Finite Group Databasebeta

← C2×#23777496 #35196492

Subgroups — C2×#23777496 #35196492

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#23777496C2132yesmaximal, direct product factor
2C2×#19099292C211yesmaximal, characteristic, complemented
3C2×#19099291C213yesmaximal, complemented
4C2×#19099286C213yesmaximal, complemented
5C2×#19099248C216yesmaximal, complemented
6C2×Q8×D316C211yesmaximal, characteristic, complemented
7C2×Q8×C22⋊D79C212yesmaximal, complemented
8C2×#19099171C216yesmaximal, complemented
9C2×#19099120C213yesmaximal, complemented
10C2×#19099097C213yesmaximal
11C2×#19099039C212yesmaximal, complemented
12C22×C4×Q8×C79C211yesmaximal, characteristic, complemented
13C22×Q8×D79C2212yes—
14#19099292C22116yescomplemented
15#19099291C22148yescomplemented
16#19099286C22148yescomplemented
17#19099248C22196yescomplemented
18Q8×D316C22116yescomplemented
19Q8×C22⋊D79C22132yescomplemented
20#19099171C22196yescomplemented
21#19099120C22148yescomplemented
22C2×#14579161C2213yes—
23C2×#14579161C2213yes—
24C2×#14579159C2216yes—
25#19099097C22148yes—
26C2×#14579156C22112yes—
27C2×#14579155C2216yes—
28C2×#14579153C2213yes—
29C2×#14579153C2216yes—
30C2×#14579152C2216yes—
31C2×#14579151C22112yes—
32C2×#14579149C2213yes—
33C2×#14579144C2216yes—
34C2×#14579141C2216yes—
35C2×#14579140C2216yes—
36C2×#14579140C2213yes—
37C2×#14579137C2216yes—
38C2×#14579135C2211yescharacteristic
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—
50C23×Q8×C79C2211yescharacteristic
51C2×C4×Q8×C79C2211yescharacteristic
52C2×C4×Q8×C79C2211yescharacteristic
53C2×C4×Q8×C79C2213yes—
54C2×C4×Q8×C79C2213yes—
55C2×C4×Q8×C79C22116yescomplemented
56C22×C4⋊C4×C79C2213yes—
57C22×C42×C79C2213yes—
58C22×Dic158C2316yes—
59C22×Dic158C2313yes—
60C2×Q8×D79—216no—
61C2×Q8×D79C23116yes—
62C22×D316C2311yescharacteristic
63C22×C22⋊D79C2312yes—
64C22×C4×D79C2316yes—
65#14579161C23124yes—
66#14579161C23124yes—
67#14579159C23148yes—
68#14579156C23196yes—
69#14579155C23148yes—
70#14579153C23148yes—
71#14579153C23124yes—
72#14579152C23148yes—
73#14579151C23196yes—
74#14579149C23124yes—
75#14579144C23148yes—
76#14579141C23148yes—
77#14579140C23148yes—
78#14579140C23124yes—
79#14579137C23148yes—
80#14579135C2318yes—
81Q8×Dic79C23116yes—
82C4×Dic158C23148yes—
83C4×Dic158C23148yes—
84C4×Dic158C23124yes—
85#14579126C23148yes—
86C4×D316C23124yes—
87C4×C22⋊D79C23148yes—
88#14579120C23148yes—
89#14579119C23148yes—
90#14579118C23148yes—
91C2×C4⋊Dic79C2316yes—
92C2×C4⋊Dic79C2316yes—
93C2×C4⋊Dic79C2316yes—
94C2×C4⋊Dic79C2313yes—
95C2×C4⋊Dic79C2311yescharacteristic
96C2×Dic79⋊C4C2316yes—
97C2×Dic79⋊C4C2316yes—
98C2×Dic79⋊C4C2316yes—
99C2×Dic79⋊C4C23112yes—
100C2×Dic79⋊C4C2316yes—
101C2×C22⋊Dic79C2316yes—
102C2×(C2×D79)⋊C4C2312yes—
103C2×(C2×D79)⋊C4C2316yes—
104C2×(C2×D79)⋊C4C2316yes—
105C2×(C2×D79)⋊C4C2316yes—
106C2×C4×Dic79C2316yes—
107C2×C4×Dic79C23112yes—
108C22×Q8×C79—28no—
109C22×Q8×C79C2318yes—
110C22×Q8×C79C2313yes—
111C22×Q8×C79C2311yescharacteristic
112C23×C4×C79C2311yescharacteristic
113C23×C4×C79C2313yes—
114C23×C4×C79C2313yes—
115C4×Q8×C79C23124yes—
116C4×Q8×C79C2318yes—
117C4×Q8×C79C23124yes—
118C4×Q8×C79C2318yes—
119C2×C4⋊C4×C79C2313yes—
120C2×C4⋊C4×C79C2313yes—
121C2×C4⋊C4×C79C2316yes—
122C2×C4⋊C4×C79C23124yes—
123C2×C42×C79C2313yes—
124C2×C42×C79C2313yes—
125C2×C42×C79C2313yes—
126C2×C42×C79C2313yes—
127C2×C42×C79C23124yes—
128C23×D79C2412yes—
129C2×Dic158—224no—
130C2×Dic158—212no—
131C2×Dic158C24124yes—
132C2×Dic158C24112yes—
133Q8×D79—264no—
134C2×D316—24no—
135C2×D316C2414yes—
136C2×C22⋊D79—28no—
137C2×C22⋊D79C2418yes—
138C22×Dic79C2416yes—
139C22×Dic79C2416yes—
140C22×Dic79C2412yes—
141C2×C4×D79—224no—
142C2×C4×D79C24124yes—
143C4⋊Dic79C24124yes—
144C4⋊Dic79C24124yes—
145C4⋊Dic79C24124yes—
146C4⋊Dic79C24112yes—
147C4⋊Dic79C2414yes—
148Dic79⋊C4C24124yes—
149Dic79⋊C4C24124yes—
150Dic79⋊C4C24124yes—
151Dic79⋊C4C24148yes—
152Dic79⋊C4C24124yes—
153C22⋊Dic79C24124yes—
154(C2×D79)⋊C4C2418yes—
155(C2×D79)⋊C4C24124yes—
156(C2×D79)⋊C4C24124yes—
157(C2×D79)⋊C4C24124yes—
158C4×Dic79C24124yes—
159C4×Dic79C24148yes—
160C24×C79C2411yescharacteristic
161C2×Q8×C79—232no—
162C2×Q8×C79—212no—
163C2×Q8×C79#4118yes—
164C2×Q8×C79C24112yes—
165C2×Q8×C79C2414yes—
166C22×C4×C79—212no—
167C22×C4×C79—212no—
168C22×C4×C79—24no—
169C22×C4×C79C2411yescharacteristic
170C22×C4×C79C2414yes—
171C22×C4×C79C2413yes—
172C22×C4×C79C2413yes—
173C22×C4×C79C24112yes—
174C22×C4×C79C2413yes—
175C22×C4×C79C24112yes—
176C22×C4×C79C2413yes—
177C22×C4×C79C2411yescharacteristic
178C4⋊C4×C79C24112yes—
179C4⋊C4×C79C24112yes—
180C4⋊C4×C79C24124yes—
181C42×C79C24112yes—
182C42×C79C24112yes—
183C42×C79C24112yes—
184C42×C79C24112yes—
185C23×C79—22no—
186C23×C79—22no—
187C23×C792−1+414yes—
188C23×C79C2512yes—
189C23×C79C2511yescharacteristic
190C22×D79—24no—
191C22×D79—24no—
192C22×D79—24no—
193C22×D79C2514yes—
194Q8×C79—224no—
195Q8×C79C2×#41116yes—
196C2×C4×C79—26no—
197C2×C4×C79—224no—
198C2×C4×C79—26no—
199C2×C4×C79—26no—
200C2×C4×C79—22no—
201C2×C4×C79—224no—
202C2×C4×C79—28no—
203C2×C4×C79C2×#41112yes—
204C2×C4×C792−1+414yes—
205C2×C4×C79C2516yes—
206C2×C4×C79C2512yes—
207C2×C4×C79C2516yes—
208C2×C4×C79C2516yes—
209C2×C4×C79C2516yes—
210C2×C4×C79C2512yes—
211C22⋊D79—216no—
212C2×Dic79—212no—
213C2×Dic79—24no—
214C2×Dic79—212no—
215C2×Dic79C25112yes—
216C2×Dic79C2514yes—
217C2×Dic79C25112yes—
218D316—28no—
219C4×D79—248no—
220Dic158—248no—
221Dic158—224no—
222C22×C79—22no—
223C22×C79—22no—
224C22×C79—28no—
225C22×C79C22×#4112yes—
226C22×C79C2×2−1+414yes—
227C22×C79C2×2−1+412yes—
228C22×C79C2×2−1+412yes—
229C22×C79C2611yescharacteristic
230C2×D79—24no—
231C2×D79—24no—
232C2×D79—24no—
233C2×D79—48no—
234C4×C79—26no—
235C4×C79—26no—
236C4×C79—26no—
237C4×C79—22no—
238C4×C79C22×#41112yes—
239C4×C79C2×2−1+414yes—
240Dic79—212no—
241Dic79—24no—
242Dic79—212no—
243C2×#3523—791nomaximal, Sylow
244C2×C79—24no—
245C2×C79#352314yescomplemented
246C2×C79C23×#4111yescharacteristic
247C2×C79C22×2−1+411yescharacteristic
248C2×C79C22×2−1+411yescharacteristic
249D79—44no—
250#3523—7932no—
251C2×#557—793no—
252C2×#557—791no—
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×Q8D7911yescharacteristic, complemented
262C79C2×#352311yescharacteristic, complemented, Sylow
263C23×Q8—792no—
264C23×Q8C2×D7911yescharacteristic
265#557—7948no—
266#557—7916no—
267#556—7948no—
268#552—7996no—
269D4×Q8—7932no—
270D4×Q8—7916no—
271#541—7996no—
272#533—7948no—
273C2×C4⋊Q8—796no—
274C2×C4⋊Q8—793no—
275C2×C4⋊Q8—793no—
276#529—7948no—
277C2×#126—7912no—
278C2×#126—796no—
279C2×#125—7912no—
280C2×#125—796no—
281C2×#125—793no—
282C2×#125—796no—
283C2×#124—793no—
284C2×#123—796no—
285C2×C22⋊Q8—796no—
286C2×C22⋊Q8—793no—
287C2×C22⋊Q8—796no—
288C2×C22⋊Q8—796no—
289C2×#121—791no—
290#519—7932no—
291C2×C4×Q8—792no—
292C2×C4×Q8—796no—
293C2×C4×Q8—796no—
294C2×C4×Q8—793no—
295C2×C4×Q8—796no—
296C2×C4×Q8C2×D7911yescharacteristic
297C2×C4×Q8C2×D7913yes—
298C2×C4×Q8C2×D7911yescharacteristic
299C2×C4×Q8C2×D7913yes—
300C2×C4×Q8C2×D79116yescomplemented
301C2×C4×D4—796no—
302C2×C4×D4—793no—
303C2×#117—796no—
304C2×#117—796no—
305C2×#117—796no—
306C22×C4⋊C4C2×D7913yes—
307C22×C42C2×D7913yes—
308C22×Q8—28no—
309C22×Q8—7916no—
310C22×Q8—793no—
311C22×Q8—796no—
312C22×Q8—15816no—
313C22×Q8C22×D7913yes—
314C22×Q8C22×D7918yes—
315C22×Q8C22×D7911yescharacteristic
316C22×D4—792no—
317C22×D4—791no—
318C23×C4—796no—
319C23×C4C22×D7913yes—
320C23×C4C22×D7913yes—
321C23×C4C22×D7911yescharacteristic
322C4⋊Q8—7948no—
323C4⋊Q8—7924no—
324C4⋊Q8—7924no—
325#126—7996no—
326#126—7948no—
327#125—7996no—
328#125—7948no—
329#125—7924no—
330#125—7948no—
331#124—7924no—
332#123—7948no—
333C22⋊Q8—7948no—
334C22⋊Q8—7924no—
335C22⋊Q8—7948no—
336C22⋊Q8—7948no—
337#121—798no—
338C4×Q8—7916no—
339C4×Q8—7948no—
340C4×Q8—7948no—
341C4×Q8—7924no—
342C4×Q8—7948no—
343C4×Q8C22×D7918yes—
344C4×Q8C22×D79124yes—
345C4×Q8C22×D7918yes—
346C4×Q8C22×D79124yes—
347C4×D4—7948no—
348C4×D4—7924no—
349#117—7948no—
350#117—7948no—
351#117—7948no—
352C2×C4⋊C4—796no—
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—791no—
360C2×C4⋊C4—7912no—
361C2×C4⋊C4—796no—
362C2×C4⋊C4C22×D7913yes—
363C2×C4⋊C4C22×D7913yes—
364C2×C4⋊C4C22×D7916yes—
365C2×C4⋊C4C22×D79124yes—
366C2×C22⋊C4—796no—
367C2×C22⋊C4—792no—
368C2×C22⋊C4—796no—
369C2×C22⋊C4—796no—
370C2×C22⋊C4—796no—
371C2×C42—7912no—
372C2×C42—796no—
373C2×C42C22×D7913yes—
374C2×C42C22×D7913yes—
375C2×C42C22×D7913yes—
376C2×C42C22×D7913yes—
377C2×C42C22×D79124yes—
378C24—792no—
379C24C23×D7911yescharacteristic
380C2×Q8—232no—
381C2×Q8—212no—
382C2×Q8—7912no—
383C2×Q8—7924no—
384C2×Q8—15864no—
385C2×Q8—15824no—
386C2×Q8—15812no—
387C2×Q8#1192962418yes—
388C2×Q8C23×D79112yes—
389C2×Q8C23×D7914yes—
390C2×D4—798no—
391C2×D4—794no—
392C2×D4—1588no—
393C2×D4—1584no—
394C22×C4—212no—
395C22×C4—212no—
396C22×C4—24no—
397C22×C4—796no—
398C22×C4—792no—
399C22×C4—796no—
400C22×C4—7924no—
401C22×C4—15824no—
402C22×C4C23×D7914yes—
403C22×C4C23×D79112yes—
404C22×C4C23×D79112yes—
405C22×C4C23×D7913yes—
406C22×C4C23×D7911yescharacteristic
407C22×C4C23×D7913yes—
408C22×C4C23×D7913yes—
409C22×C4C23×D7913yes—
410C22×C4C23×D7911yescharacteristic
411C4⋊C4—7924no—
412C4⋊C4—7924no—
413C4⋊C4—7924no—
414C4⋊C4—7948no—
415C4⋊C4—7924no—
416C4⋊C4—7924no—
417C4⋊C4—7924no—
418C4⋊C4—7924no—
419C4⋊C4—7912no—
420C4⋊C4—794no—
421C4⋊C4C23×D79112yes—
422C4⋊C4C23×D79112yes—
423C4⋊C4C23×D79124yes—
424C22⋊C4—7924no—
425C22⋊C4—798no—
426C22⋊C4—7924no—
427C22⋊C4—7924no—
428C22⋊C4—7924no—
429C42—7948no—
430C42—7924no—
431C42C23×D79112yes—
432C42C23×D79112yes—
433C42C23×D79112yes—
434C42C23×D79112yes—
435C23—22no—
436C23—22no—
437C23—794no—
438C23—1584no—
439C23—1584no—
440C23—1584no—
441C23#1457920814yes—
442C23C24×D7912yes—
443C23C24×D7911yescharacteristic
444Q8—224no—
445Q8—15824no—
446Q8—15848no—
447Q8C2×#11929624116yes—
448D4—15816no—
449D4—1588no—
450C2×C4—224no—
451C2×C4—26no—
452C2×C4—26no—
453C2×C4—22no—
454C2×C4—26no—
455C2×C4—28no—
456C2×C4—224no—
457C2×C4—7912no—
458C2×C4—794no—
459C2×C4—7912no—
460C2×C4—1584no—
461C2×C4—15812no—
462C2×C4—15812no—
463C2×C4—15848no—
464C2×C4C2×#11929624112yes—
465C2×C4#1457920914yes—
466C2×C4C24×D7916yes—
467C2×C4C24×D7912yes—
468C2×C4C24×D7916yes—
469C2×C4C24×D7916yes—
470C2×C4C24×D7912yes—
471C2×C4C24×D7916yes—
472C22—22no—
473C22—22no—
474C22—28no—
475C22—3168no—
476C22—1584no—
477C22—1584no—
478C22—1584no—
479C22C22×#1192962412yes—
480C22C2×#1457920814yes—
481C22C2×#1457920812yes—
482C22C2×#1457920912yes—
483C22C25×D7911yescharacteristic
484C4—26no—
485C4—26no—
486C4—26no—
487C4—22no—
488C4—15812no—
489C4—1584no—
490C4—15812no—
491C4C22×#11929624112yes—
492C4C2×#1457920914yes—
493C2—24no—
494C2—3164no—
495C2#2377749614yesdirect product factor
496C2C23×#1192962411yescharacteristic
497C2C22×#1457920811yescharacteristic
498C2C22×#1457920911yescharacteristic
subgroupG/H
centerC23C24×D79
commutator, cosocleC22×C79C26
FrattiniC22C25×D79
FittingC22×C4×Q8×C79C2
socleC23×C79C25
prime radicalC2×#14579135C22
hypercenterC22×C4×Q8D79
hypocenterC79C2×#3523
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #23777496C2direct (N × H)
2. C2×#19099292C2semidirect
3. C2×#19099291C2semidirect
4. C2×#19099286C2semidirect
5. C2×#19099248C2semidirect
6. C2×Q8×D316C2semidirect
7. C2×Q8×C22⋊D79C2semidirect
8. C2×#19099171C2semidirect
9. C2×#19099120C2semidirect
11. C2×#19099039C2semidirect
12. C22×C4×Q8×C79C2semidirect
14. #19099292C22semidirect
15. #19099291C22semidirect
16. #19099286C22semidirect
17. #19099248C22semidirect
18. Q8×D316C22semidirect
19. Q8×C22⋊D79C22semidirect
20. #19099171C22semidirect
21. #19099120C22semidirect
39. #19099039C22semidirect
55. C2×C4×Q8×C79C22semidirect
245. C2×C79#3523semidirect
261. C22×C4×Q8D79semidirect
262. C79C2×#3523semidirect
300. C2×C4×Q8C2×D79semidirect
495. C2#23777496direct (N × H)

Maximal subgroups

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

subgroupindex
#237774962
C2×#190992922
C2×#190992912
C2×#190992862
C2×#190992482
C2×Q8×D3162
C2×Q8×C22⋊D792
C2×#190991712
C2×#190991202
C2×#190990972
C2×#190990392
C22×C4×Q8×C792
C2×#352379