Finite Group Databasebeta

← #33095937

Subgroups — #33095937

Order 18688.

1130 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989141C211yesmaximal, characteristic
2C2×#17988609C211yesmaximal, characteristic
3#22612336C211yesmaximal, characteristic
4#22610881C212yesmaximal, complemented
5#22610852C212yesmaximal, complemented
6#22610363C212yesmaximal
7#22610361C212yesmaximal
8#22609937C211yesmaximal, characteristic, complemented
9#22609936C211yesmaximal, characteristic, complemented
10#22609933C211yesmaximal, characteristic, complemented
11C73×#1803C211yesmaximal, characteristic, complemented
12C22×#11924011C2212yes—
13#17989141—24no—
14#17989141C418yescomplemented
15C2×#14556590C2211yescharacteristic
16C2×#14556589C2211yescharacteristic
17C2×#14556581—21no—
18C2×#14556571C2211yescharacteristic
19C2×#14556569—21no—
20C2×#14556567C2211yescharacteristic
21C2×C4×Dic146—21no—
22C2×C4×Dic146C2211yescharacteristic
23C2×#14556549—21no—
24C2×#14556548—21no—
25C2×C42×D73C2211yescharacteristic
26#17988609—24no—
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556520C2212yes—
29C2×#14556516C2212yes—
30C2×#14556515C2212yes—
31C2×#14556514C2212yes—
32#17988457C2211yescharacteristic
33#17988452—21no—
34#17988444—21no—
35#17988444C2212yes—
36C4⋊C4×Dic73—21no—
37C4⋊C4×Dic73C2212yes—
38#17988350C2212yes—
39#17988349C2212yes—
40#17988349C2212yes—
41C2×C4×Q8×C73C2211yescharacteristic
42C2×C4⋊C8×C73C2211yescharacteristic
43C2×#103×C73C2212yes—
44#384×C73C2211yescharacteristic
45#340×C73C2212yes—
46C22×Dic146—21no—
47C22×Dic146C2311yescharacteristic
48C22×C4×D73D411yescharacteristic
49C22×C4×D73D411yescharacteristic
50C22×C4×D73C2311yescharacteristic
51C2×#11924011—24no—
52C2×#11924011—42no—
53C2×#11924011—42no—
54C22×#10596831C2312yes—
55#14556590—22no—
56#14556590C2×C414yes—
57#14556589—22no—
58#14556589C2×C414yes—
59#14556581—28no—
60#14556571—22no—
61#14556571C2×C414yes—
62#14556569—28no—
63#14556567—22no—
64#14556567C2×C414yes—
65C4×Dic146—28no—
66C4×Dic146—22no—
67C4×Dic146C2×C414yes—
68#14556549—44no—
69#14556548—28no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73C2312yes—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—22no—
83C2×C4×Dic73—22no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73D411yescharacteristic
86C2×C4×Dic73D411yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88C42×D73C2×C414yes—
89C42×D73D414yes—
90C4⋊#10596831—22no—
91#14556520—24no—
92#14556516—42no—
93#14556515—42no—
94#14556514—42no—
95#14556498—22no—
96#14556498—22no—
97C22×Q8×C73C2311yescharacteristic
98C22×C8×C73C2312yes—
99C4×Q8×C73—22no—
100C4×Q8×C73C2×C414yes—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73C2312yes—
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C42×C73—21no—
107C2×C42×C73C2311yescharacteristic
108C4⋊C8×C73—22no—
109#103×C73—42no—
110C23×D73C2×D411yescharacteristic
111C2×Dic146—21no—
112C2×Dic146—21no—
113C2×Dic146—21no—
114C2×Dic146—21no—
115C2×Dic146—21no—
116C2×Dic146—24no—
117C2×Dic146—42no—
118C2×Dic146—42no—
119C2×Dic146C22×C412yes—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—22no—
123C22×Dic73C2×D411yescharacteristic
124C22×Dic73C2×D411yescharacteristic
125C22×Dic73C2×D411yescharacteristic
126C2×C4×D73—22no—
127C2×C4×D73—22no—
128C2×C4×D73—21no—
129C2×C4×D73—21no—
130C2×C4×D73—21no—
131C2×C4×D73—21no—
132C2×C4×D73—22no—
133C2×C4×D73—22no—
134C2×C4×D73C22⋊C414yes—
135C2×C4×D73C22⋊C414yes—
136C2×C4×D73C22×C412yes—
137C2×C4×D73C2×D412yes—
138#11924011—48no—
139C2×#10596831—22no—
140C2×#10596831—22no—
141C2×#10596831—22no—
142C4⋊Dic73—21no—
143C4⋊Dic73—24no—
144C4⋊Dic73—22no—
145C4⋊Dic73—22no—
146C4⋊Dic73—42no—
147C4⋊Dic73—42no—
148C4⋊Dic73C22×C412yes—
149#11924007—24no—
150#11924007—24no—
151#11924007—24no—
152#11924007—24no—
153#11924007—24no—
154#11924005—24no—
155#11924005—24no—
156#11924005—44no—
157C4×Dic73—24no—
158C4×Dic73—44no—
159C4×Dic73C22⋊C414yes—
160C4×Dic73C22⋊C414yes—
161C4×Dic73C22×C412yes—
162C4×Dic73C2×D412yes—
163C2×Q8×C73—21no—
164C2×Q8×C73—21no—
165C2×Q8×C73—21no—
166C2×Q8×C73—21no—
167C2×Q8×C73—21no—
168C2×Q8×C73C22×C412yes—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—22no—
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2411yescharacteristic
175C2×C8×C73—22no—
176C2×C8×C73—22no—
177C2×C8×C73—22no—
178C4⋊C4×C73—24no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—22no—
181C4⋊C4×C73—22no—
182C4⋊C4×C73—42no—
183C4⋊C4×C73—42no—
184C4⋊C4×C73C22×C412yes—
185C42×C73—24no—
186C42×C73C22×C412yes—
187C42×C73C2×D412yes—
188C23×C73C22×D411yescharacteristic
189C22×D73—21no—
190C22×D73—21no—
191C22×D73—21no—
192C22×D73—21no—
193C22×D73—21no—
194C22×D73—21no—
195C22×D73C2×C22⋊C412yes—
196Q8×C73—22no—
197Q8×C73—22no—
198Q8×C73—41no—
199Q8×C73—41no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73—41no—
207C2×C4×C73—41no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73C2×C22⋊C412yes—
212C2×C4×C73C2×C22⋊C412yes—
213C2×C4×C73#13712yes—
214C2×C4×C73#13712yes—
215C2×C4×C73C23×C411yescharacteristic
216C2×C4×C73C22×D411yescharacteristic
217C2×C4×C73H(5,𝔽2)11yescharacteristic
218C2×C4×C73H(5,𝔽2)11yescharacteristic
219C2×C4×C732−1+411yescharacteristic
220C2×C4×C732−1+411yescharacteristic
221C2×Dic73—21no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—22no—
226C2×Dic73—22no—
227C2×Dic73—41no—
228C2×Dic73—41no—
229C2×Dic73—41no—
230C2×Dic73—41no—
231C2×Dic73—42no—
232C2×Dic73—42no—
233C2×Dic73—42no—
234C2×Dic73C2×C22⋊C412yes—
235C2×Dic73C2×C22⋊C412yes—
236C2×Dic73C2×C22⋊C412yes—
237C2×Dic73AGL(1,ℤ/8)12yes—
238C2×Dic73#13712yes—
239C4×D73—22no—
240C4×D73—22no—
241C4×D73—22no—
242C4×D73—22no—
243C4×D73—28no—
244C4×D73—28no—
245Dic146—22no—
246Dic146—22no—
247Dic146—41no—
248Dic146—41no—
249Dic146—48no—
250C8×C73—42no—
251#10596831—42no—
252C22×C73C22×C22⋊C411yescharacteristic
253C22×C73#53711yescharacteristic
254C22×C73#53711yescharacteristic
255C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C73C2×#13711yescharacteristic
257C22×C73C2×#13711yescharacteristic
258C22×C73C2×#13711yescharacteristic
259C2×D73—22no—
260C2×D73—22no—
261C2×D73—22no—
262C2×D73—41no—
263C2×D73—41no—
264C2×D73—41no—
265C2×D73—41no—
266C4×C73—22no—
267C4×C73—42no—
268C4×C73—42no—
269C4×C73—82no—
270C4×C73#41914yes—
271C4×C73#52011yescharacteristic
272C4×C73#52011yescharacteristic
273C4×C73#53511yescharacteristic
274C4×C73#53611yescharacteristic
275Dic73—22no—
276Dic73—22no—
277Dic73—42no—
278Dic73—42no—
279Dic73—82no—
280Dic73#41714yes—
281#21894—731nomaximal, Sylow
282C2×C73#231111yescharacteristic
283C2×C73C2×#41911yescharacteristic
284C2×C73C2×#41711yescharacteristic
285C2×C73#325411yescharacteristic
286C2×C73#325411yescharacteristic
287C2×C73#325511yescharacteristic
288C2×C73#325511yescharacteristic
289D73—42no—
290C2×#533—731no—
291C2×#423—731no—
292#2285—731no—
293#1988—732no—
294#1981—732no—
295#1894—732no—
296#1894—732no—
297#1803—731no—
298#1803—731no—
299#1803—731no—
300#1803D7311yescharacteristic, complemented
301C73#2189411yescharacteristic, complemented, Sylow
302C22×M16—732no—
303#533—738no—
304#533—1464no—
305C2×C4⋊Q8—731no—
306C2×C4⋊Q8—731no—
307C2×#125—1461no—
308C2×C22⋊Q8—731no—
309C2×C22⋊Q8—731no—
310C2×C22⋊Q8—1461no—
311C2×C4×Q8—731no—
312C2×C4×Q8—1461no—
313C2×C4×Q8C2×D7311yescharacteristic
314C2×#117—1461no—
315C2×#117—1461no—
316C22×C42—731no—
317#423—1464no—
318C2×C4⋊C8—732no—
319C2×C4⋊C8—731no—
320C2×C4⋊C8C2×D7311yescharacteristic
321C2×#103—732no—
322C2×#103—732no—
323C2×#103—732no—
324C2×#103C2×D7312yes—
325#384—731no—
326#384C2×D7311yescharacteristic
327#382—1461no—
328#380—732no—
329#380—1461no—
330C4×C4⋊C4—732no—
331C4×C4⋊C4—1461no—
332#340—732no—
333#340—732no—
334#340—732no—
335#340C2×D7312yes—
336C22×Q8—731no—
337C22×Q8—1461no—
338C22×Q8C22×D7311yescharacteristic
339C23×C4—731no—
340C23×C4—731no—
341C23×C4—731no—
342C2×M16—2922no—
343C2×M16—2922no—
344C2×M16—1464no—
345C22×C8—732no—
346C22×C8C22×D7312yes—
347C4⋊Q8—734no—
348C4⋊Q8—734no—
349C4⋊Q8—1462no—
350C4⋊Q8—1462no—
351#125—1468no—
352C22⋊Q8—734no—
353C22⋊Q8—734no—
354C22⋊Q8—1462no—
355C22⋊Q8—1468no—
356C22⋊Q8—1462no—
357C4×Q8—22no—
358C4×Q8—734no—
359C4×Q8—1468no—
360C4×Q8—1462no—
361C4×Q8C4×D7314yes—
362#117—2924no—
363#117—1468no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—732no—
369C2×C4⋊C4—1461no—
370C2×C4⋊C4—1461no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4C22×D7312yes—
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C22⋊C4—1462no—
380C2×C22⋊C4—1461no—
381C2×C22⋊C4—1461no—
382C2×C42—21no—
383C2×C42—731no—
384C2×C42—731no—
385C2×C42—731no—
386C2×C42—734no—
387C2×C42—734no—
388C2×C42—1462no—
389C2×C42—1461no—
390C2×C42C22×D7311yescharacteristic
391C4⋊C8—22no—
392C4⋊C8—1464no—
393C4⋊C8—1462no—
394#103—42no—
395#103—2922no—
396#103—2922no—
397#103—2922no—
398#95—1462no—
399#95—1462no—
400C24—731no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—2922no—
407C2×Q8—2922no—
408C2×Q8—732no—
409C2×Q8—1464no—
410C2×Q8—1461no—
411C2×Q8—1461no—
412C2×Q8—1461no—
413C2×Q8—1461no—
414C2×Q8—1461no—
415C2×Q8C2×C4×D7312yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—731no—
420C22×C4—731no—
421C22×C4—731no—
422C22×C4—734no—
423C22×C4—734no—
424C22×C4—732no—
425C22×C4—732no—
426C22×C4—1462no—
427C22×C4—1462no—
428C22×C4—1462no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1462no—
432C22×C4—1462no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4D4×D7311yescharacteristic
438C22×C4D4×D7311yescharacteristic
439C22×C4C23×D7311yescharacteristic
440M16—2928no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—1462no—
445C2×C8—1462no—
446C2×C8—1462no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—2922no—
454C4⋊C4—2922no—
455C4⋊C4—732no—
456C4⋊C4—1464no—
457C4⋊C4—1464no—
458C4⋊C4—1464no—
459C4⋊C4—1464no—
460C4⋊C4—1464no—
461C4⋊C4—1462no—
462C4⋊C4—1462no—
463C4⋊C4—1464no—
464C4⋊C4—1461no—
465C4⋊C4C2×C4×D7312yes—
466C22⋊C4—2924no—
467C22⋊C4—1464no—
468C22⋊C4—1464no—
469C42—24no—
470C42—2924no—
471C42—734no—
472C42—734no—
473C42—732no—
474C42—732no—
475C42—1464no—
476C42C2×C4×D7312yes—
477C42D4×D7312yes—
478C23—732no—
479C23—1461no—
480C23—1461no—
481C23—1461no—
482C23—1461no—
483C23—1461no—
484C23—1461no—
485C23C2×D4×D7311yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490Q8—2928no—
491Q8—2921no—
492Q8—2921no—
493Q8—1462no—
494Q8—1462no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—2921no—
507C2×C4—2921no—
508C2×C4—2921no—
509C2×C4—2921no—
510C2×C4—2922no—
511C2×C4—2922no—
512C2×C4—2922no—
513C2×C4—732no—
514C2×C4—732no—
515C2×C4—732no—
516C2×C4—732no—
517C2×C4—732no—
518C2×C4—1468no—
519C2×C4—1462no—
520C2×C4—1462no—
521C2×C4—1461no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4—1462no—
526C2×C4—1462no—
527C2×C4—1462no—
528C2×C4—1462no—
529C2×C4—1468no—
530C2×C4C22⋊C4×D7312yes—
531C2×C4C22⋊C4×D7312yes—
532C2×C4#1455662012yes—
533C2×C4#1455662312yes—
534C2×C4C22×C4×D7311yescharacteristic
535C2×C4C2×D4×D7311yescharacteristic
536C2×C4#1455663611yescharacteristic
537C2×C4#1455663711yescharacteristic
538C2×C4#1455663811yescharacteristic
539C2×C4#1455663911yescharacteristic
540C8—42no—
541C8—2922no—
542C22—2921no—
543C22—2921no—
544C22—2921no—
545C22—2921no—
546C22—1462no—
547C22—1462no—
548C22—1462no—
549C22C2×C22⋊C4×D7311yescharacteristic
550C22#1798916611yescharacteristic
551C22#1798916711yescharacteristic
552C22AGL(1,ℤ/8)×D7311yescharacteristic
553C22#137×D7311yescharacteristic
554C22C2×#1455662011yescharacteristic
555C22C2×#1455662311yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—2922no—
561C4—2922no—
562C4—5842no—
563C4—734no—
564C4—1462no—
565C4—1462no—
566C4#1798859414yes—
567C4#1798906711yescharacteristic
568C4#1798906811yescharacteristic
569C4#1798915311yescharacteristic
570C4#1798916011yescharacteristic
571C2—2922no—
572C2#2261252011yescharacteristic
573C2C2×#1798859411yescharacteristic
574C2#417×D7311yescharacteristic
575C2#2262131611yescharacteristic
576C2#2262131711yescharacteristic
577C2#2262132311yescharacteristic
578C2#2262132411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1803C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1803D73
hypocenterC73#21894
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #22610881C2semidirect
5. #22610852C2semidirect
8. #22609937C2semidirect
9. #22609936C2semidirect
10. #22609933C2semidirect
11. C73×#1803C2semidirect
14. #17989141C4semidirect
300. #1803D73semidirect
301. C73#21894semidirect

Maximal subgroups

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

subgroupindex
C2×#179891412
C2×#179886092
#226123362
#226108812
#226108522
#226103632
#226103612
#226099372
#226099362
#226099332
C73×#18032
#2189473