Finite Group Databasebeta

← #33056213

Subgroups — #33056213

Order 18688.

604 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619955C211yesmaximal, characteristic
2#22618905C211yesmaximal, characteristic, complemented
3#22618896C211yesmaximal, characteristic, complemented
4#22608821C211yesmaximal, characteristic, complemented
5#22608821C211yesmaximal, characteristic, complemented
6#22608817C211yesmaximal, characteristic, complemented
7#22608782C211yesmaximal, characteristic, complemented
8#22608771C211yesmaximal, characteristic, complemented
9#22608770C211yesmaximal, characteristic, complemented
10#22608759C211yesmaximal, characteristic, complemented
11#22608710C211yesmaximal, characteristic, complemented
12#22608364C211yesmaximal, characteristic, complemented
13#22608338C211yesmaximal, characteristic, complemented
14#22608213C211yesmaximal, characteristic, complemented
15C73×#1591C211yesmaximal, characteristic, complemented
16#17989124C2211yescharacteristic
17C4×#11924029C2211yescharacteristic
18#17988927C411yescharacteristic
19#17988914C411yescharacteristic
20#17988896C411yescharacteristic
21#17988890C411yescharacteristic
22#17988811C411yescharacteristic
23#17988805C411yescharacteristic
24#17988748C411yescharacteristic
25#17988735C411yescharacteristic
26#17988718C2211yescharacteristic
27Q8×#10596831C2211yescharacteristic
28C4×#11924015C2211yescharacteristic
29C4×#11924015C2211yescharacteristic
30C4×D584C2211yescharacteristic
31#17988666C2211yescharacteristic
32#17988666C2211yescharacteristic
33#17988664C2211yescharacteristic
34#17988660—21no—
35#17988660—21no—
36#17988657—21no—
37#17988652—21no—
38#17988652C2211yescharacteristic
39#17988647—21no—
40#17988643C2211yescharacteristic
41#17988639—21no—
42#17988639C2211yescharacteristic
43#17988638C2211yescharacteristic
44#17988638C2211yescharacteristic
45#17988544—21no—
46#17988539—21no—
47C4×C8×D73C2211yescharacteristic
48#17988340C2211yescharacteristic
49#17988337C2211yescharacteristic
50#17988337C2211yescharacteristic
51Q8⋊#10596831C2211yescharacteristic
52Q8⋊#10596831C2211yescharacteristic
53#17988316C2211yescharacteristic
54#17988316C2211yescharacteristic
55#17988314C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988309C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60#446×C73C2211yescharacteristic
61#446×C73C2211yescharacteristic
62C4×Q16×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65Q8⋊C8×C73C2211yescharacteristic
66C82×C73C2211yescharacteristic
67C2×#11924029C2×C411yescharacteristic
68C2×#11924015C2×C411yescharacteristic
69C2×#11924015C2×C411yescharacteristic
70C2×D584C2×C411yescharacteristic
71C2×#11924011—21no—
72C2×#11924011—21no—
73C2×#11924011—21no—
74C2×C8×D73—21no—
75C2×C8×D73D411yescharacteristic
76C2×C8×D73D411yescharacteristic
77#14556586—21no—
78#14556582C2×C411yescharacteristic
79#14556580C2×C411yescharacteristic
80#14556576—21no—
81#14556567C2×C411yescharacteristic
82#14556566C2×C411yescharacteristic
83Q8×Dic73C2×C411yescharacteristic
84C4×D292—21no—
85C4×D292—21no—
86C4×D292C2311yescharacteristic
87C4×D292C2311yescharacteristic
88#14556553—21no—
89#14556553C2×C411yescharacteristic
90#14556549—21no—
91#14556549C2×C411yescharacteristic
92#14556548—21no—
93C42×D73—21no—
94C42×D73C2311yescharacteristic
95#14556525C2×C411yescharacteristic
96#14556522C2×C411yescharacteristic
97#14556522C2×C411yescharacteristic
98C4⋊#10596831—21no—
99C4⋊#10596831—21no—
100C4⋊#10596831C2311yescharacteristic
101C4⋊#10596831C2311yescharacteristic
102#14556520—21no—
103#14556520C2×C411yescharacteristic
104#14556520C2×C411yescharacteristic
105#14556515C2×C411yescharacteristic
106#14556515C2×C411yescharacteristic
107#14556513C2×C411yescharacteristic
108#14556513C2×C411yescharacteristic
109#14556511C2×C411yescharacteristic
110#14556511C2×C411yescharacteristic
111#14556503—21no—
112#14556503—21no—
113#14556503—21no—
114#14556503—21no—
115#14556503—21no—
116#14556503—21no—
117#14556503—21no—
118#14556503C2×C411yescharacteristic
119#14556503C2×C411yescharacteristic
120#14556502—21no—
121#14556501—21no—
122C4×#10596831—21no—
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C8×Dic73—21no—
127C8×Dic73D411yescharacteristic
128C8×Dic73D411yescharacteristic
129C2×Q16×C73C2×C411yescharacteristic
130C4×Q8×C73C2311yescharacteristic
131C4×Q8×C73C2311yescharacteristic
132#107×C73C2×C411yescharacteristic
133C4⋊C8×C73—21no—
134C4⋊C8×C73—21no—
135C4⋊C8×C73C2311yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137#103×C73C2×C411yescharacteristic
138#103×C73C2×C411yescharacteristic
139#97×C73—21no—
140#97×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144#11924029—41no—
145#11924029—41no—
146C2×D292—21no—
147C2×D292—21no—
148C2×D292C22×C411yescharacteristic
149C2×D292C22×C411yescharacteristic
150C2×C4×D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—21no—
154C2×C4×D73C22×C411yescharacteristic
155C2×C4×D73C2×D411yescharacteristic
156C2×C4×D73#4111yescharacteristic
157#11924015—41no—
158#11924015—41no—
159D584—41no—
160#11924011—42no—
161#11924011—42no—
162#11924011—42no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831C22×C411yescharacteristic
167C2×#10596831C22×C411yescharacteristic
168C2×#10596831C2×D411yescharacteristic
169C2×#10596831C2×D411yescharacteristic
170C8×D73—22no—
171C8×D73—22no—
172C8×D73—42no—
173C4⋊Dic73—21no—
174C4⋊Dic73—21no—
175C4⋊Dic73C22×C411yescharacteristic
176C4⋊Dic73C22×C411yescharacteristic
177#11924007—21no—
178#11924007—21no—
179#11924005—21no—
180#11924005—21no—
181#11924005—21no—
182#11924005—21no—
183#11924005—21no—
184#11924005—21no—
185#11924005—21no—
186#11924005—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73C22×C411yescharacteristic
190C4×Dic73C2×D411yescharacteristic
191C4×Dic73#4111yescharacteristic
192C2×Q8×C73C22×C411yescharacteristic
193C2×Q8×C73C22×C411yescharacteristic
194Q16×C73—41no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73C22×C411yescharacteristic
199C2×C8×C73C22×C411yescharacteristic
200C2×C8×C73C2×D411yescharacteristic
201C2×C8×C73C2×D411yescharacteristic
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73C22×C411yescharacteristic
205C4⋊C4×C73C22×C411yescharacteristic
206C42×C73—21no—
207C42×C73—21no—
208C42×C73C2411yescharacteristic
209C22×D73—21no—
210C22×D73—21no—
211C22×D73C4×D411yescharacteristic
212Q8×C73—21no—
213Q8×C73—21no—
214Q8×C73—21no—
215Q8×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73C23×C411yescharacteristic
221C2×C4×C73C22×D411yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73C4×D411yescharacteristic
226C2×Dic73C4×D411yescharacteristic
227C2×Dic73C4×D411yescharacteristic
228D292—21no—
229D292—21no—
230D292—21no—
231D292—21no—
232D292—42no—
233D292—42no—
234C4×D73—21no—
235C4×D73—21no—
236C4×D73—21no—
237C4×D73—21no—
238C4×D73—41no—
239C4×D73—42no—
240C4×D73—42no—
241C4×D73—41no—
242C4×D73—41no—
243C4×D73—41no—
244C4×D73—41no—
245C8×C73—21no—
246C8×C73—21no—
247C8×C73—21no—
248C8×C73—41no—
249C8×C73—41no—
250C8×C73—41no—
251C8×C73#13512yes—
252#10596831—21no—
253#10596831—21no—
254#10596831—21no—
255#10596831—41no—
256#10596831—41no—
257#10596831—41no—
258#10596831#13512yes—
259C22×C73C2×C4×D411yescharacteristic
260C2×D73—21no—
261C2×D73—21no—
262C2×D73—21no—
263C2×D73—41no—
264C2×D73—41no—
265C2×D73—41no—
266C2×D73—41no—
267C2×D73—41no—
268C2×D73—41no—
269C4×C73—21no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73#56811yescharacteristic
275C4×C73#56811yescharacteristic
276C4×C73C2×#13511yescharacteristic
277C4×C73C2×#13511yescharacteristic
278Dic73—21no—
279Dic73—21no—
280Dic73—21no—
281Dic73—41no—
282Dic73—41no—
283#17932—731nomaximal, Sylow
284C2×C73#293011yescharacteristic
285C2×C73C4×#13511yescharacteristic
286C2×C73#295511yescharacteristic
287D73—41no—
288D73—81no—
289D73—81no—
290#3102—731no—
291#2966—731no—
292#2965—731no—
293#1591D7311yescharacteristic, complemented
294#1590—731no—
295#1590—731no—
296#1589—731no—
297#1584—731no—
298#1583—731no—
299#1582—731no—
300#1581—731no—
301#1572—731no—
302#1488—731no—
303#1482—731no—
304#1450—731no—
305C73#1793211yescharacteristic, complemented, Sylow
306#529—731no—
307C4×#41—731no—
308#485—731no—
309#482—731no—
310#479—731no—
311#478—731no—
312#464—731no—
313#463—731no—
314#452—731no—
315#449—731no—
316#446—731no—
317#446C2×D7311yescharacteristic
318#446C2×D7311yescharacteristic
319C8×Q8—731no—
320C4×Q16C2×D7311yescharacteristic
321C4×SD16—731no—
322C4×SD16—731no—
323C4×D8—731no—
324#436—731no—
325#436—731no—
326#436—731no—
327#435—1461no—
328#435—1461no—
329#435—1461no—
330C8×D4—731no—
331C8×D4—1461no—
332#433—1461no—
333#432—731no—
334#431—731no—
335#431—731no—
336#431—731no—
337#431—1461no—
338#405—1461no—
339C4×M16—1461no—
340C2×C4×C8—731no—
341#335—731no—
342#335C2×D7311yescharacteristic
343#334—731no—
344#334—731no—
345Q8⋊C8—731no—
346Q8⋊C8—731no—
347Q8⋊C8C2×D7311yescharacteristic
348Q8⋊C8C2×D7311yescharacteristic
349D4⋊C8—731no—
350D4⋊C8—731no—
351D4⋊C8—731no—
352D4⋊C8—731no—
353#322—731no—
354#322—731no—
355C82—731no—
356C82C2×D7311yescharacteristic
357C2×#41—731no—
358C2×Q16C4×D7311yescharacteristic
359C2×SD16—731no—
360C2×SD16—731no—
361C2×D8—731no—
362C2×M16—1461no—
363C2×M16—1461no—
364C2×M16—1461no—
365C22×C8—731no—
366C22×C8—731no—
367C22×C8—1461no—
368#126—1461no—
369#125—731no—
370#124—731no—
371#123—1461no—
372C22⋊Q8—731no—
373#121—731no—
374C4×Q8—731no—
375C4×Q8C22×D7311yescharacteristic
376C4×Q8C22×D7311yescharacteristic
377C4×D4—731no—
378C4×D4—731no—
379C4×D4—731no—
380C4×D4—1461no—
381C4×D4—1461no—
382C4×D4—1461no—
383#117—731no—
384#117—1461no—
385#117—1461no—
386C2×C42—731no—
387C2×C42—1461no—
388#107—731no—
389#107C4×D7311yescharacteristic
390#106—731no—
391#106—731no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—731no—
395C4⋊C8—731no—
396C4⋊C8—731no—
397C4⋊C8—731no—
398C4⋊C8—1461no—
399C4⋊C8—1461no—
400C4⋊C8—1461no—
401C4⋊C8C22×D7311yescharacteristic
402C4⋊C8C22×D7311yescharacteristic
403#103—731no—
404#103—731no—
405#103C4×D7311yescharacteristic
406#103C4×D7311yescharacteristic
407#102—731no—
408#102—731no—
409#102—731no—
410#102—731no—
411C22⋊C8—731no—
412C22⋊C8—731no—
413C22⋊C8—1461no—
414C22⋊C8—1461no—
415C22⋊C8—1461no—
416C22⋊C8—1461no—
417C22⋊C8—1461no—
418C22⋊C8—1461no—
419C22⋊C8—1461no—
420#97—21no—
421#97—21no—
422#97—1461no—
423#97—1461no—
424C4×C8—731no—
425C4×C8—731no—
426C4×C8—731no—
427C4×C8—731no—
428C4×C8—731no—
429C4×C8—1461no—
430C4×C8—1461no—
431C4×C8C22×D7311yescharacteristic
432C4×C8C22×D7311yescharacteristic
433C4×C8C22×D7311yescharacteristic
434#41—2921no—
435#41—2921no—
436C2×Q8C2×C4×D7311yescharacteristic
437C2×Q8C2×C4×D7311yescharacteristic
438C2×D4—731no—
439C2×D4—731no—
440C2×D4—1461no—
441C2×D4—1461no—
442C22×C4—731no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4—1461no—
449Q16—41no—
450SD16—2921no—
451SD16—2921no—
452D8—2921no—
453M16—2922no—
454M16—2922no—
455M16—2922no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—2922no—
460C2×C8—731no—
461C2×C8—731no—
462C2×C8—731no—
463C2×C8—731no—
464C2×C8—1462no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1462no—
469C2×C8C2×C4×D7311yescharacteristic
470C2×C8C2×C4×D7311yescharacteristic
471C2×C8D4×D7311yescharacteristic
472C2×C8D4×D7311yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—731no—
476C4⋊C4—731no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4C2×C4×D7311yescharacteristic
482C4⋊C4C2×C4×D7311yescharacteristic
483C22⋊C4—1461no—
484C22⋊C4—1461no—
485C22⋊C4—1461no—
486C22⋊C4—1461no—
487C22⋊C4—1461no—
488C22⋊C4—1461no—
489C22⋊C4—1461no—
490C22⋊C4—1461no—
491C42—21no—
492C42—21no—
493C42—731no—
494C42—731no—
495C42—731no—
496C42—1461no—
497C42—1461no—
498C42C23×D7311yescharacteristic
499C23—731no—
500C23—1461no—
501C23—1461no—
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8—21no—
506D4—2922no—
507D4—2922no—
508D4—1461no—
509D4—1461no—
510D4—1461no—
511D4—1461no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—2921no—
517C2×C4—2922no—
518C2×C4—2922no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—731no—
524C2×C4—731no—
525C2×C4—731no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4C22×C4×D7311yescharacteristic
533C2×C4C2×D4×D7311yescharacteristic
534C2×C4#41×D7311yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8—2921no—
542C8—2921no—
543C8—2921no—
544C8—732no—
545C8—1461no—
546C8—1461no—
547C8—1461no—
548C8#1455661412yes—
549C22—2921no—
550C22—2921no—
551C22—2921no—
552C22—2921no—
553C22—2921no—
554C22—2921no—
555C22—1461no—
556C22—1461no—
557C22—1461no—
558C22C4×D4×D7311yescharacteristic
559C4—21no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—2921no—
565C4—2921no—
566C4—1461no—
567C4—1461no—
568C4—1461no—
569C4#1798935211yescharacteristic
570C4#1798935311yescharacteristic
571C4#135×D7311yescharacteristic
572C4C2×#1455661411yescharacteristic
573C2—2921no—
574C2—5841no—
575C2—5841no—
576C2#2261863111yescharacteristic
577C2C4×#1455661411yescharacteristic
578C2#2261882911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1591C2
socleC22×C73C2×C4×D4
prime radical#17988735C4
cosocleC42×C73C24
hypercenter#1591D73
hypocenterC73#17932
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1591

Complements

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

NH ≅ G/Ntype
2. #22618905C2semidirect
3. #22618896C2semidirect
4. #22608821C2semidirect
5. #22608821C2semidirect
6. #22608817C2semidirect
7. #22608782C2semidirect
8. #22608771C2semidirect
9. #22608770C2semidirect
10. #22608759C2semidirect
11. #22608710C2semidirect
12. #22608364C2semidirect
13. #22608338C2semidirect
14. #22608213C2semidirect
15. C73×#1591C2semidirect
293. #1591D73semidirect
305. C73#17932semidirect

Maximal subgroups

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

subgroupindex
#226199552
#226189052
#226188962
#226088212
#226088172
#226087822
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