Finite Group Databasebeta

← #34258368

Subgroups — #34258368

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773471C211yesmaximal, characteristic
2#23772511C211yesmaximal, characteristic
3#23772497C211yesmaximal, characteristic, complemented
4#23762268C211yesmaximal, characteristic, complemented
5#23762254C211yesmaximal, characteristic, complemented
6#23762247C211yesmaximal, characteristic, complemented
7#23762233C211yesmaximal, characteristic, complemented
8#23762216C211yesmaximal, characteristic, complemented
9#23762209C211yesmaximal, characteristic, complemented
10#23762204C211yesmaximal, characteristic, complemented
11#23762203C211yesmaximal, characteristic, complemented
12#23762186C211yesmaximal, characteristic, complemented
13#23761804C211yesmaximal, characteristic, complemented
14#23761700C211yesmaximal, characteristic, complemented
15C79×#1488C211yesmaximal, characteristic
16#19099120C2211yescharacteristic
17#19098904C411yescharacteristic
18#19098904C411yescharacteristic
19#19098848C411yescharacteristic
20#19098843C411yescharacteristic
21#19098773C411yescharacteristic
22#19098768C411yescharacteristic
23#19098725C411yescharacteristic
24#19098725C411yescharacteristic
25#19098691—21no—
26#19098691—21no—
27#19098690C2211yescharacteristic
28#19098690C2211yescharacteristic
29Q8×C79⋊C8C2211yescharacteristic
30Q8×C79⋊C8C2211yescharacteristic
31#19098687C2211yescharacteristic
32#19098687C2211yescharacteristic
33C4×#11929614C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929612C2211yescharacteristic
36C4×#11929612C2211yescharacteristic
37D4×C79⋊C8—21no—
38D4×C79⋊C8—21no—
39D4×C79⋊C8C2211yescharacteristic
40D4×C79⋊C8C2211yescharacteristic
41#19098621—21no—
42#19098621—21no—
43#19098616—21no—
44#19098616C2211yescharacteristic
45#19098616C2211yescharacteristic
46C2×C4⋊(C79⋊C8)—21no—
47C2×C4⋊(C79⋊C8)C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098313C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098295C2211yescharacteristic
55#19098295C2211yescharacteristic
56#19098285C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C4×#41×C79C2211yescharacteristic
61#432×C79C2211yescharacteristic
62#431×C79C2211yescharacteristic
63Q8⋊C8×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66D4⋊C8×C79C2211yescharacteristic
67C2×#11929614C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C22×C79⋊C8—21no—
72C22×C79⋊C8—21no—
73C22×C79⋊C8—21no—
74#14579161C2×C411yescharacteristic
75#14579161C2×C411yescharacteristic
76#14579153—21no—
77#14579140—21no—
78#14579140C2×C411yescharacteristic
79#14579140C2×C411yescharacteristic
80C4×Dic158—21no—
81C4×Dic158C2311yescharacteristic
82C4×Dic158C2311yescharacteristic
83#14579120—21no—
84#14579094C2×C411yescharacteristic
85#14579094C2×C411yescharacteristic
86#14579092C2×C411yescharacteristic
87#14579092C2×C411yescharacteristic
88C4⋊(C79⋊C8)—21no—
89C4⋊(C79⋊C8)—21no—
90C4⋊(C79⋊C8)—21no—
91C4⋊(C79⋊C8)C2×C411yescharacteristic
92C4⋊(C79⋊C8)C2×C411yescharacteristic
93C4⋊(C79⋊C8)D411yescharacteristic
94C4⋊(C79⋊C8)D411yescharacteristic
95C4⋊(C79⋊C8)C2311yescharacteristic
96C4⋊(C79⋊C8)C2311yescharacteristic
97Dic79⋊C8—21no—
98Dic79⋊C8—21no—
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101#14579085C2×C411yescharacteristic
102#14579085C2×C411yescharacteristic
103#14579084C2×C411yescharacteristic
104#14579084C2×C411yescharacteristic
105#14579084C2×C411yescharacteristic
106#14579084C2×C411yescharacteristic
107#14579082C2×C411yescharacteristic
108#14579082C2×C411yescharacteristic
109C22⋊(C79⋊C8)—21no—
110C22⋊(C79⋊C8)—21no—
111C22⋊(C79⋊C8)C2×C411yescharacteristic
112C22⋊(C79⋊C8)C2×C411yescharacteristic
113C22⋊(C79⋊C8)D411yescharacteristic
114C22⋊(C79⋊C8)D411yescharacteristic
115#14579071—21no—
116#14579071—21no—
117C4×C79⋊C8—21no—
118C4×C79⋊C8—21no—
119C4×C79⋊C8C2311yescharacteristic
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C8×Dic79—21no—
124C8×Dic79—21no—
125C2×#41×C79C2×C411yescharacteristic
126C4×Q8×C79C2×C411yescharacteristic
127C4×Q8×C79C2311yescharacteristic
128C4×D4×C79—21no—
129C4×D4×C79—21no—
130C4×D4×C79C2×C411yescharacteristic
131C4×D4×C79C2311yescharacteristic
132#117×C79—21no—
133#117×C79C2×C411yescharacteristic
134C2×C42×C79—21no—
135C2×C42×C79C2311yescharacteristic
136C4⋊C8×C79—21no—
137C4⋊C8×C79C2311yescharacteristic
138C4⋊C8×C79C2311yescharacteristic
139C22⋊C8×C79—21no—
140C22⋊C8×C79—21no—
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×Dic158C22×C411yescharacteristic
147#11929614—22no—
148#11929614—22no—
149#11929612—22no—
150#11929612—22no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—81no—
154C2×C79⋊C8—81no—
155C2×C79⋊C8—81no—
156C2×C79⋊C8C22×C411yescharacteristic
157C2×C79⋊C8C22×C411yescharacteristic
158C2×C79⋊C8C22×C411yescharacteristic
159C2×C79⋊C8C22×C411yescharacteristic
160C2×C79⋊C8C2×D411yescharacteristic
161C2×C79⋊C8C2×D411yescharacteristic
162C2×C79⋊C8C2×D411yescharacteristic
163C2×C79⋊C8C2×D411yescharacteristic
164C4⋊Dic79—21no—
165C4⋊Dic79—21no—
166C4⋊Dic79—21no—
167C4⋊Dic79C22×C411yescharacteristic
168C4⋊Dic79C22×C411yescharacteristic
169Dic79⋊C4—21no—
170Dic79⋊C4—21no—
171Dic79⋊C4—21no—
172Dic79⋊C4—21no—
173C22⋊Dic79—21no—
174C22⋊Dic79—21no—
175C4×Dic79—21no—
176C4×Dic79—21no—
177#41×C79—22no—
178#41×C79—22no—
179C2×Q8×C79C22×C411yescharacteristic
180C2×D4×C79—21no—
181C2×D4×C79C22×C411yescharacteristic
182C22×C4×C79—21no—
183C22×C4×C79—21no—
184C22×C4×C79—21no—
185C22×C4×C79C22×C411yescharacteristic
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79#4111yescharacteristic
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79C22×C411yescharacteristic
195C4⋊C4×C79C22×C411yescharacteristic
196C22⋊C4×C79—21no—
197C22⋊C4×C79—21no—
198C22⋊C4×C79—21no—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—21no—
202C42×C79C22×C411yescharacteristic
203C42×C79C2×D411yescharacteristic
204C42×C79#4111yescharacteristic
205C42×C79C2411yescharacteristic
206C23×C79—21no—
207C23×C79C4×D411yescharacteristic
208Q8×C79—21no—
209Q8×C79#13112yes—
210D4×C79—24no—
211D4×C79—21no—
212D4×C79#13112yes—
213C2×C4×C79—24no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—81no—
225C2×C4×C79C4×D411yescharacteristic
226C2×C4×C79C4×D411yescharacteristic
227C2×C4×C79C4×D411yescharacteristic
228C2×C4×C79C23×C411yescharacteristic
229C2×C4×C79C22×D411yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235Dic158—21no—
236Dic158—21no—
237Dic158—21no—
238Dic158—21no—
239Dic158—81no—
240C8×C79—41no—
241C8×C79—41no—
242C8×C79—41no—
243C8×C79—41no—
244C79⋊C8—21no—
245C79⋊C8—21no—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C22×C79—21no—
255C22×C79—22no—
256C22×C79—21no—
257C22×C79—21no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79C2×C4×D411yescharacteristic
261C4×C79—21no—
262C4×C79—22no—
263C4×C79—21no—
264C4×C79—21no—
265C4×C79—21no—
266C4×C79—41no—
267C4×C79—41no—
268C4×C79C2×#13111yescharacteristic
269C4×C79C2×#13111yescharacteristic
270C4×C79#57711yescharacteristic
271C4×C79#57811yescharacteristic
272Dic79—41no—
273Dic79—41no—
274Dic79—41no—
275Dic79—41no—
276#17921—791nomaximal, Sylow
277C2×C79—41no—
278C2×C79—42no—
279C2×C79#292911yescharacteristic
280C2×C79#294711yescharacteristic
281C2×C79#295711yescharacteristic
282#3107—791no—
283#2988—791no—
284#2985—791no—
285#1587—791no—
286#1585—791no—
287#1584—791no—
288#1582—791no—
289#1579—791no—
290SD16⋊C8—791no—
291C8×Q16—791no—
292C8×SD16—791no—
293#1573—791no—
294#1488D7911yescharacteristic
295#1481—791no—
296#1454—791no—
297C79#1792111yescharacteristic, complemented, Sylow
298#533—791no—
299C4×#41C2×D7911yescharacteristic
300#485—791no—
301#485—791no—
302#475—791no—
303#474—791no—
304#462—791no—
305#461—791no—
306#452—791no—
307#452—791no—
308#446—791no—
309#446—791no—
310#446—1581no—
311#446—1581no—
312C8×Q8—791no—
313C8×Q8—791no—
314C8×Q8—791no—
315C8×Q8—791no—
316C4×Q16—791no—
317C4×Q16—791no—
318C4×SD16—791no—
319C4×SD16—791no—
320C8×D4—791no—
321C8×D4—791no—
322C8×D4—1581no—
323C8×D4—1581no—
324#433—1581no—
325#433—1581no—
326#432C2×D7911yescharacteristic
327#431—791no—
328#431—791no—
329#431—1581no—
330#431C2×D7911yescharacteristic
331C2×C4⋊C8—791no—
332C2×C4⋊C8—1581no—
333#335—791no—
334#335—791no—
335#334—791no—
336#334—791no—
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8C2×D7911yescharacteristic
342Q8⋊C8C2×D7911yescharacteristic
343D4⋊C8C2×D7911yescharacteristic
344D4⋊C8C2×D7911yescharacteristic
345#322—791no—
346#322—791no—
347#322—791no—
348C82—791no—
349C2×#41C4×D7911yescharacteristic
350C2×Q16—791no—
351C2×Q16—791no—
352C2×SD16—791no—
353C2×SD16—791no—
354C22×C8—1581no—
355C22×C8—1581no—
356C22×C8—1581no—
357C4⋊Q8—791no—
358C4⋊Q8—791no—
359#125—1581no—
360C22⋊Q8—791no—
361C22⋊Q8—791no—
362C22⋊Q8—1581no—
363C4×Q8—791no—
364C4×Q8—791no—
365C4×Q8—1581no—
366C4×Q8C4×D7911yescharacteristic
367C4×Q8C22×D7911yescharacteristic
368C4×D4—21no—
369C4×D4—21no—
370C4×D4C4×D7911yescharacteristic
371C4×D4C22×D7911yescharacteristic
372#117—21no—
373#117—1581no—
374#117C4×D7911yescharacteristic
375C2×C42—21no—
376C2×C42C22×D7911yescharacteristic
377#107—791no—
378#107—791no—
379#106—791no—
380#106—791no—
381C4⋊C8—21no—
382C4⋊C8—791no—
383C4⋊C8—791no—
384C4⋊C8—791no—
385C4⋊C8—791no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—1581no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8C22×D7911yescharacteristic
396C4⋊C8C22×D7911yescharacteristic
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103—791no—
402#103—791no—
403#102—791no—
404#102—791no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—791no—
408C22⋊C8—791no—
409C22⋊C8—791no—
410C22⋊C8—791no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—21no—
414#97—1581no—
415#97—1581no—
416C4×C8—791no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—1581no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8—1581no—
424C4×C8C22×D7911yescharacteristic
425C4×C8C22×D7911yescharacteristic
426#41—22no—
427#41—22no—
428C2×Q8—791no—
429C2×Q8—791no—
430C2×Q8—1581no—
431C2×Q8C2×C4×D7911yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D7911yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C2×C4×D7911yescharacteristic
438C22×C4D4×D7911yescharacteristic
439C22×C4#1192962311yescharacteristic
440Q16—1582no—
441Q16—1582no—
442SD16—1582no—
443SD16—1582no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—791no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—791no—
453C2×C8—791no—
454C2×C8—791no—
455C2×C8—791no—
456C2×C8—6321no—
457C2×C8—6321no—
458C2×C8—6321no—
459C2×C8—1581no—
460C2×C8—1581no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—791no—
464C4⋊C4—791no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4C2×C4×D7911yescharacteristic
473C4⋊C4C2×C4×D7911yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—1581no—
483C42—1581no—
484C42C2×C4×D7911yescharacteristic
485C42D4×D7911yescharacteristic
486C42#1192962311yescharacteristic
487C42C23×D7911yescharacteristic
488C23—21no—
489C23(C22⋊D79)⋊C411yescharacteristic
490Q8—21no—
491Q8—6321no—
492Q8—1581no—
493Q8—1581no—
494Q8—1581no—
495Q8—1581no—
496Q8#1457916912yes—
497D4—24no—
498D4—21no—
499D4#1457916912yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—81no—
512C2×C4—1581no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4(C22⋊D79)⋊C411yescharacteristic
517C2×C4(C22⋊D79)⋊C411yescharacteristic
518C2×C4(C22⋊D79)⋊C411yescharacteristic
519C2×C4C22×C4×D7911yescharacteristic
520C2×C4C2×D4×D7911yescharacteristic
521C2×C4C2×#1192962311yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—3161no—
527C8—3161no—
528C8—1581no—
529C8—1581no—
530C8—1581no—
531C8—1581no—
532C8—1581no—
533C8—1581no—
534C8—1581no—
535C8—1581no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22C2×(C22⋊D79)⋊C411yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—3161no—
551C4—3161no—
552C4—3161no—
553C4—3161no—
554C4C2×#1457916911yescharacteristic
555C4C2×#1457916911yescharacteristic
556C4#1909937011yescharacteristic
557C4#1909937711yescharacteristic
558C2—41no—
559C2—42no—
560C2#2377209011yescharacteristic
561C2#2377224611yescharacteristic
562C2#2377231311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1488C2
socleC22×C79C2×C4×D4
prime radicalC2×#41×C79C2×C4
cosocleC42×C79C24
hypercenter#1488D79
hypocenterC79#17921
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1488

Complements

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

NH ≅ G/Ntype
3. #23772497C2semidirect
4. #23762268C2semidirect
5. #23762254C2semidirect
6. #23762247C2semidirect
7. #23762233C2semidirect
8. #23762216C2semidirect
9. #23762209C2semidirect
10. #23762204C2semidirect
11. #23762203C2semidirect
12. #23762186C2semidirect
13. #23761804C2semidirect
14. #23761700C2semidirect
297. C79#17921semidirect

Maximal subgroups

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

subgroupindex
#237734712
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#237724972
#237622682
#237622542
#237622472
#237622332
#237622162
#237622092
#237622042
#237622032
#237621862
#237618042
#237617002
C79×#14882
#1792179