Finite Group Databasebeta

← C2×#22624275 #33993668

Subgroups — C2×#22624275 #33993668

Order 18688.

7086 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22624275C2132yesmaximal, direct product factor
2C2×#17989336C214yesmaximal, complemented
3C2×#17989317C211yesmaximal, characteristic, complemented
4C2×#17989303C211yesmaximal, characteristic, complemented
5C2×#17989298C212yesmaximal, complemented
6C2×#17989272C214yesmaximal, complemented
7C2×#17989238C214yesmaximal, complemented
8C2×#17989214C214yesmaximal, complemented
9C2×#17989211C214yesmaximal, complemented
10C2×#17989161C214yesmaximal, complemented
11C2×#17989121C212yesmaximal, complemented
12C22×C4⋊Q8×C73C211yesmaximal, characteristic, complemented
13C22×#11924028C2214yes—
14#17989336C22164yescomplemented
15#17989317C22116yescomplemented
16#17989303C22116yescomplemented
17#17989298C22132yescomplemented
18#17989272C22164yescomplemented
19#17989238C22164yescomplemented
20#17989214C22164yescomplemented
21#17989211C22164yescomplemented
22#17989161C22164yescomplemented
23C2×#14556590C2214yes—
24C2×#14556589C2214yes—
25C2×#14556589C2214yes—
26#17989121C22132yescomplemented
27C2×#14556586C2218yes—
28C2×#14556585C2214yes—
29C2×#14556583C2212yes—
30C2×#14556582C2214yes—
31C2×#14556581C2218yes—
32C2×#14556580C2214yes—
33C2×#14556580C2214yes—
34C2×#14556579C2212yes—
35C2×#14556576C2218yes—
36C2×#14556574C2214yes—
37C2×#14556574C2218yes—
38C2×#14556571C2214yes—
39C2×#14556570C2212yes—
40C2×#14556569C2218yes—
41C2×#14556567C2218yes—
42C2×#14556567C2218yes—
43C2×#14556566C2214yes—
44C2×#14556565C2212yes—
45C2×C4×Dic146C2212yes—
46C2×#14556556C2214yes—
47C2×C4×D292C2212yes—
48C2×#14556553C2214yes—
49C2×C4×C22⋊D73C2214yes—
50C2×#14556550C2214yes—
51C2×#14556549C2214yes—
52C2×#14556548C2214yes—
53C2×C4⋊C4×D73C2214yes—
54C23×Q8×C73C2212yes—
55C2×C4⋊Q8×C73C2211yescharacteristic
56C2×C4⋊Q8×C73C2212yes—
57C2×C4⋊Q8×C73C2214yes—
58C2×C4⋊Q8×C73C2211yescharacteristic
59C2×C4⋊Q8×C73C22116yescomplemented
60C22×C4⋊C4×C73C2214yes—
61C22×C42×C73C2211yescharacteristic
62C2×#11924028—416no—
63C2×#11924028C23132yes—
64C22×Dic146C2312yes—
65C22×Dic146C2314yes—
66C22×D292C2312yes—
67C22×D292C2314yes—
68C22×C22⋊D73C2314yes—
69C22×C4×D73C2314yes—
70C22×C4×D73C2318yes—
71#14556590C23132yes—
72#14556589C23132yes—
73#14556589C23132yes—
74#14556586C23164yes—
75#14556585C23132yes—
76#14556583C23116yes—
77#14556582C23132yes—
78#14556581C23164yes—
79#14556580C23132yes—
80#14556580C23132yes—
81#14556579C23116yes—
82#14556576C23164yes—
83#14556574C23132yes—
84#14556574C23164yes—
85#14556571C23132yes—
86#14556570C23116yes—
87#14556569C23164yes—
88#14556567C23164yes—
89#14556567C23164yes—
90#14556566C23132yes—
91#14556565C23116yes—
92C4×Dic146C23116yes—
93#14556556C23132yes—
94C4×D292C23116yes—
95#14556553C23132yes—
96C4×C22⋊D73C23132yes—
97#14556550C23132yes—
98#14556549C23132yes—
99#14556548C23132yes—
100C2×C4⋊Dic73C2312yes—
101C2×C4⋊Dic73C2312yes—
102C2×C4⋊Dic73C2318yes—
103C2×#11924007C2314yes—
104C2×#11924007C2314yes—
105C2×#11924007C2318yes—
106C2×#11924007C2318yes—
107C2×#11924007C2314yes—
108C2×#11924007C2314yes—
109C4⋊C4×D73C23132yes—
110C2×C22⋊Dic73C2314yes—
111C2×#11924005C2314yes—
112C2×#11924005C2314yes—
113C2×#11924005C2318yes—
114C2×#11924005C2314yes—
115C2×#11924005C2318yes—
116C2×#11924005C2314yes—
117C2×C4×Dic73C2314yes—
118C2×C4×Dic73C2314yes—
119C2×C4×Dic73C2314yes—
120C22×Q8×C73—48no—
121C22×Q8×C73C2312yes—
122C22×Q8×C73C2314yes—
123C22×Q8×C73C23116yes—
124C22×Q8×C73C2312yes—
125C23×C4×C73C2314yes—
126C23×C4×C73C2312yes—
127C23×C4×C73C2311yescharacteristic
128C4⋊Q8×C73C23132yes—
129C4⋊Q8×C73C2318yes—
130C4⋊Q8×C73C2318yes—
131C4⋊Q8×C73C23116yes—
132C2×C4⋊C4×C73C2314yes—
133C2×C4⋊C4×C73C23132yes—
134C2×C4⋊C4×C73C2314yes—
135C2×C4⋊C4×C73C2318yes—
136C2×C42×C73C2318yes—
137C2×C42×C73C2312yes—
138C2×C42×C73C2311yescharacteristic
139C2×C42×C73C2311yescharacteristic
140C23×D73C2414yes—
141#11924028—464no—
142C2×Dic146—28no—
143C2×Dic146—48no—
144C2×Dic146C2418yes—
145C2×Dic146C24116yes—
146C2×D292—28no—
147C2×D292—48no—
148C2×D292C2418yes—
149C2×D292C24116yes—
150C2×C22⋊D73—216no—
151C2×C22⋊D73C24116yes—
152C22×Dic73C2414yes—
153C22×Dic73C2414yes—
154C22×Dic73C2414yes—
155C2×C4×D73—216no—
156C2×C4×D73—416no—
157C2×C4×D73C24116yes—
158C2×C4×D73C24132yes—
159C4⋊Dic73C2418yes—
160C4⋊Dic73C2418yes—
161C4⋊Dic73C24132yes—
162#11924007C24132yes—
163#11924007C24116yes—
164#11924007C24116yes—
165#11924007C24116yes—
166#11924007C24132yes—
167#11924007C24116yes—
168C22⋊Dic73C24116yes—
169#11924005C24116yes—
170#11924005C24116yes—
171#11924005C24132yes—
172#11924005C24116yes—
173#11924005C24132yes—
174#11924005C24116yes—
175C4×Dic73C24116yes—
176C4×Dic73C24116yes—
177C4×Dic73C24116yes—
178C24×C73C2411yescharacteristic
179C2×Q8×C73—44no—
180C2×Q8×C73—48no—
181C2×Q8×C73—44no—
182C2×Q8×C73—432no—
183C2×Q8×C73C2418yes—
184C2×Q8×C73C24116yes—
185C2×Q8×C73C2418yes—
186C22×C4×C73—24no—
187C22×C4×C73—216no—
188C22×C4×C73—28no—
189C22×C4×C73C2412yes—
190C22×C4×C73C2418yes—
191C22×C4×C73C2411yescharacteristic
192C22×C4×C73C2411yescharacteristic
193C22×C4×C73C2414yes—
194C22×C4×C73C2414yes—
195C22×C4×C73C2414yes—
196C22×C4×C73C24116yes—
197C22×C4×C73C2412yes—
198C4⋊C4×C73C24116yes—
199C4⋊C4×C73C24116yes—
200C4⋊C4×C73C24132yes—
201C42×C73C2418yes—
202C42×C73C2414yes—
203C42×C73C2414yes—
204C23×C73—24no—
205C23×C73H(5,𝔽2)14yes—
206C23×C73C2512yes—
207C23×C73C2511yescharacteristic
208C22×D73—28no—
209C22×D73—28no—
210C22×D73—28no—
211C22×D73C2518yes—
212Q8×C73—48no—
213Q8×C73—416no—
214Q8×C73—48no—
215C2×C4×C73—28no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—28no—
219C2×C4×C73—28no—
220C2×C4×C73—232no—
221C2×C4×C73—24no—
222C2×C4×C73—216no—
223C2×C4×C732−1+418yes—
224C2×C4×C73C2514yes—
225C2×C4×C73C2512yes—
226C2×C4×C73C2512yes—
227C2×C4×C73C2518yes—
228C2×C4×C73C2518yes—
229C2×C4×C73C2514yes—
230C22⋊D73—232no—
231C2×Dic73—28no—
232C2×Dic73—28no—
233C2×Dic73—28no—
234C2×Dic73C2518yes—
235C2×Dic73C2518yes—
236C2×Dic73C2518yes—
237D292—216no—
238D292—416no—
239C4×D73—232no—
240C4×D73—432no—
241Dic146—216no—
242Dic146—416no—
243C22×C73—24no—
244C22×C73—28no—
245C22×C73C2×H(5,𝔽2)14yes—
246C22×C73C2×H(5,𝔽2)12yes—
247C22×C73C2×2−1+414yes—
248C22×C73C2611yescharacteristic
249C2×D73—28no—
250C2×D73—28no—
251C2×D73—28no—
252C2×D73—416no—
253C4×C73—28no—
254C4×C73—28no—
255C4×C73—22no—
256C4×C73—22no—
257C4×C73—24no—
258C4×C73C2×2−1+418yes—
259Dic73—28no—
260Dic73—28no—
261Dic73—28no—
262C2×#3544—731nomaximal, Sylow
263C2×C73—24no—
264C2×C73#354414yescomplemented
265C2×C73C22×H(5,𝔽2)11yescharacteristic
266C2×C73C22×2−1+412yes—
267D73—48no—
268#3544—7332no—
269C2×#563—734no—
270C2×Q82—731no—
271C2×#555—731no—
272C2×#554—732no—
273C2×#552—734no—
274C2×#547—734no—
275C2×#544—734no—
276C2×#542—734no—
277C2×#536—734no—
278C22×C4⋊Q8D7311yescharacteristic, complemented
279C2×#529—732no—
280C73C2×#354411yescharacteristic, complemented, Sylow
281C22×#41—734no—
282C23×Q8C2×D7312yes—
283#563—7364no—
284Q82—7316no—
285#555—7316no—
286#554—7332no—
287#552—7364no—
288#547—7364no—
289#544—7364no—
290#542—7364no—
291#536—7364no—
292C2×C4⋊Q8—734no—
293C2×C4⋊Q8—734no—
294C2×C4⋊Q8—734no—
295C2×C4⋊Q8C2×D7314yes—
296C2×C4⋊Q8C2×D7311yescharacteristic
297C2×C4⋊Q8C2×D7311yescharacteristic
298C2×C4⋊Q8C2×D7312yes—
299C2×C4⋊Q8C2×D73116yescomplemented
300#529—7332no—
301C2×#126—738no—
302C2×#126—734no—
303C2×#125—734no—
304C2×#125—738no—
305C2×#125—732no—
306C2×#124—734no—
307C2×#124—734no—
308C2×#124—732no—
309C2×#123—734no—
310C2×#123—738no—
311C2×#123—738no—
312C2×C22⋊Q8—738no—
313C2×C22⋊Q8—738no—
314C2×C22⋊Q8—732no—
315C2×C22⋊Q8—734no—
316C2×C22⋊Q8—738no—
317C2×#121—732no—
318C2×#121—734no—
319C2×C4×Q8—734no—
320C2×C4×Q8—732no—
321C2×C4×D4—734no—
322C2×C4×D4—734no—
323C2×C4×D4—732no—
324C2×#117—734no—
325C2×#117—734no—
326C2×#117—734no—
327C22×C4⋊C4—734no—
328C22×C4⋊C4C2×D7314yes—
329C22×C42C2×D7311yescharacteristic
330C2×#41—29216no—
331C2×#41—7332no—
332C22×Q8—48no—
333C22×Q8—734no—
334C22×Q8—732no—
335C22×Q8C22×D7312yes—
336C22×Q8C22×D7314yes—
337C22×Q8C22×D7312yes—
338C22×Q8C22×D73116yes—
339C22×D4—734no—
340C22×D4—734no—
341C22×D4—732no—
342C23×C4—738no—
343C23×C4—734no—
344C23×C4C22×D7314yes—
345C23×C4C22×D7312yes—
346C23×C4C22×D7311yescharacteristic
347C4⋊Q8—7332no—
348C4⋊Q8—7332no—
349C4⋊Q8—7332no—
350C4⋊Q8C22×D73132yes—
351C4⋊Q8C22×D7318yes—
352C4⋊Q8C22×D7318yes—
353C4⋊Q8C22×D73116yes—
354#126—7364no—
355#126—7332no—
356#125—7332no—
357#125—7364no—
358#125—7316no—
359#124—7332no—
360#124—7332no—
361#124—7316no—
362#123—7332no—
363#123—7364no—
364#123—7364no—
365C22⋊Q8—7364no—
366C22⋊Q8—7364no—
367C22⋊Q8—7316no—
368C22⋊Q8—7332no—
369C22⋊Q8—7364no—
370#121—7316no—
371#121—7332no—
372C4×Q8—7332no—
373C4×Q8—7316no—
374C4×D4—7332no—
375C4×D4—7332no—
376C4×D4—7316no—
377#117—7332no—
378#117—7332no—
379#117—7332no—
380C2×C4⋊C4—732no—
381C2×C4⋊C4—738no—
382C2×C4⋊C4—732no—
383C2×C4⋊C4—738no—
384C2×C4⋊C4—734no—
385C2×C4⋊C4—734no—
386C2×C4⋊C4—734no—
387C2×C4⋊C4—734no—
388C2×C4⋊C4—738no—
389C2×C4⋊C4—7332no—
390C2×C4⋊C4C22×D7314yes—
391C2×C4⋊C4C22×D7314yes—
392C2×C4⋊C4C22×D7318yes—
393C2×C4⋊C4C22×D73132yes—
394C2×C22⋊C4—734no—
395C2×C22⋊C4—734no—
396C2×C22⋊C4—738no—
397C2×C22⋊C4—734no—
398C2×C22⋊C4—738no—
399C2×C22⋊C4—734no—
400C2×C22⋊C4—734no—
401C2×C42—734no—
402C2×C42—734no—
403C2×C42—734no—
404C2×C42C22×D7312yes—
405C2×C42C22×D7311yescharacteristic
406C2×C42C22×D7311yescharacteristic
407C2×C42C22×D7318yes—
408C24—734no—
409C24C23×D7311yescharacteristic
410#41—29264no—
411C2×Q8—44no—
412C2×Q8—48no—
413C2×Q8—44no—
414C2×Q8—432no—
415C2×Q8—2928no—
416C2×Q8—7316no—
417C2×Q8—738no—
418C2×Q8—1468no—
419C2×Q8C23×D7318yes—
420C2×Q8C23×D73116yes—
421C2×Q8C23×D7318yes—
422C2×D4—2928no—
423C2×D4—7316no—
424C2×D4—7316no—
425C2×D4—738no—
426C2×D4—14616no—
427C2×D4—1468no—
428C22×C4—24no—
429C22×C4—216no—
430C22×C4—28no—
431C22×C4—29216no—
432C22×C4—7332no—
433C22×C4—734no—
434C22×C4—734no—
435C22×C4—734no—
436C22×C4—7316no—
437C22×C4—14616no—
438C22×C4C23×D7318yes—
439C22×C4C23×D7314yes—
440C22×C4C23×D73116yes—
441C22×C4C23×D7314yes—
442C22×C4C23×D7312yes—
443C22×C4C23×D7311yescharacteristic
444C22×C4C23×D7311yescharacteristic
445C22×C4C23×D7314yes—
446C22×C4C23×D7312yes—
447C4⋊C4—7332no—
448C4⋊C4—7316no—
449C4⋊C4—7316no—
450C4⋊C4—7316no—
451C4⋊C4—7316no—
452C4⋊C4—7332no—
453C4⋊C4—738no—
454C4⋊C4—738no—
455C4⋊C4—7332no—
456C4⋊C4C23×D73116yes—
457C4⋊C4C23×D73116yes—
458C4⋊C4C23×D73132yes—
459C22⋊C4—7316no—
460C22⋊C4—7316no—
461C22⋊C4—7332no—
462C22⋊C4—7316no—
463C22⋊C4—7332no—
464C22⋊C4—7316no—
465C22⋊C4—7316no—
466C42—7316no—
467C42—7316no—
468C42—7316no—
469C42C23×D7318yes—
470C42C23×D7314yes—
471C42C23×D7314yes—
472C23—24no—
473C23—738no—
474C23—1468no—
475C23—1468no—
476C23—1468no—
477C23#1455663614yes—
478C23C24×D7312yes—
479C23C24×D7311yescharacteristic
480Q8—48no—
481Q8—416no—
482Q8—48no—
483Q8—29216no—
484Q8—14616no—
485D4—29216no—
486D4—14632no—
487D4—14616no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—232no—
494C2×C4—24no—
495C2×C4—216no—
496C2×C4—29232no—
497C2×C4—738no—
498C2×C4—738no—
499C2×C4—738no—
500C2×C4—1468no—
501C2×C4—1468no—
502C2×C4—1468no—
503C2×C4—14632no—
504C2×C4#1455663918yes—
505C2×C4C24×D7318yes—
506C2×C4C24×D7314yes—
507C2×C4C24×D7312yes—
508C2×C4C24×D7312yes—
509C2×C4C24×D7318yes—
510C2×C4C24×D7314yes—
511C22—28no—
512C22—24no—
513C22—29216no—
514C22—1468no—
515C22—1468no—
516C22—1468no—
517C22C2×#1455663614yes—
518C22C2×#1455663612yes—
519C22C2×#1455663914yes—
520C22C25×D7311yescharacteristic
521C4—28no—
522C4—28no—
523C4—22no—
524C4—22no—
525C4—24no—
526C4—1468no—
527C4—1468no—
528C4—1468no—
529C4C2×#1455663918yes—
530C2—24no—
531C2—2928no—
532C2#2262427514yesdirect product factor
533C2C22×#1455663611yescharacteristic
534C2C22×#1455663912yes—
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC22×C4⋊Q8×C73C2
socleC23×C73C25
hypercenterC22×C4⋊Q8D73
hypocenterC73C2×#3544
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22624275C2direct (N × H)
2. C2×#17989336C2semidirect
3. C2×#17989317C2semidirect
4. C2×#17989303C2semidirect
5. C2×#17989298C2semidirect
6. C2×#17989272C2semidirect
7. C2×#17989238C2semidirect
8. C2×#17989214C2semidirect
9. C2×#17989211C2semidirect
10. C2×#17989161C2semidirect
11. C2×#17989121C2semidirect
12. C22×C4⋊Q8×C73C2semidirect
14. #17989336C22semidirect
15. #17989317C22semidirect
16. #17989303C22semidirect
17. #17989298C22semidirect
18. #17989272C22semidirect
19. #17989238C22semidirect
20. #17989214C22semidirect
21. #17989211C22semidirect
22. #17989161C22semidirect
26. #17989121C22semidirect
59. C2×C4⋊Q8×C73C22semidirect
264. C2×C73#3544semidirect
278. C22×C4⋊Q8D73semidirect
280. C73C2×#3544semidirect
299. C2×C4⋊Q8C2×D73semidirect
532. C2#22624275direct (N × H)

Maximal subgroups

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

subgroupindex
#226242752
C2×#179893362
C2×#179893172
C2×#179893032
C2×#179892982
C2×#179892722
C2×#179892382
C2×#179892142
C2×#179892112
C2×#179891612
C2×#179891212
C22×C4⋊Q8×C732
C2×#354473