Finite Group Databasebeta

← #34254206

Subgroups — #34254206

Order 20224.

1174 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×#19099042C211yesmaximal, characteristic
2#23772484C212yesmaximal, complemented
3#23761741C211yesmaximal, characteristic, complemented
4#23761739C212yesmaximal, complemented
5#23761732C212yesmaximal, complemented
6#23761731C212yesmaximal, complemented
7#23761719C212yesmaximal, complemented
8#23761711C212yesmaximal, complemented
9C79×#1466C211yesmaximal, characteristic, complemented
10C22×#11929623C411yescharacteristic
11#19099042—24no—
12#19099042—24no—
13C2×C4×Dic158C2211yescharacteristic
14C2×#14579126C411yescharacteristic
15C2×D4×Dic79C411yescharacteristic
16C2×C4×C22⋊D79C411yescharacteristic
17C2×C4×C22⋊D79C411yescharacteristic
18C2×(C22⋊D79)⋊C4C2211yescharacteristic
19C2×(C22⋊D79)⋊C4C2211yescharacteristic
20C2×#14579118C2211yescharacteristic
21C2×#14579117C2211yescharacteristic
22C2×#14579117C2211yescharacteristic
23C22×Dic79⋊C4C411yescharacteristic
24C22×Dic79⋊C4C411yescharacteristic
25C2×C4⋊C4×D79C411yescharacteristic
26#19098692—24no—
27#19098691—24no—
28#19098620C2212yes—
29#19098617C2214yes—
30#19098613C2214yes—
31#19098288C2214yes—
32#19098287C2212yes—
33#19098287C2212yes—
34#19098286C2214yes—
35C2×C4×D4×C79C2211yescharacteristic
36#432×C79C2212yes—
37#323×C79C2212yes—
38#323×C79C2212yes—
39C2×#11929623—24no—
40C2×#11929623—24no—
41C2×#11929623—24no—
42C22×Dic158C2×C411yescharacteristic
43C22×C22⋊D79C2×C411yescharacteristic
44C22×C22⋊D79C2×C411yescharacteristic
45C23×Dic79C2×C411yescharacteristic
46C23×Dic79C2×C411yescharacteristic
47C23×Dic79C2×C411yescharacteristic
48C23×Dic79C2×C411yescharacteristic
49C22×C4×D79C2×C411yescharacteristic
50C22×C4×D79C2×C411yescharacteristic
51C22×C4×D79C2×C411yescharacteristic
52C4×Dic158D414yes—
53C4×Dic158D414yes—
54#14579126—24no—
55D4×Dic79—24no—
56C4×C22⋊D79—22no—
57C4×C22⋊D79—22no—
58C4×C22⋊D79—22no—
59C4×C22⋊D79—22no—
60(C22⋊D79)⋊C4—24no—
61(C22⋊D79)⋊C4—24no—
62#14579118—22no—
63#14579118—22no—
64#14579117—24no—
65#14579117—24no—
66C2×C4⋊Dic79C2×C411yescharacteristic
67C2×Dic79⋊C4—22no—
68C2×Dic79⋊C4—22no—
69C2×Dic79⋊C4—22no—
70C2×Dic79⋊C4—22no—
71C2×Dic79⋊C4C2×C411yescharacteristic
72C2×Dic79⋊C4C2×C411yescharacteristic
73C2×Dic79⋊C4C2×C411yescharacteristic
74C2×Dic79⋊C4C2×C411yescharacteristic
75C2×Dic79⋊C4C2×C411yescharacteristic
76C2×Dic79⋊C4C2×C411yescharacteristic
77C2×Dic79⋊C4C2×C411yescharacteristic
78C2×Dic79⋊C4C2311yescharacteristic
79C4⋊C4×D79—24no—
80C2×C22⋊Dic79C2×C411yescharacteristic
81C2×C22⋊Dic79C2×C411yescharacteristic
82C2×(C2×D79)⋊C4C2×C411yescharacteristic
83C2×(C2×D79)⋊C4C2311yescharacteristic
84C2×C4×Dic79C2×C411yescharacteristic
85C2×C4×Dic79C2×C411yescharacteristic
86C2×C4×Dic79C2×C411yescharacteristic
87C2×C4×Dic79C2311yescharacteristic
88C2×C4×Dic79C2311yescharacteristic
89C4⋊(C79⋊C8)—22no—
90Dic79⋊C8—24no—
91Dic79⋊C8—24no—
92C22⋊(C79⋊C8)C2314yes—
93#14579072—22no—
94#14579071—24no—
95C8×Dic79—24no—
96C22×D4×C79C2×C411yescharacteristic
97C23×C4×C79C2×C411yescharacteristic
98C23×C4×C79C2×C411yescharacteristic
99C4×D4×C79—22no—
100C4×D4×C79—22no—
101C2×C4⋊C4×C79C2×C411yescharacteristic
102C2×C22⋊C4×C79C2311yescharacteristic
103C2×C22⋊C4×C79C2311yescharacteristic
104C2×C42×C79C2311yescharacteristic
105C4⋊C8×C79—22no—
106C22⋊C8×C79C2314yes—
107#97×C79—22no—
108C23×D79C22×C411yescharacteristic
109#11929623—232no—
110C2×Dic158—22no—
111C2×Dic158—22no—
112C2×Dic158C22⋊C414yes—
113C2×C22⋊D79—22no—
114C2×C22⋊D79—22no—
115C2×C22⋊D79—22no—
116C2×C22⋊D79—22no—
117C2×C22⋊D79—22no—
118C2×C22⋊D79M1614yes—
119C22×Dic79—22no—
120C22×Dic79—22no—
121C22×Dic79—22no—
122C22×Dic79—22no—
123C22×Dic79—22no—
124C22×Dic79—22no—
125C22×Dic79—22no—
126C22×Dic79—81no—
127C22×Dic79—81no—
128C22×Dic79M1614yes—
129C22×Dic79C22×C411yescharacteristic
130C22×Dic79C22×C411yescharacteristic
131C22×Dic79C22×C411yescharacteristic
132C22×Dic79C22×C411yescharacteristic
133C22×Dic79C22×C411yescharacteristic
134C22×Dic79C22×C411yescharacteristic
135C22×Dic79C22×C411yescharacteristic
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×C79⋊C8—24no—
144C4⋊Dic79C22⋊C414yes—
145Dic79⋊C4—22no—
146Dic79⋊C4—22no—
147Dic79⋊C4—22no—
148Dic79⋊C4—22no—
149Dic79⋊C4—21no—
150Dic79⋊C4—21no—
151Dic79⋊C4—22no—
152Dic79⋊C4C22⋊C412yes—
153Dic79⋊C4C22⋊C412yes—
154Dic79⋊C4C22⋊C412yes—
155Dic79⋊C4C22⋊C412yes—
156C22⋊Dic79—22no—
157C22⋊Dic79—22no—
158(C2×D79)⋊C4—21no—
159(C2×D79)⋊C4—21no—
160(C2×D79)⋊C4—22no—
161C4×Dic79—22no—
162C4×Dic79—22no—
163C4×Dic79—22no—
164C4×Dic79C2×D414yes—
165C4×Dic79C2×D414yes—
166C24×C79C22×C411yescharacteristic
167C24×C79C22×C411yescharacteristic
168C2×D4×C79—22no—
169C2×D4×C79—22no—
170C2×D4×C79—22no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79—81no—
176C22×C4×C79—81no—
177C22×C4×C79C22×C411yescharacteristic
178C22×C4×C79C22×C411yescharacteristic
179C22×C4×C79C22×C411yescharacteristic
180C22×C4×C79C22×C411yescharacteristic
181C22×C4×C79C2411yescharacteristic
182C2×C8×C79—24no—
183C4⋊C4×C79—22no—
184C22⋊C4×C79—22no—
185C22⋊C4×C79—22no—
186C42×C79C2×D412yes—
187C42×C79C2×D412yes—
188C23×C79—21no—
189C23×C79—21no—
190C23×C79—21no—
191C23×C79—21no—
192C23×C79—41no—
193C23×C79—41no—
194C23×C79—41no—
195C23×C79—41no—
196C23×C79C2×M1612yes—
197C23×C79C2×M1612yes—
198C23×C79C23×C411yescharacteristic
199C22×D79—21no—
200C22×D79—21no—
201C22×D79—41no—
202C22×D79—41no—
203C22×D79C2×M1612yes—
204D4×C79—28no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—41no—
210C2×C4×C79—41no—
211C2×C4×C79—41no—
212C2×C4×C79—41no—
213C2×C4×C79—81no—
214C2×C4×C79—81no—
215C2×C4×C79—81no—
216C2×C4×C79—81no—
217C2×C4×C79C2×C22⋊C412yes—
218C2×C4×C79C2×C22⋊C411yescharacteristic
219C2×C4×C79C2×C22⋊C411yescharacteristic
220C2×C4×C79C22×D412yes—
221C22⋊D79—28no—
222C22⋊D79—28no—
223C2×Dic79—28no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—28no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—41no—
232C2×Dic79—41no—
233C2×Dic79—41no—
234C2×Dic79—41no—
235C2×Dic79—82no—
236C2×Dic79—82no—
237C2×Dic79C2×C22⋊C412yes—
238C2×Dic79C2×C22⋊C412yes—
239C2×Dic79C2×C22⋊C412yes—
240C2×Dic79C2×C22⋊C412yes—
241C2×Dic79C2×M1612yes—
242C2×Dic79C2×M1612yes—
243C2×Dic79C2×M1612yes—
244C4×D79—28no—
245C4×D79—82no—
246C4×D79—82no—
247Dic158—28no—
248C8×C79—44no—
249C79⋊C8—44no—
250C22×C79—21no—
251C22×C79—21no—
252C22×C79—22no—
253C22×C79—22no—
254C22×C79—42no—
255C22×C79—42no—
256C22×C79—42no—
257C22×C79—42no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79C22×C22⋊C411yescharacteristic
263C22×C79C22×M1611yescharacteristic
264C22×C79C22×M1611yescharacteristic
265C2×D79—22no—
266C2×D79—42no—
267C2×D79—41no—
268C2×D79—41no—
269C2×D79—42no—
270C4×C79—22no—
271C4×C79—22no—
272C4×C79—21no—
273C4×C79—21no—
274C4×C79—41no—
275C4×C79—41no—
276Dic79—22no—
277Dic79—22no—
278Dic79—22no—
279Dic79—22no—
280Dic79—41no—
281Dic79—41no—
282Dic79#40714yes—
283#17372—791nomaximal, Sylow
284C2×C79—21no—
285C2×C79—21no—
286C2×C79—42no—
287C2×C79—42no—
288C2×C79C2×#40711yescharacteristic
289C2×C79#288411yescharacteristic
290C2×C79#288811yescharacteristic
291D79—42no—
292C2×#520—791no—
293#2983—792no—
294#1466—792no—
295#1466—791no—
296#1466D7911yescharacteristic, complemented
297#1464—792no—
298#1464—792no—
299#1460—792no—
300#1458—792no—
301C79#1737211yescharacteristic, complemented, Sylow
302C22×#41—791no—
303#520—1584no—
304#520—1584no—
305C2×C4×Q8—791no—
306C2×C4×Q8—791no—
307C2×C4×D4—791no—
308C2×C4×D4—791no—
309C2×C4×D4—791no—
310C2×C4×D4—791no—
311C2×C4×D4—791no—
312C2×C4×D4C2×D7911yescharacteristic
313C2×#117—791no—
314C2×#117—791no—
315C2×#117—791no—
316C22×C4⋊C4—791no—
317C22×C4⋊C4—791no—
318C22×C4⋊C4—791no—
319#446—1584no—
320#446—1584no—
321#432—792no—
322#432—794no—
323#432C2×D7912yes—
324#431—794no—
325#324—794no—
326#323—792no—
327#323—792no—
328#323—794no—
329#323C2×D7912yes—
330#323C2×D7912yes—
331C2×#41—1584no—
332C2×#41—1584no—
333C2×#41—1584no—
334C22×Q8—791no—
335C22×D4—791no—
336C22×D4—791no—
337C22×D4C4×D7911yescharacteristic
338C23×C4—791no—
339C23×C4—791no—
340C23×C4—791no—
341C23×C4—791no—
342C23×C4—791no—
343C23×C4—791no—
344C23×C4—791no—
345C23×C4C4×D7911yescharacteristic
346C23×C4C4×D7911yescharacteristic
347C4×Q8—794no—
348C4×Q8—794no—
349C4×Q8—1584no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—1584no—
353C4×D4—1584no—
354C4×D4—1582no—
355C4×D4—1582no—
356C4×D4—1582no—
357C4×D4—1582no—
358C4×D4—1584no—
359#117—1584no—
360#117—1584no—
361#117—1582no—
362#117—1582no—
363C2×C4⋊C4—791no—
364C2×C4⋊C4—791no—
365C2×C4⋊C4—791no—
366C2×C4⋊C4—791no—
367C2×C4⋊C4—791no—
368C2×C4⋊C4—791no—
369C2×C4⋊C4—791no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—1584no—
373C2×C4⋊C4—1582no—
374C2×C4⋊C4—1582no—
375C2×C4⋊C4—1582no—
376C2×C4⋊C4—1582no—
377C2×C4⋊C4C4×D7911yescharacteristic
378C2×C22⋊C4—791no—
379C2×C22⋊C4—791no—
380C2×C22⋊C4—791no—
381C2×C22⋊C4—791no—
382C2×C22⋊C4C22×D7911yescharacteristic
383C2×C22⋊C4C22×D7911yescharacteristic
384C2×C42—791no—
385C2×C42—791no—
386C2×C42—791no—
387C2×C42—791no—
388C2×C42—791no—
389C2×C42C22×D7911yescharacteristic
390C4⋊C8—22no—
391C4⋊C8—1584no—
392C4⋊C8—1584no—
393C4⋊C8—1582no—
394C22⋊C8—794no—
395C22⋊C8C22×D7914yes—
396#97—22no—
397#97—1584no—
398#97—1582no—
399C4×C8—1584no—
400C24—791no—
401C24C2×C4×D7911yescharacteristic
402C24C2×C4×D7911yescharacteristic
403#41—15832no—
404C2×Q8—794no—
405C2×Q8—1582no—
406C2×Q8—1582no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—794no—
411C2×D4—1582no—
412C2×D4—1582no—
413C2×D4—1582no—
414C2×D4—1582no—
415C2×D4—1582no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—791no—
423C22×C4—791no—
424C22×C4—791no—
425C22×C4—791no—
426C22×C4—791no—
427C22×C4—794no—
428C22×C4—791no—
429C22×C4—791no—
430C22×C4—6321no—
431C22×C4—6321no—
432C22×C4—6321no—
433C22×C4—6321no—
434C22×C4—1582no—
435C22×C4—1582no—
436C22×C4—1582no—
437C22×C4—1582no—
438C22×C4—1582no—
439C22×C4—1582no—
440C22×C4—1582no—
441C22×C4—1582no—
442C22×C4—1582no—
443C22×C4—1582no—
444C22×C4—1582no—
445C22×C4—1582no—
446C22×C4C2×C4×D7911yescharacteristic
447C22×C4C2×C4×D7911yescharacteristic
448C22×C4C2×C4×D7911yescharacteristic
449C22×C4C2×C4×D7911yescharacteristic
450C22×C4C23×D7911yescharacteristic
451C2×C8—24no—
452C2×C8—1584no—
453C4⋊C4—22no—
454C4⋊C4—792no—
455C4⋊C4—792no—
456C4⋊C4—792no—
457C4⋊C4—792no—
458C4⋊C4—794no—
459C4⋊C4—1582no—
460C4⋊C4—1582no—
461C4⋊C4—1582no—
462C4⋊C4—1582no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1582no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—1582no—
469C22⋊C4—1582no—
470C22⋊C4—1581no—
471C22⋊C4—1581no—
472C22⋊C4—1582no—
473C42—794no—
474C42—794no—
475C42—1582no—
476C42—1582no—
477C42—1582no—
478C42D4×D7912yes—
479C42D4×D7912yes—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—3161no—
489C23—3161no—
490C23—792no—
491C23—1581no—
492C23—1581no—
493C23M16×D7912yes—
494C23M16×D7912yes—
495C23C22×C4×D7911yescharacteristic
496Q8—1588no—
497D4—28no—
498D4—1588no—
499D4—1588no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—3161no—
513C2×C4—3161no—
514C2×C4—3161no—
515C2×C4—3161no—
516C2×C4—792no—
517C2×C4—792no—
518C2×C4—792no—
519C2×C4—792no—
520C2×C4—792no—
521C2×C4—792no—
522C2×C4—792no—
523C2×C4—6322no—
524C2×C4—6322no—
525C2×C4—6322no—
526C2×C4—6322no—
527C2×C4—1588no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1588no—
531C2×C4—1588no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4C22⋊C4×D7912yes—
537C2×C4C22⋊C4×D7911yescharacteristic
538C2×C4C22⋊C4×D7911yescharacteristic
539C2×C4C2×D4×D7912yes—
540C8—44no—
541C8—3164no—
542C22—21no—
543C22—21no—
544C22—22no—
545C22—22no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—3162no—
555C22—3162no—
556C22—3161no—
557C22—3161no—
558C22—1582no—
559C22C2×C22⋊C4×D7911yescharacteristic
560C22C2×M16×D7911yescharacteristic
561C22C2×M16×D7911yescharacteristic
562C4—22no—
563C4—22no—
564C4—21no—
565C4—21no—
566C4—41no—
567C4—41no—
568C4—3161no—
569C4—3161no—
570C4—794no—
571C4—1582no—
572C4—1582no—
573C4—1582no—
574C4—1582no—
575C2—21no—
576C2—21no—
577C2—42no—
578C2—42no—
579C2—3162no—
580C2#407×D7911yescharacteristic
581C2#2377186211yescharacteristic
582C2#2377187611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D79
commutatorC23×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1466C2
socleC22×C79C22×C22⋊C4
prime radicalC22×#11929623C4
cosocleC22×C4×C79C24
hypercenter#1466D79
hypocenterC79#17372
LCSG ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #23772484C2semidirect
3. #23761741C2semidirect
4. #23761739C2semidirect
5. #23761732C2semidirect
6. #23761731C2semidirect
7. #23761719C2semidirect
8. #23761711C2semidirect
9. C79×#1466C2semidirect
296. #1466D79semidirect
301. C79#17372semidirect

Maximal subgroups

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

subgroupindex
C2×#190990422
#237724842
#237617412
#237617392
#237617322
#237617312
#237617192
#237617112
C79×#14662
#1737279