Finite Group Databasebeta

← C2×#23777022 #35195367

Subgroups — C2×#23777022 #35195367

Order 20224.

7706 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23777022C2132yesmaximal, direct product factor
2C2×#19099292C211yesmaximal, characteristic, complemented
3C2×#19099283C213yesmaximal, complemented
4C2×#19099252C216yesmaximal, complemented
5C2×Q8×D316C211yesmaximal, characteristic, complemented
6C2×#19099220C212yesmaximal, complemented
7C2×#19099218C213yesmaximal, complemented
8C2×#19099120C213yesmaximal, complemented
9C2×#19099095C216yesmaximal, complemented
10C2×C4×#11929624C213yesmaximal, complemented
11C2×C4×Q8×D79C212yesmaximal, complemented
12C2×C4×#41×C79C211yesmaximal, characteristic, complemented
13C22×#11929624C2213yes—
14C22×Q8×D79C2212yes—
15#19099292C22116yescomplemented
16#19099283C22148yescomplemented
17#19099252C22196yescomplemented
18Q8×D316C22116yescomplemented
19#19099220C22132yescomplemented
20#19099218C22148yescomplemented
21#19099120C22148yescomplemented
22C2×#14579161C2213yes—
23C2×#14579161C2213yes—
24#19099095C22196yescomplemented
25C2×#14579154C22112yes—
26C2×#14579153C2216yes—
27C2×#14579152C2216yes—
28C2×#14579148C2216yes—
29C2×#14579143C22112yes—
30C2×#14579141C2216yes—
31C2×#14579140C2216yes—
32C2×#14579140C2216yes—
33C2×#14579132C2216yes—
34C4×#11929624C22148yescomplemented
35C2×Q8×Dic79C2212yes—
36C2×C4×Dic158C2216yes—
37C2×C4×Dic158C2213yes—
38C2×C4×Dic158C2213yes—
39C2×#14579126C2216yes—
40C4×Q8×D79C22132yescomplemented
41C2×C4×D316C2211yescharacteristic
42C2×C4×D316C2213yes—
43C2×C4×C22⋊D79C2216yes—
44C2×C4×C22⋊D79C2216yes—
45C2×#14579120C2213yes—
46C2×#14579120C2216yes—
47C2×#14579118C2216yes—
48C2×C4⋊C4×D79C2216yes—
49C2×C42×D79C2216yes—
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—
58C2×#11929624—224no—
59C2×#11929624C23124yes—
60C22×Dic158C2316yes—
61C22×Dic158C2313yes—
62C2×Q8×D79—216no—
63C2×Q8×D79C23116yes—
64C22×D316C2311yescharacteristic
65C22×C22⋊D79C2316yes—
66C22×C4×D79C2312yes—
67C22×C4×D79C2316yes—
68C22×C4×D79C2316yes—
69#14579161C23124yes—
70#14579161C23124yes—
71#14579154C23196yes—
72#14579153C23148yes—
73#14579152C23148yes—
74#14579148C23148yes—
75#14579143C23196yes—
76#14579141C23148yes—
77#14579140C23148yes—
78#14579140C23148yes—
79#14579132C23148yes—
80Q8×Dic79C23116yes—
81C4×Dic158C23148yes—
82C4×Dic158C23124yes—
83C4×Dic158C23124yes—
84#14579126C23148yes—
85C4×D316C23124yes—
86C4×D316C2318yes—
87C4×C22⋊D79C23148yes—
88C4×C22⋊D79C23148yes—
89#14579120C23124yes—
90#14579120C23148yes—
91#14579118C23148yes—
92C2×C4⋊Dic79C2316yes—
93C2×C4⋊Dic79C2311yescharacteristic
94C2×C4⋊Dic79C2313yes—
95C2×C4⋊Dic79C2316yes—
96C2×C4⋊Dic79C2316yes—
97C2×Dic79⋊C4C23112yes—
98C2×Dic79⋊C4C2316yes—
99C2×Dic79⋊C4C2316yes—
100C4⋊C4×D79C23148yes—
101C2×C22⋊Dic79C2316yes—
102C2×C22⋊Dic79C23112yes—
103C2×(C2×D79)⋊C4C2312yes—
104C2×(C2×D79)⋊C4C2316yes—
105C2×C4×Dic79C2316yes—
106C2×C4×Dic79C2316yes—
107C2×C4×Dic79C2316yes—
108C42×D79C23148yes—
109C2×#41×C79—28no—
110C2×#41×C79C2318yes—
111C22×Q8×C79C2311yescharacteristic
112C22×D4×C79C2313yes—
113C23×C4×C79C2313yes—
114C23×C4×C79C2313yes—
115C23×C4×C79C2313yes—
116C4×Q8×C79C2318yes—
117C4×Q8×C79C2318yes—
118C4×D4×C79C23124yes—
119C4×D4×C79C23124yes—
120#117×C79C23124yes—
121C2×C4⋊C4×C79C2313yes—
122C2×C4⋊C4×C79C2313yes—
123C2×C22⋊C4×C79C2316yes—
124C2×C42×C79C2311yescharacteristic
125C2×C42×C79C2313yes—
126C2×C42×C79C2313yes—
127C2×C42×C79C2313yes—
128C2×C42×C79C23124yes—
129C23×D79C2412yes—
130#11929624—296no—
131C2×Dic158—224no—
132C2×Dic158—212no—
133C2×Dic158C24124yes—
134C2×Dic158C24112yes—
135Q8×D79—264no—
136C2×D316#4118yescomplemented
137C2×D316C2414yes—
138C2×C22⋊D79—224no—
139C2×C22⋊D79C24124yes—
140C22×Dic79C2416yes—
141C22×Dic79C2416yes—
142C22×Dic79C2412yes—
143C2×C4×D79—224no—
144C2×C4×D79—224no—
145C2×C4×D79—28no—
146C2×C4×D79C24124yes—
147C2×C4×D79C24124yes—
148C2×C4×D79C2418yes—
149C4⋊Dic79C24124yes—
150C4⋊Dic79C2414yes—
151C4⋊Dic79C24112yes—
152C4⋊Dic79C24124yes—
153C4⋊Dic79C24124yes—
154Dic79⋊C4C24148yes—
155Dic79⋊C4C24124yes—
156Dic79⋊C4C24124yes—
157C22⋊Dic79C24124yes—
158C22⋊Dic79C24148yes—
159(C2×D79)⋊C4C2418yes—
160(C2×D79)⋊C4C24124yes—
161C4×Dic79C24124yes—
162C4×Dic79C24124yes—
163C4×Dic79C24124yes—
164C24×C79C2413yes—
165#41×C79—232no—
166C2×Q8×C79#4118yescomplemented
167C2×Q8×C79C2414yes—
168C2×D4×C79—212no—
169C2×D4×C79C24112yes—
170C22×C4×C79—212no—
171C22×C4×C79—212no—
172C22×C4×C79—212no—
173C22×C4×C79C24112yes—
174C22×C4×C79C2411yescharacteristic
175C22×C4×C79C2413yes—
176C22×C4×C79C2413yes—
177C22×C4×C79C24112yes—
178C22×C4×C79C24112yes—
179C22×C4×C79C2411yescharacteristic
180C22×C4×C79C2413yes—
181C22×C4×C79C2411yescharacteristic
182C4⋊C4×C79C24112yes—
183C4⋊C4×C79C24112yes—
184C22⋊C4×C79C24124yes—
185C42×C79C2414yes—
186C42×C79C24112yes—
187C42×C79C24112yes—
188C42×C79C24112yes—
189C23×C79—26no—
190C23×C79—26no—
191C23×C79—26no—
192C23×C79C2516yes—
193C23×C79C2511yescharacteristic
194C22×D79—24no—
195C22×D79—24no—
196C22×D79C2×#4118yes—
197C22×D79C2514yes—
198Q8×C79C2×#41116yescomplemented
199D4×C79—224no—
200C2×C4×C79—26no—
201C2×C4×C79—22no—
202C2×C4×C79—26no—
203C2×C4×C79—224no—
204C2×C4×C79—224no—
205C2×C4×C79—224no—
206C2×C4×C79C2×#4114yes—
207C2×C4×C79C2×#41112yes—
208C2×C4×C792−1+414yes—
209C2×C4×C79C2516yes—
210C2×C4×C79C2512yes—
211C2×C4×C79C2516yes—
212C2×C4×C79C2516yes—
213C2×C4×C79C2512yes—
214C2×C4×C79C2512yes—
215C22⋊D79—248no—
216C2×Dic79—212no—
217C2×Dic79—212no—
218C2×Dic79—24no—
219C2×Dic79C2514yes—
220C2×Dic79C25112yes—
221C2×Dic79C25112yes—
222D316C2×#41116yescomplemented
223C4×D79—248no—
224C4×D79—248no—
225C4×D79—216no—
226Dic158—248no—
227Dic158—224no—
228C22×C79—26no—
229C22×C79—26no—
230C22×C79—26no—
231C22×C79—412no—
232C22×C79C22×#4112yes—
233C22×C79C22×#4112yes—
234C22×C79C2×2−1+412yes—
235C22×C79C2611yescharacteristic
236C2×D79—24no—
237C2×D79—24no—
238C2×D79—216no—
239C2×D79C22×#4118yes—
240C4×C79—22no—
241C4×C79—26no—
242C4×C79—26no—
243C4×C79C22×#4114yes—
244C4×C79C22×#41112yes—
245C4×C79C2×2−1+414yes—
246Dic79—212no—
247Dic79—212no—
248Dic79—24no—
249C2×#3482—791nomaximal, Sylow
250C2×C79—46no—
251C2×C79#348214yescomplemented
252C2×C79C23×#4111yescharacteristic
253C2×C79C23×#4111yescharacteristic
254C2×C79C22×2−1+411yescharacteristic
255D79—28no—
256#3482—7932no—
257C2×#557—791no—
258C2×#556—793no—
259C2×#552—796no—
260C2×D4×Q8—791no—
261C2×#548—792no—
262C2×#547—793no—
263C2×#533—793no—
264C2×#529—796no—
265C2×C4×#41—793no—
266C2×C4×#41D7911yescharacteristic, complemented
267C22×C4×Q8—792no—
268C79C2×#348211yescharacteristic, complemented, Sylow
269C22×#41—793no—
270C22×#41C2×D7911yescharacteristic
271C23×Q8—792no—
272#557—7916no—
273#556—7948no—
274#552—7996no—
275D4×Q8—7916no—
276#548—7932no—
277#547—7948no—
278#533—7948no—
279C2×C4⋊Q8—793no—
280C2×C4⋊Q8—793no—
281#529—7996no—
282C2×#126—7912no—
283C2×#125—796no—
284C2×#125—796no—
285C2×#124—796no—
286C2×#123—7912no—
287C2×C22⋊Q8—796no—
288C2×C22⋊Q8—796no—
289C2×C22⋊Q8—796no—
290C2×#121—796no—
291C4×#41—7948no—
292C4×#41C2×D79116yescomplemented
293C2×C4×Q8—792no—
294C2×C4×Q8—7932no—
295C2×C4×Q8—796no—
296C2×C4×Q8—796no—
297C2×C4×Q8—793no—
298C2×C4×Q8—793no—
299C2×C4×Q8C2×D7911yescharacteristic
300C2×C4×Q8C2×D7911yescharacteristic
301C2×C4×D4—796no—
302C2×C4×D4—796no—
303C2×C4×D4—791no—
304C2×C4×D4—793no—
305C2×C4×D4C2×D7913yes—
306C2×C4×D4C2×D7913yes—
307C2×#117—793no—
308C2×#117—796no—
309C2×#117—796no—
310C2×#117C2×D7913yes—
311C22×C4⋊C4—796no—
312C22×C42—796no—
313C22×C42C2×D7913yes—
314C2×#41—28no—
315C2×#41—7924no—
316C2×#41—15824no—
317C2×#41C22×D7918yes—
318C22×Q8—7916no—
319C22×Q8—796no—
320C22×Q8—793no—
321C22×Q8—15816no—
322C22×Q8C22×D7911yescharacteristic
323C22×D4—796no—
324C22×D4—791no—
325C22×D4C22×D7913yes—
326C23×C4—796no—
327C23×C4—796no—
328C23×C4—792no—
329C23×C4C22×D7913yes—
330C23×C4C22×D7913yes—
331C23×C4C22×D7913yes—
332C4⋊Q8—7924no—
333C4⋊Q8—7924no—
334#126—7996no—
335#125—7948no—
336#125—7948no—
337#124—7948no—
338#123—7996no—
339C22⋊Q8—7948no—
340C22⋊Q8—7948no—
341C22⋊Q8—7948no—
342#121—7948no—
343C4×Q8—7916no—
344C4×Q8—7948no—
345C4×Q8—7948no—
346C4×Q8—7924no—
347C4×Q8—7924no—
348C4×Q8C22×D7918yes—
349C4×Q8C22×D7918yes—
350C4×D4—7948no—
351C4×D4—7948no—
352C4×D4—798no—
353C4×D4—7924no—
354C4×D4C22×D79124yes—
355C4×D4C22×D79124yes—
356#117—7924no—
357#117—7948no—
358#117—7948no—
359#117C22×D79124yes—
360C2×C4⋊C4—796no—
361C2×C4⋊C4—7912no—
362C2×C4⋊C4—791no—
363C2×C4⋊C4—796no—
364C2×C4⋊C4—793no—
365C2×C4⋊C4—796no—
366C2×C4⋊C4—796no—
367C2×C4⋊C4—796no—
368C2×C4⋊C4—7948no—
369C2×C4⋊C4C22×D7913yes—
370C2×C4⋊C4C22×D7913yes—
371C2×C22⋊C4—796no—
372C2×C22⋊C4—7912no—
373C2×C22⋊C4—792no—
374C2×C22⋊C4—796no—
375C2×C22⋊C4C22×D7916yes—
376C2×C42—796no—
377C2×C42—796no—
378C2×C42—796no—
379C2×C42—7948no—
380C2×C42C22×D7913yes—
381C2×C42C22×D7913yes—
382C2×C42C22×D79124yes—
383C2×C42C22×D7911yescharacteristic
384C2×C42C22×D7913yes—
385C24—792no—
386C24C23×D7913yes—
387#41—232no—
388#41—15896no—
389C2×Q8—7924no—
390C2×Q8—7912no—
391C2×Q8—15864no—
392C2×Q8—15812no—
393C2×Q8—15824no—
394C2×Q8#1192962418yescomplemented
395C2×Q8C23×D7914yes—
396C2×D4—212no—
397C2×D4—7924no—
398C2×D4—798no—
399C2×D4—794no—
400C2×D4—15824no—
401C2×D4C23×D79112yes—
402C22×C4—212no—
403C22×C4—212no—
404C22×C4—212no—
405C22×C4—792no—
406C22×C4—796no—
407C22×C4—796no—
408C22×C4—798no—
409C22×C4—7924no—
410C22×C4—7924no—
411C22×C4—15824no—
412C22×C4—15824no—
413C22×C4—1588no—
414C22×C4C23×D7913yes—
415C22×C4C23×D7911yescharacteristic
416C22×C4C23×D7913yes—
417C22×C4C23×D79112yes—
418C22×C4C23×D79112yes—
419C22×C4C23×D79112yes—
420C22×C4C23×D7911yescharacteristic
421C22×C4C23×D7913yes—
422C22×C4C23×D7911yescharacteristic
423C4⋊C4—7948no—
424C4⋊C4—7924no—
425C4⋊C4—7924no—
426C4⋊C4—7924no—
427C4⋊C4—794no—
428C4⋊C4—7912no—
429C4⋊C4—7924no—
430C4⋊C4—7924no—
431C4⋊C4C23×D79112yes—
432C4⋊C4C23×D79112yes—
433C22⋊C4—7924no—
434C22⋊C4—7948no—
435C22⋊C4—798no—
436C22⋊C4—7924no—
437C22⋊C4C23×D79124yes—
438C42—7924no—
439C42—7924no—
440C42—7924no—
441C42C23×D79112yes—
442C42C23×D7914yes—
443C42C23×D79112yes—
444C42C23×D79112yes—
445C23—26no—
446C23—26no—
447C23—26no—
448C23—794no—
449C23—798no—
450C23—1584no—
451C23—1584no—
452C23C24×D7916yes—
453C23C24×D7911yescharacteristic
454Q8—15824no—
455Q8—15848no—
456Q8C2×#11929624116yescomplemented
457D4—224no—
458D4—7916no—
459D4—15848no—
460C2×C4—26no—
461C2×C4—26no—
462C2×C4—22no—
463C2×C4—224no—
464C2×C4—224no—
465C2×C4—224no—
466C2×C4—794no—
467C2×C4—7912no—
468C2×C4—7912no—
469C2×C4—15848no—
470C2×C4—1584no—
471C2×C4—15812no—
472C2×C4—15812no—
473C2×C4—15816no—
474C2×C4—15848no—
475C2×C4#41×D7914yes—
476C2×C4C2×#11929624112yes—
477C2×C4#1457920814yes—
478C2×C4C24×D7916yes—
479C2×C4C24×D7912yes—
480C2×C4C24×D7916yes—
481C2×C4C24×D7912yes—
482C2×C4C24×D7912yes—
483C2×C4C24×D7916yes—
484C22—26no—
485C22—26no—
486C22—26no—
487C22—412no—
488C22—798no—
489C22—1584no—
490C22—1584no—
491C22—15816no—
492C22C2×#41×D7912yes—
493C22C22×#1192962412yes—
494C22C2×#1457920812yes—
495C22C25×D7911yescharacteristic
496C4—26no—
497C4—22no—
498C4—26no—
499C4—15812no—
500C4—15812no—
501C4—1584no—
502C4C2×#41×D7914yes—
503C4C22×#11929624112yes—
504C4C2×#1457920814yes—
505C2—46no—
506C2—1588no—
507C2#2377702214yesdirect product factor
508C2C22×#41×D7911yescharacteristic
509C2C23×#1192962411yescharacteristic
510C2C22×#1457920811yescharacteristic
subgroupG/H
centerC22×C4C23×D79
commutator, cosocleC22×C79C26
FrattiniC22C25×D79
FittingC2×C4×#41×C79C2
socleC23×C79C25
hypercenterC2×C4×#41D79
hypocenterC79C2×#3482
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #23777022C2direct (N × H)
2. C2×#19099292C2semidirect
3. C2×#19099283C2semidirect
4. C2×#19099252C2semidirect
5. C2×Q8×D316C2semidirect
6. C2×#19099220C2semidirect
7. C2×#19099218C2semidirect
8. C2×#19099120C2semidirect
9. C2×#19099095C2semidirect
10. C2×C4×#11929624C2semidirect
11. C2×C4×Q8×D79C2semidirect
12. C2×C4×#41×C79C2semidirect
15. #19099292C22semidirect
16. #19099283C22semidirect
17. #19099252C22semidirect
18. Q8×D316C22semidirect
19. #19099220C22semidirect
20. #19099218C22semidirect
21. #19099120C22semidirect
24. #19099095C22semidirect
34. C4×#11929624C22semidirect
40. C4×Q8×D79C22semidirect
51. C4×#41×C79C22semidirect
136. C2×D316#41semidirect
166. C2×Q8×C79#41semidirect
198. Q8×C79C2×#41semidirect
222. D316C2×#41semidirect
251. C2×C79#3482semidirect
266. C2×C4×#41D79semidirect
268. C79C2×#3482semidirect
292. C4×#41C2×D79semidirect
394. C2×Q8#11929624semidirect
456. Q8C2×#11929624semidirect
507. C2#23777022direct (N × H)

Maximal subgroups

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

subgroupindex
#237770222
C2×#190992922
C2×#190992832
C2×#190992522
C2×Q8×D3162
C2×#190992202
C2×#190992182
C2×#190991202
C2×#190990952
C2×C4×#119296242
C2×C4×Q8×D792
C2×C4×#41×C792
C2×#348279