Finite Group Databasebeta

← #33051898

Subgroups — #33051898

Order 18688.

1210 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989069C211yesmaximal, characteristic
2#22618959C212yesmaximal, complemented
3#22608275C211yesmaximal, characteristic, complemented
4#22608268C212yesmaximal, complemented
5#22608263C212yesmaximal, complemented
6#22608259C212yesmaximal, complemented
7#22608248C212yesmaximal, complemented
8#324×D73C212yesmaximal, complemented
9C73×#1467C211yesmaximal, characteristic, complemented
10C22×#11924029C411yescharacteristic
11#17989069—24no—
12#17989069—24no—
13C2×Q8×Dic73C411yescharacteristic
14C2×C4×D292C411yescharacteristic
15C2×C4×D292C411yescharacteristic
16C2×C4×D292C2211yescharacteristic
17C2×#14556553C411yescharacteristic
18C2×#14556553C2211yescharacteristic
19C2×#14556553C2211yescharacteristic
20C2×#14556548C411yescharacteristic
21C2×#14556548C411yescharacteristic
22C2×#14556548C2211yescharacteristic
23C2×C4⋊C4×D73C411yescharacteristic
24C2×C4⋊C4×D73C2211yescharacteristic
25C2×C4⋊C4×D73C2211yescharacteristic
26#17988660—24no—
27#17988657—24no—
28#17988644C2212yes—
29#17988549C2214yes—
30C22⋊C8×D73C2214yes—
31#17988313C2212yes—
32#17988313C2212yes—
33#17988312C2214yes—
34#17988310C2214yes—
35C2×C4×Q8×C73C2211yescharacteristic
36#432×C73C2212yes—
37#324×C73C2212yes—
38#324×C73C2212yes—
39C2×#11924029—24no—
40C2×#11924029—24no—
41C2×#11924029—24no—
42C22×D292C2×C411yescharacteristic
43C22×D292C2×C411yescharacteristic
44C22×D292C2×C411yescharacteristic
45C22×C4×D73C2×C411yescharacteristic
46C22×C4×D73C2×C411yescharacteristic
47C22×C4×D73C2×C411yescharacteristic
48C22×C4×D73C2×C411yescharacteristic
49C22×C4×D73C2×C411yescharacteristic
50C22×C4×D73C2×C411yescharacteristic
51C22×C4×D73C2×C411yescharacteristic
52C22×C4×D73C2311yescharacteristic
53C22×C4×D73C2311yescharacteristic
54C2×#11924011—24no—
55C2×C8×D73—24no—
56Q8×Dic73—24no—
57C4×D292—22no—
58C4×D292—22no—
59C4×D292—22no—
60C4×D292—22no—
61C4×D292D414yes—
62C4×D292D414yes—
63#14556553—24no—
64#14556553—24no—
65#14556553—24no—
66#14556548—22no—
67#14556548—22no—
68#14556548—22no—
69#14556548—22no—
70#14556548—22no—
71#14556548—22no—
72C2×C4⋊Dic73C2×C411yescharacteristic
73C2×C4⋊Dic73C2×C411yescharacteristic
74C2×C4⋊Dic73C2×C411yescharacteristic
75C2×#11924007C2×C411yescharacteristic
76C2×#11924007C2×C411yescharacteristic
77C2×#11924007C2×C411yescharacteristic
78C2×#11924007C2311yescharacteristic
79C4⋊C4×D73—24no—
80C4⋊C4×D73—24no—
81C4⋊C4×D73—24no—
82C2×#11924005C2×C411yescharacteristic
83C2×#11924005C2×C411yescharacteristic
84C2×#11924005C2×C411yescharacteristic
85C2×#11924005C2×C411yescharacteristic
86C2×#11924005C2×C411yescharacteristic
87C2×#11924005C2311yescharacteristic
88C2×C4×Dic73C2×C411yescharacteristic
89C2×C4×Dic73C2×C411yescharacteristic
90C2×C4×Dic73C2×C411yescharacteristic
91C4⋊#10596831—22no—
92C22⋊#10596831C2314yes—
93#14556503—24no—
94#14556503—24no—
95#14556502—22no—
96C22×Q8×C73C2×C411yescharacteristic
97C4×Q8×C73—22no—
98C4×Q8×C73—22no—
99C2×C4⋊C4×C73C2×C411yescharacteristic
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C4⋊C4×C73C2311yescharacteristic
102C2×C42×C73C2×C411yescharacteristic
103C2×C42×C73C2×C411yescharacteristic
104C2×C42×C73C2311yescharacteristic
105C4⋊C8×C73—22no—
106C22⋊C8×C73C2314yes—
107#97×C73—22no—
108C23×D73C22×C411yescharacteristic
109C23×D73C22×C411yescharacteristic
110C23×D73C22×C411yescharacteristic
111#11924029—232no—
112C2×D292—22no—
113C2×D292—22no—
114C2×D292—22no—
115C2×D292—22no—
116C2×D292—22no—
117C2×D292—22no—
118C2×D292—22no—
119C2×D292C22⋊C414yes—
120C2×D292M1614yescomplemented
121C22×Dic73C22×C411yescharacteristic
122C22×Dic73C22×C411yescharacteristic
123C22×Dic73C22×C411yescharacteristic
124C22×Dic73C22×C411yescharacteristic
125C22×Dic73C22×C411yescharacteristic
126C2×C4×D73—22no—
127C2×C4×D73—22no—
128C2×C4×D73—22no—
129C2×C4×D73—22no—
130C2×C4×D73—22no—
131C2×C4×D73—22no—
132C2×C4×D73—22no—
133C2×C4×D73—22no—
134C2×C4×D73—22no—
135C2×C4×D73—22no—
136C2×C4×D73—22no—
137C2×C4×D73—22no—
138C2×C4×D73—81no—
139C2×C4×D73—81no—
140C2×C4×D73—81no—
141C2×C4×D73—81no—
142C2×C4×D73—81no—
143C2×C4×D73—81no—
144C2×C4×D73M1614yescomplemented
145C2×C4×D73C2×D414yes—
146C2×C4×D73C2×D414yes—
147#11924011—84no—
148C2×#10596831—24no—
149C8×D73—84no—
150C4⋊Dic73—22no—
151C4⋊Dic73—22no—
152C4⋊Dic73C22⋊C414yes—
153#11924007—21no—
154#11924007—21no—
155#11924007—22no—
156#11924007—22no—
157#11924007—22no—
158#11924005—22no—
159#11924005—22no—
160#11924005—21no—
161#11924005—21no—
162#11924005—22no—
163#11924005C22⋊C412yes—
164#11924005C22⋊C412yes—
165#11924005C22⋊C412yes—
166#11924005C22⋊C412yes—
167C4×Dic73—22no—
168C4×Dic73—22no—
169C4×Dic73—22no—
170C2×Q8×C73—22no—
171C2×Q8×C73—22no—
172C2×Q8×C73—22no—
173C22×C4×C73C22×C411yescharacteristic
174C22×C4×C73C22×C411yescharacteristic
175C22×C4×C73C22×C411yescharacteristic
176C22×C4×C73C22×C411yescharacteristic
177C22×C4×C73C22×C411yescharacteristic
178C22×C4×C73C22×C411yescharacteristic
179C22×C4×C73C2411yescharacteristic
180C2×C8×C73—24no—
181C4⋊C4×C73—22no—
182C4⋊C4×C73—22no—
183C4⋊C4×C73—22no—
184C42×C73—21no—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—21no—
188C42×C73C2×D412yes—
189C42×C73C2×D412yes—
190C23×C73C23×C411yescharacteristic
191C22×D73—21no—
192C22×D73—21no—
193C22×D73—21no—
194C22×D73—21no—
195C22×D73—41no—
196C22×D73—41no—
197C22×D73—41no—
198C22×D73—41no—
199C22×D73—41no—
200C22×D73—41no—
201C22×D73C2×C22⋊C412yes—
202C22×D73C2×C22⋊C412yes—
203C22×D73C2×M1612yes—
204C22×D73C2×M1612yes—
205C22×D73C2×M1612yes—
206Q8×C73—28no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73C2×C22⋊C411yescharacteristic
220C2×C4×C73C2×C22⋊C411yescharacteristic
221C2×C4×C73C2×C22⋊C412yes—
222C2×C4×C73C2×M1612yes—
223C2×C4×C73C2×M1612yes—
224C2×C4×C73C22×D412yes—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—41no—
230C2×Dic73—41no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73C2×C22⋊C412yes—
234C2×Dic73C2×C22⋊C412yes—
235C2×Dic73C2×M1612yes—
236D292—28no—
237D292—28no—
238D292—28no—
239C4×D73—28no—
240C4×D73—28no—
241C4×D73—28no—
242C4×D73—44no—
243C4×D73—44no—
244C4×D73—82no—
245C4×D73—82no—
246C4×D73—82no—
247C4×D73—82no—
248C8×C73—44no—
249#10596831—44no—
250C22×C73—21no—
251C22×C73—21no—
252C22×C73C22×C22⋊C411yescharacteristic
253C22×C73C22×M1611yescharacteristic
254C22×C73C22×M1611yescharacteristic
255C2×D73—22no—
256C2×D73—22no—
257C2×D73—22no—
258C2×D73—22no—
259C2×D73—22no—
260C2×D73—22no—
261C2×D73—42no—
262C2×D73—42no—
263C2×D73—42no—
264C2×D73—42no—
265C2×D73—42no—
266C2×D73—41no—
267C2×D73—41no—
268C2×D73—42no—
269C2×D73#40714yes—
270C4×C73—22no—
271C4×C73—22no—
272C4×C73—22no—
273C4×C73—22no—
274C4×C73—21no—
275C4×C73—21no—
276C4×C73—41no—
277C4×C73—41no—
278Dic73—22no—
279Dic73—22no—
280Dic73—22no—
281Dic73—41no—
282Dic73—41no—
283#17368—731nomaximal, Sylow
284C2×C73—21no—
285C2×C73—21no—
286C2×C73C2×#40711yescharacteristic
287C2×C73#288311yescharacteristic
288C2×C73#288911yescharacteristic
289D73—24no—
290D73—42no—
291D73—42no—
292C2×#520—731no—
293#2973—732no—
294#1467—731no—
295#1467D7311yescharacteristic, complemented
296#1465—732no—
297#1464—732no—
298#1463—732no—
299#1459—732no—
300C2×#324—732no—
301C73#1736811yescharacteristic, complemented, Sylow
302C22×#41—731no—
303#520—1464no—
304#520—1464no—
305C2×C4×Q8—731no—
306C2×C4×Q8C2×D7311yescharacteristic
307C2×C4×D4—731no—
308C2×C4×D4—731no—
309C2×C4×D4—731no—
310C2×C4×D4—731no—
311C2×C4×D4—731no—
312C2×C4×D4—731no—
313C2×#117—731no—
314C2×#117—731no—
315C2×#117—731no—
316C22×C4⋊C4—731no—
317C22×C4⋊C4—731no—
318C22×C4⋊C4—731no—
319#435—1464no—
320#435—1464no—
321#432—732no—
322#432C2×D7312yes—
323#407—734no—
324C2×C22⋊C8—734no—
325#324—732no—
326#324—732no—
327#324—734no—
328#324C2×D7312yes—
329#324C2×D7312yes—
330#323—734no—
331C2×#41—1464no—
332C2×#41—1464no—
333C2×#41—1464no—
334C22×Q8C4×D7311yescharacteristic
335C22×D4—731no—
336C22×D4—731no—
337C22×D4—731no—
338C23×C4—731no—
339C23×C4—731no—
340C23×C4—731no—
341C23×C4—731no—
342C23×C4—731no—
343C23×C4—731no—
344C23×C4—731no—
345C23×C4—731no—
346C23×C4—731no—
347C2×M16—1464no—
348C22×C8—1464no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—1464no—
352C4×D4—734no—
353C4×D4—734no—
354C4×D4—1462no—
355C4×D4—1462no—
356C4×D4—1462no—
357C4×D4—1462no—
358C4×D4—1464no—
359C4×D4—1464no—
360C4×D4—1464no—
361#117—1462no—
362#117—1462no—
363#117—1462no—
364#117—1462no—
365#117—1462no—
366#117—1462no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—1464no—
375C2×C4⋊C4—1464no—
376C2×C4⋊C4—1464no—
377C2×C4⋊C4C4×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C22⋊C4—731no—
381C2×C22⋊C4—731no—
382C2×C22⋊C4—731no—
383C2×C22⋊C4—731no—
384C2×C22⋊C4—731no—
385C2×C22⋊C4—731no—
386C2×C42—731no—
387C2×C42—731no—
388C2×C42—731no—
389C2×C42C4×D7311yescharacteristic
390C2×C42C4×D7311yescharacteristic
391C2×C42C22×D7311yescharacteristic
392C4⋊C8—22no—
393C4⋊C8—1462no—
394C22⋊C8—734no—
395C22⋊C8—1464no—
396C22⋊C8—1464no—
397C22⋊C8C22×D7314yes—
398#97—22no—
399#97—1462no—
400C24—731no—
401C24—731no—
402C24—731no—
403#41—14632no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×D4—734no—
408C2×D4—734no—
409C2×D4—1462no—
410C2×D4—1462no—
411C2×D4—1462no—
412C2×D4—1462no—
413C2×D4—1462no—
414C2×D4—1462no—
415C2×D4—1462no—
416C22×C4—5841no—
417C22×C4—5841no—
418C22×C4—5841no—
419C22×C4—5841no—
420C22×C4—5841no—
421C22×C4—5841no—
422C22×C4—731no—
423C22×C4—731no—
424C22×C4—734no—
425C22×C4—734no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—734no—
430C22×C4—1462no—
431C22×C4—1462no—
432C22×C4—1462no—
433C22×C4—1462no—
434C22×C4—1462no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4C2×C4×D7311yescharacteristic
443C22×C4C2×C4×D7311yescharacteristic
444C22×C4C2×C4×D7311yescharacteristic
445C22×C4C2×C4×D7311yescharacteristic
446C22×C4C2×C4×D7311yescharacteristic
447C22×C4C2×C4×D7311yescharacteristic
448C22×C4C23×D7311yescharacteristic
449M16—5844no—
450C2×C8—24no—
451C2×C8—5844no—
452C2×C8—1464no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—734no—
457C4⋊C4—1462no—
458C4⋊C4—1462no—
459C4⋊C4—1462no—
460C4⋊C4—1462no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1462no—
464C22⋊C4—732no—
465C22⋊C4—732no—
466C22⋊C4—732no—
467C22⋊C4—732no—
468C22⋊C4—1462no—
469C22⋊C4—1462no—
470C22⋊C4—1461no—
471C22⋊C4—1461no—
472C22⋊C4—1462no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—1462no—
478C42—1462no—
479C42—1462no—
480C42D4×D7312yes—
481C42D4×D7312yes—
482C23—2921no—
483C23—2921no—
484C23—2921no—
485C23—2921no—
486C23—2921no—
487C23—2921no—
488C23—732no—
489C23—732no—
490C23—732no—
491C23—732no—
492C23—732no—
493C23—1461no—
494C23—1461no—
495C23—1461no—
496C23—1461no—
497C23C22×C4×D7311yescharacteristic
498Q8—28no—
499D4—1468no—
500D4—1468no—
501D4—1468no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—2924no—
515C2×C4—2924no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—5842no—
521C2×C4—5842no—
522C2×C4—5842no—
523C2×C4—5842no—
524C2×C4—732no—
525C2×C4—732no—
526C2×C4—732no—
527C2×C4—1468no—
528C2×C4—1468no—
529C2×C4—1468no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4C22⋊C4×D7311yescharacteristic
535C2×C4C22⋊C4×D7311yescharacteristic
536C2×C4C22⋊C4×D7312yes—
537C2×C4M16×D7312yes—
538C2×C4M16×D7312yes—
539C2×C4C2×D4×D7312yes—
540C8—44no—
541C8—2924no—
542C22—21no—
543C22—21no—
544C22—2922no—
545C22—2922no—
546C22—2922no—
547C22—2922no—
548C22—2922no—
549C22—2922no—
550C22—2921no—
551C22—2921no—
552C22—734no—
553C22—1462no—
554C22—1462no—
555C22—1462no—
556C22—1462no—
557C22—1462no—
558C22—1462no—
559C22C2×C22⋊C4×D7311yescharacteristic
560C22C2×M16×D7311yescharacteristic
561C22C2×M16×D7311yescharacteristic
562C4—22no—
563C4—21no—
564C4—21no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—2921no—
571C4—2921no—
572C4—1462no—
573C4—1462no—
574C4—1462no—
575C2—21no—
576C2—21no—
577C2—2922no—
578C2—2922no—
579C2—1464no—
580C2#407×D7311yescharacteristic
581C2#2261839011yescharacteristic
582C2#2261841111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D73
commutatorC23×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1467C2
socleC22×C73C22×C22⋊C4
prime radicalC22×#11924029C4
cosocleC22×C4×C73C24
hypercenter#1467D73
hypocenterC73#17368
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #22618959C2semidirect
3. #22608275C2semidirect
4. #22608268C2semidirect
5. #22608263C2semidirect
6. #22608259C2semidirect
7. #22608248C2semidirect
8. #324×D73C2semidirect
9. C73×#1467C2semidirect
120. C2×D292M16semidirect
144. C2×C4×D73M16semidirect
295. #1467D73semidirect
301. C73#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#179890692
#226189592
#226082752
#226082682
#226082632
#226082592
#226082482
#324×D732
C73×#14672
#1736873