Finite Group Databasebeta

← #34258859

Subgroups — #34258859

Order 20224.

594 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#23774002C211yesmaximal, characteristic
2#23772372C211yesmaximal, characteristic
3#23772329C211yesmaximal, characteristic, complemented
4#23762308C211yesmaximal, characteristic, complemented
5#23762306C211yesmaximal, characteristic, complemented
6#23762303C211yesmaximal, characteristic, complemented
7#23762299C211yesmaximal, characteristic, complemented
8C8×Dic316C211yesmaximal, characteristic, complemented
9C8×#11929614C211yesmaximal, characteristic, complemented
10C8×#11929612C211yesmaximal, characteristic, complemented
11#23762164C211yesmaximal, characteristic, complemented
12#23761871C211yesmaximal, characteristic, complemented
13#23761862C211yesmaximal, characteristic
14C8×#11929607C211yesmaximal, characteristic, complemented
15C8×SD16×C79C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17#19099095C2211yescharacteristic
18#19098970C411yescharacteristic
19#19098954C411yescharacteristic
20#19098930C411yescharacteristic
21#19098921C411yescharacteristic
22#19098822C411yescharacteristic
23#19098820C411yescharacteristic
24#19098808C411yescharacteristic
25#19098806C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098692—21no—
28#19098690C2211yescharacteristic
29#19098690C2211yescharacteristic
30C8×Dic158—21no—
31C8×Dic158C2211yescharacteristic
32C8×Dic158C2211yescharacteristic
33C4×Dic316C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929612C2211yescharacteristic
36#19098641C2211yescharacteristic
37#19098637—21no—
38C8×C22⋊D79—21no—
39#19098621—21no—
40#19098619C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098588—21no—
43#19098585C2211yescharacteristic
44#19098518—21no—
45#19098517—21no—
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098293C2211yescharacteristic
56D4⋊(C79⋊C8)C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×Dic316C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929608—21no—
71C2×#11929607—21no—
72C2×#11929607C2×C411yescharacteristic
73C2×#11929607C2×C411yescharacteristic
74C2×C8×D79D411yescharacteristic
75C2×C8×D79D411yescharacteristic
76#14579160C2×C411yescharacteristic
77#14579159C2×C411yescharacteristic
78#14579154—21no—
79#14579152C2×C411yescharacteristic
80#14579151—21no—
81#14579148C2×C411yescharacteristic
82#14579143—21no—
83#14579141C2×C411yescharacteristic
84#14579141C2×C411yescharacteristic
85#14579139—21no—
86#14579137C2×C411yescharacteristic
87#14579132C2×C411yescharacteristic
88C4×Dic158—21no—
89C4×Dic158C2311yescharacteristic
90C4×Dic158C2311yescharacteristic
91C4×C22⋊D79—21no—
92#14579120—21no—
93#14579118—21no—
94C42×D79C2311yescharacteristic
95#14579095C2×C411yescharacteristic
96#14579094C2×C411yescharacteristic
97#14579092C2×C411yescharacteristic
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107#14579086C2×C411yescharacteristic
108#14579086C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579084C2×C411yescharacteristic
112#14579082C2×C411yescharacteristic
113C22⋊(C79⋊C8)—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116#14579072—21no—
117#14579071—21no—
118#14579071—21no—
119#14579071—21no—
120#14579071C2×C411yescharacteristic
121#14579071C2×C411yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C8×Dic79—21no—
126C8×Dic79—21no—
127C8×Dic79D411yescharacteristic
128C8×Dic79D411yescharacteristic
129C2×SD16×C79C2×C411yescharacteristic
130C22×C8×C79—21no—
131C4×Q8×C79C2311yescharacteristic
132C4×D4×C79C2311yescharacteristic
133#106×C79C2×C411yescharacteristic
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#102×C79C2×C411yescharacteristic
139C22⋊C8×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×Dic158C22×C411yescharacteristic
147C2×C22⋊D79—21no—
148C2×C4×D79C22×C411yescharacteristic
149C2×C4×D79C2×D411yescharacteristic
150C2×C4×D79#4111yescharacteristic
151Dic316—22no—
152#11929614—22no—
153#11929612—22no—
154#11929608—42no—
155#11929607—22no—
156#11929607—22no—
157#11929607—42no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8C22×C411yescharacteristic
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C2×D411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C8×D79—22no—
166C8×D79—41no—
167C4⋊Dic79—21no—
168C4⋊Dic79—21no—
169C4⋊Dic79—21no—
170C4⋊Dic79C22×C411yescharacteristic
171C4⋊Dic79C22×C411yescharacteristic
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179C22⋊Dic79—21no—
180C22⋊Dic79—21no—
181C22⋊Dic79—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79C22×C411yescharacteristic
188C4×Dic79C2×D411yescharacteristic
189C4×Dic79#4111yescharacteristic
190C2×Q8×C79C22×C411yescharacteristic
191C2×D4×C79C22×C411yescharacteristic
192C22×C4×C79—21no—
193SD16×C79—22no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79—42no—
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C2×D411yescharacteristic
201C2×C8×C79C2×D411yescharacteristic
202C4⋊C4×C79—21no—
203C4⋊C4×C79C22×C411yescharacteristic
204C4⋊C4×C79C22×C411yescharacteristic
205C22⋊C4×C79—21no—
206C42×C79—21no—
207C42×C79C2411yescharacteristic
208C23×C79—21no—
209C22×D79C4×D411yescharacteristic
210Q8×C79—21no—
211Q8×C79—21no—
212D4×C79—21no—
213D4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79C23×C411yescharacteristic
220C2×C4×C79C22×D411yescharacteristic
221C2×C4×C79C2×#4111yescharacteristic
222C22⋊D79—42no—
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79C4×D411yescharacteristic
228C2×Dic79C4×D411yescharacteristic
229C2×Dic79C4×D411yescharacteristic
230C4×D79—21no—
231C4×D79—21no—
232C4×D79—21no—
233C4×D79—21no—
234C4×D79—41no—
235Dic158—21no—
236Dic158—21no—
237Dic158—21no—
238Dic158—21no—
239Dic158—42no—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—41no—
243C8×C79—41no—
244C8×C79—41no—
245C8×C79#13112yes—
246C8×C79#13712yes—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—41no—
251C79⋊C8—41no—
252C79⋊C8—41no—
253C79⋊C8#13112yes—
254C22×C79—41no—
255C22×C79—41no—
256C22×C79—41no—
257C22×C79C2×C4×D411yescharacteristic
258C2×D79—21no—
259C2×D79—21no—
260C2×D79—21no—
261C4×C79—21no—
262C4×C79—41no—
263C4×C79—41no—
264C4×C79—41no—
265C4×C79C2×#13111yescharacteristic
266C4×C79C2×#13111yescharacteristic
267C4×C79C2×#13711yescharacteristic
268C4×C79#57511yescharacteristic
269Dic79—21no—
270Dic79—21no—
271Dic79—21no—
272Dic79—41no—
273Dic79—41no—
274Dic79—41no—
275Dic79—41no—
276#17965—791nomaximal, Sylow
277C2×C79—81no—
278C2×C79#293111yescharacteristic
279C2×C79C4×#13711yescharacteristic
280C2×C79#295511yescharacteristic
281D79—41no—
282#3172—791no—
283#2966—791no—
284#2960—791no—
285#1593—791no—
286#1593—791no—
287#1592—791no—
288#1592—791no—
289C8×Q16—791no—
290C8×Q16—791no—
291C8×SD16—791no—
292C8×SD16D7911yescharacteristic, complemented
293#1569—791no—
294#1497—791no—
295#1495—791no—
296C8×M16—791no—
297C79#1796511yescharacteristic, complemented, Sylow
298#533—791no—
299#529—791no—
300#501—791no—
301#497—791no—
302#491—791no—
303#489—791no—
304#470—791no—
305#470—791no—
306#468—791no—
307#468—791no—
308#446—791no—
309#446—791no—
310#446—791no—
311#446—1581no—
312C8×Q8—791no—
313C8×Q8—791no—
314C8×Q8—1581no—
315C8×Q8C2×D7911yescharacteristic
316C4×Q16—791no—
317C4×Q16—791no—
318C4×SD16—791no—
319C4×SD16C2×D7911yescharacteristic
320#436—791no—
321#435—1581no—
322C8×D4—1581no—
323C8×D4C2×D7911yescharacteristic
324#433—1581no—
325#432—791no—
326#431—791no—
327#424—1581no—
328#423—791no—
329#405—1581no—
330#405—1581no—
331C4×M16—791no—
332C2×C4×C8—791no—
333#335—791no—
334#335—791no—
335#334—791no—
336#334C2×D7911yescharacteristic
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8C2×D7911yescharacteristic
343D4⋊C8—791no—
344D4⋊C8C2×D7911yescharacteristic
345#322—791no—
346#322—791no—
347C82—791no—
348C82C2×D7911yescharacteristic
349C2×Q16—791no—
350C2×Q16—791no—
351C2×SD16—791no—
352C2×SD16C4×D7911yescharacteristic
353C2×M16—791no—
354C2×M16—791no—
355C2×M16—1581no—
356C2×M16—1581no—
357C22×C8—21no—
358C22×C8—791no—
359C22×C8—791no—
360C4⋊Q8—791no—
361C4⋊Q8—791no—
362#126—1581no—
363#125—791no—
364#125—1581no—
365#124—791no—
366#123—1581no—
367C22⋊Q8—791no—
368C22⋊Q8—791no—
369C22⋊Q8—791no—
370C22⋊Q8—1581no—
371#121—791no—
372C4×Q8—791no—
373C4×Q8—791no—
374C4×Q8—1581no—
375C4×Q8C22×D7911yescharacteristic
376C4×D4—1581no—
377C4×D4C22×D7911yescharacteristic
378#117—1581no—
379#117—1581no—
380C2×C42—791no—
381#107—791no—
382#107—791no—
383#106—791no—
384#106C4×D7911yescharacteristic
385C4⋊C8—21no—
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#103C4×D7911yescharacteristic
403#102—791no—
404#102C4×D7911yescharacteristic
405C22⋊C8—21no—
406C22⋊C8—1581no—
407C22⋊C8—1581no—
408C22⋊C8—1581no—
409#97—791no—
410#97—791no—
411#97—1581no—
412#97—1581no—
413#97—1581no—
414#97—1581no—
415C4×C8—21no—
416C4×C8—791no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—791no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8C22×D7911yescharacteristic
424C4×C8C22×D7911yescharacteristic
425C4×C8C22×D7911yescharacteristic
426C2×Q8—791no—
427C2×Q8—791no—
428C2×Q8—1581no—
429C2×Q8C2×C4×D7911yescharacteristic
430C2×D4—1581no—
431C2×D4C2×C4×D7911yescharacteristic
432C22×C4—21no—
433C22×C4—791no—
434C22×C4—791no—
435C22×C4—791no—
436Q16—1582no—
437Q16—1582no—
438SD16—22no—
439SD16—1582no—
440M16—3162no—
441M16—3162no—
442M16—1582no—
443M16—1582no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—42no—
448C2×C8—3161no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—791no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8—1581no—
456C2×C8—1582no—
457C2×C8C2×C4×D7911yescharacteristic
458C2×C8C2×C4×D7911yescharacteristic
459C2×C8D4×D7911yescharacteristic
460C2×C8D4×D7911yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—791no—
463C4⋊C4—791no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4C2×C4×D7911yescharacteristic
475C4⋊C4C2×C4×D7911yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C22⋊C4—1581no—
482C22⋊C4—1581no—
483C42—21no—
484C42—791no—
485C42—791no—
486C42—791no—
487C42—1581no—
488C42—1581no—
489C42C23×D7911yescharacteristic
490C23—21no—
491C23—791no—
492Q8—21no—
493Q8—21no—
494Q8—3162no—
495Q8—1581no—
496Q8—1581no—
497Q8—1581no—
498Q8—1581no—
499D4—21no—
500D4—21no—
501D4—3162no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—3161no—
508C2×C4—791no—
509C2×C4—791no—
510C2×C4—791no—
511C2×C4—1581no—
512C2×C4—1581no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4C22×C4×D7911yescharacteristic
520C2×C4C2×D4×D7911yescharacteristic
521C2×C4#41×D7911yescharacteristic
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—3161no—
528C8—3161no—
529C8—3161no—
530C8—792no—
531C8—1581no—
532C8—1581no—
533C8—1581no—
534C8#1457916712yes—
535C8#1457919012yes—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—1581no—
540C22—1581no—
541C22—1581no—
542C22C4×D4×D7911yescharacteristic
543C4—21no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—3161no—
548C4—3161no—
549C4—3161no—
550C4—3161no—
551C4—1581no—
552C4—1581no—
553C4—1581no—
554C4#131×D7911yescharacteristic
555C4C2×#1457916711yescharacteristic
556C4C2×#1457919011yescharacteristic
557C4#1909936011yescharacteristic
558C2—81no—
559C2—3161no—
560C2#2377210711yescharacteristic
561C2C4×#1457919011yescharacteristic
562C2#2377229811yescharacteristic
subgroupG/H
centerC2×C8D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC8×SD16×C79C2
socleC22×C79C2×C4×D4
prime radical#14579132C2×C4
cosocleC42×C79C24
hypercenterC8×SD16D79
hypocenterC79#17965
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
3. #23772329C2semidirect
4. #23762308C2semidirect
5. #23762306C2semidirect
6. #23762303C2semidirect
7. #23762299C2semidirect
8. C8×Dic316C2semidirect
9. C8×#11929614C2semidirect
10. C8×#11929612C2semidirect
11. #23762164C2semidirect
12. #23761871C2semidirect
14. C8×#11929607C2semidirect
15. C8×SD16×C79C2semidirect
292. C8×SD16D79semidirect
297. C79#17965semidirect

Maximal subgroups

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

subgroupindex
#237740022
#237723722
#237723292
#237623082
#237623062
#237623032
#237622992
C8×Dic3162
C8×#119296142
C8×#119296122
#237621642
#237618712
#237618622
C8×#119296072
C8×SD16×C792
#1796579