×D7911yescharacteristic
446C2×D4—791no—
447C2×D4—791no—
448C2×D4—1581no—
449C2×D4—1581no—
450C2×D4—1581no—
451C2×D4—1581no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4C2×C4×D7911yescharacteristic
457C22×C4C2×C22⋊D7911yescharacteristic
458C22×C4#1192962411yescharacteristic
459SD16—3161no—
460SD16—3161no—
461SD16—1582no—
462SD16—1582no—
463M16—6321no—
464M16—6321no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—791no—
468C2×C8—791no—
469C2×C8—791no—
470C2×C8—791no—
471C2×C8—6321no—
472C2×C8—6321no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8C2×C22⋊D7911yescharacteristic
478C2×C8C2×C22⋊D7911yescharacteristic
479C2×C8C2×C22⋊D7911yescharacteristic
480C2×C8C2×C22⋊D7911yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—791no—
487C4⋊C4—791no—
488C4⋊C4—1581no—
489C4⋊C4C2×C4×D7911yescharacteristic
490C4⋊C4C2×C4×D7911yescharacteristic
491C22⋊C4—1581no—
492C22⋊C4—1581no—
493C22⋊C4—1581no—
494C22⋊C4—1581no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42C2×C4×D7911yescharacteristic
500C42C2×C22⋊D7911yescharacteristic
501C42#1192962411yescharacteristic
502C42C23×D7911yescharacteristic
503C23—1581no—
504C23—1581no—
505C23C4×C22⋊D7911yescharacteristic
506Q8—21no—
507Q8—21no—
508Q8—21no—
509Q8—42no—
510Q8—6321no—
511Q8#1457916912yes—
512D4—3162no—
513D4—6321no—
514D4—6321no—
515D4—6321no—
516D4—1581no—
517D4—1581no—
518D4—1581no—
519D4—1581no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4C4×C22⋊D7911yescharacteristic
538C2×C4C4×C22⋊D7911yescharacteristic
539C2×C4C4×C22⋊D7911yescharacteristic
540C2×C4C22×C4×D7911yescharacteristic
541C2×C4C22×C22⋊D7911yescharacteristic
542C2×C4C2×#1192962411yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8—21no—
547C8—41no—
548C8—41no—
549C8—3161no—
550C8—3161no—
551C8—3161no—
552C8—3161no—
553C8—1581no—
554C8—1581no—
555C8—1581no—
556C8—1581no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—3161no—
561C22—3161no—
562C22—3161no—
563C22—3161no—
564C22—3161no—
565C22—3161no—
566C22C2×C4×C22⋊D7911yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—22no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—3161no—
576C4—3161no—
577C4C2×#1457916911yescharacteristic
578C4#1909932911yescharacteristic
579C4#1909937111yescharacteristic
580C4#1909937111yescharacteristic
581C2—41no—
582C2—6321no—
583C2—6321no—
584C2#2377212511yescharacteristic
585C2#2377225911yescharacteristic
586C2#2377230111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#2984C2
socleC22×C79C2×C4×D4
prime radical#19099113C22
cosocleC42×C79C24
hypercenter#2984D79
hypocenterC79#18030
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2984

Complements

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

NH ≅ G/Ntype
3. #23762300C2semidirect
4. #23762290C2semidirect
5. #23762278C2semidirect
6. #23762255C2semidirect
7. #23762243C2semidirect
8. #23762234C2semidirect
9. #23762210C2semidirect
10. C8×#11929611C2semidirect
11. #23762181C2semidirect
13. #23761795C2semidirect
14. #23761700C2semidirect
15. C79×#2984C2semidirect
295. #2984D79semidirect
309. C79#18030semidirect

Maximal subgroups

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

subgroupindex
#237734312
#237724662
#237623002
#237622902
#237622782
#237622552
#237622432
#237622342
#237622102
C8×#119296112
#237621812
#237618682
#237617952
#237617002
C79×#29842
#1803079