Finite Group Databasebeta

← #34296267

Subgroups — #34296267

Order 20224.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098918C211yesmaximal, characteristic
2#23766141C211yesmaximal, characteristic, complemented
3#23766111C211yesmaximal, characteristic
4#23764369C211yesmaximal, characteristic, complemented
5#23764216C211yesmaximal, characteristic
6#23764097C211yesmaximal, characteristic
7#23764090C211yesmaximal, characteristic, complemented
8#23764083C211yesmaximal, characteristic
9#23764076C211yesmaximal, characteristic, complemented
10#23763528C211yesmaximal, characteristic, complemented
11#23763231C211yesmaximal, characteristic, complemented
12#23763226C211yesmaximal, characteristic
13C4×#14579086C211yesmaximal, characteristic, complemented
14C4×#14579083C211yesmaximal, characteristic
15C79×#1726C211yesmaximal, characteristic, complemented
16C2×#14579153C2211yescharacteristic
17C2×#14579149C2211yescharacteristic
18C2×C4×Dic158C2211yescharacteristic
19C2×C4×D316C2211yescharacteristic
20#19098918C418yescomplemented
21C2×#14579086C2211yescharacteristic
22C2×#14579086C2211yescharacteristic
23C2×#14579086C2211yescharacteristic
24C2×#14579086C2211yescharacteristic
25C2×#14579083C2211yescharacteristic
26C2×#14579083C2211yescharacteristic
27C2×#14579083C2211yescharacteristic
28C2×#14579083C2211yescharacteristic
29#19098469C2211yescharacteristic
30#19098465C2211yescharacteristic
31#19098461C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098460—21no—
34#19098460—21no—
35#19098455—21no—
36#19098455—21no—
37#19098450—21no—
38#19098450—21no—
39#19098437—21no—
40#19098437—21no—
41#19098433C2211yescharacteristic
42#19098433C2211yescharacteristic
43C4×C4⋊Dic79C2211yescharacteristic
44C4×C4⋊Dic79C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098326C2211yescharacteristic
51#19098326C2211yescharacteristic
52#19098326C2211yescharacteristic
53C2×#97×C79C2211yescharacteristic
54C2×#97×C79C2211yescharacteristic
55C2×#97×C79C2211yescharacteristic
56C2×C4×C8×C79C2211yescharacteristic
57C2×C4×C8×C79C2211yescharacteristic
58C2×C4×C8×C79C2211yescharacteristic
59#376×C79C2211yescharacteristic
60C22×Dic158C2311yescharacteristic
61C22×D316C2311yescharacteristic
62C22×C4×D79—21no—
63#14579153—22no—
64#14579153C2×C414yescomplemented
65#14579149—22no—
66#14579149C2×C414yes—
67C4×Dic158—22no—
68C4×Dic158—22no—
69C4×D316—22no—
70C4×D316—22no—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79—21no—
73C2×C4⋊Dic79—21no—
74C2×C4⋊Dic79—21no—
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×Dic79⋊C4—21no—
86C2×(C2×D79)⋊C4—21no—
87C2×(C2×D79)⋊C4—21no—
88C2×(C2×D79)⋊C4—21no—
89C2×(C2×D79)⋊C4—21no—
90C2×(C2×D79)⋊C4—21no—
91C2×C4×Dic79—21no—
92C2×C4×Dic79—21no—
93C2×C4×Dic79—21no—
94#14579086—22no—
95#14579086—22no—
96#14579086C2×C414yescomplemented
97#14579086C2×C414yescomplemented
98#14579083—22no—
99#14579083—22no—
100#14579083C2×C414yes—
101#14579083C2×C414yes—
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×C42×C79C2311yescharacteristic
113C2×C42×C79C2311yescharacteristic
114C2×C42×C79C2311yescharacteristic
115#97×C79—22no—
116#97×C79—22no—
117#97×C79—22no—
118C4×C8×C79—22no—
119C4×C8×C79—22no—
120C4×C8×C79C2×C414yescomplemented
121#95×C79D411yescharacteristic
122#95×C79D411yescharacteristic
123#95×C79D411yescharacteristic
124#95×C79D411yescharacteristic
125C23×D79—21no—
126C2×Dic158—21no—
127C2×Dic158—21no—
128C2×Dic158—21no—
129C2×Dic158—21no—
130C2×Dic158—21no—
131C2×Dic158C22×C412yes—
132C2×D316—21no—
133C2×D316—21no—
134C2×D316—21no—
135C2×D316—21no—
136C2×D316—21no—
137C2×D316C22×C412yes—
138C22×Dic79—21no—
139C22×Dic79—21no—
140C22×Dic79—21no—
141C22×Dic79—21no—
142C22×Dic79—21no—
143C22×Dic79—21no—
144C22×Dic79—21no—
145C2×C4×D79—42no—
146C2×C4×D79—81no—
147C2×C4×D79—81no—
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⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—81no—
159C4⋊Dic79—81no—
160C4⋊Dic79—81no—
161C4⋊Dic79—81no—
162C4⋊Dic79C22×C412yes—
163C4⋊Dic79C22×C412yes—
164Dic79⋊C4—24no—
165Dic79⋊C4—24no—
166Dic79⋊C4—42no—
167Dic79⋊C4—81no—
168Dic79⋊C4—81no—
169(C2×D79)⋊C4—24no—
170(C2×D79)⋊C4—24no—
171(C2×D79)⋊C4—42no—
172(C2×D79)⋊C4—81no—
173(C2×D79)⋊C4—81no—
174C4×Dic79—42no—
175C4×Dic79—81no—
176C4×Dic79—81no—
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2411yescharacteristic
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×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79C22×C412yes—
193C2×C8×C79C22×C412yes—
194C2×C8×C79#4112yes—
195C2×C8×C79#4112yes—
196C42×C79—21no—
197C42×C79—21no—
198C42×C79C22×C412yes—
199C42×C79#4112yes—
200C42×C79#4112yes—
201C42×C79#4112yes—
202C23×C79C22×D411yescharacteristic
203C22×D79—22no—
204C22×D79—41no—
205C22×D79—41no—
206C22×D79—41no—
207C22×D79—41no—
208C22×D79—41no—
209C22×D79—41no—
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#12412yes—
227C2×C4×C79#12412yes—
228C2×C4×C79C23×C411yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
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—
254C2×Dic79—41no—
255D316—22no—
256D316—22no—
257D316—41no—
258D316—41no—
259C4×D79—84no—
260Dic158—22no—
261Dic158—22no—
262Dic158—41no—
263Dic158—41no—
264C8×C79—22no—
265C8×C79—22no—
266C8×C79—22no—
267C8×C79—22no—
268C22×C79C2×C4×D411yescharacteristic
269C22×C79C2×#12411yescharacteristic
270C22×C79#53211yescharacteristic
271C22×C79#57611yescharacteristic
272C22×C79#57711yescharacteristic
273C22×C79#57711yescharacteristic
274C22×C79#57811yescharacteristic
275C2×D79—42no—
276C2×D79—42no—
277C2×D79—42no—
278C2×D79—81no—
279C2×D79—81no—
280C2×D79—81no—
281C2×D79—81no—
282C4×C79—22no—
283C4×C79—22no—
284C4×C79—22no—
285C4×C79—22no—
286C4×C79—41no—
287C4×C79—41no—
288C4×C79C4×#4111yescharacteristic
289C4×C79C4×#4111yescharacteristic
290C4×C79#52911yescharacteristic
291C4×C79#52911yescharacteristic
292Dic79—42no—
293Dic79—42no—
294Dic79—42no—
295Dic79—81no—
296Dic79—81no—
297Dic79—81no—
298Dic79—81no—
299#21643—791nomaximal, Sylow
300C2×C79C4×#12411yescharacteristic
301C2×C79#295111yescharacteristic
302C2×C79#295211yescharacteristic
303C2×C79#313611yescharacteristic
304C2×C79#313711yescharacteristic
305C2×C79#317811yescharacteristic
306C2×C79#317911yescharacteristic
307D79—82no—
308C2×#488—791no—
309#2330—791no—
310#2326—791no—
311#1992—791no—
312#1959—791no—
313#1936—791no—
314#1935—791no—
315#1934—791no—
316#1933—791no—
317#1834—791no—
318#1763—791no—
319#1762—791no—
320C4×#103—791no—
321C4×#102—791no—
322#1726D7911yescharacteristic, complemented
323C79#2164311yescharacteristic, complemented, Sylow
324C2×#125—791no—
325C2×#124—791no—
326C2×C4×Q8—791no—
327C2×C4×D4—791no—
328#488—798no—
329C2×#103—791no—
330C2×#103—791no—
331C2×#103—791no—
332C2×#103—791no—
333C2×#102—791no—
334C2×#102—791no—
335C2×#102—791no—
336C2×#102—791no—
337C2×#97C2×D7911yescharacteristic
338C2×#97C2×D7911yescharacteristic
339C2×#97C2×D7911yescharacteristic
340C2×C4×C8C2×D7911yescharacteristic
341C2×C4×C8C2×D7911yescharacteristic
342C2×C4×C8C2×D7911yescharacteristic
343#391—791no—
344#390—791no—
345#389—791no—
346#389—791no—
347#389—1581no—
348#389—1581no—
349#388—1581no—
350#388—1581no—
351#387—1581no—
352#387—1581no—
353#385—1581no—
354#385—1581no—
355#384—791no—
356#384—791no—
357C4×C4⋊C4—791no—
358C4×C4⋊C4—791no—
359#376C2×D7911yescharacteristic
360#340—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368C22×Q8—791no—
369C22×D4—791no—
370C23×C4—1581no—
371C22×C8C22×D7911yescharacteristic
372C22×C8C22×D7911yescharacteristic
373C22×C8C22×D7911yescharacteristic
374C22×C8C22×D7911yescharacteristic
375#125—794no—
376#125—1582no—
377#124—794no—
378#124—1582no—
379C4×Q8—1582no—
380C4×Q8—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—1581no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4—1581no—
395C2×C4⋊C4—1581no—
396C2×C4⋊C4—1581no—
397C2×C4⋊C4—1581no—
398C2×C22⋊C4—1581no—
399C2×C22⋊C4—1581no—
400C2×C22⋊C4—1581no—
401C2×C22⋊C4—1581no—
402C2×C22⋊C4—1581no—
403C2×C42—1581no—
404C2×C42—1581no—
405C2×C42—1581no—
406C2×C42C22×D7911yescharacteristic
407C2×C42C22×D7911yescharacteristic
408C2×C42C22×D7911yescharacteristic
409#103—794no—
410#103—794no—
411#103—1582no—
412#103—1582no—
413#102—794no—
414#102—794no—
415#102—1582no—
416#102—1582no—
417#97—22no—
418#97—22no—
419#97—22no—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8C4×D7914yescomplemented
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428#95—1581no—
429#95D31611yescharacteristic
430#95D31611yescharacteristic
431#95D31611yescharacteristic
432#95D31611yescharacteristic
433C24—1581no—
434C2×Q8—792no—
435C2×Q8—1581no—
436C2×Q8—1581no—
437C2×Q8—1581no—
438C2×Q8—1581no—
439C2×Q8—1581no—
440C2×D4—792no—
441C2×D4—1581no—
442C2×D4—1581no—
443C2×D4—1581no—
444C2×D4—1581no—
445C2×D4—1581no—
446C22×C4—3162no—
447C22×C4—6321no—
448C22×C4—6321no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4C2×D31611yescharacteristic
457C22×C4C2×D31611yescharacteristic
458C22×C4C2×D31611yescharacteristic
459C22×C4C2×D31611yescharacteristic
460C22×C4C2×D31611yescharacteristic
461C22×C4C2×D31611yescharacteristic
462C22×C4C23×D7911yescharacteristic
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×C8#1192962412yes—
474C2×C8#1192962412yes—
475C4⋊C4—3162no—
476C4⋊C4—792no—
477C4⋊C4—792no—
478C4⋊C4—6321no—
479C4⋊C4—6321no—
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—
495C4⋊C4—1584no—
496C22⋊C4—3162no—
497C22⋊C4—6321no—
498C22⋊C4—6321no—
499C22⋊C4—1584no—
500C22⋊C4—1584no—
501C42—21no—
502C42—21no—
503C42—3162no—
504C42—6321no—
505C42—6321no—
506C42C2×C4×D7912yes—
507C42#1192962412yes—
508C42#1192962412yes—
509C42#1192962412yes—
510C23—3161no—
511C23—3161no—
512C23—3161no—
513C23—3161no—
514C23—3161no—
515C23—3161no—
516C23—1582no—
517C23C22×D31611yescharacteristic
518Q8—3161no—
519Q8—3161no—
520Q8—1582no—
521Q8—1582no—
522D4—3161no—
523D4—3161no—
524D4—1582no—
525D4—1582no—
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—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
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—3161no—
553C2×C4—3161no—
554C2×C4—3161no—
555C2×C4—3161no—
556C2×C4—3161no—
557C2×C4—3161no—
558C2×C4—6324no—
559C2×C4—1582no—
560C2×C4—1582no—
561C2×C4—1582no—
562C2×C4C4×D31612yes—
563C2×C4C4×D31612yes—
564C2×C4#1457914912yes—
565C2×C4#1457914912yes—
566C2×C4C22×C4×D7911yescharacteristic
567C2×C4C2×#1192962411yescharacteristic
568C2×C4C2×#1192962411yescharacteristic
569C2×C4C2×#1192962411yescharacteristic
570C2×C4C2×#1192962411yescharacteristic
571C2×C4C2×#1192962411yescharacteristic
572C8—22no—
573C8—22no—
574C8—22no—
575C8—22no—
576C22—3162no—
577C22—3162no—
578C22—3162no—
579C22—6321no—
580C22—6321no—
581C22—6321no—
582C22—6321no—
583C22C2×C4×D31611yescharacteristic
584C22C2×#1457914911yescharacteristic
585C22#1909911311yescharacteristic
586C22#1909936811yescharacteristic
587C22#1909937511yescharacteristic
588C22#1909937511yescharacteristic
589C22#1909938011yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—3162no—
597C4—3162no—
598C4—3162no—
599C4—6321no—
600C4—6321no—
601C4—6321no—
602C4—6321no—
603C4C4×#1192962411yescharacteristic
604C4C4×#1192962411yescharacteristic
605C4#1909909711yescharacteristic
606C4#1909909711yescharacteristic
607C2—6322no—
608C2C4×#1457914911yescharacteristic
609C2#2377227711yescharacteristic
610C2#2377228511yescharacteristic
611C2#2377375611yescharacteristic
612C2#2377376711yescharacteristic
613C2#2377406811yescharacteristic
614C2#2377408311yescharacteristic
subgroupG/H
centerC23C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1726C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenter#1726D79
hypocenterC79#21643
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ #376 ◁ #1726

Complements

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

NH ≅ G/Ntype
2. #23766141C2semidirect
4. #23764369C2semidirect
7. #23764090C2semidirect
9. #23764076C2semidirect
10. #23763528C2semidirect
11. #23763231C2semidirect
13. C4×#14579086C2semidirect
15. C79×#1726C2semidirect
20. #19098918C4semidirect
64. #14579153C2×C4semidirect
96. #14579086C2×C4semidirect
97. #14579086C2×C4semidirect
120. C4×C8×C79C2×C4semidirect
322. #1726D79semidirect
323. C79#21643semidirect
422. C4×C8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190989182
#237661412
#237661112
#237643692
#237642162
#237640972
#237640902
#237640832
#237640762
#237635282
#237632312
#237632262
C4×#145790862
C4×#145790832
C79×#17262
#2164379