Finite Group Databasebeta

← #33950552

Subgroups — #33950552

Order 18688.

1878 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#22618936C218yesmaximal, complemented
2#22618927C218yesmaximal, complemented
3#22618872C211yesmaximal, characteristic, complemented
4#22618866C211yesmaximal, characteristic, complemented
5#22618649C211yesmaximal, characteristic, complemented
6#22618643C211yesmaximal, characteristic, complemented
7#22618635C212yesmaximal, complemented
8#22618626C212yesmaximal, complemented
9#22618575C212yesmaximal, complemented
10#22618569C211yesmaximal, characteristic, complemented
11#22618359C212yesmaximal, complemented
12C2×#17988543C211yesmaximal, characteristic, complemented
13C73×#3455C211yesmaximal, characteristic
14C2×#14556598C2212yes—
15C2×#14556598C2212yes—
16C22×#11924012C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18#17988718C2214yescomplemented
19#17988718C2218yescomplemented
20Q8×#10596831C2214yescomplemented
21#17988665C2218yes—
22#17988665C2214yescomplemented
23#17988659C2218yes—
24#17988659C2218yes—
25#17988659C2218yescomplemented
26D4×#10596831C2218yes—
27D4×#10596831C2214yescomplemented
28#17988648C2218yes—
29#17988648C2218yescomplemented
30#17988640C2218yescomplemented
31#17988614C2212yes—
32#17988614C2212yes—
33#17988611C2211yescharacteristic
34#17988611C2211yescharacteristic
35#17988554C2212yes—
36#17988554C2212yes—
37#17988551C2212yes—
38#17988543C2212yes—
39#17988543C2212yes—
40#17988543C2214yes—
41#17988543C2214yes—
42#17988543C2214yescomplemented
43#17988543C2214yescomplemented
44C4×#11924012C2212yes—
45C4×#11924012C2214yescomplemented
46C2×#14556502C2211yescharacteristic
47C2×#14556502C2211yescharacteristic
48C2×#14556502C2214yescomplemented
49C2×C4×#10596831C2211yescharacteristic
50C2×C4×#10596831C2211yescharacteristic
51#542×C73C414yes—
52#541×C73C412yes—
53#540×C73C412yes—
54#539×C73C414yes—
55#538×C73C414yes—
56#533×C73C2211yescharacteristic
57#532×C73C2211yescharacteristic
58#529×C73C2218yes—
59#525×C73C2212yes—
60C4×#41×C73C2212yes—
61C2×#117×C73C2211yescharacteristic
62#14556598—28no—
63#14556598—28no—
64C2×#11924012—24no—
65C2×#11924012C2312yes—
66C2×#11924012C2312yes—
67C2×#11924012C2312yes—
68C2×#11924012C2314yes—
69C2×#11924012C2312yes—
70C2×#11924012C2314yes—
71C22×#10596831—24no—
72C22×#10596831C2311yescharacteristic
73C22×#10596831C2311yescharacteristic
74C22×#10596831C2314yes—
75C22×#10596831C2314yes—
76C22×#10596831C2314yes—
77C22×#10596831C2311yescharacteristic
78C22×#10596831C2311yescharacteristic
79C4⋊#10596831C2314yes—
80C4⋊#10596831C2314yes—
81C4⋊#10596831C2318yes—
82C4⋊#10596831C2318yes—
83C22⋊#10596831C2318yes—
84C22⋊#10596831C2318yes—
85C22⋊#10596831C2318yes—
86#14556502C2314yes—
87#14556502C2314yes—
88#14556502C2314yes—
89#14556502C2314yes—
90#14556502C2312yes—
91#14556502C2312yes—
92C4×#10596831C2314yes—
93C4×#10596831C2314yes—
94C4×#10596831C2314yes—
95C4×#10596831C2312yes—
96C4×#10596831C2314yes—
97C4×#10596831C2312yes—
98C2×#41×C73C2312yes—
99C23×C4×C73C2311yescharacteristic
100C4⋊Q8×C73C2×C411yescharacteristic
101C4⋊Q8×C73C2×C412yes—
102#127×C73C2×C411yescharacteristic
103#126×C73C2×C418yes—
104#126×C73C2×C418yes—
105#125×C73C2×C418yes—
106#125×C73C2×C412yes—
107#124×C73C2×C418yes—
108#124×C73C2×C412yes—
109#123×C73C2×C414yes—
110#123×C73C2×C418yes—
111#123×C73C2×C418yes—
112C22⋊Q8×C73C2×C414yes—
113C22⋊Q8×C73C2×C418yes—
114C22⋊Q8×C73C2×C414yes—
115#121×C73C2×C414yes—
116#121×C73C2×C418yes—
117#121×C73C2×C414yes—
118#120×C73C2×C414yes—
119C4×Q8×C73C2×C414yes—
120C4×Q8×C73C2314yes—
121C4×D4×C73C2×C418yes—
122C4×D4×C73C2×C414yes—
123C4×D4×C73C2318yes—
124C4×D4×C73C2314yes—
125#117×C73C2×C412yes—
126#117×C73C2×C412yes—
127#117×C73C2312yes—
128#117×C73C2314yes—
129#117×C73C2314yes—
130C2×C4⋊C4×C73C2×C412yes—
131C2×C22⋊C4×C73C2×C412yes—
132C2×C42×C73C2311yescharacteristic
133C2×C42×C73C2314yes—
134C2×C42×C73C2311yescharacteristic
135#11924012—24no—
136#11924012—24no—
137#11924012—24no—
138#11924012—24no—
139#11924012—28no—
140#11924012—28no—
141C2×#10596831—22no—
142C2×#10596831—22no—
143C2×#10596831—28no—
144C2×#10596831—28no—
145C2×#10596831—28no—
146C2×#10596831#4114yes—
147C2×#10596831#4114yes—
148C2×#10596831C2412yes—
149C2×#10596831C2414yes—
150C2×#10596831C2414yes—
151C2×#10596831C2412yes—
152C2×#10596831C2411yescharacteristic
153C2×#10596831C2412yes—
154C2×#10596831C2411yescharacteristic
155C24×C73C22×C411yescharacteristic
156#41×C73—24no—
157#41×C73—24no—
158C2×Q8×C73C22×C412yes—
159C2×Q8×C73C22×C414yes—
160C2×D4×C73C22×C414yes—
161C2×D4×C73C22×C412yes—
162C2×D4×C73C22×C418yes—
163C2×D4×C73C22×C414yes—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73C22×C411yescharacteristic
167C22×C4×C73C22×C412yes—
168C22×C4×C73C22×C412yes—
169C22×C4×C73C22×C412yes—
170C22×C4×C73C22×C418yes—
171C22×C4×C73C2414yes—
172C22×C4×C73C2412yes—
173C22×C4×C73C2411yescharacteristic
174C4⋊C4×C73C22×C414yes—
175C4⋊C4×C73C22×C414yes—
176C4⋊C4×C73C22×C414yes—
177C4⋊C4×C73C22×C418yes—
178C4⋊C4×C73C22×C414yes—
179C4⋊C4×C73C22×C418yes—
180C4⋊C4×C73C22×C418yes—
181C4⋊C4×C73C22×C414yes—
182C22⋊C4×C73C22×C414yes—
183C22⋊C4×C73C22×C414yes—
184C22⋊C4×C73C22×C414yes—
185C22⋊C4×C73C22×C418yes—
186C22⋊C4×C73C22×C418yes—
187C22⋊C4×C73C22×C418yes—
188C22⋊C4×C73C22×C418yes—
189C42×C73C22×C412yes—
190C42×C73C22×C411yescharacteristic
191C42×C73C22×C411yescharacteristic
192C42×C73C22×C418yes—
193C42×C73C2412yes—
194C42×C73C2412yes—
195C42×C73C2414yes—
196C23×C73—22no—
197C23×C73—42no—
198C23×C73C23×C412yes—
199C23×C73C23×C414yes—
200C23×C73C23×C411yescharacteristic
201Q8×C73—24no—
202Q8×C73—44no—
203D4×C73—28no—
204D4×C73—24no—
205D4×C73—44no—
206D4×C73—48no—
207C2×C4×C73—22no—
208C2×C4×C73—24no—
209C2×C4×C73—22no—
210C2×C4×C73—22no—
211C2×C4×C73—24no—
212C2×C4×C73—24no—
213C2×C4×C73—42no—
214C2×C4×C73—48no—
215C2×C4×C73—42no—
216C2×C4×C73C23×C414yes—
217C2×C4×C73C23×C414yes—
218C2×C4×C73C23×C412yes—
219C2×C4×C73C23×C418yes—
220C2×C4×C73C23×C412yes—
221C2×C4×C73C23×C412yes—
222C2×C4×C73C23×C411yescharacteristic
223C2×C4×C73C2×#4112yes—
224C2×C4×C73C2×#4112yes—
225C2×C4×C73C2511yescharacteristic
226#10596831—22no—
227#10596831—22no—
228#10596831—24no—
229#10596831—24no—
230#10596831C2×#4114yes—
231#10596831C2×#4112yes—
232#10596831C2×#4112yes—
233C22×C73—22no—
234C22×C73—22no—
235C22×C73—24no—
236C22×C73—28no—
237C22×C73—24no—
238C22×C73—44no—
239C22×C73C4×#4112yes—
240C22×C73C24×C411yescharacteristic
241C4×C73—22no—
242C4×C73—24no—
243C4×C73—24no—
244C4×C73—22no—
245C4×C73—28no—
246C4×C73—22no—
247C4×C73C4×#4112yes—
248C4×C73C22×#4111yescharacteristic
249C4×C73C22×#4111yescharacteristic
250#60209—731nomaximal, Sylow
251C2×C73—22no—
252C2×C73—44no—
253C2×C73—42no—
254C2×C73C2×C4×#4111yescharacteristic
255C2×C73#357312yes—
256#3455D7311yescharacteristic
257#2969—738no—
258#2968—738no—
259#2961—731no—
260#2960—731no—
261#2932—731no—
262#2931—731no—
263#2930—732no—
264#2929—732no—
265#2922—732no—
266#2921—731no—
267#2875—732no—
268C2×#405—731no—
269C73#6020911yescharacteristic, complemented, Sylow
270C2×#131—732no—
271C2×#131—732no—
272C22×M16—731no—
273C23×C8—731no—
274#542Dic7314yes—
275#541Dic7312yes—
276#540Dic7312yes—
277#539Dic7314yes—
278#538Dic7314yes—
279#533C2×D7311yescharacteristic
280#532C2×D7311yescharacteristic
281#529C2×D7318yes—
282#525C2×D7312yes—
283C4×#41C2×D7312yes—
284C2×#117C2×D7311yescharacteristic
285#446—734no—
286#446—738no—
287C8×Q8—734no—
288#436—738no—
289#436—734no—
290#435—738no—
291#435—738no—
292#435—738no—
293C8×D4—738no—
294C8×D4—734no—
295#433—738no—
296#433—738no—
297#431—738no—
298#424—732no—
299#424—732no—
300#423—731no—
301#423—731no—
302#408—732no—
303#408—732no—
304#407—732no—
305#405—734no—
306#405—734no—
307#405—732no—
308#405—732no—
309#405—734no—
310#405—734no—
311C4×M16—732no—
312C4×M16—734no—
313C2×#97—731no—
314C2×#97—731no—
315C2×#97—734no—
316C2×C4×C8—731no—
317C2×C4×C8—731no—
318C2×#41C22×D7312yes—
319C23×C4C22×D7311yescharacteristic
320#131—1468no—
321#131—1468no—
322C2×M16—732no—
323C2×M16—732no—
324C2×M16—732no—
325C2×M16—732no—
326C2×M16—734no—
327C2×M16—734no—
328C2×M16—1464no—
329C22×C8—731no—
330C22×C8—731no—
331C22×C8—731no—
332C22×C8—734no—
333C22×C8—731no—
334C22×C8—734no—
335C22×C8—734no—
336C22×C8—1464no—
337C4⋊Q8C2×Dic7311yescharacteristic
338C4⋊Q8C2×Dic7312yes—
339#127C2×Dic7311yescharacteristic
340#126C2×Dic7318yes—
341#126C2×Dic7318yes—
342#125C2×Dic7318yes—
343#125C2×Dic7312yes—
344#124C2×Dic7318yes—
345#124C2×Dic7312yes—
346#123C2×Dic7314yes—
347#123C2×Dic7318yes—
348#123C2×Dic7318yes—
349C22⋊Q8C2×Dic7314yes—
350C22⋊Q8C2×Dic7318yes—
351C22⋊Q8C2×Dic7314yes—
352#121C2×Dic7314yes—
353#121C2×Dic7318yes—
354#121C2×Dic7314yes—
355#120C2×Dic7314yes—
356C4×Q8C2×Dic7314yes—
357C4×Q8C22×D7314yes—
358C4×D4C2×Dic7318yes—
359C4×D4C2×Dic7314yes—
360C4×D4C22×D7318yes—
361C4×D4C22×D7314yes—
362#117C2×Dic7312yes—
363#117C2×Dic7312yes—
364#117C22×D7314yes—
365#117C22×D7312yes—
366#117C22×D7314yes—
367C2×C4⋊C4C2×Dic7312yes—
368C2×C22⋊C4C2×Dic7312yes—
369C2×C42C22×D7311yescharacteristic
370C2×C42C22×D7311yescharacteristic
371C2×C42C22×D7314yes—
372C4⋊C8—734no—
373C4⋊C8—734no—
374C4⋊C8—738no—
375C4⋊C8—738no—
376C22⋊C8—738no—
377C22⋊C8—738no—
378C22⋊C8—738no—
379#97—732no—
380#97—732no—
381#97—734no—
382#97—734no—
383#97—734no—
384#97—734no—
385C4×C8—734no—
386C4×C8—732no—
387C4×C8—732no—
388C4×C8—734no—
389C4×C8—734no—
390C4×C8—734no—
391C24C22×Dic7311yescharacteristic
392#41—24no—
393#41—24no—
394C2×Q8C22×Dic7312yes—
395C2×Q8C22×Dic7314yes—
396C2×D4C22×Dic7314yes—
397C2×D4C22×Dic7312yes—
398C2×D4C22×Dic7318yes—
399C2×D4C22×Dic7314yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4C22×Dic7311yescharacteristic
403C22×C4C22×Dic7312yes—
404C22×C4C22×Dic7312yes—
405C22×C4C22×Dic7312yes—
406C22×C4C22×Dic7318yes—
407C22×C4C23×D7314yes—
408C22×C4C23×D7312yes—
409C22×C4C23×D7311yescharacteristic
410M16—1464no—
411M16—1464no—
412M16—1468no—
413M16—1468no—
414M16—1464no—
415M16—1464no—
416C2×C8—734no—
417C2×C8—734no—
418C2×C8—731no—
419C2×C8—731no—
420C2×C8—732no—
421C2×C8—732no—
422C2×C8—732no—
423C2×C8—734no—
424C2×C8—734no—
425C2×C8—1468no—
426C2×C8—1468no—
427C2×C8—1468no—
428C2×C8—1462no—
429C2×C8—1462no—
430C4⋊C4C22×Dic7314yes—
431C4⋊C4C22×Dic7314yes—
432C4⋊C4C22×Dic7314yes—
433C4⋊C4C22×Dic7318yes—
434C4⋊C4C22×Dic7314yes—
435C4⋊C4C22×Dic7318yes—
436C4⋊C4C22×Dic7318yes—
437C4⋊C4C22×Dic7314yes—
438C22⋊C4C22×Dic7314yes—
439C22⋊C4C22×Dic7314yes—
440C22⋊C4C22×Dic7314yes—
441C22⋊C4C22×Dic7318yes—
442C22⋊C4C22×Dic7318yes—
443C22⋊C4C22×Dic7318yes—
444C22⋊C4C22×Dic7318yes—
445C42C22×Dic7312yes—
446C42C22×Dic7311yescharacteristic
447C42C22×Dic7311yescharacteristic
448C42C22×Dic7318yes—
449C42C23×D7312yes—
450C42C23×D7312yes—
451C42C23×D7314yes—
452C23—22no—
453C23—42no—
454C23C23×Dic7312yes—
455C23C23×Dic7314yes—
456C23C23×Dic7311yescharacteristic
457Q8—24no—
458Q8—44no—
459D4—28no—
460D4—24no—
461D4—44no—
462D4—48no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—48no—
471C2×C4—42no—
472C2×C4C23×Dic7314yes—
473C2×C4C23×Dic7314yes—
474C2×C4C23×Dic7312yes—
475C2×C4C23×Dic7318yes—
476C2×C4C23×Dic7312yes—
477C2×C4C23×Dic7312yes—
478C2×C4C23×Dic7311yescharacteristic
479C2×C4#41×D7312yes—
480C2×C4#41×D7312yes—
481C2×C4C24×D7311yescharacteristic
482C8—734no—
483C8—732no—
484C8—732no—
485C8—1462no—
486C8—1464no—
487C8—1464no—
488C8—1462no—
489C22—22no—
490C22—28no—
491C22—24no—
492C22—22no—
493C22—24no—
494C22—44no—
495C22#41×Dic7312yes—
496C22C24×Dic7311yescharacteristic
497C4—24no—
498C4—22no—
499C4—28no—
500C4—22no—
501C4—22no—
502C4—24no—
503C4#41×Dic7312yes—
504C4C2×#41×D7311yescharacteristic
505C4C2×#41×D7311yescharacteristic
506C2—22no—
507C2—42no—
508C2—44no—
509C2C2×#41×Dic7311yescharacteristic
510C2#2262464312yes—
subgroupG/H
center, FrattiniC2×C4C24×D73
commutator, socleC22×C73C24×C4
Fitting, prime radicalC73×#3455C2
cosocleC2×C4×C73C25
hypercenter#3455D73
hypocenterC73#60209
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ #3455

Complements

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

NH ≅ G/Ntype
1. #22618936C2semidirect
2. #22618927C2semidirect
3. #22618872C2semidirect
4. #22618866C2semidirect
5. #22618649C2semidirect
6. #22618643C2semidirect
7. #22618635C2semidirect
8. #22618626C2semidirect
9. #22618575C2semidirect
10. #22618569C2semidirect
11. #22618359C2semidirect
12. C2×#17988543C2semidirect
18. #17988718C22semidirect
19. #17988718C22semidirect
20. Q8×#10596831C22semidirect
22. #17988665C22semidirect
25. #17988659C22semidirect
27. D4×#10596831C22semidirect
29. #17988648C22semidirect
30. #17988640C22semidirect
42. #17988543C22semidirect
43. #17988543C22semidirect
45. C4×#11924012C22semidirect
48. C2×#14556502C22semidirect
269. C73#60209semidirect

Maximal subgroups

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

subgroupindex
#226189362
#226189272
#226188722
#226188662
#226186492
#226186432
#226186352
#226186262
#226185752
#226185692
#226183592
C2×#179885432
C73×#34552
#6020973