Finite Group Databasebeta

← C2×#23777417 #35196539

Subgroups — C2×#23777417 #35196539

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#23777417C2132yesmaximal, direct product factor
2C2×Q8×Dic158C211yesmaximal, characteristic, complemented
3C2×#19099283C213yesmaximal, complemented
4C2×#19099281C216yesmaximal, complemented
5C2×#19099278C213yesmaximal, complemented
6C2×Q8×D316C211yesmaximal, characteristic, complemented
7C2×#19099220C212yesmaximal, complemented
8C2×#19099178C216yesmaximal, complemented
9C2×#19099120C213yesmaximal, complemented
10C2×#19099097C213yesmaximal, complemented
11C2×#19099039C212yesmaximal, complemented
12C2×C4×#41×C79C211yesmaximal, characteristic, complemented
13C22×Q8×D79C2212yes—
14Q8×Dic158C22116yescomplemented
15#19099283C22148yescomplemented
16#19099281C22196yescomplemented
17#19099278C22148yescomplemented
18Q8×D316C22116yescomplemented
19#19099220C22132yescomplemented
20#19099178C22196yescomplemented
21#19099120C22148yescomplemented
22C2×#14579161C2213yes—
23C2×#14579161C2213yes—
24C2×#14579160C2216yes—
25#19099097C22148yescomplemented
26C2×#14579155C2216yes—
27C2×#14579154C22112yes—
28C2×#14579153C2213yes—
29C2×#14579149C2211yescharacteristic
30C2×#14579148C2216yes—
31C2×#14579147C22112yes—
32C2×#14579144C2216yes—
33C2×#14579141C2216yes—
34C2×#14579140C2216yes—
35C2×#14579140C2216yes—
36C2×#14579140C2213yes—
37C2×#14579135C2213yes—
38C2×#14579132C2216yes—
39#19099039C22132yescomplemented
40C2×Q8×Dic79C2212yes—
41C2×C4×Dic158C2216yes—
42C2×C4×Dic158C2213yes—
43C2×C4×Dic158C2216yes—
44C2×#14579126C2216yes—
45C2×C4×D316C2213yes—
46C2×C4×C22⋊D79C2216yes—
47C2×#14579120C2216yes—
48C2×#14579119C2216yes—
49C2×#14579118C2216yes—
50C22×#41×C79C2211yescharacteristic
51C4×#41×C79C22116yescomplemented
52C2×C4×Q8×C79C2211yescharacteristic
53C2×C4×Q8×C79C2211yescharacteristic
54C2×C4×D4×C79C2213yes—
55C2×C4×D4×C79C2213yes—
56C2×#117×C79C2213yes—
57C22×C42×C79C2213yes—
58C22×Dic158C2316yes—
59C22×Dic158C2311yescharacteristic
60C22×Dic158C2316yes—
61C2×Q8×D79—216no—
62C2×Q8×D79C23116yes—
63C22×D316C2311yescharacteristic
64C22×C22⋊D79C2316yes—
65C22×C4×D79C2316yes—
66#14579161C23124yes—
67#14579161C23124yes—
68#14579160C23148yes—
69#14579155C23148yes—
70#14579154C23196yes—
71#14579153C23124yes—
72#14579149C2318yes—
73#14579148C23148yes—
74#14579147C23196yes—
75#14579144C23148yes—
76#14579141C23148yes—
77#14579140C23148yes—
78#14579140C23148yes—
79#14579140C23124yes—
80#14579135C23124yes—
81#14579132C23148yes—
82Q8×Dic79C23116yes—
83C4×Dic158C23148yes—
84C4×Dic158C23148yes—
85C4×Dic158C23124yes—
86#14579126C23148yes—
87C4×D316C23124yes—
88C4×C22⋊D79C23148yes—
89#14579120C23148yes—
90#14579119C23148yes—
91#14579118C23148yes—
92C2×C4⋊Dic79C2316yes—
93C2×C4⋊Dic79C2313yes—
94C2×C4⋊Dic79C2313yes—
95C2×C4⋊Dic79C2316yes—
96C2×Dic79⋊C4C2316yes—
97C2×Dic79⋊C4C2316yes—
98C2×Dic79⋊C4C2316yes—
99C2×Dic79⋊C4C2316yes—
100C2×C22⋊Dic79C23112yes—
101C2×C22⋊Dic79C2316yes—
102C2×(C2×D79)⋊C4C2312yes—
103C2×(C2×D79)⋊C4C2316yes—
104C2×(C2×D79)⋊C4C2312yes—
105C2×(C2×D79)⋊C4C2316yes—
106C2×C4×Dic79C2316yes—
107C2×C4×Dic79C23112yes—
108C2×#41×C79—28no—
109C2×#41×C79C2318yes—
110C22×Q8×C79C2311yescharacteristic
111C22×D4×C79C2313yes—
112C23×C4×C79C2313yes—
113C23×C4×C79C2313yes—
114C23×C4×C79C2313yes—
115C4×Q8×C79C2318yes—
116C4×Q8×C79C2318yes—
117C4×D4×C79C23124yes—
118C4×D4×C79C23124yes—
119#117×C79C23124yes—
120C2×C4⋊C4×C79C2313yes—
121C2×C4⋊C4×C79C2313yes—
122C2×C22⋊C4×C79C2316yes—
123C2×C42×C79C2311yescharacteristic
124C2×C42×C79C2313yes—
125C2×C42×C79C2313yes—
126C2×C42×C79C2313yes—
127C2×C42×C79C23124yes—
128C23×D79C2412yes—
129C2×Dic158—224no—
130C2×Dic158—24no—
131C2×Dic158—412no—
132C2×Dic158C24124yes—
133C2×Dic158C2414yes—
134C2×Dic158C24124yes—
135Q8×D79—264no—
136C2×D316—24no—
137C2×D316C2414yes—
138C2×C22⋊D79—412no—
139C2×C22⋊D79C24124yes—
140C22×Dic79C2416yes—
141C22×Dic79C2416yes—
142C22×Dic79C2412yes—
143C2×C4×D79—224no—
144C2×C4×D79C24124yes—
145C4⋊Dic79C24124yes—
146C4⋊Dic79C24112yes—
147C4⋊Dic79C24112yes—
148C4⋊Dic79C24124yes—
149Dic79⋊C4C24124yes—
150Dic79⋊C4C24124yes—
151Dic79⋊C4C24124yes—
152Dic79⋊C4C24124yes—
153C22⋊Dic79C24148yes—
154C22⋊Dic79C24124yes—
155(C2×D79)⋊C4C2418yes—
156(C2×D79)⋊C4C24124yes—
157(C2×D79)⋊C4C2418yes—
158(C2×D79)⋊C4C24124yes—
159C4×Dic79C24124yes—
160C4×Dic79C24148yes—
161C24×C79C2413yes—
162#41×C79—232no—
163C2×Q8×C79#4118yes—
164C2×Q8×C79C2414yes—
165C2×D4×C79—212no—
166C2×D4×C79C24112yes—
167C22×C4×C79—212no—
168C22×C4×C79—212no—
169C22×C4×C79—46no—
170C22×C4×C79C2413yes—
171C22×C4×C79C2411yescharacteristic
172C22×C4×C79C2411yescharacteristic
173C22×C4×C79C2413yes—
174C22×C4×C79C24112yes—
175C22×C4×C79C24112yes—
176C22×C4×C79C24112yes—
177C22×C4×C79C2413yes—
178C22×C4×C79C2411yescharacteristic
179C4⋊C4×C79C24112yes—
180C4⋊C4×C79C24112yes—
181C22⋊C4×C79C24124yes—
182C42×C79C2414yes—
183C42×C79C24112yes—
184C42×C79C24112yes—
185C42×C79C24112yes—
186C23×C79—26no—
187C23×C79—26no—
188C23×C79—26no—
189C23×C79C2516yes—
190C23×C79C2511yescharacteristic
191C22×D79—24no—
192C22×D79—24no—
193C22×D79—24no—
194C22×D79C2514yes—
195Q8×C79C2×#41116yes—
196D4×C79—224no—
197C2×C4×C79—22no—
198C2×C4×C79—26no—
199C2×C4×C79—26no—
200C2×C4×C79—22no—
201C2×C4×C79—224no—
202C2×C4×C79—224no—
203C2×C4×C79—412no—
204C2×C4×C79C2×#41112yes—
205C2×C4×C792−1+414yes—
206C2×C4×C79C2516yes—
207C2×C4×C79C2512yes—
208C2×C4×C79C2512yes—
209C2×C4×C79C2516yes—
210C2×C4×C79C2516yes—
211C2×C4×C79C2512yes—
212C22⋊D79—424no—
213C2×Dic79—212no—
214C2×Dic79—24no—
215C2×Dic79—212no—
216C2×Dic79C2514yes—
217C2×Dic79C25112yes—
218C2×Dic79C25112yes—
219D316—28no—
220C4×D79—248no—
221Dic158—248no—
222Dic158—28no—
223Dic158—424no—
224C22×C79—26no—
225C22×C79—26no—
226C22×C79—26no—
227C22×C79—412no—
228C22×C79C22×#4112yes—
229C22×C79C2×2−1+412yes—
230C22×C79#58512yes—
231C22×C79C2611yescharacteristic
232C2×D79—24no—
233C2×D79—24no—
234C2×D79—24no—
235C2×D79—48no—
236C4×C79—22no—
237C4×C79—26no—
238C4×C79—22no—
239C4×C79—26no—
240C4×C79C22×#41112yes—
241C4×C79C2×2−1+414yes—
242Dic79—212no—
243Dic79—24no—
244Dic79—212no—
245C2×#3518—791nomaximal, Sylow
246C2×C79—46no—
247C2×C79#351814yescomplemented
248C2×C79C23×#4111yescharacteristic
249C2×C79C22×2−1+411yescharacteristic
250C2×C79C2×#58511yescharacteristic
251D79—44no—
252#3518—7932no—
253C2×Q82—791no—
254C2×#556—793no—
255C2×#556—796no—
256C2×#555—793no—
257C2×D4×Q8—791no—
258C2×#548—792no—
259C2×#542—796no—
260C2×#533—793no—
261C2×#529—793no—
262C2×#519—792no—
263C2×C4×#41D7911yescharacteristic, complemented
264C79C2×#351811yescharacteristic, complemented, Sylow
265C22×#41C2×D7911yescharacteristic
266C23×Q8—792no—
267Q82—7916no—
268#556—7948no—
269#556—7996no—
270#555—7948no—
271D4×Q8—7916no—
272#548—7932no—
273#542—7996no—
274#533—7948no—
275C2×C4⋊Q8—796no—
276C2×C4⋊Q8—793no—
277C2×C4⋊Q8—793no—
278#529—7948no—
279C2×#126—7912no—
280C2×#126—796no—
281C2×#125—793no—
282C2×#124—796no—
283C2×#124—7912no—
284C2×#124—791no—
285C2×#123—796no—
286C2×C22⋊Q8—793no—
287C2×C22⋊Q8—796no—
288C2×C22⋊Q8—796no—
289C2×C22⋊Q8—796no—
290C2×#121—796no—
291C2×#121—793no—
292#519—7932no—
293C4×#41C2×D79116yescomplemented
294C2×C4×Q8—792no—
295C2×C4×Q8—796no—
296C2×C4×Q8—796no—
297C2×C4×Q8—793no—
298C2×C4×Q8—796no—
299C2×C4×Q8C2×D7911yescharacteristic
300C2×C4×Q8C2×D7911yescharacteristic
301C2×C4×D4—796no—
302C2×C4×D4—793no—
303C2×C4×D4C2×D7913yes—
304C2×C4×D4C2×D7913yes—
305C2×#117—796no—
306C2×#117—796no—
307C2×#117—796no—
308C2×#117C2×D7913yes—
309C22×C42C2×D7913yes—
310C2×#41—28no—
311C2×#41C22×D7918yes—
312C22×Q8—7916no—
313C22×Q8—796no—
314C22×Q8—791no—
315C22×Q8—796no—
316C22×Q8—15816no—
317C22×Q8C22×D7911yescharacteristic
318C22×D4—796no—
319C22×D4—791no—
320C22×D4C22×D7913yes—
321C23×C4—796no—
322C23×C4C22×D7913yes—
323C23×C4C22×D7913yes—
324C23×C4C22×D7913yes—
325C4⋊Q8—7948no—
326C4⋊Q8—7924no—
327C4⋊Q8—7924no—
328#126—7996no—
329#126—7948no—
330#125—7924no—
331#124—7948no—
332#124—7996no—
333#124—798no—
334#123—7948no—
335C22⋊Q8—7924no—
336C22⋊Q8—7948no—
337C22⋊Q8—7948no—
338C22⋊Q8—7948no—
339#121—7948no—
340#121—7924no—
341C4×Q8—7916no—
342C4×Q8—7948no—
343C4×Q8—7948no—
344C4×Q8—7924no—
345C4×Q8—7948no—
346C4×Q8C22×D7918yes—
347C4×Q8C22×D7918yes—
348C4×D4—7948no—
349C4×D4—7924no—
350C4×D4C22×D79124yes—
351C4×D4C22×D79124yes—
352#117—7948no—
353#117—7948no—
354#117—7948no—
355#117C22×D79124yes—
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⋊C4—796no—
364C2×C4⋊C4C22×D7913yes—
365C2×C4⋊C4C22×D7913yes—
366C2×C22⋊C4—7912no—
367C2×C22⋊C4—796no—
368C2×C22⋊C4—796no—
369C2×C22⋊C4—792no—
370C2×C22⋊C4—792no—
371C2×C22⋊C4—796no—
372C2×C22⋊C4C22×D7916yes—
373C2×C42—7912no—
374C2×C42—796no—
375C2×C42C22×D7913yes—
376C2×C42C22×D7913yes—
377C2×C42C22×D7913yes—
378C2×C42C22×D79124yes—
379C2×C42C22×D7911yescharacteristic
380C24—792no—
381C24C23×D7913yes—
382#41—232no—
383C2×Q8—31612no—
384C2×Q8—7924no—
385C2×Q8—794no—
386C2×Q8—7924no—
387C2×Q8—15864no—
388C2×Q8—15824no—
389C2×Q8—1584no—
390C2×Q8#1192962418yes—
391C2×Q8C23×D7914yes—
392C2×D4—212no—
393C2×D4—31612no—
394C2×D4—7924no—
395C2×D4—794no—
396C2×D4—1584no—
397C2×D4C23×D79112yes—
398C22×C4—212no—
399C22×C4—212no—
400C22×C4—46no—
401C22×C4—792no—
402C22×C4—796no—
403C22×C4—796no—
404C22×C4—7924no—
405C22×C4—15824no—
406C22×C4C23×D79112yes—
407C22×C4C23×D79112yes—
408C22×C4C23×D7913yes—
409C22×C4C23×D7911yescharacteristic
410C22×C4C23×D7911yescharacteristic
411C22×C4C23×D7913yes—
412C22×C4C23×D79112yes—
413C22×C4C23×D7913yes—
414C22×C4C23×D7911yescharacteristic
415C4⋊C4—7924no—
416C4⋊C4—7924no—
417C4⋊C4—7924no—
418C4⋊C4—7924no—
419C4⋊C4—7912no—
420C4⋊C4—7912no—
421C4⋊C4—7924no—
422C4⋊C4—7924no—
423C4⋊C4C23×D79112yes—
424C4⋊C4C23×D79112yes—
425C22⋊C4—7948no—
426C22⋊C4—7924no—
427C22⋊C4—7924no—
428C22⋊C4—798no—
429C22⋊C4—798no—
430C22⋊C4—7924no—
431C22⋊C4C23×D79124yes—
432C42—7948no—
433C42—7924no—
434C42C23×D7914yes—
435C42C23×D79112yes—
436C42C23×D79112yes—
437C42C23×D79112yes—
438C23—26no—
439C23—26no—
440C23—26no—
441C23—794no—
442C23—1584no—
443C23—1584no—
444C23—1584no—
445C23C24×D7916yes—
446C23C24×D7911yescharacteristic
447Q8—31624no—
448Q8—1588no—
449Q8—15848no—
450Q8C2×#11929624116yes—
451D4—224no—
452D4—31624no—
453D4—1588no—
454C2×C4—26no—
455C2×C4—22no—
456C2×C4—26no—
457C2×C4—22no—
458C2×C4—224no—
459C2×C4—224no—
460C2×C4—412no—
461C2×C4—794no—
462C2×C4—7912no—
463C2×C4—7912no—
464C2×C4—1584no—
465C2×C4—15812no—
466C2×C4—15812no—
467C2×C4—15848no—
468C2×C4C2×#11929624112yes—
469C2×C4#1457920814yes—
470C2×C4C24×D7916yes—
471C2×C4C24×D7912yes—
472C2×C4C24×D7912yes—
473C2×C4C24×D7916yes—
474C2×C4C24×D7912yes—
475C2×C4C24×D7916yes—
476C22—26no—
477C22—26no—
478C22—26no—
479C22—412no—
480C22—3168no—
481C22—1584no—
482C22—1584no—
483C22—1584no—
484C22C22×#1192962412yes—
485C22C2×#1457920812yes—
486C22#1909939912yes—
487C22C25×D7911yescharacteristic
488C4—22no—
489C4—26no—
490C4—22no—
491C4—26no—
492C4—15812no—
493C4—1584no—
494C4—15812no—
495C4C22×#11929624112yes—
496C4C2×#1457920814yes—
497C2—46no—
498C2—3164no—
499C2#2377741714yesdirect product factor
500C2C23×#1192962411yescharacteristic
501C2C22×#1457920811yescharacteristic
502C2C2×#1909939911yescharacteristic
subgroupG/H
centerC23C24×D79
commutator, cosocleC22×C79C26
FrattiniC22C25×D79
FittingC2×C4×#41×C79C2
socleC23×C79C25
hypercenterC2×C4×#41D79
hypocenterC79C2×#3518
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #23777417C2direct (N × H)
2. C2×Q8×Dic158C2semidirect
3. C2×#19099283C2semidirect
4. C2×#19099281C2semidirect
5. C2×#19099278C2semidirect
6. C2×Q8×D316C2semidirect
7. C2×#19099220C2semidirect
8. C2×#19099178C2semidirect
9. C2×#19099120C2semidirect
10. C2×#19099097C2semidirect
11. C2×#19099039C2semidirect
12. C2×C4×#41×C79C2semidirect
14. Q8×Dic158C22semidirect
15. #19099283C22semidirect
16. #19099281C22semidirect
17. #19099278C22semidirect
18. Q8×D316C22semidirect
19. #19099220C22semidirect
20. #19099178C22semidirect
21. #19099120C22semidirect
25. #19099097C22semidirect
39. #19099039C22semidirect
51. C4×#41×C79C22semidirect
247. C2×C79#3518semidirect
263. C2×C4×#41D79semidirect
264. C79C2×#3518semidirect
293. C4×#41C2×D79semidirect
499. C2#23777417direct (N × H)

Maximal subgroups

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

subgroupindex
#237774172
C2×Q8×Dic1582
C2×#190992832
C2×#190992812
C2×#190992782
C2×Q8×D3162
C2×#190992202
C2×#190991782
C2×#190991202
C2×#190990972
C2×#190990392
C2×C4×#41×C792
C2×#351879