Finite Group Databasebeta

← #33240421

Subgroups — #33240421

Order 18688.

710 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989018C211yesmaximal, characteristic, complemented
2C22×C16×D73C211yesmaximal, characteristic, complemented
3#22619514C211yesmaximal, characteristic, complemented
4#22619510C211yesmaximal, characteristic, complemented
5#22612028C211yesmaximal, characteristic, complemented
6#22612027C211yesmaximal, characteristic, complemented
7#22612024C211yesmaximal, characteristic, complemented
8#22612021C211yesmaximal, characteristic, complemented
9#22612020C211yesmaximal, characteristic, complemented
10#22612016C211yesmaximal, characteristic, complemented
11#22612015C211yesmaximal, characteristic, complemented
12#22611714C211yesmaximal, characteristic, complemented
13#22611637C211yesmaximal, characteristic, complemented
14#22611635C211yesmaximal, characteristic, complemented
15C73×#2217C211yesmaximal, characteristic, complemented
16C2×#14556614C2211yescharacteristic
17C2×#14556610C2211yescharacteristic
18C22×C8×D73C2211yescharacteristic
19#17989100—21no—
20#17989097—21no—
21#17989018—24no—
22C2×#14556537C2211yescharacteristic, complemented
23C2×#14556536C2211yescharacteristic, complemented
24C2×#14556532C2211yescharacteristic, complemented
25C22×#11924003C2211yescharacteristic
26C2×C16×D73—24no—
27C2×C16×D73C2211yescharacteristic
28C2×C16×D73C2211yescharacteristic
29C2×C16×D73C2211yescharacteristic
30C2×C16×D73C2211yescharacteristic
31#17988829C2211yescharacteristic
32#17988828C2211yescharacteristic
33#17988825C2211yescharacteristic
34#17988824C2211yescharacteristic
35#17988823C2211yescharacteristic
36#17988820C2211yescharacteristic
37#17988819—21no—
38#17988818—21no—
39#17988815—21no—
40#17988814—21no—
41#17988624C2211yescharacteristic
42#17988624C2211yescharacteristic
43#17988582—21no—
44#17988580—21no—
45#17988415C2211yescharacteristic, complemented
46#17988412C2211yescharacteristic, complemented
47#17988412C2211yescharacteristic, complemented
48#17988392C2211yescharacteristic, complemented
49#17988391C2211yescharacteristic, complemented
50#17988390C2211yescharacteristic, complemented
51#17988389C2211yescharacteristic, complemented
52#17988388C2211yescharacteristic, complemented
53#17988387C2211yescharacteristic, complemented
54C2×SD32×C73C2211yescharacteristic, complemented
55C22×C16×C73C2211yescharacteristic
56#467×C73C2211yescharacteristic
57#466×C73C2211yescharacteristic
58#367×C73C2211yescharacteristic, complemented
59#358×C73C2211yescharacteristic, complemented
60#357×C73C2211yescharacteristic, complemented
61C2×#11924029—21no—
62C2×#11924027—21no—
63C22×C4×D73D411yescharacteristic
64#14556614—22no—
65#14556614—22no—
66#14556610—22no—
67#14556610—22no—
68C2×Dic292C2311yescharacteristic
69C2×#11924016—21no—
70C2×#11924015—21no—
71C2×D584C2311yescharacteristic
72C22×#10596831D411yescharacteristic
73C2×C8×D73—22no—
74C2×C8×D73—22no—
75C2×C8×D73D411yescharacteristic
76C2×C8×D73D411yescharacteristic
77C2×C8×D73C2311yescharacteristic
78C2×C8×D73C2311yescharacteristic
79#14556576—41no—
80#14556573—41no—
81#14556571—21no—
82#14556570—21no—
83#14556567—21no—
84#14556566—21no—
85#14556565—21no—
86#14556562—21no—
87#14556561—41no—
88#14556559—41no—
89D4×Dic73—41no—
90#14556553—41no—
91#14556551—41no—
92#14556551—41no—
93#14556549—21no—
94#14556549—21no—
95#14556537—22no—
96#14556536—22no—
97#14556532—22no—
98C2×#11924003—22no—
99C2×#11924003C2311yescharacteristic
100C2×#11924003C2311yescharacteristic
101C16×D73—22no—
102C16×D73—22no—
103C16×D73—22no—
104C16×D73D414yescomplemented
105#14556525C2311yescharacteristic
106#14556525C2311yescharacteristic
107#14556522—21no—
108#14556522—21no—
109#14556516—21no—
110#14556515—21no—
111#14556514—21no—
112#14556513—21no—
113#14556512—21no—
114#14556511—21no—
115C2×Q16×C73C2311yescharacteristic
116C2×D8×C73C2311yescharacteristic
117C22×C8×C73C2311yescharacteristic
118C22⋊Q8×C73—21no—
119#121×C73—21no—
120SD32×C73—22no—
121C2×C16×C73—22no—
122C2×C16×C73C2311yescharacteristic
123C2×C16×C73C2311yescharacteristic
124#107×C73C2311yescharacteristic
125#107×C73C2311yescharacteristic
126#103×C73—21no—
127#102×C73—21no—
128C23×D73D811yescharacteristic
129#11924029—42no—
130#11924029—42no—
131#11924027—42no—
132#11924027—42no—
133C2×Dic146—21no—
134C2×D292—21no—
135C2×D292—41no—
136C2×D292—41no—
137C2×C22⋊D73—41no—
138C2×C22⋊D73—41no—
139C2×C22⋊D73—41no—
140C22×Dic73—41no—
141C22×Dic73D811yescharacteristic
142C2×C4×D73—22no—
143C2×C4×D73—22no—
144C2×C4×D73—41no—
145C2×C4×D73D811yescharacteristic
146C2×C4×D73D811yescharacteristic
147C2×C4×D73C2×D411yescharacteristic
148C2×C4×D73C2×D411yescharacteristic
149Dic292—21no—
150Dic292—21no—
151#11924016—42no—
152#11924015—42no—
153D584—21no—
154D584—21no—
155C2×#10596831—22no—
156C2×#10596831C2×D411yescharacteristic
157C2×#10596831C2×D411yescharacteristic
158C8×D73—22no—
159C8×D73—21no—
160C8×D73—21no—
161C8×D73—22no—
162C8×D73C2×D412yes—
163C8×D73#4112yes—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—41no—
167#11924007—41no—
168#11924007—41no—
169C22⋊Dic73—41no—
170C22⋊Dic73—41no—
171#11924005—41no—
172#11924005—41no—
173#11924005—41no—
174#11924005—41no—
175#11924005—41no—
176#11924005—41no—
177C4×Dic73—41no—
178C4×Dic73—41no—
179#11924003—21no—
180#11924003C2×D412yes—
181C2×Q8×C73—21no—
182C2×D4×C73—21no—
183C2×D4×C73—41no—
184C22×C4×C73C2×D411yescharacteristic
185Q16×C73—21no—
186Q16×C73—21no—
187D8×C73—21no—
188D8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73C2×D411yescharacteristic
192C2×C8×C73C2411yescharacteristic
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—41no—
196C22⋊C4×C73—41no—
197C22⋊C4×C73—41no—
198C16×C73—21no—
199C16×C73C2×D412yes—
200C23×C73—41no—
201C23×C73C2×D811yescharacteristic
202C22×D73—22no—
203C22×D73—22no—
204C22×D73—22no—
205C22×D73—41no—
206C22×D73C2×D811yescharacteristic
207C22×D73C2×D811yescharacteristic
208Q8×C73—41no—
209Q8×C73—41no—
210D4×C73—41no—
211D4×C73—41no—
212D4×C73—82no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73C2×D811yescharacteristic
219C2×C4×C73C22×D411yescharacteristic
220C22⋊D73—82no—
221C22⋊D73—82no—
222C22⋊D73—82no—
223C2×Dic73—22no—
224C2×Dic73—41no—
225C2×Dic73—41no—
226C2×Dic73—41no—
227C2×Dic73—82no—
228C2×Dic73C2×D811yescharacteristic
229C2×Dic73C2×D811yescharacteristic
230D292—41no—
231D292—41no—
232D292—82no—
233D292—82no—
234C4×D73—22no—
235C4×D73—21no—
236C4×D73—21no—
237C4×D73—22no—
238C4×D73—82no—
239C4×D73#12112yes—
240C4×D73#13512yes—
241Dic146—41no—
242Dic146—41no—
243C8×C73—21no—
244C8×C73C22×D411yescharacteristic
245C8×C73C2×#4111yescharacteristic
246#10596831—21no—
247#10596831#12112yes—
248C22×C73—21no—
249C22×C73—21no—
250C22×C73—81no—
251C22×C73—81no—
252C22×C73—81no—
253C22×C73#50812yescomplemented
254C22×C73C22×D811yescharacteristic
255C2×D73—28no—
256C2×D73—21no—
257C2×D73—21no—
258C2×D73—21no—
259C2×D73—21no—
260C2×D73—81no—
261C2×D73—81no—
262C2×D73—81no—
263C2×D73#46612yes—
264C2×D73#50812yescomplemented
265C4×C73—21no—
266C4×C73—81no—
267C4×C73—81no—
268C4×C73—81no—
269C4×C73C2×#12111yescharacteristic
270C4×C73C2×#13511yescharacteristic
271Dic73—21no—
272Dic73—81no—
273Dic73—81no—
274Dic73—81no—
275Dic73#46612yes—
276#33600—731nomaximal, Sylow
277C2×C73—22no—
278C2×C73—161no—
279C2×C73C2×#46611yescharacteristic
280C2×C73C2×#50811yescharacteristic
281C2×C73C2×#50811yescharacteristic
282D73—22no—
283D73—22no—
284D73—161no—
285C2×#508—731no—
286C23×C16—731no—
287#3050—731no—
288#3050—731no—
289#2217—731no—
290#2217—731no—
291#2217—731no—
292#2217D7311yescharacteristic, complemented
293#2216—731no—
294#2216—731no—
295#2215—731no—
296#2215—731no—
297#2158—731no—
298#2138—731no—
299#2138—731no—
300C73#3360011yescharacteristic, complemented, Sylow
301C2×#135—731no—
302C2×#135—731no—
303C23×C8—731no—
304#525—1461no—
305#525—1461no—
306#508—1464no—
307C2×Q32—731no—
308C2×SD32—731no—
309C2×SD32C2×D7311yescharacteristic, complemented
310C2×D16—731no—
311C22×C16—731no—
312C22×C16—731no—
313C22×C16—731no—
314C22×C16—731no—
315C22×C16—731no—
316C22×C16—1464no—
317C22×C16C2×D7311yescharacteristic
318#467—731no—
319#467—731no—
320#467—731no—
321#467C2×D7311yescharacteristic
322#466—731no—
323#466—731no—
324#466—731no—
325#466C2×D7311yescharacteristic
326#465—1461no—
327#465—1461no—
328#465—1461no—
329#465—1461no—
330#427—731no—
331#427—731no—
332#416—1461no—
333#416—1461no—
334#367—731no—
335#367C2×D7311yescharacteristic, complemented
336#366—731no—
337#366—731no—
338#358—731no—
339#358—731no—
340#358—731no—
341#358C2×D7311yescharacteristic, complemented
342#357—731no—
343#357—731no—
344#357—731no—
345#357C2×D7311yescharacteristic, complemented
346C2×#41—1461no—
347C2×#41—1461no—
348C23×C4—731no—
349#135—1462no—
350#135—1462no—
351#135—1462no—
352#135—1462no—
353C2×Q16—731no—
354C2×Q16C22×D7311yescharacteristic
355C2×SD16—1461no—
356C2×SD16—1461no—
357C2×D8—731no—
358C2×D8C22×D7311yescharacteristic
359C22×C8—731no—
360C22×C8—731no—
361C22×C8—731no—
362C22×C8—731no—
363C22×C8—731no—
364C22×C8—1462no—
365C22×C8—1462no—
366C22×C8C22×D7311yescharacteristic
367#123—2921no—
368#123—2921no—
369C22⋊Q8—21no—
370C22⋊Q8—1461no—
371C22⋊Q8—1461no—
372C22⋊Q8—1461no—
373#121—21no—
374#121—1461no—
375#121—1461no—
376#121—1461no—
377#120—2921no—
378#120—2921no—
379C4×D4—2921no—
380C4×D4—2921no—
381C4×D4—2921no—
382C4×D4—2921no—
383#117—1461no—
384#117—1461no—
385Q32—1462no—
386SD32—22no—
387SD32—1462no—
388D16—1462no—
389C2×C16—22no—
390C2×C16—731no—
391C2×C16—734no—
392C2×C16—731no—
393C2×C16—1462no—
394C2×C16—1462no—
395C2×C16—1462no—
396C2×C16—1462no—
397C2×C16C22×D7311yescharacteristic
398C2×C16C22×D7311yescharacteristic
399#107—731no—
400#107—731no—
401#107C22×D7311yescharacteristic
402#107C22×D7311yescharacteristic
403#106—1461no—
404#106—1461no—
405#103—21no—
406#103—1461no—
407#103—1461no—
408#103—1461no—
409#102—21no—
410#102—1461no—
411#102—1461no—
412#102—1461no—
413C24—731no—
414#41—2922no—
415#41—2922no—
416#41—2922no—
417#41—2922no—
418C2×Q8—21no—
419C2×Q8—1461no—
420C2×D4—21no—
421C2×D4—41no—
422C2×D4—2921no—
423C2×D4—2921no—
424C2×D4—2921no—
425C2×D4—2921no—
426C2×D4—2921no—
427C2×D4—1461no—
428C22×C4—2921no—
429C22×C4—2921no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—731no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4D4×D7311yescharacteristic
438Q16—21no—
439Q16—21no—
440Q16—1461no—
441Q16—1461no—
442SD16—2922no—
443SD16—2922no—
444D8—21no—
445D8—21no—
446D8—1461no—
447D8—1461no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—731no—
451C2×C8—732no—
452C2×C8—732no—
453C2×C8—731no—
454C2×C8—1462no—
455C2×C8—1462no—
456C2×C8—1462no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8D4×D7311yescharacteristic
460C2×C8C23×D7311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—2921no—
465C4⋊C4—2921no—
466C4⋊C4—2921no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—2921no—
472C22⋊C4—2921no—
473C22⋊C4—2921no—
474C22⋊C4—2921no—
475C22⋊C4—2921no—
476C22⋊C4—2921no—
477C22⋊C4—2921no—
478C22⋊C4—2921no—
479C42—2921no—
480C42—2921no—
481C16—21no—
482C16—732no—
483C16—1461no—
484C16D4×D7312yes—
485C23—41no—
486C23—2921no—
487C23—731no—
488C23—731no—
489C23—1462no—
490C23—1462no—
491C23—1462no—
492C23D8×D7311yescharacteristic
493Q8—41no—
494Q8—41no—
495Q8—2921no—
496Q8—2921no—
497D4—41no—
498D4—41no—
499D4—82no—
500D4—2921no—
501D4—2921no—
502D4—5842no—
503D4—5842no—
504D4—5842no—
505D4—5842no—
506D4—5842no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—2921no—
513C2×C4—2921no—
514C2×C4—2921no—
515C2×C4—5842no—
516C2×C4—5842no—
517C2×C4—731no—
518C2×C4—732no—
519C2×C4—732no—
520C2×C4—731no—
521C2×C4—1462no—
522C2×C4—1462no—
523C2×C4—1462no—
524C2×C4—1461no—
525C2×C4—1461no—
526C2×C4D8×D7311yescharacteristic
527C2×C4C2×D4×D7311yescharacteristic
528C8—21no—
529C8—732no—
530C8—1461no—
531C8C2×D4×D7311yescharacteristic
532C8#41×D7311yescharacteristic
533C22—21no—
534C22—21no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—5841no—
539C22—5841no—
540C22—5841no—
541C22—732no—
542C22—732no—
543C22—1468no—
544C22—1461no—
545C22—1461no—
546C22—1461no—
547C22—1461no—
548C22#1798901812yescomplemented
549C22C2×D8×D7311yescharacteristic
550C4—21no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—5841no—
555C4—5841no—
556C4—5841no—
557C4—732no—
558C4—1461no—
559C4#121×D7311yescharacteristic
560C4#135×D7311yescharacteristic
561C2—22no—
562C2—161no—
563C2—11681no—
564C2—1462no—
565C2—1462no—
566C2#466×D7311yescharacteristic
567C2#508×D7311yescharacteristic
568C2C2×#1798901811yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2217C2
socleC22×C73C22×D8
hypercenter#2217D73
hypocenterC73#33600
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. C2×#17989018C2semidirect
2. C22×C16×D73C2semidirect
3. #22619514C2semidirect
4. #22619510C2semidirect
5. #22612028C2semidirect
6. #22612027C2semidirect
7. #22612024C2semidirect
8. #22612021C2semidirect
9. #22612020C2semidirect
10. #22612016C2semidirect
11. #22612015C2semidirect
12. #22611714C2semidirect
13. #22611637C2semidirect
14. #22611635C2semidirect
15. C73×#2217C2semidirect
22. C2×#14556537C22semidirect
23. C2×#14556536C22semidirect
24. C2×#14556532C22semidirect
45. #17988415C22semidirect
46. #17988412C22semidirect
47. #17988412C22semidirect
48. #17988392C22semidirect
49. #17988391C22semidirect
50. #17988390C22semidirect
51. #17988389C22semidirect
52. #17988388C22semidirect
53. #17988387C22semidirect
54. C2×SD32×C73C22semidirect
58. #367×C73C22semidirect
59. #358×C73C22semidirect
60. #357×C73C22semidirect
104. C16×D73D4semidirect
253. C22×C73#508semidirect
264. C2×D73#508semidirect
292. #2217D73semidirect
300. C73#33600semidirect
309. C2×SD32C2×D73semidirect
335. #367C2×D73semidirect
341. #358C2×D73semidirect
345. #357C2×D73semidirect
548. C22#17989018semidirect

Maximal subgroups

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

subgroupindex
C2×#179890182
C22×C16×D732
#226195142
#226195102
#226120282
#226120272
#226120242
#226120212
#226120202
#226120162
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C73×#22172
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