Finite Group Databasebeta

← #33095665

Subgroups — #33095665

Order 18688.

1222 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989141C211yesmaximal, characteristic
2C2×#17988639C211yesmaximal, characteristic
3#384×D73C211yesmaximal, characteristic
4#22610890C212yesmaximal, complemented
5#22610731C212yesmaximal, complemented
6#22610237C212yesmaximal
7#22610235C212yesmaximal
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×#14556548—21no—
23C2×C4⋊C4×D73—21no—
24C2×C4⋊C4×D73—21no—
25C2×C4⋊C4×D73C2212yes—
26C2×C42×D73C2211yescharacteristic
27#17988639—24no—
28C2×C4⋊#10596831C2211yescharacteristic
29C2×#14556516C2212yes—
30C2×#14556515C2212yes—
31C2×#14556514C2212yes—
32C2×#14556503C2212yes—
33C2×C8×Dic73C2212yes—
34#17988457C2211yescharacteristic
35#17988456—21no—
36#17988456—21no—
37#17988456C2212yes—
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×D73—22no—
49C22×C4×D73D411yescharacteristic
50C22×C4×D73D411yescharacteristic
51C22×C4×D73C2311yescharacteristic
52C22×#10596831C2312yes—
53#14556590—22no—
54#14556590C2×C414yescomplemented
55#14556589—22no—
56#14556589C2×C414yescomplemented
57#14556581—28no—
58#14556571—22no—
59#14556571C2×C414yes—
60#14556569—28no—
61#14556567—22no—
62#14556567C2×C414yes—
63C4×Dic146—28no—
64C4×Dic146—22no—
65C4×Dic146C2×C414yescomplemented
66#14556548—28no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73C2312yes—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—22no—
77C2×#11924007—22no—
78C2×#11924007—22no—
79C2×#11924007—21no—
80C4⋊C4×D73—42no—
81C4⋊C4×D73—42no—
82C4⋊C4×D73—44no—
83C4⋊C4×D73—44no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73D411yescharacteristic
88C2×C4×Dic73D411yescharacteristic
89C2×C4×Dic73C2311yescharacteristic
90C42×D73C2×C414yes—
91C42×D73D414yes—
92C4⋊#10596831—22no—
93#14556516—42no—
94#14556515—42no—
95#14556514—42no—
96#14556503—24no—
97C8×Dic73—24no—
98C22×Q8×C73C2311yescharacteristic
99C22×C8×C73C2312yes—
100C4×Q8×C73—22no—
101C4×Q8×C73C2×C414yescomplemented
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73C2312yes—
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C42×C73—21no—
108C2×C42×C73C2311yescharacteristic
109C4⋊C8×C73—22no—
110#103×C73—42no—
111C23×D73C2×D411yescharacteristic
112C2×Dic146—21no—
113C2×Dic146—21no—
114C2×Dic146—21no—
115C2×Dic146—21no—
116C2×Dic146—21no—
117C2×Dic146—24no—
118C2×Dic146—42no—
119C2×Dic146—42no—
120C2×Dic146C22×C412yes—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—22no—
124C22×Dic73C2×D411yescharacteristic
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C2×C4×D73—21no—
128C2×C4×D73—21no—
129C2×C4×D73—21no—
130C2×C4×D73—21no—
131C2×C4×D73—22no—
132C2×C4×D73—22no—
133C2×C4×D73—44no—
134C2×C4×D73—44no—
135C2×C4×D73—44no—
136C2×C4×D73C22⋊C414yes—
137C2×C4×D73C22⋊C414yes—
138C2×C4×D73SD1614yes—
139C2×C4×D73Q1614yes—
140C2×C4×D73C22×C412yes—
141C2×C4×D73C2×D412yes—
142C2×#10596831—22no—
143C2×#10596831—22no—
144C2×#10596831—22no—
145C4⋊Dic73—21no—
146C4⋊Dic73—24no—
147C4⋊Dic73—22no—
148C4⋊Dic73—22no—
149C4⋊Dic73—42no—
150C4⋊Dic73—42no—
151C4⋊Dic73C22×C412yes—
152#11924007—24no—
153#11924007—24no—
154#11924007—24no—
155#11924007—24no—
156#11924007—24no—
157#11924007—44no—
158#11924007—44no—
159#11924007—44no—
160#11924005—24no—
161#11924005—24no—
162C4×Dic73—24no—
163C4×Dic73C22⋊C414yes—
164C4×Dic73C22⋊C414yes—
165C4×Dic73C22×C412yes—
166C4×Dic73C2×D412yes—
167C2×Q8×C73—21no—
168C2×Q8×C73—21no—
169C2×Q8×C73—21no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73C22×C412yes—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—22no—
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2411yescharacteristic
179C2×C8×C73—22no—
180C2×C8×C73—22no—
181C2×C8×C73—22no—
182C4⋊C4×C73—24no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—22no—
185C4⋊C4×C73—22no—
186C4⋊C4×C73—42no—
187C4⋊C4×C73—42no—
188C4⋊C4×C73C22×C412yes—
189C42×C73—24no—
190C42×C73C22×C412yes—
191C42×C73C2×D412yes—
192C23×C73C22×D411yescharacteristic
193C22×D73—21no—
194C22×D73—21no—
195C22×D73—21no—
196C22×D73—21no—
197C22×D73C2×C22⋊C412yes—
198C22×D73C2×SD1612yes—
199C22×D73C2×Q1612yes—
200Q8×C73—22no—
201Q8×C73—22no—
202Q8×C73—41no—
203Q8×C73—41no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—41no—
209C2×C4×C73—41no—
210C2×C4×C73—41no—
211C2×C4×C73—41no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73—42no—
215C2×C4×C73C2×C22⋊C412yes—
216C2×C4×C73C2×C22⋊C412yes—
217C2×C4×C73C2×SD1612yes—
218C2×C4×C73C2×Q1612yes—
219C2×C4×C73C23×C411yescharacteristic
220C2×C4×C73C22×D411yescharacteristic
221C2×C4×C73H(5,𝔽2)11yescharacteristic
222C2×C4×C73H(5,𝔽2)11yescharacteristic
223C2×C4×C732−1+411yescharacteristic
224C2×C4×C732−1+411yescharacteristic
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—22no—
228C2×Dic73—22no—
229C2×Dic73—41no—
230C2×Dic73—41no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—42no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73C2×C22⋊C412yes—
237C2×Dic73C2×C22⋊C412yes—
238C2×Dic73C2×C22⋊C412yes—
239C2×Dic73C2×SD1612yes—
240C2×Dic73C2×Q1612yes—
241C2×Dic73AGL(1,ℤ/8)12yes—
242C2×Dic73#13712yes—
243C4×D73—22no—
244C4×D73—22no—
245C4×D73—22no—
246C4×D73—22no—
247C4×D73—28no—
248C4×D73—88no—
249C4×D73#103116yescomplemented
250Dic146—22no—
251Dic146—22no—
252Dic146—41no—
253Dic146—41no—
254Dic146—48no—
255C8×C73—42no—
256#10596831—42no—
257C22×C73C22×C22⋊C411yescharacteristic
258C22×C73#53511yescharacteristic
259C22×C73#53611yescharacteristic
260C22×C73C22×SD1611yescharacteristic
261C22×C73C22×Q1611yescharacteristic
262C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C73C2×#13711yescharacteristic
264C2×D73—22no—
265C2×D73—22no—
266C2×D73—22no—
267C2×D73—22no—
268C2×D73—22no—
269C2×D73—22no—
270C2×D73C2×#10314yes—
271C4×C73—22no—
272C4×C73—42no—
273C4×C73—42no—
274C4×C73—82no—
275C4×C73C2×#10314yes—
276C4×C73#52011yescharacteristic
277C4×C73#52011yescharacteristic
278C4×C73#53711yescharacteristic
279C4×C73#53711yescharacteristic
280Dic73—22no—
281Dic73—42no—
282Dic73—42no—
283Dic73—82no—
284Dic73C2×#10314yes—
285Dic73#41714yes—
286#21860—731nomaximal, Sylow
287C2×C73#231111yescharacteristic
288C2×C73C22×#10311yescharacteristic
289C2×C73C2×#41711yescharacteristic
290C2×C73#320811yescharacteristic
291C2×C73#320911yescharacteristic
292C2×C73#322911yescharacteristic
293C2×C73#323011yescharacteristic
294D73—24no—
295C2×#533—731no—
296C2×#431—731no—
297C2×#384—731no—
298#1990—732no—
299#1956—732no—
300#1872—732no—
301#1872—732no—
302#1803—731no—
303#1803—731no—
304#1803—731no—
305#1803D7311yescharacteristic, complemented
306C73#2186011yescharacteristic, complemented, Sylow
307#533—738no—
308#533—1464no—
309C2×C4⋊Q8—731no—
310C2×C4⋊Q8—731no—
311C2×#125—1461no—
312C2×C22⋊Q8—731no—
313C2×C22⋊Q8—731no—
314C2×C22⋊Q8—1461no—
315C2×C4×Q8—731no—
316C2×C4×Q8—1461no—
317C2×C4×Q8C2×D7311yescharacteristic
318C2×#117—1461no—
319C22×C4⋊C4—732no—
320C22×C4⋊C4—1461no—
321C22×C4⋊C4—1461no—
322C22×C42—731no—
323#431—1464no—
324C2×C4⋊C8—731no—
325C2×C4⋊C8C2×D7311yescharacteristic
326C2×#103—732no—
327C2×#103—732no—
328C2×#103—732no—
329C2×#103C2×D7312yes—
330C2×C22⋊C8—732no—
331C2×C4×C8—732no—
332#384—732no—
333#384—731no—
334#384—1461no—
335#384—1461no—
336#384C2×D7311yescharacteristic
337#340—732no—
338#340—732no—
339#340—732no—
340#340C2×D7312yes—
341C22×Q8—731no—
342C22×Q8—1461no—
343C22×Q8C22×D7311yescharacteristic
344C23×C4—731no—
345C23×C4—731no—
346C23×C4—731no—
347C23×C4—1462no—
348C22×C8—732no—
349C22×C8C22×D7312yes—
350C4⋊Q8—734no—
351C4⋊Q8—734no—
352C4⋊Q8—1462no—
353C4⋊Q8—1462no—
354#125—1468no—
355C22⋊Q8—734no—
356C22⋊Q8—734no—
357C22⋊Q8—1462no—
358C22⋊Q8—1468no—
359C22⋊Q8—1462no—
360C4×Q8—22no—
361C4×Q8—734no—
362C4×Q8—1468no—
363C4×Q8—1462no—
364C4×Q8C4×D7314yescomplemented
365#117—1468no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—2922no—
370C2×C4⋊C4—2922no—
371C2×C4⋊C4—2924no—
372C2×C4⋊C4—2924no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—732no—
375C2×C4⋊C4—1462no—
376C2×C4⋊C4—1462no—
377C2×C4⋊C4—1462no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4C22×D7312yes—
387C2×C4⋊C4C22×D7311yescharacteristic
388C2×C22⋊C4—1461no—
389C2×C22⋊C4—1461no—
390C2×C42—21no—
391C2×C42—731no—
392C2×C42—731no—
393C2×C42—731no—
394C2×C42—734no—
395C2×C42—734no—
396C2×C42—1461no—
397C2×C42C22×D7311yescharacteristic
398C4⋊C8—22no—
399C4⋊C8—1462no—
400#103—42no—
401#103—2922no—
402#103—2922no—
403#103—2922no—
404C22⋊C8—1464no—
405C4×C8—1464no—
406C24—731no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—2922no—
413C2×Q8—2922no—
414C2×Q8—732no—
415C2×Q8—1464no—
416C2×Q8—1461no—
417C2×Q8—1461no—
418C2×Q8—1461no—
419C2×Q8—1461no—
420C2×Q8—1461no—
421C2×Q8C2×C4×D7312yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—2924no—
426C22×C4—2924no—
427C22×C4—2924no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—734no—
431C22×C4—734no—
432C22×C4—734no—
433C22×C4—731no—
434C22×C4—734no—
435C22×C4—732no—
436C22×C4—732no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1461no—
441C22×C4—1461no—
442C22×C4—1461no—
443C22×C4—1461no—
444C22×C4—1461no—
445C22×C4—1461no—
446C22×C4D4×D7311yescharacteristic
447C22×C4D4×D7311yescharacteristic
448C22×C4C23×D7311yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—1462no—
453C2×C8—1462no—
454C2×C8—1462no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—2922no—
462C4⋊C4—2924no—
463C4⋊C4—2924no—
464C4⋊C4—2924no—
465C4⋊C4—2922no—
466C4⋊C4—732no—
467C4⋊C4—1464no—
468C4⋊C4—1464no—
469C4⋊C4—1464no—
470C4⋊C4—1464no—
471C4⋊C4—1464no—
472C4⋊C4—1462no—
473C4⋊C4—1462no—
474C4⋊C4—1464no—
475C4⋊C4—1461no—
476C4⋊C4C2×C4×D7312yes—
477C22⋊C4—1464no—
478C22⋊C4—1464no—
479C42—24no—
480C42—734no—
481C42—734no—
482C42—732no—
483C42—732no—
484C42—1464no—
485C42C2×C4×D7312yes—
486C42D4×D7312yes—
487C23—732no—
488C23—732no—
489C23—732no—
490C23—1461no—
491C23—1461no—
492C23—1461no—
493C23—1461no—
494C23C2×D4×D7311yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—41no—
498Q8—41no—
499Q8—2928no—
500Q8—2921no—
501Q8—2921no—
502Q8—1462no—
503Q8—1462no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2922no—
520C2×C4—2922no—
521C2×C4—2922no—
522C2×C4—5848no—
523C2×C4—732no—
524C2×C4—732no—
525C2×C4—732no—
526C2×C4—732no—
527C2×C4—732no—
528C2×C4—732no—
529C2×C4—7316no—
530C2×C4—732no—
531C2×C4—1468no—
532C2×C4—1462no—
533C2×C4—1462no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1462no—
537C2×C4—1462no—
538C2×C4—1462no—
539C2×C4—1462no—
540C2×C4C22⋊C4×D7312yes—
541C2×C4C22⋊C4×D7312yes—
542C2×C4SD16×D7312yes—
543C2×C4Q16×D7312yes—
544C2×C4C22×C4×D7311yescharacteristic
545C2×C4C2×D4×D7311yescharacteristic
546C2×C4#1455663611yescharacteristic
547C2×C4#1455663711yescharacteristic
548C2×C4#1455663811yescharacteristic
549C2×C4#1455663911yescharacteristic
550C8—42no—
551C8—2922no—
552C22—734no—
553C22—1462no—
554C22—1462no—
555C22—1462no—
556C22—1462no—
557C22—1462no—
558C22—1462no—
559C22C2×C22⋊C4×D7311yescharacteristic
560C22#1798915311yescharacteristic
561C22#1798916011yescharacteristic
562C22C2×SD16×D7311yescharacteristic
563C22C2×Q16×D7311yescharacteristic
564C22AGL(1,ℤ/8)×D7311yescharacteristic
565C22#137×D7311yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—2922no—
571C4—2922no—
572C4—5842no—
573C4—734no—
574C4—734no—
575C4—1462no—
576C4#103×D7314yes—
577C4#1798906711yescharacteristic
578C4#1798906811yescharacteristic
579C4#1798916611yescharacteristic
580C4#1798916711yescharacteristic
581C2—1464no—
582C2#2261252011yescharacteristic
583C2C2×#103×D7311yescharacteristic
584C2#417×D7311yescharacteristic
585C2#2262087111yescharacteristic
586C2#2262087811yescharacteristic
587C2#2262107411yescharacteristic
588C2#2262108111yescharacteristic
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#21860
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. #22610890C2semidirect
5. #22610731C2semidirect
8. #22609937C2semidirect
9. #22609936C2semidirect
10. #22609933C2semidirect
11. C73×#1803C2semidirect
13. #17989141C4semidirect
54. #14556590C2×C4semidirect
56. #14556589C2×C4semidirect
65. C4×Dic146C2×C4semidirect
101. C4×Q8×C73C2×C4semidirect
249. C4×D73#103semidirect
305. #1803D73semidirect
306. C73#21860semidirect
364. C4×Q8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179891412
C2×#179886392
#384×D732
#226108902
#226107312
#226102372
#226102352
#226099372
#226099362
#226099332
C73×#18032
#2186073