Finite Group Databasebeta

← #34300428

Subgroups — #34300428

Order 20224.

784 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098864C211yesmaximal, characteristic
2C4×#14579153C211yesmaximal, characteristic, complemented
3#23765864C211yesmaximal, characteristic
4#23764355C211yesmaximal, characteristic, complemented
5#23764170C211yesmaximal, characteristic, complemented
6#23764143C211yesmaximal, characteristic, complemented
7#23764138C211yesmaximal, characteristic, complemented
8#23764097C211yesmaximal, characteristic
9#23764097C211yesmaximal, characteristic
10#23764028C211yesmaximal, characteristic
11#23764021C211yesmaximal, characteristic
12#23763979C211yesmaximal, characteristic, complemented
13#23763972C211yesmaximal, characteristic, complemented
14#23763411C211yesmaximal, characteristic
15C79×#1767C211yesmaximal, characteristic, complemented
16C2×#14579153C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18C2×#14579118—21no—
19#19098864C418yescomplemented
20C2×#14579095C2211yescharacteristic
21C2×#14579095C2211yescharacteristic
22C2×#14579093C2211yescharacteristic
23C2×#14579093C2211yescharacteristic
24C2×#14579083C2211yescharacteristic
25C2×#14579083C2211yescharacteristic
26C2×#14579083C2211yescharacteristic
27C2×#14579083C2211yescharacteristic
28#19098465C2211yescharacteristic
29#19098465C2211yescharacteristic
30#19098461C2211yescharacteristic
31#19098461C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098461C2211yescharacteristic
34#19098454—21no—
35#19098454—21no—
36#19098447—21no—
37#19098447—21no—
38#19098429C2211yescharacteristic
39C4×Dic79⋊C4—21no—
40C4×Dic79⋊C4—21no—
41C4×C4⋊Dic79C2211yescharacteristic
42C4×C4⋊Dic79C2211yescharacteristic
43C4×C4⋊Dic79C2211yescharacteristic
44C4×(C2×D79)⋊C4—21no—
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098326C2211yescharacteristic
51#19098326C2211yescharacteristic
52#19098326C2211yescharacteristic
53C2×C4⋊C8×C79C2211yescharacteristic
54C2×C4⋊C8×C79C2211yescharacteristic
55C2×#97×C79C2211yescharacteristic
56C2×#97×C79C2211yescharacteristic
57C43×C79C2211yescharacteristic
58#336×C79C2211yescharacteristic
59#336×C79C2211yescharacteristic
60C22×D316C2311yescharacteristic
61C22×C4×D79—21no—
62#14579153—22no—
63#14579153C2×C414yes—
64C4×D316—22no—
65C4×D316C2×C414yes—
66#14579118—44no—
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×(C2×D79)⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×(C2×D79)⋊C4—21no—
84C2×(C2×D79)⋊C4—21no—
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79—21no—
88#14579095—41no—
89#14579095C2×C414yes—
90#14579093—41no—
91#14579093C2×C414yes—
92#14579083—41no—
93#14579083—41no—
94#14579083C2×C414yes—
95#14579083C2×C414yes—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C22×C8×C79C2311yescharacteristic
106C22×C8×C79C2311yescharacteristic
107C22×C8×C79C2311yescharacteristic
108C22×C8×C79C2311yescharacteristic
109C2×C42×C79D411yescharacteristic
110C2×C42×C79D411yescharacteristic
111C2×C42×C79D411yescharacteristic
112C2×C42×C79D411yescharacteristic
113C2×C42×C79C2311yescharacteristic
114C2×C42×C79C2311yescharacteristic
115C2×C42×C79C2311yescharacteristic
116C4⋊C8×C79—22no—
117C4⋊C8×C79C2×C414yes—
118#97×C79—41no—
119#97×C79—41no—
120C23×D79—21no—
121C2×D316—21no—
122C2×D316—21no—
123C2×D316—21no—
124C2×D316—21no—
125C2×D316—21no—
126C2×D316C22×C412yes—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C2×C4×D79—24no—
135C2×C4×D79—42no—
136C2×C4×D79—42no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79C22×C412yes—
149C4⋊Dic79C22×C412yes—
150C4⋊Dic79C22×C412yes—
151Dic79⋊C4—24no—
152Dic79⋊C4—24no—
153Dic79⋊C4—42no—
154Dic79⋊C4—42no—
155Dic79⋊C4—42no—
156Dic79⋊C4—42no—
157(C2×D79)⋊C4—24no—
158(C2×D79)⋊C4—42no—
159(C2×D79)⋊C4—42no—
160(C2×D79)⋊C4—42no—
161(C2×D79)⋊C4—42no—
162C4×Dic79—42no—
163C4×Dic79—42no—
164C4×Dic79—42no—
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
172C2×C8×C79—21no—
173C2×C8×C79—21no—
174C2×C8×C79—21no—
175C2×C8×C79—21no—
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79C22×C412yes—
183C2×C8×C79C22×C412yes—
184C42×C79—22no—
185C42×C79—21no—
186C42×C79—21no—
187C42×C79—21no—
188C42×C79—21no—
189C42×C79—41no—
190C42×C79—41no—
191C42×C79—41no—
192C42×C79C22×C412yes—
193C42×C79#4112yes—
194C23×C79C22×D411yescharacteristic
195C22×D79—22no—
196C22×D79—41no—
197C22×D79—41no—
198C22×D79—41no—
199C22×D79—41no—
200C22×D79—41no—
201C22×D79—41no—
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—21no—
216C2×C4×C79—21no—
217C2×C4×C79C4×D412yes—
218C2×C4×C79C4×D412yes—
219C2×C4×C79AGL(1,ℤ/8)12yes—
220C2×C4×C79C23×C411yescharacteristic
221C2×C4×C79C2×#4111yescharacteristic
222C2×C4×C79H(5,𝔽2)11yescharacteristic
223C2×C4×C792−1+411yescharacteristic
224C2×C4×C792−1+411yescharacteristic
225C2×C4×C792−1+411yescharacteristic
226C2×Dic79—22no—
227C2×Dic79—22no—
228C2×Dic79—22no—
229C2×Dic79—41no—
230C2×Dic79—41no—
231C2×Dic79—41no—
232C2×Dic79—41no—
233C2×Dic79—41no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247D316—22no—
248D316—22no—
249D316—41no—
250D316—41no—
251C4×D79—48no—
252C8×C79—22no—
253C8×C79—22no—
254C8×C79—41no—
255C8×C79—41no—
256C22×C79C2×C4×D411yescharacteristic
257C22×C79#53611yescharacteristic
258C22×C79#53711yescharacteristic
259C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C79#57511yescharacteristic
261C22×C79#57511yescharacteristic
262C22×C79#57511yescharacteristic
263C2×D79—42no—
264C2×D79—42no—
265C2×D79—42no—
266C2×D79—81no—
267C2×D79—81no—
268C2×D79—81no—
269C2×D79—81no—
270C4×C79—22no—
271C4×C79—22no—
272C4×C79—22no—
273C4×C79—41no—
274C4×C79—41no—
275C4×C79—41no—
276C4×C79#51911yescharacteristic
277C4×C79#52011yescharacteristic
278C4×C79#54111yescharacteristic
279C4×C79#54211yescharacteristic
280Dic79—42no—
281Dic79—42no—
282Dic79—42no—
283Dic79—81no—
284Dic79—81no—
285Dic79—81no—
286Dic79—81no—
287#22120—791nomaximal, Sylow
288C2×C79#232311yescharacteristic
289C2×C79#294311yescharacteristic
290C2×C79C4×AGL(1,ℤ/8)11yescharacteristic
291C2×C79#323411yescharacteristic
292C2×C79#324011yescharacteristic
293C2×C79#325111yescharacteristic
294C2×C79#325611yescharacteristic
295D79—82no—
296C2×#478—791no—
297C4×#125—791no—
298#2293—791no—
299#1989—791no—
300#1949—791no—
301#1944—791no—
302#1943—791no—
303#1936—791no—
304#1936—791no—
305#1926—791no—
306#1925—791no—
307#1919—791no—
308#1918—791no—
309#1804—791no—
310#1767D7911yescharacteristic, complemented
311C79#2212011yescharacteristic, complemented, Sylow
312C2×#125—791no—
313C2×C4×D4—791no—
314C2×#117—1581no—
315#478—798no—
316C2×#107—791no—
317C2×#107—791no—
318C2×#106—791no—
319C2×#106—791no—
320C2×C4⋊C8C2×D7911yescharacteristic
321C2×C4⋊C8C2×D7911yescharacteristic
322C2×#102—791no—
323C2×#102—791no—
324C2×#102—791no—
325C2×#102—791no—
326C2×#97C2×D7911yescharacteristic
327C2×#97C2×D7911yescharacteristic
328#390—791no—
329#390—791no—
330#389—791no—
331#389—791no—
332#389—791no—
333#389—791no—
334#388—1581no—
335#388—1581no—
336#387—1581no—
337#387—1581no—
338#382—791no—
339C4×C4⋊C4—791no—
340C4×C4⋊C4—791no—
341C4×C4⋊C4—791no—
342C4×C4⋊C4—1581no—
343C4×C4⋊C4—1581no—
344C4×C22⋊C4—1581no—
345C43C2×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#336C2×D7911yescharacteristic
355#336C2×D7911yescharacteristic
356C22×D4—791no—
357C23×C4—1581no—
358C22×C8C22×D7911yescharacteristic
359C22×C8C22×D7911yescharacteristic
360C22×C8C22×D7911yescharacteristic
361C22×C8C22×D7911yescharacteristic
362#125—794no—
363#125—1582no—
364C4×D4—794no—
365C4×D4—1582no—
366#117—3164no—
367C2×C4⋊C4—791no—
368C2×C4⋊C4—791no—
369C2×C4⋊C4—791no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—791no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4—1581no—
377C2×C4⋊C4—1581no—
378C2×C4⋊C4—1581no—
379C2×C4⋊C4—1581no—
380C2×C22⋊C4—1581no—
381C2×C22⋊C4—1581no—
382C2×C22⋊C4—1581no—
383C2×C22⋊C4—1581no—
384C2×C22⋊C4—1581no—
385C2×C42—1581no—
386C2×C42—1581no—
387C2×C42—1581no—
388C2×C42D31611yescharacteristic
389C2×C42D31611yescharacteristic
390C2×C42D31611yescharacteristic
391C2×C42D31611yescharacteristic
392C2×C42C22×D7911yescharacteristic
393C2×C42C22×D7911yescharacteristic
394C2×C42C22×D7911yescharacteristic
395#107—3161no—
396#107—794no—
397#106—3161no—
398#106—794no—
399C4⋊C8—22no—
400C4⋊C8C4×D7914yes—
401#102—3161no—
402#102—3161no—
403#102—794no—
404#102—794no—
405#97—41no—
406#97—41no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412#95—1581no—
413#95—1581no—
414#95—1581no—
415#95—1581no—
416C24—1581no—
417C2×D4—792no—
418C2×D4—1581no—
419C2×D4—1581no—
420C2×D4—1581no—
421C2×D4—1581no—
422C2×D4—1581no—
423C22×C4—3162no—
424C22×C4—3162no—
425C22×C4—1581no—
426C22×C4—1581no—
427C22×C4—1581no—
428C22×C4—1581no—
429C22×C4—1581no—
430C22×C4—1581no—
431C22×C4—1581no—
432C22×C4—1584no—
433C22×C4C2×D31611yescharacteristic
434C22×C4C2×D31611yescharacteristic
435C22×C4C2×D31611yescharacteristic
436C22×C4C2×D31611yescharacteristic
437C22×C4C2×D31611yescharacteristic
438C22×C4C2×D31611yescharacteristic
439C22×C4C23×D7911yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8C2×C4×D7912yes—
451C2×C8C2×C4×D7912yes—
452C4⋊C4—3162no—
453C4⋊C4—3162no—
454C4⋊C4—3162no—
455C4⋊C4—3162no—
456C4⋊C4—792no—
457C4⋊C4—792no—
458C4⋊C4—792no—
459C4⋊C4—1581no—
460C4⋊C4—1581no—
461C4⋊C4—1581no—
462C4⋊C4—1581no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1584no—
471C4⋊C4—1584no—
472C22⋊C4—3162no—
473C22⋊C4—3162no—
474C22⋊C4—3162no—
475C22⋊C4—3162no—
476C22⋊C4—1584no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—22no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—3162no—
486C42—3162no—
487C42—3162no—
488C42C2×C4×D7912yes—
489C42#1192962412yes—
490C23—3161no—
491C23—3161no—
492C23—3161no—
493C23—3161no—
494C23—3161no—
495C23—3161no—
496C23—1582no—
497C23C22×D31611yescharacteristic
498D4—3161no—
499D4—3161no—
500D4—1582no—
501D4—1582no—
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—21no—
517C2×C4—3168no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—1582no—
537C2×C4—1582no—
538C2×C4—1582no—
539C2×C4C4×D31612yes—
540C2×C4C4×D31612yes—
541C2×C4#1457918912yes—
542C2×C4C22×C4×D7911yescharacteristic
543C2×C4C2×#1192962411yescharacteristic
544C2×C4#1457920611yescharacteristic
545C2×C4#1457920811yescharacteristic
546C2×C4#1457920811yescharacteristic
547C2×C4#1457920911yescharacteristic
548C8—22no—
549C8—22no—
550C8—41no—
551C8—41no—
552C22—3162no—
553C22—3162no—
554C22—3162no—
555C22—6321no—
556C22—6321no—
557C22—6321no—
558C22—6321no—
559C22C2×C4×D31611yescharacteristic
560C22#1909913911yescharacteristic
561C22#1909914411yescharacteristic
562C22C2×#1457918911yescharacteristic
563C22#1909936311yescharacteristic
564C22#1909936311yescharacteristic
565C22#1909936311yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—3162no—
573C4—3162no—
574C4—3162no—
575C4—6321no—
576C4—6321no—
577C4—6321no—
578C4—6321no—
579C4#1909903911yescharacteristic
580C4#1909904211yescharacteristic
581C4#1909917111yescharacteristic
582C4#1909917811yescharacteristic
583C2—6322no—
584C2#2376608311yescharacteristic
585C2#2377220211yescharacteristic
586C2C4×#1457918911yescharacteristic
587C2#2377459211yescharacteristic
588C2#2377464211yescharacteristic
589C2#2377476311yescharacteristic
590C2#2377480111yescharacteristic
subgroupG/H
centerC23C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1767C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenter#1767D79
hypocenterC79#22120
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
2. C4×#14579153C2semidirect
4. #23764355C2semidirect
5. #23764170C2semidirect
6. #23764143C2semidirect
7. #23764138C2semidirect
12. #23763979C2semidirect
13. #23763972C2semidirect
15. C79×#1767C2semidirect
19. #19098864C4semidirect
310. #1767D79semidirect
311. C79#22120semidirect

Maximal subgroups

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

subgroupindex
C2×#190988642
C4×#145791532
#237658642
#237643552
#237641702
#237641432
#237641382
#237640972
#237640282
#237640212
#237639792
#237639722
#237634112
C79×#17672
#2212079