Finite Group Databasebeta

← #33086990

Subgroups — #33086990

Order 18688.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989068C211yesmaximal, characteristic
2C2×#17988627C211yesmaximal, characteristic, complemented
3C2×#17988586C211yesmaximal, characteristic
4C2×#17988584C211yesmaximal, characteristic
5#22612549C211yesmaximal, characteristic, complemented
6#22610200C212yesmaximal, complemented
7#22610193C212yesmaximal, complemented
8#22610186C212yesmaximal, complemented
9#22610179C212yesmaximal, complemented
10#22609627C211yesmaximal, characteristic, complemented
11C2×#340×C73C211yesmaximal, characteristic, complemented
12C22×#11924028C2211yescharacteristic
13C22×#11924012C2211yescharacteristic
14#17989068—24no—
15#17989068—24no—
16C2×#14556556C2212yes—
17C2×#14556553C2212yes—
18C2×C4×C22⋊D73—22no—
19C2×#14556550C2211yescharacteristic
20C2×#14556550C2211yescharacteristic
21C2×#14556549—21no—
22C2×C4⋊C4×D73—21no—
23#17988627—24no—
24C2×#14556524C2212yes—
25C2×#14556522C2212yes—
26#17988586C418yescomplemented
27#17988584—24no—
28C2×#14556516C2211yescharacteristic
29C2×#14556516C2211yescharacteristic
30C2×#14556514C2212yes—
31C2×#14556513C2211yescharacteristic
32C2×#14556513C2211yescharacteristic
33C2×#14556511C2212yes—
34#17988486—21no—
35#17988486—21no—
36#17988486C2212yes—
37#17988486C2212yes—
38#17988447—22no—
39#17988341C2212yes—
40#17988341C2212yes—
41C23×C8×C73C2211yescharacteristic
42C22×C4⋊C4×C73C2211yescharacteristic
43C22×C4⋊C4×C73C2211yescharacteristic
44#340×C73C2212yes—
45#340×C73C2212yes—
46C2×#11924028—22no—
47C2×#11924028—22no—
48C2×#11924028—22no—
49C2×#11924028C2×C414yes—
50C2×#11924028D414yes—
51C2×#11924028Q814yes—
52C22×Dic146C2311yescharacteristic
53C22×D292C2311yescharacteristic
54C22×C22⋊D73—21no—
55C22×C4×D73—22no—
56C22×C4×D73—21no—
57C2×#11924012—22no—
58C2×#11924012—22no—
59C2×#11924012C2×C414yescomplemented
60C22×#10596831C2312yes—
61#14556556—24no—
62#14556556—24no—
63#14556553—24no—
64#14556553—24no—
65C4×C22⋊D73—216no—
66#14556550—22no—
67#14556550—22no—
68#14556550—22no—
69#14556550—22no—
70#14556549—28no—
71C2×C4⋊Dic73—22no—
72C2×C4⋊Dic73—22no—
73C2×C4⋊Dic73D411yescharacteristic
74C2×C4⋊Dic73D411yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—22no—
78C4⋊C4×D73—28no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73D412yes—
81C2×#11924005—22no—
82C2×C4×Dic73—22no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73C2312yes—
85#14556524—24no—
86#14556522—24no—
87#14556516—22no—
88#14556516—22no—
89#14556514C2×C418yescomplemented
90#14556513—22no—
91#14556513—22no—
92#14556511C2×C418yescomplemented
93#14556498—22no—
94#14556498—22no—
95C23×C4×C73—21no—
96C23×C4×C73—21no—
97C23×C4×C73C2311yescharacteristic
98C22×C8×C73—22no—
99C22×C8×C73—22no—
100C22×C8×C73—22no—
101C22×C8×C73C2311yescharacteristic
102C22×C8×C73C2311yescharacteristic
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—22no—
106C2×C4⋊C4×C73—22no—
107C2×C4⋊C4×C73C2×C414yescomplemented
108C2×C4⋊C4×C73D414yes—
109C2×C4⋊C4×C73C2312yes—
110C2×C4⋊C4×C73C2312yes—
111C23×D73—21no—
112#11924028—28no—
113#11924028—28no—
114#11924028—28no—
115#11924028C4⋊C4116yescomplemented
116C2×Dic146—21no—
117C2×Dic146—21no—
118C2×Dic146—21no—
119C2×Dic146C22×C412yes—
120C2×Dic146C2×D412yes—
121C2×Dic146C2×Q812yes—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292C22×C412yes—
126C2×D292C2×D412yes—
127C2×D292C2×Q812yes—
128C2×C22⋊D73—24no—
129C2×C22⋊D73—24no—
130C2×C22⋊D73—24no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73C2×D412yes—
135C22×Dic73C2×D412yes—
136C2×C4×D73—24no—
137C2×C4×D73—24no—
138C2×C4×D73—24no—
139C2×C4×D73—28no—
140C2×C4×D73—82no—
141C2×C4×D73—82no—
142#11924012—28no—
143C2×#10596831—22no—
144C2×#10596831—22no—
145C2×#10596831C22×C414yes—
146C4⋊Dic73—22no—
147C4⋊Dic73—22no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—82no—
153C4⋊Dic73—82no—
154#11924007—28no—
155C22⋊Dic73—24no—
156C22⋊Dic73—42no—
157#11924005—28no—
158C4×Dic73—28no—
159C4×Dic73—22no—
160C4×Dic73—22no—
161C4×Dic73—42no—
162C24×C73C2×D411yescharacteristic
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—24no—
166C22×C4×C73—24no—
167C22×C4×C73—81no—
168C22×C4×C73—81no—
169C22×C4×C73—81no—
170C22×C4×C73—81no—
171C22×C4×C73C22×C412yes—
172C22×C4×C73C2×D412yes—
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D412yes—
175C22×C4×C73C2×Q812yes—
176C22×C4×C73#4112yes—
177C22×C4×C73#4112yes—
178C22×C4×C73C2411yescharacteristic
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—44no—
186C4⋊C4×C73—24no—
187C4⋊C4×C73—24no—
188C4⋊C4×C73—22no—
189C4⋊C4×C73—22no—
190C4⋊C4×C73C22×C414yes—
191C4⋊C4×C73C2×D414yes—
192C23×C73—21no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73C4×D412yes—
197C23×C73#12012yes—
198C23×C73C22⋊Q812yes—
199C23×C73C22×D411yescharacteristic
200C22×D73—22no—
201C22×D73—22no—
202C22×D73—22no—
203C22×D73—41no—
204C22×D73—41no—
205C22×D73—41no—
206C22×D73—41no—
207C2×C4×C73—24no—
208C2×C4×C73—24no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—84no—
214C2×C4×C73—84no—
215C2×C4×C73C2×C4⋊C414yes—
216C2×C4×C73C4×D412yes—
217C2×C4×C73C4×D414yes—
218C2×C4×C73#12012yes—
219C2×C4×C73C22⋊Q812yes—
220C2×C4×C73C22⋊Q814yes—
221C2×C4×C73#12314yes—
222C2×C4×C73C23×C411yescharacteristic
223C2×C4×C73C22×D411yescharacteristic
224C2×C4×C73C22×D411yescharacteristic
225C2×C4×C73C22×Q811yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C73C2×#4111yescharacteristic
228C22⋊D73—216no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—22no—
233C2×Dic73—22no—
234C2×Dic73—22no—
235C2×Dic73—22no—
236C2×Dic73—22no—
237C2×Dic73—22no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244D292—22no—
245D292—22no—
246D292—22no—
247D292C2×C4⋊C414yes—
248C4×D73—216no—
249C4×D73—88no—
250Dic146—22no—
251Dic146—22no—
252Dic146—22no—
253Dic146C2×C4⋊C414yes—
254C8×C73—41no—
255C8×C73—41no—
256#10596831—24no—
257C22×C73—22no—
258C22×C73—22no—
259C22×C73—22no—
260C22×C73—22no—
261C22×C73—22no—
262C22×C73—22no—
263C22×C73#38514yes—
264C22×C73C2×C4×D411yescharacteristic
265C22×C73C2×#12011yescharacteristic
266C22×C73C2×C22⋊Q811yescharacteristic
267C22×C73#57611yescharacteristic
268C22×C73#57711yescharacteristic
269C22×C73#57711yescharacteristic
270C22×C73#57811yescharacteristic
271C2×D73—24no—
272C2×D73—42no—
273C2×D73—42no—
274C2×D73—42no—
275C2×D73—42no—
276C2×D73—42no—
277C2×D73—42no—
278C4×C73—44no—
279C4×C73—44no—
280C4×C73#38514yes—
281C4×C73C22×C4⋊C411yescharacteristic
282C4×C73C2×C4×D411yescharacteristic
283C4×C73C2×C22⋊Q811yescharacteristic
284C4×C73C2×#12311yescharacteristic
285Dic73—24no—
286Dic73—42no—
287Dic73—42no—
288Dic73—81no—
289Dic73—81no—
290#20878—731nomaximal, Sylow
291C2×C73—24no—
292C2×C73C2×#38511yescharacteristic
293C2×C73#294611yescharacteristic
294C2×C73#294711yescharacteristic
295C2×C73#301211yescharacteristic
296C2×C73#301311yescharacteristic
297C2×C73#308311yescharacteristic
298C2×C73#308411yescharacteristic
299D73—44no—
300C2×#520—731no—
301C2×#428—731no—
302C2×#417—731no—
303C2×#416—731no—
304#2316—731no—
305#1865—732no—
306#1864—732no—
307#1863—732no—
308#1862—732no—
309C2×#340D7311yescharacteristic, complemented
310#1729—731no—
311C73#2087811yescharacteristic, complemented, Sylow
312C22×#41—731no—
313C22×M16—731no—
314C23×C8C2×D7311yescharacteristic
315#520—1464no—
316#520—1464no—
317C2×C4×Q8—732no—
318C2×C4×D4—732no—
319C2×C4×D4—1462no—
320C2×#117—731no—
321C2×#117—731no—
322C2×#117—1461no—
323C22×C4⋊C4—1461no—
324C22×C4⋊C4C2×D7311yescharacteristic
325C22×C4⋊C4C2×D7311yescharacteristic
326#428—1464no—
327C2×#107—732no—
328C2×#106—732no—
329#417—738no—
330#416—1464no—
331C2×#103—732no—
332C2×#103—731no—
333C2×#103—731no—
334C2×#102—732no—
335C2×#102—731no—
336C2×#102—731no—
337#389—732no—
338#389—732no—
339#389—1461no—
340#389—1461no—
341#380—1462no—
342#340C2×D7312yes—
343#340C2×D7312yes—
344#336—732no—
345#336—732no—
346C2×#41—734no—
347C2×#41—734no—
348C2×#41—734no—
349C2×#41—1462no—
350C2×#41—1462no—
351C2×#41—1462no—
352C22×Q8—731no—
353C22×D4—731no—
354C22×D4—1461no—
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—1462no—
358C23×C4—1461no—
359C23×C4C22×D7311yescharacteristic
360C2×M16—734no—
361C2×M16—1462no—
362C2×M16—1462no—
363C22×C8—22no—
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—732no—
367C22×C8C22×D7311yescharacteristic
368C22×C8C22×D7311yescharacteristic
369C4×Q8—1464no—
370C4×Q8—1464no—
371C4×D4—1464no—
372C4×D4—14616no—
373C4×D4—1464no—
374#117—1462no—
375#117—1462no—
376#117—1468no—
377#117—1462no—
378#117—1462no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—1468no—
388C2×C4⋊C4—1462no—
389C2×C4⋊C4—1462no—
390C2×C4⋊C4—1462no—
391C2×C4⋊C4C4×D7314yescomplemented
392C2×C4⋊C4D29214yes—
393C2×C4⋊C4C22×D7312yes—
394C2×C4⋊C4C22×D7312yes—
395C2×C22⋊C4—732no—
396C2×C22⋊C4—1461no—
397C2×C22⋊C4—1462no—
398C2×C42—732no—
399C2×C42—1461no—
400C2×C42—1462no—
401#107—1464no—
402#106—1464no—
403#103—738no—
404#103—1462no—
405#103—1462no—
406#102—738no—
407#102—1462no—
408#102—1462no—
409#95—1462no—
410#95—1462no—
411C24—1461no—
412C24D4×D7311yescharacteristic
413#41—7316no—
414#41—1468no—
415#41—1468no—
416#41—1468no—
417C2×Q8—732no—
418C2×Q8—732no—
419C2×Q8—732no—
420C2×Q8—1461no—
421C2×Q8—1461no—
422C2×Q8—1461no—
423C2×D4—732no—
424C2×D4—732no—
425C2×D4—732no—
426C2×D4—1464no—
427C2×D4—1464no—
428C2×D4—1464no—
429C2×D4—1461no—
430C2×D4—1461no—
431C2×D4—1461no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—5842no—
441C22×C4—5842no—
442C22×C4—732no—
443C22×C4—732no—
444C22×C4—1468no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4—1464no—
449C22×C4—1464no—
450C22×C4—1464no—
451C22×C4C2×C4×D7312yes—
452C22×C4D4×D7312yes—
453C22×C4D4×D7311yescharacteristic
454C22×C4C2×D29212yes—
455C22×C4Q8×D7312yes—
456C22×C4#1192402712yes—
457C22×C4#1192402812yes—
458C22×C4C23×D7311yescharacteristic
459M16—1468no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—44no—
467C2×C8—734no—
468C2×C8—1462no—
469C2×C8—1462no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—5842no—
475C4⋊C4—5842no—
476C4⋊C4—1468no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1462no—
482C4⋊C4—1462no—
483C4⋊C4C2×C4×D7314yes—
484C4⋊C4C2×D29214yes—
485C22⋊C4—2922no—
486C22⋊C4—1468no—
487C22⋊C4—1464no—
488C42—2922no—
489C42—1468no—
490C42—1462no—
491C42—1462no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—2921no—
497C23—2921no—
498C23—2921no—
499C23—2921no—
500C23—1462no—
501C23—1462no—
502C23—1462no—
503C23#1455655112yes—
504C23#1455656112yes—
505C23#1455656912yes—
506C23C2×D4×D7311yescharacteristic
507Q8—734no—
508Q8—1462no—
509Q8—1462no—
510Q8—1462no—
511D4—734no—
512D4—14616no—
513D4—1462no—
514D4—1462no—
515D4—1462no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—5848no—
531C2×C4—1462no—
532C2×C4—1462no—
533C2×C4—1462no—
534C2×C4—1462no—
535C2×C4—1462no—
536C2×C4—1462no—
537C2×C4—1462no—
538C2×C4—1462no—
539C2×C4—1462no—
540C2×C4—14616no—
541C2×C4C4⋊C4×D7314yes—
542C2×C4#1455655112yes—
543C2×C4C4×D29214yes—
544C2×C4#1455656112yes—
545C2×C4#1455656912yes—
546C2×C4#1455657114yes—
547C2×C4#1455657314yes—
548C2×C4C22×C4×D7311yescharacteristic
549C2×C4C2×D4×D7311yescharacteristic
550C2×C4C22×D29211yescharacteristic
551C2×C4C2×Q8×D7311yescharacteristic
552C2×C4C2×#1192402711yescharacteristic
553C2×C4C2×#1192402811yescharacteristic
554C8—41no—
555C8—41no—
556C8—1464no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—2922no—
564C22—2922no—
565C22—2922no—
566C22—2922no—
567C22—2922no—
568C22—2922no—
569C22—1464no—
570C22#1798846114yes—
571C22C2×#1455655111yescharacteristic
572C22C2×#1455656111yescharacteristic
573C22C2×#1455656911yescharacteristic
574C22#1798938811yescharacteristic
575C22#1798939311yescharacteristic
576C22#1798939311yescharacteristic
577C22#1798940011yescharacteristic
578C4—44no—
579C4—44no—
580C4—2922no—
581C4—2922no—
582C4—5841no—
583C4—5841no—
584C4—1464no—
585C4#1798846114yes—
586C4C2×C4⋊C4×D7311yescharacteristic
587C4C2×C4×D29211yescharacteristic
588C4C2×#1455657111yescharacteristic
589C4C2×#1455657311yescharacteristic
590C2—24no—
591C2—2924no—
592C2C2×#1798846111yescharacteristic
593C2#2261875711yescharacteristic
594C2#2261876811yescharacteristic
595C2#2261924011yescharacteristic
596C2#2261925111yescharacteristic
597C2#2261978711yescharacteristic
598C2#2261979811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#340×C73C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenterC2×#340D73
hypocenterC73#20878
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C2×#17988627C2semidirect
5. #22612549C2semidirect
6. #22610200C2semidirect
7. #22610193C2semidirect
8. #22610186C2semidirect
9. #22610179C2semidirect
10. #22609627C2semidirect
11. C2×#340×C73C2semidirect
26. #17988586C4semidirect
59. C2×#11924012C2×C4semidirect
89. #14556514C2×C4semidirect
92. #14556511C2×C4semidirect
107. C2×C4⋊C4×C73C2×C4semidirect
115. #11924028C4⋊C4semidirect
309. C2×#340D73semidirect
311. C73#20878semidirect
391. C2×C4⋊C4C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890682
C2×#179886272
C2×#179885862
C2×#179885842
#226125492
#226102002
#226101932
#226101862
#226101792
#226096272
C2×#340×C732
#2087873