Finite Group Databasebeta

← C2×#23777497 #35196483

Subgroups — C2×#23777497 #35196483

Order 20224.

7022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23777497C2132yesmaximal, direct product factor
2C2×#19099292C213yesmaximal, complemented
3C2×#19099287C213yesmaximal, complemented
4C2×#19099251C216yesmaximal, complemented
5C2×Q8×D316C211yesmaximal, characteristic, complemented
6C2×Q8×D316C212yesmaximal, complemented
7C2×#19099176C216yesmaximal, complemented
8C2×#19099118C213yesmaximal, complemented
9C2×#19099100C213yesmaximal
10C2×#19099039C212yesmaximal, complemented
11C2×C4×Q8×D79C211yesmaximal, characteristic, complemented
12C2×#557×C79C211yesmaximal, characteristic, complemented
13C22×Q8×D79C2211yescharacteristic
14C22×Q8×D79C2212yes—
15#19099292C22148yescomplemented
16#19099287C22148yescomplemented
17#19099251C22196yescomplemented
18Q8×D316C22132yescomplemented
19Q8×D316C22116yescomplemented
20#19099176C22196yescomplemented
21#19099118C22148yescomplemented
22C2×#14579161C2216yes—
23C2×#14579161C2213yes—
24#19099100C22148yes—
25C2×#14579156C2216yes—
26C2×#14579156C22112yes—
27C2×#14579152C2216yes—
28C2×#14579152C22112yes—
29C2×#14579152C2213yes—
30C2×#14579150C2213yes—
31C2×#14579146C2216yes—
32C2×#14579141C2216yes—
33C2×#14579141C2216yes—
34C2×#14579141C2216yes—
35C2×#14579137C2213yes—
36C2×#14579136C2211yescharacteristic
37#19099039C22132yescomplemented
38C2×Q8×Dic79C2211yescharacteristic
39C2×Q8×Dic79C2216yes—
40C2×C4×Dic158C2213yes—
41C2×C4×Dic158C2216yes—
42C2×#14579126C2213yes—
43C2×#14579126C2216yes—
44C4×Q8×D79C22116yescomplemented
45C2×C4×D316C2216yes—
46C2×C4×D316C2213yes—
47C2×#14579119C2216yes—
48C2×#14579119C2216yes—
49C2×#14579118C2216yes—
50C2×C4⋊C4×D79C2213yes—
51C2×C42×D79C2213yes—
52#557×C79C22116yescomplemented
53C2×C4⋊Q8×C79C2213yes—
54C2×#125×C79C2216yes—
55C2×C4×Q8×C79C2211yescharacteristic
56C2×C4×Q8×C79C2212yes—
57C2×C4×Q8×C79C2213yes—
58C22×Dic158C2313yes—
59C22×Dic158C2316yes—
60C2×Q8×D79—28no—
61C2×Q8×D79—216no—
62C2×Q8×D79C2318yes—
63C2×Q8×D79C23116yes—
64C22×D316C2312yes—
65C22×D316C2311yescharacteristic
66C22×C4×D79C2311yescharacteristic
67C22×C4×D79C2313yes—
68C22×C4×D79C2313yes—
69C22×C4×D79C2316yes—
70#14579161C23148yes—
71#14579161C23124yes—
72#14579156C23148yes—
73#14579156C23196yes—
74#14579152C23148yes—
75#14579152C23196yes—
76#14579152C23124yes—
77#14579150C23124yes—
78#14579146C23148yes—
79#14579141C23148yes—
80#14579141C23148yes—
81#14579141C23148yes—
82#14579137C23124yes—
83#14579136C2318yes—
84Q8×Dic79C2318yes—
85Q8×Dic79C23148yes—
86C4×Dic158C23148yes—
87C4×Dic158C23124yes—
88#14579126C23124yes—
89#14579126C23148yes—
90C4×D316C23148yes—
91C4×D316C23124yes—
92#14579119C23148yes—
93#14579119C23148yes—
94#14579118C23148yes—
95C2×C4⋊Dic79C2313yes—
96C2×C4⋊Dic79C2316yes—
97C2×C4⋊Dic79C2313yes—
98C2×C4⋊Dic79C2316yes—
99C2×C4⋊Dic79C23112yes—
100C2×Dic79⋊C4C2316yes—
101C2×Dic79⋊C4C2316yes—
102C2×Dic79⋊C4C2316yes—
103C4⋊C4×D79C23124yes—
104C2×(C2×D79)⋊C4C2312yes—
105C2×(C2×D79)⋊C4C2316yes—
106C2×(C2×D79)⋊C4C2316yes—
107C2×(C2×D79)⋊C4C2316yes—
108C2×(C2×D79)⋊C4C2316yes—
109C2×C4×Dic79C2313yes—
110C2×C4×Dic79C2313yes—
111C2×C4×Dic79C2313yes—
112C2×C4×Dic79C23112yes—
113C42×D79C23124yes—
114C22×Q8×C79C2313yes—
115C22×Q8×C79C2311yescharacteristic
116C4⋊Q8×C79C23124yes—
117#125×C79C23148yes—
118C4×Q8×C79C2318yes—
119C4×Q8×C79C23116yes—
120C4×Q8×C79C23124yes—
121C2×C4⋊C4×C79C2311yescharacteristic
122C2×C4⋊C4×C79C2313yes—
123C2×C4⋊C4×C79C2316yes—
124C2×C4⋊C4×C79C2316yes—
125C2×C4⋊C4×C79C2316yes—
126C2×C42×C79C2316yes—
127C2×C42×C79C2313yes—
128C23×D79C2412yes—
129C23×D79C2411yescharacteristic
130C2×Dic158—212no—
131C2×Dic158—224no—
132C2×Dic158C24112yes—
133C2×Dic158C24124yes—
134Q8×D79—232no—
135Q8×D79—264no—
136C2×D316—28no—
137C2×D316—24no—
138C2×D316C2418yes—
139C2×D316C2414yes—
140C22×Dic79C2413yes—
141C22×Dic79C2416yes—
142C22×Dic79C2411yescharacteristic
143C22×Dic79C2413yes—
144C2×C4×D79—212no—
145C2×C4×D79—212no—
146C2×C4×D79—224no—
147C2×C4×D79—24no—
148C2×C4×D79C24112yes—
149C2×C4×D79C24124yes—
150C2×C4×D79C2414yes—
151C2×C4×D79C24112yes—
152C4⋊Dic79C24112yes—
153C4⋊Dic79C24124yes—
154C4⋊Dic79C24112yes—
155C4⋊Dic79C24124yes—
156C4⋊Dic79C24148yes—
157Dic79⋊C4C24124yes—
158Dic79⋊C4C24124yes—
159Dic79⋊C4C24124yes—
160(C2×D79)⋊C4C2418yes—
161(C2×D79)⋊C4C24124yes—
162(C2×D79)⋊C4C24124yes—
163(C2×D79)⋊C4C24124yes—
164(C2×D79)⋊C4C24124yes—
165C4×Dic79C24112yes—
166C4×Dic79C24112yes—
167C4×Dic79C24112yes—
168C4×Dic79C24148yes—
169C2×Q8×C79—212no—
170C2×Q8×C79#4118yes—
171C2×Q8×C79C24112yes—
172C2×Q8×C79C2414yes—
173C22×C4×C79C2412yes—
174C22×C4×C79C2413yes—
175C22×C4×C79C2416yes—
176C22×C4×C79C2413yes—
177C22×C4×C79C2411yescharacteristic
178C4⋊C4×C79C2414yes—
179C4⋊C4×C79C24112yes—
180C4⋊C4×C79C24124yes—
181C4⋊C4×C79C24124yes—
182C4⋊C4×C79C24124yes—
183C42×C79C24124yes—
184C42×C79C24112yes—
185C23×C79C2511yescharacteristic
186C22×D79—22no—
187C22×D79—22no—
188C22×D79—24no—
189C22×D79—24no—
190C22×D79—24no—
191C22×D792−1+414yes—
192C22×D79C2514yes—
193C22×D79C2512yes—
194Q8×C79—224no—
195Q8×C79C2×#41116yes—
196C2×C4×C79—26no—
197C2×C4×C79—24no—
198C2×C4×C79—212no—
199C2×C4×C79C2×#41112yes—
200C2×C4×C792−1+414yes—
201C2×C4×C79C25112yes—
202C2×C4×C79C2516yes—
203C2×C4×C79C2514yes—
204C2×C4×C79C2516yes—
205C2×C4×C79C2512yes—
206C2×Dic79—26no—
207C2×Dic79—212no—
208C2×Dic79—26no—
209C2×Dic79—22no—
210C2×Dic79C2512yes—
211C2×Dic79C2516yes—
212C2×Dic79C2516yes—
213C2×Dic79C25112yes—
214D316—216no—
215D316—28no—
216C4×D79—224no—
217C4×D79—224no—
218C4×D79—248no—
219C4×D79—28no—
220Dic158—224no—
221Dic158—248no—
222C22×C79C22×#4112yes—
223C22×C79C2×2−1+412yes—
224C22×C79C2×2−1+412yes—
225C22×C79C2611yescharacteristic
226C2×D79—22no—
227C2×D79—22no—
228C2×D79—28no—
229C2×D79—24no—
230C2×D79—24no—
231C2×D79—24no—
232C2×D79—48no—
233C2×D79C2×2−1+414yes—
234C4×C79—212no—
235C4×C79—26no—
236C4×C79—24no—
237C4×C79C22×#41112yes—
238C4×C79C2×2−1+414yes—
239Dic79—26no—
240Dic79—212no—
241Dic79—26no—
242Dic79—22no—
243C2×#3523—791nomaximal, Sylow
244C2×C79#352314yescomplemented
245C2×C79C23×#4111yescharacteristic
246C2×C79C22×2−1+411yescharacteristic
247C2×C79C22×2−1+411yescharacteristic
248D79—24no—
249D79—44no—
250#3523—7932no—
251C2×#557—793no—
252C2×#557D7911yescharacteristic, complemented
253C2×#556—793no—
254C2×#552—796no—
255C2×D4×Q8—792no—
256C2×D4×Q8—791no—
257C2×#541—796no—
258C2×#533—793no—
259C2×#529—793no—
260C2×#519—792no—
261C22×C4×Q8—791no—
262C79C2×#352311yescharacteristic, complemented, Sylow
263C23×Q8—791no—
264C23×Q8—792no—
265#557—7948no—
266#557C2×D79116yescomplemented
267#556—7948no—
268#552—7996no—
269D4×Q8—7932no—
270D4×Q8—7916no—
271#541—7996no—
272#533—7948no—
273C2×C4⋊Q8—793no—
274C2×C4⋊Q8—796no—
275C2×C4⋊Q8C2×D7913yes—
276#529—7948no—
277C2×#126—796no—
278C2×#126—7912no—
279C2×#125—793no—
280C2×#125—796no—
281C2×#125—7912no—
282C2×#125C2×D7916yes—
283C2×#124—793no—
284C2×#123—796no—
285C2×C22⋊Q8—793no—
286C2×C22⋊Q8—796no—
287C2×C22⋊Q8—796no—
288C2×C22⋊Q8—796no—
289C2×#121—791no—
290#519—7932no—
291C2×C4×Q8—791no—
292C2×C4×Q8—796no—
293C2×C4×Q8—7916no—
294C2×C4×Q8—793no—
295C2×C4×Q8—796no—
296C2×C4×Q8—793no—
297C2×C4×Q8—796no—
298C2×C4×Q8C2×D7911yescharacteristic
299C2×C4×Q8C2×D7912yes—
300C2×C4×Q8C2×D7913yes—
301C2×C4×D4—796no—
302C2×C4×D4—793no—
303C2×#117—796no—
304C2×#117—796no—
305C2×#117—796no—
306C22×C4⋊C4—793no—
307C22×C42—793no—
308C22×Q8—798no—
309C22×Q8—7916no—
310C22×Q8—793no—
311C22×Q8—796no—
312C22×Q8—1588no—
313C22×Q8—15816no—
314C22×Q8C22×D7913yes—
315C22×Q8C22×D7911yescharacteristic
316C22×D4—792no—
317C22×D4—791no—
318C23×C4—793no—
319C23×C4—793no—
320C23×C4—796no—
321C23×C4—791no—
322C4⋊Q8—7924no—
323C4⋊Q8—7948no—
324C4⋊Q8C22×D79124yes—
325#126—7948no—
326#126—7996no—
327#125—7924no—
328#125—7948no—
329#125—7996no—
330#125C22×D79148yes—
331#124—7924no—
332#123—7948no—
333C22⋊Q8—7924no—
334C22⋊Q8—7948no—
335C22⋊Q8—7948no—
336C22⋊Q8—7948no—
337#121—798no—
338C4×Q8—798no—
339C4×Q8—7948no—
340C4×Q8—7924no—
341C4×Q8—7948no—
342C4×Q8—7924no—
343C4×Q8—7948no—
344C4×Q8C22×D7918yes—
345C4×Q8C22×D79116yes—
346C4×Q8C22×D79124yes—
347C4×D4—7948no—
348C4×D4—7924no—
349#117—7948no—
350#117—7948no—
351#117—7948no—
352C2×C4⋊C4—793no—
353C2×C4⋊C4—796no—
354C2×C4⋊C4—796no—
355C2×C4⋊C4—796no—
356C2×C4⋊C4—796no—
357C2×C4⋊C4—793no—
358C2×C4⋊C4—796no—
359C2×C4⋊C4—7912no—
360C2×C4⋊C4—7924no—
361C2×C4⋊C4C22×D7913yes—
362C2×C4⋊C4C22×D7916yes—
363C2×C4⋊C4C22×D7916yes—
364C2×C4⋊C4C22×D7911yescharacteristic
365C2×C4⋊C4C22×D7916yes—
366C2×C22⋊C4—792no—
367C2×C22⋊C4—796no—
368C2×C22⋊C4—796no—
369C2×C22⋊C4—796no—
370C2×C22⋊C4—796no—
371C2×C42—793no—
372C2×C42—7912no—
373C2×C42—793no—
374C2×C42—793no—
375C2×C42—7924no—
376C2×C42C22×D7916yes—
377C2×C42C22×D7913yes—
378C24—791no—
379C24—792no—
380C2×Q8—212no—
381C2×Q8—7912no—
382C2×Q8—7924no—
383C2×Q8—15832no—
384C2×Q8—15864no—
385C2×Q8—15812no—
386C2×Q8—15824no—
387C2×Q8#1192962518yes—
388C2×Q8C23×D79112yes—
389C2×Q8C23×D7914yes—
390C2×D4—798no—
391C2×D4—794no—
392C2×D4—1588no—
393C2×D4—1584no—
394C22×C4—793no—
395C22×C4—791no—
396C22×C4—796no—
397C22×C4—793no—
398C22×C4—794no—
399C22×C4—7912no—
400C22×C4—7912no—
401C22×C4—7924no—
402C22×C4—15812no—
403C22×C4—15824no—
404C22×C4—15812no—
405C22×C4—1584no—
406C22×C4C23×D7916yes—
407C22×C4C23×D7913yes—
408C22×C4C23×D7912yes—
409C22×C4C23×D7913yes—
410C22×C4C23×D7911yescharacteristic
411C4⋊C4—7924no—
412C4⋊C4—7924no—
413C4⋊C4—7924no—
414C4⋊C4—7912no—
415C4⋊C4—7924no—
416C4⋊C4—7912no—
417C4⋊C4—7924no—
418C4⋊C4—7948no—
419C4⋊C4C23×D79112yes—
420C4⋊C4C23×D79124yes—
421C4⋊C4C23×D79124yes—
422C4⋊C4C23×D7914yes—
423C4⋊C4C23×D79124yes—
424C22⋊C4—798no—
425C22⋊C4—7924no—
426C22⋊C4—7924no—
427C22⋊C4—7924no—
428C22⋊C4—7924no—
429C42—7912no—
430C42—7912no—
431C42—7912no—
432C42—7948no—
433C42C23×D79112yes—
434C42C23×D79124yes—
435C23—794no—
436C23—792no—
437C23—794no—
438C23—1584no—
439C23—1584no—
440C23—1584no—
441C23—1582no—
442C23—1582no—
443C23C24×D7911yescharacteristic
444Q8—224no—
445Q8—15824no—
446Q8—15848no—
447Q8C2×#11929625116yes—
448D4—15816no—
449D4—1588no—
450C2×C4—26no—
451C2×C4—24no—
452C2×C4—212no—
453C2×C4—796no—
454C2×C4—792no—
455C2×C4—796no—
456C2×C4—7912no—
457C2×C4—15824no—
458C2×C4—1586no—
459C2×C4—1582no—
460C2×C4—15812no—
461C2×C4—1586no—
462C2×C4—1588no—
463C2×C4—15848no—
464C2×C4—15824no—
465C2×C4C2×#11929625112yes—
466C2×C4#1457920814yes—
467C2×C4C24×D79112yes—
468C2×C4C24×D7916yes—
469C2×C4C24×D7914yes—
470C2×C4C24×D7912yes—
471C2×C4C24×D7916yes—
472C22—3168no—
473C22—794no—
474C22—1584no—
475C22—1584no—
476C22—1584no—
477C22—1582no—
478C22—1582no—
479C22—1588no—
480C22C22×#1192962512yes—
481C222−1+4×D7912yes—
482C22C2×#1457920812yes—
483C22C25×D7911yescharacteristic
484C4—26no—
485C4—24no—
486C4—212no—
487C4—1586no—
488C4—15812no—
489C4—1586no—
490C4—1582no—
491C4C22×#11929625112yes—
492C4C2×#1457920814yes—
493C2—3164no—
494C2—1584no—
495C2#2377749714yesdirect product factor
496C2C23×#1192962511yescharacteristic
497C2C2×2−1+4×D7911yescharacteristic
498C2C22×#1457920811yescharacteristic
subgroupG/H
centerC23C24×D79
commutator, cosocleC22×C79C26
FrattiniC22C25×D79
FittingC2×#557×C79C2
socleC23×C79C25
prime radicalC2×#14579136C22
hypercenterC2×#557D79
hypocenterC79C2×#3523
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. #23777497C2direct (N × H)
2. C2×#19099292C2semidirect
3. C2×#19099287C2semidirect
4. C2×#19099251C2semidirect
5. C2×Q8×D316C2semidirect
6. C2×Q8×D316C2semidirect
7. C2×#19099176C2semidirect
8. C2×#19099118C2semidirect
10. C2×#19099039C2semidirect
11. C2×C4×Q8×D79C2semidirect
12. C2×#557×C79C2semidirect
15. #19099292C22semidirect
16. #19099287C22semidirect
17. #19099251C22semidirect
18. Q8×D316C22semidirect
19. Q8×D316C22semidirect
20. #19099176C22semidirect
21. #19099118C22semidirect
37. #19099039C22semidirect
44. C4×Q8×D79C22semidirect
52. #557×C79C22semidirect
244. C2×C79#3523semidirect
252. C2×#557D79semidirect
262. C79C2×#3523semidirect
266. #557C2×D79semidirect
495. C2#23777497direct (N × H)

Maximal subgroups

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

subgroupindex
#237774972
C2×#190992922
C2×#190992872
C2×#190992512
C2×Q8×D3162
C2×#190991762
C2×#190991182
C2×#190991002
C2×#190990392
C2×C4×Q8×D792
C2×#557×C792
C2×#352379