Finite Group Databasebeta

← #33095743

Subgroups — #33095743

Order 18688.

1086 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989141C211yesmaximal, characteristic
2C2×#17988643C211yesmaximal, characteristic
3#22612336C211yesmaximal, characteristic
4#22610902C212yesmaximal, complemented
5#22610749C212yesmaximal, complemented
6#22610242C212yesmaximal
7#22610240C212yesmaximal
8#22609937C211yesmaximal, characteristic, complemented
9#22609936C211yesmaximal, characteristic, complemented
10#22609933C211yesmaximal, characteristic, complemented
11C73×#1803C211yesmaximal, characteristic, complemented
12#17989141—24no—
13#17989141C418yescomplemented
14C2×#14556590C2211yescharacteristic
15C2×#14556589C2211yescharacteristic
16C2×#14556581—21no—
17C2×#14556571C2211yescharacteristic
18C2×#14556569—21no—
19C2×#14556567C2211yescharacteristic
20C2×C4×Dic146—21no—
21C2×C4×Dic146C2211yescharacteristic
22C2×#14556549C2212yes—
23C2×#14556548—21no—
24C2×C42×D73C2211yescharacteristic
25#17988643—24no—
26C2×C4⋊#10596831C2211yescharacteristic
27C2×#14556516C2212yes—
28C2×#14556515C2212yes—
29C2×#14556514C2212yes—
30C2×#14556503C2212yes—
31C2×#14556501C2212yes—
32#17988457C2211yescharacteristic
33#17988452C2212yes—
34#17988444—21no—
35#17988444—21no—
36C4⋊C4×Dic73—21no—
37C4⋊C4×Dic73—21no—
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
51C22×#10596831C2312yes—
52#14556590—22no—
53#14556590C2×C414yes—
54#14556589—22no—
55#14556589C2×C414yes—
56#14556581—28no—
57#14556571—22no—
58#14556571C2×C414yes—
59#14556569—28no—
60#14556567—22no—
61#14556567C2×C414yes—
62C4×Dic146—28no—
63C4×Dic146—22no—
64C4×Dic146C2×C414yes—
65#14556549—42no—
66#14556549—42no—
67#14556548—28no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73C2312yes—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924005—21no—
79C2×#11924005—21no—
80C2×#11924005—22no—
81C2×C4×Dic73—22no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73D411yescharacteristic
84C2×C4×Dic73D411yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86C42×D73C2×C414yes—
87C42×D73D414yes—
88C4⋊#10596831—22no—
89#14556516—42no—
90#14556515—42no—
91#14556514—42no—
92#14556503—24no—
93#14556501—24no—
94#14556498—22no—
95#14556498—22no—
96C22×Q8×C73C2311yescharacteristic
97C22×C8×C73C2312yes—
98C4×Q8×C73—22no—
99C4×Q8×C73C2×C414yes—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73C2312yes—
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C42×C73—21no—
106C2×C42×C73C2311yescharacteristic
107C4⋊C8×C73—22no—
108#103×C73—42no—
109C23×D73C2×D411yescharacteristic
110C2×Dic146—21no—
111C2×Dic146—21no—
112C2×Dic146—21no—
113C2×Dic146—21no—
114C2×Dic146—21no—
115C2×Dic146—24no—
116C2×Dic146—42no—
117C2×Dic146—42no—
118C2×Dic146C22×C412yes—
119C22×Dic73—21no—
120C22×Dic73—21no—
121C22×Dic73—22no—
122C22×Dic73C2×D411yescharacteristic
123C22×Dic73C2×D411yescharacteristic
124C22×Dic73C2×D411yescharacteristic
125C2×C4×D73—22no—
126C2×C4×D73—22no—
127C2×C4×D73—21no—
128C2×C4×D73—21no—
129C2×C4×D73—21no—
130C2×C4×D73—21no—
131C2×C4×D73—41no—
132C2×C4×D73—41no—
133C2×C4×D73C22⋊C414yes—
134C2×C4×D73C22⋊C414yes—
135C2×C4×D73C22×C412yes—
136C2×C4×D73C2×D412yes—
137C2×#10596831—22no—
138C2×#10596831—22no—
139C2×#10596831—22no—
140C4⋊Dic73—21no—
141C4⋊Dic73—24no—
142C4⋊Dic73—22no—
143C4⋊Dic73—22no—
144C4⋊Dic73—42no—
145C4⋊Dic73—42no—
146C4⋊Dic73C22×C412yes—
147#11924007—24no—
148#11924007—24no—
149#11924007—24no—
150#11924007—24no—
151#11924007—24no—
152#11924005—24no—
153#11924005—24no—
154#11924005—44no—
155C4×Dic73—24no—
156C4×Dic73—44no—
157C4×Dic73C22⋊C414yes—
158C4×Dic73C22⋊C414yes—
159C4×Dic73C22×C412yes—
160C4×Dic73C2×D412yes—
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73—21no—
164C2×Q8×C73—21no—
165C2×Q8×C73—21no—
166C2×Q8×C73C22×C412yes—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—22no—
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2411yescharacteristic
173C2×C8×C73—22no—
174C2×C8×C73—22no—
175C2×C8×C73—22no—
176C4⋊C4×C73—24no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—22no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—42no—
181C4⋊C4×C73—42no—
182C4⋊C4×C73C22×C412yes—
183C42×C73—24no—
184C42×C73C22×C412yes—
185C42×C73C2×D412yes—
186C23×C73C22×D411yescharacteristic
187C22×D73—21no—
188C22×D73—21no—
189C22×D73—21no—
190C22×D73—21no—
191C22×D73—21no—
192C22×D73—21no—
193C22×D73C2×C22⋊C412yes—
194Q8×C73—22no—
195Q8×C73—22no—
196Q8×C73—41no—
197Q8×C73—41no—
198C2×C4×C73—21no—
199C2×C4×C73—21no—
200C2×C4×C73—22no—
201C2×C4×C73—22no—
202C2×C4×C73—41no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73—42no—
207C2×C4×C73—42no—
208C2×C4×C73—42no—
209C2×C4×C73C2×C22⋊C412yes—
210C2×C4×C73C2×C22⋊C412yes—
211C2×C4×C73#13712yes—
212C2×C4×C73#13712yes—
213C2×C4×C73C23×C411yescharacteristic
214C2×C4×C73C22×D411yescharacteristic
215C2×C4×C73H(5,𝔽2)11yescharacteristic
216C2×C4×C73H(5,𝔽2)11yescharacteristic
217C2×C4×C732−1+411yescharacteristic
218C2×C4×C732−1+411yescharacteristic
219C2×Dic73—21no—
220C2×Dic73—21no—
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—
237C4×D73—22no—
238C4×D73—22no—
239C4×D73—22no—
240C4×D73—22no—
241C4×D73—28no—
242C4×D73—44no—
243Dic146—22no—
244Dic146—22no—
245Dic146—48no—
246Dic146—41no—
247Dic146—41no—
248C8×C73—42no—
249#10596831—42no—
250C22×C73C22×C22⋊C411yescharacteristic
251C22×C73#53511yescharacteristic
252C22×C73#53611yescharacteristic
253C22×C73C2×#13711yescharacteristic
254C22×C73C2×#13711yescharacteristic
255C22×C73#57511yescharacteristic
256C22×C73#57511yescharacteristic
257C2×D73—22no—
258C2×D73—22no—
259C2×D73—22no—
260C2×D73—41no—
261C2×D73—41no—
262C2×D73—41no—
263C2×D73—41no—
264C4×C73—22no—
265C4×C73—42no—
266C4×C73—42no—
267C4×C73—82no—
268C4×C73#41914yes—
269C4×C73#52011yescharacteristic
270C4×C73#52011yescharacteristic
271C4×C73#53711yescharacteristic
272C4×C73#53711yescharacteristic
273Dic73—22no—
274Dic73—22no—
275Dic73—22no—
276Dic73—42no—
277Dic73—42no—
278Dic73—82no—
279#21870—731nomaximal, Sylow
280C2×C73#231111yescharacteristic
281C2×C73C2×#41911yescharacteristic
282C2×C73#289611yescharacteristic
283C2×C73#321611yescharacteristic
284C2×C73#321611yescharacteristic
285C2×C73#323711yescharacteristic
286C2×C73#323711yescharacteristic
287D73—42no—
288C2×#533—731no—
289C2×#432—731no—
290#2285—731no—
291#1993—732no—
292#1960—732no—
293#1873—732no—
294#1873—732no—
295#1803—731no—
296#1803—731no—
297#1803—731no—
298#1803D7311yescharacteristic, complemented
299C73#2187011yescharacteristic, complemented, Sylow
300#533—738no—
301#533—1464no—
302C2×C4⋊Q8—731no—
303C2×C4⋊Q8—731no—
304C2×#125—1461no—
305C2×C22⋊Q8—731no—
306C2×C22⋊Q8—731no—
307C2×C22⋊Q8—1461no—
308C2×C4×Q8—731no—
309C2×C4×Q8—1461no—
310C2×C4×Q8C2×D7311yescharacteristic
311C2×#117—732no—
312C2×#117—1461no—
313C22×C42—731no—
314#432—1464no—
315C2×C4⋊C8—731no—
316C2×C4⋊C8C2×D7311yescharacteristic
317C2×#103—732no—
318C2×#103—732no—
319C2×#103—732no—
320C2×#103C2×D7312yes—
321C2×C22⋊C8—732no—
322C2×#97—732no—
323#384—731no—
324#384C2×D7311yescharacteristic
325#382—732no—
326#380—1461no—
327#380—1461no—
328C4×C4⋊C4—1461no—
329C4×C4⋊C4—1461no—
330#340—732no—
331#340—732no—
332#340—732no—
333#340C2×D7312yes—
334C22×Q8—731no—
335C22×Q8—1461no—
336C22×Q8C22×D7311yescharacteristic
337C23×C4—731no—
338C23×C4—731no—
339C23×C4—731no—
340C22×C8—732no—
341C22×C8C22×D7312yes—
342C4⋊Q8—734no—
343C4⋊Q8—734no—
344C4⋊Q8—1462no—
345C4⋊Q8—1462no—
346#125—1468no—
347C22⋊Q8—734no—
348C22⋊Q8—734no—
349C22⋊Q8—1462no—
350C22⋊Q8—1468no—
351C22⋊Q8—1462no—
352C4×Q8—22no—
353C4×Q8—734no—
354C4×Q8—1468no—
355C4×Q8—1462no—
356C4×Q8C4×D7314yes—
357#117—2922no—
358#117—2922no—
359#117—1468no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—731no—
364C2×C4⋊C4—732no—
365C2×C4⋊C4—1461no—
366C2×C4⋊C4—1461no—
367C2×C4⋊C4—1461no—
368C2×C4⋊C4—1461no—
369C2×C4⋊C4—1461no—
370C2×C4⋊C4—1461no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4C22×D7312yes—
374C2×C4⋊C4C22×D7311yescharacteristic
375C2×C22⋊C4—1462no—
376C2×C22⋊C4—1461no—
377C2×C22⋊C4—1461no—
378C2×C42—21no—
379C2×C42—731no—
380C2×C42—731no—
381C2×C42—731no—
382C2×C42—734no—
383C2×C42—734no—
384C2×C42—1462no—
385C2×C42—1461no—
386C2×C42C22×D7311yescharacteristic
387C4⋊C8—22no—
388C4⋊C8—1462no—
389#103—42no—
390#103—2922no—
391#103—2922no—
392#103—2922no—
393C22⋊C8—1464no—
394#97—1464no—
395#95—1462no—
396#95—1462no—
397C24—731no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—2922no—
404C2×Q8—2922no—
405C2×Q8—732no—
406C2×Q8—1464no—
407C2×Q8—1461no—
408C2×Q8—1461no—
409C2×Q8—1461no—
410C2×Q8—1461no—
411C2×Q8—1461no—
412C2×Q8C2×C4×D7312yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—2921no—
417C22×C4—2921no—
418C22×C4—734no—
419C22×C4—731no—
420C22×C4—731no—
421C22×C4—731no—
422C22×C4—734no—
423C22×C4—732no—
424C22×C4—732no—
425C22×C4—1462no—
426C22×C4—1461no—
427C22×C4—1461no—
428C22×C4—1462no—
429C22×C4—1462no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4D4×D7311yescharacteristic
435C22×C4D4×D7311yescharacteristic
436C22×C4C23×D7311yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—1462no—
441C2×C8—1462no—
442C2×C8—1462no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—2922no—
450C4⋊C4—2922no—
451C4⋊C4—732no—
452C4⋊C4—1464no—
453C4⋊C4—1464no—
454C4⋊C4—1464no—
455C4⋊C4—1464no—
456C4⋊C4—1464no—
457C4⋊C4—1462no—
458C4⋊C4—1462no—
459C4⋊C4—1464no—
460C4⋊C4—1461no—
461C4⋊C4C2×C4×D7312yes—
462C22⋊C4—2924no—
463C22⋊C4—1464no—
464C22⋊C4—1464no—
465C42—24no—
466C42—2924no—
467C42—734no—
468C42—734no—
469C42—732no—
470C42—732no—
471C42—1464no—
472C42C2×C4×D7312yes—
473C42D4×D7312yes—
474C23—732no—
475C23—1461no—
476C23—1461no—
477C23—1461no—
478C23—1461no—
479C23—1461no—
480C23—1461no—
481C23C2×D4×D7311yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—41no—
485Q8—41no—
486Q8—2928no—
487Q8—2921no—
488Q8—2921no—
489Q8—1462no—
490Q8—1462no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—2924no—
503C2×C4—2921no—
504C2×C4—2921no—
505C2×C4—2921no—
506C2×C4—2921no—
507C2×C4—2922no—
508C2×C4—2922no—
509C2×C4—2922no—
510C2×C4—732no—
511C2×C4—732no—
512C2×C4—732no—
513C2×C4—1462no—
514C2×C4—1462no—
515C2×C4—1461no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4—1461no—
521C2×C4—1462no—
522C2×C4—1462no—
523C2×C4—1462no—
524C2×C4—1462no—
525C2×C4—1468no—
526C2×C4C22⋊C4×D7312yes—
527C2×C4C22⋊C4×D7312yes—
528C2×C4#1455662012yes—
529C2×C4#1455662312yes—
530C2×C4C22×C4×D7311yescharacteristic
531C2×C4C2×D4×D7311yescharacteristic
532C2×C4#1455663611yescharacteristic
533C2×C4#1455663711yescharacteristic
534C2×C4#1455663811yescharacteristic
535C2×C4#1455663911yescharacteristic
536C8—42no—
537C8—2922no—
538C22—2921no—
539C22—2921no—
540C22—2921no—
541C22—2921no—
542C22—1462no—
543C22—1462no—
544C22—1462no—
545C22C2×C22⋊C4×D7311yescharacteristic
546C22#1798915311yescharacteristic
547C22#1798916011yescharacteristic
548C22C2×#1455662011yescharacteristic
549C22C2×#1455662311yescharacteristic
550C22#1798938311yescharacteristic
551C22#1798938611yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—2922no—
557C4—2922no—
558C4—5842no—
559C4—1462no—
560C4—1462no—
561C4—1462no—
562C4#1798859414yes—
563C4#1798906711yescharacteristic
564C4#1798906811yescharacteristic
565C4#1798916611yescharacteristic
566C4#1798916711yescharacteristic
567C2—2922no—
568C2#2261252011yescharacteristic
569C2C2×#1798859411yescharacteristic
570C2#2261844711yescharacteristic
571C2#2262094011yescharacteristic
572C2#2262094111yescharacteristic
573C2#2262114311yescharacteristic
574C2#2262114411yescharacteristic
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#21870
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. #22610902C2semidirect
5. #22610749C2semidirect
8. #22609937C2semidirect
9. #22609936C2semidirect
10. #22609933C2semidirect
11. C73×#1803C2semidirect
13. #17989141C4semidirect
298. #1803D73semidirect
299. C73#21870semidirect

Maximal subgroups

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

subgroupindex
C2×#179891412
C2×#179886432
#226123362
#226109022
#226107492
#226102422
#226102402
#226099372
#226099362
#226099332
C73×#18032
#2187073