Finite Group Databasebeta

← #34360995

Subgroups — #34360995

Order 20224.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23767811C211yesmaximal, characteristic
2#23764818C211yesmaximal, characteristic, complemented
3#23764795C211yesmaximal, characteristic, complemented
4#23764600C211yesmaximal, characteristic
5#23764593C211yesmaximal, characteristic
6#23764168C211yesmaximal, characteristic, complemented
7#23764130C211yesmaximal, characteristic, complemented
8#23764008C211yesmaximal, characteristic
9#23763982C211yesmaximal, characteristic, complemented
10#23763971C211yesmaximal, characteristic, complemented
11#23763654C211yesmaximal, characteristic, complemented
12#23763588C211yesmaximal, characteristic
13#23763486C211yesmaximal, characteristic
14C4×#14579085C211yesmaximal, characteristic, complemented
15C79×#2360C211yesmaximal, characteristic
16C2×#14579094C2211yescharacteristic
17C2×#14579094C2211yescharacteristic
18C2×#14579094C2211yescharacteristic
19C2×#14579092C2211yescharacteristic
20C2×C4⋊(C79⋊C8)C2211yescharacteristic
21C2×#14579085C2211yescharacteristic
22C2×#14579085C2211yescharacteristic
23C2×#14579085C2211yescharacteristic
24C2×#14579085C2211yescharacteristic
25C2×C22⋊(C79⋊C8)C2211yescharacteristic
26C2×C22⋊(C79⋊C8)C2211yescharacteristic
27C2×C4×C79⋊C8C2211yescharacteristic
28#19098504—21no—
29#19098491—21no—
30#19098461C2211yescharacteristic
31#19098461C2211yescharacteristic
32#19098451—21no—
33#19098430—21no—
34#19098429C2211yescharacteristic
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×C22⋊Q8×C79C2211yescharacteristic
49C2×C4×Q8×C79C2211yescharacteristic
50C22×C4⋊C4×C79C2211yescharacteristic
51#391×C79C2211yescharacteristic
52#389×C79—21no—
53#389×C79C2211yescharacteristic
54#387×C79—21no—
55#385×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⋊Dic79C2311yescharacteristic
64C2×C4⋊Dic79C2311yescharacteristic
65C2×C4⋊Dic79C2311yescharacteristic
66C2×C4⋊Dic79C2311yescharacteristic
67C2×Dic79⋊C4—21no—
68C2×Dic79⋊C4—21no—
69C2×C22⋊Dic79—21no—
70C2×C22⋊Dic79—21no—
71C2×C4×Dic79—21no—
72#14579094—22no—
73#14579094—22no—
74#14579094—41no—
75#14579092—22no—
76C4⋊(C79⋊C8)—22no—
77#14579085—22no—
78#14579085—22no—
79#14579085—22no—
80#14579085—41no—
81C22⋊(C79⋊C8)—22no—
82C22⋊(C79⋊C8)—41no—
83C4×C79⋊C8—41no—
84C79⋊#95—21no—
85C79⋊#95—21no—
86C79⋊#95—21no—
87C79⋊#95—21no—
88C79⋊#95—21no—
89C79⋊#95—21no—
90C79⋊#95—21no—
91C22×Q8×C79C2311yescharacteristic
92C23×C4×C79—21no—
93C23×C4×C79C2311yescharacteristic
94C22⋊Q8×C79—22no—
95C22⋊Q8×C79—22no—
96C4×Q8×C79—22no—
97C4×Q8×C79—41no—
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79—22no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79—21no—
102C2×C4⋊C4×C79—21no—
103C2×C4⋊C4×C79—41no—
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C4⋊C4×C79C2311yescharacteristic
106C2×C4⋊C4×C79C2311yescharacteristic
107C2×C22⋊C4×C79—21no—
108C2×C22⋊C4×C79D411yescharacteristic
109C2×C22⋊C4×C79D411yescharacteristic
110C2×C42×C79—21no—
111C2×C42×C79—21no—
112C2×C42×C79C2311yescharacteristic
113C2×C42×C79C2311yescharacteristic
114#95×C79—21no—
115#95×C79—21no—
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—
144Dic79⋊C4—81no—
145Dic79⋊C4—81no—
146C22⋊Dic79—42no—
147C22⋊Dic79—81no—
148C4×Dic79—42no—
149C24×C79C2×D411yescharacteristic
150C2×Q8×C79—21no—
151C2×Q8×C79—21no—
152C2×Q8×C79—21no—
153C2×Q8×C79—21no—
154C2×Q8×C79—21no—
155C2×Q8×C79#4112yes—
156C22×C4×C79—21no—
157C22×C4×C79—21no—
158C22×C4×C79—21no—
159C22×C4×C79—21no—
160C22×C4×C79—21no—
161C22×C4×C79—21no—
162C22×C4×C79—21no—
163C22×C4×C79—21no—
164C22×C4×C79—21no—
165C22×C4×C79—21no—
166C22×C4×C79—42no—
167C22×C4×C79—42no—
168C22×C4×C79—81no—
169C22×C4×C79C2×D411yescharacteristic
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2411yescharacteristic
175C4⋊C4×C79—24no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—42no—
181C4⋊C4×C79—42no—
182C4⋊C4×C79—81no—
183C4⋊C4×C79#4112yes—
184C4⋊C4×C79#4112yes—
185C22⋊C4×C79—24no—
186C22⋊C4×C79—22no—
187C22⋊C4×C79—22no—
188C42×C79—21no—
189C42×C79—21no—
190C42×C79—21no—
191C42×C79—21no—
192C42×C79—42no—
193C42×C79—81no—
194C23×C79—21no—
195C23×C79—21no—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79—21no—
199C23×C79—21no—
200C23×C79#13512yes—
201C23×C79C22×D411yescharacteristic
202Q8×C79—22no—
203Q8×C79—22no—
204Q8×C79—41no—
205Q8×C79—41no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—22no—
209C2×C4×C79—22no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—22no—
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—41no—
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—84no—
237C2×C4×C79#13712yes—
238C2×C4×C79C2×#4111yescharacteristic
239C2×C4×C79C2×#4111yescharacteristic
240C2×C4×C79C2×#4111yescharacteristic
241C2×C4×C79C2×#4111yescharacteristic
242C2×C4×C79H(5,𝔽2)11yescharacteristic
243C2×C4×C792−1+411yescharacteristic
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—
255C2×Dic79—41no—
256C79⋊C8—22no—
257C79⋊C8—41no—
258C79⋊C8—41no—
259C79⋊C8—41no—
260C22×C79—22no—
261C22×C79—22no—
262C22×C79—22no—
263C22×C79—41no—
264C22×C79—41no—
265C22×C79—41no—
266C22×C79—41no—
267C22×C79#52511yescharacteristic
268C22×C79#54611yescharacteristic
269C22×C79#54711yescharacteristic
270C22×C79C2×#13511yescharacteristic
271C22×C79C2×#13711yescharacteristic
272C22×C79#57611yescharacteristic
273C22×C79#57811yescharacteristic
274C4×C79—22no—
275C4×C79—41no—
276C4×C79—41no—
277C4×C79—42no—
278C4×C79—41no—
279C4×C79—42no—
280C4×C79—41no—
281C4×C79—42no—
282C4×C79—81no—
283C4×C79#52911yescharacteristic
284C4×C79#53911yescharacteristic
285C4×C79#55311yescharacteristic
286C4×C79#55611yescharacteristic
287Dic79—81no—
288Dic79—81no—
289Dic79—81no—
290Dic79—81no—
291#27035—791nomaximal, Sylow
292C2×C79—42no—
293C2×C79#246611yescharacteristic
294C2×C79#310811yescharacteristic
295C2×C79#312411yescharacteristic
296C2×C79#329711yescharacteristic
297C2×C79#331211yescharacteristic
298C2×C79#333611yescharacteristic
299C2×C79#335711yescharacteristic
300#2495—791no—
301#2360D7911yescharacteristic
302#2074—791no—
303#2068—791no—
304#2036—791no—
305#2035—791no—
306#1949—791no—
307#1942—791no—
308#1923—791no—
309#1920—791no—
310#1918—791no—
311#1863—791no—
312#1852—791no—
313#1823—791no—
314C4×#103—791no—
315C79#2703511yescharacteristic, complemented, Sylow
316C2×C22⋊Q8C2×D7911yescharacteristic
317C2×C4×Q8C2×D7911yescharacteristic
318C22×C4⋊C4C2×D7911yescharacteristic
319C2×#107—791no—
320C2×#107—791no—
321C2×#107—791no—
322C2×#106—791no—
323C2×C4⋊C8—791no—
324C2×#103—791no—
325C2×#103—791no—
326C2×#103—791no—
327C2×#103—791no—
328C2×C22⋊C8—791no—
329C2×C22⋊C8—791no—
330C2×C4×C8—791no—
331#400—1581no—
332#397—1581no—
333#391C2×D7911yescharacteristic
334#389—21no—
335#389—791no—
336#389—791no—
337#389C2×D7911yescharacteristic
338#388—1581no—
339#387—21no—
340#385—21no—
341#383—1581no—
342#382—791no—
343#380—791no—
344#380—791no—
345C4×C4⋊C4—791no—
346C4×C4⋊C4C2×D7911yescharacteristic
347C4×C22⋊C4—21no—
348C4×C22⋊C4C2×D7911yescharacteristic
349#340—791no—
350#340—791no—
351#340—791no—
352#340—791no—
353#340—791no—
354#340—791no—
355#340—791no—
356#340—791no—
357#336—791no—
358#336—791no—
359C22×Q8C22×D7911yescharacteristic
360C23×C4—21no—
361C23×C4C22×D7911yescharacteristic
362C22×C8—791no—
363C22×C8—791no—
364C22×C8—791no—
365C22×C8—791no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C4×Q8—22no—
369C4×Q8—41no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—41no—
376C2×C4⋊C4—791no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—791no—
379C2×C4⋊C4—791no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4C22×D7911yescharacteristic
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C4⋊C4C22×D7911yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1581no—
387C2×C22⋊C4—1581no—
388C2×C22⋊C4C22⋊D7911yescharacteristic
389C2×C22⋊C4C22⋊D7911yescharacteristic
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—1581no—
393C2×C42C22×D7911yescharacteristic
394C2×C42C22×D7911yescharacteristic
395#107—3161no—
396#107—1582no—
397#107—1582no—
398#106—1582no—
399C4⋊C8—1582no—
400#103—3161no—
401#103—1582no—
402#103—1582no—
403#103—1582no—
404C22⋊C8—3161no—
405C22⋊C8—1582no—
406C4×C8—3161no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—1581no—
412#95—1581no—
413#95—1581no—
414#95—1581no—
415#95—1581no—
416#95—1581no—
417#95—1581no—
418#95C22⋊D7911yescharacteristic
419#95C22⋊D7911yescharacteristic
420C24C2×C22⋊D7911yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1192962412yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—1581no—
441C22×C4—1581no—
442C22×C4—1581no—
443C22×C4—1581no—
444C22×C4C2×C22⋊D7911yescharacteristic
445C22×C4C2×C22⋊D7911yescharacteristic
446C22×C4C2×C22⋊D7911yescharacteristic
447C22×C4C2×C22⋊D7911yescharacteristic
448C22×C4C2×C22⋊D7911yescharacteristic
449C22×C4C23×D7911yescharacteristic
450C2×C8—792no—
451C2×C8—1581no—
452C2×C8—1581no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8—1581no—
456C2×C8—1581no—
457C2×C8—1581no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8—1581no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—6321no—
471C4⋊C4—6321no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4#1192962312yes—
481C4⋊C4#1192962412yes—
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—3162no—
486C22⋊C4—6321no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—81no—
493C42—3162no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#1457918112yes—
501C23C22×C22⋊D7911yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—84no—
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#1457919212yes—
550C2×C4#41×D7911yescharacteristic
551C2×C4C2×#1192962311yescharacteristic
552C2×C4C2×#1192962411yescharacteristic
553C2×C4C2×#1192962411yescharacteristic
554C2×C4#1457920611yescharacteristic
555C2×C4#1457920811yescharacteristic
556C8—3161no—
557C8—3161no—
558C8—3161no—
559C8—1582no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1909907411yescharacteristic
568C22#1909920711yescharacteristic
569C22#1909921211yescharacteristic
570C22C2×#1457918111yescharacteristic
571C22C2×#1457919211yescharacteristic
572C22#1909936611yescharacteristic
573C22#1909937811yescharacteristic
574C4—22no—
575C4—41no—
576C4—42no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4—6321no—
584C4—6321no—
585C4—6321no—
586C4—6321no—
587C4#1909909311yescharacteristic
588C4#1909915511yescharacteristic
589C4#1909926311yescharacteristic
590C4#1909928311yescharacteristic
591C2—42no—
592C2#2376743011yescharacteristic
593C2#2377348011yescharacteristic
594C2#2377366211yescharacteristic
595C2#2377525011yescharacteristic
596C2#2377545111yescharacteristic
597C2#2377576211yescharacteristic
598C2#2377602611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2360C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2360D79
hypocenterC79#27035
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #23764818C2semidirect
3. #23764795C2semidirect
6. #23764168C2semidirect
7. #23764130C2semidirect
9. #23763982C2semidirect
10. #23763971C2semidirect
11. #23763654C2semidirect
14. C4×#14579085C2semidirect
315. C79#27035semidirect

Maximal subgroups

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

subgroupindex
#237678112
#237648182
#237647952
#237646002
#237645932
#237641682
#237641302
#237640082
#237639822
#237639712
#237636542
#237635882
#237634862
C4×#145790852
C79×#23602
#2703579