Finite Group Databasebeta

← #34356251

Subgroups — #34356251

Order 20224.

694 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766494C211yesmaximal, characteristic, complemented
2#23764902C211yesmaximal, characteristic, complemented
3#23764636C211yesmaximal, characteristic
4#23764551C211yesmaximal, characteristic, complemented
5#23764546C211yesmaximal, characteristic
6#23764333C211yesmaximal, characteristic
7#23764093C211yesmaximal, characteristic, complemented
8#23764031C211yesmaximal, characteristic, complemented
9#23764031C211yesmaximal, characteristic, complemented
10#23764024C211yesmaximal, characteristic
11#23763975C211yesmaximal, characteristic, complemented
12#23763966C211yesmaximal, characteristic, complemented
13#23763661C211yesmaximal, characteristic
14#23763225C211yesmaximal, characteristic
15C79×#2447C211yesmaximal, characteristic
16C2×#14579153C2211yescharacteristic
17C2×#14579092C2211yescharacteristic
18C2×#14579092C2211yescharacteristic
19C2×C4⋊(C79⋊C8)C2211yescharacteristic
20C2×#14579085C2211yescharacteristic
21C2×#14579085C2211yescharacteristic
22C2×#14579085C2211yescharacteristic
23C2×#14579085C2211yescharacteristic
24C2×#14579082C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×#14579072C2211yescharacteristic
29#19098461C2211yescharacteristic
30#19098461C2211yescharacteristic
31#19098458—21no—
32#19098445—21no—
33#19098444—21no—
34C4×C4⋊Dic79C2211yescharacteristic
35C4×C4⋊Dic79C2211yescharacteristic
36C4×C4⋊Dic79C2211yescharacteristic
37#19098414—21no—
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098318C2211yescharacteristic
45#19098318C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×#124×C79C2211yescharacteristic
49C2×C4×D4×C79C2211yescharacteristic
50#397×C79—21no—
51#397×C79—21no—
52#391×C79C2211yescharacteristic
53#391×C79C2211yescharacteristic
54#390×C79C2211yescharacteristic
55#389×C79C2211yescharacteristic
56#385×C79—21no—
57#381×C79—21no—
58C4×C4⋊C4×C79C2211yescharacteristic
59C22×C79⋊C8C2311yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63#14579153—22no—
64#14579153—22no—
65C2×C4⋊Dic79C2311yescharacteristic
66C2×C4⋊Dic79C2311yescharacteristic
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×Dic79⋊C4—21no—
70C2×Dic79⋊C4—21no—
71C2×Dic79⋊C4—21no—
72C2×Dic79⋊C4—21no—
73C2×C4×Dic79—21no—
74C2×C4×Dic79—21no—
75C2×C4×Dic79—21no—
76#14579092—22no—
77#14579092—41no—
78C4⋊(C79⋊C8)—41no—
79#14579085—22no—
80#14579085—22no—
81#14579085—22no—
82#14579085—41no—
83#14579082—22no—
84#14579082—22no—
85#14579082—22no—
86#14579082—22no—
87#14579072—41no—
88C79⋊#95—21no—
89C79⋊#95—21no—
90C79⋊#95—21no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C22×Q8×C79C2311yescharacteristic
94C22×D4×C79C2311yescharacteristic
95C23×C4×C79—21no—
96#124×C79—22no—
97#124×C79—41no—
98C4×D4×C79—22no—
99C4×D4×C79—41no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79D411yescharacteristic
102C2×C4⋊C4×C79D411yescharacteristic
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×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
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—21no—
136C4⋊Dic79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79#4112yes—
142C4⋊Dic79#4112yes—
143C4⋊Dic79#4112yes—
144Dic79⋊C4—24no—
145Dic79⋊C4—24no—
146Dic79⋊C4—42no—
147Dic79⋊C4—81no—
148C4×Dic79—42no—
149C4×Dic79—81no—
150C4×Dic79—81no—
151C24×C79—21no—
152C2×Q8×C79—21no—
153C2×Q8×C79—21no—
154C2×Q8×C79—21no—
155C2×Q8×C79—21no—
156C2×Q8×C79—21no—
157C2×Q8×C79—21no—
158C2×D4×C79—21no—
159C2×D4×C79—21no—
160C2×D4×C79—21no—
161C2×D4×C79—21no—
162C2×D4×C79—21no—
163C2×D4×C79#4112yes—
164C22×C4×C79—21no—
165C22×C4×C79—21no—
166C22×C4×C79—21no—
167C22×C4×C79—42no—
168C22×C4×C79—42no—
169C22×C4×C79—81no—
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2411yescharacteristic
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—41no—
182C4⋊C4×C79—41no—
183C4⋊C4×C79—81no—
184C22⋊C4×C79—42no—
185C22⋊C4×C79—42no—
186C22⋊C4×C79—42no—
187C22⋊C4×C79—42no—
188C22⋊C4×C79—81no—
189C42×C79—21no—
190C42×C79—21no—
191C42×C79—21no—
192C42×C79—21no—
193C42×C79—21no—
194C42×C79—21no—
195C23×C79—22no—
196C23×C79—41no—
197C23×C79—41no—
198C23×C79—41no—
199C23×C79—41no—
200C23×C79—41no—
201C23×C79—41no—
202C23×C79C22×D411yescharacteristic
203Q8×C79—41no—
204Q8×C79—41no—
205Q8×C79—41no—
206Q8×C79—41no—
207D4×C79—22no—
208D4×C79—22no—
209D4×C79—41no—
210D4×C79—41no—
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—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
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—84no—
239C2×C4×C79C2×#4111yescharacteristic
240C2×C4×C79C2×#4111yescharacteristic
241C2×C4×C79C2×#4111yescharacteristic
242C2×C4×C79C2×#4111yescharacteristic
243C2×C4×C79H(5,𝔽2)11yescharacteristic
244C2×C4×C792−1+411yescharacteristic
245C2×Dic79—22no—
246C2×Dic79—22no—
247C2×Dic79—22no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C79⋊C8—41no—
258C79⋊C8—41no—
259C79⋊C8—41no—
260C79⋊C8—41no—
261C22×C79—42no—
262C22×C79—42no—
263C22×C79—42no—
264C22×C79—81no—
265C22×C79—81no—
266C22×C79—81no—
267C22×C79—81no—
268C22×C79#52511yescharacteristic
269C22×C79#54611yescharacteristic
270C22×C79#54811yescharacteristic
271C22×C79#57511yescharacteristic
272C22×C79#57511yescharacteristic
273C22×C79#57711yescharacteristic
274C22×C79#57711yescharacteristic
275C4×C79—41no—
276C4×C79—41no—
277C4×C79—41no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79—81no—
282C4×C79—81no—
283C4×C79—81no—
284C4×C79#52911yescharacteristic
285C4×C79#54211yescharacteristic
286C4×C79#55211yescharacteristic
287C4×C79#55911yescharacteristic
288Dic79—42no—
289Dic79—42no—
290Dic79—42no—
291Dic79—81no—
292#26717—791nomaximal, Sylow
293C2×C79—82no—
294C2×C79#245811yescharacteristic
295C2×C79#311111yescharacteristic
296C2×C79#312511yescharacteristic
297C2×C79#330711yescharacteristic
298C2×C79#332311yescharacteristic
299C2×C79#337111yescharacteristic
300C2×C79#337611yescharacteristic
301#2447D7911yescharacteristic
302#2370—791no—
303#2088—791no—
304#2042—791no—
305#2026—791no—
306#2025—791no—
307#1983—791no—
308#1935—791no—
309#1927—791no—
310#1927—791no—
311#1926—791no—
312#1919—791no—
313#1917—791no—
314#1864—791no—
315#1762—791no—
316C79#2671711yescharacteristic, complemented, Sylow
317C2×#125—791no—
318C2×#124C2×D7911yescharacteristic
319C2×C4×D4C2×D7911yescharacteristic
320C2×#106—791no—
321C2×#106—791no—
322C2×C4⋊C8—791no—
323C2×#103—791no—
324C2×#103—791no—
325C2×#103—791no—
326C2×#103—791no—
327C2×#102—791no—
328C2×#102—791no—
329C2×#102—791no—
330C2×#102—791no—
331C2×#97—791no—
332#397—21no—
333#397—21no—
334#391C2×D7911yescharacteristic
335#391C2×D7911yescharacteristic
336#390C2×D7911yescharacteristic
337#389—791no—
338#389—791no—
339#389—1581no—
340#389C2×D7911yescharacteristic
341#387—1581no—
342#387—1581no—
343#385—21no—
344#381—21no—
345C4×C4⋊C4—791no—
346C4×C4⋊C4—791no—
347C4×C4⋊C4—791no—
348C4×C4⋊C4—1581no—
349C4×C4⋊C4C2×D7911yescharacteristic
350#340—791no—
351#340—791no—
352#340—791no—
353#340—791no—
354#340—791no—
355#340—791no—
356#336—791no—
357#336—791no—
358#336—791no—
359#336—791no—
360C22×Q8C22×D7911yescharacteristic
361C22×D4C22×D7911yescharacteristic
362C23×C4—21no—
363C22×C8—791no—
364C22×C8—791no—
365C22×C8—791no—
366C22×C8—791no—
367#125—1582no—
368#125—1582no—
369#124—22no—
370#124—41no—
371C4×D4—22no—
372C4×D4—41no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—791no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—791no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—1581no—
379C2×C4⋊C4—1581no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4C22⋊D7911yescharacteristic
383C2×C4⋊C4C22⋊D7911yescharacteristic
384C2×C4⋊C4C22×D7911yescharacteristic
385C2×C4⋊C4C22×D7911yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—1581no—
392C2×C42—1581no—
393C2×C42—1581no—
394C2×C42C22×D7911yescharacteristic
395C2×C42C22×D7911yescharacteristic
396C2×C42C22×D7911yescharacteristic
397#106—3161no—
398#106—1582no—
399C4⋊C8—3161no—
400#103—3161no—
401#103—1582no—
402#103—1582no—
403#103—1582no—
404#102—1582no—
405#102—1582no—
406#102—1582no—
407#102—1582no—
408#97—3161no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—1581no—
415#95—1581no—
416#95—1581no—
417#95—1581no—
418#95—1581no—
419#95C22⋊D7911yescharacteristic
420#95C22⋊D7911yescharacteristic
421C24—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4#1192962312yes—
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×C4C2×C22⋊D7911yescharacteristic
450C22×C4C23×D7911yescharacteristic
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—
462C2×C8—1581no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—81no—
470C4⋊C4—3162no—
471C4⋊C4—792no—
472C4⋊C4—792no—
473C4⋊C4—792no—
474C4⋊C4—6321no—
475C4⋊C4—1584no—
476C4⋊C4—1584no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—3162no—
494C42—6321no—
495C42—6321no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C22×C22⋊D7911yescharacteristic
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
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—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
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—1582no—
550C2×C4—1582no—
551C2×C4—1582no—
552C2×C4#41×D7911yescharacteristic
553C2×C4#41×D7911yescharacteristic
554C2×C4#41×D7911yescharacteristic
555C2×C4C2×#1192962311yescharacteristic
556C2×C4#1457920611yescharacteristic
557C2×C4#1457920811yescharacteristic
558C8—3161no—
559C8—3161no—
560C8—3161no—
561C8—3161no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22#1909907011yescharacteristic
570C22#1909920711yescharacteristic
571C22#1909922011yescharacteristic
572C22#1909936111yescharacteristic
573C22#1909936111yescharacteristic
574C22#1909937111yescharacteristic
575C22#1909937411yescharacteristic
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—3162no—
586C4—3162no—
587C4—3162no—
588C4—6321no—
589C4#1909909011yescharacteristic
590C4#1909918111yescharacteristic
591C4#1909925011yescharacteristic
592C4#1909929711yescharacteristic
593C2—82no—
594C2#2376731811yescharacteristic
595C2#2377350311yescharacteristic
596C2#2377367111yescharacteristic
597C2#2377537611yescharacteristic
598C2#2377561611yescharacteristic
599C2#2377616211yescharacteristic
600C2#2377622111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2447C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
hypercenter#2447D79
hypocenterC79#26717
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2447

Complements

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

NH ≅ G/Ntype
1. #23766494C2semidirect
2. #23764902C2semidirect
4. #23764551C2semidirect
7. #23764093C2semidirect
8. #23764031C2semidirect
9. #23764031C2semidirect
11. #23763975C2semidirect
12. #23763966C2semidirect
316. C79#26717semidirect

Maximal subgroups

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

subgroupindex
#237664942
#237649022
#237646362
#237645512
#237645462
#237643332
#237640932
#237640312
#237640242
#237639752
#237639662
#237636612
#237632252
C79×#24472
#2671779