Finite Group Databasebeta

← #33958232

Subgroups — #33958232

Order 18688.

1654 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
1#22623334C211yesmaximal, characteristic, complemented
2#22623320C211yesmaximal, characteristic
3#22621224C214yesmaximal, complemented
4#22621209C214yesmaximal, complemented
5#22621194C214yesmaximal, complemented
6#22621179C214yesmaximal, complemented
7#22620455C211yesmaximal, characteristic, complemented
8#22620441C211yesmaximal, characteristic, complemented
9#22620253C211yesmaximal, characteristic, complemented
10#22619842C214yesmaximal, complemented
11#22619715C214yesmaximal, complemented
12C2×#17988649C211yesmaximal, characteristic, complemented
13C73×#3135C211yesmaximal, characteristic, complemented
14#17989217C2214yes—
15#17989216—22no—
16#17989214C2214yes—
17#17989212—22no—
18#17989206C2214yes—
19#17989198C2212yes—
20#17989191C2212yes—
21#17989182—22no—
22#17989174—22no—
23#17989144C2211yescharacteristic
24#17989144C2211yescharacteristic
25#17989141—24no—
26#17989140C2212yes—
27#17989117—24no—
28#17989094C2212yes—
29#41×Dic73C2212yes—
30C2×#14556548C2211yescharacteristic
31#117×D73C2212yes—
32#17988996C2214yescomplemented
33#17988996C2214yescomplemented
34#17988990C2212yescomplemented
35#17988986C2214yescomplemented
36#17988983C2212yescomplemented
37#17988941C2212yescomplemented
38#17988941C2212yescomplemented
39#17988937C2212yescomplemented
40#17988933C2212yescomplemented
41#17988927C2214yes—
42#17988926C2214yes—
43#17988921C2214yes—
44#17988916C2214yes—
45#17988910C2214yes—
46#17988909C2214yes—
47#17988904C2214yes—
48#17988899C2214yes—
49#17988896C2214yescomplemented
50#17988892C2214yescomplemented
51#17988889C2214yescomplemented
52#17988885C2214yescomplemented
53#17988880C2214yescomplemented
54#17988875C2214yescomplemented
55#17988870C2214yescomplemented
56#17988865C2214yescomplemented
57#17988649C2214yes—
58#17988649C2212yes—
59#17988649C2212yes—
60#17988625C2214yes—
61#17988623C2214yes—
62C2×#14556520C2212yes—
63#17988583C2214yes—
64C2×#14556503C2212yes—
65C2×C4×#10596831C2211yescharacteristic
66#533×C73C2211yescharacteristic
67#532×C73C2211yescharacteristic
68#491×C73C2212yescomplemented
69#490×C73C2212yescomplemented
70#489×C73C2214yescomplemented
71#416×C73C2214yes—
72C2×#97×C73C2211yescharacteristic
73C22×C4×D73C2311yescharacteristic
74C22×#10596831C2312yes—
75#14556591C2312yes—
76#14556591C2312yes—
77#14556590—24no—
78#14556590C2314yes—
79#14556589—24no—
80#14556586—24no—
81#14556584—24no—
82#14556583C2312yes—
83#14556583C2312yes—
84#14556582C2314yes—
85#14556581—24no—
86#14556581—24no—
87#14556581—24no—
88#14556577—24no—
89#14556575—24no—
90#14556572—22no—
91#14556572—24no—
92#14556571—24no—
93#14556570—22no—
94#14556570—22no—
95#14556569—24no—
96#14556569—24no—
97#14556568—24no—
98#14556568—24no—
99#14556567—24no—
100#14556567C2314yes—
101#14556563—24no—
102#14556562C2314yes—
103#14556559—22no—
104Q8×Dic73C2314yes—
105C4×Dic146—24no—
106C4×Dic146—22no—
107C4×Dic146—22no—
108D4×Dic73—24no—
109D4×Dic73C2314yes—
110C4×C22⋊D73—24no—
111#14556550—22no—
112#14556550C2312yes—
113#14556550C2312yes—
114#14556549C2314yes—
115#14556548—22no—
116#14556548D414yes—
117#14556548C2312yes—
118#14556548C2312yes—
119#14556547—22no—
120C2×#11924007D412yes—
121C4⋊C4×D73C2314yes—
122C2×#11924005D412yes—
123C22⋊C4×D73—22no—
124C2×C4×Dic73—22no—
125C2×C4×Dic73C2311yescharacteristic
126C42×D73—22no—
127#14556525C2314yes—
128#14556525C2314yes—
129#14556524C2314yes—
130#14556524C2314yes—
131#14556523C2314yes—
132#14556523C2314yes—
133#14556522C2314yes—
134#14556522C2314yes—
135#14556520C2314yes—
136#14556520C2314yes—
137#14556515C2314yes—
138#14556515C2314yes—
139#14556512C2314yes—
140#14556512C2314yes—
141#14556503C2314yes—
142#14556503C2314yes—
143C4×#10596831C2312yes—
144C4×#10596831C2311yescharacteristic
145C4×#10596831C2311yescharacteristic
146C2×#41×C73C2312yes—
147C22×C8×C73C2312yes—
148C4⋊Q8×C73C2311yescharacteristic
149C4⋊Q8×C73C2312yes—
150#127×C73C2311yescharacteristic
151#125×C73C2312yes—
152#124×C73C2312yes—
153C22⋊Q8×C73—22no—
154#121×C73—22no—
155C4×Q8×C73—22no—
156C4×D4×C73—22no—
157#117×C73C2312yes—
158C2×C42×C73C2311yescharacteristic
159#103×C73C2314yes—
160#103×C73C2314yes—
161#102×C73C2314yes—
162#102×C73C2314yes—
163#97×C73C2312yes—
164#97×C73C2312yes—
165C23×D73C2×D411yescharacteristic
166C2×Dic146—22no—
167C2×Dic146—22no—
168C2×Dic146—24no—
169C2×C22⋊D73—24no—
170C22×Dic73—24no—
171C22×Dic73C2×D411yescharacteristic
172C22×Dic73C2×D412yes—
173C2×C4×D73—24no—
174C2×C4×D73—24no—
175C2×C4×D73C2×D412yes—
176C2×C4×D73C2412yes—
177C2×#10596831—24no—
178C2×#10596831C2412yes—
179C2×#10596831C2412yes—
180C4⋊Dic73—22no—
181C4⋊Dic73—22no—
182C4⋊Dic73—24no—
183C4⋊Dic73—24no—
184C4⋊Dic73C2414yes—
185C4⋊Dic73C2414yes—
186#11924007—24no—
187#11924007—24no—
188#11924007—24no—
189#11924007—24no—
190#11924007—24no—
191#11924007—24no—
192#11924007C2×D414yes—
193#11924007C2×D414yes—
194C22⋊Dic73—22no—
195C22⋊Dic73—22no—
196C22⋊Dic73—24no—
197C22⋊Dic73—24no—
198#11924005—24no—
199#11924005—24no—
200#11924005C2×D414yes—
201#11924005C2×D414yes—
202C4×Dic73—22no—
203C4×Dic73—22no—
204C4×Dic73—24no—
205C4×Dic73—24no—
206C4×Dic73C2×D412yes—
207C4×Dic73C2412yes—
208#41×C73—28no—
209C2×Q8×C73—22no—
210C2×Q8×C73C2412yes—
211C2×D4×C73—22no—
212C2×D4×C73—22no—
213C2×D4×C73C2412yes—
214C22×C4×C73—22no—
215C22×C4×C73C2×D412yes—
216C22×C4×C73C2411yescharacteristic
217C2×C8×C73—24no—
218C2×C8×C73C2414yes—
219C4⋊C4×C73—22no—
220C4⋊C4×C73—24no—
221C4⋊C4×C73—22no—
222C4⋊C4×C73C2414yes—
223C22⋊C4×C73—22no—
224C22⋊C4×C73—24no—
225C42×C73—22no—
226C42×C73C2×D412yes—
227C42×C73C2411yescharacteristic
228C42×C73C2411yescharacteristic
229C23×C73—22no—
230C23×C73C22×D411yescharacteristic
231C22×D73—22no—
232C22×D73—42no—
233C22×D73C22×D412yes—
234Q8×C73—24no—
235Q8×C73—82no—
236D4×C73—24no—
237D4×C73—44no—
238D4×C73—82no—
239C2×C4×C73—24no—
240C2×C4×C73—22no—
241C2×C4×C73—42no—
242C2×C4×C73—44no—
243C2×C4×C73C22×D412yes—
244C2×C4×C73C22×D412yes—
245C2×C4×C73C22×D411yescharacteristic
246C2×C4×C732−1+414yes—
247C2×C4×C73C2511yescharacteristic
248C22⋊D73—84no—
249C2×Dic73—22no—
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—42no—
253C2×Dic73—84no—
254C2×Dic73C22×D412yes—
255C2×Dic73C22×D414yes—
256C4×D73—24no—
257C4×D73—24no—
258C4×D73—84no—
259Dic146—84no—
260Dic146—82no—
261Dic146—82no—
262C8×C73—24no—
263#10596831—22no—
264#10596831—22no—
265C22×C73—22no—
266C22×C73—44no—
267C22×C73—42no—
268C22×C73#53612yes—
269C22×C73C23×D411yescharacteristic
270C2×D73—24no—
271C2×D73—22no—
272C2×D73—82no—
273C4×C73—22no—
274C4×C73—22no—
275C4×C73—44no—
276C4×C73—42no—
277C4×C73#53612yes—
278C4×C73C2×2−1+412yes—
279Dic73—24no—
280Dic73—22no—
281Dic73—44no—
282Dic73—44no—
283#60859—731nomaximal, Sylow
284C2×C73—22no—
285C2×C73—82no—
286C2×C73C2×#53611yescharacteristic
287C2×C73#358412yes—
288D73—42no—
289#3458—731no—
290#3455—731no—
291#3244—734no—
292#3243—734no—
293#3242—734no—
294#3241—734no—
295#3162—731no—
296#3160—731no—
297#3135D7311yescharacteristic, complemented
298#3131—731no—
299#3091—734no—
300#3076—734no—
301C2×#433—731no—
302C73#6085911yescharacteristic, complemented, Sylow
303#544—734no—
304#544—734no—
305#544—1462no—
306#543—1462no—
307#542—734no—
308#541—732no—
309#540—732no—
310#539—1462no—
311#538—1462no—
312#533—732no—
313#533—731no—
314#533—731no—
315#533—1464no—
316#533C2×D7311yescharacteristic
317#532C2×D7311yescharacteristic
318#529—1464no—
319#525—732no—
320C4×#41—732no—
321C2×#117—731no—
322C2×#117—732no—
323#501—734no—
324#501—734no—
325#500—732no—
326#499—734no—
327#498—732no—
328#491C2×D7312yescomplemented
329#490C2×D7312yescomplemented
330#489C2×D7314yescomplemented
331#488—732no—
332#488—732no—
333#487—732no—
334#486—732no—
335#485—734no—
336#485—734no—
337#484—734no—
338#483—734no—
339#482—734no—
340#482—734no—
341#481—734no—
342#480—734no—
343#479—734no—
344#479—734no—
345#478—734no—
346#478—734no—
347#477—734no—
348#476—734no—
349#475—734no—
350#474—734no—
351#433—734no—
352#433—732no—
353#433—732no—
354#427—734no—
355#427—734no—
356C2×C4⋊C8—732no—
357#416—734no—
358#416C2×D7314yes—
359C2×C22⋊C8—732no—
360C2×#97C2×D7311yescharacteristic
361C2×C4×C8—731no—
362C2×#41C22×D7312yes—
363C23×C4—731no—
364C22×C8—732no—
365C22×C8C22×D7312yes—
366C4⋊Q8—734no—
367C4⋊Q8—732no—
368C4⋊Q8—732no—
369C4⋊Q8—1464no—
370C4⋊Q8—1464no—
371C4⋊Q8C22×D7311yescharacteristic
372C4⋊Q8C22×D7312yes—
373#127C22×D7311yescharacteristic
374#126—1464no—
375#126—1464no—
376#125—734no—
377#125—732no—
378#125—732no—
379#125—1464no—
380#125—1464no—
381#125—1464no—
382#125C22×D7312yes—
383#124—1464no—
384#124C22×D7312yes—
385#123—1464no—
386#123—1464no—
387#123—1462no—
388C22⋊Q8—22no—
389C22⋊Q8—734no—
390C22⋊Q8—1464no—
391C22⋊Q8—1464no—
392C22⋊Q8—1464no—
393C22⋊Q8—1464no—
394C22⋊Q8—1464no—
395C22⋊Q8—1464no—
396C22⋊Q8—1462no—
397C22⋊Q8—1462no—
398#121—22no—
399#121—734no—
400#121—1464no—
401#120—1462no—
402C4×Q8—22no—
403C4×Q8—734no—
404C4×Q8—1464no—
405C4×Q8—1462no—
406C4×Q8—1462no—
407C4×D4—22no—
408C4×D4—734no—
409C4×D4—1464no—
410C4×D4—1464no—
411#117—734no—
412#117—732no—
413#117—732no—
414#117—732no—
415#117—732no—
416#117—734no—
417#117—1462no—
418#117—1462no—
419#117—1462no—
420#117C22×D7312yes—
421C2×C4⋊C4—732no—
422C2×C4⋊C4—734no—
423C2×C22⋊C4—732no—
424C2×C22⋊C4—1462no—
425C2×C42—731no—
426C2×C42—1462no—
427C2×C42—1462no—
428C2×C42C22×D7311yescharacteristic
429#107—734no—
430#107—734no—
431#107—734no—
432#107—734no—
433#106—734no—
434#106—734no—
435#106—734no—
436#106—734no—
437C4⋊C8—734no—
438C4⋊C8—734no—
439#103—734no—
440#103—734no—
441#103C22×D7314yes—
442#103C22×D7314yes—
443#102—734no—
444#102—734no—
445#102C22×D7314yes—
446#102C22×D7314yes—
447C22⋊C8—734no—
448C22⋊C8—734no—
449#97C22×D7312yes—
450#97C22×D7312yes—
451C4×C8—732no—
452C4×C8—731no—
453C4×C8—731no—
454C24—731no—
455#41—28no—
456C2×Q8—22no—
457C2×Q8—1464no—
458C2×Q8—1462no—
459C2×Q8—1462no—
460C2×Q8C23×D7312yes—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—1464no—
464C2×D4C23×D7312yes—
465C22×C4—22no—
466C22×C4—732no—
467C22×C4—731no—
468C22×C4—732no—
469C22×C4—732no—
470C22×C4—1464no—
471C22×C4—1464no—
472C22×C4—1464no—
473C22×C4D4×D7312yes—
474C22×C4C23×D7311yescharacteristic
475C2×C8—24no—
476C2×C8—732no—
477C2×C8—732no—
478C2×C8—1464no—
479C2×C8C23×D7314yes—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—734no—
484C4⋊C4—734no—
485C4⋊C4—734no—
486C4⋊C4—734no—
487C4⋊C4—1464no—
488C4⋊C4—1464no—
489C4⋊C4—1464no—
490C4⋊C4—1464no—
491C4⋊C4—1464no—
492C4⋊C4—1464no—
493C4⋊C4—1464no—
494C4⋊C4—1464no—
495C4⋊C4—1462no—
496C4⋊C4—1462no—
497C4⋊C4C23×D7314yes—
498C22⋊C4—22no—
499C22⋊C4—24no—
500C22⋊C4—734no—
501C22⋊C4—734no—
502C22⋊C4—1464no—
503C22⋊C4—1464no—
504C22⋊C4—1464no—
505C22⋊C4—1464no—
506C22⋊C4—1462no—
507C22⋊C4—1462no—
508C42—22no—
509C42—732no—
510C42—732no—
511C42—1464no—
512C42—1464no—
513C42—1462no—
514C42—1462no—
515C42D4×D7312yes—
516C42C23×D7311yescharacteristic
517C42C23×D7311yescharacteristic
518C23—22no—
519C23—2922no—
520C23—732no—
521C23—1462no—
522C23C2×D4×D7311yescharacteristic
523Q8—24no—
524Q8—82no—
525Q8—5842no—
526Q8—5842no—
527Q8—5844no—
528D4—24no—
529D4—44no—
530D4—82no—
531D4—5844no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—2922no—
537C2×C4—5844no—
538C2×C4—5844no—
539C2×C4—734no—
540C2×C4—732no—
541C2×C4—1464no—
542C2×C4—1464no—
543C2×C4—1462no—
544C2×C4—1464no—
545C2×C4—1464no—
546C2×C4C2×D4×D7312yes—
547C2×C4C2×D4×D7312yes—
548C2×C4C2×D4×D7311yescharacteristic
549C2×C4#1455663814yes—
550C2×C4C24×D7311yescharacteristic
551C8—24no—
552C8—1462no—
553C8—1462no—
554C22—22no—
555C22—42no—
556C22—44no—
557C22—5842no—
558C22—1462no—
559C22—1464no—
560C22#1798915712yes—
561C22C22×D4×D7311yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—44no—
566C4—2924no—
567C4—2924no—
568C4—1462no—
569C4—1464no—
570C4#1798915712yes—
571C4C2×#1455663812yes—
572C2—22no—
573C2—82no—
574C2—2922no—
575C2C2×#1798915711yescharacteristic
576C2#2262470412yes—
subgroupG/H
centerC2×C4C2×D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC73×#3135C2
socleC22×C73C23×D4
prime radical#22623320C2
hypercenter#3135D73
hypocenterC73#60859
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3135

Complements

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

NH ≅ G/Ntype
1. #22623334C2semidirect
3. #22621224C2semidirect
4. #22621209C2semidirect
5. #22621194C2semidirect
6. #22621179C2semidirect
7. #22620455C2semidirect
8. #22620441C2semidirect
9. #22620253C2semidirect
10. #22619842C2semidirect
11. #22619715C2semidirect
12. C2×#17988649C2semidirect
13. C73×#3135C2semidirect
32. #17988996C22semidirect
33. #17988996C22semidirect
34. #17988990C22semidirect
35. #17988986C22semidirect
36. #17988983C22semidirect
37. #17988941C22semidirect
38. #17988941C22semidirect
39. #17988937C22semidirect
40. #17988933C22semidirect
49. #17988896C22semidirect
50. #17988892C22semidirect
51. #17988889C22semidirect
52. #17988885C22semidirect
53. #17988880C22semidirect
54. #17988875C22semidirect
55. #17988870C22semidirect
56. #17988865C22semidirect
68. #491×C73C22semidirect
69. #490×C73C22semidirect
70. #489×C73C22semidirect
297. #3135D73semidirect
302. C73#60859semidirect
328. #491C2×D73semidirect
329. #490C2×D73semidirect
330. #489C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226233342
#226233202
#226212242
#226212092
#226211942
#226211792
#226204552
#226204412
#226202532
#226198422
#226197152
C2×#179886492
C73×#31352
#6085973