Finite Group Databasebeta

← #34317137

Subgroups — #34317137

Order 20224.

1166 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098605C211yesmaximal, characteristic, complemented
2#23765783C211yesmaximal, characteristic
3C2×#19098423C211yesmaximal, characteristic
4#23764833C212yesmaximal, complemented
5#23764833C212yesmaximal, complemented
6#23764828C212yesmaximal, complemented
7#23764823C212yesmaximal, complemented
8#23763120C211yesmaximal, characteristic, complemented
9#23763120C211yesmaximal, characteristic, complemented
10#23763120C211yesmaximal, characteristic, complemented
11C2×#407×C79C211yesmaximal, characteristic
12C2×#14579120C2211yescharacteristic
13C2×#14579120C2211yescharacteristic
14C22×C4⋊Dic79C2211yescharacteristic
15C22×C4⋊Dic79C2211yescharacteristic
16C22×C22⋊Dic79—21no—
17#19098605—24no—
18C2×#14579095C2212yescomplemented
19C2×#14579093C2212yescomplemented
20#19098426—21no—
21#19098423—21no—
22#19098423—21no—
23#19098423C2212yes—
24#19098423C2212yes—
25#19098423C2211yescharacteristic
26#19098423C2211yescharacteristic
27#19098423C2211yescharacteristic
28#19098423C2211yescharacteristic
29#19098419—21no—
30C4×C22⋊Dic79—21no—
31C4×C22⋊Dic79—21no—
32C2×C79⋊#95—21no—
33#19098326C2212yescomplemented
34#19098326C2212yescomplemented
35#19098326C2212yescomplemented
36#19098326C2212yescomplemented
37#19098326C2212yescomplemented
38#19098326C2212yescomplemented
39C24×C4×C79C2211yescharacteristic
40C22×M16×C79C2211yescharacteristic
41C22×M16×C79C2211yescharacteristic
42#407×C79—24no—
43C2×C22⋊C8×C79C2212yes—
44C2×C22⋊C8×C79C2212yes—
45C23×Dic79—21no—
46C23×Dic79—21no—
47#14579120—22no—
48#14579120—22no—
49#14579120—22no—
50#14579120—22no—
51C2×C4⋊Dic79—22no—
52C2×C4⋊Dic79—22no—
53C2×C4⋊Dic79—22no—
54C2×C4⋊Dic79—22no—
55C2×C4⋊Dic79—21no—
56C2×C4⋊Dic79—21no—
57C2×C4⋊Dic79C2312yes—
58C2×C4⋊Dic79C2312yes—
59C2×C4⋊Dic79C2311yescharacteristic
60C2×C4⋊Dic79C2311yescharacteristic
61C2×C4⋊Dic79C2311yescharacteristic
62C2×C4⋊Dic79C2311yescharacteristic
63C2×C22⋊Dic79—21no—
64C2×C22⋊Dic79—21no—
65C2×C22⋊Dic79—21no—
66C2×C22⋊Dic79—21no—
67C2×C22⋊Dic79—21no—
68C2×C22⋊Dic79—21no—
69C2×C22⋊Dic79—22no—
70C2×C22⋊Dic79—22no—
71C2×C22⋊Dic79—44no—
72C2×C4×Dic79—21no—
73C2×C4×Dic79—21no—
74#14579095—24no—
75#14579093—24no—
76C79⋊#95—22no—
77C79⋊#95—22no—
78C79⋊#95—21no—
79C79⋊#95—21no—
80C79⋊#95—21no—
81C79⋊#95—21no—
82C25×C79D411yescharacteristic
83C23×C4×C79—22no—
84C23×C4×C79—22no—
85C23×C4×C79—22no—
86C23×C4×C79—22no—
87C23×C4×C79—22no—
88C23×C4×C79—22no—
89C23×C4×C79D411yescharacteristic
90C23×C4×C79D411yescharacteristic
91C23×C4×C79D411yescharacteristic
92C23×C4×C79C2311yescharacteristic
93C23×C4×C79C2311yescharacteristic
94C23×C4×C79C2311yescharacteristic
95C2×M16×C79—22no—
96C2×M16×C79—22no—
97C2×M16×C79—22no—
98C2×M16×C79—22no—
99C2×M16×C79—22no—
100C2×M16×C79D414yes—
101C22×C8×C79C2312yes—
102C22×C8×C79C2312yes—
103C22⋊C8×C79—24no—
104C22⋊C8×C79—24no—
105C22×Dic79—21no—
106C22×Dic79—21no—
107C22×Dic79—21no—
108C22×Dic79—21no—
109C22×Dic79—22no—
110C22×Dic79—22no—
111C22×Dic79—42no—
112C22×Dic79—42no—
113C22×Dic79—42no—
114C22×Dic79—42no—
115C22×Dic79—42no—
116C22×Dic79—42no—
117C4⋊Dic79—21no—
118C4⋊Dic79—22no—
119C4⋊Dic79—21no—
120C4⋊Dic79—22no—
121C4⋊Dic79—22no—
122C4⋊Dic79—22no—
123C4⋊Dic79—21no—
124C4⋊Dic79—21no—
125C4⋊Dic79—21no—
126C4⋊Dic79—21no—
127C4⋊Dic79—21no—
128C4⋊Dic79—21no—
129C4⋊Dic79—42no—
130C4⋊Dic79—42no—
131C22⋊Dic79—42no—
132C22⋊Dic79—42no—
133C22⋊Dic79—44no—
134C22⋊Dic79—42no—
135C22⋊Dic79—42no—
136C22⋊Dic79—42no—
137C22⋊Dic79—44no—
138C22⋊Dic79—42no—
139C4×Dic79—42no—
140C4×Dic79—42no—
141C24×C79—22no—
142C24×C79—22no—
143C24×C79—22no—
144C24×C79—22no—
145C24×C79—22no—
146C24×C79—41no—
147C24×C79—41no—
148C24×C79C2×D411yescharacteristic
149C24×C79C2×D411yescharacteristic
150C24×C79C2×D411yescharacteristic
151C22×C4×C79—22no—
152C22×C4×C79—22no—
153C22×C4×C79—22no—
154C22×C4×C79—22no—
155C22×C4×C79—22no—
156C22×C4×C79—22no—
157C22×C4×C79—22no—
158C22×C4×C79—22no—
159C22×C4×C79—22no—
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×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—44no—
173C22×C4×C79—44no—
174C22×C4×C79—44no—
175C22×C4×C79—44no—
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D412yes—
180C22×C4×C79#4112yes—
181C22×C4×C79#4112yes—
182C22×C4×C79#4112yes—
183C22×C4×C79#4112yes—
184C22×C4×C79#4112yes—
185C22×C4×C79C2411yescharacteristic
186M16×C79—28no—
187M16×C79—44no—
188C2×C8×C79—22no—
189C2×C8×C79—22no—
190C2×C8×C79—22no—
191C2×C8×C79—22no—
192C2×C8×C79—22no—
193C2×C8×C79C2×D414yes—
194C23×C79—21no—
195C23×C79—21no—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79—28no—
199C23×C79—21no—
200C23×C79—21no—
201C23×C79—21no—
202C23×C79—21no—
203C23×C79—21no—
204C23×C79—21no—
205C23×C79—21no—
206C23×C79—21no—
207C23×C79—21no—
208C23×C79—21no—
209C23×C79—21no—
210C23×C79—21no—
211C23×C79—44no—
212C23×C79—44no—
213C23×C79—44no—
214C23×C79—44no—
215C23×C79—44no—
216C23×C79—44no—
217C23×C79#12112yes—
218C23×C79#12312yes—
219C23×C79#12312yes—
220C23×C79AGL(1,ℤ/8)12yes—
221C23×C79#13712yes—
222C23×C79C22×D411yescharacteristic
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—44no—
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—44no—
239C2×C4×C79—44no—
240C2×C4×C79#12112yes—
241C2×C4×C79#12314yes—
242C2×C4×C79#12312yes—
243C2×C4×C79#12312yes—
244C2×C4×C79#12414yes—
245C2×C4×C79#12614yes—
246C2×C4×C79#12614yes—
247C2×C4×C79C22×D411yescharacteristic
248C2×C4×C79C2×#4111yescharacteristic
249C2×C4×C79C2×#4111yescharacteristic
250C2×C4×C79C2×#4111yescharacteristic
251C2×C4×C79C2×#4111yescharacteristic
252C2×C4×C79C2×#4111yescharacteristic
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—42no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—42no—
259C2×Dic79—42no—
260C2×Dic79—42no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79—41no—
265C2×Dic79—41no—
266C2×Dic79—41no—
267C2×Dic79—41no—
268C2×Dic79—41no—
269C2×Dic79—42no—
270C2×Dic79—42no—
271C2×Dic79—84no—
272C2×Dic79—84no—
273C8×C79—24no—
274C8×C79—42no—
275C22×C79—22no—
276C22×C79—22no—
277C22×C79—22no—
278C22×C79—22no—
279C22×C79—22no—
280C22×C79—22no—
281C22×C79—22no—
282C22×C79—22no—
283C22×C79—22no—
284C22×C79—22no—
285C22×C79—22no—
286C22×C79—22no—
287C22×C79—416no—
288C22×C79—41no—
289C22×C79—41no—
290C22×C79—41no—
291C22×C79—41no—
292C22×C79—41no—
293C22×C79—41no—
294C22×C79—41no—
295C22×C79—41no—
296C22×C79#39714yes—
297C22×C79C2×#12111yescharacteristic
298C22×C79C2×#12311yescharacteristic
299C22×C79C2×#12311yescharacteristic
300C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C79C2×#13711yescharacteristic
302C22×C79#57511yescharacteristic
303C22×C79#57511yescharacteristic
304C4×C79—41no—
305C4×C79—41no—
306C4×C79#39714yes—
307C4×C79C2×#12311yescharacteristic
308C4×C79C2×#12411yescharacteristic
309C4×C79C2×#12611yescharacteristic
310C4×C79C2×#12611yescharacteristic
311Dic79—81no—
312Dic79—81no—
313Dic79—82no—
314Dic79—81no—
315Dic79—82no—
316Dic79—81no—
317#23642—791nomaximal, Sylow
318C2×C79—24no—
319C2×C79—42no—
320C2×C79—42no—
321C2×C79C2×#39711yescharacteristic
322C2×C79#305511yescharacteristic
323C2×C79#305611yescharacteristic
324C2×C79#309211yescharacteristic
325C2×C79#309311yescharacteristic
326C2×C79#309411yescharacteristic
327C2×C79#309511yescharacteristic
328C2×#428—791no—
329C2×#407D7911yescharacteristic
330#2280—791no—
331C2×#380—791no—
332#2077—792no—
333#2077—792no—
334#2076—792no—
335#2075—792no—
336#1736—791no—
337#1736—791no—
338#1736—791no—
339C79#2364211yescharacteristic, complemented, Sylow
340C24×C4C2×D7911yescharacteristic
341C22×M16C2×D7911yescharacteristic
342C22×M16C2×D7911yescharacteristic
343C2×#117—791no—
344C2×#117—791no—
345C22×C4⋊C4—791no—
346C22×C4⋊C4—791no—
347C22×C22⋊C4—1581no—
348#428—1584no—
349C2×#107—792no—
350C2×#106—792no—
351#407—24no—
352C2×C22⋊C8C2×D7912yes—
353C2×C22⋊C8C2×D7912yes—
354#381—1581no—
355#380—791no—
356#380—792no—
357#380—791no—
358#380—792no—
359#380—791no—
360#380—791no—
361#380—1581no—
362#380—1581no—
363#379—1581no—
364C4×C22⋊C4—1581no—
365C4×C22⋊C4—1581no—
366C2×#95—1581no—
367#340—792no—
368#340—792no—
369#340—792no—
370#340—792no—
371#340—792no—
372#340—792no—
373C25D31611yescharacteristic
374C23×C4—22no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—1581no—
381C23×C4—1581no—
382C23×C4D31611yescharacteristic
383C23×C4D31611yescharacteristic
384C23×C4D31611yescharacteristic
385C23×C4C22×D7911yescharacteristic
386C23×C4C22×D7911yescharacteristic
387C23×C4C22×D7911yescharacteristic
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—22no—
391C2×M16—22no—
392C2×M16—22no—
393C2×M16C22⋊D7914yes—
394C22×C8C22×D7912yes—
395C22×C8C22×D7912yes—
396#117—1582no—
397#117—1582no—
398#117—1582no—
399#117—1582no—
400C2×C4⋊C4—792no—
401C2×C4⋊C4—792no—
402C2×C4⋊C4—791no—
403C2×C4⋊C4—791no—
404C2×C4⋊C4—791no—
405C2×C4⋊C4—791no—
406C2×C4⋊C4—1581no—
407C2×C4⋊C4—1581no—
408C2×C4⋊C4—1582no—
409C2×C4⋊C4—1582no—
410C2×C4⋊C4—1582no—
411C2×C4⋊C4—1582no—
412C2×C22⋊C4—3164no—
413C2×C22⋊C4—1582no—
414C2×C22⋊C4—1582no—
415C2×C22⋊C4—1581no—
416C2×C22⋊C4—1581no—
417C2×C22⋊C4—1581no—
418C2×C22⋊C4—1581no—
419C2×C22⋊C4—1581no—
420C2×C22⋊C4—1581no—
421C2×C42—1581no—
422C2×C42—1581no—
423#107—1584no—
424#106—1584no—
425C22⋊C8—24no—
426C22⋊C8—24no—
427#95—1582no—
428#95—1582no—
429#95—1581no—
430#95—1581no—
431#95—1581no—
432#95—1581no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—41no—
439C24—41no—
440C24C2×D31611yescharacteristic
441C24C2×D31611yescharacteristic
442C24C2×D31611yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—3162no—
469C22×C4—3162no—
470C22×C4—3162no—
471C22×C4—3162no—
472C22×C4—3162no—
473C22×C4—3162no—
474C22×C4—1581no—
475C22×C4—1581no—
476C22×C4—1581no—
477C22×C4—1581no—
478C22×C4—1582no—
479C22×C4—1582no—
480C22×C4C2×C22⋊D7912yes—
481C22×C4C2×D31611yescharacteristic
482C22×C4C2×D31611yescharacteristic
483C22×C4C2×D31611yescharacteristic
484C22×C4#1192962312yes—
485C22×C4#1192962312yes—
486C22×C4#1192962312yes—
487C22×C4#1192962312yes—
488C22×C4#1192962412yes—
489C22×C4C23×D7911yescharacteristic
490M16—28no—
491M16—44no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8C2×C22⋊D7914yes—
498C4⋊C4—3162no—
499C4⋊C4—3162no—
500C4⋊C4—1582no—
501C4⋊C4—1582no—
502C4⋊C4—1581no—
503C4⋊C4—1581no—
504C4⋊C4—1581no—
505C4⋊C4—1581no—
506C4⋊C4—1582no—
507C4⋊C4—1581no—
508C4⋊C4—1581no—
509C4⋊C4—1581no—
510C4⋊C4—1582no—
511C4⋊C4—1581no—
512C22⋊C4—3164no—
513C22⋊C4—3162no—
514C22⋊C4—3162no—
515C22⋊C4—3162no—
516C22⋊C4—3162no—
517C22⋊C4—3164no—
518C22⋊C4—3162no—
519C22⋊C4—3162no—
520C42—3162no—
521C42—3162no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—28no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23#1457913512yes—
546C23#1457914512yes—
547C23#1457914512yes—
548C23#1457918912yes—
549C23#1457919412yes—
550C23C22×D31611yescharacteristic
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—3161no—
569C2×C4—3161no—
570C2×C4—3162no—
571C2×C4—3161no—
572C2×C4—3161no—
573C2×C4—3162no—
574C2×C4—3162no—
575C2×C4—3162no—
576C2×C4—3161no—
577C2×C4—3161no—
578C2×C4—3161no—
579C2×C4—3161no—
580C2×C4—3161no—
581C2×C4—3161no—
582C2×C4—3161no—
583C2×C4—3161no—
584C2×C4—3162no—
585C2×C4—3162no—
586C2×C4—6324no—
587C2×C4—6324no—
588C2×C4#1457913512yes—
589C2×C4#1457914214yes—
590C2×C4#1457914512yes—
591C2×C4#1457914512yes—
592C2×C4#1457914814yes—
593C2×C4#1457915414yes—
594C2×C4#1457915414yes—
595C2×C4C22×C22⋊D7911yescharacteristic
596C2×C4C2×#1192962311yescharacteristic
597C2×C4C2×#1192962311yescharacteristic
598C2×C4C2×#1192962311yescharacteristic
599C2×C4C2×#1192962311yescharacteristic
600C2×C4C2×#1192962411yescharacteristic
601C8—24no—
602C8—42no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—416no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22#1909848814yes—
625C22C2×#1457913511yescharacteristic
626C22C2×#1457914511yescharacteristic
627C22C2×#1457914511yescharacteristic
628C22C2×#1457918911yescharacteristic
629C22C2×#1457919411yescharacteristic
630C22#1909936311yescharacteristic
631C22#1909936311yescharacteristic
632C4—41no—
633C4—41no—
634C4—6322no—
635C4—6321no—
636C4—6321no—
637C4—6321no—
638C4—6321no—
639C4—6322no—
640C4#1909848814yes—
641C4C2×#1457914211yescharacteristic
642C4C2×#1457914811yescharacteristic
643C4C2×#1457915411yescharacteristic
644C4C2×#1457915411yescharacteristic
645C2—24no—
646C2—42no—
647C2—42no—
648C2C2×#1909848811yescharacteristic
649C2#2377301911yescharacteristic
650C2#2377303011yescharacteristic
651C2#2377332511yescharacteristic
652C2#2377333311yescharacteristic
653C2#2377334111yescharacteristic
654C2#2377334911yescharacteristic
subgroupG/H
centerC23C22×D316
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#407×C79C2
socleC23×C79C22×D4
prime radicalC25×C79D4
hypercenterC2×#407D79
hypocenterC79#23642
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
1. C2×#19098605C2semidirect
4. #23764833C2semidirect
5. #23764833C2semidirect
6. #23764828C2semidirect
7. #23764823C2semidirect
8. #23763120C2semidirect
9. #23763120C2semidirect
10. #23763120C2semidirect
18. C2×#14579095C22semidirect
19. C2×#14579093C22semidirect
33. #19098326C22semidirect
34. #19098326C22semidirect
35. #19098326C22semidirect
36. #19098326C22semidirect
37. #19098326C22semidirect
38. #19098326C22semidirect
339. C79#23642semidirect

Maximal subgroups

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

subgroupindex
C2×#190986052
#237657832
C2×#190984232
#237648332
#237648282
#237648232
#237631202
C2×#407×C792
#2364279