Finite Group Databasebeta

← #34310295

Subgroups — #34310295

Order 20224.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098844C211yesmaximal, characteristic
2#23766489C211yesmaximal, characteristic, complemented
3#23766311C211yesmaximal, characteristic
4#23764373C211yesmaximal, characteristic, complemented
5#23764170C211yesmaximal, characteristic, complemented
6#23764138C211yesmaximal, characteristic, complemented
7#23764133C211yesmaximal, characteristic, complemented
8#23763993C211yesmaximal, characteristic, complemented
9#23763972C211yesmaximal, characteristic, complemented
10#23763664C211yesmaximal, characteristic
11#23763650C211yesmaximal, characteristic
12#23763226C211yesmaximal, characteristic
13#23763226C211yesmaximal, characteristic
14C4×#14579083C211yesmaximal, characteristic
15C79×#1771C211yesmaximal, characteristic, complemented
16C2×#14579161C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18C2×C4×D316C2211yescharacteristic
19C2×C4×D316C2211yescharacteristic
20#19098844C418yescomplemented
21C2×#14579095C2211yescharacteristic
22C2×#14579095C2211yescharacteristic
23C2×#14579095C2211yescharacteristic
24C2×#14579093C2211yescharacteristic
25C2×#14579083C2211yescharacteristic
26C2×#14579083C2211yescharacteristic
27C2×#14579083C2211yescharacteristic
28C2×#14579083C2211yescharacteristic
29#19098467—21no—
30#19098467—21no—
31#19098461C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098461C2211yescharacteristic
34#19098461C2211yescharacteristic
35#19098459—21no—
36#19098434—21no—
37#19098434—21no—
38#19098429C2211yescharacteristic
39#19098422—21no—
40C4⋊C4×Dic79—21no—
41C4×C4⋊Dic79C2211yescharacteristic
42C4×C4⋊Dic79C2211yescharacteristic
43C4×C4⋊Dic79C2211yescharacteristic
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×#97×C79C2211yescharacteristic
55C2×#97×C79C2211yescharacteristic
56C2×C4×C8×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#336×C79C2211yescharacteristic
59#336×C79C2211yescharacteristic
60C22×Dic158—21no—
61C22×D316C2311yescharacteristic
62C22×C4×D79—21no—
63C22×C4×D79—21no—
64C22×C4×D79—21no—
65#14579161—22no—
66#14579161C2×C414yescomplemented
67C4×D316—22no—
68C4×D316—22no—
69C4×D316—22no—
70C4×D316—41no—
71C4×D316—41no—
72C4×D316C2×C414yes—
73C2×C4⋊Dic79—21no—
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×C4⋊Dic79C2311yescharacteristic
79C2×C4⋊Dic79C2311yescharacteristic
80C2×C4⋊Dic79C2311yescharacteristic
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#14579095C2×C414yescomplemented
94#14579093—41no—
95#14579083—22no—
96#14579083C2×C414yes—
97#14579083C2×C414yes—
98#14579083Q814yescomplemented
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
118#97×C79—22no—
119#97×C79—41no—
120C4×C8×C79—41no—
121#95×C79D411yescharacteristic
122#95×C79D411yescharacteristic
123C23×D79—21no—
124C2×Dic158—24no—
125C2×Dic158—81no—
126C2×Dic158—81no—
127C2×D316—21no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316C22×C412yes—
131C2×D316C2×Q812yes—
132C2×D316#4112yes—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79—21no—
137C22×Dic79—21no—
138C22×Dic79—21no—
139C22×Dic79—21no—
140C2×C4×D79—24no—
141C2×C4×D79—42no—
142C2×C4×D79—42no—
143C2×C4×D79—42no—
144C2×C4×D79—42no—
145C2×C4×D79—81no—
146C2×C4×D79—81no—
147C2×C4×D79—81no—
148C2×C4×D79—81no—
149C4⋊Dic79—24no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79C22×C412yes—
161C4⋊Dic79C22×C412yes—
162C4⋊Dic79C22×C412yes—
163C4⋊Dic79C2×Q812yes—
164Dic79⋊C4—81no—
165Dic79⋊C4—81no—
166Dic79⋊C4—81no—
167Dic79⋊C4—81no—
168(C2×D79)⋊C4—24no—
169(C2×D79)⋊C4—42no—
170(C2×D79)⋊C4—42no—
171C4×Dic79—42no—
172C4×Dic79—81no—
173C4×Dic79—81no—
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2411yescharacteristic
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×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79C22×C412yes—
191C2×C8×C79C22×C412yes—
192C2×C8×C79C2×Q812yes—
193C4⋊C4×C79—22no—
194C4⋊C4×C79—41no—
195C42×C79—21no—
196C42×C79—21no—
197C42×C79—21no—
198C42×C79—21no—
199C42×C79C22×C412yes—
200C42×C79#4112yes—
201C23×C79C22×D411yescharacteristic
202C22×D79—22no—
203C22×D79—22no—
204C22×D79—22no—
205C22×D79—41no—
206C22×D79—41no—
207C22×D79—41no—
208C22×D79—41no—
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×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79C4×D412yes—
225C2×C4×C79C4×D412yes—
226C2×C4×C79#13712yes—
227C2×C4×C79C23×C411yescharacteristic
228C2×C4×C79C22×Q811yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79H(5,𝔽2)11yescharacteristic
232C2×C4×C792−1+411yescharacteristic
233C2×Dic79—22no—
234C2×Dic79—22no—
235C2×Dic79—22no—
236C2×Dic79—22no—
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—
252C2×Dic79—41no—
253C2×Dic79—41no—
254D316—22no—
255D316—22no—
256D316—22no—
257D316C4×Q814yes—
258C4×D79—84no—
259C4×D79—84no—
260C4×D79—84no—
261Dic158—84no—
262C8×C79—22no—
263C8×C79—22no—
264C8×C79—22no—
265C8×C79—41no—
266C22×C79C2×C4×D411yescharacteristic
267C22×C79#54611yescharacteristic
268C22×C79D4×Q811yescharacteristic
269C22×C79C2×#13711yescharacteristic
270C22×C79#57511yescharacteristic
271C22×C79#57611yescharacteristic
272C22×C79#57611yescharacteristic
273C2×D79—24no—
274C2×D79—42no—
275C2×D79—42no—
276C2×D79—42no—
277C2×D79—42no—
278C2×D79—42no—
279C2×D79—42no—
280C4×C79—22no—
281C4×C79—22no—
282C4×C79—22no—
283C4×C79—41no—
284C4×C79—41no—
285C4×C79—41no—
286C4×C79C2×C4×Q811yescharacteristic
287C4×C79#52011yescharacteristic
288C4×C79#55211yescharacteristic
289C4×C79#55411yescharacteristic
290Dic79—42no—
291Dic79—42no—
292Dic79—42no—
293Dic79—42no—
294Dic79—81no—
295Dic79—81no—
296Dic79—81no—
297#22979—791nomaximal, Sylow
298C2×C79#236011yescharacteristic
299C2×C79C4×#13711yescharacteristic
300C2×C79#294811yescharacteristic
301C2×C79#331411yescharacteristic
302C2×C79#331911yescharacteristic
303C2×C79#338511yescharacteristic
304C2×C79#338911yescharacteristic
305D79—44no—
306C2×#474—791no—
307#2369—791no—
308#2349—791no—
309#1993—791no—
310#1949—791no—
311#1943—791no—
312#1942—791no—
313#1921—791no—
314#1918—791no—
315#1864—791no—
316#1862—791no—
317#1771D7911yescharacteristic, complemented
318#1762—791no—
319#1762—791no—
320C4×#102—791no—
321C79#2297911yescharacteristic, complemented, Sylow
322C2×C4⋊Q8—791no—
323C2×C4×D4—791no—
324C2×C4×D4—791no—
325C2×C4×D4—791no—
326#474—798no—
327C2×#107—791no—
328C2×#107—791no—
329C2×#107—791no—
330C2×#106—791no—
331C2×C4⋊C8C2×D7911yescharacteristic
332C2×#102—791no—
333C2×#102—791no—
334C2×#102—791no—
335C2×#102—791no—
336C2×#97C2×D7911yescharacteristic
337C2×#97C2×D7911yescharacteristic
338C2×C4×C8C2×D7911yescharacteristic
339#391—1581no—
340#391—1581no—
341#389—791no—
342#389—791no—
343#389—791no—
344#389—791no—
345#389—1581no—
346#385—1581no—
347#385—1581no—
348#382—791no—
349#380—1581no—
350C4×C4⋊C4—791no—
351C4×C4⋊C4—791no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—1581no—
354C4×C4⋊C4C2×D7911yescharacteristic
355C2×#95—1581no—
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364#336C2×D7911yescharacteristic
365#336C2×D7911yescharacteristic
366C22×Q8—1581no—
367C22×D4—791no—
368C23×C4—1581no—
369C23×C4—1581no—
370C23×C4—1581no—
371C22×C8C22×D7911yescharacteristic
372C22×C8C22×D7911yescharacteristic
373C22×C8C22×D7911yescharacteristic
374C22×C8C22×D7911yescharacteristic
375C4⋊Q8—794no—
376C4⋊Q8—1582no—
377C4×D4—3161no—
378C4×D4—3161no—
379C4×D4—794no—
380C4×D4—1582no—
381C4×D4—1582no—
382C4×D4—1582no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—791no—
387C2×C4⋊C4—791no—
388C2×C4⋊C4—791no—
389C2×C4⋊C4—791no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4—1581no—
395C2×C4⋊C4D31611yescharacteristic
396C2×C4⋊C4D31611yescharacteristic
397C2×C22⋊C4—1581no—
398C2×C22⋊C4—1581no—
399C2×C22⋊C4—1581no—
400C2×C42—1581no—
401C2×C42—1581no—
402C2×C42—1581no—
403C2×C42C22×D7911yescharacteristic
404C2×C42C22×D7911yescharacteristic
405C2×C42C22×D7911yescharacteristic
406#107—3161no—
407#107—794no—
408#107—794no—
409#106—3161no—
410C4⋊C8C4×D7914yescomplemented
411#102—794no—
412#102—794no—
413#102—794no—
414#102—1582no—
415#97—22no—
416#97—41no—
417C4×C8—41no—
418#95—1581no—
419#95—1581no—
420#95—1581no—
421#95—1581no—
422#95—1581no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95D31611yescharacteristic
428#95D31611yescharacteristic
429C24—1581no—
430C2×Q8—6321no—
431C2×Q8—6321no—
432C2×Q8—1584no—
433C2×D4—792no—
434C2×D4—792no—
435C2×D4—792no—
436C2×D4—1581no—
437C2×D4—1581no—
438C2×D4—1581no—
439C22×C4—3162no—
440C22×C4—3162no—
441C22×C4—3162no—
442C22×C4—3162no—
443C22×C4—6321no—
444C22×C4—6321no—
445C22×C4—6321no—
446C22×C4—6321no—
447C22×C4—1581no—
448C22×C4—1581no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1584no—
455C22×C4C2×D31611yescharacteristic
456C22×C4C2×D31611yescharacteristic
457C22×C4C2×D31611yescharacteristic
458C22×C4C2×D31611yescharacteristic
459C22×C4C2×D31611yescharacteristic
460C22×C4C2×D31611yescharacteristic
461C22×C4C23×D7911yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8C2×C4×D7912yes—
472C2×C8C2×C4×D7912yes—
473C2×C8Q8×D7912yes—
474C4⋊C4—22no—
475C4⋊C4—41no—
476C4⋊C4—792no—
477C4⋊C4—792no—
478C4⋊C4—792no—
479C4⋊C4—792no—
480C4⋊C4—6321no—
481C4⋊C4—6321no—
482C4⋊C4—6321no—
483C4⋊C4—6321no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4—1584no—
495C22⋊C4—3162no—
496C22⋊C4—3162no—
497C22⋊C4—1584no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—3162no—
503C42—6321no—
504C42—6321no—
505C42C2×C4×D7912yes—
506C42#1192962412yes—
507C23—3161no—
508C23—3161no—
509C23—3161no—
510C23—3161no—
511C23—1582no—
512C23—1582no—
513C23—1582no—
514C23C22×D31611yescharacteristic
515Q8—6324no—
516D4—794no—
517D4—1582no—
518D4—1582no—
519D4—1582no—
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—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
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—3161no—
546C2×C4—3161no—
547C2×C4—3161no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4—6324no—
553C2×C4—6324no—
554C2×C4—6324no—
555C2×C4—1582no—
556C2×C4—1582no—
557C2×C4—1582no—
558C2×C4—1582no—
559C2×C4C4×D31612yes—
560C2×C4C4×D31612yes—
561C2×C4#1457919412yes—
562C2×C4C22×C4×D7911yescharacteristic
563C2×C4C2×Q8×D7911yescharacteristic
564C2×C4#41×D7911yescharacteristic
565C2×C4C2×#1192962411yescharacteristic
566C2×C4#1457920611yescharacteristic
567C2×C4#1457920811yescharacteristic
568C8—22no—
569C8—22no—
570C8—22no—
571C8—41no—
572C22—3162no—
573C22—3162no—
574C22—3162no—
575C22—3162no—
576C22—3162no—
577C22—3162no—
578C22—1584no—
579C22C2×C4×D31611yescharacteristic
580C22#1909920811yescharacteristic
581C22Q8×D31611yescharacteristic
582C22C2×#1457919411yescharacteristic
583C22#1909936311yescharacteristic
584C22#1909936811yescharacteristic
585C22#1909936811yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—3162no—
593C4—3162no—
594C4—3162no—
595C4—3162no—
596C4—6321no—
597C4—6321no—
598C4—6321no—
599C4C4×Q8×D7911yescharacteristic
600C4#1909904211yescharacteristic
601C4#1909925211yescharacteristic
602C4#1909926811yescharacteristic
603C2—3164no—
604C2#2376640711yescharacteristic
605C2C4×#1457919411yescharacteristic
606C2#2377225311yescharacteristic
607C2#2377548311yescharacteristic
608C2#2377555811yescharacteristic
609C2#2377631211yescharacteristic
610C2#2377636011yescharacteristic
subgroupG/H
centerC23C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1771C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenter#1771D79
hypocenterC79#22979
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #23766489C2semidirect
4. #23764373C2semidirect
5. #23764170C2semidirect
6. #23764138C2semidirect
7. #23764133C2semidirect
8. #23763993C2semidirect
9. #23763972C2semidirect
15. C79×#1771C2semidirect
20. #19098844C4semidirect
66. #14579161C2×C4semidirect
92. #14579095C2×C4semidirect
93. #14579095C2×C4semidirect
98. #14579083Q8semidirect
117. C4⋊C8×C79C2×C4semidirect
317. #1771D79semidirect
321. C79#22979semidirect
410. C4⋊C8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190988442
#237664892
#237663112
#237643732
#237641702
#237641382
#237641332
#237639932
#237639722
#237636642
#237636502
#237632262
C4×#145790832
C79×#17712
#2297979