Finite Group Databasebeta

← #34361085

Subgroups — #34361085

Order 20224.

742 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23767833C211yesmaximal, characteristic, complemented
2#23764803C211yesmaximal, characteristic, complemented
3#23764788C211yesmaximal, characteristic, complemented
4#23764579C211yesmaximal, characteristic
5#23764579C211yesmaximal, characteristic
6#23764158C211yesmaximal, characteristic, complemented
7#23764130C211yesmaximal, characteristic, complemented
8#23764008C211yesmaximal, characteristic, complemented
9#23763989C211yesmaximal, characteristic, complemented
10#23763978C211yesmaximal, characteristic, complemented
11#23763661C211yesmaximal, characteristic, complemented
12#23763599C211yesmaximal, characteristic, complemented
13#23763495C211yesmaximal, characteristic, complemented
14C4×#14579082C211yesmaximal, characteristic, complemented
15C79×#2363C211yesmaximal, characteristic
16C2×#14579094C2211yescharacteristic
17C2×#14579094C2211yescharacteristic
18C2×#14579094C2211yescharacteristic
19C2×#14579092C2211yescharacteristic
20C2×C4⋊(C79⋊C8)C2211yescharacteristic
21C2×#14579082C2211yescharacteristic
22C2×#14579082C2211yescharacteristic
23C2×#14579082C2211yescharacteristic
24C2×#14579082C2211yescharacteristic
25C2×C22⋊(C79⋊C8)C2211yescharacteristic
26C2×C22⋊(C79⋊C8)C2211yescharacteristic
27C2×C4×C79⋊C8C2211yescharacteristic
28#19098505—21no—
29#19098486—21no—
30#19098461C2211yescharacteristic
31#19098461C2211yescharacteristic
32#19098456—21no—
33#19098450—21no—
34#19098433C2211yescharacteristic
35#19098423C2211yescharacteristic
36#19098423C2211yescharacteristic
37C4×C4⋊Dic79C2211yescharacteristic
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×#121×C79C2211yescharacteristic
49C2×C4×D4×C79C2211yescharacteristic
50C2×#117×C79C2211yescharacteristic
51#390×C79—21no—
52#390×C79C2211yescharacteristic
53#389×C79C2211yescharacteristic
54#388×C79—21no—
55#386×C79—21no—
56C4×C4⋊C4×C79C2211yescharacteristic
57C4×C22⋊C4×C79—21no—
58C4×C22⋊C4×C79C2211yescharacteristic
59C22×C79⋊C8C2311yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C2×C4⋊Dic79—21no—
64C2×C4⋊Dic79—21no—
65C2×C4⋊Dic79C2311yescharacteristic
66C2×C4⋊Dic79C2311yescharacteristic
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×Dic79⋊C4—21no—
70C2×Dic79⋊C4—21no—
71C2×C22⋊Dic79—21no—
72C2×C22⋊Dic79—21no—
73C2×C4×Dic79—21no—
74#14579094—22no—
75#14579094—22no—
76#14579094—41no—
77#14579092—22no—
78C4⋊(C79⋊C8)—22no—
79#14579082—22no—
80#14579082—22no—
81#14579082—22no—
82#14579082—41no—
83C22⋊(C79⋊C8)—22no—
84C22⋊(C79⋊C8)—41no—
85C4×C79⋊C8—41no—
86C79⋊#95—21no—
87C79⋊#95—21no—
88C79⋊#95—21no—
89C79⋊#95—21no—
90C79⋊#95—21no—
91C22×D4×C79—21no—
92C22×D4×C79C2311yescharacteristic
93C23×C4×C79—21no—
94C23×C4×C79C2311yescharacteristic
95#121×C79—22no—
96#121×C79—22no—
97C4×D4×C79—22no—
98C4×D4×C79—41no—
99#117×C79—22no—
100#117×C79—41no—
101C2×C4⋊C4×C79—21no—
102C2×C4⋊C4×C79C2311yescharacteristic
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C22⋊C4×C79—21no—
106C2×C22⋊C4×C79—21no—
107C2×C22⋊C4×C79—21no—
108C2×C22⋊C4×C79—21no—
109C2×C22⋊C4×C79—21no—
110C2×C22⋊C4×C79D411yescharacteristic
111C2×C22⋊C4×C79D411yescharacteristic
112C2×C42×C79—21no—
113C2×C42×C79—21no—
114C2×C42×C79C2311yescharacteristic
115C2×C42×C79C2311yescharacteristic
116#95×C79—21no—
117#95×C79—21no—
118#95×C79D411yescharacteristic
119#95×C79D411yescharacteristic
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—21no—
124C2×C79⋊C8—21no—
125C2×C79⋊C8—21no—
126C2×C79⋊C8—21no—
127C2×C79⋊C8—21no—
128C2×C79⋊C8—21no—
129C2×C79⋊C8—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8#4112yes—
136C4⋊Dic79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—81no—
145C4⋊Dic79—81no—
146Dic79⋊C4—42no—
147Dic79⋊C4—81no—
148C22⋊Dic79—81no—
149C22⋊Dic79—81no—
150C4×Dic79—81no—
151C24×C79—21no—
152C24×C79C2×D411yescharacteristic
153C2×D4×C79—24no—
154C2×D4×C79—21no—
155C2×D4×C79—21no—
156C2×D4×C79—21no—
157C2×D4×C79—21no—
158C2×D4×C79—21no—
159C2×D4×C79—81no—
160C2×D4×C79—81no—
161C2×D4×C79#4112yes—
162C22×C4×C79—21no—
163C22×C4×C79—21no—
164C22×C4×C79—21no—
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—42no—
172C22×C4×C79—42no—
173C22×C4×C79—81no—
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2411yescharacteristic
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—81no—
185C4⋊C4×C79#4112yes—
186C4⋊C4×C79#4112yes—
187C22⋊C4×C79—24no—
188C22⋊C4×C79—22no—
189C22⋊C4×C79—22no—
190C22⋊C4×C79—42no—
191C22⋊C4×C79—42no—
192C22⋊C4×C79—42no—
193C22⋊C4×C79—81no—
194C42×C79—21no—
195C42×C79—21no—
196C42×C79—21no—
197C42×C79—21no—
198C42×C79—42no—
199C42×C79—81no—
200C23×C79—22no—
201C23×C79—21no—
202C23×C79—21no—
203C23×C79—21no—
204C23×C79—21no—
205C23×C79—21no—
206C23×C79—21no—
207C23×C79—21no—
208C23×C79—41no—
209C23×C79—41no—
210C23×C79—41no—
211C23×C79—41no—
212C23×C79—41no—
213C23×C79—41no—
214C23×C79C22×D411yescharacteristic
215D4×C79—22no—
216D4×C79—22no—
217D4×C79—41no—
218D4×C79—41no—
219D4×C79—84no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—22no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—22no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—21no—
234C2×C4×C79—21no—
235C2×C4×C79—41no—
236C2×C4×C79—41no—
237C2×C4×C79—41no—
238C2×C4×C79—41no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—41no—
242C2×C4×C79—41no—
243C2×C4×C79—41no—
244C2×C4×C79—41no—
245C2×C4×C79—41no—
246C2×C4×C79—84no—
247C2×C4×C79#13512yes—
248C2×C4×C79#13712yes—
249C2×C4×C79C2×#4111yescharacteristic
250C2×C4×C79C2×#4111yescharacteristic
251C2×C4×C79C2×#4111yescharacteristic
252C2×C4×C79C2×#4111yescharacteristic
253C2×C4×C79H(5,𝔽2)11yescharacteristic
254C2×C4×C792−1+411yescharacteristic
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79—41no—
265C2×Dic79—41no—
266C2×Dic79—41no—
267C79⋊C8—22no—
268C79⋊C8—41no—
269C79⋊C8—41no—
270C79⋊C8—41no—
271C22×C79—41no—
272C22×C79—41no—
273C22×C79—41no—
274C22×C79—42no—
275C22×C79—41no—
276C22×C79—42no—
277C22×C79—41no—
278C22×C79—42no—
279C22×C79—41no—
280C22×C79—41no—
281C22×C79—81no—
282C22×C79—81no—
283C22×C79—81no—
284C22×C79—81no—
285C22×C79#52511yescharacteristic
286C22×C79#54611yescharacteristic
287C22×C79#54711yescharacteristic
288C22×C79C2×#13511yescharacteristic
289C22×C79C2×#13711yescharacteristic
290C22×C79#57611yescharacteristic
291C22×C79#57611yescharacteristic
292C4×C79—22no—
293C4×C79—22no—
294C4×C79—41no—
295C4×C79—42no—
296C4×C79—41no—
297C4×C79—41no—
298C4×C79—42no—
299C4×C79—81no—
300C4×C79#52911yescharacteristic
301C4×C79#53911yescharacteristic
302C4×C79#55311yescharacteristic
303C4×C79#55611yescharacteristic
304Dic79—81no—
305Dic79—81no—
306Dic79—81no—
307Dic79—81no—
308#27041—791nomaximal, Sylow
309C2×C79—81no—
310C2×C79—82no—
311C2×C79#246611yescharacteristic
312C2×C79#310711yescharacteristic
313C2×C79#312211yescharacteristic
314C2×C79#330511yescharacteristic
315C2×C79#331411yescharacteristic
316C2×C79#333911yescharacteristic
317C2×C79#335711yescharacteristic
318#2497—791no—
319#2363D7911yescharacteristic
320#2071—791no—
321#2067—791no—
322#2033—791no—
323#2033—791no—
324#1947—791no—
325#1942—791no—
326#1923—791no—
327#1921—791no—
328#1919—791no—
329#1864—791no—
330#1854—791no—
331#1825—791no—
332C4×#102—791no—
333C79#2704111yescharacteristic, complemented, Sylow
334C2×#121C2×D7911yescharacteristic
335C2×C4×D4C2×D7911yescharacteristic
336C2×#117C2×D7911yescharacteristic
337C2×#107—791no—
338C2×#107—791no—
339C2×#107—791no—
340C2×#106—791no—
341C2×C4⋊C8—791no—
342C2×#102—791no—
343C2×#102—791no—
344C2×#102—791no—
345C2×#102—791no—
346C2×C22⋊C8—791no—
347C2×C22⋊C8—791no—
348C2×C4×C8—791no—
349#400—1581no—
350#396—1581no—
351#390—21no—
352#390C2×D7911yescharacteristic
353#389—791no—
354#389—791no—
355#389C2×D7911yescharacteristic
356#388—21no—
357#388—1581no—
358#387—1581no—
359#386—21no—
360#384—791no—
361#380—791no—
362#380—791no—
363C4×C4⋊C4—791no—
364C4×C4⋊C4C2×D7911yescharacteristic
365C4×C22⋊C4—21no—
366C4×C22⋊C4C2×D7911yescharacteristic
367#340—791no—
368#340—791no—
369#340—791no—
370#340—791no—
371#340—791no—
372#340—791no—
373#340—791no—
374#340—791no—
375#336—791no—
376#336—791no—
377C22×D4—21no—
378C22×D4C22×D7911yescharacteristic
379C23×C4—21no—
380C23×C4C22×D7911yescharacteristic
381C22×C8—791no—
382C22×C8—791no—
383C22×C8—791no—
384C22×C8—791no—
385#121—22no—
386#121—22no—
387C4×D4—22no—
388C4×D4—41no—
389#117—22no—
390#117—41no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—791no—
393C2×C4⋊C4—791no—
394C2×C4⋊C4—791no—
395C2×C4⋊C4—791no—
396C2×C4⋊C4—1581no—
397C2×C4⋊C4—1581no—
398C2×C4⋊C4—1581no—
399C2×C4⋊C4—1581no—
400C2×C4⋊C4C22×D7911yescharacteristic
401C2×C4⋊C4C22×D7911yescharacteristic
402C2×C4⋊C4C22×D7911yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1581no—
409C2×C22⋊C4—1581no—
410C2×C22⋊C4C22⋊D7911yescharacteristic
411C2×C22⋊C4C22⋊D7911yescharacteristic
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—1581no—
415C2×C42C22×D7911yescharacteristic
416C2×C42C22×D7911yescharacteristic
417#107—3161no—
418#107—1582no—
419#107—1582no—
420#106—1582no—
421C4⋊C8—1582no—
422#102—3161no—
423#102—1582no—
424#102—1582no—
425#102—1582no—
426C22⋊C8—3161no—
427C22⋊C8—1582no—
428C4×C8—3161no—
429#95—21no—
430#95—21no—
431#95—1581no—
432#95—1581no—
433#95—1581no—
434#95—1581no—
435#95—1581no—
436#95C22⋊D7911yescharacteristic
437#95C22⋊D7911yescharacteristic
438C24—21no—
439C24C2×C22⋊D7911yescharacteristic
440C2×D4—24no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—81no—
447C2×D4—81no—
448C2×D4#1192962412yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4C2×C22⋊D7911yescharacteristic
466C22×C4C2×C22⋊D7911yescharacteristic
467C22×C4C2×C22⋊D7911yescharacteristic
468C22×C4C2×C22⋊D7911yescharacteristic
469C22×C4C2×C22⋊D7911yescharacteristic
470C22×C4C23×D7911yescharacteristic
471C2×C8—792no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—81no—
488C4⋊C4—3162no—
489C4⋊C4—6321no—
490C4⋊C4—6321no—
491C4⋊C4—6321no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4#1192962312yes—
501C4⋊C4#1192962412yes—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C22⋊C4—6321no—
510C22⋊C4—6321no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—81no—
517C42—6321no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C22×C22⋊D7911yescharacteristic
533D4—22no—
534D4—22no—
535D4—41no—
536D4—41no—
537D4—84no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—84no—
565C2×C4—3161no—
566C2×C4—3161no—
567C2×C4—3161no—
568C2×C4—3161no—
569C2×C4—3161no—
570C2×C4—3161no—
571C2×C4—3161no—
572C2×C4—3161no—
573C2×C4—3161no—
574C2×C4—3161no—
575C2×C4—3161no—
576C2×C4—3161no—
577C2×C4#1457918112yes—
578C2×C4#1457919112yes—
579C2×C4#41×D7911yescharacteristic
580C2×C4C2×#1192962311yescharacteristic
581C2×C4C2×#1192962411yescharacteristic
582C2×C4C2×#1192962411yescharacteristic
583C2×C4#1457920611yescharacteristic
584C2×C4#1457920811yescharacteristic
585C8—3161no—
586C8—3161no—
587C8—3161no—
588C8—1582no—
589C22—41no—
590C22—41no—
591C22—42no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—41no—
597C22—42no—
598C22—41no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1909907411yescharacteristic
604C22#1909920711yescharacteristic
605C22#1909921211yescharacteristic
606C22C2×#1457918111yescharacteristic
607C22C2×#1457919111yescharacteristic
608C22#1909936611yescharacteristic
609C22#1909936611yescharacteristic
610C4—22no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—6321no—
619C4—6321no—
620C4—6321no—
621C4—6321no—
622C4#1909909311yescharacteristic
623C4#1909915511yescharacteristic
624C4#1909926311yescharacteristic
625C4#1909928311yescharacteristic
626C2—81no—
627C2—82no—
628C2#2376743011yescharacteristic
629C2#2377347111yescharacteristic
630C2#2377364311yescharacteristic
631C2#2377535011yescharacteristic
632C2#2377548111yescharacteristic
633C2#2377579611yescharacteristic
634C2#2377602011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2363C2
socleC23×C79C22×D4
prime radicalC2×#121×C79C22
hypercenter#2363D79
hypocenterC79#27041
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. #23767833C2semidirect
2. #23764803C2semidirect
3. #23764788C2semidirect
6. #23764158C2semidirect
7. #23764130C2semidirect
8. #23764008C2semidirect
9. #23763989C2semidirect
10. #23763978C2semidirect
11. #23763661C2semidirect
12. #23763599C2semidirect
13. #23763495C2semidirect
14. C4×#14579082C2semidirect
333. C79#27041semidirect

Maximal subgroups

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

subgroupindex
#237678332
#237648032
#237647882
#237645792
#237641582
#237641302
#237640082
#237639892
#237639782
#237636612
#237635992
#237634952
C4×#145790822
C79×#23632
#2704179