Finite Group Databasebeta

← #34259196

Subgroups — #34259196

Order 20224.

670 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773429C211yesmaximal, characteristic
2#23772352C211yesmaximal, characteristic, complemented
3#23772145C211yesmaximal, characteristic
4#23762287C211yesmaximal, characteristic, complemented
5#23762242C211yesmaximal, characteristic, complemented
6#23762241C211yesmaximal, characteristic, complemented
7#23762237C211yesmaximal, characteristic, complemented
8#23762234C211yesmaximal, characteristic, complemented
9#23762203C211yesmaximal, characteristic, complemented
10C8×#11929613C211yesmaximal, characteristic, complemented
11#23762169C211yesmaximal, characteristic, complemented
12#23761849C211yesmaximal, characteristic
13Q8⋊C8×D79C211yesmaximal, characteristic, complemented
14C8×#11929607C211yesmaximal, characteristic, complemented
15C79×#1590C211yesmaximal, characteristic, complemented
16C2×#14579169—21no—
17#19099110C2211yescharacteristic
18C4×Q8×D79C2211yescharacteristic
19#19098883C411yescharacteristic
20#19098880C411yescharacteristic
21#19098847C411yescharacteristic
22#19098846C411yescharacteristic
23#19098762C411yescharacteristic
24#19098761C411yescharacteristic
25#19098703C411yescharacteristic
26#19098700C411yescharacteristic
27#19098694C2211yescharacteristic
28#19098692—21no—
29#19098691—21no—
30C8×Dic158C2211yescharacteristic
31#19098687C2211yescharacteristic
32C4×#11929613C2211yescharacteristic
33C4×#11929611C2211yescharacteristic
34C4×#11929612C2211yescharacteristic
35#19098640C2211yescharacteristic
36#19098632—21no—
37#19098631—21no—
38C8×D316C2211yescharacteristic
39D4×C79⋊C8C2211yescharacteristic
40#19098614C2211yescharacteristic
41#19098585—21no—
42#19098585C2211yescharacteristic
43C4⋊C8×D79C2211yescharacteristic
44#19098528—21no—
45#19098515—21no—
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098314C2211yescharacteristic
49#19098313C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62#436×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929624—21no—
68C2×Q8×D79C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929611C2×C411yescharacteristic
72#14579169—44no—
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×#11929607D411yescharacteristic
76C2×#11929607D411yescharacteristic
77C22×C79⋊C8—21no—
78C2×C8×D79—21no—
79C2×C8×D79—21no—
80C2×C8×D79—21no—
81#14579160C2×C411yescharacteristic
82#14579157C2×C411yescharacteristic
83#14579147—21no—
84#14579136C2×C411yescharacteristic
85#14579134—21no—
86#14579132C2×C411yescharacteristic
87Q8×Dic79C2×C411yescharacteristic
88C4×Dic158—21no—
89C4×Dic158C2311yescharacteristic
90#14579126—21no—
91#14579126C2×C411yescharacteristic
92C4×D316C2311yescharacteristic
93C4×C22⋊D79—21no—
94C4⋊C4×D79—21no—
95C4⋊C4×D79C2×C411yescharacteristic
96C42×D79—21no—
97C42×D79C2311yescharacteristic
98#14579093C2×C411yescharacteristic
99#14579092C2×C411yescharacteristic
100#14579092C2×C411yescharacteristic
101C4⋊(C79⋊C8)—21no—
102C4⋊(C79⋊C8)C2311yescharacteristic
103C4⋊(C79⋊C8)C2311yescharacteristic
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108Dic79⋊C8C2×C411yescharacteristic
109Dic79⋊C8C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579085C2×C411yescharacteristic
112#14579084C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114#14579082C2×C411yescharacteristic
115#14579081C2×C411yescharacteristic
116C22⋊(C79⋊C8)—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8C2×C411yescharacteristic
120(C2×D79)⋊C8C2×C411yescharacteristic
121#14579072—21no—
122#14579071D411yescharacteristic
123#14579071D411yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79—21no—
129C8×Dic79—21no—
130C2×SD16×C79C2×C411yescharacteristic
131C2×M16×C79—21no—
132C4×Q8×C79C2311yescharacteristic
133C4×D4×C79C2311yescharacteristic
134#106×C79C2×C411yescharacteristic
135C4⋊C8×C79—21no—
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138#103×C79C2×C411yescharacteristic
139#102×C79C2×C411yescharacteristic
140C22⋊C8×C79—21no—
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145#11929624—42no—
146#11929624—42no—
147C2×Dic158—21no—
148C2×Dic158—21no—
149C2×Dic158C22×C411yescharacteristic
150Q8×D79—22no—
151Q8×D79—41no—
152C2×D316—21no—
153C2×D316C22×C411yescharacteristic
154C2×C22⋊D79—21no—
155C2×C22⋊D79—21no—
156C2×C22⋊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#11929613—22no—
164#11929612—22no—
165#11929611—41no—
166#11929607—22no—
167#11929607—42no—
168#11929607—42no—
169#11929607SD1614yescomplemented
170C2×C79⋊C8—21no—
171C2×C79⋊C8—21no—
172C2×C79⋊C8—21no—
173C2×C79⋊C8—42no—
174C2×C79⋊C8C22×C411yescharacteristic
175C2×C79⋊C8C22×C411yescharacteristic
176C2×C79⋊C8C2×D411yescharacteristic
177C2×C79⋊C8C2×D411yescharacteristic
178C8×D79—42no—
179C8×D79—42no—
180C8×D79—42no—
181C4⋊Dic79—21no—
182C4⋊Dic79C22×C411yescharacteristic
183C4⋊Dic79C22×C411yescharacteristic
184Dic79⋊C4—21no—
185Dic79⋊C4—21no—
186Dic79⋊C4—21no—
187Dic79⋊C4—21no—
188C22⋊Dic79—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191C4×Dic79—21no—
192C4×Dic79—21no—
193C4×Dic79—21no—
194C4×Dic79C22×C411yescharacteristic
195C4×Dic79C2×D411yescharacteristic
196C4×Dic79#4111yescharacteristic
197C2×Q8×C79C22×C411yescharacteristic
198C2×D4×C79C22×C411yescharacteristic
199C22×C4×C79—21no—
200SD16×C79—41no—
201M16×C79—42no—
202C2×C8×C79—21no—
203C2×C8×C79—21no—
204C2×C8×C79—21no—
205C2×C8×C79C22×C411yescharacteristic
206C2×C8×C79C22×C411yescharacteristic
207C2×C8×C79C2×D411yescharacteristic
208C2×C8×C79C2×D411yescharacteristic
209C4⋊C4×C79—21no—
210C4⋊C4×C79C22×C411yescharacteristic
211C4⋊C4×C79C22×C411yescharacteristic
212C22⋊C4×C79—21no—
213C42×C79—21no—
214C42×C79C2411yescharacteristic
215C23×C79—21no—
216C22×D79—21no—
217C22×D79C4×D411yescharacteristic
218Q8×C79—21no—
219Q8×C79—21no—
220D4×C79—21no—
221D4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79C23×C411yescharacteristic
228C2×C4×C79C22×D411yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C22⋊D79—42no—
231C22⋊D79—42no—
232C22⋊D79—81no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79C4×D411yescharacteristic
237C2×Dic79C4×D411yescharacteristic
238C2×Dic79C4×D411yescharacteristic
239D316—21no—
240D316—21no—
241D316—42no—
242C4×D79—21no—
243C4×D79—21no—
244C4×D79—41no—
245C4×D79—42no—
246C4×D79—41no—
247C4×D79—41no—
248C4×D79—81no—
249C4×D79C2×SD1612yes—
250C4×D79#13512yes—
251Dic158—21no—
252Dic158—42no—
253Dic158—42no—
254Dic158#13112yescomplemented
255C8×C79—22no—
256C8×C79—21no—
257C8×C79—21no—
258C8×C79—21no—
259C8×C79—41no—
260C8×C79—41no—
261C8×C79C2×SD1612yes—
262C79⋊C8—21no—
263C79⋊C8—21no—
264C79⋊C8—21no—
265C79⋊C8—41no—
266C79⋊C8—41no—
267C79⋊C8—41no—
268C79⋊C8C2×SD1612yes—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79—41no—
272C22×C79C2×C4×D411yescharacteristic
273C2×D79—21no—
274C2×D79—21no—
275C2×D79—41no—
276C2×D79—41no—
277C2×D79—41no—
278C2×D79C4×SD1612yes—
279C4×C79—21no—
280C4×C79—41no—
281C4×C79—41no—
282C4×C79—41no—
283C4×C79C2×#13111yescharacteristic
284C4×C79#56811yescharacteristic
285C4×C79C22×SD1611yescharacteristic
286C4×C79C2×#13511yescharacteristic
287Dic79—22no—
288Dic79—21no—
289Dic79—21no—
290Dic79—41no—
291Dic79—41no—
292Dic79C4×SD1612yes—
293#17997—791nomaximal, Sylow
294C2×C79—81no—
295C2×C79#293311yescharacteristic
296C2×C79C2×C4×SD1611yescharacteristic
297C2×C79#295611yescharacteristic
298D79—22no—
299D79—81no—
300#3103—791no—
301Q8×M16—791no—
302#2935—791no—
303#1590—791no—
304#1590D7911yescharacteristic, complemented
305#1583—791no—
306#1583—791no—
307#1582—791no—
308#1582—791no—
309C8×SD16—791no—
310C8×SD16—791no—
311#1570—791no—
312#1492—791no—
313C2×Q8⋊C8—791no—
314C8×M16—791no—
315C79#1799711yescharacteristic, complemented, Sylow
316C2×#131—1581no—
317#532—791no—
318C2×C4×Q8—791no—
319#481—791no—
320#481—791no—
321#475—791no—
322#475—791no—
323#460—791no—
324#460—791no—
325#448—791no—
326#448—791no—
327#446—791no—
328#446—1581no—
329#446—1581no—
330#446C2×D7911yescharacteristic
331C8×Q8—791no—
332C8×Q8—791no—
333C4×SD16—791no—
334C4×SD16—791no—
335C4×SD16—791no—
336C4×SD16C2×D7911yescharacteristic
337#436—791no—
338#436C2×D7911yescharacteristic
339#435—1581no—
340#435—1581no—
341C8×D4—791no—
342C8×D4—791no—
343#431—791no—
344#423—791no—
345#423—1581no—
346C2×C4⋊C8—791no—
347#408—1581no—
348C4×M16—791no—
349C4×M16—1581no—
350C2×C4×C8—791no—
351#334—791no—
352#334—791no—
353#334—791no—
354#334C2×D7911yescharacteristic
355Q8⋊C8—791no—
356Q8⋊C8—791no—
357Q8⋊C8—791no—
358Q8⋊C8C2×D7911yescharacteristic
359D4⋊C8—791no—
360D4⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8C2×D7911yescharacteristic
363#322—791no—
364#322—791no—
365C82—791no—
366C82C2×D7911yescharacteristic
367C2×#41—1581no—
368C22×Q8—791no—
369C2×SD16—791no—
370C2×SD16—791no—
371C2×SD16—791no—
372C2×SD16C4×D7911yescharacteristic
373#131—3164no—
374C2×M16—21no—
375C2×M16—791no—
376C2×M16—791no—
377C2×M16—1581no—
378C2×M16—1581no—
379C22×C8—1581no—
380C22×C8—1581no—
381C22×C8—1581no—
382C22×C8—1581no—
383C4⋊Q8—791no—
384#127—791no—
385#124—1581no—
386#121—791no—
387#121—791no—
388#121—1581no—
389C4×Q8—791no—
390C4×Q8—791no—
391C4×Q8—791no—
392C4×Q8—1581no—
393C4×Q8—1581no—
394C4×Q8C22×D7911yescharacteristic
395C4×D4—791no—
396C4×D4—1581no—
397C4×D4C22×D7911yescharacteristic
398C2×C4⋊C4—791no—
399C2×C4⋊C4—1581no—
400C2×C42—791no—
401C2×C42—1581no—
402#106—791no—
403#106—791no—
404#106—791no—
405#106C4×D7911yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—791no—
408C4⋊C8—791no—
409C4⋊C8—791no—
410C4⋊C8—791no—
411C4⋊C8—1581no—
412C4⋊C8—1581no—
413C4⋊C8—1581no—
414C4⋊C8—1581no—
415C4⋊C8—1581no—
416C4⋊C8C22×D7911yescharacteristic
417C4⋊C8C22×D7911yescharacteristic
418#103—791no—
419#103—791no—
420#103—791no—
421#103C4×D7911yescharacteristic
422#102—791no—
423#102—791no—
424#102—791no—
425#102C4×D7911yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—791no—
428C22⋊C8—791no—
429C22⋊C8—1581no—
430C22⋊C8—1581no—
431C22⋊C8—1581no—
432#97—21no—
433#97—791no—
434#97—791no—
435#97—1581no—
436C4×C8—791no—
437C4×C8—791no—
438C4×C8—791no—
439C4×C8—1581no—
440C4×C8—1581no—
441C4×C8—1581no—
442C4×C8C22×D7911yescharacteristic
443C4×C8C22×D7911yescharacteristic
444C4×C8C22×D7911yescharacteristic
445#41—3162no—
446#41—3162no—
447C2×Q8—3161no—
448C2×Q8—791no—
449C2×Q8—1581no—
450C2×Q8—1581no—
451C2×Q8—1582no—
452C2×Q8C2×C4×D7911yescharacteristic
453C2×D4—791no—
454C2×D4—1581no—
455C2×D4—1581no—
456C2×D4—1581no—
457C2×D4—1581no—
458C2×D4C2×C4×D7911yescharacteristic
459C22×C4—21no—
460C22×C4—791no—
461C22×C4—791no—
462C22×C4—791no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4—1581no—
466SD16—41no—
467SD16—3161no—
468SD16—1582no—
469SD16—1582no—
470M16—42no—
471M16—3162no—
472M16—3162no—
473M16—794no—
474M16—1582no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—3162no—
479C2×C8—3162no—
480C2×C8—3162no—
481C2×C8—3162no—
482C2×C8—791no—
483C2×C8—791no—
484C2×C8—791no—
485C2×C8—791no—
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⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4C2×C4×D7911yescharacteristic
502C4⋊C4C2×C4×D7911yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—1581no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C42—21no—
508C42—791no—
509C42—791no—
510C42—791no—
511C42—1581no—
512C42—1581no—
513C42—1581no—
514C42C23×D7911yescharacteristic
515C23—21no—
516C23—791no—
517C23—1581no—
518Q8—21no—
519Q8—21no—
520Q8—3162no—
521Q8—3162no—
522Q8—792no—
523Q8—1581no—
524D4—21no—
525D4—21no—
526D4—3162no—
527D4—3162no—
528D4—3162no—
529D4—6321no—
530D4—1581no—
531D4—1581no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—3161no—
538C2×C4—3162no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—791no—
542C2×C4—792no—
543C2×C4—792no—
544C2×C4—791no—
545C2×C4—791no—
546C2×C4—6321no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4—1581no—
552C2×C4C22×C4×D7911yescharacteristic
553C2×C4C2×D4×D7911yescharacteristic
554C2×C4#41×D7911yescharacteristic
555C8—22no—
556C8—21no—
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—3161no—
562C8—3161no—
563C8—3161no—
564C8—792no—
565C8—1581no—
566C8—1581no—
567C8—1581no—
568C8SD16×D7912yes—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—3161no—
573C22—3161no—
574C22—3161no—
575C22—792no—
576C22—1581no—
577C22—1581no—
578C22C4×D4×D7911yescharacteristic
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—3161no—
584C4—3161no—
585C4—792no—
586C4—1581no—
587C4—1582no—
588C4—1581no—
589C4#131×D7911yescharacteristic
590C4#1909932911yescharacteristic
591C4C2×SD16×D7911yescharacteristic
592C4#135×D7911yescharacteristic
593C2—81no—
594C2—6321no—
595C2—1582no—
596C2#2377212911yescharacteristic
597C2C4×SD16×D7911yescharacteristic
598C2#2377230511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1590C2
socleC22×C79C2×C4×D4
prime radical#19099110C22
cosocleC42×C79C24
hypercenter#1590D79
hypocenterC79#17997
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #23772352C2semidirect
4. #23762287C2semidirect
5. #23762242C2semidirect
6. #23762241C2semidirect
7. #23762237C2semidirect
8. #23762234C2semidirect
9. #23762203C2semidirect
10. C8×#11929613C2semidirect
11. #23762169C2semidirect
13. Q8⋊C8×D79C2semidirect
14. C8×#11929607C2semidirect
15. C79×#1590C2semidirect
169. #11929607SD16semidirect
254. Dic158#131semidirect
304. #1590D79semidirect
315. C79#17997semidirect

Maximal subgroups

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

subgroupindex
#237734292
#237723522
#237721452
#237622872
#237622422
#237622412
#237622372
#237622342
#237622032
C8×#119296132
#237621692
#237618492
Q8⋊C8×D792
C8×#119296072
C79×#15902
#1799779