Finite Group Databasebeta

← #33051909

Subgroups — #33051909

Order 18688.

1206 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×#17989066C211yesmaximal, characteristic
2#22618970C212yesmaximal, complemented
3#22608269C211yesmaximal, characteristic, complemented
4#22608266C212yesmaximal, complemented
5#22608260C212yesmaximal, complemented
6#22608259C212yesmaximal, complemented
7#22608253C212yesmaximal, complemented
8#22608242C212yesmaximal, complemented
9C73×#1465C211yesmaximal, characteristic, complemented
10C22×#11924027C411yescharacteristic
11#17989066—24no—
12#17989066—24no—
13C2×C4×Dic146C411yescharacteristic
14C2×#14556556C2211yescharacteristic
15C2×D4×Dic73C411yescharacteristic
16C2×C4×C22⋊D73C411yescharacteristic
17C2×C4×C22⋊D73C2211yescharacteristic
18C2×#14556551C411yescharacteristic
19C2×#14556551C2211yescharacteristic
20C2×#14556548C411yescharacteristic
21C2×#14556547C411yescharacteristic
22C2×#14556547C2211yescharacteristic
23C22×#11924007C411yescharacteristic
24C22×#11924007C2211yescharacteristic
25C2×C4⋊C4×D73C2211yescharacteristic
26#17988663—24no—
27#17988662—24no—
28#17988637C2214yes—
29#17988551C2212yes—
30#17988549C2214yes—
31#17988313C2212yes—
32#17988312C2214yes—
33#17988312C2214yes—
34#17988311C2212yes—
35C2×C4×D4×C73C2211yescharacteristic
36#407×C73C2212yes—
37#324×C73C2212yes—
38#323×C73C2212yes—
39C2×#11924027—24no—
40C2×#11924027—24no—
41C2×#11924027—24no—
42C22×Dic146C2×C411yescharacteristic
43C22×C22⋊D73C2×C411yescharacteristic
44C22×C22⋊D73C2×C411yescharacteristic
45C23×Dic73C2×C411yescharacteristic
46C23×Dic73C2×C411yescharacteristic
47C23×Dic73C2×C411yescharacteristic
48C23×Dic73C2×C411yescharacteristic
49C22×C4×D73C2×C411yescharacteristic
50C22×C4×D73C2×C411yescharacteristic
51C22×C4×D73C2311yescharacteristic
52C2×#11924012—22no—
53C2×#11924011—24no—
54C4×Dic146—22no—
55C4×Dic146—22no—
56#14556556—24no—
57D4×Dic73—24no—
58C4×C22⋊D73—22no—
59C4×C22⋊D73—22no—
60C4×C22⋊D73D414yes—
61C4×C22⋊D73D414yes—
62#14556551—24no—
63#14556551—24no—
64#14556548—22no—
65#14556548—22no—
66#14556547—24no—
67#14556547—24no—
68C2×C4⋊Dic73C2×C411yescharacteristic
69C2×#11924007—22no—
70C2×#11924007—22no—
71C2×#11924007—22no—
72C2×#11924007—22no—
73C2×#11924007C2×C411yescharacteristic
74C2×#11924007C2×C411yescharacteristic
75C2×#11924007C2×C411yescharacteristic
76C2×#11924007C2×C411yescharacteristic
77C2×#11924007C2×C411yescharacteristic
78C2×#11924007C2×C411yescharacteristic
79C2×#11924007C2311yescharacteristic
80C2×#11924007C2311yescharacteristic
81C4⋊C4×D73—24no—
82C2×C22⋊Dic73C2×C411yescharacteristic
83C2×C22⋊Dic73C2×C411yescharacteristic
84C2×#11924005C2×C411yescharacteristic
85C2×#11924005C2×C411yescharacteristic
86C2×C4×Dic73C2×C411yescharacteristic
87C2×C4×Dic73C2×C411yescharacteristic
88C2×C4×Dic73C2×C411yescharacteristic
89C2×C4×Dic73C2×C411yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91#14556520—24no—
92C22⋊#10596831—22no—
93C22⋊#10596831C2314yes—
94#14556503—24no—
95C8×Dic73—24no—
96C22×D4×C73C2×C411yescharacteristic
97C23×C4×C73C2×C411yescharacteristic
98C23×C4×C73C2311yescharacteristic
99C2×M16×C73—22no—
100C4×D4×C73—22no—
101C4×D4×C73—22no—
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C22⋊C4×C73C2×C411yescharacteristic
104C2×C22⋊C4×C73C2311yescharacteristic
105C2×C42×C73C2×C411yescharacteristic
106C22⋊C8×C73—22no—
107C22⋊C8×C73C2314yes—
108C23×D73C22×C411yescharacteristic
109#11924027—232no—
110C2×Dic146—22no—
111C2×Dic146—22no—
112C2×Dic146M1614yescomplemented
113C2×C22⋊D73—22no—
114C2×C22⋊D73—22no—
115C2×C22⋊D73—22no—
116C2×C22⋊D73—22no—
117C2×C22⋊D73—22no—
118C2×C22⋊D73C22⋊C414yes—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—81no—
127C22×Dic73—81no—
128C22×Dic73M1614yescomplemented
129C22×Dic73C22×C411yescharacteristic
130C22×Dic73C22×C411yescharacteristic
131C22×Dic73C22×C411yescharacteristic
132C22×Dic73C22×C411yescharacteristic
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C2×C4×D73—22no—
137C2×C4×D73—22no—
138C2×C4×D73—22no—
139C2×C4×D73—22no—
140C2×C4×D73—81no—
141C2×C4×D73—81no—
142C2×C4×D73C2×D414yes—
143#11924012—82no—
144#11924011—84no—
145C2×#10596831—24no—
146C4⋊Dic73—22no—
147#11924007—21no—
148#11924007—21no—
149#11924007—21no—
150#11924007—21no—
151#11924007—22no—
152#11924007—22no—
153#11924007—22no—
154#11924007—22no—
155#11924007—22no—
156#11924007C22⋊C412yes—
157#11924007C22⋊C412yes—
158C22⋊Dic73—22no—
159C22⋊Dic73C22⋊C414yes—
160#11924005—22no—
161#11924005C22⋊C412yes—
162#11924005C22⋊C412yes—
163C4×Dic73—22no—
164C4×Dic73—22no—
165C4×Dic73—22no—
166C4×Dic73—22no—
167C4×Dic73C2×D414yes—
168C24×C73C22×C411yescharacteristic
169C24×C73C22×C411yescharacteristic
170C2×D4×C73—22no—
171C2×D4×C73—22no—
172C2×D4×C73—22no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—81no—
176C22×C4×C73—81no—
177C22×C4×C73C22×C411yescharacteristic
178C22×C4×C73C22×C411yescharacteristic
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C22×C411yescharacteristic
181C22×C4×C73C2×D412yes—
182C22×C4×C73C2×D412yes—
183C22×C4×C73C2411yescharacteristic
184M16×C73—82no—
185C2×C8×C73—24no—
186C4⋊C4×C73—22no—
187C22⋊C4×C73—22no—
188C22⋊C4×C73—22no—
189C42×C73—21no—
190C42×C73—21no—
191C23×C73—21no—
192C23×C73—21no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73C2×C22⋊C412yes—
200C23×C73C2×M1612yes—
201C23×C73C23×C411yescharacteristic
202C22×D73—21no—
203C22×D73—21no—
204C22×D73—41no—
205C22×D73—41no—
206C22×D73C2×C22⋊C412yes—
207D4×C73—28no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—42no—
213C2×C4×C73—41no—
214C2×C4×C73—42no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—81no—
219C2×C4×C73—81no—
220C2×C4×C73C2×C22⋊C411yescharacteristic
221C2×C4×C73C2×C22⋊C411yescharacteristic
222C2×C4×C73C2×M1612yes—
223C2×C4×C73C22×D412yes—
224C22⋊D73—28no—
225C22⋊D73—28no—
226C2×Dic73—28no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—28no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—82no—
239C2×Dic73—82no—
240C2×Dic73C2×C22⋊C412yes—
241C2×Dic73C2×C22⋊C412yes—
242C2×Dic73C2×C22⋊C412yes—
243C2×Dic73C2×M1612yes—
244C2×Dic73C2×M1612yes—
245C2×Dic73C2×M1612yes—
246C2×Dic73C2×M1612yes—
247C4×D73—28no—
248C4×D73—44no—
249C4×D73—82no—
250Dic146—28no—
251C8×C73—44no—
252#10596831—44no—
253C22×C73—21no—
254C22×C73—21no—
255C22×C73—22no—
256C22×C73—22no—
257C22×C73—22no—
258C22×C73—22no—
259C22×C73—42no—
260C22×C73—42no—
261C22×C73—42no—
262C22×C73—42no—
263C22×C73—41no—
264C22×C73—41no—
265C22×C73C22×C22⋊C411yescharacteristic
266C22×C73C22×M1611yescharacteristic
267C22×C73C22×M1611yescharacteristic
268C2×D73—22no—
269C2×D73—22no—
270C2×D73—22no—
271C2×D73—42no—
272C2×D73—42no—
273C4×C73—22no—
274C4×C73—22no—
275C4×C73—21no—
276C4×C73—21no—
277C4×C73—41no—
278C4×C73—41no—
279Dic73—22no—
280Dic73—22no—
281Dic73—22no—
282Dic73—22no—
283Dic73—41no—
284Dic73—41no—
285Dic73#40714yes—
286#17370—731nomaximal, Sylow
287C2×C73—21no—
288C2×C73—21no—
289C2×C73—42no—
290C2×C73—42no—
291C2×C73C2×#40711yescharacteristic
292C2×C73#288311yescharacteristic
293C2×C73#288911yescharacteristic
294D73—42no—
295C2×#520—731no—
296#2975—732no—
297#1465—732no—
298#1465—731no—
299#1465D7311yescharacteristic, complemented
300#1463—732no—
301#1463—732no—
302#1461—732no—
303#1458—732no—
304C73#1737011yescharacteristic, complemented, Sylow
305C22×#41—731no—
306#520—1464no—
307#520—1464no—
308C2×C4×Q8—731no—
309C2×C4×Q8—731no—
310C2×C4×D4—731no—
311C2×C4×D4—731no—
312C2×C4×D4—731no—
313C2×C4×D4—731no—
314C2×C4×D4—731no—
315C2×C4×D4C2×D7311yescharacteristic
316C2×#117—731no—
317C2×#117—731no—
318C2×#117—731no—
319C22×C4⋊C4—731no—
320C22×C4⋊C4—731no—
321C22×C4⋊C4—731no—
322#436—1464no—
323#436—1464no—
324#431—734no—
325#407—732no—
326#407—734no—
327#407C2×D7312yes—
328#324—732no—
329#324—734no—
330#324—734no—
331#324C2×D7312yes—
332#323—732no—
333#323C2×D7312yes—
334C2×#41—1464no—
335C2×#41—1464no—
336C2×#41—1464no—
337C22×Q8—731no—
338C22×D4—731no—
339C22×D4—731no—
340C22×D4C4×D7311yescharacteristic
341C23×C4—731no—
342C23×C4—731no—
343C23×C4—731no—
344C23×C4—731no—
345C23×C4—731no—
346C23×C4—731no—
347C23×C4—731no—
348C23×C4C4×D7311yescharacteristic
349C23×C4C22×D7311yescharacteristic
350C2×M16—22no—
351C2×M16—1464no—
352C2×M16—1462no—
353C4×Q8—1462no—
354C4×Q8—1462no—
355C4×Q8—1464no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—734no—
359C4×D4—734no—
360C4×D4—1464no—
361C4×D4—1464no—
362C4×D4—1462no—
363C4×D4—1462no—
364C4×D4—1464no—
365#117—1462no—
366#117—1462no—
367#117—1464no—
368#117—1464no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—1464no—
379C2×C4⋊C4—1462no—
380C2×C4⋊C4—1462no—
381C2×C4⋊C4—1462no—
382C2×C4⋊C4—1462no—
383C2×C4⋊C4C22×D7311yescharacteristic
384C2×C22⋊C4—731no—
385C2×C22⋊C4—731no—
386C2×C22⋊C4—731no—
387C2×C22⋊C4—731no—
388C2×C22⋊C4C4×D7311yescharacteristic
389C2×C22⋊C4C22×D7311yescharacteristic
390C2×C42—731no—
391C2×C42—731no—
392C2×C42—731no—
393C2×C42—731no—
394C2×C42—731no—
395C2×C42C4×D7311yescharacteristic
396C4⋊C8—1464no—
397C22⋊C8—22no—
398C22⋊C8—734no—
399C22⋊C8—1464no—
400C22⋊C8—1462no—
401C22⋊C8C22×D7314yes—
402C4×C8—1464no—
403C24—731no—
404C24C2×C4×D7311yescharacteristic
405C24C2×C4×D7311yescharacteristic
406#41—14632no—
407C2×Q8—734no—
408C2×Q8—1462no—
409C2×Q8—1462no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—734no—
414C2×D4—1462no—
415C2×D4—1462no—
416C2×D4—1462no—
417C2×D4—1462no—
418C2×D4—1462no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—5841no—
424C22×C4—5841no—
425C22×C4—5841no—
426C22×C4—5841no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—734no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—734no—
434C22×C4—731no—
435C22×C4—731no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4C2×C4×D7311yescharacteristic
448C22×C4C2×C4×D7311yescharacteristic
449C22×C4C2×C4×D7311yescharacteristic
450C22×C4C2×C4×D7311yescharacteristic
451C22×C4D4×D7312yes—
452C22×C4D4×D7312yes—
453C22×C4C23×D7311yescharacteristic
454M16—82no—
455M16—5844no—
456M16—5842no—
457C2×C8—24no—
458C2×C8—1464no—
459C4⋊C4—22no—
460C4⋊C4—732no—
461C4⋊C4—732no—
462C4⋊C4—1462no—
463C4⋊C4—1462no—
464C4⋊C4—1462no—
465C4⋊C4—1462no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1462no—
471C4⋊C4—1462no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—734no—
475C22⋊C4—732no—
476C22⋊C4—732no—
477C22⋊C4—1462no—
478C22⋊C4—1462no—
479C42—21no—
480C42—21no—
481C42—734no—
482C42—1462no—
483C42—1462no—
484C42—1462no—
485C42—1462no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—2921no—
495C23—2921no—
496C23—732no—
497C23—1461no—
498C23—1461no—
499C23C22⋊C4×D7312yes—
500C23M16×D7312yes—
501C23C22×C4×D7311yescharacteristic
502Q8—1468no—
503D4—28no—
504D4—1468no—
505D4—1468no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—2924no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—5842no—
524C2×C4—5842no—
525C2×C4—5842no—
526C2×C4—732no—
527C2×C4—732no—
528C2×C4—732no—
529C2×C4—732no—
530C2×C4—732no—
531C2×C4—732no—
532C2×C4—732no—
533C2×C4—1468no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1468no—
537C2×C4—1468no—
538C2×C4—1461no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4C22⋊C4×D7311yescharacteristic
543C2×C4C22⋊C4×D7311yescharacteristic
544C2×C4M16×D7312yes—
545C2×C4C2×D4×D7312yes—
546C8—44no—
547C8—2924no—
548C22—21no—
549C22—21no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—41no—
559C22—41no—
560C22—2922no—
561C22—2922no—
562C22—1462no—
563C22—1462no—
564C22—1462no—
565C22C2×C22⋊C4×D7311yescharacteristic
566C22C2×M16×D7311yescharacteristic
567C22C2×M16×D7311yescharacteristic
568C4—22no—
569C4—21no—
570C4—21no—
571C4—22no—
572C4—41no—
573C4—41no—
574C4—2921no—
575C4—2921no—
576C4—734no—
577C4—1462no—
578C4—1462no—
579C4—1462no—
580C4—1462no—
581C2—21no—
582C2—21no—
583C2—42no—
584C2—42no—
585C2—2922no—
586C2#407×D7311yescharacteristic
587C2#2261838911yescharacteristic
588C2#2261841211yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D73
commutatorC23×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1465C2
socleC22×C73C22×C22⋊C4
prime radicalC22×#11924027C4
cosocleC22×C4×C73C24
hypercenter#1465D73
hypocenterC73#17370
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #22618970C2semidirect
3. #22608269C2semidirect
4. #22608266C2semidirect
5. #22608260C2semidirect
6. #22608259C2semidirect
7. #22608253C2semidirect
8. #22608242C2semidirect
9. C73×#1465C2semidirect
112. C2×Dic146M16semidirect
128. C22×Dic73M16semidirect
299. #1465D73semidirect
304. C73#17370semidirect

Maximal subgroups

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

subgroupindex
C2×#179890662
#226189702
#226082692
#226082662
#226082602
#226082592
#226082532
#226082422
C73×#14652
#1737073