Finite Group Databasebeta

← #33103326

Subgroups — #33103326

Order 18688.

894 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618589C211yesmaximal, characteristic, complemented
2C2×#17988581C211yesmaximal, characteristic
3#22618381C211yesmaximal, characteristic
4#22610643C212yesmaximal, complemented
5#22610577C212yesmaximal, complemented
6#22610207C212yesmaximal, complemented
7#22609967C211yesmaximal, characteristic, complemented
8#22609876C211yesmaximal, characteristic
9#22609772C212yesmaximal, complemented
10#22609709C211yesmaximal, characteristic, complemented
11C73×#1826C211yesmaximal, characteristic, complemented
12C22×#11924028C2211yescharacteristic
13C2×#14556599C2212yes—
14C23×#10596831C2211yescharacteristic
15#17988648—22no—
16#17988644—21no—
17#17988640—21no—
18C2×#14556526C2212yescomplemented
19#17988614C2212yescomplemented
20C2×#14556519C2212yes—
21C2×#14556519C2212yes—
22#17988581C411yescharacteristic, complemented
23#17988581C411yescharacteristic, complemented
24#17988581C411yescharacteristic, complemented
25#17988581C411yescharacteristic, complemented
26#17988581C2211yescharacteristic
27#17988581C2211yescharacteristic
28#17988581C2211yescharacteristic
29#17988581C2211yescharacteristic
30C2×#14556514C411yescharacteristic, complemented
31C2×#14556514C411yescharacteristic, complemented
32C2×#14556511C411yescharacteristic, complemented
33C2×#14556511C411yescharacteristic, complemented
34#17988555C2212yes—
35#17988552—22no—
36#17988551C2211yescharacteristic
37#17988549—21no—
38C2×C22⋊#10596831C2211yescharacteristic
39C22⋊C8×D73—21no—
40#17988543C2212yescomplemented
41C2×#14556502C2212yescomplemented
42#17988351C2212yescomplemented
43#17988342C2212yescomplemented
44C2×#117×C73C2211yescharacteristic
45#407×C73C2211yescharacteristic
46C4×C22⋊C4×C73C2211yescharacteristic
47#339×C73C2212yescomplemented
48#339×C73C2212yescomplemented
49C2×#11924028—24no—
50C2×#11924028—21no—
51C2×#11924028—21no—
52C2×#11924028—42no—
53C2×#11924028C2×C411yescharacteristic
54C2×#11924028C2×C411yescharacteristic
55C2×#11924028C2311yescharacteristic
56C2×#11924028C2311yescharacteristic
57C22×Dic146C2×C411yescharacteristic
58C22×D292C2×C411yescharacteristic
59C22×C22⋊D73—21no—
60C22×C4×D73—21no—
61#14556599—24no—
62#14556599—42no—
63C2×#11924012C2312yes—
64C2×#11924011—22no—
65C22×#10596831—42no—
66C22×#10596831C2×C411yescharacteristic
67C22×#10596831C2×C411yescharacteristic
68C22×#10596831C2×C411yescharacteristic
69C22×#10596831C2×C411yescharacteristic
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2312yes—
73C2×C8×D73—22no—
74#14556526—42no—
75C4⋊#10596831—22no—
76C4⋊#10596831—22no—
77#14556519—24no—
78#14556519—42no—
79#14556514C2×C411yescharacteristic
80#14556514C2×C411yescharacteristic
81#14556514C2×C411yescharacteristic
82#14556514C2×C411yescharacteristic
83#14556514C2×C411yescharacteristic
84#14556514C2×C411yescharacteristic
85#14556514C2×C411yescharacteristic
86#14556514C2×C411yescharacteristic
87#14556511C2×C411yescharacteristic
88#14556511C2×C411yescharacteristic
89#14556511C2×C411yescharacteristic
90#14556511C2×C411yescharacteristic
91#14556511C2×C411yescharacteristic
92#14556511C2×C411yescharacteristic
93#14556511C2×C411yescharacteristic
94#14556511C2×C411yescharacteristic
95C22⋊#10596831—21no—
96C22⋊#10596831—21no—
97C22⋊#10596831D411yescharacteristic
98C22⋊#10596831D411yescharacteristic
99C22⋊#10596831D412yes—
100#14556503—22no—
101#14556503—22no—
102#14556502—22no—
103#14556502—42no—
104C4×#10596831—22no—
105C23×C4×C73C2311yescharacteristic
106C2×M16×C73C2312yes—
107#117×C73—21no—
108#117×C73—21no—
109#117×C73C2×C411yescharacteristic
110#117×C73C2×C411yescharacteristic
111#117×C73C2311yescharacteristic
112#117×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2×C411yescharacteristic
114C2×C4⋊C4×C73C2×C411yescharacteristic
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C42×C73—21no—
118C2×C42×C73C2312yes—
119C22⋊C8×C73—21no—
120C22⋊C8×C73—21no—
121#95×C73—21no—
122C23×D73—21no—
123#11924028—22no—
124#11924028—22no—
125#11924028—22no—
126#11924028—22no—
127#11924028—22no—
128#11924028—41no—
129#11924028—41no—
130#11924028—44no—
131#11924028—44no—
132#11924028#4114yes—
133C2×Dic146—22no—
134C2×Dic146—21no—
135C2×Dic146—41no—
136C2×Dic146C22×C411yescharacteristic
137C2×Dic146C22×C411yescharacteristic
138C2×D292—22no—
139C2×D292—21no—
140C2×D292—41no—
141C2×D292C22×C411yescharacteristic
142C2×D292C22×C411yescharacteristic
143C2×C22⋊D73—21no—
144C2×C22⋊D73—21no—
145C2×C22⋊D73—21no—
146C2×C22⋊D73—21no—
147C2×C22⋊D73—44no—
148C22×Dic73—21no—
149C2×C4×D73—21no—
150C2×C4×D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—44no—
154#11924012—22no—
155#11924012—22no—
156#11924011—44no—
157C2×#10596831—22no—
158C2×#10596831—22no—
159C2×#10596831—22no—
160C2×#10596831—22no—
161C2×#10596831—22no—
162C2×#10596831—41no—
163C2×#10596831—41no—
164C2×#10596831—41no—
165C2×#10596831—41no—
166C2×#10596831C22×C411yescharacteristic
167C2×#10596831C22×C411yescharacteristic
168C2×#10596831C22×C411yescharacteristic
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C2×D412yes—
171C2×#10596831C2×D412yes—
172C8×D73—44no—
173C24×C73#4111yescharacteristic
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—22no—
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C22×C411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73#4111yescharacteristic
184C22×C4×C73C2411yescharacteristic
185M16×C73—22no—
186M16×C73—22no—
187C2×C8×C73—22no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73C22×C411yescharacteristic
191C4⋊C4×C73C22×C411yescharacteristic
192C4⋊C4×C73C22×C411yescharacteristic
193C4⋊C4×C73C22×C411yescharacteristic
194C22⋊C4×C73—21no—
195C22⋊C4×C73—21no—
196C22⋊C4×C73—21no—
197C22⋊C4×C73—21no—
198C22⋊C4×C73—42no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—21no—
203C42×C73—22no—
204C42×C73—22no—
205C23×C73—22no—
206C23×C73—22no—
207C23×C73—41no—
208C23×C73C4×D411yescharacteristic
209C23×C73C4×D411yescharacteristic
210C23×C73C2×#4111yescharacteristic
211C22×D73—21no—
212C22×D73—21no—
213C22×D73—42no—
214C22×D73—42no—
215C22×D73—81no—
216C2×C4×C73—22no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—42no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—81no—
232C2×C4×C73C4×D411yescharacteristic
233C2×C4×C73C4×D411yescharacteristic
234C2×C4×C73C23×C411yescharacteristic
235C2×C4×C73C22×D411yescharacteristic
236C2×C4×C73C2×#4112yes—
237C2×C4×C73H(5,𝔽2)12yes—
238C22⋊D73—24no—
239C22⋊D73—42no—
240C22⋊D73—42no—
241C22⋊D73—42no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—42no—
245D292—21no—
246D292—21no—
247D292—21no—
248D292—42no—
249D292#11712yes—
250C4×D73—24no—
251C4×D73—42no—
252C4×D73—42no—
253C4×D73—42no—
254Dic146—21no—
255Dic146—21no—
256Dic146—21no—
257Dic146—42no—
258Dic146#11712yes—
259C8×C73—42no—
260#10596831—21no—
261#10596831—21no—
262#10596831—21no—
263#10596831—21no—
264#10596831—22no—
265#10596831—22no—
266#10596831—42no—
267C22×C73—21no—
268C22×C73—21no—
269C22×C73—21no—
270C22×C73—21no—
271C22×C73—21no—
272C22×C73—21no—
273C22×C73—21no—
274C22×C73—44no—
275C22×C73#44412yes—
276C22×C73C2×C4×D411yescharacteristic
277C22×C73#55112yes—
278C2×D73—22no—
279C2×D73—41no—
280C2×D73—41no—
281C2×D73—41no—
282C2×D73—41no—
283C2×D73—41no—
284C2×D73—84no—
285C4×C73—21no—
286C4×C73—21no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—41no—
291C4×C73—82no—
292C4×C73C2×#11711yescharacteristic
293C4×C73#51811yescharacteristic
294C4×C73#54612yes—
295Dic73—22no—
296Dic73—41no—
297#22586—731nomaximal, Sylow
298C2×C73—22no—
299C2×C73—22no—
300C2×C73—41no—
301C2×C73#234211yescharacteristic
302C2×C73C2×#44411yescharacteristic
303C2×C73#295711yescharacteristic
304D73—42no—
305D73—81no—
306#2924—731no—
307C2×#416—731no—
308#2880—731no—
309#1938—732no—
310#1929—732no—
311#1866—732no—
312#1826D7311yescharacteristic, complemented
313#1810—731no—
314#1789—731no—
315#1765—732no—
316#1752—731no—
317C73#2258611yescharacteristic, complemented, Sylow
318C22×#41—731no—
319C2×#131—732no—
320C23×C8—731no—
321C2×#117C2×D7311yescharacteristic
322#433—1462no—
323#432—1461no—
324#431—1461no—
325C2×#108—732no—
326#424—732no—
327C2×#104—732no—
328C2×#104—732no—
329#416—731no—
330#416—731no—
331#416—731no—
332#416—731no—
333#416—731no—
334#416—731no—
335#416—731no—
336#416—731no—
337C2×#103—731no—
338C2×#103—731no—
339C2×#102—731no—
340C2×#102—731no—
341#408—732no—
342#408—1462no—
343#407—731no—
344#407—1461no—
345#407C2×D7311yescharacteristic
346C2×C22⋊C8—731no—
347C2×C22⋊C8—1461no—
348#405—732no—
349C2×#97—732no—
350C4×C22⋊C4C2×D7311yescharacteristic
351#341—732no—
352#339C2×D7312yescomplemented
353#339C2×D7312yescomplemented
354#336—732no—
355C2×#41—2922no—
356C2×#41—731no—
357C2×#41—731no—
358C2×#41—731no—
359C2×#41—731no—
360C2×#41—1464no—
361C2×#41—1461no—
362C2×#41—1461no—
363C22×Q8—731no—
364C22×D4—731no—
365C22×D4—1461no—
366C23×C4—1461no—
367C23×C4C22×D7311yescharacteristic
368#131—2922no—
369#131—1464no—
370C2×M16—732no—
371C2×M16—1462no—
372C2×M16C22×D7312yes—
373C22×C8—2922no—
374C22×C8—731no—
375C22×C8—731no—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8—731no—
379C22×C8—731no—
380C22×C8—732no—
381C22×C8—1462no—
382#117—21no—
383#117—21no—
384#117C4×D7311yescharacteristic
385#117C4×D7311yescharacteristic
386#117C22×D7311yescharacteristic
387#117C22×D7311yescharacteristic
388C2×C4⋊C4C4×D7311yescharacteristic
389C2×C4⋊C4C4×D7311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C42—21no—
393C2×C42C22×D7312yes—
394#108—2922no—
395C4⋊C8—1462no—
396C4⋊C8—1462no—
397#104—2922no—
398#104—1464no—
399#103—731no—
400#103—731no—
401#103—731no—
402#103—731no—
403#103—731no—
404#103—731no—
405#103—731no—
406#103—731no—
407#102—731no—
408#102—731no—
409#102—731no—
410#102—731no—
411#102—731no—
412#102—731no—
413#102—731no—
414#102—731no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—731no—
418C22⋊C8—731no—
419C22⋊C8—732no—
420C22⋊C8—1462no—
421C22⋊C8—1462no—
422C22⋊C8—1461no—
423C22⋊C8—1461no—
424#97—2922no—
425#97—1462no—
426C4×C8—1462no—
427#95—21no—
428C24—1461no—
429C24#1192402711yescharacteristic
430#41—2924no—
431#41—2921no—
432#41—2921no—
433#41—2924no—
434#41—734no—
435#41—1462no—
436#41—1462no—
437#41—1462no—
438#41—1462no—
439#41—1462no—
440C2×Q8—2921no—
441C2×Q8—731no—
442C2×Q8—731no—
443C2×Q8—1462no—
444C2×Q8—1461no—
445C2×D4—2921no—
446C2×D4—2924no—
447C2×D4—731no—
448C2×D4—731no—
449C2×D4—1462no—
450C2×D4—1461no—
451C2×D4—1461no—
452C2×D4—1461no—
453C2×D4—1461no—
454C2×D4—1461no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—2924no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4C2×C4×D7311yescharacteristic
467C22×C4C2×C4×D7311yescharacteristic
468C22×C4D4×D7311yescharacteristic
469C22×C4D4×D7311yescharacteristic
470C22×C4#1192402711yescharacteristic
471C22×C4C23×D7311yescharacteristic
472M16—22no—
473M16—22no—
474M16—2924no—
475M16—1462no—
476M16—1462no—
477C2×C8—22no—
478C2×C8—2924no—
479C2×C8—2921no—
480C2×C8—2921no—
481C2×C8—2921no—
482C2×C8—2921no—
483C2×C8—731no—
484C2×C8—731no—
485C2×C8—731no—
486C2×C8—731no—
487C2×C8—732no—
488C2×C8—732no—
489C2×C8—1462no—
490C2×C8—1462no—
491C2×C8—1462no—
492C2×C8—1462no—
493C2×C8—1462no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4C2×C4×D7311yescharacteristic
497C4⋊C4C2×C4×D7311yescharacteristic
498C4⋊C4C2×C4×D7311yescharacteristic
499C4⋊C4C2×C4×D7311yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—42no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—22no—
510C42—22no—
511C23—22no—
512C23—22no—
513C23—41no—
514C23—2922no—
515C23—2922no—
516C23—5841no—
517C23—1461no—
518C23—1461no—
519C23#1455655111yescharacteristic
520C23#1455655111yescharacteristic
521C23C2×#1192402711yescharacteristic
522Q8—2922no—
523Q8—732no—
524Q8—1461no—
525Q8—1461no—
526Q8—1461no—
527D4—2922no—
528D4—2922no—
529D4—2922no—
530D4—2922no—
531D4—732no—
532D4—1464no—
533D4—1461no—
534D4—1461no—
535D4—1461no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—81no—
552C2×C4—2922no—
553C2×C4—2922no—
554C2×C4—2922no—
555C2×C4—2922no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1464no—
559C2×C4#1455655111yescharacteristic
560C2×C4#1455655111yescharacteristic
561C2×C4C22×C4×D7311yescharacteristic
562C2×C4C2×D4×D7311yescharacteristic
563C2×C4#41×D7312yes—
564C2×C4#1455663712yes—
565C8—42no—
566C8—2922no—
567C8—1462no—
568C8—1462no—
569C8—1461no—
570C8—1461no—
571C8—1461no—
572C8—1461no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—44no—
581C22—2921no—
582C22—2921no—
583C22—2921no—
584C22—2921no—
585C22—2921no—
586C22—5844no—
587C22—1462no—
588C22#1798870612yes—
589C22C2×#1455655111yescharacteristic
590C22#1798926012yes—
591C4—21no—
592C4—21no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—82no—
598C4—2921no—
599C4—1462no—
600C4#117×D7311yescharacteristic
601C4#1798906111yescharacteristic
602C4#1798922912yes—
603C2—22no—
604C2—22no—
605C2—41no—
606C2—2922no—
607C2—5841no—
608C2#2261277311yescharacteristic
609C2C2×#1798870611yescharacteristic
610C2#2261884411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1826C2
socleC22×C73C2×C4×D4
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenter#1826D73
hypocenterC73#22586
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
1. #22618589C2semidirect
4. #22610643C2semidirect
5. #22610577C2semidirect
6. #22610207C2semidirect
7. #22609967C2semidirect
9. #22609772C2semidirect
10. #22609709C2semidirect
11. C73×#1826C2semidirect
18. C2×#14556526C22semidirect
19. #17988614C22semidirect
22. #17988581C4semidirect
23. #17988581C4semidirect
24. #17988581C4semidirect
25. #17988581C4semidirect
30. C2×#14556514C4semidirect
31. C2×#14556514C4semidirect
32. C2×#14556511C4semidirect
33. C2×#14556511C4semidirect
40. #17988543C22semidirect
41. C2×#14556502C22semidirect
42. #17988351C22semidirect
43. #17988342C22semidirect
47. #339×C73C22semidirect
48. #339×C73C22semidirect
312. #1826D73semidirect
317. C73#22586semidirect
352. #339C2×D73semidirect
353. #339C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226185892
C2×#179885812
#226183812
#226106432
#226105772
#226102072
#226099672
#226098762
#226097722
#226097092
C73×#18262
#2258673