Finite Group Databasebeta

← C2×#22624276 #33993669

Subgroups — C2×#22624276 #33993669

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#22624276C2132yesmaximal, direct product factor
2C2×#17989336C214yesmaximal, complemented
3C2×#17989303C211yesmaximal, characteristic, complemented
4C2×#17989297C212yesmaximal, complemented
5C2×#17989278C214yesmaximal, complemented
6C2×#17989240C214yesmaximal, complemented
7C2×#17989214C214yesmaximal, complemented
8C2×#17989209C214yesmaximal, complemented
9C2×#17989162C214yesmaximal, complemented
10C2×C4⋊Q8×D73C211yesmaximal, characteristic, complemented
11C2×#17989124C212yesmaximal, complemented
12C2×Q82×C73C211yesmaximal, characteristic, complemented
13C22×#11924029C2214yes—
14C22×Q8×D73C2212yes—
15#17989336C22164yescomplemented
16#17989303C22116yescomplemented
17#17989297C22132yescomplemented
18#17989278C22164yescomplemented
19#17989240C22164yescomplemented
20#17989214C22164yescomplemented
21#17989209C22164yescomplemented
22#17989162C22164yescomplemented
23C2×#14556591C2211yescharacteristic
24C2×#14556590C2214yes—
25C2×#14556589C2214yes—
26C2×#14556589C2212yes—
27C4⋊Q8×D73C22116yescomplemented
28#17989124C22132yescomplemented
29C2×#14556586C2214yes—
30C2×#14556585C2218yes—
31C2×#14556583C2214yes—
32C2×#14556582C2212yes—
33C2×#14556581C2218yes—
34C2×#14556580C2214yes—
35C2×#14556580C2212yes—
36C2×#14556579C2214yes—
37C2×#14556576C2214yes—
38C2×#14556576C2218yes—
39C2×#14556576C2218yes—
40C2×#14556569C2218yes—
41C2×#14556569C2218yes—
42C2×#14556567C2218yes—
43C2×#14556567C2214yes—
44C2×#14556567C2212yes—
45C2×#14556566C2214yes—
46C2×#14556566C2212yes—
47C2×C4×D292C2214yes—
48C2×C4×D292C2214yes—
49C2×C4×D292C2212yes—
50C2×#14556549C2214yes—
51C2×#14556548C2214yes—
52C2×#14556548C2214yes—
53C2×C4⋊C4×D73C2214yes—
54C2×C4⋊C4×D73C2214yes—
55C2×C42×D73C2211yescharacteristic
56Q82×C73C22116yescomplemented
57C2×C4⋊Q8×C73C2214yes—
58C2×C4⋊Q8×C73C2214yes—
59C2×C4⋊Q8×C73C2211yescharacteristic
60C2×C4×Q8×C73C2214yes—
61C2×C4×Q8×C73C2212yes—
62C2×#11924029—416no—
63C2×#11924029C23132yes—
64C22×Dic146C2314yes—
65C22×Dic146C2312yes—
66C2×Q8×D73—48no—
67C2×Q8×D73C23116yes—
68C22×D292C2314yes—
69C22×D292C2312yes—
70C22×D292C2314yes—
71C22×C4×D73C2311yescharacteristic
72C22×C4×D73C2314yes—
73C22×C4×D73C2312yes—
74C22×C4×D73C2314yes—
75C22×C4×D73C2318yes—
76#14556591C2318yes—
77#14556590C23132yes—
78#14556589C23116yes—
79#14556589C23132yes—
80#14556586C23132yes—
81#14556585C23164yes—
82#14556583C23132yes—
83#14556582C23116yes—
84#14556581C23164yes—
85#14556580C23116yes—
86#14556580C23132yes—
87#14556579C23132yes—
88#14556576C23132yes—
89#14556576C23164yes—
90#14556576C23164yes—
91#14556569C23164yes—
92#14556569C23164yes—
93#14556567C23116yes—
94#14556567C23132yes—
95#14556567C23164yes—
96#14556566C23116yes—
97#14556566C23132yes—
98C4×D292C23132yes—
99C4×D292C23116yes—
100C4×D292C23132yes—
101#14556549C23132yes—
102#14556548C23132yes—
103#14556548C23132yes—
104C2×C4⋊Dic73C2314yes—
105C2×C4⋊Dic73C2312yes—
106C2×C4⋊Dic73C2314yes—
107C2×#11924007C2314yes—
108C2×#11924007C2318yes—
109C2×#11924007C2318yes—
110C2×#11924007C2318yes—
111C2×#11924007C2314yes—
112C4⋊C4×D73C23132yes—
113C4⋊C4×D73C23132yes—
114C2×#11924005C2314yes—
115C2×#11924005C2314yes—
116C2×#11924005C2314yes—
117C2×#11924005C2318yes—
118C2×#11924005C2318yes—
119C2×#11924005C2314yes—
120C2×#11924005C2314yes—
121C2×C4×Dic73C2311yescharacteristic
122C2×C4×Dic73C2314yes—
123C2×C4×Dic73C2312yes—
124C42×D73C2318yes—
125C22×Q8×C73C2312yes—
126C22×Q8×C73C2312yes—
127C22×Q8×C73C2314yes—
128C4⋊Q8×C73C2318yes—
129C4⋊Q8×C73C23132yes—
130C4⋊Q8×C73C23132yes—
131C4×Q8×C73C23116yes—
132C4×Q8×C73C23132yes—
133C2×C4⋊C4×C73C2314yes—
134C2×C4⋊C4×C73C2312yes—
135C2×C4⋊C4×C73C2314yes—
136C2×C4⋊C4×C73C2318yes—
137C2×C42×C73C2311yescharacteristic
138C2×C42×C73C2314yes—
139C2×C42×C73C2314yes—
140C23×D73C2414yes—
141C23×D73C2411yescharacteristic
142#11924029—464no—
143C2×Dic146—44no—
144C2×Dic146—48no—
145C2×Dic146C2418yes—
146C2×Dic146C24116yes—
147Q8×D73—432no—
148C2×D292—216no—
149C2×D292—28no—
150C2×D292—48no—
151C2×D292C2418yes—
152C2×D292C24116yes—
153C2×D292C24116yes—
154C22×Dic73C2412yes—
155C22×Dic73C2414yes—
156C22×Dic73C2411yescharacteristic
157C22×Dic73C2414yes—
158C2×C4×D73—24no—
159C2×C4×D73—216no—
160C2×C4×D73—28no—
161C2×C4×D73—216no—
162C2×C4×D73—416no—
163C2×C4×D73C2418yes—
164C2×C4×D73C2414yes—
165C2×C4×D73C24116yes—
166C2×C4×D73C24116yes—
167C2×C4×D73C24132yes—
168C4⋊Dic73C24116yes—
169C4⋊Dic73C2418yes—
170C4⋊Dic73C24116yes—
171#11924007C24132yes—
172#11924007C24132yes—
173#11924007C24132yes—
174#11924007C24116yes—
175#11924007C24116yes—
176#11924005C24116yes—
177#11924005C24116yes—
178#11924005C24116yes—
179#11924005C24132yes—
180#11924005C24132yes—
181#11924005C24116yes—
182#11924005C24116yes—
183C4×Dic73C2418yes—
184C4×Dic73C2414yes—
185C4×Dic73C24116yes—
186C2×Q8×C73—28no—
187C2×Q8×C73—44no—
188C2×Q8×C73—48no—
189C2×Q8×C73C2418yes—
190C2×Q8×C73C24116yes—
191C2×Q8×C73C2418yes—
192C22×C4×C73C2414yes—
193C22×C4×C73C2411yescharacteristic
194C22×C4×C73C2414yes—
195C22×C4×C73C2414yes—
196C22×C4×C73C2412yes—
197C4⋊C4×C73C2418yes—
198C4⋊C4×C73C24116yes—
199C4⋊C4×C73C24116yes—
200C4⋊C4×C73C24132yes—
201C42×C73C2414yes—
202C42×C73C24116yes—
203C42×C73C24116yes—
204C23×C73C2511yescharacteristic
205C22×D73—24no—
206C22×D73—28no—
207C22×D73—28no—
208C22×D73—28no—
209C22×D73H(5,𝔽2)14yes—
210C22×D73C2518yes—
211C22×D73C2512yes—
212Q8×C73—216no—
213Q8×C73—48no—
214Q8×C73—416no—
215C2×C4×C73—28no—
216C2×C4×C73—22no—
217C2×C4×C73—28no—
218C2×C4×C73—28no—
219C2×C4×C732−1+418yes—
220C2×C4×C73C2518yes—
221C2×C4×C73C2512yes—
222C2×C4×C73C2518yes—
223C2×C4×C73C2518yes—
224C2×C4×C73C2514yes—
225C2×Dic73—28no—
226C2×Dic73—22no—
227C2×Dic73—24no—
228C2×Dic73—28no—
229C2×Dic73C2514yes—
230C2×Dic73C2518yes—
231C2×Dic73C2512yes—
232C2×Dic73C2518yes—
233D292—232no—
234D292—216no—
235D292—416no—
236C4×D73—28no—
237C4×D73—232no—
238C4×D73—216no—
239C4×D73—232no—
240C4×D73—432no—
241Dic146—416no—
242Dic146—48no—
243C22×C73C2×H(5,𝔽2)12yes—
244C22×C73C2×2−1+414yes—
245C22×C73C2611yescharacteristic
246C2×D73—24no—
247C2×D73—28no—
248C2×D73—28no—
249C2×D73—28no—
250C2×D73—28no—
251C2×D73—416no—
252C2×D73C2×H(5,𝔽2)14yes—
253C4×C73—22no—
254C4×C73—28no—
255C4×C73—28no—
256C4×C73—28no—
257C4×C73C2×2−1+418yes—
258Dic73—22no—
259Dic73—28no—
260Dic73—24no—
261Dic73—28no—
262C2×#3544—731nomaximal, Sylow
263C2×C73#354414yescomplemented
264C2×C73C22×H(5,𝔽2)11yescharacteristic
265C2×C73C22×2−1+412yes—
266D73—24no—
267D73—48no—
268#3544—7332no—
269C2×#563—734no—
270C2×Q82D7311yescharacteristic, complemented
271C2×#555—731no—
272C2×#554—732no—
273C2×#552—734no—
274C2×#547—734no—
275C2×#544—734no—
276C2×#542—734no—
277C2×#536—734no—
278C22×C4⋊Q8—731no—
279C2×#529—732no—
280C73C2×#354411yescharacteristic, complemented, Sylow
281C22×#41—734no—
282C23×Q8—732no—
283#563—7364no—
284Q82C2×D73116yescomplemented
285#555—7316no—
286#554—7332no—
287#552—7364no—
288#547—7364no—
289#544—7364no—
290#542—7364no—
291#536—7364no—
292C2×C4⋊Q8—732no—
293C2×C4⋊Q8—734no—
294C2×C4⋊Q8—7316no—
295C2×C4⋊Q8—734no—
296C2×C4⋊Q8—731no—
297C2×C4⋊Q8C2×D7311yescharacteristic
298C2×C4⋊Q8C2×D7314yes—
299C2×C4⋊Q8C2×D7314yes—
300#529—7332no—
301C2×#126—734no—
302C2×#126—738no—
303C2×#125—732no—
304C2×#125—738no—
305C2×#125—734no—
306C2×#124—732no—
307C2×#124—734no—
308C2×#124—734no—
309C2×#123—734no—
310C2×#123—738no—
311C2×#123—738no—
312C2×C22⋊Q8—732no—
313C2×C22⋊Q8—734no—
314C2×C22⋊Q8—738no—
315C2×C22⋊Q8—738no—
316C2×C22⋊Q8—738no—
317C2×#121—732no—
318C2×#121—734no—
319C2×C4×Q8C2×D7312yes—
320C2×C4×Q8C2×D7314yes—
321C2×C4×D4—734no—
322C2×C4×D4—732no—
323C2×C4×D4—734no—
324C2×#117—734no—
325C2×#117—734no—
326C2×#117—734no—
327C22×C4⋊C4—734no—
328C22×C4⋊C4—734no—
329C22×C42—731no—
330C2×#41—29216no—
331C2×#41—7332no—
332C22×Q8—2928no—
333C22×Q8—7316no—
334C22×Q8—732no—
335C22×Q8—734no—
336C22×Q8C22×D7312yes—
337C22×Q8C22×D7314yes—
338C22×Q8C22×D7312yes—
339C22×D4—734no—
340C22×D4—732no—
341C22×D4—734no—
342C23×C4—738no—
343C23×C4—734no—
344C23×C4—732no—
345C23×C4—734no—
346C23×C4—731no—
347C4⋊Q8—7316no—
348C4⋊Q8—7332no—
349C4⋊Q8—7332no—
350C4⋊Q8—738no—
351C4⋊Q8C22×D7318yes—
352C4⋊Q8C22×D73132yes—
353C4⋊Q8C22×D73132yes—
354#126—7332no—
355#126—7364no—
356#125—7316no—
357#125—7364no—
358#125—7332no—
359#124—7316no—
360#124—7332no—
361#124—7332no—
362#123—7332no—
363#123—7364no—
364#123—7364no—
365C22⋊Q8—7316no—
366C22⋊Q8—7332no—
367C22⋊Q8—7364no—
368C22⋊Q8—7364no—
369C22⋊Q8—7364no—
370#121—7316no—
371#121—7332no—
372C4×Q8C22×D73116yes—
373C4×Q8C22×D73132yes—
374C4×D4—7332no—
375C4×D4—7316no—
376C4×D4—7332no—
377#117—7332no—
378#117—7332no—
379#117—7332no—
380C2×C4⋊C4—734no—
381C2×C4⋊C4—738no—
382C2×C4⋊C4—738no—
383C2×C4⋊C4—732no—
384C2×C4⋊C4—738no—
385C2×C4⋊C4—734no—
386C2×C4⋊C4—734no—
387C2×C4⋊C4—734no—
388C2×C4⋊C4—7332no—
389C2×C4⋊C4—7332no—
390C2×C4⋊C4C22×D7312yes—
391C2×C4⋊C4C22×D7314yes—
392C2×C4⋊C4C22×D7314yes—
393C2×C4⋊C4C22×D7318yes—
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—732no—
403C2×C42—731no—
404C2×C42—738no—
405C2×C42C22×D7314yes—
406C2×C42C22×D7314yes—
407C2×C42C22×D7311yescharacteristic
408C24—731no—
409C24—734no—
410#41—29264no—
411C2×Q8—28no—
412C2×Q8—44no—
413C2×Q8—48no—
414C2×Q8—2928no—
415C2×Q8—2924no—
416C2×Q8—29232no—
417C2×Q8—738no—
418C2×Q8—7316no—
419C2×Q8C23×D7318yes—
420C2×Q8C23×D73116yes—
421C2×Q8C23×D7318yes—
422C2×D4—2928no—
423C2×D4—7316no—
424C2×D4—738no—
425C2×D4—7316no—
426C2×D4—14616no—
427C2×D4—1468no—
428C22×C4—29216no—
429C22×C4—7332no—
430C22×C4—734no—
431C22×C4—732no—
432C22×C4—731no—
433C22×C4—734no—
434C22×C4—738no—
435C22×C4—734no—
436C22×C4—7316no—
437C22×C4—7316no—
438C22×C4—1464no—
439C22×C4—14616no—
440C22×C4—14616no—
441C22×C4—1468no—
442C22×C4C23×D7311yescharacteristic
443C22×C4C23×D7314yes—
444C22×C4C23×D7314yes—
445C22×C4C23×D7314yes—
446C22×C4C23×D7312yes—
447C4⋊C4—7332no—
448C4⋊C4—7332no—
449C4⋊C4—7332no—
450C4⋊C4—7316no—
451C4⋊C4—7316no—
452C4⋊C4—7316no—
453C4⋊C4—738no—
454C4⋊C4—7316no—
455C4⋊C4C23×D7318yes—
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—738no—
467C42—734no—
468C42—7316no—
469C42C23×D7314yes—
470C42C23×D73116yes—
471C42C23×D73116yes—
472C23—734no—
473C23—732no—
474C23—738no—
475C23—1468no—
476C23—1468no—
477C23—1468no—
478C23—1464no—
479C23C24×D7311yescharacteristic
480Q8—216no—
481Q8—48no—
482Q8—416no—
483Q8—29216no—
484Q8—2928no—
485D4—29216no—
486D4—14632no—
487D4—14616no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—28no—
492C2×C4—29232no—
493C2×C4—734no—
494C2×C4—738no—
495C2×C4—732no—
496C2×C4—738no—
497C2×C4—1468no—
498C2×C4—14632no—
499C2×C4—1468no—
500C2×C4—1462no—
501C2×C4—1468no—
502C2×C4—1464no—
503C2×C4—14632no—
504C2×C4—14616no—
505C2×C4#1455663918yes—
506C2×C4C24×D7312yes—
507C2×C4C24×D7318yes—
508C2×C4C24×D7318yes—
509C2×C4C24×D7318yes—
510C2×C4C24×D7314yes—
511C22—29216no—
512C22—734no—
513C22—1468no—
514C22—1468no—
515C22—1468no—
516C22—1468no—
517C22—1464no—
518C22H(5,𝔽2)×D7312yes—
519C22C2×#1455663914yes—
520C22C25×D7311yescharacteristic
521C4—28no—
522C4—28no—
523C4—22no—
524C4—28no—
525C4—1468no—
526C4—1462no—
527C4—1464no—
528C4—1468no—
529C4C2×#1455663918yes—
530C2—2928no—
531C2—1464no—
532C2#2262427614yesdirect product factor
533C2C2×H(5,𝔽2)×D7311yescharacteristic
534C2C22×#1455663912yes—
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×Q82×C73C2
socleC23×C73C25
hypercenterC2×Q82D73
hypocenterC73C2×#3544
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. #22624276C2direct (N × H)
2. C2×#17989336C2semidirect
3. C2×#17989303C2semidirect
4. C2×#17989297C2semidirect
5. C2×#17989278C2semidirect
6. C2×#17989240C2semidirect
7. C2×#17989214C2semidirect
8. C2×#17989209C2semidirect
9. C2×#17989162C2semidirect
10. C2×C4⋊Q8×D73C2semidirect
11. C2×#17989124C2semidirect
12. C2×Q82×C73C2semidirect
15. #17989336C22semidirect
16. #17989303C22semidirect
17. #17989297C22semidirect
18. #17989278C22semidirect
19. #17989240C22semidirect
20. #17989214C22semidirect
21. #17989209C22semidirect
22. #17989162C22semidirect
27. C4⋊Q8×D73C22semidirect
28. #17989124C22semidirect
56. Q82×C73C22semidirect
263. C2×C73#3544semidirect
270. C2×Q82D73semidirect
280. C73C2×#3544semidirect
284. Q82C2×D73semidirect
532. C2#22624276direct (N × H)

Maximal subgroups

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

subgroupindex
#226242762
C2×#179893362
C2×#179893032
C2×#179892972
C2×#179892782
C2×#179892402
C2×#179892142
C2×#179892092
C2×#179891622
C2×C4⋊Q8×D732
C2×#179891242
C2×Q82×C732
C2×#354473