Finite Group Databasebeta

← C2×#23777401 #35196523

Subgroups — C2×#23777401 #35196523

Order 20224.

7854 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23777401C2132yesmaximal, direct product factor
2C2×#19099292C211yesmaximal, characteristic, complemented
3C2×#19099265C216yesmaximal, complemented
4C2×#19099233C213yesmaximal, complemented
5C2×#19099232C212yesmaximal, complemented
6C2×#19099217C213yesmaximal, complemented
7C2×#19099166C216yesmaximal, complemented
8C2×#19099111C213yesmaximal, complemented
9C2×#19099095C213yesmaximal, complemented
10C2×#19099046C212yesmaximal, complemented
11C2×C4×#11929625C211yesmaximal, characteristic, complemented
12C2×#550×C79C211yesmaximal, characteristic, complemented
13C22×#11929625C2211yescharacteristic
14C22×#11929623C2216yes—
15#19099292C22116yescomplemented
16#19099265C22196yescomplemented
17#19099233C22148yescomplemented
18#19099232C22132yescomplemented
19#19099217C22148yescomplemented
20#19099166C22196yescomplemented
21#19099111C22148yescomplemented
22C2×#14579161C2213yes—
23C2×#14579157C2216yes—
24#19099095C22148yescomplemented
25C2×#14579154C2216yes—
26C2×#14579152C2211yescharacteristic
27C2×#14579152C2216yes—
28C2×#14579148C2213yes—
29C2×#14579145C22112yes—
30C2×#14579143C2216yes—
31C2×#14579141C2213yes—
32C2×#14579135C2216yes—
33C2×#14579133C22112yes—
34C2×#14579132C2216yes—
35C2×#14579132C2213yes—
36C2×#14579132C2216yes—
37#19099046C22132yescomplemented
38C4×#11929625C22116yescomplemented
39C2×Q8×Dic79C2211yescharacteristic
40C2×Q8×Dic79C2212yes—
41C2×C4×Dic158C2213yes—
42C2×C4×D316C2213yes—
43C2×D4×Dic79C2216yes—
44C2×D4×Dic79C2216yes—
45C2×D4×Dic79C2216yes—
46C2×#14579123C2213yes—
47C2×C4×C22⋊D79C2216yes—
48C2×#14579120C2216yes—
49C2×#14579119C2213yes—
50C22×C4⋊Dic79C2216yes—
51C2×C42×D79C2213yes—
52C22×#41×C79C2212yes—
53#550×C79C22116yescomplemented
54C2×#127×C79C2213yes—
55C2×#121×C79C2216yes—
56C2×C4×Q8×C79C2211yescharacteristic
57C2×C4×D4×C79C2213yes—
58C2×#11929625—28no—
59C2×#11929625C2318yes—
60C2×#11929623—424no—
61C2×#11929623C23148yes—
62C22×Dic158C2313yes—
63C22×D316C2313yes—
64C22×C22⋊D79C2316yes—
65C22×C22⋊D79C23112yes—
66C23×Dic79C23112yes—
67C22×C4×D79C2313yes—
68C22×C4×D79C2313yes—
69C22×C4×D79C2313yes—
70#14579161C23124yes—
71#14579157C23148yes—
72#14579154C23148yes—
73#14579152C23148yes—
74#14579152C2318yes—
75#14579148C23124yes—
76#14579145C23196yes—
77#14579143C23148yes—
78#14579141C23124yes—
79#14579135C23148yes—
80#14579133C23196yes—
81#14579132C23148yes—
82#14579132C23148yes—
83#14579132C23124yes—
84Q8×Dic79C23116yes—
85Q8×Dic79C2318yes—
86C4×Dic158C23124yes—
87C4×D316C23124yes—
88D4×Dic79C23148yes—
89D4×Dic79C23148yes—
90D4×Dic79C23148yes—
91#14579123C23124yes—
92C4×C22⋊D79C23148yes—
93#14579120C23148yes—
94#14579119C23124yes—
95C2×C4⋊Dic79C23148yes—
96C2×C4⋊Dic79C2313yes—
97C2×C4⋊Dic79C2316yes—
98C2×C4⋊Dic79C2316yes—
99C2×C4⋊Dic79C2312yes—
100C2×C4⋊Dic79C2316yes—
101C2×C4⋊Dic79C2311yescharacteristic
102C2×Dic79⋊C4C2312yes—
103C2×Dic79⋊C4C2316yes—
104C2×C22⋊Dic79C2316yes—
105C2×C22⋊Dic79C2316yes—
106C2×C22⋊Dic79C2316yes—
107C2×(C2×D79)⋊C4C2316yes—
108C2×(C2×D79)⋊C4C2316yes—
109C2×C4×Dic79C2311yescharacteristic
110C2×C4×Dic79C2313yes—
111C2×C4×Dic79C2313yes—
112C2×C4×Dic79C2316yes—
113C42×D79C23124yes—
114C2×#41×C79—216no—
115C2×#41×C79C23116yes—
116C22×Q8×C79C2311yescharacteristic
117C22×D4×C79C2313yes—
118C22×D4×C79C2316yes—
119C22×D4×C79C2316yes—
120C23×C4×C79C2316yes—
121#127×C79C23124yes—
122#121×C79C23148yes—
123C4×Q8×C79C2318yes—
124C4×D4×C79C23124yes—
125C2×C4⋊C4×C79C2311yescharacteristic
126C2×C4⋊C4×C79C2313yes—
127C2×C22⋊C4×C79C2316yes—
128C2×C42×C79C2313yes—
129C23×D79C2413yes—
130#11929625—232no—
131#11929623—496no—
132C2×Dic158—212no—
133C2×Dic158C24112yes—
134C2×D316—212no—
135C2×D316C24112yes—
136C2×C22⋊D79—424no—
137C2×C22⋊D79—412no—
138C2×C22⋊D79C24148yes—
139C2×C22⋊D79C24124yes—
140C22×Dic79—424no—
141C22×Dic79C2416yes—
142C22×Dic79C2411yescharacteristic
143C22×Dic79C2411yescharacteristic
144C22×Dic79C2412yes—
145C22×Dic79C2413yes—
146C22×Dic79C24148yes—
147C2×C4×D79—212no—
148C2×C4×D79—212no—
149C2×C4×D79—46no—
150C2×C4×D79C24112yes—
151C2×C4×D79C24112yes—
152C2×C4×D79C24112yes—
153C4⋊Dic79C24112yes—
154C4⋊Dic79C24124yes—
155C4⋊Dic79C24124yes—
156C4⋊Dic79C2418yes—
157C4⋊Dic79C24124yes—
158C4⋊Dic79C2414yes—
159Dic79⋊C4C2418yes—
160Dic79⋊C4C24124yes—
161C22⋊Dic79C24124yes—
162C22⋊Dic79C24124yes—
163C22⋊Dic79C24124yes—
164(C2×D79)⋊C4C24124yes—
165(C2×D79)⋊C4C24124yes—
166C4×Dic79C2414yes—
167C4×Dic79C24112yes—
168C4×Dic79C24112yes—
169C4×Dic79C24124yes—
170C24×C79C2416yes—
171#41×C79—264no—
172C2×Q8×C79#4118yescomplemented
173C2×Q8×C79C2414yes—
174C2×D4×C79—224no—
175C2×D4×C79—212no—
176C2×D4×C79—412no—
177C2×D4×C79C24112yes—
178C2×D4×C79C24124yes—
179C2×D4×C79C24124yes—
180C22×C4×C79—224no—
181C22×C4×C79C2413yes—
182C22×C4×C79C2412yes—
183C22×C4×C79C24124yes—
184C22×C4×C79C2413yes—
185C22×C4×C79C2411yescharacteristic
186C4⋊C4×C79C2414yes—
187C4⋊C4×C79C24112yes—
188C22⋊C4×C79C24124yes—
189C42×C79C24112yes—
190C23×C79—212no—
191C23×C79—212no—
192C23×C79—212no—
193C23×C79C25112yes—
194C23×C79C2511yescharacteristic
195C22×D79—26no—
196C22×D79—26no—
197C22×D79—26no—
198C22×D79C2516yes—
199Q8×C79C2×#41116yescomplemented
200D4×C79—248no—
201D4×C79—224no—
202D4×C79—424no—
203C2×C4×C79—24no—
204C2×C4×C79—26no—
205C2×C4×C79—248no—
206C2×C4×C79C2×#41112yes—
207C2×C4×C79H(5,𝔽2)14yes—
208C2×C4×C79C2516yes—
209C2×C4×C79C2514yes—
210C2×C4×C79C2516yes—
211C2×C4×C79C2512yes—
212C22⋊D79—424no—
213C22⋊D79—448no—
214C2×Dic79—22no—
215C2×Dic79—26no—
216C2×Dic79—212no—
217C2×Dic79—22no—
218C2×Dic79—24no—
219C2×Dic79—448no—
220C2×Dic79C2512yes—
221C2×Dic79C2512yes—
222C2×Dic79C2514yes—
223C2×Dic79C2516yes—
224C2×Dic79C25112yes—
225D316—224no—
226C4×D79—224no—
227C4×D79—224no—
228C4×D79—412no—
229Dic158—224no—
230C22×C79—212no—
231C22×C79—212no—
232C22×C79—212no—
233C22×C79—424no—
234C22×C79C22×#4112yes—
235C22×C79C2×H(5,𝔽2)12yes—
236C22×C79#58512yes—
237C22×C79C2611yescharacteristic
238C2×D79—26no—
239C2×D79—26no—
240C2×D79—26no—
241C2×D79—412no—
242C4×C79—24no—
243C4×C79—26no—
244C4×C79C22×#41112yes—
245C4×C79C2×H(5,𝔽2)14yes—
246Dic79—22no—
247Dic79—212no—
248Dic79—26no—
249Dic79—24no—
250Dic79—22no—
251C2×#3517—791nomaximal, Sylow
252C2×C79—412no—
253C2×C79#351714yescomplemented
254C2×C79C23×#4111yescharacteristic
255C2×C79C22×H(5,𝔽2)11yescharacteristic
256C2×C79C2×#58511yescharacteristic
257D79—46no—
258#3517—7932no—
259C2×#557—791no—
260C2×#553—796no—
261C2×#550—792no—
262C2×#550—793no—
263C2×#550D7911yescharacteristic, complemented
264C2×#547—793no—
265C2×#540—796no—
266C2×#532—793no—
267C2×#529—793no—
268C2×#520—792no—
269C2×C4×#41—791no—
270C79C2×#351711yescharacteristic, complemented, Sylow
271C22×#41—791no—
272C22×#41—796no—
273C22×#41C2×D7912yes—
274#557—7916no—
275#553—7996no—
276#550—7932no—
277#550—7948no—
278#550C2×D79116yescomplemented
279#547—7948no—
280#540—7996no—
281#532—7948no—
282C2×C4⋊Q8—793no—
283C2×#127—796no—
284C2×#127C2×D7913yes—
285#529—7948no—
286C2×#126—796no—
287C2×#125—791no—
288C2×#125—796no—
289C2×#124—793no—
290C2×#123—796no—
291C2×#123—7912no—
292C2×C22⋊Q8—793no—
293C2×#121—793no—
294C2×#121—796no—
295C2×#121—796no—
296C2×#121—7912no—
297C2×#121—796no—
298C2×#121C2×D7916yes—
299#520—7932no—
300C4×#41—7916no—
301C2×C4×Q8—791no—
302C2×C4×Q8—792no—
303C2×C4×Q8—793no—
304C2×C4×Q8C2×D7911yescharacteristic
305C2×C4×D4—796no—
306C2×C4×D4—796no—
307C2×C4×D4—796no—
308C2×C4×D4—796no—
309C2×C4×D4—793no—
310C2×C4×D4—793no—
311C2×C4×D4C2×D7913yes—
312C2×#117—796no—
313C2×#117—793no—
314C22×C4⋊C4—796no—
315C22×C42—793no—
316C2×#41—216no—
317C2×#41—31624no—
318C2×#41—798no—
319C2×#41—7948no—
320C2×#41—1588no—
321C2×#41C22×D79116yes—
322C22×Q8—793no—
323C22×Q8C22×D7911yescharacteristic
324C22×D4—796no—
325C22×D4—7912no—
326C22×D4—793no—
327C22×D4C22×D7916yes—
328C22×D4C22×D7913yes—
329C22×D4C22×D7916yes—
330C23×C4—7912no—
331C23×C4—793no—
332C23×C4—793no—
333C23×C4—793no—
334C23×C4C22×D7916yes—
335C4⋊Q8—7924no—
336#127—7948no—
337#127C22×D79124yes—
338#126—7948no—
339#125—798no—
340#125—7948no—
341#124—7924no—
342#123—7948no—
343#123—7996no—
344C22⋊Q8—7924no—
345#121—7924no—
346#121—7948no—
347#121—7948no—
348#121—7996no—
349#121—7948no—
350#121C22×D79148yes—
351C4×Q8—798no—
352C4×Q8—7916no—
353C4×Q8—7924no—
354C4×Q8C22×D7918yes—
355C4×D4—7948no—
356C4×D4—7948no—
357C4×D4—7948no—
358C4×D4—7948no—
359C4×D4—7924no—
360C4×D4—7924no—
361C4×D4C22×D79124yes—
362#117—7948no—
363#117—7924no—
364C2×C4⋊C4—793no—
365C2×C4⋊C4—796no—
366C2×C4⋊C4—796no—
367C2×C4⋊C4—7948no—
368C2×C4⋊C4—792no—
369C2×C4⋊C4—792no—
370C2×C4⋊C4—796no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—796no—
373C2×C4⋊C4C22×D7913yes—
374C2×C4⋊C4C22×D7911yescharacteristic
375C2×C22⋊C4—796no—
376C2×C22⋊C4—796no—
377C2×C22⋊C4—796no—
378C2×C22⋊C4—796no—
379C2×C22⋊C4—796no—
380C2×C22⋊C4C22×D7916yes—
381C2×C42—793no—
382C2×C42—793no—
383C2×C42—796no—
384C2×C42—791no—
385C2×C42—7924no—
386C2×C42C22×D7913yes—
387C24—793no—
388C24C23×D7916yes—
389#41—264no—
390#41—31696no—
391#41—15832no—
392C2×Q8—7912no—
393C2×Q8—15812no—
394C2×Q8#1192962318yescomplemented
395C2×Q8C23×D7914yes—
396C2×D4—224no—
397C2×D4—212no—
398C2×D4—412no—
399C2×D4—31612no—
400C2×D4—31624no—
401C2×D4—7924no—
402C2×D4—7948no—
403C2×D4—7912no—
404C2×D4—15812no—
405C2×D4C23×D79124yes—
406C2×D4C23×D79112yes—
407C2×D4C23×D79124yes—
408C22×C4—224no—
409C22×C4—3166no—
410C22×C4—31624no—
411C22×C4—7948no—
412C22×C4—7912no—
413C22×C4—791no—
414C22×C4—791no—
415C22×C4—792no—
416C22×C4—793no—
417C22×C4—796no—
418C22×C4—7912no—
419C22×C4—7912no—
420C22×C4—15812no—
421C22×C4—15812no—
422C22×C4C23×D79124yes—
423C22×C4C23×D7913yes—
424C22×C4C23×D7912yes—
425C22×C4C23×D7913yes—
426C22×C4C23×D7911yescharacteristic
427C4⋊C4—798no—
428C4⋊C4—7924no—
429C4⋊C4—7912no—
430C4⋊C4—7924no—
431C4⋊C4—7924no—
432C4⋊C4—798no—
433C4⋊C4—7924no—
434C4⋊C4—794no—
435C4⋊C4C23×D79112yes—
436C4⋊C4C23×D7914yes—
437C22⋊C4—7924no—
438C22⋊C4—7924no—
439C22⋊C4—7924no—
440C22⋊C4—7924no—
441C22⋊C4—7924no—
442C22⋊C4C23×D79124yes—
443C42—794no—
444C42—7912no—
445C42—7912no—
446C42—7924no—
447C42C23×D79112yes—
448C23—212no—
449C23—212no—
450C23—212no—
451C23—796no—
452C23—1586no—
453C23—1586no—
454C23—1586no—
455C23C24×D79112yes—
456C23C24×D7911yescharacteristic
457Q8—15824no—
458Q8C2×#11929623116yescomplemented
459D4—224no—
460D4—248no—
461D4—424no—
462D4—31624no—
463D4—31648no—
464D4—15824no—
465C2×C4—26no—
466C2×C4—24no—
467C2×C4—248no—
468C2×C4—31612no—
469C2×C4—31648no—
470C2×C4—792no—
471C2×C4—792no—
472C2×C4—794no—
473C2×C4—796no—
474C2×C4—7912no—
475C2×C4—1586no—
476C2×C4—15812no—
477C2×C4—1582no—
478C2×C4—1584no—
479C2×C4—1582no—
480C2×C4—15824no—
481C2×C4—15824no—
482C2×C4C2×#11929623112yes—
483C2×C4#1457920714yes—
484C2×C4C24×D7916yes—
485C2×C4C24×D7914yes—
486C2×C4C24×D7912yes—
487C2×C4C24×D7916yes—
488C22—212no—
489C22—212no—
490C22—212no—
491C22—424no—
492C22—31612no—
493C22—1586no—
494C22—1586no—
495C22—1586no—
496C22C22×#1192962312yes—
497C22C2×#1457920712yes—
498C22#1909939812yes—
499C22C25×D7911yescharacteristic
500C4—24no—
501C4—26no—
502C4—1582no—
503C4—1586no—
504C4—15812no—
505C4—1582no—
506C4—1584no—
507C4C22×#11929623112yes—
508C4C2×#1457920714yes—
509C2—412no—
510C2—3166no—
511C2#2377740114yesdirect product factor
512C2C23×#1192962311yescharacteristic
513C2C22×#1457920711yescharacteristic
514C2C2×#1909939811yescharacteristic
subgroupG/H
centerC23C24×D79
commutator, cosocleC22×C79C26
FrattiniC22C25×D79
FittingC2×#550×C79C2
socleC23×C79C25
hypercenterC2×#550D79
hypocenterC79C2×#3517
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #23777401C2direct (N × H)
2. C2×#19099292C2semidirect
3. C2×#19099265C2semidirect
4. C2×#19099233C2semidirect
5. C2×#19099232C2semidirect
6. C2×#19099217C2semidirect
7. C2×#19099166C2semidirect
8. C2×#19099111C2semidirect
9. C2×#19099095C2semidirect
10. C2×#19099046C2semidirect
11. C2×C4×#11929625C2semidirect
12. C2×#550×C79C2semidirect
15. #19099292C22semidirect
16. #19099265C22semidirect
17. #19099233C22semidirect
18. #19099232C22semidirect
19. #19099217C22semidirect
20. #19099166C22semidirect
21. #19099111C22semidirect
24. #19099095C22semidirect
37. #19099046C22semidirect
38. C4×#11929625C22semidirect
53. #550×C79C22semidirect
172. C2×Q8×C79#41semidirect
199. Q8×C79C2×#41semidirect
253. C2×C79#3517semidirect
263. C2×#550D79semidirect
270. C79C2×#3517semidirect
278. #550C2×D79semidirect
394. C2×Q8#11929623semidirect
458. Q8C2×#11929623semidirect
511. C2#23777401direct (N × H)

Maximal subgroups

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

subgroupindex
#237774012
C2×#190992922
C2×#190992652
C2×#190992332
C2×#190992322
C2×#190992172
C2×#190991662
C2×#190991112
C2×#190990952
C2×#190990462
C2×C4×#119296252
C2×#550×C792
C2×#351779