Finite Group Databasebeta

← #33057385

Subgroups — #33057385

Order 18688.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619968C211yesmaximal, characteristic
2#22619029C211yesmaximal, characteristic
3#22619017C211yesmaximal, characteristic
4#22608825C211yesmaximal, characteristic, complemented
5#22608803C211yesmaximal, characteristic, complemented
6#22608801C211yesmaximal, characteristic, complemented
7#22608779C211yesmaximal, characteristic, complemented
8#22608779C211yesmaximal, characteristic, complemented
9#22608751C211yesmaximal, characteristic, complemented
10#22608735C211yesmaximal, characteristic, complemented
11#22608708C211yesmaximal, characteristic
12#22608402C211yesmaximal, characteristic
13#22608324C211yesmaximal, characteristic
14#22608232C211yesmaximal, characteristic
15C73×#1593C211yesmaximal, characteristic, complemented
16#17989141C2211yescharacteristic
17C4×Q8×D73C2211yescharacteristic
18#17988924C411yescharacteristic
19#17988921C411yescharacteristic
20#17988881C411yescharacteristic
21#17988880C411yescharacteristic
22#17988796C411yescharacteristic
23#17988795C411yescharacteristic
24#17988744C411yescharacteristic
25#17988741C411yescharacteristic
26#17988718C2211yescharacteristic
27#17988716—21no—
28#17988715—21no—
29#17988715—21no—
30#17988714C2211yescharacteristic
31Q8×#10596831C2211yescharacteristic
32C8×Dic146C2211yescharacteristic
33C8×Dic146C2211yescharacteristic
34#17988711—21no—
35#17988711C2211yescharacteristic
36C4×Dic292C2211yescharacteristic
37C4×#11924018C2211yescharacteristic
38C4×#11924018C2211yescharacteristic
39#17988649—21no—
40#17988647—21no—
41#17988643C2211yescharacteristic
42#17988642—21no—
43#17988639—21no—
44#17988639C2211yescharacteristic
45#17988638C2211yescharacteristic
46#17988638C2211yescharacteristic
47#17988638C2211yescharacteristic
48#17988339C2211yescharacteristic
49#17988339C2211yescharacteristic
50#17988338C2211yescharacteristic
51#17988319C2211yescharacteristic
52#17988319C2211yescharacteristic
53Q8⋊#10596831C2211yescharacteristic
54Q8⋊#10596831C2211yescharacteristic
55#17988317C2211yescharacteristic
56#17988317C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988308C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60#446×C73C2211yescharacteristic
61#446×C73C2211yescharacteristic
62C4×Q16×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65Q8⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×Q8×D73C2×C411yescharacteristic
68C2×Dic292C2×C411yescharacteristic
69C2×#11924018C2×C411yescharacteristic
70C2×#11924018C2×C411yescharacteristic
71#14556590C2×C411yescharacteristic
72#14556589C2×C411yescharacteristic
73#14556581—21no—
74#14556571C2×C411yescharacteristic
75#14556569—21no—
76#14556567C2×C411yescharacteristic
77Q8×Dic73C2×C411yescharacteristic
78C4×Dic146—21no—
79C4×Dic146—21no—
80C4×Dic146C2311yescharacteristic
81C4×Dic146C2311yescharacteristic
82#14556556—21no—
83#14556556C2×C411yescharacteristic
84#14556548—21no—
85C4⋊C4×D73—21no—
86C4⋊C4×D73C2×C411yescharacteristic
87C42×D73—21no—
88C42×D73C2311yescharacteristic
89#14556525C2×C411yescharacteristic
90#14556524C2×C411yescharacteristic
91#14556524C2×C411yescharacteristic
92C4⋊#10596831—21no—
93C4⋊#10596831—21no—
94C4⋊#10596831C2311yescharacteristic
95C4⋊#10596831C2311yescharacteristic
96#14556520—21no—
97#14556520—21no—
98#14556520—21no—
99#14556520—21no—
100#14556520—21no—
101#14556520C2×C411yescharacteristic
102#14556520C2×C411yescharacteristic
103#14556520D411yescharacteristic
104#14556520D411yescharacteristic
105#14556516C2×C411yescharacteristic
106#14556516C2×C411yescharacteristic
107#14556515C2×C411yescharacteristic
108#14556515C2×C411yescharacteristic
109#14556514C2×C411yescharacteristic
110#14556514C2×C411yescharacteristic
111#14556503—21no—
112#14556503—21no—
113#14556503—21no—
114#14556503C2×C411yescharacteristic
115#14556503C2×C411yescharacteristic
116#14556503D411yescharacteristic
117#14556503D411yescharacteristic
118#14556502—21no—
119#14556501—21no—
120#14556501—21no—
121C4×#10596831—21no—
122C4×#10596831C2311yescharacteristic
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C8×Dic73—21no—
126C8×Dic73—21no—
127C8×Dic73—21no—
128C8×Dic73—21no—
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
144C2×Dic146—21no—
145C2×Dic146—21no—
146C2×Dic146C22×C411yescharacteristic
147C2×Dic146C22×C411yescharacteristic
148Q8×D73—22no—
149Q8×D73—41no—
150C2×C4×D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73C22×C411yescharacteristic
153C2×C4×D73C2×D411yescharacteristic
154C2×C4×D73#4111yescharacteristic
155Dic292—22no—
156#11924018—22no—
157#11924018—41no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831C22×C411yescharacteristic
162C2×#10596831C22×C411yescharacteristic
163C2×#10596831C2×D411yescharacteristic
164C2×#10596831C2×D411yescharacteristic
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73C22×C411yescharacteristic
168C4⋊Dic73C22×C411yescharacteristic
169#11924007—21no—
170#11924007—21no—
171#11924007—21no—
172#11924007—21no—
173#11924007—21no—
174#11924007—21no—
175#11924007—21no—
176#11924007—21no—
177#11924005—21no—
178#11924005—21no—
179C4×Dic73—21no—
180C4×Dic73—21no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73C22×C411yescharacteristic
184C4×Dic73C2×D411yescharacteristic
185C4×Dic73#4111yescharacteristic
186C2×Q8×C73C22×C411yescharacteristic
187C2×Q8×C73C22×C411yescharacteristic
188Q16×C73—41no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73C22×C411yescharacteristic
193C2×C8×C73C22×C411yescharacteristic
194C2×C8×C73C2×D411yescharacteristic
195C2×C8×C73C2×D411yescharacteristic
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73C22×C411yescharacteristic
199C4⋊C4×C73C22×C411yescharacteristic
200C42×C73—21no—
201C42×C73—21no—
202C42×C73C2411yescharacteristic
203C22×D73C4×D411yescharacteristic
204Q8×C73—21no—
205Q8×C73—21no—
206Q8×C73—21no—
207Q8×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73C23×C411yescharacteristic
213C2×C4×C73C22×D411yescharacteristic
214C2×C4×C73C2×#4111yescharacteristic
215C2×Dic73—21no—
216C2×Dic73—21no—
217C2×Dic73—21no—
218C2×Dic73—21no—
219C2×Dic73C4×D411yescharacteristic
220C2×Dic73C4×D411yescharacteristic
221C2×Dic73C4×D411yescharacteristic
222C4×D73—21no—
223C4×D73—21no—
224C4×D73—21no—
225C4×D73—21no—
226C4×D73—41no—
227C4×D73—42no—
228C4×D73—42no—
229Dic146—21no—
230Dic146—21no—
231Dic146—21no—
232Dic146—42no—
233Dic146—42no—
234Dic146#13112yes—
235C8×C73—21no—
236C8×C73—21no—
237C8×C73—21no—
238C8×C73—21no—
239C8×C73—41no—
240C8×C73—41no—
241C8×C73—41no—
242#10596831—21no—
243#10596831—21no—
244#10596831—21no—
245#10596831—21no—
246#10596831—41no—
247#10596831—41no—
248#10596831—41no—
249C22×C73C2×C4×D411yescharacteristic
250C2×D73—21no—
251C2×D73—21no—
252C2×D73—21no—
253C4×C73—21no—
254C4×C73—41no—
255C4×C73—41no—
256C4×C73—41no—
257C4×C73—41no—
258C4×C73C2×#13111yescharacteristic
259C4×C73#56811yescharacteristic
260C4×C73#57811yescharacteristic
261C4×C73#57811yescharacteristic
262Dic73—22no—
263Dic73—21no—
264Dic73—21no—
265Dic73—21no—
266Dic73—41no—
267Dic73—41no—
268Dic73—41no—
269#18032—731nomaximal, Sylow
270C2×C73#293311yescharacteristic
271C2×C73#295011yescharacteristic
272C2×C73#295511yescharacteristic
273D73—41no—
274#3104—731no—
275#2986—731no—
276#2984—731no—
277#1593D7311yescharacteristic, complemented
278#1591—731no—
279#1587—731no—
280#1587—731no—
281#1584—731no—
282#1584—731no—
283#1579—731no—
284C8×Q16—731no—
285#1572—731no—
286#1497—731no—
287#1479—731no—
288#1454—731no—
289C73#1803211yescharacteristic, complemented, Sylow
290#533—731no—
291C2×C4×Q8—731no—
292#484—731no—
293#484—731no—
294#477—731no—
295#477—731no—
296#462—731no—
297#462—731no—
298#451—731no—
299#451—731no—
300#446—731no—
301#446—731no—
302#446—1461no—
303#446—1461no—
304#446—1461no—
305#446C2×D7311yescharacteristic
306#446C2×D7311yescharacteristic
307C8×Q8—731no—
308C8×Q8—731no—
309C8×Q8—731no—
310C8×Q8—731no—
311C8×Q8—1461no—
312C4×Q16—731no—
313C4×Q16—731no—
314C4×Q16—731no—
315C4×Q16C2×D7311yescharacteristic
316#433—1461no—
317#433—1461no—
318#432—731no—
319#432—1461no—
320#431—731no—
321#431—731no—
322#431—731no—
323#431—731no—
324#431—1461no—
325#335—731no—
326#335—731no—
327#334—731no—
328#334C2×D7311yescharacteristic
329Q8⋊C8—731no—
330Q8⋊C8—731no—
331Q8⋊C8—731no—
332Q8⋊C8—731no—
333Q8⋊C8—731no—
334Q8⋊C8—731no—
335Q8⋊C8C2×D7311yescharacteristic
336Q8⋊C8C2×D7311yescharacteristic
337#322—731no—
338#322—731no—
339#322C2×D7311yescharacteristic
340C82—731no—
341C22×Q8—731no—
342C2×Q16—731no—
343C2×Q16—731no—
344C2×Q16—731no—
345C2×Q16C4×D7311yescharacteristic
346C4⋊Q8—731no—
347C4⋊Q8—731no—
348#125—1461no—
349C22⋊Q8—731no—
350C22⋊Q8—731no—
351C22⋊Q8—1461no—
352C4×Q8—731no—
353C4×Q8—731no—
354C4×Q8—731no—
355C4×Q8—731no—
356C4×Q8—1461no—
357C4×Q8—1461no—
358C4×Q8—1461no—
359C4×Q8C22×D7311yescharacteristic
360C4×Q8C22×D7311yescharacteristic
361#117—1461no—
362C2×C4⋊C4—731no—
363C2×C4⋊C4—1461no—
364C2×C42—731no—
365C2×C42—1461no—
366#107—731no—
367#107—731no—
368#107—731no—
369#107C4×D7311yescharacteristic
370C4⋊C8—21no—
371C4⋊C8—21no—
372C4⋊C8—731no—
373C4⋊C8—731no—
374C4⋊C8—731no—
375C4⋊C8—731no—
376C4⋊C8—731no—
377C4⋊C8—731no—
378C4⋊C8—1461no—
379C4⋊C8—1461no—
380C4⋊C8—1461no—
381C4⋊C8—1461no—
382C4⋊C8—1461no—
383C4⋊C8—1461no—
384C4⋊C8—1461no—
385C4⋊C8C22×D7311yescharacteristic
386C4⋊C8C22×D7311yescharacteristic
387#103—731no—
388#103—731no—
389#103—731no—
390#103—731no—
391#103—731no—
392#103—731no—
393#103C4×D7311yescharacteristic
394#103C4×D7311yescharacteristic
395C22⋊C8—731no—
396C22⋊C8—731no—
397C22⋊C8—731no—
398C22⋊C8—731no—
399C22⋊C8—1461no—
400C22⋊C8—1461no—
401C22⋊C8—1461no—
402#97—21no—
403#97—21no—
404#97—1461no—
405#97—1461no—
406#97—1461no—
407C4×C8—731no—
408C4×C8—731no—
409C4×C8—731no—
410C4×C8—1461no—
411C4×C8—1461no—
412C4×C8—1461no—
413C4×C8—1461no—
414C4×C8—1461no—
415C4×C8C22×D7311yescharacteristic
416C4×C8C22×D7311yescharacteristic
417C4×C8C22×D7311yescharacteristic
418C2×Q8—2921no—
419C2×Q8—731no—
420C2×Q8—731no—
421C2×Q8—1461no—
422C2×Q8—1461no—
423C2×Q8—1462no—
424C2×Q8C2×C4×D7311yescharacteristic
425C2×Q8C2×C4×D7311yescharacteristic
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—1461no—
430C22×C4—1461no—
431Q16—41no—
432Q16—2921no—
433Q16—1462no—
434Q16—1462no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—731no—
439C2×C8—731no—
440C2×C8—731no—
441C2×C8—731no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8C2×C4×D7311yescharacteristic
446C2×C8C2×C4×D7311yescharacteristic
447C2×C8D4×D7311yescharacteristic
448C2×C8D4×D7311yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—731no—
452C4⋊C4—731no—
453C4⋊C4—1461no—
454C4⋊C4—1461no—
455C4⋊C4—1461no—
456C4⋊C4—1461no—
457C4⋊C4—1461no—
458C4⋊C4—1461no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4C2×C4×D7311yescharacteristic
464C4⋊C4C2×C4×D7311yescharacteristic
465C22⋊C4—1461no—
466C22⋊C4—1461no—
467C42—21no—
468C42—21no—
469C42—731no—
470C42—731no—
471C42—731no—
472C42—1461no—
473C42—1461no—
474C42—1461no—
475C42—1461no—
476C42C23×D7311yescharacteristic
477C23—731no—
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—2922no—
483Q8—2922no—
484Q8—732no—
485Q8—1461no—
486Q8—1461no—
487Q8—1461no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—2921no—
493C2×C4—2922no—
494C2×C4—2922no—
495C2×C4—731no—
496C2×C4—731no—
497C2×C4—731no—
498C2×C4—1461no—
499C2×C4—1461no—
500C2×C4—1461no—
501C2×C4—1461no—
502C2×C4—1461no—
503C2×C4—1461no—
504C2×C4—1461no—
505C2×C4—1461no—
506C2×C4C22×C4×D7311yescharacteristic
507C2×C4C2×D4×D7311yescharacteristic
508C2×C4#41×D7311yescharacteristic
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—2921no—
517C8—2921no—
518C8—2921no—
519C8—1461no—
520C8—1461no—
521C8—1461no—
522C8—1461no—
523C22—1461no—
524C22—1461no—
525C22—1461no—
526C22C4×D4×D7311yescharacteristic
527C4—21no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—2921no—
533C4—2921no—
534C4—2921no—
535C4—1462no—
536C4—1461no—
537C4—1461no—
538C4—1461no—
539C4#131×D7311yescharacteristic
540C4#1798935311yescharacteristic
541C4#1798940311yescharacteristic
542C4#1798940311yescharacteristic
543C2—2921no—
544C2#2261866011yescharacteristic
545C2#2261879611yescharacteristic
546C2#2261883111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1593C2
socleC22×C73C2×C4×D4
prime radicalC22×D73C4×D4
cosocleC42×C73C24
hypercenter#1593D73
hypocenterC73#18032
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1593

Complements

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

NH ≅ G/Ntype
4. #22608825C2semidirect
5. #22608803C2semidirect
6. #22608801C2semidirect
7. #22608779C2semidirect
8. #22608779C2semidirect
9. #22608751C2semidirect
10. #22608735C2semidirect
15. C73×#1593C2semidirect
277. #1593D73semidirect
289. C73#18032semidirect

Maximal subgroups

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

subgroupindex
#226199682
#226190292
#226190172
#226088252
#226088032
#226088012
#226087792
#226087512
#226087352
#226087082
#226084022
#226083242
#226082322
C73×#15932
#1803273