Finite Group Databasebeta

← #34320538

Subgroups — #34320538

Order 20224.

1278 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098718C211yesmaximal, characteristic, complemented
2C22×#14579084C211yesmaximal, characteristic, complemented
3#23766765C211yesmaximal, characteristic
4#23764538C212yesmaximal, complemented
5#23764408C212yesmaximal, complemented
6#23763824C211yesmaximal, characteristic, complemented
7#23763741C212yesmaximal, complemented
8#23763707C212yesmaximal, complemented
9#23763693C211yesmaximal, characteristic
10#23763490C211yesmaximal, characteristic, complemented
11C2×#554×C79C211yesmaximal, characteristic
12C23×Dic158C2211yescharacteristic
13C22×#11929614C2212yes—
14C23×C79⋊C8C2211yescharacteristic
15C2×#14579140C2212yes—
16#19098718—24no—
17C2×#14579085C2212yes—
18C2×#14579084—24no—
19C2×#14579084C2212yes—
20C2×#14579084C2212yes—
21C2×#14579084C2212yes—
22C2×C22⋊(C79⋊C8)C2211yescharacteristic
23C2×C22⋊(C79⋊C8)C2212yes—
24#19098483—21no—
25#19098475—21no—
26#19098469C2212yes—
27#19098443—21no—
28#19098435—21no—
29#19098423C2211yescharacteristic
30#19098325C2212yes—
31#19098325C2212yes—
32#19098325C2212yes—
33#19098318C2212yes—
34#554×C79—24no—
35#554×C79—24no—
36C2×C4⋊Q8×C79C2211yescharacteristic
37C2×#125×C79—21no—
38C2×C22⋊Q8×C79—21no—
39C2×C22⋊Q8×C79—21no—
40C2×C22⋊Q8×C79C2211yescharacteristic
41C2×C22⋊Q8×C79C2211yescharacteristic
42C2×C4×Q8×C79C2211yescharacteristic
43C2×C4×D4×C79—21no—
44C2×C4×D4×C79C2211yescharacteristic
45C22×C4⋊C4×C79C2211yescharacteristic
46C22×C4⋊C4×C79C2211yescharacteristic
47C22×Dic158—21no—
48C22×Dic158—22no—
49C22×Dic158—24no—
50C22×Dic158—22no—
51C22×Dic158—42no—
52C22×Dic158C2312yes—
53C23×Dic79—21no—
54C2×#11929614—24no—
55C2×#11929614—24no—
56C2×#11929614—42no—
57C22×C79⋊C8—22no—
58C22×C79⋊C8—42no—
59C22×C79⋊C8C2312yes—
60C22×C79⋊C8C2312yes—
61#14579140—24no—
62#14579140—42no—
63C2×C4⋊Dic79C2312yes—
64C2×Dic79⋊C4—22no—
65C2×C22⋊Dic79—21no—
66C2×C22⋊Dic79—22no—
67#14579085—42no—
68#14579084—24no—
69#14579084—24no—
70#14579084—24no—
71C22⋊(C79⋊C8)—22no—
72C22⋊(C79⋊C8)—42no—
73C79⋊#95—21no—
74C79⋊#95—22no—
75C79⋊#95—22no—
76C22×Q8×C79—21no—
77C22×Q8×C79C2311yescharacteristic
78C22×D4×C79D411yescharacteristic
79C23×C4×C79—21no—
80C23×C4×C79—21no—
81C23×C4×C79C2311yescharacteristic
82C23×C4×C79C2311yescharacteristic
83C4⋊Q8×C79—22no—
84C4⋊Q8×C79—22no—
85#125×C79—28no—
86C22⋊Q8×C79—28no—
87C22⋊Q8×C79—28no—
88C22⋊Q8×C79—22no—
89C22⋊Q8×C79—22no—
90C22⋊Q8×C79—22no—
91C22⋊Q8×C79D414yes—
92C4×Q8×C79—22no—
93C4×Q8×C79—22no—
94C4×D4×C79—28no—
95C4×D4×C79—22no—
96C4×D4×C79—22no—
97C2×C4⋊C4×C79—21no—
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79—22no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79—21no—
102C2×C4⋊C4×C79—21no—
103C2×C4⋊C4×C79—21no—
104C2×C4⋊C4×C79—22no—
105C2×C4⋊C4×C79—22no—
106C2×C4⋊C4×C79D411yescharacteristic
107C2×C4⋊C4×C79D414yes—
108C2×C4⋊C4×C79C2311yescharacteristic
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C22⋊C4×C79—21no—
112C2×C22⋊C4×C79—21no—
113C2×C22⋊C4×C79D411yescharacteristic
114C2×C22⋊C4×C79D411yescharacteristic
115C2×C42×C79—21no—
116C2×C42×C79C2311yescharacteristic
117C2×Dic158—28no—
118C2×Dic158—22no—
119C2×Dic158—22no—
120C2×Dic158—22no—
121C2×Dic158—22no—
122C2×Dic158—22no—
123C2×Dic158—48no—
124C2×Dic158—44no—
125C2×Dic158—44no—
126C2×Dic158—42no—
127C2×Dic158#4114yes—
128C22×Dic79—22no—
129C22×Dic79—22no—
130C22×Dic79—44no—
131C22×Dic79—42no—
132#11929614—48no—
133C2×C79⋊C8—22no—
134C2×C79⋊C8—22no—
135C2×C79⋊C8—22no—
136C2×C79⋊C8—22no—
137C2×C79⋊C8—22no—
138C2×C79⋊C8—22no—
139C2×C79⋊C8—44no—
140C4⋊Dic79—22no—
141C4⋊Dic79—22no—
142Dic79⋊C4—44no—
143C22⋊Dic79—44no—
144C22⋊Dic79—42no—
145C24×C79C2×D411yescharacteristic
146C24×C79C2×D411yescharacteristic
147C2×Q8×C79—24no—
148C2×Q8×C79—21no—
149C2×Q8×C79—22no—
150C2×Q8×C79—22no—
151C2×Q8×C79—82no—
152C2×Q8×C79C2×D412yes—
153C2×D4×C79—22no—
154C2×D4×C79—22no—
155C2×D4×C79Q1614yescomplemented
156C22×C4×C79—22no—
157C22×C4×C79—22no—
158C22×C4×C79—21no—
159C22×C4×C79—21no—
160C22×C4×C79—21no—
161C22×C4×C79—21no—
162C22×C4×C79—24no—
163C22×C4×C79—24no—
164C22×C4×C79—21no—
165C22×C4×C79—21no—
166C22×C4×C79—22no—
167C22×C4×C79—22no—
168C22×C4×C79—82no—
169C22×C4×C79—82no—
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D412yes—
174C22×C4×C79C2×D412yes—
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2411yescharacteristic
177C4⋊C4×C79—24no—
178C4⋊C4×C79—24no—
179C4⋊C4×C79—24no—
180C4⋊C4×C79—24no—
181C4⋊C4×C79—24no—
182C4⋊C4×C79—24no—
183C4⋊C4×C79—22no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79C2×D412yes—
188C4⋊C4×C79C2×D412yes—
189C4⋊C4×C79C2×D412yes—
190C22⋊C4×C79—24no—
191C22⋊C4×C79—24no—
192C22⋊C4×C79—22no—
193C22⋊C4×C79—22no—
194C42×C79—24no—
195C42×C79—21no—
196C42×C79—21no—
197C23×C79—22no—
198C23×C79—22no—
199C23×C79—21no—
200C23×C79—22no—
201C23×C79—21no—
202C23×C79—22no—
203C23×C79#12012yes—
204C23×C79C2×Q1612yes—
205C23×C79C2×Q1612yes—
206C23×C79#13712yes—
207C23×C79C22×D411yescharacteristic
208Q8×C79—24no—
209Q8×C79—41no—
210Q8×C79—41no—
211Q8×C79—84no—
212D4×C79—28no—
213C2×C4×C79—21no—
214C2×C4×C79—22no—
215C2×C4×C79—21no—
216C2×C4×C79—22no—
217C2×C4×C79—22no—
218C2×C4×C79—24no—
219C2×C4×C79—22no—
220C2×C4×C79—21no—
221C2×C4×C79—22no—
222C2×C4×C79—21no—
223C2×C4×C79—22no—
224C2×C4×C79—22no—
225C2×C4×C79—22no—
226C2×C4×C79—22no—
227C2×C4×C79—42no—
228C2×C4×C79—42no—
229C2×C4×C79—42no—
230C2×C4×C79—42no—
231C2×C4×C79—41no—
232C2×C4×C79—41no—
233C2×C4×C79—42no—
234C2×C4×C79—84no—
235C2×C4×C79—84no—
236C2×C4×C79#12012yes—
237C2×C4×C79C2×Q1612yes—
238C2×C4×C79C22×D411yescharacteristic
239C2×C4×C79C22×D411yescharacteristic
240C2×C4×C79C2×#4112yes—
241C2×C4×C79H(5,𝔽2)12yes—
242C2×Dic79—24no—
243C2×Dic79—48no—
244C2×Dic79—42no—
245C2×Dic79—42no—
246C2×Dic79—42no—
247C2×Dic79—42no—
248C2×Dic79—42no—
249Dic158—24no—
250Dic158—24no—
251Dic158—48no—
252Dic158—42no—
253Dic158—42no—
254C79⋊C8—42no—
255C79⋊C8—42no—
256C22×C79—22no—
257C22×C79—22no—
258C22×C79—24no—
259C22×C79—24no—
260C22×C79—22no—
261C22×C79—24no—
262C22×C79—41no—
263C22×C79—41no—
264C22×C79—42no—
265C22×C79#45114yes—
266C22×C79C2×#12011yescharacteristic
267C22×C79#54612yes—
268C22×C79C22×Q1611yescharacteristic
269C22×C79C2×#13711yescharacteristic
270C22×C79#57712yes—
271C4×C79—22no—
272C4×C79—22no—
273C4×C79—41no—
274C4×C79—42no—
275C4×C79—41no—
276C4×C79—42no—
277C4×C79—42no—
278C4×C79—42no—
279C4×C79#52511yescharacteristic
280C4×C79#53511yescharacteristic
281C4×C79#54612yes—
282Dic79—44no—
283Dic79—82no—
284#23970—791nomaximal, Sylow
285C2×C79—24no—
286C2×C79—42no—
287C2×C79#240411yescharacteristic
288C2×C79C2×#45111yescharacteristic
289C2×C79#300911yescharacteristic
290C2×C79#329912yes—
291C2×C79#331612yes—
292C2×#554D7911yescharacteristic
293C2×#451—791no—
294C22×#103—791no—
295#2409—791no—
296#2024—792no—
297#2001—792no—
298#1892—791no—
299#1878—792no—
300#1872—792no—
301#1869—791no—
302#1824—791no—
303C79#2397011yescharacteristic, complemented, Sylow
304C23×Q8—791no—
305C22×Q16—792no—
306C23×C8—791no—
307#554—24no—
308#554—24no—
309C2×C4⋊Q8C2×D7911yescharacteristic
310C2×#125—21no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8—21no—
313C2×C22⋊Q8—792no—
314C2×C22⋊Q8C2×D7911yescharacteristic
315C2×C22⋊Q8C2×D7911yescharacteristic
316C2×C4×Q8C2×D7911yescharacteristic
317C2×C4×D4—21no—
318C2×C4×D4C2×D7911yescharacteristic
319C22×C4⋊C4C2×D7911yescharacteristic
320C22×C4⋊C4C2×D7911yescharacteristic
321#451—1584no—
322C2×#103—792no—
323C2×#103—792no—
324C2×#103—792no—
325C2×#103—792no—
326C2×#103—1584no—
327C2×C22⋊C8—792no—
328C2×C22⋊C8—791no—
329#395—1581no—
330#393—1581no—
331#391—792no—
332#386—1581no—
333#385—1581no—
334#380—791no—
335#340—792no—
336#340—792no—
337#340—792no—
338#336—792no—
339C22×Q8—21no—
340C22×Q8—3162no—
341C22×Q8—792no—
342C22×Q8—1584no—
343C22×Q8—1582no—
344C22×Q8—1582no—
345C22×Q8—1581no—
346C22×Q8C22×D7911yescharacteristic
347C22×D4C22⋊D7911yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—1581no—
351C23×C4C22×D7911yescharacteristic
352C23×C4C22×D7911yescharacteristic
353C2×Q16—3162no—
354C2×Q16—1584no—
355C2×Q16—1584no—
356C22×C8—3162no—
357C22×C8—792no—
358C22×C8—792no—
359C22×C8—1582no—
360C4⋊Q8—22no—
361C4⋊Q8—22no—
362#125—28no—
363C22⋊Q8—28no—
364C22⋊Q8—28no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—3162no—
369C22⋊Q8—1584no—
370C22⋊Q8C22⋊D7914yes—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×D4—28no—
374C4×D4—22no—
375C4×D4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—792no—
386C2×C4⋊C4—1582no—
387C2×C4⋊C4C22⋊D7911yescharacteristic
388C2×C4⋊C4C22⋊D7914yes—
389C2×C4⋊C4C22×D7911yescharacteristic
390C2×C4⋊C4C22×D7911yescharacteristic
391C2×C4⋊C4C22×D7911yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1582no—
395C2×C22⋊C4—1581no—
396C2×C22⋊C4C22⋊D7911yescharacteristic
397C2×C22⋊C4C22⋊D7911yescharacteristic
398C2×C42—21no—
399C2×C42C22×D7911yescharacteristic
400#103—3162no—
401#103—1584no—
402#103—1584no—
403#103—1584no—
404C22⋊C8—3162no—
405C22⋊C8—1582no—
406#95—1581no—
407#95—1582no—
408#95—1582no—
409C24C2×C22⋊D7911yescharacteristic
410C24C2×C22⋊D7911yescharacteristic
411C2×Q8—24no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—82no—
416C2×Q8—3164no—
417C2×Q8—3168no—
418C2×Q8—3164no—
419C2×Q8—3162no—
420C2×Q8—794no—
421C2×Q8—1588no—
422C2×Q8—1582no—
423C2×Q8—1582no—
424C2×Q8—1582no—
425C2×Q8—1582no—
426C2×Q8—1582no—
427C2×Q8C2×C22⋊D7912yes—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4#1192961414yescomplemented
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—3162no—
446C22×C4—3164no—
447C22×C4—1582no—
448C22×C4—1582no—
449C22×C4C2×C22⋊D7911yescharacteristic
450C22×C4C2×C22⋊D7911yescharacteristic
451C22×C4C2×C22⋊D7911yescharacteristic
452C22×C4C2×C22⋊D7912yes—
453C22×C4C2×C22⋊D7912yes—
454C22×C4C2×C22⋊D7911yescharacteristic
455C22×C4C23×D7911yescharacteristic
456Q16—3168no—
457C2×C8—3164no—
458C2×C8—1582no—
459C2×C8—1582no—
460C2×C8—1582no—
461C2×C8—1582no—
462C2×C8—1582no—
463C2×C8—1582no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—3164no—
475C4⋊C4—1582no—
476C4⋊C4—1582no—
477C4⋊C4C2×C22⋊D7912yes—
478C4⋊C4C2×C22⋊D7912yes—
479C4⋊C4C2×C22⋊D7912yes—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—3162no—
485C22⋊C4—3164no—
486C42—24no—
487C42—21no—
488C42—21no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—21no—
493C23—21no—
494C23—22no—
495C23#1457913012yes—
496C23C2×#1192961412yes—
497C23C2×#1192961412yes—
498C23#1457919212yes—
499C23C22×C22⋊D7911yescharacteristic
500Q8—24no—
501Q8—41no—
502Q8—41no—
503Q8—84no—
504Q8—3168no—
505Q8—3162no—
506Q8—3162no—
507Q8—1584no—
508Q8—1584no—
509D4—28no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4—3168no—
534C2×C4—3162no—
535C2×C4—3162no—
536C2×C4—3162no—
537C2×C4—3162no—
538C2×C4—3162no—
539C2×C4—1584no—
540C2×C4#1457913012yes—
541C2×C4C2×#1192961412yes—
542C2×C4C22×C22⋊D7911yescharacteristic
543C2×C4C22×C22⋊D7911yescharacteristic
544C2×C4#41×D7912yes—
545C2×C4#1457920712yes—
546C8—3162no—
547C8—3162no—
548C22—22no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—24no—
553C22—22no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22#1909871914yes—
558C22C2×#1457913011yescharacteristic
559C22#1909920112yes—
560C22C22×#1192961411yescharacteristic
561C22C2×#1457919211yescharacteristic
562C22#1909937412yes—
563C4—22no—
564C4—22no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—3164no—
572C4—6322no—
573C4#1909907011yescharacteristic
574C4#1909912711yescharacteristic
575C4#1909920112yes—
576C2—24no—
577C2—42no—
578C2#2376671511yescharacteristic
579C2C2×#1909871911yescharacteristic
580C2#2377267711yescharacteristic
581C2#2377527012yes—
582C2#2377550512yes—
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#554×C79C2
socleC23×C79C22×D4
prime radicalC22×D4×C79D4
hypercenterC2×#554D79
hypocenterC79#23970
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#19098718C2semidirect
2. C22×#14579084C2semidirect
4. #23764538C2semidirect
5. #23764408C2semidirect
6. #23763824C2semidirect
7. #23763741C2semidirect
8. #23763707C2semidirect
10. #23763490C2semidirect
155. C2×D4×C79Q16semidirect
303. C79#23970semidirect
430. C2×D4#11929614semidirect

Maximal subgroups

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

subgroupindex
C2×#190987182
C22×#145790842
#237667652
#237645382
#237644082
#237638242
#237637412
#237637072
#237636932
#237634902
C2×#554×C792
#2397079