Finite Group Databasebeta

← #34346442

Subgroups — #34346442

Order 20224.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23768422C211yesmaximal, characteristic, complemented
2#23766373C211yesmaximal, characteristic
3#23764786C211yesmaximal, characteristic
4#23764772C211yesmaximal, characteristic
5#23764544C211yesmaximal, characteristic
6#23764534C211yesmaximal, characteristic
7#23764354C211yesmaximal, characteristic, complemented
8#23764350C211yesmaximal, characteristic, complemented
9#23764170C211yesmaximal, characteristic, complemented
10#23764116C211yesmaximal, characteristic, complemented
11#23763991C211yesmaximal, characteristic, complemented
12#23763984C211yesmaximal, characteristic, complemented
13#23763530C211yesmaximal, characteristic, complemented
14C4×#14579082C211yesmaximal, characteristic
15C79×#1804C211yesmaximal, characteristic
16C2×#14579160C2211yescharacteristic
17C2×#14579152C2211yescharacteristic
18C2×#14579095C2211yescharacteristic
19C2×#14579093C2211yescharacteristic
20C2×Dic79⋊C8C2211yescharacteristic
21C2×Dic79⋊C8C2211yescharacteristic
22C2×#14579082C2211yescharacteristic
23C2×#14579082C2211yescharacteristic
24C2×C4×C79⋊C8C2211yescharacteristic
25#19098497—21no—
26#19098484—21no—
27#19098467—21no—
28#19098466C2211yescharacteristic
29#19098466C2211yescharacteristic
30#19098460—21no—
31#19098459C2211yescharacteristic
32#19098459C2211yescharacteristic
33#19098456—21no—
34#19098456—21no—
35#19098429C2211yescharacteristic
36#19098426—21no—
37C4⋊C4×Dic79C2211yescharacteristic
38C4⋊C4×Dic79C2211yescharacteristic
39C4×C4⋊Dic79C2211yescharacteristic
40C4×C4⋊Dic79C2211yescharacteristic
41C4×C22⋊Dic79—21no—
42#19098326C2211yescharacteristic
43#19098326C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098317C2211yescharacteristic
51#19098317C2211yescharacteristic
52C2×C4×D4×C79C2211yescharacteristic
53C2×C4⋊C8×C79C2211yescharacteristic
54C2×#102×C79C2211yescharacteristic
55C2×#102×C79C2211yescharacteristic
56#382×C79C2211yescharacteristic
57#340×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59C22×Dic158—21no—
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62#14579160—22no—
63#14579160—41no—
64#14579152—22no—
65#14579152—41no—
66C2×C4⋊Dic79—21no—
67C2×C4⋊Dic79D411yescharacteristic
68C2×C4⋊Dic79D411yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×C22⋊Dic79—21no—
78C2×C22⋊Dic79—21no—
79C2×C22⋊Dic79—21no—
80C2×C22⋊Dic79—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79C2311yescharacteristic
85C2×C4×Dic79C2311yescharacteristic
86#14579095—22no—
87#14579093—22no—
88Dic79⋊C8—41no—
89Dic79⋊C8—41no—
90#14579082—22no—
91#14579082—22no—
92C4×C79⋊C8—22no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95D411yescharacteristic
101C79⋊#95D411yescharacteristic
102C22×D4×C79C2311yescharacteristic
103C23×C4×C79—21no—
104C22×C8×C79C2311yescharacteristic
105C22×C8×C79C2311yescharacteristic
106C4×D4×C79—22no—
107C4×D4×C79Q814yescomplemented
108C2×C4⋊C4×C79C2311yescharacteristic
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C22⋊C4×C79—21no—
112C2×C42×C79C2311yescharacteristic
113C4⋊C8×C79—22no—
114#102×C79—41no—
115#102×C79—41no—
116#95×C79—21no—
117#95×C79—21no—
118C2×Dic158—42no—
119C2×Dic158—42no—
120C2×Dic158—81no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—21no—
124C22×Dic79—21no—
125C22×Dic79C2×D411yescharacteristic
126C22×Dic79C2×D411yescharacteristic
127C22×Dic79C2×D411yescharacteristic
128C22×Dic79C2×D411yescharacteristic
129C2×C79⋊C8—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C4⋊Dic79—22no—
136C4⋊Dic79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—41no—
143C4⋊Dic79—42no—
144C4⋊Dic79#4112yes—
145C4⋊Dic79#4112yes—
146Dic79⋊C4—42no—
147Dic79⋊C4—42no—
148Dic79⋊C4—42no—
149Dic79⋊C4—81no—
150C22⋊Dic79—42no—
151C22⋊Dic79—42no—
152C22⋊Dic79—81no—
153C22⋊Dic79—81no—
154C4×Dic79—21no—
155C4×Dic79—21no—
156C4×Dic79—21no—
157C4×Dic79—21no—
158C4×Dic79—81no—
159C4×Dic79—81no—
160C4×Dic79—81no—
161C24×C79—21no—
162C2×D4×C79—21no—
163C2×D4×C79—21no—
164C2×D4×C79—21no—
165C2×D4×C79—21no—
166C2×D4×C79—21no—
167C2×D4×C79C2×Q812yes—
168C22×C4×C79—21no—
169C22×C4×C79—24no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—81no—
173C22×C4×C79—81no—
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2411yescharacteristic
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79C2×Q812yes—
189C22⋊C4×C79—24no—
190C42×C79C2×Q812yes—
191C42×C79#4112yes—
192C23×C79—22no—
193C23×C79—41no—
194C23×C79—41no—
195C23×C79—41no—
196C23×C79—41no—
197C23×C79—41no—
198C23×C79—41no—
199C23×C79C22×D411yescharacteristic
200D4×C79—22no—
201D4×C79—22no—
202D4×C79—41no—
203D4×C79—41no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—22no—
209C2×C4×C79—41no—
210C2×C4×C79—41no—
211C2×C4×C79—41no—
212C2×C4×C79—41no—
213C2×C4×C79—41no—
214C2×C4×C79—41no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79—84no—
218C2×C4×C79C22⋊Q812yes—
219C2×C4×C79C22⋊Q812yes—
220C2×C4×C79C22×Q811yescharacteristic
221C2×C4×C79C2×#4111yescharacteristic
222C2×C4×C79C2×#4111yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C2×C4×C79H(5,𝔽2)11yescharacteristic
225C2×C4×C792−1+411yescharacteristic
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—22no—
232C2×Dic79—22no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—41no—
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—
249C2×Dic79#13512yes—
250Dic158—84no—
251C8×C79—41no—
252C8×C79—41no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C22×C79—42no—
256C22×C79—42no—
257C22×C79—42no—
258C22×C79—81no—
259C22×C79—81no—
260C22×C79—81no—
261C22×C79—81no—
262C22×C79C2×C22⋊Q811yescharacteristic
263C22×C79#54611yescharacteristic
264C22×C79#54811yescharacteristic
265C22×C79C2×#13511yescharacteristic
266C22×C79#57511yescharacteristic
267C22×C79#57611yescharacteristic
268C22×C79#57711yescharacteristic
269C4×C79—22no—
270C4×C79—22no—
271C4×C79—42no—
272C4×C79—81no—
273C4×C79—81no—
274C4×C79#53311yescharacteristic
275C4×C79#54411yescharacteristic
276C4×C79#55511yescharacteristic
277C4×C79#55611yescharacteristic
278Dic79—22no—
279Dic79—42no—
280Dic79—42no—
281Dic79—41no—
282Dic79—41no—
283Dic79—41no—
284Dic79—81no—
285Dic79—81no—
286#26028—791nomaximal, Sylow
287C2×C79—82no—
288C2×C79#243911yescharacteristic
289C2×C79#307611yescharacteristic
290C2×C79#308111yescharacteristic
291C2×C79#329711yescharacteristic
292C2×C79#332611yescharacteristic
293C2×C79#336511yescharacteristic
294C2×C79#337011yescharacteristic
295#2550—791no—
296#2357—791no—
297#2067—791no—
298#2065—791no—
299#2025—791no—
300#2023—791no—
301#1989—791no—
302#1988—791no—
303#1949—791no—
304#1939—791no—
305#1921—791no—
306#1920—791no—
307#1835—791no—
308#1804D7911yescharacteristic
309C4×#102—791no—
310C79#2602811yescharacteristic, complemented, Sylow
311C2×C4⋊Q8—791no—
312C2×#125—791no—
313C2×C4×D4C2×D7911yescharacteristic
314C2×#107—791no—
315C2×#106—791no—
316C2×C4⋊C8—791no—
317C2×C4⋊C8—791no—
318C2×C4⋊C8C2×D7911yescharacteristic
319C2×#102—791no—
320C2×#102—791no—
321C2×#102C2×D7911yescharacteristic
322C2×#102C2×D7911yescharacteristic
323C2×C4×C8—791no—
324#398—1581no—
325#395—1581no—
326#391—1581no—
327#390—791no—
328#390—791no—
329#389—791no—
330#389—791no—
331#389—1581no—
332#388—1581no—
333#388—1581no—
334#382—791no—
335#382C2×D7911yescharacteristic
336#381—1581no—
337C4×C4⋊C4—791no—
338C4×C4⋊C4—791no—
339C4×C4⋊C4—791no—
340C4×C4⋊C4—791no—
341C4×C22⋊C4—1581no—
342#340—791no—
343#340—791no—
344#340—791no—
345#340—791no—
346#340—791no—
347#340—791no—
348#340—791no—
349#340—791no—
350#340C2×D7911yescharacteristic
351#340C2×D7911yescharacteristic
352#336—791no—
353#336—791no—
354C22×Q8—1581no—
355C22×D4C22×D7911yescharacteristic
356C23×C4—21no—
357C22×C8—791no—
358C22×C8—791no—
359C22×C8C22×D7911yescharacteristic
360C22×C8C22×D7911yescharacteristic
361C4⋊Q8—3161no—
362C4⋊Q8—1582no—
363#125—3161no—
364#125—1582no—
365C4×D4—22no—
366C4×D4Dic15814yescomplemented
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—1581no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4—1581no—
377C2×C4⋊C4—1581no—
378C2×C4⋊C4C22×D7911yescharacteristic
379C2×C4⋊C4C22×D7911yescharacteristic
380C2×C4⋊C4C22×D7911yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1581no—
383C2×C22⋊C4—1581no—
384C2×C22⋊C4—1581no—
385C2×C22⋊C4—1581no—
386C2×C42—791no—
387C2×C42—791no—
388C2×C42—1581no—
389C2×C42—1581no—
390C2×C42—1581no—
391C2×C42C22×D7911yescharacteristic
392#107—1582no—
393#106—1582no—
394C4⋊C8—22no—
395C4⋊C8—3161no—
396C4⋊C8—3161no—
397#102—41no—
398#102—41no—
399#102—1582no—
400#102—1582no—
401C4×C8—1582no—
402#95—21no—
403#95—21no—
404#95—791no—
405#95—791no—
406#95—1581no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412#95—1581no—
413C24—21no—
414C2×Q8—3162no—
415C2×Q8—3162no—
416C2×Q8—6321no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×Dic15812yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—791no—
430C22×C4—791no—
431C22×C4—791no—
432C22×C4—791no—
433C22×C4—1581no—
434C22×C4—1581no—
435C22×C4—1581no—
436C22×C4—1581no—
437C22×C4D4×D7911yescharacteristic
438C22×C4D4×D7911yescharacteristic
439C22×C4C23×D7911yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1581no—
447C2×C8—1581no—
448C2×C8—1581no—
449C2×C8—1581no—
450C2×C8—1581no—
451C2×C8—1581no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—3161no—
458C4⋊C4—3162no—
459C4⋊C4—3162no—
460C4⋊C4—3162no—
461C4⋊C4—3162no—
462C4⋊C4—792no—
463C4⋊C4—792no—
464C4⋊C4—6321no—
465C4⋊C4—1582no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4C2×Dic15812yes—
473C22⋊C4—24no—
474C22⋊C4—3162no—
475C22⋊C4—3162no—
476C22⋊C4—6321no—
477C22⋊C4—6321no—
478C42—6321no—
479C42—6321no—
480C42—6321no—
481C42—1581no—
482C42—1581no—
483C42—1581no—
484C42—1581no—
485C42C2×Dic15812yes—
486C42#1192962312yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C2×D4×D7911yescharacteristic
495Q8—6324no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—84no—
514C2×C4—3161no—
515C2×C4—3161no—
516C2×C4—3161no—
517C2×C4—3161no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—792no—
525C2×C4—1581no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1582no—
529C2×C4—1582no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4#1457913812yes—
539C2×C4#1457913812yes—
540C2×C4C22×Dic15811yescharacteristic
541C2×C4#41×D7911yescharacteristic
542C2×C4#41×D7911yescharacteristic
543C2×C4C2×#1192962311yescharacteristic
544C2×C4#1457920611yescharacteristic
545C2×C4#1457920911yescharacteristic
546C8—41no—
547C8—41no—
548C8—3161no—
549C8—3161no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1457913811yescharacteristic
558C22#1909921011yescharacteristic
559C22#1909922311yescharacteristic
560C22#135×D7911yescharacteristic
561C22#1909936011yescharacteristic
562C22#1909936711yescharacteristic
563C22#1909937311yescharacteristic
564C4—22no—
565C4—22no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—3162no—
570C4—3161no—
571C4—3162no—
572C4—3161no—
573C4—3161no—
574C4—6321no—
575C4—6321no—
576C4—1582no—
577C4#1909911611yescharacteristic
578C4#1909919411yescharacteristic
579C4#1909927711yescharacteristic
580C4#1909928411yescharacteristic
581C2—82no—
582C2#2376709211yescharacteristic
583C2#2377318411yescharacteristic
584C2#2377323311yescharacteristic
585C2#2377525711yescharacteristic
586C2#2377566011yescharacteristic
587C2#2377609911yescharacteristic
588C2#2377615111yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1804C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
hypercenter#1804D79
hypocenterC79#26028
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
1. #23768422C2semidirect
7. #23764354C2semidirect
8. #23764350C2semidirect
9. #23764170C2semidirect
10. #23764116C2semidirect
11. #23763991C2semidirect
12. #23763984C2semidirect
13. #23763530C2semidirect
107. C4×D4×C79Q8semidirect
310. C79#26028semidirect
366. C4×D4Dic158semidirect

Maximal subgroups

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

subgroupindex
#237684222
#237663732
#237647862
#237647722
#237645442
#237645342
#237643542
#237643502
#237641702
#237641162
#237639912
#237639842
#237635302
C4×#145790822
C79×#18042
#2602879