Finite Group Databasebeta

← #33246461

Subgroups — #33246461

Order 18688.

2094 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D73C211yesmaximal, characteristic
2#22613030C214yesmaximal, complemented
3#22613024C214yesmaximal, complemented
4#22612807C214yesmaximal
5#22612805C214yesmaximal
6#22612673C211yesmaximal, characteristic, complemented
7#22612672C211yesmaximal, characteristic, complemented
8#22612670C211yesmaximal, characteristic, complemented
9#22612546C212yesmaximal
10#22612490C212yesmaximal, complemented
11C4×#14556582C212yesmaximal, complemented
12#22612336C211yesmaximal, characteristic
13#22612285C212yesmaximal
14#22612270C211yesmaximal, characteristic
15C73×#2330C211yesmaximal, characteristic, complemented
16C2×#14556583C2211yescharacteristic
17C2×#14556582C2212yes—
18C2×#14556581C2214yes—
19#125×D73C4116yescomplemented
20C2×#14556549C2212yes—
21C2×#14556548C2211yescharacteristic
22C2×#14556548C2211yescharacteristic
23C2×C4⋊C4×D73C2212yes—
24C2×C4⋊C4×D73C2214yes—
25C2×C42×D73C2211yescharacteristic
26#17988486C2212yes—
27#17988486C2214yes—
28#17988485C2212yes—
29#17988484C2214yes—
30#17988484C2214yes—
31#17988483C2212yes—
32#17988482C2214yes—
33#17988482C2214yes—
34#17988474C2214yes—
35#17988474C2214yes—
36#17988473C2214yes—
37#17988472C2214yes—
38#17988470C2214yes—
39#17988468C2214yes—
40#17988468C2214yes—
41#17988457C2211yescharacteristic
42#17988452C2212yes—
43#17988446C2214yes—
44#17988444C2214yes—
45#17988444C2214yes—
46C4⋊C4×Dic73C2214yes—
47C4⋊C4×Dic73C2212yes—
48C4⋊C4×Dic73C2212yes—
49C4⋊C4×Dic73C2214yes—
50C4×#11924007C2214yes—
51C4×#11924007C2214yes—
52C4×C4⋊Dic73C2214yes—
53C4×C4⋊Dic73C2212yes—
54#17988438C2214yes—
55#17988438C2212yes—
56#17988438C2214yes—
57#17988438C2212yes—
58C4×#11924005C2214yes—
59#17988434C2212yes—
60#17988434C2212yes—
61#17988434C2214yes—
62#17988433C2211yescharacteristic
63C42×Dic73C2211yescharacteristic
64C42×Dic73C2211yescharacteristic
65C2×#125×C73C2211yescharacteristic
66#389×C73C2212yes—
67#389×C73C2214yes—
68#387×C73C2214yes—
69#384×C73C2211yescharacteristic
70C4×C4⋊C4×C73C2212yes—
71#376×C73C2211yescharacteristic
72C22×C4×D73C2312yes—
73C22×C4×D73C2314yes—
74C22×C4×D73C2311yescharacteristic
75#14556583C2×C418yes—
76#14556582C2×C4116yes—
77#14556581C2×C4132yes—
78#14556549—28no—
79#14556548—24no—
80#14556548Q818yes—
81C2×C4⋊Dic73C2312yes—
82C2×C4⋊Dic73C2314yes—
83C2×C4⋊Dic73C2314yes—
84C2×C4⋊Dic73C2314yes—
85C2×C4⋊Dic73C2312yes—
86C2×#11924007C2311yescharacteristic
87C2×#11924007C2311yescharacteristic
88C2×#11924007C2314yes—
89C2×#11924007C2314yes—
90C2×#11924007C2314yes—
91C2×#11924007C2314yes—
92C2×#11924007C2314yes—
93C2×#11924007C2312yes—
94C4⋊C4×D73C2×C4132yes—
95C4⋊C4×D73C2×C4116yes—
96C2×#11924005C2311yescharacteristic
97C2×#11924005C2311yescharacteristic
98C2×#11924005C2314yes—
99C2×#11924005C2312yes—
100C2×C4×Dic73C2311yescharacteristic
101C2×C4×Dic73C2314yes—
102C2×C4×Dic73C2312yes—
103C2×C4×Dic73C2312yes—
104C2×C4×Dic73C2312yes—
105C2×C4×Dic73C2312yes—
106C2×C4×Dic73C2314yes—
107C2×C4×Dic73C2314yes—
108C2×C4×Dic73C2314yes—
109C2×C4×Dic73C2311yescharacteristic
110C2×C4×Dic73C2312yes—
111C2×C4×Dic73C2311yescharacteristic
112C42×D73C2×C418yes—
113#14556498C2312yes—
114#14556498C2312yes—
115#14556498C2314yes—
116#14556498C2314yes—
117#14556498C2314yes—
118#14556498C2314yes—
119#14556498C2314yes—
120#14556498C2314yes—
121#14556498C2314yes—
122#14556498C2314yes—
123#125×C73C2×C418yes—
124C2×C4⋊C4×C73C2312yes—
125C2×C4⋊C4×C73C2314yes—
126C2×C4⋊C4×C73C2314yes—
127C2×C4⋊C4×C73C2312yes—
128C2×C4⋊C4×C73C2314yes—
129C2×C42×C73C2311yescharacteristic
130C2×C42×C73C2314yes—
131C2×C42×C73C2311yescharacteristic
132C2×C42×C73C2311yescharacteristic
133#95×C73C2314yes—
134#95×C73C2314yes—
135#95×C73C2312yes—
136#95×C73C2312yes—
137C23×D73C2411yescharacteristic
138C22×Dic73C2411yescharacteristic
139C22×Dic73C2411yescharacteristic
140C22×Dic73C2414yes—
141C22×Dic73C2412yes—
142C22×Dic73C2414yes—
143C22×Dic73C2412yes—
144C22×Dic73C2411yescharacteristic
145C2×C4×D73—42no—
146C2×C4×D73—42no—
147C2×C4×D73—44no—
148C2×C4×D73—44no—
149C2×C4×D73C22×C418yes—
150C2×C4×D73C22×C4116yes—
151C2×C4×D73C22×C414yes—
152C2×C4×D73C2×Q814yes—
153C2×C4×D73#4114yes—
154C4⋊Dic73—24no—
155C4⋊Dic73—44no—
156C4⋊Dic73—44no—
157C4⋊Dic73C22×C4116yes—
158C4⋊Dic73C22×C418yes—
159#11924007—22no—
160#11924007—22no—
161#11924007—44no—
162#11924007—44no—
163#11924007C22×C4116yes—
164#11924007C22×C4116yes—
165#11924007C22×C4116yes—
166#11924007C2×Q818yes—
167#11924005—22no—
168#11924005—22no—
169#11924005—28no—
170#11924005C2×Q818yes—
171C4×Dic73—22no—
172C4×Dic73—28no—
173C4×Dic73—42no—
174C4×Dic73—42no—
175C4×Dic73—44no—
176C4×Dic73—44no—
177C4×Dic73—42no—
178C4×Dic73—42no—
179C4×Dic73—44no—
180C4×Dic73C22×C418yes—
181C4×Dic73C22×C414yes—
182C4×Dic73C2×Q814yes—
183C22×C4×C73C2411yescharacteristic
184C22×C4×C73C2411yescharacteristic
185C22×C4×C73C2414yes—
186C22×C4×C73C2414yes—
187C22×C4×C73C2412yes—
188C22×C4×C73C2412yes—
189C22×C4×C73C2411yescharacteristic
190C4⋊C4×C73—24no—
191C4⋊C4×C73—44no—
192C4⋊C4×C73—44no—
193C4⋊C4×C73C22×C4116yes—
194C4⋊C4×C73C22×C418yes—
195C42×C73—22no—
196C42×C73—44no—
197C42×C73C22×C414yes—
198C42×C73C2×Q814yes—
199C23×C73C2511yescharacteristic
200C22×D73—22no—
201C22×D73—22no—
202C22×D73C23×C412yes—
203C22×D73C22×Q812yes—
204C22×D73C2×#4112yes—
205C2×C4×C73—22no—
206C2×C4×C73—24no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—24no—
210C2×C4×C73—22no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—22no—
216C2×C4×C73—24no—
217C2×C4×C73—22no—
218C2×C4×C73C23×C414yes—
219C2×C4×C73C23×C418yes—
220C2×C4×C73C23×C412yes—
221C2×C4×C73C22×Q814yes—
222C2×C4×C73C22×Q812yes—
223C2×C4×C73C2×#4112yes—
224C2×Dic73—22no—
225C2×Dic73—22no—
226C2×Dic73—22no—
227C2×Dic73—22no—
228C2×Dic73—24no—
229C2×Dic73—24no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—24no—
233C2×Dic73—24no—
234C2×Dic73—22no—
235C2×Dic73—22no—
236C2×Dic73—24no—
237C2×Dic73C23×C414yes—
238C2×Dic73C23×C418yes—
239C2×Dic73C23×C412yes—
240C2×Dic73C22×Q814yes—
241C2×Dic73C22×Q812yes—
242C2×Dic73C2×#4112yes—
243C4×D73—48no—
244C4×D73—416no—
245C4×D73C4×Q8116yes—
246C22×C73C24×C411yescharacteristic
247C22×C73C23×Q811yescharacteristic
248C22×C73C22×#4111yescharacteristic
249C22×C73#58512yes—
250C22×C73#58512yes—
251C2×D73—24no—
252C2×D73—24no—
253C2×D73—22no—
254C2×D73—22no—
255C2×D73C2×C4×Q814yes—
256C4×C73—28no—
257C4×C73—24no—
258C4×C73—24no—
259C4×C73—41no—
260C4×C73—44no—
261C4×C73—41no—
262C4×C73C2×C4×Q814yes—
263Dic73—24no—
264Dic73—28no—
265Dic73—24no—
266Dic73—41no—
267Dic73—41no—
268Dic73—44no—
269Dic73C2×C4×Q814yes—
270#34217—731nomaximal, Sylow
271C2×C73C22×C4×Q811yescharacteristic
272C2×C73#342812yes—
273C2×C73#350412yes—
274C2×C73#351211yescharacteristic
275C2×C73#351211yescharacteristic
276D73—24no—
277C22×#125—731no—
278#2370—734no—
279#2370—734no—
280#2345—734no—
281#2345—734no—
282#2330—731no—
283#2330—731no—
284#2330—731no—
285#2330D7311yescharacteristic, complemented
286#2316—732no—
287C4×#125—732no—
288C4×#125—732no—
289#2285—731no—
290#2274—732no—
291C4×#117—731no—
292C73#3421711yescharacteristic, complemented, Sylow
293C2×#125—7316no—
294C2×#125—732no—
295C2×#125—734no—
296C2×#125—731no—
297C2×#125C2×D7311yescharacteristic
298C2×#117—732no—
299C2×#117—731no—
300C2×#117—731no—
301C22×C4⋊C4—732no—
302C22×C4⋊C4—734no—
303C22×C42—731no—
304#389—732no—
305#389—732no—
306#389—734no—
307#389—734no—
308#389—734no—
309#389—732no—
310#389—734no—
311#389—734no—
312#389C2×D7312yes—
313#389C2×D7314yes—
314#387—734no—
315#387—734no—
316#387—734no—
317#387—734no—
318#387—734no—
319#387—734no—
320#387—734no—
321#387C2×D7314yes—
322#384—731no—
323#384C2×D7311yescharacteristic
324#382—732no—
325#380—734no—
326#380—734no—
327#380—734no—
328C4×C4⋊C4—732no—
329C4×C4⋊C4—732no—
330C4×C4⋊C4—734no—
331C4×C4⋊C4—734no—
332C4×C4⋊C4—732no—
333C4×C4⋊C4—734no—
334C4×C4⋊C4—734no—
335C4×C4⋊C4—734no—
336C4×C4⋊C4—734no—
337C4×C4⋊C4—732no—
338C4×C4⋊C4—732no—
339C4×C4⋊C4—734no—
340C4×C4⋊C4C2×D7312yes—
341C4×C22⋊C4—734no—
342C4×C22⋊C4—732no—
343C4×C22⋊C4—732no—
344C4×C22⋊C4—734no—
345#376—731no—
346#376C2×D7311yescharacteristic
347C43—731no—
348C43—731no—
349C23×C4—734no—
350C23×C4—732no—
351C23×C4—731no—
352#125—7332no—
353#125—7316no—
354#125—738no—
355#125C4×D7318yes—
356#117—738no—
357#117—1464no—
358#117—1468no—
359C2×C4⋊C4—734no—
360C2×C4⋊C4—734no—
361C2×C4⋊C4—734no—
362C2×C4⋊C4—734no—
363C2×C4⋊C4—734no—
364C2×C4⋊C4—734no—
365C2×C4⋊C4—734no—
366C2×C4⋊C4—732no—
367C2×C4⋊C4—734no—
368C2×C4⋊C4—732no—
369C2×C4⋊C4—7332no—
370C2×C4⋊C4—7316no—
371C2×C4⋊C4—732no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4C22×D7312yes—
375C2×C4⋊C4C22×D7314yes—
376C2×C4⋊C4C22×D7314yes—
377C2×C4⋊C4C22×D7312yes—
378C2×C4⋊C4C22×D7314yes—
379C2×C22⋊C4—732no—
380C2×C22⋊C4—734no—
381C2×C22⋊C4—731no—
382C2×C22⋊C4—731no—
383C2×C42—732no—
384C2×C42—732no—
385C2×C42—732no—
386C2×C42—732no—
387C2×C42—734no—
388C2×C42—734no—
389C2×C42—734no—
390C2×C42—731no—
391C2×C42—731no—
392C2×C42—732no—
393C2×C42—738no—
394C2×C42—731no—
395C2×C42—734no—
396C2×C42C22×D7311yescharacteristic
397C2×C42C22×D7314yes—
398C2×C42C22×D7311yescharacteristic
399C2×C42C22×D7311yescharacteristic
400#95—732no—
401#95—732no—
402#95—734no—
403#95—734no—
404#95—734no—
405#95—734no—
406#95—734no—
407#95—734no—
408#95—734no—
409#95—734no—
410#95C22×D7312yes—
411#95C22×D7312yes—
412#95C22×D7314yes—
413#95C22×D7314yes—
414C24—731no—
415C22×C4—2924no—
416C22×C4—2922no—
417C22×C4—2924no—
418C22×C4—2922no—
419C22×C4—738no—
420C22×C4—7316no—
421C22×C4—731no—
422C22×C4—734no—
423C22×C4—734no—
424C22×C4—734no—
425C22×C4—732no—
426C22×C4—732no—
427C22×C4—734no—
428C22×C4—731no—
429C22×C4—734no—
430C22×C4—731no—
431C22×C4C23×D7311yescharacteristic
432C22×C4C23×D7314yes—
433C22×C4C23×D7311yescharacteristic
434C22×C4C23×D7314yes—
435C22×C4C23×D7312yes—
436C22×C4C23×D7312yes—
437C22×C4C23×D7311yescharacteristic
438C4⋊C4—24no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—2924no—
442C4⋊C4—2924no—
443C4⋊C4—2924no—
444C4⋊C4—2924no—
445C4⋊C4—7316no—
446C4⋊C4—7316no—
447C4⋊C4—7316no—
448C4⋊C4—7316no—
449C4⋊C4—738no—
450C4⋊C4—738no—
451C4⋊C4—1462no—
452C4⋊C4—1462no—
453C4⋊C4—1464no—
454C4⋊C4C2×C4×D73116yes—
455C4⋊C4C2×C4×D7318yes—
456C22⋊C4—738no—
457C22⋊C4—1462no—
458C22⋊C4—1462no—
459C22⋊C4—1468no—
460C42—22no—
461C42—44no—
462C42—2922no—
463C42—2924no—
464C42—2922no—
465C42—2924no—
466C42—2924no—
467C42—2922no—
468C42—2922no—
469C42—738no—
470C42—734no—
471C42—734no—
472C42—1468no—
473C42—1462no—
474C42C2×C4×D7314yes—
475C42Q8×D7314yes—
476C23—732no—
477C23—732no—
478C23—732no—
479C23—1462no—
480C23—1462no—
481C23C24×D7311yescharacteristic
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—2928no—
496C2×C4—29216no—
497C2×C4—732no—
498C2×C4—732no—
499C2×C4—732no—
500C2×C4—7316no—
501C2×C4—734no—
502C2×C4—738no—
503C2×C4—734no—
504C2×C4—1461no—
505C2×C4—1464no—
506C2×C4—1461no—
507C2×C4—1462no—
508C2×C4—1464no—
509C2×C4—1462no—
510C2×C4—1464no—
511C2×C4—1462no—
512C2×C4—1462no—
513C2×C4—1464no—
514C2×C4—1462no—
515C2×C4—1462no—
516C2×C4—1464no—
517C2×C4C22×C4×D7314yes—
518C2×C4C22×C4×D7318yes—
519C2×C4C22×C4×D7312yes—
520C2×C4C2×Q8×D7314yes—
521C2×C4C2×Q8×D7312yes—
522C2×C4#41×D7312yes—
523C22—734no—
524C22—1464no—
525C22—1462no—
526C22—1462no—
527C22—1464no—
528C22C23×C4×D7311yescharacteristic
529C22C22×Q8×D7311yescharacteristic
530C22C2×#41×D7311yescharacteristic
531C22#1798942212yes—
532C22#1798942412yes—
533C4—24no—
534C4—28no—
535C4—24no—
536C4—41no—
537C4—44no—
538C4—41no—
539C4—2921no—
540C4—2924no—
541C4—2921no—
542C4—734no—
543C4—1464no—
544C4—1464no—
545C4—1468no—
546C4C4×Q8×D7314yes—
547C2—1464no—
548C2C2×C4×Q8×D7311yescharacteristic
549C2#2262313112yes—
550C2#2262375412yes—
551C2#2262385011yescharacteristic
552C2#2262385411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2330C2
socle, cosocleC23×C73C25
prime radicalC23×D73C24
hypercenter#2330D73
hypocenterC73#34217
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #22613030C2semidirect
3. #22613024C2semidirect
6. #22612673C2semidirect
7. #22612672C2semidirect
8. #22612670C2semidirect
10. #22612490C2semidirect
11. C4×#14556582C2semidirect
15. C73×#2330C2semidirect
19. #125×D73C4semidirect
285. #2330D73semidirect
292. C73#34217semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D732
#226130302
#226130242
#226128072
#226128052
#226126732
#226126722
#226126702
#226125462
#226124902
C4×#145565822
#226123362
#226122852
#226122702
C73×#23302
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