Finite Group Databasebeta

← #34259431

Subgroups — #34259431

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#23773443C211yesmaximal, characteristic
2#23772560C211yesmaximal, characteristic, complemented
3#23772437C211yesmaximal, characteristic, complemented
4#23762287C211yesmaximal, characteristic, complemented
5#23762282C211yesmaximal, characteristic, complemented
6#23762275C211yesmaximal, characteristic, complemented
7#23762267C211yesmaximal, characteristic, complemented
8#23762261C211yesmaximal, characteristic, complemented
9#23762260C211yesmaximal, characteristic, complemented
10#23762197C211yesmaximal, characteristic, complemented
11#23762192C211yesmaximal, characteristic, complemented
12#23762186C211yesmaximal, characteristic, complemented
13C2×#19098292C211yesmaximal, characteristic, complemented
14#23761683C211yesmaximal, characteristic, complemented
15C79×#1494C211yesmaximal, characteristic, complemented
16C2×C4×D316C2211yescharacteristic
17#19098896C411yescharacteristic
18#19098879C411yescharacteristic
19#19098860C411yescharacteristic
20#19098860C411yescharacteristic
21#19098775C411yescharacteristic
22#19098775C411yescharacteristic
23#19098713C411yescharacteristic
24#19098696C411yescharacteristic
25#19098694C2211yescharacteristic, complemented
26Q8×C79⋊C8C2211yescharacteristic, complemented
27C4×#11929611C2211yescharacteristic
28C4×#11929611C2211yescharacteristic
29C4×#11929609C2211yescharacteristic
30C4×#11929609C2211yescharacteristic
31#19098641C2211yescharacteristic, complemented
32#19098640C2211yescharacteristic
33#19098640C2211yescharacteristic
34#19098636—21no—
35#19098636—21no—
36#19098635—21no—
37D4×C79⋊C8—21no—
38D4×C79⋊C8C2211yescharacteristic, complemented
39#19098628C2211yescharacteristic
40#19098628C2211yescharacteristic
41#19098624—21no—
42#19098616C2211yescharacteristic
43#19098616C2211yescharacteristic
44#19098590—21no—
45C2×C4⋊(C79⋊C8)C2211yescharacteristic
46#19098525—21no—
47C22⋊C8×D79—21no—
48#19098315C2211yescharacteristic, complemented
49#19098315C2211yescharacteristic, complemented
50#19098313C2211yescharacteristic, complemented
51#19098313C2211yescharacteristic, complemented
52#19098292C2211yescharacteristic
53#19098292C2211yescharacteristic
54#19098292C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098284C2211yescharacteristic, complemented
57#19098284C2211yescharacteristic, complemented
58C8×C79⋊C8C2211yescharacteristic, complemented
59C8×C79⋊C8C2211yescharacteristic, complemented
60#529×C79C2211yescharacteristic
61#432×C79C2211yescharacteristic
62C2×C4×C8×C79C2211yescharacteristic
63Q8⋊C8×C79C2211yescharacteristic, complemented
64Q8⋊C8×C79C2211yescharacteristic, complemented
65D4⋊C8×C79C2211yescharacteristic, complemented
66D4⋊C8×C79C2211yescharacteristic, complemented
67C22×D316C2×C411yescharacteristic
68C22×C4×D79—21no—
69C2×#11929611C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929609C2×C411yescharacteristic
72C2×#11929609C2×C411yescharacteristic
73C2×#11929608—21no—
74C2×#11929607—21no—
75C2×#11929607—21no—
76C22×C79⋊C8—21no—
77C22×C79⋊C8—21no—
78C2×C8×D79—21no—
79C2×C8×D79—21no—
80C4×D316—21no—
81C4×D316—21no—
82C4×D316C2×C411yescharacteristic
83C4×D316C2×C411yescharacteristic
84C4×D316C2311yescharacteristic
85C4×D316C2311yescharacteristic
86C2×C4⋊Dic79C2×C411yescharacteristic
87C2×(C2×D79)⋊C4—21no—
88#14579094C2×C411yescharacteristic
89#14579094C2×C411yescharacteristic
90#14579092C2×C411yescharacteristic
91#14579092C2×C411yescharacteristic
92C4⋊(C79⋊C8)—21no—
93C4⋊(C79⋊C8)—21no—
94C4⋊(C79⋊C8)—21no—
95C4⋊(C79⋊C8)C2×C411yescharacteristic
96C4⋊(C79⋊C8)C2×C411yescharacteristic
97C4⋊(C79⋊C8)D411yescharacteristic
98C4⋊(C79⋊C8)D411yescharacteristic
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101#14579085C2×C411yescharacteristic
102#14579085C2×C411yescharacteristic
103#14579082C2×C411yescharacteristic
104#14579082C2×C411yescharacteristic
105#14579081C2×C411yescharacteristic
106#14579081C2×C411yescharacteristic
107#14579081C2×C411yescharacteristic
108#14579081C2×C411yescharacteristic
109C22⋊(C79⋊C8)—21no—
110C22⋊(C79⋊C8)—21no—
111C22⋊(C79⋊C8)C2×C411yescharacteristic
112C22⋊(C79⋊C8)C2×C411yescharacteristic
113C22⋊(C79⋊C8)D411yescharacteristic
114C22⋊(C79⋊C8)D411yescharacteristic
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119#14579072—21no—
120C4×C79⋊C8—21no—
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C22×C8×C79—21no—
126#126×C79—21no—
127#125×C79C2×C411yescharacteristic
128#124×C79C2×C411yescharacteristic
129#123×C79—21no—
130C22⋊Q8×C79C2×C411yescharacteristic
131#121×C79C2×C411yescharacteristic
132C4×Q8×C79C2311yescharacteristic
133C4×D4×C79—21no—
134C4×D4×C79C2311yescharacteristic
135#117×C79—21no—
136C2×C42×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138C4⋊C8×C79C2311yescharacteristic
139C22⋊C8×C79—21no—
140#97×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C23×D79—21no—
145C2×D316—22no—
146C2×D316—21no—
147C2×D316—41no—
148C2×D316C22×C411yescharacteristic
149C2×D316C22×C411yescharacteristic
150C22×Dic79—21no—
151C2×C4×D79—21no—
152C2×C4×D79—21no—
153C2×C4×D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—81no—
156C2×C4×D79—81no—
157#11929611—22no—
158#11929611—41no—
159#11929609—22no—
160#11929609—41no—
161#11929608—81no—
162#11929607—81no—
163#11929607—81no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8—21no—
166C2×C79⋊C8—81no—
167C2×C79⋊C8—81no—
168C2×C79⋊C8C22×C411yescharacteristic
169C2×C79⋊C8C22×C411yescharacteristic
170C2×C79⋊C8C22×C411yescharacteristic
171C2×C79⋊C8C22×C411yescharacteristic
172C2×C79⋊C8C2×D411yescharacteristic
173C2×C79⋊C8C2×D411yescharacteristic
174C2×C79⋊C8C2×D411yescharacteristic
175C2×C79⋊C8C2×D411yescharacteristic
176C8×D79—81no—
177C8×D79—81no—
178C4⋊Dic79—21no—
179C4⋊Dic79C22×C411yescharacteristic
180C4⋊Dic79C22×C411yescharacteristic
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185C2×Q8×C79C22×C411yescharacteristic
186C2×D4×C79—21no—
187C2×D4×C79C22×C411yescharacteristic
188C22×C4×C79—21no—
189C22×C4×C79C22×C411yescharacteristic
190C22×C4×C79C2×D411yescharacteristic
191C22×C4×C79#4111yescharacteristic
192C2×C8×C79—24no—
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79—21no—
200C4⋊C4×C79—21no—
201C4⋊C4×C79C22×C411yescharacteristic
202C4⋊C4×C79C22×C411yescharacteristic
203C22⋊C4×C79—21no—
204C22⋊C4×C79—21no—
205C22⋊C4×C79—21no—
206C22⋊C4×C79—21no—
207C22⋊C4×C79—21no—
208C42×C79—21no—
209C42×C79C22×C411yescharacteristic
210C42×C79C2×D411yescharacteristic
211C42×C79#4111yescharacteristic
212C42×C79C2411yescharacteristic
213C23×C79—21no—
214C23×C79C4×D411yescharacteristic
215C22×D79—21no—
216C22×D79—21no—
217C22×D79—42no—
218C22×D79—42no—
219C22×D79—81no—
220Q8×C79—21no—
221Q8×C79—21no—
222D4×C79—21no—
223D4×C79—21no—
224D4×C79—42no—
225C2×C4×C79—22no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79C4×D411yescharacteristic
236C2×C4×C79C4×D411yescharacteristic
237C2×C4×C79C4×D411yescharacteristic
238C2×C4×C79C23×C411yescharacteristic
239C2×C4×C79C22×D411yescharacteristic
240C2×C4×C79C2×#4111yescharacteristic
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—81no—
244D316—21no—
245D316—21no—
246D316—21no—
247D316—42no—
248D316#13112yes—
249C4×D79—41no—
250C4×D79—41no—
251C4×D79—41no—
252C4×D79—41no—
253C4×D79—81no—
254C4×D79—81no—
255C8×C79—22no—
256C8×C79—22no—
257C8×C79—41no—
258C8×C79—41no—
259C79⋊C8—21no—
260C79⋊C8—21no—
261C79⋊C8—21no—
262C79⋊C8—21no—
263C79⋊C8—21no—
264C79⋊C8—21no—
265C79⋊C8—21no—
266C79⋊C8—21no—
267C79⋊C8—41no—
268C79⋊C8—41no—
269C22×C79—21no—
270C22×C79—21no—
271C22×C79—41no—
272C22×C79—41no—
273C22×C79—41no—
274C22×C79#44312yes—
275C22×C79C2×C4×D411yescharacteristic
276C2×D79—22no—
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×C79—41no—
288C4×C79—41no—
289C4×C79—41no—
290C4×C79C2×#13111yescharacteristic
291C4×C79#56811yescharacteristic
292C4×C79#57611yescharacteristic
293C4×C79#57711yescharacteristic
294Dic79—41no—
295Dic79—41no—
296#18014—791nomaximal, Sylow
297C2×C79—22no—
298C2×C79—81no—
299C2×C79#293311yescharacteristic
300C2×C79#294611yescharacteristic
301C2×C79C2×#44311yescharacteristic
302D79—42no—
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#1573—791no—
316#1494D7911yescharacteristic, complemented
317C2×D4⋊C8—791no—
318#1450—791no—
319C79#1801411yescharacteristic, complemented, Sylow
320#529C2×D7911yescharacteristic
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#431—791no—
350#424—1581no—
351C2×C4⋊C8—791no—
352#407—1581no—
353C2×C22⋊C8—1581no—
354C2×C4×C8C2×D7911yescharacteristic
355#335—791no—
356#335—791no—
357#334—791no—
358#334—791no—
359Q8⋊C8C2×D7911yescharacteristic, complemented
360Q8⋊C8C2×D7911yescharacteristic, complemented
361D4⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8—791no—
364D4⋊C8—791no—
365D4⋊C8C2×D7911yescharacteristic, complemented
366D4⋊C8C2×D7911yescharacteristic, complemented
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—21no—
386#125C4×D7911yescharacteristic
387#124C4×D7911yescharacteristic
388#123—21no—
389C22⋊Q8C4×D7911yescharacteristic
390#121C4×D7911yescharacteristic
391C4×Q8C22×D7911yescharacteristic
392C4×D4—21no—
393C4×D4—791no—
394C4×D4—791no—
395C4×D4—791no—
396C4×D4—791no—
397C4×D4—1581no—
398C4×D4—1581no—
399C4×D4C22×D7911yescharacteristic
400#117—21no—
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—791no—
409C4⋊C8—791no—
410C4⋊C8—791no—
411C4⋊C8—791no—
412C4⋊C8—791no—
413C4⋊C8—791no—
414C4⋊C8—1581no—
415C4⋊C8—1581no—
416C4⋊C8—1581no—
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—791no—
431C22⋊C8—791no—
432C22⋊C8—1581no—
433C22⋊C8—1581no—
434C22⋊C8—1581no—
435C22⋊C8—1581no—
436C22⋊C8—1581no—
437C22⋊C8—1581no—
438#97—21no—
439#97—1581no—
440C4×C8—21no—
441C4×C8—791no—
442C4×C8—791no—
443C4×C8—791no—
444C4×C8—791no—
445C4×C8—1581no—
446C4×C8C22×D7911yescharacteristic
447C4×C8C22×D7911yescharacteristic
448C24—1581no—
449C2×Q8C2×C4×D7911yescharacteristic
450C2×D4—21no—
451C2×D4—3161no—
452C2×D4—791no—
453C2×D4—791no—
454C2×D4—1582no—
455C2×D4—1581no—
456C2×D4C2×C4×D7911yescharacteristic
457C22×C4—21no—
458C22×C4—6321no—
459C22×C4—6321no—
460C22×C4—1581no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4C2×C4×D7911yescharacteristic
466C22×C4D4×D7911yescharacteristic
467C22×C4#1192962311yescharacteristic
468SD16—3161no—
469SD16—1582no—
470D8—3161no—
471D8—1582no—
472M16—6321no—
473M16—6321no—
474M16—6321no—
475C2×C8—24no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—791no—
481C2×C8—791no—
482C2×C8—791no—
483C2×C8—791no—
484C2×C8—791no—
485C2×C8—791no—
486C2×C8—791no—
487C2×C8—791no—
488C2×C8—6321no—
489C2×C8—6321no—
490C2×C8—6321no—
491C2×C8—6321no—
492C2×C8—1581no—
493C2×C8—1581no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—791no—
499C4⋊C4—791no—
500C4⋊C4—1581no—
501C4⋊C4C2×C4×D7911yescharacteristic
502C4⋊C4C2×C4×D7911yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—1581no—
509C22⋊C4—1581no—
510C22⋊C4—1581no—
511C22⋊C4—1581no—
512C42—21no—
513C42C2×C4×D7911yescharacteristic
514C42D4×D7911yescharacteristic
515C42#1192962311yescharacteristic
516C42C23×D7911yescharacteristic
517C23—21no—
518C23—3162no—
519C23—3162no—
520C23—6321no—
521C23—1581no—
522C23—1581no—
523C23(C22⋊D79)⋊C411yescharacteristic
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—42no—
529D4—3162no—
530D4—792no—
531D4—1581no—
532D4—1581no—
533D4—1581no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—3161no—
548C2×C4—6321no—
549C2×C4—6321no—
550C2×C4—6321no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4(C22⋊D79)⋊C411yescharacteristic
554C2×C4(C22⋊D79)⋊C411yescharacteristic
555C2×C4(C22⋊D79)⋊C411yescharacteristic
556C2×C4C22×C4×D7911yescharacteristic
557C2×C4C2×D4×D7911yescharacteristic
558C2×C4C2×#1192962311yescharacteristic
559C8—22no—
560C8—22no—
561C8—41no—
562C8—41no—
563C8—3161no—
564C8—3161no—
565C8—1581no—
566C8—1581no—
567C8—1581no—
568C8—1581no—
569C8—1581no—
570C8—1581no—
571C8—1581no—
572C8—1581no—
573C22—21no—
574C22—21no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—3161no—
579C22—3161no—
580C22—3161no—
581C22—3161no—
582C22—3161no—
583C22—6324no—
584C22—1582no—
585C22#1909867712yes—
586C22C2×(C22⋊D79)⋊C411yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—3161no—
595C4—3161no—
596C4#131×D7911yescharacteristic
597C4#1909932811yescharacteristic
598C4#1909936511yescharacteristic
599C4#1909937011yescharacteristic
600C2—22no—
601C2—81no—
602C2—3162no—
603C2—6321no—
604C2#2377213011yescharacteristic
605C2#2377223511yescharacteristic
606C2C2×#1909867711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1494C2
socleC22×C79C2×C4×D4
prime radical#19098696C4
cosocleC42×C79C24
hypercenter#1494D79
hypocenterC79#18014
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1494

Complements

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

NH ≅ G/Ntype
2. #23772560C2semidirect
3. #23772437C2semidirect
4. #23762287C2semidirect
5. #23762282C2semidirect
6. #23762275C2semidirect
7. #23762267C2semidirect
8. #23762261C2semidirect
9. #23762260C2semidirect
10. #23762197C2semidirect
11. #23762192C2semidirect
12. #23762186C2semidirect
13. C2×#19098292C2semidirect
14. #23761683C2semidirect
15. C79×#1494C2semidirect
25. #19098694C22semidirect
26. Q8×C79⋊C8C22semidirect
31. #19098641C22semidirect
38. D4×C79⋊C8C22semidirect
48. #19098315C22semidirect
49. #19098315C22semidirect
50. #19098313C22semidirect
51. #19098313C22semidirect
56. #19098284C22semidirect
57. #19098284C22semidirect
58. C8×C79⋊C8C22semidirect
59. C8×C79⋊C8C22semidirect
63. Q8⋊C8×C79C22semidirect
64. Q8⋊C8×C79C22semidirect
65. D4⋊C8×C79C22semidirect
66. D4⋊C8×C79C22semidirect
316. #1494D79semidirect
319. C79#18014semidirect
359. Q8⋊C8C2×D79semidirect
360. Q8⋊C8C2×D79semidirect
365. D4⋊C8C2×D79semidirect
366. D4⋊C8C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734432
#237725602
#237724372
#237622872
#237622822
#237622752
#237622672
#237622612
#237622602
#237621972
#237621922
#237621862
C2×#190982922
#237616832
C79×#14942
#1801479