Finite Group Databasebeta

← C2×#22623556 #33993087

Subgroups — C2×#22623556 #33993087

Order 18688.

8418 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623556C2132yesmaximal, direct product factor
2C2×#17989317C211yesmaximal, characteristic, complemented
3C2×#17989300C214yesmaximal, complemented
4C2×#17989299C214yesmaximal, complemented
5C2×#17989258C212yesmaximal, complemented
6C2×#17989255C214yesmaximal, complemented
7C2×#17989138C211yesmaximal, characteristic, complemented
8C2×#17989136C214yesmaximal, complemented
9C2×#17989132C214yesmaximal, complemented
10C2×#17989055C214yesmaximal, complemented
11C2×C4×#11924027C212yesmaximal, complemented
12C2×#555×C73C211yesmaximal, characteristic, complemented
13C22×#11924029C2212yes—
14C22×#11924028C2214yes—
15C22×#11924027C2212yes—
16C22×#11924027C2214yes—
17#17989317C22116yescomplemented
18#17989300C22164yescomplemented
19#17989299C22164yescomplemented
20#17989258C22132yescomplemented
21#17989255C22164yescomplemented
22#17989138C22116yescomplemented
23#17989136C22164yescomplemented
24#17989132C22164yescomplemented
25C2×#14556590C2214yes—
26C2×#14556589C2214yes—
27C2×#14556588C2211yescharacteristic
28C2×#14556587C2214yes—
29C2×#14556587C2214yes—
30C2×#14556580C2212yes—
31C2×#14556579C2214yes—
32C2×#14556577C2218yes—
33C2×#14556566C2214yes—
34C2×#14556565C2214yes—
35C2×#14556564C2218yes—
36C2×#14556564C2218yes—
37C2×#14556563C2218yes—
38#17989055C22164yescomplemented
39C4×#11924027C22132yescomplemented
40C2×C4×Dic146C2212yes—
41C2×#14556556C2214yes—
42C2×#14556556C2212yes—
43C2×D4×Dic73C2212yes—
44C2×#14556553C2214yes—
45C2×#14556553C2214yes—
46C2×C4×C22⋊D73C2214yes—
47C2×#14556551C2214yes—
48C2×#14556551C2218yes—
49C2×#14556551C2218yes—
50C2×#14556549C2212yes—
51C2×#14556548C2218yes—
52C2×#14556547C2214yes—
53C22×C4×Dic73C2214yes—
54C2×C42×D73C2211yescharacteristic
55C2×C42×D73C2214yes—
56#555×C73C22116yescomplemented
57C2×C4⋊Q8×C73C2211yescharacteristic
58C2×#124×C73C2214yes—
59C2×#121×C73C2214yes—
60C2×C4×D4×C73C2212yes—
61C2×#117×C73C2214yes—
62C2×#11924029—48no—
63C2×#11924029C23116yes—
64C2×#11924028—232no—
65C2×#11924028C23132yes—
66C2×#11924027—216no—
67C2×#11924027—416no—
68C2×#11924027C23116yes—
69C2×#11924027C23132yes—
70C22×Dic146C2314yes—
71C22×Dic146C2312yes—
72C22×D292C2314yes—
73C22×D292C2314yes—
74C22×D292C2312yes—
75C22×C22⋊D73C2314yes—
76C22×C22⋊D73C2318yes—
77C22×C22⋊D73C2318yes—
78C23×Dic73C2314yes—
79C23×Dic73C2314yes—
80C22×C4×D73C2314yes—
81C22×C4×D73C2312yes—
82C22×C4×D73C2318yes—
83C22×C4×D73C2311yescharacteristic
84#14556590C23132yes—
85#14556589C23132yes—
86#14556588C2318yes—
87#14556587C23132yes—
88#14556587C23132yes—
89#14556580C23116yes—
90#14556579C23132yes—
91#14556577C23164yes—
92#14556566C23132yes—
93#14556565C23132yes—
94#14556564C23164yes—
95#14556564C23164yes—
96#14556563C23164yes—
97C4×Dic146C23116yes—
98#14556556C23132yes—
99#14556556C23116yes—
100D4×Dic73C23116yes—
101#14556553C23132yes—
102#14556553C23132yes—
103C4×C22⋊D73C23132yes—
104#14556551C23164yes—
105#14556551C23132yes—
106#14556551C23164yes—
107#14556549C23116yes—
108#14556548C23164yes—
109#14556547C23132yes—
110C2×C4⋊Dic73C2312yes—
111C2×#11924007C2314yes—
112C2×#11924007C2318yes—
113C2×#11924007C2314yes—
114C2×C22⋊Dic73C2314yes—
115C2×#11924005C2318yes—
116C2×#11924005C2318yes—
117C2×#11924005C2314yes—
118C2×C4×Dic73C2314yes—
119C2×C4×Dic73C2314yes—
120C2×C4×Dic73C23132yes—
121C2×C4×Dic73C2314yes—
122C2×C4×Dic73C2312yes—
123C2×C4×Dic73C2311yescharacteristic
124C2×C4×Dic73C2314yes—
125C42×D73C23132yes—
126C42×D73C2318yes—
127C22×Q8×C73C2312yes—
128C22×D4×C73C2312yes—
129C22×D4×C73C2314yes—
130C23×C4×C73C2314yes—
131C4⋊Q8×C73C2318yes—
132#124×C73C23132yes—
133#121×C73C23132yes—
134C4×D4×C73C23116yes—
135#117×C73C23132yes—
136C2×C4⋊C4×C73C2314yes—
137C2×C4⋊C4×C73C2312yes—
138C2×C22⋊C4×C73C2318yes—
139C2×C22⋊C4×C73C2314yes—
140C2×C42×C73C2311yescharacteristic
141C2×C42×C73C2314yes—
142C23×D73C2411yescharacteristic
143C23×D73C2414yes—
144#11924029—432no—
145#11924028—2128no—
146#11924027—264no—
147#11924027—464no—
148C2×Dic146—48no—
149C2×Dic146#41116yescomplemented
150C2×Dic146C24116yes—
151C2×Dic146C2418yes—
152C2×D292—216no—
153C2×D292—48no—
154C2×D292—44no—
155C2×D292C24116yes—
156C2×D292C2418yes—
157C2×D292C24116yes—
158C2×C22⋊D73—216no—
159C2×C22⋊D73—232no—
160C2×C22⋊D73—416no—
161C2×C22⋊D73C24132yes—
162C2×C22⋊D73C24116yes—
163C2×C22⋊D73C24132yes—
164C22×Dic73—216no—
165C22×Dic73—216no—
166C22×Dic73C2412yes—
167C22×Dic73C2414yes—
168C22×Dic73C24116yes—
169C22×Dic73C24116yes—
170C22×Dic73C2414yes—
171C22×Dic73C2411yescharacteristic
172C2×C4×D73—28no—
173C2×C4×D73—232no—
174C2×C4×D73—216no—
175C2×C4×D73—24no—
176C2×C4×D73C24132yes—
177C2×C4×D73C2418yes—
178C2×C4×D73C2414yes—
179C2×C4×D73C24116yes—
180C4⋊Dic73C2418yes—
181#11924007C24116yes—
182#11924007C24116yes—
183#11924007C24132yes—
184C22⋊Dic73C24116yes—
185#11924005C24132yes—
186#11924005C24132yes—
187#11924005C24116yes—
188C4×Dic73C24116yes—
189C4×Dic73C2418yes—
190C4×Dic73C2414yes—
191C4×Dic73C24116yes—
192C4×Dic73C24116yes—
193C4×Dic73C24116yes—
194C24×C73C2414yes—
195C2×Q8×C73—44no—
196C2×Q8×C73C2418yes—
197C2×D4×C73—28no—
198C2×D4×C73—48no—
199C2×D4×C73C2418yes—
200C2×D4×C73C24116yes—
201C22×C4×C73—216no—
202C22×C4×C73C2414yes—
203C22×C4×C73C2411yescharacteristic
204C22×C4×C73C2414yes—
205C22×C4×C73C24116yes—
206C22×C4×C73C2412yes—
207C4⋊C4×C73C2418yes—
208C4⋊C4×C73C24116yes—
209C22⋊C4×C73C24132yes—
210C22⋊C4×C73C24116yes—
211C42×C73C24116yes—
212C42×C73C2414yes—
213C23×C73—28no—
214C23×C73—28no—
215C23×C73—28no—
216C23×C73C2518yes—
217C23×C73C2511yescharacteristic
218C22×D73—24no—
219C22×D73—216no—
220C22×D73—28no—
221C22×D73—22no—
222C22×D73C2518yes—
223C22×D73C2512yes—
224Q8×C73—48no—
225D4×C73—216no—
226D4×C73—416no—
227C2×C4×C73—28no—
228C2×C4×C73—28no—
229C2×C4×C73—22no—
230C2×C4×C73—232no—
231C2×C4×C73C2×#4118yes—
232C2×C4×C73C2518yes—
233C2×C4×C73C2512yes—
234C2×C4×C73C2518yes—
235C2×C4×C73C2514yes—
236C22⋊D73—232no—
237C22⋊D73—264no—
238C22⋊D73—432no—
239C2×Dic73—24no—
240C2×Dic73—28no—
241C2×Dic73—232no—
242C2×Dic73—232no—
243C2×Dic73C2×#41116yes—
244C2×Dic73H(5,𝔽2)14yes—
245C2×Dic73C2514yes—
246C2×Dic73C2518yes—
247C2×Dic73C2518yes—
248C2×Dic73C2512yes—
249D292—232no—
250D292—416no—
251D292—48no—
252C4×D73—216no—
253C4×D73—264no—
254C4×D73—232no—
255C4×D73—28no—
256Dic146—416no—
257Dic146C2×#41132yescomplemented
258C22×C73—28no—
259C22×C73—28no—
260C22×C73—28no—
261C22×C73—416no—
262C22×C73C22×#4114yes—
263C22×C73C2×H(5,𝔽2)12yes—
264C22×C73C2611yescharacteristic
265C2×D73—216no—
266C2×D73—24no—
267C2×D73—22no—
268C2×D73—28no—
269C2×D73—44no—
270C2×D73—416no—
271C4×C73—28no—
272C4×C73—28no—
273C4×C73—22no—
274C4×C73C22×#4118yes—
275Dic73—24no—
276Dic73—28no—
277Dic73C22×#41116yes—
278Dic73C2×H(5,𝔽2)14yes—
279C2×#3483—731nomaximal, Sylow
280C2×C73—48no—
281C2×C73#348314yescomplemented
282C2×C73C23×#4112yes—
283C2×C73C22×H(5,𝔽2)11yescharacteristic
284D73—48no—
285D73—42no—
286#3483—7332no—
287C2×Q82—731no—
288C2×#555—734no—
289C2×#555—734no—
290C2×#555D7311yescharacteristic, complemented
291C2×#550—734no—
292C2×#550—732no—
293C2×#532—731no—
294C2×#532—734no—
295C2×#532—734no—
296C2×C4×#41—732no—
297C2×C4×#41—734no—
298C73C2×#348311yescharacteristic, complemented, Sylow
299C22×#41—732no—
300C22×#41—734no—
301C22×#41—732no—
302C22×#41—734no—
303Q82—7316no—
304#555—7364no—
305#555—7364no—
306#555C2×D73116yescomplemented
307#550—7364no—
308#550—7332no—
309#532—7316no—
310#532—7364no—
311#532—7364no—
312C2×C4⋊Q8—734no—
313C2×C4⋊Q8—734no—
314C2×C4⋊Q8C2×D7311yescharacteristic
315C2×#127—734no—
316C2×#127—734no—
317C2×#127—731no—
318C2×#124—732no—
319C2×#124—738no—
320C2×#124—734no—
321C2×#124C2×D7314yes—
322C2×#121—738no—
323C2×#121—734no—
324C2×#121—734no—
325C2×#121—738no—
326C2×#121—738no—
327C2×#121C2×D7314yes—
328C4×#41—7332no—
329C4×#41—7364no—
330C2×C4×Q8—732no—
331C2×C4×Q8—734no—
332C2×C4×Q8—732no—
333C2×C4×D4—732no—
334C2×C4×D4—734no—
335C2×C4×D4—734no—
336C2×C4×D4—734no—
337C2×C4×D4—734no—
338C2×C4×D4—738no—
339C2×C4×D4—738no—
340C2×C4×D4C2×D7312yes—
341C2×#117—732no—
342C2×#117—734no—
343C2×#117—738no—
344C2×#117C2×D7314yes—
345C22×C42—734no—
346C22×C42—731no—
347C22×C42—734no—
348C2×#41—29216no—
349C2×#41—2928no—
350C2×#41—7316no—
351C2×#41—7332no—
352C2×#41—7316no—
353C2×#41—7332no—
354C2×#41—14632no—
355C2×#41—14616no—
356C22×Q8—734no—
357C22×Q8—732no—
358C22×Q8C22×D7312yes—
359C22×D4—738no—
360C22×D4—734no—
361C22×D4—738no—
362C22×D4—734no—
363C22×D4—732no—
364C22×D4—734no—
365C22×D4C22×D7312yes—
366C22×D4C22×D7314yes—
367C23×C4—734no—
368C23×C4—734no—
369C23×C4—738no—
370C23×C4—732no—
371C23×C4—734no—
372C23×C4—731no—
373C23×C4C22×D7314yes—
374C4⋊Q8—7332no—
375C4⋊Q8—7332no—
376C4⋊Q8C22×D7318yes—
377#127—7332no—
378#127—7332no—
379#127—738no—
380#124—7316no—
381#124—7364no—
382#124—7332no—
383#124C22×D73132yes—
384#121—7364no—
385#121—7332no—
386#121—7332no—
387#121—7364no—
388#121—7364no—
389#121C22×D73132yes—
390C4×Q8—7316no—
391C4×Q8—7332no—
392C4×Q8—7316no—
393C4×D4—7316no—
394C4×D4—7332no—
395C4×D4—7332no—
396C4×D4—7332no—
397C4×D4—7332no—
398C4×D4—7364no—
399C4×D4—7364no—
400C4×D4C22×D73116yes—
401#117—7316no—
402#117—7332no—
403#117—7364no—
404#117C22×D73132yes—
405C2×C4⋊C4—734no—
406C2×C4⋊C4—738no—
407C2×C4⋊C4—734no—
408C2×C4⋊C4—732no—
409C2×C4⋊C4C22×D7312yes—
410C2×C4⋊C4C22×D7314yes—
411C2×C22⋊C4—734no—
412C2×C22⋊C4—738no—
413C2×C22⋊C4—734no—
414C2×C22⋊C4—738no—
415C2×C22⋊C4C22×D7314yes—
416C2×C22⋊C4C22×D7318yes—
417C2×C42—731no—
418C2×C42—734no—
419C2×C42—734no—
420C2×C42—732no—
421C2×C42—734no—
422C2×C42—734no—
423C2×C42—7332no—
424C2×C42—738no—
425C2×C42—7332no—
426C2×C42C22×D7314yes—
427C2×C42C22×D7311yescharacteristic
428C24—734no—
429C24—731no—
430C24C23×D7314yes—
431#41—29264no—
432#41—29232no—
433#41—146128no—
434#41—14664no—
435C2×Q8—44no—
436C2×Q8—2928no—
437C2×Q8—7316no—
438C2×Q8—7316no—
439C2×Q8—738no—
440C2×Q8C23×D7318yes—
441C2×D4—28no—
442C2×D4—48no—
443C2×D4—2928no—
444C2×D4—2924no—
445C2×D4—29216no—
446C2×D4—7332no—
447C2×D4—7332no—
448C2×D4—7316no—
449C2×D4—7316no—
450C2×D4—738no—
451C2×D4—7316no—
452C2×D4—14632no—
453C2×D4—14616no—
454C2×D4—14616no—
455C2×D4C23×D7318yes—
456C2×D4C23×D73116yes—
457C22×C4—216no—
458C22×C4—731no—
459C22×C4—734no—
460C22×C4—7316no—
461C22×C4—7316no—
462C22×C4—732no—
463C22×C4—734no—
464C22×C4—7332no—
465C22×C4—738no—
466C22×C4—7316no—
467C22×C4—734no—
468C22×C4—14616no—
469C22×C4—1464no—
470C22×C4—14632no—
471C22×C4—14616no—
472C22×C4—14616no—
473C22×C4—1468no—
474C22×C4C23×D73116yes—
475C22×C4C23×D7314yes—
476C22×C4C23×D7314yes—
477C22×C4C23×D7311yescharacteristic
478C22×C4C23×D7312yes—
479C4⋊C4—7316no—
480C4⋊C4—7332no—
481C4⋊C4—7316no—
482C4⋊C4—738no—
483C4⋊C4C23×D7318yes—
484C4⋊C4C23×D73116yes—
485C22⋊C4—7316no—
486C22⋊C4—7332no—
487C22⋊C4—7332no—
488C22⋊C4—7316no—
489C22⋊C4C23×D73116yes—
490C22⋊C4C23×D73132yes—
491C42—7316no—
492C42—738no—
493C42—734no—
494C42—7316no—
495C42—7316no—
496C42—7316no—
497C42C23×D73116yes—
498C42C23×D7314yes—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—738no—
503C23—732no—
504C23—14616no—
505C23—1464no—
506C23—1462no—
507C23—1468no—
508C23C24×D7318yes—
509C23C24×D7311yescharacteristic
510Q8—48no—
511Q8—29216no—
512Q8—7332no—
513D4—216no—
514D4—416no—
515D4—29216no—
516D4—2928no—
517D4—29232no—
518D4—14664no—
519D4—14632no—
520D4—14632no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—232no—
525C2×C4—7316no—
526C2×C4—734no—
527C2×C4—732no—
528C2×C4—738no—
529C2×C4—734no—
530C2×C4—738no—
531C2×C4—14632no—
532C2×C4—1468no—
533C2×C4—14664no—
534C2×C4—14632no—
535C2×C4—14632no—
536C2×C4—1464no—
537C2×C4—1468no—
538C2×C4—14616no—
539C2×C4#41×D7318yes—
540C2×C4C24×D7318yes—
541C2×C4C24×D7318yes—
542C2×C4C24×D7312yes—
543C2×C4C24×D7314yes—
544C22—28no—
545C22—28no—
546C22—28no—
547C22—416no—
548C22—29216no—
549C22—2924no—
550C22—14616no—
551C22—1464no—
552C22—1462no—
553C22—1468no—
554C22C2×#41×D7314yes—
555C22H(5,𝔽2)×D7312yes—
556C22C25×D7311yescharacteristic
557C4—28no—
558C4—28no—
559C4—22no—
560C4—7316no—
561C4—734no—
562C4—1464no—
563C4—1468no—
564C4C2×#41×D7318yes—
565C2—48no—
566C2—2928no—
567C2—2922no—
568C2#2262355614yesdirect product factor
569C2C22×#41×D7312yes—
570C2C2×H(5,𝔽2)×D7311yescharacteristic
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×#555×C73C2
socleC23×C73C25
hypercenterC2×#555D73
hypocenterC73C2×#3483
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #22623556C2direct (N × H)
2. C2×#17989317C2semidirect
3. C2×#17989300C2semidirect
4. C2×#17989299C2semidirect
5. C2×#17989258C2semidirect
6. C2×#17989255C2semidirect
7. C2×#17989138C2semidirect
8. C2×#17989136C2semidirect
9. C2×#17989132C2semidirect
10. C2×#17989055C2semidirect
11. C2×C4×#11924027C2semidirect
12. C2×#555×C73C2semidirect
17. #17989317C22semidirect
18. #17989300C22semidirect
19. #17989299C22semidirect
20. #17989258C22semidirect
21. #17989255C22semidirect
22. #17989138C22semidirect
23. #17989136C22semidirect
24. #17989132C22semidirect
38. #17989055C22semidirect
39. C4×#11924027C22semidirect
56. #555×C73C22semidirect
149. C2×Dic146#41semidirect
257. Dic146C2×#41semidirect
281. C2×C73#3483semidirect
290. C2×#555D73semidirect
298. C73C2×#3483semidirect
306. #555C2×D73semidirect
568. C2#22623556direct (N × H)

Maximal subgroups

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

subgroupindex
#226235562
C2×#179893172
C2×#179893002
C2×#179892992
C2×#179892582
C2×#179892552
C2×#179891382
C2×#179891362
C2×#179891322
C2×#179890552
C2×C4×#119240272
C2×#555×C732
C2×#348373