Finite Group Databasebeta

← #34259423

Subgroups — #34259423

Order 20224.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773449C211yesmaximal, characteristic
2#23772556C211yesmaximal, characteristic, complemented
3#23772437C211yesmaximal, characteristic, complemented
4#23762289C211yesmaximal, characteristic, complemented
5#23762284C211yesmaximal, characteristic, complemented
6#23762277C211yesmaximal, characteristic, complemented
7#23762263C211yesmaximal, characteristic, complemented
8#23762263C211yesmaximal, characteristic, complemented
9#23762256C211yesmaximal, characteristic, complemented
10#23762199C211yesmaximal, characteristic, complemented
11#23762194C211yesmaximal, characteristic, complemented
12#23761858C211yesmaximal, characteristic, complemented
13C2×#19098290C211yesmaximal, characteristic, complemented
14#23761684C211yesmaximal, characteristic, complemented
15C79×#1573C211yesmaximal, characteristic, complemented
16#19099097C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18#19098899C411yescharacteristic
19#19098882C411yescharacteristic
20#19098864C411yescharacteristic
21#19098864C411yescharacteristic
22#19098779C411yescharacteristic
23#19098779C411yescharacteristic
24#19098716C411yescharacteristic
25#19098699C411yescharacteristic
26#19098690C2211yescharacteristic, complemented
27#19098687C2211yescharacteristic, complemented
28C4×#11929613C2211yescharacteristic
29C4×#11929613C2211yescharacteristic
30C4×D632C2211yescharacteristic
31C4×D632C2211yescharacteristic
32#19098640C2211yescharacteristic
33#19098640C2211yescharacteristic
34#19098640C2211yescharacteristic, complemented
35#19098636—21no—
36#19098636—21no—
37#19098634—21no—
38#19098628C2211yescharacteristic
39#19098628C2211yescharacteristic
40#19098628C2211yescharacteristic, complemented
41C8×C22⋊D79—21no—
42#19098623—21no—
43#19098616C2211yescharacteristic
44#19098588—21no—
45#19098525—21no—
46C22⋊C8×D79—21no—
47C2×C4×C79⋊C8C2211yescharacteristic
48#19098293C2211yescharacteristic, complemented
49#19098293C2211yescharacteristic, complemented
50#19098290C2211yescharacteristic
51#19098290C2211yescharacteristic
52#19098290C2211yescharacteristic
53#19098290C2211yescharacteristic
54#19098290C2211yescharacteristic, complemented
55#19098290C2211yescharacteristic, complemented
56#19098285C2211yescharacteristic, complemented
57#19098285C2211yescharacteristic, complemented
58C8×C79⋊C8C2211yescharacteristic, complemented
59C8×C79⋊C8C2211yescharacteristic, complemented
60#432×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62C2×C4⋊C8×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic, complemented
64#335×C79C2211yescharacteristic, complemented
65#334×C79C2211yescharacteristic, complemented
66#334×C79C2211yescharacteristic, complemented
67C22×D316C2×C411yescharacteristic
68C22×C4×D79—21no—
69C2×#11929613C2×C411yescharacteristic
70C2×#11929613C2×C411yescharacteristic
71C2×D632C2×C411yescharacteristic
72C2×D632C2×C411yescharacteristic
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×#11929607—21no—
76C22×C79⋊C8—21no—
77C2×C8×D79—21no—
78C2×C8×D79—21no—
79C2×C8×D79—21no—
80#14579155—21no—
81#14579153C2×C411yescharacteristic
82#14579149C2×C411yescharacteristic
83#14579144—21no—
84#14579140C2×C411yescharacteristic
85#14579135C2×C411yescharacteristic
86C4×Dic158C2311yescharacteristic
87C4×D316—21no—
88C4×D316—21no—
89C4×D316C2×C411yescharacteristic
90C4×D316C2×C411yescharacteristic
91C4×D316C2311yescharacteristic
92C4×D316C2311yescharacteristic
93C4×D316C2311yescharacteristic
94C4×C22⋊D79—21no—
95#14579118—21no—
96C2×C4⋊Dic79C2×C411yescharacteristic
97C2×(C2×D79)⋊C4—21no—
98#14579095C2×C411yescharacteristic
99#14579095C2×C411yescharacteristic
100#14579093C2×C411yescharacteristic
101#14579093C2×C411yescharacteristic
102C4⋊(C79⋊C8)D411yescharacteristic
103C4⋊(C79⋊C8)D411yescharacteristic
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106#14579086C2×C411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579083C2×C411yescharacteristic
109#14579083C2×C411yescharacteristic
110#14579083C2×C411yescharacteristic
111#14579083C2×C411yescharacteristic
112#14579083C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114C22⋊(C79⋊C8)D411yescharacteristic
115C22⋊(C79⋊C8)D411yescharacteristic
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)⋊C8—21no—
122#14579071—21no—
123C4×C79⋊C8—21no—
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
128C8×Dic79—21no—
129C22×C8×C79—21no—
130C2×C42×C79C2311yescharacteristic
131C4⋊C8×C79—21no—
132C4⋊C8×C79C2×C411yescharacteristic
133C4⋊C8×C79C2×C411yescharacteristic
134C4⋊C8×C79C2311yescharacteristic
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138C22⋊C8×C79—21no—
139C22⋊C8×C79C2×C411yescharacteristic
140C22⋊C8×C79C2×C411yescharacteristic
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C23×D79—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×D316—22no—
147C2×D316—21no—
148C2×D316—41no—
149C2×D316C22×C411yescharacteristic
150C2×D316C22×C411yescharacteristic
151C2×D316C22×C411yescharacteristic
152C2×C22⋊D79—21no—
153C22×Dic79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—21no—
156C2×C4×D79—21no—
157C2×C4×D79—21no—
158C2×C4×D79—21no—
159C2×C4×D79—81no—
160C2×C4×D79—81no—
161#11929613—22no—
162#11929613—41no—
163D632—22no—
164D632—41no—
165#11929607—81no—
166#11929607—81no—
167#11929607—81no—
168C2×C79⋊C8—24no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8—21no—
171C2×C79⋊C8C2×D411yescharacteristic
172C2×C79⋊C8C2×D411yescharacteristic
173C2×C79⋊C8C2×D411yescharacteristic
174C2×C79⋊C8C2×D411yescharacteristic
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—
185Dic79⋊C4—21no—
186Dic79⋊C4—21no—
187C22⋊Dic79—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191(C2×D79)⋊C4—21no—
192(C2×D79)⋊C4—21no—
193(C2×D79)⋊C4—21no—
194(C2×D79)⋊C4—21no—
195(C2×D79)⋊C4—21no—
196C4×Dic79—21no—
197C22×C4×C79C22×C411yescharacteristic
198C22×C4×C79C2×D411yescharacteristic
199C22×C4×C79#4111yescharacteristic
200C2×C8×C79—21no—
201C2×C8×C79—21no—
202C2×C8×C79—21no—
203C2×C8×C79—21no—
204C2×C8×C79—81no—
205C2×C8×C79C22×C411yescharacteristic
206C2×C8×C79C22×C411yescharacteristic
207C2×C8×C79C22×C411yescharacteristic
208C2×C8×C79C22×C411yescharacteristic
209C42×C79C22×C411yescharacteristic
210C42×C79C2×D411yescharacteristic
211C42×C79#4111yescharacteristic
212C42×C79C2411yescharacteristic
213C23×C79C4×D411yescharacteristic
214C22×D79—21no—
215C22×D79—21no—
216C22×D79—21no—
217C22×D79—42no—
218C22×D79—42no—
219C22×D79—81no—
220C2×C4×C79—22no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79C4×D411yescharacteristic
226C2×C4×C79C4×D411yescharacteristic
227C2×C4×C79C4×D411yescharacteristic
228C2×C4×C79C23×C411yescharacteristic
229C2×C4×C79C22×D411yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C22⋊D79—42no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—81no—
238D316—21no—
239D316—21no—
240D316—21no—
241D316—21no—
242D316—21no—
243D316—42no—
244D316#13112yes—
245C4×D79—41no—
246C4×D79—41no—
247C4×D79—41no—
248C4×D79—41no—
249C4×D79—41no—
250C4×D79—41no—
251C4×D79—81no—
252C4×D79—81no—
253Dic158—21no—
254Dic158—21no—
255C8×C79—21no—
256C8×C79—21no—
257C8×C79—21no—
258C8×C79—21no—
259C8×C79—41no—
260C8×C79—41no—
261C8×C79—41no—
262C8×C79—41no—
263C79⋊C8—22no—
264C79⋊C8—22no—
265C79⋊C8—21no—
266C79⋊C8—21no—
267C79⋊C8—21no—
268C79⋊C8—21no—
269C22×C79—21no—
270C22×C79—21no—
271C22×C79#44312yes—
272C22×C79C2×C4×D411yescharacteristic
273C2×D79—22no—
274C2×D79—41no—
275C2×D79—41no—
276C2×D79—41no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—41no—
280C2×D79—41no—
281C2×D79—41no—
282C2×D79—84no—
283C4×C79—21no—
284C4×C79—21no—
285C4×C79—21no—
286C4×C79—21no—
287C4×C79C2×#13111yescharacteristic
288C4×C79#56811yescharacteristic
289C4×C79#57611yescharacteristic
290C4×C79#57711yescharacteristic
291Dic79—41no—
292Dic79—41no—
293Dic79—41no—
294Dic79—41no—
295Dic79—41no—
296#18014—791nomaximal, Sylow
297C2×C79—22no—
298C2×C79#293311yescharacteristic
299C2×C79#294611yescharacteristic
300C2×C79C2×#44311yescharacteristic
301D79—42no—
302D79—81no—
303D79—81no—
304#3105—791no—
305#2994—791no—
306#2974—791no—
307#1590—791no—
308#1589—791no—
309#1588—791no—
310#1586—791no—
311#1586—791no—
312#1585—791no—
313C8×SD16—791no—
314C8×D8—791no—
315#1573D7911yescharacteristic, complemented
316#1494—791no—
317C2×D4⋊C8—791no—
318#1450—791no—
319C79#1801411yescharacteristic, complemented, Sylow
320#529—791no—
321C2×C4×D4—791no—
322#484—791no—
323#481—791no—
324#478—791no—
325#478—791no—
326#463—791no—
327#463—791no—
328#450—791no—
329#447—791no—
330#446—791no—
331C8×Q8—791no—
332C4×SD16—791no—
333C4×SD16—791no—
334C4×D8—791no—
335C4×D8—791no—
336#436—791no—
337#436—791no—
338#436—791no—
339#435—1581no—
340#435—1581no—
341#435—1581no—
342C8×D4—791no—
343C8×D4—791no—
344C8×D4—791no—
345C8×D4—1581no—
346#433—1581no—
347#432C2×D7911yescharacteristic
348#431—791no—
349#431C2×D7911yescharacteristic
350#424—1581no—
351C2×C4⋊C8C2×D7911yescharacteristic
352#407—1581no—
353C2×C22⋊C8—1581no—
354C2×C4×C8—791no—
355#335C2×D7911yescharacteristic, complemented
356#335C2×D7911yescharacteristic, complemented
357#334C2×D7911yescharacteristic, complemented
358#334C2×D7911yescharacteristic, complemented
359Q8⋊C8—791no—
360Q8⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8—791no—
364D4⋊C8—791no—
365D4⋊C8—791no—
366D4⋊C8—791no—
367#322—791no—
368#322—791no—
369C82—791no—
370C82—791no—
371C22×D4—791no—
372C23×C4—1581no—
373C2×SD16—791no—
374C2×SD16—791no—
375C2×D8—791no—
376C2×D8—791no—
377C2×M16—1581no—
378C2×M16—1581no—
379C2×M16—1581no—
380C22×C8—21no—
381C22×C8—1581no—
382C22×C8—1581no—
383C22×C8—1581no—
384C22×C8—1581no—
385#126—1581no—
386#125—791no—
387#124—791no—
388#123—1581no—
389C22⋊Q8—791no—
390#121—791no—
391C4×Q8—791no—
392C4×D4—791no—
393C4×D4—791no—
394C4×D4—791no—
395C4×D4—791no—
396C4×D4—791no—
397C4×D4—1581no—
398C4×D4—1581no—
399C4×D4—1581no—
400#117—1581no—
401C2×C4⋊C4—791no—
402C2×C22⋊C4—1581no—
403C2×C42C22×D7911yescharacteristic
404#107—791no—
405#107—791no—
406#106—791no—
407#106—791no—
408C4⋊C8—21no—
409C4⋊C8—791no—
410C4⋊C8—791no—
411C4⋊C8—1581no—
412C4⋊C8—1581no—
413C4⋊C8C4×D7911yescharacteristic
414C4⋊C8C4×D7911yescharacteristic
415C4⋊C8C22×D7911yescharacteristic
416C4⋊C8C22×D7911yescharacteristic
417C4⋊C8C22×D7911yescharacteristic
418C4⋊C8C22×D7911yescharacteristic
419#103—791no—
420#103—791no—
421#102—791no—
422#102—791no—
423#102—791no—
424#102—791no—
425#102—791no—
426#102—791no—
427C22⋊C8—21no—
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⋊C8C4×D7911yescharacteristic
437C22⋊C8C4×D7911yescharacteristic
438#97—21no—
439#97—1581no—
440C4×C8—791no—
441C4×C8—791no—
442C4×C8—791no—
443C4×C8—791no—
444C4×C8—1581no—
445C4×C8—1581no—
446C4×C8C22×D7911yescharacteristic
447C4×C8C22×D7911yescharacteristic
448C24—1581no—
449C2×Q8—791no—
450C2×D4—3161no—
451C2×D4—791no—
452C2×D4—791no—
453C2×D4—791no—
454C2×D4—1582no—
455C2×D4—1581no—
456C2×D4—1581no—
457C22×C4—6321no—
458C22×C4—6321no—
459C22×C4—1581no—
460C22×C4—1581no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4C2×C4×D7911yescharacteristic
466C22×C4D4×D7911yescharacteristic
467C22×C4#1192962511yescharacteristic
468SD16—3161no—
469SD16—1582no—
470D8—3161no—
471D8—1582no—
472M16—6321no—
473M16—6321no—
474M16—6321no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—81no—
480C2×C8—791no—
481C2×C8—791no—
482C2×C8—791no—
483C2×C8—791no—
484C2×C8—6321no—
485C2×C8—6321no—
486C2×C8—6321no—
487C2×C8—1584no—
488C2×C8—1581no—
489C2×C8—1581no—
490C2×C8C2×C4×D7911yescharacteristic
491C2×C8C2×C4×D7911yescharacteristic
492C2×C8C2×C4×D7911yescharacteristic
493C2×C8C2×C4×D7911yescharacteristic
494C4⋊C4—791no—
495C4⋊C4—791no—
496C4⋊C4—791no—
497C4⋊C4—791no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4—1581no—
502C4⋊C4—1581no—
503C22⋊C4—1581no—
504C22⋊C4—1581no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C22⋊C4—1581no—
508C22⋊C4—1581no—
509C22⋊C4—1581no—
510C22⋊C4—1581no—
511C22⋊C4—1581no—
512C42—1581no—
513C42C2×C4×D7911yescharacteristic
514C42D4×D7911yescharacteristic
515C42#1192962511yescharacteristic
516C42C23×D7911yescharacteristic
517C23—3162no—
518C23—3162no—
519C23—6321no—
520C23—1581no—
521C23—1581no—
522C23—1581no—
523C23#1457912311yescharacteristic
524Q8—1581no—
525Q8—1581no—
526D4—3162no—
527D4—3162no—
528D4—792no—
529D4—1581no—
530D4—1581no—
531D4—1581no—
532D4—1581no—
533D4—1581no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—6321no—
546C2×C4—6321no—
547C2×C4—6321no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4#1457912311yescharacteristic
554C2×C4#1457912311yescharacteristic
555C2×C4#1457912311yescharacteristic
556C2×C4C22×C4×D7911yescharacteristic
557C2×C4C2×D4×D7911yescharacteristic
558C2×C4C2×#1192962511yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—1582no—
568C8—1582no—
569C8—1581no—
570C8—1581no—
571C8—1581no—
572C8—1581no—
573C22—21no—
574C22—21no—
575C22—3161no—
576C22—3161no—
577C22—3161no—
578C22—3161no—
579C22—3161no—
580C22—3161no—
581C22—3161no—
582C22—3161no—
583C22—6324no—
584C22—1582no—
585C22#1909867912yes—
586C22C2×#1457912311yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—3161no—
592C4—3161no—
593C4—3161no—
594C4—3161no—
595C4—3161no—
596C4#131×D7911yescharacteristic
597C4#1909932811yescharacteristic
598C4#1909936711yescharacteristic
599C4#1909937211yescharacteristic
600C2—22no—
601C2—3162no—
602C2—6321no—
603C2—6321no—
604C2#2377212411yescharacteristic
605C2#2377223011yescharacteristic
606C2C2×#1909867911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1573C2
socleC22×C79C2×C4×D4
prime radical#19098699C4
cosocleC42×C79C24
hypercenter#1573D79
hypocenterC79#18014
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1573

Complements

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

NH ≅ G/Ntype
2. #23772556C2semidirect
3. #23772437C2semidirect
4. #23762289C2semidirect
5. #23762284C2semidirect
6. #23762277C2semidirect
7. #23762263C2semidirect
8. #23762263C2semidirect
9. #23762256C2semidirect
10. #23762199C2semidirect
11. #23762194C2semidirect
12. #23761858C2semidirect
13. C2×#19098290C2semidirect
14. #23761684C2semidirect
15. C79×#1573C2semidirect
26. #19098690C22semidirect
27. #19098687C22semidirect
34. #19098640C22semidirect
40. #19098628C22semidirect
48. #19098293C22semidirect
49. #19098293C22semidirect
54. #19098290C22semidirect
55. #19098290C22semidirect
56. #19098285C22semidirect
57. #19098285C22semidirect
58. C8×C79⋊C8C22semidirect
59. C8×C79⋊C8C22semidirect
63. #335×C79C22semidirect
64. #335×C79C22semidirect
65. #334×C79C22semidirect
66. #334×C79C22semidirect
315. #1573D79semidirect
319. C79#18014semidirect
355. #335C2×D79semidirect
356. #335C2×D79semidirect
357. #334C2×D79semidirect
358. #334C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734492
#237725562
#237724372
#237622892
#237622842
#237622772
#237622632
#237622562
#237621992
#237621942
#237618582
C2×#190982902
#237616842
C79×#15732
#1801479