Finite Group Databasebeta

← C2×#23777412 #35196540

Subgroups — C2×#23777412 #35196540

Order 20224.

7118 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23777412C2132yesmaximal, direct product factor
2C2×Q8×Dic158C211yesmaximal, characteristic, complemented
3C2×#19099287C213yesmaximal, complemented
4C2×#19099284C216yesmaximal, complemented
5C2×#19099278C213yesmaximal, complemented
6C2×#19099224C212yesmaximal, complemented
7C2×#19099186C216yesmaximal, complemented
8C2×#19099118C213yesmaximal, complemented
9C2×#19099095C213yesmaximal, complemented
10C2×#19099041C212yesmaximal, complemented
11C2×C4×#11929625C211yesmaximal, characteristic, complemented
12C2×D4×Q8×C79C211yesmaximal, characteristic, complemented
13C22×#11929625C2211yescharacteristic
14Q8×Dic158C22116yescomplemented
15#19099287C22148yescomplemented
16#19099284C22196yescomplemented
17#19099278C22148yescomplemented
18#19099224C22132yescomplemented
19#19099186C22196yescomplemented
20#19099118C22148yescomplemented
21C2×#14579161C2213yes—
22C2×#14579160C2216yes—
23#19099095C22148yescomplemented
24C2×#14579156C22112yes—
25C2×#14579154C2216yes—
26C2×#14579152C2213yes—
27C2×#14579150C2216yes—
28C2×#14579148C2211yescharacteristic
29C2×#14579147C22112yes—
30C2×#14579143C2216yes—
31C2×#14579141C2216yes—
32C2×#14579141C2213yes—
33C2×#14579137C2216yes—
34C2×#14579135C2216yes—
35C2×#14579132C2213yes—
36#19099041C22132yescomplemented
37C4×#11929625C22116yescomplemented
38C2×Q8×Dic79C2216yes—
39C2×Q8×Dic79C2211yescharacteristic
40C2×Q8×Dic79C2212yes—
41C2×Q8×Dic79C2216yes—
42C2×C4×Dic158C2213yes—
43C2×#14579126C2216yes—
44C2×C4×D316C2213yes—
45C2×#14579123C2213yes—
46C2×C4×C22⋊D79C2216yes—
47C2×#14579120C2216yes—
48C2×#14579120C2216yes—
49C2×#14579119C2213yes—
50C2×#14579119C2216yes—
51C2×C42×D79C2213yes—
52C23×Q8×C79C2212yes—
53D4×Q8×C79C22116yescomplemented
54C2×C4⋊Q8×C79C2213yes—
55C2×C22⋊Q8×C79C2216yes—
56C2×C4×Q8×C79C2211yescharacteristic
57C2×C4×D4×C79C2213yes—
58C2×#11929625—28no—
59C2×#11929625C2318yes—
60C22×Dic158C2311yescharacteristic
61C22×Dic158C2316yes—
62C22×D316C2313yes—
63C22×C22⋊D79C2316yes—
64C22×C4×D79C2313yes—
65C22×C4×D79C2313yes—
66C22×C4×D79C2313yes—
67#14579161C23124yes—
68#14579160C23148yes—
69#14579156C23196yes—
70#14579154C23148yes—
71#14579152C23124yes—
72#14579150C23148yes—
73#14579148C2318yes—
74#14579147C23196yes—
75#14579143C23148yes—
76#14579141C23148yes—
77#14579141C23124yes—
78#14579137C23148yes—
79#14579135C23148yes—
80#14579132C23124yes—
81Q8×Dic79C23148yes—
82Q8×Dic79C23116yes—
83Q8×Dic79C23148yes—
84Q8×Dic79C2318yes—
85C4×Dic158C23124yes—
86#14579126C23148yes—
87C4×D316C23124yes—
88#14579123C23124yes—
89C4×C22⋊D79C23148yes—
90#14579120C23148yes—
91#14579120C23148yes—
92#14579119C23148yes—
93#14579119C23124yes—
94C2×C4⋊Dic79C2313yes—
95C2×C4⋊Dic79C2313yes—
96C2×C4⋊Dic79C2316yes—
97C2×C4⋊Dic79C2316yes—
98C2×C4⋊Dic79C2316yes—
99C2×Dic79⋊C4C2316yes—
100C2×Dic79⋊C4C2316yes—
101C2×C22⋊Dic79C2316yes—
102C2×C22⋊Dic79C2312yes—
103C2×C22⋊Dic79C2312yes—
104C2×(C2×D79)⋊C4C2316yes—
105C2×(C2×D79)⋊C4C23112yes—
106C2×(C2×D79)⋊C4C2316yes—
107C2×C4×Dic79C2311yescharacteristic
108C2×C4×Dic79C2313yes—
109C2×C4×Dic79C2313yes—
110C2×C4×Dic79C23112yes—
111C2×C4×Dic79C2316yes—
112C42×D79C23124yes—
113C22×Q8×C79—216no—
114C22×Q8×C79C23116yes—
115C22×Q8×C79C2316yes—
116C22×Q8×C79C2311yescharacteristic
117C22×D4×C79C2311yescharacteristic
118C23×C4×C79C2316yes—
119C4⋊Q8×C79C23124yes—
120C22⋊Q8×C79C23148yes—
121C4×Q8×C79C2318yes—
122C4×D4×C79C23124yes—
123C2×C4⋊C4×C79C2313yes—
124C2×C4⋊C4×C79C2313yes—
125C2×C4⋊C4×C79C2316yes—
126C2×C22⋊C4×C79C2316yes—
127C2×C42×C79C2313yes—
128C23×D79C2413yes—
129#11929625—232no—
130C2×Dic158—24no—
131C2×Dic158—412no—
132C2×Dic158C2414yes—
133C2×Dic158C24124yes—
134C2×D316—212no—
135C2×D316C24112yes—
136C2×C22⋊D79—412no—
137C2×C22⋊D79C24124yes—
138C22×Dic79C2416yes—
139C22×Dic79C2412yes—
140C22×Dic79C2413yes—
141C22×Dic79C2411yescharacteristic
142C22×Dic79C2411yescharacteristic
143C2×C4×D79—212no—
144C2×C4×D79—212no—
145C2×C4×D79—46no—
146C2×C4×D79C24112yes—
147C2×C4×D79C24112yes—
148C2×C4×D79C24112yes—
149C4⋊Dic79C24112yes—
150C4⋊Dic79C24112yes—
151C4⋊Dic79C24124yes—
152C4⋊Dic79C24124yes—
153C4⋊Dic79C24124yes—
154Dic79⋊C4C24124yes—
155Dic79⋊C4C24124yes—
156C22⋊Dic79C24124yes—
157C22⋊Dic79C2418yes—
158C22⋊Dic79C2418yes—
159(C2×D79)⋊C4C24124yes—
160(C2×D79)⋊C4C24148yes—
161(C2×D79)⋊C4C24124yes—
162C4×Dic79C2414yes—
163C4×Dic79C24112yes—
164C4×Dic79C24112yes—
165C4×Dic79C24148yes—
166C4×Dic79C24124yes—
167C24×C79C2412yes—
168C2×Q8×C79—264no—
169C2×Q8×C79—224no—
170C2×Q8×C79#4118yes—
171C2×Q8×C79C24124yes—
172C2×Q8×C79C2414yes—
173C2×D4×C79—24no—
174C2×D4×C79C2414yes—
175C22×C4×C79—224no—
176C22×C4×C79C2413yes—
177C22×C4×C79C2416yes—
178C22×C4×C79C24124yes—
179C22×C4×C79C2413yes—
180C22×C4×C79C2411yescharacteristic
181C4⋊C4×C79C24112yes—
182C4⋊C4×C79C24112yes—
183C4⋊C4×C79C24124yes—
184C22⋊C4×C79C24124yes—
185C42×C79C24112yes—
186C23×C79—24no—
187C23×C79—24no—
188C23×C79—24no—
189C23×C79C2514yes—
190C23×C79C2511yescharacteristic
191C22×D79—26no—
192C22×D79—26no—
193C22×D79—26no—
194C22×D79C2516yes—
195Q8×C79—248no—
196Q8×C79C2×#41116yes—
197D4×C79—28no—
198C2×C4×C79—212no—
199C2×C4×C79—26no—
200C2×C4×C79—248no—
201C2×C4×C79C2×#41112yes—
202C2×C4×C792−1+414yes—
203C2×C4×C79C25112yes—
204C2×C4×C79C2516yes—
205C2×C4×C79C2516yes—
206C2×C4×C79C2512yes—
207C22⋊D79—424no—
208C2×Dic79—22no—
209C2×Dic79—26no—
210C2×Dic79—24no—
211C2×Dic79—22no—
212C2×Dic79—212no—
213C2×Dic79C2516yes—
214C2×Dic79C2512yes—
215C2×Dic79C2512yes—
216C2×Dic79C2514yes—
217C2×Dic79C25112yes—
218D316—224no—
219C4×D79—224no—
220C4×D79—224no—
221C4×D79—412no—
222Dic158—28no—
223Dic158—424no—
224C22×C79—24no—
225C22×C79—24no—
226C22×C79—24no—
227C22×C79—48no—
228C22×C79C22×#4112yes—
229C22×C79C2×2−1+412yes—
230C22×C79#58512yes—
231C22×C79C2611yescharacteristic
232C2×D79—26no—
233C2×D79—26no—
234C2×D79—26no—
235C2×D79—412no—
236C4×C79—212no—
237C4×C79—26no—
238C4×C79C22×#41112yes—
239C4×C79C2×2−1+414yes—
240Dic79—22no—
241Dic79—24no—
242Dic79—26no—
243Dic79—212no—
244Dic79—22no—
245C2×#3518—791nomaximal, Sylow
246C2×C79—44no—
247C2×C79#351814yescomplemented
248C2×C79C23×#4111yescharacteristic
249C2×C79C22×2−1+411yescharacteristic
250C2×C79C2×#58511yescharacteristic
251D79—46no—
252#3518—7932no—
253C2×Q82—791no—
254C2×#556—793no—
255C2×#556—796no—
256C2×#555—793no—
257C2×D4×Q8D7911yescharacteristic, complemented
258C2×#548—792no—
259C2×#542—796no—
260C2×#533—793no—
261C2×#529—793no—
262C2×#519—792no—
263C2×C4×#41—791no—
264C79C2×#351811yescharacteristic, complemented, Sylow
265C22×#41—791no—
266C23×Q8C2×D7912yes—
267Q82—7916no—
268#556—7948no—
269#556—7996no—
270#555—7948no—
271D4×Q8C2×D79116yescomplemented
272#548—7932no—
273#542—7996no—
274#533—7948no—
275C2×C4⋊Q8—796no—
276C2×C4⋊Q8—793no—
277C2×C4⋊Q8C2×D7913yes—
278#529—7948no—
279C2×#126—7912no—
280C2×#126—796no—
281C2×#125—793no—
282C2×#124—796no—
283C2×#124—791no—
284C2×#124—7912no—
285C2×#123—796no—
286C2×C22⋊Q8—796no—
287C2×C22⋊Q8—793no—
288C2×C22⋊Q8—796no—
289C2×C22⋊Q8C2×D7916yes—
290C2×#121—793no—
291C2×#121—796no—
292#519—7932no—
293C4×#41—7916no—
294C2×C4×Q8—791no—
295C2×C4×Q8—796no—
296C2×C4×Q8—792no—
297C2×C4×Q8—796no—
298C2×C4×Q8—796no—
299C2×C4×Q8—793no—
300C2×C4×Q8C2×D7911yescharacteristic
301C2×C4×D4—796no—
302C2×C4×D4—793no—
303C2×C4×D4—793no—
304C2×C4×D4C2×D7913yes—
305C2×#117—796no—
306C2×#117—796no—
307C2×#117—793no—
308C2×#117—796no—
309C22×C42—793no—
310C2×#41—798no—
311C2×#41—1588no—
312C22×Q8—216no—
313C22×Q8—796no—
314C22×Q8—791no—
315C22×Q8C22×D7916yes—
316C22×Q8C22×D79116yes—
317C22×Q8C22×D7911yescharacteristic
318C22×D4—796no—
319C22×D4—793no—
320C22×D4C22×D7911yescharacteristic
321C23×C4—793no—
322C23×C4—793no—
323C23×C4—793no—
324C23×C4C22×D7916yes—
325C4⋊Q8—7948no—
326C4⋊Q8—7924no—
327C4⋊Q8C22×D79124yes—
328#126—7996no—
329#126—7948no—
330#125—7924no—
331#124—7948no—
332#124—798no—
333#124—7996no—
334#123—7948no—
335C22⋊Q8—7948no—
336C22⋊Q8—7924no—
337C22⋊Q8—7948no—
338C22⋊Q8C22×D79148yes—
339#121—7924no—
340#121—7948no—
341C4×Q8—798no—
342C4×Q8—7948no—
343C4×Q8—7916no—
344C4×Q8—7948no—
345C4×Q8—7948no—
346C4×Q8—7924no—
347C4×Q8C22×D7918yes—
348C4×D4—7948no—
349C4×D4—7924no—
350C4×D4—7924no—
351C4×D4C22×D79124yes—
352#117—7948no—
353#117—7948no—
354#117—7924no—
355#117—7948no—
356C2×C4⋊C4—793no—
357C2×C4⋊C4—793no—
358C2×C4⋊C4—796no—
359C2×C4⋊C4—796no—
360C2×C4⋊C4—796no—
361C2×C4⋊C4—796no—
362C2×C4⋊C4—796no—
363C2×C4⋊C4C22×D7913yes—
364C2×C4⋊C4C22×D7913yes—
365C2×C4⋊C4C22×D7916yes—
366C2×C22⋊C4—796no—
367C2×C22⋊C4—792no—
368C2×C22⋊C4—792no—
369C2×C22⋊C4—796no—
370C2×C22⋊C4—7912no—
371C2×C22⋊C4—796no—
372C2×C22⋊C4C22×D7916yes—
373C2×C42—793no—
374C2×C42—7912no—
375C2×C42—793no—
376C2×C42—796no—
377C2×C42—791no—
378C2×C42—7924no—
379C2×C42C22×D7913yes—
380C24—793no—
381C24C23×D7912yes—
382#41—15832no—
383C2×Q8—264no—
384C2×Q8—224no—
385C2×Q8—31612no—
386C2×Q8—7924no—
387C2×Q8—794no—
388C2×Q8—1584no—
389C2×Q8#1192962318yes—
390C2×Q8C23×D79124yes—
391C2×Q8C23×D7914yes—
392C2×D4—24no—
393C2×D4—31612no—
394C2×D4—7924no—
395C2×D4—7912no—
396C2×D4—15812no—
397C2×D4C23×D7914yes—
398C22×C4—224no—
399C22×C4—3166no—
400C22×C4—7912no—
401C22×C4—793no—
402C22×C4—791no—
403C22×C4—791no—
404C22×C4—792no—
405C22×C4—796no—
406C22×C4—7912no—
407C22×C4—7912no—
408C22×C4—15812no—
409C22×C4—15812no—
410C22×C4C23×D79124yes—
411C22×C4C23×D7916yes—
412C22×C4C23×D7913yes—
413C22×C4C23×D7913yes—
414C22×C4C23×D7911yescharacteristic
415C4⋊C4—7924no—
416C4⋊C4—7924no—
417C4⋊C4—7912no—
418C4⋊C4—7912no—
419C4⋊C4—7924no—
420C4⋊C4—7924no—
421C4⋊C4—7924no—
422C4⋊C4C23×D79112yes—
423C4⋊C4C23×D79112yes—
424C4⋊C4C23×D79124yes—
425C22⋊C4—7924no—
426C22⋊C4—798no—
427C22⋊C4—798no—
428C22⋊C4—7924no—
429C22⋊C4—7948no—
430C22⋊C4—7924no—
431C22⋊C4C23×D79124yes—
432C42—794no—
433C42—7912no—
434C42—7912no—
435C42—7948no—
436C42—7924no—
437C42C23×D79112yes—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—796no—
442C23—1586no—
443C23—1586no—
444C23—1586no—
445C23C24×D7914yes—
446C23C24×D7911yescharacteristic
447Q8—248no—
448Q8—31624no—
449Q8—1588no—
450Q8C2×#11929623116yes—
451D4—28no—
452D4—31624no—
453D4—15824no—
454C2×C4—26no—
455C2×C4—212no—
456C2×C4—248no—
457C2×C4—31612no—
458C2×C4—796no—
459C2×C4—792no—
460C2×C4—792no—
461C2×C4—794no—
462C2×C4—7912no—
463C2×C4—1586no—
464C2×C4—1584no—
465C2×C4—1582no—
466C2×C4—15812no—
467C2×C4—1582no—
468C2×C4—15824no—
469C2×C4—15824no—
470C2×C4C2×#11929623112yes—
471C2×C4#1457920914yes—
472C2×C4C24×D79112yes—
473C2×C4C24×D7916yes—
474C2×C4C24×D7912yes—
475C2×C4C24×D7916yes—
476C22—24no—
477C22—24no—
478C22—24no—
479C22—48no—
480C22—31612no—
481C22—1586no—
482C22—1586no—
483C22—1586no—
484C22C22×#1192962312yes—
485C22C2×#1457920912yes—
486C22#1909940012yes—
487C22C25×D7911yescharacteristic
488C4—212no—
489C4—26no—
490C4—1582no—
491C4—1586no—
492C4—1584no—
493C4—1582no—
494C4—15812no—
495C4C22×#11929623112yes—
496C4C2×#1457920914yes—
497C2—44no—
498C2—3166no—
499C2#2377741214yesdirect product factor
500C2C23×#1192962311yescharacteristic
501C2C22×#1457920911yescharacteristic
502C2C2×#1909940011yescharacteristic
subgroupG/H
centerC23C24×D79
commutator, cosocleC22×C79C26
FrattiniC22C25×D79
FittingC2×D4×Q8×C79C2
socleC23×C79C25
hypercenterC2×D4×Q8D79
hypocenterC79C2×#3518
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. #23777412C2direct (N × H)
2. C2×Q8×Dic158C2semidirect
3. C2×#19099287C2semidirect
4. C2×#19099284C2semidirect
5. C2×#19099278C2semidirect
6. C2×#19099224C2semidirect
7. C2×#19099186C2semidirect
8. C2×#19099118C2semidirect
9. C2×#19099095C2semidirect
10. C2×#19099041C2semidirect
11. C2×C4×#11929625C2semidirect
12. C2×D4×Q8×C79C2semidirect
14. Q8×Dic158C22semidirect
15. #19099287C22semidirect
16. #19099284C22semidirect
17. #19099278C22semidirect
18. #19099224C22semidirect
19. #19099186C22semidirect
20. #19099118C22semidirect
23. #19099095C22semidirect
36. #19099041C22semidirect
37. C4×#11929625C22semidirect
53. D4×Q8×C79C22semidirect
247. C2×C79#3518semidirect
257. C2×D4×Q8D79semidirect
264. C79C2×#3518semidirect
271. D4×Q8C2×D79semidirect
499. C2#23777412direct (N × H)

Maximal subgroups

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

subgroupindex
#237774122
C2×Q8×Dic1582
C2×#190992872
C2×#190992842
C2×#190992782
C2×#190992242
C2×#190991862
C2×#190991182
C2×#190990952
C2×#190990412
C2×C4×#119296252
C2×D4×Q8×C792
C2×#351879