Finite Group Databasebeta

← #34254178

Subgroups — #34254178

Order 20224.

1210 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099044C211yesmaximal, characteristic
2#23772426C212yesmaximal, complemented
3#23761742C212yesmaximal, complemented
4#23761737C212yesmaximal, complemented
5#23761733C212yesmaximal, complemented
6#23761728C212yesmaximal, complemented
7#23761716C212yesmaximal, complemented
8C2×#19098289C211yesmaximal, characteristic, complemented
9C2×#324×C79C211yesmaximal, characteristic, complemented
10C22×#11929624C411yescharacteristic
11#19099044—24no—
12#19099044—24no—
13C2×#14579126C411yescharacteristic
14C2×#14579126C2211yescharacteristic
15C2×#14579123C411yescharacteristic
16C2×#14579123C2211yescharacteristic
17C2×C4×C22⋊D79C411yescharacteristic
18C2×C4×C22⋊D79C411yescharacteristic
19C2×C4×C22⋊D79C2211yescharacteristic
20C2×C4×C22⋊D79C2211yescharacteristic
21C2×#14579120C411yescharacteristic
22C2×#14579119C411yescharacteristic
23C2×#14579119C2211yescharacteristic
24C2×C4⋊C4×D79C411yescharacteristic
25C2×C4⋊C4×D79C2211yescharacteristic
26#19098634—24no—
27#19098631—24no—
28#19098617C2214yes—
29#19098525C2214yes—
30C2×C22⋊(C79⋊C8)C2212yes—
31#19098289C2212yes—
32#19098289C2212yes—
33#19098288C2214yes—
34#19098286C2214yes—
35C22×C4⋊C4×C79C2211yescharacteristic
36C2×C22⋊C8×C79C2212yes—
37#324×C79C2212yes—
38#324×C79C2212yes—
39C2×#11929624—24no—
40C2×#11929624—22no—
41C2×#11929624—22no—
42C2×#11929624—24no—
43C22×Dic158C2×C411yescharacteristic
44C22×D316C2×C411yescharacteristic
45C22×C22⋊D79C2×C411yescharacteristic
46C22×C22⋊D79C2×C411yescharacteristic
47C22×C4×D79C2×C411yescharacteristic
48C22×C4×D79C2×C411yescharacteristic
49C22×C4×D79C2×C411yescharacteristic
50C22×C4×D79C2×C411yescharacteristic
51C22×C4×D79C2×C411yescharacteristic
52C22×C4×D79C2311yescharacteristic
53C2×#11929607—24no—
54C22×C79⋊C8—22no—
55#14579126—24no—
56#14579126—24no—
57#14579123—24no—
58#14579123—24no—
59C4×C22⋊D79—22no—
60C4×C22⋊D79—22no—
61C4×C22⋊D79—22no—
62C4×C22⋊D79—22no—
63C4×C22⋊D79—22no—
64C4×C22⋊D79—22no—
65C4×C22⋊D79D414yes—
66C4×C22⋊D79D414yes—
67#14579120—22no—
68#14579120—22no—
69#14579119—24no—
70#14579119—24no—
71C2×C4⋊Dic79C2×C411yescharacteristic
72C2×C4⋊Dic79C2×C411yescharacteristic
73C2×Dic79⋊C4C2×C411yescharacteristic
74C2×Dic79⋊C4C2×C411yescharacteristic
75C2×Dic79⋊C4C2×C411yescharacteristic
76C2×Dic79⋊C4C2311yescharacteristic
77C4⋊C4×D79—24no—
78C4⋊C4×D79—24no—
79C2×C22⋊Dic79C2×C411yescharacteristic
80C2×C22⋊Dic79C2×C411yescharacteristic
81C2×(C2×D79)⋊C4C2×C411yescharacteristic
82C2×(C2×D79)⋊C4C2×C411yescharacteristic
83C2×(C2×D79)⋊C4C2×C411yescharacteristic
84C2×(C2×D79)⋊C4C2311yescharacteristic
85C2×C4×Dic79C2×C411yescharacteristic
86C2×C4×Dic79C2×C411yescharacteristic
87C2×C4×Dic79C2×C411yescharacteristic
88C2×C4×Dic79C2×C411yescharacteristic
89C2×C4×Dic79C2×C411yescharacteristic
90C2×C4×Dic79C2311yescharacteristic
91Dic79⋊C8—24no—
92C22⋊(C79⋊C8)—22no—
93C22⋊(C79⋊C8)C2314yes—
94(C2×D79)⋊C8—24no—
95#14579071—24no—
96C23×C4×C79C2×C411yescharacteristic
97C23×C4×C79C2×C411yescharacteristic
98C23×C4×C79C2311yescharacteristic
99C22×C8×C79—22no—
100C2×C4⋊C4×C79—22no—
101C2×C4⋊C4×C79—22no—
102C2×C4⋊C4×C79C2×C411yescharacteristic
103C2×C4⋊C4×C79C2×C411yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C4⋊C4×C79C2311yescharacteristic
106C22⋊C8×C79—22no—
107C22⋊C8×C79C2314yes—
108C23×D79C22×C411yescharacteristic
109C23×D79C22×C411yescharacteristic
110#11929624—216no—
111#11929624—216no—
112C2×Dic158—22no—
113C2×Dic158—22no—
114C2×Dic158—22no—
115C2×D316—22no—
116C2×D316—22no—
117C2×D316—22no—
118C2×C22⋊D79—21no—
119C2×C22⋊D79—21no—
120C2×C22⋊D79—22no—
121C2×C22⋊D79—22no—
122C2×C22⋊D79—22no—
123C2×C22⋊D79C22⋊C412yes—
124C2×C22⋊D79C22⋊C412yes—
125C2×C22⋊D79M1614yescomplemented
126C22×Dic79C22×C411yescharacteristic
127C22×Dic79C22×C411yescharacteristic
128C22×Dic79C22×C411yescharacteristic
129C22×Dic79C22×C411yescharacteristic
130C22×Dic79C22×C411yescharacteristic
131C22×Dic79C22×C411yescharacteristic
132C2×C4×D79—22no—
133C2×C4×D79—22no—
134C2×C4×D79—22no—
135C2×C4×D79—22no—
136C2×C4×D79—22no—
137C2×C4×D79—22no—
138C2×C4×D79—22no—
139C2×C4×D79—22no—
140C2×C4×D79—22no—
141C2×C4×D79—81no—
142C2×C4×D79—81no—
143C2×C4×D79—81no—
144C2×C4×D79—81no—
145C2×C4×D79C2×D414yes—
146#11929607—84no—
147C2×C79⋊C8—24no—
148C2×C79⋊C8—82no—
149C4⋊Dic79—22no—
150C4⋊Dic79—22no—
151Dic79⋊C4—22no—
152Dic79⋊C4—22no—
153Dic79⋊C4—22no—
154Dic79⋊C4C22⋊C414yes—
155C22⋊Dic79—21no—
156C22⋊Dic79—21no—
157C22⋊Dic79C22⋊C412yes—
158C22⋊Dic79C22⋊C412yes—
159(C2×D79)⋊C4—22no—
160(C2×D79)⋊C4—22no—
161(C2×D79)⋊C4—22no—
162(C2×D79)⋊C4C22⋊C414yes—
163C4×Dic79—22no—
164C4×Dic79—22no—
165C4×Dic79—22no—
166C4×Dic79—22no—
167C4×Dic79—22no—
168C4×Dic79C2×D414yes—
169C24×C79C22×C411yescharacteristic
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—22no—
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79—81no—
176C22×C4×C79—81no—
177C22×C4×C79M1614yescomplemented
178C22×C4×C79C22×C411yescharacteristic
179C22×C4×C79C22×C411yescharacteristic
180C22×C4×C79C22×C411yescharacteristic
181C22×C4×C79C22×C411yescharacteristic
182C22×C4×C79C22×C411yescharacteristic
183C22×C4×C79C2×D412yes—
184C22×C4×C79C2×D412yes—
185C22×C4×C79C2411yescharacteristic
186C2×C8×C79—24no—
187C2×C8×C79—82no—
188C4⋊C4×C79—22no—
189C4⋊C4×C79—22no—
190C4⋊C4×C79—22no—
191C4⋊C4×C79—22no—
192C23×C79—41no—
193C23×C79—41no—
194C23×C79C2×C22⋊C411yescharacteristic
195C23×C79C2×C22⋊C411yescharacteristic
196C23×C79C2×M1612yes—
197C23×C79C2×M1612yes—
198C23×C79C23×C411yescharacteristic
199C22×D79—21no—
200C22×D79—21no—
201C22×D79—21no—
202C22×D79—21no—
203C22×D79—41no—
204C22×D79—41no—
205C22×D79—41no—
206C22×D79—41no—
207C22×D79C2×C22⋊C412yes—
208C22×D79C2×M1612yes—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—24no—
214C2×C4×C79—24no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—42no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—42no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—81no—
224C2×C4×C79—81no—
225C2×C4×C79C2×C22⋊C412yes—
226C2×C4×C79C2×M1612yes—
227C2×C4×C79C2×M1612yes—
228C2×C4×C79C22×D412yes—
229C22⋊D79—24no—
230C22⋊D79—24no—
231C22⋊D79—24no—
232C22⋊D79—24no—
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×Dic79C2×C22⋊C412yes—
244C2×Dic79C2×C22⋊C412yes—
245C2×Dic79C2×C22⋊C412yes—
246C2×Dic79C2×M1612yes—
247D316—28no—
248C4×D79—28no—
249C4×D79—28no—
250C4×D79—44no—
251C4×D79—82no—
252C4×D79—82no—
253C4×D79—82no—
254Dic158—28no—
255C8×C79—44no—
256C79⋊C8—44no—
257C22×C79—21no—
258C22×C79—21no—
259C22×C79—21no—
260C22×C79—21no—
261C22×C79—21no—
262C22×C79—21no—
263C22×C79—42no—
264C22×C79—42no—
265C22×C79#40712yes—
266C22×C79#40712yes—
267C22×C79C22×C22⋊C411yescharacteristic
268C22×C79C22×M1611yescharacteristic
269C22×C79C22×M1611yescharacteristic
270C2×D79—22no—
271C2×D79—22no—
272C2×D79—22no—
273C2×D79—22no—
274C2×D79—42no—
275C2×D79—42no—
276C2×D79—42no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—42no—
280C4×C79—22no—
281C4×C79—22no—
282C4×C79—22no—
283C4×C79—22no—
284C4×C79—41no—
285C4×C79—41no—
286Dic79—22no—
287Dic79—22no—
288Dic79—22no—
289Dic79—22no—
290Dic79—41no—
291Dic79—41no—
292#17368—791nomaximal, Sylow
293C2×C79—21no—
294C2×C79—21no—
295C2×C79—22no—
296C2×C79—22no—
297C2×C79C2×#40711yescharacteristic
298C2×C79#288311yescharacteristic
299C2×C79#288911yescharacteristic
300D79—42no—
301D79—42no—
302C2×#520—791no—
303#2973—792no—
304#1467—792no—
305#1465—792no—
306#1464—792no—
307#1463—792no—
308#1459—792no—
309C2×#324—791no—
310C2×#324D7911yescharacteristic, complemented
311C79#1736811yescharacteristic, complemented, Sylow
312C22×#41—791no—
313#520—1584no—
314#520—1584no—
315C2×C4×Q8—791no—
316C2×C4×Q8—791no—
317C2×C4×D4—791no—
318C2×C4×D4—791no—
319C2×C4×D4—791no—
320C2×C4×D4—791no—
321C2×C4×D4—791no—
322C2×C4×D4—791no—
323C2×#117—791no—
324C2×#117—791no—
325C2×#117—791no—
326C22×C4⋊C4—791no—
327C22×C4⋊C4—791no—
328C22×C4⋊C4C2×D7911yescharacteristic
329#435—1584no—
330#435—1584no—
331#432—794no—
332#407—794no—
333C2×C22⋊C8—792no—
334C2×C22⋊C8C2×D7912yes—
335#324—792no—
336#324—792no—
337#324—794no—
338#324C2×D7912yes—
339#324C2×D7912yes—
340#323—794no—
341C2×#41—1582no—
342C2×#41—1582no—
343C2×#41—1584no—
344C2×#41—1584no—
345C22×Q8—791no—
346C22×D4—791no—
347C22×D4—791no—
348C22×D4—791no—
349C23×C4—791no—
350C23×C4—791no—
351C23×C4—791no—
352C23×C4—791no—
353C23×C4—791no—
354C23×C4—791no—
355C23×C4C4×D7911yescharacteristic
356C23×C4C4×D7911yescharacteristic
357C23×C4C22×D7911yescharacteristic
358C2×M16—1584no—
359C22×C8—22no—
360C22×C8—1582no—
361C4×Q8—1584no—
362C4×Q8—1584no—
363C4×D4—794no—
364C4×D4—794no—
365C4×D4—1584no—
366C4×D4—1582no—
367C4×D4—1582no—
368C4×D4—1584no—
369C4×D4—1582no—
370C4×D4—1582no—
371C4×D4—1582no—
372C4×D4—1582no—
373#117—1584no—
374#117—1584no—
375#117—1582no—
376#117—1582no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—791no—
380C2×C4⋊C4—791no—
381C2×C4⋊C4—791no—
382C2×C4⋊C4—791no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—1584no—
386C2×C4⋊C4—1584no—
387C2×C4⋊C4C4×D7911yescharacteristic
388C2×C4⋊C4C4×D7911yescharacteristic
389C2×C4⋊C4C22×D7911yescharacteristic
390C2×C4⋊C4C22×D7911yescharacteristic
391C2×C22⋊C4—791no—
392C2×C22⋊C4—791no—
393C2×C22⋊C4—791no—
394C2×C22⋊C4—791no—
395C2×C22⋊C4—791no—
396C2×C22⋊C4—791no—
397C2×C42—791no—
398C2×C42—791no—
399C2×C42—791no—
400C2×C42—791no—
401C2×C42—791no—
402C2×C42—791no—
403C4⋊C8—1584no—
404C22⋊C8—22no—
405C22⋊C8—794no—
406C22⋊C8—1584no—
407C22⋊C8—1582no—
408C22⋊C8C22×D7914yes—
409#97—1584no—
410C24—791no—
411C24—791no—
412C24C2×C4×D7911yescharacteristic
413#41—15816no—
414#41—15816no—
415C2×Q8—1582no—
416C2×Q8—1582no—
417C2×Q8—1582no—
418C2×D4—792no—
419C2×D4—792no—
420C2×D4—794no—
421C2×D4—1582no—
422C2×D4—1582no—
423C2×D4—1581no—
424C2×D4—1581no—
425C2×D4—1582no—
426C2×D4—1582no—
427C2×D4—1582no—
428C2×D4—1582no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—791no—
437C22×C4—791no—
438C22×C4—794no—
439C22×C4—791no—
440C22×C4—791no—
441C22×C4—791no—
442C22×C4—791no—
443C22×C4—6321no—
444C22×C4—6321no—
445C22×C4—6321no—
446C22×C4—6321no—
447C22×C4—1582no—
448C22×C4—1582no—
449C22×C4—1582no—
450C22×C4—1582no—
451C22×C4—1582no—
452C22×C4—1582no—
453C22×C4—1582no—
454C22×C4—1582no—
455C22×C4—1582no—
456C22×C4#1192960714yescomplemented
457C22×C4C2×C4×D7911yescharacteristic
458C22×C4C2×C4×D7911yescharacteristic
459C22×C4C2×C4×D7911yescharacteristic
460C22×C4C2×C4×D7911yescharacteristic
461C22×C4C2×C4×D7911yescharacteristic
462C22×C4D4×D7912yes—
463C22×C4D4×D7912yes—
464C22×C4C23×D7911yescharacteristic
465M16—6324no—
466C2×C8—24no—
467C2×C8—82no—
468C2×C8—6322no—
469C2×C8—1584no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—794no—
475C4⋊C4—1582no—
476C4⋊C4—1582no—
477C4⋊C4—1582no—
478C4⋊C4—1582no—
479C4⋊C4—1582no—
480C22⋊C4—792no—
481C22⋊C4—792no—
482C22⋊C4—794no—
483C22⋊C4—1582no—
484C22⋊C4—1582no—
485C22⋊C4—1581no—
486C22⋊C4—1581no—
487C22⋊C4—1582no—
488C42—794no—
489C42—1582no—
490C42—1582no—
491C42—1582no—
492C42—1582no—
493C42—1582no—
494C23—41no—
495C23—41no—
496C23—3161no—
497C23—3161no—
498C23—3161no—
499C23—3161no—
500C23—792no—
501C23—792no—
502C23—1581no—
503C23—1581no—
504C23—1581no—
505C23—1581no—
506C23C22⋊C4×D7911yescharacteristic
507C23C22⋊C4×D7911yescharacteristic
508C23M16×D7912yes—
509C23C2×#1192960712yes—
510C23C22×C4×D7911yescharacteristic
511Q8—1588no—
512D4—1588no—
513D4—1584no—
514D4—1584no—
515D4—1584no—
516D4—1584no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—3164no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—792no—
539C2×C4—792no—
540C2×C4—792no—
541C2×C4—792no—
542C2×C4—6322no—
543C2×C4—6322no—
544C2×C4—6322no—
545C2×C4—1588no—
546C2×C4—1588no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4C22⋊C4×D7912yes—
554C2×C4C2×#1192960712yes—
555C2×C4C2×#1192960712yes—
556C2×C4C2×D4×D7912yes—
557C8—44no—
558C8—3164no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—42no—
566C22—42no—
567C22—3162no—
568C22—3162no—
569C22—3162no—
570C22—3162no—
571C22—3161no—
572C22—3161no—
573C22—1582no—
574C22—1582no—
575C22—1582no—
576C22—1582no—
577C22#1909852512yes—
578C22#1909852512yes—
579C22C2×C22⋊C4×D7911yescharacteristic
580C22C2×M16×D7911yescharacteristic
581C22C22×#1192960711yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—3161no—
589C4—3161no—
590C4—1582no—
591C4—1582no—
592C4—1582no—
593C4—1582no—
594C2—21no—
595C2—21no—
596C2—22no—
597C2—22no—
598C2—3162no—
599C2—3162no—
600C2C2×#1909852511yescharacteristic
601C2#2377185811yescharacteristic
602C2#2377188011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D79
commutatorC23×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#324×C79C2
socleC22×C79C22×C22⋊C4
prime radicalC22×#11929624C4
cosocleC22×C4×C79C24
hypercenterC2×#324D79
hypocenterC79#17368
LCSG ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #23772426C2semidirect
3. #23761742C2semidirect
4. #23761737C2semidirect
5. #23761733C2semidirect
6. #23761728C2semidirect
7. #23761716C2semidirect
8. C2×#19098289C2semidirect
9. C2×#324×C79C2semidirect
125. C2×C22⋊D79M16semidirect
177. C22×C4×C79M16semidirect
310. C2×#324D79semidirect
311. C79#17368semidirect
456. C22×C4#11929607semidirect

Maximal subgroups

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

subgroupindex
C2×#190990442
#237724262
#237617422
#237617372
#237617332
#237617282
#237617162
C2×#190982892
C2×#324×C792
#1736879