Finite Group Databasebeta

← C2×#23777485 #35196480

Subgroups — C2×#23777485 #35196480

Order 20224.

7758 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#23777485C2132yesmaximal, direct product factor
2C2×#19099298C213yesmaximal, complemented
3C2×Q8×Dic158C211yesmaximal, characteristic, complemented
4C2×#19099273C213yesmaximal, complemented
5C2×#19099235C211yesmaximal, characteristic, complemented
6C2×#19099224C212yesmaximal, complemented
7C2×#19099214C216yesmaximal, complemented
8C2×#19099185C216yesmaximal, complemented
9C2×#19099113C213yesmaximal, complemented
10C2×#19099097C213yesmaximal, complemented
11C2×#19099045C212yesmaximal, complemented
12C22×C4×Q8×C79C211yesmaximal, characteristic, complemented
13C22×#11929625C2212yes—
14C22×#11929624C2216yes—
15#19099298C22148yescomplemented
16Q8×Dic158C22116yescomplemented
17#19099273C22148yescomplemented
18#19099235C22116yescomplemented
19#19099224C22132yescomplemented
20#19099214C22196yescomplemented
21#19099185C22196yescomplemented
22#19099113C22148yescomplemented
23C2×#14579161C2213yes—
24C2×#14579160C2216yes—
25C2×#14579158C2213yes—
26#19099097C22148yescomplemented
27C2×#14579155C2216yes—
28C2×#14579153C2213yes—
29C2×#14579150C2216yes—
30C2×#14579149C2216yes—
31C2×#14579149C2213yes—
32C2×#14579146C22112yes—
33C2×#14579144C2216yes—
34C2×#14579140C2211yescharacteristic
35C2×#14579139C22112yes—
36C2×#14579137C2216yes—
37C2×#14579136C2216yes—
38C2×#14579135C2216yes—
39C2×#14579135C2213yes—
40#19099045C22132yescomplemented
41C2×Q8×Dic79C2212yes—
42C2×C4×Dic158C2213yes—
43C2×C4×D316C2216yes—
44C2×C4×D316C2213yes—
45C2×C4×D316C2216yes—
46C2×#14579123C2216yes—
47C2×C4×C22⋊D79C2216yes—
48C2×#14579118C2216yes—
49C2×C4⋊C4×D79C2216yes—
50C23×Q8×C79C2211yescharacteristic
51C2×C4×Q8×C79C2213yes—
52C2×C4×Q8×C79C2211yescharacteristic
53C2×C4×Q8×C79C2211yescharacteristic
54C2×C4×Q8×C79C2213yes—
55C2×C4×Q8×C79C22116yescomplemented
56C22×C4⋊C4×C79C2213yes—
57C22×C42×C79C2213yes—
58C2×#11929625—216no—
59C2×#11929625C23116yes—
60C2×#11929624—424no—
61C2×#11929624C23148yes—
62C22×Dic158C2311yescharacteristic
63C22×Dic158C2316yes—
64C22×D316C2316yes—
65C22×D316C2313yes—
66C22×D316C2316yes—
67C22×C22⋊D79C2316yes—
68C22×C4×D79C2316yes—
69C22×C4×D79C23112yes—
70#14579161C23124yes—
71#14579160C23148yes—
72#14579158C23124yes—
73#14579155C23148yes—
74#14579153C23124yes—
75#14579150C23148yes—
76#14579149C23148yes—
77#14579149C23124yes—
78#14579146C23196yes—
79#14579144C23148yes—
80#14579140C2318yes—
81#14579139C23196yes—
82#14579137C23148yes—
83#14579136C23148yes—
84#14579135C23148yes—
85#14579135C23124yes—
86Q8×Dic79C23116yes—
87C4×Dic158C23124yes—
88C4×D316C23148yes—
89C4×D316C23124yes—
90C4×D316C23148yes—
91#14579123C23148yes—
92C4×C22⋊D79C23148yes—
93#14579118C23148yes—
94C2×C4⋊Dic79C2316yes—
95C2×C4⋊Dic79C2311yescharacteristic
96C2×C4⋊Dic79C2313yes—
97C2×Dic79⋊C4C2316yes—
98C2×Dic79⋊C4C2312yes—
99C2×Dic79⋊C4C2316yes—
100C2×Dic79⋊C4C2316yes—
101C4⋊C4×D79C23148yes—
102C2×C22⋊Dic79C2312yes—
103C2×(C2×D79)⋊C4C2316yes—
104C2×(C2×D79)⋊C4C2316yes—
105C2×(C2×D79)⋊C4C2316yes—
106C2×(C2×D79)⋊C4C23112yes—
107C2×(C2×D79)⋊C4C2316yes—
108C2×C4×Dic79C2316yes—
109C22×Q8×C79—28no—
110C22×Q8×C79C2318yes—
111C22×Q8×C79C2313yes—
112C22×Q8×C79C2311yescharacteristic
113C23×C4×C79C2311yescharacteristic
114C23×C4×C79C2313yes—
115C23×C4×C79C2313yes—
116C4×Q8×C79C23124yes—
117C4×Q8×C79C2318yes—
118C4×Q8×C79C23124yes—
119C4×Q8×C79C2318yes—
120C2×C4⋊C4×C79C23124yes—
121C2×C4⋊C4×C79C2313yes—
122C2×C4⋊C4×C79C2313yes—
123C2×C4⋊C4×C79C2316yes—
124C2×C42×C79C2313yes—
125C2×C42×C79C2313yes—
126C2×C42×C79C2313yes—
127C2×C42×C79C2313yes—
128C2×C42×C79C23124yes—
129C23×D79C2416yes—
130#11929625—264no—
131#11929624—496no—
132C2×Dic158—24no—
133C2×Dic158—412no—
134C2×Dic158C2414yes—
135C2×Dic158C24124yes—
136C2×D316—224no—
137C2×D316—212no—
138C2×D316—412no—
139C2×D316C24124yes—
140C2×D316C24112yes—
141C2×D316C24124yes—
142C2×C22⋊D79—224no—
143C2×C22⋊D79C24124yes—
144C22×Dic79C2416yes—
145C22×Dic79C2412yes—
146C22×Dic79C2412yes—
147C2×C4×D79—224no—
148C2×C4×D79—424no—
149C2×C4×D79C24124yes—
150C2×C4×D79C24148yes—
151C4⋊Dic79C24124yes—
152C4⋊Dic79C2414yes—
153C4⋊Dic79C24112yes—
154Dic79⋊C4C24124yes—
155Dic79⋊C4C2418yes—
156Dic79⋊C4C24124yes—
157Dic79⋊C4C24124yes—
158C22⋊Dic79C2418yes—
159(C2×D79)⋊C4C24124yes—
160(C2×D79)⋊C4C24124yes—
161(C2×D79)⋊C4C24124yes—
162(C2×D79)⋊C4C24148yes—
163(C2×D79)⋊C4C24124yes—
164C4×Dic79C24124yes—
165C24×C79C2411yescharacteristic
166C2×Q8×C79—232no—
167C2×Q8×C79—212no—
168C2×Q8×C79#4118yescomplemented
169C2×Q8×C79C24112yes—
170C2×Q8×C79C2414yes—
171C22×C4×C79—212no—
172C22×C4×C79—212no—
173C22×C4×C79—24no—
174C22×C4×C79C2413yes—
175C22×C4×C79C24112yes—
176C22×C4×C79C2411yescharacteristic
177C22×C4×C79C2414yes—
178C22×C4×C79C2413yes—
179C22×C4×C79C2413yes—
180C22×C4×C79C24112yes—
181C22×C4×C79C2413yes—
182C22×C4×C79C2411yescharacteristic
183C4⋊C4×C79C24112yes—
184C4⋊C4×C79C24112yes—
185C4⋊C4×C79C24124yes—
186C42×C79C24112yes—
187C42×C79C24112yes—
188C42×C79C24112yes—
189C42×C79C24112yes—
190C23×C79—22no—
191C23×C79—22no—
192C23×C79H(5,𝔽2)14yes—
193C23×C79C2512yes—
194C23×C79C2511yescharacteristic
195C22×D79—212no—
196C22×D79—212no—
197C22×D79—212no—
198C22×D79C25112yes—
199Q8×C79—224no—
200Q8×C79C2×#41116yescomplemented
201C2×C4×C79—26no—
202C2×C4×C79—26no—
203C2×C4×C79—26no—
204C2×C4×C79—224no—
205C2×C4×C79—22no—
206C2×C4×C79—224no—
207C2×C4×C79—28no—
208C2×C4×C79C2×#41112yes—
209C2×C4×C792−1+414yes—
210C2×C4×C79C2512yes—
211C2×C4×C79C2516yes—
212C2×C4×C79C2516yes—
213C2×C4×C79C2516yes—
214C2×C4×C79C2516yes—
215C2×C4×C79C2512yes—
216C22⋊D79—248no—
217C2×Dic79—24no—
218C2×Dic79—212no—
219C2×Dic79—24no—
220C2×Dic79C2514yes—
221C2×Dic79C25112yes—
222C2×Dic79C2514yes—
223D316—248no—
224D316—224no—
225D316—424no—
226C4×D79—248no—
227C4×D79—448no—
228Dic158—28no—
229Dic158—424no—
230C22×C79—22no—
231C22×C79—22no—
232C22×C79—28no—
233C22×C79C22×#4112yes—
234C22×C79C2×H(5,𝔽2)14yes—
235C22×C79C2×H(5,𝔽2)12yes—
236C22×C79C2×2−1+412yes—
237C22×C79C2611yescharacteristic
238C2×D79—212no—
239C2×D79—212no—
240C2×D79—212no—
241C2×D79—424no—
242C4×C79—26no—
243C4×C79—26no—
244C4×C79—26no—
245C4×C79—22no—
246C4×C79C22×#41112yes—
247C4×C79C2×2−1+414yes—
248Dic79—24no—
249Dic79—212no—
250Dic79—24no—
251C2×#3522—791nomaximal, Sylow
252C2×C79—24no—
253C2×C79#352214yescomplemented
254C2×C79C23×#4111yescharacteristic
255C2×C79C22×H(5,𝔽2)11yescharacteristic
256C2×C79C22×2−1+411yescharacteristic
257D79—412no—
258#3522—7932no—
259C2×#559—793no—
260C2×Q82—791no—
261C2×#554—793no—
262C2×#550—791no—
263C2×#548—792no—
264C2×#547—796no—
265C2×#542—796no—
266C2×#532—793no—
267C2×#529—793no—
268C2×#520—792no—
269C22×C4×Q8D7911yescharacteristic, complemented
270C79C2×#352211yescharacteristic, complemented, Sylow
271C22×#41—792no—
272C22×#41—796no—
273C23×Q8C2×D7911yescharacteristic
274#559—7948no—
275Q82—7916no—
276#554—7948no—
277#550—7916no—
278#548—7932no—
279#547—7996no—
280#542—7996no—
281#532—7948no—
282C2×C4⋊Q8—796no—
283C2×C4⋊Q8—793no—
284C2×#127—793no—
285#529—7948no—
286C2×#126—796no—
287C2×#125—793no—
288C2×#124—796no—
289C2×#124—793no—
290C2×#124—796no—
291C2×#123—796no—
292C2×#123—7912no—
293C2×C22⋊Q8—796no—
294C2×C22⋊Q8—791no—
295C2×C22⋊Q8—7912no—
296C2×#121—793no—
297C2×#121—796no—
298C2×#121—796no—
299#520—7932no—
300C2×C4×Q8—792no—
301C2×C4×Q8—793no—
302C2×C4×Q8C2×D7911yescharacteristic
303C2×C4×Q8C2×D7913yes—
304C2×C4×Q8C2×D7911yescharacteristic
305C2×C4×Q8C2×D7913yes—
306C2×C4×Q8C2×D79116yescomplemented
307C2×C4×D4—796no—
308C2×C4×D4—796no—
309C2×C4×D4—796no—
310C2×C4×D4—793no—
311C2×C4×D4—796no—
312C2×#117—796no—
313C22×C4⋊C4—796no—
314C22×C4⋊C4C2×D7913yes—
315C22×C42C2×D7913yes—
316C2×#41—31624no—
317C2×#41—7916no—
318C2×#41—7948no—
319C2×#41—15816no—
320C22×Q8—28no—
321C22×Q8—796no—
322C22×Q8—791no—
323C22×Q8C22×D7913yes—
324C22×Q8C22×D7918yes—
325C22×Q8C22×D7911yescharacteristic
326C22×D4—796no—
327C22×D4—796no—
328C22×D4—793no—
329C22×D4—796no—
330C23×C4—7912no—
331C23×C4—796no—
332C23×C4C22×D7913yes—
333C23×C4C22×D7913yes—
334C23×C4C22×D7911yescharacteristic
335C4⋊Q8—7948no—
336C4⋊Q8—7924no—
337#127—7924no—
338#126—7948no—
339#125—7924no—
340#124—7948no—
341#124—7924no—
342#124—7948no—
343#123—7948no—
344#123—7996no—
345C22⋊Q8—7948no—
346C22⋊Q8—798no—
347C22⋊Q8—7996no—
348#121—7924no—
349#121—7948no—
350#121—7948no—
351C4×Q8—7916no—
352C4×Q8—7924no—
353C4×Q8C22×D7918yes—
354C4×Q8C22×D79124yes—
355C4×Q8C22×D7918yes—
356C4×Q8C22×D79124yes—
357C4×D4—7948no—
358C4×D4—7948no—
359C4×D4—7948no—
360C4×D4—7924no—
361C4×D4—7948no—
362#117—7948no—
363C2×C4⋊C4—796no—
364C2×C4⋊C4—796no—
365C2×C4⋊C4—792no—
366C2×C4⋊C4—791no—
367C2×C4⋊C4—793no—
368C2×C4⋊C4—796no—
369C2×C4⋊C4—796no—
370C2×C4⋊C4—7948no—
371C2×C4⋊C4C22×D7913yes—
372C2×C4⋊C4C22×D7913yes—
373C2×C4⋊C4C22×D7916yes—
374C2×C4⋊C4C22×D79124yes—
375C2×C22⋊C4—792no—
376C2×C22⋊C4—796no—
377C2×C22⋊C4—796no—
378C2×C22⋊C4—796no—
379C2×C22⋊C4—7912no—
380C2×C22⋊C4—796no—
381C2×C42—796no—
382C2×C42C22×D7913yes—
383C2×C42C22×D7913yes—
384C2×C42C22×D7913yes—
385C2×C42C22×D7913yes—
386C2×C42C22×D79124yes—
387C24—796no—
388C24C23×D7911yescharacteristic
389#41—31696no—
390#41—15864no—
391C2×Q8—232no—
392C2×Q8—212no—
393C2×Q8—31612no—
394C2×Q8—7924no—
395C2×Q8—794no—
396C2×Q8—1584no—
397C2×Q8#1192962418yescomplemented
398C2×Q8C23×D79112yes—
399C2×Q8C23×D7914yes—
400C2×D4—31612no—
401C2×D4—7924no—
402C2×D4—7924no—
403C2×D4—7912no—
404C2×D4—7924no—
405C2×D4—15824no—
406C2×D4—15824no—
407C2×D4—15812no—
408C22×C4—212no—
409C22×C4—212no—
410C22×C4—24no—
411C22×C4—31624no—
412C22×C4—7948no—
413C22×C4—792no—
414C22×C4—796no—
415C22×C4—792no—
416C22×C4—7924no—
417C22×C4—15824no—
418C22×C4C23×D7914yes—
419C22×C4C23×D79112yes—
420C22×C4C23×D79112yes—
421C22×C4C23×D7911yescharacteristic
422C22×C4C23×D7913yes—
423C22×C4C23×D7913yes—
424C22×C4C23×D7913yes—
425C22×C4C23×D7913yes—
426C22×C4C23×D7911yescharacteristic
427C4⋊C4—7924no—
428C4⋊C4—798no—
429C4⋊C4—7924no—
430C4⋊C4—7924no—
431C4⋊C4—7924no—
432C4⋊C4—794no—
433C4⋊C4—7912no—
434C4⋊C4C23×D79112yes—
435C4⋊C4C23×D79112yes—
436C4⋊C4C23×D79124yes—
437C22⋊C4—798no—
438C22⋊C4—7924no—
439C22⋊C4—7924no—
440C22⋊C4—7924no—
441C22⋊C4—7948no—
442C22⋊C4—7924no—
443C42—7924no—
444C42C23×D79112yes—
445C42C23×D79112yes—
446C42C23×D79112yes—
447C42C23×D79112yes—
448C23—22no—
449C23—22no—
450C23—7912no—
451C23—15812no—
452C23—15812no—
453C23—15812no—
454C23#1457920714yes—
455C23C24×D7912yes—
456C23C24×D7911yescharacteristic
457Q8—224no—
458Q8—31624no—
459Q8—1588no—
460Q8C2×#11929624116yescomplemented
461D4—31624no—
462D4—15848no—
463D4—15824no—
464D4—15848no—
465C2×C4—26no—
466C2×C4—26no—
467C2×C4—224no—
468C2×C4—22no—
469C2×C4—26no—
470C2×C4—28no—
471C2×C4—224no—
472C2×C4—31648no—
473C2×C4—794no—
474C2×C4—7912no—
475C2×C4—794no—
476C2×C4—15812no—
477C2×C4—1584no—
478C2×C4—1584no—
479C2×C4—15848no—
480C2×C4C2×#11929624112yes—
481C2×C4#1457920914yes—
482C2×C4C24×D7912yes—
483C2×C4C24×D7916yes—
484C2×C4C24×D7916yes—
485C2×C4C24×D7916yes—
486C2×C4C24×D7912yes—
487C2×C4C24×D7916yes—
488C22—22no—
489C22—22no—
490C22—28no—
491C22—31624no—
492C22—15812no—
493C22—15812no—
494C22—15812no—
495C22C22×#1192962412yes—
496C22C2×#1457920714yes—
497C22C2×#1457920712yes—
498C22C2×#1457920912yes—
499C22C25×D7911yescharacteristic
500C4—26no—
501C4—26no—
502C4—26no—
503C4—22no—
504C4—1584no—
505C4—15812no—
506C4—1584no—
507C4C22×#11929624112yes—
508C4C2×#1457920914yes—
509C2—24no—
510C2—31612no—
511C2#2377748514yesdirect product factor
512C2C23×#1192962411yescharacteristic
513C2C22×#1457920711yescharacteristic
514C2C22×#1457920911yescharacteristic
subgroupG/H
centerC23C24×D79
commutator, cosocleC22×C79C26
FrattiniC22C25×D79
FittingC22×C4×Q8×C79C2
socleC23×C79C25
hypercenterC22×C4×Q8D79
hypocenterC79C2×#3522
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #23777485C2direct (N × H)
2. C2×#19099298C2semidirect
3. C2×Q8×Dic158C2semidirect
4. C2×#19099273C2semidirect
5. C2×#19099235C2semidirect
6. C2×#19099224C2semidirect
7. C2×#19099214C2semidirect
8. C2×#19099185C2semidirect
9. C2×#19099113C2semidirect
10. C2×#19099097C2semidirect
11. C2×#19099045C2semidirect
12. C22×C4×Q8×C79C2semidirect
15. #19099298C22semidirect
16. Q8×Dic158C22semidirect
17. #19099273C22semidirect
18. #19099235C22semidirect
19. #19099224C22semidirect
20. #19099214C22semidirect
21. #19099185C22semidirect
22. #19099113C22semidirect
26. #19099097C22semidirect
40. #19099045C22semidirect
55. C2×C4×Q8×C79C22semidirect
168. C2×Q8×C79#41semidirect
200. Q8×C79C2×#41semidirect
253. C2×C79#3522semidirect
269. C22×C4×Q8D79semidirect
270. C79C2×#3522semidirect
306. C2×C4×Q8C2×D79semidirect
397. C2×Q8#11929624semidirect
460. Q8C2×#11929624semidirect
511. C2#23777485direct (N × H)

Maximal subgroups

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

subgroupindex
#237774852
C2×#190992982
C2×Q8×Dic1582
C2×#190992732
C2×#190992352
C2×#190992242
C2×#190992142
C2×#190991852
C2×#190991132
C2×#190990972
C2×#190990452
C22×C4×Q8×C792
C2×#352279