Finite Group Databasebeta

← #33114660

Subgroups — #33114660

Order 18688.

1102 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988615C211yesmaximal, characteristic, complemented
2C2×#17988589C211yesmaximal, characteristic
3C2×#17988581C211yesmaximal, characteristic
4#22612554C211yesmaximal, characteristic, complemented
5#22612524C211yesmaximal, characteristic
6#22611321C212yesmaximal, complemented
7#22611314C212yesmaximal, complemented
8#22611307C212yesmaximal, complemented
9#22611300C212yesmaximal, complemented
10C2×#17988349C211yesmaximal, characteristic, complemented
11C2×#428×C73C211yesmaximal, characteristic, complemented
12C22×#11924028C2211yescharacteristic
13C23×#10596831C2211yescharacteristic
14C2×#14556550C2211yescharacteristic
15C22×C4⋊Dic73C2211yescharacteristic
16#17988615—24no—
17C2×#14556520C2212yes—
18C2×#14556520C2212yes—
19#17988589—24no—
20#17988581—24no—
21C2×#14556516C2212yes—
22C2×#14556514C2211yescharacteristic
23C2×#14556514C2211yescharacteristic
24C2×#14556513C2212yes—
25C2×#14556511C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27#17988492C2212yes—
28#17988488C2212yes—
29#17988486C2212yes—
30#17988486C2212yes—
31#17988466—22no—
32#17988462—22no—
33#17988457—21no—
34#17988453—21no—
35#17988444—21no—
36#17988444—21no—
37#17988349C2212yes—
38#17988349C2212yes—
39C22×M16×C73C2211yescharacteristic
40C2×#117×C73C2211yescharacteristic
41C22×C4⋊C4×C73C2211yescharacteristic
42#428×C73—24no—
43C2×#107×C73C2212yes—
44C2×#106×C73C2212yes—
45C2×#11924028—22no—
46C2×#11924028—22no—
47C2×#11924028—22no—
48C2×#11924028—22no—
49C2×#11924028—22no—
50C2×#11924028Q814yes—
51C22×Dic146C2311yescharacteristic
52C22×D292C2311yescharacteristic
53C22×C22⋊D73—21no—
54C23×Dic73—21no—
55C22×C4×D73—21no—
56C22×#10596831—22no—
57C22×#10596831—22no—
58C22×#10596831—22no—
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61#14556550—22no—
62#14556550—22no—
63C2×C4⋊Dic73—22no—
64C2×C4⋊Dic73—22no—
65C2×C4⋊Dic73—22no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—22no—
68C2×C4⋊Dic73D411yescharacteristic
69C2×C4⋊Dic73D411yescharacteristic
70C2×C4⋊Dic73C2312yes—
71C2×C22⋊Dic73D412yes—
72C2×#11924005—22no—
73C2×#11924005—22no—
74C2×C4×Dic73C2312yes—
75#14556520—24no—
76#14556520—24no—
77#14556516—24no—
78#14556514—22no—
79#14556514—22no—
80#14556513—24no—
81#14556511—22no—
82#14556511—22no—
83#14556498—22no—
84#14556498—22no—
85#14556498—22no—
86#14556498—22no—
87C23×C4×C73—21no—
88C23×C4×C73C2311yescharacteristic
89C2×M16×C73—22no—
90C2×M16×C73—22no—
91C2×M16×C73D414yes—
92C22×C8×C73C2312yes—
93#117×C73—22no—
94#117×C73D414yes—
95C2×C4⋊C4×C73—22no—
96C2×C4⋊C4×C73—22no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73C2312yes—
99C2×C4⋊C4×C73C2311yescharacteristic
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C22⋊C4×C73—21no—
102C2×C42×C73—21no—
103#107×C73—24no—
104#106×C73—24no—
105C23×D73—21no—
106#11924028—28no—
107#11924028—28no—
108#11924028—44no—
109#11924028—44no—
110C2×Dic146—21no—
111C2×Dic146—21no—
112C2×Dic146—21no—
113C2×Dic146—21no—
114C2×Dic146—21no—
115C2×Dic146C2×Q812yes—
116C2×D292—21no—
117C2×D292—21no—
118C2×D292—21no—
119C2×D292—21no—
120C2×D292—21no—
121C2×D292C2×Q812yes—
122C2×C22⋊D73—24no—
123C2×C22⋊D73—42no—
124C2×C22⋊D73—42no—
125C22×Dic73—21no—
126C22×Dic73—22no—
127C22×Dic73—42no—
128C22×Dic73—81no—
129C22×Dic73—81no—
130C22×Dic73C2×D412yes—
131C22×Dic73C2×D412yes—
132C2×C4×D73—24no—
133C2×C4×D73—42no—
134C2×C4×D73—42no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—44no—
142C4⋊Dic73—22no—
143C4⋊Dic73—22no—
144C4⋊Dic73—22no—
145C4⋊Dic73—22no—
146C4⋊Dic73—42no—
147C4⋊Dic73—82no—
148C4⋊Dic73—82no—
149C22⋊Dic73—24no—
150#11924005—82no—
151#11924005—82no—
152C4×Dic73—22no—
153C4×Dic73—22no—
154C24×C73C2×D411yescharacteristic
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73—22no—
158C22×C4×C73—42no—
159C22×C4×C73—81no—
160C22×C4×C73—81no—
161C22×C4×C73C2×D411yescharacteristic
162C22×C4×C73C2×D412yes—
163C22×C4×C73C2×D412yes—
164C22×C4×C73C2×Q812yes—
165C22×C4×C73#4112yes—
166C22×C4×C73#4112yes—
167C22×C4×C73#4112yes—
168C22×C4×C73C2411yescharacteristic
169M16×C73—28no—
170C2×C8×C73—22no—
171C2×C8×C73—22no—
172C2×C8×C73C2×D414yes—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—22no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73—42no—
178C4⋊C4×C73C2×D412yes—
179C4⋊C4×C73C2×D412yes—
180C22⋊C4×C73—24no—
181C42×C73—24no—
182C23×C73—21no—
183C23×C73—21no—
184C23×C73—21no—
185C23×C73—21no—
186C23×C73#12112yes—
187C23×C73#12112yes—
188C23×C73C22⋊Q812yes—
189C23×C73C22×D411yescharacteristic
190C22×D73—22no—
191C22×D73—41no—
192C22×D73—41no—
193C22×D73—41no—
194C22×D73—41no—
195C22×D73—41no—
196C22×D73—41no—
197C2×C4×C73—22no—
198C2×C4×C73—22no—
199C2×C4×C73—41no—
200C2×C4×C73—41no—
201C2×C4×C73—41no—
202C2×C4×C73—41no—
203C2×C4×C73—42no—
204C2×C4×C73—42no—
205C2×C4×C73—42no—
206C2×C4×C73—84no—
207C2×C4×C73#12112yes—
208C2×C4×C73#12114yes—
209C2×C4×C73#12112yes—
210C2×C4×C73C22⋊Q814yes—
211C2×C4×C73C22⋊Q812yes—
212C2×C4×C73C22⋊Q814yes—
213C2×C4×C73#12614yes—
214C2×C4×C73C22×D411yescharacteristic
215C2×C4×C73C22×D411yescharacteristic
216C2×C4×C73C22×Q811yescharacteristic
217C2×C4×C73C2×#4111yescharacteristic
218C2×C4×C73C2×#4111yescharacteristic
219C2×C4×C73C2×#4111yescharacteristic
220C22⋊D73—48no—
221C2×Dic73—22no—
222C2×Dic73—22no—
223C2×Dic73—22no—
224C2×Dic73—22no—
225C2×Dic73—22no—
226C2×Dic73—22no—
227C2×Dic73—22no—
228C2×Dic73—41no—
229C2×Dic73—41no—
230C2×Dic73—42no—
231C2×Dic73—42no—
232C2×Dic73—42no—
233C2×Dic73—84no—
234D292—22no—
235D292—22no—
236D292—41no—
237D292—41no—
238C4×D73—48no—
239Dic146—22no—
240Dic146—22no—
241Dic146—41no—
242Dic146—41no—
243C8×C73—24no—
244#10596831—41no—
245#10596831—41no—
246C22×C73—22no—
247C22×C73—22no—
248C22×C73—22no—
249C22×C73—22no—
250C22×C73—22no—
251C22×C73—22no—
252C22×C73#39614yes—
253C22×C73C2×#12111yescharacteristic
254C22×C73C2×#12111yescharacteristic
255C22×C73C2×C22⋊Q811yescharacteristic
256C22×C73#57611yescharacteristic
257C22×C73#57711yescharacteristic
258C22×C73#57711yescharacteristic
259C22×C73#57811yescharacteristic
260C2×D73—42no—
261C2×D73—42no—
262C2×D73—42no—
263C2×D73—81no—
264C2×D73—81no—
265C2×D73—81no—
266C2×D73—81no—
267C4×C73—42no—
268C4×C73—42no—
269C4×C73—82no—
270C4×C73#39614yes—
271C4×C73C2×#12111yescharacteristic
272C4×C73C2×C22⋊Q811yescharacteristic
273C4×C73C2×C22⋊Q811yescharacteristic
274C4×C73C2×#12611yescharacteristic
275Dic73—42no—
276Dic73—42no—
277Dic73—42no—
278Dic73—82no—
279#23616—731nomaximal, Sylow
280C2×C73—24no—
281C2×C73C2×#39611yescharacteristic
282C2×C73#304511yescharacteristic
283C2×C73#304611yescharacteristic
284C2×C73#306211yescharacteristic
285C2×C73#306311yescharacteristic
286C2×C73#308311yescharacteristic
287C2×C73#308411yescharacteristic
288D73—82no—
289C2×#428D7311yescharacteristic, complemented
290C2×#424—731no—
291C2×#417—731no—
292C2×#416—731no—
293#2317—731no—
294#2311—731no—
295#2068—732no—
296#2067—732no—
297#2066—732no—
298#2065—732no—
299C2×#340—731no—
300C73#2361611yescharacteristic, complemented, Sylow
301C22×#41—731no—
302C22×M16C2×D7311yescharacteristic
303C23×C8—731no—
304C2×#117—731no—
305C2×#117C2×D7311yescharacteristic
306C22×C4⋊C4—731no—
307C22×C4⋊C4C2×D7311yescharacteristic
308#428—24no—
309C2×#107C2×D7312yes—
310C2×#106C2×D7312yes—
311#424—1464no—
312C2×C4⋊C8—732no—
313C2×C4⋊C8—732no—
314#417—1464no—
315#416—1464no—
316C2×#103—732no—
317C2×#103—731no—
318C2×#103—731no—
319C2×#102—732no—
320C2×#102—731no—
321C2×#102—731no—
322#391—732no—
323#390—732no—
324#389—732no—
325#389—732no—
326#386—1462no—
327#385—1462no—
328#384—1461no—
329#382—1461no—
330#380—1461no—
331#380—1461no—
332#340—732no—
333#340—732no—
334C2×#41—734no—
335C2×#41—1462no—
336C2×#41—1462no—
337C2×#41—1462no—
338C2×#41—1462no—
339C2×#41—1462no—
340C22×Q8—731no—
341C22×D4—731no—
342C22×D4—1461no—
343C23×C4—21no—
344C23×C4—1461no—
345C23×C4—1461no—
346C23×C4C22×D7311yescharacteristic
347C2×M16—22no—
348C2×M16—22no—
349C2×M16D29214yes—
350C22×C8—731no—
351C22×C8—731no—
352C22×C8—1462no—
353C22×C8—1462no—
354C22×C8—1462no—
355C22×C8C22×D7312yes—
356#117—22no—
357#117—1462no—
358#117—1462no—
359#117C22⋊D7314yes—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—731no—
364C2×C4⋊C4—731no—
365C2×C4⋊C4—732no—
366C2×C4⋊C4—1461no—
367C2×C4⋊C4—1462no—
368C2×C4⋊C4—1462no—
369C2×C4⋊C4—1462no—
370C2×C4⋊C4—1462no—
371C2×C4⋊C4C22×D7312yes—
372C2×C4⋊C4C22×D7311yescharacteristic
373C2×C4⋊C4C22×D7311yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—732no—
376C2×C22⋊C4—1462no—
377C2×C22⋊C4—1462no—
378C2×C42—21no—
379C2×C42—732no—
380#107—24no—
381#106—24no—
382C4⋊C8—1464no—
383C4⋊C8—1464no—
384#103—1464no—
385#103—1462no—
386#103—1462no—
387#102—1464no—
388#102—1462no—
389#102—1462no—
390#95—1462no—
391#95—1462no—
392#95—1462no—
393#95—1462no—
394C24—1461no—
395C24D4×D7311yescharacteristic
396#41—2924no—
397#41—2924no—
398#41—1468no—
399#41—1468no—
400C2×Q8—732no—
401C2×Q8—1461no—
402C2×Q8—1461no—
403C2×Q8—1461no—
404C2×Q8—1461no—
405C2×Q8—1461no—
406C2×D4—2922no—
407C2×D4—2922no—
408C2×D4—732no—
409C2×D4—1464no—
410C2×D4—1461no—
411C2×D4—1461no—
412C2×D4—1461no—
413C2×D4—1461no—
414C2×D4—1461no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—2922no—
422C22×C4—2922no—
423C22×C4—2922no—
424C22×C4—5841no—
425C22×C4—5841no—
426C22×C4—732no—
427C22×C4—732no—
428C22×C4—1462no—
429C22×C4—1461no—
430C22×C4—1464no—
431C22×C4D4×D7311yescharacteristic
432C22×C4C2×C22⋊D7312yes—
433C22×C4C2×D29212yes—
434C22×C4Q8×D7312yes—
435C22×C4#1192402712yes—
436C22×C4#1192402812yes—
437C22×C4#1192402912yes—
438C22×C4C23×D7311yescharacteristic
439M16—28no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—2924no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8C2×D29214yes—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—2922no—
456C4⋊C4—5842no—
457C4⋊C4—5842no—
458C4⋊C4—1462no—
459C4⋊C4—1462no—
460C4⋊C4—1462no—
461C4⋊C4—1462no—
462C4⋊C4C2×C22⋊D7312yes—
463C4⋊C4C2×C22⋊D7312yes—
464C22⋊C4—24no—
465C22⋊C4—5842no—
466C22⋊C4—5842no—
467C22⋊C4—1464no—
468C42—24no—
469C42—1462no—
470C42—1462no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—2921no—
476C23—2921no—
477C23—2921no—
478C23—2921no—
479C23—2921no—
480C23—2921no—
481C23—1462no—
482C23#1455656312yes—
483C23#1455656612yes—
484C23#1455656912yes—
485C23C2×D4×D7311yescharacteristic
486Q8—2921no—
487Q8—2921no—
488Q8—1462no—
489Q8—1462no—
490D4—2928no—
491D4—2921no—
492D4—2921no—
493D4—1462no—
494D4—1462no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—84no—
505C2×C4—2921no—
506C2×C4—2921no—
507C2×C4—2928no—
508C2×C4—2922no—
509C2×C4—2922no—
510C2×C4—2922no—
511C2×C4—5844no—
512C2×C4—1462no—
513C2×C4—1462no—
514C2×C4—1462no—
515C2×C4—1462no—
516C2×C4—1462no—
517C2×C4—1462no—
518C2×C4—1462no—
519C2×C4#1455656312yes—
520C2×C4#1455656514yes—
521C2×C4#1455656612yes—
522C2×C4#1455656714yes—
523C2×C4#1455656912yes—
524C2×C4#1455657114yes—
525C2×C4#1455658614yes—
526C2×C4C22×C22⋊D7311yescharacteristic
527C2×C4C22×D29211yescharacteristic
528C2×C4C2×Q8×D7311yescharacteristic
529C2×C4C2×#1192402711yescharacteristic
530C2×C4C2×#1192402811yescharacteristic
531C2×C4C2×#1192402911yescharacteristic
532C8—24no—
533C8—2921no—
534C8—2921no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—2922no—
542C22—2922no—
543C22—2922no—
544C22—5841no—
545C22—5841no—
546C22—5841no—
547C22—5841no—
548C22#1798850914yes—
549C22C2×#1455656311yescharacteristic
550C22C2×#1455656611yescharacteristic
551C22C2×#1455656911yescharacteristic
552C22#1798939111yescharacteristic
553C22#1798939611yescharacteristic
554C22#1798939711yescharacteristic
555C22#1798940311yescharacteristic
556C4—42no—
557C4—42no—
558C4—82no—
559C4—2922no—
560C4—2922no—
561C4—2922no—
562C4—5842no—
563C4#1798850914yes—
564C4C2×#1455656511yescharacteristic
565C4C2×#1455656711yescharacteristic
566C4C2×#1455657111yescharacteristic
567C4C2×#1455658611yescharacteristic
568C2—24no—
569C2—5842no—
570C2C2×#1798850911yescharacteristic
571C2#2261947411yescharacteristic
572C2#2261948511yescharacteristic
573C2#2261961211yescharacteristic
574C2#2261962311yescharacteristic
575C2#2261978111yescharacteristic
576C2#2261979211yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#428×C73C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
hypercenterC2×#428D73
hypocenterC73#23616
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#17988615C2semidirect
4. #22612554C2semidirect
6. #22611321C2semidirect
7. #22611314C2semidirect
8. #22611307C2semidirect
9. #22611300C2semidirect
10. C2×#17988349C2semidirect
11. C2×#428×C73C2semidirect
289. C2×#428D73semidirect
300. C73#23616semidirect

Maximal subgroups

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

subgroupindex
C2×#179886152
C2×#179885892
C2×#179885812
#226125542
#226125242
#226113212
#226113142
#226113072
#226113002
C2×#179883492
C2×#428×C732
#2361673