Finite Group Databasebeta

← #33056055

Subgroups — #33056055

Order 18688.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620007C211yesmaximal, characteristic
2#22619011C211yesmaximal, characteristic, complemented
3#22608807C211yesmaximal, characteristic, complemented
4#22608800C211yesmaximal, characteristic, complemented
5#22608779C211yesmaximal, characteristic, complemented
6#22608772C211yesmaximal, characteristic, complemented
7#22608748C211yesmaximal, characteristic, complemented
8#22608743C211yesmaximal, characteristic, complemented
9C8×#11924018C211yesmaximal, characteristic, complemented
10C8×#11924016C211yesmaximal, characteristic, complemented
11#22608722C211yesmaximal, characteristic, complemented
12#22608359C211yesmaximal, characteristic, complemented
13#22608339C211yesmaximal, characteristic
14#22608231C211yesmaximal, characteristic, complemented
15C73×#2993C211yesmaximal, characteristic
16C2×C4×Dic146C2211yescharacteristic
17#17988915C411yescharacteristic
18#17988898C411yescharacteristic
19#17988891C411yescharacteristic
20#17988891C411yescharacteristic
21#17988806C411yescharacteristic
22#17988806C411yescharacteristic
23#17988742C411yescharacteristic
24#17988725C411yescharacteristic
25C8×Dic146C2211yescharacteristic
26C8×Dic146C2211yescharacteristic
27C8×Dic146C2211yescharacteristic
28C8×Dic146C2211yescharacteristic
29#17988711—21no—
30#17988711—21no—
31C4×#11924018C2211yescharacteristic
32C4×#11924018C2211yescharacteristic
33C4×#11924016C2211yescharacteristic
34C4×#11924016C2211yescharacteristic
35#17988640C2211yescharacteristic
36#17988640C2211yescharacteristic
37#17988640C2211yescharacteristic
38#17988637—21no—
39#17988637—21no—
40#17988339C2211yescharacteristic
41#17988339C2211yescharacteristic
42#17988337C2211yescharacteristic
43#17988337C2211yescharacteristic
44Q8⋊#10596831C2211yescharacteristic
45Q8⋊#10596831C2211yescharacteristic
46#17988317C2211yescharacteristic
47#17988317C2211yescharacteristic
48#17988317C2211yescharacteristic
49#17988317C2211yescharacteristic
50D4⋊#10596831C2211yescharacteristic
51D4⋊#10596831C2211yescharacteristic
52#17988309C2211yescharacteristic
53#17988309C2211yescharacteristic
54#17988308C2211yescharacteristic
55C8×#10596831C2211yescharacteristic
56#529×C73C2211yescharacteristic
57#446×C73C2211yescharacteristic
58C8×Q8×C73C2211yescharacteristic
59#436×C73—21no—
60#436×C73C2211yescharacteristic
61#435×C73—21no—
62C8×D4×C73C2211yescharacteristic
63#433×C73—21no—
64#431×C73C2211yescharacteristic
65#424×C73—21no—
66#423×C73C2211yescharacteristic
67C22×Dic146C2×C411yescharacteristic
68C2×#11924018C2×C411yescharacteristic
69C2×#11924018C2×C411yescharacteristic
70C2×#11924016C2×C411yescharacteristic
71C2×#11924016C2×C411yescharacteristic
72C4×Dic146—21no—
73C4×Dic146—21no—
74C4×Dic146C2×C411yescharacteristic
75C4×Dic146C2×C411yescharacteristic
76C4×Dic146C2311yescharacteristic
77C4×Dic146C2311yescharacteristic
78C2×C4⋊Dic73C2×C411yescharacteristic
79C2×#11924007—21no—
80C2×C4×Dic73—21no—
81#14556524C2×C411yescharacteristic
82#14556524C2×C411yescharacteristic
83#14556522C2×C411yescharacteristic
84#14556522C2×C411yescharacteristic
85C4⋊#10596831—21no—
86C4⋊#10596831C2×C411yescharacteristic
87C4⋊#10596831C2×C411yescharacteristic
88C4⋊#10596831C2311yescharacteristic
89C4⋊#10596831C2311yescharacteristic
90#14556520—21no—
91#14556520—21no—
92#14556520—21no—
93#14556520—21no—
94#14556515C2×C411yescharacteristic
95#14556515C2×C411yescharacteristic
96#14556514C2×C411yescharacteristic
97#14556514C2×C411yescharacteristic
98#14556514C2×C411yescharacteristic
99#14556514C2×C411yescharacteristic
100#14556512C2×C411yescharacteristic
101#14556512C2×C411yescharacteristic
102C22⋊#10596831—21no—
103C22⋊#10596831—21no—
104C22⋊#10596831C2×C411yescharacteristic
105C22⋊#10596831C2×C411yescharacteristic
106C4×#10596831—21no—
107C4×#10596831C2311yescharacteristic
108C4×#10596831C2311yescharacteristic
109C4×#10596831C2311yescharacteristic
110C4×#10596831C2311yescharacteristic
111C8×Dic73—21no—
112C8×Dic73—21no—
113C8×Dic73—21no—
114C8×Dic73—21no—
115C2×M16×C73—21no—
116C2×M16×C73—21no—
117C22×C8×C73—21no—
118#126×C73—21no—
119#125×C73C2×C411yescharacteristic
120#124×C73C2×C411yescharacteristic
121#123×C73—21no—
122C22⋊Q8×C73C2×C411yescharacteristic
123#121×C73C2×C411yescharacteristic
124C4×Q8×C73C2311yescharacteristic
125C4×D4×C73—21no—
126C4×D4×C73C2311yescharacteristic
127#117×C73—21no—
128C2×C42×C73C2311yescharacteristic
129C4⋊C8×C73—21no—
130C4⋊C8×C73—21no—
131C4⋊C8×C73—21no—
132C4⋊C8×C73D411yescharacteristic
133C4⋊C8×C73D411yescharacteristic
134C4⋊C8×C73C2311yescharacteristic
135C4⋊C8×C73C2311yescharacteristic
136C22⋊C8×C73—21no—
137C22⋊C8×C73—21no—
138C22⋊C8×C73D411yescharacteristic
139C22⋊C8×C73D411yescharacteristic
140#97×C73—21no—
141C4×C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C2×Dic146—21no—
145C2×Dic146—22no—
146C2×Dic146—22no—
147C2×Dic146C22×C411yescharacteristic
148C2×Dic146C22×C411yescharacteristic
149C22×Dic73—21no—
150C22×Dic73—21no—
151#11924018—22no—
152#11924018—22no—
153#11924016—22no—
154#11924016—22no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831C22×C411yescharacteristic
160C2×#10596831C22×C411yescharacteristic
161C2×#10596831C22×C411yescharacteristic
162C2×#10596831C22×C411yescharacteristic
163C4⋊Dic73—21no—
164C4⋊Dic73C22×C411yescharacteristic
165C4⋊Dic73C22×C411yescharacteristic
166#11924007—21no—
167#11924007—21no—
168#11924007—21no—
169#11924007—21no—
170C4×Dic73—21no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—21no—
174C2×Q8×C73C22×C411yescharacteristic
175C2×D4×C73—21no—
176C2×D4×C73C22×C411yescharacteristic
177C22×C4×C73—21no—
178C22×C4×C73C22×C411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73#4111yescharacteristic
181M16×C73—81no—
182M16×C73—81no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—81no—
186C2×C8×C73C2×D411yescharacteristic
187C2×C8×C73C2×D411yescharacteristic
188C2×C8×C73C2×D411yescharacteristic
189C2×C8×C73C2×D411yescharacteristic
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73C22×C411yescharacteristic
195C4⋊C4×C73C22×C411yescharacteristic
196C22⋊C4×C73—21no—
197C22⋊C4×C73—21no—
198C22⋊C4×C73—21no—
199C22⋊C4×C73—21no—
200C22⋊C4×C73—21no—
201C42×C73—21no—
202C42×C73C22×C411yescharacteristic
203C42×C73C2×D411yescharacteristic
204C42×C73#4111yescharacteristic
205C42×C73C2411yescharacteristic
206C23×C73—21no—
207C23×C73C4×D411yescharacteristic
208Q8×C73—21no—
209Q8×C73—21no—
210D4×C73—21no—
211D4×C73—21no—
212D4×C73—81no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73C4×D411yescharacteristic
224C2×C4×C73C4×D411yescharacteristic
225C2×C4×C73C4×D411yescharacteristic
226C2×C4×C73C23×C411yescharacteristic
227C2×C4×C73C22×D411yescharacteristic
228C2×C4×C73C2×#4111yescharacteristic
229C2×Dic73—24no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—81no—
235Dic146—24no—
236Dic146—21no—
237Dic146—21no—
238Dic146#13112yes—
239Dic146#13112yes—
240C8×C73—21no—
241C8×C73—21no—
242C8×C73—21no—
243C8×C73—21no—
244C8×C73—41no—
245C8×C73—41no—
246#10596831—21no—
247#10596831—21no—
248#10596831—21no—
249#10596831—21no—
250#10596831—41no—
251#10596831—41no—
252#10596831—41no—
253#10596831—41no—
254C22×C73—21no—
255C22×C73—21no—
256C22×C73—21no—
257C22×C73—41no—
258C22×C73—41no—
259C22×C73—41no—
260C22×C73C2×C4×D411yescharacteristic
261C4×C73—21no—
262C4×C73—21no—
263C4×C73—21no—
264C4×C73—21no—
265C4×C73—41no—
266C4×C73—41no—
267C4×C73—41no—
268C4×C73C2×#13111yescharacteristic
269C4×C73C2×#13111yescharacteristic
270C4×C73#57711yescharacteristic
271C4×C73#57811yescharacteristic
272Dic73—22no—
273Dic73—22no—
274Dic73—41no—
275Dic73—41no—
276#17919—731nomaximal, Sylow
277C2×C73—41no—
278C2×C73—81no—
279C2×C73#292911yescharacteristic
280C2×C73#294711yescharacteristic
281C2×C73#295711yescharacteristic
282#3108—731no—
283#2993D7311yescharacteristic
284#2982—731no—
285#1588—731no—
286#1587—731no—
287#1584—731no—
288#1583—731no—
289#1579—731no—
290SD16⋊C8—731no—
291C8×Q16—731no—
292C8×SD16—731no—
293#1574—731no—
294#1487—731no—
295#1482—731no—
296#1454—731no—
297C73#1791911yescharacteristic, complemented, Sylow
298#529C2×D7311yescharacteristic
299C2×C4×Q8—731no—
300#483—731no—
301#480—731no—
302#479—731no—
303#479—731no—
304#464—731no—
305#464—731no—
306#451—731no—
307#448—731no—
308#446C2×D7311yescharacteristic
309C8×Q8—731no—
310C8×Q8—731no—
311C8×Q8—731no—
312C8×Q8—731no—
313C8×Q8—1461no—
314C8×Q8—1461no—
315C8×Q8C2×D7311yescharacteristic
316C4×Q16—731no—
317C4×Q16—731no—
318C4×SD16—731no—
319C4×SD16—731no—
320#436—21no—
321#436C2×D7311yescharacteristic
322#435—21no—
323C8×D4C2×D7311yescharacteristic
324#433—21no—
325#431—731no—
326#431—731no—
327#431—731no—
328#431—1461no—
329#431—1461no—
330#431C2×D7311yescharacteristic
331#424—21no—
332#423C2×D7311yescharacteristic
333#335—731no—
334#335—731no—
335#334—731no—
336#334—731no—
337Q8⋊C8—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343D4⋊C8—731no—
344D4⋊C8—731no—
345#322—731no—
346#322—731no—
347#322—731no—
348C82—731no—
349C22×Q8—731no—
350C2×Q16—731no—
351C2×Q16—731no—
352C2×SD16—731no—
353C2×SD16—731no—
354C2×M16—21no—
355C2×M16—21no—
356C22×C8—21no—
357#126—21no—
358#125C4×D7311yescharacteristic
359#124C4×D7311yescharacteristic
360#123—21no—
361C22⋊Q8C4×D7311yescharacteristic
362#121C4×D7311yescharacteristic
363C4×Q8—731no—
364C4×Q8—731no—
365C4×Q8—731no—
366C4×Q8—731no—
367C4×Q8—1461no—
368C4×Q8—1461no—
369C4×Q8C22×D7311yescharacteristic
370C4×D4—21no—
371C4×D4C22×D7311yescharacteristic
372#117—21no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—1461no—
375C2×C42—1461no—
376C2×C42C22×D7311yescharacteristic
377#107—731no—
378#107—731no—
379#106—731no—
380#106—731no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—731no—
385C4⋊C8—731no—
386C4⋊C8—731no—
387C4⋊C8—731no—
388C4⋊C8—1461no—
389C4⋊C8—1461no—
390C4⋊C8—1461no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8C22⋊D7311yescharacteristic
394C4⋊C8C22⋊D7311yescharacteristic
395C4⋊C8C22×D7311yescharacteristic
396C4⋊C8C22×D7311yescharacteristic
397#103—731no—
398#103—731no—
399#103—731no—
400#103—731no—
401#103—731no—
402#103—731no—
403#102—731no—
404#102—731no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—731no—
408C22⋊C8—731no—
409C22⋊C8—1461no—
410C22⋊C8—1461no—
411C22⋊C8C22⋊D7311yescharacteristic
412C22⋊C8C22⋊D7311yescharacteristic
413#97—21no—
414C4×C8—21no—
415C4×C8—731no—
416C4×C8—731no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—1461no—
420C4×C8—1461no—
421C4×C8—1461no—
422C4×C8—1461no—
423C4×C8—1461no—
424C4×C8C22×D7311yescharacteristic
425C4×C8C22×D7311yescharacteristic
426C2×Q8—731no—
427C2×Q8—731no—
428C2×Q8—1462no—
429C2×Q8—1462no—
430C2×Q8—1461no—
431C2×Q8C2×C4×D7311yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D7311yescharacteristic
434C22×C4—21no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4C2×C4×D7311yescharacteristic
438C22×C4C2×C22⋊D7311yescharacteristic
439C22×C4#1192402811yescharacteristic
440Q16—1462no—
441Q16—1462no—
442SD16—1462no—
443SD16—1462no—
444M16—81no—
445M16—81no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—81no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—731no—
452C2×C8—731no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8C2×C22⋊D7311yescharacteristic
458C2×C8C2×C22⋊D7311yescharacteristic
459C2×C8C2×C22⋊D7311yescharacteristic
460C2×C8C2×C22⋊D7311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—731no—
466C4⋊C4—731no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4C2×C4×D7311yescharacteristic
473C4⋊C4C2×C4×D7311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—21no—
480C42—1461no—
481C42—1461no—
482C42—1461no—
483C42—1461no—
484C42C2×C4×D7311yescharacteristic
485C42C2×C22⋊D7311yescharacteristic
486C42#1192402811yescharacteristic
487C42C23×D7311yescharacteristic
488C23—21no—
489C23C4×C22⋊D7311yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—732no—
493Q8—732no—
494Q8—1464no—
495Q8—1461no—
496Q8—1461no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—5841no—
511C2×C4—1461no—
512C2×C4—1461no—
513C2×C4—1464no—
514C2×C4—1461no—
515C2×C4—1461no—
516C2×C4C4×C22⋊D7311yescharacteristic
517C2×C4C4×C22⋊D7311yescharacteristic
518C2×C4C4×C22⋊D7311yescharacteristic
519C2×C4C22×C4×D7311yescharacteristic
520C2×C4C22×C22⋊D7311yescharacteristic
521C2×C4C2×#1192402811yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—2921no—
529C8—2921no—
530C8—2921no—
531C8—2921no—
532C8—1461no—
533C8—1461no—
534C8—1461no—
535C8—1461no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×C4×C22⋊D7311yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—2921no—
551C4—2921no—
552C4—1462no—
553C4—1462no—
554C4#131×D7311yescharacteristic
555C4#131×D7311yescharacteristic
556C4#1798939811yescharacteristic
557C4#1798940211yescharacteristic
558C2—41no—
559C2—81no—
560C2#2261862311yescharacteristic
561C2#2261877011yescharacteristic
562C2#2261884611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#2993C2
socleC22×C73C2×C4×D4
prime radical#121×C73C2×C4
cosocleC42×C73C24
hypercenter#2993D73
hypocenterC73#17919
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2993

Complements

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

NH ≅ G/Ntype
2. #22619011C2semidirect
3. #22608807C2semidirect
4. #22608800C2semidirect
5. #22608779C2semidirect
6. #22608772C2semidirect
7. #22608748C2semidirect
8. #22608743C2semidirect
9. C8×#11924018C2semidirect
10. C8×#11924016C2semidirect
11. #22608722C2semidirect
12. #22608359C2semidirect
14. #22608231C2semidirect
297. C73#17919semidirect

Maximal subgroups

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

subgroupindex
#226200072
#226190112
#226088072
#226088002
#226087792
#226087722
#226087482
#226087432
C8×#119240182
C8×#119240162
#226087222
#226083592
#226083392
#226082312
C73×#29932
#1791973