Finite Group Databasebeta

← #34258363

Subgroups — #34258363

Order 20224.

638 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773461C211yesmaximal, characteristic
2#23772508C211yesmaximal, characteristic, complemented
3#23772475C211yesmaximal, characteristic, complemented
4#23762280C211yesmaximal, characteristic, complemented
5#23762266C211yesmaximal, characteristic, complemented
6#23762245C211yesmaximal, characteristic, complemented
7#23762231C211yesmaximal, characteristic, complemented
8#23762225C211yesmaximal, characteristic, complemented
9#23762214C211yesmaximal, characteristic, complemented
10C8×#11929613C211yesmaximal, characteristic, complemented
11C8×D632C211yesmaximal, characteristic, complemented
12#23762185C211yesmaximal, characteristic, complemented
13#23761835C211yesmaximal, characteristic, complemented
14#23761800C211yesmaximal, characteristic, complemented
15C79×#1454C211yesmaximal, characteristic, complemented
16#19099113C2211yescharacteristic
17C4×#11929624C2211yescharacteristic
18#19098888C411yescharacteristic
19#19098888C411yescharacteristic
20#19098859C411yescharacteristic
21#19098854C411yescharacteristic
22#19098764C411yescharacteristic
23#19098759C411yescharacteristic
24#19098710C411yescharacteristic
25#19098710C411yescharacteristic
26C8×Dic158C2211yescharacteristic
27C8×Dic158C2211yescharacteristic
28C4×#11929613C2211yescharacteristic
29C4×#11929613C2211yescharacteristic
30C4×D632C2211yescharacteristic
31C4×D632C2211yescharacteristic
32#19098642C2211yescharacteristic
33#19098642C2211yescharacteristic
34#19098633—21no—
35#19098633—21no—
36C8×D316C2211yescharacteristic
37C8×D316C2211yescharacteristic
38C8×D316C2211yescharacteristic
39C8×D316C2211yescharacteristic
40(C22⋊D79)⋊C8—21no—
41(C22⋊D79)⋊C8—21no—
42#19098620C2211yescharacteristic
43#19098615—21no—
44C2×C4⋊(C79⋊C8)C2211yescharacteristic
45C4⋊C8×D79—21no—
46#19098528—21no—
47#19098528—21no—
48#19098316C2211yescharacteristic
49#19098316C2211yescharacteristic
50#19098314C2211yescharacteristic
51#19098314C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098292C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098292C2211yescharacteristic
57#19098292C2211yescharacteristic
58#19098292C2211yescharacteristic
59#19098292C2211yescharacteristic
60#431×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62#431×C79C2211yescharacteristic
63#322×C79C2211yescharacteristic
64#322×C79C2211yescharacteristic
65#322×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929624—21no—
68C2×#11929624C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929613C2×C411yescharacteristic
71C2×D632C2×C411yescharacteristic
72C2×D632C2×C411yescharacteristic
73C2×#11929607—21no—
74C2×#11929607—21no—
75C22×C79⋊C8—21no—
76C2×C8×D79—21no—
77C2×C8×D79—21no—
78C2×C8×D79—21no—
79C2×C8×D79—21no—
80#14579161C2×C411yescharacteristic
81#14579158C2×C411yescharacteristic
82#14579149—21no—
83#14579135—21no—
84#14579135C2×C411yescharacteristic
85#14579135C2×C411yescharacteristic
86C4×Dic158C2×C411yescharacteristic
87C4×Dic158C2311yescharacteristic
88C4×D316—21no—
89C4×D316C2×C411yescharacteristic
90C4×D316C2311yescharacteristic
91C4×D316C2311yescharacteristic
92C4×D316C2311yescharacteristic
93C4×C22⋊D79—21no—
94C4×C22⋊D79—21no—
95#14579120C2×C411yescharacteristic
96#14579118—21no—
97C42×D79—21no—
98#14579095C2×C411yescharacteristic
99#14579095C2×C411yescharacteristic
100#14579093C2×C411yescharacteristic
101#14579093C2×C411yescharacteristic
102C4⋊(C79⋊C8)—21no—
103C4⋊(C79⋊C8)C2311yescharacteristic
104C4⋊(C79⋊C8)C2311yescharacteristic
105C4⋊(C79⋊C8)C2311yescharacteristic
106C4⋊(C79⋊C8)C2311yescharacteristic
107Dic79⋊C8—21no—
108Dic79⋊C8—21no—
109#14579086C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579083C2×C411yescharacteristic
112#14579083C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114#14579083C2×C411yescharacteristic
115#14579083C2×C411yescharacteristic
116#14579083C2×C411yescharacteristic
117C22⋊(C79⋊C8)—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8—21no—
121(C2×D79)⋊C8—21no—
122(C2×D79)⋊C8—21no—
123(C2×D79)⋊C8—21no—
124#14579072—21no—
125C8×Dic79—21no—
126C8×Dic79—21no—
127C2×C42×C79C2311yescharacteristic
128C4⋊C8×C79—21no—
129C4⋊C8×C79C2×C411yescharacteristic
130C4⋊C8×C79C2×C411yescharacteristic
131C4⋊C8×C79D411yescharacteristic
132C4⋊C8×C79D411yescharacteristic
133C22⋊C8×C79—21no—
134C22⋊C8×C79C2×C411yescharacteristic
135C22⋊C8×C79C2×C411yescharacteristic
136C22⋊C8×C79D411yescharacteristic
137C22⋊C8×C79D411yescharacteristic
138C4×C8×C79C2311yescharacteristic
139C4×C8×C79C2311yescharacteristic
140C4×C8×C79C2311yescharacteristic
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929624—22no—
145#11929624—22no—
146#11929624—81no—
147#11929624—81no—
148C2×Dic158—21no—
149C2×Dic158C22×C411yescharacteristic
150C2×D316—21no—
151C2×D316—21no—
152C2×D316C22×C411yescharacteristic
153C2×D316C22×C411yescharacteristic
154C2×D316C22×C411yescharacteristic
155C2×C22⋊D79—21no—
156C2×C22⋊D79—21no—
157C2×C22⋊D79—21no—
158C2×C4×D79—21no—
159C2×C4×D79—21no—
160C2×C4×D79—21no—
161C2×C4×D79—21no—
162C2×C4×D79—21no—
163#11929613—22no—
164#11929613—22no—
165D632—22no—
166D632—22no—
167#11929607—81no—
168#11929607—81no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8—21no—
171C2×C79⋊C8—21no—
172C2×C79⋊C8—21no—
173C2×C79⋊C8—81no—
174C8×D79—81no—
175C8×D79—81no—
176C8×D79—81no—
177C8×D79—81no—
178C4⋊Dic79—21no—
179C4⋊Dic79C22×C411yescharacteristic
180C4⋊Dic79C22×C411yescharacteristic
181C4⋊Dic79C22×C411yescharacteristic
182C4⋊Dic79C22×C411yescharacteristic
183Dic79⋊C4—21no—
184Dic79⋊C4—21no—
185C22⋊Dic79—21no—
186(C2×D79)⋊C4—21no—
187(C2×D79)⋊C4—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191(C2×D79)⋊C4—21no—
192C4×Dic79—21no—
193C4×Dic79—21no—
194C4×Dic79—21no—
195C22×C4×C79C22×C411yescharacteristic
196C22×C4×C79C2×D411yescharacteristic
197C22×C4×C79#4111yescharacteristic
198C2×C8×C79—21no—
199C2×C8×C79—21no—
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C22×C411yescharacteristic
202C2×C8×C79C22×C411yescharacteristic
203C2×C8×C79C22×C411yescharacteristic
204C2×C8×C79C2×D411yescharacteristic
205C2×C8×C79C2×D411yescharacteristic
206C2×C8×C79C2×D411yescharacteristic
207C2×C8×C79C2×D411yescharacteristic
208C42×C79C22×C411yescharacteristic
209C42×C79C2×D411yescharacteristic
210C42×C79#4111yescharacteristic
211C42×C79C2411yescharacteristic
212C23×C79C4×D411yescharacteristic
213C22×D79—21no—
214C22×D79—21no—
215C22×D79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—41no—
221C2×C4×C79C4×D411yescharacteristic
222C2×C4×C79C4×D411yescharacteristic
223C2×C4×C79C4×D411yescharacteristic
224C2×C4×C79C23×C411yescharacteristic
225C2×C4×C79C22×D411yescharacteristic
226C2×C4×C79C2×#4111yescharacteristic
227C22⋊D79—24no—
228C22⋊D79—81no—
229C22⋊D79—81no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235D316—21no—
236D316—21no—
237D316—21no—
238D316—21no—
239D316—21no—
240D316—81no—
241D316—81no—
242D316#13112yes—
243C4×D79—24no—
244C4×D79—41no—
245C4×D79—41no—
246C4×D79—41no—
247C4×D79—41no—
248C4×D79—41no—
249C4×D79—41no—
250C4×D79—81no—
251Dic158—21no—
252Dic158—81no—
253Dic158#13112yes—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79—21no—
257C8×C79—21no—
258C8×C79—21no—
259C8×C79—21no—
260C8×C79—21no—
261C8×C79—21no—
262C8×C79—41no—
263C8×C79—41no—
264C79⋊C8—41no—
265C79⋊C8—41no—
266C79⋊C8—41no—
267C79⋊C8—41no—
268C22×C79—21no—
269C22×C79—21no—
270C22×C79—21no—
271C22×C79C2×C4×D411yescharacteristic
272C2×D79—22no—
273C2×D79—41no—
274C2×D79—41no—
275C2×D79—41no—
276C2×D79—41no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—41no—
280C2×D79—41no—
281C4×C79—21no—
282C4×C79—21no—
283C4×C79—21no—
284C4×C79—21no—
285C4×C79C2×#13111yescharacteristic
286C4×C79C2×#13111yescharacteristic
287C4×C79#57611yescharacteristic
288C4×C79#57711yescharacteristic
289Dic79—22no—
290Dic79—41no—
291Dic79—41no—
292Dic79—41no—
293Dic79—41no—
294#17920—791nomaximal, Sylow
295C2×C79—41no—
296C2×C79#292911yescharacteristic
297C2×C79#294611yescharacteristic
298C2×C79#295711yescharacteristic
299D79—42no—
300D79—81no—
301D79—81no—
302#3106—791no—
303#2988—791no—
304#2981—791no—
305#1588—791no—
306#1586—791no—
307#1583—791no—
308#1581—791no—
309#1580—791no—
310SD16⋊C8—791no—
311C8×SD16—791no—
312C8×D8—791no—
313#1573—791no—
314#1488—791no—
315#1480—791no—
316#1454D7911yescharacteristic, complemented
317C79#1792011yescharacteristic, complemented, Sylow
318#532—791no—
319C4×#41—791no—
320#482—791no—
321#482—791no—
322#477—791no—
323#476—791no—
324#460—791no—
325#459—791no—
326#449—791no—
327#449—791no—
328C8×Q8—791no—
329C8×Q8—791no—
330C4×SD16—791no—
331C4×SD16—791no—
332C4×D8—791no—
333C4×D8—791no—
334#436—791no—
335#436—791no—
336#435—1581no—
337#435—1581no—
338C8×D4—791no—
339C8×D4—791no—
340C8×D4—791no—
341C8×D4—791no—
342C8×D4—1581no—
343C8×D4—1581no—
344#432—791no—
345#431—1581no—
346#431C2×D7911yescharacteristic
347#431C2×D7911yescharacteristic
348#431C2×D7911yescharacteristic
349C2×C4⋊C8—791no—
350C2×C4⋊C8—1581no—
351#408—1581no—
352#408—1581no—
353#335—791no—
354#335—791no—
355#334—791no—
356#334—791no—
357Q8⋊C8—791no—
358Q8⋊C8—791no—
359D4⋊C8—791no—
360D4⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8—791no—
364D4⋊C8—791no—
365#322C2×D7911yescharacteristic
366#322C2×D7911yescharacteristic
367#322C2×D7911yescharacteristic
368C82C2×D7911yescharacteristic
369C2×#41—791no—
370C2×#41—1581no—
371C2×SD16—791no—
372C2×SD16—791no—
373C2×D8—791no—
374C2×D8—791no—
375C2×M16—1581no—
376C2×M16—1581no—
377C22×C8—1581no—
378C22×C8—1581no—
379C22×C8—1581no—
380C22×C8—1581no—
381C22×C8—1581no—
382C4⋊Q8—791no—
383#127—791no—
384#124—1581no—
385#121—791no—
386#121—791no—
387#121—1581no—
388C4×Q8—791no—
389C4×Q8—791no—
390C4×D4—791no—
391C4×D4—791no—
392C4×D4—791no—
393C4×D4—791no—
394C4×D4—1581no—
395C4×D4—1581no—
396C4×D4—1581no—
397#117—791no—
398#117—1581no—
399C2×C42—1581no—
400C2×C42C22×D7911yescharacteristic
401#107—791no—
402#107—791no—
403#106—791no—
404#106—791no—
405C4⋊C8—21no—
406C4⋊C8—791no—
407C4⋊C8—791no—
408C4⋊C8—791no—
409C4⋊C8—791no—
410C4⋊C8—1581no—
411C4⋊C8—1581no—
412C4⋊C8—1581no—
413C4⋊C8C4×D7911yescharacteristic
414C4⋊C8C4×D7911yescharacteristic
415C4⋊C8D31611yescharacteristic
416C4⋊C8D31611yescharacteristic
417#103—791no—
418#103—791no—
419#102—791no—
420#102—791no—
421#102—791no—
422#102—791no—
423#102—791no—
424#102—791no—
425C22⋊C8—21no—
426C22⋊C8—1581no—
427C22⋊C8—1581no—
428C22⋊C8—1581no—
429C22⋊C8—1581no—
430C22⋊C8—1581no—
431C22⋊C8—1581no—
432C22⋊C8—1581no—
433C22⋊C8C4×D7911yescharacteristic
434C22⋊C8C4×D7911yescharacteristic
435C22⋊C8D31611yescharacteristic
436C22⋊C8D31611yescharacteristic
437#97—1581no—
438C4×C8—1581no—
439C4×C8—1581no—
440C4×C8C22×D7911yescharacteristic
441C4×C8C22×D7911yescharacteristic
442C4×C8C22×D7911yescharacteristic
443C4×C8C22×D7911yescharacteristic
444C4×C8C22×D7911yescharacteristic
445C4×C8C22×D7911yescharacteristic
446#41—6321no—
447#41—6321no—
448#41—1582no—
449#41—1582no—
450C2×Q8—791no—
451C2×Q8—1581no—
452C2×D4—791no—
453C2×D4—791no—
454C2×D4—791no—
455C2×D4—1581no—
456C2×D4—1581no—
457C2×D4—1581no—
458C2×D4—1581no—
459C2×D4—1581no—
460C22×C4—1581no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4C2×C4×D7911yescharacteristic
466C22×C4C2×D31611yescharacteristic
467C22×C4#1192962411yescharacteristic
468SD16—1582no—
469SD16—1582no—
470D8—1582no—
471D8—1582no—
472M16—6321no—
473M16—6321no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—6321no—
477C2×C8—6321no—
478C2×C8—6321no—
479C2×C8—6321no—
480C2×C8—6321no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C2×C8C2×C4×D7911yescharacteristic
486C2×C8C2×C4×D7911yescharacteristic
487C2×C8C2×C4×D7911yescharacteristic
488C2×C8C2×C4×D7911yescharacteristic
489C2×C8C2×D31611yescharacteristic
490C2×C8C2×D31611yescharacteristic
491C2×C8C2×D31611yescharacteristic
492C2×C8C2×D31611yescharacteristic
493C4⋊C4—791no—
494C4⋊C4—791no—
495C4⋊C4—791no—
496C4⋊C4—791no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C22⋊C4—1581no—
501C22⋊C4—1581no—
502C22⋊C4—1581no—
503C22⋊C4—1581no—
504C22⋊C4—1581no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C42—1581no—
508C42—1581no—
509C42—1581no—
510C42C2×C4×D7911yescharacteristic
511C42C2×D31611yescharacteristic
512C42#1192962411yescharacteristic
513C42C23×D7911yescharacteristic
514C23—1581no—
515C23—1581no—
516C23—1581no—
517C23C4×D31611yescharacteristic
518Q8—792no—
519Q8—6321no—
520Q8—1581no—
521D4—792no—
522D4—6321no—
523D4—6321no—
524D4—6321no—
525D4—6321no—
526D4—1584no—
527D4—1581no—
528D4—1581no—
529D4—1581no—
530D4—1581no—
531D4—1581no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—6321no—
544C2×C4—1584no—
545C2×C4—1581no—
546C2×C4—1581no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4C4×D31611yescharacteristic
551C2×C4C4×D31611yescharacteristic
552C2×C4C4×D31611yescharacteristic
553C2×C4C22×C4×D7911yescharacteristic
554C2×C4C22×D31611yescharacteristic
555C2×C4C2×#1192962411yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—21no—
564C8—41no—
565C8—41no—
566C8—3161no—
567C8—3161no—
568C8—3161no—
569C8—3161no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—3161no—
574C22—3161no—
575C22—3161no—
576C22—3161no—
577C22—3161no—
578C22—3161no—
579C22—3161no—
580C22—3161no—
581C22—1582no—
582C22C2×C4×D31611yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—3161no—
588C4—3161no—
589C4—3161no—
590C4—3161no—
591C4—1582no—
592C4#131×D7911yescharacteristic
593C4#131×D7911yescharacteristic
594C4#1909936811yescharacteristic
595C4#1909937511yescharacteristic
596C2—41no—
597C2—3162no—
598C2—6321no—
599C2—6321no—
600C2#2377209611yescharacteristic
601C2#2377223411yescharacteristic
602C2#2377231811yescharacteristic
subgroupG/H
centerC2×C4C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1454C2
socleC22×C79C2×C4×D4
prime radical#23773461C2
cosocleC42×C79C24
hypercenter#1454D79
hypocenterC79#17920
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
2. #23772508C2semidirect
3. #23772475C2semidirect
4. #23762280C2semidirect
5. #23762266C2semidirect
6. #23762245C2semidirect
7. #23762231C2semidirect
8. #23762225C2semidirect
9. #23762214C2semidirect
10. C8×#11929613C2semidirect
11. C8×D632C2semidirect
12. #23762185C2semidirect
13. #23761835C2semidirect
14. #23761800C2semidirect
15. C79×#1454C2semidirect
316. #1454D79semidirect
317. C79#17920semidirect

Maximal subgroups

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

subgroupindex
#237734612
#237725082
#237724752
#237622802
#237622662
#237622452
#237622312
#237622252
#237622142
C8×#119296132
C8×D6322
#237621852
#237618352
#237618002
C79×#14542
#1792079