Finite Group Databasebeta

← C2×#22623551 #33993083

Subgroups — C2×#22623551 #33993083

Order 18688.

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#22623551C2132yesmaximal, direct product factor
2C2×#17989310C213yesmaximal, complemented
3C2×#17989275C216yesmaximal, complemented
4C2×Q8×D292C211yesmaximal, characteristic, complemented
5C2×#17989249C212yesmaximal, complemented
6C2×#17989240C213yesmaximal, complemented
7C2×#17989142C213yesmaximal, complemented
8C2×#17989122C216yesmaximal, complemented
9C2×C4×#11924029C213yesmaximal, complemented
10C2×C4×#11924029C211yesmaximal, characteristic, complemented
11C2×C4×Q8×D73C212yesmaximal, complemented
12C2×#557×C73C211yesmaximal, characteristic, complemented
13C22×#11924029C2213yes—
14C22×#11924029C2211yescharacteristic
15C22×Q8×D73C2212yes—
16#17989310C22148yescomplemented
17#17989275C22196yescomplemented
18Q8×D292C22116yescomplemented
19#17989249C22132yescomplemented
20#17989240C22148yescomplemented
21#17989142C22148yescomplemented
22C2×#14556591C2213yes—
23#17989122C22196yescomplemented
24C2×#14556586C22112yes—
25C2×#14556582C2216yes—
26C2×#14556579C2216yes—
27C2×#14556576C22112yes—
28C2×#14556571C2216yes—
29C2×#14556571C2216yes—
30C2×#14556567C2216yes—
31C2×#14556566C2216yes—
32C4×#11924029C22116yescomplemented
33C4×#11924029C22148yescomplemented
34C2×Q8×Dic73C2211yescharacteristic
35C2×Q8×Dic73C2213yes—
36C2×C4×Dic146C2216yes—
37C2×#14556556C2216yes—
38C4×Q8×D73C22132yescomplemented
39C2×C4×D292C2213yes—
40C2×C4×D292C2213yes—
41C2×C4×D292C2216yes—
42C2×#14556553C2211yescharacteristic
43C2×#14556553C2213yes—
44C2×#14556553C2216yes—
45C2×#14556549C2213yes—
46C2×#14556549C2216yes—
47C2×#14556549C2213yes—
48C2×#14556549C2216yes—
49C2×C4⋊C4×D73C2216yes—
50C2×C42×D73C2216yes—
51C2×C42×D73C2213yes—
52#557×C73C22116yescomplemented
53C2×C4⋊Q8×C73C2213yes—
54C2×#125×C73C2216yes—
55C2×C4×Q8×C73C2212yes—
56C2×C4×Q8×C73C2211yescharacteristic
57C2×C4×Q8×C73C2213yes—
58C2×#11924029—224no—
59C2×#11924029—28no—
60C2×#11924029C23124yes—
61C2×#11924029C2318yes—
62C22×Dic146C2316yes—
63C2×Q8×D73—216no—
64C2×Q8×D73C23116yes—
65C22×D292C2316yes—
66C22×D292C2313yes—
67C22×D292C2311yescharacteristic
68C22×C4×D73C2316yes—
69C22×C4×D73C2316yes—
70C22×C4×D73C2313yes—
71C22×C4×D73C2313yes—
72C22×C4×D73C2312yes—
73C22×C4×D73C2313yes—
74#14556591C23124yes—
75#14556586C23196yes—
76#14556582C23148yes—
77#14556579C23148yes—
78#14556576C23196yes—
79#14556571C23148yes—
80#14556571C23148yes—
81#14556567C23148yes—
82#14556566C23148yes—
83Q8×Dic73C23124yes—
84Q8×Dic73C2318yes—
85C4×Dic146C23148yes—
86#14556556C23148yes—
87C4×D292C23124yes—
88C4×D292C23124yes—
89C4×D292C23148yes—
90#14556553C23124yes—
91#14556553C23148yes—
92#14556553C2318yes—
93#14556549C23148yes—
94#14556549C23124yes—
95#14556549C23148yes—
96#14556549C23124yes—
97C2×C4⋊Dic73C2313yes—
98C2×C4⋊Dic73C2316yes—
99C2×C4⋊Dic73C2313yes—
100C2×C4⋊Dic73C2316yes—
101C2×#11924007C23112yes—
102C4⋊C4×D73C23148yes—
103C2×#11924005C2312yes—
104C2×#11924005C2316yes—
105C2×#11924005C2316yes—
106C2×#11924005C2316yes—
107C2×#11924005C23112yes—
108C2×C4×Dic73C2313yes—
109C2×C4×Dic73C2311yescharacteristic
110C2×C4×Dic73C2313yes—
111C2×C4×Dic73C2316yes—
112C2×C4×Dic73C2316yes—
113C42×D73C23148yes—
114C42×D73C23124yes—
115C22×Q8×C73C2313yes—
116C22×Q8×C73C2311yescharacteristic
117C4⋊Q8×C73C23124yes—
118#125×C73C23148yes—
119C4×Q8×C73C23116yes—
120C4×Q8×C73C2318yes—
121C4×Q8×C73C23124yes—
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2313yes—
124C2×C4⋊C4×C73C2316yes—
125C2×C4⋊C4×C73C2316yes—
126C2×C4⋊C4×C73C2316yes—
127C2×C42×C73C2313yes—
128C2×C42×C73C2316yes—
129C23×D73C2413yes—
130C23×D73C2412yes—
131#11924029—232no—
132#11924029—296no—
133C2×Dic146—224no—
134C2×Dic146C24124yes—
135Q8×D73—264no—
136C2×D292—224no—
137C2×D292—212no—
138C2×D292#4118yescomplemented
139C2×D292C24124yes—
140C2×D292C24112yes—
141C2×D292C2414yes—
142C22×Dic73C2413yes—
143C22×Dic73C2411yescharacteristic
144C22×Dic73C2416yes—
145C22×Dic73C2411yescharacteristic
146C2×C4×D73—224no—
147C2×C4×D73—224no—
148C2×C4×D73—212no—
149C2×C4×D73—212no—
150C2×C4×D73—28no—
151C2×C4×D73—212no—
152C2×C4×D73C2418yes—
153C2×C4×D73C24124yes—
154C2×C4×D73C24124yes—
155C2×C4×D73C24112yes—
156C2×C4×D73C24112yes—
157C2×C4×D73C24112yes—
158C4⋊Dic73C24112yes—
159C4⋊Dic73C24112yes—
160C4⋊Dic73C24124yes—
161C4⋊Dic73C24124yes—
162#11924007C24148yes—
163#11924005C2418yes—
164#11924005C24124yes—
165#11924005C24124yes—
166#11924005C24124yes—
167#11924005C24148yes—
168C4×Dic73C24112yes—
169C4×Dic73C2414yes—
170C4×Dic73C24112yes—
171C4×Dic73C24124yes—
172C4×Dic73C24124yes—
173C2×Q8×C73—212no—
174C2×Q8×C73#4118yescomplemented
175C2×Q8×C73C24112yes—
176C2×Q8×C73C2414yes—
177C22×C4×C73C2412yes—
178C22×C4×C73C2413yes—
179C22×C4×C73C2416yes—
180C22×C4×C73C2413yes—
181C22×C4×C73C2411yescharacteristic
182C4⋊C4×C73C2414yes—
183C4⋊C4×C73C24112yes—
184C4⋊C4×C73C24124yes—
185C4⋊C4×C73C24124yes—
186C4⋊C4×C73C24124yes—
187C42×C73C24124yes—
188C42×C73C24112yes—
189C23×C73C2511yescharacteristic
190C22×D73—24no—
191C22×D73—24no—
192C22×D73—26no—
193C22×D73—26no—
194C22×D73—26no—
195C22×D73C2×#4118yes—
196C22×D73C2516yes—
197C22×D73C2514yes—
198Q8×C73—224no—
199Q8×C73C2×#41116yescomplemented
200C2×C4×C73—212no—
201C2×C4×C73—24no—
202C2×C4×C73—26no—
203C2×C4×C73C2×#4114yes—
204C2×C4×C73C2×#41112yes—
205C2×C4×C73C2514yes—
206C2×C4×C73C2516yes—
207C2×C4×C73C25112yes—
208C2×C4×C73C2516yes—
209C2×C4×C73C2512yes—
210C2×Dic73—26no—
211C2×Dic73—22no—
212C2×Dic73—212no—
213C2×Dic732−1+414yes—
214C2×Dic73C2516yes—
215C2×Dic73C2512yes—
216C2×Dic73C25112yes—
217C2×Dic73C2512yes—
218D292—248no—
219D292—224no—
220D292C2×#41116yescomplemented
221C4×D73—248no—
222C4×D73—248no—
223C4×D73—224no—
224C4×D73—224no—
225C4×D73—216no—
226C4×D73—224no—
227Dic146—248no—
228C22×C73C22×#4112yes—
229C22×C73C22×#4112yes—
230C22×C73C2×2−1+412yes—
231C22×C73C2611yescharacteristic
232C2×D73—24no—
233C2×D73—24no—
234C2×D73—216no—
235C2×D73—26no—
236C2×D73—26no—
237C2×D73—26no—
238C2×D73—412no—
239C2×D73C22×#4118yes—
240C4×C73—212no—
241C4×C73—24no—
242C4×C73—26no—
243C4×C73C22×#4114yes—
244C4×C73C22×#41112yes—
245Dic73—212no—
246Dic73—26no—
247Dic73—22no—
248Dic73C2×2−1+414yes—
249C2×#3482—731nomaximal, Sylow
250C2×C73#348214yescomplemented
251C2×C73C23×#4111yescharacteristic
252C2×C73C23×#4111yescharacteristic
253C2×C73C22×2−1+411yescharacteristic
254D73—28no—
255D73—46no—
256#3482—7332no—
257C2×#557D7311yescharacteristic, complemented
258C2×#556—733no—
259C2×#552—736no—
260C2×D4×Q8—731no—
261C2×#548—732no—
262C2×#547—733no—
263C2×#533—733no—
264C2×#529—736no—
265C2×C4×#41—731no—
266C2×C4×#41—733no—
267C22×C4×Q8—732no—
268C73C2×#348211yescharacteristic, complemented, Sylow
269C22×#41—733no—
270C22×#41—731no—
271C23×Q8—732no—
272#557C2×D73116yescomplemented
273#556—7348no—
274#552—7396no—
275D4×Q8—7316no—
276#548—7332no—
277#547—7348no—
278#533—7348no—
279C2×C4⋊Q8—733no—
280C2×C4⋊Q8C2×D7313yes—
281#529—7396no—
282C2×#126—7312no—
283C2×#125—736no—
284C2×#125C2×D7316yes—
285C2×#124—736no—
286C2×#123—7312no—
287C2×C22⋊Q8—736no—
288C2×C22⋊Q8—736no—
289C2×C22⋊Q8—736no—
290C2×#121—736no—
291C4×#41—7348no—
292C4×#41—7316no—
293C2×C4×Q8—731no—
294C2×C4×Q8—733no—
295C2×C4×Q8—7332no—
296C2×C4×Q8—736no—
297C2×C4×Q8—736no—
298C2×C4×Q8C2×D7312yes—
299C2×C4×Q8C2×D7311yescharacteristic
300C2×C4×Q8C2×D7313yes—
301C2×C4×D4—731no—
302C2×C4×D4—736no—
303C2×C4×D4—733no—
304C2×C4×D4—733no—
305C2×C4×D4—733no—
306C2×C4×D4—736no—
307C2×#117—733no—
308C2×#117—736no—
309C2×#117—733no—
310C2×#117—736no—
311C22×C4⋊C4—736no—
312C22×C42—736no—
313C22×C42—733no—
314C2×#41—7324no—
315C2×#41—738no—
316C2×#41—14624no—
317C2×#41—1468no—
318C22×Q8—7316no—
319C22×Q8—736no—
320C22×Q8—14616no—
321C22×Q8C22×D7313yes—
322C22×Q8C22×D7311yescharacteristic
323C22×D4—731no—
324C22×D4—736no—
325C22×D4—733no—
326C23×C4—736no—
327C23×C4—736no—
328C23×C4—733no—
329C23×C4—733no—
330C23×C4—733no—
331C23×C4—732no—
332C4⋊Q8—7324no—
333C4⋊Q8C22×D73124yes—
334#126—7396no—
335#125—7348no—
336#125C22×D73148yes—
337#124—7348no—
338#123—7396no—
339C22⋊Q8—7348no—
340C22⋊Q8—7348no—
341C22⋊Q8—7348no—
342#121—7348no—
343C4×Q8—738no—
344C4×Q8—7324no—
345C4×Q8—7348no—
346C4×Q8—7348no—
347C4×Q8C22×D73116yes—
348C4×Q8C22×D7318yes—
349C4×Q8C22×D73124yes—
350C4×D4—738no—
351C4×D4—7324no—
352C4×D4—7348no—
353C4×D4—7324no—
354C4×D4—7324no—
355C4×D4—7348no—
356#117—7324no—
357#117—7348no—
358#117—7324no—
359#117—7348no—
360C2×C4⋊C4—733no—
361C2×C4⋊C4—7312no—
362C2×C4⋊C4—736no—
363C2×C4⋊C4—733no—
364C2×C4⋊C4—736no—
365C2×C4⋊C4—7348no—
366C2×C4⋊C4C22×D7316yes—
367C2×C4⋊C4C22×D7311yescharacteristic
368C2×C4⋊C4C22×D7313yes—
369C2×C4⋊C4C22×D7316yes—
370C2×C4⋊C4C22×D7316yes—
371C2×C22⋊C4—732no—
372C2×C22⋊C4—736no—
373C2×C22⋊C4—736no—
374C2×C22⋊C4—736no—
375C2×C22⋊C4—7312no—
376C2×C42—733no—
377C2×C42—736no—
378C2×C42—736no—
379C2×C42—731no—
380C2×C42—733no—
381C2×C42—7348no—
382C2×C42—7324no—
383C2×C42C22×D7313yes—
384C2×C42C22×D7316yes—
385C24—732no—
386C24—733no—
387#41—14696no—
388#41—14632no—
389C2×Q8—212no—
390C2×Q8—7324no—
391C2×Q8—14664no—
392C2×Q8—14624no—
393C2×Q8#1192402918yescomplemented
394C2×Q8C23×D73112yes—
395C2×Q8C23×D7314yes—
396C2×D4—738no—
397C2×D4—734no—
398C2×D4—7324no—
399C2×D4—7312no—
400C2×D4—14624no—
401C2×D4—14612no—
402C22×C4—731no—
403C22×C4—733no—
404C22×C4—731no—
405C22×C4—736no—
406C22×C4—738no—
407C22×C4—7324no—
408C22×C4—7324no—
409C22×C4—7312no—
410C22×C4—7312no—
411C22×C4—7312no—
412C22×C4—14612no—
413C22×C4—1468no—
414C22×C4—14624no—
415C22×C4—14612no—
416C22×C4—14624no—
417C22×C4—14612no—
418C22×C4C23×D7312yes—
419C22×C4C23×D7313yes—
420C22×C4C23×D7316yes—
421C22×C4C23×D7311yescharacteristic
422C22×C4C23×D7313yes—
423C4⋊C4—7348no—
424C4⋊C4—7312no—
425C4⋊C4—7324no—
426C4⋊C4—7312no—
427C4⋊C4—7324no—
428C4⋊C4C23×D73124yes—
429C4⋊C4C23×D7314yes—
430C4⋊C4C23×D73112yes—
431C4⋊C4C23×D73124yes—
432C4⋊C4C23×D73124yes—
433C22⋊C4—738no—
434C22⋊C4—7324no—
435C22⋊C4—7324no—
436C22⋊C4—7324no—
437C22⋊C4—7348no—
438C42—734no—
439C42—7312no—
440C42—7312no—
441C42—7324no—
442C42—7324no—
443C42C23×D73112yes—
444C42C23×D73124yes—
445C23—738no—
446C23—734no—
447C23—736no—
448C23—1466no—
449C23—1466no—
450C23—1466no—
451C23—1464no—
452C23—1464no—
453C23C24×D7311yescharacteristic
454Q8—224no—
455Q8—14648no—
456Q8C2×#11924029116yescomplemented
457D4—7316no—
458D4—14648no—
459D4—14624no—
460C2×C4—212no—
461C2×C4—24no—
462C2×C4—26no—
463C2×C4—734no—
464C2×C4—732no—
465C2×C4—736no—
466C2×C4—732no—
467C2×C4—7312no—
468C2×C4—14624no—
469C2×C4—14616no—
470C2×C4—14648no—
471C2×C4—1466no—
472C2×C4—1462no—
473C2×C4—14612no—
474C2×C4—14624no—
475C2×C4—14648no—
476C2×C4—14624no—
477C2×C4#41×D7314yes—
478C2×C4C2×#11924029112yes—
479C2×C4C24×D7314yes—
480C2×C4C24×D7316yes—
481C2×C4C24×D73112yes—
482C2×C4C24×D7312yes—
483C2×C4C24×D7316yes—
484C22—29212no—
485C22—738no—
486C22—1466no—
487C22—1466no—
488C22—1466no—
489C22—14616no—
490C22—1464no—
491C22—1464no—
492C22C2×#41×D7312yes—
493C22C22×#1192402912yes—
494C222−1+4×D7312yes—
495C22C25×D7311yescharacteristic
496C4—212no—
497C4—24no—
498C4—26no—
499C4—734no—
500C4—1466no—
501C4—1462no—
502C4—14612no—
503C4C2×#41×D7314yes—
504C4C22×#11924029112yes—
505C2—2926no—
506C2—1468no—
507C2#2262355114yesdirect product factor
508C2C22×#41×D7311yescharacteristic
509C2C23×#1192402911yescharacteristic
510C2C2×2−1+4×D7311yescharacteristic
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×#557×C73C2
socleC23×C73C25
hypercenterC2×#557D73
hypocenterC73C2×#3482
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. #22623551C2direct (N × H)
2. C2×#17989310C2semidirect
3. C2×#17989275C2semidirect
4. C2×Q8×D292C2semidirect
5. C2×#17989249C2semidirect
6. C2×#17989240C2semidirect
7. C2×#17989142C2semidirect
8. C2×#17989122C2semidirect
9. C2×C4×#11924029C2semidirect
10. C2×C4×#11924029C2semidirect
11. C2×C4×Q8×D73C2semidirect
12. C2×#557×C73C2semidirect
16. #17989310C22semidirect
17. #17989275C22semidirect
18. Q8×D292C22semidirect
19. #17989249C22semidirect
20. #17989240C22semidirect
21. #17989142C22semidirect
23. #17989122C22semidirect
32. C4×#11924029C22semidirect
33. C4×#11924029C22semidirect
38. C4×Q8×D73C22semidirect
52. #557×C73C22semidirect
138. C2×D292#41semidirect
174. C2×Q8×C73#41semidirect
199. Q8×C73C2×#41semidirect
220. D292C2×#41semidirect
250. C2×C73#3482semidirect
257. C2×#557D73semidirect
268. C73C2×#3482semidirect
272. #557C2×D73semidirect
393. C2×Q8#11924029semidirect
456. Q8C2×#11924029semidirect
507. C2#22623551direct (N × H)

Maximal subgroups

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

subgroupindex
#226235512
C2×#179893102
C2×#179892752
C2×Q8×D2922
C2×#179892492
C2×#179892402
C2×#179891422
C2×#179891222
C2×C4×#119240292
C2×C4×Q8×D732
C2×#557×C732
C2×#348273