Finite Group Databasebeta

← #34329638

Subgroups — #34329638

Order 20224.

746 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×#19098855C211yesmaximal, characteristic, complemented
2C2×#19098624C211yesmaximal, characteristic
3#23766082C211yesmaximal, characteristic
4#23764831C211yesmaximal, characteristic
5#23764826C211yesmaximal, characteristic
6#23764603C211yesmaximal, characteristic
7#23764596C211yesmaximal, characteristic
8#23764353C211yesmaximal, characteristic, complemented
9#23764166C211yesmaximal, characteristic, complemented
10#23764140C211yesmaximal, characteristic, complemented
11#23764130C211yesmaximal, characteristic, complemented
12#23764089C211yesmaximal, characteristic, complemented
13#23764075C211yesmaximal, characteristic, complemented
14#23763412C211yesmaximal, characteristic, complemented
15C79×#2628C211yesmaximal, characteristic
16C2×#14579140C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18#19098855—24no—
19#19098624—24no—
20C2×#14579094C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×C4⋊(C79⋊C8)C2211yescharacteristic
26C2×#14579084C2211yescharacteristic
27C2×#14579084C2211yescharacteristic
28C2×#14579084C2211yescharacteristic
29C2×#14579084C2211yescharacteristic
30C2×C22⋊(C79⋊C8)C2211yescharacteristic
31C2×C22⋊(C79⋊C8)C2211yescharacteristic
32C2×#14579072C2211yescharacteristic
33C2×C4×C79⋊C8C2211yescharacteristic
34#19098469C2211yescharacteristic
35#19098461C2211yescharacteristic
36#19098458—21no—
37#19098457—21no—
38#19098451—21no—
39#19098444—21no—
40#19098423C2211yescharacteristic
41C4×C4⋊Dic79C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50C2×C4⋊Q8×C79C2211yescharacteristic
51C2×#117×C79C2211yescharacteristic
52#400×C79—21no—
53#399×C79—21no—
54#395×C79—21no—
55#393×C79—21no—
56#389×C79C2211yescharacteristic
57#389×C79C2211yescharacteristic
58#382×C79C2211yescharacteristic
59#380×C79C2211yescharacteristic
60#380×C79C2211yescharacteristic
61C22×Dic158C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66#14579140—22no—
67#14579140—22no—
68C4×Dic158—22no—
69C4×Dic158—22no—
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×C22⋊Dic79—21no—
77C2×C22⋊Dic79—21no—
78C2×C4×Dic79—21no—
79C2×C4×Dic79—21no—
80#14579094—22no—
81#14579094—22no—
82#14579094—22no—
83#14579092—22no—
84C4⋊(C79⋊C8)—22no—
85C4⋊(C79⋊C8)D414yes—
86#14579084—22no—
87#14579084—22no—
88#14579084—22no—
89#14579084—22no—
90C22⋊(C79⋊C8)—22no—
91C22⋊(C79⋊C8)—22no—
92#14579072—22no—
93C4×C79⋊C8—22no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C22×Q8×C79—21no—
100C23×C4×C79C2311yescharacteristic
101C4⋊Q8×C79—22no—
102C4⋊Q8×C79—22no—
103#117×C79—22no—
104#117×C79—22no—
105C2×C4⋊C4×C79—21no—
106C2×C4⋊C4×C79—21no—
107C2×C4⋊C4×C79—21no—
108C2×C4⋊C4×C79D411yescharacteristic
109C2×C4⋊C4×C79D411yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C22⋊C4×C79—21no—
115C2×C22⋊C4×C79—21no—
116C2×C22⋊C4×C79D411yescharacteristic
117C2×C22⋊C4×C79D411yescharacteristic
118C2×C42×C79C2311yescharacteristic
119C2×C42×C79C2311yescharacteristic
120#95×C79—21no—
121#95×C79—21no—
122#95×C79—21no—
123#95×C79—21no—
124#95×C79—21no—
125#95×C79—21no—
126C2×Dic158—21no—
127C2×Dic158—21no—
128C2×Dic158—21no—
129C2×Dic158—21no—
130C2×Dic158—21no—
131C2×Dic158#4112yes—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8C2×D412yes—
147C2×C79⋊C8C2×D412yes—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79#4112yes—
154Dic79⋊C4—24no—
155Dic79⋊C4—42no—
156Dic79⋊C4—81no—
157C22⋊Dic79—24no—
158C22⋊Dic79—81no—
159C4×Dic79—42no—
160C4×Dic79—81no—
161C24×C79C2×D411yescharacteristic
162C2×Q8×C79—42no—
163C2×Q8×C79—81no—
164C2×Q8×C79—81no—
165C22×C4×C79—21no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2411yescharacteristic
181C4⋊C4×C79—22no—
182C4⋊C4×C79—22no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—42no—
192C4⋊C4×C79—81no—
193C4⋊C4×C79—81no—
194C22⋊C4×C79—22no—
195C22⋊C4×C79—22no—
196C22⋊C4×C79—81no—
197C22⋊C4×C79—81no—
198C42×C79—21no—
199C42×C79—21no—
200C42×C79C2×D412yes—
201C42×C79#4112yes—
202C23×C79—21no—
203C23×C79—21no—
204C23×C79—21no—
205C23×C79—21no—
206C23×C79—21no—
207C23×C79—21no—
208C23×C79—21no—
209C23×C79C22×D411yescharacteristic
210Q8×C79—84no—
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×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79—41no—
237C2×C4×C79—41no—
238C2×C4×C79—41no—
239C2×C4×C79—41no—
240C2×C4×C79#12112yes—
241C2×C4×C79#12112yes—
242C2×C4×C79C22×D411yescharacteristic
243C2×C4×C79C2×#4111yescharacteristic
244C2×C4×C79C2×#4111yescharacteristic
245C2×C4×C79C2×#4111yescharacteristic
246C2×C4×C792−1+411yescharacteristic
247C2×C4×C792−1+411yescharacteristic
248C2×Dic79—22no—
249C2×Dic79—22no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260Dic158—22no—
261Dic158—22no—
262Dic158—41no—
263Dic158—41no—
264C79⋊C8—22no—
265C79⋊C8—22no—
266C79⋊C8—41no—
267C79⋊C8—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79—41no—
272C22×C79—41no—
273C22×C79—41no—
274C22×C79—41no—
275C22×C79C2×#12111yescharacteristic
276C22×C79#52511yescharacteristic
277C22×C79#53611yescharacteristic
278C22×C79#57611yescharacteristic
279C22×C79#57711yescharacteristic
280C22×C79#57811yescharacteristic
281C22×C79#57811yescharacteristic
282C4×C79—22no—
283C4×C79—22no—
284C4×C79—41no—
285C4×C79—41no—
286C4×C79—41no—
287C4×C79—81no—
288C4×C79—81no—
289C4×C79—81no—
290C4×C79—81no—
291C4×C79#54711yescharacteristic
292C4×C79#54811yescharacteristic
293C4×C79#55211yescharacteristic
294C4×C79#55611yescharacteristic
295Dic79—42no—
296Dic79—42no—
297Dic79—81no—
298Dic79—81no—
299#24714—791nomaximal, Sylow
300C2×C79—81no—
301C2×C79#241511yescharacteristic
302C2×C79#305911yescharacteristic
303C2×C79#306211yescharacteristic
304C2×C79#312111yescharacteristic
305C2×C79#312411yescharacteristic
306C2×C79#324211yescharacteristic
307C2×C79#324311yescharacteristic
308C2×#477—791no—
309C2×#433—791no—
310#2628D7911yescharacteristic
311#2323—791no—
312#2076—791no—
313#2075—791no—
314#2036—791no—
315#2035—791no—
316#1988—791no—
317#1949—791no—
318#1944—791no—
319#1942—791no—
320#1935—791no—
321#1933—791no—
322#1805—791no—
323C79#2471411yescharacteristic, complemented, Sylow
324C2×C4⋊Q8C2×D7911yescharacteristic
325C2×C22⋊Q8—791no—
326C2×C4×Q8—791no—
327C2×#117C2×D7911yescharacteristic
328#477—1584no—
329#433—1584no—
330C2×#107—791no—
331C2×#107—791no—
332C2×#107—791no—
333C2×#106—791no—
334C2×C4⋊C8—791no—
335C2×C4⋊C8—791no—
336C2×#103—791no—
337C2×#103—791no—
338C2×#103—791no—
339C2×#103—791no—
340C2×C22⋊C8—791no—
341C2×C22⋊C8—791no—
342C2×#97—791no—
343C2×C4×C8—791no—
344#400—21no—
345#399—21no—
346#395—21no—
347#393—21no—
348#391—791no—
349#389—791no—
350#389—1581no—
351#389C2×D7911yescharacteristic
352#389C2×D7911yescharacteristic
353#388—1581no—
354#388—1581no—
355#387—1581no—
356#382C2×D7911yescharacteristic
357#380—791no—
358#380C2×D7911yescharacteristic
359#380C2×D7911yescharacteristic
360C4×C4⋊C4—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340—791no—
369C22×Q8—21no—
370C22×Q8—791no—
371C23×C4C22×D7911yescharacteristic
372C22×C8—791no—
373C22×C8—791no—
374C22×C8—791no—
375C22×C8—791no—
376C4⋊Q8—22no—
377C4⋊Q8—22no—
378C22⋊Q8—1582no—
379C22⋊Q8—1582no—
380C4×Q8—1582no—
381C4×Q8—1582no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
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⋊C4C22⋊D7911yescharacteristic
394C2×C4⋊C4C22⋊D7911yescharacteristic
395C2×C4⋊C4C22×D7911yescharacteristic
396C2×C4⋊C4C22×D7911yescharacteristic
397C2×C4⋊C4C22×D7911yescharacteristic
398C2×C4⋊C4C22×D7911yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1581no—
402C2×C22⋊C4—1581no—
403C2×C22⋊C4C22⋊D7911yescharacteristic
404C2×C22⋊C4C22⋊D7911yescharacteristic
405C2×C42—1581no—
406C2×C42—1581no—
407C2×C42C22×D7911yescharacteristic
408C2×C42C22×D7911yescharacteristic
409#107—1582no—
410#107—1582no—
411#107—1582no—
412#106—1582no—
413C4⋊C8—794no—
414C4⋊C8—1582no—
415#103—1582no—
416#103—1582no—
417#103—1582no—
418#103—1582no—
419C22⋊C8—1582no—
420C22⋊C8—1582no—
421#97—1582no—
422C4×C8—1582no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—1581no—
430#95—1581no—
431#95—1581no—
432#95—1581no—
433#95—1581no—
434C24C2×C22⋊D7911yescharacteristic
435C2×Q8—42no—
436C2×Q8—81no—
437C2×Q8—81no—
438C2×Q8—792no—
439C2×Q8—1581no—
440C2×Q8—1581no—
441C2×Q8—1581no—
442C2×Q8—1581no—
443C2×Q8—1581no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4—1581no—
457C22×C4—1581no—
458C22×C4C2×C22⋊D7911yescharacteristic
459C22×C4C2×C22⋊D7911yescharacteristic
460C22×C4C2×C22⋊D7911yescharacteristic
461C22×C4C2×C22⋊D7911yescharacteristic
462C22×C4C2×C22⋊D7911yescharacteristic
463C22×C4C23×D7911yescharacteristic
464C2×C8—792no—
465C2×C8—792no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8—1581no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—3162no—
490C4⋊C4—792no—
491C4⋊C4—6321no—
492C4⋊C4—1584no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C22⋊C4—6321no—
503C22⋊C4—1584no—
504C42—21no—
505C42—21no—
506C42—3162no—
507C42—6321no—
508C42D4×D7912yes—
509C42#1192962312yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23C22×C22⋊D7911yescharacteristic
518Q8—84no—
519Q8—3161no—
520Q8—3161no—
521Q8—1582no—
522Q8—1582no—
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—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—3161no—
555C2×C4—3161no—
556C2×C4—3161no—
557C2×C4—3161no—
558C2×C4—3161no—
559C2×C4—3161no—
560C2×C4—3161no—
561C2×C4—3161no—
562C2×C4—1582no—
563C2×C4—1582no—
564C2×C4#1457913312yes—
565C2×C4#1457913312yes—
566C2×C4C2×D4×D7911yescharacteristic
567C2×C4#41×D7911yescharacteristic
568C2×C4#41×D7911yescharacteristic
569C2×C4C2×#1192962311yescharacteristic
570C2×C4#1457920911yescharacteristic
571C2×C4#1457920911yescharacteristic
572C8—3161no—
573C8—3161no—
574C8—1582no—
575C8—1582no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1457913311yescharacteristic
584C22#1909907011yescharacteristic
585C22#1909913711yescharacteristic
586C22#1909936611yescharacteristic
587C22#1909937411yescharacteristic
588C22#1909937811yescharacteristic
589C22#1909937811yescharacteristic
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—3162no—
600C4—3162no—
601C4—6321no—
602C4—6321no—
603C4#1909921511yescharacteristic
604C4#1909922311yescharacteristic
605C4#1909924711yescharacteristic
606C4#1909928411yescharacteristic
607C2—81no—
608C2#2376683011yescharacteristic
609C2#2377305211yescharacteristic
610C2#2377307611yescharacteristic
611C2#2377362311yescharacteristic
612C2#2377365611yescharacteristic
613C2#2377466811yescharacteristic
614C2#2377468311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2628C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2628D79
hypocenterC79#24714
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
1. C2×#19098855C2semidirect
8. #23764353C2semidirect
9. #23764166C2semidirect
10. #23764140C2semidirect
11. #23764130C2semidirect
12. #23764089C2semidirect
13. #23764075C2semidirect
14. #23763412C2semidirect
323. C79#24714semidirect

Maximal subgroups

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

subgroupindex
C2×#190988552
C2×#190986242
#237660822
#237648312
#237648262
#237646032
#237645962
#237643532
#237641662
#237641402
#237641302
#237640892
#237640752
#237634122
C79×#26282
#2471479