Finite Group Databasebeta

← C2×#22624017 #33994196

Subgroups — C2×#22624017 #33994196

Order 18688.

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#22624017C2132yesmaximal, direct product factor
2C2×#17989322C213yesmaximal, complemented
3C2×#17989298C213yesmaximal, complemented
4C2×#17989259C211yesmaximal, characteristic, complemented
5C2×#17989249C212yesmaximal, complemented
6C2×#17989240C216yesmaximal, complemented
7C2×#17989211C216yesmaximal, complemented
8C2×#17989138C213yesmaximal, complemented
9C2×#17989124C213yesmaximal, complemented
10C2×#17989069C212yesmaximal, complemented
11C2×C4×Q8×D73C211yesmaximal, characteristic, complemented
12C2×Q82×C73C211yesmaximal, characteristic, complemented
13C22×#11924029C2212yes—
14C22×#11924029C2216yes—
15C22×Q8×D73C2211yescharacteristic
16#17989322C22148yescomplemented
17#17989298C22148yescomplemented
18#17989259C22116yescomplemented
19#17989249C22132yescomplemented
20#17989240C22196yescomplemented
21#17989211C22196yescomplemented
22#17989138C22148yescomplemented
23C2×#14556588C2213yes—
24#17989124C22148yescomplemented
25C2×#14556586C2216yes—
26C2×#14556582C2213yes—
27C2×#14556580C2213yes—
28C2×#14556580C2216yes—
29C2×#14556579C2216yes—
30C2×#14556576C22112yes—
31C2×#14556576C2216yes—
32C2×#14556571C2216yes—
33C2×#14556567C22112yes—
34C2×#14556567C2211yescharacteristic
35C2×#14556566C2216yes—
36C2×#14556566C2216yes—
37C2×#14556566C2213yes—
38#17989069C22132yescomplemented
39C2×Q8×Dic73C2211yescharacteristic
40C2×C4×Dic146C2213yes—
41C2×#14556556C2213yes—
42C4×Q8×D73C22116yescomplemented
43C2×C4×D292C2216yes—
44C2×C4×D292C2216yes—
45C2×C4×D292C2213yes—
46C2×#14556553C2216yes—
47C2×#14556553C2216yes—
48C2×#14556548C2216yes—
49C2×C4⋊C4×D73C2213yes—
50C2×C4⋊C4×D73C2216yes—
51C2×C42×D73C2213yes—
52Q82×C73C22116yescomplemented
53C2×C4⋊Q8×C73C2216yes—
54C2×C4⋊Q8×C73C2213yes—
55C2×C4×Q8×C73C2213yes—
56C2×C4×Q8×C73C2211yescharacteristic
57C2×C4×Q8×C73C2212yes—
58C2×#11924029—216no—
59C2×#11924029—424no—
60C2×#11924029C23116yes—
61C2×#11924029C23148yes—
62C22×Dic146C2313yes—
63C2×Q8×D73—28no—
64C2×Q8×D73C2318yes—
65C22×D292C2313yes—
66C22×D292C2316yes—
67C22×D292C2316yes—
68C22×D292C2316yes—
69C22×C4×D73C2313yes—
70C22×C4×D73C2313yes—
71C22×C4×D73C2316yes—
72C22×C4×D73C2311yescharacteristic
73C22×C4×D73C23112yes—
74#14556588C23124yes—
75#14556586C23148yes—
76#14556582C23124yes—
77#14556580C23124yes—
78#14556580C23148yes—
79#14556579C23148yes—
80#14556576C23148yes—
81#14556576C23196yes—
82#14556571C23148yes—
83#14556567C2318yes—
84#14556567C23196yes—
85#14556566C23124yes—
86#14556566C23148yes—
87#14556566C23148yes—
88Q8×Dic73C2318yes—
89C4×Dic146C23124yes—
90#14556556C23124yes—
91C4×D292C23148yes—
92C4×D292C23124yes—
93C4×D292C23148yes—
94#14556553C23148yes—
95#14556553C23148yes—
96#14556548C23148yes—
97C2×C4⋊Dic73C2313yes—
98C2×C4⋊Dic73C2311yescharacteristic
99C2×C4⋊Dic73C2316yes—
100C2×#11924007C2316yes—
101C2×#11924007C2316yes—
102C2×#11924007C2312yes—
103C4⋊C4×D73C23124yes—
104C4⋊C4×D73C23148yes—
105C2×#11924005C2316yes—
106C2×#11924005C2316yes—
107C2×#11924005C23112yes—
108C2×#11924005C2312yes—
109C2×#11924005C2316yes—
110C2×#11924005C2316yes—
111C2×C4×Dic73C2313yes—
112C2×C4×Dic73C2313yes—
113C2×C4×Dic73C2313yes—
114C42×D73C23124yes—
115C22×Q8×C73C2311yescharacteristic
116C22×Q8×C73C2316yes—
117C22×Q8×C73C2311yescharacteristic
118C4⋊Q8×C73C23124yes—
119C4⋊Q8×C73C23148yes—
120C4×Q8×C73C23124yes—
121C4×Q8×C73C23116yes—
122C4×Q8×C73C2318yes—
123C2×C4⋊C4×C73C2316yes—
124C2×C4⋊C4×C73C2313yes—
125C2×C4⋊C4×C73C2313yes—
126C2×C4⋊C4×C73C2316yes—
127C2×C42×C73C2313yes—
128C2×C42×C73C2316yes—
129C23×D73C2416yes—
130C23×D73C2411yescharacteristic
131#11924029—264no—
132#11924029—496no—
133C2×Dic146—212no—
134C2×Dic146C24112yes—
135Q8×D73—232no—
136C2×D292—224no—
137C2×D292—224no—
138C2×D292—212no—
139C2×D292—412no—
140C2×D292C24112yes—
141C2×D292C24124yes—
142C2×D292C24124yes—
143C2×D292C24124yes—
144C22×Dic73C2411yescharacteristic
145C22×Dic73C2413yes—
146C22×Dic73C2413yes—
147C22×Dic73C2412yes—
148C2×C4×D73—212no—
149C2×C4×D73—212no—
150C2×C4×D73—224no—
151C2×C4×D73—24no—
152C2×C4×D73—424no—
153C2×C4×D73C2414yes—
154C2×C4×D73C24112yes—
155C2×C4×D73C24112yes—
156C2×C4×D73C24124yes—
157C2×C4×D73C24148yes—
158C4⋊Dic73C24112yes—
159C4⋊Dic73C2414yes—
160C4⋊Dic73C24124yes—
161#11924007C24124yes—
162#11924007C24124yes—
163#11924007C2418yes—
164#11924005C24124yes—
165#11924005C24124yes—
166#11924005C24148yes—
167#11924005C2418yes—
168#11924005C24124yes—
169#11924005C24124yes—
170C4×Dic73C24112yes—
171C4×Dic73C24112yes—
172C4×Dic73C24112yes—
173C2×Q8×C73—24no—
174C2×Q8×C73—412no—
175C2×Q8×C73#4118yescomplemented
176C2×Q8×C73C2414yes—
177C2×Q8×C73C24124yes—
178C2×Q8×C73C2414yes—
179C22×C4×C73C2412yes—
180C22×C4×C73C2413yes—
181C22×C4×C73C2416yes—
182C22×C4×C73C2413yes—
183C22×C4×C73C2411yescharacteristic
184C4⋊C4×C73C24124yes—
185C4⋊C4×C73C24112yes—
186C4⋊C4×C73C24112yes—
187C4⋊C4×C73C24124yes—
188C42×C73C24112yes—
189C42×C73C24124yes—
190C23×C73C2511yescharacteristic
191C22×D73—22no—
192C22×D73—22no—
193C22×D73—212no—
194C22×D73—212no—
195C22×D73—212no—
196C22×D73H(5,𝔽2)14yes—
197C22×D73C25112yes—
198C22×D73C2512yes—
199Q8×C73—28no—
200Q8×C73—424no—
201Q8×C73C2×#41116yescomplemented
202C2×C4×C73—26no—
203C2×C4×C73—212no—
204C2×C4×C73—24no—
205C2×C4×C73C2×#41112yes—
206C2×C4×C732−1+414yes—
207C2×C4×C73C2514yes—
208C2×C4×C73C2516yes—
209C2×C4×C73C25112yes—
210C2×C4×C73C2516yes—
211C2×C4×C73C2512yes—
212C2×Dic73—24no—
213C2×Dic73—26no—
214C2×Dic73—26no—
215C2×Dic73—22no—
216C2×Dic73C2512yes—
217C2×Dic73C2516yes—
218C2×Dic73C2516yes—
219C2×Dic73C2514yes—
220D292—248no—
221D292—248no—
222D292—224no—
223D292—424no—
224C4×D73—224no—
225C4×D73—224no—
226C4×D73—248no—
227C4×D73—28no—
228C4×D73—448no—
229Dic146—224no—
230C22×C73C22×#4112yes—
231C22×C73C2×H(5,𝔽2)12yes—
232C22×C73C2×2−1+412yes—
233C22×C73C2611yescharacteristic
234C2×D73—22no—
235C2×D73—22no—
236C2×D73—28no—
237C2×D73—212no—
238C2×D73—212no—
239C2×D73—212no—
240C2×D73—424no—
241C2×D73C2×H(5,𝔽2)14yes—
242C4×C73—26no—
243C4×C73—212no—
244C4×C73—24no—
245C4×C73C22×#41112yes—
246C4×C73C2×2−1+414yes—
247Dic73—26no—
248Dic73—24no—
249Dic73—26no—
250Dic73—22no—
251C2×#3522—731nomaximal, Sylow
252C2×C73#352214yescomplemented
253C2×C73C23×#4111yescharacteristic
254C2×C73C22×H(5,𝔽2)11yescharacteristic
255C2×C73C22×2−1+411yescharacteristic
256D73—24no—
257D73—412no—
258#3522—7332no—
259C2×#559—733no—
260C2×Q82D7311yescharacteristic, complemented
261C2×#554—733no—
262C2×#550—731no—
263C2×#548—732no—
264C2×#547—736no—
265C2×#542—736no—
266C2×#532—733no—
267C2×#529—733no—
268C2×#520—732no—
269C22×C4×Q8—731no—
270C73C2×#352211yescharacteristic, complemented, Sylow
271C22×#41—736no—
272C22×#41—732no—
273C23×Q8—731no—
274#559—7348no—
275Q82C2×D73116yescomplemented
276#554—7348no—
277#550—7316no—
278#548—7332no—
279#547—7396no—
280#542—7396no—
281#532—7348no—
282C2×C4⋊Q8C2×D7313yes—
283C2×C4⋊Q8C2×D7316yes—
284C2×#127—733no—
285#529—7348no—
286C2×#126—736no—
287C2×#125—733no—
288C2×#124—733no—
289C2×#124—736no—
290C2×#124—736no—
291C2×#123—736no—
292C2×#123—7312no—
293C2×C22⋊Q8—731no—
294C2×C22⋊Q8—7312no—
295C2×C22⋊Q8—736no—
296C2×#121—733no—
297C2×#121—736no—
298C2×#121—736no—
299#520—7332no—
300C2×C4×Q8—731no—
301C2×C4×Q8—7316no—
302C2×C4×Q8—733no—
303C2×C4×Q8—733no—
304C2×C4×Q8C2×D7311yescharacteristic
305C2×C4×Q8C2×D7312yes—
306C2×C4×Q8C2×D7313yes—
307C2×C4×D4—736no—
308C2×C4×D4—736no—
309C2×C4×D4—736no—
310C2×C4×D4—736no—
311C2×C4×D4—733no—
312C2×#117—736no—
313C22×C4⋊C4—733no—
314C22×C4⋊C4—736no—
315C22×C42—733no—
316C2×#41—29224no—
317C2×#41—7348no—
318C2×#41—7316no—
319C2×#41—14616no—
320C22×Q8—738no—
321C22×Q8—733no—
322C22×Q8—1468no—
323C22×Q8C22×D7311yescharacteristic
324C22×Q8C22×D7316yes—
325C22×Q8C22×D7311yescharacteristic
326C22×D4—736no—
327C22×D4—733no—
328C22×D4—736no—
329C22×D4—736no—
330C23×C4—7312no—
331C23×C4—733no—
332C23×C4—736no—
333C23×C4—733no—
334C23×C4—731no—
335C4⋊Q8C22×D73124yes—
336C4⋊Q8C22×D73148yes—
337#127—7324no—
338#126—7348no—
339#125—7324no—
340#124—7324no—
341#124—7348no—
342#124—7348no—
343#123—7348no—
344#123—7396no—
345C22⋊Q8—738no—
346C22⋊Q8—7396no—
347C22⋊Q8—7348no—
348#121—7324no—
349#121—7348no—
350#121—7348no—
351C4×Q8—738no—
352C4×Q8—7324no—
353C4×Q8—7324no—
354C4×Q8C22×D7318yes—
355C4×Q8C22×D73116yes—
356C4×Q8C22×D73124yes—
357C4×D4—7348no—
358C4×D4—7348no—
359C4×D4—7348no—
360C4×D4—7348no—
361C4×D4—7324no—
362#117—7348no—
363C2×C4⋊C4—733no—
364C2×C4⋊C4—736no—
365C2×C4⋊C4—736no—
366C2×C4⋊C4—732no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—736no—
369C2×C4⋊C4—7324no—
370C2×C4⋊C4—7348no—
371C2×C4⋊C4C22×D7316yes—
372C2×C4⋊C4C22×D7313yes—
373C2×C4⋊C4C22×D7313yes—
374C2×C4⋊C4C22×D7316yes—
375C2×C22⋊C4—736no—
376C2×C22⋊C4—736no—
377C2×C22⋊C4—732no—
378C2×C22⋊C4—736no—
379C2×C22⋊C4—7312no—
380C2×C22⋊C4—736no—
381C2×C42—733no—
382C2×C42—733no—
383C2×C42—733no—
384C2×C42—7324no—
385C2×C42C22×D7316yes—
386C2×C42C22×D7313yes—
387C24—731no—
388C24—736no—
389#41—29296no—
390#41—14664no—
391C2×Q8—24no—
392C2×Q8—412no—
393C2×Q8—7312no—
394C2×Q8—14632no—
395C2×Q8—14612no—
396C2×Q8#1192402918yescomplemented
397C2×Q8C23×D7314yes—
398C2×Q8C23×D73124yes—
399C2×Q8C23×D7314yes—
400C2×D4—29212no—
401C2×D4—7324no—
402C2×D4—7312no—
403C2×D4—7324no—
404C2×D4—7324no—
405C2×D4—14624no—
406C2×D4—14624no—
407C2×D4—14612no—
408C22×C4—29224no—
409C22×C4—7348no—
410C22×C4—731no—
411C22×C4—733no—
412C22×C4—733no—
413C22×C4—732no—
414C22×C4—734no—
415C22×C4—7312no—
416C22×C4—7312no—
417C22×C4—7324no—
418C22×C4—14612no—
419C22×C4—14612no—
420C22×C4—14624no—
421C22×C4—1464no—
422C22×C4C23×D7313yes—
423C22×C4C23×D7312yes—
424C22×C4C23×D7316yes—
425C22×C4C23×D7313yes—
426C22×C4C23×D7311yescharacteristic
427C4⋊C4—7324no—
428C4⋊C4—7324no—
429C4⋊C4—738no—
430C4⋊C4—7312no—
431C4⋊C4—734no—
432C4⋊C4—7324no—
433C4⋊C4C23×D73124yes—
434C4⋊C4C23×D73112yes—
435C4⋊C4C23×D73112yes—
436C4⋊C4C23×D73124yes—
437C22⋊C4—7324no—
438C22⋊C4—7324no—
439C22⋊C4—738no—
440C22⋊C4—7324no—
441C22⋊C4—7348no—
442C22⋊C4—7324no—
443C42—7312no—
444C42—7312no—
445C42—7312no—
446C42C23×D73112yes—
447C42C23×D73124yes—
448C23—734no—
449C23—732no—
450C23—7312no—
451C23—14612no—
452C23—14612no—
453C23—14612no—
454C23—1462no—
455C23—1462no—
456C23C24×D7311yescharacteristic
457Q8—28no—
458Q8—424no—
459Q8—14624no—
460Q8C2×#11924029116yescomplemented
461D4—29224no—
462D4—14648no—
463D4—14648no—
464D4—14624no—
465C2×C4—212no—
466C2×C4—26no—
467C2×C4—24no—
468C2×C4—29248no—
469C2×C4—732no—
470C2×C4—736no—
471C2×C4—736no—
472C2×C4—734no—
473C2×C4—14624no—
474C2×C4—1464no—
475C2×C4—1466no—
476C2×C4—1466no—
477C2×C4—1462no—
478C2×C4—14624no—
479C2×C4—14648no—
480C2×C4—1468no—
481C2×C4C2×#11924029112yes—
482C2×C4#1455663914yes—
483C2×C4C24×D7316yes—
484C2×C4C24×D7314yes—
485C2×C4C24×D73112yes—
486C2×C4C24×D7316yes—
487C2×C4C24×D7312yes—
488C22—29224no—
489C22—734no—
490C22—14612no—
491C22—14612no—
492C22—14612no—
493C22—1468no—
494C22—1462no—
495C22—1462no—
496C22C22×#1192402912yes—
497C22H(5,𝔽2)×D7312yes—
498C22C2×#1455663912yes—
499C22C25×D7311yescharacteristic
500C4—212no—
501C4—26no—
502C4—24no—
503C4—1464no—
504C4—1466no—
505C4—1466no—
506C4—1462no—
507C4C22×#11924029112yes—
508C4C2×#1455663914yes—
509C2—29212no—
510C2—1464no—
511C2#2262401714yesdirect product factor
512C2C23×#1192402911yescharacteristic
513C2C2×H(5,𝔽2)×D7311yescharacteristic
514C2C22×#1455663911yescharacteristic
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×Q82×C73C2
socleC23×C73C25
hypercenterC2×Q82D73
hypocenterC73C2×#3522
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. #22624017C2direct (N × H)
2. C2×#17989322C2semidirect
3. C2×#17989298C2semidirect
4. C2×#17989259C2semidirect
5. C2×#17989249C2semidirect
6. C2×#17989240C2semidirect
7. C2×#17989211C2semidirect
8. C2×#17989138C2semidirect
9. C2×#17989124C2semidirect
10. C2×#17989069C2semidirect
11. C2×C4×Q8×D73C2semidirect
12. C2×Q82×C73C2semidirect
16. #17989322C22semidirect
17. #17989298C22semidirect
18. #17989259C22semidirect
19. #17989249C22semidirect
20. #17989240C22semidirect
21. #17989211C22semidirect
22. #17989138C22semidirect
24. #17989124C22semidirect
38. #17989069C22semidirect
42. C4×Q8×D73C22semidirect
52. Q82×C73C22semidirect
175. C2×Q8×C73#41semidirect
201. Q8×C73C2×#41semidirect
252. C2×C73#3522semidirect
260. C2×Q82D73semidirect
270. C73C2×#3522semidirect
275. Q82C2×D73semidirect
396. C2×Q8#11924029semidirect
460. Q8C2×#11924029semidirect
511. C2#22624017direct (N × H)

Maximal subgroups

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

subgroupindex
#226240172
C2×#179893222
C2×#179892982
C2×#179892592
C2×#179892492
C2×#179892402
C2×#179892112
C2×#179891382
C2×#179891242
C2×#179890692
C2×C4×Q8×D732
C2×Q82×C732
C2×#352273