Finite Group Databasebeta

← #34310116

Subgroups — #34310116

Order 20224.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098864C211yesmaximal, characteristic
2#23766494C211yesmaximal, characteristic, complemented
3#23766311C211yesmaximal, characteristic
4#23764365C211yesmaximal, characteristic, complemented
5#23764170C211yesmaximal, characteristic, complemented
6#23764133C211yesmaximal, characteristic, complemented
7#23764128C211yesmaximal, characteristic, complemented
8#23763993C211yesmaximal, characteristic, complemented
9#23763986C211yesmaximal, characteristic, complemented
10#23763664C211yesmaximal, characteristic
11#23763650C211yesmaximal, characteristic
12C4×#14579083C211yesmaximal, characteristic
13C4×#14579083C211yesmaximal, characteristic
14C4×#14579083C211yesmaximal, characteristic
15C8×C4⋊C4×C79C211yesmaximal, characteristic, complemented
16C2×#14579153C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18C2×C4×D316C2211yescharacteristic
19C2×C4×D316C2211yescharacteristic
20#19098864C418yescomplemented
21C2×#14579095C2211yescharacteristic
22C2×#14579095C2211yescharacteristic
23C2×#14579093C2211yescharacteristic
24C2×#14579093C2211yescharacteristic
25C2×#14579083C2211yescharacteristic
26C2×#14579083C2211yescharacteristic
27C2×#14579083C2211yescharacteristic
28C2×#14579083C2211yescharacteristic
29#19098461C2211yescharacteristic
30#19098461C2211yescharacteristic
31#19098461C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098458—21no—
34#19098445—21no—
35#19098444—21no—
36#19098434—21no—
37#19098434—21no—
38#19098429C2211yescharacteristic
39#19098422—21no—
40C4×C4⋊Dic79C2211yescharacteristic
41C4×C4⋊Dic79C2211yescharacteristic
42C4×C4⋊Dic79C2211yescharacteristic
43#19098414—21no—
44#95×D79—21no—
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098326C2211yescharacteristic
51#19098326C2211yescharacteristic
52#19098326C2211yescharacteristic
53C2×C4⋊C8×C79C2211yescharacteristic
54C2×C4×C8×C79C2211yescharacteristic
55C2×C4×C8×C79C2211yescharacteristic
56C2×C4×C8×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#336×C79C2211yescharacteristic
59#336×C79C2211yescharacteristic
60C22×D316C2311yescharacteristic
61C22×C4×D79—21no—
62C22×C4×D79—21no—
63C22×C4×D79—21no—
64#14579153—22no—
65#14579153C2×C414yescomplemented
66C4×D316—22no—
67C4×D316—22no—
68C4×D316—22no—
69C4×D316—41no—
70C4×D316—41no—
71C4×D316C2×C414yes—
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×C4⋊Dic79C2311yescharacteristic
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×Dic79⋊C4—21no—
82C2×Dic79⋊C4—21no—
83C2×Dic79⋊C4—21no—
84C2×Dic79⋊C4—21no—
85C2×(C2×D79)⋊C4—21no—
86C2×(C2×D79)⋊C4—21no—
87C2×(C2×D79)⋊C4—21no—
88C2×C4×Dic79—21no—
89C2×C4×Dic79—21no—
90C2×C4×Dic79—21no—
91#14579095—41no—
92#14579095C2×C414yescomplemented
93#14579093—41no—
94#14579093C2×C414yescomplemented
95#14579083—22no—
96#14579083C2×C414yes—
97#14579083C2×C414yes—
98#14579083Q814yes—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C22×C8×C79C2311yescharacteristic
109C22×C8×C79C2311yescharacteristic
110C22×C8×C79C2311yescharacteristic
111C22×C8×C79C2311yescharacteristic
112C2×C4⋊C4×C79D411yescharacteristic
113C2×C4⋊C4×C79D411yescharacteristic
114C2×C42×C79C2311yescharacteristic
115C2×C42×C79C2311yescharacteristic
116C2×C42×C79C2311yescharacteristic
117C4⋊C8×C79C2×C414yescomplemented
118C4×C8×C79—22no—
119C4×C8×C79—41no—
120C4×C8×C79—41no—
121#95×C79D411yescharacteristic
122#95×C79D411yescharacteristic
123C23×D79—21no—
124C2×D316—21no—
125C2×D316—21no—
126C2×D316—21no—
127C2×D316C22×C412yes—
128C2×D316C2×Q812yes—
129C2×D316#4112yes—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79—21no—
137C2×C4×D79—24no—
138C2×C4×D79—42no—
139C2×C4×D79—42no—
140C2×C4×D79—42no—
141C2×C4×D79—42no—
142C2×C4×D79—81no—
143C2×C4×D79—81no—
144C2×C4×D79—81no—
145C2×C4×D79—81no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79C22×C412yes—
157C4⋊Dic79C22×C412yes—
158C4⋊Dic79C22×C412yes—
159C4⋊Dic79C2×Q812yes—
160Dic79⋊C4—24no—
161Dic79⋊C4—24no—
162Dic79⋊C4—42no—
163Dic79⋊C4—81no—
164Dic79⋊C4—81no—
165Dic79⋊C4—81no—
166(C2×D79)⋊C4—24no—
167(C2×D79)⋊C4—42no—
168(C2×D79)⋊C4—42no—
169C4×Dic79—42no—
170C4×Dic79—81no—
171C4×Dic79—81no—
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2411yescharacteristic
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79C22×C412yes—
189C2×C8×C79C22×C412yes—
190C2×C8×C79C2×Q812yes—
191C4⋊C4×C79—22no—
192C4⋊C4×C79—22no—
193C42×C79—21no—
194C42×C79—21no—
195C42×C79—21no—
196C42×C79—21no—
197C42×C79C22×C412yes—
198C42×C79#4112yes—
199C23×C79C22×D411yescharacteristic
200C22×D79—22no—
201C22×D79—22no—
202C22×D79—22no—
203C22×D79—41no—
204C22×D79—41no—
205C22×D79—41no—
206C22×D79—41no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79C4×D412yes—
222C2×C4×C79C4×D412yes—
223C2×C4×C79#13512yes—
224C2×C4×C79#13512yes—
225C2×C4×C79C23×C411yescharacteristic
226C2×C4×C79C22×Q811yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C2×C4×C79C2×#4111yescharacteristic
229C2×C4×C79H(5,𝔽2)11yescharacteristic
230C2×C4×C792−1+411yescharacteristic
231C2×Dic79—22no—
232C2×Dic79—22no—
233C2×Dic79—22no—
234C2×Dic79—22no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252D316—22no—
253D316—22no—
254D316—22no—
255D316C4×Q814yes—
256C4×D79—84no—
257C4×D79—84no—
258C4×D79—84no—
259C8×C79—22no—
260C8×C79—22no—
261C8×C79—22no—
262C8×C79—41no—
263C22×C79C2×C4×D411yescharacteristic
264C22×C79#54611yescharacteristic
265C22×C79D4×Q811yescharacteristic
266C22×C79C2×#13511yescharacteristic
267C22×C79C2×#13511yescharacteristic
268C22×C79#57611yescharacteristic
269C22×C79#57711yescharacteristic
270C2×D79—24no—
271C2×D79—42no—
272C2×D79—42no—
273C2×D79—42no—
274C2×D79—42no—
275C2×D79—42no—
276C2×D79—42no—
277C4×C79—22no—
278C4×C79—22no—
279C4×C79—22no—
280C4×C79—22no—
281C4×C79—41no—
282C4×C79—41no—
283C4×C79C2×C4×Q811yescharacteristic
284C4×C79#52011yescharacteristic
285C4×C79#55211yescharacteristic
286C4×C79#55411yescharacteristic
287Dic79—42no—
288Dic79—42no—
289Dic79—42no—
290Dic79—42no—
291Dic79—81no—
292Dic79—81no—
293Dic79—81no—
294#22966—791nomaximal, Sylow
295C2×C79#236011yescharacteristic
296C2×C79C4×#13511yescharacteristic
297C2×C79#294611yescharacteristic
298C2×C79#329611yescharacteristic
299C2×C79#329711yescharacteristic
300C2×C79#338011yescharacteristic
301C2×C79#338111yescharacteristic
302D79—44no—
303C2×#478—791no—
304#2370—791no—
305#2349—791no—
306#1991—791no—
307#1949—791no—
308#1942—791no—
309#1941—791no—
310#1921—791no—
311#1920—791no—
312#1864—791no—
313#1862—791no—
314C8×C4⋊C4D7911yescharacteristic, complemented
315C4×#102—791no—
316C4×#102—791no—
317C4×#102—791no—
318C79#2296611yescharacteristic, complemented, Sylow
319C2×#125—791no—
320C2×C4×D4—791no—
321C2×C4×D4—791no—
322C2×C4×D4—791no—
323#478—798no—
324C2×#107—791no—
325C2×#107—791no—
326C2×#106—791no—
327C2×#106—791no—
328C2×C4⋊C8C2×D7911yescharacteristic
329C2×#102—791no—
330C2×#102—791no—
331C2×#102—791no—
332C2×#102—791no—
333C2×C4×C8C2×D7911yescharacteristic
334C2×C4×C8C2×D7911yescharacteristic
335C2×C4×C8C2×D7911yescharacteristic
336#389—791no—
337#389—791no—
338#389—791no—
339#389—791no—
340#389—1581no—
341#387—1581no—
342#387—1581no—
343#385—1581no—
344#385—1581no—
345#382—791no—
346#380—1581no—
347C4×C4⋊C4—791no—
348C4×C4⋊C4—791no—
349C4×C4⋊C4—791no—
350C4×C4⋊C4—1581no—
351C4×C4⋊C4C2×D7911yescharacteristic
352C2×#95—1581no—
353#340—791no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#336C2×D7911yescharacteristic
362#336C2×D7911yescharacteristic
363C22×D4—791no—
364C23×C4—1581no—
365C23×C4—1581no—
366C23×C4—1581no—
367C22×C8C22×D7911yescharacteristic
368C22×C8C22×D7911yescharacteristic
369C22×C8C22×D7911yescharacteristic
370C22×C8C22×D7911yescharacteristic
371#125—794no—
372#125—1582no—
373C4×D4—3161no—
374C4×D4—3161no—
375C4×D4—794no—
376C4×D4—1582no—
377C4×D4—1582no—
378C4×D4—1582no—
379C2×C4⋊C4—791no—
380C2×C4⋊C4—791no—
381C2×C4⋊C4—791no—
382C2×C4⋊C4—791no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—1581no—
387C2×C4⋊C4—1581no—
388C2×C4⋊C4—1581no—
389C2×C4⋊C4—1581no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4D31611yescharacteristic
393C2×C4⋊C4D31611yescharacteristic
394C2×C22⋊C4—1581no—
395C2×C22⋊C4—1581no—
396C2×C22⋊C4—1581no—
397C2×C42—1581no—
398C2×C42—1581no—
399C2×C42—1581no—
400C2×C42C22×D7911yescharacteristic
401C2×C42C22×D7911yescharacteristic
402C2×C42C22×D7911yescharacteristic
403#107—3161no—
404#107—794no—
405#106—3161no—
406#106—794no—
407C4⋊C8C4×D7914yescomplemented
408#102—794no—
409#102—794no—
410#102—794no—
411#102—1582no—
412C4×C8—22no—
413C4×C8—41no—
414C4×C8—41no—
415#95—1581no—
416#95—1581no—
417#95—1581no—
418#95—1581no—
419#95—1581no—
420#95—1581no—
421#95—1581no—
422#95—1581no—
423#95—1581no—
424#95D31611yescharacteristic
425#95D31611yescharacteristic
426C24—1581no—
427C2×D4—792no—
428C2×D4—792no—
429C2×D4—792no—
430C2×D4—1581no—
431C2×D4—1581no—
432C2×D4—1581no—
433C22×C4—3162no—
434C22×C4—3162no—
435C22×C4—3162no—
436C22×C4—3162no—
437C22×C4—6321no—
438C22×C4—6321no—
439C22×C4—6321no—
440C22×C4—6321no—
441C22×C4—1581no—
442C22×C4—1581no—
443C22×C4—1581no—
444C22×C4—1581no—
445C22×C4—1581no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4—1584no—
449C22×C4C2×D31611yescharacteristic
450C22×C4C2×D31611yescharacteristic
451C22×C4C2×D31611yescharacteristic
452C22×C4C2×D31611yescharacteristic
453C22×C4C2×D31611yescharacteristic
454C22×C4C2×D31611yescharacteristic
455C22×C4C23×D7911yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8C2×C4×D7912yes—
466C2×C8C2×C4×D7912yes—
467C2×C8Q8×D7912yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—3162no—
471C4⋊C4—792no—
472C4⋊C4—792no—
473C4⋊C4—792no—
474C4⋊C4—792no—
475C4⋊C4—6321no—
476C4⋊C4—6321no—
477C4⋊C4—6321no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1584no—
489C4⋊C4—1584no—
490C22⋊C4—3162no—
491C22⋊C4—3162no—
492C22⋊C4—1584no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—3162no—
498C42—6321no—
499C42—6321no—
500C42C2×C4×D7912yes—
501C42#1192962412yes—
502C23—3161no—
503C23—3161no—
504C23—3161no—
505C23—3161no—
506C23—1582no—
507C23—1582no—
508C23—1582no—
509C23C22×D31611yescharacteristic
510D4—794no—
511D4—1582no—
512D4—1582no—
513D4—1582no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—6324no—
546C2×C4—6324no—
547C2×C4—6324no—
548C2×C4—1582no—
549C2×C4—1582no—
550C2×C4—1582no—
551C2×C4—1582no—
552C2×C4C4×D31612yes—
553C2×C4C4×D31612yes—
554C2×C4#1457918312yes—
555C2×C4#1457918312yes—
556C2×C4C22×C4×D7911yescharacteristic
557C2×C4C2×Q8×D7911yescharacteristic
558C2×C4#41×D7911yescharacteristic
559C2×C4C2×#1192962411yescharacteristic
560C2×C4#1457920611yescharacteristic
561C2×C4#1457920811yescharacteristic
562C8—22no—
563C8—22no—
564C8—22no—
565C8—41no—
566C22—3162no—
567C22—3162no—
568C22—3162no—
569C22—3162no—
570C22—3162no—
571C22—3162no—
572C22—1584no—
573C22C2×C4×D31611yescharacteristic
574C22#1909920811yescharacteristic
575C22Q8×D31611yescharacteristic
576C22C2×#1457918311yescharacteristic
577C22C2×#1457918311yescharacteristic
578C22#1909936811yescharacteristic
579C22#1909937511yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—3162no—
587C4—3162no—
588C4—3162no—
589C4—3162no—
590C4—6321no—
591C4—6321no—
592C4—6321no—
593C4C4×Q8×D7911yescharacteristic
594C4#1909904211yescharacteristic
595C4#1909925211yescharacteristic
596C4#1909926811yescharacteristic
597C2—3164no—
598C2#2376640711yescharacteristic
599C2C4×#1457918311yescharacteristic
600C2#2377223411yescharacteristic
601C2#2377523711yescharacteristic
602C2#2377525211yescharacteristic
603C2#2377626811yescharacteristic
604C2#2377627711yescharacteristic
subgroupG/H
centerC23C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC8×C4⋊C4×C79C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenterC8×C4⋊C4D79
hypocenterC79#22966
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #23766494C2semidirect
4. #23764365C2semidirect
5. #23764170C2semidirect
6. #23764133C2semidirect
7. #23764128C2semidirect
8. #23763993C2semidirect
9. #23763986C2semidirect
15. C8×C4⋊C4×C79C2semidirect
20. #19098864C4semidirect
65. #14579153C2×C4semidirect
92. #14579095C2×C4semidirect
94. #14579093C2×C4semidirect
117. C4⋊C8×C79C2×C4semidirect
314. C8×C4⋊C4D79semidirect
318. C79#22966semidirect
407. C4⋊C8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190988642
#237664942
#237663112
#237643652
#237641702
#237641332
#237641282
#237639932
#237639862
#237636642
#237636502
C4×#145790832
C8×C4⋊C4×C792
#2296679