Finite Group Databasebeta

← #34356279

Subgroups — #34356279

Order 20224.

680 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#23767198C211yesmaximal, characteristic
2#23764891C211yesmaximal, characteristic, complemented
3#23764641C211yesmaximal, characteristic
4#23764548C211yesmaximal, characteristic, complemented
5#23764543C211yesmaximal, characteristic
6#23764332C211yesmaximal, characteristic
7#23764096C211yesmaximal, characteristic
8#23764041C211yesmaximal, characteristic, complemented
9#23764020C211yesmaximal, characteristic
10#23764020C211yesmaximal, characteristic
11#23763992C211yesmaximal, characteristic, complemented
12#23763966C211yesmaximal, characteristic, complemented
13#23763653C211yesmaximal, characteristic, complemented
14#23763233C211yesmaximal, characteristic, complemented
15C79×#2370C211yesmaximal, characteristic, complemented
16C2×#14579149C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18C2×#14579094C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×C4⋊(C79⋊C8)C2211yescharacteristic
21C2×#14579084C2211yescharacteristic
22C2×#14579084C2211yescharacteristic
23C2×#14579084C2211yescharacteristic
24C2×#14579084C2211yescharacteristic
25C2×#14579081C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×#14579081C2211yescharacteristic
28C2×#14579081C2211yescharacteristic
29C2×#14579072C2211yescharacteristic
30#19098498—21no—
31#19098492—21no—
32#19098469C2211yescharacteristic
33#19098465C2211yescharacteristic
34#19098465C2211yescharacteristic
35#19098461C2211yescharacteristic
36#19098453—21no—
37#19098452—21no—
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098318C2211yescharacteristic
45#19098318C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×#125×C79C2211yescharacteristic
49#389×C79—21no—
50#389×C79C2211yescharacteristic
51#389×C79C2211yescharacteristic
52#387×C79—21no—
53#387×C79—21no—
54C4×C4⋊C4×C79—21no—
55C4×C4⋊C4×C79C2211yescharacteristic
56C4×C4⋊C4×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C4⋊C4×C79C2211yescharacteristic
59C22×Dic158C2311yescharacteristic
60C22×D316C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65#14579149—22no—
66#14579149—41no—
67C4×Dic158—22no—
68C4×Dic158—41no—
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×Dic79⋊C4—21no—
72C2×Dic79⋊C4—21no—
73C2×(C2×D79)⋊C4—21no—
74C2×(C2×D79)⋊C4—21no—
75C2×(C2×D79)⋊C4—21no—
76C2×(C2×D79)⋊C4—21no—
77C2×(C2×D79)⋊C4—21no—
78C2×(C2×D79)⋊C4—21no—
79C2×C4×Dic79—21no—
80#14579094—22no—
81#14579094—41no—
82C4⋊(C79⋊C8)—41no—
83#14579084—22no—
84#14579084—22no—
85#14579084—22no—
86#14579084—22no—
87#14579081—22no—
88#14579081—22no—
89#14579081—22no—
90#14579081—41no—
91#14579072—41no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95#125×C79—22no—
96#125×C79—22no—
97C2×C4⋊C4×C79—21no—
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79D411yescharacteristic
102C2×C4⋊C4×C79D411yescharacteristic
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C4⋊C4×C79C2311yescharacteristic
106C2×C4⋊C4×C79C2311yescharacteristic
107C2×C42×C79—21no—
108C2×C42×C79—21no—
109C2×C42×C79—21no—
110C2×C42×C79C2311yescharacteristic
111C2×C42×C79C2311yescharacteristic
112C2×C42×C79C2311yescharacteristic
113#95×C79—21no—
114#95×C79—21no—
115#95×C79—21no—
116#95×C79—21no—
117#95×C79—21no—
118#95×C79D411yescharacteristic
119#95×C79D411yescharacteristic
120C23×D79—21no—
121C2×Dic158—21no—
122C2×Dic158—21no—
123C2×Dic158—21no—
124C2×Dic158—21no—
125C2×Dic158—21no—
126C2×Dic158#4112yes—
127C2×D316—21no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316—21no—
131C2×D316—21no—
132C2×D316—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⋊C8—21no—
147C2×C79⋊C8—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152Dic79⋊C4—42no—
153Dic79⋊C4—81no—
154(C2×D79)⋊C4—42no—
155(C2×D79)⋊C4—42no—
156(C2×D79)⋊C4—42no—
157(C2×D79)⋊C4—42no—
158(C2×D79)⋊C4—81no—
159(C2×D79)⋊C4—81no—
160C4×Dic79—42no—
161C22×C4×C79—21no—
162C22×C4×C79—21no—
163C22×C4×C79—21no—
164C22×C4×C79—21no—
165C22×C4×C79C2×D411yescharacteristic
166C22×C4×C79C2×D411yescharacteristic
167C22×C4×C79C2×D411yescharacteristic
168C22×C4×C79C2×D411yescharacteristic
169C22×C4×C79C2×D411yescharacteristic
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2411yescharacteristic
172C4⋊C4×C79—24no—
173C4⋊C4×C79—24no—
174C4⋊C4×C79—21no—
175C4⋊C4×C79—21no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—41no—
180C4⋊C4×C79—41no—
181C4⋊C4×C79—42no—
182C4⋊C4×C79—81no—
183C4⋊C4×C79#4112yes—
184C4⋊C4×C79#4112yes—
185C4⋊C4×C79#4112yes—
186C42×C79—21no—
187C42×C79—21no—
188C42×C79—21no—
189C42×C79—21no—
190C42×C79—21no—
191C42×C79—21no—
192C42×C79—42no—
193C42×C79—81no—
194C42×C79—81no—
195C23×C79C22×D411yescharacteristic
196C22×D79—41no—
197C22×D79—41no—
198C22×D79—41no—
199C22×D79—41no—
200C22×D79—41no—
201C22×D79—41no—
202C22×D79—41no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—22no—
206C2×C4×C79—22no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—22no—
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×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×C4×C79H(5,𝔽2)11yescharacteristic
238C2×C4×C792−1+411yescharacteristic
239C2×Dic79—22no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248D316—41no—
249D316—41no—
250D316—41no—
251D316—41no—
252Dic158—22no—
253Dic158—22no—
254Dic158—41no—
255Dic158—41no—
256C79⋊C8—41no—
257C79⋊C8—41no—
258C79⋊C8—41no—
259C79⋊C8—41no—
260C22×C79#52511yescharacteristic
261C22×C79#54611yescharacteristic
262C22×C79#54811yescharacteristic
263C22×C79#57511yescharacteristic
264C22×C79#57511yescharacteristic
265C22×C79#57611yescharacteristic
266C22×C79#57811yescharacteristic
267C2×D79—81no—
268C2×D79—81no—
269C2×D79—81no—
270C2×D79—81no—
271C2×D79—81no—
272C2×D79—81no—
273C2×D79—81no—
274C4×C79—41no—
275C4×C79—42no—
276C4×C79—42no—
277C4×C79—41no—
278C4×C79—42no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79—41no—
282C4×C79—41no—
283C4×C79—81no—
284C4×C79#52911yescharacteristic
285C4×C79#54211yescharacteristic
286C4×C79#55211yescharacteristic
287C4×C79#55911yescharacteristic
288Dic79—42no—
289Dic79—81no—
290Dic79—81no—
291#26719—791nomaximal, Sylow
292C2×C79#245811yescharacteristic
293C2×C79#311111yescharacteristic
294C2×C79#312411yescharacteristic
295C2×C79#330911yescharacteristic
296C2×C79#332711yescharacteristic
297C2×C79#337011yescharacteristic
298C2×C79#337711yescharacteristic
299D79—161no—
300#2448—791no—
301#2370D7911yescharacteristic, complemented
302#2086—791no—
303#2042—791no—
304#2026—791no—
305#2025—791no—
306#1983—791no—
307#1936—791no—
308#1928—791no—
309#1925—791no—
310#1925—791no—
311#1921—791no—
312#1917—791no—
313#1863—791no—
314#1763—791no—
315C79#2671911yescharacteristic, complemented, Sylow
316C2×#125C2×D7911yescharacteristic
317C2×#124—791no—
318C2×C4×Q8—791no—
319C2×#107—791no—
320C2×#107—791no—
321C2×C4⋊C8—791no—
322C2×#103—791no—
323C2×#103—791no—
324C2×#103—791no—
325C2×#103—791no—
326C2×#102—791no—
327C2×#102—791no—
328C2×#102—791no—
329C2×#102—791no—
330C2×#97—791no—
331#399—1581no—
332#397—1581no—
333#391—791no—
334#390—791no—
335#390—791no—
336#389—21no—
337#389—791no—
338#389C2×D7911yescharacteristic
339#389C2×D7911yescharacteristic
340#388—1581no—
341#388—1581no—
342#387—21no—
343#387—21no—
344C4×C4⋊C4—21no—
345C4×C4⋊C4C2×D7911yescharacteristic
346C4×C4⋊C4C2×D7911yescharacteristic
347C4×C4⋊C4C2×D7911yescharacteristic
348C4×C4⋊C4C2×D7911yescharacteristic
349#340—791no—
350#340—791no—
351#340—791no—
352#340—791no—
353#340—791no—
354#340—791no—
355#336—791no—
356#336—791no—
357#336—791no—
358#336—791no—
359C22×Q8—791no—
360C22×D4—791no—
361C22×C8—791no—
362C22×C8—791no—
363C22×C8—791no—
364C22×C8—791no—
365#125—22no—
366#125—22no—
367#124—3161no—
368#124—1582no—
369C4×Q8—3161no—
370C4×Q8—1582no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—791no—
377C2×C4⋊C4—1581no—
378C2×C4⋊C4—1581no—
379C2×C4⋊C4C22⋊D7911yescharacteristic
380C2×C4⋊C4C22⋊D7911yescharacteristic
381C2×C4⋊C4C22×D7911yescharacteristic
382C2×C4⋊C4C22×D7911yescharacteristic
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C4⋊C4C22×D7911yescharacteristic
385C2×C22⋊C4—1581no—
386C2×C22⋊C4—1581no—
387C2×C22⋊C4—1581no—
388C2×C22⋊C4—1581no—
389C2×C22⋊C4—1581no—
390C2×C22⋊C4—1581no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—1581no—
395C2×C42C22×D7911yescharacteristic
396C2×C42C22×D7911yescharacteristic
397C2×C42C22×D7911yescharacteristic
398#107—3161no—
399#107—1582no—
400C4⋊C8—3161no—
401#103—1582no—
402#103—1582no—
403#103—1582no—
404#103—1582no—
405#102—3161no—
406#102—1582no—
407#102—1582no—
408#102—1582no—
409#97—3161no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1581no—
416#95—1581no—
417#95—1581no—
418#95C22⋊D7911yescharacteristic
419#95C22⋊D7911yescharacteristic
420C24—1581no—
421C2×Q8—792no—
422C2×Q8—1581no—
423C2×Q8—1581no—
424C2×Q8—1581no—
425C2×Q8—1581no—
426C2×Q8—1581no—
427C2×D4—1581no—
428C2×D4—1581no—
429C2×D4—1581no—
430C2×D4—1581no—
431C2×D4—1581no—
432C2×D4—1581no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—1581no—
438C22×C4—1581no—
439C22×C4—1581no—
440C22×C4C2×C22⋊D7911yescharacteristic
441C22×C4C2×C22⋊D7911yescharacteristic
442C22×C4C2×C22⋊D7911yescharacteristic
443C22×C4C2×C22⋊D7911yescharacteristic
444C22×C4C2×C22⋊D7911yescharacteristic
445C22×C4C2×C22⋊D7911yescharacteristic
446C22×C4C23×D7911yescharacteristic
447C2×C8—1581no—
448C2×C8—1581no—
449C2×C8—1581no—
450C2×C8—1581no—
451C2×C8—1581no—
452C2×C8—1581no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8—1581no—
456C2×C8—1581no—
457C2×C8—1581no—
458C2×C8—1581no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—3162no—
471C4⋊C4—6321no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4#1192962412yes—
477C4⋊C4#1192962412yes—
478C4⋊C4#1192962512yes—
479C22⋊C4—3162no—
480C22⋊C4—3162no—
481C22⋊C4—3162no—
482C22⋊C4—3162no—
483C22⋊C4—6321no—
484C22⋊C4—6321no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—81no—
493C42—81no—
494C42—3162no—
495C23—3161no—
496C23—3161no—
497C23—3161no—
498C23—3161no—
499C23—3161no—
500C23—3161no—
501C23—3161no—
502C23C22×C22⋊D7911yescharacteristic
503Q8—3161no—
504Q8—3161no—
505Q8—1582no—
506Q8—1582no—
507D4—3161no—
508D4—3161no—
509D4—3161no—
510D4—3161no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
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—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
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—1582no—
550C2×C4#41×D7911yescharacteristic
551C2×C4C2×#1192962411yescharacteristic
552C2×C4C2×#1192962411yescharacteristic
553C2×C4C2×#1192962511yescharacteristic
554C2×C4#1457920711yescharacteristic
555C2×C4#1457920911yescharacteristic
556C8—3161no—
557C8—3161no—
558C8—3161no—
559C8—3161no—
560C22—6321no—
561C22—6321no—
562C22—6321no—
563C22—6321no—
564C22—6321no—
565C22—6321no—
566C22—6321no—
567C22#1909907411yescharacteristic
568C22#1909920111yescharacteristic
569C22#1909922411yescharacteristic
570C22#1909936111yescharacteristic
571C22#1909936111yescharacteristic
572C22#1909936611yescharacteristic
573C22#1909937811yescharacteristic
574C4—41no—
575C4—42no—
576C4—42no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—42no—
583C4—81no—
584C4—3162no—
585C4—6321no—
586C4—6321no—
587C4#1909910011yescharacteristic
588C4#1909918511yescharacteristic
589C4#1909925411yescharacteristic
590C4#1909929811yescharacteristic
591C2—12641no—
592C2#2376731611yescharacteristic
593C2#2377350911yescharacteristic
594C2#2377366211yescharacteristic
595C2#2377540011yescharacteristic
596C2#2377566211yescharacteristic
597C2#2377615311yescharacteristic
598C2#2377624211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2370C2
socleC23×C79C22×D4
prime radicalC22×D316C23
hypercenter#2370D79
hypocenterC79#26719
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
2. #23764891C2semidirect
4. #23764548C2semidirect
8. #23764041C2semidirect
11. #23763992C2semidirect
12. #23763966C2semidirect
13. #23763653C2semidirect
14. #23763233C2semidirect
15. C79×#2370C2semidirect
301. #2370D79semidirect
315. C79#26719semidirect

Maximal subgroups

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

subgroupindex
#237671982
#237648912
#237646412
#237645482
#237645432
#237643322
#237640962
#237640412
#237640202
#237639922
#237639662
#237636532
#237632332
C79×#23702
#2671979