Finite Group Databasebeta

← #34270985

Subgroups — #34270985

Order 20224.

596 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773982C211yesmaximal, characteristic, complemented
2#23773188C211yesmaximal, characteristic
3#23772367C211yesmaximal, characteristic
4#23762984C211yesmaximal, characteristic, complemented
5#23762980C211yesmaximal, characteristic, complemented
6#23762977C211yesmaximal, characteristic, complemented
7#23762973C211yesmaximal, characteristic, complemented
8#23762926C211yesmaximal, characteristic, complemented
9#23762921C211yesmaximal, characteristic, complemented
10#23762892C211yesmaximal, characteristic, complemented
11#23762887C211yesmaximal, characteristic, complemented
12C2×#19098313C211yesmaximal, characteristic, complemented
13#23762058C211yesmaximal, characteristic, complemented
14#23762057C211yesmaximal, characteristic, complemented
15C79×#3076C211yesmaximal, characteristic
16#19099120C2211yescharacteristic
17#19098966C2211yescharacteristic
18#19098942C2211yescharacteristic
19#19098915C2211yescharacteristic
20#19098913C2211yescharacteristic
21#19098870—21no—
22#19098863—21no—
23#19098858C2211yescharacteristic
24#19098853C2211yescharacteristic
25#19098848C2211yescharacteristic
26#19098843C2211yescharacteristic
27#19098802—21no—
28#19098802—21no—
29#19098694C2211yescharacteristic
30#19098694C2211yescharacteristic
31C4×#11929614—21no—
32C4×#11929614—21no—
33#19098641C2211yescharacteristic
34#19098641C2211yescharacteristic
35#19098635—21no—
36#19098635—21no—
37#19098616—21no—
38#19098616C2211yescharacteristic
39#19098599—21no—
40C2×C4⋊(C79⋊C8)C2211yescharacteristic
41#19098559—21no—
42C4×#11929608—21no—
43C2×C4×C79⋊C8C2211yescharacteristic
44#19098313C2211yescharacteristic
45#19098313C2211yescharacteristic
46#19098313C2211yescharacteristic
47#19098313C2211yescharacteristic
48#19098312C2211yescharacteristic
49#19098312C2211yescharacteristic
50#19098311C2211yescharacteristic
51#19098311C2211yescharacteristic
52#19098309C2211yescharacteristic
53#19098309C2211yescharacteristic
54#19098308C2211yescharacteristic
55#19098308C2211yescharacteristic
56#533×C79C2211yescharacteristic
57C4×#41×C79C2211yescharacteristic
58#479×C79—21no—
59#478×C79—21no—
60#477×C79C2211yescharacteristic
61#476×C79C2211yescharacteristic
62#475×C79C2211yescharacteristic
63#474×C79C2211yescharacteristic
64#427×C79—21no—
65C2×C4⋊C8×C79C2211yescharacteristic
66#416×C79—21no—
67C2×#11929614—21no—
68C2×#11929614—21no—
69C2×#11929608—21no—
70C2×#11929608—21no—
71C22×C79⋊C8—21no—
72C22×C79⋊C8D411yescharacteristic
73C22×C79⋊C8D411yescharacteristic
74#14579161C2311yescharacteristic
75#14579161C2311yescharacteristic
76#14579153—21no—
77#14579140—21no—
78#14579140—21no—
79C4×Dic158—21no—
80C4×Dic158—21no—
81#14579120—21no—
82#14579094—21no—
83#14579094—21no—
84#14579094—21no—
85#14579094—21no—
86#14579092—21no—
87#14579092—21no—
88C4⋊(C79⋊C8)—21no—
89C4⋊(C79⋊C8)—21no—
90C4⋊(C79⋊C8)—21no—
91C4⋊(C79⋊C8)C2311yescharacteristic
92C4⋊(C79⋊C8)C2311yescharacteristic
93C4⋊(C79⋊C8)C2311yescharacteristic
94C4⋊(C79⋊C8)C2311yescharacteristic
95#14579085—21no—
96#14579085—21no—
97#14579085—21no—
98#14579085—21no—
99#14579084—21no—
100#14579084—21no—
101#14579082—21no—
102#14579082—21no—
103C22⋊(C79⋊C8)—21no—
104C22⋊(C79⋊C8)—21no—
105C22⋊(C79⋊C8)—21no—
106#14579072—21no—
107#14579072—21no—
108C4×C79⋊C8—21no—
109C4×C79⋊C8D411yescharacteristic
110C4×C79⋊C8D411yescharacteristic
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C2×#41×C79D411yescharacteristic
114C22×C8×C79—21no—
115C4⋊Q8×C79C2311yescharacteristic
116C4⋊Q8×C79C2311yescharacteristic
117#125×C79—21no—
118C22⋊Q8×C79—21no—
119C22⋊Q8×C79—21no—
120C4×Q8×C79—21no—
121C4×Q8×C79—21no—
122C4×Q8×C79D411yescharacteristic
123C4×Q8×C79C2311yescharacteristic
124C4×D4×C79—21no—
125C4×D4×C79—21no—
126C4×D4×C79D411yescharacteristic
127C4×D4×C79C2311yescharacteristic
128#117×C79—21no—
129#117×C79—21no—
130#117×C79D411yescharacteristic
131C2×C42×C79—21no—
132C2×C42×C79C2311yescharacteristic
133#107×C79—21no—
134#107×C79—21no—
135#106×C79—21no—
136#106×C79—21no—
137C4⋊C8×C79—21no—
138C4⋊C8×C79C2311yescharacteristic
139C4⋊C8×C79C2311yescharacteristic
140#103×C79—21no—
141#103×C79—21no—
142#102×C79—21no—
143#102×C79—21no—
144C2×Dic158—21no—
145C2×Dic158—21no—
146#11929614—42no—
147#11929614—42no—
148#11929608—42no—
149#11929608—42no—
150C2×C79⋊C8—22no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8—41no—
156C2×C79⋊C8—42no—
157C2×C79⋊C8C2×D411yescharacteristic
158C2×C79⋊C8C2×D411yescharacteristic
159C2×C79⋊C8C2×D411yescharacteristic
160C2×C79⋊C8C2×D411yescharacteristic
161C4⋊Dic79—21no—
162C4⋊Dic79—21no—
163C4⋊Dic79—21no—
164C4⋊Dic79—21no—
165Dic79⋊C4—41no—
166Dic79⋊C4—41no—
167C22⋊Dic79—41no—
168C4×Dic79—41no—
169#41×C79—22no—
170#41×C79—22no—
171C2×Q8×C79—21no—
172C2×Q8×C79—21no—
173C2×Q8×C79C2×D411yescharacteristic
174C2×D4×C79—21no—
175C2×D4×C79C2×D411yescharacteristic
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79#4111yescharacteristic
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79—42no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—41no—
192C4⋊C4×C79—41no—
193C4⋊C4×C79C2×D411yescharacteristic
194C4⋊C4×C79C2×D411yescharacteristic
195C22⋊C4×C79—21no—
196C22⋊C4×C79—21no—
197C22⋊C4×C79—21no—
198C22⋊C4×C79—41no—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—21no—
202C42×C79—41no—
203C42×C79C2×D411yescharacteristic
204C42×C79C2×D411yescharacteristic
205C42×C79#4111yescharacteristic
206C42×C79C2411yescharacteristic
207C23×C79—21no—
208C23×C79#12111yescharacteristic
209Q8×C79—21no—
210Q8×C79—21no—
211Q8×C79—41no—
212Q8×C79—41no—
213Q8×C79—81no—
214D4×C79—21no—
215D4×C79—21no—
216D4×C79—42no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—41no—
223C2×C4×C79—42no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—42no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79#12111yescharacteristic
230C2×C4×C79#12111yescharacteristic
231C2×C4×C79#12111yescharacteristic
232C2×C4×C79#13712yes—
233C2×C4×C79#13712yes—
234C2×C4×C79C22×D411yescharacteristic
235C2×C4×C79C22×D411yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×Dic79—41no—
238C2×Dic79—41no—
239Dic158—41no—
240Dic158—41no—
241Dic158—81no—
242C8×C79—41no—
243C8×C79—41no—
244C79⋊C8—21no—
245C79⋊C8—21no—
246C79⋊C8—21no—
247C79⋊C8—41no—
248C79⋊C8—41no—
249C79⋊C8—41no—
250C79⋊C8—41no—
251C79⋊C8#13512yes—
252C22×C79—21no—
253C22×C79—21no—
254C22×C79—41no—
255C22×C79—41no—
256C22×C79—41no—
257C22×C79#47012yes—
258C22×C79C2×#12111yescharacteristic
259C4×C79—21no—
260C4×C79—21no—
261C4×C79—21no—
262C4×C79—41no—
263C4×C79—41no—
264C4×C79—41no—
265C4×C79—81no—
266C4×C79—81no—
267C4×C79#47012yes—
268C4×C79C2×#13511yescharacteristic
269C4×C79C2×#13711yescharacteristic
270C4×C79C2×#13711yescharacteristic
271C4×C79#57511yescharacteristic
272Dic79—81no—
273Dic79—81no—
274#19158—791nomaximal, Sylow
275C2×C79—22no—
276C2×C79—81no—
277C2×C79#303511yescharacteristic
278C2×C79C2×#47011yescharacteristic
279C2×C79#306711yescharacteristic
280#3170—791no—
281#3076—791no—
282#3076D7911yescharacteristic
283#2965—791no—
284#1704—791no—
285#1704—791no—
286#1703—791no—
287#1703—791no—
288#1696—791no—
289#1695—791no—
290#1690—791no—
291#1689—791no—
292C2×#334—791no—
293#1542—791no—
294#1542—791no—
295C79#1915811yescharacteristic, complemented, Sylow
296#533—791no—
297#533C2×D7911yescharacteristic
298C4×#41C2×D7911yescharacteristic
299#500—791no—
300#494—791no—
301#487—791no—
302#487—791no—
303#479—21no—
304#479—1581no—
305#478—21no—
306#478—1581no—
307#477—791no—
308#477C2×D7911yescharacteristic
309#476—791no—
310#476C2×D7911yescharacteristic
311#475—791no—
312#475C2×D7911yescharacteristic
313#474—791no—
314#474C2×D7911yescharacteristic
315#467—1581no—
316#467—1581no—
317#446—791no—
318#446—791no—
319C4×Q16—1581no—
320C4×Q16—1581no—
321#436—791no—
322#436—791no—
323#435—1581no—
324#435—1581no—
325#431—791no—
326#431—1581no—
327#427—21no—
328#427—1581no—
329C2×C4⋊C8—791no—
330C2×C4⋊C8C2×D7911yescharacteristic
331#416—21no—
332#416—1581no—
333C4×M16—1581no—
334C2×C4×C8—791no—
335#334—791no—
336#334—791no—
337#334—791no—
338#334—791no—
339#333—791no—
340#333—791no—
341#333—791no—
342#333—791no—
343#332—791no—
344#332—791no—
345#332—791no—
346#332—791no—
347C2×#41C22⋊D7911yescharacteristic
348C2×Q16—1581no—
349C2×Q16—1581no—
350C2×M16—1581no—
351C2×M16—1581no—
352C22×C8—21no—
353C22×C8—791no—
354C22×C8—791no—
355C22×C8—1581no—
356C4⋊Q8—791no—
357C4⋊Q8—791no—
358C4⋊Q8C22×D7911yescharacteristic
359C4⋊Q8C22×D7911yescharacteristic
360#125—21no—
361#125—1581no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—1581no—
365C22⋊Q8—1581no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—1581no—
369C4×Q8—1581no—
370C4×Q8C22⋊D7911yescharacteristic
371C4×Q8C22×D7911yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C22⋊D7911yescharacteristic
375C4×D4C22×D7911yescharacteristic
376#117—21no—
377#117—21no—
378#117—1581no—
379#117C22⋊D7911yescharacteristic
380C2×C42—21no—
381C2×C42C22×D7911yescharacteristic
382#107—21no—
383#107—21no—
384#107—1581no—
385#107—1581no—
386#107—1581no—
387#107—1581no—
388#106—21no—
389#106—21no—
390#106—1581no—
391#106—1581no—
392C4⋊C8—21no—
393C4⋊C8—791no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—1581no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8C22×D7911yescharacteristic
401C4⋊C8C22×D7911yescharacteristic
402#103—21no—
403#103—21no—
404#103—1581no—
405#103—1581no—
406#103—1581no—
407#103—1581no—
408#103—1581no—
409#103—1581no—
410#102—21no—
411#102—21no—
412#102—1581no—
413#102—1581no—
414C22⋊C8—1581no—
415C22⋊C8—1581no—
416C22⋊C8—1581no—
417#97—1581no—
418#97—1581no—
419C4×C8—791no—
420C4×C8—791no—
421C4×C8—791no—
422C4×C8—791no—
423C4×C8—1581no—
424#41—22no—
425#41—22no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—1581no—
429C2×Q8—1581no—
430C2×Q8C2×C22⋊D7911yescharacteristic
431C2×D4—21no—
432C2×D4C2×C22⋊D7911yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4D4×D7911yescharacteristic
437C22×C4C2×C22⋊D7911yescharacteristic
438C22×C4#1192962311yescharacteristic
439Q16—3162no—
440Q16—3162no—
441M16—3162no—
442M16—3162no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—42no—
446C2×C8—3162no—
447C2×C8—3161no—
448C2×C8—791no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—1581no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8—1581no—
456C2×C8—1582no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—3161no—
466C4⋊C4—3161no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4C2×C22⋊D7911yescharacteristic
472C4⋊C4C2×C22⋊D7911yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—41no—
477C22⋊C4—3161no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—3161no—
483C42D4×D7911yescharacteristic
484C42C2×C22⋊D7911yescharacteristic
485C42#1192962311yescharacteristic
486C42C23×D7911yescharacteristic
487C23—21no—
488C23#1457913311yescharacteristic
489Q8—21no—
490Q8—21no—
491Q8—41no—
492Q8—41no—
493Q8—81no—
494Q8—3161no—
495Q8—3161no—
496Q8—6321no—
497D4—21no—
498D4—21no—
499D4—42no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—3161no—
513C2×C4—3161no—
514C2×C4#1457913311yescharacteristic
515C2×C4#1457913311yescharacteristic
516C2×C4#1457913311yescharacteristic
517C2×C4#1457919112yes—
518C2×C4#1457919212yes—
519C2×C4C2×D4×D7911yescharacteristic
520C2×C4C22×C22⋊D7911yescharacteristic
521C2×C4C2×#1192962311yescharacteristic
522C8—41no—
523C8—41no—
524C8—3161no—
525C8—3161no—
526C8—3161no—
527C8—3161no—
528C8—792no—
529C8—1581no—
530C8—1581no—
531C8—1581no—
532C22—21no—
533C22—21no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22#1909881912yes—
538C22C2×#1457913311yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—6321no—
548C4—6321no—
549C4#1909881912yes—
550C4#135×D7911yescharacteristic
551C4C2×#1457919111yescharacteristic
552C4C2×#1457919211yescharacteristic
553C4#1909936011yescharacteristic
554C2—22no—
555C2—81no—
556C2#2377285111yescharacteristic
557C2C2×#1909881911yescharacteristic
558C2#2377311711yescharacteristic
subgroupG/H
centerC2×C4#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#3076C2
socleC22×C79C2×#121
prime radicalC2×#41×C79D4
hypercenter#3076D79
hypocenterC79#19158
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #23773982C2semidirect
4. #23762984C2semidirect
5. #23762980C2semidirect
6. #23762977C2semidirect
7. #23762973C2semidirect
8. #23762926C2semidirect
9. #23762921C2semidirect
10. #23762892C2semidirect
11. #23762887C2semidirect
12. C2×#19098313C2semidirect
13. #23762058C2semidirect
14. #23762057C2semidirect
295. C79#19158semidirect

Maximal subgroups

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

subgroupindex
#237739822
#237731882
#237723672
#237629842
#237629802
#237629772
#237629732
#237629262
#237629212
#237628922
#237628872
C2×#190983132
#237620582
#237620572
C79×#30762
#1915879