Finite Group Databasebeta

← #34258356

Subgroups — #34258356

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#23773458C211yesmaximal, characteristic
2#23772510C211yesmaximal, characteristic, complemented
3#23772474C211yesmaximal, characteristic, complemented
4#23762275C211yesmaximal, characteristic, complemented
5#23762267C211yesmaximal, characteristic, complemented
6#23762244C211yesmaximal, characteristic, complemented
7#23762228C211yesmaximal, characteristic, complemented
8#23762225C211yesmaximal, characteristic, complemented
9C8×#11929611C211yesmaximal, characteristic, complemented
10C8×#11929609C211yesmaximal, characteristic, complemented
11#23762184C211yesmaximal, characteristic, complemented
12#23761833C211yesmaximal, characteristic, complemented
13#23761797C211yesmaximal, characteristic, complemented
14#23761701C211yesmaximal, characteristic, complemented
15C79×SD16⋊C8C211yesmaximal, characteristic, complemented
16#19099110C2211yescharacteristic
17C4×#11929625C2211yescharacteristic
18#19098889C411yescharacteristic
19#19098885C411yescharacteristic
20#19098856C411yescharacteristic
21#19098852C411yescharacteristic
22#19098761C411yescharacteristic
23#19098757C411yescharacteristic
24#19098711C411yescharacteristic
25#19098707C411yescharacteristic
26Q8×C79⋊C8C2211yescharacteristic
27C4×#11929611C2211yescharacteristic
28C4×D632C2211yescharacteristic
29C4×#11929609C2211yescharacteristic
30#19098641C2211yescharacteristic
31#19098640C2211yescharacteristic
32#19098637—21no—
33#19098634—21no—
34C8×D316—21no—
35C8×D316C2211yescharacteristic
36C8×D316C2211yescharacteristic
37#19098628—21no—
38#19098628C2211yescharacteristic
39#19098619C2211yescharacteristic
40#19098614—21no—
41#19098614C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098614C2211yescharacteristic
44C4⋊C8×D79—21no—
45C4⋊C8×D79C2211yescharacteristic
46#19098529—21no—
47#19098528—21no—
48#19098315C2211yescharacteristic
49#19098314C2211yescharacteristic
50#19098313C2211yescharacteristic
51Q8⋊(C79⋊C8)C2211yescharacteristic
52#19098292C2211yescharacteristic
53#19098292C2211yescharacteristic
54D4⋊(C79⋊C8)C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929625C2×C411yescharacteristic
68C2×#11929624—21no—
69C2×#11929611C2×C411yescharacteristic
70C2×D632C2×C411yescharacteristic
71C2×#11929609C2×C411yescharacteristic
72C2×#11929608—21no—
73C2×#11929607—21no—
74C2×C8×D79—21no—
75C2×C8×D79—21no—
76C2×C8×D79—21no—
77C2×C8×D79—21no—
78#14579160C2×C411yescharacteristic
79#14579157C2×C411yescharacteristic
80#14579147—21no—
81#14579136C2×C411yescharacteristic
82#14579134—21no—
83#14579132C2×C411yescharacteristic
84Q8×Dic79C2×C411yescharacteristic
85C4×D316—21no—
86C4×D316C2311yescharacteristic
87C4×D316C2311yescharacteristic
88#14579123—21no—
89#14579123C2×C411yescharacteristic
90C4×C22⋊D79—21no—
91#14579119—21no—
92#14579119C2×C411yescharacteristic
93C42×D79—21no—
94C42×D79C2311yescharacteristic
95#14579095C2×C411yescharacteristic
96#14579094C2×C411yescharacteristic
97#14579092C2×C411yescharacteristic
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
105Dic79⋊C8D411yescharacteristic
106Dic79⋊C8D411yescharacteristic
107#14579085C2×C411yescharacteristic
108#14579083C2×C411yescharacteristic
109#14579083C2×C411yescharacteristic
110#14579082C2×C411yescharacteristic
111#14579081C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113C22⋊(C79⋊C8)—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊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(C2×D79)⋊C8D411yescharacteristic
123(C2×D79)⋊C8D411yescharacteristic
124#14579071—21no—
125C4×C79⋊C8—21no—
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
128C4×C79⋊C8C2311yescharacteristic
129C2×SD16×C79C2×C411yescharacteristic
130C22×C8×C79—21no—
131C4×Q8×C79C2311yescharacteristic
132C4×D4×C79C2311yescharacteristic
133#106×C79C2×C411yescharacteristic
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#102×C79C2×C411yescharacteristic
139C22⋊C8×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929625—22no—
145#11929625—22no—
146#11929624—81no—
147#11929624—81no—
148C2×Dic158—21no—
149C2×D316—21no—
150C2×D316—21no—
151C2×D316C22×C411yescharacteristic
152C2×D316C22×C411yescharacteristic
153C2×C22⋊D79—21no—
154C2×C22⋊D79—21no—
155C2×C22⋊D79—21no—
156C2×C4×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#11929611—22no—
164D632—22no—
165#11929609—22no—
166#11929608—81no—
167#11929607—81no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8—21no—
171C2×C79⋊C8C22×C411yescharacteristic
172C2×C79⋊C8C22×C411yescharacteristic
173C2×C79⋊C8C2×D411yescharacteristic
174C2×C79⋊C8C2×D411yescharacteristic
175C8×D79—81no—
176C8×D79—81no—
177C8×D79—81no—
178C8×D79—81no—
179C4⋊Dic79—21no—
180C4⋊Dic79C22×C411yescharacteristic
181C4⋊Dic79C22×C411yescharacteristic
182Dic79⋊C4—21no—
183C22⋊Dic79—21no—
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—
189C4×Dic79—21no—
190C4×Dic79—21no—
191C4×Dic79C22×C411yescharacteristic
192C4×Dic79C2×D411yescharacteristic
193C4×Dic79#4111yescharacteristic
194C2×Q8×C79C22×C411yescharacteristic
195C2×D4×C79C22×C411yescharacteristic
196C22×C4×C79—21no—
197SD16×C79—22no—
198C2×C8×C79—21no—
199C2×C8×C79—21no—
200C2×C8×C79—21no—
201C2×C8×C79—81no—
202C2×C8×C79C22×C411yescharacteristic
203C2×C8×C79C22×C411yescharacteristic
204C2×C8×C79C2×D411yescharacteristic
205C2×C8×C79C2×D411yescharacteristic
206C4⋊C4×C79—21no—
207C4⋊C4×C79C22×C411yescharacteristic
208C4⋊C4×C79C22×C411yescharacteristic
209C22⋊C4×C79—21no—
210C42×C79—21no—
211C42×C79C2411yescharacteristic
212C23×C79—21no—
213C22×D79—21no—
214C22×D79—21no—
215C22×D79C4×D411yescharacteristic
216Q8×C79—21no—
217Q8×C79#13112yes—
218D4×C79—21no—
219D4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79C23×C411yescharacteristic
226C2×C4×C79C22×D411yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C22⋊D79—81no—
229C22⋊D79—81no—
230C22⋊D79—81no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79C4×D411yescharacteristic
234C2×Dic79C4×D411yescharacteristic
235C2×Dic79C4×D411yescharacteristic
236D316—24no—
237D316—21no—
238D316—21no—
239D316—21no—
240D316—81no—
241D316#13112yes—
242C4×D79—24no—
243C4×D79—21no—
244C4×D79—21no—
245C4×D79—21no—
246C4×D79—21no—
247C4×D79—41no—
248C4×D79—41no—
249C4×D79—41no—
250C4×D79—41no—
251C4×D79—41no—
252C4×D79—81no—
253Dic158—81no—
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—22no—
273C2×D79—21no—
274C2×D79—21no—
275C2×D79—21no—
276C2×D79—41no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—41no—
280C2×D79—41no—
281C4×C79—22no—
282C4×C79—21no—
283C4×C79—41no—
284C4×C79—41no—
285C4×C79C2×#13111yescharacteristic
286C4×C79C2×#13111yescharacteristic
287C4×C79#57611yescharacteristic
288C4×C79#57711yescharacteristic
289Dic79—21no—
290Dic79—21no—
291Dic79—21no—
292Dic79—41no—
293Dic79—41no—
294#17920—791nomaximal, Sylow
295C2×C79—81no—
296C2×C79#292911yescharacteristic
297C2×C79#294611yescharacteristic
298C2×C79#295711yescharacteristic
299D79—41no—
300D79—42no—
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⋊C8D7911yescharacteristic, complemented
311C8×SD16—791no—
312C8×D8—791no—
313#1573—791no—
314#1488—791no—
315#1480—791no—
316#1454—791no—
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×Q8C2×D7911yescharacteristic
330C4×SD16—791no—
331C4×SD16C2×D7911yescharacteristic
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—1581no—
342C8×D4—1581no—
343C8×D4C2×D7911yescharacteristic
344#432—791no—
345#431—791no—
346#431—791no—
347#431—791no—
348#431—1581no—
349C2×C4⋊C8—791no—
350C2×C4⋊C8—1581no—
351#408—1581no—
352#408—1581no—
353#335—791no—
354#335C2×D7911yescharacteristic
355#334—791no—
356#334—791no—
357Q8⋊C8—791no—
358Q8⋊C8C2×D7911yescharacteristic
359D4⋊C8—791no—
360D4⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8—791no—
364D4⋊C8C2×D7911yescharacteristic
365#322—791no—
366#322—791no—
367#322C2×D7911yescharacteristic
368C82—791no—
369C2×#41—791no—
370C2×#41—1581no—
371C2×SD16—791no—
372C2×SD16C4×D7911yescharacteristic
373C2×D8—791no—
374C2×D8—791no—
375C2×M16—1581no—
376C2×M16—1581no—
377C22×C8—21no—
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×Q8C22×D7911yescharacteristic
390C4×D4—791no—
391C4×D4—791no—
392C4×D4—791no—
393C4×D4—1581no—
394C4×D4—1581no—
395C4×D4—1581no—
396C4×D4C22×D7911yescharacteristic
397#117—791no—
398#117—1581no—
399C2×C42—791no—
400C2×C42—1581no—
401#107—791no—
402#107—791no—
403#106—791no—
404#106C4×D7911yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—791no—
407C4⋊C8—791no—
408C4⋊C8—791no—
409C4⋊C8—791no—
410C4⋊C8—791no—
411C4⋊C8—791no—
412C4⋊C8—1581no—
413C4⋊C8—1581no—
414C4⋊C8—1581no—
415C4⋊C8C22×D7911yescharacteristic
416C4⋊C8C22×D7911yescharacteristic
417#103—791no—
418#103C4×D7911yescharacteristic
419#102—791no—
420#102—791no—
421#102—791no—
422#102—791no—
423#102—791no—
424#102C4×D7911yescharacteristic
425C22⋊C8—21no—
426C22⋊C8—791no—
427C22⋊C8—791no—
428C22⋊C8—791no—
429C22⋊C8—791no—
430C22⋊C8—1581no—
431C22⋊C8—1581no—
432C22⋊C8—1581no—
433C22⋊C8—1581no—
434C22⋊C8—1581no—
435C22⋊C8—1581no—
436C22⋊C8—1581no—
437#97—1581no—
438C4×C8—21no—
439C4×C8—791no—
440C4×C8—791no—
441C4×C8—791no—
442C4×C8—1581no—
443C4×C8C22×D7911yescharacteristic
444C4×C8C22×D7911yescharacteristic
445C4×C8C22×D7911yescharacteristic
446#41—6321no—
447#41—6321no—
448#41—1582no—
449#41—1582no—
450C2×Q8—1581no—
451C2×Q8C2×C4×D7911yescharacteristic
452C2×D4—791no—
453C2×D4—791no—
454C2×D4—1581no—
455C2×D4—1581no—
456C2×D4—1581no—
457C2×D4—1581no—
458C2×D4—1581no—
459C2×D4C2×C4×D7911yescharacteristic
460C22×C4—21no—
461C22×C4—791no—
462C22×C4—791no—
463C22×C4—791no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4—1581no—
467C22×C4—1581no—
468SD16—22no—
469SD16—1582no—
470D8—1582no—
471D8—1582no—
472M16—6321no—
473M16—6321no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—81no—
478C2×C8—791no—
479C2×C8—791no—
480C2×C8—791no—
481C2×C8—791no—
482C2×C8—6321no—
483C2×C8—6321no—
484C2×C8—6321no—
485C2×C8—6321no—
486C2×C8—1581no—
487C2×C8—1581no—
488C2×C8—1581no—
489C2×C8C2×C4×D7911yescharacteristic
490C2×C8C2×C4×D7911yescharacteristic
491C2×C8D4×D7911yescharacteristic
492C2×C8D4×D7911yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—791no—
495C4⋊C4—791no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4C2×C4×D7911yescharacteristic
499C4⋊C4C2×C4×D7911yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—1581no—
502C22⋊C4—1581no—
503C22⋊C4—1581no—
504C22⋊C4—1581no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C42—21no—
508C42—791no—
509C42—791no—
510C42—791no—
511C42—1581no—
512C42—1581no—
513C42C23×D7911yescharacteristic
514C23—21no—
515C23—791no—
516C23—1581no—
517C23—1581no—
518Q8—21no—
519Q8—6321no—
520Q8#1457916712yes—
521D4—21no—
522D4—21no—
523D4—792no—
524D4—6321no—
525D4—6321no—
526D4—6321no—
527D4—6321no—
528D4—1584no—
529D4—1581no—
530D4—1581no—
531D4—1581no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—791no—
543C2×C4—791no—
544C2×C4—791no—
545C2×C4—6321no—
546C2×C4—1584no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4C22×C4×D7911yescharacteristic
554C2×C4C2×D4×D7911yescharacteristic
555C2×C4#41×D7911yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—21no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—3161no—
564C8—3161no—
565C8—3161no—
566C8—1581no—
567C8—1581no—
568C8—1581no—
569C8—1581no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—3161no—
574C22—3161no—
575C22—3161no—
576C22—3161no—
577C22—3161no—
578C22—1581no—
579C22—1581no—
580C22—1582no—
581C22—1581no—
582C22C4×D4×D7911yescharacteristic
583C4—21no—
584C4—22no—
585C4—41no—
586C4—41no—
587C4—3161no—
588C4—3161no—
589C4—1581no—
590C4—1581no—
591C4—1581no—
592C4#131×D7911yescharacteristic
593C4C2×#1457916711yescharacteristic
594C4#1909936711yescharacteristic
595C4#1909937211yescharacteristic
596C2—81no—
597C2—3162no—
598C2—3161no—
599C2—6321no—
600C2#2377208911yescharacteristic
601C2#2377223111yescharacteristic
602C2#2377231411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×SD16⋊C8C2
socleC22×C79C2×C4×D4
prime radical#23773458C2
cosocleC42×C79C24
hypercenterSD16⋊C8D79
hypocenterC79#17920
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #23772510C2semidirect
3. #23772474C2semidirect
4. #23762275C2semidirect
5. #23762267C2semidirect
6. #23762244C2semidirect
7. #23762228C2semidirect
8. #23762225C2semidirect
9. C8×#11929611C2semidirect
10. C8×#11929609C2semidirect
11. #23762184C2semidirect
12. #23761833C2semidirect
13. #23761797C2semidirect
14. #23761701C2semidirect
15. C79×SD16⋊C8C2semidirect
310. SD16⋊C8D79semidirect
317. C79#17920semidirect

Maximal subgroups

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

subgroupindex
#237734582
#237725102
#237724742
#237622752
#237622672
#237622442
#237622282
#237622252
C8×#119296112
C8×#119296092
#237621842
#237618332
#237617972
#237617012
C79×SD16⋊C82
#1792079