Finite Group Databasebeta

← #34361387

Subgroups — #34361387

Order 20224.

672 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766425C211yesmaximal, characteristic
2#23764819C211yesmaximal, characteristic, complemented
3#23764778C211yesmaximal, characteristic, complemented
4#23764603C211yesmaximal, characteristic
5#23764596C211yesmaximal, characteristic
6#23764166C211yesmaximal, characteristic, complemented
7#23764135C211yesmaximal, characteristic, complemented
8#23764015C211yesmaximal, characteristic
9#23763989C211yesmaximal, characteristic, complemented
10#23763985C211yesmaximal, characteristic, complemented
11#23763653C211yesmaximal, characteristic, complemented
12#23763597C211yesmaximal, characteristic
13#23763492C211yesmaximal, characteristic
14#23763233C211yesmaximal, characteristic, complemented
15C79×#2494C211yesmaximal, characteristic
16C2×#14579140C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18C2×#14579120C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×C4⋊(C79⋊C8)C2211yescharacteristic
24C2×#14579084C2211yescharacteristic
25C2×#14579084C2211yescharacteristic
26C2×#14579084C2211yescharacteristic
27C2×#14579084C2211yescharacteristic
28C2×C22⋊(C79⋊C8)C2211yescharacteristic
29C2×C22⋊(C79⋊C8)C2211yescharacteristic
30C2×#14579072C2211yescharacteristic
31#19098469C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098457—21no—
34#19098451—21no—
35#19098445—21no—
36C4×C4⋊Dic79C2211yescharacteristic
37#19098414—21no—
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48#401×C79—21no—
49#397×C79—21no—
50#389×C79C2211yescharacteristic
51#389×C79C2211yescharacteristic
52#388×C79—21no—
53#387×C79—21no—
54#382×C79C2211yescharacteristic
55#380×C79C2211yescharacteristic
56#380×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C22⋊C4×C79C2211yescharacteristic
59C22×Dic158C2311yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64#14579140—22no—
65#14579140—22no—
66C4×Dic158—22no—
67C4×Dic158—41no—
68#14579120—22no—
69#14579120—41no—
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—
80C2×C4×Dic79—21no—
81#14579094—22no—
82#14579094—22no—
83#14579092—22no—
84#14579092—41no—
85C4⋊(C79⋊C8)—22no—
86#14579084—22no—
87#14579084—22no—
88#14579084—22no—
89#14579084—41no—
90C22⋊(C79⋊C8)—22no—
91C22⋊(C79⋊C8)—41no—
92#14579072—41no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C23×C4×C79C2311yescharacteristic
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79C2311yescharacteristic
101C2×C4⋊C4×C79C2311yescharacteristic
102C2×C4⋊C4×C79C2311yescharacteristic
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C22⋊C4×C79—21no—
105C2×C22⋊C4×C79—21no—
106C2×C22⋊C4×C79D411yescharacteristic
107C2×C22⋊C4×C79D411yescharacteristic
108C2×C42×C79—21no—
109C2×C42×C79C2311yescharacteristic
110C2×C42×C79C2311yescharacteristic
111#95×C79—21no—
112#95×C79—21no—
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
120C2×Dic158—21no—
121C2×Dic158—21no—
122C2×Dic158—21no—
123C2×Dic158—21no—
124C2×Dic158—21no—
125C2×Dic158#4112yes—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—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—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8#4112yes—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79#4112yes—
147C4⋊Dic79#4112yes—
148Dic79⋊C4—24no—
149Dic79⋊C4—42no—
150Dic79⋊C4—42no—
151C22⋊Dic79—24no—
152C22⋊Dic79—42no—
153C4×Dic79—42no—
154C4×Dic79—42no—
155C4×Dic79—42no—
156C24×C79C2×D411yescharacteristic
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—21no—
167C22×C4×C79C2×D411yescharacteristic
168C22×C4×C79C2×D411yescharacteristic
169C22×C4×C79C2×D411yescharacteristic
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2411yescharacteristic
173C4⋊C4×C79—21no—
174C4⋊C4×C79—21no—
175C4⋊C4×C79—21no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—42no—
182C4⋊C4×C79—81no—
183C22⋊C4×C79—41no—
184C22⋊C4×C79—41no—
185C22⋊C4×C79—42no—
186C22⋊C4×C79—81no—
187C42×C79—21no—
188C42×C79—21no—
189C42×C79—21no—
190C42×C79—21no—
191C42×C79—81no—
192C23×C79—21no—
193C23×C79—21no—
194C23×C79—21no—
195C23×C79—21no—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79—21no—
199C23×C79C22×D411yescharacteristic
200C2×C4×C79—21no—
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
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×C79C2×#4111yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79H(5,𝔽2)11yescharacteristic
236C2×C4×C792−1+411yescharacteristic
237C2×Dic79—22no—
238C2×Dic79—22no—
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—
248C2×Dic79—41no—
249Dic158—22no—
250Dic158—22no—
251Dic158—41no—
252Dic158—41no—
253C79⋊C8—22no—
254C79⋊C8—41no—
255C79⋊C8—41no—
256C79⋊C8—41no—
257C22×C79—41no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79—41no—
263C22×C79—41no—
264C22×C79#52511yescharacteristic
265C22×C79#54611yescharacteristic
266C22×C79#54711yescharacteristic
267C22×C79#57511yescharacteristic
268C22×C79#57511yescharacteristic
269C22×C79#57711yescharacteristic
270C22×C79#57811yescharacteristic
271C4×C79—41no—
272C4×C79—41no—
273C4×C79—41no—
274C4×C79—41no—
275C4×C79—41no—
276C4×C79—81no—
277C4×C79—81no—
278C4×C79—81no—
279C4×C79—81no—
280C4×C79#52911yescharacteristic
281C4×C79#53911yescharacteristic
282C4×C79#55311yescharacteristic
283C4×C79#55611yescharacteristic
284Dic79—42no—
285Dic79—42no—
286Dic79—42no—
287Dic79—81no—
288#27061—791nomaximal, Sylow
289C2×C79—81no—
290C2×C79#246611yescharacteristic
291C2×C79#311111yescharacteristic
292C2×C79#312111yescharacteristic
293C2×C79#330711yescharacteristic
294C2×C79#332711yescharacteristic
295C2×C79#334911yescharacteristic
296C2×C79#335011yescharacteristic
297#2494D7911yescharacteristic
298#2362—791no—
299#2074—791no—
300#2066—791no—
301#2036—791no—
302#2035—791no—
303#1949—791no—
304#1943—791no—
305#1924—791no—
306#1921—791no—
307#1920—791no—
308#1863—791no—
309#1854—791no—
310#1825—791no—
311#1763—791no—
312C79#2706111yescharacteristic, complemented, Sylow
313C2×C22⋊Q8—791no—
314C2×C4×Q8—791no—
315C2×#117—791no—
316C2×#107—791no—
317C2×#107—791no—
318C2×#106—791no—
319C2×#106—791no—
320C2×C4⋊C8—791no—
321C2×#103—791no—
322C2×#103—791no—
323C2×#103—791no—
324C2×#103—791no—
325C2×C22⋊C8—791no—
326C2×C22⋊C8—791no—
327C2×#97—791no—
328#401—21no—
329#397—21no—
330#391—791no—
331#389—791no—
332#389C2×D7911yescharacteristic
333#389C2×D7911yescharacteristic
334#388—21no—
335#388—1581no—
336#388—1581no—
337#387—21no—
338#387—1581no—
339#382C2×D7911yescharacteristic
340#380C2×D7911yescharacteristic
341#380C2×D7911yescharacteristic
342C4×C4⋊C4—791no—
343C4×C4⋊C4—1581no—
344C4×C4⋊C4C2×D7911yescharacteristic
345C4×C22⋊C4C2×D7911yescharacteristic
346#340—791no—
347#340—791no—
348#340—791no—
349#340—791no—
350#340—791no—
351#340—791no—
352#340—791no—
353#340—791no—
354#336—791no—
355#336—791no—
356C22×Q8—791no—
357C23×C4C22×D7911yescharacteristic
358C22×C8—791no—
359C22×C8—791no—
360C22×C8—791no—
361C22×C8—791no—
362C22⋊Q8—1582no—
363C22⋊Q8—1582no—
364C4×Q8—3161no—
365C4×Q8—1582no—
366#117—3161no—
367#117—1582no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—1581no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4C22×D7911yescharacteristic
377C2×C4⋊C4C22×D7911yescharacteristic
378C2×C4⋊C4C22×D7911yescharacteristic
379C2×C4⋊C4C22×D7911yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1581no—
383C2×C22⋊C4—1581no—
384C2×C22⋊C4C22⋊D7911yescharacteristic
385C2×C22⋊C4C22⋊D7911yescharacteristic
386C2×C42—21no—
387C2×C42—1581no—
388C2×C42—1581no—
389C2×C42—1581no—
390C2×C42C22×D7911yescharacteristic
391C2×C42C22×D7911yescharacteristic
392#107—1582no—
393#107—1582no—
394#106—3161no—
395#106—1582no—
396C4⋊C8—1582no—
397#103—3161no—
398#103—1582no—
399#103—1582no—
400#103—1582no—
401C22⋊C8—3161no—
402C22⋊C8—1582no—
403#97—3161no—
404#95—21no—
405#95—21no—
406#95—21no—
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#95C22⋊D7911yescharacteristic
416#95C22⋊D7911yescharacteristic
417C24C2×C22⋊D7911yescharacteristic
418C2×Q8—792no—
419C2×Q8—1581no—
420C2×Q8—1581no—
421C2×Q8—1581no—
422C2×Q8—1581no—
423C2×Q8—1581no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—1581no—
435C22×C4—1581no—
436C22×C4—1581no—
437C22×C4—1581no—
438C22×C4C2×C22⋊D7911yescharacteristic
439C22×C4C2×C22⋊D7911yescharacteristic
440C22×C4C2×C22⋊D7911yescharacteristic
441C22×C4C2×C22⋊D7911yescharacteristic
442C22×C4C2×C22⋊D7911yescharacteristic
443C22×C4C23×D7911yescharacteristic
444C2×C8—792no—
445C2×C8—1581no—
446C2×C8—1581no—
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—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4—3162no—
467C4⋊C4—3162no—
468C4⋊C4—792no—
469C4⋊C4—792no—
470C4⋊C4—1584no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—3162no—
480C22⋊C4—1584no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—81no—
486C42—3162no—
487C42—3162no—
488C42—3162no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23C22×C22⋊D7911yescharacteristic
497Q8—3161no—
498Q8—3161no—
499Q8—1582no—
500Q8—1582no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
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—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—1582no—
542C2×C4—1582no—
543C2×C4—1582no—
544C2×C4#41×D7911yescharacteristic
545C2×C4#41×D7911yescharacteristic
546C2×C4#41×D7911yescharacteristic
547C2×C4#41×D7911yescharacteristic
548C2×C4#1457920711yescharacteristic
549C2×C4#1457920811yescharacteristic
550C8—3161no—
551C8—3161no—
552C8—3161no—
553C8—1582no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#1909907011yescharacteristic
562C22#1909920111yescharacteristic
563C22#1909921211yescharacteristic
564C22#1909936111yescharacteristic
565C22#1909936111yescharacteristic
566C22#1909937411yescharacteristic
567C22#1909937811yescharacteristic
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—3162no—
578C4—3162no—
579C4—3162no—
580C4—6321no—
581C4#1909909611yescharacteristic
582C4#1909915411yescharacteristic
583C4#1909926411yescharacteristic
584C4#1909928011yescharacteristic
585C2—81no—
586C2#2376743211yescharacteristic
587C2#2377350311yescharacteristic
588C2#2377362311yescharacteristic
589C2#2377537011yescharacteristic
590C2#2377566211yescharacteristic
591C2#2377592411yescharacteristic
592C2#2377593911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2494C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2494D79
hypocenterC79#27061
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2494

Complements

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

NH ≅ G/Ntype
2. #23764819C2semidirect
3. #23764778C2semidirect
6. #23764166C2semidirect
7. #23764135C2semidirect
9. #23763989C2semidirect
10. #23763985C2semidirect
11. #23763653C2semidirect
14. #23763233C2semidirect
312. C79#27061semidirect

Maximal subgroups

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

subgroupindex
#237664252
#237648192
#237647782
#237646032
#237645962
#237641662
#237641352
#237640152
#237639892
#237639852
#237636532
#237635972
#237634922
#237632332
C79×#24942
#2706179